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Physics of thin-film ferroelectric oxides

M. Dawber∗

DPMC, University of Geneva, 24 quai Ernest-Ansermet, CH-1211, Geneva 4, Switzerland

K.M. Rabe†

Dept of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Rd, Piscataway, NJ 00854-8019, USA

J.F. Scott‡

Dept of Earth Sciences, University of Cambridge, Downing St, Cambridge CB2 3EQ, UK

This review covers the important advances in recent years in the physics of thin film ferroelectricoxides, the strongest emphasis being on those aspects particular to ferroelectrics in thin filmform. We introduce the current state of development in the application of ferroelectric thin filmsfor electronic devices and discuss the physics relevant for the performance and failure of thesedevices. Following this we cover the enormous progress that has been made in the first principlescomputational approach to understanding ferroelectrics. We then discuss in detail the importantrole that strain plays in determining the properties of epitaxial thin ferroelectric films. Finally,we look at the emerging possibilities for nanoscale ferroelectrics, with particular emphasis onferroelectrics in non conventional nanoscale geometries.

Contents

I. INTRODUCTION 1

II. FERROELECTRIC ELECTRONIC DEVICES 2A. Ferroelectric Memories 2B. Future prospects for non-volatile Ferroelectric Memories3

C. Ferroelectric FET’s 4D. Replacement of gate oxides in DRAMs 5

III. FERROELECTRIC THIN FILM DEVICE PHYSICS6

A. Switching 61. Ishibashi-Orihara Model 62. Nucleation models 73. The scaling of coercive field with thickness 74. Mobility of 90o domain walls 75. Imaging of domain wall motion 8

B. Electrical Characterization 91. Standard Measurement techniques 92. Interpretation of dielectric permittivity data 113. Schottky barrier formation at metal-ferroelectric junctions12

4. Conduction mechanisms 15C. Device Failure 16

1. Electrical Breakdown 162. Fatigue 173. Retention failure 19

IV. FIRST PRINCIPLES 19A. DFT studies of bulk ferroelectrics 20B. First-principles investigation of ferroelectric thin films22

1. First principles methodology for thin films 222. Overview of systems 243. Studies of individual one component-systems 254. Studies of individual heterostructures 29

∗Electronic address: matthew.dawber@physics.unige.ch†Electronic address: rabe@physics.rutgers.edu‡Electronic address: jsco99@esc.cam.ac.uk

5. First principles modelling: methods and lessons 316. Challenges for first principles modelling 33

V. STRAIN EFFECTS 34

VI. NANOSCALE FERROELECTRICS 38A. Quantum confinement energies 38B. Coercive fields in nanodevices 38C. Self Patterned nanoscale ferroelectrics 39D. Non-planar geometries:Ferroelectric nanotubes 40

VII. Conclusions 41

References 41

I. INTRODUCTION

The aim of this review is to provide an account of theprogress in the understanding of the physics of ferroelec-tric thin film oxides, particularly the physics relevant topresent and future technology that exploits the charac-teristic properties of ferroelectrics. An overview of thecurrent state of ferroelectric devices is followed by iden-tification and discussion of the key physics issues thatdetermine device performance. Since technologically rel-evant films for ferroelectric memories are typically thickerthan 120 nm, characterization and analysis of these prop-erties can initially be carried out at comparable lengthscales. However, for a deeper understanding, as well asfor the investigation of the behavior of ultrathin filmswith thickness on the order of lattice constants, it is ap-propriate to re-develop the analysis at the level of atomicand electronic structure. Thus, the second half of this re-view is devoted to a description of the state of the art infirst principles theoretical investigations of ferroelectricoxide thin films, concluding with a discussion of experi-ment and theory of nanoscale ferroelectric systems.

As a starting point for the discussion, it is helpful tohave a clear definition of ferroelectricity appropriate to

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thin films and nanoscale systems. Here we consider aferroelectric to be a pyroelectric material with two ormore stable states of different nonzero polarization (un-like electrets, ferroelectrics have polarization states thatare thermodynamically stable, not metastable). Further-more, it must be possible to switch between the two statesby the application of a sufficiently strong electric field,the threshold field being designated the coercive field.This field must be less than the breakdown field of thematerial, or the material is merely pyroelectric and notferroelectric. Because of this switchability of the sponta-neous polarization, the relationship between, the electricdisplacement D, and the electric field E is hysteretic.

For thin film ferroelectrics the high fields that must beapplied to switch the polarisation state can be achievedwith low voltages, making them suitable for integratedelectronics applications. The ability to create highdensity arrays of capacitors based on thin ferroelectricfilms has spawned an industry dedicated to the com-mercialization of ferroelectric computer memories. Theclassic textbooks on ferroelectricity(Lines and Glass,1977),(Fatuzzo and Merz, 1967), though good, are nowover twenty years old, and pre-date the shift in emphasisfrom bulk ceramics and single crystals towards thin-filmferroelectrics. While much of the physics required to un-derstand thin-film ferroelectrics can be developed fromthe understanding of bulk ferroelectrics, there is also be-havior specific to thin films that cannot be readily un-derstood in this way. This is the focus of the presentreview.

One of the points that will become clear in the course ofthis review is that a ferroelectric thin film cannot be con-sidered in isolation, but rather the measured propertiesreflect the entire system of films, interfaces, electrodesand substrates. We also look in detail at the effects ofstrain on ferroelectrics. All ferroelectrics are grown onsubstrates which can impose considerable strains, mean-ing that properties of ferroelectric thin films can oftenbe considerably different from those of their bulk parentmaterial. The electronic properties also have a charac-teristic behavior in thin-film form. While bulk ferroelec-tric materials are traditionally treated as good insulators,as films become thinner it becomes more appropriate totreat them as semiconductors with a fairly large bandgap.These observations are key to understanding the poten-tial and the performance of ferroelectric devices, and tounderstanding why they fail when they do.

In parallel with the technological developments in thefield, the power of computational electronic structure the-ory has increased dramatically, giving us new ways of un-derstanding ferroelectricity. Over the last fifteen years,more and more complex systems can be simulated withmore accuracy; and as the length scales of experimentalsystems decrease, there is now an overlap in size betweenthe thinnest epitaxial films and the simulated systems.It is therefore an appropriate and exciting time to reviewthis work, and to make connections between it and theproblems considered by experimentalists and engineers.

Finally we look at some issues and ideas in nano-scaleferroelectrics, with particular emphasis on new geome-tries for ferroelectric materials on the nanoscale such asferroelectric nanotubes and self-patterned arrays of fer-roelectric nano-crystals.

We do not attempt to cover some of the issues whichare of great importance but instead refer readers to re-views by other authors. Some of the more important ap-plications for ferroelectrics make use of their piezoelec-tric properties, for example in actuators and microsen-sors; this topic has been reviewed by Muralt (2000). Re-laxor ferroelectrics in which ferroelectric ordering occursthrough the interaction of polar nanodomains induced bysubstitution are also of great interest for a number of ap-plications and have recently been reviewed by Samara(2003).

II. FERROELECTRIC ELECTRONIC DEVICES

A. Ferroelectric Memories

The idea that electronic information can be stored inthe electrical polarization state of a ferroelectric materialis a fairly obvious one; however it’s realization is not sostraightforward. The initial barrier to the developmentof ferroelectric memories was the necessity to make themextremely thin films, because the coercive voltage of fer-roelectric materials is typically of the order of severalkV/cm, requiring sub-micron thick films to make devicesthat work on the voltage scale required for computing(all Si devices work at ≤ 5V). With today’s depositiontechniques this is no longer a problem, and now high-density arrays of non-volatile ferroelectric memories arecommercially available. However, reliability remains akey issue. The lack of good device models means thatdesign of ferroelectric memories is expensive and thatit is difficult to be able to guarantee that a device willstill operate ten years into the future. Because compet-ing non-volatile memory technologies exist, ferroelectricmemories can succeed only if these issues are resolved.

A ferroelectric capacitor, while capable of storing in-formation, is not sufficient for making a non-volatile com-puter memory. A pass-gate transistor is required so thata voltage above the coercive voltage is only applied tothe capacitor when a voltage is applied to both the wordand bit line (this is how one cell is selected from an arrayof memories). The current (measured through a smallload resistor in series with the capacitor) is compared tothat from a reference cell that is poled in a definite di-rection. If the capacitor being read is in a different statethe difference in current will be quite large (the displace-ment current associated with switching accounts for thedifference). If the capacitor does not switch because itis already in the reference state, the difference in cur-rent between the capacitor being read and the referencecapacitor is zero.

Most memories use either a 1T-1C design (1

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FIG. 1 a)1T-1C memory design. When a voltage is appliedto both the word and bit line, the memory cell is addressed.Shown also is the voltage applied to the capacitor and the cur-rent output, depending on whether a one or a zero is stored.The current for the zero state is pure leakage current and bycomparison to a reference capacitor can be removed. (b) A2T-2C memory cell in which the reference capacitor is part ofthe memory cell

Transistor-1 Capacitor) or a 2T-2C (2 Transistor-2 Ca-pacitor) design (Fig. 1). The important difference is thatthe 1T-1C design uses a single reference cell for the en-tire memory for measuring the state of each bit, whereasin the 2T-2C there is a reference cell per bit. A 1T-1Cdesign is much more space-effective than a 2T-2C design,but has some significant problems, most significantly thatthe reference capacitor will fatigue much faster than theother capacitors, and so failure of the device occurs morequickly. In the 2T-2C design the reference capacitor ineach cell fatigues at the same rate as its correspond-ing storage capacitor, leading to better device life. Aproblem with these designs is that the read operation isdestructive, so every time a bit is read it needs to bewritten again. A ferroelectric field effect transistor, inwhich a ferroelectric is used in place of the metal gate ona field effect transistor, would both decrease the size ofthe memory cell and provide a non-destructive read out;however, no commercial product has yet been developed.Current efforts seem to run into serious problems withdata retention.

An example of a real commercially available memory isthe Samsung lead zirconate titanate based 4 Mbit 1T-1Cferroelectric memory. The SEM cross-section (Fig. 2)of the device gives some indication of the complexity ofdesign involved in a real ferroelectric memory.

Lead zirconate titanate (PZT) has long been theleading material considered for ferroelectric memories,though strontium bismuth tantalate (SBT), a layeredperovskite, is also a popular choice due to its superiorfatigue resistance and the fact that it is lead free(Fig. 3).However it requires higher temperature processing, whichcreates significant integration problems. Some recentprogress has been made in optimizing precursors. Un-til recently the precursors for Sr, Bi, and Ta/Nb did notfunction optimally in the same temperature range, butlast year Inorgtech developed Bi(mmp)3 – a 2-methoxy-2-propanol propoxide that improves reaction and lowersthe processing temperature for SBT, its traditional maindisadvantage compared to PZT. This material also sat-urates the bismuth coordination number at 6. Recently

FIG. 2 Cross-sectional SEM image of the Samsung 4Mbit 1T-1C 3 metal FRAM

several other layered perovskites, for example bismuthtitanate, have also been considered.

FIG. 3 (a)ABO3 cubic perovskite structure, (b) StrontiumBismuth Tantalate (layered perovskite structure)

As well as their applications as FRAM’s, ferroelectricmaterials have potential use in DRAM’s because of theirhigh dielectric constant in the vicinity of the ferroelec-tric phase transition, a topic which has been reviewed byKingon, Maria and Streiffer (2000). Barium strontiumtitanate (BST) is one of the leading materials in thisrespect since by varying composition a transition tem-perature just below room temperature can be achieved,leading to a high dielectric constant over the operatingtemperature range.

B. Future prospects for non-volatile Ferroelectric Memories

There are two basic kinds of ferroelectric random ac-cess memories in production today: (1) The free-standingRAMs and (2) fully embedded devices (usually a CPU,which may be a CMOS EEPROM (complementary metaloxide semiconductor electrically erasable programmableread-only memory, the current generation widely usednon-volatile memory technology), plus an FRAM (fer-roelectric random access memory), and an 8-bit micro-processor). The former have reached 4 Mbit both at

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Samsung (using PZT) and Matsushita (using SBT). TheSamsung device is not yet, as far as the authors know, incommercial production for real products, but the NECFRAM is going into full-scale production this year inToyama (near Kanazawa). Fujitsu clearly leads in theactual commercial use of its embedded FRAMs. The Fu-jitsu embedded FRAM is that used in the SONY Playsta-tion 2. It consists of 64 Mbit of EEPROM plus 8 kbit ofRAM, 128 kbit ROM, and a 32-kbit FRAM plus securitycircuit. The device is manufactured with a 0.5-micronCMOS process. The capacitor is 1.6 x 1.9 microns andthe cell size is either 27.3 square microns for the 2T-2Cdesign or 12.5 square microns for the 1T-1C.

The leading competing technologies in the long termfor non volatile computer memories are FRAM (ferro-electric random access memories) and MRAM (magneticrandom access memories). These are supposed to replaceEEPROMs (electrically erasable programable read-onlymemories) and ”Flash” memories in devices such as dig-ital cameras. Flash, though proving highly commerciallysuccessful at the moment, is not a long term technology,suffering from poor long term endurance and scalability.It will be difficult for Flash to operate as the silicon logiclevels decrease from 5V at present to 3.3V, 1.1V, and0.5V in the near future. The main problem for ferro-electrics is the destructive read operation, which meansthat each read operation must be accompanied by a writeoperation leading to faster degradation of the device. Theoperation principle of MRAMs is that the tunneling cur-rent through a thin layer sandwiched between two fer-romagnetic layers is different depending on whether theferromagnetic layers have their magnetization parallel oranti-parallel to each other. The information stored inMRAMs can thus be read non-destructively, but theirwrite operation requires high power which could be ex-tremely undesirable in high density applications. Wepresent a summary of the current state of developmentin terms of design rule and speed of the two technologiesin the following table.

Company Design Rule Speed

(Feature Size) (Access Time)

MRAMs

NEC/Toshiba 1Mb

IBM 16 Mb

Matsushita 4Mb

Sony 8kb 0.18 microns

Cypress 256 kb 70 ns

State of the art 16 Mb 0.09 microns 25 ns

FRAMs

Fujitsu 32 kb 100 ns

Samsung 32 Mb 0.18 microns 60 ns

Matsushita 4 Mb 60 ns

Laboratory 800 ps

Some clarification of the numbers in this table is re-quired. The size of the Fujitsu FRAM memory may seem

small but it is for an actual commercial device in largescale use (in every Playstation 2), whereas the othersare figures from internal sampling of unreleased devicesthat have not been commercialized. No MRAMS existin any commercial device, giving FRAMs a substantialedge in this regard. The most recent commercial FRAMproduct actually shipped is a large-cell-area six-transistorfour-capacitor (6T-4C) memory for smart credit cardsand radio frequency identification tags (RF-ID) and fea-tures non-destructive read out (Masui et al., 2003). Atotal of 200 million ferroelectric memories of all typeshave been sold industry-wide. The Sony MRAM, thoughsmall, has sub-micron design rules, meaning that in prin-ciple a working device could be scaled up to Mb size.

Partly in recognition of the fact that are distinctadvantages for both ferroelectrics and ferromagnets,there has been a recent flurry of activity in the fieldof multiferroics, i.e. materials that display both ferro-electric and magnetic ordering, the hope being that onecould develop a material with a strong enough couplingbetween the two kinds of ordering to realize a devicethat can be written electrically and read magnetically.In general multi-ferroic materials are somewhat rare,and certainly the conventional ferroelectrics like PbTiO3

and BaTiO3 will not display any magnetic behaviouras the Ti-O hybridisation required to stabilize theferroelectricity in these compounds will be inhibited bythe partially filled d-orbitals that would be requiredfor magnetism (Hill , 2000). However there are othermechanisms for ferroelectricity and in materials whereferroelecticity and magnetism co-exist there can becoupling between the two. For example, in BaMnF4

the ferroelectricity is actually responsible for changingthe antiferromagnetic ordering to a weak canted fer-romagnetism (Fox et al., 1980) In addition, the largemagnetoelectric coupling in these materials causeslarge dielectric anomalies at the Neel Temperatureand at the in-plane spin ordering temperature (Scott ,1977),(Scott , 1979). More recent theoretical andexperimental efforts have focused on BiMnO3,BiFeO3

(Seshadri and Hill , 2001),(Moreira de Santos et al.,2002),(Wang et al., 2003) and YMnO3 (Van Aken et al.,2004),(Fiebig et al., 2002).

C. Ferroelectric FET’s

It has been known for some time that replacing themetal gate in a field effect transistor (FET) by a fer-roelectric could produce a device with non-destructiveread-out (NDRO), in which the polarization of thegate (+ for ”1” and - for ”0”) could be sensed sim-ply by monitoring the source-drain current magnitude.Thus such a device requires no reset operation aftereach READ and will experience very little fatigue ina normal frequent-read, occasional-write usage. Theearly ferroelectric FETs utilized gates of lithium nio-bate (Rice Univ.) (Rabson et al., 1995) or BaMgF4

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(Westinghouse) (Sinharoy et al., 1991),(Sinharoy et al.,1992),(Sinharoy et al., 1993). An example of a ferroelec-tric FET device as fabricated by Mathews et al. (1997)is shown in Fig. 4

LaAlO3

Pt contacts

Source

Gate

DrainPZT

LCMO

Vappl

I

VSD

SD

FIG. 4 Mathews et al. (1997) Schematic diagram of an all-perovskite ferroelectric FET and measurement circuit

The optimum parameters for such a ferroelectric gatematerial are extremely different from those for pass-gateswitched capacitor arrays; in particular, the latter re-quire a remanent polarization ca. 10 µC/cm2, whereasthe ferroelectric-gated FETs can function well with 50xless (0.2 µC/cm2). However, the switched capacitor array(FRAM) is very tolerant of surface traps in the ferroelec-tric (which may be ca. 1020 cm−3 in the interface regionnear the electrode), since the ferroelectric makes contactonly with a metal (or metal-oxide) electrode. By com-parison, the ferroelectric gate in an FET contacts theSi substrate directly (MOSFET channel - metal-oxide-semiconductor field-effect-transistor channel). Thus itmust be buffered from the Si to prevent charge injection.Unfortunately, if a thin buffer layer of a low-dielectricmaterial such as SiO2 is used, most of the applied volt-age will drop across the buffer layer and not the ferro-electric gate, making it impossible to switch the gate.As a result, much of the ferroelectric FET research hasemployed buffer layers with relatively high dielectric con-stants, or else rather thick buffer layers, for example, thefirst BaMnF4 FET made at Symetrix (Scott, 1998) useda buffer layer of ca. 40 nm of SiO2. Subsequent studiesoften used PZT,(Kalkur et al., 1994) although the largeremanent polarization in this case (ca. 40 µC/cm2) isactually undesirable for a ferroelectric FET gate.

As pointed out by Yoon and Ishiwara (2001), the de-polarization field in a ferroelectric gate is inevitably gen-erated when the gate is grounded, and this makes itvery difficult to obtain > 10 year data retention in anFE-FET. Their solution is to utilize a 1T-2C capaci-tor geometry in which this depolarization field is sup-pressed by poling the two capacitors in opposite direc-tions. With this scheme Ishiwara and his colleaguesachieved an on/off source-drain current ratio of >1000for a 150 nm thick SBT film in a 5 x 50 micron MOS-FET channel, with Pt electrodes on the SBT capacitor.

Note that the direct contact of the ferroelectric ontoSi produces a semiconductor junction that is quite differ-ent from the metal-dielectric interface discussed above.

The Schottky barrier heights for this case have been cal-culated by Peacock and Robertson (2002). The electronscreening length in the Si will be much greater than in thecase of metal electrodes; in particular this will increasethe minimum ferroelectric film thickness required to sta-bilize the device against depolarization instabilities. Al-though this point was first emphasized by Batra and Sil-verman (1973), it has been neglected in the more recentcontext of ferroelectric FETs. In our opinion, this depo-larization instability for thin ferroelectric gates on FETsis a significant source of the observed retention failurein the devices but has not yet been explicitly modeled.If we are correct, the retention problem in ferroelectricFETs could be minimized by making the ferroelectricgates thicker and the Si contacts more conducting (e.g.,p+ rather than p). See Scott (Microelectron. Eng. 2005,in press) for a full discussion of all-perovskite FETs.

Table I lists a number of the most promising gate ma-terials under recent study, together with the buffer lay-ers employed in each case. Studies of the I(V) charac-teristics of such ferroelectric FETs have been given byMacleod and Ho (2001) and a disturb-free programmingscheme described by Ullman et al. (2001).

FET Gate Buffer Layer Reference

LiNbO3 none Rabson et al. (1995)

SBT SrTa2O6 Ishiwara (1993),

Ishiwara et al. (1997),

Ishiwara (2001)

SBT CeO2 Shimada et al. (2001),

Haneder et al. (2001)

SBT SiO2 Okuyuma et al. (2001)

SBT ZrO2 Park and Oh (2001)

SBT Al2O3 Shin et al. (2001)

SBT Si3N4 Han et al. (2001)

SBT Si3N4/SiO2 Sugiyama et al. (2001)

SBT poly-Si + Y2O3 Kalkur and Lindsey (2001)

Pb5Ge3O11 none Li and Hsu (2001)

YMnO3 Y2O3 Cheon et al. (2001),

Choi et al. (2001)

Sr2(Ta2xNb2−2x)O7 none Kato (2001)

PZT CeO2 Xiaohua et al. (2001)

BST(strained) YSZ(zirconia) Jun and Lee (2001)

BaMnF4 SiO2 Scott (1998),

Kalkur et al. (1994)

Beyond its use in modulating the current in a semicon-ductor channel the ferroelectric field effect can also beused to modify the properties of more exotic correlatedoxide systems.(Ahn,Triscone and Mannhart, 2003)

D. Replacement of gate oxides in DRAMs

At present there are three basic approaches to solvingthe problem of SiO2 gate oxide replacement for DRAMs:The first is to use a high-dielectric (”high-k”) mate-rial such as SrTiO3 (k = 300 is the dielectric constant;ǫ = k − 1 is the permittivity; for k >> 1 the termsare nearly interchangeable) deposited by some form ofepitaxial growth. This is the technique employed at Mo-torola, but the view elsewhere is that it is too expen-

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sive to become industry process-worthy. The second ap-proach is to use a material of moderate k (of order 20),with HfO2 favored but ZrO2 also a choice. Hafniumoxide is satisfactory in most respects but has the sur-prising disadvantage that it often degrades the n-channelmobility catastrophically (by as much as x10,000). Re-cently ST Microelectronics decided to use SrTiO3 butwith MOCVD deposition from Aixtron, thus combininghigh dielectric constant and cheaper processing.

The specific high-k integration problems are four: (1)depletion effects in the polysilicon gate; (2) interfacestates; (3) strain effects; and (4) etching difficulties (HfO2

is hard to wet-etch). The use of a poly-Si gate instead ofa metal gate produces grain boundary stress in the poly,with resultant poor conductivity. This mobility degrada-tion is only partly understood. The general view is thata stable amorphous HfO2 would be a good strain-free so-lution. Note that HfO2 normally crystallizes into two orthree phases, one of which is monoclinic (Morrison et al.,2003). Hurley in Cork has been experimenting witha liquid injection system that resembles Isobe’s earlierSONY device for deposition of viscous precursors withflash evaporation at the target.

III. FERROELECTRIC THIN FILM DEVICE PHYSICS

We now turn to some of the physics questions whichare relevant to ferroelectric thin film capacitors.

A. Switching

In the ferroelectric phase ferroelectric materials formdomains where the polarisation is all aligned in the samedirection, in an effort to minimise energy. When a fieldis applied the ferroelectric switches by the nucleation ofdomains and the movement of domain walls and not bythe spontaneous reorientation of all of the polarisationin a domain at once. In contrast to ferromagnets whereswitching usually occurs by the sideways movement ofexisting domain walls, ferroelectrics typically switch bythe generation of many new reverse domains at particularnucleation sites, which are not random; i.e., nucleationis inhomogeneous. The initial stage is nucleation of op-posite domains at the electrode, followed by fast forwardpropagation of domains across the film, and then slowerwidening of the domains(Fig. 5). In perovskite oxidesthe final stage of the switching is usually much slowerthan the other two stages, as first established by Merz(1954). In other materials nucleation can be the slowest(rate-limiting) step.

1. Ishibashi-Orihara Model

For many years the standard model to de-scribe this process has been the Ishibashi-Orihara

Electric field applied

Stage-I: Nucleation

Stage-II: Forward Growth

Stage-III: Sideways Growth

Domain reversal complete

Domain wall

Ferroelectric domain

FIG. 5 The three phases of domain reversal, I. Nucleation(fast) II. Forward growth (fast) III. Sideways growth (slow)

model(Orihara et al., 1994) based on Kolomogorov-Avrami growth kinetics. In this model one considersa nucleus formed at time t’ and then a domain prop-agating outwards from it with velocity V. In theIshibashi-Orihara model the velocity is assumed to bedependent only on the electric field E, and not on thedomain radius r(t). This makes the problem analyticallytractable but gives rise to unphysical fitting parameters,such as fractional dimensionality D. The fractional Dis not related to fractals. It is an artifact that arisesbecause domain wall velocity V is actually proportionalto 1/r(t) for each domain and is not a constant atconstant E. The volume of a domain at time t is givenby

C(t, t′) = CD[

∫ t

t′V (t′′)dt′′]D (1)

where D is the dimensionality of the growth andCD is a constant which depends on the dimensional-ity. It is also assumed within this model that thenucleation is deterministic and occurs at pre-definedplaces; i.e., this is a model of inhomogeneous nucleation.This is an important point since some researchers stilluse homogeneous nucleation models. These are com-pletely inappropriate for ferroelectrics (where the nucle-ation is inhomogeneous, as is demonstrated by imag-ing experiments(Ganpule et al., 2001; Shur et al., 2000;Shur, 1996)).

