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Preface
Heterostructures consist of combinations of different materials that are in con-tact through at least one interface. Heterostructuring can occur naturally, asin some phase-segregated systems, or artificially, as due to a layering processduring growth. The excitement surrounding artificial heterostructures is thatthey can embrace new and unusual physical properties that do not otherwiseexist in nature. They can introduce new periodicities and they can be usedto create composite materials out of components whose properties tend tobe mutually exclusive. This book is devoted to magnetic heterostructures.Magnetic heterostructures share the virtues of being both fascinating to basicresearchers and potentially useful in many practical applications. Examplesare ferromagnet/semiconductor, ferromagnet/superconductor, and ferromag-net/antiferromagnet. These combinations display unique physical propertiesthat differ from their individual building blocks. Interlayer exchange coupling,exchange bias (EB) effect, proximity effects, giant magneto-resistance (GMR),tunneling magneto-resistance (TMR), spin injection, and spin transport areexamples of new physical phenomena that rely on combinations of variousfunctional layers that include metals, semiconductors, and oxides. Such het-erostructures are generated by stackwise deposition of layers of these materialsand/or by laterally fabricating them via lithographic means.
The history of magnetic films research has been traced back over 150 yearsby Grunberg, in his article on GMR published in Physics Today [1]. Modernmagnetic multilayer research started to emerge in the late 1970s and early1980s at multiple research institutions. Notable early examples include multi-layers for applications in magnetic recording [2] and as neutron spin filters [3],and to study elastic properties [4]. Exchange coupling in rare earth contain-ing superlattice systems represented a turning point in tailoring fundamentalproperties and gaining new insights into magnetic coupling phenomenon. Butthe watershed event, in hindsight, was the exploration of exchange couplingin transition-metal superlattices by Grunberg, which led to the discovery ofGMR and its practical applications.
VI Preface
The chapters of this book are written 21 years after the discovery of ex-change coupling in transition metal superlattices by Grunberg and cowork-ers [5], 50 years after the discovery of the exchange bias effect by Meiklejohnand Bean [6], and 30 years after spin tunneling was first observed by Jullierein a magnetic tunnel junction [7]. Therefore, it is fair to say that the fieldof magnetic heterostructures has reached a certain stage of maturity. Never-theless, magnetic heterostructures are still a timely topic. The reason for thisis the much higher chemical purity, layer precision, and interface sharpnessthat can be reached today as compared to 20 or 30 years ago. Because of thisincreased interface definition, TMR values observed today are up to 400% atroom temperature as compared to 14% at 4.2 K some 30 years ago [8], theoscillatory exchange coupling can be observed with monolayer precision, andthe exchange bias can be “designed” to display a well-defined exchange biasfield and coercivity. In addition, there are a number of new topics that haveemerged in recent years, including (i) the spin injection from ferromagneticmetal electrodes into semiconductors, (ii) spin-transfer torque effects leadingto current-driven magnetization switching in nanostructures, and (iii) prox-imity effects between superconducting and ferromagnetic films.
The literature on magnetic heterostructures is widely spread and highlyspecialized. This calls for a book that provides an overview on the basics andthe state-of-the-art aspects of magnetic heterostructures. This book attemptsto present a comprehensive overview of an exciting and fast-developing fieldof research that has already resulted in numerous applications and that servesas a basis for future spintronic applications. Both young researchers enteringthis field and those more experienced should benefit from these overviewson the present status and future challenges of various aspects of magneticheterostructures.
Before starting to investigate the physical properties of magnetic het-erostructure, there is always the issue of how to grow artificial heterostruc-tures. The choice of the growth method depends on the material combinationand other factors. In the early 1980s, the growth of metal superlattices withmolecular beam epitaxy (MBE) was the most advanced method [9, 10]. Fol-lowing the growth of semiconductor superlattices, Nb/Ta and rare earth su-perlattices were grown for the investigation of superconducting properties [11],hydrogen density modulations [12], and the exchange coupling of complex spinstructures in rare-earth metals across interlayers [13, 14]. As the structuralquality and chemical purity have steadily improved over the years, sputter-ing methods are now considered as competitive to MBE methods. This is inparticular true for complex material combinations of metals and oxides orwhen alloy layers of complex stoichiometry are to be grown. Another impor-tant topic is the choice of a proper substrate before metal deposition, andthe proper techniques to be chosen for in situ and ex situ characterization ofthe structural properties of the deposited layers. There is a tendency to gofrom the growth of “simple” elemental magnetic and non-magnetic transitionmetals to more complex oxides with rich functional properties. The growth
Preface VII
of these oxide layers is particulary challenging. These and other topics aretreated in the first chapter by Hjorvarsson and Pentcheva.
Magnetic systems always display an intrinsic magnetic anisotropy. Howthis anisotropy is affected by film thickness, surfaces, interfaces, and temper-ature is the topic of the second chapter by Lindner and Farle. This chapterfocuses on the discussion of single-element ferromagnetic metallic epitaxialfilms on single crystalline substrates. Even though the scope of the chapter isrestricted, it shows how with a few epitaxial layers the magnetic anisotropycan artificially be controlled, adding to an improved understanding of theunderlying physical mechanisms. After discussion of the main sources of themagnetic anisotropy energy, ferromagnetic resonance (FMR) is introduced asthe main and most accurate source of information on the magnetic anisotropyof thin films. This chapter concludes with a tutorial description of the mag-netic anisotropy of three prototypical systems: Fe/MgO(001), Fe/GaAs(001),and Ni/Cu(001).
Particularly important heterostructures are combinations of ferro- and an-tiferromagnetic materials. When cooled through the Neel temperature in anexternal field, Meijkeljohn and Bean [6] discovered in the early 1950s that thehysteresis of the ferromagnet is characteristically shifted, indicating an inter-action with the antiferromagnet across the common interface. This exchangebias effect has become very important in recent years for the fabrication ofspin-valve devices. Therefore, much effort has been spent to understand andcontrol the EB effect. In the third chapter, Radu and Zabel provide an in-troduction to the basic physical mechanism of the EB effect. They focus onnumerical calculations and analytical treatment of some basic models that arepredicated on the Stoner–Wohlfarth model [15] and that describe the coherentmagnetization reversal process. Subsequently, other models for the EB effectare also discussed that consider various defects and domain walls in the anti-ferromagnet, and spin glass–type disorder at the ferromagnet/antiferromagnetinterface. In the last part, these models are compared with recent experimen-tal data.
The phenomenon of exchange coupling is well known in the context ofmagnetic impurities in non-magnetic metal hosts and for explaining the richspin order in rare-earth metals. But it was not until the discovery of inter-layer exchange coupling that this effect has been analyzed in such a systematicfashion. In the fourth chapter, by Heinrich, the phenomenological and the fun-damental aspects of exchange coupling are described. Particular emphasis isgiven to the discussion of exchange coupling in Fe/Cr and Fe/Ag superlat-tices, which serve as model systems. In the last part of this chapter, aspectsof the dynamic exchange coupling across spacer layers are discussed. Theseaspects, which were first investigated by FMR techniques and which are nowbeing investigated by time-resolved and element-specific x-ray magnetic cir-cular dichroism techniques, have received much attention in recent years.
The communication of spin across superconducting and ferromagnetic in-terfaces is an exquisite and most enlightening physical problem. When in
VIII Preface
contact, Cooper pairs penetrate into the ferromagnetic layer while ferromag-netic spins diffuse into the superconducting film. The mutual proximity effectbetween feromagnetic and superconducting layers has evoked not only the ideaof a re-entrant superconducting state when the thickness of the ferromagneticlayer is varied, known as the LOFF state [16], but also the generation of anodd-in-frequency spin triplet superconductivity at the interface that can pen-etrate deep into the ferromagnetic layer, and vice versa a ferromagnetic stateinduced in the superconducting layer when cooled below the superconductingtransition temperature Tc. The latter effect has been termed the “inverse prox-imity effect.” These fundamental issues of the proximity effect are describedin the fifth chapter by Efetov, Garifullin, Volkov, and Westerholt. They alsoreview and discuss the most recent experimental investigations relevant to thetheoretical predictions.
For spin-polarized tunneling in magnetic heterostructures, we need a sub-strate and a ferromagnetic layer that is pinned via exchange bias, an oxidebarrier, and a ferromagnetic counter electrode that can be readily rotated in anexternal magnetic field. The preceding chapters provide the basic knowledgefor these ingredients. The sixth chapter builds on this information and pro-vides the latest up-to-date information on magnetic tunnel junctions, whichare either very challenging to fabricate and/or provide the highest TMR effectsto date. At present the most important tunnel barriers are Al-O and MgO;as magnetic electrodes Fe, Co, and/or FeCoB are preferred, but recently alsoHeusler alloys are starting to move to center stage. Reiss and coauthors pro-vide a brief discussion of the fundamental aspects of the TMR effect. Themain part is focused on applications of the TMR effect in read heads of harddisk drives, in storage cells of magnetoresistive random access memory de-vices, in field-programmable logic circuits, and in biochips. The success ofTMR devices in modern spintronic products depends on the scalability of thejunction. There is a fundamental problem, which could possibly be overcomeby current-induced magnetization switching. Therefore, the last section of thischapter is devoted to a discussion of spin-transfer torque phenomena, whichmay lead to a flip of the “free” ferromagnetic layer, thereby replacing theswitching by external magnetic fields.
Quantum size effects will become crucial in semiconductor devices overthe next 20 years with further decrease of structure size. In this limit thebest solution for coping with the quantum limit may be the transport of spinrather than the transport of charge via electrons and holes. In the seventhchapter by Hofmann and Oestreich, the increasing attractiveness of utilizingelectron and hole spin transfer for future semiconductor devices is described.Starting from the spin transistor as the prototype spintronic device, proposed17 years ago by Datta and Das, the authors continue with a discussion of theprincipal problem of spin injection across interfaces. Resistive models versustunneling across Schottky barriers are discussed as the basic mechanisms forspin injection.
Preface IX
Clearly in the present book not all aspects of magnetic heterostructurescan be covered because of size restrictions. Therefore, we refer to other re-views and topical monographs for further important topics that could not becovered in this book. We refer to some books that have recently been pub-lished and that cover topics related to those in the present book: MagneticNanostructures, edited Tagirov et al. [17], Ultrathin magnetic films, Vol. I-IV,edited by Heinrich and Bland [18], and Advanced Magnetic Nanostructures,edited by Sellmyer and Skomski [19].
Finally, we thank our colleagues in the research community for their stim-ulating activities, which continue to heighten interest in this ever-fascinatingtopic. Simultaneously, we take responsibility for all mistakes and omissionsthat could have been corrected. Many omissions will undoubtedly blossominto topics for future books as the story of magnetic heterostructures has notentered anywhere near its final chapter of development.
Bochum - Argonne, Hartmut ZabelJune 2007 Samuel D. Bader
1 Department of Physics, Uppsala University, Box 530, 75121 Uppsala, [email protected]
2 Department of Earth and Environmental Sciences, Section Crystallography,University of Munich, Theresienstrasse 41, 80333 Munich, [email protected]
Abstract. The growth and characterization of magnetic materials has progressedsubstantially during the last decades. In this chapter we give a brief overview of thisvastly growing field of research. We highlight some of the relevant growth techniquesfor different materials classes but we do not intend to be complete with respect totechnical details or materials systems. We also outline some of the concepts andtheories of the growth of modern magnetic materials, emphasizing the role of firstprinciples calculations in providing microscopic understanding of the growth mech-anisms. We discuss the growth of metallic and oxide single crystal films, multilayersand superlattices and the influence of thickness, strain, crystallinity, structure andmorphology on the resulting magnetic properties.
1.1 Growth and Characterization
The growth and characterization of magnetic materials has progressed sub-stantially during the last decades. In this chapter we give a brief overview ofthis vastly growing field of research. We highlight some of the relevant growthtechniques for different materials classes but we do not intend to be completewith respect to technical details or materials systems. We also outline someof the concepts and theories of the growth of modern magnetic materials,emphasizing the role of first principles calculations in providing microscopicunderstanding of the growth mechanisms. We discuss the growth of metallicand oxide single crystal films, multilayers and superlattices and the influenceof thickness, strain, crystallinity, structure and morphology on the resultingmagnetic properties.
1.1.1 Concepts: The Thermodynamic Versus Kinetic Picture
The first concepts used to describe crystal growth were based on generalthermodynamic considerations [1]. The resulting structures were assumed to
B. Hjorvarsson and R. Pentcheva: Modern Growth Problems and Growth Techniques, STMP
be in thermal equilibrium and therefore determined by the minimum of thefree energy. Depending on the balance between the surface energies of thesubstrate material, γS, the adsorbate layer, γA, and the interface energy, γI,three different growth modes are distinguished (shown in Fig. 1.1): .
Δγ = γS − γA − γI . (1.1)
If Δγ > 0, wetting of the substrate by the deposited material is expected,resulting in layer-by-layer growth mode (Franck van der Merwe mode). Inthis case, a new layer starts to grow only after the first one is completed.In the opposite case, Δγ < 0, the formation of three dimensional islandsis likely to occur (Vollmer-Weber growth mode). An intermediate situationcan appear when Δγ > 0 and the growing film is strained. Initially the filmgrows in a layer-by-layer mode, up to a critical thickness where the growthbecomes island like. This mode is often referred to as Stransky-Krastanovgrowth mode. The transition from a layer-by-layer growth to island like growthcan be viewed as governed by the strain state and the elastic properties of thegrowing material.
Although useful, this classification has its limitations, it does, for example,not include surface alloying. Furthermore in the homoeptiaxial case, i.e. whensubstrate and adlayer are of the same material, it predicts a layer-by-layergrowth. This is not generally valid and different growth modes are observedfor homoepitaxial systems. For example, Ag forms three dimensional islandson strained Ag(111) [2, 3]. Furthermore, ferromagnetic materials (e.g. Co, Fe)have a higher surface energy than the noble metals (e.g. Cu, Ag). Accordningto this simplified view, Co and Fe should not grow in a layer-by-layer mode onCu and Au substrates. However, Co deposited on Cu(001) grows up to twentymonolayers (ML) in a layer by layer mode [4].
These are just a few examples illustrating the possibilities and limitationsin the thermodynamic description of growth of materials. The main reasonfor the failures is that the growth process is by definition a non−equilibriumprocess and the thermodynamic equilibrium condition is thereby not fulfilled.In this non-equilibrium kinetic process one or more steps can be rate limiting.By understanding the underlying processes, the growth procedures can be
Vollmer-Weber Stranski-KrastanovFrank-van der Merwe
γAγS γI
Fig. 1.1. The illustration on the left hand side describes the basis for the thermo-dynamic description of growth. The relative energy of the interface (I), the surfaceof the growing layer (A) and the substrate surface (S) is assumed to determinethe resulting growth mode. Schematic illustration of the three basic types are alsoincluded in the figure
1 Modern Growth Problems and Growth Techniques 3
used to control the morphology and the crystallinity and thereby some of theemerging material properties.
In an atomistic approach, the growth of a film can be described as aresult of a number of microscopic processes such as adsorption, diffusion anddesorption of adatoms (cf. Fig. 1.2). Diffusion of atoms can take place on flatregions of the substrate, along or across step edges or around island corners.Therefore, besides adatom diffusion also the adatom-adatom and adatom-step interaction determine island nucleation and growth. In the framework oftransition state theory (TST) [5], surface diffusion is described by diffusionrates D which are determined by diffusion barriers, Ed, and prefactors, D0.
D = D0e− Ed
kBT . (1.2)
Time scales relevant for sample growth are of the order of seconds andminutes and the length scales of kinetically controlled structures and islandsare of the order of 100 A and involve a large number of atoms (> 105).On the other hand the detailed quantum mechanical description of atomisticprocesses is currently restricted to relatively small system sizes, up to about104 atoms. Typical time scales of e.g. ab initio molecular dynamics are ofthe order of picoseconds which limits its application to the determination ofpossible processes, probable paths, diffusion barriers and attempted jump rate(prefactors) of the adatoms.
A phenomenological or statistical description of growth can for examplebe obtained by using nucleation theory [6] or kinetic Monte Carlo simulations.These methods are often based on empirical or semiempirical parameters andtheir predictive power is therefore limited. In nucleation theory, growth isdescribed by rate equations, yielding the time evolution of the adatom andisland density. When the desorption rate is negligible a simple relation be-tween the saturation island density nx, deposition rate R, diffusion rate Dand temperature is obtained [6]:
nx ∝ (R/D)i/(i+2) . (1.3)
deposition
diffusion via hopping
substitutional adsorption
adsorption
Fig. 1.2. Atomistic picture of growth, including different processes like deposition,adsorption, diffusion of adatoms on the terrace, incorporation into existing islands,as well as incorporation via exchange in the substrate layer
4 B. Hjorvarsson and R. Pentcheva
Typically, the critical island size corresponds to i = 1, which implies thattwo adatoms form a stable configuration. The linear dependence that followsbetween ln nx and 1/T is often used to extract the diffusion barrier from theisland density at a constant deposition rate. It can also be used to determinethe critical island size i from the deposition rate dependence of island densityat a constant temperature. The graphical representation of ln nx(1/T ) is oftenreferred to as Arrhenius plots.
The rate equations express the time evolution of the average adatom andisland density. Because it is a mean field approach, immediate and constantadatom density in the vicinity the growing islands is assumed, while in realitythe islands have “depletion zones” with lower adatom densities. For systemswhere medium-range interactions are important (e.g. on strained surfaces),island densities predicted from nucleation theory can differ by as much as anorder of magnitude when only short range interactions are considered. Thiswas shown in a DFT-KMC-study [7] and emphasizes the need for includingstress and strain in the theoretical considerations. For a review of the micro-scopic view on metal homoepitaxy, see [8].
Nucleation theory is restricted to adatoms forming islands on the surfaceand an exchange with the underlying material is not taken into account. How-ever, exchange processes between adsorbate and substrate can significantlyinfluence the resulting heteroepitaxial growth modes. An attempt to includeexchange mechanism, within nucleation theory, was obtained by introducingsystems with a critical island size of zero, i = 0 (see e.g. [9]). Some furtheraspects of metal heteroepitaxy will be discussed in Sect. 1.2.1.
When describing the growth of thin films within the framework of statis-tical mechanics, the exact motion of an adatom is irrelevant. The motion istreated as a random process, while the probability as a functional of the en-ergy of a particular configuration is exact. The kinetic Monte Carlo approachprovides a statistical description of the evolution with time, enabling realisticdescription of a non-equilibrium processes. Combining DFT results on diffu-sion barriers and chemical interactions on the atomic scale with a statisticaldescription of the time evolution in a kinetic Monte Carlo simulation (ab ini-tio kMC) is therefore a promising route to bridge the gap between the timeand length scales of first principle calculations and experimental observations[10, 11, 12]
1.1.2 Growth Techniques
After this brief overview on the theoretical description of growth, we nowconsider some of the experimental aspects of the involved processes. The mostcommonly used deposition techniques for growing magnetic materials are:
• molecular beam epitaxy (MBE);• magnetron sputtering• ion beam sputtering
1 Modern Growth Problems and Growth Techniques 5
• pulsed laser deposition (PLD);• metal organic chemical vapor depostition (MOCVD);
These deposition techniques have many similarities, but do also differ sub-stantially with respect to the underlying physical processes. In this section,the complementary aspects will be emphasized and general requirements forsuccessful deposition of different materials will be addressed.
The crystal coherency and the chemical purity are the most importantparameters describing the quality of as grown samples. Surface oxidation andintermixing at the substrate interface are issues of concern for single films,while interface mixing and thickness variation become important when dis-cussing multilayered structures. The chemical purity of the as grown structuresdepends strongly on the purity of the ambient. For this purpose, the vacuumconditions can be used as a qualifier. Under ultra high vacuum (UHV) condi-tions, corresponding to 10−9 mbar or lower, the impinging rate of the residualgas is below 10−3/s. Thus, with a growth rate of 1 monolayer (ML) per second,the impurity level of the samples originating from the vacuum environmentcan be below 10−3 (atomic ratio), which is comparable to a representativepurity of commercially available deposition material.
The residual gas in a tight UHV system is typically governed by H2. Thepartial pressure of water (pH2O) is typically one or two orders of magnitudebelow that of H2. pO2 is often below the detection limit of most residual gasanalyzers (10−13 mbar) and can be ignored in this context. Although pH2O
is orders of magnitude below that of pH2 , the influence of H2O can dominatethe impurity levels. The sticking coefficient of H2O is close to unity whilethat of H2 is typically � 1 at 300 K. Furthermore, the sticking coefficient isstrongly dependent on the chemical composition of the surface as well as thetemperature and has therefore to be considered with extreme care.
When using magnetron sputtering, the pressure during growth is typicallyin the 10−3 to 10−1 mbar range. Using UHV growth chambers and ultra puregases, the partial pressure of impurities in the sputtering gas can be in thesame range as in e.g. MBE systems. Impurity levels of bottled gases are atthe ppm level at best which can result in adequate chemical purity, if the gasis not contaminated in the gas handling system.
The composition of the low pressure ambient can influence the physicalproperties of the growing material. For example, the presence of water willinevitability cause H impurities, significantly altering e.g. optical and elasticproperties of oxide films [13, 14]. Furthermore, when growing metallic layers,this will result in both H and O impurities in the films. The partial pressureof active gases in the sample environment is therefore a good measure of thepossibility to grow samples with high chemical purity. The noble gases arenot to be considered in this context, due to their inertness with respect tochemical reactions.
The use of all metal connections and thoroughly outgassing the gas linesas well as the deposition system (baking) can be a simple route to improve the
6 B. Hjorvarsson and R. Pentcheva
sample quality. This approach reduces the contamination of the process gas,however, the purity of the commercially available gases can be insufficient. Theuse of gas purifiers is therefore sometimes needed for obtaining the requiredpurity levels. Getter materials and cold traps based on molecular sieves canbe used for this purpose. The combination of all-metal bakeable gas lines andgas purifiers allows significant reduction of impurity levels.
Although the sputtering gases are chemically inert, the inclosure of thesputtering gases can cause significant contributions to the chemical composi-tion. Typically the sputtering gases (Ar, Kr, etc.) are found in voids createdduring growth [15]. This effect can be profound in the context of high growthrates of polycrystalline materials but is almost always negligible in single crys-tal materials.
The substrate temperature determines the sticking coefficient, the surfacemobility and the desorption rates, and is therefore one of the most importantprocess parameters. To determine the actual growth temperature is experi-mentally challenging and the quoted numbers are typically rather inaccurate.This is of special importance with respect to the growth of heteroepitaxialmaterials. This is important especially in the case of heteroepitaxial growth.Interdiffusion and crystallization are competing processes which can result ina narrow temperature range available for the required growth processes. How-ever, reproducible temperatures can easily be obtained in any growth system,which enables reproducibility of the growth while using the same setup. Theconcern is therefore the transfer of experimental procedures between growthsystems, as temperature calibrations are often crude and are material andsubstrate dependent.
Substrates
Oxides and semicoductors are frequently used as substrates for growing mag-netic films, multilayers as well as superlattices. The benefits from this choiceare many. First of all, the availability and price. Single crystal MgO, Al2O3,SrTiO3, Si, Ge, GaAs, etc., with different orientations, are commercially avail-able. The surface and bulk crystalline quality varies and the influence of ex-posure to ambient air differs substantially. The variation is not limited todifferent suppliers, large differences in substrate quality can be found fromone and the same vendor. This is clearly seen in Fig. 1.3, which illustratesrocking curves around the MgO(002) peaks from two different substrates.One of the substrates shows a well defined peak (solid curve) consistent witha single crystal structure, while the second one exhibits number of peaks cor-responding to crystallites which are only partially oriented. These substrateswere obtained from the same vendor and represent two batches of the samematerial.
The pre-treatment of substrates are both performed ex- and in-situ, de-pending on the purpose of the treatment. For example, ex-situ heat treatmentof Al2O3 (1500◦C for 1−5 hours, see Fig. 1.4) and SrTiO3 are commonly
1 Modern Growth Problems and Growth Techniques 7
0
5000
10000
15000
20000
0
50
100
150
200
250
300
350
400
20,5 21 21,5 22 22,5
Cou
nts C
ounts
Angle (deg.)
Fig. 1.3. Rocking curve from a MgO(001) sample representing a good (solid) and abad (dashed line) batch from the same supplier. The measurements were performedusing Cu Kα radiation around the (002) peak of MgO [17]
used, increasing both the terrasse width and crystalline quality. Correspond-ing heat treatment of MgO destroys the surface completely, easily identifiedby an opaque appearance.
In-situ treatment almost always involves extended annealing, removingwater adsorbed at the surfaces. The required temperature and time dependson the adsorption energy of water on the surface, which can vary substan-tially between different materials and crystallographic orientations [16]. Hightemperature annealing of semiconductor substrates can result in inward dif-fusion of the near surface oxides. Pre-sputtering is required to avoid this, butpost annealing is required for obtaining smooth surface after sputter cleaning.Pre-sputtering is normally not utilized when working with oxide substrates.
Fig. 1.4. AFM pictures showing the development of a step pattern during theannealing in air of sapphire substrates for six hours at different temperatures [18]
8 B. Hjorvarsson and R. Pentcheva
Molecular Beam Epitaxy and Sputtering
Molecular Beam Epitaxy (MBE) and sputtering techniques are commonlyutilized for growing magnetic thin films and superlattices. The main dif-ferences between these two techniques is the energy of the material flux,reaching the substrate and the different conditions of the evaporating ma-terial. In MBE growth, the material is heated to a temperature giving thedesired vapor pressure, which can be either below or above the melting tem-perature of the material. For example, Mg has a high sublimation rate farbelow the melting temperature. Mg is therefore typically not melted dur-ing an evaporation process. This involves both limitations and possibilities,as in situ cleaning/outgassing of e.g. Mg becomes difficult. Secondly, theevaporation temperature of the material defines the kinetic energy of theatoms reaching the substrate and later the growing film. Thus, the energyof the impinging atoms are typically far below 0.2 eV in an evaporationprocess.
In UHV based evaporation systems such as MBE, the mean free path ismuch larger than the system size. In magnetron based sputtering processes,the typical mean free path is of the order of centimeters. The flux from theevaporation source can therefore be regarded as highly directional, while theflux from magnetron sources is close to random at the substrate surface (cov-ering 2π). This has profound implications on the possibilities of using masksto obtain patterned growth [19], for which e.g. MBE is much better suited ascompared to magnetron sputtering.
There are two main routes for evaporating materials, namely through di-rect heating as accomplished in effusion cells (through radiation or directheating from heating elements) and by direct bombardment using high en-ergy electron beam (e-beam). Effusion cells have much better stability withrespect to the flux of the evaporated material and is therefore the method ofchoice when well defined layer thicknesses are required. The stability is criti-cal when growing, for example multilayers and superlattices, where the layerthicknesses have to be extremely well defined. The instability in the materialflux from an e-beam source originates from the dynamics of the melted re-gion. Scanning the electron beam across the target material often increasesthe stability, but the resulting fluctuations in the flux are still much largerthan obtained from effusion cells. The limitations in the use of effusion cellsoriginate from the chemical and thermal stability of the crucibles which are indirect contact with the evaporating material. Typical crucible materials areAl2O3, BeO, C (pyrolytic graphite), Ta and W.
For the growth of materials consisting of more than one element froma single source, the use of evaporation techniques can be inferior to thatof sputtering. As the flux of the evaporating material is determined by thevapor pressure of the elements, the growing film is likely to have significantlydifferent chemical composition, as compared to the composition of the sourcematerial. In a sputtering process, the surface composition changes initially but
1 Modern Growth Problems and Growth Techniques 9
eventually compensates the different cross sections of the elements, resultingin a close matching of the film composition to that of the source (targetmaterial).
Sputtering techniques can be viewed as complementary to MBE, not al-lowing any in situ treatment of the target material, but having wide flexibilitywith respect to the kinetic energy of the material flux. The kinetic energiesrelevant in the sputtering process must be divided into two classes, neutralsand charged particles. The energy of these can, in principle, be adjusted inde-pendently. The neutrals as well as the charged particles are directly affectedby collisions with the residual gas, while a bias of the sample only affects theratio of the kinetic energy of positive and negatively charged particles. Pos-itive bias retards the positively charged sputtering gas and target material,while it accelerates the electrons and negatively charged atoms of the tar-get material toward the growing film. This technique has been used to alterthe morphology as well as the crystallinity of thin films and superlattices.The simplest and often used route to utilize the electric potential, is to keepthe sample at a floating bias.
An alternative mode of operation is reactive sputtering. By introducingreactive gases in the growth chamber, these will readily react with e.g. metalsin an ionic or a neutral state. Reactive sputtering can be used to grow ox-ides, nitrides or any material where one of the components can be introducedin the gas phase feeding the sputtering process. The plasma chemistry canalso be used for obtaining phases which can not be synthesized by regularchemistry under normal conditions. There are two main modes of sputter-ing, namely Radio Frequency sputtering (RF-sputtering) and Direct Current(DC-sputtering). Although the DC mode can be used for reactive sputtering,target poisoning poses a severe challenge, demanding high degree of controlof the pressure and the electric potential at the target. RF sputtering is onepossible route to remedy this, even allowing the use of an insulating targetmaterial. The corresponding MBE approach is denoted Oxygen Plasma As-sisted Molecular Beam Epitaxy (OPAMBE), and is used for oxide growth asthe name indicates.
Although sputtering has a wide range of applications, there are some ma-terials that cannot successfully be deposited using this technique. As an exam-ple, growth of high purity actinide and lanthanide films is typically restrictedto ultra high vacuum evaporation. The chemical purity of the purchased ma-terials is specified for all elements but hydrogen, which is typically high. Theprocessing required for obtaining good quality films, involves therefore ex-tended outgassing prior to deposition for reducing the hydrogen content. Thisprocessing is not compatible with standard sputtering techniques. The growthof the actinides and lanthanides is therefore typically restricted to UHV evapo-ration. Although great precaution is taken with respect to in-situ purification,substantial amounts of hydrogen is inevitably present in rare earth films, aswill be discussed later.
10 B. Hjorvarsson and R. Pentcheva
Other Techniques
Both MOCVD and PLD are widely used for growing oxides and semicon-ductors. In the first technique the chemistry in the reaction chamber can betailored, while some restrictions apply with respect to the most reactive ele-ments such as Sc, Y, Lu, Lr and the rare earth elements. The use of in-situtools is restricted due to the high pressures in the reaction chamber. In themagnetic community MOCVD is mainly used for the growth of doped ferro-magnetic semiconductors.
The use of ion beam techniques for growth of magnetic samples for researchpurposes, has increased substantially. The main advantage of this technique isthe versatility, allowing large number of target materials in the one and samesetup. Although the use has increased, it is still not widely used.
Pulsed-laser deposition (PLD) has been utilized for the growth of singlefilms and superlattices. This technique is often favored for the growth of com-plex oxides, allowing complete control of the ambient gas. Here, a laser beamis focussed on a target in an UHV chamber. Material ablated by the laserpulses is deposited on the substrate. This technique has two major advantages:First, the target has already the desired stoichiometry making the oxidationstep superfluous and second, the amount of deposited material can be con-trolled/calibrated by the number of pulses, allowing high degree of precision ofthe thicknesses of the layers while growing thin films and superlattices [20, 21].A finite oxygen pressure is often required to obtain stoichiometric oxides.
1.1.3 Characterization Techniques
In situ Characterization
Both in-situ and ex-situ characterization tools are used for obtaining infor-mation on the composition and structure of the as-grown materials. Thesecan be viewed, in many respects, as complementary. The big advantage of thein-situ techniques is the absence of the limitations invoked by the exposure toambient, while the advantage of ex-situ characterization lies in the versatility.
Some aspects of the structural quality of the growing materials can bedetermined in-situ, however, most of the relevant work is done ex-situ, whenoperating a production device for thin films and heterostructures. The uti-lization of in-situ characterization is demanding and significantly increasesthe complexity level of the operation. Therefore many production systemsare only equipped with a limited amount of in-situ possibilities. Examples ofcommon in-situ equipment are typical surface characterization tools such asAuger spectroscopy, Reflective High Energy Electron Diffraction (RHEED),Low Energy Electron Diffraction (LEED) as well as Scanning Probes such asAtomic Force Microscopy (AFM) and Scanning Tunneling Microscopy (STM).
The most widely used probe for in-situ measurements is the electron. Forexample, probing the energy of the ejected Auger electrons gives information
1 Modern Growth Problems and Growth Techniques 11
about the near surface composition, diffraction from the surface yields the sur-face structure and electron energy loss spectra (EELS) even yields informationon the oxidation state.
When determining the surface structure, there are two basic configura-tions, the electrons arrive to the surface almost parallel to the surface normal(LEED) or close to parallel to the surface plane (RHEED). Consequently,these techniques probe different directions and lengthscales, RHEED is highlysensitive with respect to steps and island growth while LEED yields informa-tion on the atomic distances and periodicity in the near surface region ofthe sample. An advantage of LEED with respect to other structural tech-niques such as e.g. XRD, is the higher sensitivity to the oxygen positions andthe lower penetration depth making the method sensitive to the near surfacestructure. Thus, LEED is used to determine the crystal quality, in-plane lat-tice parameters, and the development of superstructures. By simulations ofthe LEED I − V curves, information on the atomic positions and interlayerdistances of the outermost layers can be obtained.
RHEED is mainly used to monitor the thickness of evolving films andthe growth quality. Oscillations in the RHEED intensity correspond to theformation of complete layers (closing of layers) and thus can be directly relatedto the thickness of the deposited film.
The use of STM and AFM allows the visualization of the real space surfacemorphology, thereby serving as a complementary tool to reciprocal techniquessuch as LEED and RHEED. Both techniques have been used to follow the dif-fusion of adatoms on surfaces as well as monitoring the surface quality of thinfilms and heterostructures. Also diffusion parameters from island density mea-surements using Arrhenius plots are described in the literature. The combina-tion of these techniques allows contact microscopy, enabling identification ofchemical elements with near to atomic resolution. Thus, these scanning probetechniques are exceedingly valuable tools on all growth systems.
There are also techniques which have to be classified as in-situ techniques,although these are not used as a routine equipment for monitoring growth inconventional growth systems. One of the most important ones is the force ionmicroscopy (FIM) which allows the study of individual diffusion processes anddetermination of diffusion barriers. For a comprehensive review see Tsong [22].
Ex situ Characterization Techniques
As mentioned above, some aspects of the structural quality of the growing ma-terials is preferably determined in-situ, however, when operating a productiondevice for thin films and heterostructures most of the routine characterizationis performed ex-situ. Two of the most important parameters describing theresulting films are the chemical composition and the crystalline structure.
The composition of thin films is conveniently determined by Ion BeamAnalysis techniques (IBA), such as Rutherford Back Scattering (RBS) and
12 B. Hjorvarsson and R. Pentcheva
Nuclear Resonance Analysis (NRA). These techniques have been used to de-termine the composition of wide variety of materials such as metals [23, 24, 25],oxides [14], nitrides [26], carbides [27] and hydrides [28, 29, 30]. The signal ina RBS experiment scales as Z2, thus, the sensitivity increases strongly withincreasing atomic number. High energy scattering can be used for changingthe scattering cross section, when a better sensitivity is required for low Z ma-terials. The basic idea is to overcome the Coulomb barrier, entering a regionwhere the cross section is highly varying with the energy of the impinging ions.The presence of resonances can even be used to obtain isotope selective depthprofiling in materials. For example, the resonance between α and O16 can beused for depth profiling of oxygen [31] with a resolution and sensitivity whichis far better than most other techniques. When the depth resolution is lessimportant than the detection limit, Elastic Recoil Detection Analysis (ERDA)can be utilized [32]. These techniques can all be operated in an absolute mode,counting the number of ions hitting the sample. This in combination with theknown scattering cross section allows absolute determination of the concen-tration with an accuracy only limited by the determined stopping power ofthe ions. Typical accuracies are below few atomic percent, while the precisioncan be much better.
Most of the structural information is obtained by x-ray scattering usingconventional laboratory sources. The combination of reflectivity and diffrac-tion allows the probing of all relevant length scales, from the overall thicknessof the film to the the atomic distances. The x-ray scattering yields the crys-tal coherency of the material, the sharpness of the chemical modulations i nmultilayers and superlattices as well as the thickness variation at all relevantlengthscales [33, 34, 35]. However, interdiffusion and thickness variation (seeFig. 1.5) in a superlattice can not be separated by solely performing specularscattering experiments. Only by combining off-specular and specular scatter-ing the relative weight of the components can be separated. Simulations andfitting routines for off-specular scattering are currently not generally available.
The combination of x-ray scattering and Transmission Electron Microscopy(TEM) can be useful. For example, when investigating combinations of oxidesand metals, the difference in the scattering cross section is often large enoughto obtain near atomic resolution of the interface composition. This combinedwith large contrast in the x-ray scattering allows detailed comparison givingunique insight in the actual local and global variation in the chemical compo-sition in samples. However, TEM is highly destructive technique. The samplepreparation involves slicing and thinning of a cross sectional part of the sam-ple. Consequently, the investigated samples can not be used for any othermeasurements. Examples from investigations using most of these techniqueswill be given in the following sections.
Investigations of the structural quality is not restricted to the use of x-rayscattering. For example, the use of ion beam channeling can be highly reward-ing. When the ions channel through a single crystal film, the back scatteringyield is small. If the film and the substrate have a coherent interface, the yield
1 Modern Growth Problems and Growth Techniques 13
Fig. 1.5. Illustration of interdiffusion (left) and roughness (right) at interfaces. Ifit is possible to insert a surface which separates all the atoms of the two types, thereis no interdiffusion
will remain small. On the other hand, if the interface is incoherent, therewill be substantial back scattering from the region corresponding to the in-terface between the film and the substrate. Ion beam scattering can therebygive highly relevant information about the interface quality as illustrated inthe investigations of the initial growth of Cr on Fe [37]. Not only is the epi-taxial relation between the substrate and the film established, the thermalvibrations of the individual components can also be extracted. This has beendemonstrated using high quality sputter deposited Fe/Cr(001) superlatticesby Ruders et al. [23].
One important aspect of the use of nuclear scattering is the isotope selec-tivity. The cross section is unique for the isotope combination in the scatteringprocess, allowing investigations of e.g. O in oxides by growing isotope layers ofO16 and O18. This region in the scattering cross section corresponds to scat-tering above the Coulomb barrier, at which the two particles can be viewedas a compound nucleus in the moment of scattering. For a comprehensiveintroduction to different nuclear scattering techniques see for example [38]and [39].
The isotope selectivity is also prominent in neutron scattering experiments.Furthermore, the scattering cross section is strongly varying with the atomicnumber and the choice of isotopes, which makes neutron scattering extremelyuseful as a complement to x-ray scattering. Neutron reflectivity has been usedto determine the composition variation in multilayers and superlattices as wellas the magnetic profiles in thin films, multilayers and superlattices. Neutronreflectivity is one of the few methods that allow the full determination of themagnetic structure in materials [40, 41]. Full determination of the magneticorder can be obtained using polarization analysis. This includes the deter-mination of the magnetic ordering in layered magnets as well as stripes andislands.
Magneto Optical Kerr Effect (MOKE) is one of the most used ex- and in-situ technique to determine changes in magnetization. For a review of the useof MOKE, see e.g. [42]. The MOKE is one of the most versatile approaches to
14 B. Hjorvarsson and R. Pentcheva
probe the changes in magnetization with temperature, to measure anisotropiesas well as the susceptibility. The absolute moment can not be determined byMOKE. Superconducting Quantum Interference Devices (SQUID) are there-fore often used for calibrating MOKE results.
The magnetic properties of the material can serve as a qualifier with re-spect to the structural quality. For example, the ordering temperature of ex-tremely thin layers is strongly depending on the thickness. Consequently, ifthe thickness is changing substantially at lengthscales equal or larger than themagnetic interaction, this would result in a ill defined ordering temperature.In Fig. 1.6, the magnetization of 3.4 ML Fe on GaAs (001) is displayed. Theresults clearly support the presence of highly uniform layer thickness at allbut extremely short length scales and thereby carries information about theuniformity of the grown film.
Fig. 1.6. Remanent Kerr rotation (a) and magnetic susceptibility (b) versus tem-perature for 3.4 ML Fe on GaAs (001)–(2× 6). Almost no tailing is observed in themagnetization and the susceptibility is narrow and well defined. From [36]
1 Modern Growth Problems and Growth Techniques 15
1.2 Growth of Metals
The first step towards fabricating a multilayer or superlattice stack is thegrowth of a thin film. In this section we discuss the growth of transition metalson a noble metal substrate and of rare earths. The growth of transition metalson other transition metals as well as oxides is covered in Sect. 1.4.
1.2.1 Transition Metals on a Noble Metal Substrate
Growth of thin magnetic films on a nonmagnetic substrate provides the pos-sibility to design materials that do not exist in the bulk phase: e.g. Co (hcpin bulk) grows fcc on a Cu substrate. Similarly, Fe (bcc in bulk) was inferredto adapt a fcc structure on Cu(001) up to a thickness of 10 ML, while thestructure is transformed into a body centered tetragonal phase at thicknessesabove 10 ML [43, 44, 45, 46]. Three (0–4 ML, 4–10 ML and> 10 ML) thicknessregions of Fe on Cu(001) are identified exhibiting widely different magneticproperties, closely connected with the structure of the films. Recently the lo-cal atomic structure of the first iron layers has been revisited and there is stilla substantial controversy whether the structure is bct as determined usingSTM, LEED and DFT-calculations [47, 48, 49] or fcc [50].
The obtained structure is a result of an intricate balance between latticemismatch, strain and bond strength between adsorbate and substrate, versusadsorbate-adsorbate. We will discuss some specific aspects of heteroepitaxialversus homoepitaxial growth using Co on Cu(001) as an example.
Co and Cu are immiscible in the bulk and have only a small lattice mis-match (2%). Still, intermixing influences the obtained interface quality sub-stantially. In the initial stages of metal homoepitaxy, nucleation theory [6]predicts the logarithm of the island density to decreases linearly with increas-ing temperature (cf. 1.3).
Instead of a linear dependence, a complex N -shaped non-Arrhenius behav-ior of the island density (illustrated in Fig. 1.7) is obtained for Co/Cu(001)from an ab initio thermodynamics kinetic Monte Carlo study, using diffusionbarriers from DFT [51, 52]. At low temperatures, the heteroepitaxial case re-sembles homoepitaxial growth with adatoms diffusing on the surface, formingnearly square islands (see STM image at 295 K in Fig. 1.7). At approximately340 K the activation of atomic exchange leads to a minimum in ln nx and asubsequent increase of island density due to pinning at substitutionally ad-sorbed Co adatoms [53]. At higher temperatures, the exchange mechanismand diffusion of the substrate material on the surface results in a bimodalisland size distribution with large Cu islands decorated with Co as well as alarge number of small predominantly Co islands (see STM image at 415 K inFig. 1.7). Island densities obtained from He-scattering experiments [51] con-firm the predicted N -shape of ln nx(1/T ) (cf. Fig. 1.7). This unusual behaviorwas also observed in STM-measurements [53, 54, 55]. Experimental results for
16 B. Hjorvarsson and R. Pentcheva
2.0 2.5 3.0 3.5 4.0 4.5 5.0
400K 300K 200K
1011
1012
1013
1014
340 380 420
1x1013
2x1013
ntot
nCo
nCu
isla
nd
de
nsit
y n
x [cm
-2]
1000/T[K-1]
T=415K T=293 K
Fig. 1.7. N-shaped non-Arrhenius behavior of island density in the initial growthof Co on Cu(001) obtained from DFT-kMC simulations (solid diamonds) and ionscattering experiments (solid circles) for F = 0.0045 ML/s [51]. Empty diamonds:theoretical results for F = 0.1 ML/s. Open circles: island densities derived fromSTM-images (shown above) for F = 0.0033 ML/s [53]. Inset: linear plot of islanddensity between 340 and 410 K for Co (nCo
x ) (solid triangles) and Cu (nCux ) (open
triangles) islands. Experimental error bars comprise statistical and, for high and lowtemperatures, possible systematic errors
Fe/Cu(001) [9], Fe/Au(001) [56], Ni/Cu(001) [57], and Co on Ag(001) [58] im-ply, that the scenario described above could be relevant for a broader class ofmaterials.
Epitaxial growth of Co films up to 20 ML has been reported in the litera-ture. However, bilayer growth was observed for the first two layers [4, 54], andthe second layer starts to grow before the first one is completed. At elevatedtemperatures a Cu capping layer is formed [4, 59]. To explain these experimen-tal observations, DFT calculations were performed for different configurationssuch as monolayers, bilayers and sandwich structures [60]. The correspondingformation energy for 1 ML of Co is shown in Fig. 1.8. The tendency towards
1 Modern Growth Problems and Growth Techniques 17
Fig. 1.8. Formation energy of different ferromagnetically ordered configurationsfor a total cobalt coverage of 1 ML as a function of the cobalt island thickness N .The structures consist of clean Cu(001) and a compact island with N Co layers (◦)or N Co-layers capped by copper. The area covered by the cobalt islands is 1
Nof
the whole surface. Especially for the copper terminated systems the separation inhigher than bilayer cobalt islands is unlikely because of a negligible energy gain
the growth of bilayer islands results from the affinity of Co to maximize thenumber of Co-Co bonds versus Co-substrate bonds. This is also expressed ina strong relaxation (Δd/dCu = −13.4%) of the interlayer spacing between thetwo Co-layers. A Co double layer capped by 1 ML of Cu was found to bethe thermodynamically most stable configuration. Similar results were foundfor Co films grown on Cu(111) [61]. A recent molecular dynamics simulationbased on Tight Binding Second Moment Approximation (TBSMA) [62] iden-tifies an upward diffusion mechanism at island edges as the origin of bilayergrowth [63].
The interaction between ultra thin transition metal films and noble metalsubstrates is relatively weak. Consequently, the surface strain of such filmsmay be much larger that the one suggested from the bulk phases. For examplein the extreme case of a free standing Co monolayer (missing interaction withthe substrate) DFT-calculations predict an equilibrium lattice constant 12.2%smaller than the one for Cu, while the lattice constant of bulk fcc Co is only2% smaller than Cu [60]. Related to this is the so called mesoscopic strain:based on ab initio calculations Stepanyuk et al. [64] found that the shape ofthe growing islands and the underlying substrate is strongly deformed by theinhomogeneous stress field around Co islands.
1.2.2 Rare Earth
The rare earth (RE) elements (actinides and lanthanides) are extremely re-active and therefore difficult to purify and to maintain impurity free. Thesecan be purchased in a reasonably pure form with respect to all elements but
18 B. Hjorvarsson and R. Pentcheva
hydrogen. Typically, RE materials are refined in situ, by extensive annealingand outgassing procedures to minimize the hydrogen content. This severelylimits the possible deposition techniques and MBE appears to be the tech-nique of choice for successful deposition. Here, we regard Y as representativefor RE materials due to the similarities with respect to the outermost electronstates, yielding similar chemical properties.
The reactivity of the RE materials influences the sample design and mostresearchers use diffusion barriers to hinder e.g. oxygen transport from the sub-strate to the film, forming RE oxide. For example, the growth of RE materialson Al2O3 often involves a Nb layer serving simultaneously as a diffusion bar-rier and a seeding layer, as described by Kwo et al. [65]. This aspect will bediscussed further when addressing the growth of RE superlattices. A compre-hensive review of the growth of Nb (110) on Al2O3 is found in [67].
The choice of capping layer is important for hindering deterioration of thematerial. The capping has to wet the RE film and form a stable continuouslayer hindering reactions with the ambient atmosphere. Even here, Nb hasbeen used successfully, as Nb forms a self passivating oxide layer at ambientconditions [68]. A good counterexample is the use of gold as a capping layer. Atfirst glance, gold appears to be the ideal material choice for capping, it is inertand it is possible to form what appears to be continuous films. However, Auis highly unsuitable as a capping layer as the RE films deteriorate rapidly[66].The root of the deterioration is the adsorption of H2O on the Au surface.Water diffuses readily on grain boundaries, reaching the underlying film where
[H]/[Y
] (Atom
ic Ratio)
Yie
ld (
Cou
nts/
2μC
)
Energy (MeV)
GoldGoldGoldGoldYOxHzNiobiumSapphire
Mixed ! -Yand fcc YH2
(a) (b)
Fig. 1.9. Illustration of the deterioration of a representative film by H2O. (a) Thehydrogen content was determined by nuclear resonance analysis using the N-15method, and the oxygen content by Rutherford back scattering spectrometry. Thehydrogen diffuses deep into the film, while the oxygen forms oxides in the near sur-face region. Notice the absence of hydrogen in the Nb layers. An illustration of thededuced structure is shown in (b) [66]
1 Modern Growth Problems and Growth Techniques 19
-10
0
10
20
30
40
50
60
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
6.2 6.4 6.6 6.8 7 7.2 7.4
No
rmal
ized
Yie
ld (
cou
nts
/μC
)
Energy (MeV)
H/Y
(atomic ratio)
YHxNb Nb Al2O3
Fig. 1.10. Illustration of the hydrogen content of a Y film. The Nb capping is hin-dering the reaction of H2O with the Y, however, substantial hydrogen concentrationis still found in the film
it dissociates. This results in the formation of oxides, hydroxides and hydrides.This is illustrated in Fig. 1.9 [66].
Although Nb appears to be an excellent capping material, substantialamount of hydrogen is still found in Nb capped MBE grown RE materials.Typical procedures for the MBE growth of RE materials involves extended(days) outgassing of the target materials, followed by evaporation onto a sub-strate at elevated temperatures. In-situ capping and subsequent dry oxidationof the Nb capping still results in significant hydrogen content of the grownfilms. Representative results on the determined hydrogen content are illus-trated in Fig. 1.10 [69]. The hydrogen content of Y and RE films are typi-cally in the same range, as expected from the similar chemical properties, asconfirmed by measurements on Y, Gd and Ho. The hydrogen content of REmaterials is frequently ignored, which seriously influences the reliability of thededuced film properties.
1.3 Growth of Magnetic Oxidesand Magnetic Semiconductors
While detailed atomistic models for the homoepitaxial growth of metals havebeen put forward and attempts to incorporate some aspects of heteroepitaxialgrowth have been made (see discussion in Sect. 1.2.1), a kinetic descriptionof oxide growth is largely lacking. One of the reasons is the complexity ofthe oxide structures, the different nature of bonding and the variety of chem-ical species that are involved, leading to a multitude of potentially relevantdiffusion processes.
Besides the temperature and the deposition rate, the partial pressureof oxygen is an important parameter in the epitaxial growth of oxides.
20 B. Hjorvarsson and R. Pentcheva
Post-growth treatment (e.g. annealing) in vacuum may lead to reduction ofthe oxide. Vice versa, post-growth annealing in oxygen atmosphere can helpto reduce oxygen vacancies. Concerning the characterization of the grownfilm, most of the surface science techniques require ultra high vacuum (UHV)conditions. The film properties and structure may be altered by the ambi-ent, which imposes a problem. Furthermore, the insulating nature of oxideshampers the application of imaging techniques, such as scanning tunnelingmicroscopy which requires a reasonably conducting sample. This limitationcan be circumvented by using thin oxide films grown on a metal support,providing sufficient conductivity.
The surface stoichiometry and structure has important consequences forthe reactivity but also for the magnetic and electronic properties of the ma-terial. Oxides and their surfaces are typically classified according to electro-static considerations. The most commonly used are Tasker’s scheme [70] andthe autocompensation rule [71]. Originating from semiconductor physics and acovalent picture of bonding, the autocompensation rule states that on a stablesurface all anion- (cation-) derived dangling bonds have to be filled (empty).Tasker’s classification, which emphasizes the ionic nature of bonding, is shownin Fig. 1.11. Here, oxide surfaces are divided in three groups, according to thecharge of the layers Q and the dipole moment μ perpendicular to the surface.Systems of type one have neutral layers and no dipole moment perpendicu-lar to the surface (Q = 0, μ = 0). Systems of type two and three consist ofcharged layers. In type two systems the repeat unit has no dipole momentperpendicular to the surface, while in type three it has a nonvanishing dipolemoment perpendicular to the surface. It should be noted that depending onthe termination, surfaces of the same orientation can be polar or non-polar.For type three surfaces both the scheme of Tasker and the autocompensa-tion rule postulate a diverging surface energy (“polar catasptrophy”) thatcan only be compensated by strong changes of the surface stoichiometry ei-ther by reconstructions or by faceting. Although usefull, these concepts havetheir limitations, because both models rely on the bonding and valence statein the bulk which may substantially be altered at the surface.
Type 1: Q=0, µ=0 Type 2: Q=0, µ=0 Type 3: Q=0, µ=0
Fig. 1.11. Classification of polar oxide surfaces after the scheme of Tasker [70].See text for details
1 Modern Growth Problems and Growth Techniques 21
DFT calculations have shown that lattice relaxations, where electroniccharge redistribution often leads to metallization of the surface, can be aneffective mechanisms to reduce and even compensate surface polarity (see re-view of Noguera [72]). Other mechanisms emerging from the correlated natureof transition metal oxides will be discussed in Sect. 1.4.3. The developmentof ab initio thermodynamics [73, 74, 75] contributed substantially to identifycases where simple electrostatic arguments fail. In ab initio thermodynam-ics density functional theory is combined with concepts from thermodynam-ics to describe the surface stability at ambient pressures and temperatures.The main idea is that the most favorable surface configuration minimizes thesurfaces energy. The latter depends on the Gibbs free energy of the systemGslab
MxOy(001) as well as on the chemical potentials of the constituents.
γ(T, p) =1
2A
[Gslab
MxOy(001) −NMμM (T, p) −NOμO(T, p)]. (1.4)
When the entropic contributions are small or cancel out, one can susbtituteGslab
MxOy(001) with the total energy from DFT-calculations. The chemical poten-tials μM (T, p) and μO(T, p) are not independent of each other. The conditionthat the surface is not only in equilibrium with the gas reservoir (e.g. oxygenpressure in the atmosphere) but also with the bulk oxide MxOy results in onlyone independent variable, the oxygen chemical potential which can be trans-lated into partial pressures at a particular (growth) temperature. For furtherdetails, see e.g. [75].
A further aspect that has to be noted is that the treatment of transitionmetal oxides represents a challenge for DFT- methods due to the correla-tion effects in the d states, localized oxygen orbitals and magnetism. For suchsystems all electron methods provide the highest accuracy and for the treat-ment of on-site Coulomb repulsion methods that go beyond the local den-sity approximation (LDA) or the generalized gradient approximation (GGA)like e.g. the LDA+U method [77] are gaining importance. Such methodshave mainly been applied to bulk systems and only recently to surfaces andinterfaces.
1.3.1 Binary Oxides
In this section we limit the discussion to few examples of the growth of ox-ides with ferro- or ferrimagnetic coupling. Prominent examples are the half-metallic ferromagnets, Fe3O4 and CrO2.
Fe3O4
Magnetite is the oldest known magnetic material and its importance rangesfrom geology to magnetic recording. The predicted half metallic behavior [78]paired with a high Curie temperature makes it a prospective material forspintronics applications. This has generated substantial research activities onmagnetite.
22 B. Hjorvarsson and R. Pentcheva
Magnetite is a ferrimagnet, that crystallizes in the inverse spinel structurewhere oxygen ions form a slightly distorted fcc lattice. Trivalent iron ionsoccupy one fourth of the tetrahedral sites (FeA), while 50% of the octahe-dral sites are occupied by mixed valence FeB-ions. While a p(1× 1)-structurehas been reported on the (111)-surface, Fe3O4(001) shows a (
√2×√
2)R45◦-reconstruction. The origin of the latter was subject of a controversial debatein the literature. Various models for a compensated (001)-surface were pro-posed, where the surface reconstruction was understood as an ordering ofsurface defects [79, 80, 81, 82, 83, 84]. The surface phase diagram, obtainedwithin the framework of ab initio thermodynamics (shown in Fig. 1.12a) forall possible surface models, revealed that a modified bulk termination yieldsthe lowest energy over the entire range of accessible oxygen pressures. In thisso called modified B-layer (top and side view displayed in Fig. 1.12b) the sur-face reconstruction is a result of a wavelike Jahn-Teller-distortion and not anordering of surfaces vacancies as in previous models. This termination doesnot fulfill the electrostatic models and was therefore ignored in the structuralanalysis so far. Experimental evidence for this structure is obtained fromXRD [85], LEED and STM measurements [76]. The wave like pattern in thesurface layer is clearly visible in the STM image of the Fe3O4(001)-surfaceand the STM simulation in Tersoff-Hamann model using the charge densityfrom the DFT-calculation shown in Fig. 1.12c). Both the DFT and spinpolar-ized photoemission measurements [76] show that the surface stabilization isaccompanied by strong changes in the electronic properties: e.g. a half-metalto metal transition takes place from bulk to the surface.
Besides natural samples synthetic single crystals as well as epitaxial filmsare used in experiment. Koltun et al. [86] grew synthetic Fe3O4 crystals usingthe floating zone technique from pre-sintered magnetite bars prepared fromiron oxalate. After crystallization the samples were annealed at 1473 K for20 h in a partial oxygen pressure of 3.2 × 10−6 mbar.
a) b) c)
Fig. 1.12. (a)Surface phase diagram of the Fe3O4(001)-surface; (b) side andtop view of the modified B-termination, oxygen, tetrahedral and octahedral ironare marked by white, grey and dark grey circles, respectively; (c) STM image ofthe Fe3O4(001)-surface [76] together with an STM simulation of the modified B-termination both showing the wave like structure in the 110-direction
1 Modern Growth Problems and Growth Techniques 23
Epitaxial films were grown on a variety of substrates. Due to the nearly per-fect lattice match (0.31%), MgO is an excellent candidate for the growth of epi-taxial magnetite films. Fe3O4-films in the (001)-orientation are typically grownon MgO(001). Other susbtrates used are SrTiO3(100) [87](lattice mismatch-7.1%), MgAl2O4(100) [88] (lattice mismatch–3.8%), and GaAs(100) [89]. Thelatter substrate is particularly interesting for the incorporation of Fe3O4 inspintronic devices. Two main techniques are used for the synthesis of Fe3O4:(i) oxidation of Fe films with oxidizing agents as O2, NO2 [84] or oxygenplasma; (ii) oxidation during deposition of Fe in an oxygen rich environment.
On MgO(100), Fe3O4-films were grown using O2 assisted MBE. With asubstrate temperature of 525 K and a growth rate of 22.5 A /min good qualityfilms were obtained [90]. Oxygen plasma assisted MBE was used by Kimet al. to grow Fe3O4(001) on MgO(001) [92]. The best quality was obtainedunder oxygen poor conditions (pO2 = 4 × 10−6 mbar) and an iron depositionrate of 0.6 A/s with an electron-cyclotron resonance (ECR) plasma sourcerunnung at 200 W. Voogt et al. [84] used NO2 as an oxidizing agent and asimilar substrate temperature of 525 K. The growth was monitored by RHEEDand the time to form a ML of Fe3O4(001) was estimated to be 46 s. Thethreshold of Mg interdiffusion at 625–675 K represents an upper limit for thegrowth temperature.
For spintronics application it is desirable to combine the half-metallicoxides with semiconductor devices. Lu et al. [91] grew Fe3O4(001) onGaAs(100): initially a bcc epitaxial Fe-layer was grown on GaAs(100), sup-pressing the formation of secondry phases (e.g. FeAs) to avoid the develop-ment of a magnetically dead interface layer. Subsequently the Fe-film wasoxidized at pO2 = 5 × 10−5 mbar. An aspect that needs further investigationis whether the half-metallic behavior is preserved at the interface to MgO(001)or GaAs(001).
For the growth of Fe3O4(111), different substrates have been used, e.g. MgO(111) [94, 95] and Al2O3 [96, 97], as well as metallic Pt(111). Weiss et al. [98]repeatedly deposited and oxidized layers of iron on Pt(111). The LEED anal-ysis of well ordered films suggested a termination with 1/4 monolayer of Feover a distorted hexagonal oxygen layer. Dedkov et al. oxidized a Fe(110)film grown on W(110) [99]. Depending on the oxygen pressure and the postgrowth annealing procedures, lattice parameters corresponding to the forma-tion of FeO and Fe3O4(111) were obtained. A FeO(111)-surface was formedafter 100 L oxygen exposure and post-annealing at 525 K. The FeO-film wastransformed into a Fe3O4(111) film after subsequent exposure to 200 L oxy-gen and post-annealing at 525 K. Alternatively, Fe3O4(111) was obtained af-ter an extended exposure to 900 L oxygen. Fonin et al. [93] grew Fe3O4(111)by oxidizing a Fe(110)-film grown on Mo(110)/Al2O3(1120) at 700◦C andpO2 = 5×10−6 mbar. A TEM cross section of the film demonstrating the fourdifferent regions of the sample with sharp interfaces is shown in Fig. 1.13a). Alower spin-polarization of 60% was measured for this sample, as compared tothe nearly fully spin-polarized (80%) one grown on W(110) [99]. This result
24 B. Hjorvarsson and R. Pentcheva
is consistent with DFT calculations [100] of bulk magnetite showing that uni-axial strain reduces the degree of spin-polarization.
A 100 A thick Fe3O4(111)-film was grown on Al2O3(0001) [97] by co-deposition of Fe from an effusion cell and atomic O using a plasma source, ata substrate temperature of 450 K and postannealing at 900 K. Such interfacesare interesting as magnetic tunneling junctions (MTJ). For Fe3O4(111) filmsgrown on an (1 × 1)-OH terminated MgO(111)-surface [94] and Al2O3(0001)[96] a phase separation in Fe and FeO nanoinclusions were observed at the in-terface. Unlike MgO(100), the MgO(111)-surface is polar. Substrate polaritywas identified as the driving mechanism towards phase separation at the inter-face. In regions between the Fe crystalites atomically abrupt interfaces wereobserved. A HRTEM image is shown in Fig. 1.13. DFT-GGA calculations [95]find that these are stabilized through electronic screening and metallization atthe interface in contrast to the stoichiometry change expected from classicalelectrostatic models.
CrO2
Since the prediction of half metallic behavior of CrO2 [101], this oxide has at-tracted attention as a potential material for spintronic devices. Unfortunately,rutile CrO2 is unstable at room temperature and transforms irreversibly into
a)a) b)b)
MgO
Fig. 1.13. TEM cross section micrograph of a) the Fe3O4(111)/Fe(110/Mo(110)/Al2O3(1120) system [93] with sharp interfaces and b) Fe3O4(111) grown onMgO(111) showing the formation of Fe(110) nanocrystals at the interface [94]
1 Modern Growth Problems and Growth Techniques 25
Cr2O3, which is an insulating antiferromagnet. In order to stabilize the rutilestructure, epitaxial films were grown on TiO2(001) and Al2O3(0001) usingchemical vapor deposition CVD [102]. On Al2O3(0001) a 400 A Cr2O3-layeris formed prior to the growth of CrO2(001). A 1000 A thick CrO2(001) film ofgood crystal quality was obtained on TiO2 with no Cr2O3 formation. The mag-netic ordering temperature of this film was 385 K. Heterostructures based onthe insulating TiO2 as a barrier are interesting for magnetic tunnel junctions.On the other hand, metallic RuO2-spacers are interesting as GMR-elements.CrO2 (a = 4.421 A, c = 2.916 A) and RuO2 (a = 4.499 A, c = 3.107 A)have also a good lattice match. Upon deposition of RuO2 on CrO2/TiO2 theCr2O3 termination of the surface is transformed to CrO2 but despite the re-stored conductivity a relatively low magnetoresistence of the sample indicatessusbtantial chemical and magnetic disorder associated with this transforma-tion [103]. Besides TiO2 and RuO2, also SnO2 was used as a substrate, howeverthe measured magnetoresistance was still relatively low.
w-NiO
Analogous to the growth of ferromagnetic materials on a non-magnetic sub-strate, an attractive possibility is opened by the growth of oxides on semicon-ductors where the oxide adopts the structure of the substrate, which does notexist in the bulk. Recently, Wu et al. [104] predicted, using LDA+U calcula-tions, that NiO in the wurzite structure (w-NiO) should be halfmetallic andon substrates like ZnO or GaN the ferromagnetic coupling should be morestable than the antiferromagnetic.
1.3.2 Ferromagnetic Semiconductors
The design of semiconductors with magnetic and/or spin-related propertiesand high Curie temperature for spintronic devices is a demanding and by farnot completely resolved issue. In this subsection we will briefly summarizethe current knowledge on one of the most intensively studied III-V semicon-ductors, Mn doped GaAs (TC ∝ 170 K), and then discuss several systemswhere the prediction of room temperature ferromagnetism (e.g. Co:TiO2, p-type Mn:ZnO, Mn:GaN) has envigorated a lot of research in the last years.A major issue in the fabrication of doped semiconductors is the incorpora-tion of dopants in the lattice and whether a homogeneous distribution can beachieved.
Mn:GaAs
Unlike II-VI semiconductors where there is no solubility limit for 3d dopands,the solubility limit is very low for III-V systems (e.g. 0.1% for Mn). Thereforea secondary phase like MnAs is formed under typical growth conditions. Toavoid this, MBE growth is performed at low temperatures (LT-MBE) of 473–573 K. The Mn ions primarily occupy cation sites (MnGa) where they act as
26 B. Hjorvarsson and R. Pentcheva
acceptors introducing both magnetic moments and holes. The current under-standing of the underlying mechanism is that of hole-mediated ferromagnetismin p-type materials. Since the growth proceeds at highly non-equilibrium con-ditions where kinetic effects dominate, there are two main types of defectsthat form and influence the magnetic properties and conductivity: the Asantisites (AsGa) and the Mn-interstitials (MnI). Both act as donors, i.e. re-duce the number of holes. Additionally, MnI couple antiferromagnetically toMnGa and thus suppress ferromagnetism. Yu et al. [105] provided evidencethat the reduction of TC is directly related to formation of MnI and the latterdepends on the doping of the barrier layer on which Mn:GaAs is deposited.Therefore the reduction of both types of defects is the main path towardsobtaining a higher TC. This is done through a control of the As flux dur-ing growth and a post-deposition low temperature (473–573 K) annealingstep. As shown by polarized neutron reflectometry on as grown and annealedsamples, annealing improves substantially the homogeneity of Mn and thusthe magnetic properties [106]. For further reading the reader is referred toseveral reviews e.g. [107, 108, 109].
Doped Thin Film Oxides
Co:TiO2: The interplay of growth conditions (temperature, deposition rate),but also the partial pressure of oxygen plays a decisive role on the quality ofthe samples. For example PLD growth of Co:TiO2 (anatase) from a mixedmetal oxide target may result in Co -nanoinclusions within the TiO2 matrixif the O2-pressure is too low or if the Co:Ti ratio is too high [110, 111, 112].A continuous epitaxial film with no signs for Co enrichment was obtained fordepostion rates of 0.1 A/s and T = 650◦C. On the other hand OPA-MBEmaterial is found to be FM at RT for xCo ≈ 5−7% (1.1 μB/Co) [113]. Similarmorphologies but a lower saturation magnetization of 0.6 μB/Co is obtainedfor Ar-sputtering of Ti and Co metal targets at T ≈ 650◦C using water as anoxidant [114]. The incorporation of hydrogen could be of importance in thiscontext, as discussed in Sect. 1.1.2 above.
Another issue, similar to the ones arising at oxide interfaces, is the mech-anism of charge compensation: e.g. in Co:TiO2 Co2+ substitutes for Ti4+.DFT-GGA calculations predict Co2+-segregation together with the forma-tion of an oxygen vacancy [115]. Annealing in vacuum is not likely to leadto Co-oxidation. For example, XANES measurements revealed metallic Co atT > 750◦C [116]. Post-growth annealing results in enrichment of Co at sur-faces, grain boundaries and interfaces. Also the origin of room temperatureferromagnetism is not well understood, e.g. in Co:TiO2 crystallographicallyperfect samples were obtained for T ≈ 550◦C and R = 0.014 A/s whichturned out to be nearly nonmagnetic [117]. This leads to the assumption thatdefect formation at surfaces and interfaces plays a significant role in triggeringmagnetism.
1 Modern Growth Problems and Growth Techniques 27
Cr:TiO2: PLD growth of Cr: TiO2 (rutile) on Al2O3(012) resulted inconducting films (reflecting a finite density of O-vacancies) and a saturationmoment of 2.9 μB/Co for x = 0.07. OPAMBE films grown on TiO2 (110) wereon the other hand insulating and nonmagnetic. XPS showed that Cr3+ substi-tutes for Ti4+. Post growth annealing in vacuum reduced the film and madeit n-type, showing weak room temperature ferromagnetism with 1 μB/Co.
Cr:TiO2 (anatase): LaAlO3 or SrTiO3 were chosen as substrates forOPAMBE growth. The in-plane lattice mismatch between anatase and theperovskite structure along (001)-direction differs by an order of magnitudefor LaAlO3 and SrTiO3(–0.26 vs. –3.1%), respectively. For R ≈ 0.1 A/s andT ≈ 650◦C the as grown films exhibited room temperature ferromagnetismwhich was enhanced after annealing in vacuum. The films exhibit a high Curietemperature (Tc=690 K). An improved crystalline quality is obtained forR ≈ 0.015 A/s and T ≈ 550◦C [117] but again as for Co:TiO2 magneticproperties deteriorate. In summary, room temperature ferromagnetism in Coor Cr:TiO2 appears to be driven by defects and is not an intrinsic prop-erty of the material.
Co:ZnO: The appearance of room temperature ferromagnetism in Co:ZnOdepends critically on electron doping. Epitaxial films of Co:ZnO grown onAl2O3(012) by MOCVD [118] were found paramagnetic and insulating. How-ever, the interdiffusion of atomic Zn in these samples (Zn occupies interstitialsites) results in a weakly ferromagnetic semiconducting sample [119].
Ti:Fe2O3: A nontraditional candidate for room temperature ferromag-netism is Ti doped Fe2O3. The host is a (canted) antiferromagnetic insula-tor at room temperature, but the incorporation of Ti in the lattice is ex-pected to lead to ferrimagnetism. Chambers and collaborators [116, 120] grewa 710 A α-Fe2O3-film on Al2O3 using a 130 A thick buffer layer of Cr2O3 toreduce the in-plance lattice mismatch (5.8%). The OPAMBE growth was per-formed at Tsub = 550◦C with growth rate of 0.25 A/s at an oxygen pressure of1.5× 10−5 mbar. XRD measurements suggested a high degree of crystallinity.The magnetic signal depends sensitively on whether Ti is incorporated onone spin sublattice or randomly distributed on both spin sublattices. Themeasured saturation magnetization is however much lower (approximately0.5 μB/Ti) than the expected 5 μB/Ti indicating that only about 12% of Ticontributes to a FM ordered phase and the the majority of Ti is randomlydistributed in the lattice.
1.4 Multilayers and Superlattices
1.4.1 General Considerations
A multilayer is a general term describing one dimensional variation in compo-sition. A multilayer can be single crystalline, polycrystalline, amorphous or acombination of (poly-)crystalline and amorphous. When a multilayer is singlecrystalline and has many repetitions, it is denoted superlattice, see Fig. 1.14.
28 B. Hjorvarsson and R. Pentcheva
:!
Lc
LSL
c
Substrate
Capping
Seed layer
LA
LB
Fig. 1.14. Illustration of a typical superlattice structure. A seed layer is grown ona substrate, followed by the growth of the superlattice. The structure is thereaftertypically covered by a capping layer, hindering the deterioration of the superlattice
A superlattice can therefore be thought of as a single crystal multilayer, withwell defined atomic distances as well as chemical repeat distance (Λ). Λ de-fines the unit cell in the growth direction, where Λ=La+Lb where La and Lb
are the thicknesses of layer a and b, respectively. The variation in the chemicalcomposition is a route to create a modulation in the electronic states, formingnew material classes with unique properties. However, formation of a super-lattice is only the first step. Altering the chemical composition in 3 dimensionswould allow much larger degree of freedom, forming unique electronic statesdefined by the extension and the compositional variation in the material. Thiscan be viewed as the ultimate task of materials processing of today.
The growth of superlattices bears large similarities to the growth of singlefilms. However, there are also significant differences both with respect to thegrowth procedures as well as the analysis of the samples. The inherent latticeparameters of the constituents have a special role, which is often used tojudge the possibility for the growth of high quality superattices. The basicideas resemble in many ways the criteria for the growth of single films on asubstrate, as will be apparent below.
1.4.2 Metallic Superlattices
We will use the combination of Fe, Mo and V as examples for the possibilitiesand limitations of the growth of metallic superlattices. All these elements arebcc with a bulk lattice parameter of 2.86, 3.16 and 3.02 A, respectively andcan therefore, in principle, form congruent single crystals. The difference inthe lattice parameter of Fe and V is 5%, Mo and V is 4% and finally Fe and
1 Modern Growth Problems and Growth Techniques 29
Mo it is close to 10%. The initial growth of V on Fe has been investigatedusing RHEED [121] following the in-plane lattice parameter changes of theV, in a wide temperature range. A 200 ML Fe (001) layer was initially grownon a MgO(001) substrate, on which the V was deposited. Between 300 and800 K, RHEED oscillations were observed up to 9 ML, consistent with a welldefined layer by layer growth of V. However, the in-plane lattice parameterat the V surface was observed to relax from that of Fe when the number of Vlayers exceeded 7 ML. At 800 K, no RHEED oscillations were observed, whichindicates the presence of strong island formation and an upper boundary for awell defined growth. Thus, it is possible to grow V on Fe up to 7 ML, althoughthe difference of the lattice parameter is as large as 5%. When the number ofrepeats of the superlattice period is large, the average in-plane lattice param-eter reflects the thickness ratio of the constituents. Thus, when growing e.g.Fe/V superlattices, the average in-plane lattice parameter can be estimateddirectly from the thickness ratio, as the elastic constants of Fe and V are rathersimilar. For example, when the layers have equal thicknesses (LFe=LV), theaverage in-plane lattice parameter will be close to the average lattice param-eter of the constituents. This reduces the in-plane strain from 5 to below 3%,which should be reflected in an increased critical thickness for the growth ofcoherent layers. This was observed in RHEED investigations of Fe(3)/V(x)(001) superlattices in which the critical layer thickness was determined tobe around 16 ML [122]. The growth of Fe/V(001) superlattices has beenthoroughly investigated by a number of authors (see for example [123, 124]).The temperature dependence of the growth on MgO (001) was established byIsberg et al. [124] where the quality of the SL were shown to depend stronglyon the growth temperature. The best crystalline quality was obtained witha substrate temperature in the temperature range 570–600 K. The thicknessvariation of the layers was also at its best in the same range and was deter-mined to be 1 A. This variation in layer thickness represents the lower limitfor a non phase locked growth, resulting in incomplete formation of the indi-vidual Fe and V layers. Thickness variation corresponding to one monolayerof both the Fe and V layers is thus inevitable. Representative X-ray resultsare shown in Figs. 1.15 and 1.16.
The optimal growth temperature of Mo/V superlattices is close to 1000 K,which is substantially higher than for Fe/V superlattices. This correlates withthe substantially higher melting temperature of Mo compared to Fe. Thegrowth of this material combination was pioneered by the group of Fisher[125, 126, 127], demonstrating both a (001) and (110) growth on MgO(001)and Al2O3(1120) respectively. The influence of the critical thickness of thelayers in Mo/V superlattices on the superconducting properties was discoveredby Karkut et al. [125]. A critical thickness of 16 ML was inferred for boththe (001) and (110) grown Mo/V superlattices, when the ratio of the layerthicknesses was close to unity. The difference of the lattice parameters of Moand Fe with respect to V is similar, but with different sign. Thus, a comparablecritical thickness for epitaxial growth is observed for compressive and tensile
30 B. Hjorvarsson and R. Pentcheva
Fig. 1.15. Representative reflectivity data from a Fe/V(001) superlattice with Λ =25.1 A. From [122]
biaxial strain in V, as expected from symmetry reasons. A compressive strainin one layer is balanced by a tensile strain in the second. Thus, the sign of thestrain appears to be irrelevant, and a hint of the relation between the latticemismatch and the critical thickness of the layers emerges.
Above the critical thickness, the formation of dislocations and other de-fects results in buckling and increases therefore the variation in the layerthicknesses [126, 127, 128]. This leads to relaxation of the in plane lattice pa-rameter, where the variation is accomplished by the presence of defects. Thethickness variation is governed by the V layers, because V atoms have largersurface mobility at the actual growth temperature [128].
Fig. 1.16. Representative diffraction data from a Fe/V(001) superlattice with Λ =25.1 A˙ The inset highlights the crystalline quality of the superlattice structure. Thepresence of Laue oscillations implies a interference between the scattering from thefirst and the last monolayer of the Fe/V(001) stack. From [122]
1 Modern Growth Problems and Growth Techniques 31
Let us now consider the growth of Fe on single crystal Mo(110), with alattice mismatch of close to 10%. The growth results in a complicated defectgeneration already in the first monolayer. The first layer grows pseudomor-phically [129] followed by a pronounced relaxation in the third layer. Thus,the difference in the lattice parameters of Fe and Mo appear to be beyond thelimit for epitaxial growth. The growth and characterization of Fe/Mo multi-layers is described in the literature [130, 131], but no reports are found onsuccesful growth of Fe/Mo superlattices.
The choice of growth temperature is a compromise between two con-straints, surface mobility and interdiffusion. Surface mobility of adatoms isincreased with increased temperature, but so is the interdiffusion. This limi-tation is clearly seen in many of the transition metal superlattices and consti-tutes therefore a substantial challenge for optimizing the growth. One routeto circumvent this limitation is the use of surfactants. The basic idea isto decrease the activation energy of the surface diffusion and thereby in-creasing the surface mobility. As seen in (1.2) the weight of the change inactivation energy (Ed) has the same influence on the diffusion rate (D) asthe change in temperature.
One possible surfactant is hydrogen. The influence of hydrogen on thediffusion of Pt adatoms on Pt(111) was investigated by STM [133]. A clearincrease of the diffusion rate was demonstrated. However, the presence of hy-drogen has also been found to inhibit surface mobility [134]. The main benefitof the use of hydrogen in this context, is the compatibility with the vacuumprocesses. Hydrogen is simply removed by evacuation from the depositionchamber. The use of hydrogen as a surfactant was recently demonstrated byRemhof et al. [132], where a substantial increase of the quality of Fe/V(001)superlattices was obtained. Representative x-ray reflectivity results are illus-trated in Fig. 1.17.
Co/Cu
Superlattices of Co/Cu have been widely studied during the last decades.Successful growth of an epitaxial Co/Cu on a single crystal Cu(001) underUHV conditions was demonstrated by Cebollada et al. [135]. The Cu sub-strates were cut from a single crystal bar and oriented within 0.3◦of the [001]direction using Laue diffraction. The substrates were cleaned in-situ by cyclesof Ar+ sputtering and annealing to 1000 K. The composition of the sur-face was investigated using Auger electron spectroscopy and the crystallinequality was investigated by thermal-energy atom scattering (TEAS). The re-sulting surface consisted of, on average, 300 A wide flat terraces separated bymonoatomic steps. The authors used extremely low deposition rate, 0.01 A/sas compared to few A/s, to reduce the amount of imperfections. The thick-ness of the evolving layers was monitored and determined by counting thenumber of oscillations in the TEAS signal. The thickness obtained this waywas confirmed by calibrated quartz balance. The samples were all covered by
32 B. Hjorvarsson and R. Pentcheva
Fig. 1.17. Small angle x-ray reflectivity scans recorded at E = 8.048 keV. The uppercurve displays the reflectivity of the sample sputter deposited at pH2 = 2×10−6 mbarat T = 320◦C. The lower curve shows the reflectivity of a reference sample grownwithout the presence of hydrogen. The scans are shifted along the intensity axisfor clarity. The inset shows the diffuse scattering recorded at the first superlatticepeak [132]
1000 A Cu, to hinder oxidation of the underlying superlattice. The quality ofthe resulting structure was established by neutron reflectivity and diffraction.
Concerning the growth of Co/Cu superlattices, we tie up to discussion ofthe initial growth of Co on Cu(001) in Sect. 1.2.1. The lattice parameter of fccCo at room temperature is 3.548 A, while the lattice parameter of Cu is 3.615A under the same conditions. The lattice mismatch is therefore slightly below2%. When growing thin Co layers on Cu(001) single crystal, the Co adaptsthe in-plane lattice parameter of Cu. This results in a tetragonal distortion ofthe fcc lattice, where the out-of-plane lattice parameter is contracted and thein plane lattice parameter is expanded, as compared to the bulk value [57].The restoring force at the interface between the Co and the Cu substrate isfixed but the strain energy associated to the tetragonal distortion increaseslinearly with the thickness of the Co film. Thus, at some critical thickness, itwill be energetically more favorable to form defects, relieving the strain andforming a non distorted fcc lattice. In the LEED study of Navas et al. [136]Co was found to grow in the strained state up to at least 10 monolayers. Theinherent electronic structure of the Co layers is therefore significantly differentfrom bulk Co. The relation between the structural and magnetic properties ofsingle Co films and Co/Cu superlattices is discussed by de Miquel et al. [137].
The subsequent growth of Cu on the thin Co films retained the bulk latticeparameter of Cu. Thus, while growing Co/Cu(001) superlattices on a Cu(001)
1 Modern Growth Problems and Growth Techniques 33
substrate, only the Co layer should exhibit a tetragonal distortion, as long asthe superlattice is grown in a coherent mode.
Fe/Cr
The work on Fe/Cr superlattices was pioneered using evaporation techniques.For example, Etienne and collaborators used effusion cells for the growth ofboth Fe and Cr on GaAs under UHV conditions [138]. The surface structureof the substrate was improved by growing a 3000–5000 A GaAs(001) bufferat ∼ 1000 K. The surface reconstructed from 2 × 4 to 2 × 6 after a briefGa-exposure at 725 K. The metallic superlattices were grown on such bufferlayers, starting with Fe. The growth temperatures of the superlattice werein the range 225−325 K, the drift reflecting the thermal balance with thecooling and heating devices in the MBE system. The presence of layering wasconfirmed by a combination of Auger spectroscopy and sputtering and thesurface crystallinity was investigated using RHEED. The chemical purity wasinvestigated by Auger spectroscopy and the samples were found to be free ofboth oxygen and carbon. The authors did not present any x-ray data. Theresulting giant magnetoresistance (GMR) of these structures was in the rangeof 100%, taking the high field resistance as a reference.
Epitaxial growth of Fe/Cr(001) by sputtering on MgO(001) was laterdemonstrated by Fullerton et al. [139]. In this case, the authors used a Crbuffer layer grown at 873 K which leads to an improved surface flatness andwetting of the initial layers of the superlattice. The bcc Cr is rotated withrespect to the MgO substrate in the same way as discussed for Fe and V onMgO. The Fe/Cr superlattice was grown at ∼450 K and the resulting struc-ture was investigated by x-ray reflectivity and diffraction. Typical rockingcurves of the (002) diffraction peak yielded a FWHM of 0.7◦ and a FWHMof 0.2◦ in 2θ, using Cu Kα radiation. The authors did not dwell on the choiceof growth temperatures. This approach resulted in a 150% GMR, which issubstantially higher than obtained by Etienne and collaborators.
The structural quality is of large importance for the resulting physicalproperties, such as GMR. This is clearly seen when comparing the results fromsingle crystal structures with polycrystalline samples. Parkin and York [140]grew polycrystalline Fe/Cr multilayers by sputtering using Si substrates. Thestructure was a combination of (110) and (001) textured crystallites, of whichthe (001) increased in weight with increasing temperature in the range 300 Kto 475 K. These samples yielded a maximum 40% GMR effect and the resis-tivity of the optimized samples was 4.4 μΩ cm at 4.2 K, as compared with14 μΩ cm for single crystal superlattices [139].
The growth of Fe/Cr(001) on SrTiO3 using evaporation, was discussedby Ono and Shinjo [141]. The crystalline quality was substantially worse ascompared with the results of Fullerton et al. [139], although great care wastaken concerning the surface flatness of the substrates. Chemical etching wasused to obtain better flatness and in-situ RHEED was used to investigate
34 B. Hjorvarsson and R. Pentcheva
the substrate as well as the surface quality of the resulting film. A stronginfluence of the substrate quality on the GMR was obtained, the chemicallyetched substrate yielded twice as large values for GMR.
The x-ray contrast between Fe and Cr is poor due to the similarity inthe electron density of these elements. This was a major obstacle for ob-taining higher order Fourier components in, for example, x-ray reflectivitymeasurements. Bai et al. overcame this difficulty using resonant scatteringtechniques [142], allowing detailed simulations of the composition profile aswell as the roughness of the samples. The experiments were performed onsputtered superlattices, using MgO(001) as a substrate.
Rare Earth Superlattices
The main part of the magnetic moment in Rare Earth materials (RE, in shortfor actinides and lanthanides) stems from the localized f-electrons. In contrastto itinerant magnets, due to this localization one can often safely ignore theeffect of hybridization on the magnetic moment. Kwo et al. [65] demonstratedthe growth of high quality Gd/Y superlattices and related the changes in mo-ment and ordering temperature to the repeat distance in the samples. Thegrowth of the superlattices was done in an UHV MBE system, with a basepressure in the 10−11 mbar range. A buffer layer of Nb (011) was grown onan Al2O3(1120) at ≈1170 K, hindering the transport of oxygen from the sub-strate to the RE film. A seeding layer of Y(0001) was subsequently grownat ≈970 K, resulting in a flat surface, enabling the growth of high qualityGd/Y(0001) superlattices at ≈470 K. Lower growth temperature is chosenfor the superlattice to limit the interdiffusion of the constituents. An alloyinterface region of 2 monolayers was established by combined x-ray and mag-netic analysis. The interface region was inferred to be a GdY alloy, withoutmagnetic ordering, even at low temperatures. The magnetic susceptibility islarge, which confirms the proposed model of the compositional modulation.The growth and characterization of other RE superlattices was establishedby a number of groups most of which followed the ideas of Kwo et al. withrespect to diffusion barrier for oxygen and a seeding layer. McMorrow andcollaborators investigated the chemical structure of Ho/Lu and Ho/Y super-lattices [143] using high resolution x-ray scattering to address the nature ofthe interface imperfections. By detailed investigations of the shape of the su-perlattice Bragg peaks, the presence of conformal roughness of the interfaceswas established. The basic idea behind these investigations was to establishnot only the type of roughness, but to separate roughness and intermixing.This is of primary interest as the physical properties are highly dependenton the type of interface imperfections. Intermixing can for example resultin an absence of ferromagnetic ordering as discussed above [65], while bothconformal and uncorrelated roughness will give rise to local anisotropy fields.A thorough description of the growth and structural characterization of rareearth superlattices is given by Majkrzak et al. [144].
1 Modern Growth Problems and Growth Techniques 35
1.4.3 Metal-Oxide Superlattices and Magnetic Tunnel Junctions
Magnetic tunnel junctions containing an oxide barrier sandwiched betweentwo ferromagnetic layers are a prototypic system to achieve high tunnel mag-netoresistance (TMR) values. In this subsection we discuss several issues con-cerning the quality of the interface that have impact on the measured TMRvalue. For structurally perfect interfaces a TMR value of several 1000% waspredicted theoretically for Fe/MgO/Fe(001) [145]. Experimentally, TMR val-ues up to 188% at room temperature (RT) were achieved [146] for a junctionwhere Fe was grown using MBE. The MgO barrier was epitaxially grown us-ing electron-beam evaporation of a stoichiometric source material. To avoidthe oxdation of the bottom Fe layer the first MgO layer was deposited atpO2 = 2.5 × 10−5 mbar. The top Fe electrode was deposited at 473 K sub-strate temperature. A measured asymmetric current-voltage characteristicwas attributed to an assymmetry of the interface structure. This is rational-ized by thermodynamic arguments (cf. (1.1): the lower surface energy of MgO(1.1 J/m2) versus Fe (2.9 J/m2)) suggests layer-by-layer growth of MgO on Febut not vice versa. Tusche et al. [147] found a substantial effect of the oxygenatmosphere on the quality of the interface. Starting with 2 MLs of MgO de-posited on Fe(001) at a rate of R=0.125 ML/min by electron bombardment ofa polycrystalline MgO rod under UHV conditions they deposited 8 ML of Feusing MBE at R=0.25 ML/min. In the sample where the Fe film was depositedat UHV conditions, 30% of the interface layer was found to be FeO with asubsequent disordered Fe layer. In a second sample where the initial 0.5 MLFe were deposited in oxygen pressure of 10−7mbar, surface x-ray diffraction(SXRD) measurements and quantitative analysis revealed that a coherent Felayer is formed attributed to the formation of a nearly complete FeO layer be-tween the MgO spacer and the Fe film. In a thermodynamic picture, the role ofthe FeO layer is understood as to reduce the interface energy (cf. (1.1)). DFTcalculations found strong dependence of the transport properties on the struc-ture of the interface [148]. Only a symmetric Fe/FeO/MgO/FeO/Fe junctionwas predicted to give rise to a giant TMR [147].
