3Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg,Maryland 20899-6202, USA
4Maryland NanoCenter, University of Maryland, College Park, Maryland, 20742, USAReceived 25 September 2007; revised manuscript received 29 December 2007; published 2 April 2008
A detailed first-principles analysis of the transport properties of different magnetic electrode materials forMgO tunnel junctions is performed to elucidate the microscopic origin of the tunneling magnetoresistanceTMR effect. The spin-dependent transport properties of the magnetic materials are analyzed separately fromthe particular interface geometry with the tunneling barrier. We use the bulk properties of the barrier to identifythe important tunneling states. For MgO these are 1-like states. From the analysis of this effective spinpolarization we can predict the potential of certain magnetic materials to create a high TMR ratio in a tunneljunction. This polarization is as high as 98 and 86 % for Fe and Co, respectively for only a few monolayers,but is very small and negative, −7 %, for amorphous Fe. This explains the finding that for crystalline Co andFe one monolayer next to the MgO barrier is sufficient to reach TMR ratios higher than 500 % independent ofwhether the crystalline monolayer is coupled to a non-magnetic or to an amorphous lead. However, in directcontact with MgO amorphous Fe reduces the TMR ratio drastically to 44 %.
The effect of tunneling magnetoresistance TMR hasbeen a focus of research since its rediscovery by Moodera etal.1 and Miyazaki et al.2 in 1995. Julliere,3 who analyzed thiseffect in 1975, observed that the tunneling current through alayered system consisting of two ferromagnetic electrodesseparated by an insulating barrier depends on the relativeorientation of the magnetization of the ferromagnetic layersto each other. Different currents are measured for parallel Pand for antiparallel AP alignment of the lead magnetiza-tions. To quantify this effect different TMR ratios are definedin the literature. The most common one is the optimisticTMR ratio given by
RAP − RP
RP=
gP − gAP
gAP, 1
where gP RP and gAP RAP are the conductances resis-tances in the parallel and antiparallel configuration. Typicalapplications exploiting the TMR effect are hard disk readheads4,5 and magnetic random access memories MRAM.6,7
Three properties of a tunnel junction are important for indus-trial usage. First, a high TMR ratio is necessary. Second, thetunnel junction has to have a practicable signal to noise ratioand third, the device design has to allow for a large-scaleproduction at low costs. In the last decade the TMR ratio oftunnel junctions with crystalline MgO barriers was increasedremarkably and reached values above 300 % Refs. 8–11 atroom temperature. In contrast, the values for systems withamorphous aluminum oxide Al2O3 barriers are still lessthan 100 %.12,13 In theoretical investigations of systems withcrystalline MgO barriers TMR ratios beyond 1000 % werepredicted.14,15 The discrepancy between experimental andtheoretical results could be explained in terms of structuralimperfections in the experimental systems in contrast to theideal systems of the theoretical investigations. Further struc-
tural analysis9,16,17 of tunnel junctions showed that even withpartial structural disorder in the magnetic layers consisting ofCoFeB, a TMR ratio of 230 % Ref.16 is realizable. Inaddition, a crystalline magnetic CoFe layer forms betweenthe MgO barrier and the otherwise amorphous electrode un-der the influence of annealing.9 Recent ab initio calculationsused semi-infinite ideal leads but do not consider partialstructural disorder and finite thickness of the ferromagneticelectrode layers.14,15,18–24 Only disorder at the Fe /MgOinterface18,22 or in the MgO barrier25 was analyzed. The aimof this article is to present a detailed discussion of the mi-croscopic origin of tunneling under the influence of struc-tural disorder and finite thickness of the ferromagnetic elec-trodes. In the present paper we expand on our earlieranalysis26 and consider additional lead materials.
In this work three important experimental results concern-ing the structure of a tunnel junction are taken into accountas sketched in Fig. 1. First, the tunnel junction is embeddedbetween semi-infinite nonmagnetic reservoirs to account forthe finite thicknesses of the magnetic layers. Second, we in-clude a layer with structural disorder represented by amor-phous iron a-Fe. Third, there is a crystalline magnetic layerof finite thickness next to the barrier represented by a finitenumber of Fe or Co monolayers in a bcc structure.