The result of the model is that the fraction of switchedcharge as a function of field and frequency may be ex-pressed as

Q(E, f) = 1 − exp(−f−DΦ(E)) (2)

where Φ(E) depends on the waveform used for switching.After some consideration and the substitution Φ(E) =Ek one obtains a useful relationship for the dependenceof the field on frequency:

Ec = fDk (3)

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This relationship has been used to fit data fairly wellin TGS (Hashimoto et al., 1994), PZT and SBT (Scott,1996). More recently however Tsurumi et al. (2001) andJung et al. (2002) have found that over larger frequencyranges the data on several materials is better fitted bythe nucleation limited model of Du and Chen (1998).Tagantsev et al. (2002) have also found that over largetime ranges the Ishibashi-Orihara model is not a good de-scription of switching current data and that a nucleationlimited model is more appropriate. It is quite possibleof course that domain wall-limited switching (Ishibashi)is operative in one regime of time and field but that inanother regime the switching is nucleation-limited.

2. Nucleation models

Some of the earliest detailed studies of switching in fer-roelectrics developed nucleation limited switching modelswhere the shape of the nucleus of the reversed domainwas very important. In the work of Merz (1954) andWieder (1956),Wieder (1957) a nucleation-limited modelwas used in which when dagger-shaped nuclei were as-sumed, the correct dependence of the switching currenton electric field could be derived. This approach leads tothe concept of an activation field for nucleation (some-what different from the coercive field). Activation fieldsin thin film PZT capacitors were measured by Scott et al.

(1988); very recently Jung et al. (2004) have studied theeffects of micro-geometry on the the activation field inPZT capacitors.

The switching model of Tagantsev et al. (2002) is a dif-ferent approach in which a number of non-interacting el-ementary switching regions are considered. These switchaccording to a broad distribution of waiting times.

3. The scaling of coercive field with thickness

For the last forty years the semi-empirical scal-ing law,(Janovec, 1958; Kay and Dunn, 1962) Ec(d) ∝

d−2/3, has been used successfully to describe thethickness-dependence of the coercive field in ferroelectricfilms ranging from 100 microns to 200 nanometers.(Scott,2000) In the ultrathin PVDF films of Bune et al. (1998) adeviation from this relationship was seen for the thinnestfilms(Ducharme et al., 2000). Although they attributethis to a new kind of switching taking place (simultane-ous reversal of polarisation, as opposed to nucleation andgrowth of domains), Dawber et al. (2003b) have shown,to the contrary, that if the effects of a finite depolar-ization field due to incomplete screening in the electrodeare taken into account, then the scaling law holds over sixdecades of thickness and the coercive field does not devi-ate from the value predicted by the scaling law (Fig. 6).Recently Pertsev et al. (2003) measured coercive fields invery thin PZT films. Although they have used a differentmodel to explain their data, it can be seen that in fact

the scaling law describes the data very well.

FIG. 6 The scaling of coercive field with thickness in ferro-electrics; from mm to nm scale, from Dawber et al. (2003b)

4. Mobility of 90o domain walls

The mobility of domain walls, especially 90-degreewalls, depends upon their width. In this respect thequestion has been controversial, with some authors claim-ing very wide widths (hundreds of angstroms) and im-mobile walls. Some recent papers show experimentallythat 90-degree domain walls in perovskite ferroelectricsare extremely narrow (Foeth et al., 1999),(Tsai et al.,1992),(Stemmer et al., 1995),(Floquet et al., 1997). InPbTiO3 they are 1.0 ± 0.3 nm wide. This connects thegeneral question of how wide they are and whether theyare immobile. The review by Floquet and Valot (1999)is quite good. They make the point that in ceramicsthese 90-degree walls are 14.0 nm wide (an order of mag-nitude wider than in single crystals). This could be whytheory and experiment disagree, i.e. that something spe-cial in the ceramics makes them 10-15 times wider (andless mobile?). The latter point is demonstrated clearlyin experiments on KNbO3, together with a theoreticalmodel that explains geometrical pinning in polyaxial fer-

8

roelectrics in terms of electrostatic forces. In this respectthe first principles study of Meyer and Vanderbilt (2002)is extremely interesting. Not only do they show that90 degree domain walls in PbTiO3 are narrow and formmuch more easily than 180 degree domain walls, but thatthey should be much more mobile as well, the barrier formotion being so low they predict thermal fluctuation ofabout 12 unit cells at room temperature, which couldperhaps explain why they appear to be wide.

Some experimental studies using atomic force mi-croscopy (AFM) have attempted to answer the ques-tion of whether 90-degree domain walls were mobileor not. In certain circumstances they were immobile(Ganpule et al., 2000),(Ganpule et al., 2000), but in an-other study (Nagarajan et al., 2003) the motion of 90-degree domain walls under an applied field was directlyobserved. It seems that in principle 90 degree domainwalls can move, but this depends quite strongly on thesample conditions.

5. Imaging of domain wall motion

The direct imaging of ferroelectric domain walls is anexcellent method for understanding domain wall motionand switching. At first this was carried out in materialswhere the domains were optically distinct such as leadgermanate(Shur et al., 1990), but more recently atomicforce microscopy (AFM) has become a powerful tool forobserving domain wall motion. The polarisation at apoint can be obtained from the piezoresponse detected bythe tip, and the tip itself can be used to apply a field tothe ferroelectric sample and initiate switching. It is thuspossible to begin switching events and watch their evolu-tion over time. AFM domain writing of ferroelectric do-mains can also be used to write extremely small domainstructures in high density arrays(Paruch et al., 2001) orother device-like structures, such as surface acoustic wavedevices(Sarin Kumar et al., 2004).

The backswitching studies of Ganpule et al. (2001)show two very interesting effects (Fig. 7). The first isthe finding that reverse domains nucleate preferentiallyat antiphase boundaries. This was studied in more de-tail subsequently by Roelofs et al. (2002) who invoked adepolarisation field mediated mechanism to explain theresult. Another explanation might be that the strain isrelaxed at these antiphase boundaries, resulting in favor-able conditions for nucleation. Secondly, the influenceof curvature on the domain wall relaxation is accountedfor within the Kolomogorov-Avrami framework. The ve-locity of the domains is dependent on curvature; and asthe relaxation proceeds, the velocity decreases and thedomain walls become increasingly faceted. In Fig. 7the white polarisation state is stable, whereas the blackis not. The sample is poled into the black polarisationstate and then allowed to relax back.

A different kind of study was undertaken byTybell et al. (2002), in which they applied a voltage pulse

FIG. 7 Ganpule et al. (2001) Piezoresponse scans of a singlecell in PbZr0.2Ti0.8O3, (b)-(d) illustrate the the spontaneousreversal of polarization within this region after wait times of(b) 1.01 x 103, (c) 1.08 x 105, (d) 1.61 x 105, and (d) 2.55 x 105

s. Faceting can be seen in (c),(d),and (e). (f) Transformation-time curve for the data in (b)-(e)

to switch a region of the ferroelectric using an AFM tipand watched how the reversed domain grew as a functionof pulsewidth and amplitude. They were able to showthat the process was well described by a creep mech-anism, thought to arise due to random pinning of do-main walls in a disordered system (Fig. 8). Thoughthe exact origin of the disorder was not clear, it is sug-gested that it is connected to oxygen vacancies, whichalso play a role in pinning the domain walls during afatigue process due to their ordering (Park and Chadi,1998),(Scott and Dawber, 2000).

FIG. 8 Tybell et al. (2002) (a) Domain size increases loga-rithmically with pulse widths longer than 20 µs and saturatesfor shorter times as indicated by the shaded area. (b) Domainwall speed as function of the inverse applied electric field for290, 370, and 810 A thick samples. The data fit well thecharacteristic velocity field relationship of a creep process.

An unsolved puzzle is the direct observa-tion via AFM of domain walls penetrating grainboundaries(Gruverman et al., 1997). This is contraryto some expectations and always occurs at non-normalincidence, i.e. at a small angle to the grain boundary.

One of the interesting things to come out of the work inlead germanate (where ferroelectric domains are opticallydistinct due to electrogyration) by Shur et al. (1990) is

9

that at high applied electric fields (15kV cm−1) tiny do-mains are nucleated in front of the moving domain wall(Fig. 9). A very similar effect is seen in ferromagnets asobserved by Randoshkin (1995) in a single crystal irongarnet film.

FIG. 9 Shur et al. (1990) Nucleation of nanodomains in frontof domain wall in lead germanate at high electric field. Blackand white represent the two directions of polarisation.

However in ferromagnets the effect is modelled by aspin wave mechanism.(Khodenkov, 1975) This mecha-nism is based on the gyrotropic model of domain wall mo-tion in uniaxial materials(Walker, 1963). When a strongmagnetic driving field (exceeding the Walker Threshold)acts upon a domain wall, the magnetization vectors inthe domain wall begin to precess with a frequency γH ,where γ is the effective gyromagnetic ratio. By relat-ing the precession frequency in the domain wall with thespin wave frequency in the domain, good predictions canbe made for the threshold fields at which the effect oc-curs. We note that the domains nucleated in front of thewall may be considered as vortex-like skyrmions. Thesimilarity between these effects is thus quite surprisingand suggests that perhaps there is more in common be-tween ferroelectric domain wall motion and ferromagentdomain wall motion than is usually considered. How-ever, whereas Democritov et al. (1988) have shown thatmagnetic domain walls can be driven supersonically (re-sulting in a phase-matched Cerenkov-like bow wave ofacoustic phonon emission), there is no direct evidenceof supersonic ferroelectric domains. Processes such asthe nano-domain nucleation described above seem to oc-cur instead when the phase velocity of the domain wallmotion approaches the speed of sound. Of course themacroscopic electrical response to switching can arriveat a time t < v/d where v is the sound velocity and d,the film thickness, simply from domain nucleation withinthe interior of the film between cathode and anode.

B. Electrical Characterization

1. Standard Measurement techniques

Several kinds of electrical measurements are made onferroelectric capacitors. We briefly introduce them here

before proceeding to the following chapters where we dis-cuss in detail the experimental results obtained by usingthese techniques.

a. Hysteresis One of the key measurements is naturallythe measurement of the ferroelectric hysteresis loop.There are two measurement schemes commonly used.Traditionally a capacitance bridge as first described bySawyer and Tower(Sawyer and Tower, 1930) was used(Fig. 10). Although this is no longer the standardway of measuring hysteresis the circuit is still useful(and very simple and cheap) and we have made sev-eral units which are now in use in the teaching labsin Cambridge for a demonstration in which studentsare able to make and test their own ferroelectric KNO3

capacitor(Dawber et al., 2003a).

FIG. 10 Sawyer and Tower (1930)(a)The original Sawyer-Tower circuit, (b) Hysteresis in Rochelle salt measured usingthis circuit by Sawyer and Tower at various temperatures

This method is not very suitable in practice formany reasons, for example the need to compensatefor dielectric loss and the fact that the film is be-ing continuously cycled. Most testing of ferroelec-tric capacitors is now carried out using commercialapparatus from one of two companies, Radiant Tech-nologies (http://www.ferrodevices.com/) and AixAcct(http://www.aixacct.com/). Both companies’ testerscan carry out a number of tests and measurements, andboth machines use charge or current integration tech-niques for measuring hysteresis. Both machines also offerautomated measurement of characteristics such as fatigueand retention.

In measuring P(E) hysteresis loops several kinds of ar-tifacts can arise. Some of these are entirely instrumen-tal, and some arise from the effects of conductive (leaky)specimens.

Hysteresis circuits do not measure polarization P di-rectly. Rather, they measure switched charge Q. For anideal ferroelectric insulator

Q = 2PrA (4)

where Pr is the remanent polarization and A is elec-trode area is a parallel plate capacitor. For a somewhatconductive sample

10

Q = 2PrA + σEat (5)

where σ is the electrical conductivity; Ea, is the appliedfield and t, the measuring time. Thus Q in a pulsedmeasuring system depends on the pulse width.

The four basic types of apparent hysteresis curves thatare artifacts are shown in Fig.11.

(Breakdown)

(a) (b)

(c) (d)

FIG. 11 Common hysteresis artifacts: (a) Dead short, (b)linear lossy dielectric, (c) saturated amplifier, (d) non-linearlossy dielectric.

Fig11 a is a dead short in a Sawyer Tower circuit ormodern variant and is discussed in the instruction docu-mentation for both the AixAcct1 and Radiant2 testers.

Fig 11 b shows a linear lossy dielectric. The pointswhere the loop crosses Va = 0 are often misinterpretedas Pr values. Actually this curve is a kind of Lissajousfigure. It can be rotated out of the page to yield astraight line (linear dielectric response). Such a rota-tion can be done electrically and give a “compensated”curve. Here “compensation” means to compensate thephase shift caused by dielectric loss.

Fig 11 c is more subtle. Here are two seemingly per-fect square hysteresis loops, obtained on the same ornominally equivalent specimens at different maximumfields. The smaller loop was run at an applied voltageof Va = 10V , and yields Pr = 6µCcm−2 and the largerat Va = 50V and yields Pr = 100µCcm−2. Note thatboth curves are fully saturated (flat tops). This is im-possible, if the dipoles of the ferroelectric are saturatedat Pr = 6µCcm−2 then there are no additional dipolesto produce Pr = 100µCcm−2 in the larger loop at highvoltage. What actually occurs in the illustration is sat-uration of the amplifier in the measuring system, notsaturation of the polarization in the ferroelectric. The

1 TF Analyzer 2000 FE-module instruction manual2 http://www.ferroelectrictesters.com/html/specs.html#tut

figure is taken from Jaffe, Cook and Jaffe (1971) wherethis effect is discussed (p. 39). It will be a serious prob-lem if conductivity is large in Eq. 5. “Large” in this senseis σ > 10−6(Ωcm)−1 and “small” is σ < 10−7(Ωcm)−1.This is probably the source of Pr > 150µCcm−2 reportsin BiFeO3 where σ can exceed 10−4(Ωcm)−1.

Finally Fig 11 is a nonlinear lossy dielectric. If it isphase compensated it still resembles real hysteresis. Onecan verify whether it is real or an artifact only by varyingthe measuring frequency. Artifacts due to dielectric lossare apt to be highly frequency dependent. Figs 11 b andd are discussed in Lines and Glass (1977) (p.104).

No data resembling Figs 11 a-d should be published asferroelectric hysteresis.

b. Current measurements Another measurement of im-portance which is carried out in an automated way bythese machines is the measurement of the leakage cur-rent. This is normally discussed in terms of a current-voltage (I-V) curve, where the current is measured at aspecified voltage. It is important however that sufficienttime is allowed for each measurement step that the cur-rent is in fact true steady state leakage current and notrelaxation current, and for this reason current-time (I-t)measurements can also be important(Dietz and Waser,1995). Relaxation times in ferroelectric oxides such asbarium titanate are typically 1000 s at room tempera-ture.

c. Dielectric permittivity An impedance analyser mea-sures the real and the imaginary parts of the impedanceby use of a small-amplitude AC signal which is appliedto the sample. The actual measurement is then madeby balancing the impedance of the sample with a set ofreferences inside the impedance analyser. From this thecapacitance and loss can be calculated (all this is doneautomatically by the machine). It is possible at the sametime to apply a DC bias to the sample, so the signal isnow a small AC ripple superimposed on a DC voltage.Ferroelectric samples display a characteristic “butterflyloop” in their capacitance voltage relationships, becausethe capacitance is different for increasing and decreas-ing voltage. The measurement is not exactly equivalentto the hysteresis measurement. In a capacitance voltagemeasurement a static bias is applied and the capacitancemeasured at that bias, whereas in a hysteresis measure-ment the voltage is being varied in a continuous fashion.Therefore it is not strictly true that C(V) = (d/dV)P(V)[area taken as unity] as claimed in some texts, since thefrequency is not the same in measurements of C andP. Impedance spectroscopy (where the frequency of theAC signal is varied) is also a powerful tool for analy-sis of films, especially as it can give information on thetimescales at which processes operate. Interpretation ofthese results must be undertaken carefully, as artifactscan arise in many circumstances and even when this is

11

not the case many elements of the system (e.g., elec-trodes, grain boundaries, leads etc.) can contribute tothe impedance in complicated ways.

2. Interpretation of dielectric permittivity data

a. Depletion charge vs intrinsic response Before lookingfor ferroelectric contributions to a system’s electricalproperties one should make sure there are not contri-butions due to the properties of the system unrelatedto ferroelectricity. Although much is sometimes madeof the dependence of capacitance on voltage, it is worthnoting that metal-semiconductor-metal systems have acharacteristic capacitance voltage which arises from theresponse of depletion layers to applied voltage (Fig. 12).

FIG. 12 Current voltage and capacitance voltage re-lationship of Pt-Si-Pt punch-through diode (Sze et al.,1971),characteristics very similar to those obtained in metal-ferroelectric-metal systems

Essentially the problem boils down to the fact thatthere are two possible sources of the dependence of ca-pacitance on applied field, either changes in depletionwidth or changes in the dielectric constant of the mate-rial, i.e.

C(E)

A=

ǫ(E)

d(E)(6)

Several groups have assumed, that all the change inthe capacitance with field C(E) comes from change indepletion width d(E) and that (E) is changing negligi-bly. The first to suggest this was Joe Evans (IEEE-ISAF Meeting, Urbana, Ill. 1990), who found d = 20nm in PZT. Later Sandia claimed that there was no de-pletion (d=0 or d=infinity)(Miller et al., 1990). Severalauthors have assumed that d(E) is responsible for C(E)(Brennan, 1992), (Mihara et al., 1992), (Hwang et al.,1998), (Hwang, 1998) ,(Scott et al., 1992), (Sayer et al.,1992)

In contrast to the approach of explaining these char-acteristics using semiconductor models Basceri et al.

(1997) account for their results on the basis of a Landau-Ginzburg style expansion of the polarisation (Fig. 13).

The change in with field due to its nonlinearity has alsobeen calculated by both Outzourhit et al. (1995) andDietz et al. (1997). The real problem is that both pic-tures are feasible. One should not neglect the fact thatthe materials have semiconductor aspects; but at thesame time it is not unreasonable to expect that the knownnonlinearity of the dielectric response in these materialsshould be expressed in the capacitance voltage character-istic. Probably the best approach is to avoid making anyconclusions on the basis of these kinds of measurementsalone, as it is quite possible that the relative sizes of thecontributions will vary greatly from sample to sample,or even in the same sample under different experimentalconditions.

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

10

20

30

40

50

60

70

Bias (V)

C/A

(fF

/m

)

m2

t = 24 nm

t = 40 nm

t = 80 nm

t = 160 nm

FIG. 13 (Capacitance v applied bias for BST thin films. Thedata are due to Basceri et al. (1997)

b. Domain wall contributions Below the coercivefield there are also contributions to the permittivityfrom domain walls, as first pointed out by Fouskovain 1965(Fouskova, 1965),(Fouskova and Janousek,1965). In PZT the contributions of domainwall pinning to the dielectric permittivity hasbeen studied in detail by Damjanovic andTaylor(Damjanovic, 1997),(Taylor and Damjanovic,1997),(Taylor and Damjanovic, 1998), who showed thatthe sub-coercive field contributions of the permittivitywere described by a Raleigh law with both reversibleand irreversible components, the irreversible componentbeing due to domain wall pinning.

c. Dielectric measurements of phase transitions One of themost common approaches to measuring the transitiontemperature of a ferroelectric material is naturally tomeasure the dielectric constant and loss. However, inthin films there are significant complications. In bulkthe maximum in the dielectric constant is fairly well cor-related with the transition temperature, but this does not

12

always seem to be the case in thin films. As pointed outby Vendik and Zubko (2000), a series capacitor modelis required to extract the true transition temperature,which in the case of BST has been shown to be indepen-dent of thickness(Lookman et al., 2004), in contrast tothe temperature at which the permittivity maximum oc-curs, which can depend quite strongly on thickness (Fig.14).

FIG. 14 (Lookman et al. (2004) Comparison between the ap-parent Curie temperature in BST taken from Curie-Weissplots of raw data (empty circles) and intrinsic data after cor-rection for interfacial capacitance (filled squares) had beenperformed. The intrinsic Curie temperature appears to be in-dependent of film thickness

3. Schottky barrier formation at metal-ferroelectric junctions

In general, since ferroelectric materials are good insu-lator the majority of carriers are injected from the elec-trode. When a metal is attached to a ferroelectric ma-terial, a potential barrier is formed if the metal workfunction is greater than the electron affinity of the ferro-electric. This barrier must be overcome if charge carriersare to enter the ferroelectric, On the other hand if theelectron affinity is greater than the work function, thenan ohmic contact is formed. For the usual applicationsof ferroelectrics (capacitors) it is desirable to have thelargest barrier possible. If a metal is brought into contactwith an intrinsic pure ferroelectric and surface states donot arise (i.e., the classic metal-insulator junction), thenthe barrier height is simply

φb = φm − χ (7)

In this case the fermi level of the metal becomes thefermi level of the system, as there is no charge within theinsulator with which to change it. On the other hand,if there are dopants or surface states, then there can bea transfer of charge between the metal and ferroelectric,which allows the ferroelectric to bring the system fermilevel towards its own fermi level.

Although single crystals of undoped ferroelectric ti-tanates tend to be slightly p-type, simply because thereare greater abundances of impurities with lower valencesthan those of the ions for which they substitute (Na+for Pb+2; Fe+3 for Ti+4) (Smyth, 1984),(Chan et al.,1981),(Chan et al., 1976) in reality most ferroelectric ca-pacitors are fine grained polycrystalline ceramics and arealmost always oxygen deficient. Typically the regionsof the capacitor near to interface are more oxygen de-ficient than the bulk. Oxygen vacancies act as donorions, and this means there can be a transition from n-type behaviour at the interface to p-type behaviour inthe centre of the film as is evident in the kelvin probestudy of Nowotny and Rekas (1994), who found thatin bulk BaTiO3 with Pt electrodes a change in work-function from 2.5 ± 0.3eV for surfaces and 4.4 ± 0.4eVin the bulk of the material. The nature of the materialnear the surface is important since it determines whethera blocking or ohmic contact is formed. It has been shownby Dawber and Scott (2002) that the defect concentra-tion profile as measured by capacitance voltage techniquemay be explained by a model of combined bulk and grainboundary diffusion of oxygen vacancies during the hightemperature processing of a film.

Regardless of the p-type or n-type nature of the mate-rial , in most oxide ferroelectrics on elemental metal elec-trodes the barrier height for electrons is significantly lessthan the barrier height for holes(Robertson and Chen,1999), and so the dominant injected charge carriers areelectrons. As the injected carriers dominate the conduc-tion, leakage currents in ferroelectrics are electron cur-rents and not hole currents, contrary to the suggestion ofStolichnov and Tagantsev (1998b).

The first picture of barrier formation in semiconduc-tors is due to Schottky (1938) and Mott (1938). In thispicture the conduction band and valence band bend suchthat the vacuum levels at the interface are the same andthe fermi level is continuous through the interface, butdeep within the bulk of the semiconductor retains its orig-inal value relative to the vacuum level. This is achievedby the formation of a depletion layer which shifts theposition of the fermi level by altering the number of elec-trons within the interface.

Motivated by the experimental observation that manySchottky barrier heights seemed to be fairly indepen-dent of the metal used for the electrode Bardeen (1947)proposed a different model of metal-semiconductor junc-tions. In this picture the fermi level of the semiconductoris “pinned” by surface states to the original charge neu-trality level. These states, as first suggested by Heine(1965), are not typically real surface states but ratherstates induced in the band-gap of the semiconductor bythe metal.

Most junctions lie somewhere between the Schottkyand Bardeen limits. The metal induced gap states(MIGS) can accommodate some but not all of the differ-ence in the fermi level between the metal and the semi-conductor, and so band bending still occurs to some ex-

13

tent. The factor S = dφb

dφmis used to define this, with

S = 1 being the Schottky limit and S = 0 represent-ing the Bardeen limit. The value of S is determined bythe nature of the semiconductor; originally experimen-tal trends linking this to the covalency or ionicity of thebonding in the material were observed (with covalent ma-terials developing many more MIGS. than ionic materi-als) (Kurtin et al., 1969). However better correlation wasfound between the effective band gaps (dependent on theelectronic dielectric constant ǫ∞) (Schluter, 1978), with(Monch, 1986)

(1

S− 1) = 0.1(ǫ∞ − 1)2. (8)

Although SrTiO3 was invoked as one of the materi-als that violated the electronegativity rule by Schluter(1978), it is omitted from the plot against (ǫ∞ − 1).The experimental value for S in SrTiO3 can be measuredfrom Dietz’s data as approximately 0.5. This does notagree well with what one would expect from Monch’sempirical relation, which gives S = 0.28 (as used byRobertson and Chen (1999)). Note that the use of theionic trap-free value S=1 for BST gives a qualitative er-ror: It predicts that BST on Al should be ohmic, whereasin actuality it is a blocking junction; an S-value of ap-proximately 0.3 predicts a 0.4 eV Schottky barrier height,in agreement with experiment.(Scott, 2000)

We can extract the penetration depth for Pt states intoBaTiO3 from the first principles calculation of Rao et al.

(1997) by fitting the density of states (DOS) of platinumstates in the oxygen layers to an exponential relationshipto extract the characteristic length as 1.68 angstroms.

Cowley and Sze (1965) derived an expression for thebarrier height for junctions between the two extremes.In this approach the screening charges in the electrodeand the surface states are treated as delta functions ofcharge separated by an effective thickness δeff . This ef-fective thickness takes into account both the Thomas-Fermi screening length in the metal and the penetra-tion length of the MIGS, and is essentially an air-gapapproach.

The expression for the barrier height is

φb = S(φm − χ) + (1 − S)(Eg − φ0) + ζ (9)

ζ =S2C

2− S

3

2 [C(φm − χ) + (1 − S)(Eg − φ0)C

S

−C

S(Eg − Ef + kT ) +

C2S

4]1

2 (10)

In the above S = 11+q2δeff Ds

,C = 2qεsNDδ2eff . When

εs ≈ 10ε0 and ND < 1018 cm−3 C is of the order of0.01 eV and it is reasonable to discard the term ζ asCowley and Sze (1965) did. Neglecting this term, ashas been pointed out by Rhoderick and Williams (1988),

FIG. 15 Energy band diagram of a metal n-type semiconduc-tor contact after Cowley and Sze.

is equivalent to neglecting the charge in the depletionwidth. In the systems under consideration here this termshould not be neglected as it can be quite large. Todemonstrate the effect on the barrier height we calculatethe barrier height for a Pt-SrTiO3 barrier over a widerange of vacancy concentrations (Fig. 16).