Parkin and coworkers measured 220% TMR at RT in a FeCo(001)/MgO(001)/(Fe70Co30)80B20 sample where the bottom electrode was a poly-crystalline bcc FeCo-layer with a (001)-texture [149]. Djayaprawira et al. [150]found a significantly higher RT TMR using amorphous CoFeB electrodes ascompared to polycrystalline CoFe (values of TMR were 230% vs. 62%). Inthis experiment the metal reference and the free layer were deposited by dcmagnetron sputtering, while rf sputtering was used for the MgO film. Af-ter annealing at 593 K a partial crystallization at the interface was observedin HRTEM. Microstructural analysis of the MgO(001) spacer shows a goodcrystal quality with a fibre texture.
The role of the recrystallization after annealing and the distribution of Bis still not well understood. First principles calculations find a preference forB to reside at the interface [151], which is expected to suppress the TMR.
36 B. Hjorvarsson and R. Pentcheva
The authors conclude that inhibiting B segregation at the interface duringprocessing is likely to enhance TMR. X-ray photoemission studies [152] onCoFeB/MgO bilayers find evidence for CoFeB oxidation during MgO depo-sition, while annealing in vacuum leads to B interdiffusion into MgO andMgBxOy formation. To avoid this a Mg-buffer layer is introduced betweenCoFeB and MgO.
A Co/MgO/Co MTJ where Co is stabilized in the bcc structure was pre-dicted from DFT to have an even higher MR than Fe/MgO/Fe [153]. In-deed, MBE grown Co-based MTJ showed a MR of 410% [154]. To retain themetastable bcc structure the thickness of the Co layers on both sides of theinsulating MgO barrier were limited to 4 ML and grown at RT. The authorsreported that Co does not wet the MgO(001) spacer but grows in a 3D man-ner. The subsequent annealing step at 525 K for 30 min was used to improvethe crystalinity of the sample.
Oxide Superlattices
Transition metal oxide superlattices open new possibilities to make artificialmaterials with magnetic and electronic properties that differ from the bulkcomponents. Analogous to oxide surfaces (cf. Sect. 1.3.1) the question of po-larity and disruption of charge neutrality arises also at oxide/oxide interfaces.For example perovskites possess a natural charge modulation in the [001]-direction, e.g. in LaTiO3 positively charged (LaO)+ alternate with negativelycharged (TiO2)−, while in SrTiO3 both the SrO and TiO2-layers are neutral.Thus the interface between these two insulators represents a simple realiza-tion of a polar discontinuity. Using PLD from a single crystal STO targetand a polycrystalline La2Ti2O7, Ohtomo et al. [155] fabricated superlatticesof LaTiO3 and SrTiO3 with an atomically controlled number of layers of eachmaterial which therefore are refered to as ”digital”. The films were grown at970 K at an oxygen pressure of ∼ 10−5 mbar to stabilize both valence statesof Ti and subsequently annealed at 670 K to fill residual oxygen vacancies.An anular dark field TEM image is shown in Fig. 1.18. RHEED oscillationswere used to monitor growth. Ohtomo et al. found that although the parentcompounds are a Mott and a band insulator respectively, the heterostructureis conducting with electron energy loss spectra suggesting mixed Ti-valencein the interface region. Based on Hubbard models, Okamoto and Millis [156]proposed an “electronic reconstructuion” of this interface.
Another system showing unexpected behavior are heterostructures of thetwo simple band insulators LaAlO3 and SrTiO3. Here, both the A and B sub-lattice cations in the perovskite structure change across the interface givingrise to two different types of interfaces: an n-type between a LaO and a TiO2-layer that was found conducting with a high electron mobility and a p-typebetween a SrO and an AlO2-layer that showed insulating behavior despitethe charge mismatch [157]. Using PLD, the n-type LAO/STO IF was grown
1 Modern Growth Problems and Growth Techniques 37
Ti3+
Ti4+
a) b)
Fig. 1.18. a) an anular dark field STEM image of bright LTO layers in a STOhost. The boosted up view above shows a 1× 5 LTO/STO superlattice [155]; b) 45◦
checkerboard charge density distribution of the occupied 3d states in the charge andorbitally ordered TiO2 layer at the LTO/STO-IF. The positions of O-, Ti3+ andTi4+-ions are marked by white, black and grey circles, respectively [159]
on a TiO2-terminated SrTiO3-substrate. To grow a p-type LAO/STO inter-face a SrO-layer was deposited on the SrTiO3-substrate prior to growth ofLaAlO3. The oxygen pressure is quite an important growth parameter in thesesystems that controls the oxygen stoichiometry and the underlying proper-ties, e.g. STO alone can change from a wide band insulator to a metal withpO2 [20]. Nakagawa, Hwang and Muller [158] discussed recently that the p-typeLAO/STO IF is ionically stabilized with an enhanced roughness attributedto oxygen vacancies while the n-type interface is electronically stabilized andhence sharp.
In correlated materials with multivalent ions correlation driven charge or-der offers an additional degree of freedom to accommodate the charge imbal-ance. To this end LDA+U calculations predict a compensation mechanismby a charge disproportionation: a charge and orbitally ordered IF-layer isfound for the LTO/STO and the n-type LAO/STO interface with Ti3+ andTi4+ ordered in a checkerboard manner [159, 160]. Such a correlation drivencompensation mechanism is not present e.g. at polar semiconductor interfaces.Moreover, although both LaAlO3 and SrTiO3 are nonmagnetic and LaTiO3 isan antiferromagnet of G-type, the diluted layer of Ti3+ in the IF layer has aslight preference to couple antiferromagnetically with a magnetic moment of0.71 μB. Recent experiments give first indication for localized magnetic mo-ments at the n-type LAO/STO interface [161]. Thus the violation of chargeneutrality at interfaces of transition metal oxides can be used to generatenovel charge and magnetically ordered phases that do not exist in the bulk.
38 B. Hjorvarsson and R. Pentcheva
1.5 Conclusions and Outlook
Growth of thin films, multilayers and superlattices has provided extremelyfruitful routes to realize systems with novel structural and physical propertiesdistinct from the bulk phases. Within the field of magnetism, thin film process-ing is the single most important factor for the rapid development during thelast decades, including the discovery of oscillatory exchange coupling, giantmagneto resistance and tunneling magneto resistance. Thin film technologywill certainly continue to be of large importance, not least for the processingof materials for patterning devices, addressing both fundamental and appliedresearch questions. In this chapter we have focussed on the modulation of thechemical composition in one dimension, the remaining two define the finalsteps in the current paradigm of materials growth. Self assembly and orga-nization which is certainly another important route to create materials withunusual properties goes beyond the scope of this chapter.
We would like to summarize here some aspects we consider important inthe growth of magnetic heterostructures: the window of optimal growth con-ditions in heteroepitaxial growth is much narrower and thus more challengingthan the homoepitaxial case as higher tempertures lead to intermixing anda substantial interface roughness even for systems that are immiscible in thebulk. When the bonding to the substrate is weak (as on nobel metal sub-strates) ultrathin magnetic films may experience a much stronger strain thanwhat the lattice mismatch of the bulk phases would suggest. This may have astrong effect on lattice relaxations and the strain field in small islands (meso-scopic strain). The growth of oxide films and heterostructures is intricatelyrelated to the question of surface and interface polarity. However, transitionmetal oxides provide a much richer variety of mechanisms to compensate ex-cess charges than e.g. semiconductor systems: besides atomic reconstruction,electron redistribution via metallization, lattice distortion or even charge dis-porportionation can lead to novel properties.
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2
Magnetic Anisotropy of Heterostructures
Jurgen Lindner and Michael Farle
Fachbereich Physik and Center for Nanointegration, Universitat Duisburg-Essen,Lotharstr. 1, 47048 Duisburg, [email protected]
Abstract. The chapter provides a detailed introduction to magnetic anisotropy offerromagnetic ultrathin films and its analysis by ferromagnetic resonance on a tuto-rial level. While the microscopic origins of the magnetic anisotropy as well as recentdevelopments in its theoretical description are shortly discussed, emphasis is put ona phenomenological description using the free energy of the system together withits symmetries. The formalism is used to describe ferromagnetic resonance exper-iments which present an extremely sensitive method to experimentally investigatemagnetic anisotropy in thin film heterostructures. Expressions for the free energyand the resonance equations are derived for the most widely used crystal symmetriessuch as cubic, tetragonal and hexagonal. The general equations are illustrated bygiving selected examples of current research on thin metallic films on different kindsof substrates (MgO, GaAs and Cu).
2.1 Introduction
Magnetic thin films have provided a highly successful test ground for un-derstanding the microscopic mechanisms which determine macroscopicallyobservable quantitities like the magnetization vector, different types of mag-netic order (ferro-, ferri- and antiferromagnetism), magnetic anisotropy andordering temperatures (Curie, Neel temperature). The success has beenbased on the simultaneous development of the following techniques: a) thepreparation of single-crystalline mono- and multilayers on different types ofsubstrates in ultrahigh vacuum systems, b) the development of vacuum com-patible, monolayer-sensitive magnetic analysis techniques, c) the advance incomputing power to provide first-principles calculations of magnetic groundstate properties [1]. Aside from these basic research orientated investigationsthe technological exploitation of thin film magnetism has lead to huge in-creases in the hard disk’s magnetic data storage capacities [2] and new typesof magneto-resistive angle and position sensitive sensors in the automotiveindustry, for example.
J. Lindner and M. Farle: Magnetic Anisotropy of Heterostructures, STMP 227, 45–96 (2007)
The purpose of this article is twofold: a) an introductory level overview onmagnetic anisotropy, b) characteristic examples of current research using mag-netic resonance techniques to quantitatively determine magnetic anisotropyand explore its microscopic sources.Various aspects of ultrathin film magnetism [3] have been discussed in ex-tensive reviews and book chapters over the last few years (see for example[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). There is no way to summarize allthese issues in such limited space and the reader is referred to the reviewsmentioned before. For an overview on the technological relevance or the manyexperimental techniques and methods that have been developed to investi-gate magnetic heterostructures the reader is referred to the series of booksby Heinrich and Bland [6, 7, 8]. This review also excludes laterally structuredsamples (for excellent reviews see e.g. [9, 10, 11]) and epitaxially grown filmscomprised of two or more elements (double- tri- and multilayers, see the articleof B. Heinrich in this book or [6, 7, 8, 14]) in which coupling effects lead tophenomena like the tunneling magneto-resistance (TMR), the giant magneto-resistance (GMR) or spin current related effects (see e.g. [16, 17, 18] for adetailed discussion) such as current induced switching [19], current induceddomain wall movement or spin torque induced magnetic damping [20, 21, 22].The examples which will be discussed here are strictly restricted to the thick-ness and temperature dependent magnetic anisotropy of single element ferro-magnetic metallic monolayers on different kind of single crystalline substrates(metals, semiconductors and insulators). It will be shown that epitaxial filmsconsisting of few atomic layers provide an interesting playground for artifi-cially controlling magnetic properties and hence improving the understandingof the underlying physical mechanisms.This chapter is divided into two basic sections. While the first will shortly dis-cuss the sources of magnetic anisotropy energy (MAE), explain FerromagneticResonance (FMR) and introduce a phenomenological description of the MAEand its influence on the FMR resonance equations in terms of the magneticpart of the free energy of the system, the second section will give examples ofFMR investigations of heterostructures. In the framework of a tutorial descrip-tion, the prototype systems Fe/MgO(001), Fe/GaAs(001) and Ni/Cu(001)were chosen.
2.2 Origin of Magnetic Anisotropy
Magnetic anisotropy describes the fact that the energy of the ground state ofa magnetic system depends on the direction of the magnetization. The effectoccurs either by rotations of the magnetization vector with respect to theexternal shape of the specimen (shape anisotropy) or by rotations relative tothe crystallographic axes (intrinsic or magneto-crystalline anisotropy). Thedirection(s) with minimum energy, i.e. into which the magnetization pointsin the absence of external fields are called easy directions. The direction(s)
2 Magnetic Anisotropy of Heterostructures 47
with maximum energy are called hard direction. The MAE between twocrystallographic directions is given by the work WMAE needed to rotatethe magnetization from an easy direction into the other direction. TheMAE is a small contribution on the order of a few μeV/atom to the to-tal energy (several eV/atom) of a bulk crystal. To estimate the magnitudeof the MAE one can use as a rule of thumb, that the lower the sym-metry of the crystal or of the local electrostatic potential (crystal or lig-and field) around a magnetic moment, the larger the MAE is. This be-comes evident, if one remembers that in a crystal field of cubic symmetrythe orbital magnetic moment is completely quenched in first approximation[23]. Only by calculating in higher order (2nd) or by allowing a slight dis-tortion of the cubic crystal a small orbital magnetic moment, i.e. a non-vanishing expectation value of the orbital momentum’s z component is re-covered. Without the presence of the orbital momentum which couples thespin degrees of freedom to the spatial degrees of freedom the MAE wouldbe zero, since the exchange interaction is isotropic. One should also note,that the easy axis can deviate from crystallographic directions as for ex-ample in the case of Gd whose easy axis is temperature dependent andlies between the c-axis and the basal plane at T = 0 K [24]. Table 2.1gives an overview about easy and hard axes and on the magnitude of theMAE for some elementary ferromagnetic materials with different crystalsymmetry.There are fundamentally two sources of magnetic anisotropy: (i) spin-orbit(LS)interaction and (ii) the magnetic dipole-dipole interaction. From the pointof view of quantummechanics both interactions are relativistic correctionsto the Hamilton-operator of the system that lift the rotational invarianceof the quantization axis. Despite the fact that the dipole-dipole interactionas well as the LS coupling are much weaker than the exchange interaction(≈ 1 − 100 μeV/atom compared to ≈ 0.1 eV/atom), they link the magnetic
Table 2.1. Anisotropic orbital moments, direction of easy axis of the magnetization,and magnetic anisotropy energy at T = 0 K for the four elemental ferromagnets astaken from standard references like Landolt-Bornstein [32]. ΔμL is the difference ofthe orbital magnetic moment measured along the easy and hardest magnetizationdirection. μtot is the sum of orbital and effective spin moment. The latter includesthe so-called 〈Tz〉 contribution entering the sum rule analysis of x-ray magneticcircular dichroism measurements
Material Easy axis |MAE| (μeV/atom) ΔμL/μtot
bcc Fe 〈100〉 1.4 1.7 · 10−4
hcp Co 〈0001〉 65 4.5 · 10−4
fcc Co 〈111〉 1.8fcc Ni 〈111〉 2.7 1.8 · 10−4
hcp Gd tilted hcp 50 ≈ 10−3
48 J. Lindner and M. Farle
moments to position space. In the following we will use the term magneticanisotropy energy or MAE for the magnetic anisotropy energy density, re-sulting from spin-orbit interaction which includes the so-called magneto-crystalline and also the magneto-elastic contributions [15]. Contributions orig-inating from the magnetic dipole-dipole interaction will be called shape ormagneto-static anisotropy.We note that the important aspect of magnetic domain formation is not dis-cussed here, since we restrict our discussion to the intrinsic contributions tothe magnetic anisotropy and the main technique which will be discussed indetail in this chapter is ferromagnetic resonance (FMR). This technique is –in most cases – performed in large enough fields that drive the magnetic filminto a single domain state. The reader should keep in mind, however, that theinterplay of magnetostatic energy, exchange energy and magnetic anisotropyin general leads to an energetically favored multi-domain state. Especially,the analysis of hysteresis loops is complicated due to domain formation atsmall magnetic fields. Special imaging techniques have been developed [25] toobtain quantitative understanding of the many domain configurations. Due tothe single-domain description used in this chapter, aspects of configurationalmagnetic anisotropy [26] which appear in submicron sized nanomagnets dueto small deviations from the uniform state will not be discussed. Similarly,so-called exchange anisotropies which may arise from different exchange cou-pling constants along different crystallographic directions in a crystal will notbe specifically addressed, since the experimental observations can be well de-scribed by the phenomenological approach presented in Sect. 2.3.2. Also adiscussion of unidirectional anisotropy or exchange bias is beyond the scopeof this chapter. Excellent overviews can be found in [27, 28, 29, 30]. Finally, wenote that the aspect of a non- homogeneous magnetization across the thick-ness of a several nanometer thick film does not enter the following discussion.An excellent overview on the magnetization profile across a thin film and itsdependence on the film’s morphology has been recently given by Jensen andBennemann [31].
2.2.1 Spin-orbit Interaction
In a classical picture the orbital motion of the electrons in a perfect crystal isdefined by the potential that is predetermined by the crystal lattice. In casethat there is an interaction between the orbital motion and the spin of theelectrons (i.e. when spin-orbit interaction is present), the spins and thus themagnetization become coupled to the lattice. By using perturbation theory inwhich the LS coupling is described as perturbation of the exchange splittingBruno [33] showed that the energy correction and the orbital moment of theminority spins are related as ΔELS ∝ − 1
4ξS ·L↓. Here ξ is the radial part ofthe spin-orbit interaction, S is the unit vector along the spin direction, deter-mining the magnetization direction and L↓ the orbital angular momentum of
2 Magnetic Anisotropy of Heterostructures 49
the minority spin band, so that S ·L↓ is the projection of L↓ on the magneti-zation direction. The magnetic anisotropy energy for uniaxial symmetry, forwhich the energy differs between the directions parallel (‖) and perpendicular(⊥) to one anisotropy axis, is given by the anisotropy of the orbital angularmomentum ΔL↓ = L↓⊥ −L↓ ‖ (or, respectively, the anisotropy of the orbitalmoment Δμ↓
L = μ↓⊥L − μ↓ ‖
L ):
MAE = ΔE⊥LS − ΔE‖
LS ∝ − ξ4ΔL↓ = − ξ
4μB
Δμ↓L . (2.1)
According to Bruno the easy axis is the one, where the orbital moment islargest. This fact which is experimentally often overlooked yields informationon the intrinsic origin of the macroscopically measured MAE by straight-forward SQUID magnetometry measurements along different crystallographicaxes. The saturation magnetizations along the easy and hard axes are dif-ferent! The effect is very small, in bulk crystals–on the order of 10−4, butmeasurable, and well documented (see Table 2.1 and [34]). Here, one shouldnote that shape anisotropy is not involved. That is to say, that when takingthe shape anisotropy (see Sect. 2.3.2) into account, the equilibrium (easy)direction of magnetization in zero field may be a hard magnetocrystallineanisotropy direction with the smaller orbital moment. While Bruno assumeda fully occupied majority spin band in his model (exchange splitting muchlarger than the bandwidth), this restriction was dropped in the later work ofvan der Laan [35], who extended Bruno’s relation by including the majorityspin band orbital moment μ↑
L:
MAE ∝ − ξ
4μB
(Δμ↓
L − Δμ↑L
). (2.2)
Although approaches that employ perturbation theory have the advantageof being less complex, they often yield wrong results on a quantitative ba-sis (in most cases too large values). Ab initio theories that consider the LScoupling within a fully relativistic ansatz lead to a clear improvement. Theyare, however, much more elaborate as the precision of the calculation of thetotal energy of the system has to be very high. The reason is that the over-all total energy is of the order of 1 eV/atom, while the MAE is very muchsmaller and of the order of several μeV/atom. Nevertheless, a considerableprogress on ab-initio calculations of the MAE was achieved within the last10−15 years (see e.g. [31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]).The correlation between the anisotropy of the orbital angular momentum andthe MAE becomes strikingly evident in experimental results on ultrathin films[50] and few atom nanostructures of Co [51], for which orbital anisotropies upto ΔμL/μL ≈ 20% have been experimentally confirmed. In general, however,one should note that a direct proportionality between ΔμL and the MAE isonly correct for ΔμL → 0 [41].
50 J. Lindner and M. Farle
2.2.2 Dipole-Dipole Interaction
The magnetic field produced by a dipole μi at the position ri is given by:
Hi (ri) =3 (ri · μi) · ri
r5i− μi
r3i. (2.3)
Due to this field a second dipole μj in the distance rij with respect to thefirst has the energy EDip = −μj · Hi:
EDip =μi · μj
r3ij− 3 (rij · μi) · (rij · μj)
r5ij. (2.4)
Since the dipoles are placed on periodic positions within a crystal lattice, theaxis rij connecting the two dipoles is linked to the crystallographic directionsand, in fact, the interaction energy is connected to the relative orientation ofthe crystallographic axes and the direction of the magnetic moments. This inturn leads to magnetic anisotropy.
2.3 Models of Magnetic Anisotropy
Phenomenologically, the crystallographic easy axis of the magnetization isdetermined by the minimum of the free energy F 1. Before the explicit ex-pressions of F for various crystal symmetries are discussed, the microscopicorigins that explain magnetic anisotropy will be shortly described.
2.3.1 Single Ion Anisotropy
The single ion anisotropy is determined by the interaction between the orbitalstate of a magnetic ion and the surrounding crystalline field, when the crystalfield is very strong. The anisotropy is the result of the quenching of the orbitalmoment by the crystalline field. As this field has the symmetry of the crystallattice, the orbital moments can be strongly coupled to the lattice. This inter-action is transferred to the spin moments via the spin-orbit coupling, giving aweaker electron coupling of the spins to the crystal lattice. When an externalfield is applied the orbital moments may remain coupled to the lattice whilstthe spins are more free to turn. The magnetic energy depends on the orienta-tion of the magnetization relative to the crystal axes.
1 In the chapter by B. Heinrich in this book, the Gibbs free energy U is used insteadof F . As long as no external energy contributions are incorporated into F , thefree energy is the appropriate thermodynamic potential. Upon inclusion of theZeeman energy of the external field, however, U is the relevant potential. However,to avoid confusion, we will use the term F throughout the whole chapter.
2 Magnetic Anisotropy of Heterostructures 51
In a magnetic layer, the single-ion anisotropy is present throughout its volume,and contributes in general to the volume part of the MAE. In transition met-als this contribution is usually much smaller than the shape anisotropy butcan be comparable in magnitude in rare earth metals. However, in some casesalso for 3d metals or alloys with small magnetization (and consequently shapeanisotropy) the single ion contribution might overcome the shape anisotropy,leading to perpendicular magnetic anisotropy, for which the easy axis of mag-netization is aligned perpendicular to the film surface.The single-ion anisotropy can also contribute to the surface anisotropy viaNeel interface anisotropy [52], where the reduced symmetry at the interfacestrongly modifies the anisotropy at the interface compared to the rest of thelayer [53]. One may question the relevance of this purely phenomenological ap-proach, which has also been extended in terms of two-ion anisotropy contribu-tions. The importance lies in the fact that a simple model for the temperaturedependence of the MAE and its correlation to the temperature dependenceof the magnetization can be established. More recently, Mryasov et al. [54]have put this model on solid ground by showing that a model of magneticinteractions on the basis of first-principles calculations of non-collinear mag-netic configurations in FePt effectively contains the observed single-ion andtwo-ion contributions and explains the observed unusual scaling exponent Γbetween the magnetization and the MAE (K(T ) ∝M(T )Γ, see also Sect. 2.4).Later on, in Sect. 2.5, we discuss that for ultrathin Fe films a single-ion model(Γ = 3) yields an excellent explanation for the measured correlation of M(T )and K(T ).
2.3.2 Free Energy Density
As stated above the magnetic anisotropy energy is the work WMAE neededto rotate the magnetization between two different directions. If this rotationis performed at constant temperature T , the MAE is given by the differenceof the free energy F of the system with the magnetization pointing alongthe two directions. This is easy to see when one considers that for a closedsystem (no exchange of particles) dF = −dW −SdT , S being the entropy, atconstant T reduces to dF = −dW . Setting dW ≡ dWMAE this in turn yieldsF2 − F1 =
∫ 2
1dWMAE = MAE, where 1 and 2 denote the initial (e.g. easy)
and the final direction of the magnetization. Provided that an expression forthe magnetic part of F is given for the system under consideration, it can beused to interpret the FMR data on thin magnetic films. The phenomenologicalexpression for F is usually found by symmetry considerations. In the followingwe will summarize the expressions for F of the most often used symmetries.We discuss cubic, tetragonal as well as hexagonal symmetry as these are widelyfound in thin film systems. As special case of hexagonal symmetry the uniaxialone is introduced, which in form of shape anisotropy always occurs in thin filmsand which due to stray field minimization favors an easy in-plane alignment ofthe magnetization. We will use anisotropy constants having suffixes according
52 J. Lindner and M. Farle
to the symmetry they describe, e.g. K2 for uniaxial (second order) and K4
for cubic symmetry. This seems for us to be more transparent than to justnumber the constants in a row. Note that the latter numbering is used by otherauthors, so that care has to be taken when comparing results of anisotropyconstants. A first order cubic anisotropy constant (denoted as K4 within ournomenclature) is often denoted as K1 by other authors.An expression for the anisotropy field follows from considering the torqueexerted on the magnetization by the effective magnetic field within the sample.We assume for simplicity that the free energy of the system only dependson the angle θ of the magnetization with respect to an anisotropy axis. IfθB is the angle of the external field relative to this axis, the free energycan be written as F = Fa −M · B cos(θ − θB), where the first term is theanisotropy energy and the second the Zeeman contribution of the externalfield. The equilibrium angle of M can be found from ∂F
∂θ = 0 = ∂Fa
∂θ + M ·B sin(θ − θB) = ∂Fa
∂θ + |M × B|. This equation means that in equilibriumthe torque M × B due to the external field is balanced by the torque due tothe magnetic anisotropy field given by −∂Fa
∂θ (the opposite sign indicates thatthe torques are antiparallel). When B causes a turn of M of δθ, the torquedue to the anisotropy field is proportional to δθ and given by −∂Fa
∂θ = c · δθ.Thus, for δθ → 0 c = −∂2Fa/∂θ2
∣∣δθ=0
and the equilibrium condition becomesc ·δθ+M ·B sin δθ ≈ − ∂2Fa/∂θ2
∣∣δθ=0
·δθ+M ·Bδθ = 0. From this equationthe anisotropy field is found to be
Ba = − 1M
· ∂2Fa
∂θ2
∣∣∣∣δθ=0
. (2.5)
Note that the derivative has to be taken at the equilibrium angle, for whichδθ = 0.It is a quite common though not the only possibility to expand the free energyof a magnetically saturated single crystal (i.e. no domain walls in the crystal)as a series of the direction cosines αi of the magnetization vector relative toa rectangular Cartesian system of coordinate axes. The direction cosines arethe projections of M onto the three unit vectors defining the crystal latticeand given by αi = M/M · ei (i = 1, 2, 3) where the ei are the unit vectors.According to Birss one can write [55]:
This series is a direct consequence of Neumann’s principle stating that anytype of symmetry which is exhibited by the point group of the crystal, i.e.by the group of symmetry operations that describe the symmetry of theunit cell of the crystal, is possesed by every physical property tensor. Thus,the limitations of crystal symmetry must be reflected by the tensors bijk....Note, that the higher order terms make F to oscillate rapidly with the an-gular orientation of the direction of magnetization. Since this is contray to
2 Magnetic Anisotropy of Heterostructures 53
experimental observations the higher order terms must be very small. Thecomponents of the tensors bijk... transform under a rotation of coordinateaxes according to the relations bijk...n = lipljqlkr . . . lnubpqr...u (in many casesthis transformation of a tensor is used as definition of the tensor as a phys-ical object). Note that in the equation the Einstein notation is used, i.e.when a letter occurs as a suffix twice in the same term on one side of theequation, summation with respect to that suffix is to be understood, i.e.bijk...n =
∑3p=1
∑3q=1
∑3r=1 . . .
∑3u=1 lipljq lkr . . . lnubpqr...u. The matrix [lip]
describes the symmetry operation. As special case the equation contains thetransformation of a vector given by bi =
∑3p=1 lipbp = lipbp. As example a
right-handed rotation of 180◦ about the z-axis is described by the matrix
[lip] =
⎛⎝
cos 180◦ 0 00 cos 180◦ 00 0 cos 0◦
⎞⎠ =
⎛⎝
−1 0 00 −1 00 0 1
⎞⎠ . (2.7)
With this it follows that the requirement that bijk... is a property tensorand invariant under all permissible symmetry operation appropriate to theparticular crystal class is equivalent to the requirement that the componentsbijk...n satisfy the set of equations:
bijk...n = σipσjqσkr . . .σnubpqr...u . (2.8)
All the matrices [σ] correspond to permissible symmetry operations. It can beshown that there are only 9 so-called generating matrices that are needed todescribe all crystal classes, i.e. all symmetry operations of the point groups canbe described by these matrices and multiplications of them. The matrices are:
[σ(unit)
]=
⎛⎝
1 0 00 1 00 0 1
⎞⎠ [
σ(inv)]
=
⎛⎝
−1 0 00 −1 00 0 −1
⎞⎠
[σ(2⊥z)
]=
⎛⎝
−1 0 00 1 00 0 −1
⎞⎠ [
σ(2‖z)]
=
⎛⎝
−1 0 0−1 0 0
0 0 1
⎞⎠
[σ(2⊥z)
]=
⎛⎝
1 0 00 −1 00 0 1
⎞⎠
[σ(2‖z)
]=
⎛⎝
1 0 00 1 00 0 −1
⎞⎠ (2.9)
[σ(3‖z)
]=
⎛⎝
− 12
12
√3 0
− 12
√3 − 1
2 00 0 1
⎞⎠[σ(4‖z)
]=
⎛⎝
0 1 0−1 0 0
0 0 1
⎞⎠
[σ(4‖z)
]=
⎛⎝
0 −1 01 0 00 0 −1
⎞⎠
[σ(3dia)
]=
⎛⎝
0 1 00 0 11 0 0
⎞⎠ .
The first matrix is the unity matrix, the second describes a point inversionthrough the unit cell of the crystal. The next two matrices describe a twofold
54 J. Lindner and M. Farle
rotation parallel to the z-axis and to an axis perpendicular to the z-axis,respectively. The other matrices describe in analogy fourfold and threefoldrotational axes (a bar on top denoting a rotation followed by a point inversion).Finally, the last matrix describes a threefold rotation parallel to a cube-bodydiagonal. Other symmetry operations can be written as multiplications of theabove matrices. E.g. a six-fold rotation parallel to the z-axis can be describedby[σ(inv)
] [σ(2‖z)
] [σ(3‖z)
].
In the following expressions for the free energy in uniaxial, hexagonal,cubic and tetragonal crystals are derived. The cartesian coordinate systemchosen to describe the crystal is shown in Fig. 2.1. The system is chosen sothat the z-axis coincides with the film normal. Consequently, the x- and y-axesare located within the film plane. To obtain expressions that are a functionof the external magnetic field B0 and the magnetization M , the polar anglesθB and θ as well as the azimuthal angles ϕB and ϕ are introduced. In caseof an additional distinguished direction (like the direction of step edges), theangle δ is defined with respect to the x-axis.
Cubic Symmetry
For crystals of cubic symmetry the generating matrices are:[σ(inv)
],[σ(4‖[001])]
and[σ(4‖[111])]. While the first matrix describes the fact that the cubic unit
cell is centro-symmetric, the second and third matrix describe fourfold ro-tational axes parallel to a cube-body edge and diagonal, respectively. Us-ing
[σ(inv)
]within (2.8) yields bijk...n = −bijk...n for all tensors of odd
rank (n is an odd number), thus making them vanish. We note that this
z
y
x
B0q
B
jB
j
qM
d
Fig. 2.1. Cartesian coordinate system used to derive the expressions for the freeenergy for the different crystal symmetries
2 Magnetic Anisotropy of Heterostructures 55
also follows directly from time inversion symmetry, which is valid as magne-tocrystalline anisotropy is a static property. Using the matrices
[σ(4‖[001])]
and[σ(4‖[111])] leads to relations showing that for the tensor bij all compo-
nents, for which i �= j vanish, while b11 = b22 = b33 which expresses therequirement that for cubic symmetry the energy must not change upon ex-changing two αi (change of equivalent cubic axes). The first non-vanishingterm is thus given by bijαiαj = b11α2
x + b11α2y + b11α2
z . Similarly one ob-tains the allowed terms for bijkl. Using again the symmetry matrices one getsb1111 = b2222 = b3333 and b1122(6) = b2233(6) = b1133(6) ((6) means the 6 per-mutation that can be made from the term). Then, one has bijklαiαjαkαl =b1111
(α4
x + α4y + α4
z
)+6b1122
(α2
xα2y + α2
yα2z + α2
zα2x
). The factor 6 arises from
the multiplicity implicit in the second relations of the bijkl. The first non-vanishing contributions for cubic crystals therefore are:
Fcub = b11(α2
x + α2y + α2
z
)+ b1111
(α4
x + α4y + α4
z
)
+ 6b1122(α2
xα2y + α2
yα2z + α2
zα2x
)+ . . . . (2.10)
Using the relations α2x + α2
y + α2z = 1 and 1 − 2
(α2
xα2y + α2
xα2z + α2
yα2z
)=
α4x + α4
y + α4z yields:
Fcub = K0 +K4
(α2
xα2y + α2
xα2z + α2
yα2z
)+K6α2
xα2yα2
z . . . . (2.11)
Here the anisotropy constants K0 = b11 + b1111 and K4 = 6b1122 −2b1111 weredefined. Note that also the next higher order term was introduced, for whichK6 = 3b111111 − 45b111122 + 90b112233 (see [55] for details).
Cubic Crystals with (001)-orientation
Considering the coordinate system of Fig. 2.1 the direction cosines can bewritten as αx = sin θ cosϕ, αy = sin θ sinϕ, αz = cos θ. In the following weidentify the z-axis with the [001]-direction, the x(y)-axis with the [100]([010])-direction of the cubic crystal. Inserting the expressions for the αi into (2.11)one obtains:
This expression can be transformed into another equivalent form:
F001 = K4
(sin2 θ cos2 θ + sin4 θ cos2 ϕ sin2 ϕ
)(2.13)
= K4
[sin2 θ
(1 − sin2 θ
)+ sin4 θ cos2 ϕ
(1 − cos2 ϕ
)]
= K4
[sin2 θ − sin4 θ
(cos4 ϕ− cos2 ϕ+ 1
)]
56 J. Lindner and M. Farle
[001]
[100]
[110]
[001]
[100]
[110]
[001]
[110]
Fig. 2.2. Free energy for a cubic crystal with (001)-orientation with the 〈100〉-(left panel), the 〈111〉- (middle panel) and the 〈110〉-axes (right panel) being theeasy ones
= K4
[sin2 θ − sin4 θ
(18
cos 4ϕ+12
cos 2ϕ+38− 1
2− 1
2cos 2ϕ
)]
= K4 sin2 θ − 18K4 (cos 4ϕ+ 7) sin4 θ.
The equation shows that for K4 > 0 the 〈100〉-directions are the easy ones,whereas for K4 < 0 the 〈111〉-directions are the easy ones. This is visualizedin Fig. 2.2, where the free energy has been plotted as polar plot using thecoordinates defined by Fig. 2.1 (the z-axis coincides with the [001]-direction).For the case that K6 can not be neglegted, the 〈110〉-directions can be the easyones (see right panel in Fig. 2.2). Table 2.2 shows the combinations of K4 andK6 and the corresponding easy, intermediate and hard directions. A graphicalrepresentation in the form of stability or flow diagramms of K4 versus otheranistropy constants have been published by several authors [56, 57, 58, 59].
Cubic Crystals with (011)-orientation
To derive an expression for F in the case of a cubic crystal with (011)-orientation, one needs to rotate the (x, y, z)-coordinate system, for whichthe axes are parallel to the 〈100〉-directions, yielding a new system (x′, y′, z′)
Table 2.2. Conditions for easy, intermediate and hard axes in cubic symmetry
with its z′-axis parallel to any of the {110}-planes. As example in Fig. 2.3
this rotation is shown for the case that the z′-axis becomes parallel to the
[011]-direction. Still the direction cosines in the new systems are given byαx′ = sin θ cosϕ, αy′ = sin θ sinϕ, αz′ = cos θ. θ is now defined withrespect to the z′-axis ([011]-direction), ϕ is measured against the x′-axis([100]-direction). For (2.11), however, the direction cosines with respect tothe (x, y, z)-coordinate system are needed, i.e., the direction cosines withinthe (x, y, z)-system have to be expressed by means of the direction cosineswithin the (x′, y′, z′)-system. From Fig. 2.3 the direction cosines with respectto the axes of the two systems can be deduced. The relation are summa-rized in Table 2.3. This yields αx = αx′ = sin θ cosϕ, αy = 1/
√2 αy′ +
1/√
2 αz′ = 1/√
2 (sin θ sinϕ+ cos θ) , αz = −1/√
2 αy′ + 1/√
2 αz′ =1/
√2 (− sin θ sinϕ+ cos θ). Using these direction cosines within (2.11) one
derives for the free energy of an (011) oriented cubic crystal:
F011 =K4
4
(cos4 θ + sin4 θ
[sin4 ϕ+ sin2 (2ϕ)
]+ sin2 (2θ)
[cos2 ϕ− sin2 ϕ
2
]).
(2.14)
We note that this equation describes the same polar plot that (2.12) does withthe difference that it has been rotated such that the [011]-direction forms the
Table 2.3. Direction cosines between (001) and (011)-coordinate system
x y z
x′
cos 0◦ = 1 cos 90◦ = 0 cos 90◦ = 0
y′
cos 90◦ = 0 cos 45◦ = 1/√
2 cos 135◦ = −1/√
2
z′
cos 90◦ = 0 cos 45◦ = 1/√
2 cos 45◦ = 1/√
2
58 J. Lindner and M. Farle
[001]
[110][100]
[100]
[111]
[110]
[011]
[100]
[011]
[110]
[112]
Fig. 2.4. Free energy for a cubic crystal with (001)-orientation (uppermost panel),(111)-orientation (middle panel) and (111)-orientation (lowest panel) for K4 > 0, i.e.for the 〈100〉-directions being the easy ones. The right column shows a cut of thepolar plots on the left side, for which the [001]−, [111]− and [011]−direction pointout of the paper plane. In this projection the in-plane symmetry can be better seen
film normal (see lowest panel of Fig. 2.4). The azimuthal plane according tothe coordinates introduced in Fig. 2.1 shows a dominating twofold symmetryas shown in the lowest panel (right plot) of Fig. 2.4.
Cubic Crystals with (111)-orientation
For (111) oriented cubic crystals one changes into a coordinate system((x, y, z) → (x′, y′, z′)), for which the z′-direction is oriented parallel to the[111]-direction (see Fig. 2.3). This time θ denotes the polar angle with
2 Magnetic Anisotropy of Heterostructures 59
Table 2.4. Direction cosines between the (001) and (111)-coordinate system
x y z
x′
cos 114.1◦ = −1/√
6 cos 114.1◦ = −1/√
6 cos 35.3◦ =√
2/√
3
y′
cos 45◦ = 1/√
2 cos 135◦ = −1/√
2 cos 90◦ = 0
z′
cos 54.7◦ = 1/√
3 cos 54.7◦ = 1/√
3 cos 54.7◦ = 1/√
3
respect to the z′-axis ([111]-direction), ϕ the one with respect to the x′-axis([112]-direction). The direction cosines between the two systems are given inTable 2.4. In analogy to the (011)-plane one gets expressions for the directioncosines within the (x, y, z)-system as function of the α’s within the rotatedsystem:
αx = − 1√6
αx′ +1√2
αy′ +1√3
αz′ = − sin θ cosϕ√6
+sin θ sinϕ√
2+
cos θ√3
=1√3
[cos θ −√
2 sin θ sin(ϕ+
(π6
))]
αy = − 1√6
αx′ − 1√2
αy′ +1√3
αz′ = − sin θ cosϕ√6
− sin θ sinϕ√2
+cos θ√
3
=1√3
[cos θ −
√2 sin θ cos
(ϕ+
(π3
))]
αz =√
2√3
αx′ +1√3
αz′ =√
2√3
sin θ cosϕ+1√3
cos θ
(2.15)
=1√3
[cos θ +
√2 sin θ cosϕ
].
Here the trigonometric expressions for sin (x± y) = sinx cos y ± cosx sin y,cos (x± y) = cosx cos y ∓ sinx sin y and sin (π/6) = cos (π/3) = 1/2 andcos (π/6) = sin (π/3) =
√3/2 have been used. Equation (2.11) yields for the
free energy of a (111)-oriented cubic crystal:
F111 = K4
(13
cos4 θ +14
sin4 θ −√
23
sin3 θ cos θ cos 3ϕ
). (2.16)
F111 is visualized in Fig. 2.4 (middle panel). Now the [111]-direction coin-cides with the z-axis from Fig. 2.1. In the azimuthal plane the free energy isisotropic (see right plot of the middle panel of Fig. 2.4). An angular depen-dence within the azimuthal plane does only occur, when the next higher orderterm (K6-term in (2.11)) is considered (see also Sect. 2.3.4 for the azimuthaldependence of F ).
60 J. Lindner and M. Farle
Cubic Films with In-plane Magnetization
For thin layers of a cubic material and for the case that the magnetizationis confined to the film plane, the direction cosines for the (001)-, (011)- and(111)-plane take the forms listed in Table 2.5. For the free energy also listed inthe table in addition to theK4-term the next higher order term (K6) accordingto (2.11) was considered. Table 2.5 shows that for the (001)-orientation afourfold symmetry in the plane exists, while the (011)-orientation shows alsotwofold terms. For (111)-oriented films the in-plane anisotropy in first ordervanishes and only to the next higher order shows a sixfold symmetry, which,however, in most cases is very small.
Tetragonal Symmetry
In case of tetragonal symmetry only the (001)-orientation will be discussedas this is the one most widely dicussed one in literature. The transforma-tion to other orientations can be performed in the same way as discussedfor cubic symmetry. The symmetry matrices for tetragonal systems are:[σ(inv)
],[σ(2⊥z)
]and
[σ(4‖z)
]. The rotational axis perpendicular to the [001]-
direction now presents only a twofold symmetry and thus the terms in theexpansion of the free energy reflect this lowering of symmetry. The first al-lowed term is b11
(α2
x + α2y
)+ b33α2
z and thus, the twofold symmetry does notvanish as for cubic symmetry. The first terms are given by:
Ftet = b11(α2
x + α2y
)+ b33α2
z + b1111(α4
x + α4y
)+ b3333α4
z +
+ 6b1122α2xα2
y + 6b1133(α2
xα2z + α2
yα2z
)+ . . . . (2.17)
Table 2.5. Direction cosines and free energy for the case that the magnetization isconfined in the film plane
Plane Direction cosines Free energy
(001)αx = cos ϕαy = sin ϕαz = 0
F001 = K4 cos2 ϕ sin2 ϕ
= K44
sin2 2ϕ
= K48
(1 − cos 4ϕ)
(011)
αx = cos ϕαy = 1√
2sin ϕ
αz = −1√2
sin ϕ
F011 = K44
(sin4 ϕ + 4 sin2 ϕ cos2 ϕ
)+K6
4sin4 ϕ cos2 ϕ
= K44
(sin4 ϕ + sin2 2ϕ
)+K6
16sin2 ϕ · sin2 2ϕ
= K432
(7 − 4 cos 2ϕ − 3 cos 4ϕ)+ 1
128K6 (2 − cos 2ϕ − 2 cos 4ϕ + cos 6ϕ)
(111)
αx = − cos ϕ√6
+ sin ϕ√2
αy = − cos ϕ√6
− sin ϕ√2
αz =√
3√2
cos ϕ
F111 = K44
+ K654
(9 cos2 ϕ − 24 cos4 ϕ + 16 cos6 ϕ
)= K4
4+ K6
108(1 + cos 6ϕ)
2 Magnetic Anisotropy of Heterostructures 61
One can see that for b11 = b33, b1111 = b3333 and b1122 = b1133, i.e. for thecase where x-,y- and z-axes are equivalent, the expression for cubic symmetryis retained. Using the relations α2
z = 1 − (α2x + α2
y
)and
(α2
xα2z + α2
yα2z
)=
12 − 1
2
(α4
x + α4y + α4
z
)− α2xα2
y the equation transforms to:
Ftet = K ′0 −K ′
2⊥α2z +K ′
4‖(α4
x + α4y
)+K ′′
4‖α2xα2
y +K ′4⊥α4
z + . . . , (2.18)
with K ′0 = b11 + 3b1133, K ′
2⊥ = b11 − b33, K ′4‖ = b1111 − 3b1133, K ′′
4‖ =6 (b1122 − b1133) and K ′
4⊥ = b3333 − 3b1133. A further simplification is madethrough the relation α4
4⊥ − K ′′4‖ = −2b3333 + 6b1133 − 6 (b1122 − b1133). Using the
polar coordinates according to Fig. 2.1 finally yields:
Ftet = −K2⊥ cos2 θ − 12K4⊥ cos4 θ − 1
8K4‖ (3 + cos 4ϕ) sin4 θ . (2.20)
Uniaxial and Hexagonal Symmetry
The symmetry matrices for hexagonal systems are[σ(inv)
],[σ(2⊥z)
],[σ(2‖z)
]and
[σ(3‖z)
]. The matrix σ(3‖z) describes threefold rotational symmetry about
the z-axis (note that the sixfold symmetry of the hexagonal unit cell can bedescribed by combinations of the matrices). As for the other cases, we have acentrosymmetrical unit cell (due to
[σ(inv)
]) and thus all odd rank tensors van-
ish. The use of[σ(2⊥z)
]within (2.8) further shows that the σ‘s are separately
non-zero only if i = p or j = q or . . . In addition, the product σipσjqσkr . . .σip
is –1 when the subscript 2 appears an odd number of times (note that thenumber of σ’s within the product must be even). Since this means thatdijkl... = −dijkl... all coefficients, in which the subcript 2 appears an oddnumber of times, must vanish. Similarly, if
[σ(2‖z)
]is used, it can be shown
that all coefficients, in which the subscript 3 appears must vanish. These tworestrictions then imply that the coefficients, in which any subscript appears anodd number off times vanish. Thus, the first non-vanishing term is the sameas for tetragonal symmetry, i.e. bijαiαj = b11
(α2
x + α2y
)+ b33α2
z . The last ma-trix yields several relations between the remaining dijkl...’s, leading finally tobijklαiαjαkαl = b1122
(α2
x + α2y
)2 + 6b1133(α2
x + α2y
)α2
z + b3333α4z (see [55] for
details of the calculation). Taking even the next higher order term one obtains:
Fhex = K0 +K2⊥(α2
x + α2y
)+K4⊥
(α2
x + α2y
)2+K6⊥
(α2
x + α2y
)3+
+ K6‖(α2
x − α2y
) (α4
x − 14α2xα2
y + α4y
). (2.21)
62 J. Lindner and M. Farle
One sees that for K4 << Ki<4 there is no in-plane anisotropy and the hexag-onal symmetry converts to an uniaxial one. Using polar coordinates yields:
Figure 2.5 shows a polar plot according to the coordinate system of Fig. 2.1for (left panel) K2⊥ < 0, for which the [0001]-direction is the hard one andfor (right panel) K2⊥ > 0, for which the [0001]-direction is the easy one. K6‖leads to the sixfold anisotropy within the azimuthal plane.
Shape Anisotropy
At each lattice point within a cubic lattice the dipole fields of all neighborscancel out. This is, however, only true for an infinite system. As soon assurfaces are present, magnetic poles develop and thus, the dipole-dipole in-teraction leads to anisotropy. Since the shape of the sample determines thisanisotropy, one usually calls it shape anisotropy. In systems with reduced di-mension such as thin films the shape anisotropy might be even the dominatingcontribution to the overall MAE. For a thin disc (realized by the thin film),the shape anisotropy has, phenomenologically, the form of uniaxial anisotropy(being a special case of hexagonal anisotropy):
F shapeuni =
μ0
2(N⊥ −N‖
)M2 cos2 θ , (2.23)
where N⊥ and N‖ are the demagnetizing factors parallel and perpendicularto the film plane. In the limit of a homogeneous disc with infinite diameterN⊥ = 1 and N‖ = 0 holds and thus shape anisotropy always favors an easyin-plane orientation of the magnetization. The only exception occurs for rough
[0001] [0001]
Fig. 2.5. Polar plot of the free energy surface for a hexagonal system for (leftpanel) easy axis parallel to the hexagonal axis and (right panel) perpendicular tothe hexagonal axis. For uniaxial systems (K6i = 0) the polar plot is the same withthe only difference that no sixfold symmetry in the basal plane perpendicular to theuniaxial axis is present
2 Magnetic Anisotropy of Heterostructures 63
films. This situation was theoretically studied by Bruno [60], who modeled theroughness by two parameters σ and ξ, the former being the mean vertical de-viation from a reference plane, the latter describing the average lateral size offlat areas. The calculation shows that the stray field energy for perpendicularmagnetization of a rough film is given by:
F shaperough = 0.5μ0M
2Sd
[1 − σ
2df
(σξ
)]. (2.24)
Here d is the average thickness, i.e. the thickness of the reference plane. MS
is the saturation magnetization. The function f has the value 1 for a flat film,σ = 0, and it approaches 0 for increasing roughness, σ/ξ → 1. This modelshows that film roughness gives rise to a small dipolar surface anisotropy con-tribution of magnitude ∝ σ/ξ that favors an out-of-plane easy axis of magne-tization. Such a contribution was indeed found in [61] for rough Ni films onCu(001).Another point raised by Heinrich and Cochran [4] and already earlier byBenson and Mills [62] is that for very thin films of only a few atomic layers thecontinuum approximation fails to describe the dipolar shape anisotropy. Thediscreteness of the atomic moments results in a variation of the dipolar fieldacross the sample which depends on the number of atomic layers involved. Thedipolar field of a given layer decreases exponentially away from its surface witha decay length corresponding to the in-plane lattice spacing. Thus, the dipo-lar field inside the film decreases when approaching the sample surface frominside the film and the value of the average dipolar field decreases stronglywhen the thickness of the film is reduced towards the monolayer regime. Thelarger the lattice spacing, the stronger this effect will be. The reduced dipolarfield will appear as a reduced shape anisotropy and the reduction can be writ-ten as a reduced demagnetizing factor. For bcc(001) films the demagnetizingfactor is given as N⊥ = 1 − 0.425
N , while for the more densely packed fcc(001)surface N⊥ = 1− 0.234
N , N being the number of atomic planes of the film. Thecase of hcp structure is discussed in [24]. Except for very thin films with athickness of few monolayers this correction is less than 1% and will thus notbe considered in the following.
Uniaxial Symmetry – Surface Anisotropy
In thin films the presence of the symmetry breaking surface and interface tothe substrate also introduces an uniaxial anisotropy term that can be writtenin the form:
F⊥uni = K2⊥ sin2 θ = K0 −K2⊥ cos2 θ = K0 −K2⊥α2
z , (2.25)
with K0 = 1 2. When an uniaxial distortion of the crystal lattice is presentin the volume of the material, such a term may also contribute to volume2 Note that as a potential energy contribution F is defined as energy difference, so
that adding constant (angular independent) terms has no influence on the value
64 J. Lindner and M. Farle
anisotropy. In general, magnetic anisotropies of thin films can be decomposedin volume contributions, that are independent of thickness and surface stateand can be explained as a superposition of shape, magnetocrystalline andresidual strain anisotropies, and surface contributions, which scale with 1/dand depend sensitively on the state of the surface. In some cases Neel’s phe-nomenological anisotropy model has provided a useful connection betweendifferent components of surface anisotropies. For example, the role of epi-taxial strain for the anisotropies of ultrathin films was demonstrated for thecase of Co(0001) films on W(110) by measurements of anisotropies using tor-sion oscillation magnetometry combined with measurement of the strain byhigh-angular-resolution low-energy electron diffraction. Up to a thickness ofd = 2 nm, the films were found to grow in a state of constant strain, governedby pseudomorphism with a growth relation [1100]Co‖ [110]W, which resultsin a true volume-type strain anisotropy. Above 2 nm, a relaxation of strainis observed which scales roughly with 1/d and, therefore, results in an appar-ent surface-type contribution to strain anisotropy, superimposed on a reducedvolume contribution [63].Besides uniaxial anisotropy parallel to the film normal uniaxial contributionsfrequently appear along a direction in the film plane. This can be caused e.g.by preferential interactions due to oriented hybridization at the film-substrateinterface or an uniaxial in-plane distortion in the volume. Such a term can bedescribed by:
F‖uni = K2‖ sin2 θ cos2 (ϕ− δ) , (2.26)
where ϕ is measured with respect to the [100]-direction (x-axis). To includeany possible in-plane easy axis the angle δ was defined as shown in Fig. 2.1.One should note the following: The anisotropy due to F⊥
uni, Fshapeuni (and in
many cases also F ‖uni) are direct results of the fact that the specimen has the
shape of a thin film with interfaces that break the translational symmetry ofthe system. Thus, all these contributions are inherently connected with thefilm surface itself and do not depend on the crystallographic direction of thefilm normal. Consequently, for other crystallographic orientations of the film,no transformation of F⊥
uni, Fshapeuni and F
‖uni has to be made, in contrast to
the case of crystalline anisotropy resulting from the volume symmetry of thefilm material as discussed for cubic symmetry in Sect. 2.3.2.
In order to separate the different anisotropy contributions into volume andsurface contributions one needs also to perform thickness dependent measure-ments. The thickness dependence of each anisotropy constant can be fittedby a constant term representing a volume contribution (Kv
i ) and an effectivesurface/interface contribution (Ks,eff
i ) being proportional to 1/d where d isthe thickness of the film.
of F . Therefore, the constant term K0 does not contribute to magnetic anisotropyand can be made to vanish upon normalizing F (i.e. subtracting the constant K0).
2 Magnetic Anisotropy of Heterostructures 65
Ki = Kvi +
Ks,effi
di = 2, 4. (2.27)
Magnetoelastic Contributions
pt Once one realizes that the MAE is a quantity describing the interactionbetween the electron spin and the lattice, it is intuitively clear that changes ofthe lattice constant will affect the magnetic properties. In a magnetized bodyone has energy terms that depend both on the strain and the magnetizationdirection: the magneto-elastic energy. Although resulting from the same ori-gin, namely the spin-orbit interaction, magneto-elastic anisotropies only existwhen stress is exerted on the magnetic system. For instance in iron, the effectof tension on a single crystal is to create a preferred direction of magnetizationparallel to the direction of stress. The experimentally obtained magneto-elasticconstants are significantly larger than the crystalline anisotropy constants [64].As a consequence, even small strains may give rise to an important anisotropycontribution. Moreover, this phenomenon may be of importance in epitaxialstructures, where considerable strains may result from the epitaxial growth ofthe film on a substrate or adjacent layers having a different lattice parame-ter. With respect to the film material, the strain in epitaxial films is given byη = (asub − afilm) /afilm, i.e. the misfit is determined by the lattice constantsai, which describe the atomic distances on the relevant surface orientation. Ifthe lattice mismatch is not too large, below a critical thickness dc (coherentregime), the misfit is accommodated by introducing a tensile strain η in onelayer and a compressive strain in the other such that both adopt the samein-plane lattice magnetic anisotropy parameter. A reasonable epitaxial matchof the film lattice with respect to the substrate one can also be achieved bythe rotation of the two lattices against each other. This happens e.g. for bcc-Fe(001) growing on fcc-Ag(001), for which the lattices are rotated by 45◦ withrespect to each other. For relatively thin films the strain and the magneto-elastic coupling are independent of thickness. Above the critical thickness dc,it becomes energetically more favorable to introduce misfit dislocations, whichpartially accommodate the lattice misfit, allowing the uniform strain to be re-duced (incoherent regime). In the incoherent regime, the contribution to themagneto-elastic energy contains a reciprocal thickness dependence [65]. Thefilm strain can be isotropic in the plane of the film (i.e. ε11 = ε22 = η, wherethe εii are the in-plane component of the strain tensor along two axes that areperpendicular to each other) or also anisotropic (ε11 �= ε22) The latter mightoccur when preferentially oriented misfit dislocations have formed or a stronginterface hybridization between the film and substrate along specific directionsis present. According to continuum elasticity the in-plane strain leads to anout-of-plane variation of the lattice. From the requirement of a minimum of theelastic energy one can calculate the strain component ε33 perpendicular to thefilm plane. Table 2.6 lists the results for several symmetries of the film lattice.The corresponding change in volume is given by ΔV/V = (ε11 + ε22 + ε33).
66 J. Lindner and M. Farle
Table 2.6. Calculated out-of-plane strain ε33 as function of in-plane strain compo-nents ε11 , ε22 (from [15]). The elastic stiffness constants cij are tabulated in [66]
The magneto-elastic part to the magnetic anisotropy for a cubic system canbe written as [15]:
F cubMEL = B1
(ε11α2
x + ε22α2y + ε33α2
z
)(2.28)
+ 2B2 (αxαyε12 + αyαzε23 + αxαzε31) + . . . .
For hexagonal systems the magneto-elastic contribution is [15]:
FhexMEL = B1
(ε11α2
x + ε22α2y + ε12αxαy
)
+ B2
(1 − α2
z
)ε33 +B3
(1 − α2
z
)(ε11 + ε22) (2.29)
+ B4 (αyαzε23 + αxαzε13) + . . . ,
where the εij (i, j = 1, 2, 3) are the strain components, the Bi the magneto-elastic coupling constants and the αi the direction cosines (see coordinatesystem in Fig. 2.1) given by α1 = sin θ cosϕ, α2 = sin θ sinϕ and α3 = cos θ.One should note that while in general the Bi of ultrathin films are differentthan in the respective bulk material one finds in the case of Ni films that thestrain induced anisotropy contributions can be explained – even as a functionof temperature – by the respective bulk values [5]. In Sect. 2.5 we will, however,show that the Bi-values for Fe films on GaAs differ from the ones of Fe bulk.In order to calculate the magneto-elastic part one needs to measure the Bi
(volume values for 3d ferromagnets are found in e.g. in [15]). In (2.29) if i = j,the strain is along the cubic 〈100〉 axes, for i �= j the strain is along the 〈110〉axes. While the former type of strain leads to a change of the volume of theunit lattice cell, the latter is equivalent to a shearing of the lattice keepingthe volume constant.
2.3.3 Landau-Lifshitz Equation of Motion and General ResonanceEquation
Magnetic excitations from the ground state that occur in the microwaveregime and are detected within an FMR experiment (see Sect. 2.5 for de-tails on the FMR technique) are usually described within the Landau-Lifshitzformalism. Due to the high number of spins that take part in the absorptionprocess and the large quantum numbers associated with it classical and quan-tummechanical description lead to identical results [67] and thus a classicalformulation of the process is usually considered.
2 Magnetic Anisotropy of Heterostructures 67
Upon considering the total angular momentum Jn as a classical vector withcontinuous possibilities of adjustments, the time dependence of Jn is – accord-ing to Newton’s law – given by the torque Dn = dJn
dt . In our case the torqueis given by a magnetic field acting on the magnetic moment μn that resultsfrom Jn. Considering a magnetic part of the rf-field due to the microwaveexcitation brf
μn, an external field B0 and also an internal field Bint
μnwithin the
sample (e.g. magnetic anisotropy fields) as contributions to the overall mag-netic field, the torque is given by μn ×
(Bint
μn+ B0 + brf
μn
). Taking the two
expression for the torque together and using μn = −γJn with γ = gJ μB
�being
the gyromagnetic ratio (gJ : Lande g-factor) one obtains:
1γ
dμn
dt= −μn ×
(bint
μn+ b0 + brf
μn
). (2.30)
Summing up all magnetic moments yields the macroscopic internal fieldBint =
∑n Bint
μn, that acts on the total magnetic moment μt =
∑μn. Taking
into account that the magnetization M is defined as magnetic moment perunit volume (μt/V ) and assuming a homogeneous microwave field brf over thesample, one has:
dM
dt= −γ
(M × Beff
). (2.31)
Here the abbreviation Beff = Bint + B0 + brf was used. The latter equationis known as the Landau-Lifshitz(LL)-equation. We note that it can be alsoderived in the framework of quantummechanics [68]. The LL equation canbe extended to include magnetic damping, leading to a finite linewidth ofthe FMR signal. However, throughout this paper, which focusses on magneticanisotropy and thus on the field needed for resonance (resonance field) only,damping will be neglected.
A straightforward but complex way to describe FMR is to solve the LLequation for given anisotropy fields. There is, however, an alternative route,which uses the LL equation to formulate a general equation on the basis ofthe magnetic part of the free energy of the system. As this approach is rathergeneral, it is described in some detail in the following. The method to calculatethe resonance frequency or, equivalently, the resonance field for the uniformFMR mode (collective precession of all magnetic moments) was introducedby Smit and Beljers and independently by Suhl ([69, 70]). In this formalismthe equation of motion is described by the free energy F . The same resultwas obtained by Gilbert by solving a Lagrange-Equation for the motion of M[71]. The magnetization is considered as classical gyroscope with moment ofinertia I. Figure 2.6 shows the transformation from the laboratory (x, y, z)-coordinate system to another cartesian one (x
′, y
′, z
′), in which the z
′-axis
rotates with the magnetization. The transformation is uniquely given by thethree Euler Angles ϕ, θ, ψ.
The kinetic energy of the system is given by Ekin = I2
(ψ + ϕ cos θ
)2
.From this the Lagrangian function of the system follows to beL = Ekin − Epot (ϕ, θ), Epot being the potential energy. The Langrangian
68 J. Lindner and M. Farle
z
y
x
z’
M
�
�
y’
x’�
Fig. 2.6. Euler Angles to describe the rotation of the coordinate system
equation of motion then is ddt
∂L∂qi
− ∂L∂qi
= 0, where the qi are the generalizedcoordinates, which in our case are given by ϕ, θ and ψ. This approach yieldsthe following three equations:
ddt
[I(ψ + ϕ cos θ
)cos θ
]+∂Epot
∂ϕ= 0
I(ψ + ϕ cos θ
)ϕ sin θ +
∂Epot
∂θ= 0 (2.32)
ddt
[I(ψ + ϕ cos θ
)]= 0.
As e.g. shown in [72] the term I(ψ+ ϕ cos θ) describes the angular momentumof the magnetization within the (x
′, y
′, z
′)-system and thus, the last equation
shows the time invariance of the angular momentum. Considering the LLGequation of motion the angular momentum of the magnetization is given byMS/γ. Therefore, the two remaining Lagrangian equations yield:
MS
γθ sin θ =
∂F
∂ϕ(2.33)
−MS
γϕ sin θ =
∂F
∂θ.
Here it was taken into account that Epot is given by the free energy F ofthe system. We now assume that the precession angle of the magnetization issmall, so that only small variations δθ and δϕ with respect to the equilibriumorientation
(θ0, ϕ0
)occur. This approach must be modified for high power
microwave excitations which cause large precession angles and non-linear re-sponses. In the small angle regime we have θ = θ0 + δθ, ϕ = ϕ0 + δϕ and onecan expand the first derivatives of F around the equilibrium position into aseries of δϕ and δθ, in which only the linear terms have to be considered:
Fθ = Fθθδθ + Fθϕδϕ , Fϕ = Fϕθδθ + Fϕϕδϕ . (2.34)
2 Magnetic Anisotropy of Heterostructures 69
The derivatives have to be taken at the equilibrium positions, i.e. Fθθ∣∣θ0 and
so on. Periodic solutions δθ, δϕ∝ exp(iωt) have to be found, as the deviationsaround the equilibrium position are driven by the periodic excitation due tothe microwave field with a frequency ω. This yields θ = iωδθ and ϕ = iωδϕ.Taking into account that for small deviations sin
(θ0 + δθ
)= sin θ0 cos δθ +
cos θ0 sin δθ ≈ sin θ0 + cos θ0δθ ≈ sin θ0, one has from (2.33):(iωMS
γsin θ0 − Fϕθ
)δθ − Fϕϕδϕ = 0 (2.35)
(− iωMS
γsin θ0 − Fϕθ
)δϕ− Fθθδθ = 0 . (2.36)
In matrix formulation this can witten as:(Fϕθ − iω
γ ·MS sin θ0 Fϕϕ
Fθθ Fϕθ + iωγ ·MS sin θ0 .
)·(
δθδϕ
)= 0 . (2.37)
The condition for a solution is F 2θϕ
−FθθFϕϕ +ω2γ−2M2S sin2 θ0 = 0, yielding
the following equation for the resonance frequency:
(ω
γ
)2
−(FθθFϕϕ − F 2
θϕ
)
M2S sin2 θ0 = 0 ⇒ ω
γ=
1MS sin θ0
√(FθθFϕϕ − F 2
θϕ
).
(2.38)
If one has an expression for the free energy, from which also the equilibriumangles ϕ0 and θ0 can be determined, (2.38) yields the resonance conditionω(B) or an expression for the resonance field Bres as function of the anglesof the external field for a fixed frequency. Equation (2.38) shows that FMR issensitive to the curvature of the free energy surface. As this surface stronglydepends on the anisotropy fields, FMR is a very useful tool to quantitativelydetermine magnetic anisotropy. We finally note that (2.38) presents a sin-gularity at the angle θ0 = 0. This problem is removed from the resonanceequation by adding the first derivatives as discussed in [73], the improvedform being:
(ω
γ
)2
=1M2
S
[Fθθ
(Fϕϕ
sin2 θ0 +cos θ0
sin θ0 Fθ
)−(Fθϕ
sin θ0 − cos θ0
sin θ0
Fϕ
sin θ0
)2].
(2.39)
For θ0 = π/2, i.e. for the in-plane configuration (see Fig. 2.1) the latter res-onance equation has the same form as the original one, since the prefactorcos θ0/ sin θ0 vanishes. Also for other angles, except θ0 = 0, the original formis still numerically correct, since at equilibrium the first derivatives Fθ andFϕ are zero. The original equation is, however, not convenient anymore as the
70 J. Lindner and M. Farle
different terms of the free energy are mixed, covering the symmetries. Theform by Baselgia et al. [73] is therefore favorable compared to the older ver-sion of the resonance equation, in particular when out-of-plane dependenciesare considered.
2.3.4 Resonance Equations for Angular and FrequencyDependent FMR
In the following the resonance equations for tetragonal, cubic and hexagonalfilms will be explicitly given. As for thin films shape anisotropy and uniaxial in-plane terms play an important role, they are included (see (2.26) and (2.23)).Moreover, the Zeeman energy FZee = −M · B due to the presence of the ex-ternal field is included. Only the equations for the so-called ‘saturated’ modesare given, being solutions, for which the precessional motion of the magneti-zation is mainly determined by the external field. So called ‘unsaturated’ (ornot aligned) modes are solutions, for which the motion of the magnetizationvector is strongly influenced by internal fields (e.g. anisotropy fields). Suchmodes usually occur for small external magnetic fields and only a numericaldescription is possible. One should keep in mind that in the following res-onance equations not aligned modes are excluded. This implies that we setϕB = ϕ0 for the out-of-plane geometries (meaning that the magnetization isconfined to the same plane, in which the external magnetic field is varied) andθB = θ0 for the in-plane geometry.
Cubic and Tetragonal Symmetry: Out-of-plane Geometry
For the out-of-plane geometry the external magnetic field is varied in a planethat comprises the film normal and one principal in-plane crystallographicaxis (see also Fig. 2.1 for the coordinate system in use).
(001)-Orientation
The free energy used to derive the resonance equations for tetragonal films(of which cubic ones are a special case) is (see Sect. 2.3.2):
F = − MB0 (sin θ sin θB cos (ϕ− ϕB) + cos θ cos θB)
+ K2‖ sin2 θ cos2 (ϕ− δ) −(μ0
2(N⊥ −N‖
)M2 −K2⊥
)sin2 θ
− 12K4⊥ cos4 θ − 1
8K4‖ (3 + cos 4ϕ) sin4 θ. (2.40)
The equation includes cubic systems, as can be seen by setting K4⊥ = K4‖ =K4. Except for a constant term, which does not lead to anisotropy this yieldsthe expression for cubic symmetry as given in Sect. 2.3.2. Then, (2.39) for
2 Magnetic Anisotropy of Heterostructures 71
the out-of-plane geometry, for which the external field is varied from the filmnormal [001] to the [100]-direction (ϕ0 = ϕB = 0) yields:
(ω
γ
)2
=[Bres
⊥ cosΔθ +(Meff +
K4⊥M
− K4‖M
)cos 2 θ0
+(K4⊥M
+K4‖M
)cos 4 θ0 + u1
](2.41)
·[Bres
⊥ cosΔθ +(Meff − 4K4‖
M
)cos2 θ0
+(
2K4⊥M
+2K4‖M
)cos4 θ0 +
2K4‖M
+ u2
]− u3,
with Δθ = θ0−θB and Meff = 2K2⊥M −μ0
(N⊥ −N‖
)M denoting the effective
out-of-plane anisotropy field. For Meff < 0 (> 0) the easy axis of the systemlies in (normal to) the film plane. Note that according to Fig. 2.1 the anglesθ of the magnetization and θB of the external field are measured with respectto the film normal, while the in-plane angles ϕ and ϕB were defined withrespect to the [100]-direction. The terms ui resulting from an uniaxial in-planeanisotropy are listed in Table 2.7. The set of the ui being appropriate to agiven out-of-plane geometry is determined by the in-plane angle of the externalmagnetic field ϕB (being equal to the equilibrium angle of the magnetizationϕ0 for the reasons mentioned at the beginning of Sect. 2.3.4). For cubic systemsone has to set K4⊥ = K4‖ = K4 within the equation.For the out-of-plane geometry, for which the external field is varied from thefilm normal [001] to the [110]-direction (ϕ0 = ϕB = −π/4) the followingequation results:
(ω
γ
)2
=[Bres
⊥ cosΔθ +(Meff +
K4⊥M
− K4‖2M
)cos 2 θ0
+(K4⊥M
+K4‖2M
)cos 4 θ0 + u1
](2.42)
·[Bres
⊥ cosΔθ +(Meff +
K4‖M
)cos2 θ0
+(
2K4⊥M
+K4‖M
)cos4 θ0 − 2K4‖
M+ u2
]− u3.
Again the replacement K4⊥ = K4‖ = K4 leads to the special case of cubicsymmetry.
(011) and (111)-Orientation
In this case the same free energy expression is used as for the (001)-orientation(2.40) with the only difference that the cubic anisotropy contribution beingproportional toK4 is now given by (2.14) for (011)-oriented films and by (2.16)
72 J. Lindner and M. Farle
in case of (111)-orientation. This yields for the case that the external field isvaried from the film normal ([011]- or [111]-direction) to the in-plane direction([011] or ϕB = 90◦ in case of (011)-orientation and [110] or ϕB = 90◦ for the(111)-orientation):
(ω
γ
)2
=[Bres
⊥ cosΔθ +Meff cos 2θ0 + a+ u1
](2.43)
· [Bres⊥ cosΔθ +Meff cos2 θ + b + u2
]− u3.
While the ui are the same as given by Table 2.7, the terms a and b are listedin Table 2.8. For the case that the external field is varied from the film normal([011]-direction) to the [100]-direction (ϕB = 90◦) different values for a and bresult which are also given in Table 2.8.
Cubic and Tetragonal Symmetry: In-plane Geometry
Using the free energy expressions according to Table 2.5 within the generalresonance equation ((2.39)), one obtains for the case that the magnetizationis restricted to the film plane:
(ω
γ
)2
=[(Bres
‖ cosΔϕ−Meff + a− u1
)] [Bres
‖ cosΔϕ+ b− u2
]− c2 ,
(2.44)
with Δϕ = ϕ0−ϕB. The relations for a, b and c are summarized in Table 2.9. Ifan uniaxial in-plane anisotropy is present, the terms u1 = 2K2‖
M cos2(ϕ0 − δ
)
and u2 = 2K2‖M cos 2
(ϕ0 − δ
)have to be added.
One should note that the angles ϕ and θ are measured with respect todifferent crystallographic axes for the different orientations, i.e. θ is measuredeither against the [111]-, the [011]- or the [001]-direction, ϕ against the [100]-direction in case of the (011)- and (001)-orientation and with respect to the[112]-direction in case of the (111)-orientation.
Table 2.7. Uniaxial in-plane terms contributing to the resonance equations for atetragonal (cubic) thin film with (001)-orientation. The equilibrium angle θ0 fromminimizing the free energy given by (2.40)
ϕB =ϕ0 u1 u2 u3
02K2‖
Mcos2 δ cos 2θ0 2K2‖
M
(cos2δ cos2θ0 − cos 2δ
) K22‖
M2 cos2θ0sin22δ∣∣ π4
∣∣ 2K2‖M
cos2( π
4+δ)cos 2θ0 2K2‖
M
(cos2( π
4+δ)cos2θ0+sin 2δ
) K22‖
M2 cos2θ0cos22δ∣∣ π2
∣∣ 2K2‖M
cos2 δ cos 2θ0 2K2‖M
(sin2δ cos2θ0 + cos 2δ
) K22‖
M2 cos2θ0sin22δ
2 Magnetic Anisotropy of Heterostructures 73
Table 2.8. Cubic terms within the resonance equations for (011)- and (111)-orientedcubic systems. The equilibrium angle θ0 from minimizing the free energy given by(2.40)
Plane a b
[111]to[110]
− K43M
cos θ0(16
√2 sin θ0 cos2 θ0
−10√
2 sin θ0 + 28 cos3 θ0
−27 cos θ0) − K4
M
− K43M
cos θ0(4√
2 sin θ0 cos2 θ0
−10√
2 sin θ0 + 7 cos3 θ0 − 3 cos θ0)
[011]to[100]
K4M
(12 cos4 θ0 − 13 cos2 θ0 + 2
)K4M
(3 cos4 θ0 − 7 cos2 θ0 + 2
)
[011]to[011]
− 2K4M
(8 cos4 θ − 8 cos2 θ + 1
) −K4M
(3 cos4 θ − 3 cos2 θ − 1
)
Hexagonal Symmetry
For hexagonal symmetry and (0001)-oriented films the resonance equation forthe in-plane variation of the external field (in the plane perpendicular to thec-axis) is given by the same equation as for tetragonal symmetry ((2.44)) witha and b listed in Table 2.9. For the out-of-plane geometry one can in most casesneglect the very small sixfold anisotropy in the azimuthal plane given by K6‖and only consider the out-of-plane constant of highest order (K2‖). Then, theresonance equation has the form of (2.42) and (2.43) when one sets K4i = 0.
2.4 Temperature Dependence of Magnetic Anisotropy
The macroscopic anisotropy energy density is temperature dependent. Thisstatement holds for the anisotropy contributions due to dipole-dipole and spin-orbit interaction. The shape anisotropy which is proportional to the squareof the magnetization (see Sect. 2.3.2) vanishes at the Curie temperature TC .
Table 2.9. Resonance equations for a cubic thin film with different crystallographicorientations as well as for an (001)-oriented tetragonal and a (0001)-oriented hexag-onal system. The equilibrium angle ϕ0 from minimizing the free energy given by(2.40)
Plane a b c
(001) K42M
(cos 4ϕ0 + 3
)2K4M
cos 4ϕ0 0(001)tetra.
K4‖2M
(cos 4ϕ0 + 3
) 2K4‖M
cos 4ϕ0 0
(011) K4M
(3 cos4 ϕ0 + cos2 ϕ0 − 2
)K4M
(12 cos4 ϕ0− 11 cos2 ϕ0 + 1
)0
(111) −K4M
0 −√2K4
Msin3ϕ0
(0001)hex.
− 4K4⊥+6K6⊥+6K6‖ sin 6ϕ0
M− 36K6‖
Msin 6ϕ0 0
74 J. Lindner and M. Farle
Also, the intrinsic magneto-crystalline (spin-orbit) MAE is temperature de-pendent (see e.g. [5, 12, 34, 55]) and vanishes at TC . This has been often over-looked in the comparison of theoretical studies (usual performed at T = 0 K)and experimental investigations (usually conducted at room temperature). Inregard to the microscopic origin of MAE, i.e. the anisotropy of the orbital mo-ment, it is surprising to experimentally measure a temperature dependence ofthe MAE. When one considers that the spin-orbit interaction (approximately70 meV in 3d ferromagnets) is temperature independent, and smearing outthe exchange split states at the Fermi level (order of eV) [38] does not affectthe easy direction of the magnetization, one has to conclude that the differ-ence of the orbital magnetic moment along the easy and the hard magneticaxis persists above TC . Unfortunately, there is no direct evidence for this,since the magnetic moment fluctuates too vividly in space and time aboveTC . Most techniques will measure an averaged magnetic moment only. How-ever, susceptibility measurements and paramagnetic resonance measurements(which actually measure the susceptibility at microwave frequencies) proofthe existence of atomic magnetic moments above TC even in an intinerantferromagnet like Ni. As the magnetic moment above TC is the same (exceptfor polarization of the conduction electrons) as the one measured below TC
(for T = 0 K), it is reasonable to conclude that the orbital magnetic momentis unchanged in the paramagnetic state. A direct proof of the existence ofthe orbital magnetic moment and its anisotropy in the paramagnetic stateis obtained by angular dependent measurements in the paramagnetic phaseof a ferromagnet in magnetic fields of several kOe. To our knowledge suchmeasurements can be performed by electron spin resonance (ESR, EPR) only[74]. Here, the deviations of the spectroscopic splitting factor which is pro-portional to the ratio of orbital to spin magnetic moment was found to bedifferent for different crystallographic directions and could be well describedin the framework of crystal field theory.How can one resolve the conceptual problem that the macroscopically mea-sured MAE is temperature dependent while its microscopic origin is not? Theclassical theory of the temperature dependence of the intrinsic anisotropy(see for example [75] and references therein) was worked out based on theassumption that around each lattice site there exists a region of short-rangemagnetic order in which the local anisotropy constants are temperature in-dependent. Due to thermal motion, the local instantaneous magnetizationsof these regions will be distributed randomly, and they produce the averagemagnetization of the crystal as a whole which vanishes at TC . This does notmean that the magnetic moment vector vanishes, but it fluctuates so quicklyand uncorrelated to other moments that the spatially and timely averagedmoment vanishes. Hence, also the macroscopically measurable MAE vanishes,it averages out above TC . This hand-waving argument has been quantifiedby expanding the MAE in a series of spherical harmonics Ylm(θ, ϕ), whichreflects the role of crystal field and spin-orbit interaction with temperaturedependent coefficients k2l,m(T ):
2 Magnetic Anisotropy of Heterostructures 75
F =∞∑l=0
2l∑m=−2l
k2l,mY2l,m . (2.45)
From a theoretical point of view the advantage of using spherical harmonicsis the fact that they are orthogonal. As anisotropy is an even function of themagnetization, polynomials off odd degree vanish in the expansion. Accord-ingly, the expansion for cubic systems is given by [55]
with the Y2l,m listed in Table 2.10. The equation for hexagonal symmetryincludes the special case of uniaxial symmetry. For uniaxial symmetry per-pendicular to the film plane, the equation is also valid when one sets k6,6 = 0.For uniaxial anisotropy in the film plane (see (2.26)) and perpendicular to thefilm plane (see (2.25)) the expression to first order is:
Funi = k0,0Y0,0 + k2,0Y2,0 + k2,2Y2,2 (2.48)
=1
105(70K2⊥ + 35K2‖
)Y0,0 − 1
21(14K2⊥ + 7K2‖
)Y2,0 +
16K2‖Y2,2 .
Table 2.10. Spherical harmonics used for the expansion of the free energy of cubic,uniaxial and hexagonal systems
Y0,0 = 1Y1,0 = αz
Y1,1 = αx
Y1,−1 = αy
Y2,0 = 12
(3α2
z − 1)
Y2,2 = 3(α2
x − α2y
)Y2,−2 = 3 (2αxαy)
Y4,0 = 18
(35α4
z − 30α2z + 3
)Y4,4 = 105
(α4
x + α4y − 6α2
xα2y
)Y4,−4 = 105 (4αxαy)
(α2
x − α2y
)
Y6,0 = 116
(231α6
z − 315α4z + 105α2
z − 5)
Y6,4 = 9452
(α4
x + α4y − 4αxαy
) (11α2
z − 1)
Y6,−4 = 9452
(4αxαy)(α2
x − α2y
) (11α2
z − 1)
Y6,6 = 10395(α2
x − α2y
) (α4
x − 14α2xα2
y + α4y
)Y6,−6 = 10395 (2αxαy)
(3α2
x − α2y
) (α2
x − 3α2y
)
76 J. Lindner and M. Farle
The relationship between the temperature variations of the anisotropy coeffi-cients ki and the magnetization M was theoretically [76] and experimentally[77] found to have the form:
k2l,m(T )k2l,m(0)
∝ M(T )Γ
M(0), (2.49)
where Γ = l(l + 1)/2, l being the order of the sperical harmonics. This givesfor example k2,m ∝ M(T )3, k4,m ∝ M(T )10. The Callen-Callen model doesnot identify the microscopic origin of the anisotropy coefficients, but includesthe contributions from magneto-elastic as well as magnetostrictive proper-ties entering into the spin hamiltonian through the combination of spin-orbitcoupling and crystal field splitting.
As the relation above holds for the anisotropy coefficients k2l,m, one has tobe careful when comparing to temperature dependencies of the experimentallymeasured anisotropy constants Ki. The relations between the k2l,m and theKi can be found from (2.46), (2.47) and 2.48). Assuming a typical tempera-ture dependence of the magnetization one can plot the anisotropy coefficientsas shown in Fig. 2.7. One sees that the k2,0 and k4,0 decrease monotonicallywith increasing temperature and vanish at TC . If one confuses these tem-perature dependent ki,0 with the usual magnetic anisotropy parameters Ki,one would draw the conclusion that a temperature change of the easy axis ofmagnetization is not possible [5]. However, one finds that if one rearrangesthe cos and sin terms in the Spherical harmonics in terms of increasing pow-ers that the new parameters Ki (the ones used in the experimental analysis)can vary in sign so that their temperature dependent change of sign in Co
0 100 200 300 400-2
0
2
TC
K2
k4
k2
K4�
Temperature (K)
Ki /
arb.units
K2(T) = 1.47 k2 - 3.3 k4(T)
K4�(T) = 3.85 k4(T)
K2
K(a
rb.
un
its)
i
Fig. 2.7. Temperature dependence of the coefficients k2l,m used when expanding thefree energy into spherical harmonics and the expected experimental Ki anisotropyparameters that are coefficients of an expansion into direction cosines (reproducedfrom [5])
2 Magnetic Anisotropy of Heterostructures 77
or Gd can be quantitatively understood. An illustrative example is plotted inFig. 2.7 showing that K2⊥ changes its sign. This behavior becomes also clearfrom (2.47), where one can see that k2,0 is a linear combination of K2⊥ andK4⊥ (for K6⊥ = 0).
There are only few experimental results on the power law dependenceof the second order normalized MAE on the magnetization. Γ = 2.6(5)for 5.6 monolayer (ML) Fe on Cu(100) [78] and Γ = 6.5 for W(110)/Fe6 nm/W(110) [79] was reported. While the value for Fe on Cu(100) showsreasonable agreement with the theory, the reason for the large value in thecase of Fe on W(110) is unclear. One may speculate that higher order con-tributions of the anisotropy were not properly accounted for. An unusualexponent Γ = 2.1 was also reported for bulk like FePt films [80, 81] andfound to be the result of delocalized induced Pt moments leading to a two-ionanisotropy. This result has been explained by ab initio electronic structure the-ory for L10 ordered FePt [47, 54]. The importance of separating higher orderterms from second order terms for this type of analysis was recently shown byZakeri et al. [83]. Here, the thickness of Fe layers on GaAs(001) was tuned toa critical thickness so that higher order anisotropies present in thicker layersvanished. In this case perfect agreement was found within the error bar of theexperiment with the theoretical prediction Γ = 3 (Fig. 2.8). A linear power lawcorrelation with Γ = 2.9 is observed in the experiment. The inset shows thedeviations from the linear behaviour, i.e. deviations from Γ = 2.9, for otherfilm thicknesses in which K4 contributions become important. Similarly, the
arb
.units
Fig. 2.8. Temperature dependence of the uniaxial out-of-plane anisotropy K2⊥(filled circles) and the magnetization M (open squares) for 5 ML Fe/GaAs(001).The inset shows the dependence for Fe films from 5 ML to 20 ML (‘Reprintedfigure with permission from Kh. Zakeri et al., Phys. Rev. B, Vol. 73, 052405 (2006).Copyright (2006) by the American Physical Society’)
78 J. Lindner and M. Farle
temperature dependence of the surface anisotropy and its relation to the sur-face magnetization following a different temperature dependence than the bulkone have been analyzed [5].
2.5 Selected Experimental Results
Before discussing the experimental results, a typical FMR-signal ist shortlyexplained. Within an FMR experiment, the specimen is placed in a cavity, intowhich microwaves are coupled that excite the magnetic system. To generateresonant absorption from the microwave field inside the cavity, the experi-ment is performed in an external magnetic dc-field that is varied while themicrowave frequency is kept constant. Detailed descriptions of various setupsmay be found elsewhere [4, 5, 7, 14]. FMR absorption spectroscopy measuresthe imaginary part of the high frequency susceptibility χ = mrf/hrf . mrf
is the dynamic contribution of the magnetization that is created due to thehigh frequency magnetic field hrf of the microwaves and, thus, χ determinesthe response of the magnetic system to the excitation (see [4, 5, 7, 14] fordetails).A typical FMR signal from a thin film measured at a microwave frequency
of 9 GHz is shown in Fig. 2.9. While the main plot shows the derivative ofthe signal obtained from the lock-in detection procedure, the inset shows theintegral, i.e. the absorption signal itself. Three pieces of information can bedirectly extracted: (i) The resonance field Bres that includes information onthe internal fields, such as anisotropy fields. (ii) The linewidth ΔB that yieldsinformation on magnetic damping and the distribution of internal magnetic
250 300 350 400 450 500 550 600-0.6
-0.4
-0.2
0
0.2
0.4
0.6
B (mT)0||
d’’/
dB
(arb
.u
nits)
c
DBpp
BresA300 400 500 600
0B (mT)0||
c’’
(arb
.u
nits)
Bres
DBpp
Iso
tro
pic
reso
na
nce
fie
ld
Fig. 2.9. Typical FMR spectrum of a thin film. The spectrum is measured asderivative of the high frequency susceptibility with respect to the external magneticfield The inset shows the integral of the spectrum (reproduced from [82])
2 Magnetic Anisotropy of Heterostructures 79
fields and (iii) the intensity of the signal that is proportional to the number ofmagnetic moments taking part in the resonance absorption. In the followingwe focus only on the analysis of the resonance field, as this quantity is theone used to investigate the MAE in thin film systems. From Fig. 2.9 it canbe seen that the resonance field is moved away from the so-called isotropicresonance field (dotted vertical line in Fig. 2.9), which is the field in case thatno anisotropy field is present. For a given microwave frequency ω = 2πν, theisotropic field is given by ω/γ = B0. This follows directly from (2.39) whenone uses the expression for the free energy F without anisotropy terms ((2.40)with Ki = N⊥ = N‖ = 0).
2.5.1 Films on Semiconducting and Insulating Substrates
In the following sections some examples of different systems are given, forwhich FMR was used to determine the MAE. Only metallic thin films werechosen that were epitaxially grown on different kind of single crystal sub-strates.
Fe on MgO(001)
Fe films on MgO(001) are known to be a prototype system for epitaxial growthof a metal on an insulating substrate. The main reasons for this are therather simple preparation of monoatomically flat MgO substrates and thefact that the interface hybridization and intermixing between Fe and MgO isvery small [84, 85]. The lattice mismatch of the Fe and the MgO lattice isreduced due to a rotation of the Fe lattice by 45◦ with respect to the MgO(meaning that the 〈110〉-directions of Fe are parallel to the 〈100〉-directions ofthe MgO) [86]. With the lattice constants of Fe (aFe = 0.287 nm) and MgO(aMgO = 0.421 nm) this leads to a lattice misfit of aMgO−√
2aF e
aMgO= 3.6%.