The paper is organized as follows. After a short introduc-tion of our theoretical methods, the following sections are
. . . . . .
bcc-Fe/bcc-Co
MgO
a-Fe a-Fe
non-magnaticreservoir
non-magnaticreservoir
FIG. 1. Color online Schematic structure of the investigatedtunnel junctions.
separated in two parts. The first part Sec. II covers theresults of the amorphous iron. In particular, the atomic struc-ture simulation, the electronic and magnetic structure calcu-lation, and the analysis of the transport properties of amor-phous iron is discussed in this part.
The second part of the article includes all sections con-cerning the TMR effect. In Sec. III we review the basic prin-ciples of the microscopic origin of the TMR effect. We ex-tend this part by the introduction of special spin polarizationsof the current P1
and Peff to discuss the potential of dif-ferent magnetic materials to lead to high TMR ratios in atunnel junction with crystalline MgO in the following sec-tion. In addition we present results of conductance calcula-tions of different magnetic materials embedded between non-magnetic reservoirs to analyze the different currentpolarizations. As magnetic materials we consider crystallineiron bcc-Fe, crystalline cobalt in a iron structure bcc-Coand amorphous iron a-Fe. In Sec. V we compare theseresults to our ab initio calculations of tunnel junctions usingthese magnetic materials.
I. METHODS
The theoretical method used for the electronic structurecalculation is a screened Kohn-Korringa-Rostoker KKRGreen’s function method.27,28 It is based on density func-tional theory DFT in the local spin density approximationLSDA. The potentials were treated in the atomic sphereapproximation ASA using a cutoff for the angular momen-tum expansion of the Green’s function of lmax=3. The mag-netic moments are forced to be collinear. For the conduc-tance calculations the Baranger-Stone formalism29 in theKKR Green’s function expression30 was used. This formal-ism is equivalent to the Landauer approach31 and the conduc-tance is expressed in the zero bias limit as
g = g0
T = g0 d2kTk ,
gPAP = g↑↑↑↓ + g↓↓↓↑, 2
where T is the spin-dependent transmission probability, g0=e2 /h denotes the universal conductance quantum, and isthe spin index. The in-plane wave vector k is used becauseof the translational invariance parallel to the interface. Theintegration has to be performed over the surface Brillouinzone. The transmission, which is energy dependent in gen-eral, has to be taken at the common Fermi level. The spin-dependent conductance in the parallel magnetic configura-tion of the tunnel junction is labeled by g↑↑ and g↓↓ for themajority and minority spin channel, respectively. For the an-tiparallel configuration the conductances are labeled g↑↓ andg↓↑. For the conductance calculations the structure was em-bedded between nonmagnetic semi-infinite leads to accountfor the open boundary conditions. This is done by the deci-mation technique implemented in the KKR code.32,33 For thecalculation of the conductance we use the same angular mo-mentum cutoff of lmax=3 as in the self-consistent calcula-tions.
II. AMORPHOUS IRON
In this section we present the results of the structuralsimulation and the corresponding electronic properties ofamorphous Fe a-Fe. In addition, we show that a-Fe is anOhmic conductor on the analyzed length scale. There havebeen a number of experimental investigations of amorphousFe.34–41 Since the structure is not stable as bulk phase it wasinvestigated as a thin film,35,36 or stabilized by Co and Bimpurities,34,41 or as a substitutional alloy with B.40 There-fore, the experimental results differ strongly. For example,the average magnetic moment ranges between 1.0 B and1.5 B.
Ab initio electronic structure calculations of a-Fe arerare42–46 since typical ab initio methods are well suited forcrystalline systems but not for the description of amorphousmaterials. A possible way to treat amorphous systems is thesupercell method. The numerical expense of this procedure ishigh and restricts the size of the supercells.
To describe the atomic short range order of a-Fe the paircorrelation function
Gr =1
N
Nr4r2r
3
is used,35,36,41 where N is the average particle density andNr the number of atoms in the spherical shell between rand r+r, if one atom is situated in the center of the sphere.For comparison we use the experimentally obtained pair cor-relation function from the analysis of a thin film35 with adensity similar to the crystalline system 7.9 g cm−3. Thecalculated pair correlation function of large periodic systemscontaining 16 to 108 atoms per unit cell was fitted to theexperimental one by a Monte Carlo algorithm. Figure 2shows Gr of a supercell of 16, 27, and 108 atoms in com-parison to the experimental result given by Ichikawa.35 Weconclude that a supercell with 16 atoms is well suited todescribe the structural properties of the amorphous system
0 2 4 6 8 100
1
2
3
4
r ( )Å
G(r
)
experiment [34]
16 atoms
27 atoms
108 atoms
FIG. 2. Color online Calculated pair correlation function Grfor supercells with 16, 27, and 108 atoms in comparison with anexperimental one Ref. 35.