FIG. 16 Schottky barrier height of Pt-SrTiO3 as a functionof oxygen vacancy concentration, note that this may explainthe variation of experimental values from ca. 0.7 eV to 1.0eV

It can be seen that the effect of vacancies on bar-rier height becomes important for typical concentra-tions of vacancies encountered in ferroelectric thin films.Dawber et al. (2001) have addressed this issue and alsothe effect of introduced dopants on barrier heights. De-spite their omission of the term discussed above the workof Robertson and Chen is valuable because of their calcu-lation of the charge neutrality levels for several ferroelec-tric materials, an essential parameter for the calculationof metal-ferroelectric barrier heights.

In a ferroelectric thin film this distribution of chargesat the interface manifests itself in more ways than sim-ply in the determination of the Schottky barrier height.Electric displacement in the system is screened over theentire charge distribution.

In measuring the small -signal capacitance against

14

thickness there is always a non-zero intercept, which hasbeen typically associated with a ”dead layer” at the metalfilm interface. However, in most cases this interfacialcapacitance can be understood by recognizing that a fi-nite potential exists across the charge at the interface.In the simplest approximation one neglects any chargein the ferroelectric and uses a Thomas-Fermi screeningmodel for the metal. This was initially considered byKu and Ullman (1964) and first applied to high k di-electrics by Black and Welser (1999). In their work theyuse a large value for the dielectric constant of the oxidemetal, considering it as the dielectric response of the ionsstripped of their electrons. This may seem quite reason-able but is not however appropriate. In general we thinkof metals not being able to sustain fields, and in the bulkthey certainly cannot, but the problem of the penetra-tion of electric fields into metals is actually well knownin a different context, that of the microwave skin depth.It is very instructive to go through the derivation as anAC current problem and then find the DC limit whichwill typically apply for our cases of interest.

We describe the metal in this problem using the DrudeFree Electron Theory:

σ =σ0

1 + iωτ(11)

There are three key equations to describe the chargedistribution in the metal.

Poisson’s equation for free charges:

ρ(z) =1

4π

∂E(z)

∂z(12)

The continuity equation:

− iωρ(z) =∂j(z)

∂z(13)

The Einstein transport equation:

j = σE − D∂ρ

∂z(14)

These are combined to give:

∂2ρ

∂z2=

4πσ

D(1 +

iω

4πσ)ρ(z) (15)

This tells us that if at a boundary of the metal thereexists a charge it must decay with the metal exponen-tially with characteristic screening length λ:

λ = (4πσ

D(1 +

iω

4πσ))−1/2 (16)

In the DC limit (which applies for most frequencies ofour interest) this length is the Thomas-Fermi screeninglength;

λ0 = (4πσ0

D)−1/2 (17)

So it becomes clear that the screening charge in themetal may be modelled by substituting a sheet of chargedisplaced from the interface by the Thomas-Fermi screen-ing length, but that in calculating the dielectric thicknessof this region the effective dielectric constant that mustbe used is 1, consistent with the derivation of the screen-ing length. Had we used a form of the Poisson equationthat had a non-unity dielectric constant, i.e.,

ρ(z) =ǫ

4π

∂E(z)

∂z(18)

then our screening length would be

λ0 = (4πσ0

ǫD)−1/2 (19)

which is not the Thomas-Fermi screening length. Thusthe use of a non-unity dielectric constant for the metal isnot compatible with the use of the Thomas-Fermi screen-ing length.

Measurements on both sol-gel and CVD lead zirconate-titanate (PZT) films down to ca. 60 nm thickness showthat reciprocal capacitance 1/C(d) versus thickness d ex-trapolates to finite values at d=0, demonstrating an in-terfacial capacitance. However whereas the value for thesol-gel films is consistent with the Thomas Fermi screen-ing approach (.05nm), the value of interfacial thickness(0.005 nm) for the CVD films is only 10% of the in-terfacial capacitance that would arise from the knownFermi-Thomas screening length of 0.05 nm in the Pt elec-trodes (Dawber et al., 2003b). That is, if this result wereinterpreted in terms of a ”dead layer”, the dead layerwould have negative width. This result may arise from acompensating ”double-layer” of space charge inside thesemiconducting PZT dielectric; the Armstrong-Horrocks(Armstrong and Horrocks , 1997) semiconductor formal-ism form of the earlier Helmholtz and Gouy-Chapmanpolar-liquid models of the double layer can be used. Sucha double layer is unnecessary in PVDF because that ma-terial is highly insulating (Moreira , 2002). This explainsquantitatively the difference (x8) of interfacial capaci-tance in sol-gel PZT films compared with CVD PZT filmsof the same thickness. The magnitude of the electroki-

netic potential (or zeta-potential) ζ = σd′

ǫǫ0that develops

from the Helmholtz layer can be estimated without ad-justable parameters from the oxygen vacancy gradientdata of Dey for a typical oxide perovskite, SrTiO3; us-ing Dey’s surface charge density σ of 2.8 x 1018 e / m2,a Gouy screening length in the dielectric d’ = 20 nm,and a dielectric constant of ǫ = 1300 yields ζ = 0.78 eV;since this is comparable to the Schottky barrier height, itimplies that much of the screening is provided internallyby mobile oxygen vacancies. [Here σ(τ, µ) is a functionof time τ and mobility µ for a bimodal (a.c.) switchingprocess.]

15

4. Conduction mechanisms

In general conduction is undesirable in memory devicesbased on Capacitors, and so the understanding and min-imization of conduction has been a very active area ofresearch over the years. Many mechanisms have beenproposed for the conduction in ferroelectric thin films.

a. Schottky injection Perhaps the most commonly ob-served currents in ferroelectrics are due to thermionic in-jection of electrons from the metal into the ferroelectric.The current-voltage characteristic is determined by theimage force lowering of the barrier height when a poten-tial is applied. A few points should be made about Schot-tky injection in ferroelectric thin films. The first is aboutthe dielectric constant appropriate for use. In ferro-electrics the size of the calculated barrier height loweringdepends greatly on which dielectric constant, the static orthe electronic, is used. The correct dielectric constant isthe electronic one (∼ 5.5), as discussed by Scott and usedby Dietz, and by Zafar. Dietz and Waser (1997) used themore general injection law of Murphy and Good (1956)to describe charge injection in SrTiO3 films. They foundthat for lower fields the Schottky expression was valid,but at higher fields numerical calculations using the gen-eral injection law were required. They did not howeverfind that Fowler-Nordheim Tunneling was a good descrip-tion of any of the experimental data.

It has been shown by Zafar et al. (1998) that in factthe correct form of the Schottky equation that shouldbe used for ferroelectric thin films is the diffusion lim-ited equation of Simmons. Furthermore, very recentlyDawber and Scott (2004) have shown that when one con-siders the ferroelectric capacitor as a metal-insulator-metal system with diffusion-limited current (as opposedto a single metal- insulator junction), the leakage cur-rent is explained well; in addition, a number of unusualeffects,,such as the negative differential resistivity ob-served by Watanabe et al. (1998) and the PTCR effectobserved by Hwang et al. (1997),Hwang et al. (1998) areaccounted for.

b. Poole-Frenkel One of the standard ways of identifyinga Schottky regime is to plot log( J

T ) against V1

2 . In thiscase the plot will be linear if the current injection mech-anism is Schottky injection. Confusion can arise becausecarriers can also be generated from internal traps by thePoole-Frenkel effect, which on the basis of this plot isindistinguishable from Schottky injection. However, ifthe I-V characteristic is asymmetric with respect to pos-itive and negative voltages (as is usually the case) thenthe injection process is most probably Schottky injection.There are however some papers that show symmetricalI-V curves and correctly explain their data on the basisof a Poole-Frenkel conduction mechanism (Chen et al.,1998).

c. Fowler-Nordheim Tunneling Many researchers havediscussed the possibility of tunnelling currents in ferro-electric thin film capacitors. For the most part they arenot discussing direct tunneling through the film, whichwould be impossible for typical film thicknesses, but in-stead tunneling through the potential barrier at the elec-trode. The chief experimental evidence that it might in-deed be possible is due to Stolichnov, Tagantsev et al.

(1998) who have seen currents that they claim to beentirely tunneling currents in PZT films 450 nm thickat temperatures between 100-140 K. It should be notedhowever that they only observed tunneling currentsabove 2.2MV/cm, below which they were unable to ob-tain data. The narrowness of the range of fields for whichthey have collected data is a cause for concern, since thedata displayed in their paper go from 2.2-2.8 MV/cm.We conducted leakage current measurements on a 70 nmBST thin film at 70 K and found that the leakage cur-rent, while of much lower magnitude, was still well de-scribed by a Schottky injection relationship, although ifone fitted this data to a similarly narrow field region itdid appear to satisfy the Fowler-Nordheim relationshipwell (Fig. 17).

FIG. 17 Leakage current data from Au-BST-SrRuO3 film atroom temperature and at T=70K

The effective masses for tunneling obtained inthe studies of Stolichnov, Tagantsev et al. (1999) andBaniecki et al. (2001) also seem to be at odds withthe normal effective masses considered for these mate-rials. Whereas they use effective masses of 1.0, the ef-fective masses in perovskite oxides seem to be some-what larger, from 5-7 me for barium titanate and stron-tium titanate.(Scott et al., 2003). Although the tun-neling mass and the effective (band) mass need not bethe same in general, if the tunneling is through thick-nesses of > 2 nm, they are nearly so. [Schnupp (1967);Conley and Mahan (1967), also find that the tunnellingmass due to light holes in GaAs fits the band mass verywell. ].”

d. Space Charge Limited Currents The characteristicquadratic relationship between current and voltage thatis the hallmark of space charge limited currents are often

16

seen in ferroelectrics. Sometimes it is observed that spacecharge limited currents are seen when a sample is biasedin one direction, whereas for the opposite bias Schottkyinjection dominates.

e. Ultra-Thin Films - Direct Tunneling Very recentlyRodriguez Contreras et al. (2003) have succeeded inproducing metal-PZT-metal junctions sufficiently thin(6nm) that it appears that direct tunneling or phononassisted tunneling (in contrast to Fowler Nordheim tun-neling)through the film may occur, though this resultrequires more thorough investigation, since the authorsnote the barrier heights extracted from their data usinga direct tunelling model are much smaller than expected.The principal result of this paper is resistive switching,which may be of considerable interest in device appli-cations, but also requires more thorough investigation.This very interesting experimental study raises impor-tant questions about the way that metal wave functionspenetrating from the electrode and ferroelectric polarisa-tion interact with each other in the thinnest ferroelectricjunctions.

f. Grain boundaries Grain boundaries are often consid-ered to be important in leakage current because of theidea that they will provide conduction pathways throughthe film.

Gruverman’s results suggest that this is not the case inSBT. In his experiment an AFM tip is rastered across thesurface of a polycrystalline ferroelectric film. The imagedpattern records the leakage current at each point: whiteareas are high-current spots; dark areas, low current. Ifthe leakage were predominantly along grain boundaries,we should see dark polyhedral grains surrounded by whitegrain boundaries, which become brighter with increasingapplied voltage. In fact, the opposite situation obtains:This indicates that the grains have relatively low resis-tivity, with high-resistivity grain boundaries. The secondsurprise is that the grain conduction comes in a discretestep; an individual grain suddenly “turns on” (like a lightswitch). Smaller grains generally conduct at lower volt-ages (in accord with Maier’s theory of space charge effectsbeing larger in small grains with higher surface-volumeratios(Lubomirsky et al., 2002)).

C. Device Failure

1. Electrical Breakdown

The process of electrical shorting in ferroelectric PZTwas first shown by Plumlee (1967) to arise from dendrite-like conduction pathways through the material, initiatedat the anodes and/or cathodes. These were manifest asvisibly dark filamentary paths in an otherwise light ma-terial when viewed through an optical microscope. They

have been thought to arise as “virtual cathodes” via thegrowth of ordered chains of oxygen-deficient material.This mechanism was modeled in detail by Duiker et al(Duiker , 1990),(Duiker and Beale , 1990),(Duiker et al.,1990).

To establish microscopic mechanisms for beakdown inferroelectric oxide films one must show that the depen-dences of breakdown field EB upon film thickness d,ramp rate, temperature, doping, and electrodes are sat-isfied. The dependence for PZT upon film thickness ismost compatible with a low power-law dependence orpossibly logarithmic (Scott et al., 2003). The physicalmodels compatible with this include avalanche (logarith-mic), collision ionization from electrons injected via fieldemission from the cathode (Forlani and Minnaja, 1964),which gives

EB = Ad−w (20)

with 14 < w < 1

2 , or the linked defect model ofGerson and Marshall (1959), which has d = 0.3. Thedependence on electrode material arises from the elec-trode work function and the ferroelectric electron affinitythrough the resultant Schottky barrier height. FollowingVon Hippel (1935) we have (Scott (2000), p62.)

eEBλ = h(ΦM − ΦFE) (21)

where ΦM and ΦFE are the work functions of the metaland of the semiconducting ferroelectric,; is electron meanfree path; and h is a constant of order unity.

Even in films for which there is considerable Poole-Frenkel limitation of current (a bulk effect), the Schot-tky barriers at the electrode interfaces will still dominatebreakdown behavior.

In general electrical breakdown in ferroelectric oxidesis a hybrid mechanism (like spark discharge in air) inwhich the initial phase is electrical but the final stageis simple thermal run-away. This makes the dependenceupon temperature complicated.

There are at least three different contributions to thetemperature dependence. The first is the thermal prob-ability of finding a hopping path through the material.Following Gerson and Marshall and assuming a randomisotropic distribution of traps, Scott (Scott, 1995) showedthat

EB = G −kBT

Blog A (22)

which gives both the dependence on temperature Tand electrode area A in agreement with practically allexperiments on PZT, BST, and SBT.

In agreement with this model the further assumption ofexponential conduction (non-ohmic) estimated to occurfor applied field E > 30 MV/m (Scott, 2000)

17

σ(T ) = σ0 exp−b

kBT(23)

in these materials yields the correct dependence ofbreakdown time tB upon field

log tB = c1 − c2EB (24)

as well as the experimentally observed dependence ofEB on rise time tc of the applied pulse:

EB = c3t− 1

2

c (25)

Using the same assumption of exponential conduction,which is valid for

aeE ≪ kBT (26)

where a is the lattice nearest neighbor oxygen-site hop-ping distance (approximately a lattice constant) and e,the electron charge, Scott (Scott, 2000) shows that thegeneral breakdown field expression

CVdT

dt−∇(K · ∇T ) = σE2

B (27)

in the impulse approximation (in which the secondterm in the above equation is neglected) yields

EB(T ) = [3CV K

σ0btc]1

2 T exp(b

2kBT) (28)

which suffices to estimate the numerical value of break-down field for most ferroelectric perovskite oxide films;values approximating 800 MV/m are predicted and mea-sured.

A controversy has arisen regarding the temperaturedependence of EB(T ) and the possibility of avalanche(Stolichnov et al., 2001) In low carrier concentration sin-gle crystals, especially Si, avalanche mechanisms give atemperature dependence that is controlled by the meanfree path of the injected carriers. This is physically be-cause at higher temperatures the mean free path λ de-creases due to phonon scattering and thus one must applya higher field EB to achieve avalanche conditions.

λ = λ0 tanh(EB

kT) (29)

However, this effect is extremely small even for lowcarrier concentrations (10% change in EB between 300Kand 500K for n = 1016 cm−3) and negligible for higherconcentrations. The change in EB in BST between 600Kand 200K is > 500% and arises from Eq. 22, not Eq.

29). Even if the ferroelectrics were single crystals, with1020 cm−3 oxygen vacancies near the surface, any T-dependence from Eq. 29 would be unmeasurably small;and for the actual fine-grained ceramics (40 nm grain di-ameters), the mean free path is ca. 1 nm and limited bygrain boundaries (T-independent). Thus the conclusionof Stolichnov et al. (2001) regarding avalanche is quali-tatively and quantitatively wrong in ferroelectric oxides.

2. Fatigue

Polarisation fatigue, which is the process whereby theswitchable ferroelectric polarisation is reduced by repet-itive electrical cycling is one of the most serious devicefailure mechanisms in ferroelectric thin films. It is mostcommonly a problem when Pt electrodes, desirable be-cause of their high work-functions, are used.

P2.5 x 10 Cycles

6

Virgin

Initial Trace

+5 V-5 V

FIG. 18 Change in the polarisation hysteresis loop with fa-tigue Scott and Pouligny (1988)

Importantly fatigue occurs through the pinning of do-main walls, which pins the polarization in a particulardirection, rather than any fundamental reduction of thepolarisation. Scott and Pouligny (1988) demonstrated inKNO3 via Raman spectroscopy that only a very smallpart of the sample was converted from the ferroelectricto non ferroelectric phase with fatigue, thus implyingthat fatigue must be caused by pinning of the domainwalls. They also demonstrated that the domain wallscould be de-pinned via the application of a large field, asshown in Fig.17. The pinning of domain walls has alsobeen observed directly by Atomic Force microscopy byGruverman et al. (1996) and by Colla et al. (1998).

There is a fairly large body of evidence that oxygen va-cancies play some key part in the fatigue process. Augerdata of Scott et al. (1991) show areas of low oxygen con-centration in a region near to the metal electrodes, imply-ing a region of greater oxygen vacancy data. Scott et alalso reproduced Auger data from Troeger (unpublished)for a film that had been fatigued by 1010 cycles show-ing an increase in the width of the region with depletedoxygen near the platinum electrode (Fig. 19). There ishowever no corresponding change at the gold electrode.Gold does not form oxides, this might be an explana-tion of the different behaviour at the two electrodes. Al-

18

though some researchers believe that platinum also doesnot form oxides, the adsorption of oxygen onto Pt sur-faces is actually a large area of research, because of theimportant role platinum plays as a catalyst in fuel cellelectrodes. Oxygen is not normally adsorbed onto goldsurfaces but can be if there is significant surface rough-ness (Mills et al., 2003).

SPACE CHARGE

METAL

SPACE CHARGE

REGION OF LOW OXYGENCONCENTRATION(APPROX. N-TYPE)

200-400Ao

REGION OF LOW OXYGENCONCENTRATION(APPROX. N-TYPE)

METAL

SPACE CHARGE

SPACE CHARGE

NEARLY STOCHIOMETRICREGION(P-TYPE)

0

20

40

60

80

100

120

14020 40 60 80 10010 30 50 70 90

METALAu

OXYGEN

METALPt

Zr

Pb

Ti

DE

PT

H (

nm

)

STOICHIOMETRIC PERCENTAGE

SURFACELAYER ANDCONTACTINTERFACE

Au Pt

OXYGEN(INITIAL)

(FINAL)10 CYCLES

10UN

-NO

RM

ALIZ

ED

AT

OM

ICP

ER

CE

NTA

GE

DEPTH (nm)

100

80

60

40

20

0

0 50 100 150

FIG. 19 Scott et al. (1991)(a) Auger depth profile of PZTthin film capacitor (b) Effect of fatigue on oxygen concentra-tion near the electrode

It has also been found experimentally that films fa-tigue differently in atmospheres containing different oxy-gen partial pressures. (Brazier et al., 1999). Pan et al.

(1996) claim to have seen oxygen actually leaving a fer-roelectric sample during switching, though we note thatNuffer et al. (2001) claim this to be an experimental arte-fact. The results of Schloss et al. (2002) are very inter-esting in that they show directly by O18 tracer studiesthat the oxygen vacancies redistribute themselves duringvoltage cycling. In their original paper they concludedthat the redistribution of oxygen vacancies this is notthe cause of fatigue, because they did not see redistribu-tion of O18 when the sample has been annealed, thoughthe sample still fatigues, however in a more recent pub-lication they conclude the reason they could not see theoxygen tracer distribution was more probably due to achange in the oxygen permeability of the electrode afterannealing (Schloss et al., 2004).

It has been known for some time that the fatigue ofPZT films can be improved by the use of oxide electrodes,such as iridium oxide or ruthenium oxides. Araujo et al.

(1995) explain the improved fatigue resistance by the factthat oxides of iridium and platinum can reduce or re-oxidise reversibly and repeatedly without degradation.It is for the same reason that iridium is preferred to plat-inum an a electrode materials for medical applicationswhere this property was originally studied by Robblee(1986). This property does make the leakage currentproperties of these electrodes more complicated, and gen-erally films with Ir/IrO2 or Ru/RuO2 electrodes havehigher leakage currents than those with platinum elec-trodes. Unless carefully annealed at a certain temper-ature RuO2 electrodes will have elemental Ru metallicislands. Since the work function for Ru is 4.65 eV andthat for RuO2 is 4.95 eV, almost all the current will passthrough the Ru islands, producing hot spots and occa-sional shorts (Hartmann et al., 2000). By contrast, al-though one expects that there will be similar issues with

mixtures of Ir and IrO2 in iridium based electrodes, whenmetallic Ir is oxidized to IrO2 its workfunction decreasesto 4.23 eV (Chalamala et al., 1999).

The idea that planes of oxygen vacancies perpendicu-lar to the polarisation direction could pin domain wallsis originally due to Brennan (1993). Subsequently, in atheoretical microscopic study of oxygen vacancy defectsin PbTiO3 Park and Chadi (1998) showed that planesof vacancies are much more effective at pinning do-main walls than single vacancies. In bulk ferroelectricsArlt and Neumann (1988) have discussed how underrepetitive cycling the vacancies can move from theiroriginally randomly distributed sites in the perovskitestructure to sites in planes parallel to the ferroelectric-electrode interface. We suspect that while this may ac-count for fatigue in bulk ferroelectrics it is not the oper-ative mechanism in thin films. Scott and Dawber (2000)have suggested that in thin films the vacancies can reachsufficiently high concentrations that they order them-selves into planes in a similar way as occurs in Fe-dopedbulk samples and on the surfaces of highly reduced spec-imens. Direct evidence that this occurs in bulk PZT wasfound using Atomic force microscope imagery of PZTgrains by Lupascu and Rabe (2002) (Fig. 20). Recentlyevidence of oxygen vacancy ordering has also been foundin barium titanate reduced after an accelerated life test(Woodward et al., 2004).

FIG. 20 Atomic force microscopy images (Lupascu) (a) beforeand (b) after, cycling showing evidence of planes of oxygenvacancies in the fatigued sample. Image width = 10µm.

While most researchers acknowledge that oxygen va-cancies play a role in fatigue, it should be noted thatTagantsev et al. (2001) has aggressively championed amodel of charge injection; however since this model isnot developed into a quantitative form it is very hard to

19

verify or falsify it. Charge injection probably does play arole in fatigue, an idea at least in part supported by thedetailed experimental study of Du and Chen (1998), butin the model of Dawber and Scott (2000), (which drawsupon the basic idea of Yoo and Desu (1992) that fatigueis due to the electromigration of oxygen vacancies) it isnot included, nevertheless most of the experimental re-sults in the literature may be accounted for. The modelof Dawber and Scott (2000) basically shows that in a fer-roelectric thin film under an AC field there is in fact a netmigration of vacancies towards the interface and it is thehigh concentration of vacancies in this region that resultsin ordering of the vacancies and pinning of domain walls.The interfacial nature of fatigue in thin films has beendemonstrated by Colla et al. (1998) and by Dimos et al.

(1994).To really understand fatigue better what is needed is

more experiments that try to look at the problem in novelways, standard electrical measurements alone probablycannot shed a great deal of additional light on the prob-lem, especially given the problems between comparingsamples grown in different labs using different techniques.Very recently a very interesting study was undertaken byDo et al. (2004) using X-Ray microdiffraction to studyfatigue in PZT. Using this technique they were able tosee how regions of the film stopped switching as it fa-tigued. One of the key findings of this study was thatthere appears to be two fatigue effects operative, a re-versible effect that occurs when low to moderate fieldsare used for switching and an irreversible effect whichoccurs under very high fields.

3. Retention failure

Clearly a non-volatile memory that fails to retain theinformation stored in it will not be a successful device.Furthermore producers of memories need to be able toguarantee retention times over much longer periods oftime then they can possibly test. A better understand-ing of retention failure is thus required so that modelscan used that allow accelerated retention tests to be car-ried out, the work of Kim et al. (2001) is a step in thisdirection.

It seems that imprint and retention failure are closelylinked phenomena, ie. if a potential builds up acrossthe system of the time it can destabilize the ferroelec-tric polarisation state and thus cause loss of information.Further comparison of retention-time data and fatiguedata suggest quite strongly a link between the two ef-fects. DC degradation of resistance in BST seems also tobe a related effect (Zafar et al., 1999). The electromigra-tion of oxygen vacancies under an applied field in a Fe-doped SrTiO3 single crystal has been directly observedvia electro-chromic effects (Waser et al., 1990). Oxygenvacancy redistribution under applied field has also beeninvoked to explain a slow relaxation of the capacitancein BST thin films (Boikov et al., 2001). It would seem to

make sense that whereas fatigue relates to the cumulativemotion of oxygen vacancies under an AC field, resistancedegradation is a result of their migration under an ap-plied DC field and retention failure is a result of theirmigration under the depolarisation field or other built-infields in the material.