We do not want to give an overview over the magnetism of epitaxial Fe filmson MgO(001). For details we refer to some of the many publications on thissystem [85, 87, 88, 89]. In this chapter we will focus on how FMR is used toextract the MAE in form of the anisotropy constants that were introduced inthe preceeding sections.In Fig. 2.10 (a) FMR spectra of 10 nm Fe on MgO(001) are shown that weretaken at room temperature and at constant microwave frequencies of 9.2 and24 GHz. The external magnetic field B0 was applied along the direction givenin the Figure (note that the directions refer to the Fe lattice, see also the insetof Fig. 2.10 (b), where a hard sphere model of the Fe/MgO(001) system isshown). While at 24 GHz only one peak can be measured for a given angle ofthe external field, at 9.2 GHz two peaks are detectable. These two peaks onlyappear for external field angles close to the Fe 〈110〉-directions, while for otherangles no signal is observed. This can be seen better in Fig. 2.10 (b) and (d),where the complete in-plane angular dependence of the resonance fields mea-sured at the two frequencies is plotted (the angle ϕ is measured with respect
80 J. Lindner and M. Farle
0 100 200 300 400 5000
5
10
15
20
25
30
�( G
Hz)
B0 (mT)
B0|| [110] (hard in-plane axis)
B0|| [100] (easy axis)
0 100 200 300
0
B0|| [100]
24 GHz@RT
d�'
'/dB
0(a
rb.units
)
B0 (mT)
9.2 GHz@RT
B0|| [110] B
0|| [110]
-90 -60 -30 0 30 60 90
220
240
260
280
300
320
340
saturated mode
�B(°)
B(m
T)
res
Fe[110]
Fe[100]
Fe[110]
f=24GHz
T=RT
a)
-60 -30 0 30 6020
40
60
80
100
�B(°)
B(m
T)
res
saturated modeunsaturated mode
Fe[110]Fe[110]
f=9.2GHz
T=RT
[110]
(001)[110
]
Fe
MgO
b)
c) d)
-90 -60 -30 0 30 60 900
0.5
1.0
1.5
2.0
2.5
3.0
q (°)B
B(T
)re
s
out of field range
sat. mode [110]sat. mode [100]
Fe[001]
-90 -60 -30 0 30 60 900
0.5
1.0
1.5
2.0
2.5
qB(°)
B(T
)re
s
saturated modeunsaturated mode
out of field range
B
B(T
)re
s
Fe[110]
Fe[001]
f=9.2GHz
T=RT f=24GHz
T=RT
e) f)
Fe[110]
Fig. 2.10. In-plane angular dependence for 10 nm Fe/MgO(001) at room temper-ature at (b) 9.2 GHz and (d) 24 GHz. The solid lines are fits to the data (see text).a) shows typical FMR spectra measured along the given directions of the externalfield, c) shows the calculated dispersion relation. In (e) and (f) the out-of-planedependence is shown for the two frequencies
to the Fe [100]-direction). The two signals at 9.2 GHz appear within an anglerange of only 10◦ around the Fe 〈110〉-directions. At 24 GHz one clearly ob-serves a fourfold in-plane symmetry of Bres (note that only half of the wholedependence is shown) that is slightly disturbed by a twofold one. The latterleads to somewhat smaller resonance fields along the Fe [110]-direction com-
2 Magnetic Anisotropy of Heterostructures 81
pared to Fe [110]. The same symmetry is observed at 9.2 GHz. The minimumof the dependence at 24 GHz indicates that the Fe 〈100〉-directions are theeasy ones of the film.When the external field is varied in the plane given by either the Fe [110]-direction and the film normal ([001]-direction) or the Fe [100]-direction and thefilm normal, the situation is different. Figure 2.10 (e) shows the experimentalresult for 9.2 GHz, where (two) resonance signals are visible only within the[110]/[001]-plane, while in Fig. 2.10 (f) shows the data for 24 GHz, where theangular dependence within the [110]/[001]-plane (filled circles) as well as inthe [100]/[001]-plane (open circles) can be measured. All out-of-plane angulardependencies show a twofold symmetry with a minimum resonance field in theplane of the film. This immediately implies that the film normal is the hardaxis of the system. The twofold symmetry results from the effective magneti-zation Meff that is the sum of the demagnetizing field μ0M and the intrinsicuniaxial out-of-plane anisotropy field 2K2⊥/M (see Sect. 2.3.2).The solid lines within the dependencies shown in Fig. 2.10 (b) (d)–(f) arefits of the measured dependence according to (2.42) and (2.43) in case of theout-of-plane variation and according to (2.44) in case of the in-plane depen-dence. The equations were used for cubic(001) systems, including the effectivemagnetization Meff , a fourfold term K4 as well as an uniaxial in-plane termK2‖ as fitting parameters. For the gyromagnetic ratio γ a g-factor of 2.09(bulk Fe) was used. The equilibrium positions for a given angle of the ex-ternal field was found from numerically minimizing the free energy. All fourangular dependencies could be fitted with the same set of parameters thatare Meff = 2.1 T, 2μ0K4/M = 55 mT and 2μ0K2‖/M = 2 mT. The firsttwo contributions are very close to the Fe bulk value, showing that the filmhas a dominating fourfold volume anisotropy with an in-plane easy directiondue to shape anisotropy. The surprising contribution is the twofold in planeone. From the fit one can conclude that its easy direction is parallel to Fe[110] or parallel to MgO [100]. As MgO substrates preferentially exhibit stepsparallel to this direction, the small anisotropy with twofold symmetry canbe attributed to steps on the substrate. The example shows that FMR isa very sensitive tool to disentangle and quantitatively determine even smallanisotropy contributions.The nature of the modes detected at the two frequencies can be finally deducedfrom plotting the resonance dispersion ν(B0) using the anisotropy constantsthat have been determined from the angular dependencies. The resultingdispersion is shown in Fig. 2.10 (c). The expected resonance fields can beextracted from the intersections of the horizontal dashed lines with the dis-persion branches. The upper frequency branch is the one for the case thatthe external field is aligned parallel to the easy 〈100〉-directions, while thelower branch gives the dispersion for the case that the external field is alignedparallel to the hard in-plane axes (〈110〉-directions). One sees that at 24 GHzone resonance signal along the easy as well as hard direction is expected,while for the lower frequency of 9.2 GHz no signal along the easy axes can be
82 J. Lindner and M. Farle
measured as the dispersion for this case starts at frequencies above the oneused for the FMR experiment. The behavior of the dispersion along the hardaxes leads to the occurence of the two resonances that have been observedin experiment. The change of a negative to a positive dispersion results fromthe fact that at small fields a not aligned mode occurs, for which the magne-tization precesses around the internal anisotropy field. As the external fieldstrength increases, this mode becomes an aligned mode that precesses aroundthe external magnetic field.
Fe on GaAs(001)
Fe/GaAs(001) can be considered as a prototype system for metallic films on asemiconducting substrate and was investigated by many groups, mainly dueto its possible application within spintronic devices. A good review about thedifferent approaches used to grow high quality thin films and determine theirmagnetic properties may be found in [90]. In [91] Fe films grown on the Garich 4 × 6-reconstructed GaAs(001) surface were investigated using in situultrahigh vacuum (UHV)-FMR. This technique can be used within the UHVenvironment and thus allows for the study of films without protective layersthat usually have to be deposited on top of the film to avoid contamination.UHV-FMR is thus well suited for the analysis of surface anisotropy that mightstrongly differ from the volume contribution.In the insets of Fig. 2.11 typical FMR spectra at 9.3 GHz with the exter-
nal field parallel to the [110]-direction are shown for (a) a 5 ML and (b) a20 ML thick Fe film grown on 4 × 6-reconstructed GaAs(001). Note the verysmall linewidth ΔB=1.8 mT of the bulk like 20 ML film shown in the in-sets of Fig. 2.11 (b) indicating excellent magnetic homogeneity of the films.Figure 2.11 shows the polar angular dependence of the saturated resonancesignal, Bres, for (a) 5 ML Fe and (b) 20 ML Fe measured at room tem-perature. The solid lines are fits using (2.43) for cubic symmetry (i.e. forK4‖ = K4⊥) with the parameters given in Table 2.11. The maximum of theresonance field along the film normal indicates that the magnetization of thefilms favors an in-plane alignment at both thicknesses. For the 20 ML film thedifference between the resonance field in parallel and perpendicular configu-ration is larger than for the 5 ML film implying that the thin film has a largeranisotropy.The fits directly yield the anisotropy fields 2Ki/M . The g-value in (2.43) waschosen to be g = 2.09, which is the Fe-bulk value. From the anisotropy fieldsthe anisotropy constants were extracted using the bulk saturation magnetiza-tion (M=1.71× 106 A/m). The results for the magnetocrystalline anisotropyconstants are listed in Table 2.11 for different film thicknesses (d=5–20 ML).The value of K2⊥ in the first column of Table 2.11 does not dominate overthe shape anisotropy given by 1
2μ0M2 (i.e. Meff < 0). Consequently, the
magnetization lies in the film plane for all films. As K4 is positive for filmsabove 11 ML, the magnetization is aligned along the [100]-direction within
2 Magnetic Anisotropy of Heterostructures 83
0.8
-80° -60°-40°-20° 0° 20° 40° 60° 80°
-80° -60°-40°-20° 0° 20° 40° 60° 80°
0.0
0.2
0.4
0.6
1.0
1.2
-1
225 250
0
1 5 ML9.24 GHz
d�'
'/dB
[a.u
.]
B [mT]
9.24 GHz
4.03 GHz
polar angle
Bre
s(T
)
��
0.9
0.0
0.3
0.6
1.2
0 20 40 100 120-2
-1
0
120 ML
9.24 GHz
x32d
�''/d
B[a
.u.]
B [mT]
polar angle�
�
9.24 GHz
4.03 GHz
Bre
s(T
)a) 5 ML
b) 20 ML
Fig. 2.11. Out-of-plane angular dependence of the resonance field for (a) 5 MLFe and (b) 20 ML Fe/GaAs(001) measured at room temperature at microwavefrequencies of (open squares) 4 GHz and (open circles) 9.2 GHz. The inset showstypical FMR spectra measured in the film plane (θB = 90◦) (from [91])
the thicker films, which is the easy axis of bulk-Fe. For thinner films d≤7 MLa strong thickness dependent in-plane uniaxial anisotropy K2‖ is observed(Table 2.11). The interplay between K4 and K2‖ leads to a change of the easyaxis from the [100]- towards the [110]-direction around d≈7 ML. Qualitatively,the strong uniaxial in-plane anisotropy may be understood by considering thetwofold surface symmetry of the Fe-GaAs interface due to the 4×6 reconstruc-tion. The rectangular surface cell is supposed to be directly connected to theFe-Ga and Fe-As bonds at the interface and thus to the atomic configuration[92]. The uniaxial anisotropy could therefore be related to an uniaxial stresswithin the Fe film or to a change of the Fe band structure at the interface dueto hybridization. We come back to this point later.To investigate the in-plane anisotropy in more detail the in-plane angular de-pendence of the resonance field close to the [110]-direction was investigated.The result is presented in Fig. 2.12 (a), where the resonance field as a function
84 J. Lindner and M. Farle
Table 2.11. Magnetic anisotropy constants of Fe on GaAs(001) for different thick-ness at room temperature. The conversion to μeV/atom is given by 1×105 J/m3
of the in-plane angle is plotted for the 5 ML thick Fe film. With rotation of themagnetic field in the film plane the saturated resonance mode (open squares)moves to lower fields, whereas the unsaturated resonance mode (solid circles)moves to higher field values and within 2◦ of rotation the FMR signal dis-appears. Upon comparison to the case of Fe/MgO(001) (see Sect. 2.5.1) thisdirectly shows that the [110]-direction is the hard (in-plane) axis of the systemand that the lower field mode in an unsaturated one. Using the anisotropyconstants, which have been determined by the out-of-plane angular dependentmeasurements the fits for both saturated (closed circles) as well as unsatu-rated (open circles) modes reproduce the in-plane angular dependence aroundthe hard direction very well.
0.03 0.06 0.09 0.12 0.15
-20
-15
-10
-5
0
5
10
15
-20
-15
-10
-5
0
5
10
Ki (
eV
/ato
m)
�d(nm-1)
KI
(10
5J/
m3)
x20
(ML)6820 15 11 d
.
7 5
K4
K2II
K2�110
120
130
140
-47° -46°
Bre
s(m
T)
in-plane angle jB
[1 1 0] -44° -43°
a) b)
Fig. 2.12. (a) In-plane angular dependence of a 5 ML thick Fe film on GaAs(001)measured at a microwave frequency of 9.3 GHz. (b) Surface and volume anisotropiesfor Fe/GaAs(001). The open triangles denote the uniaxial out-of-plane anisotropy,the open circles the uniaxial in-plane anisotropy and the filled squares the fourfoldanisotropy (from [91])
2 Magnetic Anisotropy of Heterostructures 85
Table 2.12. Surface/interface and volume contributions to the magnetic constantsof Fe on GaAs(001) at room temperature. The conversion to μeV/atom is given by1×10−3 J/m2=515 μeV/atom
In order to separate volume and surface/interface anisotropy contribu-tions we plot in Fig. 2.12 (b) the anisotropy constants as a function of filmthickness according to (2.27). The volume and surface/interface anisotropyconstants resulting from this analysis are summarized in Table 2.12. We startthe discussion with the uniaxial out-of-plane contribution K2⊥. The volumecontribution Kv
2⊥ = –1.7±0.8×105 J/m3 (–13±6 μeV/atom) is very small. Itis close to the value for Au capped Fe/GaAs reported by McPhail et al. (Kv
2⊥= 1.2 ±0.7×105 J/m3 = 9±6 μeV/atom) [93]. The negative sign of Kv
2⊥ indi-cates a preferential alignment of the magnetization in the film plane due to thisvolume term, which is enhanced by the larger contribution to Meff , mainlyresulting from the shape anisotropy. The much larger surface/interface termKs,eff
2⊥ =1.17±0.1×10−3 J/m2 (600±50 μeV/atom) is a superposition of the Fe-vacuum surface and Fe-GaAs interface anisotropy. As shown by the positivesign of Ks,eff
2⊥ the interface anisotropy favors an easy axis out-of-plane, whichis well known also to be the case for thin Fe films on Cu(001) [94]. As the in-terface contribution gets more important for thinner films, the reduced valueof Meff at lower film thicknesses is a direct result of the interface anisotropyof second order.The analysis of Fig. 2.12 (b) shows that the fourfold anisotropy vanishes below7 ML indicating a transition from cubic to predominantly uniaxial symmetry.This transition has also been observed by Brockmann et al. at a film thick-nesses of 8±1 ML [95]. The thickness dependence of the fourfold anisotropyconstant (Fig. 2.12 (b)) yields a negative surface/interface contribution (Ks,eff
4
= –6.1±0.1×10−5J/m2=–31±1 μeV/atom and a positive volume contribu-tion (Kv
4 = 0.66±0.1×105 J/m3 = 4.8±0.75 μeV/atom) close to bulk iron(K4,bulk = 0.47×105 J/m3=3.5 μeV/atom). These values indicate that theinterior part of the thinner films exhibits a rather moderate strain. Kv
4 is re-sponsible for the alignment of the magnetization parallel to the [100]-direction(d>7 ML), which is the easy axis for bulk-Fe. The decrease of K4 at smallerfilm thickness results from the negative interface contribution Ks,eff
4 . The neg-ative sign indicates that the 〈110〉 directions are the favored easy axes, whichis different from the bulk easy axes 〈100〉 (positive K4).
86 J. Lindner and M. Farle
Similar to the uniaxial out-of-plane anisotropy, the thickness dependenceof the uniaxial in-plane anisotropy K2‖ shows a very small value for thevolume contribution Kv
2‖ = 0.18±0.25×105J/m3(1.3±1.85 μeV/atom), prov-ing that the uniaxial anisotropy is an interface effect. It should be noted thatwithin the error bar this volume value of Kv
2‖ is approximately zero. The sur-face/interface contribution is, however, large. From the fit in Fig. 2.12 (b)one gets Ks,eff
2‖ = –8.9 ±0.4×10−5 J/m2 (–46.3±2 μeV/atom). This value
was also found for Au capped samples studied by McPhail et al. (Ks,eff2‖
= 10 ±1×10−5J/m2=52±6 μeV/atom) [93] and also by Brockmann et al.(Ks,eff
2‖ = 12 ±2×10−5J/m2=62±11 μeV/atom) [95]. The negative sign of
Ks,eff2‖ shows that the easy axis given by the uniaxial in-plane anisotropy is
the [110]-direction, whereas the [110] is the hard in-plane direction of our Fefilms. One should note that some confusion concerning the identification of thecrystallographic in-plane directions occurred in the literature (see e.g. [96]),where [110]- and [110]-direction were erroneously exchanged. The correct de-scription is given in [97]. The change of the easy axis for thicker films towardsthe [100]-direction, which occurs at about 7 ML results therefore from theincreasing influence of the volume part of the fourfold anisotropy Kv
4 .
Using a magneto-elastic model the anisotropy constants can be investi-gated in more detail. As shown by quantitative studies of the stress evolutionduring Fe deposition [98] the Fe films were found to present a compressivestress of –3.5 GPa at the initial stage of growth (first 2–3 ML). This stress iseven larger than the one resulting from an ideal coherent growth, for whichthe stress would be given by the 1.36% misfit between Fe (a =0.2866 nm) andGaAs (a/2 =0.2827 nm) yielding a compressive stress of –2.8 GPa [98]. Thisenhancement was explained in terms of surface stress changes when the sub-strate reconstruction changes to the new interface consisting of Fe, Ga and Asatoms. Within the model used in [91] Fe is assumed to be uniaxially strainedat the interface. The compression of the lattice parallel to the [110]-directionresults in a contribution of the magneto-elastic energy F cub
MEL per unit volumeto the overall free energy density (see Sect. 2.3.2 for the discussion of magneto-elastic contributions to the free energy). For Fe/GaAs(001), the strain isgiven by ε12 parallel to the (110)-direction. This leads to a contribution ofF cub
MEL given by 2B2α1α2ε12. Note that ε12 is negative in our case due tothe compressive stress [98]. In [99] the magneto-elastic constants of Fe onGa-terminated GaAs(001) were measured. For 25 nm thick Fe films valuesof B1 = 3.5 × 106 J/m3 and B2 = 7.2 × 106 J/m3 were obtained, while theFe-bulk values are given by B1 = −3.44×106 J/m3 and B2 = 7.62×106 J/m3,respectively. Using the direction cosines α1 = sin θ cosφ and α2 = sin θ sinφone gets FMEL = +B2ε12 along the (110)-direction (α1 = α2 =
√2/2) and
FMEL = −B2ε12 along the (110)-direction (α1 = −α2 =√
2/2). With B2 > 0and ε12 < 0 a total energy reduction along the (110)-direction results, whereas
2 Magnetic Anisotropy of Heterostructures 87
the (110)-direction becomes a hard one in excellent agreement to our re-sults for the films with d<7 ML. The contribution of the magneto-elasticanisotropy is directly given by the energy difference for the two directions, i.e.KMEL
2‖ = FMEL,110 − FMEL,110 = 2B2ε12. For ε12 = −1.7% (corresponding toa stress of –3.5 GPa)KMEL
2‖ = −2.6 × 105 J/m3 results. This has to be com-
pared to the experimental value for the uniaxial interface anisotropy Ks,eff2‖ /d
= –6.2×105J/m3, where d is the thickness of the monolayer (d = 0.1433 nm).The value is of the correct order of magnitude and thus strain at the interfaceplays a crucial role for the uniaxial anisotropy in the film plane that domi-nates at small film thicknesses below ≈ 7 ML. The fact that the experimentalvalue is larger than the estimated one due to strain indicates, however, thatother sources possibly contribute to the uniaxial interface anisotropy as well.One reason could be hybridization of the Fe bands with the ones of GaAs,in particular when interface intermixing is present. However, to verify thismodel, theoretical support is needed.While the interface obviously exhibits a strain of uniaxial character, in thevolume of the films no uniaxial strain persists as can be concluded from thevanishing value of Kv
2‖. The experiments in [98] showed that the Fe films staycompressed even for film thicknesses up to 20 nm. The magnitude of this stressis small due to the incorporation of misfit dislocations. An x-ray absorptionfine structure experiment in [100] showed that a 10 ML thick film is strainedby –1.1%. Since the experiments in [91] show that no uniaxial strain is presentin the volume, the strain in the volume is governed by compressive strain ofbiaxial character. This is consistent with a nearly cubic bcc environment ofthe Fe in the inner part of the films, and – unlike at the interface – the twofoldsymmetry of the reconstructed GaAs substrate has no effect. Hence, the in-plane biaxial strain in the volume is given by ε11 = ε22 ≡ ε. In [15] it hasbeen shown that a biaxial strain does not lead to an in-plane anisotropy forcubic (001) systems. It leads, however, to an uniaxial anisotropy contributionperpendicular to the film surface given by B1 (ε− ε33). The strain componentperpendicular to the surface ε33 can be calculated from elastic theory to be−2 c12
c11ε [15]. cij are the elastic constants of the material. For Fe c12 ≈ c11, so
that ε33 ≈ −2ε and one obtains a (perpendicular) uniaxial anisotropy termgiven by Kv
2⊥ ≈ 3B1ε. Using the experimental value for B1 = 3.5× 106 J/m3
from [99] and ε ≈ −1% according to [100] Kv2⊥ ≈ –1.0×105 J/m3 results.
This is within the error bar the same value as the experimental one (Kv2⊥
=–1.7±0.8×105 J/m3) with the same negative sign showing that strain is theorigin for Kv
2⊥ . The fact that the film strain becomes smaller for thicker filmsis confirmed by the observation of the fourfold volume anisotropy term Kv
4
that is very close to the bulk Fe value.
88 J. Lindner and M. Farle
2.5.2 Films on Metallic Surfaces
The growth of metallic films on metallic substrates will be discussed using theexample of the system Ni/Cu(001). This system became the focus of manystudies, as it presents an out-of-plane easy axis of magnetization (see [5]). Usu-ally the shape anisotropy overcomes the magneto-crystalline contribution, sothat the magnetization will be aligned in the film plane. Due to the small mag-netic moment of Ni, shape anisotropy for Ni films is small. Such spin reorienta-tion transitions (SRTs) of the magnetization in ultrathin ferromagnetic filmshave attracted much interest, experimentally [101, 102, 103, 104, 105, 106] aswell as theoretically [57, 107, 108, 109, 110, 111, 112]. One reason for the inter-est is that temperature driven SRTs may become of technological importance,leading to new magnetic thin film sensors or switches. From a basic researchpoint of view, one should note that the existence of a perpendicular magnetiza-tion is non trivial since the demagnetizing energy and the entropy of disorder[9] favors in-plane magnetization in thin films. In the case of Ni/Cu(001) it isnot the surface contribution to the intrinsic anisotropy, but the volume part,that lead to the easy axis out of the film plane. The complicated interplaybetween all contributions can be experimentally manipulated yielding directaccess to the thickness of the reorientation transition from in to out-of-plane.
Ni/Cu(001)
Ni films on Cu(001) with thicknesses below 10–11 ML exhibit a film mag-netization with easy axis in the film plane. For larger thicknesses Ni filmsshow a spin reorientation transition (change in sign of Meff ), which drivesthe magnetization out of the film plane [5]. This transition is shifted downto about 7.5 ML when capping the films with Cu [113]. In addition, the TC
is reduced by a Cu overlayer. The complete out-of-plane angular dependentmeasurement for Ni at room temperature is shown in Fig. 2.13, where Bres
as a function of θB for 8 ML of Ni/Cu(001) (a) and 9 ML of Ni/Cu(001) (b)
-45° 0° 45° 90°q B
0
2
4
6
8
B(1
0m
T)
res
2
M = 1.336(8) kGeff,8
M = -4.80(5) kGeff,8
a)
-90° 0° 90°q B
0
1
3
5
B(1
0m
T)
res
2
M = -1.9(1) kGeff,9
M = 2.56(4) kGeff,9
b)
45°-45°
Fig. 2.13. Out-of-plane angular dependence for (left panel) 8 ML and (right panel)9 ML Ni/Cu(001) before (filled circles) and after (open circles) capping the filmswith 4 ML of Cu (reproduced from [14])
2 Magnetic Anisotropy of Heterostructures 89
is plotted. In each panel we show the results before (filled circles) and aftercapping the film with 4 ML of Cu (open circles). For the bare Ni films thenegative values of Meff indicate an easy axis in-plane. The smaller value ofthe Ni9 film3 shows that this film is closer to the spin reorientation at about10–11 ML, where Meff is close to zero and thus, the higher order anisotropyconstants become important [5]. After capping the two films with Cu the min-imum of Bres moves from θB = 90◦ to θB = 0◦. This means that the easyaxis has switched from in the film plane towards the film normal. This changeof the easy axis is also reflected in the positive sign of Meff . Besides the signof Meff its value is larger for the Cu4Ni9 film compared to the Cu4Ni8 film,which means an increase of the out-of-plane anisotropy within the Cu cappedNi films.
This trend is continued with increasing Ni thickness (Fig. 2.14). Figure 2.14(c) shows the dispersion curves according to (2.42) with K4i = K2‖ = 0 asdetermined from experimentally measured Bres values for Ni films with thick-nesses ranging between 8 and 12 ML already capped with Cu. The externalfield was aligned along the in-plane direction (θB = 90◦). The points of in-tersection of the 9 GHz line with the dispersion yield the resonance fields.Figure 2.14 (a) shows the original FMR spectra for the smallest (8 ML) andthe largest (12 ML) Ni thickness. For 12 ML two resonances are observed(unsaturated and saturated mode, see Sect. 2.5.1 for details). This behav-ior can be explained with an increase in the perpendicular anisotropy givenby K2⊥ and corresponds to the case of a film having an out-of-plane easyaxis of magnetization. The complete analysis presented in Fig. 2.14 (c) indi-cates that the positive values of Meff increase with Ni thickness. Therefore,two resonances are observed for thicknesses larger than 10 ML. The transi-tion between observing one and two signals is observed for the 10 ML film,which was investigated at different temperatures in the range 300–400 K.The transition between observing one and two signals can be seen directlyin Fig. 2.14 (b), where the measured resonance fields of the film are plot-ted as a function of the temperature. At room temperature one observestwo signals, which are also shown in the inset for T = 313 K. The reso-nance field for the signal at low field values moves towards smaller fieldsuntil it vanishes at about T = 375 K. Above 375 K only one resonance isobserved as shown in the inset for T = 380 K, which stays even up to thehighest temperature of 400 K. As Fig. 2.14 (d) shows, this behavior can beexplained quantitatively with the help of the dispersion calulations. Due tothe decrease of Meff at higher temperatures the left branch of the dispersioncurve moves below the 9 GHz line, so that only one signal is observed at largerT values.We have seen that capping the Ni films with Cu induces a shift of the thick-ness, at which (for constant temperature) the reorientation of the easy axisof the magnetization occurs. Cu tends to decrease the critical thickness. This
3 In the following the number of monolayers is given as subscript.
90 J. Lindner and M. Farle
0 200 400 600 8000
10
20
Microwavefrequency
d)
T=310K, M =372(5) mT
360K, 302(5) mT380K, 220(5) mT400K, 206(5) mT
eff
B (mT)0||
(Ghz)
c)
0 200 400 600 800 10000
20
10
B (mT)0||
Cu Ni /Cu(001), M =532(5) mT
Cu Ni 403(5) mT
Cu Ni 256(4) mT
Cu Ni 133.6(8) mT
4 12 eff
4 10
5 9
4 8
(Ghz)
300 320 340 360 380 4000
2
4
6
T (K)
Cu Ni /Cu(001), f=9GHz, B II [110]4 10 0
B (mT)0||
200 400 600
0
T=313 K
T=380 K
d/d
B(a
rb)
c¢¢
.units
B(1
0m
T)
res
2
b)a)
B (mT)0||
300 400 500 600 700
0
Cu Ni /Cu(001)4 8
Cu Ni /Cu(001)4 12
d/d
B(a
rb.units)
c¢¢
a)
� �
Fig. 2.14. a) Spectra for 8 ML and 12 ML thick Ni films capped by 4 ML of Cu.b) Resonance fields as a function of the temperature for the two signals observed ina 10 ML thick Ni film capped by 4 ML of Cu. One of the signal vanishes at highertemperatures. The inset shows the spectra at two temperatures. c) Microwave reso-nance frequency as a function of the external field for Ni films with thicknesses from8–12 ML. d) Microwave frequency as a function of the external field for various tem-peratures calculated for the same film shown in a). The calculation was performedsuch that the observed resonance fields are best reproduced by the intersections ofthe calculated curve with the 9 GHz line being the experimental frequency. Fromthe calculation the anisotropy is derived despite the fact that not the whole angulardependence could be measured (reproduced from [14])
effect was the motivation by several groups to investigate capping layers ofdifferent kind and their influence on the reorientation thickness. Upon cappingthe Ni films with Cu and also investigating the same film without a protectivelayer within an UHV environment, one can disentangle the Ni/vacuum fromthe Ni/Cu interface anisotropy. In [113] the influence of CO on the interfaceanisotropy and in [114] the Ni films were grown on O-reconstructed Cu(001).The latter procedure interestingly leads to a chemically bonded O layer float-ing on top of the Ni film. The changes of the interface anisotropy are listedin Table 2.13. One sees that the growth on the O-reconstructed surface ismost effective. Using H2 has the advantage that the procedure is reversible as
2 Magnetic Anisotropy of Heterostructures 91
Table 2.13. Interface anisotropies for Ni films on Cu(001) with various cappinglayers. The error for Kint
2⊥ is 10% and 15% for O/Ni. The values for CO and H2 arefrom [113]. Note that the measurements in [113] are performed at room temperature
by heating the hydrogen can be driven out of the film. This example showsthat adsorbates and (or) chemically bonded layers can be effectively used toinfluence interface anisotropies and thus taylor magnetic properties of thinfilms.
The reason for the out-of-plane reorientation of Ni films can be deducedfrom thickness dependent measurements. The result, plotted as function ofthe reciprocal Ni film thickness, for bare Ni/Cu(001) (filled circles), Cucapped films (open circles) and films grown on a O-reconstructed Cu(001)surface (open squares, see [114] for details) are shown in Fig. 2.15. Accord-ing to (2.27) one extracts a negative surface anisotropy (that is the sum ofthe two interfaces) and a positive volume anisotropy that even overcomes the
0 0.05 0.10 0.15 0.20-20
-15
-10
-5
0
5
10
15
20
25
Ni/Cu(001)
CuNi/Cu(001)
Ni on O/Cu(001)
t=0.75
M2
1/d (1/ML)
K(µ
eV
/ato
m)
2^
0
2
Fig. 2.15. Surface and volume uniaxial out-of-plane anisotropy for (filledcircles) Ni/Cu(001), (open circles) Cu/Ni/Cu(001) and (open squares) Ni onO-reconstructed Cu(001) (reproduced from [82])
92 J. Lindner and M. Farle
shape anisotropy for thicknesses above the critical thickness of the reorienta-tion. Within the error the volume contribution is not affected by capping theNi film. A negative surface anisotropy favors an in-plane easy alignment and,thus, it is the volume anisotropy that leads to the out-of-plane easy axis. Thisis contrary to many other systems, where the surface MAE favors an out-of-plane easy axis (as e.g. for Fe/GaAs(001), see Table 2.12). In such a case,for a thickness of only a few ML, the surface contribution does not dominateanymore and the easy axis turns into the film plane. The fact that Ni filmson Cu(001) show the opposite effect make them a very interesting system. In[115] it was shown that the positive volume anisotropy is a direct consequenceof the tetragonal distortion of the Ni films, showing again that the structureand symmetry of the system plays a major role for the MAE.
Acknowledgments
We thank all our past and present coworkers who have contributed to theresults presented here. Although we have tried to include all the relevantliterature of the many groups working in this field, the list of references mustbe incomplete due to the large number of publications and we apologize forany missing citation. This work was supported by the DFG, Sfb 290 and 491.
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Abstract. The exchange bias effect, discovered more than fifty years ago, is afundamental interfacial property, which occurs between ferromagnetic and antifer-romagnetic materials. After intensive experimental and theoretical research overthe last ten years, a much clearer picture has emerged about this effect, which is ofimmense technical importance for magneto-electronic device applications. In this re-view we start with the discussion of numerical and analytical results of those modelswhich are based on the assumption of coherent rotation of the magnetization. Thebehavior of the ferromagnetic and antiferromagnetic spins during the magnetizationreversal, as well as the dependence of the critical fields on characteristic parameterssuch as exchange stiffness, magnetic anisotropy, interface disorder etc. are analyzedin detail and the most important models for exchange bias are reviewed. Finallyrecent experiments in the light of the presented models are discussed.
3.1 Fundamental Aspects of Exchange Bias: Introduction
The exchange bias (EB) effect was discovered 50 years ago by Meiklejohn andBean [1]. Meanwhile the EB effect has become an integral part of modernmagnetism with implications for basic research and for numerous device ap-plications. The EB effect was the first of its kind which relates to an interfaceeffect between two different classes of materials, here between a ferromagnetand an antiferromagnet. Later on the interlayer exchange coupling betweenferromagnets interleaved by paramagnet layers was discovered [2], and theproximity effect between ferromagnetic and superconducting layers was de-scribed [3, 4]. Recent reviews on these topics can be found in [5, 6, 7] and alsoin this book. The EB effect manifests itself in a shift of the hysteresis loopto negative or positive direction with respect to the applied field. Its originis related to the magnetic coupling across the common interface shared by aferromagnetic (F) and an antiferromagnetic (AF) layer. Extensive research isbeing carried out to unveil the details of this effect, which has resulted in more
F. Radu and H. Zabel: Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures,
then 600 publications in the last five years and since the last comprehensivereviews by Nogues and Schuller [8], Kiwi [9], Berkowitz [10], and Stamps [11].
An EB bilayer consists of two key elements, with rather different magneticand structural properties: the ferromagnetic layer and the antiferromagneticlayer. While the ferromagnetic layer can be studied in detail by using labo-ratory equipment like SQUID, MOKE and MFM, this is not the case for themagnetic interface and for the antiferromagnetic layer. The interface embed-ded in between the F and AF layer has low volume, therefore it is difficult toseparate its contribution from the F layer. Still, for the exchange bias effect theinterfacial magnetic properties are essential for understanding the effect. Forthis purpose polarized neutron scattering and soft-xray magnetic scatteringtechniques can reveal some of the key magnetic properties of the interface. TheAF layer has in principle no macroscopic magnetization, even so the magneticmoment of individual atoms is rather high. The magnetic properties of the AFmaterials are traditionally studied by neutron diffraction. In thin films, due tothe reduced AF volume available for scattering, this method is rather difficultto apply. Here, soft-xray magnetic scattering through the linear dichroism canreveal information about the magnetic properties of the AF layer, thereforeproviding useful insights into the origin of the EB effect.
From the application point of view the situation appears to be less com-plex. The effect is being used in spin valves with one pinned and one freeferromagnetic layer which are embedded in devices such as storage media,readout sensors, and magnetic random access memory(MRAM). Neverthe-less, robust, reliable and easy to control functional elements based on theexchange bias phenomenon require more understanding of the fundamentalsof the effect, which is further motivating research in this field.
Because of recent experimental verifications of the existence of interfaciallayers by several groups [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], earliermodels of EB need to be revisited and eventually modified to take into accountthe effects, which are introduced by this layer.
Here we first review some of the basic models for exchange bias. We focuson numerical calculations and analytical treatment of those models whichare based on the Stoner-Wohlfarth model [24, 25]. This has the advantagethat analytical expressions can be derived and a numerical analysis can bemuch more efficiently performed as compared to micromagnetic simulations.It has, however, the disadvantage that only coherent magnetization reversalprocesses are described within this formalism. Nevertheless, for a large fractionof experimental situations the Stoner-Wolhfarth approach is adequate. Thebehavior of the F and AF spins during the magnetization reversal, as wellas the dependence of the critical fields on the parameters of the F and AFlayer are analyzed in detail. The Meiklejohn and Bean [1, 26, 27] model andthe Mauri model [28] are revisited and numerical and analytical expressionsare compared. We continue with describing the main features of the RandomField model (RF) of Malozemoff [29, 30, 31] and the Domain State (DS)model [32, 33, 34, 35, 36, 84, 38, 39]. Then, we review the Kim and Stamp
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 99
approach [40], which focuses on a spring-like behavior of the AF layers andcoercivity enhancement. We continue with one of the most recent models forexchange bias, the Spin Glass (SG) model [41]. Assuming a realistic state of theinterface between the F and AF layers, the SG model describes well most of theimportant features of EB heterostructures, including azimuthal dependence ofexchange bias field and coercivity, AF and F thickness dependence, the inverselinear dependence on the lateral extension, and training effects. Finally we willdiscuss recent experiments in the light of the presented models. However, themain emphasis of this review is to describe the basic models in a systematicfashion and to compare them with recent experimental results.
3.2 Stoner-Wohlfarth Model
The term anisotropy refers to the orientation of the magnetic moments withrespect to given geometrical directions. In bulk materials the crystal axes arethe reference directions, while in thin films other reference systems becomeimportant. In order to account for the orientation of the magnetic momentsin magnetic materials, the minimum energy state is provided by analysisof the different contributing terms to the total magnetic energy: Zeemanterm, anisotropy terms, and exchange coupling terms. This evaluation is per-formed by minimization of the total magnetic energy with respect to variousparameters.
In the following we use the simplest possible expression for the total mag-netic energy for a ferromagnetic thin film and calculate the magnetic hysteresisloops. We assume that all spins are confined in the film plane and that thefilm has a uniaxial anisotropy. The response of the magnetization to an ap-plied magnetic field is then uniform. Therefore the spins will coherently rotateduring the variation of the external field. The direction of the magnetizationcan be described by only one parameter, namely the (θ − β) angle definingthe direction of the magnetization vector with respect to the applied field (seeFig. 3.1). Many complexities of the magnetization reversal are neglected inthis approach. Nevertheless, the Stoner-Wohlfarth (SW) model [24, 25, 42],named after the investigators who developed it for treating the magnetizationreversal of a small single domain, is used with considerable success for variousmagnetic thin films and heterostructures.
The total magnetic energy per unit volume of a ferromagnetic film within-plane uniaxial anisotropy reads:
EV (β) = −μ0HMF cos(θ − β) +KF sin2(β) , (3.1)
where the first term is the Zeeman energy contribution and the second term isthe magnetic crystalline anisotropy (MCA), here assumed to have an uniaxialsymmetry. The other parameters are: H for the applied field, MF for thesaturation magnetization of the ferromagnet, KF for the volume anisotropy
100 F. Radu and H. Zabel
H
MF
�
�
KF
Ferromagnetic film
Easy
ax
is
Hard axis
Fig. 3.1. Definition of angles and vectors used in Stoner-Wohlfarth type modelcalculations. The reference direction of the film is along the unidirectional anisotropy
constant of the ferromagnet, and θ for the orientation of the applied magneticfield with respect to the uniaxial anisotropy direction, and β the orientationof the magnetization vector during the magnetization reversal.
The minimization of the total magnetic energy with respect to the angleβ and the stability equation:
By solving (3.3) with the condition imposed by the (3.4) one obtains theangle β, which determines the longitudinal component (m|| = cos(β − θ))and the transverse component (m⊥ = sin(β − θ)) of the magnetization. Bothcomponents are plotted in Fig. 3.2 for different in-plane orientations (θ). Theevolution of the hysteresis loops for different angles θ between the appliedmagnetic field and the orientation of the uniaxial anisotropy is shown in theleft column of Fig. 3.2 and reflects the typical behavior of thin films within-plane uniaxial anisotropy. Along the easy axis the hysteresis loop is squareshaped and the transverse component is zero. When the applied field is per-pendicular to the anisotropy axis (hard axis), the hysteresis loop has a linearslope, whereas the transverse component is ovally shaped.
The expression for the coercive field can easily be inferred from (3.3):
Hc (θ) =2KF
μ0MF| cos θ| . (3.5)
For the hysteresis loops shown in Fig. 3.2, the coercive field follows in detailthe expression above. At the position of the easy axis (θ = 0, π) the coercive
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 101
Fig. 3.2. The longitudinal (left column) and transverse (right column) componentsof the magnetization for a film with in-plane uniaxial anisotropy. The curves aregenerated by numerical evaluation of (3.3)
field is equal to the anisotropy field Ha = 2KF/μ0MF , whereas along thehard axis (θ = ±π/2) the coercive field is zero. Experimentally, this is an oftenencountered situation. For instance, polycrystalline magnetic films grown ona-plane sapphire substrates show such uniaxial and growth induced anisotropydue to steps at the substrate surface.
Aside from the coercive field dependence as a function of the azimuthalangle θ, another critical field can be recognized in Fig. 3.2. This is the fieldwhere the magnetization changes irreversibly (i.e. where the hysteresis opens).This irreversible switching field Hirr can be extracted by solving both (3.3)and (3.4). Expressing the applied magnetic field H by its components alongthe easy and hard axis directions: H = (Hx, Hy ) = (H cos θ, H sin θ ),the solution of the system of (3.3) and (3.4) gives: Hx = −Ha cos3 β andHy = +Ha cos3 β. Eliminating β from the previous two equations we obtainthe asteroid equation [43, 44, 42]:
|Hx|2/3 + |Hy|2/3 = H2/3a . (3.6)
102 F. Radu and H. Zabel
Now, introducing back into the equation above the expression for the fieldcomponents we obtain for the irreversible switching field the following expres-sion [42, 43, 44]:
Hirr
Ha=
1[(sin2 θ)2/3 + (cos2 θ)2/3]3/2
. (3.7)
This field is plotted in the right panel of Fig. 3.3. At the position ofthe easy axis (θ = 0, π) the irreversible field is equal to the anisotropyfield Ha, whereas at θ = π/4, 3π/4, the irreversible field is equal to halfof the anisotropy field (Hirr (π/4) = Ha/2). The irreversible switching fieldcan be experimentally extracted from the transverse components of themagnetization, whereas the coercive field is extracted from the longitudinalcomponent of the magnetization(see Fig. 3.2).
Figure 3.6 shows the so called asteroid curve which defines stability criteriafor the magnetization reversal (3.6). The asteroid method refers to an elegantgeometrical solution of (3.1) introduced by Slonczewski [43]. An extendedanalysis can be found in [44]. The field measured in units of Ha appears asa point in Fig. 3.4. Given a field P1 outside the asteroid curve, two solutionscan be found by drawing tangent lines to the critical curve. Only one is astable solution and is given by the tangent closest to the easy axis, orientingthe magnetization towards the field. For fields inside the asteroid curve (P2)four tangents leading to four solutions can be drawn. Two solutions are stableand the other two are unstable. The magnetization is stable oriented alongthe corresponding tangent [43, 44].
Fig. 3.3. a) The azimuthal dependence of the normalized coercive field of a ferro-magnetic film with uniaxial anisotropy. The curve is calculated with (3.5). b) Theazimuthal dependence of the normalized irreversible switching field of a ferromag-netic film with uniaxial anisotropy. The curve is calculated using (3.7)
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 103
Fig. 3.4. The asteroid curve for a film with uniaxial anisotropy. Two situationsare depicted for finding a geometrical solution to the (3.6): a) The magnetic fieldrepresented as a point P1 lying outside the asteroid region exhibits one stable solu-tion (solid line with filled arrow) and one unstable one (solid line with open arrow).b) A magnetic field P2 within the asteroid curve exhibits four solutions (see thetangent lines): two of them are stable (solid line with filled arrow) and the othertwo are unstable (solid line with open arrow). The dotted line show tangents for theunstable solutions [42, 43, 44]
3.3 Discovery of the Exchange Bias Effect
The exchange bias (EB) effect, also known as unidirectional anisotropy, wasdiscovered in 1956 by Meiklejohn and Bean [1, 26, 27] when studying Co parti-cles embedded in their native antiferromagnetic oxide CoO. It was concludedfrom the beginning that the displacement of the hysteresis loop is broughtabout by the existence of an oxide layer surrounding the Co particles. Thisimplies that the magnetic interaction across their common interface is essen-tial in establishing the effect. Being recognized as an interfacial effect, thestudies of the EB effect have been performed mainly on thin films consistingof a ferromagnetic layer in contact with an antiferromagnetic one. Recently,however, the lithographically prepared structures as well as F and AF particlesare studied with renewed vigor.
104 F. Radu and H. Zabel
Fig. 3.5. a) Hysteresis loops of Co-CoO particles taken at 77◦K. The dashed lineshows the loop after cooling in zero field. The solid line is the hysteresis loop mea-sured after cooling the system in a field of 10 kOe. b) Torque curve for Co particlesat 300 K showing uniaxial anisotropy. b) Torque curve of Co-CoO particles taken at77◦K showing the unusual unidirectioal anisotropy. d) The torque magnetometer.The main component is a spring which measures the torque as a function of the θangle on a sample placed in a magnetic field. [1, 26]
In Fig. 3.5 the original figures from [1, 26] show the shift of a hystere-sis loop of Co-CoO particles. The system was cooled from room temperaturedown to 77 K through the Neel temperature of CoO (TN (CoO) = 291 K).The magnetization curve is shown in Fig. 3.5a) as a dashed line. It is sym-metrically centered around zero value of the applied field, which is the generalbehavior of ferromagnetic materials. When, however, the sample is cooled ina positive magnetic field, the hysteresis loop is displaced to negative values(see continuous line of Fig. 3.5a)). Such displacement did not disappear evenwhen extremely high applied fields of 70 000 Oe were used.
In order to get more insight into this unusual effect, the authors studiedthe anisotropy behavior by using a self-made torque magnetometer schemati-cally shown in Fig. 3.5d). It consists of a spring connected to a sample placedin an external magnetic field. Generally, torque magnetometry is an accu-rate method for measuring the magnetocrystalline anisotropy (MCA) of singlecrystal ferromagnets. The torque on a sample is measured as a function of theangle θ between certain crystallographic directions and the applied magneticfield. In strong external fields, when the magnetization of the sample is almostparallel to the applied field (saturation), the torque is equal to:
T = −∂E(θ)∂θ
,
where E(θ) is the MCA energy. In the case of Co, which has a hexagonal struc-ture, the torque about an axis perpendicular to the c-axis follows a sin(2θ)function as seen in Fig. 3.5b). The torque and the energy density can then bewritten as:
T = −K1 sin(2θ)
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 105
EV =∫K1 sin(2θ) dθ = K1 sin2(θ) +K0 ,
where K1 is the MCA anisotropy and K0 is an integration constant. It isclearly seen from the energy expression that along the c-axis, at θ = 0 andθ = 180◦, the particles are in a stable equilibrium. This typical case of auniaxial anisotropy is seen for the Co particles at room temperature, wherethe CoO is in a paramagnetic state. At 77◦K, after field cooling, the CoO isin an antiferromagnetic state. Here, the torque curve of the Co-CoO systemlooks completely different as seen in Fig. 3.5c). The torque curve is a functionof sin(θ):
T = −Ku sin(θ) , (3.8)
hence,
EV =∫Ku sin(θ) dθ = −Ku cos(θ) +K0 . (3.9)
The energy function shows that the particles are in equilibrium for oneposition only, namely θ = 0. Rotating the sample to any angle, it tries toreturn to the original position. This direction is parallel to the field coolingdirection and such anisotropy was named unidirectional anisotropy.
Now, one can analyze whether the same unidirectional anisotropy observedby torque magnetometry is also responsible for the loop shift. In Fig. 3.1 areshown schematically the vectors involved in writing the energy per unit volumefor a ferromagnetic layer with uniaxial anisotropy having the magnetizationoriented opposite to the field. It reads:
EV = −μ0HMF cos(−β) +KF sin2(β) , (3.10)
where H is the external field, MF is the saturation magnetization of theferromagnet per unit volume, and KF is the MCA of the F layer. The twoterms entering in the formula above are the Zeeman interaction energy of theexternal field with the magnetization of the F layer and the MCA energy ofthe F layer, respectively. Now, writing the stability conditions and assumingthat the field is parallel to the easy axis, we find that the coercive field is:
Hc = 2KF/(μ0MF ) . (3.11)
Next step is to cool the system down in an external magnetic field and tointroduce in (3.10) the unidirectional anisotropy term. The expression for theenergy density then becomes:
EV = −μ0HMF cos(−β) +KF sin2(β) −Ku cos(β) . (3.12)
We notice that the solution is identical to the previous case (3.10) with thesubstitution of an effective field: H ′ = H+Ku/MF . This causes the hysteresisloop to be shifted by −Ku/(μ0MF ). Thus, Meiklejohn and Bean concludedthat the loop displacement is equivalent to the explanation for the unidirec-tional anisotropy.
106 F. Radu and H. Zabel
Besides the shift of the magnetization curve and the unidirectionalanisotropy, Meiklejohn and Bean have observed another effect when measur-ing the torque curves. Their experiments revealed an appreciable hysteresis ofthe torque (see Fig. 3.9 and Fig.3.10 of [26] and Fig. 3.2 of [27] ), indicatingthat irreversible changes of the magnetic state of the sample take place whenrotating the sample in an external magnetic field. As the system did not dis-play any rotational anisotropy when the AF was in the paramagnetic state,this provided evidence for the coupling between the AF CoO shell and theF Co core. Such irreversible changes were suggested to take place in the AFlayer.
3.4 Ideal Model of the Exchange Bias: Phenomenology
The macroscopic observation of the magnetization curve shift due to uni-directional anisotropy of a F/AF bilayer can qualitatively be understoodby analyzing the microscopic magnetic state of their common interface.Phenomenologically, the onset of exchange bias is depicted in Fig. 3.6.A ferromagnetic layer is in close contact to an antiferromagnetic one. Theircritical temperatures should satisfy the condition: TC > TN , where TC is theCurie temperature of the ferromagnetic layer and TN is the Neel temperatureof the antiferromagnetic layer. At a temperature which is higher than the Neeltemperature of the AF layer and lower than the Curie temperature of the fer-romagnet (TN < T < TC), the F spins align along the direction of the appliedfield, whereas the AF spins remain randomly oriented in a paramagnetic state(see Fig 3.6(1)). The hysteresis curve of the ferromagnet is centered aroundzero, not being affected by the proximity of the AF layer. Next, we saturatethe ferromagnet by applying a high enough external field HFC and then, with-out changing the magnitude or direction of the applied field, the temperatureis decreased to a finite value lower then TN (field cooling procedure).
After field cooling the system, due to exchange interaction at the interfacethe first monolayer of the AF layer will align parallel (or antiparallel) to theF spins. The next monolayer of the antiferromagnet will align antiparallel tothe previous layer as to complete AF order, and so on (see Fig 3.6(2)). Notethat the spins at the AF interface are uncompensated, leading to a finite netmagnetization of this monolayer. It is assumed that both the ferromagnet andthe antiferromagnet are in a single domain state and that they will remainin this single domain state during the magnetization reversal process. Whenreversing the field, the F spins will try to rotate in-plane to the opposite di-rection. Being coupled to the AF spins, it takes a bigger force and therefore astronger external field to overcome this coupling and to rotate the ferromag-netic spins. As a result, the first coercive field is higher than it used to be atT > TN , where the F/AF interaction is not yet active (Fig 3.6(3)). On theway back from negative saturation to positive field values (Fig 3.6(4)), the Fspins require a smaller external force in order to rotate back (Fig 3.6(5)) to
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 107
AF
F
M
M
HFC
H TN
FC
TC
0
H
HHEB
1)
2)3)
4) 5)
Fig. 3.6. Phenomenological model of exchange bias for an AF-F bilayer. 1) The spinconfiguration at a temperature which is higher than TN and smaller than TC . TheAF layer is in a paramagnetic state while the F layer is ordered. Its magnetizationcurve (top-right) is centered around zero value of the applied field. Panel 2): the spinconfiguration of the AF and F layer after field cooling the system through TN of theAF layer in a positive applied magnetic field (HF C). Due to uncompensated spinsat the AF interface, the F layer is coupled to the AF layer. Panel 4): the saturatedstate at negative fields. Panel 3) and 5) show the configuration of the spins duringthe remagnetization, assuming that this takes place through in-plane rotation of theF spins. The center of magnetization curve is displaced at negative values of theapplied field by Heb. (The description is in accordance with [1, 8, 26])
the original direction. A torque is acting on the F spins for all other angles,except the stable direction which is along the field cooling direction (unidirec-tional anisotropy). As a result, the magnetization curve is shifted to negativevalues of the applied field. This displacement of the center of the hysteresisloop is called exchange bias field, and is negative in relation to the orientationof the F spins after field cooling (negative exchange bias). It should be notedthat in this simple description the AF spins are considered to be rigid andfixed to the field cooling direction during the entire reversal process.
108 F. Radu and H. Zabel
3.5 The Ideal Meiklejohn-Bean Model: QuantitativeAnalysis
Based on their observation about the rotational anisotropy, Meiklejohn andBean proposed a model to account for the magnitude of the hysteresis shift.The assumptions made are the following [1, 8, 42]:
• The F layer rotates rigidly, as a whole;• Both the F and AF are in a single domain state;• The AF/F interface is atomically smooth;• The AF layer is magnetically rigid, meaning that the AF spins remain
unchanged during the rotation of the F spins;• The spins of the AF interface are fully uncompensated: the interface layer
has a net magnetic moment;• The F and the AF layers are coupled by an exchange interaction across
the F/AF interface. The parameter assigned to this interaction is the in-terfacial exchange coupling energy per unit area Jeb;
• The AF layer has an in-plane uniaxial anisotropy.
In general, for describing the coherent rotation of the magnetization vec-tor the Stoner-Wohlfarth [24, 25] model is used. Different energy terms canbe added as needed and to best account for the quantitative and qualita-tive behavior of the macroscopic magnetization reversal. In Fig. 3.7 is shown
H
MF
�
KFKAF ,
Ideal Meiklejohn-Bean Model
�
Fig. 3.7. Schematic view of the angles and vectors used in the ideal Meiklejohn andBean model. The AF layer is assumed to be rigid and no deviation from its initiallyset orientation is allowed. KAF and KF are the anisotropy of the AF layer andF layer, respectively, which are assumed to be parallel oriented to the field coolingdirection. β is the angle between magnetization vector MF of the F layer and theanisotropy direction of the F layer. This angle is variable during the magnetizationreversal. H is the external magnetic field which can be applied at any direction θwith respect to the field cooling direction at θ = 0 (see [1, 8, 26])
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 109
schematically the geometry of the vectors involved in the ideal Meiklejohn andBean model. H is the applied magnetic field, which makes an angle θ withrespect to the field cooling direction denoted by θ = 0, KF and KAF are theuniaxial anisotropy directions of the F and the AF layer, respectively. Theyare assumed to be oriented parallel to the field cooling direction. MF is themagnetization orientation of the F spins during the magnetization reversal. Itis assumed that the AF spins are fixed to their orientation defined during thefield cooling procedure (rigid AF). In the analysis below the angle (θ = 0) forthe applied field is assumed to be parallel to the field cooling direction. Thiscondition refers to the direction along which the hysteresis loops is measured,whereas θ �= 0 is used for torque measurements or for measuring the azimuthaldependence of the exchange bias field.
Within this model the energy per unit area assuming coherent rotation ofthe magnetization, can be written as [1, 8, 11]:
EA = −μ0HMF tF cos(−β) +KF tF sin2(β) − Jeb cos(β) , (3.13)
where Jeb [J/m2] is the interfacial exchange energy per unit area, and MF isthe saturation magnetization of the ferromagnetic layer. The interfacial ex-change energy can be further expressed in terms of pair exchange interactions:Eint =
∑ij Ji jS
AFi SF
j , where the summation includes all interactions withinthe range of the exchange coupling [29, 31, 45, 46].
The stability condition ∂EA / ∂θ = 0 has two types of solutions: one isβ = cos−1[(Jeb − μ0HMF tF )/(2KF )] for μ0HMF tF − Jeb ≤ 2KF ; theother one is β = 0, π for μ0HMF tF − Jeb ≥ 2KF , corresponding to positiveand negative saturation, respectively. The coercive fields Hc1 and Hc2 areextracted from the stability equation above for β = 0, π:
Hc1 = −2KF tF + Jeb
μ0MF tF(3.14)
Hc2 =2KF tF − Jeb
μ0MF tF. (3.15)
Using the expressions above, the coercive field Hc of the loop and thedisplacement Heb can be calculated according to:
Hc =−Hc1 +Hc2
2and Heb =
Hc1 +Hc2
2(3.16)
which further gives:
Hc =2KF
μ0MF, (3.17)
andHeb = − Jeb
μ0MF tF. (3.18)
Equation (3.18) is the master formula of the EB effect. It gives the expectedcharacteristics of the hysteresis loop for an ideal case, in particular the linear
110 F. Radu and H. Zabel
dependence on the interfacial energy Jeb and the inverse dependence on theferromagnetic layer thickness tF . Therefore this equation serves as a guidelineto which experimental values are compared. In the next section we will discusssome predictions of the model above.
3.5.1 The Sign of the Exchange Bias
Equation (3.18) predicts that the sign of the exchange bias is negative. Al-most all hysteresis loops shown in the literature are shifted oppositely to thefield cooling direction. The positive or negative exchange coupling across theinterface produces the same (negative) sign of the exchange bias field. Thereare, however, exceptions. Positive exchange bias was observed for CoO/Co,FexZn1−xF2/Co and Cu1−xMnx/Co bilayers when the measuring temperaturewas close to the blocking temperature [14, 47, 48, 49, 50]. At low tempera-tures positive exchange bias was observed in Fe/FeF2 [163] and Fe/MnF2 [51]bilayers. Specific of the last two systems is the low anisotropy of the anti-ferromagnet and the antiferromagnetic type of coupling between the F andAF layers. It was proposed that, at high cooling fields, the interface layer ofthe antiferromagnet aligns ferromagnetically with the external applied fieldand therefore ferromagnetically with the F itself. As the preferred orientationbetween the interface spins of the F layer and AF layer is the antiparallel one(AF coupling), the EB becomes positive. Further theoretical and experimentaldetails of the positive exchange bias mechanism are presented in [52, 53, 54].In the original Meiklejohn and Bean model the interaction of the cooling fieldwith the AF spins is not taken into account. However this interaction canbe easily introduced in their model. The positive exchange bias could also beaccounted for in the M&B model by simply changing the sign of Jeb in (3.13)from negative to positive.
3.5.2 The Magnitude of the EB
Often the exchange coupling parameter Jeb is identified with the exchangeconstant of the AF layer (JAF ). For various calculations a value ranging fromJAF to JF was assumed. For CoO, JAF = 21.6K = 1.86 meV [55]. Using thisvalue, the expected exchange coupling constant Jeb of a CoO(111)/F layercan be estimated as [29, 56]:
Jeb = NJAF /A = 4mJ/m2 , (3.19)
where N = 4 is the number of Co2+ ions at the uncompensated CoO interfaceper unit area A=
√(3) a2, and a = 4.27 A is the CoO lattice parameter. With
this number we would expect for a 100 A thick Co layer, which shares aninterface with a CoO AF layer, an exchange bias of:
Heb [Oe] =Jeb [J/m2]
MF [kA/m] tF [A]1011 (3.20)
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 111
Heb =0.004
1460× 1001011 = 2740 Oe .
This exchange bias field is by far bigger than experimentally observed. Sofar an ideal magnitude of the EB field as predicted by the equation (3.19)has not yet been observed, even so for some bilayers high EB fields weremeasured (see Table 3.1). We encounter here two problems: first, we do notknow how to evaluate the real coupling constant Jeb at the interfaces withvariable degrees of complexity, and the second, in reality interfaces are neveratomically smooth. The unknown interface was nicely labelled by Kiwi [9] as“a hard nut to crack”. Indeed, the features of the interfaces may be complexregarding the structure, the roughness, the magnetic properties, and domainstate of the AF and F layers.
In Table 3.1 are listed some EB data of systems with CoO as the AF layer.We focus on experimentally determined interfacial exchange coupling con-stants using Jeb = −Heb μ0MF tF . The observed exchange coupling constantis usually smaller then the expected value of 4 mJ/m2 for CoO/Co bilayersby a factor ranging from 3 to several orders of magnitude. One anomaly isseen for the multilayer system Co/CoO which is actually ∼3 times higher then
Table 3.1. Experimental values related to Co/CoO exchange bias systems. Thesymbols used in the table are: ebe-electron-beam evaporation, rsp-reactive sputter-ing, msp-magnetron sputtering, mbe-molecular beam epitaxy, F-ferromagnet, AF an-tiferromagnet, tAF -the thickness of the AF, tF -the thickness of the F, Heb-measuredexchange bias field, Hc-measured coercive field, TB-measured blocking temperature,Tmes-the measuring temperature, Jeb-the coupling energy extracted from the exper-imental value of exchange bias field (Jeb = −Heb (μ0MF tF ))
AF F tAF tF Heb Hc TB Tmes Jeb Ref[A] [A] [Oe] [Oe] [K] [K] [mJ/m2]
the expected value of 4mJ/m2, and to our knowledge is the highest value ob-served experimentally [60]. Such a variation of the experimental values for theinterfacial exchange coupling constant is motivating further considerations ofthe mechanisms controlling the EB effect.
3.5.3 The 1/tF Dependence of the EB Field
Equation (3.18) predicts that the variation of the EB field is proportional tothe inverse thickness of the ferromagnet:
Heb ≈ 1tF
. (3.21)
This dependence was subject of a large number of experimental investigations[8], because it is associated with the interfacial nature of the exchange biaseffect. For the CoO/Co bilayers no deviation was observed [61], even for verylow thicknesses (2 nm) of the Co layer [14]. For other systems with thin Flayers of the order of several nanometers it was observed that the 1/tF lawis not closely obeyed [8]. It was suggested that the F layer is no longer lat-erally continuous [8]. Deviations from 1/tF dependence for the other extremewhen the F layer is very thick were observed as well [8]. For this regime itis assumed that for F layers thicker than the domain wall thickness (500 nmfor permalloy), the F spins may vary appreciably across the film upon themagnetization reversal [64].
3.5.4 Coercivity and Exchange Bias
According to (3.17) the coercivity of the magnetic layer is the same withand without exchange bias effect. This contradicts experimental observations.Usually an increase of the coercive field is observed.
3.6 Realistic Meiklejohn and Bean Model
In [26] a new degree of freedom for the AF spins was introduced: the AF is stillrigid, but it can slightly rotate during the magnetization reversal as a wholeas indicated in Fig. 3.8. This parameter was introduced in order to account forthe rotational hysteresis observed during the torque measurements. Allowingthe AF layer to rotate is not in contradiction to the rigid state of the AFlayer, because it is allowed only to rotate as a whole. Therefore, the fourth as-sumption of the ideal M&B model in Sect. 3.5 is removed. The new conditionfor the AF spins is: α �= 0. With this new assumption, the equation (3.13)reads [8, 27]:
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 113
H
MAF
MF
��
KFKAF ,
Meiklejohn-Bean Model
Fig. 3.8. Schematic view of the angles and vectors used for the Meiklejohn and Beanmodel, allowing a rotation α of the AF layer as a whole with respect to the initiallyset orientation. M AF is the sublattice magnetization of the AF layer. KAF and KF
are the anisotropy of the AF layer and F layer, respectively, which are assumed to beparallel oriented to the field cooling direction. β is the angle between F magnetizationvector MF and the anisotropy direction of the F layer. This angle is variable duringthe magnetization reversal. H is the external magnetic field which can be appliedat any direction θ with respect to the field cooling direction at θ = 0 ([1, 8, 26, 27])
where tAF is the thickness of the antiferromagnet, and KAF is the MCA ofthe AF layer per unit area. The new energy term in the equation above ascompared to (3.13) is the anisotropy energy of the AF layer.
Equation (3.22) above can be analyzed numerically by minimization ofthe energy in respect to the α and the β angles. Below we will perform anumerical analysis of (3.23) and highlight a few of the conclusions discussedin [26, 27, 42]. The minimization with respect to α and β leads to a system oftwo equations:
H
H∞eb
sin(θ − β) + sin(β − α) = 0
R sin(2 α) − sin(β − α) = 0 . (3.23)
whereH∞
eb ≡ − Jeb
μ0MF tF(3.24)
114 F. Radu and H. Zabel
is the value of the exchange bias field when the anisotropy of the AF is in-finitely large, and
R ≡ KAF tAF
Jeb, (3.25)
is the parameter defining the ratio between the AF anisotropy energy andthe interfacial exchange energy Jeb. As we will see further below, exchangebias is only observed, if the AF anisotropy energy is bigger than the exchangeenergy. The unknown variables α and β are numerically extracted as a functionof the applied field H . Note that for clarity reasons the anisotropy of theferromagnet was neglected (KF = 0) in the system of equation above. As aresult the coercivity, which will be discussed further below, is not related tothe F layer anymore, but to the AF layer alone. Also, in order to simplify thediscussion we consider first the case θ = 0, which corresponds to measuring ahysteresis loop parallel to the field cooling direction.
Numerical evaluation of the (3.23) yields the angles:
• α of the AF spins as a function of the applied field during the hysteresismeasurement
• β of the F spins which rotate coherently during their reversal
The β angle defines completely the hysteresis loop and at the same time thecoercive fields Hc1 and Hc2. These fields, in turn, define the coercive field Hc
and the exchange bias field Heb (see equation (3.16)). The α angle influencesthe shape of the hysteresis loops when the R-ratio has low values, as we willsee below. For high R values the rotation angle of the antiferromagnet is closeto zero, giving a maximum exchange bias field equal to H∞
eb .The properties of the EB system originate from the properties of the AF
layer, which are accounted for by one parameter, the R-ratio. We will considerthe effect of the R-ratio on the angels β and α which, as stated above, definethe macroscopic behavior and the critical fields of the EB systems.
Numerical simulations of (3.23) as a function of R-ratio are shown inFig. 3.9 and in Fig. 3.10. We distinguish three physically distinct regions [26,27, 42, 65]:
• I. R ≥ 1In this region the coercive field is zero and the exchange bias field is fi-nite, decreasing from the asymptotic value H∞
eb to the lowest finite valueat R = 1. The AF spins rotate reversibly during the complete reversalof the F spins. The α angle has a maximum value as a function of theR-ratio, ranging from approximatively zero for R = ∞ to α = 45◦ atR = 1. Notice that as the maximum angle of the AF spins increases, aslight decrease of the exchange bias field is observed. When the R-ratioapproaches the critical value of unity, the exchange bias has a minimum.In the phase diagram, only range I can cause a shift of the magnetizationcurve. Simulated hysteresis loops for three different values of the R-ratioare shown in Fig. 3.10. One notices that not only the size of the exchange
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 115
Fig. 3.9. Left: The phase diagram of the exchange bias field and the coercivefields as a function of the Meiklejohn-Bean parameter R. Right: Typical behaviorof the antiferromagnetic angle α for the three different regions of the phase diagram.Only region I can lead to a shift of the hysteresis loop. In the other two regions acoercivity is observed but no exchange bias field
bias field decreases when the R-ratio approaches unity, but also the shapeof the hysteresis curve is changing. At high R-ratios the reversal is rathersharp, whereas for R-ratios close to unity it become more extended, almostresembling a spring-like behavior.
• II. 0.5 ≤ R < 1Characteristic for this region is that the AF spins are no more reversible.They follow the F spins and they change direction irreversibly, causinga coercive field at the expense of the exchange bias field, which becomeszero. Furthermore, depending on the field sweeping direction, there is ahysteresis-like behavior of the AF spin rotation. At a critical angle β ofthe F spin rotation, the AF spins cannot withstand the torque by thecoupling to the F spins and they jump in a discontinuous fashion to anotherangle (jump angle). The hysteresis loops corresponding to this region (seeFig. 3.10) are drastically different from the previous case. The coercivityshows a strong dependence as function of the R-ratio and they are notshifted at all. Moreover, the AF jump angles are clearly visible as kinks inthe hysteresis during the reversal.
• III. R < 0.5This region preserves the features of the previous one with one excep-tion, namely that the AF spins follow reversibly the F spins, withoutany jumps. Therefore, no hysteresis-like behavior of the α angle is seen.The exchange bias field is zero and the coercive field is finite, dependingon the R-ratio. The hysteresis loops shown in Fig. 3.10 are quite similarto a ferromagnet with uniaxial anisotropy. Within the Stoner-Wolhfarthmodel the resultant coercive field can be roughly approximated as [42]:Hc ≈ 2KAF tAF /(μ0MF tF ).
116 F. Radu and H. Zabel
Fig. 3.10. Simulation of several hysteresis loops and antiferromagnetic spin orienta-tions during the magnetization reversal. For the simulation we used the Meiklejohn-Bean formalism. Top row shows three hysteresis loops calculated for different Rratios within region I. The graph in the last column to the right shows the α angleof the AF layer for the three R values. The middle raw shows corresponding hystere-sis loops and α angles for R values in region II. The bottom row shows simulationsfor region III. Note that the scales for the α angle in the top panel is enlargedcompared to those in the lower two panels
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 117
It is easy to recognize that allowing the AF to rotate as a whole leads toan impressively rich phase diagram of the EB systems as a function of the pa-rameters of the AF layer (and F layer). The R-ratio can be varied across thewhole range from zero to infinity by changing the thickness of the AF layer [8],by varying its anisotropy (dilution of the AF layers with non-magnetic impu-rities [32, 66, 67]), or by varying the interfacial exchange energy Jeb (low doseion bombardment [68, 69, 70]). Recently [71], an almost ideal M&B behaviorhas been observed in Ni80Fe20/Fe50Mn50 bilayers. At high thicknesses of theAF layer the hysteresis loop is shifted to negative values and the coercivityis almost zero, whereas for reduced AF thicknesses a strong increase of thecoercive field is observed together with a drastic decrease of exchange bias.
3.6.1 Analytical Expression of the Exchange Bias Field
First we calculate analytically the expression of the exchange bias field forθ = 0 and KF = 0. The exact analytical solution is obtained by solving thesystem of (3.23) for β = 0, which leads to:
Heb =
⎧⎪⎨⎪⎩H∞
eb
√1 − 1
4R2 R ≥ 1
0 R < 1. (3.26)
This equation retains the 1/tF dependence of the exchange bias field, but atthe same time provides new features. The most important one is an addi-tional term, which effectively lowers the exchange bias field when the R-ratioapproaches the critical value of one. The R-ratio has three terms. One of themincludes the thickness of the AF layer. The analytical expression for the ex-change bias field (3.26) predicts that there is a critical AF thickness tcr
AF belowwhich the exchange bias cannot exist. This is [72]:
tcrAF =
Jeb
KAF. (3.27)
Below this critical thickness the interfacial energy is transformed into coerciv-ity. Above the critical thickness the exchange bias increases as a function ofthe AF layer thickness, reaching the asymptotic (ideal) value H∞
eb when tAF
is infinite. Most recent observation of an AF critical thickness can be foundin [73].
A similar expression as (3.26) was derived by Binek et al. [127] using aseries expansion of (3.22) with respect to α=0 : Heb ≈ H∞
eb (1 − 18R2 ) for
1/R≥0. Note that close to the critical value of R=1, the α angle could reachhigh values up to 45◦. Therefore, the series expansion with respect to α=0 isa good approximation for R>5.
118 F. Radu and H. Zabel
3.6.2 Azimuthal Dependence of the Exchange Bias Field
In the following we consider the exchange bias field in region I, where itacquires non vanishing values. The coercive fields and the exchange bias fieldare extracted from the condition β = θ+π/2 for both Hc1 and Hc2. This givesHc1 = Hc2 = (−Jeb/μ0MF tF ) cos(α(R, θ + π/2) − θ), where α(R, θ + π/2)is the value of the rotation angle of the AF spins at the coercive field. Withthe notation: α0 ≡ α(R, θ + π/2), and using the expression 3.16, the angulardependence of the exchange bias field becomes:
Heb(θ) =−Jeb
μ0MF tFcos( α0 − θ) . (3.28)
The equation above can be also written as:
Heb(θ) = −KAF tAF
μ0MF tFsin(2 α0) . (3.29)
Interestingly, the exchange coupling parameter Jeb in (3.28) is missing in (3.29),leaving instead an explicit dependence of the exchange bias field on the pa-rameters of the antiferromagnet and the ferromagnet. The exchange couplingconstant and the θ angle are accounted for by the AF angle α0.
Equations (3.28) and (3.29) are the most general expressions for an ex-change bias field. They include both, the influence of the rotation of the AFlayer and the influence of the azimuthal orientation of the applied field. More-over, the anisotropy and the thickness of the AF layer are explicitly shownin (3.29). To illustrate their generality we consider below two special cases forthe (3.28):
• θ = 0In this case the hysteresis loop is measured along the field cooling direction(θ = 0) and (3.28) becomes equivalent to the (3.26).
• R → ∞When R is very large (R � 1), α approximates zero, i.e. the rotation ofthe AF - layer becomes negligible. This is actually the original assumptionof the Meiklejohn and Bean model. Such a condition (R → ∞) is approx-imately satisfied for large thicknesses of the AF layer. Then the exchangebias field as function of θ can be written as [94, 127, 165]:
Hα=0eb (θ) =
−Jeb
μ0MF tFcos(θ) . (3.30)
In order to get more insight into the azimuthal dependence of the ex-change bias field, we show in Fig. 3.11 the normalized exchange bias fieldHeb(θ) /|H∞
eb (0)| as a function of the θ angle, according to (3.28) and (3.29),and for three different values of the R ratio (R = 1.1, R = 1.5, R = 20).The α0 angle (see Fig. 3.10) was obtained by numerically solving the system
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 119
Fig. 3.11. Azimuthal dependence of exchange bias as a function of the θ angle. Thecurves are calculated by the (3.28) and (3.29)
of (3.23). For large values of R, the azimuthal dependence of the exchange biasfield follows closely a cos(θ) unidirectional dependence. When, however, theR-ratio takes small values but larger then unity, the azimuthal behavior of theEB field deviates from the ideal unidirectional characteristic. There are twodistinctive features: one is that at θ = 0 the exchange bias field is reduced, andthe other one is that the maximum of the exchange bias field is shifted fromzero towards negative azimuthal angle values. According to (3.28) this shiftangle is equal to α0. In other words, the exchange bias field is not maximumalong the field cooling direction. Another striking feature is that the shiftedmaximum of the exchange bias field with respect to the azimuthal angle θdoes not depend on thickness and anisotropy of the AF layer:
HMAXeb = − Jeb
μ0MF tF. (3.31)
Summarizing we may state that, within the Meiklejohn and Bean model,a reduced exchange bias field is observed along the field cooling directiondepending on the parameters of the AF layer (KAF and tAF ). However, forR ≥ 1 the maximum value for the exchange bias field which is reached at θ �= 0does not depend on the anisotropy constant (KAF ) and thickness (tAF ) of theAF layer. The azimuthal characteristic of the exchange bias allows to extractall three essential parameters defining the exchange bias field: Jeb,KAF and,tAF .
The condition for extracting the Hc1 and Hc2 from the same β angle hidesan important property of the magnetization reversal of the ferromagneticlayer. This will be described next.
120 F. Radu and H. Zabel
3.6.3 Magnetization Reversal
A distinct feature of exchange bias phenomena is the magnetization reversalmechanism. In Fig. 3.12 is shown the parallel component of the magnetizationm|| = cos(β) versus the perpendicular component m⊥ = sin(β) for severalR-ratios and for θ = 30◦. The geometrical conventions are the ones shownin Fig. 3.8. We see that for R < 1 the reversal of the F spins is symmetric,similar to typical ferromagnets with uniaxial anisotropy. Although the regions0.5 ≤ R < 1 and R < 0.5 exhibit different reversal modes of the AF spins (seeFig. 3.10), there is little difference with respect to the F spin rotation. Forboth regions 0.5 ≤ R < 1 and R < 0.5 the F spins do make a full rotation,similar to the uniaxial ferromagnets. At the steep reversal branches one wouldexpect magnetic domain formation.
When R ≥ 1 another reversal mechanism is observed. The ferromagneticspins first rotate towards the unidirectional axis as lowering the field frompositive to negative values, and then the rotation proceeds continuously untilthe negative saturation is reached. On the return path, when the field isswept from negative to positive values, the ferromagnetic spins follow thesame path towards the positive saturation. The rotation is continuous withoutany additional steps or jumps. A similar behavior was observed theoreticallywithin the domain state model [36, 39]. The magnetization reversal modes canbe accessed experimentally by using the Vector-MOKE technique [74, 75, 76,41] which allows to follow both the magnetization vector and its angle duringthe reversal process.
Fig. 3.12. Magnetization reversal for several values of the R ratio. The parallelcomponent of the magnetization vector m|| = cos(β) is plotted as a function of theperpendicular component of the magnetization m⊥ = sin(β). The reversal for R < 1resembles the typical reversal of ferromagnets with uniaxial anisotropy. For R ≥ 1the reversal proceeds along the same path for the increasing and decreasing branchof the hysteresis loop. The angle θ = 30◦ is chosen arbitrarily
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 121
3.6.4 Rotational Hysteresis
We briefly discuss again the rotational hysteresis deduced from torque mea-surements [1, 26, 27], now in the light of the analysis provided above. Thetorque measurements were carried out in a strong applied magnetic field H.Therefore the applied field H and the magnetization MF can be assumed tobe parallel (β = θ). The torque is given by:
T = −∂E(θ)∂θ
= Jeb sin(θ − α(θ)).
This expression differs from (3.8) for the ideal model by the rotation of theAF spins through the α angle. However, this does not explain the energyloss during the torque measurements, as observed in the experiment (3.6(b)).The torque curve would only be a bit distorted but completely reversible.The integration of the energy curve predicts a rotational hysteresis Wrot = 0.In order to account for a finite rotational hysteresis, one can assume that afraction p of particles at the F-AF interface are uniaxially coupled behavingas in region II, whereas the remaining fraction (1 − p) of the F-AF particlesare coupled unidirectionally, having the ideal behavior as described in regimeI. As seen in Fig. 3.9(left), when the R-ratio of the uniaxial particle is in therange 0.5 ≤ R < 1, the AF spins will rotate irreversibly, showing hysteresis-like behavior due to α jumps indicated in the Fig. 3.9 (right). A rotationalhysteresis is not expected for unidirectional particles with R ≥ 1 because theAF structure changes reversibly with θ. With this assumption the uniaxialparticles will contribute to the energy loss during the torque measurements,while the unidirectional particles are responsible for the unidirectional featureof the torque curve. This argument was used by Meiklejohn and Bean [27]when studying the exchange bias in core-Co/shell-CoO. A fraction p = 0.5was inferred from the torque curves shown in Fig. 3.5.
3.7 Neel’s AF Domain Wall–Weak Coupling
Both concepts, the rigid AF spin state and rigidly rotating AF spins impose arestriction on the behavior of the antiferromagnetic spins, namely that the AForder is preserved during the magnetization reversal. Such restriction impliesthat the interfacial exchange coupling is found almost entirely in the hysteresisloop either as a loop shift or as coercivity. Experimentally, however, the size ofthe exchange bias does not agree with the expected value, being several ordersof magnitude lower then predicted. In order to cope with such loss of couplingenergy, one can assume that a partial domain wall develops in the AF layerduring the magnetization reversal. This concept was introduced by Neel [77, 9]when considering the coupling between a ferromagnet and a low anisotropyantiferromagnet. The AF partial domain wall will store an important fractionof the exchange coupling energy, lowering the shift of the hysteresis loop.
122 F. Radu and H. Zabel
Neel has calculated the magnetization orientation of each layer througha differential equation. The weak coupling is consistent with a partial AFdomain wall which is parallel to the interface (Neel domain wall). His modelpredicts that a minimum AF thickness is required to produce hysteresis shift.More importantly the partial domain wall concept forms the basis for furthermodels which incorporate either Neel wall or Bloch wall formation as a wayto reduce the observed magnitude of exchange bias.
3.8 Malozemoff Random Field Model
Malozemoff (1987) proposed a novel mechanism for exchange anisotropypostulating a random nature of exchange interactions at the F-AF inter-face [29, 30, 31]. He assumed that the chemical roughness or alloying at theinterface, which is present for any realistic bilayer system, causes lateral vari-ations of the exchange field acting on the F and AF layers. The resultantrandom field causes the AF to break up into magnetic domains due to theenergy minimization. By contrast with other theories, where the unidirec-tional anisotropy is treated either microscopically [78, 79, 80] or macroscopi-cally [1, 26, 28], the Malozemoff approach belongs to models on the mesoscopicscale for surface magnetism.
The general idea for estimating the exchange anisotropy is depicted inFig. 3.13, where a domain wall in an uniaxial ferromagnet is driven by anapplied in-plane magnetic field H [29]. Assuming that the interfacial energyin one domain (σ1) is different from the energy in the neighboring domain(σ2), then the exchange field can be estimated by the equilibrium conditionbetween the applied field pressure 2HMF tF and the effective pressure fromthe interfacial energy Δσ:
Heb =Δσ
2MF tF, (3.32)
where MF and tF are the magnetization and thickness of the ferromagnet.When the interface is treated as ideally “compensated”, then the exchangebias field is zero. On the other hand, if the AF/F interface is ideally uncom-pensated there is an interfacial energy difference Δσ = 2Ji/a
2, where Ji is
AF
F
Domain
WallM F
���
H
tF
tAF
Fig. 3.13. Schematic side view of a F/AF bilayer with a ferromagnetic wall drivenby an applied field H [29]
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 123
b) c)
x x xxx x x x
UNCOMPENSATED COMPENSATED
�= - J/a2
�= + J/a2
�= 0 �= 0
a) d)
Fig. 3.14. Schematic view of possible atomic configurations in a F-AF bilayer withideal interfaces. Frustrated bonds are indicated by crosses. Compensated configura-tion a) will result in configuration b) by reversing the ferromagnetic spins throughdomain wall movement. It gives an exchange bias field of Heb = Ji/a2MF tF . Thecompensated configuration c) will result in the compensated configuration d). Theexchange bias field for this case is zero (Heb = 0). [29]
the exchange coupling constant across the interface, and a is the lattice pa-rameter of a simple cubic structure assigned to the AF layer. The EB field isHeb = Ji/a
2MF tF (see Fig. 3.14)1.Estimating numerically the size of the EB field using the equation above
for an ideally uncompensated interface, results in a discrepancy of severalorders of magnitude with respect to the experimental observation . Therefore,a novel mechanism based on random fields at the interface acting on the AFlayer is proposed as to drastically reduce the resulting exchange bias field.
By simple and schematic arguments Malozemoff describes how roughnesson the atomic scale of a “compensated” AF interface layer can lead to un-compensated spins required for the loop to shift. An atomic rough interfacedepicted in Fig. 3.15a) containing a single mono-atomic bump in a cubic inter-face gives rise to six net antiferromagnetic deviations from a perfect compen-sation. A bump shifted by one lattice spacing as shown in Fig. 3.15b), whichis equivalent to reversing the F spins, provides six net ferromagnetic devia-tions from perfect compensation. Thus a net energy difference of ziJi withzi = 12 acts at the interface favoring one domain orientation over the other.
x
xx
xx
x
a)
x x x
b) c)
Fig. 3.15. Schematic side view of possible atomic moment configurations for a non-planar interface. The bump should be visualized on a two-dimensional interface.Configuration c) represents the lower energy state of a). The configuration b) is en-ergetically equivalent to flipping the ferromagnetic spins of a). The x signs representfrustrated bonds. [29]
1 For this example the energy is calculated as Ekl = −JiSkSl per pair of nearestneighbor spins kl at the interface.
124 F. Radu and H. Zabel
Note that for an ideally uncompensated interface the energy difference is only8Ji when reversing the F spins. This implies that an atomic step roughnessat a compensated interface leads to a higher exchange bias field as comparedto the ideally compensated interface.
The estimates of this local field can be further refined assuming a moredetailed model. For example, by inverting the spin in the bump shown inFig. 3.15c, the interfacial energy difference is reduced by 5 × 2Ji at the costof generating one frustrated pair in the AF layer just under the bump. Thisfrustrated pair increases the energy difference by 2JA, where JA is the AFexchange constant. Thus the energy difference between the two domains be-comes 2Ji + 2JA or roughly 4J if Ji ≈ JA ≈ J . If one allows localized cantingof the spins, one expects the energy difference to be reduced somewhat further.
Each interface irregularity will give a local energy difference between do-mains whose sign depends on the particular location of the irregularity andwhose magnitude is on the average 2zJ , where z is a number of order unity.Furthermore, for an interface which is random on the atomic scale, the lo-cal unidirectional interface energy σl = ±zJ/a2 will also be random andits average σ in a region L2 will go down statistically as σ ≈ σl/
√N , where
N = L2/a2 is the number of sites projected onto the interface plane. Thereforethe effective AF-F exchange energy per unit area is given by:
Jeb ≈ 1√NJi ≈ 1
LJi,
where Ji is the exchange energy of a fully uncompensated AF-surface.Given a random field provided by the interface roughness and assuming a
region with a single domain of the ferromagnet, it is energetically favorablefor the AF to break up into magnetic domains, as shown schematically inFig. 3.16. A perpendicular domain wall is the most preferable situation. Thisperpendicular domain wall is permanently present in the AF layer. It should bedistinguished from a domain wall parallel to AF/F interface, which accordingto the Mauri model [28] develops temporarily during the rotation of the Flayer.
By further analyzing the stability of the magnetic domains in the presenceof random fields, a characteristic length L of the frozen-in AF domains andtheir characteristic height are obtained: L ≈ π
√AAF /KAF and h = L/2,
whereAAF is the exchange stiffness and h is the characteristic height of the AF
x xx
Fig. 3.16. Schematic view of a vertical domain wall in the AF layer. It appears asan energetically favorable state of F/AF systems with rough interfaces [29, 81]
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 125
domains. Once these domains are fixed, flipping the ferromagnetic orientationcauses an energy change per unit area of Δσ = 4zJ/πaL, which further leadsto the following expression for the EB field [29]:
Heb =2 z
√AAFKAF
π2MF tF. (3.33)
Assuming a CoO/Co(100 A) film, the calculated exchange bias using the(3.33) is:
Heb =2 × 1
√0.0186×1.6×10−19[J]
4.27×10−10[m] 2.5 × 107 [J/m3]
π2 × 1460 [kA/m] 100× 10−10× 10
= 580Oe. (3.34)
For the estimations above we used for the exchange stiffness the followingvalue: AAF = JAF /a, where a is the lattice parameter of CoO (a = 4.27A)and JAF = 1.86 meV is the exchange constant for CoO [55].
The characteristic length of the AF domains is for CoO:
The height of the AF domains is h = L/2 = 8.3 A. Comparing this valueto the experimental data on CoO(25A)/Co studied in [82], we notice thatthe calculated EB field agrees well with the value observed experimentally.For example, the exchange bias field for CoO(25 A)/Co(119 A) is 557 Oe andthe theoretical value calculated with (3.33) is 487 Oe. Also, the length andthe height of the AF domains have enough space to develop. The differencebetween theory and experiment is, however, that experimentally AF domainscan occur and vary size and orientation after the very first magnetizationreversal, whereas within the Malozemoff model the AF domains are assumedto develop during the field cooling procedure. Nevertheless, the agreementappears to be excellent.
3.9 Domain State Model
The Domain State model (DS) introduced by Nowak and coworkers [32, 33,35, 36, 83, 84, 38] is a microscopic model in which disorder is introduced viamagnetic dilution not only at the interface but also in the bulk of the AFlayer as in Fig. 3.17. The key element in the model is that the AF layer isa diluted Ising antiferromagnet in an external magnetic field (DAFF) whichexhibits a phase diagram like the one shown in Fig. 3.18 [35]. In zero field the
126 F. Radu and H. Zabel
Fig. 3.17. Sketch of the domain state model with one ferromagnetic layer and threediluted antiferromagnetic layers. The dots mark defects [35]
system undergoes a phase transition from a disordered, paramagnetic state toa long-range-ordered antiferromagnetic phase at the dilution dependent Neeltemperature. In the low temperature region, for small fields, the long-rangeinteraction phase is stable in three dimensions. When the field is increased atlow temperature the diluted antiferromagnet develops a domain state phasewith a spin-glass-like behavior. The formation of the AF domains in the DSphase originates from the statistical imbalance of the number of impurities ofthe two AF sublattices within any finite region of the DAFF. This imbalanceleads to a net magnetization which couples to the external field. A spin reversalof the region, i. e., the creation of a domain, can lower the energy of the system.The formation of a domain wall can be minimized if the domain wall passesthrough nonmagnetic defects at a minimum cost of exchange energy.