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up to the third-nearest neighbor shell in agreement with othertheoretical investigations.46 Therefore, we use this supercellin our electronic structure calculations.
Figure 3 compares the calculated density of states DOSof the supercell with 16 atoms with the DOS of crystallinebulk Fe. The main features of the DOS of bulk Fe are broad-ened because of the structural disorder and the weights at theFermi level are changed. These results are in very goodagreement with the calculations of Turek et al.42 who used alarger supercell with 64 atoms. The conductance is deter-mined by the states at the Fermi level and it can be con-cluded from the DOS that the transport properties of a-Fewill be different from that of crystalline Fe.
The average magnetic moment of a-Fe is 1.7 B which ishigher than the experimental results between 1.0 B and1.5 B.37,39,41 Turek et al.42 found a lower magnetic momentof 1.3 B, even though they applied also the LSDA. Thereason for different average magnetic moments is the collin-ear treatment which leads to self-consistent results sensitiveto the used atomic configuration and several energy minima.In Fig. 4 the distribution of the local magnetic moments ispresented. The collinear magnetic ground state is character-ized by moments around 2 B and only a few negative mag-netic moments which agrees with the results of Turek et al.42
and is comparable to noncollinear calculations.43,44,46 Al-though the average magnetic moments in collinear and non-collinear configurations are quite similar43,44,46 the transportproperties can differ significantly47 and have to be investi-gated separately. For the tunnel junction we will show thata-Fe in direct contact to the barrier reduces the TMR effectstrongly because of a drastic decrease in the spin polarizationof the tunneling states. Introducing noncollinear magneticorder leads to a further reduction of the spin polarization andtherefore to a lowering of the magnetoresistance effect.48,49
For the aim of our investigation it is not mandatory to takenon-collinear magnetic order into account because the struc-tural disorder itself causes already low TMR ratios withamorphous Fe as electrode material.
For the calculations of the electronic transport throughamorphous Fe, the potential of an amorphous layer with 5 aFe
thickness was calculated self-consistently in a supercellwhere aFe=2.87 Å is the lattice constant of crystalline Fe.First of all, the conductance through this cell embedded be-tween semi-infinite crystalline Fe electrodes was computed.The insertion of the amorphous layer decreases the conduc-tance of pure crystalline Fe. Figure 5 shows the conductancemap Tk of both spin channels for the crystalline Fe incomparison to the system with one and seven amorphouslayers. The Brillouin zone BZ in this representation is de-termined by the in-plane dimension of the supercell 2aFe2aFe. The existence of an in-plane BZ is an artifact of thesupercell description and does not exist for a real amorphoussystem. Nevertheless, the analysis of this artificial BZ re-veals the underlying physics. The arguments of Sec. V willbe even stronger for a real amorphous system which has avanishing BZ. The structural disorder of the supercell leadsto a broadening of the well defined integer transmission val-ues of the crystalline system given by the number of Blochstates at a certain k point. For the amorphous structure thecontributions are noninteger values which is expected sincethe matching is disturbed with respect to the crystalline sys-tem by structural disorder. In a next step we calculate thethickness dependence of the resistance given by the inverseconductance R=1 /g. No thickness dependence exists for apure crystalline system without interfaces since the numberof Bloch states is conserved and only the Sharvin resistanceR0 is obtained.31 Structural disorder should, however, causea thickness dependence of the resistance. To reduce the nu-merical effort n cells of the self-consistent potential of theamorphous supercell were embedded between semi-infinitecrystalline Fe electrodes. The results presented in Fig. 6show that the resistance area product for both spin channelsincreases nearly linear with increasing number of amorphouslayers on this length scale. The same behavior is obtained forthe total resistance area product. Fitting a linear function tothe calculated values the resistivity is given by 0.6 m,which is in good agreement with experimental values of1 m.36–38 The conclusion of this section is that a smallsupercell of 16 atoms is large enough to describe the mainproperties of amorphous Fe. Within this length scale this
-6 0 6
-2
0
2
E-E (eV)F
bcc-Fe
a-Fe
majority spin
minority spin
DO
S(e
Vat
om
)-1
-1
FIG. 3. Color online Calculated spin-dependent DOS foramorphous Fe a-Fe simulated by a supercell with 16 atoms incomparison with bcc-bulk Fe.