IV. FIRST PRINCIPLES

With continuing advances in algorithms and computerhardware, first principles studies of the electronic struc-ture and structural energetics of complex oxides can nowproduce accurate, material-specific information relevantto the properties of thin-film ferroelectrics. In this sec-tion, we focus on first-principles studies that identifyand analyze the characteristic effects specific to thinfilms. First, we briefly review the relevant methodologi-cal progress and the application of these methods to bulkferroelectric materials. Next, we will survey the first-principles investigations of ferroelectric thin films andsuperlattices reported in the literature. It will be seenthat the scale of systems that can be studied directly byfirst principles methods is severely limited at present bypractical considerations. This can be circumvented bythe construction of nonempirical models with parame-ters determined by fitting to the results of selected first-principles calculations. These models can be parame-terized interatomic potentials, permitting molecular dy-namics studies of nonzero temperature effects, or first-principles effective Hamiltonians for appropriate degreesof freedom (usually local polarization and strain). Theform of the latter strongly resembles that of a Landau-Devonshire theory, providing a connection between first-principles approaches and the extensive literature on phe-nomenological models for the behavior of thin film ferro-electrics. The advantages and disadvantages of using firstprinciples results rather than experimental data to con-struct models will be considered. In addition to allowingthe study of systems far more complex than those thatcan be considered by first principles alone, this modellingapproach yields physical insight into the essential differ-ences between bulk and thin film behavior, which will bediscussed at greater length in subsection IV.B.5. Finally,it will be seen that despite practical limitations, the com-plexity of the systems for which accurate calculations canbe undertaken has steadily increased in recent years, tothe point where films of several lattice constants in thick-ness can be considered. While this is still far thinner thanthe films of current technological interest, concommitentimprovements in thin film synthesis and characterizatonhave made it possible to achieve a high degree of atomicperfection in comparable ultrathin films in the context ofresearch. This progress has led to a true relevance of cal-culational results to experimental observations, opening ameaningful experimental-theoretical dialogue. However,this progress in some ways only serves to highlight thefull complexity of the physics of real ferroelectric films: as

20

questions get answered, more questions, especially aboutphenomena at longer length scales and about dynamics,are put forth. These challenges will be discussed in sub-section IV.B.6.

A. DFT studies of bulk ferroelectrics

In parallel with advances in laboratory synthesis, thepast decade has seen a revolution in the atomic-scaletheoretical understanding of ferroelectricity, especiallyin perovskite oxides, through first-principles density-functional-theory (DFT) investigations. The central re-sult of a DFT calculation is the ground state energycomputed within the Born-Oppenheimer approximation;from this the predicted ground state crystal structure,phonon dispersion relations, and elastic constants aredirectly accessible. The latter two quantities can beobtained by finite-difference calculations, or more effi-ciently, through the direct calculation of derivatives ofthe total energy through density-functional perturbationtheory (DFPT) (Baroni et al., 2001).

For the physics of ferroelectrics, the electric polar-ization and its derivatives, such as the Born effectivecharges and the dielectric and piezoelectric tensors, areas central as the structural energetics, yet proper for-mulation in a first-principles context long proved to bequite elusive. Expressions for derivatives of the polar-ization corresponding to physically measurable quanti-ties were presented and applied in DFPT calculations inthe late 1980’s(de Gironcoli et al., 1989). A key concep-tual advance was establishing the correct definition ofthe electric polarization as a bulk property through theBerry phase formalism of King-Smith, Vanderbilt andResta (King-Smith and Vanderbilt, 1993; Resta, 1994).With this and the related Wannier function expression(King-Smith and Vanderbilt, 1994), the spontaneous po-larization and its derivatives can be computed in a post-processing phase of a conventional total-energy calcula-tion, greatly facilitating studies of polarization-relatedproperties.

For perovskite oxides, the presence of oxygen andfirst row transition metals significantly increases thecomputational demands of density functional total en-ergy calculations compared to those for typical semi-conductors. Calculations for perovskite oxides havebeen reported using essentially all of the available first-principles methods for accurate representation of theelectronic wavefunctions: all-electron methods, mainlylinearized augmented plane wave (LAPW) and full-potential linearized augmented plane wave (FLAPW),linear muffin-tin orbitals (LMTO), norm-conservingand ultrasoft pseudopotentials, and projector-augmentedwavefunction (PAW) potentials. The effects of dif-ferent choices for the approximate density functionalhave been examined; while most calculations are car-ried out with the local-density approximation (LDA),for many systems the effects of the generalized-gradient

approximation (GGA) and weighted-density approx-imation (WDA)(Wu et al., 2004) have been investi-gated, as well as the alternative use of the Hartree-Fock approach. Most calculations being currentlyreported are performed with an appropriate stan-dard package, mainly VASP(Kresse and Furthmuller,1996; Kresse and Hafner, 1993), with ultrasoft pseu-dopotentials and PAW potentials, ABINIT (ABI),with norm-conserving pseudopotentials and PAW po-tentials, PWscf (Baroni et al.), with norm-conservingand ultrasoft pseudopotentials, SIESTA (Soler et al.,2002), with norm-conserving pseudopotentials, WIEN97(FLAPW) (Blaha et al., 1990) and CRYSTAL (Hartree-Fock)(Saunders et al., 1999).

To predict ground state crystal structures, the usualmethod is to minimize the total energy with respect tofree structural parameters in a chosen space group, ina spirit similar to that of a Rietveld refinement in anexperimental structural determination. The space groupis usually implicitly specified by a starting guess for thestructure. For efficient optimization, the calculation offorces on atoms and stresses on the unit cell is essentialand is by now included in every standard first-principlesimplementation following the formalism of Hellmann andFeynman (Feynman, 1939) for the forces and Nielsen andMartin (Nielsen and Martin, 1985) for stresses.

The accuracy of DFT for predicting theground state structures of ferroelectrics wasfirst investigated for the prototypical cases ofBaTiO3 and PbTiO3 (Cohen and Krakauer, 1990;Cohen, 1992; Cohen and Krakauer, 1992), andthen extended to a larger class of ferroelectricperovskites(King-Smith and Vanderbilt, 1994). Ex-tensive studies of the structures of perovskite oxidesand related ferroelectric oxide structures have sincebeen carried out (Resta, 2003). The predictive powerof first-principles calculations is well illustrated by theresults of Singh for PbZrO3 (Singh, 1995), in which thecorrect energy ordering between ferroelectric and anti-ferroelectric structures was obtained and furthermore,comparisons of the total energy resolved an ambiguityin the reported space group and provided an accuratedetermination of the oxygen positions.

It is important to note, however, that there seem tobe limitations to the accuracy to which structural pa-rameters, particularly lattice constants, can be obtained.Most obvious is the underestimate of the lattice con-stants within the LDA, typically by about 1% (GGAtends to shift lattice constants upward, sometimes sub-stantially overcorrecting). Considering that the calcula-tion involves no empirical input whatsoever, an error assmall as 1% could be regarded not as a failure, but as asuccess of the method. Moreover, the fact that the under-estimate varies little from compound to compound meansthat the relative lattice constants and thus the type of lat-tice mismatch (tensile/compressive) betwen two materi-als in a heterostructure is generally correctly reproducedwhen using computed lattice parameters. However, for

21

certain questions, even a 1% underestimate can be prob-lematic. The ferroelectric instability in the perovskiteoxides, in particular, is known to be very sensitive topressure (Samara, 1987) and thus to the lattice constant,so that 1% can have a significant effect on the ferroelec-tric instability. In addition, full optimization of all struc-tural parameters in a low symmetry space group can insome cases, PbTiO3 being the most well studied example,lead to an apparently spurious prediction, though fixingthe lattice constants to their experimental values leadsto good agreement for the other structural parameters(Saghi-Szabo et al., 1998). Thus, it has become accept-able, at least in certain first-principles contexts, to fix thelattice parameters or at least the volume of the unit cell,to the experimental value, when this value is known.

In a first-principles structural prediction, the initialchoice of space group may appear to limit the chancethat a true ground state structure will be found. In gen-eral, once a minimum is found, it can be proved (or not)to be a local minimum by computation of the full phonondispersion and of the coupling, if allowed by symmetry,between zone-center phonons and homogeneous strain(Garcia and Vanderbilt, 1996). Of course, this does notrule out the possibility of a different local minimum withlower energy with an unrelated structure.

For ferroelectrics, the soft-mode theory of ferroelec-tricity provides a natural conceptual framework for theidentification of low-symmetry ground state structuresand for estimating response functions. The starting pointis the identification of a high-symmetry reference struc-ture. For perovskite compounds, this is clearly the cubicperovskite structure, and for layered perovskites, it isthe nonpolar tetragonal structure. The lattice instabil-ities of the reference structure can be readily identifiedby the first-principles calculation of phonon dispersionrelations (Ghosez et al., 1999; Sai and Vanderbilt, 2000;Waghmare and Rabe, 1997a), this being especially effi-cient within density functional perturbation theory. Insimple ferroelectric perovskites, the ground state struc-ture is obtained to lowest order by freezing in the mostunstable mode (a zone-center polar mode). This pic-ture can still be useful for more complex ground statestructures that involve the freezing in of two or morecoupled modes (e.g. PbZrO3) (Cockayne and Rabe,2000; Waghmare and Rabe, 1997b), as well as foridentifying low-energy structures that might be sta-bilized by temperature or pressure (Fennie and Rabe;Stachiotti et al., 2000) The polarization induced by thesoft polar mode can be obtained by computation ofthe Born effective charges, yielding the mode effectivecharge. The temperature-dependent frequency of thesoft polar mode and its coupling to strain are expectedlargely to determine the dielectric and piezoelectric re-sponse of ferroelectric and near-ferroelectric perovskiteoxides; this idea has been the basis of several calcula-tions (Cockayne and Rabe, 1998; Garcia and Vanderbilt,1998).

Minimization of the total energy can similarly be

used to predict atomic arrangements and forma-tion energies of point defects (Astala and Bristowe,2004; Betsuyaku et al., 2001; Man and Feng, 2002; Park,2003; Park and Chadi, 1998; Poykko and Chadi, 2000;Robertson, 2003), domain walls (He and Vanderbilt,2003; Meyer and Vanderbilt, 2002; Poykko and Chadi,2000) and non-stoichiometric planar defects suchas antiphase domain boundaries (Li et al., 2002;Suzuki and Fujimoto, 2001),in bulk perovskite oxides.The supercell used must accommodate the defect ge-ometry, and generally must contain many bulk primi-tive cells to minimize the interaction of a defect with itsperiodically-repeated images. Thus, these calculationsare extremely computationally intensive, and many im-portant questions remain to be addressed.

While much of the essential physics of ferroelectricsarises from the structural energetics, the polarization andthe coupling between them, there has been increasinginterest in ferroelectric oxides as electronic and opticalmaterials, for which accurate calculations of the gap anddipole matrix elements are important. Furthermore, aswe will discuss in detail below, the bandstructures enterin an essential way in understanding the charge trans-fer and dipole layer formation of heterostructures involv-ing ferroelectrics, other insulators, and metals. Whiledensity functional theory provides a rigorous foundationonly for the computation of Born-Oppenheimer ground-state total energies and electronic charge densities, itis also often used for investigation of electronic struc-ture. In the vast majority of density functional im-plementations, calculation of the ground state total en-ergy and charge density involves the computation of abandstructure for independent electron states in an ef-fective potential, following the work of Kohn and Sham(Kohn and Sham, 1965). This bandstructure is gener-ally regarded as a useful guide to the electronic structureof materials, including perovskite and layered perovskiteoxides (Cohen, 1992; Robertson et al., 1996; Tsai et al.,2003). It should be noted that it is consistently foundthat with approximate functionals, such as the LDA,the fundamental bandgaps of insulators and semiconduc-tors, including perovskite ferroelectrics, are substantiallyunderestimated. While for narrow gap materials, thesystem may even be erroneously found to be metallic,for wider gap systems such as most of the simple ferro-electric perovskite compounds considered here, the bandgap is still nonzero and thus the structural energetics inthe vicinity of the ground state structure is unaffected.While this error might be considered to be an insuperablestumbling block to first-principles investigation of elec-tronic structure and related properties, there is at presentno widely available, computationally tractable, alterna-tive (there are, though, some indications that the use ofexact-exchange functionals can eliminate much of this er-ror (Piskunov et al., 2004); it has long been known thatHartree-Fock, i.e. exact exchange only, leads to overesti-mates of the gaps). The truth is that, as will be discussedfurther below, these results can, with care, awareness of

22

the possible limitations and judicious use of experimen-tal input, be used to extract useful information about theelectronic structure and related properties in individualmaterial systems.

At present, the computational limitations of full first-principles calculations to 70-100 atoms per supercell havestimulated considerable interest in the development anduse in simulations of effective models, from which asubset of the degrees of freedom have been integratedout. Interatomic shell-model potentials have been de-veloped for a number of perovskite oxide systems, elim-inating most of the electronic degrees of freedom ex-cept for those represented by the shells (Heifets et al.,2000b; Sepliarsky et al., 2004; Tinte et al., 1999). Amore dramatic reduction in the number of degrees offreedom is performed to obtain effective Hamiltonians,in which typically one vector degree of freedom decribesthe local polar distortion in each unit cell. This ap-proach has proved useful for describing finite temperaturestructural transitions in compounds and solid solutions(Bellaiche et al., 2000; Rabe and Waghmare, 1995, 2002;Waghmare and Rabe, 1997a; Zhong et al., 1994). To theextent that the parameters appearing in these potentialsare determined by fitting to selected first-principles re-sults (structures, elastic constants, phonons), these ap-proaches can be regarded as approximate first-principlesmethods. They allow computation of the polarization aswell as of the structural energetics, but not, however, ofthe electronic states. Most of the effort has been focusedon BaTiO3, though other perovskites, including PbTiO3

and KNbO3, SrTiO3 and (Ba,Sr)TiO3 and Pb(Zr,Ti)O3

have been investigated in this way, and useful results ob-tained.

B. First-principles investigation of ferroelectric thin films

The fascination of ferroelectric thin films and superlat-tices lies in the fact that the properties of the system as awhole can be so different from the individual propertiesof the constituent material(s). Empirically, it have beenobserved that certain desirable bulk properties, such asa high dielectric response, can be degraded in thin films,while in other investigations there are signs of novel in-teresting behavior obtained only in thin film form.

Theoretical analysis of the observed properties of thinfilms presents a daunting challenge. It is well known thatthe process of thin film growth itself can lead to non-trivial differences from the bulk material, as observedin studies of homoepitaxial oxide films such as SrTiO3

(Klenov et al., 2004). However, as synthetic methodshave developed, the goal of growing nearly ideal, atom-ically ordered, single crystal films and superlattices iscoming within reach, and the relevance of first-principlesresults for perfect single crystal films to experimental ob-servations emerging.

As will be clear from the discussion in the rest of thissection, an understanding of characteristic thin film be-

havior can best be achieved by detailed quantitative ex-amination of individual systems combined with the con-struction of models incorporating various aspects of thephysics, from which more general organizing principlescan be identified. In first principles calculations, thereis a freedom to impose constraints on structural param-eters and consider hypothetical structures that goes farbeyond anything possible in a real system being studiedexperimentally. This will allow us to isolate and examinevarious influences on the state of a thin film: epitaxialstrain, macroscopic electric fields, surfaces and interfaces,characteristic defects associated with thin-film growth,and “true” finite size effects, and how they change thechanges in atomic arrangements, electronic structure, po-larization, vibrational properties, and responses to ap-plied fields and stresses. While the main focus of thisreview is on thin films, this approach also applies nat-urally to multilayers and superlattices. Extending ourdiscussion to include these latter systems will allow us toconsider the effects of the influencing factors in differentcombinations, for example the changing density of inter-faces, the degree of mismatch strain, and the polarizationmismatch. These ideas also are relevant to investigatingthe behavior of bulk layered ferroelectrics, which can beregarded as natural short-period superlattices.

Within this first-principles modelling framework, wecan more clearly identify specific issues and results forinvestigation and analysis. The focus on modelling isalso key to the connection of first-principles results to theextensive literature on phenomenological analysis and toexperimental observations. This makes the most effectiveuse of first-principles calculations in developing a concep-tual and quantitative understanding of characteristic thinfilm properties, as manifested by thickness dependence aswell as by the dependence on choice of materials for thefilm, substrate and electrodes.

1. First principles methodology for thin films

The fundamental geometry for the study of thin films,surfaces and interfaces is that of an infinite single-crystalline planar slab. Since three-dimensional periodic-ity is required by most first-principles implementations,in those cases the slab is periodically repeated to pro-duce a supercell; a few studies have been carried outwith electronic wavefunction basis sets that permit thestudy of an isolated slab. The variables to be specifiedinclude the orientation, the number of atomic layers in-cluded, choice of termination of the surfaces, and widthof vacuum layer separating adjacent slabs. As in thefirst-principles prediction of bulk crystal structure, choiceof a space group for the supercell is usually establishedby the initial structure; relaxations following forces andstresses do not break space group symmetries. The di-rection of the spontaneous polarization is constrained bythe choice of space group, allowing comparison of unpo-larized (paraelectric) films with films polarized along the

23

normal or in the plane of the film.

As we will see below, in most cases the slabs are verythin (ten atomic layers or fewer). It is possible to re-late the results to the surface of a semi-infinite systemor a coherent epitaxial film on a semi-infinite substrateby imposing certain constraints on the structures con-sidered. In the former case, the atomic positions forinterior layers are fixed to correspond to the bulk crys-tal structure. In the latter, the in-plane lattice parame-ters of the supercell are fixed to the corresponding bulklattice parameters of the substrate, which is not other-wise explicitly included in the calculation. More sophis-ticated methods developed to deal with the coupling ofvibrational modes at the surface with bulk modes of thesubstrate(Lewis and Rappe, 1996) could also be appliedto ferroelectric thin films, though this has not yet beendone.

The LDA underestimate of the equilibrium atomic vol-ume will in general also affect slab calculations, and sim-ilar concerns arise about the coupling of strain and theferroelectric instability. As in bulk crystal structure pre-diction, it may in some cases be appropriate to fix certainstructural parameters according to experimental or bulkinformation. In the case of superlattices and supercellsof films on substrates, it may on the other hand be agood choice to work consistently at the (compressed) the-oretical lattice constant, since the generic underestimateof the atomic volume ensures that the lattice mismatchand relative tensile/compressive strain will be correctlyreproduced. This applies for example to the techniquementioned in the previous paragraph, in which the ef-fects of epitaxial strain are investigated by performingslab calculations with an appropriate constraint on thein-plane lattice parameters.

As in first-principles predictions of bulk crystal struc-ture, the initial choice of space group constrains, to alarge extent, the final “ground-state” structure. If thesupercell is constructed by choosing a bulk termination,the energy minimization based on forces and stresses willpreserve the initial symmetry, yielding information aboutsurface relaxations of the unreconstructed surface. Alower-energy structure might result from breaking addi-tional point or translational symmetries to obtain a sur-face reconstruction. This type of surface reconstructioncould be detected by computing the Hessian matrix (cou-pled phonon dispersion and homogeneous strain) for therelaxed surface. More complex reconstructions involv-ing adatoms, vacancies or both would have to be stud-ied using appropriate starting supercells. Information re-garding the existence and nature of such reconstructionsmight be drawn from experiments and/or from knownreconstructions in related materials.

One very important consideration in the theoreticalprediction of stable surface orientations and termina-tions and of favorable surface reconstructions is thatthese depend on the relative chemical potential of theconstituents. Fortunately, since the chemical poten-tial couples only to the stoichiometry, the prediction of

the change of relative stability with chemical potentialcan be made with a single total-energy calculation foreach structure (see, for example, (Meyer et al., 1999)).Because of the variation in stoichiometry for different(001) surface terminations, what is generally reported isthe average surface energy of symmetric AO and BO2-terminated slabs.

A problem peculiar to the study of periodically re-peated slabs with polarization along the normal is theappearance of electric fields in the vacuum. As shown inFig. 21, this occurs because there is a nonzero macro-scopic depolarizing field in the slab and thus a nonzeropotential drop between the two surfaces of the slab. Asthe potential drop across the entire supercell must bezero, this inevitably leads to a nonzero electric field in thevacuum. Physically, this can be interpreted as an exter-nal field applied uniformly across the unit cell which actspartially to compensate the depolarizing field. An anal-ogous situation arises for an asymmetrically terminatedslab when the two surfaces have different work functions.To eliminate the artificial field in the vacuum, one tech-nique is to introduce a dipole layer in the mid-vacuumregion far away from the slab (Bengtsson, 1999). Thiscan accommodate a potential drop up to a critical value,at which point electrons begin to accummulate in an ar-tificial well in the vacuum region (see Fig. 21). Thisapproach can be also be used to compensate the depo-larization field in a perpendicular polarized film, thoughit may happen that the maximum field that can be ap-plied is smaller than that needed for full compensation.Alternatively, by using a first-principles implementationwith a localized basis set, it is possible to perform com-putations for isolated slabs and thus avoid not only thespurious electric fields, but also the interaction betweenperiodic slab images present even for symmetric nonpo-lar slabs. Comparison between results obtained with thetwo approaches is presented in (Fu et al., 1999).

In determining the properties of an ultrathin film,the film-substrate and/or film electrode interface playsa role at least as important as the free surface. In first-principles calculations, the atomic and electronic struc-ture of the relevant interface(s) is most readily obtainedby considering a periodic multilayer geometry identicalto that used for computing the structure and propertiesof superlattices. To simulate a semi-infinite substrate,the in-plane lattice constant should be fixed to that ofthe substrate bulk. The relaxation of a large number ofstructural degrees of freedom requires substantial com-puter resources, and some strategies for efficiently gener-ating a good starting structure will be described in thediscussion of specific systems in Section IV.B.3.

Calculations of quantities characterizing the electronicstructure are based on use of the Kohn-Sham one-electron energies and wavefunctions. Band structures for1x1 (001) slabs are generally displayed in the 2D surfaceBrillouin zone for ksupercell,z=0. One way to identifysurface states is by comparison with the appropriatelyfolded-in bulk bandstructure. Another analysis method

24

E(z)

(z)

(z)per

(z)dip

E=0

ext

E

E

(a)

(b)

(c)

(d)

c0

v

v

v

v

z

z

z

z

FIG. 21 Schematic picture of the planar-averaged potentialv(z) for periodically repeated slabs: (a) with periodic bound-ary conditions, (b) potential of the dipole layer, (c) dipole-corrected slabs with vanishing external electric field, and (d)dipole-corrected slabs with vanishing internal electric field.From Meyer and Vanderbilt, 2001.

is to compute the partial density of states projected ontoeach atom in the slab. From these plots, the domi-nant character of a state at a particular energy can befound. In addition, an estimate of valence band offsetsand Schottky barriers at an interface can be obtainedby analyzing the partial density of states for a superlat-tice of the two constituents. This is done comparing theenergies of the highest occupied states in the interior ofthe relevant constituent layers (because of the band gapproblem, the positions of the conduction bands are com-puted by using the experimental bulk bandgaps). Thisestimate can be refined, as described in Junquera et al.,2003, by computation of the average electrostatic energydifference between the relevant constituent layers.

As the computational resources required for full first-principles calculations even of the simplest slab-vacuumsystem are considerable, there is a strong motivation toturn to interatomic potentials and effective Hamiltoni-ans. Interatomic potentials based on shell models fit-ted to bulk structural energetics are generally directlytransferred to the isolated slab geometry, with no changesfor the undercoordinated atoms at the surface. As willbe discussed further below, this approach seems to besuccessful in reproducing the relaxations observed in fullfirst-principles studies and has been applied to far largersupercells and superlattices. In the case of effective

Hamiltonians, it is, at least formally, possible simply toperform a simulation by removing unit cells and use bulkinteraction parameters for the unit cells in the film. Fora more accurate description, modification of the effectiveHamiltonian parameters for the surface layers is advis-able to restore the charge neutrality sum rule for thefilm (Ghosez and Rabe, 2000; Ruini et al., 1998). In ad-dition, the effects of surface relaxation would also resultin modified interactions at the surface.

2. Overview of systems

In this section, we present a list of the materials andconfigurations that have been studied, followed by a briefoverview of the quantities and properties that have beencalculated in one or more of the reported studies. A moredetailed description of the work on individual systems isprovided in the following subsection.

The configurations that have been considered todate in first principles studies can be organized intoseveral classes. The simplest configuration is a slabof ferroelectric material alternating with vacuum;this can be used to investigate the free surface ofa semi-infinite crystal, an unconstrained thin film,or an epitaxial thin film constrained to match thelattice constant of an implicit substrate. Specificmaterials considered included BaTiO3 ((001) surfaces(Cohen, 1996, 1997; Fu et al., 1999; Heifets et al.,2001a; Krcmar and Fu, 2003; Meyer and Vanderbilt,2001) and (110) surfaces(Heifets et al., 2001a))SrTiO3 (Heifets et al., 2002a, 2001a, 2002b;Kubo and Nozoye, 2003; Padilla and Vanderbilt, 1998),PbTiO3 (Bungaro and Rabe; Ghosez and Rabe, 2000;Meyer and Vanderbilt, 2001), KNbO3 (Heifets et al.,2000a) and KTaO3 (Li et al., 2003a). Another typeof configuration of comparable complexity is ob-tained by replacing the vacuum by a second material.If this is another insulating perovskite oxide, thecalculation can yield information about ferroelectric-dielectric (e.g BaTiO3/SrTiO3 (Neaton and Rabe,2003) and KNbO3/KTaO3 (Sepliarsky et al., 2002,2001)) or ferroelectric-ferroelectric interfaces and su-perlattices. This configuration can also be used tostudy the interface between the ferroelectric and adielectric (nonferroelectric) oxide (e.g. BaTiO3/BaOand SrTiO3/SrO (Junquera et al., 2003)). Replace-ment of the vacuum by a conductor simulates a filmwith symmetrical top and bottom electrodes, e.g.BaTiO3/SrRuO3(Junquera and Ghosez, 2003). Morecomplex multilayer geometries including two or moredifferent materials as well as vacuum layers have beenused to simulate ferroelectric thin film interactions withthe substrate (e.g. PbTiO3/SrTiO3/PbTiO3/vacuum(Johnston and Rabe), and with realistic electrodes(e.g. Pt/BaTiO3/Pt/vacuum(Rao et al., 1997) andPt/PbTiO3/Pt/vacuum (Sai and Rappe), as well as thestructure of epitaxial alkaline-earth oxide on silicon,

25

TiO terminated2BaO/PbO terminated

23

∆ <0d12

layer dipole

>0∆

Ba/Pb Ti O

η

η

η

1

2

3

d

2

3

1

moments

FIG. 22 Schematic illustration of the structure of the firstthree surface layers. From Meyer and Vanderbilt, 2001.

used as a buffer layer for growth of perovskite oxidefilms (McKee et al., 2003).