Nowak et al. [35] further argue that during the field cooling below theirreversibility line Ti(B), in an external field and in the presence of the in-terfacial exchange field of the ferromagnet, the AF develops a frozen domainstate with an irreversible surplus of magnetization. This irreversible surplusmagnetization controls then the exchange bias.
The F layer is described by a classical Heisenberg model with the nearest-neighbor exchange constant JF . The AF is modelled as a magnetically dilutedIsing system with an easy axis parallel to that of the F. The Hamiltonian ofthe system is given by [35]:
Fig. 3.18. Schematic phase diagram of a three-dimentional diluted antiferromag-net. AF is the antiferromagnetic phase, DS the domain state phase, and PM theparamagnetic phase [35]
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 127
H = −JF
∑<i,j>εF
Si.Sj −∑iεF
(dzS2iz + dxS
2ix + μBSi)
−JAF
∑<i,j>εAF
εiεjσiσj −∑iεAF
μBzεiσi
−JINT
∑<iεAF,jεF>
εiσiSjz , (3.36)
where the Si and σi are the classical spin vectors at the ith site of the Fand AF, respectively. The first line contains the energy contribution of theF, the second line describes the diluted AF layer, and the third line includesthe exchange coupling across the interface between F and DAFF, where it isassumed that the Ising spins in the topmost layer of the DAFF interact withthe z component of the Heisenberg spins of the F layer.
In order to obtain the hysteresis loop of the system, the Hamiltonianin (3.36) is treated by Monte Carlo simulations. Typical hysteresis loops areshown in Fig. 3.19 [84], where both the magnetization curve of the F layerand of the interface monolayer of the DAFF are shown. The coercive field ex-tracted from the hysteresis curve depends on the anisotropy of the F layer, butit is also influenced by the DAFF. It, actually, depends on the thickness andanisotropy of the DAFF layer. The coercive field decreases with the increasingthickness of the DAFF layer [84], which can be understood as follows: the inter-face magnetization tries to orient the F layer along its direction. The coercivefield has to overcome this barrier, and the higher the interface magnetizationof the DAFF, the stronger is the field required to reverse the F layer. The
Fig. 3.19. Simulated hysteresis loops within the domain state model. The tophysteresis belongs to the ferromagnetic layer and the bottom hysteresis belongs toan AF interface monolayer [84]
128 F. Radu and H. Zabel
interface magnetization decreases with increasing DAFF thickness due to acoarsening of the AF domains accompanied by smoother domain walls.
The strength of the exchange bias field can be estimated from the (3.36)using simple ground state arguments. Assuming that all spins in the F remainparallel during the field reversal and some net magnetization of the interfacelayer of the DAFF remains constant during the reversal of the F, a simplecalculation gives the usual estimate for the bias field [35]:
lμBeb = JINTmINT , (3.37)
where l is the number of the F layers and mINT is the interface magneti-zation of the AF per spin. Beb is the notation for the exchange bias fieldin [35] and is equivalent to Heb in this chapter. For an ideal uncompensatedinterface (mINT = 1) the exchange bias is too high, whereas for an ideallycompensated interface the exchange bias is zero. Within the DS model theinterface magnetization mINT < 1 is neither a constant nor is it a simplequantity [35]. Therefore, it is replaced by mIDS , which is a measure of theirreversible domain state magnetization of the DAFF interface layer and isresponsible for the EB field. With this, an estimate of exchange bias field forl = 9, JINT = −3.2 × 10−22J , and μ = 1.7 μB gives a value of about 300 Oe.
The exchange bias field depends also on the bulk properties of the DAFFlayer as shown by Miltenyi et al. [32]. There the AF layer was diluted bysubstituting non-magnetic Mg in the bulk part and away from the interface.The representative results are shown in Fig. 3.20. It was shown experimentallythat the EB field depends strongly on the dilution of the AF layer. As afunction of concentration of the non-magnetic Mg impurities, the EB evolvesas following: at zero dilution the exchange bias has finite values, whereas byincreasing the Mg concentration, the EB field increases first, showing a broadpeak-like behavior, and then, when the dilution is further increased the EBfield decreases again. Simulations within the DS model showed an overallgood qualitative agreement. The peak-like behavior of the EB as a functionof the dilution is clearly seen in the simulations (see Fig. 3.20). However, itappears that at zero dilution, the DS gives vanishing exchange bias whereasexperimentally finite values are observed. The exchange bias is missing at lowdilutions because the domains in the AF cannot be formed as they would costtoo much energy to break the AF bonds. This discrepancy [35, 85] is thoughtto be explained by other imperfections, such as grain boundaries in the AFlayer which is similar to dilution and which can also reduce the domain-wallenergy, thus leading to domain formation and EB even without dilution of theAF bulk.
An important property of the kinetics of the DAFF is the slow relaxation ofthe remanent magnetization, i.e., the magnetization obtained after switchingoff the cooling field [35]. It is known that the remanent magnetization of theDS relaxes nonexponentially on extremely long-time scales after the field isswitched off or even within the applied field. In the DS model the EB is relatedto this remanent magnetization. This implies a decrease of EB due to slow
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 129
Fig. 3.20. In the right side of the figure is shown the film structure used to studythe dilution influence on the exchange bias field. a) EB field as function of the Mgconcentration x in the Co1−xMgxO layer for several temperatures. b) EB field as afunction of different dilutions of the AF volume. [32]
relaxation of the AF domain state. The reason for the training effect can beunderstood within the DS model from Fig. 3.19 bottom panel, where it isshown that the hysteresis loop of the AF interface layer is not closed on theright hand side. This implies that the DS magnetization is lost partly duringthe hysteresis loop due to a rearrangement of the AF domain structure. Thisloss of magnetization clearly leads to a reduction of the EB.
The blocking temperature 2 within the DS model can be understood byconsidering the phase diagram of the DAFF shown in Fig. 3.17. The frozenDS of the AF layer occurs after field cooling the system below the irre-versibility temperature Ti(b). Within this interpretation, the blocking tem-perature corresponds to Ti(b). Since Ti(b) < TN , the blocking tempera-ture should be always below the Neel temperature and should be depen-dent on the strength of the interface exchange field. The simulations withinthe DS model shows that EB depends linearly on the temperature, as ob-served experimentally in some Co/CoO systems, but no reason is givenfor this behavior [35]. In [86] the blocking temperature of a DAFF sys-tem (Fe1−xZnxF2(110)/Fe/Ag with x=0.4) exhibits a significant enhancement
2 The blocking temperature of an exchange bias system is the temperature wherethe hysteresis loop acquires a negative or positive shift with respect to the fieldaxis. It is always lower then the Neel temperature of the AF layer.
130 F. Radu and H. Zabel
with respect to the global ordering temperature TN=46.9 K, of the bulk an-tiferromagnet Fe0.6Zn0.4F2.
Overall, it is believed that strong support for the DS model is given byexperimental observations where nonmagnetic impurities are added to the AFlayer in a systematic and controlled fashion [32, 87, 69, 88, 66, 85, 67]. Also,good agreement has been observed in [89], where the dependence of the EB asa function of AF thickness and temperature for IrMn/Co was analyzed. Theasymmetry of the magnetization reversal mechanisms [36, 39] is shown to bedependent on the angle between the easy axis of the F and DAFF layers. It wasfound that either identical or different F reversal mechanisms (domain wallmovement or coherent rotation) can occur as the relative orientation betweenthe anisotropy axis of the F and AF is varied. This is discussed in more detailin Sect. 3.12.4 and 3.14.1.
3.10 Mauri Model
The model of Mauri et al. [28] renounces the assumption of a rigid AF layerand proposes that the AF spins develop a domain wall parallel to the interface.The motivation to introduce such an hypothesis was to explore a possiblereduction of the exchange bias field resulting from the Meiklejohn and Beanmodel.
The assumptions of the Mauri model are:
• both the F and AF are in a single domain state;• the F layer rotates rigidly, as a whole;• the AF layer develops a domain wall parallel to the interface;• the AF interface layer is uncompensated (or fully compensated);• the AF layer has a uniaxial anisotropy;• the cooling field is oriented parallel to the uniaxial anisotropy of the AF
layer;• the AF and F spins rotate coherently, therefore the Stoner-Wohlfarth
model is used to describe the system.
Schematically the spin configuration within the Mauri model is shown inFig. 3.21. The F spins rotate coherently, when the applied magnetic field isswept as to measure the hysteresis loop. The first interfacial AF monolayeris oriented away from the F spins making an angle α with the field coolingdirection and with the anisotropy axis of the AF layer. The next AF monolay-ers are oriented away from the interfacial AF spins as to form a domain wallparallel to the interface. The spins of only one AF sublattice are depicted, thespins of the other sublattice being oppositely oriented for completing the AForder. At a distance ξ at the interface, a ferromagnetic layer of thickness tFfollows. Using the Stoner-Wohlfarth model, the total magnetic energy can bewritten as [28]:
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 131
Fig. 3.21. Mauri model for the interface of a thin ferromagnetic film on a antifer-romagnetic substrate [28]
where the first term is the Zeeman energy of the ferromagnet in an appliedmagnetic field, the second term is the anisotropy term of the F layer, the thirdterm is the interfacial exchange energy and, the forth term is the energy of thepartial domain wall. The new parameter in the equation above is the exchangestiffness AAF . As in the case of the Meiklejohn and Bean model, the interfacialexchange coupling parameter Jeb [J/m2] is again undefined, assuming that itranges between the exchange constant of the F layer to the exchange constantof the AF layer divided by an effective area (see Sect. 3.5.2).
The total magnetic energy can be written in units of 2√AAFKAF , which
is the energy per unit surface of a 90◦ domain wall in the AF layer [28]:
e = k (1 − cos(β)) + μ cos(β)2
+λ [1 − cos(α − β)] + (1 − cos(α)), (3.39)
where λ = Jeb /( 2√AAFKAF ), is the interface exchange, with Jeb being re-
defined as Jeb ≡ A12 /ξ, where A12 is the interfacial exchange stiffness andξ is the thickness of the interface (see Fig. 3.21), μ = KF tF / 2
√AAFKAF
is the reduced ferromagnet anisotropy, and k = μ0HMF tF / 2√AAFKAF is
the reduced external magnetic field.Mauri et al. [28] have calculated the magnetization curves by numerical
minimization of the reduced total magnetic energy (3.39). Several values of theλ and μ parameters were considered providing quite realistic hysteresis loops.Their analysis highlights two limiting cases with the following expressions forthe exchange bias field:
Heb =
{− (A12 / ξ)/ μ0MF tF for λ� 1− 2
√AAFKAF/ μ0MF tF for λ� 1
. (3.40)
132 F. Radu and H. Zabel
In the strong coupling limit λ � 1, the expression for the exchange biasfield is similar to the value given by the Meiklejohn and Bean model. Forthis situation, practically no important differences between the predictions ofthe two models exist. When the coupling is weak (λ � 1), the Mauri modeldelivers a reduced exchange bias field which is practically independent of theinterfacial exchange energy. It depends on the domain wall energy and theparameters of the F layer. In either case the “1/tF ” dependence is preservedby the Mauri-model.
3.10.1 Analytical Expression of Exchange Bias Field
In order to compare the predictions of the Mauri model and the Meiklejohnand Bean approach, we reconsider the analysis of the free energy. Startingfrom the expression of the free energy (3.38), the minimization with respectto the α and β angles leads to the following system of equations:
⎧⎪⎨⎪⎩
H
− Jebμ0MF tF
sin(θ − β) + sin(β − α) = 0
2√
AAF KAF
Jebsin(α) − sin(β − α) = 0 .
(3.41)
Similar to the Meiklejohn and Bean model we define the parameters P ≡2√
AAF KAF
JebandH∞
eb ≡ − Jeb
μ0 MF tF. Also we set θ = 0, meaning that the applied
field is swept along the easy axis of the AF layer. Also, we do not take intoaccount the anisotropy of the ferromagnet (KF = 0), for two reasons. For onethe coercive fields of the exchange bias systems are usually much higher thenthe coercive field of the isolated F layer, and secondly it is easier to comparethe results of the Mauri model and the M&B model when the anisotropy ofthe F layer is disregarded. From inspection of the (3.41) and (3.23) one canclearly see that the first equations of the two systems are identical, while thesecond ones are different in two respects. The first difference is related to theterm P , which includes the domain wall energy instead of the AF anisotropyterm in the R ratio. The second difference is that instead of a sin(2 α) termin the second equation of (3.23), the Mauri model has a sin(α) term, whichinfluences strongly the phase diagram shown in Fig. 3.22.
The analytical expression for the exchange bias are obtained by solv-ing the system of (3.41). First step in solving the (3.41) is to extract theα angle (α = ± arccos(± P−cos(β)√
1+P 2−2 P cos(β))) from the second equation and to
introduce it in the first equation. Next we use the condition that at the coer-cive field β = −π/2 to obtain both coercive fields Hc1 = Hc2. Then, insertingthem into the general expressions of (3.16), the coercive field Hc is zero andthe exchange bias field becomes [11]:
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 133
Fig. 3.22. Left: The phase diagram of the exchange bias field and the coercive fieldsgiven by the Mauri formalism. Right: Typical behavior of the antiferromagnetic angleα for the two different regions of the phase diagram. In both regions I and II a shiftof the hysteresis loop can exist. The coercive field is zero in both regimes
Heb = − Jeb
μ0MF tF
2√AAF KAF√
J2eb + 4AAF KAF
= − Jeb
μ0MF tF
P√1 + P 2
. (3.42)
This equation is plotted as a function of H/|HEB| and for different P valuesranging from P = 0 to P = 5. (compare Fig. 3.22 left panel). The behaviorof the EB field according to the (3.42) is monotonic with respect to the stiff-ness and anisotropy of the AF spins. At P � 1 the exchange bias is equalto HP→∞
eb = − Jeb
μ0 MF tF, which is the well known expression given by M&B
model. When, however, the P -ratio approaches low values, the exchange biasdecreases, vanishing at P = 0, provided that the thickness of the AF layeris sufficiently thick to allow a 180◦ wall. With some analytical analysis ofthe (3.42) one can easily reach the limiting cases of weak coupling (P � 1)and strong coupling (P � 1) discussed by Mauri et al. [28] (see (3.40)). InFig. 3.22 right column is shown the representative behavior of the α angle ofthe first interfacial AF monolayer as a function of the β orientation of the Fspins during the magnetization reversal and for two representative P values(see the discussion below).
In Fig. 3.23 the hysteresis loops (m|| = cos(β)) and the correspondingAF angle rotation during the magnetization reversal are plotted for severalP -ratios. They were obtained by solving numerically the system of (3.41). Forall the values of the P -ratio the magnetization curves are shifted to negativevalues of the applied magnetic field. We distinguish two different regions withrespect to the behavior of the α angle of the first AF monolayer. In the firstregion, for P ≥ 1 (region I), the AF monolayer in the proximity of the F layerbehaves similar to the Meiklejohn and Bean, namely the α angle deviates
134 F. Radu and H. Zabel
Fig. 3.23. Several hysteresis loops and antiferromagnetic spin orientations as plot-ted during the magnetization reversal. For the simulation we used the Mauri for-malism. Top row shows three hysteresis loops calculated for different P ratios of theregion I shown in Fig. 3.22. The right hand panel in the top row shows the α angleof the antiferromagnetic layer for the three P parameters of the hysteresis loop. Thebottom raw are the corresponding hysteresis loops and α angles for the P valueswithin region II
reversibly from the anisotropy direction as function of β. The maximum valueof the α angle acquired during the rotation of the F layer is two times higherfor the Mauri model as compared to the M&B model, reaching a maximumvalue of 90◦ at P = 1. The coercive field in this region is zero. The angle α inthe region II where P < 1 has a completely different behavior. It rotates withthe ferromagnet following the general behavior depicted in Fig. 3.23. Noticethat α follows monotonically the rotation of the F spins, with no jumps orhysteresis-like behavior in contrast to the M&B model(see Fig. 3.9). Very im-portantly, the exchange bias field does not vanish in this region and thereforeno additional coercive field related to the AF is observed, provided that the
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 135
AF layer is sufficiently thick to allow for a domain wall as shown in Fig. 3.21.In this region (II) the EB field is smaller as compared to the M&B model. Thisreduction is more clearly seen further below, when analyzing the azimuthaldependence of the EB field within the Mauri model.
Comparing the phase diagram of the Mauri model in Fig. 3.22 left to thecorresponding one given by the M&B model in Fig. 3.9 left one can clearlysee that region I of both models is very similar with respect to the quali-tative behavior of the exchange bias field as a function of the R-ratio and,respectively, P -ratio. However, we can compare those curves only when ac-counting for the variation of the EB field as a function of the anisotropyof the AF layer. Both models predict that the EB field depends on theanisotropy of the AF layer in a similar qualitative manner. Additionally,within the M&B model the EB field includes also the dependence on thethickness of the AF layer, which is not visible in the Mauri model. Theother regions of both phase diagrams are completely different. Within theMauri model, the exchange bias does not vanish at P < 1, but it contin-uously decreases, whereas the M&B model predicts that the exchange biasfield vanishes for R < 1 leading to enhanced coercivity. Also note that forthe weak coupling region (II) of the Mauri model, the exchange bias wouldstrongly depend on the temperature through the anisotropy constant of theAF layer [90].
3.10.2 Azimuthal Dependence of the Exchange Bias Field
Next we analyze the azimuthal dependence of the EB field by deriving ananalytical expression of the EB field as a function of the rotation angle θ.By solving the second equation of the system of (3.41) with respect to β, onefinds the angle α as function of β. Using the condition for the coercive fieldas β = θ − π/2, and introducing it in the first line of (3.41), one obtains thecoercive fields Hc1 = Hc2. It follows that the coercive field Hc(θ) = 0 and theexchange bias field as function of the azimuthal angle is [11]:
Heb(θ) = − Jeb
μ0MF tF
2√AAFKAF cos(θ)√
J2eb + 4AAFKAF − 4Jeb
√AAFKAF sin(θ)
. (3.43)
In Fig. 3.24 is plotted the EB bias field calculated by the expression abovefor different values of the P -ratio, which was also confirmed numerically. Inregion I the EB field is maximum parallel to the field cooling directions (θ = 0)only for very large P -ratios. When P approaches the unity, the maximum ofthe EB field is shifted away from θ = 0, to higher azimuthal angles and hasthe value:
HMAX,P≥1eb = − Jeb
μ0MF tF. (3.44)
This expression is identical to (3.31) of the M&B model and shown in Fig. 3.11.The shape of the curves evolves from an ideal unidirectional shape at P → ∞to a skewed shape at P ≥ 1.
136 F. Radu and H. Zabel
Fig. 3.24. Azimuthal dependence of exchange bias as a function of the θ angle. Thedotted line for P = 20 can be considered an “ideal” case. The curves are plottedaccording to (3.43)
In region II (Fig. 3.24 left) a drastic change as compared to region I is seenfor the maximum of the exchange bias field as a function of azimuthal angle.Its value decreases monotonically towards zero according to the following ex-pression:
HMAX,P<1eb = −P = −2
√AAF KAF
Jeb. (3.45)
In this region the shape of the curves is also skewed for P -ratios close to unity,but as P decreases towards zero, the curves acquire a more ideal unidirectionalbehavior. Similar to region I, the maximum EB value is shifted away from thefield cooling direction to higher azimuthal angles, whereas in M&B it is shiftedto lower azimuthal angles.
In the limit of strong R and P -ratios (R,P � 1) the Mauri and M&Bmodels give similar results. The differences appear for the R and P -ratioswhich are close to but higher than one. This region (0 < R,P ≤ 5) can beexperimentally explored in order to decide in favor of one or the other model.In order to distinguish between the Mauri and the M&B model, the azimuthaldependence of the exchange bias offers an excellent tool because it is visiblydifferent for the two models (shift to higher or lower azimuthal angles).
The presence of the planar domain wall as described by Mauri et al. ap-pears to have been confirmed experimentally for first time in [16]. Althoughthe Co/NiO system studied by Scholl et al. did not exhibit a hysteresis loopshift, the rotation angle α of the AF planar wall was deduced as function offield. Also, a recent publication by Gornakov et al. [91] show experimentalresults that are similar to the characteristic curves for the α angle shown inFig. 3.22. An AF critical thickness which is often observed experimentally donot appear explicitly in the Mauri model. To account for the AF thickness
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 137
dependence, Xi and White [92] proposed a model which assumes a helicalstructure for the AF spins during the magnetization reversal. The temper-ature dependence of EB is accounted for by the Mauri model through theanisotropy of the AF (HEB ≈ √
KAF ). Since the anisotropy constant is ratherdifficult to measure for thin films, the evidence for the predicted temperaturedependence remains elusive [90]. One insufficiency of the Mauri model is theinability to predict any changes in coercivity. The domain wall produces onlyshifted reversible magnetization curves [40]. Therefore, further refinements ofthe model were introduced which is described in the next section.
3.11 Kim-Stamps Approach – Partial Domain Wall
The approach of Kim and Stamps [11, 93, 94, 95, 96, 40] follows from the workof Neel and Mauri et al. extending the model of an extended planar domainwall to the concept of a partial domain wall in the AF layer. Biquadratic(spin-flop) and bilinear coupling energies are used to describe exchange biasedsystem, where the bias is created by the formation of partial walls in theAF layer. The model applies to compensated, partially compensated, anduncompensated interfaces.
A typical hysteresis loop indicative for a partial domain wall in the AFlayer is shown in Fig. 3.25 [40]. In saturation at h > 0, the F spins are
Fig. 3.25. (a) Magnetization curve for the ferromagnet/antiferromagnet system.(b) Calculated spin structure at three different points of the magnetization curve.The creation of a partial antiferromagnet domain wall can be seen in (iii). Only thespins close to the F/AF interface are shown. From [40]
138 F. Radu and H. Zabel
aligned with the external field while the AF spins are in a perfect Neel state,collinear with the easy axis. The interface spins are antiparallel to the Flayer due to a presumably antiparallel coupling. As the field is reduced andreversed, the AF pins the F layer by interfacial exchange coupling until thecritical value of the reversal field is reached at hc, where the magnetizationbegins to rotate. When it is energetically more favorable to deform the AF,rather than breaking the interfacial coupling, a partial wall twists up as theF rotates. The winding and unwinding of the partial domain wall in the AFis reversible, therefore the magnetization is reversible (no coercivity). Thismechanism is only possible if the AF is thick enough to support a partialwall. The magnitude of the exchange bias is similar to the one given by theMauri model. Neiter the partial-wall theory nor the Mauri model account,however, for the coercivity enhancement that accompanies the hysteresis loopshift in single domain materials, which is usually observed in experiments.
The enhanced coercivity observed experimentally, is proposed to be relatedto the domain wall pinning at magnetic defects. The presence of an attractivedomain-wall potential in the AF layer, arising from magnetic impurities canprovide an energy barrier for domain-wall processes that controls coercivity.Following the treatment of pinning in magnetic materials by Braun et al. [97],Kim and Stamps examined the influence of a pointlike impurity at an arbitraryposition in the AF layer. As a result, the AF energy acquires, besides thedomain wall energy, another term which depends on the concentration of themagnetic defects. These defects decrease the anisotropy locally and lead toan overall reduction of the AF energy. This reduction of the AF energy givesrise to a local energy minimum for certain defect positions relative to theinterface. The domain walls can be pinned at such positions and contributeto the coercivity.
Kim and Stamps argue that irreversible rotation of the ferromagnet dueto a combination of wall pinning an depinning transitions, give rise to asym-metric hysteresis loops. Some examples are given in Fig. 3.26 [40]. The loopsare calculated with an exchange defect at xL = 5, for three different values ofdefect concentration ρJ , where xL denotes the defect positions in the antifer-romagnet, with xL = 0 corresponding to the interface layer and xL = tAF − 1being the free surface. At low defect concentrations, the pinning potential is in-sufficient to modify partial-wall formation. The resulting magnetization curve,as shown in Fig. 3.26(a), is reversible and resembles the curve obtained withthe absence of impurities. Pinning of the partial wall occurs during reversalfor moderate concentrations, which appears as a sharp rotation of the magne-tization at negative fields, as shown in Fig. 3.26(b). During remagnetizationthe wall is released from the pinning center at a different field, thus resultingin an asymmetry in the hysteresis loop. The release of the wall is indicatedby a sharp transition in M. The energy barrier between wall pinning and re-lease increases with defect concentration, resulting in a larger coercivity andreduced bias as in Fig. 3.26(c).
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 139
Fig. 3.26. Defect-induced asymmetry in hysteresis loops. The hysteresis loopsare shown for a reduced exchange defect at xL = 5 for three concentrations: (a)ρJ = 0.15,. (b) ρJ = 0.45, and (c) ρJ = 0.75. The components of magnetizationparallel (M||) (dots) and perpendicular (M⊥) (open circles) to the field directionare shown. The arrows indicate the directions for reversal and remagnetization. Thespin configuration near the interface is shown for selected field values below thehysteresis curves [40]
Within this model, the asymmetry of the hysteresis loops is interpretedin terms of domain-wall pinning processes in the antiferromagnet. This ex-planation appears to be consistent with some recent work by Nikitenko et al.and Gornakov et al. on a NiFe/FeMn system [98, 91] who concluded that thepresence of an antiferromagnetic wall at the interface is necessary to explaintheir hysteresis measurements.
3.12 The Spin Glass Model of Exchange Bias
To overcame the theoretical difficulties in explaining interconnection betweenthe exchange bias and coercivity, in [41, 63] is considered a magnetic stateof the interface between F and AF layer which is magnetically disorderedbehaving similar to a spin glass system. The assumptions of the spin glass(SG) model are:
• the F/AF interface is a frustrated spin system (spin-glass like);• frozen-in uncompensated AF spins are responsible for the EB shift;• low anisotropy interfacial AF spins contribute to the coercivity.
140 F. Radu and H. Zabel
AF INT FKAF
Frozen-in AF Spins
Rotatable AF Spins
KAF
Training Effect
f Jeb
Jeb
Fig. 3.27. Schematic view of the SG Model. At the interface between the AF andthe F layer the AF anisotropy is assumed to be reduced leading to two types ofAF states after field cooling the system: frozen-in AF spins and rotatable AF spins.After reversing the magnetic field, the rotatable spins follows the F layer rotationmediating coercivity. The frozen-in spins remain largely unchanged in moderatefields. But some of them will also deviate from the original cooling state. This couldlead to training effects and also to an open loop in the right side of the hysteresisloop. At larger applied fields in the negative direction, the frustrated frozen-in spinscan further reverse leading to a slowly decreasing slope of the hysteresis loop. A morecomplex antiferromagnetic state consisting of frozen magnetic domains or/and AFgrains can be also reduced to the basic concepts depicted here
Within this model, the AF layer is assumed to contain, in a first approxi-mation, two types of AF states (see Fig. 3.27). One part has a large anisotropywith the orientation ruled by the AF spins and another part with a weakeranisotropy which allows some spins to rotate together with the F spins. Thisinterfacial part of the AF is a frustrated region (spin-glass-like) and gives riseto an increased coercivity. The presence of a low anisotropy AF region canbe rationalized as follows: the interface between the F and AF layer is neverperfect, therefore one may assume chemical intermixing, deviations from sto-ichiometry, structural inhomogeneities, low coordination, etc, at the interfaceto take place. This leads to the formation of a transition region from the pureAF state to a pure F state. On average, the anisotropy of such an interfacialregion is reduced. In addition, structural and magnetic roughness can providea weak AF interface region. Therefore, we assume that a fraction of the frus-trated interfacial spins do rotate almost in phase with the F spins and thatthey mediate enhanced coercivity. We describe them by an effective uniaxialanisotropy Keff
SG , because they are coupled to the presumably uniaxial AFlayer.
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 141
Generally, one can visualize a spin glass system [37] as a collection ofspins which remains in a frozen disordered state even at low-temperatures.In order to achieve such a state, two ingredients are necessary: a) there mustbe a competition among the different interactions between the moments, inthe sense that no single configuration of the spins is uniquely favored by allinteractions (this is commonly called ‘frustration’); b) these interactions mustbe at least partially random. This partial random state will be introduced inthe M&B model as an effective uniaxial anisotropy.
Adding this effective anisotropy to the M&B model, the free energyreads [41]:
where, KeffSG is an effective uniaxial SG anisotropy related to the frustrated
AF spins with reduced anisotropy at the interface, Jeffeb is the reduced interfa-
cial exchange energy, and γ is the average angle of the effective SG anisotropy.KAF (tAF ) is the anisotropy constant of AF layer. To avoid further compli-cations for the numerical simulations, we neglect the thickness dependenceof the KAF anisotropy (KAF (tAF ) ≡ KAF ). We mention though, that thisdependence could become important for low AF thicknesses due to finite sizeeffects and due to structurally non-ideal very thin layers. From now on, theMCA anisotropy of the ferromagnetic layer (KF = 0) will also be neglectedas to highlight more clearly the influences of AF layer and the SG interfaceonto the general properties of the EB systems. Note that the Zeeman ener-gies of the ferromagnetic-like AF interfacial spins are neglected in the modelsince they are usually much smaller as compared to Zeeman energy of the Flayer. Nevertheless, they can be seen as a vertical shift of the hysteresis loop(frozen-in AF spins in Fig. 3.27) and as an additional contribution to the totalmagnetization (rotatable AF spins in Fig. 3.27).
The model is depicted schematically in Fig. 3.27. At the interface tworather distinct AF phases are assumed to occur in an EB system: the rotatableAF spins, depicted as open circles and frozen-in AF spins shown as filledcircles. After field cooling, a presumably collinear arrangement is depicted inthe right hand panel. After reversing the magnetic field, the rotatable AFspins follow the F layer rotation mediating coercivity. The frozen-in spinsremain largely unchanged in moderate fields. But some of them could alsodeviate from the original cooling state. Irreversible changes of the frustratedAF spins lead to training effects and also to an open loop in the right side ofthe hysteresis loop. At larger negative applied fields, the frustrated frozen-inspins could further reverse leading to a slowly decreasing slope of the hysteresisloop. A more complex antiferromagnetic state consisting of frozen magneticdomain state or/and AF grains can be also reduced to the basic two spin
142 F. Radu and H. Zabel
components depicted in Fig. 3.27. Basically, additional frozen-in spins canoccur in the AF layer extending to the interface and, therefore, leading toeven more interfacial disorder.
Next, we evaluate numerically the resulting hysteresis loops and azimuthaldependence of the exchange bias within the SG model. When the Keff
SG pa-rameter is zero, the system behaves ideally as described by the M&B modeldiscussed in Sect. 3.6: the coercive field is zero and the exchange bias is finite.In the other case, when the interface is disordered we relate the SG effectiveanisotropy to the available interfacial coupling energy as follows:
Keff = (1 − f)Jeb
Jeffeb = f Jeb , (3.47)
where Jeb is the total available exchange energy and f is a conversion factordescribing the fractional order at the interface, with f = 1 for a perfectinterface and f = 0 for perfect disorder. Some basic models to calculate theavailable exchange energy were discussed in the previous sections. For evenmore complicated situations when the AF consists of AF grains and/or AFdomains the exchange energy can be further estimated as described in [46].
With these notations we write the system of equations resulting from theminimization of the (3.46) with respect to the angles α and β:
h sin(θ − β) +(1 − f)f
sin(2 (β − γ)) + sin(β − α) = 0
R sin(2 α) − sin(β − α) = 0 , (3.48)
where,
h =H
− Jeffeb
μ0 MF tF
=H
− f Jeb
μ0 MF tF
,
is the reduced applied field and
R ≡ KAF tAF
Jeffeb
=KAF tAF
f Jeb,
is the R-ratio defining the strength of the AF layer.The system of equations above can easily be solved numerically, but it does
not provide simple analytical expressions for the exchange bias. Numericalevaluation provides the α and β angles as a function of the applied magneticfield H . The reduced longitudinal component of magnetization along the fieldaxis follows from m|| = cos(β − θ) and the transverse component from m⊥ =sin(β − θ).
With the assumptions made above the absolute value of the exchange biasfield is directly proportional to f . The parameter f can be called conversionfactor, as it describes the conversion of interfacial energy into coercivity. Forexample, in the M&B phase diagram in the region II and III corresponding
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 143
to reduced R-ratios, the exchange bias field is zero and the coercive field isenhanced as a result of such a conversion of the interfacial energy into coercivefield.
The idea of reduced interfacial anisotropy at the interface can be tracedback to the the Neel weak ferromagnetism at the surface of AF particles.Neel [77] discussed the training effect as a tilting of the superficial magnetiza-tion of the AF domains. Later, Schlenker et al. [99] suggested that successivereversals of the F magnetization could lead to changes of the interface un-compensated AF magnetization and therefore provide means of going fromone ground state to another. Such multiple interface configurations are sim-ilar to a spin glass system. The spin-glass interface is further discussed byother authors [100, 101, 102, 103, 104, 105, 106, 107, 50]. Exchange bias hasbeen observed recently for spin-glass/F system [108]. The interfacial mag-netic disorder was observed through hysteresis loop widening below a criticaltemperature point [109]. Non-collinearity have been observed at the AF/F inremanence [110] and even in saturation [111, 60, 112]. The frozen spins atthe interface were also observed by MFM [113]. Using element specific tech-niques such as soft x-ray resonant magnetic dichroism (XMCD) and of x-rayresonant magnetic scattering (XRMS), both frozen and rotatable AF spinscan be studied [13, 114, 115, 116, 15, 117, 17, 19, 20]. The frozen-in spinsappear as a shift of the hysteresis loop along magnetization axis, whereas theAF rotatable spins exhibit a hysteresis loop. Moreover, an evidence for SGbehavior is recently reported in thin films [118] and AF nanoparticles [119].Therefore, we believe that there is enough experimental evidence to considerthe interface between the AF/F layer as a disordered state behaving similarto a spin-glass system.
3.12.1 Hysteresis Loops as a Function of the Conversion Factor f
If Fig. 3.28 we show simulations of several hysteresis loops as a function of theconversion factor f . We assume a strong antiferromagnet in contact with aferromagnet, where the interface has different degrees of disorder depicted inthe right column of Fig. 3.28. For the R ratio we assume the following value:R = KAF tAF
fJeb= 62.5/f which corresponds to a 100 A thick CoO antiferromag-
netic layer. The field cooling direction and the measuring field direction areparallel to the anisotropy axis of the AF. The anisotropy of the ferromagnetis neglected in the simulations below. For the interface we have chosen a SGanisotropy oriented 10 degrees away from the unidirectional anisotropy orien-tation (γ = 10◦). On the abscissa the reduced exchange bias field h = H/|H∞
eb |(H∞
eb ≡ − Jeffeb
μ0 MF tF) is plotted, which then can easily be compared to the M&B
model. With this assumption the system of (3.48) was solved numerically.The left column shows the longitudinal component of the magnetization
(parallel to the measuring field direction) (m|| = cos(β)) whereas the middlecolumn shows the transverse component of the magnetization (m⊥ = sin(β)).
144 F. Radu and H. Zabel
Fig. 3.28. Longitudinal (m|| = cos(β)) and transverse (m⊥ = sin(β)) componentsof the magnetization for a F(KF = 0)/AF(R = 62.5/f, γ = 10◦,θ = 0) bilayer fordifferent values of the conversion factor f (f = 80%, 60%, 20%). We observe, thatwhen the AF layer is strong (R 1), the hysteresis loops are symmetric whenmeasured along the field cooling direction. The hysteresis loops are simulated bysolving numerically the (3.48). In the right column is schematically depicted thelayer structure, here the emphasis is given to the disorder state at the interface.The AF layer is depicted as consisting of magnetic domains which also contributeto interface disorder
We observe that with decreasing conversion factor f the exchange biasvanishes linearly. The reduction of the EB field is accompanied by an increasedcoercivity. The shape of the hysteresis loop is close to the results found inliterature. For instance the hysteresis loop with f = 60% is similar to thedata shown in [120, 61]. The hysteresis loop with f = 20% is similar tothe data shown in [14, 109]. The longitudinal and transverse componentsof the magnetization show that the reversal mechanism is symmetric. Thesymmetry is directly related to the strength of the AF layer, when no trainingeffect is involved. For the examples depicted in Fig. 3.28, the R-ratio is muchlarger than 1 (R � 1), and therefore the hysteresis loops are symmetric whenmeasured along the field cooling direction and along the anisotropy axis of theAF layer. The asymmetry of the hysteresis loops is discussed further below.
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 145
3.12.2 Phase Diagram of Exchange Bias and Coercive FieldWithin the Spin Glass Model
In this section we discuss the phase diagram for the exchange field and coer-cive field as a function of the R-value within the SG-model. The additionalparameter is the conversion factor f . In Fig. 3.29 phase diagrams are shownfor reduced exchange bias and reduced coercive fields as a function of theR-ratio for four different values of the conversion factor f . This allows usto compare directly the behavior of exchange bias fields as predicted in theSG model and the M&B model. The reduced exchange bias field plotted inFig. 3.29 (left panel) is defined:
heb =Heb
Jeffeb
μ0 MF tF
=Heb
f H∞eb
,
where the Heb is the absolute value of the exchange bias within the SG modeland the denominator term Jeb
μ0 MF tfis the exchange bias field within the ideal
M&B model.The reduced coercive field shown in Fig. 3.29 is defined:
hc =HSG
c
Jeffeb
μ0 MF tF
,
where HSGc is the absolute value of the coercive field within the SG model. It
has no relation to the coercive field of the M&B model because the coercive
Fig. 3.29. The dependence of the reduced exchange bias field heb and the reducedcoercive field hc as a function of the R-ratio for four different values of the conversionfactor f
146 F. Radu and H. Zabel
field within the M&B model is considered to be a constant when the exchangebias is finite.
3.12.3 Azimuthal Dependence of Exchange Bias and CoerciveField Within the Spin Glass Model
In this section, the azimuthal dependence of exchange bias and coercivefields within the SG model are discussed and compared with experimentalresults of polycrystalline Ir17Mn83(15nm)/Co70Fe30(30nm) exchange biassystem [41].
In Fig. 3.30 calculated magnetization components are plotted togetherwith the experimental data points, and in Fig. 3.31b the azimuthal dependenceof the coercive field and exchange bias field are plotted and compared tothe experimental data in Fig. 3.31a. The hysteresis loops were calculated bynumerical minimizing the expressions in (3.48). The parameters used in thesimulation f = 80%, R = 5.9/f, γ = 20◦ do best reproduce the experimentaldata. Furthermore, it is assumed that the AF layer has a uniaxial anisotropy.The MCA anisotropy of the F layer is neglected (KF = 0). Therefore, thecoercivity which appears in the simulations is not related to the F properties,but to the interfacial properties of the F/AF bilayer.
First we discuss the hysteresis loops shown in Fig. 3.30. The system iscooled down in a field oriented parallel to the AF anisotropy direction. Thehysteresis loops (solid lines) are simulated for different azimuthal angles θof the applied field in respect to the field cooling orientation. In Fig. 3.30representative hysteresis curves are shown for the longitudinal (m||) andtransverse (m⊥) magnetization. At θ = 0◦, the magnetization curves are sym-metric and shifted to negative fields. At θ = 180◦, the magnetization curvesare also symmetric but shifted to positive fields. At θ = 3◦, however, thelongitudinal hysteresis loop becomes asymmetric. The first reversal at Hc1 issharp and the reversal at Hc2 is more rounded. This asymmetry is also seen inthe transverse component of the magnetization. The F spins rotate asymmet-rically: the values of the β angle depend on the external field scan direction,being different for swaps from negative to positive saturation as compared withswaps from positive to negative saturation. As the azimuthal angle increases,the coercive field becomes zero. For instance, at θ = 20◦, 90◦ and 160◦ thereis almost no coercivity. Also, the transverse component of the magnetizationshows that the F spins do not follow a 360◦ path, but they rotate within the180◦ angular space.
In Fig. 3.31 the coercive field and the exchange bias field are extractedfrom the experimental and simulated hysteresis loops using (3.16). We dis-tinguish the following characteristics of the Hc and Heb: the unidirectionalbehavior (≈ cos(θ)) of the Heb as a function of the azimuthal angle is (seeFig. 3.11) clearly visible; additionally, the behavior of the Heb as a function ofthe azimuthal angle shows sharp modulations close to the orientation of theAF uniaxial anisotropy; the coercive field Hc has a peak-like behavior close
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 147
Fig. 3.30. Experimental (open circles)and simulated hysteresis loops (black lines)for different azimuthal angles. The simulated curves are calculated by the (3.48)with the following parameters: f = 80%, R = 5.9/f, γ = 20◦ [41]
to the orientation of the AF uniaxial anisotropy, at θ = 0◦ and θ = 180◦.In all cases we find an astounding agreement between calculated curves andexperimental data. It is remarkable, that the EB field and the coercive fieldare completely reproduced by the SG model
Experimentally the azimuthal dependence of the exchange bias field wasfirst explored for NiFe/CoO bilayers [121]. It was suggested that the experi-mental results can be better described with a cosine series expansions, withodd and even terms for Heb and Hc , respectively, rather than being a simplesinusoidal function as initially suggested by Meiklejohn and Bean [1, 26].
The simulations shown in this section are different with respect to the pre-vious reports on the angular dependence of exchange bias field [121, 122, 123,
148 F. Radu and H. Zabel
Fig. 3.31. a) The experimental coercive field (open symbols ) and exchange bias(filled symbols) field as a function of the azimuthal angle θ.d) Simulated coercive field(dotted line) and exchange bias field (continuous line) as a function of the azimuthalangle [41]
124, 125]. One difference is that the MCA anisotropy of the F is supposed tobe negligible when compared with the coercive fields obtained experimentally,and the sharp features of the Heb are reproduced numerically rather thenbeing described by cosine series expansions. Recently, Camarero et al. [76]reports on very similar azimuthal dependent hysteresis loops as shown here.There, an elegant way based on asteroid curve is used to describe the intrinsicasymmetry of the hysteresis loops close to the 0◦ and 180◦ azimuthal angle. Aunidirectional anisotropy displaces the asteroid critical curve from the origin.Therefore, if the applied field is not parallel to the unidirectional anisotropy,the field sweep line does not pass through the symmetry center of the as-teroid critical curve leading to inequivalent switching fields and consequentlyasymmetric reversals [76].
3.12.4 Dependence of Exchange Bias Field on the Thicknessof the Antiferromagnetic Layer
After a short inspection of the phase diagram of EB and coercive field(Fig. 3.29) we notice that there is a critical value for the R-ratio at which
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 149
the exchange bias vanishes and the coercive field is enhanced. This criticalvalue R=1 depends on four parameters: the anisotropy of the antiferromagnet,the interfacial exchange coupling parameter, the thickness of the ferromagnet,and the conversion factor. The conversion factor further depends on the AFdomain and/or AF grain size, if any.
In Fig. 3.32 the normalized exchange bias field is plotted as a function ofthe AF thickness and for several conversion factors. The anisotropy of the AFlayer (KAF ) and the Jeb parameter are assumed to be constant.We notice twomain characteristics of the EB field dependence on the AF thickness: whenf has high values close to unity, the EB field decreases with decreasing AFthickness. However, when f is reduced, a completely different behavior of theEB is observed. The EB field increases as the thickness of the AF decreases,developing a peak-like feature. This peak-like behavior for the EB field atcritical AF thickness is a result of enhanced coercivity which is accounted forby the f-factor. Also, an essential parameter is the α angle, which describesthe rotation of the AF spins during the magnetization reversal. The criticalthickness is preserved by the SG model, but it differs in magnitude as com-pared to the M&B model. Since some interfacial coupling energy is dissipatedas coercivity, the critical thickness within SG model is lower as compared tothe corresponding one given by the M&B model. The critical AF thicknesswithin the SG model follows from the condition R = 1:
Fig. 3.32. The normalized exchange bias heb/f = Heb/H∞eb as a function of the
tAF KAF /Jeb for different values of the conversion factor f . The anisotropy constantKAF and Jeb are assumed to be constant as the AF thickness is being varied foreach f . An asymmetric peak like behavior of the exchange bias field develops forhigh values of the conversion factor
150 F. Radu and H. Zabel
tSGAF,cr =
f Jeb
KAF= f tMB
AF,cr , (3.49)
where tSGAF,cr and tMB
AF,cr are the AF critical thickness predicted by the SG andM&B models, respectively.
Experimentally the R-ratio can be tuned by changing the thickness of theAF and keeping the other three parameters constant. As a result one observesa critical thickness of the AF layer for which the EB disappears [126, 72, 127,73, 128, 81, 89]. This AF critical thickness can be qualitatively understoodwithin the M & B model. When the hardness of the AF layer is reduced,the AF spins will rotate under the torque created by the F layer trough theinterfacial coupling constant. The shape of the EB as function of AF thick-ness, however, can be different from one system to another depending on theother three parameters. The most prominent experimental feature of the EBdependence on the AF thickness is the development of a peak close to the crit-ical thickness. Several proposals were made to describe the peculiar shapes ofEB field dependence on AF thickness. According to the Malozemoff model, achange in the AF domain size as function of AF thickness results in a changeof exchange bias magnitude. Other influences on the AF dependence of theEB and coercive field are of structural origin [126, 73, 18]. It has been shownby Kuch et al. [18] that at the microscopic level, the coupling between theAF an F layers depends on the atomic layer filling and on the morphology ofthe interface. The AF-F coupling was observed to vary by a factor of two be-tween filled and half-filled interface. Moreover, islands and vacancy islands atthe interface lead to a quite distinct coupling behavior. Therefore, structuralconfigurations are indeed contributing to the EB-dependence as function ofAF thickness.
The peak-like behavior of the EB field as function of the AF thickness isstrongly dependent on temperature. An almost complete set of curves, showinga monotonous development of the AF peak from high to low temperatures wasmeasured by Ali et al. [89]. The data is reproduced in Fig. 3.33 together withthe simulations based on DS model. Although the DS model does describewell some experimental observed features, some discrepancies still exists. Forinstance the development of the AF peak as well as the critical AF thicknessas function of temperature are more pronounced in the experimental data ascompared to the DS simulations.
The SG model, through the conversion factor f, appears to be ableto describe the evolution of the critical AF thickness(see Fig. 3.32 andFig. 3.33 (left)). Also the shape of the EB dependence on the AF thick-ness, from an almost ideal M&B type at T=290 K to a pronounced peakedcurve at T=2 K is qualitatively reproduced. Although not considered so far,the conversion factor seems to be temperature dependent. This can be un-derstood if we consider the basic assumption of the SG model, namely thefrustration at the interface. Temperature fluctuations acting on metastablespin states cause a variation of the SG anisotropy as function of tempera-
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 151
Fig. 3.33. Left: IrMn thickness dependence of the exchange bias field Heb for anumber of temperatures. Lines between the points are a guide to the eye. Right:Prediction of the DS model for the AF thickness dependence of the exchange biasfield from the stability analysis of the interface AF domains at different tempera-tures. (from [89])
ture. We will not further speculate on the exact temperature dependence ofthe f-factor, but we mention that a numerical analysis for the EB dependen-cies shown in Fig. 3.33 would allow to untangle all the parameters in theR-ratio. The conversion factor is given by the shape of the EB curves, fJeb
can be extracted from the temperature dependence of the EB field at highAF thicknesses, and finally, the anisotropy constant will be deduced from thecritical AF thickness. Note also, that the SG model has the potential to evendescribe the different temperature dependent shapes of the EB field, namelylinear dependence versus more rounded shape: for an AF thickness close tothe critical region, the temperature dependence of the EB bias will be clearlysteeper (linear-like) as compared to the corresponding one at higher AF thick-nesses(more rounded).
3.12.5 The Blocking Temperature for Exchange Bias
Experimentally it is found that the temperature where the exchange bias effectfirst occurs is usually lower than the Neel temperature of the AF layer (TN ) [8].This lower temperature is called blocking temperature (TB). For thick AFlayers TB ≤ TN , whereas for thin AF layers TB � TN [8]. Furthermore,the coercive field increases starting just below TN (with some exceptions) incontrast to the EB field, which appears only below TB.
These three experimentally observed characteristic features can qualita-tively be explained within the M&B and SG models. In order to have a non-vanishing EB field in the region with R ≥ 1, the following condition has to bebe fulfilled:
KAF >fJeb
tAF= KAF,crit , (3.50)
152 F. Radu and H. Zabel
where KAF,crit is the critical AF anisotropy for the onset of the EB field. Fora fixed AF layer thickness, the condition above sets a critical value for the AFanisotropy for which the EB can exist. Considering that the AF anisotropy in-creases steadily below TN , for large AF layer thicknesses the condition of (3.50)is fulfilled just below TN , whereas for thinner AF layers this condition is ful-filled at a correspondingly lower temperature TB. It is clear from the phasediagrams in Fig. 3.29 and in Fig. 3.32 that there is a region of anisotropy0 < KAF < Kcrit
AF where the EB field is zero and the coercive field is enhanced.It follows that the enhancement of the coercive field should be observed abovethe blocking temperature and below the Neel temperature of the AF. Thissituation is indeed observed experimentally. For the case of CoO(25 A)/Colayers the coercive field increases starting from the TCoO
N = 291 K, whereasthe exchange bias field first appears below TB = 180 K [14].
Further possible causes for a reduced blocking temperature and for the be-havior of the EB and coercive fields as a function of temperature are discussedelsewhere: finite size effects [129], stoichiometry [130] or multiple phases [131],AF grains [132] and diluted AF [38].
3.13 Training Effect
The training effect refers to the dramatic change of the hysteresis loop whensweeping consecutively the applied magnetic field of a system which is in abiased state. The coercive fields and the resulting exchange bias field ver-sus n, where n is the nth measured hysteresis loop, displays a monotonicdependence [133, 134, 134, 99]. The absolute value of Hc1 and of the EB fielddecreases from an initial value at n = 1 to an equilibrium value at n = ∞.The absolute value of the coercive field Hc2, however, displays an oppositebehavior, i.e. it increases with n. These features of the training effect is re-ferred to as Type I by Zhang et al. [135]. The other case when both |Hc1|and |Hc2| decrease is called Type. II. In this section we deal only with theso-called Type I training effect. Several mechanisms were suggested as a pos-sible cause of the effect. While it is widely accepted that the training effect isrelated to the unstable state of the AF layer and/or F/AF interface preparedby field cooling procedure, it is not yet well established what mechanisms aredominantly contributing to the training effect.
Neel [77] discussed the training effect as a tilting of the superficial magne-tization of the AF domains. This would lead to a Type I training effect. Neelalso discussed that a creeping effect could lead to a Type II training effect.
Micromagnetic simulations within the DS model [32, 35] show that thehysteresis curve is not closed after a complete loop. The lost magnetizationis directly related to a partial loss of the superficial magnetization of the AFdomains, which further leads to a decreased exchange bias.
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 153
Zhang et al. [136] suggested that the training effect can be explained by in-corporating into the Fulcomer and Charap’s model [137] positive and negativeexchange coupling between the grains constituting the AF layers.
In [138], the authors found direct evidence for the proportionality betweenthe exchange bias and the total saturation moment of the heterostructure.The findings were related to the prediction of the phenomenological M&Bapproach, where a linear dependence of the exchange bias on the AF interfacemagnetization is expected.
Binek [139] suggested that the phenomenological origin of the trainingeffect is a deviation of the AF interface magnetization from its equilibriumvalue. Analytical calculations in the framework of non-equilibrium thermody-namics leads to a recursive relation accounting for the dependence of the Heb
field on n.Hoffmann [140] argues that only biaxial AF symmetry can lead to training
effects, reproducing important features of the experimental data, while sim-ulation with uniaxial AF symmetry show no difference between the first andsecond hysteresis loops.
Experiments performed by PNR, AMR and Kerr Microscopy [111, 47,14, 141, 59, 112, 142, 143] also support the irreversible changes taking placeat the F/AF interface and in the AF layer. It has been observed that duringthe very first reversal at Hc1, interfacial magnetic domains are formed andthey do not disappear even in positive or negative “saturation”. The interfa-cial domains serve as seeds for the subsequent magnetization reversals. Theseferromagnetic domains at the interface have to be intimately related to theAF domain state [144]. Therefore, the irreversible changes of the AF domainstate are responsible for the training effect. Furthermore measurements havedetected out-of-plane magnetic moments [82, 59] hinting at the existence ofperpendicular domain walls in the AF layer, as originally suggested by Mal-ozemoff. Therefore, irreversible changes of the AF magnetic domains and ofthe interfacial domains during the hysteresis loops play an important role forthe training effect.
3.13.1 Interface Disorder and the Training Effect
In the following we analyze the AF domains and interface contributions to thetraining effect. We assume that a gradual increase of the interfacial disorder ofthe F/AF system leads to a training effect. Within the SG model, the magneticstate of the F/AF interface can be mimiced through a unidirectional inducedanisotropy Keff , which is allowed to have an average direction γ, where γ isrelated to the spin disorder of the interface. Also, we will consider the influenceof a progressive rotation of an AF domain anisotropy during the reversal. Bothsituations will be treated below.
In Fig. 3.34a) and b) we show first and second hysteresis loops (longitu-dinal and transverse components of magnetization) calculated with the helpof (3.48). In these calculations we set the conversion factor to f = 60% and
154 F. Radu and H. Zabel
Fig. 3.34. Simulations of training effect within the SG model. Longitudinal a)and transverse b) components of magnetization for a scenario (see text) involvingirreversible changes of the AF domain angle θAF during the magnetization rever-sal. Longitudinal c) and transverse d) components of magnetization for the casewhen only the interface disorder parameter γ increases and the AF state remainsunchanged
R = 62.5/f . We consider a drastic change of an AF domain which progres-sively rotates its anisotropy axis during the magnetization reversal. This situ-ation can be accounted for in the SG model by replacing the α angle in (3.48)with α− θAF , where the θAF is the orientation of the AF domain anisotropy.We also set the γ angle to be almost zero.
Following closely the experimental observations [111, 14], before the firstreversal θAF is zero, and just after the first reversal θAF increases towards anequilibrium value. The first branch of the hysteresis loop appears rather sharp,therefore we assume that the AF spins and F spins are collinear immediatelyafter cooling in a field (θAF = 0◦). For the second branch of the 1st hysteresisloop we consider that θAF = 20◦, therefore the second leg appears morerounded. The transition from θAF = 0◦ to θAF = 20◦ is assumed to happenright after or during the first reversal at Hc1. This is in accordance with theobservation that for thin CoO layers [111, 14] where the disordered interfaceappears after the first reversal at Hc1. Now, the first branch of the secondhysteresis loop is simulated with θAF = 20◦. At the third reversal, we again
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 155
assume that the AF domain angle further increases. Therefore, the secondbranch of the hysteresis loop is simulated assuming a new value of θAF = 30◦.
The hysteresis loops bear all the features observed experimentally [99,136, 135, 58, 61, 111, 60, 138]. More strikingly, the transverse component ofmagnetization shows a small step increase at Hc1 and larger increase at Hc2,on the reverse path. These are typical features observed experimentally byPNR [111, 14], AMR [47, 143, 112] and MOKE [141]. Moreover, the trans-verse component at saturation behaves very close to recent observations ofBrems et al. [112] on Co/CoO bilayers and lithographically nanostructuredwires. After field cooling and before passing through the first magnetizationreversal in the descending field branch, the resistance in saturation (whichis proportional to the square of the orientation of transverse component ofmagnetization) reaches its maximum because all spins are oriented along thecooling field. After going through a complete hysteresis loop, the resistanceat saturation is reduced, indicating that spins in the F are rotated away fromthe cooling field. After reversing the field back to positive saturation the re-sistance does not recover its initial value. Moreover, the untrained state canbe partially reinduced by changing the orientation of the applied magneticfield [112] which can be interpreted as a further indication of AF domainrotation during the reversal.
Next, we consider only the interface disorder through a progressive changeof γ angles. In Fig. 3.34b) and c) we show first and second hysteresis loops cal-culated with the help of (3.48). In these calculations we consider that the AFis strong, R = 62.5/f . For the conversion factor we take a value of f = 60%.Also, we assume the average AF orientation to be parallel to the field coolingorientation (θAF = 0). Following closely the experimental observations, be-fore the first reversal γ is zero, while just after the first reversal, γ increasestowards an equilibrium value. The first branch of the hysteresis loop appearsrather sharp, therefore we assume that the AF spins and F spins are collinearimmediately after cooling in a field (γ = 0). For the second branch of the 1sthysteresis loop we consider that γ = 10◦, therefore the second leg appearsrather rounded. The transition from γ = 0 to γ = 10◦ is assumed to happenright after or during the first reversal at Hc1. The first branch of the secondhysteresis loop is simulated with γ = 10◦. At the third reversal, we again as-sume that the disorder of the interface increases. Therefore, the second branchof the hysteresis loop (and right after during during the reversal at Hc1 of thesecond loop) is simulated assuming a new value of γ = 20◦.
The simulations above implies a viscosity-like behavior to the disorderedinterface [145]. For example, when the F magnetization acquires an angle withrespect to the unidirectional anisotropy, the torque exerted on the interfacialspins will drag them away from the initial direction set by the field cooling.Reversing the magnetization back to positive directions the Keff spins willnot follow (completely), they remain close to this position (viscosity). This isbecause the maximum torque exerted by the F spins was already acting atnegative fields, while for positive fields it is much reduced. When measuring
156 F. Radu and H. Zabel
again the hysteresis loop, at negative coercive field, the Keff spins will rotateeven further and so on. Therefore, the angle of the Keff anisotropy and/orof the AF domains increases after each hysteresis loop, similar to a rachet,causing a decreased exchange bias field.
Comparing the hysteresis curves shown in Fig. 3.34a) with the ones inFig. 3.34c) one notices the same qualitative characteristics. This is not thecase for the transverse magnetization curve. The F spins rotate only on thepositive side for the first case, whereas for the second case the F magnetizationrotation covers the entire 360◦ angular range. The anisotropic magnetoresis-tance (AMR) and PNR hides the chirality of the ferromagnetic spin rotation asthey provide sin2(β) information, whereas MOKE is sensitive to the chiralityas it provides sin(β) information. Therefore, measuring both hysteresis com-ponents by MOKE, can help to distinguish between the dominant influenceon the training effect: AF domain (or/and grain) rotation versus SG interfaceinstability. The training effect is discussed furthermore in Sect. 3.13.2.
3.13.2 Empirical Expression for the Training Effect
The very first empirical expression for training [133, 134] effects suggested apower law dependence of the coercive fields and the exchange bias field as afunction of cycle index n:
Hneb = H∞
eb +k√n, (3.51)
where k is an experimental constant. This expression follows well the experi-mental dependence of the EB field for n ≥ 2, but when the very first point isincluded to the fit, then the agreement is poor.
Binek [139] has shown by using non-equilibrium thermodynamics that us-ing a recursive relation, the evolution of the EB field as a function of n, can bewell reproduced for all cycle indexes (n ≥ 1). The recursive expression reads:
Hn+1eb −Hn
eb = −γ (Hneb −H∞
eb )3 , (3.52)
where γ is a physical parameter which, for n� 1, was directly related to thek parameter of (3.51). It was shown that a satisfactory agreement betweenthe (3.51) and (3.52) is achieved for n ≥ 3. Therefore the approach of Binekappears to provide the phenomenological origin of the hitherto unexplainedpower-law decay of the EB field with increasing loop index n > 1. The ana-lytic expression (3.52) was further tested for temperature dependent trainingeffect [146].
More recently the equation (3.52) has been further refined by extendingthe free energy expansion with a correction of the leading term. The newequation for training effect reads [147]:
Hn+1eb −Hn
eb = −γb (Hneb −H∞
eb )3 − γc (Hneb −H∞
eb )5 , (3.53)
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 157
where the new γc parameter results from the higher order expansion of the freeenergy and hence γc � γb. The γb parameter is similar to the γ in (3.52). Bothparameters γb and γc exhibits a exponential dependence on the sweep rate formeasuring the hysteresis loop. A number of three fit parameters is requiredfor both (3.52) and (3.53), but the last equation provides better fitting resultsfor moderate sweep rates.
In the following we analyze another type of expression which reproducesthe dependence of the coercive field and exchange bias field as a function ofthe loop index n and for different temperatures. It is based on the simulationsshown in the previous section. There it was argued that the training effectis related to the interfacial spin disorder. With each cycle the spin disorderincreases slightly, thereby decreasing the exchange bias field. Additional effectsare related to the AF domain size that also affects the magnitude of the EBand Hc fields. Both contributions cause a gradual decrease of exchange biasas a function of cycle n. They can be treated probabilistically. We suggest thefollowing expression to simulate the decrease of the EB as a function of n:
Hneb = H∞
eb +Af exp(−n/Pf ) +Ai exp(−n/Pi) , (3.54)
where, Hneb is the exchange bias of the nth hysteresis loop, Af and Pf are
parameters related to the change of the frozen spins, Ai and Pi are param-eters related to the evolution of the interfacial disorder. The A parametershave dimension of Oersted while the P parameters have no dimension butthey are similar to a relaxation time, where the continuous variable “time” isreplaced by a discrete variable n. We expect that the interfacial contributionsharply decreases with n as the anisotropy of the interfacial spins is reduced(low AF anisotropy spins), while the contribution from the “frozen” AF spinsbelonging to the AF domains (“frozen-in” uncompensated spins) appear asa long decreasing tail as they are intimately embedded into a much stifferenvironment.
In the following we show fits to the “trained” exchange bias field. InFig. 3.35a) the EB field of thirteen consecutive hysteresis loops were mea-sured at T = 10 K and are plotted as a function of loop index for aCoO(40 A)/Fe(150 A)/Al2O3 bilayer. Three fits are shown: one using theempirical relation (3.51), the second one is a fit performed by Binek [148]using the equation (3.53), and the third one using (3.54). We observe thatthe fit using the (3.51) (H∞
eb = 146.6Oe, k = 44Oe) follows well the exper-imental curve for n ≥ 2. However, the best fits are obtained using (3.53)and (3.54). The best fit parameters using (3.53) are [148]: H∞
eb = 148.632Oe,γbeb = 0.00029472, and γc
eb = −2.772 10−8. The parameters obtained fromfits to the data using the (3.54) are: H∞
eb = 158Oe,Af = 25.87Oe, Pi =4.33, Ai = 739.14Oe, Pi = 0.39. Within the SG approach, we distinguish, in-deed, a sharp contribution due to low anisotropy AF spins at the interfaceand a much weaker decrease from the “frozen-in” uncompensated spins.
The temperature dependence of the training effect for a epitaxialFe(150 A)(110)/CoO(300 A)(111)/Al2O3 bilayer [149, 62, 63] is shown in
158 F. Radu and H. Zabel
Fig. 3.35. a) Exchange bias as function of the loop index n. The gray line is thebest fit to the data using (3.51). The open circles are the best fit [148] to the datausing (3.53). The black line is the best fit to the data using (3.54). b)Temperaturedependence of the training effect. The lines are the best fit to the data using (3.54)
Fig. 3.35b). The sample was field cooled in saturation to the measuring tem-perature where 31 consecutive hysteresis loops were measured. The fits to thedata using (3.54) are shown as continues lines in Fig. 3.35b).
We distinguish three main characteristics related to the temperature de-pendence of the training effect:
• each curve shows two regimes, a fast changing one and a slowly decreas-ing tail;
• the “relaxation times” (Pi and Pf ) do not visibly depend on the temper-ature;
• the interface transition towards the equilibrium state is approximativelyten times faster then the transition of the “frozen” spins towards theirstable configuration.
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 159
3.14 Further Characteristics of EB-systems
3.14.1 Asymmetries of the Hysteresis Loop
A curios characteristic of EB systems is the often observed asymmetry be-tween the two branches of the hysteresis loop for descending and ascendingmagnetic fields. The hysteresis loop shape of an isolated ferromagnet is withno exceptions symmetric with respect to the field and magnetization axis.This is not the case for an exchange bias system, where the unidirectionalanisotropy and the stability of the AF can result in asymmetries of the hys-teresis loops and of the magnetization reversal modes. One can distinguishtwo different classes of the hysteresis loop asymmetries. One of them can beassigned to intrinsic properties of the EB systems which lacks training effects,and the another one is intimately related to irreversible changes of the AFdomain structure during the magnetization reversal.i) In the first category we encounter four different situations of asymmetryicmagnetization reversal all related to a stable interface without training effect:a) the first branch of the hysteresis loop is much extended compared to theascending branch (see Fig. 3.36). This asymmetric hysteresis loop was ob-served in FeNi/FeMn bilayers [98]. The underlying mechanism is related to aMauri type mechanism for exchange bias where a parallel domain wall (ex-change spring) is formed in the AF layer. The reversal is understood in termsof domain wall pinning in the antiferromagnet [98, 40, 91].
b) coherent rotation during the first reversal at Hc1, domain wall nucleationand propagation at Hc2 (see Fig. 3.37). Such asymmetric magnetization rever-sal has been observed by PNR in Fe/FeF2 and Fe/MnF2 systems [150]. Thisasymmetry depends on the relative orientation of the field cooling directionwith respect to the twin structure of the AF layer. The reversal asymmetrymentioned above takes place when the FC is parallel oriented with a directionbisecting the anisotropy axes of the two AF structural domains. When thefield cooling orientation is parallel to the anisotropy axes of one AF domain,
Fig. 3.36. a) Asymmetric hysteresis loop (a) of a NiFe/FeMn bilayer and schematicsof domain structure at different stages of magnetization reversal [98]
160 F. Radu and H. Zabel
Fig. 3.37. a) Asymmetric hysteresis loop (b) of a Fe/MnF2 bilayer and and thecorresponding neutron reflectivity curves: a) the PNR curves recorded at the firstcoercive field, Hc1 and c) the PNR curve recorded at Hc2. The lack of SF reflec-tivity at Hc2 suggests that the reversal proceeds by domain wall nucleation andpropagation, whereas at Hc1 the magnetization reverses by rotation [150, 90]
the reversal mechanism is symmetric, i.e. for both branches of the hysteresisloop magnetization rotation prevails.
c) sharp reversal on the descending branch and rounded reversal on theascending one (see Fig. 3.38). This asymmetry has recently been clarified bystudies of the azimuthal dependence of exchange bias in IrMn/F bilayers [76,41, 151]. It is an intrinsic property of the EB bilayer systems and it takes placewhenever the measuring external field is offset with respect to the field coolingorientation. By simply analyzing the geometrical asteroid solutions, it becomesobvious that the sweep line does not symmetrically cross the shifted asteroidwhen the field is not parallel to the unidirectional anisotropy. Actually, thisis a peculiar case of asymmetry which can be understood even within thephenomenological model for exchange bias [76] which assumes a rigid AF spinstructure. Within the SG and M&B model [41] such asymmetric reversalscan be simulated over a wide range of AF thicknesses and anisotropies. Also,
Fig. 3.38. Asymmetric hysteresis loop of a IrMn/CoFe bilayer along an offsetθ = 3◦ angle [41]
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 161
within the DS model [36, 39], the effect of an offset measuring field axis withrespect to the anisotropy axes of the AF and F layer can result in asymmetricreversal modes.
d) the descending part is steeper, while the ascending branch is morerounded (see Fig. 3.39). This asymmetry of the hysteresis loop needs to bedistinguished from the previous ones, since it occurs when the external field isoriented parallel with respect to the anisotropy axis. It is observed in EB bilay-ers with thin antiferromagnetic layers or for systems containing low anisotropyAF layers. We call these antiferromagnets weak antiferromagnets and char-acterize them by the R-ratio. When the R-ratio is slightly higher than one(weak AF layers), then the asymmetry of the hysteresis loop can be repro-duced within the SG model. When the R-ratio is much higher than one (strongAF layers), then the hysteresis loops are symmetric as shown in Fig. 3.28. Toaccount for the asymmetry we consider the following example where it is es-sential that the AF layer is weak but the R ratio is higher than 1: R = 1.1,f = 60%, γ = 5◦, and θ = 0◦. For these values the minimization of thefree energy is evaluated numerically. The results are plotted in Fig. 3.39. Thelongitudinal and transverse components of the magnetization vector is shownas a function of the reduced field h = H/|H∞
eb |. We clearly recognize thatthe hysteresis loop is asymmetric: steeper on the descending leg and morerounded on the ascending leg. The asymmetry is due to the large rotation an-gle of the AF spins during the F magnetization reversal. This asymmetry hasnot received experimental recognition so far, therefore it remains a predictionof the SG model.ii) The second class of asymmetric hysteresis loop is directly related to thetraining effect, and therefore to the stability of the AF layer and AF/F mag-netic interface during the magnetization reversal.
Fig. 3.39. Longitudinal (left) and transverse components (right) of the magnetiza-tion vector for an week antiferromagnet: R = 1.1. The hysteresis loop is asymmetric:the descending part is steeper than the ascending part. The asymmetry is clearlyseen also in the transverse component of the magnetization
162 F. Radu and H. Zabel
a) sharp reversal at Hc1 and more rounded reversal on the ascendingbranch, at Hc2 (see Fig. 3.40). Measuring subsequent hysteresis loops, therounded reversal character does not change but appears also at Hc1. Thistype of asymmetry, being related to the training effect, is frequently re-ported in different exchange bias systems [133, 99, 61, 58, 136, 152, 135,111, 60, 138, 141, 59, 153, 154, 142]. Its underlying microscopic origin, hasbeen recently demonstrated to stem in irreversible changes that occurs inthe AF layer. Polarised Neutron Reflectivity measurements have revealedthat at Hc1 the reversal proceeds by domain wall nucleation and propa-gation [111, 155, 60, 156, 59, 153, 154, 142]. At the second coercive field,magnetization rotation is the reversal mechanism. Moreover, by analyzing theAF/F interface [111, 60], it has been observed experimentally that a transi-tion from a collinear state to an non-collinear disorder state occurs. It suggeststhat the AF layer in (CoO thin layer)/F evolves from a single AF to a multipleAF domain state. In a AF layer that exhibits a domain state the anisotropyorientation in different domains is laterally distributed causing a reduced co-ercive and exchange bias field. Note that a thick CoO film is suggested to be
Fig. 3.40. (a) MOKE hysteresis loop of a CoO/Co bilayer after field cooling to50 K in an external field of 2000 Oe. The black dots denote the first hysteresis loop,the dotted line the second loop. Any further loops are not significantly differentfrom the second. (b) and (c) Hysteresis loops recorded by polarized neutrons fromthe same sample but at 10 K. I++, I−−, I+− and I−+ refer to non-spin flip andspin-flip intensities as a function of external magnetic field. [111]
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 163
already in a domain state [35] after field cooling, therefore one would expecta variation of the loop asymmetry (and training effect) with respect to thethickness of the AF layer.
3.14.2 Temperature Dependence of the Rotatable AF Spins(Coercivity)
We discuss here the temperature dependence of the interfacial properties.Experimentally this can best be studied by an element selective method[114, 115, 117, 17, 19] to distinguish between the hysteresis of the F and AFlayer. Element specific hysteresis loops have been studied for Fe/CoO [19],which highlights the behavior of the rotatable interfacial AF spins.
The exchange bias hysteresis loops measured at the L3 absorption edgesof Co (E=780 eV, closed symbols) and Fe (E=708.2 eV, open symbols) andfor different temperatures are shown in Fig. 3.41. After FC to 30 K, several
Fig. 3.41. The temperature dependence of the exchange biased hysteresis loopsmeasured at L3 absorption edges of Co (E=780 eV, closed symbols) and Fe (E=708.2eV, open symbols). Scattering angle is 2θ = 32◦ [19]
164 F. Radu and H. Zabel
hysteresis loops were measured in order to eliminate training effects. Subse-quentially the temperature was raised stepwise, from low to high T. For eachtemperature an element-specific hysteresis loop at the energies correspondingto Fe and CoO, respectively, was measured. The hysteresis loops of Fe as afunction of temperature show a typical behavior. At low temperatures an in-creased coercive field and a shift of the hysteresis loop is observed. As thetemperature is increased, the coercive field and the exchange bias decreaseuntil the blocking temperature is reached. Here, the exchange bias vanishesand the coercive field shows little changes as the temperature is further in-creased.
A ferromagnetic hysteresis loop corresponding to the CoO layer is observedfor all temperatures, following closely the hysteresis loop of Fe, with somenotable differences. It appears that the ferromagnetic components of CoOdevelop higher coercive fields than Fe below the blocking temperature. This isan essential indication that the AF rotatable spins mediate coercivity betweenthe AF layer and the F one, justifying the conversion factor introduced in theSG model. After careful analysis of the element specific reflectivity data [157,19], one can conclude that a positive exchange coupling across the Fe/CoOinterface. The ferromagnetic moment of CoO is present also above the Neeltemperature. Here, the AF layer is in a paramagnetic state, therefore thecoercive fields for the Fe and CoO rotatable spins are equal.
3.14.3 Vertical Shift of Magnetization Curves (Frozen AF Spins)
A vertical shift of magnetization has been observed frequently [53, 158, 138,66, 35, 159, 15, 85, 20, 160, 161, 160, 119, 162] and is considered to haveseveral origins related to the different mechanisms for exchange bias. Withinthe M&B model a AF monolayer in contact to the F layer is assumed to beuncompensated, but still being part of the AF lattice. At most one could ex-pect a contribution to the macroscopic or microscopic magnetization equal tothat of the net magnetization of an AF monolayer and this only by probingan AF layer consisting of an odd number of monolayers. The Mauri mech-anism for exchange bias is not likely to result in a vertical shift of the hys-teresis loop, since the AF interface is compensated. Within the SG, Mal-ozemoff, and DS models for EB, a small vertical shift is intrinsic. At theinterface between the AF and F layer, a number of frozen AF spins will beuncompensated due to the proximity of the F layer. Their orientation is ei-ther parallel or antiparallel oriented with respect to the F spins, depend-ing on the type of coupling (direct or indirect exchange). They contributeto the magnetization of the system. In case of an indirect exchange cou-pling mechanism, the hysteresis loop should be shifted downwards [163, 158],whereas in case of direct exchange coupling the magnetization curve shouldbe shifted upwards. The other part of the interface magnetization, namelythe rotatable AF spins, do not cause any shift since they rotate in phasewith the F layer. Within the DS model AF domains cause an additional
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 165
shift to the macroscopic magnetization curve along the magnetization axis[35, 66, 85].
To demonstrate the shift of the magnetization we discuss a recent experi-ment by Ohldag et al. [20] using XMCD [164]. The evolution of the dichroicsignal as function of the magnetic field, providing the element specific hys-teresis loops, is shown Fig. 3.42.
The sample structure is Pd(2 nm)/Co(2.8 nm)/FeF2(68 nm)(110)/MgF2(110)/substrate and has been grown by via molecular beam epitaxy.The F layer is a polycrystalline Co whereas the AF layer is a FeF2(110)untwined single crystalline layer. The system exhibits a positive exchange biasat large cooling fields [163]. For weak cooling fields the exchange bias curve isshifted to negative fields, as usual for all EB systems. The microscopic originof the positive exchange bias is an antiferromagnetic coupling at the F/AFinterface [163]. Although, the mechanism was clearly demonstrated by mag-netometery measurements, the microscopic investigation of the AF interfaceprovides more detailed information.
The hysteresis loop of Co is typical and appears symmetric with respectto the magnetization axis (see Fig. 3.42). This is not the case for the AF in-terface magnetization which shows a twofold behavior: a) some interfacial AFspins are parallel oriented with respect to the F spins and both are rotatingalmost in phase; the AF hysteresis loop is shifted downwards with respectto the magnetization axis. This seems to be a direct proof of the preferredantiparallel coupling between the F Co and the AF Fe magnetization at theAF-F interface. The interfacial AF Fe moments are aligned during FC by theexchange interaction which acts as an effective field on the uncompensated
Fig. 3.42. Element-specific Co (black) and Fe (gray) hysteresis loops acquired atT= 15 K after field cooling in +200 Oe along the FeF2 [001] axis, parallel to the AFspin axis. The direction of the cooling field and the vertical shift of the Fe loop atT = 15 K is indicated by arrows. Reproduced from [20]
166 F. Radu and H. Zabel
spins, most likely confined to the interface [20]. An AF uncompensated mag-netization in FeF2 can also occur due to piezomagnetism, which is allowedby symmetry in rutile-type AF compounds and may be induced by stressesoccurring below the Neel temperature [165, 148].
These experimental observations can also be described by the SG model.During the field cooling procedure the frozen-in spins depicted in Fig. 3.27will be pointing opposite to the F spins as to fulfill the indirect exchangecondition at the F/AF interface. The rotatable ones remain unchanged inthe figure, being parallel aligned with the F spins. The orientation of thefrozen spins is governed by the exchange interaction at the F/AF interface,whereas the rotatable spins are aligned by the F layer and the external field.Upon reversal the rotatable spins will follow the ferromagnet whereas thefrozen spins remain pinned, leading to a shift downwards of the interfacialAF hysteresis loop. As the system described above does not exhibit trainingeffects, no irreversible changes occur during the magnetization reversal. Inhigh enough cooling fields, the frozen-in spins will align parallel to the coolingfield becoming also parallel with the rotatable AF and F spins. This willcause a positive shift of the hysteresis loop (positive exchange bias), since theorientation of the AF frozen spins is negative with respect to the couplingsign [163].
The experimental results described above could most likely be describedalso by the DS model and the Malozemoff model if magnetic domains in theAF layer will be confirmed.
A relative vertical shift of the magnetization related to training effects isdescribed by Hochstrat et al. [138] for a NiO(0001)/Fe(110) exchange biassystem. The antiferromagnet is a single crystal NiO whereas the ferromag-netic material is an Fe layer deposited under ultrahigh vacuum condition.Upon successive reversals of the F layer a decrease of the total magnetizationwas observed by SQUID magnetometery. The variation of the vertical shiftas a function of the hysteresis loop index n was correlated to a decrease ofthe AF magnetization. A linear correlation between the AF magnetization,deduced from the vertical shift, and the exchange bias field during trainingwas found, suggesting that the training effect may be related to a reduced AFmagnetization along the measuring field axis and as a function of the loopindex.
In Fig. 3.43 we show another situation where a vertical shift is visible inmacroscopic magnetization curves measured by SQUID magnetometery [166].The system is a polycrystalline CoO(2.5 nm)/Co(15 nm) bilayer grown bymagnetron sputtering [112]. The hysteresis loop at room temperature andalong the easy magnetization axis of the F layer is symmetric with respectto the magnetization axis and shows a low coercive field. Upon field coolingthe system through the Neel temperature t of the CoO layer (TN = 291 K)to the measuring temperature T=4.2 K the system is set in an exchangebias state. Then, a hysteresis loop is measured by sweeping the field from2000 Oe to –2000 Oe and back. By comparing this hysteresis loop with the
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 167
Fig. 3.43. Hysteresis loops of CoO(25 A)/Co(180 A). The dashed line is thehysteresis loop recorded at 300 K which is above the Neel temperature of the CoOlayer. After field cooling the system to 4.2 K, a hysteresis loop is recorded sweepingthe applied field from 9000 to –9000 Oe and back to 9000 Oe (black line, long loop).A second loop is measured under the same conditions as the previous one, but thefield range is shorter, namely between 2000 Oe to –2000 Oe and back to 2000 Oe.The cooling field was positive. A vertical shift of the hysteresis loop is clearly visiblefor the short loop, whereas the long loop appears to be centered with respect tomagnetization axis. [166]
one measured at room temperature one clearly observes a vertical shift upalong the magnetization axis. Moreover, this up-shift is due to the ferromag-netic Co spins which do not fully saturate at –2000 Oe. Previous studieshave shown that after the reversal at Hc1, the AF CoO layer breaks into AFdomains exhibiting an anisotropy distribution. Due to the strong coupling be-tween the F and the AF domains, the F layer cannot easily be saturated. Next,after repeating the field cooling procedure, another hysteresis loop was mea-sured by sweeping the external field to a much larger negative value, namelyto –9000 Oe. Now one observes that the hysteresis loop becomes more sym-metric with respect to the magnetization axis. The saturation field, where thehysteresis loops closes, is about –8000 Oe.
The example above shows that a vertical shift of magnetization can berelated to a non homogeneous state of the F layer due to non-collinearitiesat the AF/F interface. The relation to the training effect is clearly seen asdifferent coercive fields Hc2 depending on the strength of the applied fields.When a stronger field is applied in the negative direction, the exchangebias decreases due to a larger degree of irreversible changes into the AFlayer.
168 F. Radu and H. Zabel
A distinct class of vertical shifts is found in exchange bias systems where adiluted antiferromagnet [167, 66, 35, 85] acts as a pinning layer. As function ofdilution of a CoO layer by Mg impurities, as well as, partial oxygen pressureduring the deposition, the AF layer acquires an excess magnetization seen asa vertical shift of the hysteresis loop. This strongly supports the fact that thedomain state in the AF layer as well as the EB effect is caused and controlledby the defects [66, 85].
Vertical shifts in nanostructures and nanoparticles are often observed dueto uncompensated AF spins. For a detailed discussion we refer to recent re-views by Nogues et al. [161] and Iglesias et al. [160] and also recent papers onAF nanoparticles [119, 162].
3.15 Further Evidence for Spin-Glass Like BehaviorObserved in Finite Size Systems
Although nanoparticles are not covered in this review, we, nevertheless, dis-cuss two recent instances which in AF nanoparticles confirm the SG behavior.One is Co3O4 nanowires [119] and the other is CoO granular structure [118].
3.15.1 AF Nanoparticles
Nanoparticles of antiferromagnetic materials have been predicted by Neel [168]to have a small net magnetic moment due to an unequal number of spins onthe two sublattices as a result of the finite size [169]. Hysteresis loops ofAF nanoparticles have been observed and several suggestions were made toaccount for their weak ferromagnetism [169]. One important finite size effect ofAF magnetic nanoparticle is the breaking of a large number of exchange bondsfor surface atoms. This can have a particularly strong effect on ionic materials,since the exchange interactions are superexchange interactions. The deficit ofexchange bonds could lead to a spin disordered shell exhibiting spin-glass likebehavior. The last effect is demonstrated most recently by Salabas et al. [119]and discussed further below.
In Fig. 3.44 two hysteresis loops of Co3O4 nanoparticles (8 nm diameter)prepared by a nanocasting route are shown [119]. Both curves were measuredat T = 2 K after cooling the system in zero field (ZFC hysteresis loop) andin an external applied field of +4 T (FC hysteresis loop). Whereas the ZFCloop shows typical weak ferromagnetic properties specific to AF particles,after FC a completely different behavior is observed. The hysteresis loop isexchange biased, it is vertically shifted and shows training effect. Moreover,the temperature dependence (not shown) of the coercive field and exchangebias field increases with decreasing temperature, which is also a usual behaviorof EB systems. More strikingly the FC hysteresis loop does not close on theright side at positive fields. This open hysteresis loop is a direct indicationof a spin-glass like behavior, similar to the loops observed in pure spin glass
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 169
Fig. 3.44. Field cooled and zero field cooled hysteresis loops of Co3O4 nanowires.From [119]
systems [37]. This behavior appears to support the SG model (Fig. 3.27),where a reduced anisotropy is assumed to occur at the F/AF interface. Bothfrozen-in spins and rotatable spins are directly seen in the FC hysteresis loop.Moreover, the irreversible changes of the surface AF spins are causing the openloop. A similar open loop is also predicted by the DS model. Within the DSmodel the hysteresis loop of the AF interface layer is not closed on the righthand side because the DS magnetization is lost partly during F reversal dueto a rearrangement of the AF domain structure. The AF particles, however,are supposed to be single AF domains, therefore irreversible changes are dueto surface effects rather than caused by AF domain kinetics. Certainly, atvery large cooling fields, the bulk structure of the AF particle should be alsoaffected. The spin-glass like behavior was also recently observed for cobaltferrite nanoparticles [162], for (Mn,Fe)2O3−t nanograins [170], and for a ε-Fe3NCrN nanocomposite system [171].
3.15.2 Extended Granular AF Film
Another direct experimental evidence of a spin-glass like behavior was ob-served by Gruyters [118] on CoO/Au multilayers, CoO/Cu/Fe trilayers, andCoO/Fe bilayers with granular structure. In Fig. 3.45a) the ZFC hysteresisloop of a CoO/Au multilayer and for different temperature is shown. Onenotices that the coercive field is enhanced at low temperatures, but no hys-teresis shift develops. The FC hysteresis loop (see Fig. 3.45b)), however, isalmost completely shifted to one side of the field axis. These observationsare explained by Gruyters as an effect of the uncompensated AF spins of thegranular film structure. The saturation magnetization of this granular CoOfilm is about 60–62 emu/gCoO which would result in an enormous amount ofuncompensated spins equivalent to 22% for the observed particles volume. Nostoichiometry influences are considered to contribute to this value. Althoughthe uncompensation level is unclear, the evidence of a spin-glass behavior
170 F. Radu and H. Zabel
Fig. 3.45. a) ZFC hysteresis loops of [CoO [x A]/Au (60 A)] multilayers. b) FChysteresis loop of a [CoO [20 A]/Au (60 A)] multilayer. c) ZFC and FC magnetizationcurves as a function of increasing temperature T in different fields for a [CoO [15A]/Au (60 A)] multilayer. d) Field dependence of Tirr raised to the 2/3 power fortwo different multilayers. ([118])
is well demonstrated. In Fig. 3.45c) ZFC and FC magnetization curves aremeasured as function of temperature and for different external fields. Twomain characteristic features are observed for these curves: an irreversibilitytemperature Tirr, where the ZFC and FC branches of MCoO(T) coalesce, anda pronounced peak due to superparamagnetic blocking in the ZFC magneti-zation. The direct evidence of a spin-glass behavior is the field dependenceof Tirr shown in Fig. 3.45d). The existence of critical lines spanned by thevariables temperature and magnetic field can be explained by mean-field the-ory. One of these lines has been predicted by de Almeida and Thouless forIsing spin systems [118]. The Tirr raised to 2/3 power as function of fieldexhibits a linear dependence in agreement with the predictions of Almeidaand Thouless.
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 171
3.15.3 Dependence of the Exchange Bias Field on Lateral Sizeof the AF Domains
The relation between the exchange bias and the reduced size effects duenano-structuring of the AF-F systems is important from both fundamentaland technological points of view. From a fundamental point of view, the re-duced lateral size of both F and AF objects induces significant changes ofthe exchange bias field, coercive fields and also the asymmetry of the hystere-sis loops [172, 173, 174, 175, 176, 180, 179, 177, 182, 181, 178, 154, 183]. Insome systems an increased exchange bias field occurs for reduced lateral sizes,whereas in other cases an opposite behavior is reported, the exchange biasfield decreases with decreasing the lateral length scales [161]. We refer in thefollowing to the last situation.
In Fig. 3.46 are depicted schematically several lateral systems which arecommonly used to study the influences of the nano-structuring onto the ex-change bias properties. A reduced lateral size of the ferromagnet (Fig. 3.46b)and Fig. 3.46c)) gives rise to additional shape anisotropies for the ferromag-net leading to a change of both coercive and exchange bias fields as well asa change of the hysteresis shape. In Fig. 3.46a) these additional anisotropiesare minimized and therefore the dependence of exchange bias as function ofthe AF lateral size is more transparent. For all three situations we can as-sume that at the borders defined by the geometrical nanostructures, there isan additional disorder extending to the interface. Even for the case b) wherethe F is nano-structured we may expect that the lithographic process willnot always be stopped exactly at the interface but also affecting the AF layeraround the dot.
Due to nano-structuring it is natural to expect that at the edges of thedot there are AF spins with reduced anisotropy. These spins will contribute tothe coercivity at the expense of the interfacial exchange energy. The effectiveinterfacial exchange energy can be written as: Jeff
eb = FfJeb, where F isa conversion factor related to size effects, similar to the f defined for theinterface. It is easy to estimate the F -parameter, as the fraction of the outershell area divided by the total dot area: F ≈ A1/A2 ≈ (π(D − d)2)/(πD2) =1− 2d/D+ d2/D2, where d is the lateral thickness of the outer shell and D isthe diameter of the dot itself. Assuming that d << D and that d is constant,
AF
F AF
F F
AF
a) b) c)
Fig. 3.46. A schematic view of different nanostructered systems used to study thefinite size effects on the exchange bias properties
172 F. Radu and H. Zabel
we obtain the general expression for the variation of the exchange bias fieldas D:
−Heb +HD→∞eb
HD→∞eb
≈ 1 − F ≈ 1D, (3.55)
where HD→∞eb is the exchange bias of an extended film. Within this simple
model the exchange bias field decreases as the lateral size of the AF structuredecreases. This 1/D dependence is often observed experimentally [161].