-2 -1 0 1 20
1
2
3
4
5
6
7
8
m ( )lok B
n
FIG. 4. Color online Histogram of the local magnetic momentsof amorphous Fe a-Fe within a supercell of 16 atoms.
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material represents an Ohmic conductor with a calculatedresistivity comparable to experimental results. The contribu-tions Tk in the conductance map are broadened with re-spect to crystalline Fe by structural disorder. In the followingwe will use this material characterized by one 16-atom su-percell to introduce structural disorder in magnetic tunneljunctions. We restrict ourselves to only one configuration ofthis 16-atom supercell because the averaging over a numberof supercells is numerically very demanding. The generalconclusion is expected to be very similar. We evaluate theelectronic structure of different amorphous supercells andgot comparable results for the density of states.
III. BASIC PRINCIPLES
In this section we review the current understanding of theTMR effect. For this purpose we summarize briefly two dif-ferent approaches to describe the TMR effect.
The first approach is the Julliere model3 which is onlyvalid in the diffusive limit of transport. In this limit thek-parallel conservation in planar tunnel junctions is stronglyviolated. For the Julliere model the spin polarizationP=
n↑−n↓n↑+n↓
of the density of tunneling states in the electrodesis used to estimate the TMR ratio, where n↑ and n↓ are usu-ally approximated by the density of states at the Fermi level
FIG. 5. Color online Spin-dependent con-ductance maps Tk of crystalline Fe a, withone b, and seven c layers of amorphous ironembedded between crystalline Fe.
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for majority and minority electrons, respectively. Using Eq.1 the TMR ratio is given by
2P2
1 − P2 4
assuming identical leads. Typical spin polarizations of tun-neling currents for ferromagnetic materials through an Al2O3barrier into a superconducting electrode are limited to valuesof 44 % Refs. 50 and 51 which causes a TMR ratio of lessthan 50 %. This is the case for tunnel junctions with amor-phous barriers. However, the MgO barrier can be grown epi-taxially and in this case the spin polarizations of Fe onlycannot be used. The important states for the tunneling currentthrough crystalline MgO have to be selected. Parkin et al.have performed measurements for the polarizations of tun-neling currents through crystalline MgO into a superconduct-ing electrode and have found polarizations as high as 74 and85 % for Fe and CoFe as the magnetic electrode,respectively.17 From these values a quite high TMR ratio upto 520 % is derived within the Julliere model.
In contrast, theoretical predictions for systems with crys-talline MgO barriers are even higher than 1000 %.15,14 Forthe explanation of such high values coherent tunnelingthrough the MgO barrier has to be considered. As Butler etal.15 and Heiliger et al.52 discussed the complex band struc-ture of MgO leads to a symmetry selection of the barrier. The
description of Butler et al.15 is only valid at the point,where the states can be classified by their symmetry charac-ter. Heiliger et al.52 have extended this description to the fullBZ. We adopt their nomenclature to call the band with a state
of 1 symmetry at the point 1-like band. We call thestates belonging to this band 1-like states. Only these statescan tunnel around the center of the BZ effectively throughthe barrier. States of other bands and a different symmetry
character at the point experience a stronger damping by theMgO. A high TMR effect is expected if the electrodes trans-form the symmetry selection of the barrier via their exchangesplitting into a spin-filtering effect. The ideal case would be
to have 1-like states around the center of the BZ for one
spin channel only. For Fe at the this is full filled in 100direction.53,54 For a finite part of the BZ it is illustrative toanalyze the k-resolved transmission probability Tkthrough the magnetic materials for this property.
The details of the conductance maps Tk for a realtunnel junction depend on the interface between the mag-netic electrodes and the barrier. Even relatively smallchanges of the interface structure can influence the corre-sponding tunneling states strongly.20 Especially interfacestates or resonances can affect the tunneling current but theyare essentially important for MgO thicknesses smaller thansix monolayers.21 As discussed in the literature the influenceof these states strongly depends on the position relative tothe Fermi level.14,15,21,23,24,55 In this article we do not want toanalyze their influence.