In each class of configurations, there are correspondingquantities and properties that are generally calculated.For the single slab-vacuum configuration, for each ori-entation and surface termination the surface energy isobtained. While in some of the initial studies the atomicpositions were fixed according to structural informationfrom the bulk (Cohen, 1996), in most current studiesrelaxations are obtained by energy minimization proce-dures. For the most commonly studied perovskite (001)slab, the relaxation geometry is characterized by changesin interplane spacings and rumplings quantified as shownin Fig. 22 Most studies assume bulk periodicity in theplane. For the study of surface reconstructions, it is nec-essary to expand the lateral unit cell, leading to a sub-stantial additional cost in computational resources. Mostattention has been focused on the paraelectric SrTiO3,although recently studies have been carried out as wellfor PbTiO3. A byproduct of the total energy calcula-tion that is often though not universally presented is thebandstructure and or density of states; the local densityof states at the surface is of particular interest.

The two-component superlattice configuration can betaken to model a film on a substrate and/or with elec-trode layers, or to model an actual superlattice such asthat which can be obtained by techniques such as MBE.In these studies, the interface and coupling between thetwo constituents is of primary interest; combinationsmost considered to date are ferroelectric + paraelectric,ferroelectric + dielectric, and ferroelectric + metal, whilethe combination of two ferroelectrics or ferroelectric +ferromagnetic material has been less intensively investi-gated. The main questions of interest are the structuralrearrangements at the interface and the change in thestructure, polarization, and related properties of individ-ual layers relative to the bulk resulting from the interac-tion with the other constituents. Analysis of the trendswith varying thickness(es) of the ferroelectric film and, inthe superlattice, other constituents is particularly useful.The electronic structure of these systems can be mostreadily characterized by the band offset between the twomaterials, which should also control the charge transferacross the interface, formation of a dipole layer, and the

potential difference between the two constituents. In thecase of a metal, this will determine the type of contact.The existence of interface states is also very relevant tothe physical behavior of the system.

At a considerable increase in computational expensebut also in realism, a system with three or more compo-nents can be studied; e.g. the combination of a substrate,a film, and vacuum. The main questions of interest in thefew such studies to date are the analysis of the ferroelec-tric instability in the film, and the film-induced changesin the substrate layers closest to the interface. As in two-component heterostructures, the partial density of statesand the layer-average electrostatic potential are also use-ful in extracting the electronic behavior of the system.

In all of these studies, one of the main questions isthat of the structure and polarization of the ferroelec-tric layer compared to that of the bulk. Certainly, thechange of environment (electrical and mechanical bound-ary conditions) and the finite dimensions (film thickness,particle size) are expect strongly to affect the structureand perhaps to eliminate the ferroelectric instability en-tirely. Relevant quantities to examine include the rela-tive stability of lower and higher symmetry phases, spa-tial variation in polarization, changes in the average po-larization magnitude and direction, and the depth andshape of the ferroelectric double-well potential. Thesechanges can also be expected to lead to changes in thedielectric and piezoelectric response of thin films and su-perlattices, which can be studied theoretically and com-pared with experiments. The implications of the variousfirst-principles studies included in this review will be de-scribed at further length below.

3. Studies of individual one component-systems

In this section, we describe a representative sampling offirst principles studies and their results. Most of the lit-erature has concentrated on BaTiO3, providing a usefulcomparative test of various first-principles implementa-tions, as well as a benchmark for evaluation and analysisof results on other systems. We first consider the calcu-lations for single slabs of pure material that focus on theproperties of surfaces: surface relaxation, surface recon-structions and surface states. Depending on the struc-tural constraints, these calculations are relevant eitherto the surface of a semi-infinite bulk, for a free-standingthin film, or for a thin film epitaxially constrained by asubstrate. This will be followed by discussions of studiesof systems with two or more material components.

a. BaTiO3 For BaTiO3, full first-principles results havebeen reported primarily for the (001) orientation, with afew results for the (110) and (111) orientations. In theslab-vacuum configuration, systems up to 10 atomic lay-ers have been considered. The unpolarized slab is com-pared with slabs with nonzero polarization, in the plane

26

and/or along the normal. After reviewing the results onstructures, we will describe the results of extension tolarger-scale systems through the use of interatomic po-tentials. The discussion of BaTiO3 will be concluded bydescription of the first-principles results for surface elec-tronic structure.

First-principles FLAPW calculations for the BaTiO3

(001) and (111) slabs were first presented in 1995 (Cohen,1996), and later extended using the LAPW+LO method(Cohen, 1997). The supercells contained 6 and 7 atomiclayers, corresponding to asymmetric termination and twosymmetric terminations (BaO and TiO2), and an equalvacuum thickness. The central mirror plane symmetryz→ -z symmetry is broken for the asymmetric termina-tion even in the absence of ferroelectric distortion. Theprimitive cell lattice constant was fixed at the experimen-tal cubic value 4.01 A. Total energies of several selectedparaelectric and ferroelectric structures were computedand compared: the ideal paraelectric slab and ferroelec-tric slabs with displacements along z corresponding tothe experimental tetragonal structure. For the asymmet-rically terminated slab, both the ferroelectric structurewith polarization towards the BaO surface (+) and theother towards the TiO2 surface (-) were considered. Thesurface layers were relaxed for the ideal and (+) ferroelec-tric asymmetrically-terminated slabs and the ferroelec-tric BaO-BaO slab. It was found that the depolarizationfield of the ferroelectric slabs indeed strongly destabilizesthe ferroelectric state, as expected, even taking into ac-count the energy lowering due to the surface relaxation.In all slabs considered, this consists of an inward-pointingdipole arising from the relative motion of surface cationsand oxygens. The average surface energy of the idealBaO and TiO2 surfaces is 0.0574 eV/A2=0.923 eV persurface unit cell.

In Padilla and Vanderbilt, 1997, ultrasoft pseudopo-tential calculations with fully relaxed atomic coordinateswere reported for symmetrically terminated (both BaOand TiO2) 7-layer BaTiO3 (001) slabs separated by twolattice constants of vacuum. The in-plane lattice con-stant was set equal to the theoretical equilibrium lat-tice constant a computed for the bulk tetragonal phase(a = 3.94 A). The average surface energy of the idealBaO and TiO2 surfaces is 1.358 eV per surface unit cell;at least part of the difference relative to Cohen, 1996could be due to the different lattice constant. Relax-ations were reported for unpolarized slabs and for polar-ized slabs with polarization along (100) (in the plane ofthe slab). Deviations from the bulk structure were con-fined to the first few atomic layers. The surface layerrelaxes substantially inwards, and rumples such that thecation (Ba or Ti) moves inward relative to the oxygens,as in Fig. 22. While the relaxation energy was found tobe much greater than the ferroelectric double well depth,the in-plane component of the unit-cell dipole momentwas seen to be relatively insensitive to the surface relax-ation, with a modest enhancement at the TiO2 termi-nated surface and a small reduction at the BaO termi-

nated surface. The relative stability of BaO and TiO2

terminations were compared and both found to be stabledepending on whether the growth was under Ba-rich orTi-rich conditions.

This investigation was extended inMeyer and Vanderbilt, 2001 to 7-layer and 9-layerpolarized slabs with polarization along the normal.The problem of the artificial vacuum field in thisperiodically repeated slab calculation was addressedby the techniques of introducing an external dipolelayer in the vacuum region of the supercell describedabove in Section IV.B.1. This technique can also beused to generate an applied field that partially or fullycompensates the depolarization field for BaTiO3 slabs.As a function of applied field, the change in structurecan be understood as arising from oppositely-directedelectrostatic forces on the positively charged cationsand negatively charged anions, leading to correspondingchanges in the rumplings of the atomic layers andfield-induced increases of the layer dipoles. Analysis ofthe internal electric field as a function of the appliedfield allows a determination of whether the slab isparaelectric or ferroelectric. The BaO-terminated slabis clearly ferroelectric, with vanishing internal electricfield at an external field of 0.05 a.u. and a polarizationof 22.9 µCcm−2, comparable to the bulk spontaneouspolarization. The ferroelectric instability is suppressedin the TiO2-terminated slab, which appears to bemarginally paraelectric.

The Hartree-Fock method was used inCora and Catlow (1999) and Fu et al. (1999). InCora and Catlow (1999), a detailed analysis of thebonding was performed using tightbinding parametriza-tion. For the 7-layer BaO-terminated slab, the reporteddisplacement of selected Ti and O atoms is in goodagreement with the results of Padilla and Vanderbilt(1997), and these calculations were extended to slabs ofup to 15 layers. Fu et al. (Fu et al., 1999) performedHartree-Fock calculations for slabs of two to eight atomiclayers, with symmetric and asymmetric terminations.Using a localized basis set, they were able to performcalculations for isolated slabs as well as periodicallyrepeated slabs. Calculations of the macroscopicallyaveraged planar charge density, surface energy andsurface dynamical charges were reported as a functionof thickness and termination for a cubic lattice constantof 4.006 A. The relative atomic positions were fixedto their bulk tetragonal structure values (note thatthis polarized structure in both isolated and periodicboundary conditions has a very high electrostatic energyand is not the ground state structure). This wouldsignificantly affect the comparison of the computedsurface properties with experiment. In particular, it ispresumably responsible for the high value of the averagesurface energy reported (1.69 eV per surface unit cell).However, a useful comparison between isolated andperiodic slabs is possible. It was found that the surfacecharge and surface dipoles of isolated slabs converge

27

quite rapidly as a function of slab thickness and can beused, combined with a value of ǫ∞ taken from the bulk,to extract a spontaneous polarization of 0.245 C/m2

(corrected to zero field using the electronic dielectricconstant ǫ∞ = 2.76), to be compared with 0.240 C/m2

from a Berry-phase calculation. This is only slightlyless than the bulk value of 0.263 C/m2 taken fromexperiment. The average surface energy for the twoterminations of symmetrically terminated slabs is 0.85eV per surface unit cell. Surface longitudinal dynamicalcharges differ considerably from bulk values, satisfyinga sum rule that the dynamical charges at the surfaceplanes add up to half of the corresponding bulk value(Ruini et al., 1998). Convergence of all quantities withslab and vacuum thickness of periodically repeated slabswas found, in comparision, to be slow, with significantcorrections due to the fictitious field in the vacuum (forpolarized slabs) and the interaction between slab images.

The isolated slab was also the subject of an FLAPWstudy (Krcmar and Fu, 2003). The symmetric TiO2-TiO2 (9-layer) and asymmetric TiO2-BaO (10-layer)slabs were considered in a paraelectric structure with afixed to 4.00 A and a polar tetragonal structure witha and c = 4.00 and 4.04 A, respectively. For the cu-bic TiO2-terminated slab, displacements in units of cfor the surface Ti, surface O, subsurface Ba and sub-surface O are -0.021, +0.007, +0.022 and -0.009 c, to becompared with the results(Padilla and Vanderbilt, 1997)(-0.0389,-0.0163,+0.0131,-0.0062) for a periodically re-peated 7-layer slab with lattice constant 3.94 A. Thetetragonal phase was relaxed to a convergence criterionof 0.06 eV/A on the atomic forces; the rumplings ofthe layers follow overall the same pattern as that re-ported in Meyer and Vanderbilt, 2001, with an inward-pointing surface dipole arising from surface relaxation,though the reduction of the rumpling in the interior isnot as pronounced as for the zero-applied field case inMeyer and Vanderbilt, 2001. The energy difference be-tween the paraelectric and ferroelectric slabs was not re-ported.

With interatomic potentials, it is possible to studyadditional aspects of surface behavior in BaTiO3 thinfilms and nanocrystals. The most important feature ofinteratomic studies of thin films relative to full first-principles calculations is the relative ease of extendingthe supercell in the lateral direction, allowing the forma-tion of 180 degree domains and molecular dynamics stud-ies of finite temperature effects. In Tinte and Stachiotti(2001), a 15-layer TiO2-terminated slab periodically re-peated with a vacuum region of 20 A was studied, usinginteratomic potentials that had previously been bench-marked against first-principles surface relaxations andenergies(Tinte and Stachiotti, 2000). The unconstrained(stress-free) slab is found to undergo a series of phasetransitions with decreasing temperature, from a paraelec-tric phase to ferroelectric phases, first with polarizationin the plane along (100), and then along (110). Enhance-ment of the surface polarization at low temperatures ap-

FIG. 23 Cell-by-cell out-of-plane (top panel) and in-plane(bottom panel) polarization profiles of a randomly chosenchain perpendicular to the slab surface for different misfitstrains η at T = 0 K. In the in-plane polarization profilespy = px. From Tinte and Stachiotti, 2001.

pears to be linked to the existence of an intermediatetemperature regime of surface ferroelectricity. For slabswith a strongly compressive epitaxial strain constraint (η= -2.5%), there is a transition to a ferroelectric state with180 degree domains and polarization along the surfacenormal. At the surface, the polarization has a nonzero xcomponent and a reduced z-component, giving a rotationat the surface layer. The width of the stripe domains can-not be determined, as it is limited by the lateral supercellsize at least up to 10x10. Reducing the compression to η= -1.0% also gives a 180 degree domain structure in the zcomponent of the polarization, combined with a nonzerocomponent along [110] in the interior of the film as wellas the surface, as can be see from examination of Fig.23. The thickness dependence of the transition temper-ature to this ferroelectric domain phase was studied atη = -1.5%, with Tc decreasing from the 15-layer film tothe 13-layer and lowest Tc 11-layer film. At and above acritical thickness of 3.6 nm, stress-free films exhibit thesame ferroelectric-domain ground-state structure.

Heifets and coworkers (Heifets et al., 2001a, 2000b)studied BTO (001) and (110) surfaces of an isolated slabusing the shell model of Heifets et al., 2000b. Between 1and 16 atomic planes were relaxed in the electrostaticpotential of a a rigid slab of 20 atomic planes whoseatoms were fixed in their perfect (presumably cubic) lat-tice sites. The relaxed structures of the two (001) termi-nations are in good agreement with other calculations,except for the sign of the surface dipole in the relax-ation of the BaO terminated surface, which is found to

28

be positive (though small). The (110) surfaces are foundto have much higher surface energy, except for the re-laxed “asymmetric O-terminated” surface where everysecond surface O atom is removed and the others occupythe same sites as in the bulk structure. In this struc-ture, displacements of cations parallel to the surface arefound substantially to lower the surface energy. A similarstudy of the KNbO3 (110) surface (Heifets et al., 2000a),which is the surface of most experimental interest forthis 1+/5+ perovskite, showed very strong relaxationsextending deep below the surface, consistent with sug-gestions that this surface has a complicated chemistry.

Next, we consider results on the electronic structureof BaTiO3 films, particularly the surface states. In theFLAPW study of Cohen, the band gap of the ideal slabis is found to be reduced from the bulk. A primarily Op occupied surface state on the TiO2 surface was identi-fied at M (Cohen, 1996), with a primarily Ti d surfacestate near the bottom of the conduction band. Analysisof the ferroelectric BaO-terminated slab showed that themacroscopic field resulted in a small charge transfer tothe subsurface Ti d states from the O p and Ba p states atthe other surface, making the surfaces metallic. Furtherstudy of this effect in Krcmar and Fu, 2003 showed thatfor symmetric TiO2 9-layer slab the ferroelectric distor-tion similarly shifts the top surface Ti states and bottomsurface O state toward the bulk midgap as in Fig. 24, re-sulting in a small charge transfer and a metallic characterfor the surfaces.

b. PbTiO3 In contrast to the numerous papers on cal-culations on the surfaces of BaTiO3, there are rela-tively few for the related material PbTiO3. Regard-ing surface relaxations and energies, it was found inMeyer and Vanderbilt, 2001 that the two compounds arequite similar. In perpendicularly polarized films, it seemsthat both terminations give ferroelectric films if the de-polarization field is compensated, consistently with thestronger ferroelectric instability of PbTiO3 and the mi-croscopic model analysis of Ghosez and Rabe, 2000, thelatter not including the effects of surface relaxation. Be-cause of the larger spontaneous polarization of PbTiO3,it is not possible fully to compensate the depolarizationfield using the dipole-layer technique.

There are important differences between A-site Ba andPb, which are evident even for the bulk. While the po-larization in BaTiO3 is dominated by the Ti displace-ments, Pb off-centering contributes substantially to thespontaneous polarization of PbTiO3; this can be linkedto the much richer chemistry of Pb oxides compared toalkaline-earth oxides. One downside is that it is morechallenging to construct accurate interatomic potentialsfor perovskites with Pb than with alkaline earth A cations(Sepliarsky et al., 2004). In the surface, the character-istic behavior of Pb leads to an antiferrodistortive sur-face reconstruction of the (001) PbO-terminated surface(Bungaro and Rabe; Munkholm et al., 2002). Specifi-

FIG. 24 (a) Paraelectric phase energy-band structure of nine-layer slab of BaTiO3. The surface states in (or near) the bandgap are highlighted. (b) As in (a), for the ferroelectric nine-layer slab. From Krcmar and Fu, 2003.

cally, first principles calculations (Bungaro and Rabe)show that the reconstruction in the subsurface TiO2 layeroccurs only for the PbO termination and not for TiO2

termination, and also that if the Pb in the surface layeris replaced by Ba, the reconstruction is suppressed.

c. SrBi2Ta2O9 A first-principles study of an isolatedBiO2-terminated slab of SrBi2Ta2O9 (SBT) one latticeconstant thick (composition SrBi2Ta2O11 was reportedin Tsai et al., 2003. Spin-polarized calculations showedthat such a film would be ferromagnetic as well as fer-roelectric. This intriguing possibility suggests further in-vestigation.

d. SrTiO3 and KTaO3 In addition to the work on trueferroelectrics, there has been quite a lot of interest infirst-principles studies of surfaces and heterostructuresof incipient ferroelectrics, mainly SrTiO3 and, to a lesserextent, KTaO3. As these have closely related propertiesthat can illuminate issues in the ferroelectric perovskites,we include a description of a few representative resultsdrawn from the very extensive literature on this subject.

Many of the studies of BaTiO3 discussed above in-

29

cluded analogous calculations for SrTiO3. As alreadynoted in the discussion of PbTiO3, the (001) surface re-laxations and energies of the nonpolar slab are very sim-ilar for all three materials. First principles surface re-laxation for the SrO surface is reported (in units of a= 3.86A) for surface Sr, surface O, subsurface Ti andsubsurface O as -0.057, 0.001, 0.012 and 0.0, respectively(Padilla and Vanderbilt, 1998), to be compared with thevalues -0.071, 0.012, 0.016, 0.009 (computed using a shellmodel with the experimental lattice constant 3.8969 A).Similarly, first principles surface relaxation for the TiO2

surface is reported for surface Ti, surface O, subsurfaceSr and subsurface O as -0.034, -0.016, +0.025 and -0.005,respectively (Padilla and Vanderbilt, 1998), to be com-pared with the values -0.030, -0.017, +0.035 and -0.021.The average energy of the two surfaces is found to be1.26 eV per surface unit cell. A detailed comparison ofvarious Hartree-Fock and density-functional implemen-tations showed generally good agreement for the sur-face relaxation (Heifets et al., 2001b). Inward surfacedipoles due to relaxation are found for both termina-tions, with the TiO2 termination to be smaller in mag-nitude than for BaTiO3 (Heifets et al., 2000b). The pos-sibility of an in-plane ferroelectric instability at the sur-face was examined and it was found to be quite weak(Padilla and Vanderbilt, 1998). In these studies, the an-tiferrodistortive instability exhibited by bulk SrTiO3 atlow temperatures was suppressed by the choice of a 1x1in-plane unit cell. The surface electronic band structuresshow a behavior highly similar to that of BaTiO3 de-scribed above.

For SrTiO3, there is considerable evidence for a widevariety of surface reconstructions of varying stoichiom-etry, depending on conditions such as temperature andoxygen partial pressure as well as the relative chemicalpotentials of TiO2 and SrO. Candidate structures canbe obtained by creating vacancies on the surface (forexample, missing rows of oxygen) and adding adatoms(Kubo and Nozoye, 2003). More drastic rearrangementsof the surface atoms have also been proposed, for examplea (2x1) Ti2O3 reconstruction (Castell, 2002) and a (2x1)double-layer TiO2 reconstruction with edge-sharing TiO6

octahedra (Erdman et al., 2002). As in the case of semi-conductor surface reconstructions, first-principles calcu-lations of total energies are an essential complement toexperimental structural determination, and can also beused to predict scanning tunnelling microscope imagesfor comparison with experiment (Johnston et al., 2004).Even so, there are still many open questions about theatomic and electronic structure of SrTiO3 surfaces un-der various conditions. The same applies to KTaO3;the structures and lattice dynamics of a variety of (1x1)and (2x1) surface structures were studied in (Li et al.,2003a)).

As previously mentioned, the theoretical and experi-mental literature on SrTiO3 is so extensive that it wouldrequire a review paper in its own right to cover it fully.Since, if anything, we expect the ferroelectric instability

in related systems such as BaTiO3 and PbTiO3 to makethe physics more, not less, complicated, this suggests thatwe have only scratched the surface in developing a com-plete understanding of the surfaces of perovskite ferro-electrics and the resulting effects on thin film properties.

4. Studies of individual heterostructures

Now we turn to the description of studies of systemswith two or more material components. The main struc-tural issues are the rearrangements at the interface, thechange in electrical and mechanical boundary conditionsfelt by each constituent layer, and how these changesmodify the ferroelectric instability exhibited by the sys-tem as a whole. This geometry also allows the calculationof band offsets and/or Schottky barriers, crucial in prin-ciple to understanding the electronic behavior (thoughwith the caveat that the measured Schottky barrier inreal systems will be strongly influenced by effects such asoxidation of the electrodes that are not included in thehighly idealized geometries studied theoretically). Thecurrent state of knowledge, derived from experimentalmeasurements, is described in Section. III.B.3.

The first combination we discuss is that of a ferro-electric thin film with metallic electrodes. Transitionmetal interfaces with nonpolar BaO-terminated layers ofBaTiO3 were studied in Rao et al., 1997, specifically sys-tems of 3 and 7 atomic layers of BaTiO3, with latticeconstant set to the bulk value of 4.00 A, combined withtop and bottom monolayers of Ta, W, Ir and Pt rep-resenting the electrodes. The preferred absorption sitefor the metal atoms was found to be above the O site,with calculated metal oxygen distances ranging from 2.05A for Ta to 2.11 A for Pt. The BaTiO3 slabs were as-sumed to retain their ideal cubic structure. Analysis ofthe partial density of states of the heterostructure showsthat the Pt and Ir Fermi energies lie in the gap of theBaTiO3 layer at 0.94 eV and 0.64 eV, respectively, abovethe top of the valence band (this is, fortunately, smallerthan the underestimated computed gap of 1.22 eV forthe BaTiO3 slab). Using the experimental gap of 3.13eV, a Schottky barrier height of 2.19 eV for Pt and 2.49eV for Ir is thus obtained. Experimentally, however, theSchottky barrier is known to be substantially lower for Irthan for Pt, illustrating the limitations mentioned in theprevious paragraph.

In Robertson and Chen, 1999, first principles calcula-tions of the charge neutrality levels were combined withexperimental values of the band gap, electronic dielectricconstant ǫ∞, the electron affinity, and the empirical pa-rameter S, described above in Section III.B 3. Values forSrTiO3 and PZT were reported for Pt, Au, Ti, Al andthe conduction and valence bands of Si.

To explore how the electrodes affect the ferroelectricinstability of the film, Junquera and Ghosez considereda supercell of 5 unit cells of metallic SrRuO3 and 2-10 unit cells of BaTiO3, with a SrO/TiO2 interface be-

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tween the SrRuO3 and the BaTiO3. A SrTiO3 substratewas treated implicitly by constraining the in-plane lat-tice constant of the supercell to that of bulk SrTiO3. Foreach BaTiO3 thickness, the system was relaxed assum-ing a nonpolar state for the BaTiO3, and then the energyof the bulk-like tetragonal distortion was computed as afunction of overall amplitude of the distortion. Above acritical thickness of six unit cells, this distortion loweredthe energy, demonstrating the development of a ferro-electric instability. As will be discussed further in sectionIV.B.5, this finite size effect can be largely understood byconsidering the imperfect screening in the metal layers.

Considerable first-principles effort has been devotedto investigating various aspects of epitaxial ferroelectricthin films on Si. As perovskite oxides cannot be growndirectly on Si, an approach developed by McKee andWalker (McKee et al., 1998, 2001) is to include a AObuffer layer, which apparently also results in the forma-tion of a silicide interface phase. The constituent layersof this heterostructure should thus be considered to beSi/ASi2/AO/ABO3. The full system has not been sim-ulated directly, but first-principles approaches have beenused to investigate individual interfaces. The importanceof relaxations, the additional role of the buffer layer inchanging the band offset, and the analysis of electronicstructure within the LDA are illustrated by the following.

In (McKee et al., 2003), first-principles results are pre-sented for the atomic arrangements and electronic struc-ture in the Si/ASi2/AO system, in conjunction with anexperimental study. A strong correlation is found be-tween the valence band offset and the dipole associatedwith the A-O bond linking the A atom in the silicide tothe O atom in the oxide, shown in Fig. 25. It is thusseen that the structural rearrangements in the interfaceare a key determining factor in the band offset.