The general behavior of the hysteresis loops as a function of the f -parameter is depicted in Fig. 3.28. We see that for high f corresponding to alarge diameter AF dot, the coercive field is small and the exchange bias is high.When, however, the lateral size of a AF object becomes smaller, the f ratio de-creases leading to an increased coercive field and reduced exchange bias field.Transition from the top hysteresis to the bottom hysteresis of Fig. 3.28 areusually observed due to nanostructuring of the AF layers [172, 177, 178, 161].
3.16 Conclusions
The results presented in this chapter provides an overview of the physics ofa F/(Interface)/AF exchange bias systems. Fundamental properties of theunidirectional anisotropy are considered and discussed. The Meiklejohn andBean model as well as the Mauri model are considered in detail and compared,both analytically and numerically. The Neel approach, Malozemoff model,Domain State model and a model of Kim and Stamps are discussed as theyprovide novel and fundamental ideas on the EB phenomenon. The Spin Glassmodel is discussed in even more details. Further experimental results touchingthe fundamentals of exchange bias are described.
We distinguish several outcomes of our overview:
• The exchange bias is an interface effect, as clearly proven by the 1/tF
dependence. Deviations from this law were observed in the literature, butfundamentally this expression is clearly well established.
• The AF anisotropies in the bulk of the AF layer and at the interface toa soft ferromagnetic layer give rise to an impressively rich behavior ofthe magnetic properties: the hysteresis loops can be shifted along bothfield and magnetization axis and in both positive and negative directions,the azimuthal dependence of exchange bias exhibits non-intuitive behaviorsuch as a shift of its maximum with respect to the field cooling orientation,the hysteresis loops are asymmetrically shaped, etc. Some of the exchangebias characteristics mentioned above are not fully and consistently revealedexperimentally which leaves an open window for further quests.
• The analytical formulae for the exchange bias within the M&B modeldepends mainly on the properties of the AF layer, namely on its anisotropyand thickness.
3 Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures 173
• The peak-like increase of the EB close to the critical thickness of the AFlayer can be reproduced by the SG model. This effect is also accounted forby the DS model.
• The azimuthal dependence of the exchange bias effect could help to dis-tinguish between two ideal mechanisms for exchange bias: Mauri versusM&B models.
• The SG model describes the conversion of coupling into coercivity.
List of acronyms
AMR Anisotropic Magneto-ResistanceFC Field CoolingZFC Zero Field CoolingEB Exchange BiasDS Domain State ModelSG Spin Glass ModelSW Stoner-Wohlfarth ModelM&B Meiklejohn and Bean ModelMCA Magneto-Crystalline AnisotropyMOKE Magneto-Optical Kerr EffectPNR Polarized Neutron ReflectivitySQUID Superconducting Quantum Interference DeviceAF AntiferromagnetF FerromagnetMFM Magnetic Force MicroscopyMRAM Magnetic Random Access MemoryDAFF Diluted Antiferromagnets in an External Magnetic FieldXMCD Soft X-ray Resonant Magnetic DichroismXRMS Soft X-ray Resonant Magnetic Scattering
Acknowledgments
We gratefully acknowledge support by the DFG Sonderforschungsbereich 491:Magnetic Heterostructures: spin structure and spin transport, and by BESSY.
We have benefitted from discussions with: Werner Keune, Wolfgang Klee-mann, Kurt Westerholt, Ulrich Nowak, Klaus D. Usadel, Christian Binek,Kristiaan Temst, Ivan K. Schuller, Robert L. Stamps.
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4
Exchange Coupling in Magnetic Multilayers
Bretislav Heinrich
Physics Department Surion Fraset University Burnaby, BC, V5AIS6, Canada
Abstract. Spintronics devices employ a wide range of magnetic multilayerstructures. In this chapter the coupling between the magnetic layers throughnon-magnetic spacers will be reviewed. It starts with phenomenology of interlayermagnetic coupling. This is followed by experimental techniques and a detailed in-terpretation of measurements. The main emphases will be given to the static and rfmeasurement techniques. Theory of magnetic coupling covers the basic ideas of inter-layer exchange coupling and dipolar coupling. The additional contributions causedby imperfections in realistic samples are discussed in sessions on orange peel cou-pling, pinhole coupling, and biquadratic exchange coupling. Experimental resultsand their interpretation were carried out to emphasize the richness of magneticinteractions in a wide range of systems. A particular attention was given to theFe/Ag,Au/Fe, Co/Cu/Co and Fe/Cr,Mn,Pd/Fe structures allowing one to demon-strate the importance of interfaces on magnetic coupling in multilayer systems. Fi-nally, it is shown that the time retarded response of interlayer exchange interactionleads to an entirely new coupling which is based on spin pumping and spin sinkconcepts. This contribution arises only during spin dynamics and compared to thestatic interlayer exchange coupling is long ranged allowing one in principle move in-formation by means of a spin current via a mechanism that does not directly involvea net transport of electron charge.
4.1 Introduction
Spintronics and high density magnetic recording employ a number of magneticlayers separated by non magnetic spacers. The magnetic layers are coupledthrough a non-magnetic spacer. The coupling between the magnetic layerscan be caused by intrinsic and extrinsic mechanisms and be of static anddynamic origin. The purpose of this Chapter is to review the basic conceptsof magnetic coupling in 3d transition elements. A number of excellent reviewarticles have been published by Hathaway [1], Slonczewski [2], Stiles [3], andEdwards and Umerski [4]. An extensive review of the behavior of spin densitywaves in Fe/Cr/Fe trilayers and multilayers is provided by Fishman [5].
B. Heinrich: Exchange Coupling in Magnetic Multilayers, STMP 227, 185–250 (2007)
The Chapter is organized in the following way: Section 4.2 describesthe phenomenology of magnetic coupling. Section 4.3 deals with the basictechniques employed in the study of magnetic coupling. A brief summaryof theoretical concepts of magnetic coupling is carried out in Sect. 4.4. Asummary of experimental results on Fe/Ag,Au/Fe(001), Co/Cu/Co(001), andFe/Cr,Mn,Pd/Fe(001) systems is presented in Sect. 4.5. The dynamic couplingwill be described in Sect. 4.6.
4.2 Phenomenology of Magnetic Coupling
The description of magnetic coupling will be mostly restricted to ultrathinmagnetic films. In ultrathin films magnetic variations across the thicknessof the film are mainly suppressed. This means that the magnetic momentson lattice sites across the film thickness are nearly parallel to each other.This is not exactly correct, but greatly simplifies the treatment of magneticproperties. The limits of this concept are described in [6] and [7]. The filmcan be fairly well considered as ultrathin when its thickness does not muchexceed the exchange length δ, see [6],
δ =(
A
2πM2s
)0.5
, (4.1)
where A is the exchange stiffness coefficient (for Fe A =2×10−6 ergcm−1) andMs is the saturation magnetization. The exchange length in Fe is 3.2 nm.
The simplest form of magnetic coupling per unit area of an ultrathin filmtrilayer structure of two ferromagnetic films (FM) separated by a normal metal(NM) spacer, FM1/NM/FM2, can be described by a bilinear form
E1 = −J1n1 · n2, (4.2)
where J1 is the exchange coupling coefficient in erg cm−2 and n1 and n2
are unit vectors along the magnetic moments in layers 1 and 2, respectively.Another common coupling equation has a biquadratic form
E2 = +J2 (n1 · n2)2 , (4.3)
where J2 describes the strength of biquadratic coupling. Cases of the bilinearcoupling J1 are found having either a + or a–sign. For a + sign the minimumof the bilinear energy term is reached for a parallel orientation of the magneticmoments; for a–sign the minimum energy corresponds to antiparallel magneticmoments. J2 is almost exclusively found to be positive and the minimumof the biquadratic energy term is reached when the magnetic moments areoriented perpendicularly to each other. A detailed description of bilinear andbiquadratic coupling terms will be carried out in Sect. 4.4.
The equilibrium of the magnetic moments and their dynamic response canbe found by using the Landau-Lifshitz-Gilbert (L.L.G) equation of motion [8]
4 Exchange Coupling in Magnetic Multilayers 187
1γ1,2
∂M1,2
∂t= − [M1,2 × Heff ,1,2] +
G1,2
γ21,2Ms,1,2
[M1,2 × ∂n1,2
∂t
], (4.4)
where γ1,2, M1,2, and G1,2 are the the absolute values of the electron gyro-magnetic ratios, magnetic moments per unit area , and Gilbert damping pa-rameters of layers 1 and 2, respectively. The damping parameter is expressedoften in a dimensionless parameter α = G/γMs. The first term on the right-hand side of (4.4) represents the precessional torque per unit area and thesecond term represents the well known Gilbert damping torque per unit area.The effective fields Heff ,1,2 are given by the derivatives of the magnetic Gibbsenergy density, F, with respect to the components (Mx,1,2,My,1,2,Mz,1,2) ofthe magnetization vector densities M1,2, see [9],[10],[6], and [11],
Heff = − ∂F∂M
. (4.5)
F includes the Zeeman energy of the dc applied magnetic field, demagnetiz-ing fields Hdem, rf magnetic field h, magnetic anisotropies, and the inter-layermagnetic coupling energy. In order to carry out appropriate partial derivativesin (4.5) the magnetic bilinear and biquadratic coupling terms (4.2) and (4.3)are rewritten by replacing n1,2 by M1,2/Ms,1,2. In thick films (∼100 nm) theinternal rf magnetic field has to be evaluated by using Maxwell’s equations inthe presence of the externally applied rf magnetic field. This means that therf field and rf magnetization vary across the film thickness, see e.g. [7, 12]. Inthe ultrathin film approximation the variations across the film thickness areneglected.
For small precession angles (| m |� Ms corresponding to an angle ofprecession less than a few Degrees) the magnetization vector can be linearizedby setting M = m+Ms, where Ms and m are the longitudinal and transversecomponents of M, see Fig. 4.1. This allows one to linearize the equation ofmotion (4.4).
4.3 Experimental Techniques for Magnetic CouplingMeasurements
The magnetic coupling can be determined by measuring the dependence ofthe net magnetic moment on the applied dc field. That can be done using lowand high-frequency techniques. SQUID, sample vibrating magnetometer, andMagneto Optical Kerr Effect (MOKE)[13] are the most common techniquesallowing one to measure the total magnetic moment as a function of the ap-plied field. X-ray Magnetic Circular Diochroism (XMCD)[14] is an elementsensitive technique allowing one to follow the field dependence of a particularlayer in a multilayer structure. The neutron scattering is a very powerful toolfor investigating the structural and magnetic properties of multilayer films[15, 16]. Polarized Neutron Reflection (PNR) measurements allow one to in-vestigate the spatial distribution of the magnetic moment inside the sample
188 B. Heinrich
Fig. 4.1. The coordinate system for the instantaneous magnetization M and exter-nal field H. X,Y, and Z axes are oriented along the principle crystallographic axes. θand ϕ are the polar and azimuthal angles with respect to the crystallographic axes.The x,y,z coordinate system is oriented along the magnetization M with the x axisdirected along the saturation magnetization moment. For a small precessional anglethe components My , Mz � Mx
[17, 18, 19]. Photo Electron Emission Microscopy (PEEM) [20], Low EnergyElectron Microscopy (LEEM) [21, 22] and Secondary Electron Microscopywith Polarization Analysis (SEMPA) [23] are other techniques suitable for thestudy of magnetic coupling. Ferromagnetic Resonance [10] and Brillouin LightScattering (BLS) [24] are rf techniques allowing one to determine quantita-tively all magnetic macroscopic parameters including interlayer and intralayerexchange couplings.
In this Section the discussion will be limited to the most common dc andrf techniques: SQUID, vibrating sample magnetometer, MOKE, FMR, andBLS. The experimental details can be found in the above reference. Here thediscussion will be limited to a quantitative interpretation of the experimentalmeasurements. SQUID and vibrating sample magnetometer measure the totalsample response. At FMR the microwave penetration (skin) depth is ∼ 100nm. In MOKE and BLS the total depth of studies is given by the depthof penetration of the laser beam into metallic samples. For visible light thisis approx. 15 nm. The depth resolution of MOKE signal was investigatedby Hamrle et al. [25] using magnetic multilayers. They determined both therotational and elliptical contributions to MOKE as a function of depth.
4 Exchange Coupling in Magnetic Multilayers 189
4.3.1 Dc Magnetometry
In SQUID and vibrating sample magnetometers the total magnetic momentis measured, usually along the direction of the applied field. In MOKE themagnetic moment is commonly measured in the polar and longitudinal con-figurations. In the longitudinal configuration the dc field is applied parallelto the film surface and it is usually applied in the plane of incidence of thelaser beam. In the longitudinal configuration one is sensitive to three contribu-tions. The first contribution arises from the magnetic moment that is parallelto the dc field (longitudinal Kerr effect), the second contribution comes fromthe magnetic component perpendicular to the film surface (polar Kerr effect),and in a non collinear configuration of magnetic moments the third contri-bution arises from the product of the parallel and perpendicular componentsof the magnetic moments with respect to the applied field. This contributionis called the quadratic magneto-optic effect. It arises from the second ordermagneto-optical effect [26]. This effect is the reflection analogue of the Voigteffect [27]. The magnetization components in the quadratic magneto-opticeffect are confined to the film surface. The quadratic magneto-optic effectcontribution often becomes important in magnetization measurements usingeither antiferromagneticaly coupled films or with the field applied along thehard magnetic axis. The quadratic magneto-optic contribution is a nuisancein MOKE measurements. The quadratic magneto-optic effect is an even func-tion of the applied magnetic field. It does not change its sign with reversalof the magnetic moment and creates a profound asymmetry in magnetizationmeasurements as a function of applied field. It can be removed by adding themeasured MOKE signal to its inverted counterpart. The signal is invertedaround a point that is the intersection of the MOKE signal axis with a lineparallel to the field axis and located midway between the saturated MOKEsignals for positive and negative magnetic fields [6]. The even part, propor-tional to the quadratic effect, can be obtained by subtracting the measuredand inverted signals.
The discussion in this Section will be limited to the micromagnetics oftrilayer structures consisting of a crystalline ultrathin film FM1/NM/FM2structure. For simplicity these calculations will be limited to cubic materialswith the film surface oriented in the (001) plane, see Fig. 4.1. The films of3d transition group elements are often but not exclusively grown on (001)templates. This is a particular case but it involves all the ingredients neededto formulate calculations for any other configurations involving an arbitrarydirection of the applied field and crystalline orientation. The discussion belowshould be viewed as an example taken from a user’s manual.
The total Gibbs energy per unit area for ultrathin films in a distortedcubic material with the saturation magnetic moments, Ms1 and Ms2, andthe applied field H can be written in the following form
190 B. Heinrich
F =∑
i=1,2
[− K
‖1,eff,i
2(α4
X,i + α4Y,i
)− K⊥1,eff,i
2α4
Z,i −
− K‖u,eff,i
(nu,i ·Mi)2
M2s,i
−K⊥u,eff,iα2
Z,i − Mi ·H]di − J1m1 · m2 +
+ J2 (m1 · m2)2 , (4.6)
where αX,i,αY,i and αZ,i are the directional cosines between the magne-tization vector Mi and the crystallographic axes [100], [010], and [001], re-spectively. K‖
1,eff,i,K⊥1,eff,i, K
‖u,eff,i, and K⊥
u,eff,i are the parameters of thein-plane effective four-fold (cubic), perpendicular effective four-fold, in-planeuniaxial, and perpendicular uniaxial anisotropies, respectively. nu,i are the di-rections of the in-plane uniaxial axes and di are the film thicknesses. m1 andm2 are unit vectors directed along the magnetizations of the coupled films:J1 and J2 are the bilinear and biquadratic coupling coefficients. The indicesi =1 and 2 describe the properties of the layers FM1 and FM2, respectively.In ultrathin films the magnetic moments across the film are locked togetherby intralayer exchange coupling and they can be considered to be giant mag-netic molecules [6]. For ultrathin films the role of the interface anisotropiesof a uniformly magnetized sample can be included in the effective anisotropyparameter,
K‖1,eff = K
‖1,bulk +
K‖1,s
d
K‖u,eff = K
‖u,bulk +
K‖u,s
d
K⊥u,eff = −2πM2
s +K⊥
u,s
d, (4.7)
where the subscript s represents the interface contributions. The units of theinterface anisotropies are erg/cm2, see [6]. The energy expression in (4.6) isvalid for a wide range of magnetic ultrathin films such as Fe on Ag(001) [6]and GaAs(001) [28, 29, 30] templates. One can easily generalize it by using theappropriate film symmetry An example for an arbitrary orientation of mag-netic moments can be found in [31]. The expression for K⊥
1,eff is not includedin (4.7). It originates in variations of the perpendicular uniaxial anisotropy,K⊥
u,s, across the film surface. It leads to a higher order dependence on 1/d,see [32] and the end of subsection Biquadratic exchange coupling.
The field dependence of the magnetic moment can be found by minimizingthe total Gibbs energy. The static equilibrium is found by minimizing the totalenergy with respect to the angles ϕ1, ϕ2, θ1, and θ2 for the given angles ϕH
and θH , see Fig. 4.1. In this Chapter the calculations will be restricted to thein-plane geometry, this means θ1 = θ2 = θH = π/2.
There are a number of minimization procedures available and they are usu-ally implemented by individual groups according to their liking. One should
4 Exchange Coupling in Magnetic Multilayers 191
realize that minimum energy solutions can exhibit metastable states. Usuallyone looks for the lowest energy state. This means no hysteresis is present inmagnetization measurements. This is not often the case in experiments es-pecially those using films grown on GaAs(001) substrates [29]. Therefore itis imperative to carry out MOKE measurements in the lowest energy state.The lowest energy state for a given magnetic field can be achieved by cyclingthe magnetic state at the given applied field with a transverse ac magneticfield which increases to some preselected maximum and then the amplitude isgradually decreased to zero. This has to be repeated for each applied dc field.An example of such procedure can be found in [29] using exchange coupledGaAs/Fe/Au/Fe/Au(001) structures.
Several examples of hysteresis loops using exchange coupled magnetic bi-layers FM1/NM/FM2 are shown in Figs. 4.2 and 4.3. In order to avoid com-plexities associated with a general direction of the applied field one shouldemploy two magnetic films having different thicknesses and with the externalfield applied along the magnetic easy axis of the thicker film. In that case thethicker film remains oriented close to the easy axis. It is the thinner film thatundergoes a full angular dependence, see the insets in Fig. 4.2. Since the twomagnetic moments are different it is easy to identify the contributions fromthe individual layers.
The family of curves describing the role of biquadratic magnetic couplingis shown in Fig. 4.2. When the biquadratic magnetic coupling becomes com-parable to the bilinear coupling one is not able to achieve an antiparallelconfiguration of magnetic moments in low magnetic fields. The biquadraticcoupling can lead in zero field to an angle between the two magnetic momentsranging from–180 to 90 degrees. The biquadratic magnetic coupling can alsoaffect the approach to saturation. For even small J2 the saturation point isreached without a jump in the magnetic moment; this is also true even in thepresence of magnetic anisotropies, see Fig. 4.2. The interpretation of the datawhen there is no discontinuity in the magnetization becomes quite simple.The deviation from saturation can be treated using a small angle expansion.It is easy to show that the saturation field along the easy direction can bedescribed by the simple expression
Hsat +2Keff
Ms= −J1 − 2J2
Ms
(1d1
+1d2
), (4.8)
where it has been assumed that the saturation magnetization Ms is the samefor both films. Keff is an effective anisotropy field obtained from minimiza-tion of (4.6), and d1 and d2 are the thicknesses of the layers FM1 and FM2,respectively.
In samples where one is not able to get both easy axes along the samedirection the magnetization curves can be kept relatively simple if one orientsthe field along the easy axis of the film having a large in-plane anisotropy. Anexample is shown in Fig. 4.3. Notice that the thinner film 8Fe with a large
192 B. Heinrich
Fig. 4.2. Simulations of magnetization curves using a set of exchange couplingparameters. The lines 1, 2, 3, 4 and 5 correspond to J2=0.0, 0.01, 0.04, 0.06,and 0.08 erg/cm2, respectively. The bilinear coupling was kept constant, J1=–0.1. The calculations were carried out for the magnetic parameters obtainedin GaAs/8Fe/Au/16Fe/Au(001) structures [30], where the integers represent the
number of atomic layers. 16Fe: K‖1,eff=3.1×105 erg/cm3 ; 8Fe:K
‖1,eff=1.33×105
erg/cm3. 4πMs=21.5 kOe. In-plane uniaxial anisotropies were omitted in order tokeep the easy magnetic axis in both films along the [100] crystallographic direction.The applied field was oriented along the magnetic easy axis [100]. The inset showsthe field dependence of the magnetization angle for J1=–0.1 erg/cm2 and J2=0. Thedashed line corresponds to the 16Fe film and solid line to the 8Fe film. Note thatthe first jump brings the magnetic moment of the thinner 8Fe film from the parallelorientation over the first hard axis, [110], close to the second easy axis, [010]. Thesecond jump corresponds to pulling the magnetic moment over the second hard axis,[110] resulting in an antiparallel configuration of the magnetic moments
in-plane uniaxial anisotropy rotates from the easy axis by a small angle, whilethe thick 16Fe film undergoes a large angular rotation.
In FM1/NM/FM2 structures one can measure by MOKE only negativebilinear J1 (antiferromagnetic coupling). One is able to measure a posi-tive (ferromagnetic) bilinear coupling by using spin “engineered structures”.An additional FM0 layer with a normal metal spacer creating a large an-tiferromagnetic coupling between FM0 and FM1 is needed [33, 34], e.g.FM0/NM0/FM1/NM/FM2. In these structures the magnetic moments in
4 Exchange Coupling in Magnetic Multilayers 193
Fig. 4.3. Simulations of magnetization curves for the magnetic anisotropies cor-responding to GaAs/8Fe/Au/16Fe/Au(001) structures [30], the integers repre-sent the number of atomic layers. The following magnetic parameters were used.16Fe: K
‖1,eff=3.1×105 erg/cm3, K
‖u,eff=3.3×104erg/cm3; 8Fe: K
‖1,eff=1.33×105
erg/cm3, K‖u,eff=–1.14×106 erg/cm3. The hard axis of the in-plane uniaxial
anisotropy in the 8Fe film is oriented along the [110] direction. 4πMs=21.5 kOe.The solid, dashed, and dotted lines in (a) correspond to J1=0.0, –0.1, and –0.3erg/cm2, respectively. The applied field was oriented along the magnetic easy axis,[110], of the 8Fe film. Note that this film has a large in-plane uniaxial anisotropy.This means that for the 16Fe film the applied field was oriented along the hard axis.(b) shows the field dependence of the magnetization angle with respect to the [100]axis for J1=–0.3 erg/cm2. The dotted line corresponds to the 16Fe film and the solidline to the 8Fe film. Note that below 1.5 kOe the magnetic moments are orientedantiparallel to each other with the magnetic moment of the 8Fe film oriented alongits easy magnetic axis [110](45 Degree), and the magnetic moment of the 16Fe filmoriented along its hard magnetic axis [110] (–135 Degree)
FM1 and FM2 in zero applied field are oriented antiparallel to that in FM0.One needs to apply a dc field along the direction of the magnetic moment ofthe FM0 layer to overcome the ferromagnetic coupling and thus to orient themoments in FM1 and FM2 in an antiparallel configuration. Strong antifer-romagnetic coupling can be achieved by using ultrathin Ru spacers [33], seeSect. 4.5.
4.3.2 FMR and BLS Techniques
The magnetic coupling can be measured by rf techniques, see [6, 10, 24, 35, 36,37]. In FMR one usually sets the microwave frequency and sweeps the field.
194 B. Heinrich
However in this age of network analyzers (NA) this is not limitation; one canset the field and sweep the frequency. In BLS one sets the applied field andsweeps the frequency. When the field is held constant the angle between themagnetic moments is fixed. This is a simpler situation compared to a regu-lar FMR measurement (holding constant frequency and sweeping the field)where the angle between the magnetic moments changes in non-collinear con-figurations. For a saturated sample the difference between constant field andconstant frequency sweeps is minimal. The main difference is that in BLS onemeasures spinwaves having an in-plane wave-vector q-component corespond-ing to that of the in-plane wavelength of the incoming laser light whereas inFMR the measured spinwave mode corresponds to q�0 (homogeneous mode).This means that the ferromagnetic films in BLS are always coupled by a dipo-lar interaction, see Subsection 4.2. Interpretation of FMR and BLS results canbe carried out by using rf solutions of (4.4). Let us briefly outline a methodfor setting up the equations of motion for the rf magnetization components.The coordinate system for an arbitrary orientation of the magnetic momentwith respect to the crystallographic axes is shown in Fig. 4.1.
Further discussion will be limited to the in-plane configuration where thesaturation moment and external field are oriented in the X-Y plane, θi = π/2.For a noncollinear configuration with static magnetic moments in the filmplane the directional cosines αX,i,αY,i, and αZ,i with respect to the crystal-lographic axes are given by
αX,i =Mx,i
Mscos(ϕi) − My,i
Mi,ssin(ϕi)
αY,i =Mx,i
Ms,isin(ϕi) +
My,i
Mscos(ϕi)
αZ,i =Mz,i
Ms, (4.9)
where Mx,i,My,i and Mz,i are the instantaneous magnetization componentsin the coordinate systems with the x-axis parallel to the saturation magneti-zation. The effective fields in the frame of the moments FM1 and FM2 can beobtained by inserting (4.9) into (4.6) and using (4.5). However (4.6) is not thedensity of energy. This requires conversion of the interlayer coupling energyto the energy density. This conversion is well described on pp. 569–570 in [6].Assuming an even sharing of the interlayer coupling by all atomic layers insidethe film the density of energy, Ui, for the individual layers FM1 and FM2 aregiven by
Ui = −K‖1,eff,i
2(α4
X,i + α4Y,i
)−K‖u,eff,i
(ni ·Mi)2
M2s,i
+
+ K⊥u,eff,iα
2Z,i − MiH−
(J1m1 ·m2 − J2 (m1 · m2)2
)/di
Heff,i = − ∂Ui
∂Mi. (4.10)
4 Exchange Coupling in Magnetic Multilayers 195
The partial derivatives are taken with respect to Mx,i,My,i and Mz,i. InFMR and BLS the rf transversal components are negligible compared to themagnetization component parallel to the x-axis (static effective fields). Thereplacement of the x-component of the magnetization by the saturation mag-netization Ms can be done only after the partial derivatives have been carriedout. The equations of motion (torque equations) for FM1 and FM2 are ob-tained by using the above effective fields in the LLG (4.4). One can ignore allterms which are quadratic in the small rf components. Notice that the perpen-dicular 4-fold anisotropy,K⊥
1,eff,i, is not included in the parallel configuration;it leads to 3rd order terms in the small rf magnetic componentMz,i. This leadsto a coupled system of equations for the transverse magnetization componentsfor FM1 and FM2. The z-components of the torque include also expressionswhich contain only large static magnetization Ms,i components. These ex-pressions correspond to static equilibrium and are equal zero. They providesolutions for ϕi(H) for the applied field H. The static equilibrium can bealso obtained from minimizing the total energy given by (4.6). The coupledequations for the rf magnetization components leads to two solutions. Theprecessional motions are coupled and result in an acoustic mode in which themagnetic moments in the two layers precess in phase and in an optical mode inwhich the magnetic moments precess in antiphase. The simplest interpretationof the coupling can be obtained in the saturated case when the dc magneticmoments are parallel to the applied field. The isolated films FM1 and FM2must have different resonant frequencies (fields) in order to be able to observeacoustic and optical modes. If the resonance frequencies (fields) of the twofilms are exactly same than the strength of the optical mode is zero. Differ-ent resonant fields are easy to establish by choosing different film thicknessesfor the films FM1 and FM2. The interface anisotropies scale with 1/d andin consequence the isolated films exhibit two separate resonance frequencies(fields) and the optical mode becomes observable. The sign of coupling canbe determined from the relative positions of the acoustic and optical modes.In FMR the microwave frequency is usually fixed. For antiferromagnetic cou-pling (J1 < 0) and ferromagnetic coupling (J1 > 0) the optical modes arelocated at higher and lower fields than the acoustic modes, respectively. Thepositions and intensities of these two modes are nontrivial functions of themagnetic anisotropies and the strength of the interlayer coupling. Howeverthe resonance spectrum can be easily evaluated by using the coupled L.L.G.equations of motion, see Fig. 4.4.
In the saturated state (collinear magnetic moments) the overall strengthof the interlayer coupling, Jeff is given by the superposition of bilinear andbiquadratic interlayer couplings,
Jeff = J1 − 2J2 . (4.11)
The optical mode has its magnetization components out of phase and con-sequently a homogeneous rf driving field inside the ultrathin films makes
196 B. Heinrich
Fig. 4.4. Simulations of acoustic and optical resonance peaks at f=36 GHzas a function of bilayer exchange coupling in an FM1/NM/FM2 structure. Inpanel (a) J1=0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 ergs/cm2. In panel (b) J1=0.0,–0.1, –0.2, –0.3, and –0.4 ergs/cm2. Note that the antiferromagnetic interlayer cou-pling moves the resonant peaks to larger fields. For the antiferromagnetic cou-pling the acoustic and optical peaks move to higher magnetic fields at a fixedFMR frequency. The acoustic peaks keep increasing their intensity with increasingcoupling while the optic peaks get weaker with increasing coupling. The acous-tic peaks gradually approach a fixed point which is located between the reso-nance peaks of the uncoupled films. Calculations were carried out for the mag-netic parameters obtained in GaAs/8Fe/Au/16Fe/Au(001) structures [30], wherethe integers represent the number of atomic layers. The following magnetic pa-rameters were used: 16Fe: K
‖1,eff=3.1×105 erg/cm3, K⊥
u,s=0.88 erg/cm2, and
K‖u,eff=3.3×104erg/cm3; 8Fe:K
‖1,eff=1.33×105 erg/cm3, K⊥,s=0.82 erg/cm2, and
K‖u,eff=–1.14×106erg/cm3. 4πMs=21.5 kG, g=2.09, and α=0.009. The in-plane uni-
axial easy axes for the 16Fe and 8Fe films were along the [110] and [110] directions,respectively. The applied field was oriented along the [110] crystallographic axis.The damping parameter was increased approx. 3 fold, compared to the measuredvalues, to make the FMR lines wide for easy viewing
excitation of optical modes ineffective. The optical mode signal rapidly de-creases with the strength of the interlayer coupling, see Fig. 4.4. It is relativelyeasy to measure the strength of the interlayer coupling up to 0.5 ergs/cm2
[38]. In the saturated state one is not able to measure the interlayer couplingstrength if the two films have the same magnetic properties. The difference inthickness does not help. However in a non-collinear configuration of the mag-netic moments one can measure the exchange coupling even in films havingthe same magnetic properties. In that case in FMR one gets only one resonantmode which depends strongly on the exchange coupling, see Fig. 4.5. This isstrictly only true for the rf field oriented perpendicular to the dc applied field.
4 Exchange Coupling in Magnetic Multilayers 197
Fig. 4.5. The dependence of the FMR absorption peak on the bilinear magneticcoupling. Simulations were carried out at 10 GHz for a FM/NM/FM structure. Themagnetic films were of the same thickness. The magnetic anisotropies were assumedto be zero,4πMs=21.5 kG, g=2.09, and α=0.009. The numbers above the absorptionpeaks represent the strength of bilinear magnetic coupling. One needs to use a lowenough microwave frequency to bring the FMR resonance to low fields where themagnetic moments are not parallel (the unsaturated state). In the saturated statethe FMR signal does not depend on the interlayer coupling
The effectiveness of the coupling between a homogeneous rf field and theoptical mode can be increased if the magnetic moments in the two films arenoncollinear, see Fig. 4.6. It was shown by Z. Zhang et al. [31] that for therf field oriented parallel to the dc field one gets the projected rf field compo-nents in phase with the optical rf magnetization components resulting in anenhancement of the optical resonance. Note in Fig. 4.6 that the acoustic peakis completely absent for the rf field parallel to the dc field while the opticalpeak reaches its maximum. The effective rf field components (perpendicularto the dc magnetic moments) in the magnetic layers are antiparallel. This wayone is not coupled to the acoustic mode but the optical peak is fully excited.
The strength of biquadratic coupling can not be measured independentlyin the saturated state, see (4.11). However in a non-collinear state the con-tributions of bilinear and biquadratic interlayer couplings in FMR and BLSmeasurements can be separated, see Fig. 4.7 and [39].
There is an alternative approach to evaluate the resonance modes using theSmit and Beljers method which is based on the partial derivatives of the Gibbsenergy with respect to the magnetization angles. The details of this approachcan be found in [35]. An excellent theoretical treatment of rf excitations ina wide range of multiayers with complex spin configurations can be found inthe review article by Camley and Stamps [40].
198 B. Heinrich
Fig. 4.6. The dependence of the FMR signal on magnetic coupling in a non-collinearconfiguration. Simulations were carried out at 10 GHz for a FM/NM/FM structure.The magnetic films were of the same thickness. The magnetic anisotropies wereassumed to be zero, 4πMs=21.5 kG, g=2.09, and α=0.009. (a) J1=0.0 (b) J1=–0.4erg/cm2. The rf magnetic field is perpendicular to the applied dc field. Only theacoustic mode is excited. (c) J1=–0.4 erg/cm2. The rf field is oriented 45 Degreeswith respect to the dc applied field. Note that with this rf driving one can seeboth the acoustic and optical modes. (d) J1=–0.4 erg/cm2. The rf field is orientedparallel to the dc applied dc field. Only the optical mode is excited. For (b), (c) and(d) the magnetic moments are non-collinear. Their magnetic moments are cantedsymmetrically away from the dc magnetic field. The FMR signal in (a) is 4.5x largerthan those in (b), (c) and (d)
In order to obtain a reliable interpretation of the magnetic coupling inMOKE and FMR measurements one needs to know reasonably well the mag-netic anisotropies. Independent measurements of the magnetic anisotropies inindividual films are extremely useful. The interlayer coupling parameters arethen the only parameters left to fit the measured data obtained for a pair ofcoupled thin films.
4.4 Theory
4.4.1 Interlayer Exchange Coupling
The first successful model of interlayer exchange coupling was introduced byMathon, Villeret and Edwards [41]. They correctly pointed out that exchangecoupling is primarily a property of the normal metal (NM) spacer and is
4 Exchange Coupling in Magnetic Multilayers 199
Fig. 4.7. BLS spectra for an FM1/NM/FM2 structure. The in-plane magneticanisotropies are assumed to be zero. The upper curves correspond to acousticpeaks, and the lower curves correspond to optical peaks. The calculations werecarried out for J1=–0.2 ergs/cm2 (solid line), and J1=–0.1 ergs/cm2 and J2=0.05ergs/cm2 (dashed line) providing an identical coupling in the saturated state toJ1=–0.2 ergs/cm2, see (4.11). Note that the resonant frequencies are indeed iden-tical for fields greater than that required to align the magnetizations in the twofilms (H=1.38 kOe). However significant differences in resonant frequencies occur inthe non-collinear state allowing one to separate the contributions from the bilinearand biquadratic exchange couplings. Similar behavior would be obtained for FMRmeasurements carried out as a function of microwave frequency at fixed magneticfield
related to the confinement of Fermi surface electrons in the NM. This modelwas quickly extended to include the spin dependent electron reflectivity at theFM/NM interfaces [42, 43, 44]. One has to include the itinerant nature of the3d, 4sp electrons in the FM layers. The interlayer bilinear coupling, J, is givenby the difference in energy between the antiparallel and parallel alignment ofthe magnetic moments in FM/NM/FM structure,
J =1
2A(E↑↓ − E↑↑) , (4.12)
where A is the area of the film. Calculations of energy differences are simpli-fied by using the force theorem. The main problem is how to treat electroncorrelations self consistently. The force theorem says that the energy differ-ence between the two configurations is well accounted for by taking the dif-ference in single particle energies. It is adequate to take an approximate spindependent potential and to calculate the single particle energies in the parallel
200 B. Heinrich
and antiparallel configurations. This difference in energy is very close to thatobtained from self-consistent calculations, see the further discussion in [3]. Infact this section follows closely Stiles’s Sect. 4.4 in [3]. This procedure signifi-cantly simplifies the calculation of exchange coupling and interface magneticanisotropies. In calculations of the interlayer exchange coupling one does notcreate a big error by neglecting spin orbit interactions, while in calculationsof the interface anisotropies spin orbit coupling is the crucial ingredient. Sin-gle particle energy calculations require one to evaluate the electron energyin four quantum well states (QWS), see Fig. 4.8. For thick FM layers onefinds large energy contributions. However these large contributions cancel outin the difference (4.12). In order to avoid mistakes in this procedure it isbetter to calculate the cohesive energy of the QWS by subtracting the bulkcontributions,
ΔEQWS = Etot − EFMVFM − ENMVNM , (4.13)
where VFM,NM and EFM,NM are the total volumes and bulk energies for FMand NM layers, respectively.
Quantum interference
Let us consider a simple one dimensional model in which an electron with awave vector k⊥ travels inside the NM spacer and is partially reflected at theFM/NM (interface A) and NM/FM (interface B) interfaces with reflectioncoefficients RA,B = rA,Bexp(iφA,B). After multiple interference the electrondensity of states (EDS) changes. The phase of the wavefunction after a roundtrip changes by
Δφ = 2k⊥d+ φA + φB . (4.14)
Ferromagnet Spacer
E
EF
k
Unoccupiedstates
k
↑
ParallelAlignment
AntiparallelAlignment
Spin ↑M ↑M ↑ M ↓ M ↑
EF
Spin ↓M ↑ M ↑M ↑ M ↓
EF
↓
M ↑
Fig. 4.8. Quantum wells employed in the calculation of the interlayer ex-change coupling. These spin dependent potentials correspond reasonably well toa Co/Cu/Co(001) system. On the left side the four panels show quantum wellsfor spin up and spin down electrons for parallel and antiparallel alignment of themagnetic moments. The grey regions show the occupied states
4 Exchange Coupling in Magnetic Multilayers 201
The amplitude after multiple reflections is given by a sum of round trips is
∞∑1
[rArBeiΔφ]n =rArBe
iΔφ
1 − rArBeiΔφ. (4.15)
The denominator becomes small when one obtains a constructive interfer-ence Δφ=2nπ. For energies less than the potential barrier at the interfacerA=rB=1 and one gets perfect QWS. For energies above the barrier hight theQWS become broader resonances by partly transmitting its amplitude to thesurrounding FM layers. By changing the NM spacer thickness these states passthrough the Fermi energy, see Fig. 4.9, which leads to an oscillatory behaviorof the cohesive energy and consequently to an oscillatory interlayer exchangecoupling. The first clear experimental observation of QWS was presented byHimpsel and Ortega [45, 46] using photoemission and inverse photoemissionusing a nonmagnetic layer on top of a magnetic layer.
In first approximation the change in the density of states due to interfer-ence, Δ n(ε), should be proportional to rArBcos (Δφ) and to the spacer widthd and the density of states per unit length, (2/π)(dk⊥/dε) [44]. Therefore Δn(ε) per spin can be written as
Δn(ε) � 2dπdk⊥dε
rArBcos(Δφ) =1πIm
(i2d
dk⊥dε
rArBeiΔφ
). (4.16)
For multiple scattering one has to use the expression in (4.15). It is relativelyeasy to show that 4.16 can be generalized to [47]
Δn(ε) = −1πIm
d
dε
[ln(1 − rArBe
iΔφ)], (4.17)
note that (4.17) equals to (4.16) for small reflection coefficients.The cohesive energy is than given by
Ecoh = −∫ EF
−∞dε(ε− EF )Δn(ε) . (4.18)
D + 2π 2kFD + π / 2kFD
E F
/
Fig. 4.9. Evolution of quantum well (QW) states as a function of the film thickness.The solid lines represent bound states (localized in the QW) and resonance statesare shown in “fuzzy ellipses”. EF is the Fermi energy
202 B. Heinrich
Using integration by parts one gets
Ecoh =1πIm
∫ EF
−∞dεln(1 − rArBe
iΔφ) . (4.19)
For fixed thickness d the integral oscillates rapidly as a function of k⊥. Onlycontributions close to the Fermi level will leave non-zero contributions, seeFig. 4.10. It can be shown that in these regions for large d [3]
Ecoh =�vF
2πd
∑n
1nRe((rArB)neinΔφ(kF )
). (4.20)
For small reflection coefficients
Ecoh � �vF
2πdrArBcos(2kF d+ φA + φB) . (4.21)
The interlayer exchange energy, J, is then given by adding all cohesive energiesin Fig. 4.9, assuming the same reflection coefficients at the A and B interfaces
J � �vF
4πdRe(R↑R↓ +R↓R↑ −R2
↑ −R↓2)ei2kF d = −�vF
4πdRe(R↑ −R↓)2ei2kF d .
(4.22)
The exchange coupling in this simple one dimensional limit is inverselyproportional to the film thickness, d, and its oscillatory period is given bythe Fermi spanning vector 2kF . In 3D space one has to take into account thek-vectors parallel to the surface. These k-vectors due to the lattice periodicityare conserved in going from FM to NM regions. In this 3D case the onedimensional QWS have additional k-wave-vectors parallel to the interface.The total cohesive energy per unit area involves integration of the QWS overthe interface Brillouin zone. In the limit of large d [3] (asymptotic limit)
Ecoh � �vF
2πd
∫
IBZ
d2k
(2π)2Re(ei2kF z(k)dRR(k)RL(k)
), (4.23)
where RR(k), RL(k) are the reflectivity coefficients at the right and left in-terfaces, and kF2 kFz is the perpendicular k-vector at the Fermi surface. Theintegrand in (4.23) oscillates rapidly with the argument 2kFz(k)d except onareas of the Fermi surface where opposite sheets of the Fermi surface arenearly parallel, see Fig. 4.10.
The vector connecting these parts of the Fermi surface are called criticalspanning vectors. The spanning k-vectors for (001) interfaces for simple metalssuch as Cu and complex spin density Cr are shown in Figs. 4.14 and 4.21.
The exchange coupling involves the difference in cohesive energies forparallel and antiparallel configuration of magnetic moments. In its asymp-totic form this coupling can be written as
4 Exchange Coupling in Magnetic Multilayers 203
Inte
gran
d
Thickness
2kF
2π/2kF
Fig. 4.10. The right hand side shows a slice through a spherical Fermi surface. Thesolid double headed arrows represent spanning k vectors. The left hand side showstheir oscillatory contributions to the cohesive energy. The sum of these contributionsis shown by a heavy line. The constructive interference (heavy line) comes mostlyfrom the belly area of the Fermi surface. Note that this constructive interferencedecreases with increasing film thickness
J �∑
i
�vi⊥κ
i
16π2d2Re((Ri
↑ −Ri↓)
2eiqi⊥deiχi
), (4.24)
where the vi⊥ are Fermi velocities at the spanning vectors, qi
⊥ is the lengthof a critical spanning wave-vector, κi is the phase associated with the type ofthe critical point, and Ri
↑, Ri↓ are corresponding reflectivities. The periods of
the observed exchange coupling oscillations as the film thickness is varied arein good agreement with those obtained in de Haas-van-Alphen measurements,see Table 4.1 in [3]. A detailed discussion of calculations of exchange couplingfor Co/Cu/Co(001), Fe/Au/Fe(001) and Fe/Ag/Fe(001) systems can be foundin [3]. The quantitative agreement for the exchange coupling between theoryand experiment is far from being good. The main reason is that the interfacesin real samples are far from being ideal and measurements are often not carriedout in the asymptotic limit.
4.4.2 Dipolar Coupling
In measurements involving an inhomogeneous distribution of magnetizationone has to include dipolar coupling. BLS measurements in the backscatteredconfiguration [24, 37, 48] represents perhaps one of the best defined cases ofdipolar couplings. In this case the in-plane k-vector of the rf magnetization isgiven by k = 2qcos(θ), where q is is the length of the laser wave-vector, and θ isthe angle of the incoming laser beam with respect to the film surface. For a filmwith its normal oriented along the z-axis, the in-plane dc magnetic momentoriented along the x-axis, and the rf magnetization components in form ofmyexp(i(k‖x+k⊥y)), mzexp(i(k‖x+k⊥y)), the spatially averaged dipolar fieldcomponents inside the film in the limit as kd→ 0 are given by
204 B. Heinrich
Table 4.1. Comparison of oscillation periods measured in magnetic multilayers withthose expected from the critical spanning extracted from Fermi surfaces measuredin de Haas-van Alphen measurements (dHvA). This Table is a copy of Table 4.1in reference [3], see further references contained therein
interface Period (ML) Period (ML) Technique
Ag/Fe(100) 2.382.37±0.07
5.585.73±0.05
dHvA SEMPA
Au/Fe(100) 2.512.48±0.05
8.68.6±0.3
dHvA SEMPA
Cu/Co(100) 2.562.6±0.052.58 to 2.77
5.888.0±0.56.0 to 6.17
dHvA MOKE SEMPA
Cr/Fe(100) 11.112±112.5
dHvA SEMPA MOKE
Cr/Fe(112) 14.415.4
dHvA MOKE
hx = −2πmy
(k‖k⊥k2
)kdei(k‖x+k⊥y)
hy = −2πmy
(k⊥k
)2
kdei(k‖x+k⊥y)
hz = (−4πmz + 2πmzkd) ei(k‖x+k⊥y) , (4.25)
and k = (k2‖ + k2
⊥)0.5
Outside the film for z ≥ d:
hx = −[2πmy
(k‖k⊥k2
)+ 2πimz
(k‖k
)]kdei(k‖x+k⊥y)e−k(z−d)
hy = −[2πmy
(k⊥k
)2
+ 2πimz
(k⊥k
)]kdei(k‖x+k⊥y)e−k(z−d)
hz = −[2πimy
(k⊥k
)− 2πmz
]kdei(k‖x+k⊥y)e−k(z−d) . (4.26)
Outside the film for z ≤ 0.:
4 Exchange Coupling in Magnetic Multilayers 205
hx = −[2πmy
(k‖k⊥k2
)− 2πimz
(k‖k
)]kdei(k‖x+k⊥y)ekz
hy = −[2πmy
(k⊥k
)2
− 2πimz
(k⊥k
)]kdei(k‖x+k⊥y)ekz
hz =[2πimy
(k⊥k
)+ 2πmz
]kdei(k‖x+k⊥y)ekz . (4.27)
k‖ and k⊥ being in-plane wave-vector components parallel and perpendicularto the saturation magnetization, see Fig. 4.11. Notice that the dipolar fieldoutside the film decays exponentially with the decay length of 1/k. A gen-eral treatment of dipolar field can be found in [49, 50]. Dipolar fields play animportant role in rf measurements using coplanar and slotted transmissionlines. The distribution of k-vectors depends on the geometry of the trans-mission line. These inhomogeneous dipolar fields lead to both a shift of theresonant frequency and a broadening of the FMR line [51, 52].
Orange peel coupling
Rough interfaces lead to a surface magnetic charge density and consequently todipolar coupling. This coupling was introduced by Neel [53]. It is often called“orange peel” coupling [54, 55]. It leads to an additional dipolar magneticcoupling. Figure 4.12 indicates that the interface roughness creates a lowerenergy for the parallel orientation of the film magnetic moments than that forthe antiparallel configuration. Usually the surface roughness is slowly varyingand its amplitude is much smaller than the film thickness. Calculations thenbecome simple. The surface charge can be distributed over a flat surface [3].Assuming that the surfaces vary along the x-direction as zs = Δzcos(2πx/L)
Fig. 4.11. The coordinate system and the film geometry corresponding to dipolarfields generated by a spinwave with the wave vector k. The magnetic layer has itsnormal in the z-direction. d is the layer thickness. The saturation magnetization andspin wave vector k are oriented in the film surface. The k−vector propagates withthe angle ϕ with respect to the saturation magnetization Ms
206 B. Heinrich
----- - - - --
+++++----
++++-
+ +
++++++++++
-----+++++
-----+
- -
(b)
++++++++++
-----+++++
----+
- -
++++++++++
-----+++++
-----+
- -++++++++++
-----+++++
-----+
- -
Fig. 4.12. A schematic picture demonstrating the presence of interface effectivemagnetic charges for an in-phase corrugated interface roughness. The solid shortarrows represent the local induced magnetic dipoles. For the parallel orientation ofthe film magnetic moments the magnetic dipoles form a closed magnetic pattern. Inthe antiparallel configuration this pattern changes to a head to head and tail to tailconfiguration
and zs = d+Δzcos(2πx/L), see Fig. 4.12. For the magnetization perpendicularto corrugation the ferromagnetic coupling strength is given by [3]
J1,dip ∼ 4πM2s
Δz2
Le−2πd/L . (4.28)
When the interface roughness is completely uncorrelated the bilinear dipolarexchange coupling goes to zero.
Pinhole coupling
Magnetic coupling can arise from pinholes. Basically parts of the FM filmsare in a direct contact that results in an overall ferromagnetic coupling [56].However this coupling is not homogeneously distributed over the surface. Fluc-tuations of pinhole coupling over the film surface can result in a contributionto biquadratic exchange coupling.
4.4.3 Biquadratic Exchange Coupling
The presence of biquadratic exchange coupling was observed at the sametime by Heinrich et al. [57] on Co/Cu/Co(001) and by Ruehrig et al. [58] onFe/Cr/Fe(001). The evidence for biquadratic exchange coupling in [57] wasfound in the magnetization loops. In order to properly explain the observedcritical fields one needed to use an angular dependent bilinear exchange cou-pling in the form of
J(θ) = J1 − J2cos(θ) , (4.29)
where θ is the angle between the magnetic moments. Consequently the corre-sponding exchange energy was given by
Ruehrig et al. observed a perpendicular orientation of the magnetic momentsin an Fe/wedge Cr/Fe(001) sample in which the Cr interlayer was grownwith a linearly variable thickness. They explained the observed perpendicularconfiguration by using
Clearly these two concepts are identical. Slonczewski soon after that pro-posed a theoretical interpretation [59]. He realized that fluctuations in theinterlayer thickness could result in an additional coupling term. The ferro-magnetic layers at different parts of the sample have different thicknesses andconsequently different strengths of coupling, see Fig. 4.13. Short-wavelengthoscillations can even result in changing the coupling from ferromagnetic toantiferromagnetic. His model is applicable when lateral variations in the bi-linear coupling strength are on a shorter length scale than the lateral ex-change length. This means that local angular variations from the averagedirection of the magnetic moments are small so that in this case the prob-lem can be treated by perturbation theory. The magnetic moments are frus-trated across the film surface by a variable interlayer coupling. Consequentlythere is an additional energy term which prefers to orient the magneticmoments in the FM layers perpendicularly to each other. This additionalcoupling has then an angular dependence given by cos2(θ), for which the
Fig. 4.13. A schematic picture demonstrating variations of the local magnetic mo-ment across a film surface. The local magnetic moments (solid black arrows) arepartly rotated away from the average direction of the magnetic moment (light greyarrows) in an attemp to decrease the overall interlayer exchange coupling energy. Asa result, moments in FM coupled regions rotate a little towards each other whereasin AFM coupled regions the magnetizations rotate away from each other
208 B. Heinrich
name “biquadratic exchange coupling” was coined. Its strength is given bythe competition between variations in the interlayer exchange coupling field,ΔJ1/Msd , and the in-plane intralayer exchange field, 2Ak2/Ms. The lengthof the k-vector, k, is given by the average lateral variations of the inter-layer exchange coupling, and ΔJ1 is the average variation of the interlayerexchange coupling. Slonczewski has shown that the exchange coupling fluc-tuations are decreased by a factor due to exchange averaging. In the sim-plest form the strength of the biquadratic exchange coupling can be ex-pressed by
J2 =4πΔJ1
ΔJ1Msd
2Ak2
Ms
. (4.32)
Notice that the large fraction describes the exchange averaging effect. A moregeneral description can be found in [59]. The above expression shows thatbiquadratic coupling has only a positive sign that encourages a perpendicularorientation of the magnetic moments. The angle between the magnetic mo-ments is given by a competition between the bilinear, biquadratic magneticcouplings, and the magnetic anisotropies, see Sect. 4.3.1 In zero field this anglecan range continuously from 0 to π.
It is often believed that biquadratic exchange coupling occurs only fromshort wavelength exchange coupling oscillations where the exchange couplingchanges its sign between two subsequent atomic terraces. This is not cor-rect. Any lateral variations in magnetic coupling strength (including ferro-magnetic coupling) will result in biquadratic exchange coupling. Once themagnetic moments are in a non-collinear state the magnetic frustrations dueto an inhomogeneous magnetic coupling strength result in biquadratic mag-netic coupling.
Further details about biquadratic exchange coupling can be found inDemokritov’s review article on “Biquadratic exchange coupling in layeredmagnetic systems” [60].
The Slonczewski idea of additional energy terms due to imperfect inter-faces is more general. It was shown [32] that “in any system that exhibits alateral inhomogeneity, one can expect additional energy terms. It originatesfrom intrinsic magnetic energy terms that fluctuate in strength across thesample interface. These additional terms have a next higher angular powercompared with that of the intrinsic term, and they should have only one sign.The power of a higher order angular term has to satisfy the requirements ofsample symmetry including time inversion symmetry. Variations of the in-terlayer exchange coupling results in a cos2(θ) angular term; variations in auniaxial interface anisotropy results in an angular dependent term having theform cos4(ϑ), where ϑ is the angle between the magnetic moment and the filmnormal.
4 Exchange Coupling in Magnetic Multilayers 209
4.5 Experimental Studies
Early studies
The early stages of interlayer coupling are well described in review articles[3, 6]. The first measurements of interlayer coupling were carried out by Gru-enberg’s group [61]. Using BLS measurements they showed that Cr can coupleFe layers antiferromagnetically. This result was expected considering that Crcontains a spin-density wave having a period of approximately 2 ML. It wasnot clear that one could expect antiferromagnetic coupling through simplemetal spacers such as Cu. The first indication that the exchange couplingthrough Cu could be antiferromagnetic was found by Cellobata et al. [62]in superlattices of fcc [6Co/8Cu](001) and [9Co/5Cu](001) using spin polar-ized neutron scattering and magnetometric techniques. Soon after that sev-eral FMR and BLS experiments established a cross-over from ferromagneticto antiferromagnetic coupling through bcc Cu(001). The first cross-over wasobserved at 8 ML of Cu and the first antiferromagnetic maximum at 11 ML[63, 64]. These measurements were quickly followed by a number of measure-ments that identified exchange coupling terms having long range oscillatoryperiods of 10 ML and 5.5–8 ML for bcc and fcc Cu(001) [38, 65, 66, 67, 68, 69],respectively.
Systematic studies of multilayers grown by means of sputtering revealedoscillatory couplings having oscillation periods in the range of 0.9 nm to 1.2nm for V,Nb,Mo,Rh,Ru,Ta,W,Re and Ir spacer layers [70, 71, 72, 73, 74]. TheCo/Ru/Co and Co/Rh/Co systems became very useful in forming antipar-allel pinned spin valves that are employed in GMR read out heads [75], andMagnetic Random Access Memories(MRAM) using the spin tunelling effect.In Co/Ru/Co and Co/Rh/Co structures the first antiferromagnetic couplingwas found at 0.3 and 0.8 nm with the strength of 5 and 1.6 ergs/cm2 forRu and Rh, respectively [70]. These results were obtained by monitoring theGiant Magneto Resistance (GMR) effect. The resistance of an FM/NM/FMstructure increases for the antiparallel orientation of the magnetic moments(antiferromagnetic coupling) due to the GMR effect. By following the maximaof the resistance one can determine the regions of antiferromagnetic couplingas a function of the spacer layer thickness [76]. The multilayer structures forGMR studies are mostly prepared by means of sputtering. In the work carriedout by the Strasbourg group [72] crystalline Co/Ru/Co(0001) hcp films wereprepared using MBE. The interlayer exchange coupling strength was investi-gated using FMR. The strongest coupling was found to be 6 ergs/cm2 observedat 0.5 nm of Ru at RT. The period of oscillations in the coupling strengthwas found to be 1.2 nm. Preparation of the films using sputtering resulted ina weaker exchange, see above. This indicates that smoother interfaces resultin a stronger coupling.
Ru is used in antiferromagnetic coupled multilayers which are attractive foruse as high density recording media. [Co(0.4)/Pt(0.7)]X−1 multilayer struc-tures, where X represents the number of repetitions, and the numbers are
210 B. Heinrich
in nm, possess a strong perpendicular uniaxial anisotropy that allows theCo magnetic moment to be oriented perpendicular to the film surface. In[[Co(0.4)/Pt(0.7)]X/Co(0.4/Ru(0.9)]]N structures one can create vertical andlaterally coherent antiferromagnetic films by changing X [77].
The strong antiferromagnetic coupling through Ru requires large appliedfields to saturate FM/Ru/FM trilayers. For a Py(5 nm)/Pd(.5 nm)/Py(5 nm)structure the magnetic field required to achieve saturation of the magneticmoments is in excess of 10 kOe at RT, [70, 76]. A FM/Ru/FM film havingzero net magnetic moment can be effectively pinned by an exchange biasfield from an antiferromagnet (AFM) [75, 76]. Such a hard magnetic layercomposed of AFM/Fe/Ru/Fe is extensively used in spin valve structures.
The presence of short wavelength oscillations in the exchange couplingwere observed for the first time using perfect single crystals of Fe whiskers assubstrates. Whiskers were prepared by means of chemical vapor deposition us-ing FeCl2 and H2 as a transport gas. Under correct conditions which requireda proper temperature and a proper flow of hydrogen gas one could sometimesget single crystals of Fe in the form of whiskers having {001} crystalline facets.Whiskers were usually between several mm to 1–2 cm long and 10–100 μmacross. The facets were smooth with atomic terrace sizes in excess of severalμm. Fe whiskers proved to be ideal templates for the observation of shortwavelength oscillations. Approximately at the same time the NIST group ofUnguris et al. [78], and Purcell et al. [79] (Philips group), observed short wave-length oscillations having a period of 2 ML. The exchange coupling throughspin-density wave Cr will be highlighted in detail in Sect. 4.5.3. After realizingthat short wavelength oscillations can be observed in carefully prepared sam-ples a large number of papers were devoted to Co/Cu/Co films oriented alongall the main (001), (110) and (111) crystallographic orientations. A detailedaccount of this work can be found in [6].
In the following Section the emphasis will be put on several prototypes ofexchange coupling covering the simple metal spacers Cu, Ag, Au, Cr and Mn.
4.5.1 Simple Interlayers: Cu, Ag and Au
The observation of short wavelength oscillations required a very smooth inter-face. Convincing evidence of short-wavelength oscillations was presented bythe Philips group [80]. Fcc Co/Cu/Co(001) grown on Cu(001) bulk substratesand bcc Fe/Cu/Fe(001) grown on Fe whiskers were investigated by means ofMOKE. The thickness dependence of the exchange coupling was achieved byusing a Cu wedge grown between the ferromagnetic layers. The spacer thick-ness varied continuously from 4 to 19 atomic layers. In the fcc Co/Cu/Co(001)system the variation of the magnetic coupling with Cu thickness was describedby a superposition of two oscillatory terms having periods of 2 and 8 atomiclayers.
In bcc Fe/Cu/Fe(001) grown on a Ag(001) crystal the observed interlayercoupling oscillated with a period of 2 atomic layers. One does not have to use
4 Exchange Coupling in Magnetic Multilayers 211
Fe whiskers to be able to observe two monolayer exchange coupling oscillationsusing bcc Cu(001) spacers. The interface smoothness of Fe/Cu/Fe(001) sys-tems was significantly improved by growing an Fe film on a Ag(001) singlecrystal substrate at 415 K [81]. The terrace size was increased from 3 to 15nm and resulted in the presence of short wavelength oscillations. The FMRand MOKE data were interpreted by the simultaneous use of bilinear and bi-quadratic exchange coupling terms [6, 81]. The exchange coupling was foundto have maxima of ferromagnetic coupling at 9,11 and 13 atomic layers. Themaxima for antiferromagnetic coupling occurred for 10 and 12 atomic layers ofCu. No ferromagnetic coupling was observed in the Philips data. The maximafor antifferomgnetic coupling were observed at 12,14 and 16 atomic layers inthe Philips work. There was an obvious difference reported in the phase ofthe coupling between the Phillips and B. Heinrich et al. (SFU groups).
Comprehensive studies of exchange coupling and its relationship to quan-tum well states, QW, were carried out by the Qiu group at the University ofCalifornia at Berkeley [82] (see references within) using wedged Cu Co/Cu/Co(001)structures grown on Cu(001) single crystal substrates. This systemwas particularly convenient for such studies because Cu has a simple Fermisurface whose sp bands can be easily separated from the other energy bands,see the Fermi surface of Cu in Fig. 4.14. Cu and Co can be grown in the (001)orientation with atomically flat interfaces. Angular resolved PhotoemissionSpectroscopy (ARPES) of QW states was carried out at the Advanced LightSource (ALS) of the Lawrence Berkeley National Lab: the orientation of the
Fig. 4.14. The (110) cross-section of the fcc Cu Fermi surface. The hexagon ofstraight lines outlines the first Brillouin zone. The solid dots represent reciprocal k-vectors. All three important orientations are present. The critical spanning vectorsin the extended Brillouin zone are outlined by the solid arrows. Along the [001]direction the two critical spanning vectors are located at the belly and neck of theFermi surface
212 B. Heinrich
magnetic moments was determined using Magnetic X-ray Linear Diochroism(MXLD) from the Co 3p level, and the the coupling strength was determinedby means of the MOKE technique. The density of states (DOS) is signifi-cantly increased at energies corresponding to the QW states, see also [46].This allows one to follow the QW states as a function of energy for differentCu thicknesses.
In Fig. 4.15 ARPES measurements show the formation of QW states cor-responding to the belly direction of the fcc Cu Fermi surface, see Fig. 4.14.The study was carried out for 20 ML thick Co grown on a Cu(001) substrateand with a Cu wedge grown on top of the Co layer. The ARPES oscillationshave clearly shown the QW states corresponding to the sp electrons in the Culayer. The periodicity of the oscillations was found to be 5.88 atomic layersat the Fermi level and this is exactly the periodicity of the long period ofthe interlayer exchange coupling in Co/Cu/Co(001) systems. Photoemissionintensity along the belly and the neck directions (with k vectors oriented 30Degrees with respect to the film surface) of the Cu Fermi surface are shownin Fig. 4.16.
The belly, 5.88, and neck, 2.67, atomic layer periodicities can be explainedby employing the extended Brillouin zone picture, see the arrowed solid linesin Fig. 4.14. In this case one subtracts from the regular spanning vector insidethe first Brillouin zone a k vector with the atomic layer periodicity (4π/a).The oscillatory period in Cu is given by
2kedCu − φA − φB = 2πn , (4.33)
Fig. 4.15. Photoemission spectra obtained along the surface normal correspondingto the belly direction of the fcc Cu Fermi surface [82]. Oscillations in intensity as afunction of the Cu layer thickness and electron energy demonstrate the formationof quantum well states (QW)
4 Exchange Coupling in Magnetic Multilayers 213
Fig. 4.16. Photoemission intensity along the belly direction (a) and neck direction(b) of the fcc Cu Fermi surface, see Fig. 4.14, as a function of the film thickness andelectron energy below the Fermi level. Two distinct oscillatory periods are present.The dotted curves are calculated using the phase accumulation method [82]
where ke = kBZ − k, kBZ = 2π/a is a Brillouin-zone vector, n is an integer,and φA,B are the phase shifts of electron wavefunctions upon reflection atthe two boundaries of the potential well formed by the Cu layer surroundedby Co and vacuum, and a is the lattice spacing of Cu. Equation (4.33) ex-plains the long and short wavelength oscillation periods by the belly and neckspanning k vectors, respectively. It is caused by evaluating the strength ofthe exchange coupling at the discrete atomic layer separations. This is of-ten called aliasing effect. Simple calculations using an image potential andthe work function at the Cu/vacuum interface allow one to determine thephase shift at the Cu/vacuum interface. The phase shift at the Co/Cu inter-face is determined by the confinement of Cu electrons by the minority spinenergy band of Co. The Cu sp conduction band can be approximated by anearly-free-electron model. The solutions of (4.33) are shown by dotted linesin Fig. 4.16. Clearly this simple model can account well for the QW states inCu. The QW states are the underlying basis for the presence of the interlayerexchange coupling. To insure the direct comparison of the exchange couplingperiodicity with the QW states as a function of the Cu spacer thickness a halfof the wedge sample was covered by 3 ML thick Co. MXLD measurementsare only surface sensitive [83] and consequently the FM and AFM couplingcan be determined by monitoring the MXLD signal coming from the 3 MLthick Co. Images of the DOS (by photo-emission measurements) at the bellyand neck of the Fermi surface were obtained by scanning the photon beamacross the Cu wedge on the Co/Cu side of the wedge. Figure 4.17c shows theobserved MXLD signal with maxima and minima intensities correspondingto AFM and FM couplings respectively. Clearly long and short wavelengthoscillations are easily visible.
214 B. Heinrich
Fig. 4.17. (a) QW states at the belly of the Cu Fermi surface. (b) QW states atthe neck of the Cu Fermi surface. (c) XMLD from the top 3 atomic layers of Coevaporated over the Cu wedge spacer. See further details in [82]. The dark and lightregions correspond to ferromagnetic and antiferromagnetic coupling. (d) Calculatedinterlayer coupling. Notice remarkable agreement between theoretical predictionsand experiment for the sign of the exchange coupling
The coupling between the layers is determined by the energy differencebetween the parallel (P) and antiparallel (AP) alignment of the magneticmoments
2J ∼ EP − EAP =∫ EF
−∞EΔDdE , (4.34)
where ΔD = DP − DAP is the difference of the DOS between P and APalignment of the magnetic moments. For the P configuration of the magneticmoments the minority spins are confined and form well defined QW states.At the neck of the Fermi surface the minority spins are completely confinedby the spin potential of Co. At the belly of the Fermi surface they are onlypartially confined. Whenever the energy of a QW state crosses the Fermi levelit adds energy to EP making the P configuration of the magnetic momentsunfavorable. Fitting of the MXLD data with
J = −A1
d2sin
(2πΛ1
+ Φ1
)− A2
d2sin
(2πΛ2
+ Φ2
), (4.35)
resulted in Λ1=5.88 ML, Λ2=2.67 ML, A1/A2=1.2, Φ1=–86π, and Φ2=64π.MXLD is not able to determine the strength of the coupling. The couplingstrength was investigated using MOKE [82]. Only saturation fields were givenallowing one to estimate of the strength of the AFM coupling in (4.35). At d=6
4 Exchange Coupling in Magnetic Multilayers 215
ML JAFM �0.1 erg/cm2. Surprisingly this is a weak coupling considering thatthe QW states were so well defined. In addition the MOKE results mostly haveshown oscillations with a periodicity of Λ1=5.88 ML. The short wavelengthoscillations are very weakly present with saturation fields less than 100 Oeimplying that J<0.06 erg/cm2. Clearly the interface roughness is a big factorin the exchange coupling measurements but is much less pronounced in theQW state studies.
The recent studies of interlayer exchange coupling in Cu/Ni/Cu/Ni/Cu(001)and Ni/Cu/Co/Cu(001) structures by FMR were carried out by J. Lindnerand K. Baberschke [36]
Ag and Au:
Ag, Au, and Cu have similar Fermi surfaces. Excellent work using Ag and Auspacers was carried out by the NIST group using Fe whiskers as substrates[84, 85]. They used wedged samples with the Au and Ag spacers ranging inthickness from 0 to 15 nm (equivalent of 75 atomic layers). The NM spacerswere covered by 1–2 nm of Fe. Again these structures were oriented along(001) and displayed an excellent crystalline quality and large atomic terraces.The magnetic state of the top Fe film was monitored using scanning electronmicroscopy with polarization analysis (SEMPA). SEMPA is a surface sensitivetechnique that allows one to measure all three magnetization components [23].
The exchange coupling was found to oscillate between FM and AFM cou-pling over a range of 50–65 atomic layers.
This long range of oscillations allowed one to determine with a high degreeof accuracy the short and long wavelength periods. SEMPA measurements re-vealed the short and long wavelength periodicities 2.38, 5.73 ML and 2.48, 8.6ML for the Fe/Ag/Fe(001) and Fe/Au/Fe(001) films, respectively. See furtherdetails in Table 4.1 in Stiles review chapter [3]. These oscillatory periods are invery good agreement with those measured using de Haas-van Alphen (dHvA)and cyclotron resonance measurements of the Fermi surface extremal areas[3, 86]. The strength of the bilinear coupling in the Fe whisker/Au/Fe(001)system as a function of the Au spacer thickness is shown in Fig. 4.18
Theoretical estimates of the asymptotic value of the coupling at 5 ML ofAu is about a factor 3 higher than that measured. Stiles discusses this dif-ference in detail in his review article [3]. One should note that the measuredexchange coupling is always substantially smaller than theoretical predictions.In my view the interlayer coupling is very sensitive to interface structure andthis is the origin of the discrepancy between theory and experiment. Averagingexchange coupling by taking a statistical distribution of terraces only partlyexplains this difference. Interface mixing can even affect in a profound way thesign of the coupling, see Sect. 4.5.3. The heights of the atomic steps on Au(001)are poorly matched to the Fe(001)interlayer spacing, thus atomic steps lead toa strong vertical mismatch. The aliasing mechanism is no longer effective andcan even wipe out long wavelength oscillations [88]. In Fe/Ag/Fe(001) trilayers
216 B. Heinrich
0 5 10 15 20 25 30Au spacer thickness (layers)
- J
(
mJ/
m )
avg
2
10.0
1.00
0.10
0.01
0.001
best fit
measurement
Fig. 4.18. (a) The solid circules represent the measured values of the antiferromag-netic interlayer exchange coupling in Fe whisker/Au/Fe(001)using MOKE and thegrey areas show the best fit function over a wide range of coupling strength andspacer layer thickness using short, λ1=2.48 ML, and long wavelength, λ1=8.6 ML,oscillatory periods [87]
entirely grown at room temperature (RT) the ferromagnetic exchange couplingdecreased rapidly with increasing thickness. It reached zero at 7 ML [38].Growing the Fe layers at elevated temperatures, Tsub ∼ 180◦, significantlydecreased the density of atomic steps at the Fe/Ag interface and recovered theoscillatory behavior of the exchange coupling [89]. Good quantitative agree-ment with theory is very hard to reach. The NIST group results are unique;they were able to establish a well defined oscillatory behavior of the interlayerexchange coupling over a wide range of thicknesses. This allowed them to de-termine the true exchange coupling periods and showed that well defined QWstates exist in simple metals even for large spacer thicknesses. Quantitativeagreement with theory proves to be more elusive.
The spin-dependent potential in multilayer films creates electron confine-ment and resonant states which are responsible for the oscillatory behaviorof the exchange coupling. According to theoretical calculations [42, 90] suchstates are not restricted to the nonmagnetic spacers, but are also present insidethe ferromagnetic layers, and the coupling cannot be entirely described as aninteraction that is localized at the interfaces. The energy terms coming fromthe electron confinement in the ferromagnetic layers due to multiple reflectionsand the interference of such states with the states inside the spacer layer re-sults in a variation of the interlayer exchange coupling with the ferromagneticlayer thickness. Calculations and experimental studies on the Co/Cu/Co(001)system [74] have shown that the exchange coupling contains a component thatoscillates as a function of the ferromagnetic layer thickness. However, the os-cillatory part is smaller than the total strength of the exchange coupling sothat the sign of the coupling is determined by the thickness of the nonferro-magnetic spacer layer.
4 Exchange Coupling in Magnetic Multilayers 217
One can expect that the quantum well states affect also other magneticproperties. Bayreuther et al. [91] using Ni(0.73 nm) covered by a Au over-layer with a variable thickness (from 0.4 to 8 nm), have shown that the Curiepoint Tc in the Ni layer oscillates with the Au thickness overlayer in agree-ment with QW states in Au. The observed variations in Tc were as largeas ±20 K. Oscillations in Tc of 40 K were also observed by Ney et al. [92]in 2.8Co/2-4Cu/4.8Ni trilayers. The integers represent the number of atomiclayers. These results are supported by theoretical calculations by Pajda et al.[93] of the ordering temperature Tc for a single atomic layer of Fe or Co cov-ered by a variable thickness of Cu(001). The Green’s function method usingthe Random Phase Approximation have shown that the QW states in Cusignificantly affect Tc. The amplitude of oscillations in Tc could be as largeas 30 K which is 10 percents of the average value of Tc. The thickness depen-dence of Tc is particularly spectacular in fcc Fe(001) grown on fcc Cu(001).In these studies by Vollmer et al. [94] the Fe and Cu layer thicknesses werecontinuously changed between 0 to 12 and 0 to 9 atomic layers, respectively,using a double wedge Fe/Cu(001) structure. It is interesting to note that theoscillations in Tc were completely suppressed after the lattice transformationat 10 atomic layers from the fcc structure to the stable bcc Fe structure.
4.5.2 Temperature Dependence of the Exchange Coupling
The temperature dependence of the interlayer exchange coupling provides agreat theoretical challenge. It is always difficult to account correctly for therole of spin fluctuations in any system and nanosystems provide a great palateranging from 1D to 3D behavior including topological complexities relatedto the interface roughness. The temperature dependence of the bilinear andbiquadratic exchange coupling has usually been investigated for magnetic tri-layer FM/NM/FM systems. These studies were usually carried out in systemshaving a significant degree of interface roughness such that the bilinear cou-pling was substantially smaller than that in samples having atomically flatinterfaces, see above. This roughness can be expected to affect the role ofspin fluctuations at the FM/NM and NM/FM interfaces. Small values of theinterlayer exchange coupling implies a strong cancellation of short and longwavelength oscillatory exchange contributions. It can be expected under thesecircumstances that the the bilinear exchange coupling would be particularlyaffected by thermal fluctuations having a wavelength comparable to the aver-age separation between atomic terraces. The temperature dependence of theexchange coupling was measured over the interval from 77 to 400 K by Celin-ski et al. [95] using the system Ag-substrate/9Fe/10,12bccCu/16Fe(001) andby Lindner and Baberschke [35] over the temperature range from 55 to 350 Kusing the system fcc Cu-substrate/7Ni/5,9Cu/2Co(001). The integers repre-sent the number of atomic layers. It was found that the strength of the bilinearexchange coupling was increased by a factor of 3 to 4 times to a maximum of0.18 erg/cm2 in the Fe/Cu/Fe system as the temperature was reduced from
218 B. Heinrich
400 K to 77 K, [95]. In the Ni/Cu/Co system the bilinear exchange couplingstrength increased by a factor of 4 to a maximum of 0.06 erg/cm2 as the tem-perature was reduced from 350 K to 55 K; moreover, this increase was foundto scale with an inverse 3/2 power law in temperature [35]. It is interestingto note that in the Ag-substrate/9Fe/10,12bccCu/16Fe(001)samples no evi-dence of a 3/2 power law was found. In the Cu-substrate/7Ni/5,9Cu/2Co(001)samples the Co layer was only 2 atomic layers thick and therefore a strongdependence on thermal fluctuations would be expected. In fact, in a recentpaper by Schwieger et al. [96] it has been shown that an extended Heisenbergmodel accounts for 75% of the reduction of the bilinear exchange couplingstrength from 0 K to RT. However a similar increase in the bilinear exchangecoupling strength with decreasing temperature using relatively thick Fe lay-ers [95] suggests that there are some other contributions which are significantand not caused by thermal fluctuations. In fact for such thick Fe films onecan’t expect to have a strong temperature dependence since the saturationmagnetization changes by less than 10%. This view is even more supportedby measurements by Milton et al. [97] on the Fe-whisker/Cr/Fe(001) sampleshaving the best available interfaces. Here the temperature dependence of thebilinear coupling strength is a linear function of the temperature and it in-creased by a factor of 2 over the temperature interval from 300 K to 110 K. Inthe Ag-substrate/9Fe/10,12bccCu/16Fe(001) samples the temperature depen-dence was measured independently for the bilinear and biquadratic couplingstrengths. Surprisingly the biquadratic coupling strength was found to be lin-early dependent on the bilinear exchange coupling strength over the abovetemperature interval. This result does not explicitly follow the prediction bySlonczewski’s formula 4.32. In this formula the biquadratic exchange couplingstrength is proportional to the square of the mean deviation of variations of thebilinear coupling strength and therefore one would expect that the biquadraticexchange coupling strength should follow a quadratic power law in the bilinearexchange coupling strength. The biquadratic exchange coupling is definitelycaused by lateral variations in the bilinear coupling strength. In many samplesit has been found that the biquadratic exchange coupling strength decreasessignificantly more rapidly with decreasing terrace size than the bilinear cou-pling strength, see [6, 81], in agreement with formula 4.32. Therefore theobserved linear dependence of the biquadratic exchange coupling strength ontemperature implies that the exchange averaging factor in (4.32) is nearlytemperature independent.
Co(3.2 nm)/Ru(0.9 nm)/Co(3.2 nm)(0001) trilayer structures prepared bymeans of MBE by Zhang et al. [31] have provided good samples for testing thetemperature dependence of the exchange strength in samples having a largeinterlayer exchange coupling strength. In these samples the exchange couplingstrength increased from RT to He temperatures by 30%. This temperaturedependence was well described by the theoretical predictions of Edwards [98]and Bruno [86]. Agreement with theory indicated that the interlayer exchange
4 Exchange Coupling in Magnetic Multilayers 219
coupling through Ru was closely related to the spatial confinement of d holesin the Ru layer due to spin splitting of the d band in the Co layers.
The magnetic order in a [2Fe/13(VHx)]x200 superlattice was investigatedby Leiner et al. [99] as a function of the hydrogen content. It was shown thatthe interlayer exchange coupling strength could be continuously and reversiblychanged via absorption of hydrogen into the nonmagnetic V. The magneticphase diagram of a [2Fe/13(VHx)]x200 superlattice exhibited a critical concen-tration xc �0.022 at which the magnetic order changed from antiferromagneticto ferromagnetic order. In the vicinity of xc �0.022 the transition temperaturestrongly depended on x and decreased continuously as the concentration ap-proached xc �0.022. The hydrogenation of V was used by Paernaste et al. [100]to investigate the magnetic order in a 3Fe/14.4VHx/3Fe trilayer as a func-tion of the oscillatory interlayer exchange coupling strength through the VHx
spacer. A pure V interlayer showed a 2D FM (X-Y) behavior which is consis-tent with a weak interlayer coupling strength. As a result of the introductionof H into the V the critical point Tc exhibited an oscillatory dependence onan increasing H content. The minima and maxima of Tc were expected to bedirectly associated with an oscillatory interlayer exchange coupling strength.An FM exchange coupling could be expected to be associated with an increasein Tc, and AF exchange coupling associated with a decrease of Tc.
4.5.3 Spin Density Wave (SDW) in Cr
Fe-whisker/Cr/Fe(001) wedge structures, see Fig. 4.19, have played a crucialrole in the study of interlayer exchange coupling.
Unguris, Celotta, and Pierce studies [78, 101] using scanning electronmicroscopy with polarization analysis (SEMPA) on Fe-whisker/Cr/Fe(001)samples showed that the exchange coupling oscillates with a short-wavelengthperiod of �2 monolayers ( ML). The SEMPA images revealed in a very explicitway the presence of both short-wavelength and long-wavelength oscillationsin the thickness range from 5 to 80 ML, see Fig. 4.20.
The period of the short-wavelength oscillations, λ=2.11 ML, was found tobe slightly incommensurate with the Cr lattice spacing; the period of the longwavelength oscillations was found to be 12 ML. These two basic periods areexpected for the complex Cr Fermi surface, see Fig. 4.21. The incommensuratenature of the short wavelength oscillations in Cr, λ=2.11 ML, results in thephase slips of exchange coupling at 24 and 44 atomic layers of Cr, see Fig. 4.20and further details in [23].
Heinrich and co-workers (SFU group) carried out quantitative exchangecoupling measurements on Fe-whisker/Cr/Fe(001) samples [6, 102, 103]. Theobjective of the SFU group was to grow samples having the best available in-terfaces, to measure quantitatively the strength of the exchange coupling, andto compare these coupling strengths with ab initio calculations that explic-itly include the presence of spin-density waves in the Cr. The requirementof smooth interfaces limited our study to samples which were grown on
220 B. Heinrich
Fig. 4.19. A schematic view of Fe whisker/Cr/Fe(001) sample containing a wedgedCr spacer employed by the NIST group in SEMPA measurements. The arrows showthe directions of the magnetization. The Fe whisker is demagnetized by a single 180degree domain wall. The direction of the magnetization in the Fe film follows thesign of the local interlayer exchange coupling
Fe-whisker templates with the Cr spacers terminated at an integral numberof Cr atomic layers. It was found that the strength of the exchange couplingthrough the Cr(001) spacer even for these smooth Cr layers was extremelysensitive to small variations in the growth conditions. Cr ultrathin films donot possess a robust spin-density wave; the spatial variation of the spin mo-ments is extremely sensitive to the interfaces. In our studies we concentratedon samples for which the Cr thickness ranged from 4 to 13 atomic layerswhere the role of interfaces was most pronounced. The measured exchangecoupling was found to be reproducible only in those samples that exhibitedtrue layer by layer growth. The existence of unattenuated RHEED inten-sity oscillations of the RHEED specular spot during the growth of the Crspacer did not guarantee reproducible values of the exchange coupling be-tween a thin Fe(001) film and the whisker(001) substrate. It was necessary toestablish conditions such that new layers were initiated at a repeatable pat-tern of nucleation sites and subsequent attachment of adatoms to the atomicsteps of the newly formed atomic islands. This condition was monitored byexamining RHEED intensity oscillation amplitudes together with the widthof the RHEED specular spot profile. The best results were obtained whenthe RHEED intensity oscillations exhibited sharp cusps at the RHEED in-tensity maxima (filled atomic layers)and the RHEED spot profile oscillatedrepeatedly between narrow peaks (filled atomic layers) and wider, split inten-sity peaks (half filled layers), see [103]. These requirements were achieved bymaintaining the substrate temperature in a narrow range of temperatures, 280C<Ts,opt <320 C. The presence of the specular spot splitting in a direction
4 Exchange Coupling in Magnetic Multilayers 221
Fig. 4.20. The direction of the magnetic moment in the Fe film grown over a Crwedge is shown in the second SEMPA image. The top SEMPA image shows thepresence of a magnetic moment in the Cr wedge using an Fe whisker/Cr(001) sam-ple. White and black contrast indicate parallel and antiparallel orientations of themagnetic moment with respect to the Fe whisker magnetization. Note in the bottomSEMPA image that at the boundaries between the parallel and antiparallel orienta-tion of the magnetic moments a perpendicular component of the magnetic momentis present. This is caused by magnetic frustrations due to a partial completion ofthe top Cr/Fe interface which results in a strong biquadratic exchange coupling
parallel with the RHEED streaks for half filled atomic layers indicated thatnew atomic layers were formed from nucleation centers which were separatedby a well defined mean distance of ∼ 80–90 nm. In most depositions of Fe lay-ers the first 5 ML were grown at 100 C. The substrate temperature was thenincreased to 120–240 C. All samples prior to their removal from the vacuumsystem were covered by a 20-ML-thick epitaxial Au (001) layer. The growth ofAu exhibited well defined RHEED intensity oscillations and the surface wasterminated by a 5×1 reconstruction typical for the Au(001) surface.
Magnetic studies
A great review of magnetic studies on Fe/Cr/Fe systems that includes exper-iments and underlying theories can be found in the review paper by Fishman[5]. Here I will summarize mostly studies on the best available samples whichwere prepared on Fe whiskers.
Quantitative BLS studies of the exchange coupling in Fe-whisker/Cr/Fe(001) have been discussed in [6, 103, 104]: the main results are shown inFig. 4.22. These studies showed that the exchange coupling through Cr(001)contains both oscillatory bilinear J1 and positive biquadratic J2 exchange
222 B. Heinrich
k100
k010
k100
k010
Fig. 4.21. The left panel shows a representation of the Cr Fermi surface in theparamagnetic state. The gray shaded areas are ellipsoids centered at the N points inthe bcc reciprocal lattice. The right panel shows a slice through the Fermi surface asindicated in the left panel. The gray shaded arrows are the critical spanning vectorsat the N-centered ellipsoids and the white arrows indicate the nested parts of theFermi surface that give rise to spin density wave antifferomagnetism [3]
Fig. 4.22. The thickness dependence of the bilinear J1 (•) and biquadratic J2 (+)exchange coupling. The biquadratic coupling can be measured only for AFM coupledsamples. The values of J2 for FM coupled samples (10 and 12 ML) were assumed tobe the same as those for the AFM coupled samples having 9,11, and 13 ML of Cr.Note that the coupling becomes AFM for thicknesses greater than 4 ML and thethickness dependence of J1 has a broad maximum around 7 ML of Cr. This behavioris caused by long wavelength oscillations. Large short wavelength oscillations appearfor Cr thicknesses greater than 9 ML
4 Exchange Coupling in Magnetic Multilayers 223
coupling terms. The exchange coupling first becomes antiferromagnetic at 4ML of Cr. For Cr spacer thicknesses dCr <8 ML the strength of the short-wavelength oscillations is quite weak, ∼ 0.1 ergs/cm2. The exchange cou-pling in this range is antiferromagnetic. This is due to the presence of anantiferromagnetic (AFM) long-wavelength bias. This AFM bias is peaked inthe range between 6 to 7 ML. It is interesting to note that the strength ofthe long-wavelength AFM bias is very nearly the same as that observed inFe/Cr/Fe(001) epitaxial multilayers prepared by sputtering where relativelylarge interface roughness annihilates the presence of the shortwavelength oscil-lations [105]. Exchange coupling in the sputtered films showed long-wavelengthoscillations with a rapidly decreasing strength of the coupling for thicknessesgreater than 10 ML of Cr. For Cr spacers thicker than 8 ML, dCr >8 ML, theexchange coupling in films grown on a whisker substrate is dominated by theshort-wavelength oscillations. In this thickness range the samples are antiferro-magnetically coupled for an odd number of Cr atomic layers, Jtot = |J1-2J2| ∼1.5 erg/cm2, and ferromagnetically (FM) coupled for an even number of Cratomic layers. It is interesting to note that the strength of the exchange cou-pling, 1 erg/cm2 in our samples, is very close to that measured by the NISTgroup, see Fig. 4.23, indicating that the Fe-whisker treatment and growthprocedure carried out by the SFU and NIST groups were similar.
MOKE measurements on Fe/wedgeCr/Fe(001) samples were carried outby the NIST group [106]. They covered a wide range of Cr thicknesses, seeFig. 4.23, showing that antiferromagnetic coupling (AFM) was observed atan odd number of atomic layers of Cr. This changes after the first phase slipbetween 24–25 ML of Cr.
Interface atom exchange (interface alloying)
The coupling between the Fe and Cr atoms at the Fe/Cr interface is expectedto be strongly antiferromagnetic [107] and consequently the spin-density wavein Cr is locked to the orientation of the Fe magnetic moments. Since the periodof the short-wavelength oscillations is close to 2 ML one would expect AFcoupling for an even number of Cr atomic layers and FM coupling for an oddnumber of Cr atomic layers. For the period 2.11 ML the first phase slip inthe shortwavelength coupling is predicted to occur at 20 ML. Surprisingly theBLS and SEMPA measurements showed clearly that the phase of the short-wavelength oscillations is exactly opposite to that expected.
It is also important to note that the strength of the exchange coupling wasfound to be much less than that obtained from the first-principles calculations,J1 ∼30 ergs/cm2 [108]. Our studies showed that the strength of the bilinearexchange coupling J1 is very sensitive to the initial growth conditions: a lowerinitial substrate temperature resulted in a larger exchange coupling strength.The bilinear exchange coupling could be changed by as much as a factor of5 by varying the substrate temperature during the growth of the first Cratomic layer [109]. This behavior led us to believe that the atomic formation
224 B. Heinrich
Fig. 4.23. The saturation field as a function of Cr thickness using an Fe-whisker/Crwedge/15Fe(001) sample. The integer represents the number of atomic layers. Thesemeasurements were carried out using MOKE measurements [106]. These measure-ments were possible only for AFM coupling. Note that antiferromagnetic couplingcorresponds to an odd number of Cr atomic layers in agreement with the SFU mea-surements. No quantitative interpretation of the bilinear and biquadratic couplingwas provided
of the Cr layer was more complex than had been previously acknowledged.Angular resolved Auger spectroscopy (ARAES) [110, 111], STM [112], andproton induced Auger electron spectroscopy (ARAES) [113] have shown thatthe formation of the Fe/Cr(001) interface is strongly affected by an interfaceatom exchange mechanism (interface alloying).