However, to prove the ability of a magnetic system toserve as a lead material in combination with the MgO barrierin a tunnel junction, an analysis that combines the Jullieremodel with the symmetry selection of the barrier is useful.This means first, to restrict the polarization in the Jullieremodel to the current-carrying 1-like states P1
and sec-ond, to restrict the analysis to states close to the center of theBZ Peff. The latter point takes into account that only statesaround the center of the BZ contribute to the tunneling cur-rent through a MgO barrier.
To estimate the size of the effective tunneling regionaround the BZ center, one has to analyze the complex bandstructure of MgO as done by Butler et al.15 and Heiliger etal.52 The transmission probability Td at a thickness d isdetermined by the imaginary part of the complex wave vec-tor in the transport direction Imk. The band with thesmallest imaginary part is the 1-like band. In combinationwith the well known expression
Td e−2Imkd 5
for the transmission probability through a tunneling barrierthe area contributing to the tunneling current can be esti-mated.
Figure 7 shows the k dependence of the transmission
0 1 2 3 4 5 6 7 8
RAR AR A(R-R )A= l n0 0ρ
(R-R
)A
(mm
)0
Ωµ
2
d/l0
20
0
4
8
12
16
FIG. 6. Color online Thickness dependence of the resistancearea product RA of amorphous Fe embedded between semi-infinite bcc-Fe electrodes thickness of one amorphous cell l0=5aFe, R0 the Sharvin resistance Ref. 31, and the resistivity.
FIG. 7. Color online Transmission probability for six mono-layers of MgO calculated from the 1-like band of the complexband structure of MgO via Eq. 5; the black circle marks the regionwith k 0.1 2
aFewhere 94 % of the transmission takes place.
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probability through six monolayers of MgO for the 1-likeband. The black ring indicates the region where 94 % of thetransmission occurs and has the radius of 0.1 2
aFe. The effec-
tive polarization Peff is calculated in this region.Both polarizations P1
and Peff can be used to estimatethe TMR ratio in the Julliere model and the best magneticmaterial can be selected. This analysis may be applied to anycrystalline barrier material provided that the important tun-neling states are known.
IV. TRANSPORT PROPERTIES OF POSSIBLEELECTRODE MATERIALS
In the following we apply the analysis described in thelast section to estimate the ability of three different magneticmaterials to create a high TMR ratio in FM /MgO /FM tunneljunctions. One is crystalline bcc iron bcc-Fe which will becompared with amorphous iron a-Fe and bcc cobalt bcc-Co with the lattice structure of bcc-Fe aFe=2.87 Å. Toestimate the polarization of the 1-like states P1
and toinvestigate the influence of the finite thickness of the mag-netic material we attach an effective nonmagnetic material tothe magnetic electrodes. This material has to serve as anelectron reservoir with 1-like states. For simplicity bcc-Cuis used with a lattice constant of bcc-Fe. The correspondingDOS shown in Fig. 8a indicates that the d states are occu-pied and the band structure in the current direction see Fig.8b shows a 1 state at the Fermi level. Therefore, bcc-Cuis well suited to serve as a nonmagnetic free-electron-likereservoir with 1-like states close to the center of the BZwhich causes optimal matching to the 1-like states of theMgO barrier. Tunneling via 1-like states is also importantwith respect to future applications since it guarantees a lowerresistance area product than states of other symmetry at thesame MgO thickness.
In addition to analyzing bcc-Fe and a-Fe as we did in ourprevious article26 we also embed bcc-Co between bcc-Cusee Fig. 9 to compare the transport properties. The resultsfor the conductance calculations depending on the thicknessof the magnetic layers are presented in Fig. 10. The thicknessof the magnetic layer is measured in aFe /2 which is themonolayer spacing in crystalline Fe. Both crystalline mag-nets show a similar decay of the conductance with the thick-ness of the magnetic layer which is different from that of
amorphous Fe. While the conductance density remainsnearly constant for the majority spin channel for bcc-Fe andbcc-Co the values for the minority spin decrease drasticallywith increasing number of crystalline monolayers.