A detailed examination of the interface between theperovskite oxide and the alkaline oxide buffer layer,specifically BaO/BaTiO3 and SrO/SrTiO3, was carriedout in (Junquera et al., 2003). A periodic 1x1x16 su-percell was chosen with stacking of (001) atomic layers:(AO)n-(AO-TiO2)m, with n = 6 and m = 5. Two mir-ror symmetry planes were fixed on the cental AO andBO2 layers, and the in-plane lattice constant chosen forperfect matching to the computed LDA lattice constantof Si (this epitaxial strain constraint is the only effect ofthe Si substrate included in the calculation). Relaxationswithin the highest-symmetry tetragonal space group con-sistent with this supercell were performed. Analysis ofthe partial density of states showed no interface-inducedgap states. The main effect observed for relaxations wasto control the size of the interface dipole, which in turnwas found to control the band offsets, shown here in Fig-ure 26. As in the studies described above, the conduc-tion band offset is obtained from the computed valenceband offset by using the bulk experimental band gap.These results were combined with offsets reported for theother relevant interfaces to estimate band alignments forSi/SrO/SrTiO3/Pt and Si/BaO/BaTiO3/SrRuO3 het-

FIG. 25 The image illustrates three layers of the alkalineearth oxide on the (001) face of silicon observed in a crosssection at the [110] zone axis (blue, alkaline earth metal; yel-low, oxygen; and green, silicon). A distinct interface phasecan be identified as a monolayer structure between the oxideand the silicon in which the charge density in interface statesis strongly localized around the silicon atoms in the interfacephase. The dipole in the ionic A-O bond between the alka-line earth metal in the silicide and the oxygen in the oxidebuffers the junction against the electrostatic polarization ofthe interface states localized on silicon. The electron den-sity of this valence surface state at the center of the Brillouinzone is shown with the purple isosurface (0.3 10−3 e). FromMcKee et al., 2003

FIG. 26 Schematic representation of the valence-band off-set (VBO) and the conduction-band offset (CBO) forBaO/BaTiO3 (a) and SrO/SrTiO3 (b) interfaces. Ev, Ec,and Eexpt

gap stand for the top of the valence band, the bot-tom of the conduction band, and the experimental band gap,respectively. Values for Ev , measured with respect to theaverage of the electrostatic potential in each material, are in-dicated. The solid curve represents the profile of the macro-scopic average of the total electrostatic potential across theinterface. ∆V stands for the resulting lineup. The in-planelattice constant was set up to the theoretical one of Si (5.389 ).The size of the supercell corresponds to n=6 and m=5. FromJunquera et al., 2003

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erostructures, confirming that the AO layer introducesan electrostatic barrier of height greater than 1 eV. Thisis sufficient to eliminate the carrier injection from theSi into the conduction band states of the perovskitethat would be occur if the two were in direct contact(Robertson and Chen, 1999).

5. First principles modelling: methods and lessons

As discussed above, the analysis and prediction of thebehavior of ferroelectric thin films and heterostructurescan be carried out with direct first-principles simulationsonly for highly idealized configurations. However, it ispossible to consider more complex and realistic situationsby constructing models that incorporate certain physicalideas about the nature of these systems, with material-specific parameters determined by fitting to the results offirst-principles calculations carefully selected for a com-bination of informativeness and tractability. This mod-elling approach also has the advantage of providing aconceptual framework for organizing the vast amount ofmicroscopic information in large-scale first-principles cal-culations, and communicating those results, particularlyto experimentalists. This will not diminish in importanceeven as such calculations become easier with continuingprogress in algorithms and computer hardware.

For successful modeling of measurable physical proper-ties, the film must be considered as part of a system (sub-strate+film+electrode) as all components of the systemand their interaction contribute to determine propertiessuch as the switchable polarization and the dielectric andpiezoelectric response. We start by considering the classof first-principles models in which the constituent layers(film, superlattice layers, electrodes and substrate) areassumed to be subject to macroscopic electric fields andstresses resulting from the combination of applied fieldsand stresses and the effects of the other constituents, withthe responses of the layers being given to lowest order bythe bulk responses. For systems with constituent layerthicknesses as low as one bulk lattice constant, it seemsat first unlikely that such an approach could be useful,but in practice it has been found to be surprisingly suc-cessful.

One simple application of this approach has beenused to predict and analyze the strain in nonpolarthin films and multilayers. In the construction ofthe reference structure for the AO/ABO3 interfaces in(Junquera et al., 2003), macroscopic modelling of thestructure with bulk elastic constants for the constituentlayers yielded accurate estimates for the lattice constantsalong the normal direction. In cases of large lattice mis-match, very high strains can be obtained in very thinfilms and nonlinear contributions to the elastic energycan become important. These can be computed witha slightly more sophisticated though still very easy-to-implement method that has been developed to study theeffects of epitaixial strain more generally on the structure

and properties of a particular material, described next.

As is discussed further in Section V, the effects ofepitaxial strain in ultrathin films and heterostructureshave been identified as a major factor in determiningpolarization-related properties, and have been the sub-ject of intense interest in both phenomenological and firstprinciples modelling. In particular, for ferroelectric per-ovskite oxides it has long been known that there is astrong coupling between strain (e.g. pressure-induced)and the ferroelectric instability, as reflected by the fre-quency of the soft mode and the transition temperature.In both phenomenological and first-principles studies, ithas become common to study the effects of epitaxialstrain induced by the substrate by studying the struc-tural energetics of the strained bulk. Specifically, twoof the lattice vectors of a bulk crystal are constrainedto match the substrate and other structural degrees offreedom are allowed to relax. as described in the pre-vious paragraph. In most cases, these calculations areperformed for zero macroscopic electric field, as wouldbe the case for a film with perfect short-circuited elec-trodes. Indeed, it is often the case (Junquera et al., 2003;Neaton and Rabe, 2003; Pertsev et al., 1998) that thestrain effect is considered to be the dominant effect of thesubstrate, which is otherwise not included (thus greatlysimplifying the calculation). At zero temperature, thesequence of phases and phase boundaries can be readilyidentified as a function of in-plane strain directly throughtotal-energy calculations of the relaxed structure subjectto the appropriate constraints. Atomic-scale informationcan be obtained for the precise atomic positions, band-structure, and phonon frequencies and eigenvectors. Thetemperature axis in the phase diagram can be includedby using effective Hamiltonian (or interatomic potential)simulations. Results for selected perovskite oxides arediscussed in Section V; a similar analysis was reportedfor TiO2 in (Montanari and Harrison, 2004).

This modelling is based on the assumption that thelayer stays in a single-domain state. As discussed inSection V, the possibility of strain relaxation throughformation of multidomain structures must be allowedfor. While this cannot be readily done directly in first-principles calculations, first-principles data on structuralenergetics for large misfit strains could be used to refineLandau parameters for use in calculations such as thosein Alpay and Roytburd, 1998; Bratkovsky and Levanyk,2001; Li et al., 2003a; Speck and Pompe, 1994. The ef-fects of inhomogeneous strain due to misfit dislocationsthat provide elastic relaxation in thicker films have beenalso been argued to be significant.

Next we consider the application of these “continuum”models to analyzing structures in which the macroscopicfield is allowed to be nonzero. Macroscopic electrostaticsis applied to the systems of interest by a coarse-grainingover a lattice-constant-scale window to yield a value forthe local macroscopic electric potential. Despite the factthat this is not strictly within the regime of validity ofthe classical theory of macroscopic electrostatics, which

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requires slow variation over many lattice constants, thisanalysis turns out to be remarkably useful in the analysisof first principles results. In the simplest example, the po-larization of an polar BaTiO3 slab (periodically repeatedin a supercell with vacuum) is accurately reproduced us-ing bulk values for the bulk spontaneous polarization andelectronic dielectric constant even for slabs as thin as twolattice constants (Junquera and Rabe).

We have already mentioned in the previous sectionthat perpendicular (to the surface) polarization can leadto a nonzero macroscopic field that opposes the po-larization (the depolarizing field). Unless compensatedby fields from electrodes or applied fields, this stronglydestabilizes the polarized state. In systems with twoor more distinct constituent layers, this condition in

the absence of free charge favors states with ∇ · ~P =0. For example, in the first-principles calculations ofshort-period BaTiO3/SrTiO3 superlattices, the local po-larization along [001] is found to be quite uniform inthe two layers (Johnston and Rabe; Neaton and Rabe,2003), though the in-plane component can be very dif-ferent (Johnston and Rabe). To the extent that layeredferroelectrics such as SBT can be treated in this macro-scopic framework, one similarly expects that polarizationalong c will tend to be energetically unfavorable, since thelayers separating the polarized perovskite-like layers typ-ically have low polarizability (Fennie and Rabe). Thisobservation provides a theoretical framework for evalu-ating claims of large ferroelectric polarization along c inlayered compounds (Chon et al., 2002); unless there isan unusually high polarizability for the non-perovskitelayers, or a strong competing contribution to the energydue, for example, to the interfaces to help stabilize a highc polarization, other reasons for the observations need tobe considered (Garg et al., 2003).

These considerations become particularly importantfor ultrathin films with metal electrodes. The limitingcase of complete screening of the depolarizing field byperfect electrodes is never realized in real thin film sys-tems. The screening charge in real metal electrodes isspread over a characteristic screening length and is as-sociated with a voltage drop in the electrode. For thickfilms, this can be neglected, but the relative size of thevoltage drop increases as the film thickness decreases.This has been identified as a dominant contribution tothe relation between the applied field and the true fieldin the film for the thinnest films (Dawber et al., 2003b).One way this shows up is in the thickness dependence ofthe apparent coercive field; it is found that the true coer-cive field scales uniformly down to the thinnest films. Ef-fects are also expected on the structure and polarization.While films with partial compensation of the depolariza-tion field may still exhibit a ferroelectric instability, thepolarization and the energy gain relative to the nonpolarstate are expected to decrease. This simple model was de-veloped and successfully used in (Junquera and Ghosez,2003) to describe the thickness dependence of the fer-roelectric instability in a BaTiO3 film between SrRuO3

electrodes. This analysis identified the thickness depen-dence of the residual depolarization field as the prin-cipal source of thickness dependence in this case. InLichtensteiger et al., the reduction of the uniform polar-ization by the residual field and its coupling to tetragonalstrain was suggested to be the cause of the decrease intetragonality with decreasing thickness of PbTiO3 ultra-thin films.

It is well known that 180 degree domain formation pro-vides an effective mechanism for compensating the depo-larization field, and is expected to be favored when thescreening available from electrodes is poor or nonexistent(for example, on an insulating substrate (Streiffer et al.,2002)). Instability to domain formation is discussedin Bratkovsky and Levanyuk, 2000 as the result of anonzero residual depolarization field due to the presenceof a passive layer. Similarly, a phase transition from auniform polarized state to a 180 degree domain state withzero net polarization is expected to occur with decreasingthickness (Junquera et al.).

Despite the usefulness of macroscopic models, it shouldnot be forgotten that they are being applied far outsidethe regime of their formal validity (i.e length scales ofmany lattice constants) and that atomistic effects canbe expected to play an important role, especially at thesurfaces and interfaces. The structural energetics couldbe substantially altered by relaxations and reconstruc-tions (atomic rearrangements) at the surfaces and inter-faces. These relaxations and reconstructions are also ex-pected to couple to the polarization (Bungaro and Rabe;Meyer and Vanderbilt, 2001), with the possibility of ei-ther enhancing or suppressing the switchable polariza-tion. The surfaces and the interfaces will also be primar-ily responsible for the asymmetry in energy between upand down directions for the polarization. For ultrathinfilms, the surface and interface energy can be importantenough to dominate over elastic energy, leading to a pos-sible tradeoff between lattice matching and atomic-scalematching for favorable bonding at the interface. Thesesurface and interface energies could even be large enoughto stabilize non-bulk phases with potentially improvedproperties. This should be especially significant for in-terfaces between unlike materials.

Electronic states associated with surfaces and inter-faces will also contribute to determining the equilibriumconfiguration of electric fields and polarizations. In thesimple example of periodically repeated slabs separatedby vacuum, as the slab gets thicker, a breakdown is ex-pected where the conduction band minimum on one sur-face of the slab falls below the valence band maximum onthe other. In this case, charge will be transferred acrossthe slab, with the equilibrium charge (for fixed atomic po-sitions) being determined by a combination of the macro-scopic electrostatic energy and the single particle densityof states. This tends partially to screen the depolariza-tion field. The role of interface states in screening thedepolarization field in the film has been discussed in amodel for BaTiO3 on Ge (Reiner et al.). The presence of

33

surface and interface states can be established by exam-ination of the bandstructure and PDOS, as discussed inthe previous section.

Finally, we turn our attention to an analysis of whatthe discussion above tells us about finite-size effects inferroelectric thin films. We have seen that many factorscontribute to the thickness dependence of the ferroelec-tric instability: the thickness dependence of the depolar-ization field, the gradual relaxation of the in-plane latticeconstant from full coherence with the substrate to its bulkvalue and the changing weight of the influence of surfacesand interfaces. The “true” finite-size effect, i.e the mod-ification of the collective ferroelectric instability due tothe removal of material in the film relative to the infinitebulk. could possibly be distangled from the other factorsby a carefully-designed first principles calculation, butthis has not yet been done. We speculate that this effectdoes not universally act to suppress ferroelectricity, butcould, depending on the material, enhance ferroelectric-ity (Ghosez and Rabe, 2000).

6. Challenges for first principles modelling

First principles calculations have advanced tremen-dously in the last decade, to the point where systemsof substantial chemical and structural complexity can beaddressed, and a meaningful dialogue opened up betweenexperimentalists and theorists. With these successes, thebar gets set ever higher, and the push is now to make thetheory of ferroelectrics truly realistic. The highest long-term priorities include making finite temperature calcu-lations routine, proper treatment of the effects of defectsand surfaces, and the description of structure and dy-namics on longer length and time scales. In addition,there are specific issues that have been raised that maybe addressable in the shorter term through the interac-tion of theory and experiment, and the rest of this sectionwill highlight some of these.

Many applications depend on the stability of films witha uniform switchable polarization along the film nor-mal. This stability depends critically on compensation ofthe depolarization field. Understanding and controllingthe compensation mechanism(s) are thus the subjects ofintense current research interest. There are two mainclasses of mechanism: compensation by “free” charges (inelectrodes/substrate or applied fields) and compensationby the formation of polarization domains. On insulat-ing substrates, this latter alternative has been observedand characterized in ultrathin films (Fong et al., 2004;Streiffer et al., 2002); it has been proposed that domainformation will occur in films on conducting substrates atvery low thicknesses as well where the finite screeninglength in realistic electrodes inhibits that mechanism ofcompensation (Junquera et al.). The critical thicknessfor this instability depends on the domain wall energy.This is expected to be different in thin films than in bulk,one factor being that the bulk atomic plane shifts across

the domain walls.

Compensation of the depolarization field by freecharges appears to be the dominant mechanism in filmson conducting substrates (even relatively poor conduc-tors), with or without a top electrode. In the latter casethere must be free charge on the surface; the challenge isto understand how the charge is stabilized. There are alsounresolved questions about how the charge is distributedat the substrate-film interface, and how this couples tolocal atomic rearrangements. Asymmetry of the com-pensation mechanism may prove to be a significant con-tribution to the overall up-down asymmetry in the filmdiscussed in the previous section. A better understandingcould lead to the identification of system configurationswith more complete compensation and thus an enhance-ment of stability.

The study of the behavior of ferroelectrics in appliedelectric fields also promises progress in the relatively nearfuture. Recently with the solution of long-standing ques-tions of principle, it has become possible to perform DFTcalculations for crystalline solids in finite electric fields(Souza et al., 2002). In ferroelectrics, this allows the in-vestigation of nonlinearities in structure and polarizationat fields relevant to experiments, and the possibility ofmore accurate modelling of constituent layers of thin filmand superlattice systems subject to nonzero fields. It isalso of interest to ask what the intrinsic breakdown fieldwould be in the absence of defects, though the questionis rather academic with respect to real systems.

The nonzero conductivity of real ferroelectrics becomesparticularly important for thinner films, both since ahigher concentration of free carriers is expected to beassociated with characteristic defects in the film, andbecause a given concentration of free carriers will havea more significant impact as thickness decreases. Freecarriers can at least partially screen macroscopic elec-tric fields. At the macroscopic level, the concepts ofband bending and space charge arising in semiconductorphysics can be applied to thin film ferroelectrics, whilea correct atomic-scale treatment of this effects could beimportant to describing the behavior of ultrathin films.

The physics of switching presents a significant chal-lenge, requiring description of structure and dynamicson long length and time scales. The questions of whatchanges, if any, occur in switching as films become thin-ner continue to be debated. The possibility of switch-ing as a whole rather than via a domain-wall mechanismhas been raised for ultrathin films of PVDF (Bune et al.,1998), while a different interpretation has been offered inDawber et al., 2003b. Some progress has been made us-ing interatomic potentials for idealized defect-free films,though real systems certainly are affected by defects re-sponsible for such phenomena as imprint and fatigue.Ongoing comparison of characteristics such as coercivefields, time scales, material sensitivity, and thickness de-pendence of domain wall nucleation, formation energy,and motion with experimental studies promise that atleast some of these issues will soon be better understood.

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To conclude this section, we emphasize that it is notvery realistic to expect first-principles calculations quan-titatively to predict all aspects of the behavior of chem-ically and structurally complex systems such as ferro-electric thin films, although successful predictions shouldcontinue to become increasingly possible and frequent.Rather, the quantitative microscopic information and thedevelopment of a useful conceptual framework contributein a close interaction with experiment to build an under-standing of known phenomena and to propel the fieldinto exciting new directions.

V. STRAIN EFFECTS

Macroscopic strain is a important factor in determin-ing the structure and behavior of very thin ferroelectricfilms. The primary origin of homogeneous film strain islattice mismatch between the film and the substrate. Inaddition, defects characteristic of thin films can produceinhomogeneous strains that can affect the properties ofthicker relaxed films of technological relevance. Becauseof the strong coupling of both homogeneous and inhomo-geneous strains to polarization, these strains have a sub-stantial impact on the structure, ferroelectric transitiontemperatures and related properties such as the dielectricand piezoelectric responses, which has been the subjectof extensive experimental and theoretical investigation.

The largest effects are expected in coherent epitaxialfilms. These films are sufficiently thin that the areal elas-tic energy density for straining the film to match thesubstrate at the interface is less than the energy costfor introducing misfit dislocations to relax the lattice pa-rameters back towards their unconstrained equilibriumvalues. (We note that for ultrathin films, the relaxed in-plane lattice constant will not in general not be the sameas the bulk lattice constant, and the former is more ap-propriate for computing lattice mismatch (Rabe).) Veryhigh homogeneous strains, of the order of 2%, are achiev-able. For example, BTO films on STO, with a bulk mis-match of 2.2%, remain coherent in equilibrium up to acritical thickness of 2-4 nm (Sun et al., 2004). With low-temperature growth techniques, the formation of misfitdislocations is kinetically inhibited and coherent films canbe grown to thicknesses of two to three times the criticalthickness (Choi et al., 2004). Even these films, however,are much thinner than the minimum 120 nm thickness forfilms used in contemporary applications, and thus muchof the discussion in this section is at present primarily offundamental rather than technological interest.

The structure of a coherent film can be a single-crystalmonodomain structure, a polydomain structure, or evenpossibly multiphase. We discuss the simplest single-crystal monodomain case first. The phase diagram asa function of in-plane strain will in general include lowersymmetry phases due to the symmetry-breaking charac-ter of the epitaxial constraint. A nomenclature for thesephases of perovskites on a surface with square symmetry

FIG. 27 Phase diagram of a (001) single-domain PbTiO3 thinfilms epitaxially grown on different cubic substrates providingvarious misfit strains um in the heterostructures. The second-and first-order phase transitions are shown by thin and thicklines, respectively. From Pertsev et al., 1998.

(e.g. a perovskite (001) surface)) has been establishedin Pertsev et al., 1998. For example, a ferroelectric per-ovskite rhombohedral phase will be lowered to monoclinicsymmetry (called the r-phase). For highly compressivein-plane strains, coupling between strain and the polar-ization tends to favor the formation of a tetragonal phasewith polarization along c (the c-phase) for highly com-pressive strains. Conversely, highly tensile strains lead toan orthorhombic phase with polarization along the cubeface diagonal perpendicular to the normal (the aa-phase)or along the in-plane cartesian direction (the a phase).As a result of the added constraint, it is possible in prin-ciple to stabilize perovskite-derived phases not observedin bulk, for example, the monoclinic r-phase and the or-thorhombic aa-phase in PbTiO3. This mechanism alsoplays a role in the more general phenomenon of epitaxialstabilization, discussed, for example, in Gorbenko et al.,2002.

As discussed in Section IV.B.5, theoretical analysis ofthe in-plane strain phase diagrams has focused on iso-lating the effects of strain by computing bulk single-crystal monodomain phase diagrams under the epitaxialconstraint and zero macroscopic electric field, using phe-nomenological Landau theory or first principles methods.Phenomenological analysis based on Landau-Devonshiretheory for a number of perovskite oxides has beenpresented by Pertsev and co-workers (Koukhar et al.,2001; Pertsev et al., 2003, 2000a,b, 1998, 1999), withtemperature-strain diagrams for BaTiO3, PbTiO3 (seeFig. 27), SrTiO3, and Pb(Zr,Ti)O3 (PZT), the lattergeneralized to include nonzero stress. First-principlesmethods have been used to construct a temperature-strain diagram for BaTiO3 (Dieguez et al., 2004) and

35

-3 -2 -1 0 1 2 3mistfit strain [%]

0

0.1

0.2

0.3

0.4

0.5Po

lari

zatio

n [C

/m2 ]

FIG. 28 Polarization of SrTiO3 as a function of in-planestrain. Solid circles and squares denote polarization along[001] and [110], respectively. From Antons et al..

a zero-temperature strain diagram for PbTiO3 andordered PZT (Bungaro and Rabe, 2004) and SrTiO3

(Antons et al.). These theoretical phase diagrams havesome notable features. In particular, compressive in-plane strain is found to elevate the ferroelectric(c)-paraelectric transition temperatures in BaTiO3 andPbTiO3, and tensile in-plane strain elevates the ferroelec-tric (aa)-paraelectric transition temperatures. In bothcases, the transition is second order, in contrast to thefirst-order transition in bulk. To eliminate a possiblesource of confusion, we comment that zero misfit strainas defined in (Pertsev et al., 1998) is not equivalent toan unconstrained film (the low-temperature bulk phasesare in general not cubic, and the constrained dielectricand piezoelectric responses are clamped, as will be dis-cussed further below). The nearly vertical morphotropicphase boundary characteristic of bulk PZT is substan-tially modified (Li et al., 2003b; Pertsev et al., 2003). InSrTiO3, ferroelectricity is found to be induced by bothsufficiently compressive and tensile strains, with a corre-sponding direction for the spontaneous polarization (c-type and aa-type) (Antons et al.; Pertsev et al., 2000a),as shown in Fig. 28. The enhancement of the polariza-tion in the c-phase by compressive in-plane strain hasbeen noted for BaTiO3 (Neaton and Rabe, 2003) andPZT (Pertsev et al., 2003). For both strained SrTiO3

and strained BaTiO3, the ferroelectric Tc’s are predictedto increase as the strain magnitudes increase (Choi 2004,Haeni 2004), as shown in Fig. 29.

The use of phenomenological bulk Landau parame-ters yields a very accurate description for small strainsnear the bulk Tc. However, different parameter setshave been shown, for example in the case of BaTiO3,to extrapolate to qualitatively different phase diagramsat low temperatures (Pertsev et al., 1998, 1999). Quan-titatively, the uncertainty in predicted phase boundaries,produced by the fitting of the Landau theory parameterincreases with increasing misfit strain as shown in Fig.29 (Choi et al., 2004). The Landau analysis is thus well-

Paraelectric

Rangeof

transition

FerroelectricDyScO3 GdScO3

Te

mp

era

ture

( C

)o

In-plane strain (%)es

200

400

600

800

1000

1200

-1.8 -1.2 -0.6 0 0.6 1.2 1.8(a)

Range offerroelectric

transition

Range offerroelectric

transition

DyScO3

Paraelectric phase

Ferroelectric phase

LSATP

P

In-plane strain es

Te

mp

era

ture

(K)

0

100

200

300

400

-0.012 -0.008 -0.004 0.000 0.004 0.008 0.012(b)

FIG. 29 Expected Tc of (a) (001) BaTiO3 (from Choi et al.(2004)) and (b) (001) SrTiO3 (from Haeni et al. (2004)) basedon thermodynamic analysis. The range of transition repre-sents the uncertainty in the predicted Tc resulting from thespread in reported property coefficients.

complemented by first-principles calculations, which canprovide very accurate results in the limit of zero temper-ature. In the case of BaTiO3, the ambiguity in the low-temperature phase diagram (Pertsev et al., 1998, 1999)has been resolved in this way (Dieguez et al., 2004), inthe case of PbTiO3, the phenomenological result is con-firmed (Bungaro and Rabe, 2004).

The epitaxial-strain induced changes in structure andpolarization are also expected to have a substantial ef-fect on the dielectric and piezoelectric responses. Whileoverall, the dielectric and piezoelectric responses shouldbe reduced by clamping to the substrate (Canedy et al.,2000; Li et al., 2001), these responses will tend to divergenear second-order phase boundaries (Bungaro and Rabe,2004; Pertsev et al., 2003) Responses will also be largein phases, such as the r-phase, in which the directionof the polarization is not fixed by symmetry, so thatan applied field or stress can rotate the polarization(Wu and Krakauer, 1992). This polarization rotationhas been identified as a key mechanism in the colossalpiezoresponse of single-crystal relaxors (Fu and Cohen,

36

7 7.2 7.4 7.6 7.8 8a

|| (a.u.)

0

0.02

0.04

0.06

0.08

ε-1(0

)c-phase r-phase aa-phase

c)

FIG. 30 Inverses of the eigenvalues of the phonon contribu-tion to the dielectric tensor as a function of the in-plane lat-tice constant for the [001]-(PbTiO3)1(PbZrO3)1 superlattice.The lines are a guide to the eye. From Bungaro and Rabe,2004.

2000; Park and Shrout, 1997). The sensitivity of thezero-field responses should also be reflected in the non-linear response; thus the electric-field tunability of thedielectric response can be adjusted by changing the mis-fit strains (Chen et al., 2003).