The above studies revealed that the Cr adatoms undergo interface mix-ing when the substrate temperature is adjusted for optimum growth. TheARAES data showed that the interface mixing was confined mainly to twoFe atomic layers; see Fig. 4.24, and interface alloying during the growth al-ready starts at low substrate temperatures, Tsub ∼100 C. Using temperaturesless than Tsub ∼100 C one was not able to establish a proper layer by layergrowth even by subsequently raising the substrate temperature. The inter-face alloying increases with increasing substrate temperature; see Fig. 4.24. Itshould be noted that interface alloying due to the atom exchange mechanismis not, in general symmetric; it occurs chiefly at one interface [114]. Interface
4 Exchange Coupling in Magnetic Multilayers 225
Fig. 4.24. The substrate temperature dependence of the fraction of Cr atoms inthe first layer deposited on an Fe whisker substrate (�), the fraction of Cr atomscontained in the whisker surface layer (•), and the fraction of Cr atoms in the firstwhisker subsurface layer (�). The fractional coverages were obtained from fitting theangular dependence of the Auger Cr (529 eV) peak intensity using angular resolvedAuger electron spectroscopy [110]
alloying is driven by the difference in binding energies between the substrateand adatoms. The binding energies are proportional to the melting pointsof the solids. Interface alloying has been observed in systems for which thesubstrates have lower melting points than do the adatom solids [114, 115].
Freyss, Stoeffler, and Dreysse [116] investigated the phase of the exchangecoupling for intermixed Fe/Cr interfaces. The calculations were carried outusing a tight-binding d-band Hamiltonian and a real-space recursive methodfor two mixed layers: Fe(001)/CrxFe1−x /Cr1−xFex /Crn, where n representsthe number of pure Cr atomic layers. This simulates our experimental stud-ies which were carried out on specimens for which the first few atomic Crlayers were grown at lower substrate temperatures where the surface alloy-ing was mainly confined to the two interface atomic layers. The calculationswere able to account for two important experimental observations. First, thecrossover to antiferromagnetic coupling and onset of short-wavelength oscilla-tions was predicted to occur at 4 to 5 ML of Cr, in good agreement with ourobservations, see Fig. 4.22, and in agreement with the NIST studies using theSEMPA imaging technique. Second, the phase of the coupling was reversedfor a concentration x>0.2. AFM and FM coupling was obtained at an oddand an even number of Cr atomic layers, respectively, in perfect agreementwith experiment.
We tested the role of interface mixing by terminating the Cr growth withthe co-deposition of Cr plus Fe to produce a Cr-Fe alloy. Two alloy concen-trations were used: Cr 85%-Fe 15% and Cr 65%-Fe 35%. BLS and MOKEmeasurements revealed that the sign of the exchange coupling between theiron film and the whisker substrate was not changed by this alloyed atomic
226 B. Heinrich
layer. The sign was given by the number of Cr atomic layers deposited with-out any Fe. The system behaved as if the Cr alloy layer formed part of theiron film. This result strongly supports the idea that interface alloying atthe Fe-whisker/Cr interface leads to the observed phase reversal of the short-wavelength oscillations in the strength of the exchange coupling relative to asystem having perfect interfaces.
A large number of measurements showed that a part of J2 was propor-tional to the measured value of J1. We found that J2 ∼0.1+0.16|J1 |. Thisis an interesting result that requires a brief comment. Stoeffler and Gautierpredicted the presence of a biquadratic exchange coupling term for Fe layerscoupled through Cr(001)[108]. This could indicate that the main part of thebiquadratic coupling is of an intrinsic origin. However in subsequent calcu-lations by the Strasburg group [117] for a spin system that was completelyrelaxed the exchange coupling through Cr(001) spacers with ideal interfaceswas formally similar to Slonczewski
′s proximity (torsion) model [118] in which
the exchange energy increases quadratically with the angle between the mag-netic moments of the Fe layers:
E = J(Δϕ− π)2 , (4.36)
where in zero field Δϕ=0 and 180◦ for the odd and even number of Cr atomiclayers, respectively. For AF coupling ϕ varies between 180◦ and 0◦, conse-quently the total magnetic moment approaches saturation gradually; thereis no torque free solution in high fields. The BLS and MOKE measurementsusing Fe-whisker/Cr/Fe(001) samples having a low density of atomic stepsexhibited a much weaker coupling than that predicted by theory and at thesame time it was possible to fully saturate the magnetic moments in thesesamples in sufficiently large external fields. All our MOKE and BLS mea-surements carried out on Fe-whisker/Cr/Fe(001) samples that were preparedwith the best possible interfaces (low density of atomic steps) are consistentwith the use of bilinear and biquadratic exchange coupling terms. In partic-ular, we found that in MOKE and BLS measurements for fields somewhatabove the critical field H2, see Fig. 4.26, the iron film and whisker momentswere parallel, whereas for fields slightly less than the critical field H1 thethin film and whisker moments were antiparallel. The discrepancy betweenexperiment and theory was very likely caused by the presence of interfacealloying at the Fe/Cr interface. Interface alloying not only significantly de-creases the exchange coupling but even changes the angular dependence ofthe coupling.
Neutron-diffraction and MOKE studies by the Bochum group [119] usingFe/Cr/Fe(001) samples having a high density of atomic steps, showed that theground state in Fe/Cr/ Fe(001) multilayers exhibited a noncollinear orienta-tion of the magnetic moments for which the approach to saturation could bedescribed by the Slonczewski magnetic proximity model in which the exchangeenergy depends quadratically on the angle between the magnetic moments ofthe ferromagnetic layers.
4 Exchange Coupling in Magnetic Multilayers 227
However one should realize that dc magnetometry can be affected by sam-ple inhomogeneities. The approach to saturation at the critical field H2 asobserved using MOKE, and the onset of the antiferromagnetic configurationat the critical field H1 , see Fig. 4.25, is more gradual than is expected usingthe sum of bilinear and biquadratic exchange coupling terms; see Fig. 4.26,curve A.
The calculated approach to saturation clearly exhibits a well defined kinkat the field H2, while the experimental measurements usually show a con-cave gradual approach to saturation; see Fig. 4.25. According to the calcu-lations the antiferromagnetic configuration of the Fe magnetic moments isreached via a first-order phase jump; the experimental measurements show amore gradual s-shaped change. However these experimental MOKE featurescan be explained by an inhomogeneous distribution of the exchange couplingstrength. A 10% variation in the exchange coupling across the measured areawould result in hysteresis loops that are very similar to those observed usingMOKE; compare Fig. 4.25 with Fig. 4.26. In fact an inhomogeneous distribu-tion of the exchange coupling also explains the observed differences betweenvalues of the critical fields H2 that have been obtained using the MOKE andthe BLS techniques. The BLS measurements on Fe/Cr/Fe(001) always yieldlower values for the critical field H2, with corresponding lower values of theexchange coupling strength, compared with those obtained using MOKE. TheBLS thin-film resonant modes were also visibly broadened for external fieldsgreater than the saturation field H2 , where the Fe film magnetic momentis parallel to the Fe-whisker moment. The Fe film resonance mode was ob-served to be much narrower at a field midway between H1 and H2 where theFe film and whisker magnetizations are nearly orthogonal [104]. The differ-
Fig. 4.25. The longitudinal MOKE signal for the sample Fe-whisker/11Cr/20Fe(001) as a function of applied field.The integers represent the number of atomiclayers. The saturation field measured using MOKE was 300 Oe higher than thatmeasured using BLS. The difference between the MOKE and BLS measurementswas caused by lateral inhomogeneities in the exchange coupling [103]
228 B. Heinrich
Fig. 4.26. Calculated MOKE signal for a 20 ML Fe(001) film exchange coupledto a bulk Fe(001) substrate and assuming an inhomogeneous distribution of thebilinear coupling strength J1. The biquadratic coupling strength J2 was set to 0.3ergs/cm2 for all curves. The spread of bilinear coupling was assumed to satisfyGauss’s distribution with 〈J1〉=–1 ergs/cm2 , and the distribution width Δ J. CurveA Δ J=0, B Δ J=0.2 ergs/cm2, and C Δ J=0.5 ergs/cm2
ence between critical fields measured using BLS and MOKE, as well as thebroadening of the BLS signal for fields greater than H2 , can be explainedby inhomogeneous distributions of J1 and J2. The BLS technique measuresthe frequencies of the rf resonance modes. The mode corresponding to thethin Fe film covering the Cr spacer exhibits a resonance at a field that isthe algebraic average of local inhomogeneous resonance fields, correspond-ing to the distribution of local exchange coupling strengths. The broadeningof this BLS resonance peak is related to the distribution of the local reso-nance fields. On the other hand, in the MOKE studies a complete saturationis observed only after the external field reaches the value corresponding tothe maximum value of the H2 critical field distribution. The observed differ-ences between the MOKE and BLS measurements clearly indicate that thecoupling through Cr is not even homogeneous across the small area only afew micrometers in diameter corresponding to the laser spot size in the BLSmeasurements.
Role of multiple scattering
The role of electron interface scattering was tested using heterogeneous Crspacers. Two specimens were grown with a Cu interface layer between the Crspacer and the Fe thin film: Fe-whisker/ 11Cr/1Cu/20Fe(001)/20Au and Fe-whisker/11Cr/2Cu/20Fe(001)/20Au, where the integers represent the number
4 Exchange Coupling in Magnetic Multilayers 229
of atomic layers. Two specimens were grown with a silver interface layer: Fe-whisker/11Cr/1Ag/20Fe(001)/20Au and Fe-whisker/11Cr/2Ag/20Fe(001)/20Au. Measurements carried out using BLS and MOKE revealed that theexchange coupling in the Fe-whisker/ 11Cr/1−2Cu/Fe(001) was found to in-crease twofold compared to that observed in samples having a simple 11 ML Crspacer layer. No significant difference was found for the samples having a Aginterlayer where the exchange coupling decreased by 20%. Mirbt and Johans-son [120] presented calculations that are in accord with these results. Their cal-culations showed that the enhanced coupling strength in Fe/Cr/Cu/Fe(001)samples is due to a change in the spin dependent reflectivity of the Cr spacerelectrons at the Cr/Cu/Fe interface. The presence of the Cu atoms changesthe spin dependent interface potential due to hybridization of the Cu electronstates with the Fe electron states. Since the Fe majority spin band lies closestto the Fermi level the effect of the hybridization will be most pronounced forthe majority spin Fe band. The hybridization with Cu results in a downwardenergy shift that moves the Fe majority spin band below the Fermi level. Anenergy gap is created at the Cu/Fe interface, and consequently the majorityspin electrons in Cr undergo a nearly perfect reflection. The states for minorityspin electrons are very little affected by the Cu, and therefore their reflectivityis left unchanged. It follows that the spin reflection asymmetry is increasedleading to an increased coupling. The effect of a Ag spacer on the couplingin Fe/Cr/Ag/Fe films is less dramatic. Calculations show that the spin asym-metry in reflectivity is somewhat decreased leading to an overall decrease inthe exchange coupling. The theoretical calculations of Mirbt and Johanssonsuggested that a proper model for exchange coupling through spin-densitywaves in Cr has to include two contributions: (a) a spin dependent potentialdue to the magnetic moments on the antiferromagnetic Cr atoms; (b) a spindependent potential at the Fe/Cr and Cr/Fe interfaces. The first contribu-tion for Cr layers thinner than 24 ML can be described by a Heisenberg-likeHamiltonian [108] with AFM coupling between the Cr magnetic moments onadjacent (001) planes and a strong AFM coupling between the Cr and Featomic moments at the interfaces. The experimental result implies that thespin-density and multiple scattering contributions act in phase to increase thetotal exchange coupling.
Dependence of exchange coupling on the thickness of the Fe layer
Okuno and Inomata [121, 122] reported a strong oscillatory dependence oniron thickness of the exchange coupling in Fe/Cr/ Fe(001) multilayer speci-mens. The period of the oscillation was 6 ML. However, the specimens usedin their work were multilayers characterized by rough interfaces, and the ex-change coupling exhibited no short period ∼2 ML variations with Cr thick-ness. We have measured the dependence of the exchange coupling strengthon iron film thickness using an Fe(001) whisker substrate, an 11 ML layerof Cr, and a wedge-shaped Fe film deposited on the Cr. This structure,
230 B. Heinrich
Fe-whisker/11Cr/wedge Fe(001), was then capped by 20 ML layers of gold.The Cr was deposited using optimum conditions for a smooth growth. Noevidence was found for either a short or a long wavelength oscillatory de-pendence of the coupling strength on the iron film thickness. We concludedthat Fe/Cr/Fe/Au(001) samples having a low density of interfacial steps,and that exhibit 2 ML oscillations in the exchange coupling as a function ofthe Cr layer thickness, displayed no measurable variations of the exchangecoupling strength as a function of the Fe film thickness. It appears thatthe exchange coupling between Fe layers separated by a Cr spacer can beascribed to interactions that are localized to the interfaces. However, oursubsequent experiments did indicate that the exchange coupling is sensi-tive to the Fe film thickness when the iron film is capped with Cr. For awhisker/11Cr/20Fe/20Au structure J1 =–0.82 erg/cm2, J2 =0.3 erg/cm2;and for a whisker/11Cr/20Fe/11Cr/20Au(001) J1 =–1.6 erg/cm2, J2 =0.33erg/cm2. Clearly the insertion of Cr between the Fe and Au films caused asubstantial increase in the coupling strength. Note that the biquadratic cou-pling term is essentially the same for both specimens. Since the exchangecoupling for Cr/Fe/Cr films depends upon the Fe film thickness, it followsthat the Fe/Cr interfaces support the formation of electron resonance statesin the Fe films. On the other hand, the Fe/Au interfaces tend to suppresselectron resonance states.
Neutron studies
Extensive work on Cr has been carried out by a large number of groups. Iwould like particularly to point out the work done using neutron scattering.Neutron studies have been carried out on thick Cr layers ranging from 2.0nm to 300 nm and over a wide range of temperatures down to cryogenictemperatures where one can study the Cr SDW phase diagram. The conclu-sion of these studies is that the SDW magnetism in Cr films is complex andexquisite. The SDW in Cr films is not a rigid property but depends on thefilm thickness, the temperature, and is crucially affected by interface rough-ness, and interface exchange coupling. Theoretical and experimental investi-gations have been mainly limited to the (001) orientation. The first reliableSDW data in Cr films were reported in papers by Schreyer et al. [123, 124]on [Fe/Cr/Fe](001) superlattices prepared using MBE. Soon after neutronstudies on superlattices grown in the [001], [011], and [111] crystallographicorientations and prepared by sputtering were performed by Fullerton et al.[125] and Adenwalla et al. [126]. The results on the MBE and sputtered sam-ples agree on a few points. The MBE grown superlattices exhibited bothincommensurate collinear (I) and commensurate spiral SDW (C) magneticstates for a Cr spacer thicker than 3.5 nm [127]. The non-collinear commen-surate C state results from magnetic frustration at the Fe/Cr interfaces. Thismode is equivalent to the H SDW in Fishman’s review article, see Fig. 4.15in [5]. Schreyer et al. suggested that the exchange interlayer coupling energy
4 Exchange Coupling in Magnetic Multilayers 231
in C follows the proximity (torque) model suggested by Slonczewski [118],see (4.36) and (4.37). This torque model resulted in a spiral configurationof the Fe magnetic moments. For dCr <4.5 nm only the C state exists. Thetransition temperature from I to C starts from 0 K at 4.5 nm and then grad-ually increases to 300 K with increasing dCr. The Neel ordering temperatureTN �500 K was found to be independent of the Cr thickness, dCr. See thefull phase diagram in [127].
The studies by Fullerton et al. on sputtered specimens [125] yielded differ-ent results. They found a collinear I SDW state with the SDW nodes near theFe/Cr and Cr/Fe interfaces. No C and H SDW states were observed. The pres-ence of SDW nodes at the interfaces can result in the loss of magnetic contactwith the neighboring Fe layers. The Cr layer is either in an incommensuratestate for thick enough layers or becomes paramagnetic with decreasing filmthickness or with increasing temperature. One needs at least one SDW periodto create a collinear I SDW. The SDW period was found to be nearly thesame as the bulk value. The Neel ordering temperature TN vanished for filmshaving a thickness less than a critical thickness of about 30 ML [128].
Clearly the different results obtained using MBE and sputtered samplesis due to the different interface roughness. The sputtered samples have mostlikely a greater interface roughness. The SDW behavior in Cr is very adaptableand easily affected by the interfaces.
The spin flip transition from a transverse to a longitudinal SDW observedin bulk Cr at T=123 K is not observed in superlattices in the thickness range sofar studied. Although in thick Cr layers (above 30 nm) covered by an Fe(001)layer the transverse SDW with the q-vector parallel to the film surface andwith the magnetic moment of Cr perpendicular the film surface was observed[129, 130]. In this case the interface frustrations are minimized by flippingthe magnetic moment of Cr perpendicular to the Fe magnetic moment. Thisis analogous to the biquadratic exchange coupling mechanism in interlayerexchange coupling.
Layer specific measurements of the magnetization reversal in AFM cou-pled Fe/Cr/7Fe/Cr/Fe(001) layers were carried out by L’abbe et al. [131] usingthe isotope (7Fe) and nuclear resonant scattering of circularly polarized syn-chrotron radiation. This work provided interesting results. The total magnetichysteresis loop with the field applied along the magnetic easy axes indicated atypical antiferromagnetic coupling in the presence of 4-fold in-plane magneticanisotropy, compare Fig. 4.14b in [131] with line 3 in Fig. 4.2. The absence ofa jump in order to reach full saturation indicated that biquadratic exchangecoupling was present, but the existence of a well defined jump at low fieldsindicated that the bilinear coupling was strong enough to orient the magneticmoments antiparallel to each other and parallel to the magnetic easy axes.However the resonant scattering results in [131] suggested that the magneticmoment in the central Fe film was oriented away from the magnetic easyaxis and this observation indicated a non-trivial configuration of magneticmoments.
232 B. Heinrich
Magnetic domains in the remanent state in [Fe/Cr] superlattices wereinvestigated by means of Polarized Neutron Reflectivity (PNR) [132] and Syn-chrotron Moessbauer spectroscopy techniques [133].
It is not possible to discuss these studies in greater detail within the spaceavailable for this chapter. There are excellent review articles by Zabel [134],Fullerton et al. [135], Boedeker et al. [129], and Fishman [5] that cover boththeory and experiment including SWD phase diagrams in Cr films; see alsothe references contained therein.
4.5.4 Antiferromagnetic Mn
The growth of Mn is challenging by virtue of its ability to adopt various formsof crystalline structures ranging from the bulk αMn (with many atoms perunit cell) to bcc and fcc structures stabilized by alloying or by epitaxial growthon suitable substrates [136]. Kim et al. [137] found that Mn grows on Fe(001)in a strained body-centered tetragonal (bct) structure with lattice spacing pa-rameters a=0.2866 nm and c=0.3228 nm for thicknesses greater than 4 atomiclayers of Mn. The measured RT values of the Mn magnetic moment for a thinfilm on Fe range from 1.7 to 4.5 μB [138, 139]. The magnetic state of Mn filmsthinner than 5 ML is rather complicated. Magnetic Circular X-ray Dichro-ism (MCXD indicates that the fully filled atomic layer of Mn is magneticallycompensated [139, 140]. For submonolayer coverage the Mn moments wereoriented antiparallel to the Fe magnetic moments [140]. This observation is inagreement with calculations by Wu and Freeman [141]. The ground state of aMn atomic layer is a c(2 × 2) magnetically compensated state. The situationis simpler for Mn films thicker than 4 atomic layers. SEMPA measurementsby the NIST group showed that Mn(001) planes are ferromagnetically orderedwith the adjacent atomic layers oriented antiferromagneticaly, thus the mag-netization exhibits 2 ML oscillations [142]. This conclusion is also supportedby spin polarized STM (spSTM) studies [143, 144, 145]. Schlickum et al. [145]found that in regions where Mn overgrows Fe substrate steps a frustration ofthe antiferromagnetic order results in a 180◦ domain wall in the Mn film. Thewidth of the domain wall increases linearly with the Mn layer thickness andreaches 7 nm in a 20 ML thick film of Mn. The ability of spSTM to observethe ferromagnetic order in the top Mn layer is not obvious: it is a weak effect(a few %). Calculations indicate that in ballistic transport the spin currentasymmetry can be attributed mainly to the symmetry breaking at the surface[146].
The exchange coupling in Fe whisker/Mn/Fe(001) was investigated by theNIST group using SEMPA. The SEMPA images indicated that for the firstthree Mn atomic layers the magnetization of the top Fe layer pointed in thesame direction as the Fe whisker magnetic moment, ie. FM coupling. For Mnthicknesses greater than 4 ML the magnetic moment of the Fe film was notcollinear with the magnetic moment of the Fe whisker. Between 4 to 8 layersof Mn the direction of the Fe film magnetic moment lay at an angle of 60–80◦
4 Exchange Coupling in Magnetic Multilayers 233
relative to the magnetization of the Fe whisker. Beginning at the 9th Mn layer,the direction of the coupling oscillated with a two ML period between 90◦−φand 90◦+φ, where φ was sample dependent. Values of φ were found between10 to 30◦. Similar behavior was found by means of MOKE measurementsusing Fe/Mn/Fe wedged samples deposited on a GaAs/Fe/Ag(001) template[147]. For Mn thicknesses greater than 1.2 nm (7 ML) a non-collinear con-figuration (φ=0) was observed. Hysteresis loops in these measurements werewell described by Slonczewski′s proximity magnetism model which accountsfor interface roughness in strong antiferromagnets. The total coupling energyE is given by
E = C+θ2 + C−(θ − π)2 , (4.37)
where θ is the angle between the magnetic moments of the two Fe layers.The strength of such proximity coupling showed weak oscillations having a2 ML period. It should be pointed out that Mn can not be grown with in-terfaces as smooth as Cr, and therefore it is not surprising that Mn spacersprovide an excellent system where the consequences of magnetic frustrationare fully demonstrated. It is known that even small concentrations of Mn inCr results in a strong and commensurate antiferromagnetism [148]. It wastherefore thought to be of interest to investigate the phase and the strengthof the exchange coupling between Fe layers separated by layers of a Cr-Mnalloy. To this end the SFU group attempted to grow Fe/Cr-Mn/Fe structures.Unfortunately we found that the Mn atoms have a very strong tendency tosegregate on the surface of Cr during the growth. It was necessary to maintaina substrate temperature greater than 200 C in order to obtain a good layer bylayer growth. At this temperature the top surface layer contained a stronglyenhanced concentration of Mn (∼50%). In view of this surface segregation,and since the interfaces play a very crucial role in exchange coupling Heinrichet al. [103] decided to grow pure layers of Mn between the Cr and the Felayers. Eleven and twelve ML of Cr were grown on an Fe(001) whisker usinggrowth conditions optimizing layer-by-layer growth. Mn layers were depositedon the Cr at a substrate temperature of 120◦C. The substrate was allowedto cool to room temperatures and 20 ML of Fe(001) were deposited on theMn layer, and a protective layer of 20 ML of Au(001) were deposited on theiron. At 120 C the deposition of the first two atomic layers of Mn proceededin a good layer by layer growth with large RHEED intensity oscillations atthe second anti-Bragg scattering condition. At substrate temperatures wellbelow 100 C the Mn does not segregate on Fe. In this way one is able togrow well defined Fe/Cr/Mn/Fe(001) structures having smooth and abruptinterfaces. It should be pointed out that the top Cr surface atomic layer issmooth, with large atomic terraces corresponding to those of the Fe whisker,and is unaffected by interface alloying during the deposition of the Mn. There-fore variations in the exchange coupling due to the addition of the Mn layersare primarily due to the presence of the Mn atomic layers and their magneticstate. There is no intermediate Cr-Mn mixed region similar to the Cr-Fe mixed
234 B. Heinrich
region that occurs at the Fe-whisker/Cr interface. The following samples werestudied: Fe(001)/11,12Cr/ 1,2,3Mn/20Fe(001)/20Au. This selection of sam-ples allowed one to determine the effect of Mn on the exchange coupling. Thecoupling was found to be AFM for the 11 ML Cr spacer and FM for the 12ML Cr spacer. Therefore the phase of the coupling was only determined bythe thickness of the Cr layer. Assuming that the Mn is an AFM state with anuncompensated magnetic moment in the (001) plane the coupling should havechanged its sign as each Mn atomic layer was added. Such a phase reversalwas not observed. The exchange coupling oscillated with the same phase andthe same periodicity of 2 ML as was observed using pure Cr spacer layers. Theassumption that the magnetic state of Mn in Fe/Cr/ Mn/Fe(001) structurescan be described by commensurate antiferromagnetism with uncompensated(001) planes is not necessarily correct. In recent calculations by Krueger et al.[149] the magnetic structure of bct Mn in bulk was studied as function ofc/a, a measure of the tetragonal distortion. The calculations showed that bctMn grown on Fe(001) is in a magnetic state that is at the border line be-tween the AFM1 configuration, in which the magnetic moments are parallelin the (001) planes, and the AFM3(110) configuration, having ferromagneticplanes oriented along (110), and fully compensated (001) planes having zeronet magnetic moment. The AFM3(110) state would not lead to an alternatingsign of exchange coupling with increasing Mn thickness in agreement with ourexperimental observations.
4.5.5 Loose Spins
Fe atoms can be dispersed in NM interlayers [150, 151] and result in additionalcoupling between two ferromagnetic films. Slonczewski developed a modelincorporating the exchange interaction between the FM layers and “loosespins” inside the NM spacer [152]. He assumed that the exchange field onthe “loose spins” is given by H1,2 = H1,2(z, d− z)m1,2, where m1,2 are unitvectors along the magnetic moments of layers FM1 and FM2 and z, and d− zdetermine the distance of a loose spin from the interfaces. The strength of theexchange field, H1,2(z) can be estimated from the thickness dependence of theinterlayer exchange coupling. The energy levels of a loose spin are given byεm = −Um/S with m=–S, –S+1,...S, where
U(θ) =(U2
1 + U22 + 2U1U2cos(θ)
)1/2. (4.38)
U1,2 are energies associated with the exchange fields H1,2. U is expected tobe weak and the total energy of a loose spin is given by thermal excitations.The total energy is then multiplied by the concentration of loose spins. Thisresults in bilinear and biquadratic contributions. Heinrich et al. [153] andSchaefer et al. [154] investigated the role of loose spins in Fe/Cu/Fe(001) andFe/Ag/Fe(001) structures, respectively, by inserting less than 1 ML of Fe inwell prescribed positions inside the NM spacer. The strengths of U1 and U2
4 Exchange Coupling in Magnetic Multilayers 235
were estimated from the thickness dependence of the interlayer coupling usingpure Cu spacers. We found that the presence of an alloyed Cu-Fe ML inside theCu spacer significantly decreased the strength of the bilinear coupling whileleaving the biquadratic coupling almost unchanged. This decrease in the bi-linear coupling can then result in the coupled Fe magnetic moments becomingoriented in mutually perpendicular directions. Both groups have shown thatthe consideration of loose spins to describe the data requires the inclusion ofclusters of Fe atoms inside the NM. The application of Slonczewski’s theoryin the work by Heinrich et al. [153] required the inclusion of a molecular ex-change field in the alloyed atomic layer in order to account for the measuredtemperature dependence of the bilinear exchange coupling.
Recently the magnetic state of Cr spacers in [Fe/Cr/Fe](001) superlatticeswas investigated by means of perturbed angular correlation (PAC) spectrom-etry [155]. These measurements revealed that above the blocking temperatureof Cr the Cr magnetic moment exhibits superparamagnetic spin fluctuations.In this temperature regime the interlayer coupling in these samples is mostlygiven by the biquadratic contribution. It was suggested [155] that the su-perparamagnetic clusters acted as “loose spins” and were responsible for thepresence of biquadratic exchange coupling. In my view this is a rather strongclaim. “Loose spins” as treated using the Slonczewski model create a strongbilinear coupling term that surpasses the strength of the biquadratic contri-bution. In order to obtain a dominant biquadratic exchange coupling contri-bution one needs to compensate the bilinear coupling contribution by someother mechanism. The absence of interlayer coupling below the Neel orderingtemperature in [Fe/Cr/Fe] superlattices is caused by the nodes in the Cr mag-netic moment around the Fe-Cr interfaces, see the above section on Neutronstudies. In the Cr paramagnetic regime one can get a direct coupling as iscommonly observed in other simple metal spacers. In that case the interfaceroughness will significantly decrease the bilinear exchange coupling strengthin paramagnetic Cr and will introduce biquadratic coupling due to interfacemagnetic frustrations. Most likely the biquadratic coupling strength observedin [Fe/Cr/Fe] superlattices is a combination of all of these contributions.
We found a typical loose spin-like behavior in Co/Cu/Co(001)/Fe andFe/Pd/Fe(001) systems. The sample 4Co/6Cu/4Co(001) grown on Cu(001)was found to be coupled antiferromagnetically. With the addition of a 3 MLfilm of Fe deposited on the top of the second Co layer the top Fe surface devel-oped a strong reconstruction and the exchange coupling became ferromagnetic[38, 57]. Moreover, the temperature dependence of the exchange coupling fol-lowed a Curie-Weiss type of dependence, proportional to 1/T . In our viewthe strong lattice reconstruction resulted in lattice defects and subsequentpenetration of Co atoms into the Cu spacer. These Co atoms then acted as“loose spins” inside the Cu spacer. These loose spins were subjected to theexchange field of the surrounding FM Co layers and their contribution to theoverall exchange coupling scaled with their magnetic moments. The magneticmoments of true loose spins are expected to follow a Curie-Weiss law.
236 B. Heinrich
4.5.6 Lattice Strained Pd in Fe/Pd/Fe(001) Structures
In the Fe/Pd(001) system, a lattice strain results from the 4.2% mismatchbetween the bcc Fe(001) and the fcc Pd(001) surface nets. Ultrathin Pd(001)films grown on Fe(001) are expanded laterally to match the Fe mesh. Thestructure and magnetism of fcc Pd in Fe/Pd/Fe (001) trilayers grown onAg(001) substrates were studied in [156, 157]. Using RHEED and x-ray diffrac-tion, Fullerton et al. [157], have shown that the Pd ultrathin films grew onFe(001) with a 4.2% latterly expanded lattice accompanied by an out-of-planecontraction of 7.2% (c/a=0.89). The ultrathin films of Pd grown on Fe(001)had a pronounced face centered tetragonal (fct) structure. Theoretical ab ini-tio calculations of ultrathin film Pd layers on an Fe(001) template have shownthat the ground state of the lattice strained Pd(001) has a fct structure suchthat the bulk Pd atomic volume is maintained [157]. This theory is in goodagreement with the results of RHEED and x-ray diffraction measurements. Po-larized neutron reflectivity measurements on an Fe(5.6 ML)/Pd(7 ML)/Au(20ML) sample determined the average moment per Fe atom to be 2.66 μB [157].Ab initio spin density calculations for the same structure showed that thisvalue is consistent with the observed induced Pd polarization. It is interestingto point out that a 4.2% lattice expansion of bulk metallic fcc Pd would resultin long range ferromagnetic order. The main conclusion of the magnetic mea-surements and calculations was that the Pd was ferromagnetic only for twoadjacent Pd atomic layers at the Fe/Pd and Pd/Fe interfaces. By increasingthe thickness of the Pd by one additional atomic layer (a total thickness of5 ML) the long range ferromagnetic order in Pd was lost. Clearly the latticevertical relaxation has to be taken into account in order to explain the realmagnetic properties of strained epitaxial structures. The exchange couplingthrough Pd exhibited an oscillatory behavior as a function of the Pd thickness[156]. The period of oscillations was 4 ML. This period is close to the 3 MLperiod corresponding to the large Pd 4d belly Fermi surface sheets [158]. Thetemperature dependence of the coupling for 5 ML thick Pd showed a perfectCurie Weiss dependence. A strong temperature dependence is also observedfor 6 ML Pd. For thicker Pd layers the temperature dependence is weak. Thisagain indicates the presence of fluctuating magnetic moments in the Pd layersin the range of thicknesses from 5 to 6 ML. In my view the above two systems,Co/Cu/Co/Fe(001) and Fe/Pd/Fe(001), represent the best examples of selfassembled loose spins.
4.6 Time Dependent Exchange Coupling
4.6.1 Multilayers
The interlayer exchange coupling had so far no explicit time dependent be-havior. The spin dynamics studies in magnetic FM/NM/FM trilayers revealedthat the coupling between the magnetic layers can also have a purely dynamic
4 Exchange Coupling in Magnetic Multilayers 237
character. The role of interface damping has been investigated in high qualitycrystalline Au/Fe/Au/Fe(001) structures grown on GaAs(001) substrates, seethe details in [30]. The in-plane FMR experiments were carried out using 10,24, and 36 GHz systems [159]. The in-plane resonance fields and resonancelinewidths were measured as a function of the azimuthal angle ϕ between theexternal dc magnetic field and the Fe in-plane cubic axes.
Single Fe ultrathin films with thicknesses of 8, 11, 16, 21, and 31 mono-layers (ML) were grown directly on GaAs(001). They were covered by a 20ML thick Au(001) cap layer for protection under ambient conditions. FMRmeasurements were used to determine the in-plane four-fold and uniaxial mag-netic anisotropies, K‖
1,eff and K‖u,eff , and the effective demagnetizing field
perpendicular to the film surface, 4π Meff (=4πMs-2K⊥u,s/Ms), as a function
of the film thickness d [30, 160]. The magnetic anisotropies were well describedby a linear dependance on 1/d. The constant and linear terms represent thebulk and interface magnetic properties, respectively. The ultrathin Fe filmsgrown on GaAs(001) have magnetic properties nearly equal to those of bulkFe, modified only by sharply defined interface anisotropies; this indicates thatthe Fe layers are of a high crystalline quality with well defined interfaces.The lineshapes of the FMR peaks are Lorentzian and the FMR linewidthsare small (less than 100 Oe for our microwave frequencies) and only weaklydependent on the film thickness. The reproducible magnetic anisotropies andsmall FMR linewidths provided an excellent opportunity for the investigationof non-local relaxation processes in magnetic multilayer films.
The ultrathin Fe films which were studied in the single layer structureswere regrown as a part of magnetic double layer structures. The thin Fe film(F1) was separated from the second thicker layer (F2), 40 ML thick, by a 40ML thick Au spacer. The magnetic double layers were covered by a 20 MLAu(001) layer for protection under ambient conditions. The thickness of theAu spacer layer was much smaller than the electron mean free path in gold(38 nm) [161], and hence ballistic spin transfer between the magnetic layersis allowed.
The interface magnetic anisotropies separated the FMR fields of F1 andF2 by a large margin ( 1 kOe), see [30]. That separation allowed us to carryout FMR measurements on F1 without exciting a large response in F2: theangle of precession in F2 was negligible compared to that in F1. The FMRlinewidths in single and double layer structures were only weakly dependenton the azimuthal angle ϕ of the saturation magnetization with respect to thein-plane crystallographic axes.
The 16Fe film (F1) in the single and double layer structures had the sameFMR field showing that the interlayer exchange coupling [6] through the 40ML thick Au spacer was negligible, and the magnetic properties of the Fe filmsgrown by MBE on well prepared GaAs(001) substrates were fully reproducible.
The FMR linewidth in the thin films always increased in the presence ofthe thick layer F2. The additional FMR linewidth, ΔHadd, followed an inverse
238 B. Heinrich
Fig. 4.27. The FMR linewidth measurements (half width half maximum (HWHM))in the parallel configuration. (a) The dependence of the additional FMR linewidthΔHadd = ΔHd − ΔHs on 1/d at f = 36 GHz. ΔHd and ΔHs represent the FMRlinewidths for the Fe films in the double and single layer magnetic structures, re-spectively. d is the thickness of the F1 layer in a GaAs/nFe/40 Au/40Fe/20Au(001)structure. The integers in the structure notation represent the number of atomic lay-ers. (b) The frequency dependence of the FMR linewidth in the 16 ML Fe layer: ΔHd
(•) in a GaAs/16Fe/40Au/40Fe/20Au(001) structure; ΔHs (◦) in a single magneticlayer GaAs/16 Fe/20Au(001) structure; and the additional FMR linewidth ΔHadd
is shown in ()
dependence on the thin film thickness d, see Fig. 4.27a. The non-local dampingoriginates at the film interface (F1/Au). The linear dependence of ΔHadd onthe microwave frequency for both the parallel and perpendicular configurationwith negligible zero-frequency offset, see Fig. 4.27b is equally important. Thismeans that the additional contribution to the FMR linewidth can be describedby an interface Gilbert damping. The additional Gilbert damping for the 16Fefilm was found to be weakly dependent on the crystallographic direction,Gadd = 1.2 × 108 s−1 along a cubic axis.
Discussion of the interface torque
Tserkovnyak, Brataas and Bauer [162, 163] showed that interface dampingcan be generated by a spin current flowing from a ferromagnet into adjacentnormal metallic layers. The spin current is generated by a precessing magneticmoment in F1. The spin current was calculated using Brouwer’s scatteringmatrix [164] which evolves under a time dependent parameter (phase angleof precession). Normal metal (NM) layers surrounding a magnetic layer wereviewed as reservoirs in common thermal equilibrium as result of contact withan infinite thermal bath. The calculations were carried out assuming that thereservoirs acted as ideal spin sinks. The resulting spin current per unit areais given by
jspin =�
4π
(ReA↑↓n× ∂n
∂t+ ImA↑↓ ∂n
∂t
), (4.39)
4 Exchange Coupling in Magnetic Multilayers 239
where n is a unit vector along the magnetic moment M, and A↑↓ is the complexspin-pumping conductance per unit area given by the difference between thereflection (gr
↑↓) and transmission (gt↑↓) mixing conductances per unit area,
A↑↓ = gr↑↓ − gt
↑↓ . (4.40)
The spin dependent reflection and transmission matrices for the ferromagneticfilm are given by
gr↑↓ =
∑m,n
(δm,n − r↑m,nr
↓∗m,n
)
gt↑↓ =
∑m,n
t↑m,nt↓∗m,n , (4.41)
where r↑↓m,n are the reflection parameters for spin up and spin down elec-trons in the NM reservoirs, and t↑↓m,n are the transmission parameters into thereservoirs. The indices m and n in (4.40) label the modes (channels) corre-sponding to k‖,⊥ wave-vectors (parallel and perpendicular to the interface)at the Fermi energy. The Gilbert damping is given by the conservation of thetotal spin momentum per unit area
jspin − 1γ∂Mtot
∂t= 0 , (4.42)
where Mtot is the total magnetic moment of F1. After simple algebraic stepsone obtains an expression for the dimensionless damping parameter
α =G
γ1Ms
=(
αbulk +gμB
4πMs
1d1Re(A↑↓)
)
1γeff
=1γ
(1 − gμB
4πMs
1d1Im(A↑↓)
), (4.43)
where d1 is the thickness of the ferromagnetic layer F1. The imaginary partof A↑↓ arises from the
∑m,n r
↑m,nr
↓∗m,n and
∑m,n t
↑m,nt
↓∗m,n sums and is very
close to zero due to cancellation of phases. Therefore spin-pumping mostlyeffects the damping, the renormalization of the gyromagnetic ratio γ due tospin pumping is very small. The inverse dependence of the Gilbert dampingon the film thickness clearly testifies to its interfacial origin. The layer F1 actsas a spin pump. Now another important point has to be answered: how isthe generated spin current dissipated? This answer can be found in the recentarticle by Stiles and Zangwill in Anatomy of spin transfer torque [165], seeSect. 4.4.1. They showed that the transverse component of the spin currentin a normal layer (NM) is entirely absorbed at the NM/FM interface. Forsmall precessional angles the spin current jspin is almost entirely transverse.This means that the N/F2 interface acts as an ideal spin sink, and providesan effective spin brake for F1, see Fig. 4.28(a). For ferromagnetic layers that
240 B. Heinrich
Fig. 4.28. A cartoon representing the dynamic coupling between two magneticlayers which are separated by a non-magnetic spacer. (a) represents two magneticlayers with different FMR fields. F1 is at resonance, and F2 is nearly stationary. Abow like arrow in the normal spacer describes the direction of the spin current. Thedashed line represents the instantaneous direction of the spin momentum. F1 actsas a spin pump, F2 acts as a spin sink, and consequently F1 acquires an additionalGilbert damping. (b) represents a situation when F1 and F2 resonate at the samefield. Both layers act as spin pumps and spin sinks. In this case the net spin mo-mentum transfer across each interface is zero. No additional damping is present asthe precession is in phase
are thicker than the spin lateral coherence length( 2–4 A), and the electronscattering at the interfaces is partly diffuse, the coefficient A↑↓ is nearly equalto the number of transverse channels in the normal metal NM,
∑m,n δm,n,
see [166], [167], [168]. In simple metals this sum per unit area is given by
A↑↓ =k2
F
4π= 0.85n2/3 , (4.44)
where kF is the Fermi wavevector and n is the density of electrons per spinin the normal metal N. The spin current has the form of Gilbert damping. Itcan be shown that the coefficient A↑↓ is proportional to the interface mixingconductance, A↑↓ � h
e2 gcond↑↓ , see [166], [167].
The layers F1 and F2 act as mutual spin pumps and spin sinks. Theequation of motion for F1 can be written as
1γ∂M1
∂t= − [M1 × Heff,1] +
G1
γ2M2s,1
[M1 × ∂M1
∂t
]
+�
4πd1g↑↓,1n1 × ∂n1
∂t− �
4πd1g↑↓,2n2 × ∂n2
∂t, (4.45)
4 Exchange Coupling in Magnetic Multilayers 241
where g↑↓,1,2 are the real parts of A↑↓,1,2 for the layers F1 and F2. M1 is themagnetization vector of F1, n1,2 are the unit vectors along M1,2, and d1,d2
are the thicknesses of layers F1 and F2. The exchange of spin currents is asymmetric concept and the equation of motion for the layer F2 is obtained byinterchanging the indices 1 ⇔ 2.
Equation (4.45) can be tested by investigating the FMR linewidth aroundan accidental crossover of the resonance fields for F1 and F2 [169]. In thiscase the resonant field of F1 approaches the resonant field of F2. When theyreach the same resonant field the rf magnetization components of F1 and F2are parallel with each other. Each precessing magnetization creates its ownspin current which is pumped across the NM spacer. The electron mean freepath in Au thick films is 38 nm [161], and consequently the spin transportis purely ballistic even in an 80 ML thick Au spacer. At the same time bothinterfaces F1/NM and NM/F2 act as spin sinks. It follows that the net flowof the spin current through each interface can be zero, and the contributionof spin pumping to the FMR linewidth can disappear, see Fig. 4.28(b). Thebulk Gilbert damping in F1 and F2 were very nearly equal which resulted inthe marked disappearance of the additional FMR linewidth at the crossoverof the resonance fields, see Fig. 4.29. The good agreement between theoryand experiment clearly shows the validity of the spin pumping theory basedon (4.45). Even in the absence of static interlayer exchange coupling the mag-netic layers are coupled by dynamic interlayer exchange. The increase in theadditional FMR linewidth for 16Fe in Fig. 4.29 which appears for angles nearthe accidental crossover in the resonance fields shows that the spin pumpingeffect can be enhanced when the rf magnetic moments are partly out of phase.This effect is present in ferromagnetic films that are exchange coupled. In theexchange coupled case the optical mode exhibits a larger linewidth than thatexpected from simple spin pumping in which the spin sink has a negligibleprecessional amplitude, see [170, 171].
The quantitative comparison between experiment and spin pumping the-ory is very good [170]. First principles electron band calculations [168] resultedin g↑↓ ≈1.1×1015cm−2 for a clean Cu/Co(111) interface. By scaling this valueto Au using (4.44) one obtains a value for the Gilbert damping parameterGsp,cal =1.4×108s−1 for a 16 ML thick Fe film; this value is very close tothe experimentally observed value Gsp,exp =1.2×108s−1 measured at RT us-ing FMR. This agreement is amazing considering the fact that calculationsof the intrinsic damping in bulk metals have been carried out over the lastthree decades and have not been able to produce a comparable agreementwith experiment.
The spin pumping can be also found in single Fe films surrounded bynormal metal layers provided that the spin current diffuses away from theFM/NM interface. Interface damping was observed in NM/Py/NM sand-wiches by Mizukami et al. [172] where NM=Pt,Pd and Ta non-magnetic layerswere surrounding a permalloy (Py) layer.
242 B. Heinrich
Fig. 4.29. The FMR field and linewidth at 24 GHz as a function ofthe angle ϕ of the applied dc field. The measurements were done onGaAs/16Fe/40Au/40Fe/20Au(001), where the integers represent the number ofatomic layers. The upper Figure shows the FMR fields, the symbols () correspondto 40Fe and (◦) symbols to 16Fe. Notice that the 16Fe film grown on GaAs(001)exhibits a strong in-plane uniaxial anisotropy. The presence of the in-plane uniaxialanisotropy allows one to get an accidental crossover of the resonance fields at theangle ϕ=112◦. The lower Figure shows the FMR linewidth. The solid lines wereobtained from calculations using (4.45). The symbols (•) correspond to F1 (16Fe)and () correspond to F2(40 ML). Note that the FMR linewidth for the thinnersample, 16Fe, first increases before it reaches its minimum value corresponding toits single GaAs/16Fe/20Au(001) layer structure. Note also that the additional linebroadening due spin pumping scales inversely with the film thickness
Spin pumping allows one to take a new look at the new field of spintronics.One can in principle move information by means of a spin current at frequen-cies in the GHz range via a mechanism that does not directly involve a nettransport of electron charge. This potentially represents an approach to elec-tronics that is truly different from that employed using semiconductors.
The spin pump model is a rather exotic theory for those who are usedto magnetic studies. One would expect that there should be a more di-rect connection with common concepts used to understand the behavior
4 Exchange Coupling in Magnetic Multilayers 243
of magnetic multilayers. The obvious choice would be a generalization ofinterlayer exchange coupling. One would expect that a dynamic part of ex-change coupling could create magnetic damping. A ferromagnetic sheet sur-rounded by a normal metal can be investigated using a contact exchangeinteraction between the ferromagnetic spins and the electrons in the normalmetal [173, 174]. A similar model was used by Yafet [175] in order to calculatethe static interlayer coupling. One can extend Kubo’s linear response theory[176] in order to treat a slow precessional motion using a linear approximationfor the retarded magnetic moment,
S(t− τ) ∼= S(t) − τ∂S(t)∂t
, (4.46)
where S(t) is the spin moment of the magnetic sheet at the instantaneoustime t and τ is the time delay of the retarded response. The induced momentin the N metal at the F/N interface results in an effective damping field whichis proportional to the imaginary part of the rf transverse susceptibility of Nand the time derivative of the magnetic moment
Hsddamp ∼
[∂
∂ω
∫ ∞
−∞
dq
2πImχ(q, ω)
]
ω→0
dM(t)dt
. (4.47)
This damping term also satisfies the phenomenology of Gilbert damping. Byusing the same interaction potential it was shown [173, 174] that the Gilbertdamping in dynamic interlayer exchange coupling, Gs−d, is identical to thatcalculated using the spin-pumping theory [162] combined with a perfect spinsink. This leads to an important conclusion: The spin pumping theory is di-rectly related to the dynamic response of the interlayer exchange coupling.
Acknowledgements
The author would like to thank his colleagues Professor J.F. Cochran, Mr.B. Kardasz, and Mr. O. Mosendz for stimulating discussions and invaluablehelp in the preparation of this Chapter. I also owe a debt of gratitude to allthose with whom I have held extensive discussions and who provided valuablehelp during the preparation of this manuscript: Dr. M. Stiles, Dr. J. Unguris,Professor Z. Qiu, Dr. E. Fullerton, Professor H. Zabel, and Referees of thisChapter. Without this help and these discussions the Chapter would have beenless complete. I would like to thank Professor Dr. M. Stiles (Figs. 4.8, 4.9, 4.10and 4.21), Dr. J. Unguris (Figs. 4.18, 4.19, 4.20 and 4.23), Professor Z. Qiu(Figs. 4.15, 4.16 and 4.17), Dr. G. Woltersdorf (Fig. 4.1), and Prof. P. Bruno(Fig. 4.14) for allowing me to include their Figures in this chapter. I thankDr. Stiles for allowing me to include some of his presentation which beautifullydemonstrates theory of interlayer exchange coupling (section Quantum in-terference). The field of magnetic coupling is a very diversified field covering
244 B. Heinrich
at least 2–3 decades. It is not possible for a single Chapter to describe all thecomplexities of this field. I apologize to those who contributed to this fieldand whose results are not explicitly included in this chapter. The omissionis not intentional but merely reflects my inability to include all studies in aChapter that has to be constrained to a reasonable length.
I would like to thank especially the Natural Sciences and Engineering Re-search Council of Canada (NSERC), and the Canadian Institute for AdvancedResearch (CIAR) for continued research funding which makes my work pos-sible. I would also like to express my thanks to the Alexander von HumboldtFoundation and Professor J. Kirschner, Max Planck Institute in Halle, for pro-viding me with generous support during my recent summer research semestersin Germany where this manuscript was partly prepared.
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Proximity Effectsin Ferromagnet/Superconductor
Heterostructures
Konstantin B. Efetov1,2, Ilgiz A. Garifullin3, Anatoly F. Volkov1,4
2 L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia3 Zavoisky Physical-Technical Institute RAS, 420029 Kazan, Russiailgiz [email protected]
4 Institute for Radioengineering and Electronics RAS, 125009 Moscow, [email protected]
Abstract. We review the present status of the experimental and theoretical re-search on the proximity effect in heterostructures composed of superconducting (S)and ferromagnetic (F) thin films. First, we discuss traditional effects originating fromthe oscillatory behavior of the superconducting pair wave function in the F-layer.Then, we concentrate on recent theoretical predictions for S/F layer systems. Theseare a) generation of odd triplet superconductivity in the F-layer and b) ferromag-netism induced in the S-layer below the superconducting transition temperature Tc
(inverse proximity effect). The second part of the review is devoted to discussionof experiments relevant to the theoretical predictions. In particular, we present re-sults of measurements of the critical temperature Tc as a function of the thicknessof F-layers and we review experiments indicating the existence of the odd tripletsuperconductivity, cryptoferromagnetism and inverse proximity effect.
5.1 Introduction
If a superconducting layer S is brought into contact with a non supercon-ducting metallic layer N, the superconducting critical temperature Tc of Sdecreases with increasing the thickness of the N-layer and the supercon-ducting condensate penetrates into the N-layer over a long distance. Thisphenomenon, the conventional proximity effect, has been studied since the1960’s (see reviews [1, 2]).
K. B. Efetov et al.: Proximity Effects in Ferromagnet/Superconductor Heterostructures,
Although attractive electron-electron interactions may be absent in theN-layer, the condensate wave-function (or the Cooper pair wave function)f(t− t′) penetrates into N over a distance ξN , much exceeding the interatomicspacing. In a metal with a high impurity concentration (τT/� << 1, where τis the momentum relaxation time), ξN , the correlation length, is given by ξN =√D/2πT , whereD = vF l/3 is the diffusion coefficient, vF is the Fermi velocity
and l = vF τ is the mean free path of the conduction electrons. Magneticimpurities or a magnetic field significantly reduce the values of ξN .
An impressive manifestation of the induced superconductivity in a nor-mal metal is the Josephson effect in S/N/S junctions. If the thickness of theN-layer L is of the order of ξN , the critical current jc decays exponentiallywith L as jc ∼ exp(−L/ξN), which means that the characteristic length ofthe decay is ξN and not interatomic distances. Due to this effect the Joseph-son critical current can still be observed even if the thickness of the N-layerexceeds 1 μm.
Replacing the normal metallic layer N in an S/N proximity effect struc-ture by a metallic ferromagnetic layer F, one basically has the same effect:The pair wave function from S penetrates into F and makes the F-layer su-perconducting. However, there are important differences, rendering the S/Fproximity an interesting subject on its own.
The first important difference is that the penetration depth of the pairwave function into the F-layer is drastically reduced as compared to the N-layer. As will be explained in the theoretical sections below, in the diffusivelimit the penetration depth into the ferromagnet is given by the correlationlength ξF =
√DF /2h with the diffusion coefficient of the ferromagnetDF and
the exchange field in the ferromagnet h. For strong ferromagnets like Fe, Coand Ni, the length ξF has a typical value of 0.7 nm, i.e. the superconductingpairing function decreases in F exponentially on a nearly atomic length scale.The basic physical reason for this is that the exchange field in the F-layerh tends to align the spins of a Cooper pair and this leads to a strong pairbreaking effect.
However, a faster decay of the superconducting condensate in the fer-romagnet is not the only difference in comparison with the normal metals.Actually, there are other novel features of the S/F proximity effect that areless obvious and they are the main subject of the present review.
Since in the F-layer the spin-up and spin-down bands are split by the ex-change field h, the electrons of a Cooper pair at the Fermi energy have neces-sarily different k-vectors for the up and down spins and thus the Cooper pairsacquire a finite momentum Δk. As a consequence, the condensate function inthe ferromagnet oscillates in space. As described in the theoretical chapter be-low, this leads to oscillations of the superconducting transition temperature Tc
as a function of the Fe-layer thickness. For the same reason, in Josephson junc-tions with an S/F/S structure, where the insulating barrier of a conventionaltunnel junction is replaced by a ferromagnetic layer, the condensate functionmay change sign when crossing the F-layer, which leads to π-type coupling of
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 253
the two S-layers. This effect has been predicted long ago theoretically [3] butonly recently confirmed experimentally [4, 5].
Following the same line it has been demonstrated that, for a supercon-ducting film covered from both sides by ferromagnetic layers (F/S/F- trilayerstructure), the critical temperature Tc depends on the mutual orientation ofthe ferromagnetic layers [6, 7]. This is the superconducting spin valve effect,the latter term originating from the possibility to switch the resistivity be-tween zero and a finite value by changing the mutual magnetization directionof the two ferromagnetic films. First experiments demonstrating this effectwill be reviewed in the subsequent experimental section.
A fascinating new aspect of S/F structures discovered recently is a pos-sibility of generation of a new unconventional superconducting pairing state.The original Bardeen-Cooper-Schrieffer (BCS) theory [8] leads to a conven-tional s-wave pairing and for several decades this type of superconductivityhad been the only one observed experimentally. On the other hand, the super-conductivity in high-Tc cuprates discovered later shows a d-wave symmetryor a mixture of s- and d-wave components of the order parameter [9]. Boththe s-wave and d-wave types of the symmetries of the order parameter usuallyimply singlet pairing, which means that the total spin of the Cooper pair iszero. In this case, the order parameter Δαβ has the form Δαβ = Δ (k) ·(iσ2)αβ,where σ2 is the second Pauli matrix in the spin space and Δ (k) is a func-tion of the momentum k. As the spin part is antisymmetric with respect totransposition of the spin indices, the antisymmetricity of the order parameterfollowing from the Pauli principle is fulfilled provided the function Δ (k) iseven (Δ (k) = Δ (−k)).
Another type of pairing, spin-triplet superconductivity, has been discov-ered in materials with strong electronic correlations, namely, in heavy fermionintermetallic compounds as well as in organic materials (for a review see [10]).Recently, much work has been devoted to studying superconducting propertiesof Sr2RuO4 and convincing experimental data in favor of the triplet p-wave su-perconductivity have been obtained. We refer the reader to the review articles[11] and [12].
In contrast to singlet superconductivity, the spin part of the order param-eter for triplet superconductivity is symmetric with respect to exchanging thespin indices. Assuming that the order parameter (or the condensate function)does not depend on frequency (which is a standard assumption) one comes tothe conclusion that the order parameter must be antisymmetric with respectto the inversion of the momentum or, equivalently, to the transposition of thecoordinates. That is the type of superconductivity which has been observede.g. in heavy fermions.
Still, there is one more, non-trivial possibility for triplet pairing first pre-dicted in [42] that may be realized in S/F systems. It turns out that tripletpairing is also possible when the condensate is an even function of momen-tum and an odd function of the Matsubara frequency. As will be described inthe theoretical sections below, a corresponding component of the condensate
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function can be generated at the S/F interface by a spatially inhomogeneousmagnetization. First experimental evidence in favor of the existence of suchan odd triplet superconductivity has been reported recently [13, 14, 15]. Ahallmark of triplet superconductivity in S/F systems is its large penetrationdepth in the F-layer, which follows from the fact that the exchange field inthe ferromagnet does not break the triplet pairs. Moreover, in contrast tothe “conventional” triplet pairing, the odd triplet superconductivity is notsensitive to non-magnetic disorder and therefore is very robust.
Last but not least, in S/F layer systems the conventional proximity ef-fect is not the only interesting phenomenon caused by the mutual influenceof ferromagnetic and superconducting order. As will be explained below, theferromagnetic state of the F-layer can in turn be strongly modified by thepresence of the superconductor, an effect that is usually referred to as crypto-ferromagnetism [16, 17, 18].
Remarkably, not only supercoductivity can penetrate ferromagnets butalso the S-layers can become ferromagnetic [19]. For the latter phenomenonthe term inverse proximity effect was coined.
Thus, one can see that there is not only one proximity effect in S/F struc-tures but there are many of them. This makes these systems extremely in-teresting for both theorists and experimentalists. In the following sections weanalyze the proximity effects in S/F systems from both theoretical and ex-perimental points of view. Although several reviews on the proximity effectshave been recently published [20, 21, 22, 23, 24, 25], we emphasize novel de-velopments and open problems. At the same time, we focus mainly on ourown recent work.
5.2 Proximity Effect: Theory
5.2.1 S/F Structures: Uniform Magnetization of the Ferromagnet
In this section we review the main theoretical results on what happens if onereplaces the normal metal N in a N/S proximity structure by a ferromagneticmetal F. The effective ferromagnetic exchange field acts on spins of the con-duction electrons in the ferromagnet resulting in an additional term Hex forthis interaction in the total Hamiltonian Htot describing the proximity effect:
Htot = H + Hex, Hex = −∫d3rψ+
α (r)(h (r)σαβ
)ψβ (r) dr , (5.1)
where ψ+ (ψ) are creation and destruction operators, h is the exchange field,σαβ are Pauli matrices, and α, β are spin indices. The operator H standsfor a non-magnetic part of the Hamiltonian. It includes the kinetic energy,impurities, external potentials and is sufficient to describe all properties ofthe system in the absence of the exchange field h.
The energy of spin-up electrons differs from the energy of spin-downelectrons by the Zeeman energy 2h. All functions, including the condensate
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 255
Green’s function f , become matrices in the spin space with non-zero, diagonaland off-diagonal elements. In this subsection we consider the case of a homo-geneous magnetization M . In this situation the matrix f is diagonal and canbe represented in the form
f = f3σ3 + f0σ0 , (5.2)
where f3 is the amplitude of the singlet component and f0 is the amplitudeof the triplet component with zero projection of the magnetic moment of theCooper pairs on the z axis (S = 0). Note that for S/N structures the con-densate function has a singlet structure only, i.e. it is proportional to σ3. Thepresence of the exchange field h leads to the appearance of the triplet termproportional to σ0. In the general (non-homogeneous magnetization) case, thematrix f contains not only the matrices σ0,3 but also the matrices σ1,2.
The amplitudes of the singlet and triplet components are related to thecorrelation functions
One can see that a permutation of spins does not change the function f3(0),whereas such a permutation leads to a change of the sign of f0 (0). This meansthat the amplitude of the triplet component taken at equal times is zero inagreement with the Pauli exclusion principle. Later we will see that in thecase of a non-homogeneous magnetization all triplet components including〈ψ↑(t)ψ↑(0)〉 and 〈ψ↓(t)ψ↓(0)〉 differ from zero.
Let us begin with a discussion of properties of F/S systems with homo-geneous magnetization. The exchange interaction tends to align the spins ofthe free electrons in one direction, whereas the superconducting correlationsresult in formation of Cooper pairs consisting of electrons with opposite spins.Therefore, the superconducting transition temperature Tc of an S/F bilayersystem should be considerably reduced in S/F structures provided the inter-face transparency is high. In this case the electron can travel freely from theS to the F side and vice versa. However, it turns out that the dependence ofTc on the exchange field h and on the thickness of S- or F-layers is nontrivial:Tc may vary with increasing dF in a non-monotonic way.
The critical temperature for S/F bilayer and multilayer structures wascalculated in many works [22, 28, 29, 30, 31, 32]. In most theoretical papersit is assumed that the transition to the superconducting state is of a secondorder, i.e. the order parameter Δ varies continuously from zero to a finitevalue with decreasing temperature T . However, generally this is not the case.
If the phase transition is of the second order, one can linearize the cor-responding equations (the Eilenberger or Usadel equations) for the matrixGreen’s function f assuming that T is close to Tc. This is this case thatwas considered in most papers on this topic. The critical temperature of anS/F structure can be calculated using an equation obtained from the self-consistency condition. Close to Tc, the self-consistency condition can be lin-earized in Δ, and in the Matsubara representation it acquires the form (see,e.g. [24] and references therein)
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lnTc
T ∗c
= (πT ∗c )∑ω
(1
|ωn| − ifωΔ) , (5.4)
where Tc is the critical temperature in the absence of the proximity effect andT ∗
c is the critical temperature when the proximity effect is taken into account.The condensate function in (5.4) is the (11) element of the matrix Green’sfunction fω. In the diffusive case the matrix condensate function obeys thelinearized Usadel equation
DF∂2fF /∂x
2 − 2(|ω|σ0 + ihωσ3)fF = 0 , (5.5)
in the F-layer and the equation
DS∂2fS/∂x
2 − 2(|ω|σ0fS + iσ3Δ) = 0 , (5.6)
in the S-layer; where DF,S is the diffusion coefficient in the F- or S-layer,hω = hsgnω, h is the value of the exchange field. These equations should besupplemented by boundary conditions that near Tc have the form (see, forexample, [24, 25] and references therein)
γF,S∂fF,S/∂x = −(fS − fF ) , (5.7)
where γF,S = 2RbσF,S , Rb is the S/F interface resistance per unit area, σF,S
are the conductivities of the F- and S-films in the normal state. The Usadelequation is applicable to systems with a short mean free path l, which, inother words, means that the inverse momentum relaxation time τ−1 shouldbe larger than max{h, 2πT } in the ferromagnet and τ−1 should be larger thanTc in the superconductor. If these conditions are not met, one has to solve themore complicated Eilenberger equation.
At the first glance, (5.5) and (5.6) look simple and seem to allow a straight-forward solution. However this is not so, because the order parameter Δ de-pends on x : Δ = Δ(x). In order to solve these equations, a single-modeapproximation has been introduced in several papers [6, 22, 29, 30]. Usingthis approximation one comes to an interesting non-monotonic dependenceof Tc on dF . A more refined multi-mode method leads to a change of thisdependence that can be significant for some values of parameters [33].
If the interface transparency is low, (5.5) and (5.6) can be solved. Thelow transparency limit means that the condition |κhγF | << 1 is fulfilled,where κ2
h = 2(|ω| + ihω)/DF . In this case the condensate function in S isnot affected in the main approximation by the proximity effect and is equalto fS = −iσ3Δ/|ω|, where Δ is approximately constant in space. A solutionfor (5.5) can be found
f±(x) = ± fS
γF,Sκ±exp(−κ±x) , (5.8)
where κ± =√
2(|ω| ± ihω)/DF , fS = –iΔ/|ω| and f±(x) are the (11) and(22) elements of the matrix fF .
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 257
Usually the exchange energy h is much larger than the temperature T andthe expression for κ± shows that the condensate function f± (x) decays in astrong ferromagnet (h >> T ) on a rather short length ξF =
√DF /2h and
experiences oscillations with the same period (to be more precise, the periodof oscillations is 2πξF ). This oscillatory behavior of the condensate functionf(x) in F leads to a non-monotonic dependence of Tc on dF and to oscillationsof the critical current in S/F/S Josephson junctions (see below).
We would like to emphasize here an important point. The characteristiclength of the oscillations and decay of the condensate function is equal toξF =
√DF /2h only in the diffusive limit (hτ << 1). In the opposite limit
(hτ >> 1) the situation is different: the period of the condensate oscillationsin the ferromagnet is 2πvF /h, whereas the decay length is of the order of themean free path l [34, 35].
One can see that the singlet component f3 = (f+ − f−)/2 is an evenfunction of ω and the triplet component f0 = (f+ +f−)/2 with zero total spin(S = 0) is an odd function of ω. However, both these components, singletand triplet with S = 0, coexists in the ferromagnet over a short length of theorder of ξF . In the next Section we will see that in case of a non-monotonicmagnetization a triplet component with S = ±1 arises and penetrates theferromagnet over much larger distance of order ξN .
Of course, in the more complicated F/S/F structure the critical tempera-ture is also suppressed. However, this suppression depends also on the mutualorientation of the magnetization in the left and right side ferromagnets. Thisproperty has led to the idea to switch the system between the superconductingand normal states by varying the magnetization orientation, [6, 7].
Qualitatively, this effect can be understood as follows. Consider an F/S/Fstructure with thin F- and S-layers. Assuming that the S/F interfaces arehighly transparent, one can average the Usadel equation over the thickness. Inthis case the condensate function is continuous across the S/F interfaces, andafter averaging one can obtain an equation with an effective order parameterΔeff = Δds/d and an effective exchange field [36] heff = (hl + hr)dF /d,where d = 2dF +dS is the total thickness of the trilayer and hl and hr are theexchange fields from the left and from the right, respectively. Thus, the F/S/Fstructure is similar to a magnetic superconductor with an effective exchangefield heff . It is known that the critical temperature of this superconductordecreases with increasing heff and may be even a multivalued function of heff
[37]. If the magnetizations of the left and right side ferromagnet are oppositeto each other, we obtain heff = 0 and therefore Tc is larger than in the caseof parallel orientations of the magnetization when heff �= 0. In order to findTc in the general case, one has to solve (5.5) and (5.6) with the boundaryconditions (5.7). These calculations have been performed in [6, 7].
The oscillations of the condensate function in the ferromagnet, (5.8), leadto interesting peculiarities not only in the dependence Tc(dF ) but also for theJosephson effect in the S/F/S junctions. It turns out that under certain con-ditions the Josephson critical current Ic changes sign and becomes negative.
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In this case the energy of the Josephson coupling EJ = (�Ic/e)[1− cosϕ] hasa minimum in the ground state when the phase difference ϕ is equal not to 0,as in conventional Josephson junctions, but to π (the so called π−junction).
This effect was predicted for the first time by Bulaevskii et al. [3]. Theauthors considered a Josephson junction consisting of two superconductorsseparated by a region containing magnetic impurities. Later on, the Josephsoncurrent was calculated for a S/F/S junction [38]. Recently, this interestingtheoretical prediction has been confirmed experimentally [4, 5, 39, 40].
5.2.2 Exotic Superconductivity in S/F Structures
As mentioned in the Introduction, there is strong experimental evidence forthe realization of non-BCS type of superconducting states in several highlycorrelated electron systems like the high-Tc cuprates, heavy fermions and Sr-Ruthenates. In this section, we will demonstrate that under certain conditionstriplet pairing is expected also in S/F systems for an arbitrary superconduc-tor S. In other words, the triplet component can be artificially generated bythe exchange field. Before presenting explicit calculations, let us summarizecertain general features of superconducting condensates using symmetry ar-guments.
Due to anticommutation of the fermionic creation ψ+ and annihilationψ operators, the condensate function < ψα(r, t)ψβ(r′, t′) > must be at equaltimes, t = t′, an odd function with respect to permutations α � β, r � r′. Thetriplet pairing means that the spins of the Cooper pairs are parallel to eachother and the transposition of the spin indices does not change the condensatefunction. Provided this function remains finite at t = t′ it must change thesign under transposition of the coordinates r and r′. So, the triplet Cooperpair has to be an odd function of the orbital momentum or, in other words,the orbital angular momentum L is an odd number.
The dependence of the condensate function on the direction in the spacemakes such a superconductivity very sensitive to disorder. The p-wave conden-sate (as well as d-wave pairing, etc.) is strongly suppressed by non-magneticimpurities. Of course, then order parameter Δαβ =
∑k Δαβ(kF) ∼ ∑
k <ψα(r, t)ψβ(r′, t) >k is also suppressed. The s-wave (L = 0) singlet conden-sate is an exception, because it is a scalar and therefore is not destroyed bynon-magnetic impurities (Anderson theorem).
At first glance, any non-singlet pairing should be suppressed by impurities,which makes an experimental observation very difficult. However, one morenon-trivial possibility for triplet pairing exists. The previous conclusion aboutthe antisymmetricity of the orbital part of the condensate function remainsfinite at equal times, which excluds functions antisymmetric in t− t′.
At the same time, nothing forbids the function < ψα(r, t)ψβ(r′, t) > tochange sign under the transposition t � t′. In the frequency representa-tion, this property is realized if the correlator < ψα(r, τ)ψβ(r′, τ ′) >k,ω is
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 259
an odd function of the Matsubara frequency ω. However, if the conden-sate function is odd in frequency, it may be even in the momentum andwe have the triplet pairing again. In this case, the correlation function< ψα(r, τ)ψβ(r′, τ ′) >k,ωequals zero at coinciding times (the sum over allfrequencies is zero) and therefore the Pauli principle for the equal-time corre-lators is not violated.
This type of pairing was suggested by Berezinskii [41] as a possible mech-anism of superfluidity of 3He. He assumed that the order parameter Δ(ω) isan odd function of ω : Δ(ω) = −Δ(−ω). However experiments on superfluid3He have shown that the Berezinskii’s state was not realized in this system.Now it is well known that the condensate in 3He is antisymmetric in the mo-mentum space and symmetric (triplet) in spin space. Thus, the Berezinskiihypothetical pairing mechanism remained unrealized for a few decades.
Recent theoretical studies have shown that a superconducting state similarto the one suggested by Berezinskii might be induced in S/F systems due tothe proximity effect [42, 43]. In the next sections we will analyze this new typeof superconductivity with triplet pairing that is odd in frequency and, in thediffusive limit, even in momentum. This can be s-wave pairing and thereforethis type of superconductivity is not sensitive to impurities.
We note, however, that there is a qualitative difference between this newsuperconducting state in S/F structures and the one proposed by Berezinskii.In S/F structures both singlet and triplet types of the condensate coexistand the order parameter Δ existing only in the S region (we assume that thesuperconducting coupling in the F region is zero) is determined solely by thesinglet part of the condensate.
Note that, while theories of unconventional superconductivity often implystrongly correlated systems, the triplet state induced in S/F structures can bederived within the framework of the BCS theory valid in the weak-couplinglimit. This fact not only drastically simplifies theoretical considerations butalso helps in designing experiments, since well known elemental superconduc-tors prepared under controlled growth procedures may be used in order todetect the triplet superconductivity.
To finish this subsection, let us summarize the properties of this new typeof superconductivity that we call odd triplet superconductivity:
• It contains a triplet component. In particular the components with pro-jection S = ±1 are insensitive to the presence of an exchange field andtherefore long-range proximity effects arise in S/F structures.
• In the dirty limit it has an s-wave symmetry. The condensate functionis even in p and therefore, contrary to unconventional superconductorswith triplet pairing, is not destroyed by the presence of non-magneticimpurities.
• The triplet condensate function is odd in frequency.
Before we turn to a more detailed theoretical analysis of triplet supercon-ductivity, we remark that in the F-regions of the S/F structures no attractive
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electron-electron interaction exists, and thereforeΔ = 0 in F. This means thatonly the superconducting condensate function f in the ferromagnet exists and,as it will become clear later, it arises only due to the proximity effect.
5.2.3 Triplet Odd Superconductivity Inducedby an Inhomogeneous Magnetization in S/F Structures
As discussed in Sect. 5.2.1, the presence of an exchange field results in the for-mation of the triplet component of the condensate function. In a homogeneousexchange field, only the component with the projection S = 0 is induced. Thenthe natural question arises: Can the other components with S = ±1 also beinduced? If they could, this would lead to a long range penetration of thesuperconducting correlations into the ferromagnet because these componentscorrespond to the correlations of the type < ψ↑ψ↑ > with parallel spins andthey are not as sensitive to the exchange field as the other ones.
In what follows, we analyze a few examples of S/F structures in which allthe projections of the triplet component are induced. The common feature ofthese structures is that the magnetization should be non-homogeneous.
F/S/F Trilayer Structure
We start with considering the F/S/F system shown schematically in Fig. 5.1.The structure consists of one S-layer and two F-layers with the magnetizationinclined at the angle ±α with respect to the z-axis (in the yz plane). Aswe have seen in the previous section, each of the layers generates the tripletcomponent with the zero total projection of the spin, S = 0, in the directionof the exchange field. If the magnetic moments of the layers are collinear(parallel or antiparallel), the total projection remains zero. However, if themoments of the ferromagnetic layers are not collinear, the superposition ofthe triplet components coming from the different layers should have all thepossible projections of the total spin.
dSdS dS dF+dS dF+( )−
FF S
x
−α α
−
Fig. 5.1. Trilayer geometry. The magnetization of the left (right) side F-layer makesan angle α (−α) with the z-axis
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 261
From this qualitative argument we can really expect the non-trivial effectof the generation of the triplet components with all the projections of the totalspin, provided the thickness of the S-layer is not too large. The point is thatthe triplet component decays in S on a length of the order of the coherencelength ξS ≈ √DS/πTc, (5.6). We assume that the thickness of the S-layerdoes not much exceed this length.
In order to find all types of the condensate (singlet and triplet), one hasto solve the linearized Usadel equation in the F-region (we assume a weakproximity effect) [43] for the condensate function f that is a 4 × 4 matrix inthe particle-hole and spin spaces
(∂2f /∂x2) − κ2ω f + iκ2
h{ ˆτ0⊗[σ3, f ]+ cos α ± ˆτ3⊗[σ2, f ] sin α} = 0 , (5.9)
where [σ3, f ]+ = σ3 ⊗ f + f ⊗ σ3.The wave vectors κω and κh entering (5.9) have the form
κ2ω = 2|ω|/DF , κ2
h = 2hsgn(ω)/DF . (5.10)
The magnetization vector M lies in the (y, z)-plane and has the compo-nents: M = M{0,± sinα, cos α}. The sign “+” (“-”) corresponds to the right(left) side F-film. We consider here the simplest case of a highly transparentS/F interface and temperatures close to Tc. In this case the function f , beingsmall, obeys a linear equation in S similar to (5.6).
The boundary conditions at the S/F interfaces are obtained by a general-ization of (5.7) (see [43]).
A solution for (5.9) can be found. The matrix f can be represented as
f = iτ2 ⊗ f2 + iτ1 ⊗ f1 , (5.11)
where f1 = b1(x)σ1, f2 = b3(x)σ3 + b0(x)σ0.For the left side F-layer the functions bk(x) are to be replaced by bk(x).
For simplicity we assume that the thickness of the F-films dF exceeds ξF (thecase of an arbitrary dF was analyzed in [43]). Using the representation, (5.11),we find the functions bi(x) and bi(x). They are decaying exponential functionsand can be written as
bk(x) = bk exp(−κ(x− dS)), bk(x) = bk exp(κ(x+ dS)) . (5.12)
Substituting (5.12) into (5.9), we obtain a set of linear equations for the coef-ficients bk that should be complemented by expressions for the eigenvalues κ.
In the limit of large exchange energy h ({T,Δ} << h, but h << τ−1), theeigenvalues κ are equal to
κ = κω, κ± ≈ (1 ± i)κh . (5.13)
We see from (5.13) that the solutions κω and κ± are completely different.The roots κ± proportional to κh are very large and therefore the corresponding
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solutions bk (x) decay very fast (similar to the singlet component). This is thesolution that exists for a homogeneous magnetization (collinear magnetizationvectors).
In contrast to the roots κ±, the value for κω given by (5.13) does notdepend on the exchange energy h and is much smaller. It is this eigenvaluethat leads to a slow decay of the superconducting correlations. The solutionscorresponding to the root κω describe the long-range penetration of the tripletcomponent into the ferromagnetic region. The function b1(x) is the amplitudeof the triplet component penetrating the F-region over a long distance of theorder of κ−1
ω ∼ ξN . Its value, as well as the values of the other functions bk(x),is to be found from the boundary conditions at the S/F interfaces.
Matching the solutions in S and F at the S/F interfaces, we obtain thecoefficients bk and bk. Note that b3± = b3± and bω = −bω. Although thesolution can be found for arbitrary parameters entering the equations, wepresent here the expressions for b3± and bω in some limiting cases only. Forexample, if the parameter γκh/κS is small, the amplitudes of the long-rangetriplet component bω and singlet components b3± can be written in a rathersimple form
bω ≈ −2ΔEω
(γκh
κS)sin α cos2 αsinh(2ΘS)
, b3+ ≈ b3− ≈ Δ
2iEω, (5.14)
where Θ = κSdS and Eω =√ω2 +Δ2, γ = σF /σS and σF (σS) is the conduc-
tivity in the ferromagnet (superconductor).At the S/F interface the amplitude of the triplet component bω is small
compared to the magnitude of the singlet one b3+. However, the triplet compo-nent decays over a long distance ξN , while the singlet one vanishes at distancesexceeding the short length ξF . The amplitudes bω and b3± become comparableif the parameter γκh/κS is of the order of unity.
It follows also from (5.14) that the amplitude of the triplet component bω iszero in the case of collinear vectors of magnetization, what means at α = 0 orα = π/2. It reaches the maximum at the angle αm for which sin αm = 1/
√3.
Therefore the maximum angle-dependent factor in (5.14) is sin αm cos2 αm =2/3
√3 ≈ 0.385.
One can see from (5.14) that bω becomes exponentially small if thethickness dS of the S-films significantly exceeds the coherence length ξS ≈√DS/πTc. This means that in order to have a considerable penetration of
the superconducting condensate into the ferromagnet, one should not makethe superconducting layer too thick. On the other hand, if the thickness dS
is too small, Tc is suppressed. In order to avoid this suppression, one has touse, for instance, a F/S/F structure with a small thickness of the F-films. InFig. 5.2 we plot the spatial dependence of the singlet and triplet componentsin F/S/F structure.
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 263
0
1
-dS-(dS+dF) dS+dF dS
Fig. 5.2. The spatial dependence of Im(b3(x)) (dashed line) and the long-rangepart of Re(b(x)) (solid line). We have chosen σF /σS = 0.2, h/Tc = 50, σF Rb/ξF =0.05, dF
√Tc/DS = 2, dS
√Tc/DS = 0.4 and α = π/4. The discontinuity of the
triplet component at the S/F interface is because the short-range part is not shownin this figure. Taken from [43]
Domain Wall at the S/F Interface and Helical Ferromagnets
Now we consider another example of an S/F structure in which the long-rangetriplet component (LRTC) also arises. This structure is shown schematically inFig. 5.3. It consists of an S/F bilayer with a non-homogeneous magnetizationin the F-layer. We assume for simplicity that the magnetization vector M =M(0, sin α, cos α) rotates in the F-film starting from the S/F interface (x = 0),and the rotation angle has a simple, piece-wise x-dependence: α(x) = Qx, inthe domain wall for 0 < x < w and α(x) = Qw for w < x. This means that theM vector is directed parallel to the z-axis at the S/F interface and rotates bythe angle α(w) over the length w (w may be the width of a domain wall). Atx > w the orientation of the vector M is fixed. This structure was considered
0 w L
αw
α
S
S
FF
FF
S
Fig. 5.3. S/F structure with a domain wall in the region 0 < x < w. In this regionα = Qx, where Q is the wave vector which describes the spiral structure of thedomain wall. For x > w it is assumed that the magnetization is homogeneous, i.e.,α = Qw
264 K. B. Efetov et al.
first in [42] and later in [44]. The Usadel equation for this case has been solvedin [42]. The solution is found in the region of the domain wall 0 < x < w andin the region of a constant magnetization: w < x <∞.
For this configuration of the magnetic moment the LRTC arises in thedomain wall and spreads into the ferromagnet over a long distance. The char-acteristic decay length of the LRTC inside the domain wall is
ξQ = (Q2 + κ2ω)−1/2 , (5.15)
whereas its value equals κ−1ω outside the domain wall. The singlet component
penetrates the ferromagnet over a short length of the order ξF . Although theamplitude of the LRTC at the S/F interface may be comparable with theamplitude of the singlet component, the decay length of the LRTC is muchlarger (see Fig. 5.4). One more system where the LRTC arises is a helicalferromagnet [45] (see Fig. 5.15). Such a structure is realized, for example, inseveral heavy rare earth metals. In this ferromagnet the magnetization vectorrotates around the z-axis and has a non-zero projection Hz on this axis.
It was shown that in this case the LRTC penetrates the ferromagnet overa length of the order of ξQ, (5.15). What is interesting, the monotonic decayof the LRTC in this case occurs only if the the cone angle θ is less thansin−1(1/3) ≈ 19◦. At larger θ the decay of the LRTC is accompanied byoscillations. In the quasi-ballistic case (hτ > 1), the characteristic length ofthe LRTC penetration into the ferromagnet changes.
In the case of Neel-type domain walls the LRTC vanishes provided themagnetization vector M rotates continuously [46]. However, in an S/F struc-ture with several Neel domain walls (the vector M rotates only inside thedomain walls) the LRTC arises at the domain walls and decays in the do-mains over a large distance [47].
0.4
0.3αw = π
αw = π/5
0 x=w x=L .
Fig. 5.4. Spatial dependence of amplitudes of the singlet (dashed line) and triplet(solid line) components of the condensate function in the F wire for different valuesof αw. Here w = L/5, ε = ET , and h/ET = 400. ET = DF /L2 is the Thoulessenergy, ε = iω is the energy (From [42])
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 265
5.2.4 Other Proximity Effects in S/F Structures
Up to now we implicitly assumed that the proximity effect in S/F structureschanges the superconducting properties but leaves the magnetization of theF-layer unchanged. However, this is not always true and experiments per-formed by [48] and [49] indicate that the ferromagnetic magnetization of S/Fbilayers may decrease when lowering the temperature below Tc. At that timeit was not quite clear what physical mechanism causes this decrease of themagnetization. Here we review two different and independent mechanismsthat may explain the effect.
5.2.5 Cryproferromagnetism
In a classic paper Anderson and Suhl [16] proposed an idea that at some cir-cumstances superconductivity might coexist with a non-homogeneous mag-netic ordering. They called this magnetic non-homogeneously ordered statecryptoferromagnetic. The basic reasoning leading to this suggestion was thatsuperconductivity could survive in a ferromagnetic background, if the magne-tization direction varied on a scale smaller than the superconducting coherencelength. The cryptoferromagnetic state in S/F structures was considered firstin [17] in the case of a weak ferromagnet.
In a more recent theoretical paper on cryptoferromagnetism in S/F bilay-ers [18] a more realistic case of a strong ferromagnet was considered. It wasshown that even if the exchange field is large; the cryptoferromagnetic stateis still possible provided the ferromagnetic film is sufficiently thin. A phasediagram containing the cryptoferromagnetic state has been drawn depend-ing on the stiffness of the ferromagnet J , the thickness of the F-film dF andthe exchange field h of the system. This phase diagram (a, λ) for the S/Fsystem is represented in Fig. 5.5, where a = 2h2d2
F /(DFTcη2), η = vF /vS ,
λ = (J dF /NF
√2TcD3
F )(7ς(3)/2π2), and NF is the density-of-states (DOS)in the ferromagnet. Estimates of the parameters (J , h and dF ) for the samplesused in the experiments [49] in which a reduction of the effective magnetizationwas observed show that the results of [18] agree with the experimental data.
The calculations show that the proximity effect may lead to a magneticspiral structure in the F-film even if the exchange energy h is much largerthan the characteristic energy of Tc. This cryptoferromagnetic ordering isrelated to the existence of low lying states in the ferromagnet. The spiralstructure increases the magnetic energy only by a small amount, whereas theenergy of interaction between the exchange field and the superconductivitycan essentially be reduced.
At the same time, there exists another mechanism that can reduce the totalmagnetization in S/F structures and it is also due to the proximity effect. Thisis the inverse proximity effect describing the situation when the orientation ofthe magnetization remains unchanged, while its magnitude changes both inthe S- and F-layers.
266 K. B. Efetov et al.
0 0.5 1 1.5 2 2.5 30
0.005
0.01
0.015
λ
a
|τ|=0.2
|τ|=0.4
|τ|=0.6
Fig. 5.5. Phase diagrams (λ, α) for different values of |τ | = (Tc − T )/Tc. The areaabove (below) the curves corresponds to the F (CF) state
5.2.6 Inverse Proximity Effect
The inverse proximity effect is due to a contribution of free electrons both inthe ferromagnet (δMF ) and in the superconductor (MS) to the total magne-tization. On one hand, the DOS in the F-film is reduced due to the proximityeffect, thus decreasing of the magnetization in F by δMF . On the other hand,the Cooper pairs in S are polarized in the direction opposite to MF , givingrise to a magnetization (MS) with a direction opposite to MF . So, the S-layerbecomes ferromagnetic and this is the reason for calling this effect the inverseproximity effect. For a more detailed qualitative explanation of this mecha-nism we consider the S/F structure with a thin F-layer in Fig. 5.6. We assumethat the exchange field of F is homogeneous and directed along the z-axis.
d− S
S F
x0 Fd
Fig. 5.6. S/F structure and schematic representation of the inverse proximity effect.The dashed curves show the local magnetization
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 267
If temperature exceeds Tc, the total magnetization of the system Mtot
equals M0dF , where dF is the thickness of the F-layer. When the temperatureis lowered below Tc, the S-layer becomes superconducting and Cooper pairswith a size of the order of ξS appear in the superconductor. Due to the prox-imity effect the Cooper pairs cross the interface and penetrate into the ferro-magnet. In the case of a homogeneous magnetization in F the Cooper pairsare composed of electrons with opposite spins, such that the total magneticmoment of the pair equals zero. The exchange field is assumed to be not toostrong, otherwise the Cooper pairs would be destroyed.
It is clear from this simple picture that pairs located entirely in the super-conductor cannot contribute to the magnetic moment of the superconductor.However, some pairs are distributed in space in a more complicated manner:one of the electrons of the Cooper pair stays in the superconductor, whereasthe other one enters the ferromagnet. These are these pairs that create themagnetic moment in the superconductor.
Energetically it is favorable for the electron of the Cooper pair with thespin parallel to the magnetization of the ferromagnet to have a higher prob-ability density in F. This means that the electron with opposite spin has ahigher probability density in S. This is the reason why these pairs form amagnetic moment in the S-layer. As a result, the ferromagnetic order is cre-ated in the S-layer with a direction of the magnetic moment opposite to thedirection of M in F-layer. The induced magnetic moment penetrates the su-perconductor over the size of the Cooper pairs, which may be much largerthan dF .
Using similar arguments we can predict a related effect: the magnetic mo-ment in the ferromagnet should be reduced in the presence of superconductiv-ity because some of the electrons located entirely in the ferromagnet condenseinto Cooper pairs and do not contribute to the magnetization.
From this qualitative, simplified picture one can expect that the totalmagnetization of an S/F system will be reduced for temperatures below Tc.A quantitative analysis based on the Usadel equation (diffusive case) [19] oron the Eilenberger equation (quasiballistic case) [50] supports the qualitativepicture. It turned out that at low temperatures the magnetic moment MF
in F is screened completely by the spin-polarized Cooper pairs in S if MF
is due to free electrons (ideal itinerant ferromagnet) i.e. MS = −MF . Thisconclusion is valid in the limit h < DF /d
2F .
With increasing the exchange energy h the induced magnetic momentdecreases monotonically in the diffusive limit [19] or non-monotonically in theclean limit [50].
It should be stressed that both the mechanism discussed here and thatof the last section lead to a decrease of the total magnetization. The spinpolarization of Cooper pairs in the superconductor in F/S/F structure witha non-collinear magnetization in F was studied in [51].
268 K. B. Efetov et al.
5.3 S/F Proximity Effect: Experiments
5.3.1 Superconducting Transition Temperature in F/S Systems
Following the theoretical predictions (see Sect. 5.2.2), a ferromagnetic filmdeposited on a superconducting film should drastically suppress the super-conducting Tc. In experimental systems, however, this is often not the case,the Tc- suppression appears rather moderate. This is due to two differentreasons. First, in real thin film systems there is often an intermediate alloylayer caused by interdiffusion that is weakly magnetic or even non-magnetic.This is the case, e.g., in Fe/Nb [52, 53] and, probably, in Gd/Nb [54]. Thisinterlayer prevents the direct contact between the F- and S-layer and weak-ens the suppression of Tc. Second, the quantum mechanical transparency of areal S/F interface is often quite small, i.e. the coefficient γF,S in (5.8) is smalland the Tc-suppression is much weaker than that with an ideally transparentinterface.