This results in a spin polarization of the currents P1through bcc-Fe and bcc-Co as high as 70 % see Table Iwith four magnetic monolayers only. The index 1 indicatesthat the current carrying states are 1-like triggered by thebcc-Cu reservoir. The value for P1
is in good agreementwith the experimentally found polarization PexpMgO Ref.17 for the tunneling current from bcc-Fe through MgO intoa superconductor. In the amorphous Fe system the conduc-tance values in the majority and the minority spin channelare comparable to each other and lie in between the values ofthe spin channels for the crystalline magnetic materials. Thecorresponding polarization P1
is as small as 14 %.The TMR ratios calculated from P1
by the Julliere modelEq. 4 are presented in Table I. For a-Fe the TMR ratio isnegligible but with bcc-Fe and bcc-Co the TMR ratio reachesnearly 200 % with only four magnetic monolayers. This es-timated TMR ratio is higher than the one obtained from thepolarization based on the total DOS at the Fermi level50,51
PexpAl2O3, but is still drastically smaller than the pre-dicted ratios for bcc-Fe Refs. 14 and 15 and bcc-Co Ref.19 electrodes. The shortcoming of the estimation is that allk states in the BZ contribute with equal weight since g is notin the tunneling regime. To account for the decay of thetunneling currents with increasing k due to the complexband structure of the MgO barrier, the estimation is furtherreduced to the states close to the center of the BZ. In par-
Ener
gy
rela
tive
toE
(eV
)F
H10
-5
0
5
10
E-E (eV)F
DO
S(e
Vat
om
)-1
-1
a) b)
-8 -6 -4 -2 0 20
1
2
3 EF
FIG. 8. Color online a Density of states of bcc-Cu with thelattice constant aFe of bcc-Fe, b band structure of bcc-Cu alongH current direction.
bcc-Fe/
bcc-Co/
a-Fe
. . . . . .bcc-Cu bcc-Cu
FIG. 9. Color online Junction geometry for the calculations ofthe transport properties through different magnetic electrodematerials.
g/A
(mm
)Ω
µ−1
-2
d/(a /2)Fe
0 1 2 3 4 5 6 7 80.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Co-upCo-down
Fe-upFe-down
a-Fe-upa-Fe-down
FIG. 10. Color online Conductance densities g /A for the ana-lyzed magnetic materials embedded between semi-infinite bcc-Cuvs thickness of the magnetic layer d in units of aFe /2.
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ticular, we use the effective polarization Peff introduced inSec. III and Fig. 7.
To illustrate these regions where Peff is calculated Fig. 11shows the transmission maps of the crystalline junctions withfour magnetic monolayers of bcc-Fe and bcc-Co in compari-son to bcc-Cu for the BZ of the size 2 /aFe 2 /aFewhich corresponds to the bcc001 surface. The conductancemap for bcc-Cu is identical in both spin channels since Cu isnonmagnetic. The results for bcc-Fe and bcc-Co are similarto each other. The contributions of the majority spin directionare comparable to the values of bcc-Cu. In contrast, the con-tributions of the minority spin direction are very small andindicate especially around the center of the Brillouin zone, alow transmission probability as in a tunneling regime. This isconsistent with the band structure of bcc-Fe and bcc-Co in abulk structure as discussed by Zhang et al.19 and Yuasa etal.10
For a quantitative analysis, the effective spin polarizationPeff of the current is presented in Table I. With four mono-layers, the calculated polarization is as high as 98 and 86 %for bcc-Fe and bcc-Co, respectively. Applying Eq. 4 TMRratios of 4850 and 568 % are expected already for thin mag-netic layers of bcc-Fe and bcc-Co. These ratios are signifi-cantly higher than the values based on the spin polarizationof all 1-like states P1
and are comparable to the ab initiocalculated TMR ratios of systems with semi-infinite mag-netic electrodes.14,15,19
For a comparison of these results to a system with a-Fe asthe magnetic electrode it is important to consider the differ-ent size of the BZ. As mentioned before the BZ of the amor-phous system has a size of /aFe /aFe because of thelarger unit cell. In principle, the calculations for the crystal-line system can be performed with a unit cell of larger size toenable comparison of the conductance maps Tk in thereduced BZ of the amorphous system. Figure 12a illustratesthe down-folding of the larger BZ left into the smaller one /aFe /aFe where two additional points of the BZboundary are folded into the BZ center. This procedure ismathematically correct but information concerning the trans-
mission, in particular the symmetry selection, got lost. Obvi-ously, one can no longer distinguish between the states origi-
nally occurring at the BZ center and the ones down-foldedfrom the BZ boundary. But in a crystalline tunnel junctionwith a MgO barrier only those states which are originally at
the point can match to the states with the lowest decayrates of the MgO. For a-Fe see Fig. 12b the symmetry isreduced and all states at the BZ center can match to the MgOstates with the lowest decay rate.