While the phase diagrams thus derived are quite rich,even the optimal single-crystal monodomain structuremay be unfavorable with respect to formation of polydo-main (Bratkovsky and Levanyk, 2001; Roytburd et al.,2001; Speck and Pompe, 1994) or multiphase structures,which allow strain relaxation on average and reduce elas-tic energy. The evaluation of the energies of polydomainstructures requires taking both strain and depolarizationfields into account. A recent discussion of PbTiO3 usinga phase-field analysis (Li et al., 2003b) suggested that in-cluding the possibility of polydomain structure formationsignificantly affects the phase diagram for experimentallyrelevant misfit strains and temperatures, as shown in Fig.31. For very thin films, additional effects associated withthe interface between the film and substrate are also ex-pected to contribute significantly. For example, althoughpolydomain formation can accommodate misfit strain av-eraged over different variants, each domain will be mis-matched to the substrate at the atomic level, with a cor-responding increase in interface energy. Furthermore, theenergy of a domain wall perpendicular to the substratewill be higher than the energy of the corresponding wallin the bulk, due to the geometrical constraint on the al-lowed shifts of the atomic planes across the domain wall(as found for the bulk in Meyer and Vanderbilt, 2002)imposed by the planar interface. Different domain wallswill in general be affected differently by the constraint,possibly changing the relative energy of different polydo-main configurations.

With recent advances in thin film synthesis, it is pos-

FIG. 31 (a)Phase diagram of PZT film under in-plane ten-sile strain of 0.005 obtained using thermodynamic calcula-tions assuming a single - domain state. There are onlytwo stable ferroelectric phases. The solid lines representthe boundaries separating the stability fields of the paraelec-tric and ferroelectric phases, or the ferroelectric orthorhom-bic and distorted rhombohedral phases. (b) Superpositionof the phase diagram of a PZT film under in-plane tensilestrain of 0.005 from the phase-field approach (scattered sym-bols) and from thermodynamic calculations assuming a sin-gle domain (solid lines). There are three stable ferroelectricphases: tetragonal-”square,” orthorhombic-”circle,” and dis-torted rhombohedral-”triangle” according to the phase-fieldsimulations. The scattered symbols represent the ferroelectricdomain state obtained at the end of a phase-field simulation.The shaded portion surrounded by the scattered symbols la-bel the stability regions of a single ferroelectric phase, and thenonshaded region shows a mixture of two or three ferroelectricphases. From Li et al., 2003b.

sible to grow and characterize high-quality films thatare sufficiently thin to be coherent or partially relaxed.Here, we give a few examples of experimental observa-tions of changes in structure and polarization in verythin films. Tc elevation in strained PbTiO3 films hasbeen reported and analyzed in Rossetti et al. (1991) andStreiffer et al. (2002). The strain-induced r-phase hasbeen observed in PZT films thinner than 150 nm on Ir-electroded Si wafers (Kelman et al., 2002). An antifer-roelectric to ferroelectric transition has been observedin thin films of PbZrO3 (Ayyub et al., 1998), thoughwhether it is induced by strain or some other thin-filmrelated effect is considered an open question. Large po-larization enhancements have been observed in epitax-ially strained BaTiO3 films (Choi et al., 2004). Mostdramatically, room-temperature ferroelectricity has beenachieved for SrTiO3 under biaxial tensile strain inducedby a DyScO3 substrate (Haeni et al., 2004). For thickerfilms, observation of polydomain structures is reported in

37

FIG. 32 Evolution of a⊥ as a function of filmthickness for Ba0.6Sr0.4TiO3 thin films grown on0.29(LaAlO3):0.35(Sr2TaAlO6) substrates. .Also shownis the theoretical curve, given by the open circles. Thestraight dashed line represents the lattice parameter of theceramic target (a = 0.39505 nm). From Canedy et al., 2000.

Roytburd et al., 2001. It has been observed that domainformation may be suppressed by rapid cooling throughthe transition (Ramesh et al., 1993).

For thicker films grown at high temperature, misfitdislocations will form at the growth temperature par-tially or completely to relax misfit strain. The degreeof relaxation increases with increasing thickness, un-til, for thick enough films, the epitaxial strain is neg-ligible. This behavior has been studied theoretically(Matthews and Blakeslee, 1974) and observed experi-mentally, for example as in Fig. 32. Additional straincan arise during cooling from the growth temperature ifthere is differential thermal expansion between the filmand the substrate and the formation of misfit dislocationsis kinetically inhibited. A detailed theoretical study ofstrain relaxation in epitaxial ferroelectric films, with dis-cussion of the interplay of misfit dislocations, mixed do-main formation and depolarizing energy, was undertakenby Speck and Pompe (Speck and Pompe, 1994). It wasassumed that for rapid cooling from the growth tempera-ture, the effect of misfit dislocations can be incorporatedby using an effective substrate lattice parameter, while inthe limit of slow cooling, the system optimally accommo-dates misfit strain with dislocations. (This assumptionis valid for films with thickness of order 1 µ, while thetreatment needs to be slightly modified for intermediatethicknesses where the equilibrium concentration of misfitdislocations leads to only partial strain relaxation). Elas-tic domains form to relax any residual strain. Below Tc,depolarizing energy can change the relative energetics ofdifferent arrangements of polarized domains and misfitdislocations. It was suggested that the electrostatic en-ergy should be more of a factor for smaller tetragonalitysystems (BaTiO3 vs PbTiO3) where the strain energy isless, though this could at least be partially balanced bythe fact that the polarization is smaller as well. A typical

e = 0.0125

h = 83 AC

o

e = 0.067

h = 5 AC

o

e = 0.034

h = 18.5 AC

o

e = 0.088

h = 140 AC

o

c f

a

a/c/a

c/a/c

ca f

Tetragonal

Cubic

Ce

ll D

ime

nsio

ns (

A)

o

Temperature ( C)o

0 100 200 300 400 500 600 700 800 900 1000

3.90

3.85

3.95

4.00

4.05

4.10

4.15

4.20

4.25MgO

Au

MgAl O2 4

PtSrTiO

3

ap

FIG. 33 Coherent temperature dependent domain stabilitymap for PbTiO3 including the cubic lattice parameter forseveral common single crystal oxide substrates. The misfitstrains for epitaxial growth of PbTiO3 at 600o C are includedin the insets along with the critical thickness h for misfit dis-location formation. From Speck and Pompe, 1994.

coherent diagram is shown in Fig. 33.With transmission electron microscopy (TEM) it is

possible to make detailed studies of the types and ar-rangements of misfit dislocations in perovskite thin films.Recent studies of high-quality films include Suzuki et al.,1999 and Sun et al., 2004. While strain-relaxing defects,such as misfit dislocations, reduce or eliminate the elas-tic energy associated with homogeneous strain, these andother defects prevalent in films do generate inhomoge-neous strains. As mentioned at the beginning of this sec-tion, the inhomogeneous strains couple strongly to thepolarization, and it has been shown by phenomenologi-cal analysis (Balzar and Popa, 2004; Balzar et al., 2002)that their effects on Tc can be significant. They havealso been argued to contribute to the degradation of thedielectric response in thin films relative to bulk values(Canedy et al., 2000).

Strains and their coupling to polarization are also cen-tral to the properties exhibited by short-period superlat-tices of lattice-mismatched constituents. As the resultof recent work on artificial superlattices of ferroelectricmaterials, there are some indications that improved ferro-electric properties and/or very large dielectric constantscan be achieved. The most studied system at presentis BaTiO3/SrTiO3 (Ishibashi et al., 2000; Jiang et al.,2003; Nakagawara et al., 2000; Neaton and Rabe, 2003;Rios et al., 2003; Shimuta et al., 2002; Tabata et al.,1994; Tian et al.). In BaTiO3/SrTiO3 superlatticeslattice-matched to a SrTiO3 substrate, the compressivein-plane strain on the BaTiO3 layer substantially raisesits polarization. Theoretical studies suggest that theSrTiO3 layer is polarized (and the polarization in theBaTiO3 layer is reduced) by electrostatic energy consid-erations, which favor continuity of the component of thepolarization along the normal. Overall the polarization is

38

enhanced above that of bulk BaTiO3, though not as highas that of a pure coherent BaTiO3 film if it were possi-ble to suppress the formation of strain-relaxing defects.While the natural lattice constant of BaTiO3/SrTiO3 isintermediate between the two endpoints, so that on aSrTiO3 substrate the superlattice is under compressivein-plane stress, it has been suggested that the multilayerstructure tends to inhibit the formation of misfit disloca-tions so that a thicker layer of coherent superlattice ma-terial can be grown. As the superlattice material thick-ness increases, there will be strain relaxation via misfitdislocations and the in-plane lattice constant should in-crease, putting the SrTiO3 layer under in-plane tensilestrain. In this case the SrTiO3 layer is observed to have acomponent of polarization along [110] (Jiang et al., 2003;Rios et al., 2003), consistent with theoretical studies ofepitaxially strained SrTiO3 (Antons et al.; Pertsev et al.,2000a) and of the BaTiO3/SrTiO3 superlattice with ex-panded in-plane lattice constant (Johnston and Rabe).

The real appeal of short-periodicity ferroelectricmultilayers is the potential to make “new” artificiallystructured materials with properties that could open thedoor to substantial improvements in device performanceor even radically new types of devices. Perovskitesare particularly promising, as individual materialspossess a wide variety of structural, magnetic, andelectronic properties, while their common structureallows matching at the interface to grow superlattices.Beyond the prototypical example of BaTiO3/SrTiO3

discussed in the previous paragraph, there has beenwork on other combinations such as KNbO3/KTaO3

(Christen et al., 1996; Sepliarsky et al., 2002, 2001;Sigman et al., 2002), PbTiO3/SrTiO3 (Jiang et al.,1999), PbTiO3/PbZrO3(Bungaro and Rabe, 2002,2004), La0.6Sr0.4MnO3/La0.6Sr0.4FeO3(Izumi et al.,1999), CaMnO3/CaRuO3(Takahashi et al., 2001),LaCrO3-LaFeO3(Ueda et al., 1998, 1999a),and LaFeO3-LaMnO3(Ueda et al., 1999b). In nearly all cases,strain plays an important role in understanding theaggregate properties of these short-period multilayersand superlattices. In addition to lattice mismatch,the layers also interact through the mismatch inpolarization along the layer normal, which leads tomutual influences governed by considerations of elec-trostatic energy and nonzero macroscopic electricfields. With three or more constituents, it is possibleto break inversion symmetry to obtain superlatticematerials with possibly favorable piezoelectric prop-erties. This idea was first proposed theoretically(Sai et al., 2000), leading to experimental studies ofCaTiO3/SrTiO3/BaTiO3 (Warusawithana et al., 2003)and LaAlO3/(La,Sr)MnO3/SrTiO3 (Kimoto et al.,2004; Ogawa et al., 2003; Yamada et al., 2002). Also,perovskite superlattices combining ferroelectric andferromagnetic layers offer a path to the development ofmultiferroic materials. The identification, synthesis andcharacterization of further combinations remains thesubject of active research interest.

VI. NANOSCALE FERROELECTRICS

A. Quantum confinement energies

Confinement energies are a trendy topic in nanoscalesemiconductor microelectronics devices.(Petroff et al.,2001) The basic idea is that in a system in which the elec-tron mean free path is long with respect to the lateral di-mension(s) of the device, a quantum-mechanical increasein energy (and of the bandgap) in the semiconductor willoccur. In general confinement energies exist only in theballistic regime of conduction electrons, that is, where theelectron mean free path exceeds the dimensions of thecrystal. This usually requires a high-mobility semicon-ductor at ultra-low temperatures. Such effects are bothinteresting and important in conventional semiconduc-tors such as Si or Ge, GaAs and other III-Vs, and perhapsin II-VIs. However, despite the fact that the commonlyused oxide ferroelectrics are wide-bandgap p-type semi-conductors (3.0 eV < Eg < 4.5 eV),(Waser and Smyth,1996) neither their electron nor hole mean free paths aresufficiently long for any confinement energies to be mea-sured. Typically the electron mean free path in an ABO3

ferroelectric perovskite is 0.1 to 1.0 nm,(Dekker , 1954)depending on applied electric field E, whereas the de-vice size d is at least 20 nm. Therefore any confinementenergy (which scales as d−2) might be a meV or two,virtually unmeasurable, despite a few published claims(Yu et al., 1997),(Kohiki et al., 2000),(J.F. Scott , 2000)reporting extraordinarily large effects. In the case ofBi2O3 and SrBi2Ta2O9 (SBT) these effects may arisefrom two-phase regions at the sample surfaces.(Zhou ,1992),(Switzer et al., 1999) This is theoretically inter-esting and very important from an engineering devicepoint of view; if it were not true the contact poten-tial at the electrode interface in a 1T-1C device, or atthe ferroelectric-Si interface in a ferroelectric-gate FET,would depend critically on the cell size, which would adda very undesirable complication to device design.

B. Coercive fields in nanodevices

One of the most pleasant surprises in the researchon small-area ferroelectrics is the observation, shown inFig. 34 , that the coercive field is independent of lateralarea.(Alexe et al., 1999) Coercive fields in nanophase fer-roelectric cells have generally been measured via atomicforce microscopy (AFM).(Gruverman et al., 1996) Do-main structures, polarization and coercive fields ofnanoscale particles of BaTiO3 have been studied theoret-ically using interatomic potentials (Stachiotti , 2004) anda first-principles effective Hamiltonian (Fu and Bellaiche,2003)

39

FIG. 34 Lack of significant dependence of coercive field onlateral area in nanoscale ferroelectrics Alexe et al. (1999)

C. Self Patterned nanoscale ferroelectrics

One approach to producing nanoscale ferroelectrics isto attempt to produce self patterned arrays of nanocrys-tals, in which ordering is produced by interactions be-tween islands through the substrate. This approach couldbe used to produce arrays of metallic nanoelectrodes ontop of a ferroelectric film or alternatively arrays of crys-tals from the ferroelectric materials themselves. The firstscheme was suggested by Alexe et al. (Alexe et al., 1998)who found that a bismuth oxide wetting layer on top ofa bismuth titanate film formed an array of metallic bis-muth oxide nanocrystals on top of the film, which werepartially registered along the crystallographic directionsof the underlying substrate (Fig. 35). These nanocrys-tals were used successfully as electrodes to switch regionsof the film(Alexe et al., 1999). In the second approachone might use a material such as PbTiO3 on a SrTiO3

substrate, which was first demonstrated to form islandswhen grown epitaxially at very thin film thicknesses bySeifert et al. (Seifert et al., 1996) In the context of selfpatterning of oxide materials a recent work by Vasco etal studies the growth of self organised SrRuO3 crystalson LaAlO3 (Vasco et al., 2003).

When small amounts of materials are deposited onsubstrates where there is some degree of mismatch be-tween the two materials, islands form and the repulsiveinteractions between them are mediated via strain fieldsin the substrate as first suggested by Andreev.(Andreev,1981) This idea has been developed into a detailed the-ory by Shchukin and Bimberg;(Shchukin and Bimberg,1999) however this theory is a zero-temperature the-ory, whereas a thermodynamic theory is required to de-scribe the crystallization processes which occur at quitehigh temperatures. An extension of the theory to finitetemperatures has been carried out by Williams and co-

FIG. 35 Sample of Alexe et al(Alexe et al., 1998), (a) TEMcross-section showing underlying layers and bismuth oxide na-noeelectrodes, (b) Semi-registered array of nanoelectrodes tak-ing their orientation from the underlying Si substrate

workers.(Williams et al., 2000),(Rudd et al., 2003) Thechief result of this theory are the prediction of three dif-ferent kinds of structures (pyramids, domes and super-domes), a volume distribution for a particular species ofstructure, and a shape map to describe relative popula-tions of structures as a function of coverage and crystal-lization temperature. One interesting result from exper-iment is that similar shaped structures are observed inboth the Volmer-Weber (VW) and Stranski-Krastanow(SK) growth modes, but on different size scales. In thework of Williams the thickness above which dome pop-ulations occur is of the order of 4-5 monolayers, cor-responding to the critical thickness for misfit disloca-tions for Ge on Si(100). On the other hand Capellini etal. (Capellini et al., 1997) studied via atomic force mi-croscopy the growth of Ge on Si(100) in the SK growthmode and found a much larger critical structure heightof 50 nm at which dislocations were introduced and thestructures changed from being pyramidal in geometry todomelike. The large increase in critical thickness is dueto a substantial part of the misfit strain being taken upby the substrate in the SK growth mode, as described byEaglesham and Cerrulo (Eaglesham and Cerullo, 1990).The description of self patterned ferroelectric nanocrys-tals by the models of Schukin and Williams has recentlybeen undertaken by Dawber et al. (2003c).

Prior to this two groups have grown PbTiO3 nanocrys-tals on Pt/Si(111) substrates to measure size effectsin ferroelectricity. (Roelofs et al., 2003),(Shimizu et al.,2003) These works both show a lack of piezoresponsein structures below 20nm in lateral size (Fig. 36),though we expect that this is connected to mechanicalconstraints rather than any fundamental limiting size forferroelectric systems. Chu et al. (2004) have highlightedthe role that misfit dislocations can play in hamperingferroelectricity in small structures. Interestingly in thework of Roelofs et al. (2003) and Shimizu et al. (2003)because of the (111) orientation of their substrates, in-stead of square-based pyramids they obtain triangularbased structures that display hexagonal rather than cu-bic registration (an analogous result is observed whenGe is grown on Si(111). (Capellini et al., 1999)). Thegrowth and analysis of PZT nanocrystals on SrTiO3 has

40

FIG. 36 Roelofs et al. (2003)(a) topographic image of grainsfrom 100 nm down to 20 nm in lateral size (b) piezoresponseimage of same grains showing the absence of peizoresponse forthe grains below 20nm

been carried out by Szafraniak et al. (2003)). A reviewon size effects in ferroelectric nanocrystals is currently inpreparation by Ruediger et al. (2004).

Although there is potential to produce self patternedarrays with greater registration by better choice of ma-terials and processing conditions our general conclusionis that highly registered memory arrays will not occurspontaneously in the absence of a pre-patterned field.

D. Non-planar geometries:Ferroelectric nanotubes

Almost all recent work on ferroelectric oxide films haveinvolved planar geometries. However, from both a deviceengineering point of view and from theoretical consider-ations, it is now appropriate to analyze carefully non-planar geometries, especially nanotubes.

Nanotubes made of oxide insulators have a varietyof applications for pyroelectric detectors, piezoelectricink-jet printers, and memory capacitors that cannotbe filled by other nanotubes (Herzog and Kattner,1985),(Sakamaki et al., 2001),(Sajeev and Busch,2002),(Gnade et al., 2000),(Averdung et al., 2001). Inthe drive for increased storage density in FRAM andDRAM devices, complicated stacking geometries, 3Dstructures and trenches with high aspect ratios arealso being investigated to increase the dielectric surfacearea. The integration of ferroelectric nanotubes into Sisubstrates is particularly important in construction of3D memory devices beyond the present stacking andtrenching designs, which according to the internationalULSI schedule 3 must be achieved by 2008. Templatesynthesis of nanotubes and wires is a versatile and in-expensive technique for producing nanostructures. The

3 International Technology Roadmap for Semi-conductors (ITRS) 2002 (available athttp://public.itrs.net/Files/2002Update/Home.pdf)

size, shape and structural properties of the assembly aresimply controlled by the template used. Using carbonnanotubes as templates, tubular forms of a number ofoxides including V2O5, SiO2, Al2O3 and ZrO2 havebeen generated (Patzke et al., 2002). Much larger (>20-micron diameter) ferroelectric micro-tubes have beenmade by sputter deposition around polyester fibres (Fox,1995),(Pokropivny, 2001) - Fox has made them fromZnO and PZT, with 23 µm inside diameter, about 1000xlarger than the smallest nano-tubes reported in thepresent paper. Porous sacrificial templates as opposedto fibres have also been used. Porous anodic aluminahas a polycrystalline structure with ordered domains ofdiameter 1-3 µm, containing self-organised 2D hexagonaltubular pore arrays with an interpore distance of 50-420nm (Li et al., 1998). This nano-channel material cantherefore be used as a template for individual nanotubesbut is not suitable for making an ordered array of tubesover length scales greater than a few mm. Many oxidenanotubes, such as TiO2, In2O3, Ga2O3, BaTiO3 andPbTiO3, as well as nanorods of MnO2, Co3O4 andTiO2, have been made using porous alumina membranesas templates (Patzke et al., 2002)]. Hernandez et al.

(2002), used a sol-gel template synthesis route to prepareBaTiO3 and PbTiO3 nanotube bundles by dipping alu-mina membranes with 200 nm pores into the appropriatesol. The BaTiO3 and PbTiO3 nanotubes were shownto be cubic (paraelectric) and tetragonal (ferroelectric)by x-ray diffraction, although Raman studies indicatedsome non-centrosymmetric phase on a local scale inthe BaTiO3. Porous silicon materials are also availableas suitable templates. Mishina et al. (2002) used asol-gel dipping technique to fill nanoporous silicon witha PbZr1−xTixO3 (PZT) sol producing nanograins andnanorods 10-20 nm in diameter. The presence of theferroelectric PZT phase was shown by second harmonicgeneration (SHG) measurements. In this instancethe porous silicon does not have a periodic array ofpores (Smith and Collins, 1992) and as in the case forthose produced by Hernandez et al, we emphasise thatthose nanotubes are not ordered arrays, but insteadspaghetti-like tangles of nano-tubes that cannot beused for the Si device embodiments. A second type ofporous Si templates, however, consist of a very regularperiodic array of pores with very high aspect ratios. Bya combination of photolithography and electrochemicaletching hexagonal or orthogonal arrays of pores withdiameters 400 nm to a few mm and up to 100 µm deepcan be formed in single crystal Si wafers (Schilling et al.,2001),(Ottow et al., 1996). These crystals were origi-nally developed for application as 2D photonic crystals,but also find applications as substrates for templatedgrowth and integration of oxides nanostructures withSi technology. Luo et al. (2003) recently used suchcrystals to produce individual, free standing PZT andBaTiO3 ferroelectric nanotubes by a polymeric wettingtechnique. Morrison et al. (2003) described the useof liquid source misted chemical deposition (LSMCD)

41

to fill such photonic Si crystals with SBT precursor.During deposition, the SBT precursor was shown to coatthe inside of the pores. After etching of the photoniccrystal with pore diameter 2 µm for 30 seconds with aq.HF/HNO3 the interface between the Si substrate andSBT coating is dissolved, exposing the uniform SBTtube, Fig. 37a. The tube walls are very uniform with athickness of ca. 200 nm. The same sample is shown incross-sectional view after complete removal of the hostSi walls between pores, Fig. 37b. The result is a regulararray of tubes attached to the host Si matrix only at thetube base. Although these tubes have suffered damageduring handling, it is clear that the pores have beenfilled uniformly to the bottom, a depth of ca 100 µm.

FIG. 37 SEM micrograph indicating a plan view of a regulararray of SBT tubes in host silicon substrate with diameterca. 2 µm and wall thickness ca. 200 nm (a). SBT tubes incross sectional view indicating coating to bottom of pore (b).Micrograph of free-standing array of tubes with diameter ca.800 nm (c) and wall thickness < 100 nm (d).

The second photonic crystal with pore diameter 800nm underwent fewer depositions and after etching re-vealed a regular array of uniform tubes of diameter 800nm, Fig. 37c. The wall thickness is uniform and < 100nm, Fig. 37d. The tubes are ca. 100 µm long, completelydiscrete and are still attached to the host Si matrix, creat-ing a perfectly registered hexagonal array. Free-standingtubes may be produced by completely dissolving the hostSi matrix. As yet, no one has applied cylindrical elec-trodes to the tubes; however, Steinhart et al. (2003) re-cently used porous anodic alumina templates to growpalladium nanotubes. Using a similar method it maybe possible to alternately deposit Pd or Pt and SBT toproduce a concentric electrode/FE/elctrode structure ineach nanotube. The use of the photonic crystal templatewith a regular array of pores has significant benefits overother porous substrates in that the coatings/tubes pro-duced are also in a registered array ordered over severalmm’s or even cm’s. This facilitates addressing of such anarray for device applications. DRAMs utilise high sur-

face area dielectrics, and high aspect ratio SBT coatingssuch as these embedded in Si could increase storage den-sity. Current state-of-the-art deep trenched capacitorsare 0.1 mm diameter by 6 mm deep, aspect ratio 60:1.Using SBT (or other FE oxide) nanotubes of wall thick-ness <100 nm, a trench (or array of trenches) of 0.1 µmdiameter and 100 µm deep, an aspect ratio of > 1000:1 ispossible. Applying and addressing electrodes to an arrayof FE nanotubes could generate 3D FRAM structuresoffering high storage density with improved read/writecharacteristics compared to conventional planar stacks.On removal of the Si walls, the piezoelectric response(expansion/contraction under an applied field) of suchan array of nanotubes could be utilized for a numberof MEMs applications. These could include: (1) ink-jetprinting - delivery of sub-picolitre droplets for lithogra-phy free printing of submicron circuits; (2) biomedicalapplications - nanosyringes, inert drug delivery implants;(3) micropositioners or movement sensors.

Almost no theoretical work has been published on thephysics of ferroelectric nanotubes. Analytical solutionsfor the effects on the dij piezoelectric coefficients of hol-low tubes have been given for both the case in which po-larization P is along the length z (Ebenezer and Ramesh,2003) and for P radial (Ebenezer and Abraham, 2002),they did not however solve the azimuthal case where po-larization goes around the tube. It it this latter casewhich been measured as hysteresis by Luo et al. (2003)with a tube lying on a bottom electrode with a semicir-cular sputtered top electrode. Important matters such asthe dependence of TC upon tube diameter have also notbeen examined.

VII. CONCLUSIONS

In this review we have sought to cover the importantadvances in recent years in the physics of thin film ferro-electric oxides. At the present point of time ferroelectricthin films memory devices have reached a point of ma-turity where they are beginning to appear in real com-mercial devices. At the same time new directions such asthe drive to faster, smaller, nanoscale devices and non-planar geometries are evolving and new levels of physicalunderstanding will be required. Over the next years it isexpected that first principles computational approacheswill continue to develop, suggesting a new synergy be-tween the computational modeling and experimental re-alizations of ferroelectric systems with new and excitingproperties.