An interesting feature of the S/F proximity effect that has recently beenunder intensive discussion in the literature, is oscillation of the superconduct-ing transition temperature as a function of the F-layer thickness dF .
There are quite different physical mechanisms that may cause Tc(dF ) os-cillations or a non monotonic Tc(dF ) behavior. An indirect mechanism, notdirectly related to the proximity effect, has been observed in Fe/Nb bilay-ers [52, 53] (Fig. 5.7c). Here an alloying at the interface leads to a non-ferromagnetic NbFe interlayer of about 0.7 nm thickness and therefore theminimum in Tc(dF ) just correlates with the onset of ferromagnetism. Theexplanation of this phenomenon is that strong longitudinal spin fluctuationsexist in the NbFe interlayer with a concentration close to the onset of fer-romagnetic long range order. They are responsible for the strong initial Tc-suppression when increasing the Fe-thickness from 0 to 0.7 nm in Fig. 5.7c.When the first ferromagnetic Fe layer appears above dFe =0.7 nm, the spinfluctuations in the NbFe interlayer are suppressed by the exchange field of theFe-layer and result in an increase of Tc. This is a rather indirect influence ofthe ferromagnetic Fe-layer on the superconductivity of the Nb layer.
Now, coming to oscillations in Tc(dF ) induced by the S/F proximity effect,we recall that, as discussed in Sect. 5.2.2, oscillations of the condensate func-tion fs in space may directly lead to a non-monotonic Tc -dependence on h ordF . Actually the reason for this non monotonic behavior may be different forS/F bilayers and S/F/S trilayers, since the boundary conditions in these twocases are different. For bilayers, only one side of the F-layer is in contact withthe superconductor, whereas in the S/F/S trilayers the F-layer is in contactwith the superconducting layers on the both sides.
For the case of trilayers (but not for bilayers) oscillations of Tc may be dueto the appearance (or disappearance) of Josephson π-coupling. As mentionedin Sect. 5.2.3, due to the oscillation of the superconducting pairing function inthe F-layer, the phase difference in the superconducting pairing function on
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 269
(c)
dNb= 60 nm
(a)
dNb= 50 nm
(b)
06.0
6.5
7.0
6.5
7.0
7.5
8.0
41 2 3dGd (nm)
dNb= 40 nm
0 2.50.5 1 1.5 2dFe (nm)
Tc
(K)
7
3
4
5
6
Tc
(K)
Fig. 5.7. Dependence of superconducting transition temperature on the ferromag-netic layer thickness in (a and b) two series of Nb/Gd multilayers (Jiang et al. [54]),and (c) Fe/Nb/Fe trilayers (Muhge et al. [52, 53])
both sides of the F-layer may have opposite phases at certain F-layer thick-nesses. This means that the phase difference between the neighboring S-layersmay be equal to π. Radovic et al. [28] concluded from their calculations that Tc
for π-coupling most probably is higher than for the vanishing phase difference.Jiang et al. [54] claimed that the observed oscillations of Tc(dGd) (Fig. 5.7a
and 5.7b) are due to this type of the Josephson π-coupling. Several other workson S/F multilayers have reported a single peak in Tc(dF ), and have attributedthis feature to “π-switching” (see, e.g., [55, 56, 57]).
Whereas the role of π-coupling for the non monotonous Tc(dF ) in S/Fmultilayers has not been finally clarified and an alternative explanation exists(see below), a clear experimental evidence for π-coupling across an F-layer
270 K. B. Efetov et al.
comes from the study of Josephson junctions using F-layers as barriers [4, 39].Tunnelling spectroscopy revealed damped oscillations of the superconduct-ing order parameter induced in the F-film by the proximity effect [39].Ryazanov et al. [4] performed measurements of the critical current in Joseph-son junctions consisting of superconducting Nb and weakly ferromagnetic in-terlayers and found that the character of the junction changed from 0-phaseat high temperatures to π-phase at low temperatures. This result was laterconfirmed by Blum et al. [5]. A different phase sensitive experiment [58] alsogave evidence for the oscillatory behavior of the critical supercurrent of S/Flayered system when varying the F-layer thickness.
Without invoking π-coupling, oscillations of Tc(dF ) can simply originatefrom oscillations of the condensate amplitude in space within the F-layer.As shown theoretically by [28, 30, 59], due to these oscillations and takingthe boundary conditions for the pairing wave function at the S/F interfacesinto account, the Tc(dF )-curve may have an oscillatory character with theoscillation period of the order ξh = vF /h (see Sect. 5.2.2).
The physical origin of the oscillatory character of Tc(dF ) can qualitativelybe traced back to the propagating character of the superconducting pairingwave function in the ferromagnet. If the thickness of the F-layer is smaller thanthe penetration depth of the pairing wave function, this function, when trans-mitted through the S/F interface into the F-layer, will interfere with the wavereflected from the opposite surface of the ferromagnet. As a result, the flux ofthe pairing wave function crossing the S/F interface varies with the thicknessof the F-layer dF .
Then, the coupling between the electrons of the ferromagnet and the super-conductor will be modulated and Tc will oscillate with dF . If the interferenceat the S/F interface is essentially constructive (this corresponds to a minimaljump of the pairing function amplitude at the S/F interface), the coupling isweak, and one expects Tc to be maximal. When the interference is destructive,the coupling is maximized and Tc(dF ) is minimal.
It should be noted that this model explaining Tc(dF ) oscillations appliesto the case of the S/F bilayers as well as to F/S/F trilayers or S/F multilayers,whereas the π-coupling concept does not apply for bilayers.
Aarts et al. [60] studied V/V1−xFex multilayers without interdiffusionat the interface. They showed that Tc strongly depends on the interfacetransparency and presented experimental evidence for an intrinsically reducedinterface transparency. From the dependence of Tc on the magnetic layer thick-ness they calculated the penetration depth of Cooper pairs into the F-layerand found it to be inversely proportional to the effective magnetic moment perFe atom. For the interpretation of the observed peculiarities they introduceda finite transparency of the S/F interface and argued, based on their experi-mental data, that with an increasing the exchange splitting of the conductionband in the F-layer the transparency of the S/F interface for Cooper pairsdecreases.
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 271
Lazar et al. [61] experimentally studied the role of the interface trans-parency in the Fe/Pb/Fe system and, for comparison, in the Fe/V/Fe system,too [62]. In contrast to the case of Fe/Nb/Fe discussed above, in Fe/Pb/Fe andFe/V/Fe the intermixing at the interfaces is much weaker. Figure 5.8 showsthe dependence of Tc on the thicknesses of the Fe layers for Fe/Pb/Fe tri-layers. A theoretical analysis of the curves using model calculations revealedthat the experimental results can only be described assuming Pb/Fe inter-faces that are not perfectly transparent. The Tc for the case of S/F interfaceswith a non-perfect transparency has been calculated by Golubov [60] andTagirov [30].
A fit to the experimental points using the model calculations [30] is plottedon Fig. 5.8 as a solid line. The quality of the fit is satisfactory and reproducesthe details of the Tc(dFe)-curve. The most important parameter obtained fromthis fit is the value of Tm, characterizing the transparency of the interface.The fit gives Tm =0.4. This value corresponds to a quantum mechanical trans-mission coefficient T = Tm/(1 + Tm) = 0.3 [61] that is considerably reducedas compared to the ideally transparent interface with T = 1.
Lazar et al. [61] concluded that the exchange splitting of the conductionband of the F-layer is the main physical reason for the strongly reduced inter-face transparency. In principle, the calculation of the interface transparencyis a standard quantum mechanical problem of reflection and transmission ofelectrons at the interface of two metals with different Fermi energies. It is ob-vious that two electrons with opposite spins forming a Cooper pair can nevermatch the Fermi momenta of the exchange-split subbands of a ferromagnetsimultaneously and there will always be a Fermi vector mismatch reducing
0 1 2 3 4
1
2
3
4
5
6
7
dPb
=73 nm
Tc
(K)
dFe
(nm)
Fig. 5.8. Tc dependence on the Fe thickness at fixed value dPb = 73 nm for Fe/Pb/Fetrilayers. The dashed line is obtained by the Radovic et al. [28] theory which supposesan ideally transparent interface. The solid line takes a finite transparency of theinterface into account [30]
272 K. B. Efetov et al.
the transmission. Additionally, a decrease of T is expected due to a chemicalmismatch of Pb and Fe giving rise to a contact potential barrier at the inter-face. The barrier height should be larger for immiscible metals like Pb and Fein comparison to metals that form solid solutions in the whole concentrationrange as, e.g., V and Fe.
Measurements of dependence of Tc on the thickness were performed alsofor Fe/V/Fe trilayers [62]. For two series of samples at small iron thicknessesthe Tc drops sharply when increasing dFe up to 0.5 nm. Then, at dFe ∼0.7 nmfor the series with dV = 31 and 29 nm, a clear minimum of Tc is observed. Thedeepness of this minimum increases with decreasing dV . For these two seriesthe residual resistivity ratio RRR� 4, meaning that the mean free path of theconduction electrons in the S-layer is lS ∼ 4 nm [63]. The parameters resultingfrom a theoretical fit of these curves are the superconducting coherence lengthξS =
√ξ0lS/3.4 = 4 nm (here ξ0 = 44 nm is the BCS coherence length),
as estimated from the resistivity data. For the transparency parameter oneobtains Tm = 1.6 and for the exchange length in the Fe film ξh = 0.7 nm.
Using the theoretical model calculations [30] as a guideline, one can ex-tract the important physical parameters necessary for an observation of thetheoretically predicted rather spectacular re-entrant behavior of the super-conductivity, i.e. superconductivity vanishing for a certain range of dF andcoming back for larger dF . The system should possess a large electron meanfree path in the F- as well as in the S-layer, a high quantum-mechanicaltransparency of the S/F interface and a geometrically flat interface withoutintroducing too much diffuse scattering of the electrons. The last two condi-tions are well fulfilled in Fe/V/Fe trilayers, so one could try to further increasethe electron mean free path lF or lS . Whereas this is hardly possible for theF-layers since lF is limited by the very small layer thickness dF , improvedgrowth conditions of the V-layer is a promising perspective to increase lSThis was accomplished by samples prepared on single crystalline MgO (100)with nearly epitaxial quality and an RRR-value of the order of 10. For this setof samples re-entrant superconductivity was observed (Fig. 5.9) for the firsttime. From the RRR-value for this series we estimate the mean free path ofthe conduction electrons lS ∼ 12 nm and the corresponding coherence lengthξS ∼ 13 nm. The latter was used for the theoretical fit of the Tc(dFe)-curvein Fig. 5.9. From the transparency parameter Tm = 1.6 the average quantummechanical transmission coefficient T [61] can be estimated to be T � 0.6.This value of T is about twice as large as the T -value for the Pb/Fe interfacediscussed above [61]. This relatively high transparency of the Fe/V interfaceis an essential ingredient for observing re-entrance behavior. As mentionedabove, a highly transparent S/F interface is difficult to achieve with strongferromagnets, since problems with the matching of the Fermi momentum nec-essarily occur for at least one spin direction.
An even higher transparency of the S/F interface can, in principle,be achieved combining a superconductor with a ferromagnet weakened by
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 273
0 1 2 3
1
2
3
4
5d
V=33.9 nm
TC
(K)
dFe
(nm)
Fig. 5.9. Superconducting transition temperature vs Fe thickness at fixed V thick-ness for the Fe/V/Fe series with dV = 33.9 nm. The drawn line is a theoretical curvewith the parameters given in the text
dilution. Recently [64], the re-entrant superconductivity has been observedfor the Nb/Cu1−xNix bilayers.
It is important to note that the results described above for the Fe/V systemwere fitted using the values of the mean free path of the conduction electronsin the F- and S-layers calculated from the resistivity data. In contrast, forthe Nb/Cu1−xNix system Zdravkov et al. [64] had to use surprisingly largevalues for the mean free path for the conduction electrons in the F-layer intheir fitting procedure.
Finishing this section, we would like to mention also study of the in-terplay between magnetism and superconductivity in epitaxial structures ofhalf metal-colossal magnetoresistive La2/3Ca1/3MnO3 (LCMO) and high-Tc
superconducting YBa2Cu3O7−δ (YBCO) [65, 66, 67, 68, 69]. Jacob et al.[65] demonstrated the possibility of preparation of hybrid perovskite high-Tc superconductor/ferromagnet superlattices. The superlattices consisting ofYBCO and LBMO (La2/3Ba1/3MnO3) layers with the thickness of a few unitcells, showed both strong colossal magnetoresistance at room temperature andsuperconductivity at low temperatures.
Yeh et al. [66] reported phenomena manifesting nonequilibrium supercon-ductivity induced by spin-polarized quasiparticles in F/I/S (I is insulator)structures. Sefrioui et al. [67] based on their measurements of Tc vs S- andF-layer thickness speculate that the injection of spin-polarized carriers fromLCMO into YBCO may add a new source of superconductivity suppression:pair breaking by spin-polarized carriers. This pair breaking effect extends overthe spin diffusion length into the S, which can be very long (it can be as longas 8 nm for YBCO). As a result in the YBCO layer superconductivity is sup-pressed by the presence of manganite layers with a characteristic length scalemuch longer than the one predicted by existing theories of the S/F proximityeffect.
274 K. B. Efetov et al.
The same result has been obtained by Holden et al. [68] using ellipsom-etry measurements of the far-infrared dielectric properties of superlatticescomposed of thin layers of YBCO and LCMO. Finally, Soltan et al. [69] stud-ied the role of spin-polarized self injection from LCMO into the YBCO layer.They concluded that the nearly full spin polarization at the Fermi level ofLCMO leads to quenching of the proximity effect since it prevents the Cooperpairs to tunnel into the magnetic layer. Thus, one can see that the results forsuperlattices consisting of YBCO and LCMO presented above are somewhatcontradicting each other. Nevertheless, they provide an avenue for future the-oretical studies of the F/S proximity effect in presence of the spin-polarizedferromagnets.
5.3.2 Superconducting Spin Valve
In recent years much attention has been devoted to experimental realization ofthe superconducting spin valve. As described in Sect. 5.2.1, a consequence ofthe S/F proximity effect is that the superconducting transition temperatureof a F/S/F sandwich depends on the mutual orientation of the magnetizationof the two F-layers, the antiparallel orientation having a higher Tc than theparallel one [6]. In an ideal superconducting spin valve the superconductivityof the S-layer can be switched on and off by rotating the magnetization ofone of the F-layers relative to the other, giving an infinite magnetoresistancefor the switching field. The device is similar to the well known conventionalspin valve F/N/F system with a normal metallic layer N interleaved betweentwo ferromagnetic layers F. In this device the antiparallel magnetization stateusually has a larger resistance than the parallel one.
It turned out that the realization of a superconducting spin valve is difficultexperimentally and the effects obtained until now are quite small. There aretwo recent reports in the literature on the successful realization a F/S/Fsuperconducting spin valve. In the CuNi/Nb/CuNi trilayers system [70, 71] themaximum shift is only 6 mK of the superconducting transition temperatureTc by changing the mutual orientation of the two ferromagnetic layers fromparallel to antiparallel. Actually, such a small shift may also be due to changesof the domain structure of the ferromagnetic layers under the influence of theexternal magnetic field [72].
Pena et al. [14] measured the magnetoresistance of F/S/F trilayers com-bining the ferromagnetic manganite La0.7Ca0.3MnO3 with the high-Tc super-conductor YBa2Cu3O7. They observed a magnetoresistance in excess 1000%for the superconducting state of YBa2Cu3O7 that vanished in the normalstate.
There is another possible design for the realization of the superconductingspin valve effect proposed by Sungjun Oh et al. [73] that found less attentionuntil now. It has the layer structure S/F1/N/F2, i.e. two ferromagnetic layersF1 and F2 separated by a nonmagnetic (N) layer are deposited on the one
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 275
side of the superconductor with F1 and N thin enough to allow the super-conducting pair wave function to penetrate into F2. The authors have shownthat changing the mutual magnetization direction of F1 and F2 from parallelto antiparallel results in a substantial difference ΔTc when the microscopicparameters for S- and F-films are optimized.
For the realization of the F/S/F spin valve design [6] it would be opti-mal to use a system where the re-entrant Tc(dF )-behavior is observed. Asdiscussed above, Fe/V/Fe fulfills this criterion (see Fig. 5.9). However, anacceptable performance of the spin valve with a sizable shift of Tc can onlybe expected if the S-layer thickness dS is close to the superconducting co-herence length ξS . The studies of the Fe/V/Fe system however revealed thatthe superconductivity vanishes typically already at dS < 3ξS . A possibil-ity to overcome this problem and maintain superconductivity at dS ∼ ξSis to introduce very thin non-ferromagnetic layers between the S- and F-layers that should screen to some extent the very strong exchange field of theF-layers.
A proper Fe/Cr/V/Cr/Fe system, where the Cr layers play the role of suchscreening layers, has been studied in detail [74]. In Fig. 5.10 the Tc valuesmeasured for the samples from series with a fixed dFe = 5 nm and dCr variedare plotted. In the other three series dCr has been kept constant at dCr =1.5,2.8 and 4.7 nm and the thickness of the Fe-layer was varied. The results forthe transition temperatures of these series are reproduced in Figs. 5.11b–5.11dand compared to previous results on Fe/V/Fe trilayers [62] (Fig. 5.11a). Thesalient features of the results shown in Fig. 5.11 are as follows.
Fig. 5.10. The superconducting transition temperature as a function of the Cr-layerthickness for all samples from the series (5.1). The solid line is a theoretical curve(see main text)
276 K. B. Efetov et al.
Fig. 5.11. Superconducting transition temperature as a function of the Fe-layerthickness for samples from series with dCr =1,5 nm (b); with dCr =2.8 nm (c); withdCr = 4.7 (d). The corresponding curve for Fe/V/Fe trilayers is taken from [62] andshown in (a) for comparison. The solid lines are calculations according to a modelexplained in the main text
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 277
(1) The overall shape of the Tc (dFe )-curve is similar to that obtained forFe/V/Fe.
(2) The amplitude of the initial drop in Tc decreases with increasing the thick-ness of the interleaved Cr-layer.
(3) At dCr = 4.7 nm in Fig. 5.11d the Fe-layers have virtually no influenceon Tc anymore, indicating that the amplitude of the pair wave functionin the Fe-layer is negligible. This allows the estimation of the penetrationdepth of the pair wave function in Cr of about 4 nm, consistent with theresults on Cr/V/Cr trilayers [75]. These features are due to the expectedscreening effect of the Cr-layer, since with increasing dCr the Cooper pairdensity reaching the Fe-layer is continuously reduced and the effect of thestrong exchange field in Fe on the superconductivity is weakened.
The results of the model calculations are shown by the solid lines inFigs. 5.10 and 5.11a. The complications caused by the spin density wave(SDW) state of antiferromagnetic Cr are neglected in these calculations. Thestandard procedure described in the literature (see, e.g., [75] and referencestherein) was applied and the proximity effect of the V/Cr interface was treatedby the conventional theory for S/N metal films originally developed by deGennes [1]. In addition, pair breaking scattering of Abrikosov-Gor’kov type[76] at magnetic defects in the Cr-layer is characterized by a spin-flip scatter-ing time τs i.e. Cr is treated as a paramagnetic (P) layer.
Theory of the proximity effect for S/P/F layer systems has been developedby Vodopyanov et al. [77]. With certain assumptions [74] and the microscopicparameters known from studies of Fe/V/Fe trilayers [62] all data points inFigs. 5.10 and 5.11 have been fitted simultaneously, with τs being the onlyfitting parameter. All curves can be best described with τs = 5 · 10−13 s.The overall shape of the curves is well reproduced, including the penetrationdepth of about 4 nm for the superconducting pairing wave function in Cr. Thisremarkably small penetration depth in Cr is thus clearly proven to result fromstrong inelastic pair breaking scattering leading to an exponential damping ofthe pair wave function amplitude within the Cr-layer.
There is, however, an additional interesting experimental detail in Fig. 5.10,which the applied model fails to describe even in qualitative terms. This isthe drop of Tc(dCr) for dCr ≥ 4 nm, clearly seen in Fig. 5.10. This feature wasattributed to a transition of the entire Cr-layer from a non-magnetic state toan incommensurate SDW state at dCr ∼ 4 nm. The assumption of a strongsuppression of the Cooper pair density by the transition of the Cr layer froma non-magnetic to a spin density wave state is plausible by the following rea-son. BCS-ordering and SDW-ordering in the same region of the Fermi surfacecan be considered as competing electronic ordering phenomena. In a theoret-ical paper on studying this problem (see, e.g., [78]), it was shown that thoseparts of the Fermi surface where the nesting feature leads to a SDW state theformation of the BCS-gap is suppressed and the superconducting transitiontemperature is reduced.
278 K. B. Efetov et al.
The study of the superconducting proximity effect in Fe/Cr/V/Cr/Fe gavenew results concerning the magnetic phase transition in the Cr-layer, demon-strated a strong screening of the ferromagnetic exchange field of Fe by theinterleaved Cr-layers and allowed an estimate of the upper limit of the thick-ness of the screening Cr-layers for a spin valve to operate.
A novel approach for a realization of the superconducting spin valve designoriginally proposed by Sungjun et al. [73] was also undertaken recently. Theidea [79] for the realization of such a device was to choose as the non-magneticinterlayer N in the S/F1/N/F2/ layer scheme an interlayer with a thicknesscorresponding exactly to an antiferromagnetic interlayer exchange couplingbetween F1 and F2 [80]. Then, one can rotate the relative magnetizationdirection of F1 and F2 from antiparallel to parallel in an external field andobserve the accompanying shift of Tc.
The experimental system of choice was the epitaxial superlattice systemMgO(100)/[Fe2V11]20/V (dV ). (The index denotes the number of monolayers.)There are several reasons that make the choice of the epitaxial (V/Fe)-systemfavorable for demonstrating the superconducting spin valve effect: First, it isthe superior quality of the Fe/V interface in the superlattice [81, 82, 83, 84]that guarantees a high interface transparency and weak diffusive pair breakingscattering at the interface. Second, the Fe2 layers have a thickness dF of about0.3 nm only, whereas for the decay length of the superconducting pair densityξF ∼ 0.7 nm holds (see, e.g., [61]). Thus the pair wave function within theFe2-layer will only be weakly damped and the condition dF /ξF < 0.5 optimalfor observing the superconducting spin valve effect will be fulfilled [73]. InFig. 5.12 we reproduce the magnetization curve of a [Fe2V11]20- superlatticemeasured at 10 K. The shape of the hysteresis shows that the interlayer ex-change coupling is antiferromagnetic with a ferromagnetic saturation field ofHsat = 6 kOe. The upper critical magnetic field for the field direction paralleland perpendicular to the film plane is plotted in Fig. 5.13 for several samples.
Fig. 5.12. Magnetization hysteresis loop of the sample [Fe2V11]20/V(18 nm)measured at 10 K
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 279
1.0 1.5 2.0 2.5 3.0 3.5 4.00
5
10
15
dV=30 nm
dV=22 nm
dV=16 nm
Hc2
(kO
e)
T (K)
Fig. 5.13. Upper critical magnetic field versus temperature with the field appliedparallel and perpendicular to the film plane for three samples [Fe2V11]20/V(dV ).The thickness dV is given in the figure, the open symbols refer to the magneticfield direction parallel to the film plane, the solid symbols refer to the directionperpendicular to the plane
For a two dimensional (2D) thin film with the magnetic field perpendicular orparallel to the film plane the classic result for the upper critical field is [85]:
Hperpc2 =
Φ0
2πξ2(0)
(1 − T
Tc
), (5.16)
Hparc2 =
Φ0
2πξ(0)
√12ds
√(1 − T
Tc
), (5.17)
with the flux quantum Φ0, the thickness of the film dS and the Ginzburg-Landau correlation length ξ(0) related to Pippard’s correlation length ξs asξ(0) = 1.6ξs.
The measurements of the upper critical field for Fe/V/Fe trilayers forparallel orientation of the magnetic field relative to the film plane is per-fectly described by (5.16), as it was observed earlier [86, 87]. In Figs. 5.14aand 5.14b the square of parallel upper critical field are plotted together withthe straight line that describes the temperature dependence for fields above6 kOe perfectly. Below H = 6 kOe there is an increasing deviation from thestraight line. From the extrapolation of the straight line one gets a Tc thatis more than 0.1 K below the true transition temperature measured at zerofield. A comparison with the magnetization curve in Fig. 5.12 shows that theferromagnetic saturation field of 6 kOe is correlated with the first deviationof H2
c2(T ) from the straight line in Fig. 5.14a and 5.14b. From this one caninfer that the deviation of the upper critical field from the 2D-behavior inFig. 5.14 is caused by the gradual change of the sublattice magnetizationdirection of the [Fe2/V11]20-superlattice from parallel above 6 kOe to antipar-allel in zero field. For the sample with dV = 16 nm in Fig. 5.14a Tc = 1.78 K
280 K. B. Efetov et al.
-0.1 0.00.00
0.25
2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.70
10
20
30
40
50
1.4 1.5 1.6 1.7 1.80
20
40
60
80
100
M( μμ μμ
ΒΒ ΒΒ/F
e-A
tom
)
ΔΔΔΔTc
(K)
ΔΔΔΔTc
(b)
dV=30 nm
(Hc2
par )2
(kO
e)2
T (K)
ΔΔΔΔTc
(a)
dV=16 nm
(Hc2
par )2
(kO
e)2
T (K)
Fig. 5.14. Square of the parallel upper critical magnetic field versus temperaturefor the sample [Fe2/V11]20/V(16 nm) (a) and [Fe2/V11]20/V(30 nm) (b). The fullstraight line is the extrapolation of the linear temperature dependence for higherfields, the dashed line is the theoretical curve expected if the magnetization of thesuperlattice would not change. ΔTc is the shift of the superconducting transitiontemperature between the superlattice in the antiferromagnetic state and in ferro-magnetic saturation. The inset in panel (a) depicts the shift of the superconductingtransition temperature with the magnetization of the [Fe2/V11]20 superlattice
in antiferromagnetic state, while in ferromagnetic saturation it extrapolatesto Tc = 1.67 K. The temperature difference ΔTc = 0.11 K is the anticipatedsuperconducting spin valve effect.
These experiments demonstrate that the superconducting transition tem-perature of the V-film reacts sensitively to the mutual magnetization ori-entation of the Fe2 layers of an antiferromagnetically coupled [Fe2V11]20superlattice. Actually the ferromagnetic layers in this system cannot beswitched from the parallel to the antiparallel state, since the parallel stateneeds the application of a strong external magnetic field. At the same time,it should in principle be possible to construct a switching device by replacing
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 281
the antiferromagnetically coupled [Fe/V] superlattice by a conventional spinvalve trilayer system.
5.3.3 Odd Triplet Superconductivity in S/F/S Structures
In Sect. 5.2.3 a theoretical model predicting a possible robust triplet proximityeffect in S/F structures has been described. The mechanism is operational inthe presence of a rotating magnetization at the S/F interface. Recently Sosninet al. [13] presented the first experimental indication of this type of proximityeffect using an Andreev type of interferometer and an S/F/S mesoscopic thinfilm structure. The design of the interferometer is depicted intergerometer inFig. 5.15a. It consists of a superconducting Al-loop with an area of 20 μm2
with a narrow gap bridged by a ferromagnetic Ho-stripe. The distance betweenthe two Al/Ho contact points was more than one order of magnitude largerthan the singlet magnetic coherence length ξF0. A rotating magnetization atS/F interface is established here by the intrinsic conical ferromagnetism of Ho(see Fig. 5.15b). The essential experimental finding is that below the Tc of Althe resistance of the Ho wire exhibits oscillations as a function of the super-conducting phase difference between the two interfaces of the Ho-stripe withthe superconducting Al ring, as shown in Fig. 5.16a. The phase difference wasgenerated by varying the magnetic flux penetrating the Al-loop. The period ofthe oscillations corresponds to the flux quantum Φ0 = 2 ·10−7 Gcm2 and givesrise to the sharp peaks in the Fourier spectrum of oscillations (Fig. 5.16b).Estimates show that for the relative amplitude of the conductance oscillationsΔR/RF � 10−4 ( RF is the resistivity of the ferromagnetic wires) is expected.These oscillations were observed for the samples with a distance between the
c-axis
α
5.62 Å
M
θ=30º
α=80º
θ
θ
(a) (b)
1 μm
I2
I1
V2
V1
B
Al
Ho
Fig. 5.15. a) Experimental set-up and SEM micrograph of S/F/S junction areaprepared by shadow evaporation. b) Magnetic structure of Ho: magnetization Mrotates by 30◦ each atomic layer along c-axis at an angle of 80◦ to this axis
282 K. B. Efetov et al.
0.5 1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
-4 -2 0 2 4
-2
0
2
(b)
(a)
(Φ / Φ0)-1
Am
plit
ude
(a.u
.)ΔR
(m
Ω)
Φ / Φ0
Fig. 5.16. a) Magnetoresistance oscillations of the sample shown in Fig. 5.15ameasured at T=0.27 K as a function of normalized external flux through the loop.The sample resistance is 94.3 Ω. b) Fourier spectrum of the oscillations confirmingthe hc/2e periodicity
Al/Ho contact points interfaces of up to LF = 160 nm. Such a long-rangephase coherence cannot be explained by the proximity effect involving thepenetration of the ordinary singlet pairs, since the upper limit for the singletpenetration depth ξF0 is equal to the electron mean free path l which forwas l ≈ 6 nm. Thus, the observed oscillations of the magnetoresistance seemto originate from the long-range penetration of a helical triplet component ofsuperconductivity generated in a ferromagnetic conductor and induced by thepresence of a rotating magnetization.
Recently Keizer et al. [15] studied lateral S/F/S Josephson junctions com-bining the strong ferromagnet CrO2 that belongs to the group of half-metalswith full spin polarization of the electrons at the Fermi level and the conven-tional s-wave superconductor NbTi. They observed a Josephson supercurrentprevailing over very long length scales up to ∼ 1 μm. This is orders of mag-nitude larger than expected for singlet correlations, which is of the order of1 nm. In addition to the long-range penetration of the superconducting pairdensity into CrO2, they found that the supercurrent strongly depended onthe magnetization direction in the ferromagnet. On the basis of these findings
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 283
Keizer et al. attributed the long-range supercurrent to the triplet correla-tions. In this case of a half-metallic ferromagnet it is reasonable to assumethat the LRTC is created at the S/F interface where spin-flip processes mayhappen [88].
Hints of the realization of the triplet proximity effect also came from recentmagnetization data on hybrid structures consisting of multilayers of mangan-ites [La0.33Ca0.67MnO3/La0.60Ca0.40iMnO3]15 in contact with a low-Tc Nbsuperconductor [89].
5.3.4 Other Proximity Effects
It seems natural and actually it is theoretically well established that the pen-etration of superconductivity from the S- into the F-layers is not the onlypossible proximity effect in S/F systems (see Sect. 5.2.5 and 5.2.6). The prox-imity effect can also work in the reverse direction, i.e. the ferromagnetismfrom the F-layer can leak into the S-layer (inverse proximity effect) or theS-layer can modify the ferromagnetic state of the F-layer (cryptoferromag-netism). However, these effects are more subtle from the experimental pointof view and are still less well established.
Cryptoferromagnetism in S/F Layers
As shown in Sect. 5.2.5, under certain conditions the ferromagnetic order inF-layers may be reconstructed by the action of the S-layer into a new mag-netic domain state [17]) or a cryptoferromagnetic state [18]. The basic physicalreason for this behavior is that the destructive influence of the ferromagneticexchange field on the superconductivity can be considerably reduced if the fer-romagnetic state is modified in such a manner that the exchange field cancelswhen averaged over the superconducting coherence length.
The first hint in favor of a reconstruction of the ferromagnetic state belowthe superconducting transition temperature was obtained from the anoma-lous temperature dependence of the effective magnetization extracted fromthe ferromagnetic resonance (FMR) line position in epitaxial Fe/Nb bilayersbelow Tc [48]. However, a quantitative estimate using the theory of Buzdin andBulaevskii [17] rises doubts about this interpretation, since the effect in Fe/Nbshould only occur at an Fe-layer thickness an order of magnitude smallerthan observed experimentally. Later Bergeret et al. [18] studied theoreticallythe possibility of a non-homogeneous magnetic order of a ferromagnetic filmplaced on top of a bulk superconductor. They also concluded that due tothe large magnetic stiffness constant in Fe, the cryptoferromagnetic state canhardly be realized using pure Fe films. These considerations suggested thatthe tendency to a reconstruction of the ferromagnetic state observed experi-mentally in Fe/Nb might be caused by a granular structure of the very thinFe layers.
284 K. B. Efetov et al.
Quantitative estimates by Bergeret et al. [18] showed that the transitionfrom the ferromagnetic to the cryptoferromagnetic state should be observablein a ferromagnet with a magnetic stiffness constant an order of magnitudesmaller than that of pure Fe. This can be achieved by dilution of Fe in suitablealloy systems, a favorable choice being Pd1−xFex at small x due to its low andtunable Curie temperature.
In an FMR study for a series of samples V/Pd1−xFex the temperature de-pendence of the effective magnetization 4πMeff = 4πM−(2Ku/M) (M is thesaturation moment of the ferromagnet and Ku is the perpendicular anisotropyconstant) was measured [49]. The low-temperature part of 4πMeff (T ) is de-picted in Fig. 5.17. One observes a decrease of the effective magnetization4πMeff below Tc for the sample 2 (Fig. 5.17) but not for sample 1. A de-crease of 4πMeff can be caused by a decrease of the saturation magnetizationM or by an increase of the perpendicular uniaxial anisotropy constant Ku. Acomparison of FMR results for films with different thickness of the ferromag-netic layers leads to the conclusion that Ku is very small and the decreaseof 4πMeff must be caused by a decrease of the saturation magnetization M .This suggests that the decrease of the saturation magnetization below Tc iscaused by a reconstruction of the ferromagnetic state.
An estimate following the phase diagram by Bergeret et al. [18] (Fig. 5.5)gave the parameters a ∼ 1.2 and λ ∼ 1.3 ·10−3 for sample 2 with dM ∼1.2 nmand TCurie ∼ 100 K. In accordance with the phase diagram of Bergeret et al.(Fig. 5.5) this implies that starting from τ ∼ 0.2 (T ∼ 3.2 K) a transitionfrom the ferromagnetic to the cryptoferromagnetic state should take place,as it is actually observed experimentally. For the sample 1 with dM ∼ 4.4nm and TCurie ∼ 250 K we have a ∼ 20 and λ ∼ 1.4 · 10−2. With theseparameter values the ferromagnetic state should be stable at any temperature,in agreement with the experimental result.
0 1 2 3 4 5
1
2
3
4
5
TC
TC
sample 1 sample 2
4πM
eff (
kG)
T (K)
Fig. 5.17. Low-temperature parts of 4πMeff (T ) for the sample 1 with dV = 37.2nm, dPd−F e = 4.4 nm, Tc = 4.0 K and sample 2 with dV = 40 nm, dPd−F e = 1.2nm, Tc = 4.2 K. The arrows show the Tc-values at the resonance field H0
5 Proximity Effects in Ferromagnet/Superconductor Heterostructures 285
Thus, these estimates support the conclusion that a phase transition fromthe ferromagnetic state to the cryptoferromagnetic state occurs in sample 2.However, one cannot completely exclude that the anomalous temperature de-pendence of Meff might be due to the screening of the magnetic momentsof the ferromagnetic layer by the polarized Cooper pairs, as discussed inSect. 5.2.6 and in the next section.
Inverse Proximity Effect
Up to now any unequivocal experimental evidence for the penetration of themagnetization from the ferromagnetic side into the superconducting side of anS/F bilayer, as discussed in Sect. 5.2.6, does not exist. First interpretations inthis direction have been published recently [89, 90]. Stahn et al. [90] studiedthe magnetization profile of [YBa2Cu307/La2/3Ca1/3MnO3] multilayers usingneutron reflectometry. From a change of the Stahn et al. [90] argue in favorof the first possibility, but the situation is not yet settled.
Stamopoulos et al. [89] presented magnetization measurements on multi-layers of manganites [La0.33Ca0.67MnO3/La0.60Ca0.40MnO3]15 in contact witha low-Tc superconductor. They came to the conclusion that the superconduc-tor below Tc becomes ferromagnetically coupled to the multilayer. Since itis expected that for singlet pairing the magnetization of F penetrates intoS antiferromagnetically, the authors conclude that a spin-triplet supercon-ducting component forms and penetrates into the F-layer thus inducing theferromagnetic coupling observed experimentally.
5.4 Summary and Conclusions
The main purpose of the present paper was to review the status of researchon proximity effects in S/F layer systems from the experimental as well asfrom the theoretical point of view.
Peculiarities of the S/F proximity effect originating from the penetrationof the condensate function into the ferromagnet that have been discussedcontroversially in the beginning seem to be well established now. The S/F/SJosephson junctions with π-coupling are, e.g., even suggested as basic unitsfor the realization of Q-bits for quantum computing [91].
It has become traditional in the field of the S/F proximity effect that the-ory is somewhat ahead of experiment. The situation persists and intriguingtheoretical predictions are still waiting for the first experimental verificationsor further experimental support. One of these predictions concerns the un-conventional superconductivity in S/F systems. The experimental realizationis difficult, since the unconventional superconductivity expected here, namelyodd triplet superconductivity, can only be generated by a rotating magne-tization at the interface. Nevertheless, the first experimental indications ofits existence have already been reported. What is important, the odd triplet
286 K. B. Efetov et al.
superconductivity is insensitive to scattering on non-magnetic impurities andthis is certainly helpful for an experimental observation.
We should also mention further recent ideas on how to identify the oddtriplet superconductivity [92, 93, 94].
The inverse proximity effect, i.e. the penetration of the magnetic order pa-rameter into a superconductor, has not been clearly observed experimentallyyet. However, indications on the closely related effect, namely the decreasingof the total ferromagnetic moment below Tc, already exist. Yet, it is not easyto clarify to what extent the non-homogeneous distribution of MF producedin the ferromagnet below Tc contributes to the effect. The best way to observethe spin screening of MF is either probing directly the spatial distribution ofthe magnetic field using neutron scattering or by measuring muon spin reso-nance. Since the magnetic moment M varies on the macroscopic length ξS , itshould be possible to detect it.
A considerable work still remains to be done on the experimental side.Only careful material selection, optimization of film preparation and devicedesign will enable a clarification of all the complex phenomena that may occurin the S/F proximity systems. The study of F/S structures with compara-ble ferromagnetic Curie and superconducting transition temperatures seemsvery promising. Combining elemental superconductors with elemental ferro-magnets, as was done in the majority of papers on the S/F proximity effectpublished until now, is not the best way for the observation of the proximity ef-fects because the ferromagnetic exchange energy is orders of magnitude largerthan the superconducting condensation energy. In this case the ferromagneticstate can hardly be modified by the superconductor. Rare earth based ferro-magnetic compounds with low Curie temperatures would, in principle, be abetter choice. Combining high-Tc superconductors with ferromagnetic oxidesis another promising option.
Acknowledgement
The authors are grateful for the support by the Deutsche Forschungsgemein-schaft (DFG) within SFB 491. One of us (AFV) would like to acknowledgefinancial support from DFG within the project
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6
Magnetic Tunnel Junctions
Gunter Reiss1, Jan Schmalhorst1, Andre Thomas1, Andreas Hutten1,and Shinji Yuasa2
1 Thin Films and Physics of Nanostructures, Department of Physics,Bielefeld University, Bielefeld, [email protected]
2 National Institute of Advanced Industrial Science and Technology (AIST),Nanoelectronics Research Institute, Tsukuba, [email protected]
Abstract. In magnetoelectronic devices large opportunities are opened by the spindependent tunneling resistance, where a strong dependence of the tunneling currenton the relative orientation of the magnetization of the electrodes is found. Withina short time, the amplitude of the resistance change of the junctions increased dra-matically. We will cover Al-O and MgO based junctions and present highly spin-polarized electrode materials such as Heusler alloys. Furthermore, we will give ashort overview on applications such as read heads in hard disk drives, storage cellsin MRAMs, field programmable logic circuits and biochips. Finally, we will discussthe currently growing field of current induced magnetization switching.
6.1 Introduction
Tunneling is a quantum mechanical phenomenon based on the wave characterof particles [1]. An, e.g., electron impinging on a wall of potential energy ofenergy height Φ and width d will be reflected with a certain probability R andtransmitted with T = 1 − R. In classical mechanics, R will be 1 (i.e. T = 0)if the kinetic energy of the electron is smaller than Φ and 0 otherwise. Inquantum mechanics, however, the wave character of the electron produces anon-vanishing probability for transmission already for a kinetic energy smallerthan Φ. This effect – called tunneling – forms the basis for the so-called mag-netic tunnel junctions (MTJs). Whereas tunneling is usually demonstrated ina trilayer system consisiting of two simple metallic (non-magnetic) electrodesseparated by a thin insulator, in MTJs the two electrodes are ferromagnetssuch as Fe or Co. In this case, the tunneling current obtained through suchstructures not only depends on the averaged density of electronic states inthe two electrodes but additionally on the relative direction of the magnetiza-tions in the two ferromagnets. In a simple theoretical description, one obtainsfor the transmission probability for the majority (↑) and minority carriers
G. Reiss et al.: Magnetic Tunnel Junctions, STMP 227, 291–333 (2007)
(↓), respectively: T↑,↓ ∝ exp(−k↑,↓Φ1/2d), i.e. the transmission becomes spindependent [2] due to the exchange splitting of the band structure in the fer-romagnetic metalls (k↑,↓ : Complex wave vector of the electrons).
However, it was not before 1975 that a spin dependence of the tunnelingcurrent was observed experimentally: Julliere [3] found in the tunnelingtransport properties of Fe/Ge/Co MTJs at low temperature an unambigoussignature of the magnetic states of the two ferromagnetic electrodes. Again,it took a long time, until this effect was also measured at room temper-ature. Moodera and Miyazaki published simultaneously [4, 5] results forCo/Al2O3/NiFe MTJs, where they observed a significant change of around15% of the tunneling resistance between the parallel (R↑↑)and anti-parallel(R↑↓) alignment of the magnetizations of the electrodes, if one defines theTMR ratio as TMR = (Rmax − Rmin)/Rmin. Some years before, a similareffect was found in all metallic structures such as Fe/Cr/Fe or Co/Cu/Co[6, 7] (GMR, Giant Magnetoresistance), which triggered enormous researchactivities on spin dependent transport properties.
First attempts to form reliable MTJ’s used one relatively hard and onerelatively soft ferromagnetic electrode. This leads typically to minor loops fora tunnel junction with about 10 μm × 10 μm size as shown in Fig. 6.1 [8].Here, only the soft magnetic electrode switches its magnetization, whereasthat of the hard electrode is supposed to stay constant. As can be already seenfrom Fig. 6.1, the dependence of the resistance on the external field shows acurved crossover to the saturation values and therefore does not correspondto 100% remanence of the magnetization. Moreover, it exhibits small kinkswhich additionally are not reproducible. This simple hard / soft architectureturned out to be not stable with respect to magnetically cycling the softelectrode, because the domain splitting of the soft electrode causes large strayfields which induce a deterioration of the hard magnetic material [9]. The
Fig. 6.1. Resistance as a function of an external magnetic field (minor loop) fora tunneling junction Co / Al2O3 / Ni80Fe20 from 1999 showing about 30% TMRamplitude
6 Magnetic Tunnel Junctions 293
same remains true, if the hard electrode is additionally stabilized by an anti-ferromagnetically RKKY-coupled [10, 11, 12] trilayer as, e.g., Co / Cu / Co[13]. Moreover, the Cu turned out to diffuse to the barrier between Co andAl2O3 giving rise to a rapid decrease of the TMR while maintaing ‘good’tunneling properties.
For a further stabilization of the hard magnetic electrode, a direct contactwith a natural antiferromagnet can be used. Due to the exchange interac-tion between antiferro- and ferromagnet, an annealing of this structure inan external field leads to a unidirectional anisotropy in the ferromagnet [14]with a shift of its hysteresis of several 100Oe. Thus, the nowadays standardstack design is a sequence of conductor- and seed layers followed by a nat-ural anitferromagnet. The subsequent artificial anti-ferromagnet usually is aCoFe / Ru / CoFe trilayer in the second maximum of the anti-ferromagneticcoupling. This combination of exchange bias and anti-ferromagnetic couplingfurther both stabilizes the hard electrode and gives the possibilty to tailor itsnet magnetic moment. The tunneling barrier is ususally made by depositingan Al film with a thickness between 0.6 nm and 1.6 nm which then is oxidizedby a mild treatment in an oxygen- or an Ar-O plasma. From intensive investi-gations, it is known, that the energy of the ions impinging on the film duringthis process should be kept well below 50 eV in order to avoid O implantationin the underlying ferromagnet.
A considerable increase of this amplitude was reached by replacing thecrystalline electrode deposited prior to the barrier layer by an amorphous Co-Fe-B ferromagnet [15]. For these amorphous Co-Fe-B electrodes, TMR valuesof more than 70% have been shown (see Fig. 6.2). The reason for this largeTMR being not yet known causes some uncertainty concerning the maximumTMR reachable with conventional 3d ferromagnets.
The next step in the preparation routine is then the initialization of theexchange bias which is usually done by annealing the film stack in a magneticfields at a typical temperature of about 300◦C for some minutes.
Fig. 6.2. TMR major loop of a layer sequence Ta5 / Cu30 / Ta5 / Cu5 / Mn-Ir12 /Co-Fe-B3.5 / Al1.2 + oxidation Co-Fe-B3.5 / Ni-Fe3 / Ta5 / Cu20 / Au50 (subscripts:layer thickness in nm) with the amorphous ferromagnet Co-Fe-B (12% B)
294 G. Reiss et al.
Considerable further improvement of the theoretical understanding ofthe spin dependence of the tunneling was obtained by taking into accountthe complex band structure of the insulator and the lattice match betweenferromagnets and barrier material.
In this contribution, we will highlight some of the major aspects of thedevelopment of MTJs. The discussion starts with some of the present andpossible future applications of MTJs which put forward some challenges forthese new devices such as amplitude and magnetic switching. In the nextsections, possible answers to these challenges will be sketched.
6.2 Applications
Magnetoresistance has been used for many years in conventional devices forsensing magnetic fields. These sensors use the so called Anisotropic Magne-toresistance (AMR) which produces a change of the resistance of the orderof a few percent during magnetization switching. One mass market for theseproducts was the application as read heads in hard disk drives. For storinginformation, magnetic devices also played a major role from the mid-1950sto around 1970 as magnetic core memory. Afterwards, however, the semicon-ductor technology produced rapid advances in down-scaling, performance andprice so that – although devices such as bubble memories have been developedsimultaneously – storing data by magnetics was essentially restricted to harddisks and tapes [16].
The discovery of the giant magnetoresistance in magnetic multilayers andsandwiches in 1986 [6, 7] and the development of the related spin valve device[17], then showed great promise for read-head sensors for hard disk drives andthus boosted the development of magnetoresistive devices. Spin valve basedprinciples have been also proposed for storing information, but it was thediscovery of a large tunneling magnetoresistance at room temperature [4, 5],which opened the field of data storage in the so called Magnetic RandomAccess Memory (MRAM).
For this purpose, first a ‘cross point architecture’ was proposed. This con-sist just of two arrays of conducting wires running perpendicularly to eachother on a chip. The wire arrays are separated by an insulating film. At thecrossing points, magnetic tunnel junctions connect the wires (see Fig. 6.3).If the high resistance state of the junctions is related with, e.g., logic 1 and
Fig. 6.3. A sketch of the crossed wire architecture of a random access memory withmagnetic tunnel junctions
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the low resistive with logic 0, one bit of information can be stored in thisdevice. It, however, turned out that the feasibility of this simple crossed wirearchitecture is questionable, because the selected path between two wires isshunted by the other crossing point MTJs. Thus, an additional select transis-tor has to be integrated for reliably selecting one MTJ.
The downscaling of the MTJs to sizes used in nowadays microelectronicsturned out to be a challenge for the development of the technology. The pos-sible benefits of non-volatility of the information, unrestricted read- and writeaccess and potential down-scaling capabilities down to the 32 nm node, how-ever, initiated a participation of several major companies of the electronicindustry in the development of the MRAM. For the related downscaling,magnetostatic coupling by stray fields [18, 19], a homogeneous magnetizationswitching and the reproducible preparation of the insulating barrier turnedout to be critical issues for the applicability of the MRAM. Whereas strayfield effects can be minimized by tailoring the net magnetization of the an-tiferromagnetically coupled hard electrode, the homogeneity of the switchingand the barrier resistance are challenging for both the film deposition and thelithography.
By using elliptically shaped MTJ cells with smooth edges, the distributionof the switching fields was successfully optimized. Also, the soft magnetic layeris often replaced by an additional antiferromagnetically coupled trilayer inorder to drastically reduce the total stray field of the device. The tunnelingbarriers can be deposited nowadays with a homogeniety of around 2% on12 inch wafers [20], leading to a variance of the MTJ resistance of around 5%.In Fig. 6.4, we show the distribution of the resistance values of MTJs foundboth for the high resistive (antiparallel) and the low resistive state.
A further application related with MRAM is the use of arrays of MTJsto perform logic operations [21, 22, 23]. The realization of a logic based onthe same principles and technologies as the – potentially universal – memoryMRAM is of large interest, because it opens the way to a common technology
Fig. 6.4. The distribution of the area resistance products of MTJs in the high- andlow resistive states, respectively. The gap between these two distributions is largeenough for a reliable operation in an MRAM matrix
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Fig. 6.5. Bridge configuration of MTJs. The input/programming lines produce amagnetic field that rotates the soft magnetic electrode’s magnetization, changingthe output voltage Vout, which represents the logic function of the inputs
platform for both storing as well as computing data. Moreover, magnetic logicgate arrays can be field programmable, leading to field programmable logicgate arrays (FPGAs). Such FPGAs are programmable ‘on the fly’ and, thus,open also a path to fast reconfigurable computing [24]. In Fig. 6.5, we showthe principle of the operation of an FPGA based on four MTJs. Here, theinput is represented by currents on two input lines, which can change themagnetization state of the MTJs soft electrodes. Two neighbouring lines areused to set the resistance states of the other two MTJs which ‘program’,i.e. define the value Vout obtained as logic function of the two inputs. Forsuch logic operations, however, the requirements concerning the quality of theMTJs are more stringent than for the MRAM. An estimation of the yield ofproducing logic gate arrays with for MTJs shows [25], that at least a TMReffect amplitude of 100% is necessary in order to enable a production withrealistic tolerances concerning the variances of resistance and TMR.
Thus, although these challenges have been successfully solved, there areprototype MRAM chips available only up to a density of 16Mb in the 180nmnode technology [26]. One of the main reasons for this lack in density is thecurrent need to perform the magnetization switching on the chip: At thelocation of the MTJ two perpendicular magnetic fields are applied by runningcurrent through perpendicular metallization wires in two metallization levelsbelow the MTJ level. The two related field pulses, one in hard- and one in easyaxis direction of the soft electrode then switch the magnetization, because thesum of the field vectors is large enough to overcome the Stoner-Wohlfarthastroid [18]. The currents needed for this are, however, in the range of somemA. Thus both the wires in the metallization levels below the MTJ as well asthe transistors for switching these large currents have to be much larger thanthe mimimum feature size in these cells.
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Fig. 6.6. Principle of the magnetic biochip (left): Biotin marked DNA-moleculeshybridize with complementary strands attached to the surface, streptavidin coatedbeads then bind to the biotin. By applying a magnetic field perpendicular to thesurface, only the in plane components of the dipole stray field of the beads aredetected. A TMR sensor surface covered by magnetic beads (middle) and the TMRsignal measured during applying a magnetic field perpendicular to the sensor surfaceand an in plane field which is close to the switching field of the soft electrode(right) [27]
As last application related example, we now turn to a completely differentfield: In biotechnology and medical applications, molecules like DNA or pro-teins are frequently marked by magnetic spheres called ‘beads’ [28]. This opensthe possibility to measure the presence or absence of these biomolecules by de-tecting the magnetic beads with an MTJ. Baselt et al. [29] already describedthis technique using Giant Magnetoresistance sensors. Figure 6.6 shows theprinciple of this method and the result of the measurement of different beadconcentrations with a 100 μm wide TMR cell [27]. As can be seen in Fig. 6.6,reasonable signals as in dependence of the perpendicular field are obtained ata surface coverage of only a few percent, if an in plane field is additionallyapplied which brings the soft electrode close to switching. Comparisons withthe established optical method of marking with fluorescent molecules showed,that the magnetic biosensor can be more sensitive at low concentrations ofthe analyte molecule, which is the most interesting area of application. Thedetection of such magnetic markers, however, requires both a high sensitivityand a non-hysteretic answer of the sensor. Thus again very high TMR ampli-tudes are needed, but now in devices responding in an unambiguous mannerto an external magnetic field. At the moment, the use of TMR sensors withvery soft magnetic free layers and a strong shape anisotropy seems to be themost promising approach. Work towards a single molecule detection usingsuch systems is currently in progress.
In summary, these and other applications puts forward challenges for thefurther development of magnetic tunnel junctions. We have used the MRAM,magnetic logic and a magnetic biosensor to point out the major fields, wherethe properties of the MTJs need improvements:
• The TMR ratio of around 75% obtained for standard Al2O3 based MTJsis enough to prepare some working FPGAs in the laboratory, but it is not
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satisfying the needs of a high yield production. Here, the lower limit is atabout 100% TMR.
• The currents in the clock- and wordlines needed to switch the tunnelingcells by crossed field pulses is too large and does not scale down withthe size of the MTJs. This is regarded to be the most severe obstacle forthe MRAM and applications in logic, because a successful introduction ofa product in this field must have perspectives well below around 50 nmminimum feature size.
• If the sensors which can be realized with magnetic tunneling cells arenot only used as counters then they also need a higher sensitivity overa relatively broad range of external fields. Moreover, for most sensor ap-plications, the hysteretic answer of a TMR cell to an external field is adisadvantage, because it limits the accuracy of the field detection.
6.3 Current Induced Magnetization Switching
As addressed in the foregoing section, the scalability of the MTJs is a crucialpoint for their successful introduction in microelectronic products. Especiallythe traditional switching mechanism for the soft layer’s magnetization by twoperpendicular field pulses is not scalable, because the current creating thefields does not go down as the TMR cell size shrinks.
The solution for this obstacle could be current induced magnetizationswitching [30, 31] which was recently demonstrated by several groups [31].In this technique, a spin polarized current is driven into a ferromagnetic thinfilm. If the spin of the incoming electrons and the magnetization of the fer-romagnet is not alligned, these two systems will exchange torque, which canabove a critical current density lead to a flip of the magnetization of the ferro-magnet. This method has a high potential in magnetic tunneling cells, becausethe current needed to obtain the switching scales down in the same way as theTMR cell shrinks. Mandatory, however, are fast switching and critical currentdensities lower than both the breakdown properties of the tunnel junctions aswell as the electromigration threshold which lies between 106 A/cm2 and 107
A/cm2.The torque exterted by the spin polarized current on the soft electrode’s
magnetization MFs is included in its equation of motion (Landau-Lifshitz-
Gilbert equation) by an additional term introduced by Slonzewski [30].
1γdMF
S
dt= MF
S × [H − (α
|MFS |M
FS × (H +HS)] (6.1)
HS = JAε�
2eMFS α
(6.2)
JCr ∝ αMFS tF (H +HK + 2πMS)
ε. (6.3)
6 Magnetic Tunnel Junctions 299
The classical equation of motion (6.1) describes both the precession ofthe magnetization MF
S upon applying a magnetic field H and the momentumtransfer from a spin polarized electrical current with a density J, which occursin the LLG equation (6.1) as an additional field term HS (6.2). If a criticalcurrent density Jcr is applied to the device at zero temperature (6.3), the softmagnetization layer should switch, i.e., MF
S changes its direction. Furtherimportant parameters are the gyromagnetic ratio γ, the damping parameter αand the spin torque efficiency ε which contains the microscopic description ofthe transfer of momentum between the incoming spin polarized electrons andthe magnetization of the soft layer. If directly related with the TMR effect,ε should strongly decrease with decreasing ΔR; for large TMR values, thisdecrease should be nearly linearly [31].
For common materials and tunnel junctions,this results in some 106
A/cm2–107 A/cm2 critical current density of the injected electrons. In or-der to prepare samples capable of current induced magnetization switching(CIMS), however, both a large TMR as well as a very low resistive tunnelingbarrier are required. For these purposes, we prepared MgO based [32, 33] tun-nel junctions with a stack sequence of Pt-Mn20 nm / Co-Fe2 nm / Ru0.75 nm
/ Co-Fe-B2 nm / MgO1.3 nm / Co-Fe-B3 nm both with low and a high arearesistance product (6.5 Ωμm2 and 51 Ωμm2, respectively) and a tunnelingmagnetoresistance ratio between 120% and 140% by sputter deposition in aSingulus TIMARIS PVD tool.
In Fig. 6.7, we show a CIMS curves of the low and high resistive tunneljunctions. This is recorded by applying first a voltage (i.e. current-) pulse of100msec and then measuring the sample’s resistance at a low bias voltage(20mV). For the low resistive junctions, switching occurs at a bias voltagebetween 0.5V and 0.7V at a current density of around 107 A/cm2, which iswell within the expected range but too large as compared with the electromi-gration threshold. For the samples with the larger area resistance product of
Fig. 6.7. Magnetoresistance of a 100 nm × 200 nm large tunneling junction asa function of the amplitude of the current pulses applied prior to the individualresistance measurements for the low and high resistive (area resistance product:6.5 Ωμm2 and 51 Ωμm2) MgO based magnetic tunnel junctions
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51 Ωμm2, where switching occurs between 0.8V and 1.2V bias pulse voltage,the shape of the switching curves are very similar. A siginificant difference,however, occurs for the current density needed to switch the magnetization.Whereas for the low resistive samples around 107 A/cm2 are necessary, thehigh resistive tunnel junctions already switch between 1 × 106 A/cm2 and2 × 106 A/cm2, i.e., at a value about five times lower.
During switching, the role of thermal heating also needs to be considered,because both the coercive field as well as the saturation magnetization aretemperature dependent. In this case, however, the power dissipated by thecurrent is considerably lower for the high resistive samples. Thus, although thetemperature during current pulsing is not reliably known, it should be largerfor the low resistive samples and, therefore, these results cannot be explainedby the difference in the power dissipation in the two types of samples.
In terms of the theoretical description of current induced magnetizationswitching this means, that the spin torque efficiency ε in (6.3) is at least5 times larger for the spin current injected at 1 V bias compared to thatat 0.6V. Whereas the physical origin for this behavior is not yet known,this effect, nevertheless, opens a way to further lower the current densityneeded for switching such MTJs. If it is possible to enhance the scatteringrate by tailoring the band structures of the involved materials, it would enablenanoscale MTJ electronics.
6.4 Heusler Alloys
6.4.1 Introduction to Highly Spin Polarized Materials
Two complementary avenues are currently followed so as to realize the visionof spin electronics: to employ MTJs with TMR-effect amplitudes of up toseveral thousand percent. One of which is determined by evaluating differenttunnel barrier materials such as Al2O3 and MgO and the other is aiming at100% spin polarized magnetic electrodes in MTJs.
Hence, one of the crucial issues today in the development of spin electronicdevices is the search for new materials that exhibit large carrier spin polariza-tion [34]. Potential candidates include half-metallic ferromagnets oxides [35] orferromagnetic half-metals such as Heusler alloys [36]. Characteristic for thesemagnetic materials named ‘half-metallic’ [37] is their unusual band structurewith only one spin direction being metallic. Electrons of the opposite spin havea gap in their density of states (DOS) at the Fermi level (EF ) and hence areinsulating. Consequently with only one spin band present at EF half-metallicferromagnets are 100% spin-polarized and allow the transport of only onespin carrier across an interface or tunnel barrier into an adjacent material.Therefore, the spectacular theoretical prediction of 100% spin polarizationin an entire class of materials, the half-Heusler XYZ [37, 38, 39, 40] as wellas full-Heusler X2Y Z [41, 42, 43, 44] alloys is currently the driving force for
6 Magnetic Tunnel Junctions 301
evaluating the potential of MTJs with at least one magnetic electrode madeof a Heusler alloy.
However, the earlier experimental efforts to realize MTJs did not showany evidence for a true enhancement of the TMR-effect amplitude when usingHeusler alloys as magnetic electrodes. The spin polarization of the half-Heuslercompound NiMnSb which has been integrated in a MTJ [45] was measured tobe 25% at 4.2K and is related to a TMR-effect amplitude of 19.5%. The corre-sponding TMR value at RT was found to be 9% only. Using a full-Heusler alloyof type Co2Cr0.6Fe0.4Al as one of the magnetic electrodes in a MTJ slightlylarger TMR values of 16% at RT and 26% at 4.2K could be realized [46].
Nevertheless, the list of promising half- and full-Heusler alloys is long[40, 44]. One interesting candidate is Co2MnSi which is also characterizedby a 100% spin polarization as was predicted [41] from band structure calcu-lations. In addition, Co2MnSi possesses with TC = 985 K [47] a large Curietemperature identifying it to be an excellent candidate for technological appli-cations; materials with large Curie temperatures should have a high remnantmagnetization at RT.
Recently, the spin polarization of Co2MnSi was determined to be 54%at 4.2K using point contact Andreev reflection spectroscopy [48] which isfairly comparable with the spin polarization of the 3d-based magnetic elementsor their alloys, i.e., at RT a maximum spin polarization of 44% was foundfor Co50Fe50, whereas that of Ni80Fe20 reached 53% at 10K [49]. By thattime the potential of Co2MnSi integrated as one ferromagnetic electrode intechnological relevant magnetic tunnel junctions had yet to be proven. To copewith this challenging task magnetic tunnel junctions had been fabricated [50]consisting of one Co70Fe30 and one Co2MnSi electrode separated by a verythin insulating AlOx barrier so as to determine spin polarization of Co2MnSiand the resulting TMR-effect amplitude. The TMR-effect amplitude achievedin these MTJs is 94.6% at 20K [51] applying a bias voltage of 1 mV. Thiscorresponds to a 65.5% spin polarization of Co2MnSi which clearly exceedsthat of the 3d-based magnetic elements or their alloys but is also well belowthe predicted 100%. These results have triggered enormous research activitieson the integration of Co2MnSi as magnetic electrodes in MTJs. Today weare looking on an impressive development [52, 53] which is crowned by theTMR ratio of 570% at 2 K [54], which is the largest value so far reported incombination with an amorphous AlOx tunneling barrier.
6.4.2 Structure and Ordering
The key to achieve large TMR-effect amplitudes in MTJs with integrated mag-netic Co2MnSi electrodes is to induce a high degree of atomic order withinthese Co2MnSi layers. From a crystallographic point of view, the evolution ofthe long-range order parameter as a function of preparation conditions suchas annealing time and temperature can be determined by monitoring the in-tensity change of superlattice reflections in x-ray or neutron or transmission
302 G. Reiss et al.
electron diffraction patterns of thin polycrystalline or epitaxial Co2MnSi lay-ers. Thus, the question of interest to be answered in the following is about theexperimental possibility to employ these diffraction techniques so as to deter-mine the disorder-order transition in Co2MnSi. The unit cell of Co2MnSi [55]contains 16 atoms which are located on four interpenetrating fcc sublatticesA, B, C and D where each of the sublattices is occupied by atoms of oneelement: A by Co, B by Mn, C by Co and D by Si when fully ordered. Thisarrangement corresponds to the L21 structure type giving rise to non-zeroBragg reflections only when the Miller indices of the corresponding scatteringplanes are either all even or all odd. The planes with all even Miller indicescan further be divided in those for which (h + k + l)/2 is odd and those forwhich (h+ k+ l)/2 is even. The intensities of these allowed reflections can becalculated by the squares of the relevant structure factors as given by [56]:
h, k, l all odd : for example F 2(111) = 16[(fA − fC)2 + (fB − fD)2]h+ k + l
2= 2n+ 1 for example : F 2(200) = 16(fA − fB + fC − fD)2 (6.4)
h+ k + l
2= 2n for example : F 2(220) = 16(fA + fB + fC + fD)2,
where the fi are the averaged atomic scattering factors for the four sublat-tices sites. The squared structure factor F 2(220) is characterized by the sumof all averaged atomic scattering factors for the four sublattices sites. Hence,even in the presence of complete disorder the resulting reflection intensity willremain unchanged and identifies this class of scattering planes as the principalreflections. In contrast, the intensities of the two other classes of scatteringplanes F 2(111) and F 2(200) are very sensitive to any ordering / disorderingprocess whereby one atom of one sublattice is interchanged by one atom ofanother sublattice and identify the superlattice reflections.
Since Co2MnSi is a ternary alloy it is not possible to describe the state oforder by one single order parameter as is usually done or binary alloyed phases.On the contrary, the possible ways of disorder have to be associated with acertain disorder parameter α which defines the fraction of either x or y atomsnot on the correct sublattice. The four sublattices present in Co2MnSi allowsix most probable types of disorder as is summarized in Fig. 6.8. The samerelations are also valid for the iso-structural Co2FeSi when Mn is replaced byFe. Inserting these fi into (6.4) enables to calculate the intensity change of thesuperlattice reflections (111) and (200) for all six probable ways of disorder asa function of the disorder parameter α. The results are summarized in Fig. 6.9.
As it can be seen the unique (111) and (200) intensity changes with in-creasing atomic disorder can experimentally be used as a fingerprint so as todetermine which way the atomic disorder is following upon annealing, despitethe fact that way 2 and 4 are identical. Nevertheless, to quantify the state ofatomic ordering using X-ray or electron diffraction is quite demanding since
6 Magnetic Tunnel Junctions 303
(6)random disorder
in allsublattices
(5)Co(A)-
Si(D)-Co(C)Interchange
(4)Co(A)-
Mn(B)-Co(C)Interchange
(3)Mn(B)-Si(D)interchange
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Scattering factor fA ofsublattice C
Scattering factor fA ofsublattice C
Scattering factor fA ofsublattice B
Scattering factor fA ofsublattice A
Way of disorder:
(1 ) Co Mnf fα α− + (1 ) Mn Cof fα α− + Cof Cof
(1 ) Co Sif fα α− + (1 ) Si Cof fα α− +Mnf Cof
Cof (1 )− +Mn Sif fα α Cof (1 )− +Si Mnf fα α
(1 )2 2
− +Co Mnf fα α
(1 )− +Mn Cof fα α (1 )2 2
− +Co Mnf fα α
Sif
(1 )2 2
− +Co Sif fα α
Mnf (1 )2 2
− +Co Sif fα α
(1 )− +Si Cof fα α
2(1 ) ( )
3 3− + +Co Mn Sif f f
αα (1 ) (2 )3
− + +Mn Co Sif f fαα 2
(1 ) ( )3 3
− + +Co Mn Sif f fαα (1 ) (2
3− + +Si Co Mnf f f
αα )
Fig. 6.8. Scattering factors of the four sublattices A, B, C and D after [56] takenfrom [57]. fCo, fMn and fSi are the corresponding atomic scattering factors
Fig. 6.9. Summary of the evolution of the disorder parameters of the six most prob-able ways to disorder the L21 structure of Co2MnSi taken from [57]. The intensitiesare normalized to the intensity value at αi = 0 the state of full atomic order
304 G. Reiss et al.
the crystallographic texture of the Co2MnSi layer imposes an additional con-strain on these techniques. However, employing electron diffraction patternstaken from cross section samples in HRTEM would benefit from an overalltextured Co2MnSi layer with (100) orientation in growth direction. This en-sures a high probability to find columnar Co2MnSi grains close to a [58] typeof zone axis which in turn contains strong superstructure reflection as is pre-dicted by calculation and experimentally demonstrated in Fig. 6.10. UsingX-ray diffraction patterns would also require textured Co2MnSi layers with(100) orientation in growth direction so as to compare the intensity ratios ofthe (h00) family type of reflections. To extract information about the atomicordering measured X-ray diffraction patterns could be fitted to atomic or-dering models taking into account all six possible ways of disorder discussedabove by performing a Rietveld analysis. However, both techniques wouldreveal only qualitative results unless a Co2MnSi single crystal can be usedas a reference. Nevertheless, taking all these obstacles towards uncoveringthe ordering mechanism in these full Heusler alloys is worth while and willdefinitely contribute to a better understanding and hopefully tuning of themicrostructural TMR-properties relationships in the near future.
6.4.3 Transport Properties of Heusler Alloy Based MagneticTunnel Junctions
In this section the transport properties as well as chemical and magnetic in-terface properties of MTJs with a Heusler alloy electrode and Al-O barrier arediscussed. A large TMR effect is anticipated for many applications, whereas
Fig. 6.10. HRTEM micrograph of a Co2MnSi grain in [58] zone axis orientation:(a) Calculated electron [58] diffraction pattern. (b) Live FFT of the HRTEM mi-crograph. Clearly visible are the superstructure reflection of [2] type family
6 Magnetic Tunnel Junctions 305
the TMR amplitude is connected with the Julliere spin polarization [3] Pa,b ofelectrode a and b: TMR = 2PaPb/[1−PaPb]. The ultimate magnetoelectronicmaterial should have a gap in the minority (or majority) electron densityof states at the Fermi energy EF and thus 100% spin polarization. Thesematerials are called ferromagnetic half-metals. This property has been pre-dicted theoretically for some half and full Heusler compounds [37, 41, 44, 59].
Starting in 1999 with Tanaka et al. [45] (NiMnSb), different Heusler com-pounds have been implemented in magnetic tunnel junctions, whereas half-metallicity has not been demonstrated so far in these structures. The largestreported spinpolarizations are found for Co2Cr1−xFexAl [60] and Co2MnX(X=Si, Al or Ge) [61, 62, 63] single layer electrodes and very recently for[Co2MnSi / Co2FeSi]10x multilayer electrodes [64].
To achieve high effective spin-polarization for Co2MnSi based MTJs an in-situ annealing procedure applied after barrier formation and before depositingthe top electrode of the junction has been developed [50]. The in-situ anneal-ing forces the atomic ordering of the Co2MnSi thin films and, hence, increasestheir spin-polarization. This preparation techniques has also been applied suc-cessfully to MTJs with Co2FeSi single layer and [Co2MnSi / Co2FeSi]10x mul-tilayer electrodes [64]. These MTJs will be addressed in more detail in the fol-lowing: We will compare the temperature and bias voltage dependent TMR ofour Heusler alloy based junctions with more conventional Co62Fe26B12 / AlOx
/ Co62Fe26B12 and Co70Fe30 / AlOx / Ni80Fe20 MTJs to show up the specificcharacteristics of the Heusler alloy based MTJs. The influence of the tempera-ture dependent magnetic moments at the Heusler alloy / barrier interfaces aswell as electronic band structure effects will be discussed. The experimentalresults for the Co2MnSi based MTJs will also be compared to bandstructurecalculations for perfectly L21-ordered Co2MnSi bulk material (lattice constant0.565 nm) obtained using the SPR-KKR program package [65].
The MTJs were deposited at room temperature by DC- and RF-magnetronsputteringfrom stoichiometric targets on thermally oxidized Si(100) wafers.On a 40 nm thick V buffer we deposited a magnetically soft electrode(Co2MnSi100 nm and Co2FeSi100 nm single layers or a [Co2MnSi5 nm /Co2FeSi5 nm]x10 multilayer). The subsequently deposited thin Al film witha typical thickness of 1.5 nm was plasma oxidized to form a tunnel barrier.To achieve high spin-polarization this bottom electrode including the barrierwas in-situ annealed for 40–60 min. at 450◦C (Co2MnSi) and 380◦C (Co2FeSiand [Co2MnSi5 nm / Co2FeSi5 nm]x10), respectively. Afterwards, the barrierwas shortly oxidized again to remove contamination of the barrier surfaceand covered with the top electrode (Co70Fe30
5 nm). The Co-Fe is pinned byMn83Ir1710 nm and finally covered by the upper conduction leads (Ta / Cu /Ta / Au). Then, the layer stacks were ex-situ vacuum annealed at 275◦C in amagnetic field of 0.1 T to set the exchange bias of the pinned electrode. Thejunctions were patterned by optical lithography and ion beam etching. Thetransport properties of the junctions were measured as a function of magneticfield, bias voltage and temperature by conventional two-probe DC technique.
306 G. Reiss et al.
For probing the structural and magnetic properties of the Heusler alloy basedelectrodes and at the Heusler alloy / AlOx interfaces, half junctions with-out top electrode were also fabricated and investigated by an AlternatingGradient Field Magnetometer (AGM), x-ray absorption spectroscopy (XAS),x-ray magnetic circular dichroism (XMCD) and x-ray photoemission spec-troscopy (XPS). Temperature dependent XAS and XMCD were performedat beamline 4.0.2 and 7.3.1.1 of the Advanced Light Source, Berkeley, USA.The Co-, Fe- and Mn-L edges and the Si-K edge were investigated. Surface-sensitive total electron yield (TEY) [66] as well as bulk-sensitive fluorescenceyield (FY) spectra [66] were recorded.
Now, we give an overview on the characteristic transport properties ofMTJs with Heusler alloy electrode. The low temperature TMR majorloopsof Co2FeSi single layer and [Co2MnSi / Co2FeSi]×10 multilayer based MTJsand their room temperature TMR amplitude as a function of Al thickness isshown in Fig. 6.11. An TMR of up to 114% has been observed correspondingto a spin polarization 0.74 [67]. The low temperature TMR majorloops ofCo2MnSi based MTJs prepared with in-situ at 450◦C (‘CMS100’) and with-out in-situ annealing (‘CMS100ag’) are shown in Fig. 6.12a. Sample ‘CMS100’shows up to 95% TMR at 20 K /1mV and a sharp magnetization reversal of theCo2MnSi around zero magnetic field. Without in-situ annealing (‘CMS100ag’)the TMR is strongly reduced to maximal 1.3% at 16K / 20mV (±2000Oefield range). The Co2MnSi electrode shows superparamagnetic behavior andis not saturated at ±2000Oe. The pinned Co-Fe top electrode of both samplesshows an exchange bias of about –650Oe. The TMR amplitude of 95% foundfor ‘CMS100’ corresponds to effective spin polarizations of PCo2MnSi = 66%assuming PCo70Fe30 = 49% [61]. The low temperature bias voltage dependence(±500mV range) of the junctions is shown in Fig. 6.13a. ‘CMS100’ shows aconsiderably stronger bias voltage dependence than more conventional Co-Fe-B / Al-O / Co-Fe-B (‘CoFeB’) and Co-Fe / Al-O / Ni-Fe (‘CoFe-NiFe’) MTJswith a maximum TMR amplitude of 114% (‘CoFeB’, PCo62Fe26B12 = 60%)[68] and 71% (‘CoFe-NiFe’) [68, 69], respectively. The same holds for the tem-
0.5 1.0 1.5 2.0 2.50
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R[%
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Fig. 6.11. (a) Low temperature TMR majorloops of Co2FeSi single layer (“SL”,black curve, 16 K) and [Co2MnSi / Co2FeSi]×10 multilayer (“ML”, red curve, 17 K)based MTJs measured with 10 mV bias for 1.5 nm thick Al. (b) Room tempera-ture TMR amplitude (10 mV bias) of MTJs with Co2FeSi single and [Co2MnSi /Co2FeSi]×10 multilayer electrode as a function of Al thickness. Taken from [67]
6 Magnetic Tunnel Junctions 307
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20
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"CMS100"
TM
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Fig. 6.12. TMR majorloops for MTJ ‘CMS100’ (20 K / 1mV) and ‘CMS100ag’(16 K / 20 mV). Taken from [68]
perature dependence of the TMR (Fig. 6.13b), which is identical for MTJs‘Co-Fe-B’ and ‘MTJ-NiFe’, although their maximum TMR amplitudes are sig-nificantly different. Especially, their TMR(T)-curves are concave. In contrast,the Co2MnSi based junction show a convex TMR(T)-dependence. A convexTMR(T) dependence is also found by Oogane et al. [62] for junctions withepitaxial Co2MnSi electrodes. The Co2MnSi based MTJs prepared without in-situ annealing (‘CMS100ag’) show no TMR above 200K. Especially, a linearbias voltage dependence of the TMR is observed for ‘CMS100’, ‘CoFeB’ and‘CoFe-NiFe’ in the bias voltage (V) range of a few 10mV. Remarkably, thein-situ annealed junctions ‘CMS100’ show an inversed TMR of up to –6.3%at room temperature for large negative bias voltage, i.e., when the electronsare tunneling from the Co-Fe into the Co2MnSi electrode (see Fig. 6.13c). Forpositive bias voltage, when the electrons are tunneling from Co2MnSi into Co-Fe, the TMR remains positive. This inversion of the TMR is not observed forthe other three junction types ‘CMS100ag’, ‘CoFe-NiFe’ and ‘CoFeB’. Theirmonotonic decrease of the positive TMR with increasing bias voltage shownin Fig. 6.13 for the ±500mV bias voltage range just goes on up to the di-electric breakdown of the junctions. E.g., the Co-Fe /AlOx / Ni-Fe junctions[69] ‘MTJ-NiFe’ still show a room temperature TMR of about +10% for ±1200mV bias voltage. For ‘CMS100ag’ junctions, which were not in-situ an-nealed during layer deposition, the TMR at 16K is +0.2% for ± 1200mVbias voltage. The inversion of the TMR turned out to be very characteristicfeature for Heusler alloy based MTJs. It has been also observed for junctionswith Co2FeSi single layer and Co2MnSi/Co2FeSi multilayer electrode [64] andfor Co2MnGe / MgO / Co-Fe junctions [63].
Before these results can be interpreted, we have to focus on the chemi-cal, magnetic and electronic properties at the Heusler alloy / Al-O interface.The case of a Co2MnSi / AlOx interface will be discussed exemplarily: Theelement specific properties of Co, Mn and Si in ‘half’ junctions correspond-ing to the full MTJ stacks ‘CMS100’ and ‘CMS100ag’ were probed by x-rayabsorption spectroscopy. Bulk sensitive FY spectra of Co and Mn are shownin Fig. 6.14. Pronounced XANES oscillations indicating the atomic orderingthroughout the Co2MnSi layers are found for the annealed sample ‘CMS100’
308 G. Reiss et al.
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Fig. 6.13. (a) Typical bias voltage dependence at low temperature for ‘CMS100’(measured at 20 K), and ‘CoFeB’ (30 K) and for ‘MTJ-NiFe’ (10K). All data isnormalized to the maximum TMR at low bias. The inset shows the TMR am-plitude for ‘CMS100ag’ (16 K) (b) Typical normalized temperature dependenceof ‘CMS100’ (measured at 10mV), ‘CoFeB’ (10mV), ‘MTJ-NiFe’ (10mV) and‘CMS100ag’ (20 mV). (c) Typical bias voltage dependence in the ±1500 mV biasvoltage range of a ‘CMS100’ junction, measured at room temperature. The insetsshow TMR minor loops measured at +10mV and –1300 mV, when only the mag-netization of the soft Co2MnSi electrode is switched by the external magnetic field.The magnetizations of Co2MnSi and Co-Fe are aligned parallel (antiparallel) formagnetic fields of –60 Oe (+60Oe). Taken from [68]
(see Fig. 6.14). Additionally, this sample shows shoulders at about 4 eV abovethe maximum intensities of the Mn- and Co-L2,3 resonances (marked by ‘A’ inFig. 6.14) which are not present for ‘CMS100ag’. These shoulders reflect smallpeaks in the density of unoccupied d-like states [65] of the ordered Co2MnSi.The TEY-spectra probing the Co2MnSi / AlOx interface show these finger-prints of the atomic ordering in the annealed sample ‘CMS100’ only for Co(see arrows in Fig. 6.15b). MnO identified by its characteristic XAS multiplett
6 Magnetic Tunnel Junctions 309
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[arb
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ts]
"A""A"
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(a)
Fig. 6.14. XAS-FY spectra at (a) Mn- and (b) Co-L edge of ‘CMS100ag’ and‘CMS100’. The spectra were measured at 15K. The XANES oscillations found forthe annealed ‘CMS100’ are marked by arrows and are also visible at 300 K. Theintensities of the L2,3-resonances are reduced because of saturation effects [66]. Takenfrom [68]
structure [70] (see peaks ‘A1’–‘B2’ in Fig. 6.15c) is present at the interfacefor both samples and masks the fingerprints of ordering for interfacial Mn insample ‘CMS100’. Although the saturation effects change the shape of thebulk sensitive Mn FY-spectra shown in Fig. 6.14a, it is obvious that the MnOmultiplett structure is not present in the Mn FY-spectra. In addition to the
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Co @ 15K: "CMS100ag" "CMS100"
Fig. 6.15. (a) Magnetization loop of the as grown sample ‘CMS100ag’ measured at15K (open circles). The black line is a fit by the Langevin function (see text). (b)X-ray absorption spectra and magnetic circular dichroism in TEY-mode at the Co-L edge of ‘CMS100’ measured at 15 K. (c) X-ray absorption spectra and magneticcircular dichroism in TEY-mode at the Mn-L edge of ‘CMS100’ measured at 15K.The red curve δ(hν) is the difference of the Mn XMCD asymmetries of ‘CMS10’measured at 15 K and 300 K, respectively. The inset shows differences of the CoXMCD asymmetry at 15 K between the ordered and disordered sample ‘CMS100’and ‘CMS100ag’, the photon energy is defined with respect to the energy positionof maximum intensity of the Co-L3 resonance. Taken from [68]
310 G. Reiss et al.
interfacial MnO we found some SiO2 at the interface in the Si-K edge XASspectra.
The temperature dependence of the interfacial magnetic moments of Coand Mn is adressed now. For these atoms the magnetic moments are governedby the 3d-electrons which can be probed by x-ray absorption at the L2,3-edges.According to the sum rules analysis [71] the XMCD asymmetries measuredat 15K correspond to a ratio of the spin magnetic moments mMn
s for Mn andmCo
s for Co of mMns /mCo
s = 1.4–2.1. The large uncertainty of this value resultsfrom the not precisely known factor to correct the jj-mixing for Mn [72].However, this value is significantly smaller than mMn
s /mCos =2.9 as expected
from bandstructure calculations [44]. The reduced magnetic moment ratio isreasonable, because of the Mn-O formation at the interface.
Compared to ‘CMS100’ the XMCD asymmetries of the as grown sample‘CMS100ag’ are by a factor of 7 (Co) and 9 (Mn) smaller at 15K (appliedfields ±0.55 T), accordingly their magnetic moments are strongly reduced.Furthermore, the magnetization loop (Fig. 6.15a) calculated [73] from its Coand Mn XMCD asymmetries in TEY mode shows a typical superparamagneticbehavior. Compared to the 15K values the XMCD asymmetries are reduced at300K by a factor of 14.0±1.3 for Co and 13.5±0.9 for Mn, respectively, whichresults from the intrinsic temperature dependence of the superparamagneticclusters and from an additional reduction of mean magnetic moments of eachcluster by a factor of 1.8.
In contrast, the temperature dependence of the interfacial magnetic mo-ments of the ordered sample ‘CMS100’ is much smaller than for ‘CMS100ag’,their magnetic moments of Co and Mn are only reduced to 94% and 90% ofthe low temperature values. Assuming a Bloch-like temperature dependence ofthe interfacial magnetic moments,m(T )/m(0) = (1−αT 3/2), this correspondsto spin wave parameters α of 1.17× 10−5K−3/2 for Co and 1.95× 10−5K−3/2
for Mn. These values are 4 (Co) and 6.7 (Mn) times larger than the spin waveparameter αbulk = 2.81 ± 0.21 × 10−6K−3/2 of bulk Co2MnSi measured byRitchie et al. [74].
Whereas the majority of the interfacial Mn2+ ions remains paramagneticor becomes antiferromagnetically ordered at low temperature, the generallylarger temperature dependence of the Mn magnetic moment can be attributedto the metallic Mn in not perfectly ordered Co2MnSi and might be a resultof the antiferromagnetic Mn-Mn coupling in the alloy.
Compared to ‘CMS100’, the temperature dependence of the interfacialmagnetic moments of sample ‘CoFeB’ is weaker. Co and Fe magnetic momentsat the Co-Fe-B / AlOx interface are only reduced to 98% and 97% of the lowtemperature moments, when the temperature increases from 25K to 300K.Therefore, thermal magnon excitation at the barrier–electrode interface ismuch more pronounced in the Co2MnSi based junctions than in our MTJswith Co-Fe-B electrodes.
Finally, we will discuss the observed transport properties on the base of thechemical and magnetic interface properties. The superparamagnetic behavior
6 Magnetic Tunnel Junctions 311
of the Co2MnSi electrode found in the major loops of ‘CMS100ag’ junctions(Fig. 6.12) is consistent with the temperature dependent magnetic propertiesat the Co2MnSi / AlOx interface. The superparamagnetism of the electrodecan reduce its interfacial magnetization at ±2 kOe with raising temperatureso strongly, that the TMR vanishes nearly at room temperature. In general,the temperature and bias voltage dependent transport properties of MTJs arerather complex because of the variety of different contributions to the totalconductance like direct tunneling including bandstructure effects [75, 76] andthe shape of the tunnel barrier [77], thermally induced changes of the inter-facial magnetization [78], magnon [79, 80, 81] and phonon assisted tunneling[80], unpolarized conductance via defect states in the barrier [78, 82], spin scat-tering on paramagnetic ions in the barrier[83, 84] and possible spin flip scat-tering on interfacial antiferromagnons [84] in the MnO. Because of the MnOformation at the lower barrier interface spin scattering on paramagnetic Mn2+
ions in the barrier [83, 84] has to be taken into account. For the small appliedmagnetic fields in our transport measurements (≤ 0.2 T) the Zeeman energyof paramagnetic Mn-ions is much smaller than the smallest thermal energy(kBT≈1meV at 10K) and bias voltage (1mV). Accordingly, spin scatteringon paramagnetic ions is a quasi-elastic process on our energy scale and shouldnot influence the bias voltage or temperature dependence [84]. However, thisprocess can reduce the effective spin polarization of the Co2MnSi, which is in-deed PCo2MnSi = 66% for the in-situ annealed ‘CMS100’ junctions instead oftheoretically expected 100%. The same holds for the unpolarized conductancevia one defect state in the barrier [82], which can partly reduce the effectivepolarization but does not have an influence on the bias voltage or temperaturedependence. Please note, that the partial oxidation of Mn and Si at the barrierinterface should disturb the ordering process of the interfacial Co2MnSi duringthe in-situ annealing which should also contribute to the reduction of the ef-fective spin-polarization of the Co2MnSi. If MnO orders antiferromagneticallyat low temperature, spin flip scattering on interfacial antiferromagnons shouldresult in a decrease of the TMR with the square of the bias voltage in first-order approximation as shown by Guinea [84]. Phonon assisted tunneling [80]and unpolarized hopping conductance via two or more defect states [82] shouldalso result in a non-linear bias voltage dependence of the TMR. In contrast,a linear bias voltage dependence as observed for ‘CMS100’ in the bias voltage(V) range of a few 10mV has been predicted for magnon assisted tunneling[80, 81] assuming an energy independent band structure around the Fermienergy and surface magnon excitation. Compared with ‘CoFeB’ and ‘CoFe-NiFe’ junctions TMR(V) is considerably larger, which would imply, that themagnon assisted tunneling is more pronounced in ‘CMS100’ junctions. Thatis consistent with the temperature dependent XAS and XMCD investigationswhich showed, that thermal magnon excitation is considerably larger at theCo2MnSi / AlOx than at the Co-Fe-B / AlOx interface. Additionally to thelinear bias voltage dependence, the convex TMR temperature dependenceof ‘CMS100’ junctions is a characteristic feature for magnon-assisted tunnel-
312 G. Reiss et al.
ing as proposed by Han et al. [81]. Taking only direct and magnon-assistedtunneling into account the resulting selfconsistent fits of the bias voltage de-pendence of the TMR and the temperature dependence of the area resistanceproduct for parallel (RP ) and antiparallel (RAP ) alignment are very good.Especially, for antiparallel alignment of the Co2MnSi and Co-Fe electrodes,when magnon-assisted tunneling would be most important. Therefore, fromthe comparison of the transport properties and the temperature-dependentinterfacial magnetic moments we suggest, that the TMR temperature and(low) bias voltage dependence of our in-situ annealed ‘CMS100’ junctions isstronger compared with conventional Co-Fe-B or Co-Fe / Ni-Fe based MTJsbecause of enhanced magnon-assisted tunneling. The inversion of the TMRfor larger bias voltage is an astonishing feature for the Heusler alloy basedMTJs which will be explained in the next paragraph.