As a result of the structural disorder the transmission ofall k states is comparable for both spin directions. The TMRratio deduced from the effective spin polarization Peff is 1 %.The effective spin polarization of −7 % of the lead materialsee Table I leads to a larger conductance for the minorityspin channel in comparison with the majority spin channel inthe parallel configuration of the real tunnel junction.
Our calculations predict that subnanometer bcc-Fe andbcc-Co layers are efficient spin-filters for 1-like states andpromising candidates to yield a high TMR ratio in contactwith a MgO barrier. In the next section we compare ourpredictions based on the spin polarizations of currentsthrough different magnetic materials to our results of ab ini-tio calculations of special tunnel junctions.
V. TUNNEL JUNCTIONS WITH FINITEFERROMAGNETIC LAYERS
In this section we investigate different tunnel junctions toquantify the conclusions from the properties of the electrodematerials in the last section. This means that now the role ofthe interface between magnetic layer and barrier and thewhole complex band structure of the barrier is included in abinitio calculations. To take the finite thickness of the mag-netic layer into account, the reservoirs are again semi-infinitebcc-Cu electrodes as sketched in Fig. 13. The barrier consistsof six monolayers of MgO which acts as a symmetry filter
TABLE I. Transport properties of bcc-Fe, bcc-Co, and amorphous Fe with a thickness of 2aFe embedded between semi-infinite bcc-Cu,and corresponding TMR ratios. The following different polarizations are used to estimate the TMR ratio in the Julliere model using Eq. 4.Pexp Al2O3: spin polarization of a ferromagnet FM measured in a FM/amorphous Al2O3/superconductor tunnel junction Refs. 50 and51. Pexp MgO: spin polarization of a ferromagnet FM measured in a FM /MgO /superconductor tunnel junction Ref. 17. P1
: spinpolarization of 1-like states this article by using a reservoir which provides only 1-like states. Peff: effective spin polarization of 1-likestates in a region of the BZ with k 0.1 2
aFemarked with circles in Fig. 11 where the main contribution to the total tunneling current is
expected see Fig. 7. Tunnel junctions ab initio: ab initio calculation of the TMR ratio of the full tunnel junctions see Sec. V. Thegeometries used are 3 and 4 in Fig. 13 for bcc-Fe and bcc-Co with four monolayers. For the amorphous Fe the junction geometry 1 isused without any crystalline magnetic layer.
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for 1-like states as mentioned above. The junction geometry1 is given by amorphous ferromagnetic layers with thethickness of 2 aFe adjacent to the barrier. In a second step, afinite number of bcc-Fe monolayers is inserted between thebarrier and the amorphous Fe denoted as junction 2 in Fig.13. The influence of the amorphous Fe in a junction withrelatively many bcc-Fe monolayers is discussed by omittingthe amorphous layer getting the junction geometry 3. Thisjunction will be compared to a junction with bcc-Co insteadof bcc-Fe sketched as 4. The effective Kohn-Sham poten-tials of all structures are calculated self-consistently.
As expected from the transport properties of amorphousFe see Table I, the TMR effect is very small with onlyamorphous iron as magnetic electrode material. In the Jul-
liere model we estimate about 1 %. In the ab initio calcula-tion we calculate a TMR ratio of 44 %. The small TMR ratiois due to the very low spin polarization P1
=14 % and thesimilar behavior for both spin channels around the center ofthe BZ. Obviously, the self-consistent treatment of the inter-face structure leads to a higher TMR ratio but the lack of ahigh Peff drastically restricts the achievable TMR ratio. Thisresult is comparable to previous works where interfaceroughness between a ferromagnet and a tunneling barrier18,22
or a semiconductor56 was introduced. Additional disorderleads to a lowering of the TMR effect and a smaller effi-ciency of spin injection into the semiconductor. Note that theactual value of the TMR ratio can differ with respect to theused configuration of the amorphous supercell. However, the
FIG. 11. Color online Transmission mapsTk for different crystalline magnetic electrodematerials thickness of the magnetic layer 2aFe:a bcc-Cu/bcc-Cu, b bcc-Cu/bcc-Fe/bcc-Cu, cbcc-Cu/bcc-Co/bcc-Cu. Red circles indicate theregion where the effective spin polarization Peff
is calculated.
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structural disorder leads to the low polarization P1and Peff
in any configuration. Therefore, we expect similar results forother configurations.
As discussed in our previous article26 it turned out thatinserting two monolayers of crystalline Fe between the MgOsee Fig. 13 2 and the amorphous layers is sufficient torecreate a TMR ratio as high as 2900 %, which is compa-rable to the system with semi-infinite crystalline Fe elec-trodes see Fig. 14. In such a system the crystalline Fecauses a high effective spin polarization Peff for the 1-likestates close to the center of the BZ as discussed in the pre-vious section. There we estimate the TMR ratio with fourcrystalline magnetic layers to be 4850 %. Now we calculate8800 % see Table I for the real tunnel junction whichshows that the estimation works well.
The increase of the polarization from a-Fe to bcc-Fe givesrise to these high TMR ratios. Therefore, the question of theinfluence of a-Fe arises when there are a couple of bcc-Fe
monolayers between the barrier and a-Fe. For this purposewe compare in Fig. 14 the junctions with and without ana-Fe layer. The influence of the amorphous Fe is negligible ifmore than two crystalline Fe monolayers are inserted.
For a system with bcc-Co electrodes structural disorder isnot considered and the results are similar to the case of crys-talline Fe. For both systems a high TMR ratio has been foundwith magnetic electrodes of only one crystalline monolayerthickness and with three monolayers the TMR ratio is com-parable to the results of semi-infinite electrodes. The strongeroscillations with the thickness of the ferromagnetic layer inthe bcc-Co system result from sharp resonance effects in theminority spin channel of the parallel configuration.
In contrast to other theoretical results19 we found lowervalues for the TMR ratio 2000 % for semi-infinite bcc-Co
FIG. 12. Color online Comparison of thetransmission maps of bcc-Fe and a-Fe for theanalysis as magnetic electrode materials thick-ness of the magnetic layer: 2 aFe:a bcc-Cu/bcc-Fe/bcc-Cu, b bcc-Cu /a-Fe /bcc-Cu. Whitecircles indicate the region where the effectivespin polarization Peff is calculated see Table I.
FIG. 14. Color online TMR ratio as a function of the numberof crystalline Fe layers next to the barrier for the model tunneljunctions sketched in Fig. 13.
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electrodes than in a system with bcc-Fe 6000 %. The rea-son can be attributed to different structural data of the inter-face. In the present paper the experimentally obtainedbcc-Fe /MgO /bcc-Fe structure57,58 was used for both thebcc-Fe and the bcc-Co system. The explanation for the high
TMR ratio given by Zhang et al.19 is restricted to the pointonly. There the majority spin channel of bcc-Co has only 1symmetry and no other states occur, whereas in bcc-Fe stateswith 5 and 2 symmetry exist. This leads for bcc-Co to atotal reflection of all states in the antiparallel configuration at
the point. This behavior occurs also in the present calcu-lation where no contributions in both spin channels of the
antiparallel configuration at the point exist. However,
around the point non-negligible contributions have beenfound, which leads to a finite, quite a high TMR ratio.
VI. CONCLUSION
The results of this paper demonstrate that the magnitudeof the TMR effect can strongly be manipulated by structureand geometry of the ferromagnetic electrode. We showedthat a detailed analysis of the spin-dependent transport prop-erties of the magnetic layers considering the symmetry selec-tion properties of MgO is sufficient to estimate the expected
TMR ratio of a tunnel junction. If the symmetry selection ofthe barrier, following from the complex band structure of thebulk material, is known, a similar procedure can be used forother tunnel junctions with different crystalline barriers. Weestimate the TMR ratio of a MgO based tunnel junction bycalculating the spin-dependent transport properties throughdifferent magnetic layers a-Fe, bcc-Fe, bcc-Co embeddedbetween nonmagnetic bcc-Cu reservoirs. In particular, weanalyze the spin filter effect of the magnetic layers by calcu-lating an effective polarization. We verify these estimationsby ab initio calculations of the complete tunnel junctions. Itturned out that bcc-Co and bcc-Fe as thin magnetic layersshow similar behavior and even 1 monolayer is sufficient tocreate a high TMR ratio. The influence of an amorphouslayer in addition to a crystalline ferromagnetic layer is neg-ligible. In contrast, with an electrode of amorphous Fe indirect contact with MgO the TMR effect is suppressed.
ACKNOWLEDGMENTS
This work has been supported in part by the NIST-CNST/UMD-NanoCenter Cooperative Agreement. Financial sup-port by the DFG Grant No. FG 404 is kindly acknowl-edged.
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