6.4.4 Band Structure Calculations of Heusler Alloys
Band structure calculations were performed using the SPR-KKR package [65]to investigate the theoretical behavior of Heusler based magnetic tunnel junc-tions and to get a better understanding of the properties that lead to high spinpolarization and, thus, high magnetoresistance. A better understanding alsoresults in in new possibilities of tailoring the properties of magnetic tunneljunctions.
The calculations were carried out in the framework of local spin densityapproximation (LSDA), where the KKR-method for calculating the electronicstructure was used. The Green‘s function is determined by treating the solidas a multi stray system, where the pertubating potentials of every atom areassumed to be spherically symmetric without overlapping (atomic sphere ap-proach, ASA). The system was analysed taking relativistic effect into account,so the used Dirac-Hamiltonian for the spin-polarized system is given by
(�
icα · ∇ + βmc2 + Veff (r) + βσ · Beff (r)
)Ψi(r) = εiΨi(r)
with Beff (r) = Bext(r) +δExc [n,m]
δm(r),
where α,β are the Dirac matrices. Within this approach it can be shown, thatGreen’s function may be written as
G(r, r ′, E) =∑ΛΛ′
ZnΛ(r, E)τnn′
ΛΛ′ (E)Zn′×Λ′ (r ′, E)
−∑Λ
(ZnΛ(r, E)Jn×
Λ (r ′, E)Θ(r′ − r)
+ JnΛ(r, E)Zn×
Λ (r ′, E)Θ(r − r′))δnn′ ,
where Z and J are the regular and the irregular solution of the Dirac equationfor a spin-polarized potential well, respectively and τnn′
ΛΛ′ gives the scattering
6 Magnetic Tunnel Junctions 313
path operator. Green’s function provides all the information about the system,e.g. the charge density and magnetization are given by
n(r) = −1π� tr
∫ EF
dE G(r, r, E)
m(r) = −1π� tr
∫ EF
dE βσzG(r, r, E).
Figure 6.16 shows the calculated bandstructure of a Co2MnSi compoundin L21 phase. The viewgraph illustrates the contributions of the different Ele-ments at different energies. To put the calculation in relation to the expectedtunnel magnetoresistance, the density of states is reduced to the DOS of thes-electrons around the Fermi-level. The s-electrons are believed to be the maincontribution to the tunneling current in Alumina based tunnel barriers [86],although, the tunnel barrier material [87] and structure [58] can select differ-ent electrons in different tunneling structures. The left side of Fig. 6.17 showsonly the s-electrons close to the Fermi-level to take this into account. Theright side shows the Co-Fe counter electrode to complete the tunnel junction.One can see the gap just below the Fermi-level in the total DOS as well as inthe s-electron DOS, followed by a sharp increase just above the Fermi-energy.The definition of the effective spin polarization is applied:
P =N(E)↑ −N(E)↓N(E)↑ +N(E)↓
, (6.5)
where N(E)↑ and N(E)↓ are the DOS of majority and minority s-electrons,respectively. The spin polarization of Co2MnSi has positive values at and
Fig. 6.16. Bandstructure calculations using the SPR-KKR package [65] for aCo2MnSi compound in L21 phase. Please note the Fermi-energy at the edge ofthe bandgap and the pronounced increase next to the edge
314 G. Reiss et al.
Fig. 6.17. Bandstructure of the tunneling s-electrons in L21 Co2MnSi and Co-Feat bias-voltages of 500, 0 and –500 mV. Since mainly the density of states Figuretaken from [85]
below the Fermi-energy and negative values in a region above it. To get theeffective spin polarization at a certain applied bias voltage in a tunnel junction,one has to integrate the DOS at that bias value (taking temperature Fermismearing into account). Then it becomes obvious that the resulting effectivespin polarization is negative only for energies well above the minority peak.
The TMR effect is generally interpreted by Julliere‘s expression to calcu-late the TMR ratio out of a given spin polarization:
ΔR
R=RAP −RP
RP=
2P1P2
1 − P1P2, (6.6)
where resistance in anti-parallel (RAP ) and parallel state (RP ) are relatedto the spin polarizations P1 and P2 of the two electrodes, PCo−Fe and PCo2MnSi
in our case. Therefore, a negative effective spin polarization of only one elec-trode at a certain bias voltage leads to a negative resulting TMR-effect.
And exactly this behavior is seen in the bias voltage dependence of theTMR ratio at RT (Fig. 6.13c). A negative bias voltage applied to a Co2MnSi/ AlOx / Co70Fe30 MTJ drives s-like electrons from occupied states at EF
of Co-Fe through the Alumina tunnel barrier into unoccupied s-like states ofthe Co2MnSi above its Fermi-level. Figure 6.13c shows a characteristic kinkin the bias voltage dependence but no crossover at about 400mV, because ofthe integration and Fermi-smearing the actual crossover is shifted to highervoltages. The (integrated) number of majority spins is higher at all energiesfor the Co-Fe counter electrode, in other words the spin polarization remainspositive. Therefore, positive bias voltages lead to a positive TMR effect am-plitude which decreases with increasing bias voltage [88]. Consequently, wecan select negative and positive magnetoresistance by applying different bias
6 Magnetic Tunnel Junctions 315
voltages, although, the originally large TMR ratios have not been achievedat these voltages so far. However, in principle this opens up new ways forapplications, e.g., in programmable magnetic logic devices [85].
6.5 Magnetic Tunnel Junctions with a CrystallineMgO(001) Tunneling Barrier
As described in Sect. 6.1, magnetic tunnel junctions (MTJs) with an amor-phous Al-O tunnel barrier have been extensively studied since the discoveryof the room-temperature TMR effect [4, 5]. In Al-O-based MTJs, magne-toresistance (MR) ratios of up to 70% have been achieved experimentally atroom temperature (RT) [89] as shown in Fig. 6.18. However, these relativelysmall MR ratios are considered to severely limit the feasibility of applicationsof spintronic devices. For example, MR ratios significantly higher than 70%at room temperature are indispensable for developing high-density magne-toresistive random-access-memory (MRAM, see Fig. 6.3). According to theJulliere’s model [3] the MR ratio of an MTJ can be expressed by spin polar-ization P of ferromagnetic electrodes as MR = 2P1P2/(1 − P1P2). Here, P1
and P2 are spin polarizations of the two ferromagnetic electrodes defined asP ≡ (N↑(EF ) − N↓(EF ))/(N↑(EF ) + N↓(EF )), where N↑(EF ) and N↓(EF )are the density of states (DOS) of the electrode at Fermi energy (EF ) formajority-spin and minority-spin bands, respectively. Spin polarization of aferromagnet (FM) at low temperature can be directly measured using FM/Al-O/superconductor tunnel junctions [90]. According to this kind of measure-ment, spin polarizations of 3d-ferromagnetic metals and alloys based on iron(Fe), nickel (Ni) and cobalt (Co) are usually in the range 0 < P < 0.6 at lowtemperature [90, 91]. Julliere’s model with these spin polarizations yields amaximum MR ratio of about 100% at low temperature. MR ratio of about70% at RT is therefore close to the Julliere’s limit for the 3d- ferromagnetic-
1995 2000 20050
50
100
150
200
250
300
350
400
450
oitarR
M(%
)T
Rta
Year
”1”: CNRS-Thales 61
“5”: Anelva–AIST28
Tohoku
Tohoku 84
“6”: AIST113
Nancy 52
“2”: AIST14
“3”: AIST114
“4”: IBM77
AIST
MgO(001) barrier
AmorhousAl-O barrier
Tohoku 68
MIT 70
NVE 104
Fig. 6.18. History of improvement in MR ratio at room temperature (RT). Solidcircles denote MTJs with an amorphous Al-O barrier; open circles represent MTJswith a crystalline MgO(001) barrier
316 G. Reiss et al.
alloy electrodes if a reduction in P at finite temperature (due to thermalspin fluctuations) is taken into account. One way to achieve a much higherMR ratio is the use of special kinds of ferromagnetic materials called “halfmetals”, which have a full spin polarization (| P |= 1) and, therefore, aretheoretically expected to exhibit a huge MR ratio (up to infinity, in principle)when used as the electrodes of MTJs. As candidates for such half metals,there are several types of materials: CrO2, Heusler alloys such as Co2MnSi,Fe3O4, and manganese-perovskite oxides such as La1−xSrxMnO3. Very highMR ratios above several hundred percent have been achieved at low temper-ature in La1−xSrxMnO3 / SrTiO3 / La1−xSrxMnO3 MTJs [92] and Co2MnSi/ Al-O / Co2MnSi MTJs [54]. However, such high MR ratios have never beenobserved at RT because of the large temperature dependence of MR ratio inthese MTJs [93]. Another way to achieve a very high MR ratio is to use coher-ent spin-dependent tunneling in an epitaxial MTJ with a crystalline tunnelbarrier such as MgO(001). This is the main subject of Sect. 6.5.
Before going into the details of coherent tunneling, an incoherent tun-neling process through the amorphous Al-O tunnel barrier is explained first.Figure 6.19 a schematically illustrates a tunneling process in the MTJ with anamorphous Al-O barrier. Here, the top electrode layer is Fe(001) as an exampleof a 3d-ferromagnet. Various Bloch electronic states with different symmetriesof wave functions exist in the electrode. Because the tunnel barrier is amor-phous, there is no crystallographic symmetry in the tunnel barrier. Becauseof this non-symmetric structure, Bloch states with various symmetries havefinite tunneling probabilities. This tunneling process can be regarded as anincoherent tunneling. In 3d-ferromagnetic metals and alloys, Bloch states withΔ1 symmetry (mainly s-like states) usually have a large positive spin polar-ization at EF , while Bloch states with Δ2 symmetry (mainly d-like states)usually have a negative spin polarization at EF . Julliere’s model assumesthat tunneling probabilities are equal for all Bloch states. This assumptioncorresponds to a completely incoherent tunneling, in which none of the mo-mentum and coherency of Bloch states are conserved. However, even in the
(a) (b)
k//
kZ
Fe(001)
Al-O
Δ2 Δ5 Δ1 Fe(001)
Fe(001)
Δ2 Δ5Δ1
Δ1
Δ1
MgO(001)
Fig. 6.19. Schematic illustrations of electron tunneling through (a) an amorphousAl-O barrier and (b) a crystalline MgO(001) barrier
6 Magnetic Tunnel Junctions 317
case of an amorphous Al-O barrier, this assumption is not valid. Althoughspin polarization P defined with band structure is negative for Co and Ni,the experimentally observed P is positive for these materials when combinedwith the Al-O barrier [90, 91]. This discrepancy indicates that the tunnelingprobability in actual MTJs depends on the symmetry of Bloch states. The ac-tual tunneling process is explained in the following way. The Δ1 Bloch stateswith large P are considered to have higher tunneling probabilities than theother Bloch states [94, 95]. This results in a positive net spin polarizationof the ferromagnetic electrode. Because the contribution of the other Blochstates, such as Δ2 states (P < 0), to the tunneling current is not negligible,the spin polarization of the electrode is reduced below 0.6 in the case of usual3d-ferromagnetic metals and alloys. If only the highly spin-polarizedΔ1 statescoherently tunnel (Fig. 6.19b), a very high spin polarization and, thus, a veryhigh MR ratio are expected to appear. Such an ideal coherent tunneling inan epitaxial MTJ with a crystalline MgO(001) tunnel barrier is theoreticallypredicted as explained below. It should be noted here that the actual tunnel-ing through the amorphous Al-O barrier is considered to be an intermediateprocess between the completely incoherent tunneling represented by Julliere’smodel and the coherent tunneling illustrated in Fig. 6.19 b.
6.5.1 Theory of Coherent Tunneling Through a MgO(001)Tunnel Barrier
As shown in Fig. 6.20, a crystalline MgO(001) barrier layer can be epi-taxially grown on a bcc Fe(001) layer with a relatively small lattice mis-match of about 3%. This amount of lattice mismatch can be reduced bylattice distortions in the Fe and MgO layers and/or absorbed by formingdislocations at the interface. A coherent tunneling transport in epitaxial
Fe[010]
Fe[100]
Fe[001] MgO[001]
MgO[010] MgO[100]
(a) (b)
Fe[001] ,MgO[001]
Fe[010] ,MgO[110]
Fe[100] ,MgO[110]
0.14 nm
0.21 nm
aFe
aMgO
: Mg
: O
: Fe
Fig. 6.20. Crystallographic relationship and interface structure of epitaxial bccFe(001) / NaCl-type MgO(001): (a) top view and (b) cross-sectional view. aF e
and aMgO denote the lattice constants of bcc Fe and NaCl-type MgO unit cells,respectively
318 G. Reiss et al.
Fe(001) / MgO(001) / Fe(001) MTJ is schematically illustrated in Fig. 6.19 b.In the case of ideal coherent tunneling, Fe-Δ1 states are theoretically expectedto dominantly tunnel through the MgO(001) barrier by the following mecha-nism [96, 97]. For the k‖ = 0 direction ([001] direction in this case), in whichthe tunneling probability is the highest, three kinds of evanescent states, Δ1,Δ5 and Δ2′ , exist in the band gap of MgO(001). When the symmetries of tun-neling wave functions are conserved, Fe-Δ1 Bloch states couple with MgO-Δ1
evanescent states, as shown in Fig. 6.21 a. Figure 6.21b shows partial DOS(obtained by first-principle calculations) of the decaying evanescent states ina MgO barrier layer in the case of parallel magnetic alignment obtained byfirst-principle calculation [96]. Among these states, the Δ1 evanescent stateshave the slowest decay (and the longest decay length) of partial DOS in theMgO barrier. The dominant tunneling channel for parallel magnetic state isthis Fe-Δ1 ↔ MgO-Δ1 ↔ Fe-Δ1. Band dispersion of bcc Fe for the [001](k‖ = 0) direction is shown in Fig. 6.22a. The net spin polarization of Fe issmall because both majority-spin and minority-spin bands have many statesat EF . However, the Fe-Δ1 states are fully spin-polarized at EF (P = 1). Avery large TMR effect in the epitaxial Fe(001) / MgO(001) / Fe(001) MTJ istherefore expected. It should also be noted that even for anti-parallel magneticstates, a finite tunneling current flows. Tunneling probability as a function ofk‖ wave vectors (kx and ky) is shown in Fig. 6.23 [96]. For the majority-spinconductance in parallel magnetic state (P state) (Fig. 6.23a), tunneling takesplace dominantly at k‖ = 0 owing to the coherent tunneling of majority-spinΔ1 states. For the minority-spin conductance in P state (Fig. 6.23b) and theconductance in anti-parallel magnetic state (AP state) (Fig. 6.23c), spikes oftunneling probability appear at finite k‖ points called “hot spots”. The originof this “hot-spot tunneling” is resonant tunneling between interface resonantstates [96]. Although a finite tunneling current flows in AP state, the tunneling
(a) (b)
MgO(001)Fe(001) Fe(001)
2 3 4 5 6 7 8 109 11 12 13 14 15
1
10 5
10 10
10 15
10 20
10 25
Layer Number
Den
sity
of S
tate
s
Δ1 (spd)
Δ5 (pd)
Δ2’ (d)
MgO(001)
Δ1
Δ5
Δ2’
Δ1
Δ5
Δ2
Fe(001)
Δ1
Δ5
Δ2
Fe(001)
k//
kZ
Fig. 6.21. (a) Coupling of wave functions between the Bloch states in Fe and theevanescent states in MgO for k‖ = 0 direction. (b) Tunneling DOS of majority-spinstates for k‖ = 0 in Fe(001) / MgO(001)(8 ML) / Fe(001) with parallel magneticstate [96]
6 Magnetic Tunnel Junctions 319
0
1.0
2.0
1.0
Δ1↑Δ1↓ Δ1↑Δ1↓
Δ2’↑
Δ5↑
EF
EF
Γ Η Γ Ηkz kz
Ene
rgy
( eV
)
(a) (b)
Δ2’↑
Δ5↑
Fig. 6.22. (a) Band dispersion of bcc Fe in the [001] (Γ -H) direction. (b) Banddispersion of bcc Co in the [001] (Γ -H) direction redrawn from Bagayako et al. [98].Solid and dotted lines represent majority-spin and minority-spin bands, respectively.Thick solid and dotted lines represent majority-spin and minority-spin Δ1 bands,respectively. EF denotes Fermi energy. To compare relative levels of EF in bccFe and bcc Co with respect to the majority-spin Δ1 band, bottom edges of themajority-spin Δ1 band in (a) and (b) are aligned at the same energy level
conductance in P state is much larger than that in AP state, resulting in avery high MR ratio.
It should be noted that the Δ1 Bloch states are highly spin-polarizednot only in bcc Fe(001) but also in many other bcc ferromagnetic met-als and alloys based on Fe and Co. For example, band dispersion of bccCo(001) (a metastable structure) is shown in Fig. 6.22b. The Δ1 states in
Transmission
0.06
0.04
0.02
0-0.6 -0.4 -0.2 0
0.60.40.2-0.6
-0.4-0.2
0
0.60.4
0.2
0.03
0.02
0.01
0-0.6 -0.4 -0.2 0
0.60.40.2-0.6
-0.4-0.2
0
0.60.4
0.2
0.004
0.003
0.002
0-0.6 -0.4 -0.2 0
0.60.40.2-0.6
-0.4-0.2
0
0.60.4
0.2
0.001
Transmission
Transmission
kx
ky
kx
ky
kx
ky
(a)
(b)
(c)
P stateMajority spin
P stateMinority spin
AP state
Fig. 6.23. Tunneling probability in Fe(001) / MgO(001)(4 ML) / Fe(001) MTJ asa function of kx and ky wave vectors [96]. (a) Majority-spin conductance in parallelmagnetic state (P state), (b) minority-spin conductance in P state, (c) conductancein anti-parallel magnetic state (AP state)
320 G. Reiss et al.
bcc Co are fully spin-polarized at EF as in the case of bcc Fe. Accord-ing to first-principle calculations, a Co(001) / MgO(001) / Co(001) MTJexhibits TMR effect even larger than that of an Fe(001) / MgO(001) /Fe(001) MTJ [99]. Note also that a very large TMR effect is theoreticallyexpected not only for the MgO(001) barrier but also for other crystallinetunnel barriers such as ZnSe(001) [100] and SrTiO3(001) [101]. However, alarge TMR effect has never been experimentally observed in MTJs with thesecrystalline barriers, except for MgO(001), because of experimental difficultiesin fabricating high-quality MTJs without pin-holes and inter-diffusion at theinterfaces.
6.5.2 Giant TMR Effect in Epitaxial Magnetic Tunnel Junctionswith a Single-crystal MgO(001) Barrier
Since the theoretical predictions of a very large TMR effect in Fe / MgO /Fe MTJs [96, 97], there have been some experimental attempts to fabricatefully epitaxial Fe(001) / MgO(001) / Fe(001) MTJs [102, 103, 104]. Bowenet al. first succeeded in observing a relatively high MR ratio of 30% at RT(“1” in Fig. 6.18) [103]. However, the MR ratios did not exceed the highestvalue for the Al-O-based MTJs (70% at RT) as shown in Fig. 6.18. The maindifficulty at the early stage of experimental attempts was the fabrication ofan ideal interface structure like that shown in Fig. 6.20b. It was experimen-tally observed that Fe atoms at the Fe(001) / MgO(001) interface were easilyover-oxidized [105]. Results of first-principle calculations on the ideal inter-face and the over-oxidized interface are shown in Fig. 6.24 [106]. In the caseof the ideal interface (Fig. 6.24a), where there are no O atoms in the first Femonolayer at the interface, the Fe-Δ1 Bloch states effectively couple with theMgO-Δ1 evanescent states in the k‖ = 0 direction. In the case of the over-oxidized interface (Fig. 6.24b), where excessive oxygen atoms are located inthe interfacial Fe monolayer, the Fe-Δ1 states do not couple with the MgO-Δ1
: O : Mg : Fe
(a) (b)
Fe[001],MgO[001]
Fe[010],MgO[110]
Fe[100],MgO[110]
LDOS ofΔ1↑ states
ExcessiveO atom at interface
Fig. 6.24. Partial density of states at EF due to the majority-spin Δ1 states near theFe(001) / MgO(001) interface, redrawn from Zhang et al. [106]. (a) Ideal interfaceand (b) over-oxidized interface
6 Magnetic Tunnel Junctions 321
states effectively; this prevents coherent tunneling of Δ1 states and results ina significant reduction in the MR ratio. In this way, coherent tunneling is verysensitive to the structure of barrier / electrode interfaces.
Yuasa et al. fabricated high-quality fully epitaxial Fe(001) / MgO(001)/ Fe(001) MTJs by using MBE growth [33, 107]. A cross-section transmis-sion electron microscope (TEM) image of the MTJ is shown in Fig. 6.25.Single-crystal lattices of MgO(001) and Fe(001) are clearly identified in theTEM image. They succeeded in observing very high MR ratios of up to 180%at RT (“2” and “3” in Fig. 6.18), which exceeded the highest MR ratio forAl-O-based MTJs for the first time. Magnetoresistance curves of the epitax-ial Fe(001) / MgO(001) / Fe(001) MTJ are shown in Fig. 6.26a. The keyfor achieving such high MR ratios is considered to be clean Fe / MgO in-terfaces without over-oxidation. X-ray absorption spectroscopy (XAS) andx-ray magnetic circular dichroism (XMCD) studies revealed that interfacialFe atoms adjacent to the MgO(001) layer are not oxidized at all and have alarge magnetic moment, indicating that there are no oxygen atoms in the firstFe monolayer at the interface [108]. Parkin et al. fabricated highly-orientedpoly-crystalline (or textured) Fe-Co(001) / MgO / Fe-Co(001) MTJs usingsputtering deposition on a SiO2 substrate with a TaN seed layer, which wasused to orient the entire MTJ stack in the (001) plane, and they achieved highMR ratios of up to 220% at RT (“4” in Fig. 6.18) [32]. It should be noted thatthe fully epitaxial MTJs and the textured MTJs are basically the same froma microscopic point of view, if structural defects such as grain boundaries donot have a strong influence on transport properties. Fully epitaxial Co(001)/ MgO(001) / Co(001) MTJs with metastable bcc Co(001) electrodes werealso fabricated using MBE and were observed to show even higher MR ratios,i.e., above 400%, at RT (Fig. 6.26b, “6” in Fig. 6.18) [109] than those in theFe(001) / MgO(001) / Fe(001) MTJs as theoretically predicted by Zhang et al.[99]. The very large TMR effect in the MgO-based MTJs is called the “giantTMR effect”.
MgO(001)
Fe(001)
2 nm
Fe(001)
Fig. 6.25. Cross-section transmission electron microscope (TEM) image of epitax-ial Fe(001) / MgO(001)(1.8 nm) / Fe(001) MTJ [33]. The vertical and horizontaldirections correspond respectively to the MgO[001] (Fe[001]) axis and MgO[100](Fe[110]) axis
322 G. Reiss et al.
-200 -100 0 100 2000
100
200
300
MR
rat
io (
% )
H ( Oe )
293 K
20 K
MR = 245 %
MR = 180 %
-200 -100 0 100 2000
100
200
300
400
500
600
MR
rat
io (
% )
H ( Oe )
290 K
20 K
MR = 507 %
MR = 410 %
(a) (b)
Fig. 6.26. Magnetoresistance curves at room temperature and 20 K at bias voltageof 10 mV for (a) epitaxial Fe(001) / MgO(001) / Fe(001) MTJ [33] and (b) epitaxialCo(001)/MgO(001)/Co(001) MTJ with metastable bcc Co(001) electrodes [109].Arrows represent magnetization alignments. In these MTJs, the top ferromagneticelectrode layer is exchange-biased by an antiferromagnetic Ir-Mn layer
6.5.3 Other Phenomena Observed in Epitaxial Magnetic TunnelJunctions with a MgO(001) Barrier
The epitaxial MTJs with single-crystal MgO(001) barrier are a model systemin studying the physics of coherent spin-dependent tunneling because of thewell-defined crystalline structure with atomically flat interfaces. In additionto the giant TMR effect, other interesting phenomena, as explained below,have been observed in epitaxial MTJs.
The MR ratio of the epitaxial Fe(001) / MgO(001) / Fe(001) MTJswas observed to oscillate with respect to MgO barrier thickness, tMgO [33].Figure 6.27a shows the tMgO-dependence of a resistance-area product, RA,namely, tunneling resistance for a unit junction area (in units of Ω · μm2).The tunneling resistance increases exponentially with respect to the barrier
1.0 1.5 2.0 2.5 3.0101
102
103
104
105
106
107
P state (RP)
AP state (RAP)
RA
(Ω
·μm
2)
tMgO ( nm )1.0 1.5 2.0 2.5 3.0
0
50
100
150
200
250
293 K
3.0 Å(1.5 ML)M
R r
atio
( %
)
tMgO ( nm )
20 K
(a) (b)
Fig. 6.27. (a) MgO thickness (tMgO)-dependence of resistance-area product (RA)in parallel magnetic state (P state) and anti-parallel magnetic state (AP state) at20 K for epitaxial Fe(001) / MgO(001) / Fe(001) MTJs. (b) tMgO-dependence ofMR ratio at 20 K and 293 K for epitaxial Fe(001) / MgO(001) / Fe(001) MTJs
6 Magnetic Tunnel Junctions 323
thickness (tMgO), which is a typical tunneling character. The tMgO-dependenceof the MR ratio is shown in Fig. 6.27b. Surprisingly, the MR ratio was ob-served to oscillate as a function of tMgO with a single oscillation period of 0.30nm at both low temperature and room temperature. It should be noted thatsuch an oscillation of transport property has never been observed in regardsto MTJs with an amorphous Al-O barrier. It might be thought that the TMRoscillation is a result of the layer-by-layer epitaxial growth of MgO(001), inwhich the growth of one monolayer is almost completed before the growth of anew monolayer begins. This growth could cause an oscillation in the MR ratiobecause the interface morphology (atomic step density) changes periodically,layer by layer. This cannot be the origin of the observed TMR oscillation,however, because the oscillation period (0.30 nm) is not the thickness of themonoatomic MgO(001) layer (0.21 nm).
As an origin of the TMR oscillation, Butler et al. proposed an interferencebetween tunneling states [96]. Regarding the evanescent states at EF in MgO,an interference between two states, which correspond to Δ1 and Δ5 at k‖ = 0,could cause an oscillation of tunneling conductance as a function of tMgO. Thisoscillation is explained as follows. These states have complex wave vectors,the perpendicular components (z-components) of which are expressed as k1 =kr1 + iκ1 and k2 = kr
2 + iκ2. When k‖ ·Δz > 0.59 (Δz is an interlayer spacingof MgO(001)), kr
1 �= kr2 and κ1 = κ2 = κ [96], the tunneling transmittance T
The tunneling transmittance thus oscillates as a function of tMgO with a pe-riod proportional to 2π/(kr
1 − kr2). This transmittance oscillation could cause
an oscillation in the MR ratio. It should be noted that (kr1 − kr
2) is a functionof k‖. The oscillation, therefore, in principle could not have a single oscilla-tion period. However, it is also theoretically predicted that “hot spots” existunder both the P state and AP state (as shown in Fig. 6.23). The oscillation,therefore, could have a single period for k‖ of the hot spots. Although thedetailed mechanism is still not clear, the observed TMR oscillation could beevidence of coherent spin-dependent tunneling through MgO(001).
The epitaxial Fe(001) / MgO(001) / Fe(001) MTJ is also a model sys-tem used in studying interlayer exchange coupling (IEC) between two fer-romagnetic (FM) layers via an insulating non-magnetic (NM) spacer. IECin a FM / NM / FM structure with a metallic NM spacer has been exten-sively studied and is well known to show oscillations as a function of spacerthickness [110]. IEC for a metallic spacer is induced by conduction electronsat EF . Although similar IEC is also theoretically expected in a FM / NM/ FM structure with an insulating NM spacer (MTJ structure) [111], suchintrinsic IEC has not been observed in MTJs with an amorphous Al-O bar-rier. Faure-Vincent et al. first succeeded in observing intrinsic antiferromag-netic IEC in an epitaxial Fe(001) / MgO(001) / Fe(001) MTJ structure, in
324 G. Reiss et al.
which an extrinsic ferromagnetic magnetostatic coupling was superposed [112].Katayama et al. then obtained a more refined experimental result, in whichlittle extrinsic magnetostatic coupling was exhibited, as shown in Fig. 6.28[113]. Antiferromagnetic coupling was observed for tMgO < 0.8 nm. Withincreasing tMgO, the sign of IEC reverses at tMgO = 0.8 nm and graduallyapproaches to zero. Because of the atomically flat barrier / electrode interfaces(see Fig. 6.25), no extrinsic magnetostatic coupling was observed. The IEC fora MgO(001) spacer is, therefore, considered to be mediated by spin-polarizedtunneling electrons.
6.5.4 Giant TMR Effect in CoFeB/MgO/CoFeBMagnetic Tunnel Junctions
As explained in Sect. 6.5.2, fully epitaxial MTJs with a single-crystal MgO(001)barrier and textured MTJs with a (001)-oriented poly-crystal MgO barrier ex-hibit the giant TMR effect at room temperature, which is a desirable propertyfor spintronic applications such as MRAM and the read head of a hard diskdrive (HDD). However, the epitaxial and textured MTJ structures are notcompatible with the manufacturing processes of these devices for the follow-ing reason. MTJs for practical applications need to have the following basicstacking structure: seed layer / AF / SyF / tunnel barrier / free FM. Here,AF denotes an antiferromagnetic layer for exchange biasing. Ir-Mn, Pt-Mn,or related alloys are used as the AF layer. Free FM denotes the top ferromag-netic electrode layer, which acts as a free layer of a spin-valve. SyF denotes asynthetic ferrimagnetic structure (i.e., an antiferromagnetically coupled FM/ NM / FM trilayer such as Co-Fe / Ru / Co-Fe), which acts as a pinnedlayer of a spin-valve. This type of bottom structure is indispensable not onlyfor a robust exchange-bias on the bottom pinned layer but also for reducinga stray magnetic field acting on the top free layer. In practical MTJs, thebottom AF / SyF layers are based on a fcc structure with (111)-orientation.The problem concerning with this structure is that a NaCl-type MgO(001)
0.6 0.8 1.0 1.2 1.4 1.6 1.8-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
J(
erg
/ cm
2)
tMgO ( nm )
AF coupling
F coupling
Fig. 6.28. Interlayer exchange coupling (IEC) in epitaxial Fe(001) / MgO(001) /Fe(001) at room temperature [113]
6 Magnetic Tunnel Junctions 325
barrier and bcc(001)-oriented ferromagnetic electrode layers cannot be grownon the fcc(111)-oriented AF / SyF structure because of mismatch in structuralsymmetry.
To solve this growth problem, Djayaprawira et al. developed a new MTJstructure, CoFeB / MgO / CoFeB, by using sputtering deposition [114]. Cross-section TEM images of the MTJ are shown in Fig. 6.29. As seen in the high-resolution TEM image (Fig. 6.29a), the bottom and top CoFeB electrodelayers have an amorphous structure in an as-grown state. Surprisingly, theMgO barrier layer grown on the amorphous CoFeB is (001)-oriented poly-crystalline. Because the bottom CoFeB electrode is amorphous, the CoFeB /MgO / CoFeB MTJ can be grown on any kinds of underlayers by sputteringdeposition at room temperature. As shown in Fig. 6.29b, for practical appli-cations, the CoFeB / MgO / CoFeB MTJ can be grown on a standard AF/ SyF bottom structure. After post-annealing at 360◦C, this CoFeB / MgO/ CoFeB MTJ exhibited a giant MR ratio of 230% at RT (“5” in Fig. 6.18).Because this CoFeB / MgO / CoFeB structure is highly compatible with man-ufacturing processes, recent research and development on spintronic devicessuch as MRAM and HDD read head is based on this MTJ structure. Up tonow, MR ratios above 350% at RT have been achieved in CoFeB / MgO /CoFeB - MTJs [115].
The mechanism of the giant TMR effect in CoFeB / MgO / CoFeB MTJsis explained below. As illustrated in Figs. 6.19b and 6.21a, four-fold sym-metry of the crystalline structure in both the MgO barrier and the elec-trodes is essential for coherent tunneling under the Δ1 states. The giant TMReffect is therefore not theoretically expected for amorphous CoFeB electrodes.
(a)
(b)
SyF structure
Free layer
Pinned layer
AF layer forexchange bias
Fig. 6.29. Cross-section TEM images for CoFeB / MgO / CoFeB MTJ with asynthetic ferromagnetic (SyF) pinned layer and an antiferromagnetic (AF) layer forexchange-biasing underneath the MTJ part. (a) is a magnification of (b)
326 G. Reiss et al.
Yuasa et al. experimentally observed that the amorphous CoFeB adjacent tothe MgO(001) layer crystallizes in the bcc(001) structure by annealing above250◦C, as illustrated in Fig. 6.30 [116]. This type of crystallization processis known as “solid phase epitaxy”, in which the MgO(001) layer acts as atemplate for crystallizing the amorphous CoFeB layers due to the good lat-tice matching between MgO(001) and bcc CoFeB(001). The observed giantTMR effect in CoFeB / MgO / CoFeB can therefore be interpreted within theframework of the theory for epitaxial MTJs because the microscopic structureof bcc CoFeB(001) / MgO(001) / bcc CoFeB(001) MTJs is basically the sameas that of epitaxial MTJs.
6.5.5 Applications of the Giant TMR Effect
As explained in the previous section, the CoFeB / MgO / CoFeB MTJs show-ing the giant TMR effect are compatible with manufacturing process for spin-tronics devices because they can be fabricated on any kinds of underlayers bysputtering deposition at RT followed by ex-situ post-annealing. Besides giantMR ratios and manufacturing compatibility, industrial applications requirethe MTJs to satisfy many other factors such as small bias-voltage dependenceof MR ratio, high break-down voltage, and appropriate resistance-area (RA)products. CoFeB / MgO / CoFeB MTJs basically satisfy all the major require-ments for applications. As an example, the RA products of the MTJs are de-scribed below. Impedance matching in electronic circuits is indispensable forhigh-speed operations of devices. The RA product of MTJ should therefore beadjusted to an appropriate value to satisfy the impedance matching. MRAMapplications require RA in the range from 50 Ω · μm2 to 10 kΩ · μm2 depend-ing on the lateral MTJ size (i.e., areal density) of MRAM. In this RA range,giant MR ratios can be easily obtained, as shown in Fig. 6.27. On the otherhand, a very low RA, below about 1 Ω · μm2, is required for a read head of ahigh-density HDD with an areal recording density above 500 Gbit/inch2 (seeFig. 6.31). A high MR ratio above 50%, which is desirable for this application,
Amorphous CoFeB
MgO(001)
Amorphous CoFeB
As-grown MTJ
Post - annealingabove 250 ºC
After annealing
bcc CoFeB(001)
MgO(001)
bcc CoFeB(001)
(a) (b)Fig. 6.30. Schematic illustration of the structure of a CoFeB / MgO / CoFeB MTJin (a) as-grown state and (b) after annealing above 250◦C
6 Magnetic Tunnel Junctions 327
Recording densitiesabove 500 Gbit / inch2
0.1 1 100
50
100
150
200
250
MR
rat
io (
%)
at R
T
RA ( Ω·μm2 )
CPP-GMRAl-O MTJsfor HDD head
MgO MTJs
Fig. 6.31. MR ratio at room temperature vs. resistance-area (RA) product. Opencircles represent CoFeB / MgO / CoFeB MTJs [117]. Black areas represent Al-O-based MTJs and CPP-GMR devices. The Required zone for hard disk drives withrecording density above 500 Gbit/inch2 is indicated by the gray area
has never been realized using conventional Al-O-based MTJs and CPP-GMRdevices with ultra-low RA < 1 Ω ·μm2, as shown in Fig. 6.31. Using CoFeB /MgO / CoFeB MTJs, Nagamine et al. achieved both ultra-low RA, i.e., downto 0.4 Ω·μm2, and high MR ratios, above 50%, as shown in Fig. 6.31 [117]. Theultra-low resistance MgO-based MTJ is thus considered a promising candidatefor next-generation HDD read head.
The giant TMR effect is also useful in developing MRAM. In conventionalMRAM, the writing process (i.e. magnetization reversal of a free layer) usesa magnetic field generated by pulse currents, and the read-out process usesa resistance change between parallel and anti-parallel magnetic states (i.e.,TMR effect). The giant TMR effect enables high-speed read-out because thehigh MR ratio yields a high output signal for read-out [118]. In the conven-tional MRAM, however, the writing pulse currents increase by shrinking thelateral size of MTJs, which makes it difficult to develop gigabit-scale high-density MRAM. In a new type of MRAM, the so-called spin-transfer MRAM,on the other hand, the writing process uses the current-induced magnetiza-tion switching (CIMS) [30] (see also Sect. 6.3). This phenomenon is especiallyimportant in developing high-density MRAM because the writing pulse cur-rent flowing through the MTJ can be reduced by shrinking the lateral MTJsize. CIMS was first experimentally demonstrated in CPP-GMR devices andlater in Al-O-based MTJs [119]. CIMS in MgO-based MTJs, which is espe-cially important for MRAM, has been successfully demonstrated in CoFeB/ MgO / CoFeB MTJs [120, 121, 122]. An example of CIMS is shown inFig. 6.32. Switching between parallel and anti-parallel magnetic states is in-duced not only by applying a magnetic field (Fig. 6.32a) but also by sendinga pulse current through an MTJ (Fig. 6.32b). Based on the giant TMR effectand CIMS in MgO-based MTJs, a prototype spin-transfer MRAM was devel-oped as shown in Fig. 6.33, and reliable read-out and writing operations weredemonstrated [122]. At present, the intrinsic critical current density, Jc0, or
328 G. Reiss et al.
600
500
400
300
R(
Ω)
-2 -1 0 1 2
Ipulse ( mA )
P state
AP state600
500
400
300
R(
Ω)
-400 0 400
H ( Oe )
P state
AP state
(a) (b)
Fig. 6.32. (a) Magnetoresistance curve (R − H loop) and (b) current-inducedmagnetization reversal (CIMS) curve (R− I loop) at room temperature in a CoFeB/ MgO / CoFeB MTJ with a lateral size of 70 x 160 nm [120]
a pulse-current density with a pulse duration of 1 nsec necessary for CIMS(about 2× 106A/cm2) [31, 121, 122] is not small enough for developing high-density MRAMs. If Jc0 is reduced to about 5× 105A/cm2), it will be possibleto develop gigabit-scale high-density MRAM.
The MgO-based MTJs also have the potential for microwave-device appli-cations. Tulapurkar et al. demonstrated that a DC voltage is generated be-tween the two electrodes when an AC current with a microwave frequency ispassed through a CoFeB / MgO / CoFeB MTJ [123]. This phenomenon named“spin-torque diode effect” can be used for detecting microwaves. The spin-torque diode effect originates from the giant TMR effect and a resonant pre-cession of free-layer magnetic moment induced by spin-transfer torque [123].
1 nm
MTJ Bit Line
Transistors
10 nm
Pt-Mn
Buffer layer
Capping layer
MgO
CoFeB(pinned)
CoFeB(free)
Ru
CoFe
SyF
Fig. 6.33. Cross-section TEM images of 4 kbit spin-transfer MRAM using CoFeB/ MgO / CoFeB MTJs (courtesy of Sony Corp.) [122]
6 Magnetic Tunnel Junctions 329
6.6 Summary and Conclusions
Magnetic tunnel junctions are the basis for many magnetoelectronic applica-tions. Their magnetoresistance effect amplitude resulting from spin-dependenttunneling increased dramatically within a short time. MTJs based on amor-phous Al-O barriers and 3d-transition metal alloy electrodes such as Ni-Fe,Co-Fe and Co-Fe-B reach a TMR amplitude of around 70% at room tempera-ture. To increase the TMR further fully spin-polarized half-metallic electrodematerials are very attractive. Half-metallicity is suggested for a large numberof Heusler alloys (e.g., Co2MnSi, Co2FeSi and Co-Cr-Fe-Al). However, MTJsbasing on these electrode materials and Al-O barriers generally show a largetemperature and bias voltage dependence of the TMR. So far, the superior lowtemperature transport properties of the Heusler based MTJs with Aluminabarrier compared to MTJs with conventional 3d-alloy electrodes can not beconserved up to room temperature. By utilizing a new barrier material, a crys-talline MgO(001), giant TMR ratios above 400% at room temperature havebeen reached. It has been suggested, that coherent tunneling is responsible forthis strongly enhanced magnetoresistance. Furthermore, very low area resis-tance values can be reached with MgO based MTJs, which enables to switchthe magnetization of the electrodes not only by external magnetic fields butalso by spin-polarized tunnel currents transfering a spin torque (CIMS). Inapplications, output performance is roughly proportional to MR ratio. The im-provements of MTJs, especially the giant TMR effect in MTJs with MgO(100)barrier is thus expected not only to extend the applications of existing devicesbut also to realize novel spintronic applications such as high-density MRAMusing CIMS, field programmable logic circuits, high-performance magneticsensor for next-generation hard disk drives and biochips including magne-toresistive sensors and magnetic manipulation systems for sensing DNA orproteins.
Acknowledgment
Some of the authors (G. R., A. H., A. T. and J. S.) acknowledge the financialsupport by the Deutsche Forschugnsgemeinschaft (DFG) and the EuropeanUnion (EU). J.S. gratefully acknowledges the opportunity to work at theAdvanced Light Source, Berkeley, which is supported by the U.S. Departmentof Energy under Contract No. DE-AC03-76SF00098. S.Y. acknowledges thefinancial support by the New Energy and Industrial Technology DevelopmentOrganization (NEDO) of Japan.
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Abstract. We review the recent progress for spin injection into semiconductors.After discussing the physical background on the basis of the optical selection ruleswe describe spin injection via magnetic semiconductors and ferromagnetic metalcontacts. The concepts for optical detection of spin injection and the problem ofspin relaxation in the semiconductor are analyzed before we finally address spin -optoelectronic devices, namely the spin light-emitting diode (spin-LED) and thespin vertical-cavity surfaceemitting laser (spin-VCSEL).
7.1 Introduction
The principle of semiconductor devices will drastically change in the next 20years since quantum size effects will dominate with further decrease of struc-ture size. The semiconductor industry has recognized this upcoming revolu-tion but so far nobody knows the optimal concept for future quantum devices.Classical semiconductor devices use the exact control of charge by moving elec-trons and holes in the conduction and valence band but the charge might notbe the best adapted quantity for quantum devices since the coherence time ofthe electron space wave function is extremely short. In addition to the electri-cal charge, the electron also has a spin. This spin is from the principle pointof view particularly suitable for quantum devices since the spin is a quantummechanical entity that is under certain conditions extremely stable even atroom temperature. At the same time, the required energy for changing thespin orientation is in principle very small while the Coulomb charging energyper electron increases dramatically with decreasing structure size.
The utilization of the electron and hole spin for future semiconductor de-vices is a rapidly growing research field known as semiconductor spintronics.In 1990, Datta and Das suggested the first spintronic device which is nowa-days known as spin transistor [1]. Figure 7.1 schematically depicts its prin-
M. R. Hofmann and M. Oestreich: Ferromagnet/Semiconductor Heterostructures and
Fig. 7.1. Schematics of a spin transistor [1]. Spin polarized electrons (red arrows) areinjected from a ferromagnetic source contact into a semiconductor channel where thespin orientation is controlled by the electric field of a non magnetic gate contact andanalyzed by a ferromagnetic drain contact. The blue arrows schematically depict theeffective magnetic field that results from the electric gate field and the wave vectorof the electrons (from Ref. [2])
ciple of operation. Spin polarized electrons are injected from a ferromagneticsource contact. The spin orientation of the electrons is controlled during theirtransport to the drain contact by the electrical field of a gate contact whichintroduces a spin precession due to the Rashba Hamiltonian. The conductiv-ity of the device is high/low if the spin orientation of the electrons at thedrain contact is parallel/antiparallel to the spin orientation of the ferromag-netic drain contact at the Fermi energy. Thereby, the spin transistor switchesbetween high and low output solely by the control of the spin orientation.This kind of spin transistor is at the moment not applicable as real devicedue to several reasons but it shows nicely the principles and the prerequisitesof spintronic devices. One of the main prerequisites, which will be discussedin the following, is obviously the efficient injection and especially the efficientelectrical injection of spin polarized carriers from a ferromagnetic injector intoa semiconductor.
7.2 Spin Injection
7.2.1 Theory
The optical injection of spin polarized electrons into direct semiconductorswith finite spin orbit coupling is extensively studied in the literature andnearly 100% spin injection efficiency has been experimentally demonstrated[3]. Figure 7.2 shows schematically the optical selections rules for GaAs andoptical excitation in growth direction [4]. In bulk material, heavy and lighthole are degenerate and excitation with circularly polarized light yields in thissimplified picture 50% electron spin polarization P0 = ↑−↓
↑+↓ = 3−13+1 since the
transition matrix element of the heavy hole is three times stronger than the
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 337
T=3
T=1 T=1
T=3
3/2
1/2-
1/2-
1/2-
+ 3/2-
1/2+1/2+
1/2+
LH
HH
2D
3D
CB
VB
�-
�- �+
�+
Fig. 7.2. (left) Strongly simplified schematics of the optical selection rules in bulkGaAs (3D) and GaAs/AlGaAs quantum wells (2D) for right (σ−) and left (σ+)circularly polarized excitation in growth direction. Circular polarization of light andspin polarization are inevitable connected in this configuration. (right) Degree ofspin polarization for a 20 nm GaAs quantum well calculated by k ·p-theory. For highexcitation energies or high confinement energies in thin quantum wells, very fast spinrelaxation due to the large spin splitting of the conduction band can strongly reducethe spin polarization in the first picoseconds after excitation. (from [4])
transition matrix element of the light hole. In principle the holes are polar-ized, too but spin relaxation of free holes is usually extremely fast, i.e. onthe time scale of the momentum relaxation time, so that free holes can beconsidered as unpolarized in most experiments. In quantum wells, the degreeof spin polarization depends strongly on the excitation energy and accuratevalues must be calculated by an elaborate k · p-theory including excitonic ef-fects [4]. The right part of Fig. 7.2 depicts the calculated energy dependenceof the degree of spin polarization for a 20 nm GaAs quantum well and ex-citation in growth direction. The degree of polarization reaches for resonantexcitation of the lowest heavy hole transition nearly 100%, changes sign forresonant excitation of the light hole, and has a well pronounced structure dueto excitonic effects even at high energies. The calculated spin polarization isfor low excitation densities in excellent agreement with experiment. For highexcitation densities, bleaching of the absorption reduces the degree of spin po-larization [4]. The selection rules are valid for excitation and recombination,so that the degree of circular polarization of the photoluminescence from theheavy hole transition is a very good approximation to the degree of the elec-tron spin polarization, if unpolarized holes are assumed. The selection rulesdepend strongly on the direction of excitation and recombination. The degreeof circular polarization of the heavy hole transition in unstrained GaAs quan-tum wells is for example for in-plane excitation/recombination independentof the electron and hole spin polarization, i.e., the heavy hole quantum wellside emission is not suitable for measuring electron or hole spin polarization.
The electrical injection of spin polarized carriers from a ferromagneticmetal into a semiconductor is much more challenging than the optical spininjection, since the interface between metal and semiconductor plays a crucial
338 M. R. Hofmann and M. Oestreich
role. Schmidt and coworkers pointed out that the basic obstacle for spin injec-tion from a ferromagnetic metal into a semiconductor is the huge conductivitymismatch between these two materials [5]. They showed that the degree ofspin polarization in the semiconductor is nearly negligible for Ohmic contactsand typical device geometries and concluded that contacts with almost 100%spin polarization are needed for efficient spin injection. Only four monthslater, Rashba published a more detailed picture of electrical spin injectionpointing out correctly that tunnel contacts can solve the problem of electricalspin injection from a ferromagnetic metal into a semiconductor [6].
7.2.2 Experiment
Two major concepts have been suggested to realize electrical injection of spinpolarized carriers into semiconductors. The most obvious concept is to replacethe commonly used nonmagnetic metal contacts by ferromagnetic metals suchas Fe or Co. This concept with its potential and limitations will be discussedin the second paragraph. First, we address an alternative that is completelysemiconductor based and uses dilute magnetic semiconductors for spin injec-tion or spin alignment.
Spin Injection by Dilute Magnetic Semiconductors
As a consequence of the enormous progress in material science, the fabricationof new semiconductor materials with designed properties is a fast growing andsuccessful research field. In particular, the realization of semiconductor ma-terials with magnetic properties has become feasible by introducing magneticcomponents as, for example, Mn. [7] So called dilute magnetic semiconduc-tors are particularly attractive for spin injection because the conductivitymismatch problem discussed above does not appear. Two approaches havebeen successfully realized for spin injection with dilute magnetic semiconduc-tors: spin injection out of ferromagnetic semiconductors [8] and spin alignmentwith paramagnetic semiconductors [9].
The idea to use ferromagnetic semiconductors for spin injection is straight-forward but depends on the availability of ferromagnetic semiconductors ofsufficient quality. It has been shown that the introduction of a few % of Mninto GaAs is indeed possible and that GaMnAs becomes ferromagnetic undercertain conditions [10]. Ohno et al. have used a ferromagnetic GaMnAs layeron a GaInAs/GaAs light emitting diode (LED) structure to investigate spininjection. Figure 7.3a shows their sample structure. The GaMnAs is used asthe p-contact since GaMnAs is usually p-doped [7]. Accordingly spin polarizedholes are injected.
Figure 7.3b shows the experimental results at a temperature of 6 K andan external magnetic field of 1000 Oe (0.1 T) applied in in-plane direction.The degree of circular polarization of the LED emission out of the cleavededge of the sample (i.e. perpendicular to the growth direction) is spectrally
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 339
Fig. 7.3. (a) Schematic structure for spin injection with GaMnAs spin injector and(b) electroluminescence intensity and polarization degree as a function of energy(from Ref. [8])
analyzed in order to investigate the degree of spin injection. The degree of cir-cular polarization exhibits a peak coincident with the maximum of the LEDemission and at the quantum well ground state transition energy [8]. Theauthors report a background polarization which they attribute to a combi-nation of sample strain and experimental geometry [8]. In order to eliminatethese background effects, the changes in polarization are also investigated asa function of magnetic field.
Figure 7.4 shows the relative change in polarization as a function of mag-netic field for different temperatures. The maximum change in polarizationat T=6 K is of the order of 1%. For low temperatures a hysteresis behavior
Fig. 7.4. Relative change in light polarization degree as a function of magnetic fieldfor different temperatures. The inset shows the maximum polarization degree andthe magnetization of the GaMnAs layer (determined from SQUID measurements)as a function of temperature (from Ref. [8])
340 M. R. Hofmann and M. Oestreich
is observed which disappears for temperatures above 52 K. This disappear-ance is consistent with the Curie temperature of the GaMnAs layer whichwas determined to be 52 K from SQUID measurements. A comparison with areference structure without magnetic GaMnAs layer supports the claim of theauthors to observe spin injection. However, these results have been controver-sially discussed for several reasons. First, the sample geometry does not allowa clear interpretation of the results via the optical selection rules. In quantumwell structures, the selection rules as shown in Fig. 7.2 only hold for verticaldirection (i.e. with light emission and spin orientation parallel or antiparallelto the growth direction) and in the vicinity of k = 0. In addition, the GaMn-As contact injects holes instead of electrons but the hole spin relaxation isextremely fast.
Besides the serious problems concerning the interpretation of these results,an additional problem is the low Curie temperature of the GaMnAs layer ofonly 52 K. The need of cryogenic cooling is not attractive for future applica-tions and therefore, large material development effort has to be invested todevelop ferromagnetic semiconductors with a Curie temperature TC consid-erably above room temperature. Dietl and Ohno have analyzed the potentialcandidates for ferromagnetic semiconductors in detail [11]. Though TC canexceed 100 K in GaMnAs, a TC above room temperature can most probablynot be reached with this material [11].
Among the various candidates for ferromagnetic semiconductors currentlythe Nitrides like GaMnN [11, 12] and MnAs clusters in GaAs environment [13]seem to have the highest application potential due to their high Curie temper-atures and compatibility to existing optoelectronic semiconductor technology.The latter might be a critical issue for other ferromagnetic semiconductorswith TC above room temperature as, for example, ZnCrTe [14], Cr-dopedIn2O3 [15], CdMnGeP2 [16], or ZnMnO [11].
Instead of using ferromagnetic semiconductors for spin injection, one mightalso use paramagnetic semiconductors with large Zeeman splitting for spinalignment [17]. Fiederling et al. have successfully followed this approach [9].The left part of Fig. 7.5 shows their sample geometry. They use an GaAs/Al-GaAs LED structure with a paramagnetic BeMnZnSe layer in the n region.The huge g-factor due to super exchange interaction leads in BeMnZnSe to alarge Zeeman splitting in an external field at low temperatures [9]. Accord-ingly, the occupation of states with one particular spin orientation (dependingon the direction of the magnetic field) is strongly preferred and the electronspins are aligned when they pass this layer.
The right part of Fig. 7.5 shows the measured degree of circular polariza-tion of the light emitted in growth direction as a function of magnetic fieldin the same direction [9]. A maximum polarization degree of 43% is reportedwith a thick (300 nm) BeMnZnSe layer. The value is lower for a thin (3 nm)layer and disappears for a non-polarizing BeMgZnSe layer.
Though this successful proof for spin alignment with considerable polar-ization degrees of the optical emission, this concept also suffers from two
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 341
Fig. 7.5. (left) Device geometry of a BeMnZnSe spin aligner LED. (right) Thedegree of circular polarization of the electroluminescence for a 300 nm thick Be-MnZnSe spin aligner (squares), a 3 nm thick BeMnZnSe spin aligner (circles) andwith a non-polarizing BeMnZnSe layer (triangles). The crosses show the intrinsicpolarization degree of the GaAs layer (from Ref. [9])
drawbacks. First, the g-factor in BeMnZnSe dramatically decreases with in-creasing temperature such that room-temperature operation is unrealistic [9].Second, and more severe, this spin aligner concept requires large magneticfields of the order of 1–5 T to ensure considerable Zeeman splitting. Suchfields usually require external superconducting magnets which are a majorbarrier for practical applications.
Spin Injection by Ferromagnetic Metals
Spin injection from ferromagnetic metals is an alternative to using dilute mag-netic semiconductors and has other advantages and drawbacks. In contrast tothe magnetic semiconductors, most of the ferromagnetic metal injectors haveCurie temperatures above room temperature and thus spin injection at roomtemperature is feasible. However, the injection efficiencies with ferromagneticmetal contacts are usually considerably smaller than those of magnetic semi-conductor based injectors.
The first successful spin injection with ferromagnetic contacts at roomtemperature was published by Zhu et al. [18]. They used an 20 nm thick Fe filmepitaxially grown onto a GaAs/GaInAs LED structure in one MBE machinewith a separate chamber for the Fe growth. The device structure is shownschematically in the left part of Fig. 7.6 [18]. The spin injection efficiency wasinvestigated by analyzing the polarization degree and orientation of the LEDemission. However, the magnetization of the thin Fe layer is in plane withoutexternal field due to the shape anisotropy. As mentioned above, the selectionrules require a vertical orientation of the spins in the active layer in orderto achieve an unambiguous connection between spin polarization and circularpolarization of the light emission (which is also in direction perpendicularto the layer plane). That means that the magnetization of the ferromagneticmetal contact also has to be perpendicular to the layer plane. Accordingly,an external magnetic field has to be applied to tilt the magnetization in the
342 M. R. Hofmann and M. Oestreich
Fig. 7.6. (left) Spin LED with Fe injector. (right) Circular polarization degreeas a function of external magnetic field with (squares) and without (triangles) Fecontact at 25 K (from Ref. [18])
injector into vertical orientation. For Fe, a field of about 2 T is required toachieve complete vertical magnetization of the injector.
To investigate the spin injection, one usually analyzes the degree of circularpolarization of the LED emission. The right part of Fig. 7.6 shows the degreeof circular polarization of the spin LED at 25 K [18] as a typical example forsuch experiments. The effects due to spin injection are small and superim-posed by other contributions. To unambiguously isolate the contribution ofspin injection, the authors additionally analyzed a reference sample withoutferromagnetic Fe spin injector. This reference sample also exhibits circularlypolarized light contributions to the LED emission for nonzero magnetic field.These contributions are due to Zeeman splitting in the semiconductor anddon’t have anything to do with spin injection. Due to the Zeeman splitting ofthe spin up and spin down states there is a difference in occupation of stateswith different spin orientation, and thereby a slight preference for the emis-sion of light with one circular polarization occurs. Accordingly, even withoutferromagnetic contacts circularly polarized light is emitted in the presence ofa magnetic field. In the regime of interest, the degree of circular polarizationdue to Zeeman splitting varies linearly with magnetic field [18]. However, thisZeeman contribution is also present in structures with ferromagnetic spin in-jectors and will be superimposed to the spin injection component. This canclearly be seen in the right part of Fig. 7.6. Even without Fe contacts, the circu-lar polarization increases linearly with magnetic field and reaches a maximumvalue of below 1%. With Fe contacts, an additional contribution due to spininjection is superimposed. It increases strongly in the regime between –2 Tand 2 T and saturates at magnetic fields above ±2 T. At zero external fieldthe circular polarization degree is zero showing that indeed a high external
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 343
field is needed to achieve detectable spin injection in this geometry. Fields ofthe order of 2 T, however, require a superconducting magnet which introducesconsiderable complexity, in particular with regard to future applications. Themaximum circular polarization degree due to the spin injection componentis in this experiment 2%. It should be noted, however, that this value doesnot directly correspond to the spin injection efficiency. The spin injection ef-ficiency can be supposed to be higher because a considerable fraction of thespin oriented carriers injected into the semiconductor will loose their spin ori-entation due to spin relaxation effects. We will address this issue which is ofparticular relevance at higher temperatures approaching room temperaturein more detail below. However, Zhu et al. observe spin injection with circu-lar polarization degrees of about 2% (in saturation) up to room temperature.The circular polarization degree at room temperature measured as a functionof magnetic field is shown in Fig. 7.7. In particular for such small circularpolarization degrees, it is essential to unambiguously prove that the observedsignatures really arise from spin injection and not from artefacts as, e.g. mag-netoptic effects [19, 20] at the contacts. Zhu et al. therefore analyze the LEDemission at different spectral positions corresponding to the electron to heavyhole and electron to light hole transitions, respectively.
Considering the optical selection rules shown in Fig. 7.2 and assumingthat, for example, the +1/2 electron state is preferentially occupied due tospin injection, the heavy and light hole transitions from this state necessarilyhave opposite circular polarization: the transition into the +3/2 heavy hole isσ+ polarized and the transition into the –1/2 light hole is σ− polarized. Bothpolarizations change sign when the direction of the magnetic field is invertedand the electron –1/2 state is preferentially occupied. Accordingly, the heavy
Fig. 7.7. Circular polarization degree as a function of external magnetic field atroom temperature. The open sqares correspond to the light hole transition and thefilled squares to the heavy hole transition, respectively. (from Ref. [18])
344 M. R. Hofmann and M. Oestreich
and light hole transitions have opposite field dependence. This behavior isexactly seen in Fig. 7.7 and finally proves the spin injection.
Though this was the first successful realization of spin injection from aferromagnetic metal into a semiconductor, the concept still suffers from a fewdrawbacks. The first is the need for high external magnetic fields. This issuewill be addressed at the end of this section. The second is the Fe/GaAs inter-face. The growth conditions for the iron layer are difficult because at growthtemperatures considerably above room temperature interfacial componentsbetween Fe and GaAs might form which destroy the magnetic functionalityof the injector. Any heating of the device when it is operated might alsolead to the formation of such interfacial components and to a degradation ofthe device. Most of the groups working with Fe/GaAs or Fe/AlGaAs contactshave taken great care on the preparation of the interface by growing the wholestructure including the Fe layer in one single MBE machine [18, 21]. Though itcould be shown that successful spin injection can also be realized growing thesemiconductor and the ferromagnetic metal layer in separate machines witha well controlled transfer procedure,[22] large effort has been invested to findalternatives for the spin injectors which are more stable than the Fe/GaAs orFe/AlGaAs interfaces. A first attempt was to use MnAs as the ferromagneticmetal for spin injection [23]. Similar injection efficiencies as with the Fe injec-tors could be achieved. The problem of this approach is that MnAs only hasa Curie temperature of 40◦C. The use of Fe3Si provides injection efficienciesin the same range as MnAs and Fe but seems to be more promising becauseof its favorable thermal stability [24].
These first proofs for spin injection from ferromagnetic metals into a semi-conductor rebut the prediction by Schmidt et al. [5] who thought this conceptfor spin injection to be impossible. According to the analysis of Rashba [6], itis now commonly assumed that the spin polarized carriers tunnel through theSchottky barrier from the ferromagnetic Fe layer into GaAs. Detailed anal-ysis of the transport process over the Fe/GaAs interface by Hanbicki et al.supports the assumption that the carriers predominantly tunnel through theSchottky barrier [25]. Consequently, the quality and shape of the Schottkybarrier is crucial for the spin injection [26]. Figure 7.8 shows schematicallythe flat band conduction band structure for an Fe film on a GaAs/AlGaAsstructure [21].
The Schottky barrier has a triangular shape and its thickness is determinedby the doping profile in the semiconductor. Hanbicki et al. have optimized theSchottky barrier using an adapted doping profile in the semiconductor andsucceeded in considerably higher injection efficiencies [21]. They reported apolarization degree of the emission of more than 10% at low temperatures butobserved a strong decrease with rising temperature. Their analysis yields spininjection efficiencies of up to about 30% in their sample though this analysisand the exact values have been controversially discussed [27].
However, it was obvious from these first publications on spin injection withferromagnetic metal contacts that the interface between the ferromagnet and
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 345
Fig. 7.8. Schematic flat band conduction band structure of Fe on a GaAs/AlGaAsLED structure
the semiconductor and, in particular, the shape of the tunnel barrier is crucialfor the spin injection. Instead of using Schottky contacts as the tunnel barriers,tunneling through insulator based tunnel contacts has also been investigated[19]. A comparison between Fe Schottky contacts and Fe/Al2O3 tunnel barriercontacts provided slightly better results for the latter concept [28]. This findingis strongly supported by the work of Jiang et al. [29] who report a roomtemperature spin injection efficiency of 32% using a CoFe/MgO (001) tunnelinjector into GaAs/AlGaAs LED structures. The data of Jiang et al. are shownin Fig. 7.9.
At low temperatures, the polarization degree of the LED emission is above50% and even at room temperature 32% polarization was reported. In addi-tion to spin injection effects, the authors observe linear contributions to thepolarization as a function of magnetic field which they attribute to Zeemansplitting and to a magnetic field dependent spin relaxation process [29]. Thedata by Jiang et al. currently represent the published record value for spininjection at room temperature. It should be noted that postgrowth thermalannealing of the structures provided a substantial improvement of the spininjection efficiency [30]. This again confirms the enormous importance of theinterfaces for spin injectors.
Though considerable spin injection efficiencies have been achieved by op-timization of the tunnel injectors, one major drawback for applications stillremains. In all the work mentioned in this section so far, spin injection couldonly be achieved in the presence of significant external magnetic fields of theorder of 1 T to 2 T. This problem is due to the fact that the magnetization ofall injectors discussed so far is in plane whereas the selection rules require per-pendicular magnetization. With these common ferromagnetic injectors, thiscan only be achieved with external magnetic fields that turn the magnetiza-tion out of plane. For Fe contacts, for example, about 2 T are required toachieve maximum perpendicular orientation. An alternative which is partic-ularly attractive for applications is to use injector materials that exhibit amagnetization with a strong out of plane component even for zero externalfield. Such injectors could enable spin injection in remanence which would be a
346 M. R. Hofmann and M. Oestreich
Fig. 7.9. Magnetic field dependence of the electroluminescence of the GaAs /Al-GaAs LED with the CoFe/MgO (001) tunnel injector for 100 K (a) and 290 K (b).Sample I contains 8% Al, sample II 16%. The crosses in (a) show the linear Zee-man contribution. In (c) and (d) the linear contribution is subtracted from the dataof (a) and (b), respectively. The solid lines show the scaled SQUID magnetometerresults for the contacts (from [29])
remarkable progress because no superconducting magnet is necessary for spininjection in that case. Gerhardt et al. have recently suggested to use Fe/Tbmultilayers to turn the magnetization of the injecting Fe layer out of plane[22]. Figure 7.10 shows the device structure they investigated.
The LED structure used is a GaInAs/GaAs quantum well structure [22]similar to that used by Zhu et al. [18]. The spin injection is via a Schottkybarrier contact between a thin epitaxial Fe layer and GaAs. In contrast tothe work by Zhu et al. [18] the semiconductor structure and the Fe layer weregrown in separate MBE machines with a well controlled transfer procedure be-tween the machines [31]. On top of the Fe layer, an Fe/Tb multilayer sequenceis grown which causes a partially vertical orientation of the magnetization inthe injecting Fe layer [31].
Figure 7.11 shows the polarization degree of the LED emission at 90 Kas measured with a Stokes polarimeter [22] in comparison to the SQUIDmeasurement of the contact. The data on the polarization degree are correctedfor the linear Zeeman contribution [31]. The polarization degree reaches amaximum value of 0.8% for magnetic fields above 0.5 T. This value is lower
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 347
Fig. 7.10. Schematic structure of a Fe/Tb spin injector on a GaInAs/GaAs LED [22]
than that reported by Zhu et al. who showed polarizations of 2% [18]. However,one has to consider that the structures by Zhu et al. most probably exhibit amore perfect interface structure between Fe and GaAs due to the growth inonly one MBE machine.
The most important results of the work of Gerhardt et al. is that evenfor zero field, i.e. in remanence, a considerable polarization is observed [22].This confirms the first spin injection from a ferromagnetic metal contact into asemiconductor in remanence. The dependence of the polarization on magneticfield reproduces well the hysteresis loop of the magnetic layers as measuredby a SQUID magnetometer. The switching from one saturated magnetizationstate to the opposite takes place in less than 0.1 T, indicating an almost sin-gle domain state at low temperatures. To exclude artifacts, Gerhardt et al.also investigated the spectral dependence of the polarization. Like Zhu et al.,they proved spin injection with the inversion of the polarization orientationwhen detecting at the light hole transition instead of the heavy hole transi-tion [31]. Even at higher temperatures up to room temperature they reporthysteresis behavior of the emitted polarization degree confirming spin injec-tion in remanence up to room temperature. Recently, van‘t Erve et al. havealso reported spin injection in remanence but in a different geometry [32].They use an edge emitting AlGaAs/GaAs LED with an Fe contact for spininjection and demonstrate circular polarization degrees of about 5% in re-manence but at a low temperature of 20 K only. Though they observe aclear hysteresis behavior the interpretation of these data is rather difficultbecause the selection rules do not provide a clear connection between cir-cular polarization degree and spin polarization for quantum wells in thatgeometry.
348 M. R. Hofmann and M. Oestreich
-2 -1 0 1 2 3-1.5-1.0-0.50.00.51.01.5
(a)
(b)
Magn.moment(10
-4emu)
Magnetic field (T)
-0.8
-0.4
0.0
0.4
0.8
Circ.pol.degree(%)
Fig. 7.11. Polarization of the LED emission as a function of magnetic field (top) andSQUID measurement of the contacts (bottom) at 90 K. The circles are for the firstmeasurement series with the field varied from zero to –2 T, the inverted trianglesfor the field varied from –2 T to 2 T, and the triangles for the field varied from 2 Tto –2 T. The linear Zeeman contribution was determined with a reference sampleand subtracted [31]
7.3 Spin Relaxation
Optical detection is the most common tool for the analysis of spin injection likein the examples discussed in the last paragraph. However, one has to be awareof that the detection introduces a conceptional problem to the interpretationof the spin injection data. After spin injection, the spin polarized carriershave to be transported over distances of a few hundred nanometers to theactive region of the LED. This is crucial because the spin relaxation in thesemiconductor is so strong that, in particular at room temperature, all holesand also a considerable fraction of the spin polarized electrons have lost theirorientation before they recombine. Accordingly, the measured polarizationdegree is usually lower than the real spin injection efficiency. A quantitativeanalysis to determine the real injection efficiency is not straightforward andhas introduced controversial discussions [21, 27]. Moreover, when spintronicdevices are considered, electron spin relaxation processes always have to betaken into account. Therefore, we discuss this issue in more detail in thissection.
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 349
Electron spin relaxation in semiconductors is governed by five most rel-evant spin relaxation mechanisms named D’yakonov-Perel’ (DP), intersub-band spin relaxation (ISR), Bir-Aronov-Pikus (BAP), Elliott-Yafet, and thehyperfine-interaction mechanism. The D’yakonov-Perel’ spin relaxation mech-anism emerges in systems lacking inversion symmetry due to spin-orbit cou-pling and arises therefore for example in GaAs and ZnSe but not in Si [33].The starting point for the theoretical description of the DP spin relaxation isthe Dresselhaus-Hamiltonian for binary semiconductors
Hspin = Γ∑
i
σiki(k2i+1 − k2
i+2) , (7.1)
where i = x, y, z are the principal crystal axes with i + 3 → i, Γ is thespin-orbit coefficient for the bulk semiconductor, σi are the Pauli spin ma-trices, and k is the wave vector of the electron [34]. The comparison of theDresselhaus-Hamiltonian with the Hamiltonian for a free electron in a mag-netic field, H = 1
2
∑i μBσiBi, directly shows that the k-dependent spin split-
ting in (7.1) can be interpreted as a k-dependent magnetic field. The wavevector of the electron scatters randomly in time, leading to an effective mag-netic field that changes randomly in amplitude and direction. This randommagnetic field destroys the average of the spin orientation of an ensemble ofdiffusive electrons irrecoverable since individual spins precess around differentdirections and the momentum scattering annihilates the memory about theeffective magnetic field. The DP increases in bulk with increasing temperatureand in quantum wells with increasing confinement energy due to the occupa-tion of higher k states with larger spin splitting. The faster spin relaxation athigher temperatures is partially reduced due to motional narrowing effects, i.e.the spin lifetime is inversely proportional to the momentum scattering timedue to the faster change of the precession direction. As a consequence theDP mechanism is less efficient in low mobility samples. At room temperaturethe DP mechanism is usually the most efficient spin relaxation mechanism insemiconductors without inversion symmetry.
The direction of the effective magnetic field from the DP-mechanism isspecial for (110)-oriented quantum wells. The Hamiltonian from 7.1 reads in(110)-quantum wells
HDP = −Γσzkx[12〈k2
z〉 −12(k2
x − k2y)] , (7.2)
where z ‖ [110] is the confinement direction, x ‖ [110], y ‖ [001], 〈k2z〉 =∫ |∇Ψz|2dz, and Ψz denotes the z part of the electron wave function. The re-
sulting magnetic field fluctuates randomly in amplitude but points for all kin growth direction. Therefore, a spin in z or −z direction does not precess inthe random magnetic field, does not relax by the DP mechanism, and exhibitsvery long spin relaxation times at room temperature. Ohno et al. measuredspin relaxation times of several nanoseconds in 7.5 nm (110)-GaAs quantum
350 M. R. Hofmann and M. Oestreich
Fig. 7.12. (left) Temperature dependence of the electron spin lifetime τs for B =0 T (closed circles, spin along z) and B = 0.6 T (open circles, spin precesses aroundx) in a 20 nm (110)-GaAs quantum well. (from [36]). (right) Comparison of thespin relaxation rates due to the DP and the ISR mechanism measured in the same20 nm n-doped (110)-GaAs quantum well. The open circles show the spin relaxationrates for in-plane spins where the DP mechanism is dominant. The filled circlesshow the spin relaxation rates of for spins in z-direction, where the DP mechanism isabsent and ISR is dominant. The solid lines are calculations with a single momentumrelaxation time as fit parameter (from [37]). The ISR spin relaxation rate drops byone order of magnitude for a quantum well width decreased by a factor of two
wells at room temperature [35]. Figure 7.12 shows the temperature depen-dence of the spin relaxation in an n-doped 20 nm (110)-GaAs quantum wellmeasured by time-resolved photoluminescence [36]. The spin relaxation timeτs increases between 5 K and 120 K with increasing temperature since thecoupling of electrons and photo-created holes decreases. This spin relaxationdue to electron hole coupling is known as BAP spin relaxation and is dis-cussed in the next paragraph. Above 120 K the spin relaxation time decreasesagain with increasing temperature due to the intersubband spin relaxation(ISR) mechanism. The ISR is based on the Dresselhaus spin splitting andresulting spin flip transitions at intersubband transitions. The ISR spin relax-ation rate decreases drastically with decreasing quantum well width since theintersubband transition probability decreases.
The decrease of the electron spin relaxation rate with decreasing QW widthis only observed in n-doped (110)-GaAs QWs where the ISR mechanism isthe dominant spin relaxation mechanism. In photo-excited and especially inp-doped semiconductors, scattering in combination with exchange interactionbetween electrons and holes yields an efficient spin relaxation mechanism forconduction band electrons, as first pointed out by Bir, Aronov, and Pikus [38].Figure 7.13(a) depicts the room-temperature τs versus the QW confinementenergy. In contrast to ISR, τs strongly decreases with decreasing quantumwell width since the exchange interaction between electrons and holes, whichis governed by the Hamiltonian
H = AS · Jδ(r) , (7.3)
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 351
Fig. 7.13. (a) Room-temperature spin relaxation time τs of undoped (110)-GaAsquantum wells versus the confinement energy of the conduction electron Ee1 for lowexcitation density. (b) Room-temperature τs vs excitation density for a 7 nm andan 8.5 nm wide (110)-GaAs QW. (from [40])
increases with decreasing QW width. Here, A is proportional to the exchangeintegral between electron and hole, J is the angular momentum operator forthe holes, S is the electron-spin operator, and r is the relative position ofelectrons and holes [39]. Figure 7.13(b) depicts τs versus excitation density.The spin relaxation time increases with increasing excitation density since theelectron hole exchange interaction decreases with increasing density, i.e. theexcitonic electron hole coupling is less pronounced in a high density electronhole plasma.
Another important electron spin relaxation mechanism in semiconductorsand metals was pointed out by Elliott [41] and studied in detail by Yafet. Thesingle electron Bloch wavefunctions in a semiconductor are in the presenceof spin-orbit coupling no eigenstates of the Pauli matrix σz but mixturesof Pauli spin-up | ↑> and spin-down | ↓> states. These spin-up and spin-down states couple in the case of momentum scattering which leads to spinrelaxation. The EY spin relaxation time increases with momentum relaxationtime and can be approximated for conduction electrons with energy Ek inIII-V semiconductors by
1τs(Ek)
= A
(δso
Eg + δso
)2(Ek
Eg
)2 1τp(Ek)
, (7.4)
where Eg is the band gap energy and δso is the spin-orbit splitting. The numer-ical factorA depends on the dominant electron scattering mechanism (phonon,electron-electron, electron-hole, impurity). Equation (7.4) shows that the EY
352 M. R. Hofmann and M. Oestreich
mechanism is most important in small band gap semiconductors with largespin orbit coupling.
In most semiconductors also the nuclei have a finite spin and the magneticmoments of the electrons interact with the magnetic moments of the nuclei.This hyperfine interaction is extremely weak for free electrons since the elec-tron wave function averages over many nuclei. Electrons localized on donorsor in quantum dots average on the other hand only over a finite number oftypically 104−106 nuclei and hyperfine spin relaxation becomes important.Typical Larmor precession periods of these localized electrons due to the vari-ance of the nuclear magnetic field Bn are in GaAs at thermal equilibrium andfinite temperatures around 1 ns.[42]. These precessions can be reversed byspin-echo experiments. Temporal fluctuations of Bn lead to irreversible spindephasing, i.e. spin relaxation.
Spin relaxation in semiconductors is a multi-layered problem and dependson the semiconductor material, temperature, internal and external electricand magnetic fields, electron and hole density, confinement energy, disorder,scattering, the nuclear polarization, etc.. The spin relaxation times range fromfemtoseconds for free holes in semiconductors with large spin orbit couplingover a few ps in InGaAs quantum wells [43] or several nanoseconds in (110)GaAs quantum wells for electron spins at room temperature [35] up to nearlyseconds for electron spins in 28Si at low temperatures [44]. Knowledge of thespin relaxation mechanisms is therefore important for the development ofspin optoelectronic devices and yields the opportunity to design specific spinrelaxation times for specific spintronic devices.
7.4 Spin Optoelectronic Devices
The spin relaxation in the semiconductor is a major problem and makes thetransport of spin information over distances longer than a few micrometerin most semiconductors difficult or even impossible. The orientation of the(optical) polarization of a light field, in contrast, is much more stable thanthe spin orientation in a semiconductor. Accordingly, spin information couldpotentially be transported optically over long distances. Therefore, it mightbe attractive to use optoelectronics not only as a tool for detecting spin in-jection into semiconductors but to consider spin optoelectronic devices as analternative to pure electrical spintronic devices. In this section we will discussthe potential of two specific spin optoelectronic devices: the spin LED and thespin laser.
7.4.1 Spin LED
Figure 7.5 shows a typical spin LED device design. A spin LED is the moststraightforward concept for a spin optoelectronic device. The achievementsdiscussed above make room temperature operation of a spin LED feasible
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 353
when ferromagnetic metal contacts are used. Spin injectors based on dilutemagnetic semiconductors do not yet operate at room temperature and aretherefore not attractive for applications. The state of the art for room tem-perature spin injection concerning efficiency is defined by the work of Jianget al. [29]. They achieved an injection efficiency of 32% but required highexternal magnetic fields due to the in plane magnetization of their contacts.Combining their MgO (001) tunnel injector with the Fe/Tb multilayer fer-romagnetic contact by Gerhardt et al. [22] would potentially provide a spinLED with considerable injection efficiency and without need for high exter-nal magnetic fields. Polarization degrees of the order of 30% in remanencecombined with switching of the polarization orientation with less than 0.1 Tshould be available without major further development effort. However, muchhigher polarization degrees cannot be expected since the magnetic polariza-tion degree of the ferromagnetic metal injectors is considerably below 100%and the conversion from spin polarized carriers to polarized light in an LEDdoes not involve an intrinsic amplification process. But polarization degrees ofthe order of 30% will most probably be too low for practical applications, e.g.in communication technology. Such applications require effects close to 100%which, based on the current state of the art, cannot be reached with spinLEDs. Moreover, LEDs in general exhibit only very moderate modulationspeed and are therefore not attractive for high end information technologyapplications.
7.4.2 Spin Laser
Spin controlled semiconductor lasers have higher application potential thanspin LEDs but their realization still faces severe technological problems. Wewill address these problems at the end of this section and first discuss theconcept of a spin laser in principle.
The selection rules require a vertical geometry for spin controlled lasers.Therefore, the concepts for spin controlled lasers are all based on verticalcavity surface emitting lasers (VCSELs). Hallstein et al. showed that spinprecession in an external magnetic field can modulate the emission of an opti-cally pumped VCSEL structure, even with extremely high modulation speed[45]. The control of the emission of an optically pumped VCSEL by the pho-ton spin was also reported by Ando et al. [46]. These reports highlight thegeneral potential of spin VCSELs and accordingly, a device concept for anelectrically pumped spin VCSEL was suggested by Oestreich et al. [47]. Thisconcept is shown in Fig. 7.14 The idea is to combine a VCSEL with the spininjectors discussed above. If two injectors with opposite vertical magnetizationare used, one might be able to control the polarization of the VCSEL emissionsimply by switching between the two contacts. Before discussing details andpossible complications of this concept we analyze in more detail the generaladvantages of a spin VCSEL over a spin LED and over a conventional VCSEL.
354 M. R. Hofmann and M. Oestreich
Fig. 7.14. Schematic depiction of a spin VCSEL [47]. The circular polarization ofthe stimulated laser emission is switched between σ+ and σ− by electrical injectionof either spin-up or spin-down electrons
The fundamental advantage of a spin VCSEL over a spin LED is dueto the fact that a spin VCSEL is a highly nonlinear device, in particular atthreshold. Accordingly, a small spin polarization in the active region due tospin injection, i.e. a slight difference in carriers with opposite spin orientations,has different consequences than in an LED structure. In particular, wheninversion is reached, this difference leads to differences in the optical gainspectra for σ+ and σ− polarization as shown schematically in Fig. 7.15. Inthe particular situation of Fig. 7.15, the gain for the σ+ polarization reaches
Fig. 7.15. Gain spectra for σ+ and σ− polarization under spin injection. The dashedline shows the loss level
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 355
the loss level, which is equivalent to the laser threshold, while the gain forthe σ− polarization is below threshold. Accordingly, the laser starts operatingin σ+ polarization and the opposite σ− polarization is suppressed. Note thatthis, in principle, leads to a 100% polarization of the laser output with onlya small spin polarization in the active region.
Various all optical test experiments have been performed to analyze thisconcept in more detail. For that purpose, a VCSEL is excited optically withcircularly polarized light. According to the optical selection rules, one canachieve up to 100% spin polarization in the active region in a quantum wellsystem if only the heavy hole to conduction band transition is excited. How-ever, because of the stopband of the Bragg reflectors, one usually excites withconsiderable excess energy such that one heavy hole and one light hole transi-tion are excited with circular polarized excitation (see Fig. 7.2). In this case,one can achieve a theoretical maximum of about 50% spin polarization in theactive region because the heavy hole transition is three times stronger thanthe light hole transition. But one has to be aware that spin relaxation willreduce the maximum spin polarization considerably at room temperature.
However, by variation of the degree and orientation of the polarization ofthe excitation, one can practically control the spin polarization in the activeregion between zero and an upper limit of 50%. Hovel et al. and Gerhardtet al. have analyzed the conditions for spin control of an optically pumpedVCSEL in detail [48, 49]. They varied the polarization of the excitation with apulsed Ti:sapphire laser and analyzed the polarization degree and orientationof the VCSEL emission. A typical room temperature result of this study isshown in Fig. 7.16.
Fig. 7.16. Circular polarization degree (squares) of an optically excited VCSELversus input polarization orientation (stars). The solid line shows the theoreticalmaximum for the spin polarization in the active region. The experiment was doneat room temperature
356 M. R. Hofmann and M. Oestreich
The VCSEL polarization is obviously controlled by the polarization of theexcitation. Moreover, the VCSEL polarization is in most cases even higherthan the polarization of the excitation confirming that the nonlinearity of thelaser leads to an effective amplification of spin information. In detail, less than30% spin polarization in the active region is required to achieve a polariza-tion degree of the VCSEL emission of 100% and only 8% spin polarizationis sufficient to achieve an output polarization degree of 50% [48]. Note that30% spin polarization is already achievable by electrical room temperaturespin injection [29]. Gerhardt et al. have recently shown that spin control of aVCSEL also works with continuous wave excitation at room temperature, i.e.under realistic device conditions [49].
Before considering the technical aspects of an electrically pumped spinVCSEL it is worthwhile to think about further advantages and applicationpotential of the spin VCSEL concept in addition to the pure spin controlledemission discussed so far. First, a spin VCSEL can operate more efficientlythan a conventional VCSEL. For circularly polarized emission, only electronsof one spin orientation are required in the active region. For linear or unpolar-ized emission, in contrast, electrons with both orientations are required. Withother words, circularly polarized, spin controlled emission only requires halfof the electrons as compared to unpolarized or linearly polarized emission.Accordingly, the threshold of a spin VCSEL should be considerably reduced.Indeed, Rudolph et al. have demonstrated this effect [50, 51, 52]. The thresh-old reduction is significant at low temperatures. At room temperature, thereis still a threshold reduction for a spin laser as shown in Fig. 7.17 but thereduction is less pronounced so far due to the fast room temperature spinrelaxation in (100) GaAs QWs.
A further important aspect is that spin controlled VCSELs should en-able extremely high modulation speed. This holds in particular when mod-ulation between the opposite circular polarizations in envisioned. Hallsteinet al. demonstrated that modulation frequencies of up to 120 GHz might
Fig. 7.17. Integrated VCSEL emission as a function of average pump power forrandom spins (triangles) and aligned spins (circles) [50]
7 Ferromagnet/Semiconductor Heterostructures and Spininjection 357
be accessible with spin control. In their particular experiment, this wasachieved at low temperatures by fast spin precession in an optically pumpedVCSEL [45].
To make use of all these advantages for real applications, optical excitationhas to be replaced by electrical spin injection. Though, in principle, sufficientspin injectors are available as discussed in earlier sections, several technicalcomplications arise in the details of the realization of an electrically pumpedspin VCSEL. A first major problem for the realization of electrically pumpedspin VCSELs is that the output polarization of a VCSEL is mostly determinedby geometrical factors. Though electrically pumped VCSEL structures usu-ally have cylindrical shape, and thus no particular polarization orientation ispreferred, the output polarization of most devices is linear and exhibits com-plicated dynamics including, for example, polarization switching. This com-plicated behavior is due to inhomogeneities, strain, or birefringence caused bythe internal electrical fields [53, 54]. For spin controlled VCSELs a first butmajor step is to remove or balance these sources for polarization imprint toenable polarization control only by the spin orientation of the carriers in theactive region.
The second and even more severe technological problem is the long trans-port path of the carriers (electrons) from the spin injector into the active re-gion. In electrically pumped VCSELs, the injection is usually either throughthe Bragg mirror layers or through lateral contacts circumventing the mir-rors. The first concept will most probably fail for spin VCSELs because thetransport path is a few micrometers long and passes several heterointerfaces.Accordingly, spin relaxation will be so strong that no usable spin orientationreaches the active region. Lateral contacts, in contrast, remove the need forpassing many interfaces but still, the transport paths are in the few μm range.Clever device concepts will therefore have to be developed to really access thehigh application potential of electrically pumped spin VCSELs. A possibleconcept is to use semiconductor spin aligners which can be introduced intothe semiconductor with small enough separation from the active region. Butthe state of the art spin aligners do not operate at room temperature. Thatreduces the attractiveness of this particular approach considerably though afirst attempt to realize a spin VCSEL was based on this concept [55]. How-ever, this device operated at low temperature with high external magneticfield and low efficiency only. Moreover, it has been controversially discussedwhether the observed polarization is due to injection of spin polarized holesor due to reabsorption effects in the GaMnAs spin aligner [56].
7.5 Summary and Outlook
Electrical spin injection into semiconductors is an intricate but most impor-tant problem in semiconductor spintronics. While optical spin injection is wellunderstood and yields spin injection efficiencies of up to 100%, electrical spin
358 M. R. Hofmann and M. Oestreich
injection is still a challenge. Paramagnetic semiconductors yield electrical spininjection efficiencies which are comparable to optical efficiencies but only atlow temperatures and high magnetic fields. Spin injection by ferromagneticdilute magnetic semiconductors is from the device point of view probably thebest approach, but high degrees of electron spin polarization have not beendemonstrated so far. Several dilute magnetic semiconductors or ferromagneticmetal clusters in semiconductors exhibit apparently high Currie temperaturesbut either the electrons responsible for transport are not spin polarized so faror holes mediate the ferromagnetism and the spin relaxation times of freeholes is much too fast for most devices. Metallic ferromagnet/semiconductorheterostructures yield at the moment the highest degrees of electron spin po-larization at room temperature and are very promising for several spintronicdevices. However, spin relaxation constrains the distance between spin injec-tor and the active spintronic region and metallic spin injection contacts arein first approximation limited to the semiconductor surface. Thereby metal-lic spin injectors are less flexible than their potential counterpart, the ferro-magnetic semiconductor spin injector, and are therefore for several spintronicdevice geometries not of practical interest.
Acknowledgement
The authors thank N.C. Gerhardt and S. Hovel for stimulating discussions.M.R.H. thanks the German Science Foundation for support within the SFB491. M.O. thanks the German Science Foundation and the BMBF for financialsupport.
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