References

ABINIT is a common project of the Universite Catholique deLouvain, Corning Incorporated, and other contributors; seethe URL http://www.mapr.ucl.ac.be/ABINIT.

Ahn, C.H., J.-M. Triscone, and J. Mannhart, 2003, Nature424, 1015.

42

Alexe, M., J.F. Scott, C. Curran, N.D. Zakharov, D. Hesse,and A. Pignolet 1998, Appl. Phys. Lett. 73, 1592.

Alexe, M., A. Gruverman, C. Harnagea, N.D. Zakharov, A.Pignolet, D. Hesse, and J.F. Scott, 1999, Appl. Phys. Lett.75, 1158.

Alpay, S. P., and A. L. Roytburd, 1998, J. Appl. Phys. 83,4714.

Andreev, A.F., 1981, JETP Lett. 53, 1063.Antons, A., J. B. Neaton, K. M. Rabe, and D. Vanderbilt,

Unpublished, Tunability of the dielectric response of epi-taxially strained srtio3 from first principles.

Araujo, C.A.-Paz de, J.D. Cuchiaro, L.D. Mc Millan, M.C.Scott, and J.F. Scott, 1995, Nature 374, 627.

Arlt, G., and H. Neumann, 1988, Ferroelectrics 87, 109.Armstrong, R.D., B.R. Horrocks, 1997, Sol. St. Ionics 94, 181.Astala, R. K., and P. D. Bristowe, 2004, Model. Simul. Mater.

Sc. 12, 79.Averdung, J et al. 2 January 2001, German Patent Number

DE10023456.Ayyub, P., S. Chattopadhyay, R. Pinto, and M.S. Multani,

1998, Phys. Rev. B 57, R5559.Balzar, D., and N. C. Popa, 2004, Thin Sol. Films 450, 29.Balzar, D., P. A. Ramakrishnan, P. Spagnol, and et al., 2002,

Jap. J. Appl. Phys. 41, 6628.Baniecki, J.D., R.B. Laibowitz, T.M. Shaw, C. Parks, J. Lian,

H. Xu, Q.Y. Ma, 2001, J. Appl. Phys. 89, 2873.Bardeen, J., 1947, Phys. Rev. 71, 717.Baroni, S., A. dal Corso, S. de Gironcoli, and P. Giannozzi,

Unpublished, http://www.pwscf.org.Baroni, S., S. de Gironcoli, A. D. Corso, and P. Giannozzi,

2001, Rev. Mod. Phys. 73, 515.Basceri, C., S.K. Streiffer, A.I. Kingon, and R. Waser, 1997,

J. Appl. Phys. 82, 2497.Bellaiche, L., A. Garcia, and D. Vanderbilt, 2000, Phys. Rev.

Lett. 84, 5427.Bengtsson, L., 1999, Phys. Rev. B 59, 12301.Betsuyaku, K., H. Tanaka, H. Katayama Yoshida, and

T. Kawai, 2001, Jap. J. Appl. Phys. 40, 6911.Black, C.T., and J.J. Welser, 1999, IEEE Trans. Elec. Dev.

46, 776.Blaha, P., K. Schwarz, P. Sorantin, and S. B. Trickey, 1990,

Comput. Phys. Commun. 59, 399.Boikov, Yu.A., B.M. Goltsman, V.K. Yamarkin, V.V. Le-

manov, 2001, Appl. Phys. Lett. 78, 3866.Bratkovsky, A. M., and A. P. Levanyuk, 2000, Phys. Rev.

Lett. 84, 3177.Bratkovsky, A. M., and A. P. Levanyk, 2001, Phys. Rev. Lett.

86, 3642.Brazier, M., S. Mansour and M. McElfresh 1999, Appl. Phys.

Lett. 74, 4032.Brennan, C.J., 1992, Integ Ferroelec 2, 73.Brennan, C., 1993, Ferroelectrics 150, 199.Bune, A.V., V.M. Fridkin, S. Ducharme, L.M. Blinov, S.P.

Palto, A.V. Sorokin, S.G. Yudin, and A. Zlatkin, 1998,Nature 391, 1998.

Bungaro, C., and K. M. Rabe, Unpublished, Antiferrodis-tortive reconstruction of the (001) pbtio3 surface from firstprinciples.

Bungaro, C., and K. M. Rabe, 2002, Phys. Rev. B 65, 224106.Bungaro, C., and K. M. Rabe, 2004, Phys. Rev. B 69, 184101.Canedy, C. L., H. Li, S. P. Alpay, L. Salamanca-Riba, A. L.

Roytburd, and R. Ramesh, 2000, Appl. Phys. Lett. 77,1695.

Capellini, G., L. Di Gaspare, F. Evangelisti, 1997, Appl. Phys.

Lett 70, 493.Capellini, G., N. Motta, A. Sgarlata, R. Calarco, 1999, Sol.

Stat. Comm. 112, 145.Castell, M. R., 2002, Surface Science 505, 1.Chalamala, B.R., Y. Wei, R.H. Reuss, S. Aggarwal, B.E.

Gnade, R. Ramesh, J.M. Bernhard, E.D. Sosa, and D.E.Golden 1999, Appl. Phys. Lett. 74, 1394.

Chan, N.-H., R.K. Sharma, and D.M. Smyth, 1981, J. Amer.Ceram. Soc. 64, 556.

Chan, N.-H., R.K. Sharma, and D.M. Smyth, 1976, J. Elec-trochem. Soc. 123, 1584.

Chen, H-M, S-W Tsaur, and Y-M Lee J., 1998, Jap. J. Appl.Phys 137, 4056.

Chen, L., V. Nagarajan, R. Ramesh, and A. L. Roytburd,2003, J. Appl. Phys. 94, 5147.

Cheon, C.I., K.Y. Yun, J.S. Kim, and J.H. Kim, 2001, Integ.Ferroelec. 34, 1513.

Choi, K.-J., W-C. Shin, S.-G. Yoon, 2001, Integ. Ferroelec.34, 1541.

Choi, K. J., M. Biegalski, Y. L. Li, and A. Sharan, Unpub-lished, Giant enhancement of ferroelectricity in strained ba-tio3 films.

Chon, U., H. M. Jang, M. G. Kim, and C. H. Chang, 2002,Phys. Rev. Lett. 89, 087601.

Christen, H.-M., L. A. Boatner, J. D. Budai, M. F. Chisholm,L. A. Ga, P. J. Marrero, , and D. P. Norton, 1996, Appl.Phys. Lett. 68, 1488.

Chu, M.-W., I. Szafraniak, R. Scholz, C. Harnagea, D. Hesse,M. Alexe, , and U. Gosele, 2004, Nature Materials 3, 87.

Cockayne, E., and K. M. Rabe, 1998, Amer. Inst. Phys. Conf.Proc. 436, 61.

Cockayne, E., and K. M. Rabe, 2000, J. Phys. Cond. Matt.61, 305.

Cohen, R. E., and H. Krakauer, 1990, Phys. Rev. B 42, 6416.Cohen, R.E., 1992, Nature 358, 136.Cohen, R. E., 1996, J. Chem. Phys. Solids 57, 1393.Cohen, R. E., 1997, Ferroelectrics 194, 323.Cohen, R. E., and H. Krakauer, 1992, Ferroelectrics 136, 65.Colla, E.L., S. Hong, D.V. Taylor, A.K. Tagantsev, and N.

Setter, 1998, Appl. Phys. Lett. 72, 2763.Conley, J.W., and G.D. Mahan, 1967, Phys. Rev. 161, 681.Cora, F., and C.R.A. Catlow, 1999, Faraday Discussions 114,

421.Cowley, A.M., and S.M. Sze, 1965, J. Appl. Phys. 36, 3212.Damjanovic, D., 1997, Phys. Rev. B 55, R649.Dawber, M., and J.F. Scott, 2000, Appl. Phys. Lett. 76, 1060;

Appl. Phys. Lett. 76, 3655.Dawber, M., J.F. Scott, and A.J. Hartmann, 2001, J. Europ.

Ceram. Soc. 21, 1633.Dawber, M., and J.F. Scott, 2002, Jap. J. Appl. Phys. 41,

6848.Dawber, M., I. Farnan and J.F. Scott, 2003, American J.

Physics 71, 819.Dawber, M., P. Chandra, P.B. Littlewood, and J.F. Scott,

2003, J. Phys. C. 15, L393.Dawber, M., I. Szafraniak, M. Alexe and J.F. Scott 2003, J.

Phys. C. 15, L667.Dawber, M., and J.F. Scott, 2004, Unpublished: cond-

mat/0407428Dekker, A.J., 1954, Phys. Rev. 94, 1179.Democritov, S.O., and N.M. Kreines, 1988, JETP Lett. 48,

294Dieguez, O., S. Tinte, A. Antons, C. Bungaro, J. B. Neaton,

K. M. Rabe, and D. Vanderbilt, 2004, Phys. Rev. B 69,

43

212101.Dietz, G.W., and and R. Waser, 1995, Integ. Ferro. 9, 317.Dietz, G.W., and R. Waser, 1997, Thin Solid Films 299, 53.Dietz, G.W., M. Schumacher, R. Waser, S.K. Streiffer, C.

Basceri, and A.I. Kingon 1997, J Appl Phys 82, 2359.Dimos, D., W.L. Warren, M.B. Sinclair, B.A. Tuttle, and

R.W. Schwartz, 1994, J. Appl. Phys 76 4305.Do, D-H., P.G. Evans, E.D. Issacs, D.M. Kim, C.B. Eom, and

E.M. Dufresne 2004, Nature Materials 3, 365.Du, X. and I.W. Chen, 1998, J. Appl. Phys. 83, 7789.Du, X., and I.W. Chen, 1998, Mat. Res. Soc. Proc 493, 311.Ducharme, S., V.M. Fridkin, A.V. Bune, S.P. Palto, L.M.

Blinov, Nn. Petukhova, and S.G. Yudin, 2000, Phys. Rev.Lett. 84, 175.

Duiker, H.M., 1990, PhD thesis, Univ. ColoradoDuiker, H.M., and P.D. Beale, 1990, Phys. Rev. B 41, 490.Duiker, H.M., P.D. Beale, J.F. Scott, C.A. Paz de Araujo,

B.M. Melnick, and J.D. Cuchario 1990, J. Appl. Phys. 68,5783.

Eaglesham, D.J., and M. Cerullo, 1990, Phys. Rev. Lett. 64,1943.

Ebenezer, D.D., and P. Abraham, 2002, Current Science 83,981.

Ebenezer, D.D., and R. Ramesh, 2003, Current Science 85,1173.

Erdman, N., K. R. Poeppelmeier, M. Asta, O. Warschkow,D. E. Ellis, and L. D. Marks, 2002, Nature 419, 55.

Fatuzzo, E., and W.J. Merz, 1967, Ferroelectricity (North-Holland, Amsterdam)

Fennie, C. J., and K. M. Rabe, UnpublishedFeynman, R. P., 1939, Phys. Rev. 56, 340.Fiebig, M., Lottermoser T., Frohlich D., Goltsev A.V. and

Pisarev R.V., 2002, Nature 419, 818.Floquet, N., C.M. Valot, M.T. Mesnier, J.C. Niepce, L. Nor-

mand, A. Thorel, and R., Kilaas 1997, J Phys III France7, 1105.

Floquet, N. and C. Valot, 1999, Ferroelectrics 234, 107.Foeth, M., A. Sfera, P. Stadelmann, and P-A. Buffat, 1999,

J. Electron Microscopy 48, 717.Fong, D. D., G. B. Stephenson, S. K. Streiffer, J. A. Eastman,

O. Auciello, P. H. Fuoss, and C. Thompson, 2004, Science304, 1650.

Forlani, F., and N. Minnaja, 1964, Phys. Stat. Sol. 4, 311.Fouskova, A., 1965, J. Phys. Soc. Jpn. 20, 1625.Fouskova, A., and V. Janousek, 1965, J. Phys. Soc. Jpn 20,

1619.Fox, D.L., D.R. Tilley, J.F. Scott, and H.J. Guggenheim,

1980, Phys. Rev. B 21, 2926.Fox, G.R., 1995, J. Mater. Sci. Lett. 14, 1496.Fu, H., and L. Bellaiche, 2003, Phys. Rev. Lett 91, 257601.Fu, H. X., and R. E. Cohen, 2000, Nature 403, 281.Fu, L., E. Yaschenko, L. Resca, and R. Resta, 1999, Phys.

Rev. B 60, 2697.Ganpule, C.S., V. Nagarajan, H. Li, A.S. Ogale, D.E. Stein-

hauser, S. Aggarwal, E. Williams, R. Ramesh and P. DeWolf 2000, Appl. Phys. Lett. 77, 292.

Ganpule, C.S., V. Nagarajan, S.B. Ogale, A.L. Roytburd,E.D. Williams, and R. Ramesh 2000, Appl. Phys. Lett. 77,3275.

Ganpule, C.S, A.L. Roytburd, V. Nagarajan, B.K. Hill, S.B.Ogale, E.D. Williams, R. Ramesh, and J.F. Scott 2001,Phys. Rev. B. 65, 014101.

Garcia, A., and D. Vanderbilt, 1996, Phys. Rev. B 54, 3817.Garcia, A., and D. Vanderbilt, 1998, Appl. Phys. Lett. 72,

2981.Garg, A., Z. H. Barber, M. Dawber, J. F. Scott, A. Snedden,

and P. Lightfoot, 2003, Appl. Phys. Lett. 83, 2414.de Gironcoli, S., S. Baroni, and R. Resta, 1989, Phys. Rev.

Lett. 62, 2853.Gerson, R., and T.C. Marshall, 1959, J. Appl. Phys. 30, 1650.Ghosez, P., E. Cockayne, U. V. Waghmare, and K. M. Rabe,

1999, Phys. Rev. B 60, 836.Ghosez, P., and K. M. Rabe, 2000, Appl. Phys. Lett. 76, 2767.Gnade, B et al. 7 March 2000, US Patent Number US6033919.Gorbenko, O. Y., S. V. Samoilenkov, I. E. Graboy, and A. R.

Kaul, 2002, Chem. Mat. 14, 4026.Gruverman, A., O. Auciello, and H. Tokumoto, 1996, Appl.

Phys. Lett. 69, 3191.Gruverman, A., H. Tokumoto, A.S. Prakash, S. Aggarwal, B.

Yang, M. Wuttig, R. Ramesh, O. Auciello, and T. Venkate-san, 1997 Appl. Phys. Lett. 71, 3492.

Haeni, J.H. et al. 2004, Nature 430, 758.Haneder, T.P., W. Hoenlein, H. Bachhofer, H. Von Philips-

born, and R. Waser, 2001, Integ. Ferroelec. 34, 1487.Han, J.-P., X. Guo, C.C. Broadbridge, T.P. Ma, M. Cantoni,

J.-M. Sallese, and P. Fazan, 2001, Integ. Ferroelec 34, 1505.Hartmann , A.J., M. Neilson, R.N. Lamb, K. Watanabe, and

J.F. Scott, 2000, Applied Physics A 70, 239.Hashimoto, S. , H. Orihara, and Y. Ishibashi 1994, J. Phys.

Soc. Japan 63, 1601.He, L. X., and D. Vanderbilt, 2003, Phys. Rev. B 68, 134103.Heifets, E., R. I. Eglitis, and E. A. K. et al., 2002a, Surf. Sci.

513, 211.Heifets, E., R. I. Eglitis, E. A. Kotonin, and G. Borstel, 2001a,

Amer. Inst. Phys. Conf. Proc. 582, 201.Heifets, E., R. I. Eglitis, E. A. Kotonin, and G. Borstel, 2002b,

Amer. Inst. Phys. Conf. Proc. 626, 285.Heifets, E., R. I. Eglitis, E. A. Kotonin, J. Maier, and G. Bors-

tel, 2001b, Phys. Rev. B 64, 235417.Heifets, E., E. Kotomin, and P. W. M. Jacobs, 2000a, Thin

Sol. Films 375, 64.Heifets, E., E. A. Kotonin, and J. Maier, 2000b, Surf. Sci.

462, 19.Heine, V., 1965, Phys. Rev. 138, A1689.Hernandez, B.A., K-S Chang, E.R. Fisher, and P.K. Dorhout,

2002, Chem. Mater. 14, 481.Herzog, K, and E Kattner, 12 March 1985, US Patent number

4504845.Hill, N.A., 2000, J. Phys. Chem. B. 104, 6694.Huffmann, M., 1995, Integr. Ferroelectrics 10, 39.Hwang, C.S., B.T Lee, H-J Cho, K.H. Lee, C.S Kang, H.

Horii, S.I. Lee, and M.Y. Lee, 1997, Appl. Phys. Lett 71,371.

Hwang, C.S., B.T Lee, C.S Kang, J.W. Kim, K.H. Lee, H-JCho, H. Horii, W.D. Kim, S.I. Lee, Y.B. Roh, and M.Y.Lee 1988, J. Appl. Phys. 83, 3703.

Hwang, C.S., 1998, Mater Sci & Eng B 56, 178.Ishibashi, Y., N. Ohashi, and T. Tsurumi, 2000, Jap. J. Appl.

Phys. 39, 186.Ishiwara, H., 2001, Integ. Ferroelec. 34, 1451.Ishiwara, H., 1993, Jpn. J. Appl. Phys. 32, 442.Ishiwara, H., T. Shimamura, and E. Tokumitsu, 1997, Jpn. J.

Appl. Phys. 36, 1655.Izumi, M., Y. Murakami, Y. Konishi, T. Manako,

M. Kawasaki, and Y. Tokura, 1999, Phys. Rev. B 60, 1211.Jaffe, B., W.R Cook, and H. Jaffe, 1971, Piezoelectric ceram-

ics (Academic Press, London)Janovec, V., 1958, Czech. J. Phys. 8, 3.

44

Jiang, J. C., X. Q. Pan, W. Tian, C. D. Theis, and D. G.Schlom, 1999, Appl. Phys. Lett. 74, 2851.

Jiang, A. Q., J. F. Scott, H. Lu, and Z. Chen, 2003, J. Appl.Phys. 93, 1180.

Johnston, K., M. R. Castell, A. T. Paxton, and M. W. Finnis,2004, Phys. Rev. B 70(8), 085415.

Johnston, K., and K. M. Rabe, Unpublished, First principlesstudy of orthorhombic structure in batio3/srtio3 superlat-tices.

Jun, S., and J. Lee, 2001, Integ. Ferroelec. 34, 1579.Jung, D.J., F.D. Morrison, M. Dawber, H.H. Kim, K. Kim,

and J.F. Scott 2004, J. Appl. Phys. 95, 4968.Jung, D.J., M. Dawber, J.F. Scott, L.J. Sinnamon, and J.M

Gregg, 2002, Integ. Ferro. 48, 59.Junquera, J., and P. Ghosez, 2003, Nature 422, 506.Junquera, J., M. Zimmer, P. Ordejon, and P. Ghosez, 2003,

Phys. Rev. B 67, 155327.Junquera, J., P. Ghosez, and K. M. Rabe, UnpublishedJunquera, J., and K. M. Rabe, UnpublishedKalkur, T.S., B. Jacob and G. Argos, 1994, Integ. Ferroelec.

5, 177.Kalkur, T.S., and J. Lindsey, 2001, Integ. Ferroelec. 34, 1587.Kato, K., 2001, Integ. Ferroelec. 34, 1533.Kay, H.F., and J.W. Dunn, 7, Phil. Mag. 7, 1962.Kelman, M.B., L.F. Schloss, P.C. McIntyre, B.C. Hendrix,

S.M Bilodeau, and J.F. Roeder, 2002, Appl. Phys. Lett.80, 1258.

Khodenkov, G.E., 1975, Fiz. Met. Metalloved. 39, 466.Kim, J, C.J. Kim, and I. Chung, 2001, Integ. Ferro. 33, 133.Kim, M., G. Duscher, N.D. Browning, K. Sohlberg, S.T. Pan-

telides, and S.J. Pennycock, 2001, Phys. Rev. Lett. 86,4056.

Kimoto, K., Y. Matsui, H. Yamada, H. Yamada, and H. Ya-mada, 2004, Appl. Phys. Lett. 84, 5374.

King-Smith, R. D., and D. Vanderbilt, 1993, Phys. Rev. B47, 1651.

King-Smith, R. D., and D. Vanderbilt, 1994, Phys. Rev. B49, 5828.

Kingon, A.I., J.P. Maria, and S.K. Streiffer, 2000, Nature406, 1032.

Klenov, D. O., T. R. Taylor, and S. Stemmer, 2004, J. Mat.Res. 19, 1477.

Kohiki, S., S. Takada, A. Shmizu, K. Yamada, and M. Mitone,2000, J. Appl. Phys. 87, 474.

Kohn, W., and L. J. Sham, 1965, Phys. Rev. Ser. A 140,1133.

Koukhar, V. J., N. A. Pertsev, and R. Waser, 2001, Phys.Rev. B 64, 214103.

Krcmar, M., and C. L. Fu, 2003, Phys. Rev. B 68, 115404.Kresse, G., and J. Furthmuller, 1996, Phys. Rev. B 54, 11169.Kresse, G., and J. Hafner, 1993, Phys. Rev. B 47, 558.Krishnan, A., M.M.J. Treacy, M.E. Bisher, P. Chandra, and

P.B. Littlewood, 2002, Integ. Ferroelec. 43, 31.Ku, H.Y., and F.G. Ullman, 1964, J. Appl. Phys. 35, 265.Kubo, T., and H. Nozoye, 2003, Surf. Sci. 542, 177.Kurtin, S., T.C. McGill, and C.A. Mead, 1969, Phys. Rev.

Lett. 22, 1433.Lewis, S. P., and A. M. Rappe, 1996, Phys. Rev. Lett. 77,

5241.Li, A.P., F Muller, A Birner, K Nielsch, and U Gosele, 1998,

J. Appl. Phys. 84, 6023.Li, T., S.T. Hsu, 2001, Integ. Ferroelec. 34, 1495.Li, H., A. L. Roytburd, S. P. Alpay, T. D. Tran, L. Salamanca-

Riba, and R. Ramesh, 2001, Appl. Phys. Lett. 78, 2354.

Li, H., H. Zheng, L. Salamanca-Riba, R. Ramesh, I. Naumov,and K. Rabe, 2002, Appl. Phys. Lett. 81, 4398.

Li, J. A., E. A. Akhadov, J. Baker, L. A. Boatner, D. Bonart,and S. A. Safron, 2003a, Phys. Rev. B 68, 045402.

Li, Y. L., S. Choudhury, Z. K. Liu, and L. Q. Chen, 2003b,Appl. Phys. Lett. 83, 1608.

Lichtensteiger, C., J. M. Triscone, J. Junquera, andP. Ghosez, 2005, Phys. Rev. Lett. 94, 047603.

Lines, M.E, and A.M. Glass, 1967, Principles and applicationsof ferroelectrics and related materials (Clarendon Press,Oxford)

Lookman, A., R.M. Bowman, J.M. Gregg, J. Kut, S. Rios, M.Dawber, A. Ruediger, and J.F. Scott 2004, J. Appl. Phys.96, 555.

Lubomirsky, I., J. Fleig and J. Maier, 2002, J. Appl. Phys.92, 6819.

Luo, Y., I. Szafraniak, N.D. Zakharov, V. Nagarajan, M.Steinhart, R.B. Wehrspohn, J.H. Wendorff, R Ramesh, andM. Alexe, 2003, Appl. Phys. Lett 83, 440.

Lupascu, D.C. and U. Rabe, Phys. Rev. Lett. 89, 187601.Macleod, T.C., and F.D. Ho, 2001, Integ. Ferroelec. 34, 1461.Man, Z. Y., and X. Q. Feng, 2002, Sol. St. Comm. 123, 333.Masui, S., 2003, IEEE Custom Integrated Circuits Conf. ab-

stract book 403.Mathews, S., R. Ramesh, T. Venkatesan, and J. Benedetto,

1997, Science 276, 238.Matthews, J. W., and A. E. Blakeslee, 1974, J. Cryst. Growth

27, 118.McKee, R. A., F. J. Walker, and M. F. Chisholm, 1998, Phys.

Rev. Lett. 81, 3014.McKee, R. A., F. J. Walker, and M. F. Chisholm, 2001, Sci-

ence 293, 468.McKee, R. A., F. J. Walker, M. B. Nardelli, W. A. Shelton,

and G. M. Stocks, 2003, Science 300, 1726.McMillan, L.D., C.A. Paz De Araujo, T. Roberts, J. Cuchiaro,

M.C. Scott, J.F. Scott, 1992, Integr. Ferroelectrics 2, 351.Meyer, B., J. Padilla, and D. Vanderbilt, 1999, Faraday Dis-

cussions 114, 395.Meyer, B., and D. Vanderbilt, 2001, Phys. Rev. B 63, 205426.Meyer, B. E., and D. Vanderbilt, 2002, Phys. Rev. B 65,

104111.Merz, W.J., 1954, Phys. Rev. 95, 690.Miller, SL, RD Nasby, JR Schwank, MS Rodgers, PV

Dressendorfer, and 1990, J. Appl. Phys. 68, 6463.Mills, G., M.S. Gordon, and H. Metiu, 2003, J. Chem. Phys.

118, 4198.Mihara, T., H. Watanabe, H. Yoshimori, C.A. Paz De Araujo,

B. Melnick, and L.D. McMillan 1992, Integ. Ferroelec. 1,269.

Mishina, E.D., K.A Vorotilov, V.A. Vasil’ev, A.S. Sigov, N.Ohta, and S. Nakabayashi, 2002, J. Exp. Theor. Phys. 95,502.

Monch, W., 1986, Phys. Rev. Lett. 58, 1260.Montanari, B., and N. M. Harrison, 2004, J. Phys. Cond.

Matt. 16, 273.Moreira, R.L, 2002, Phys. Rev. Lett. 88, 179701.Moriera de Santos, A., Parashar S., A.R. Raju, Y.S. Zhao,

A.K. Cheetham, and C.N.R. Rao 2002, Sol. Stat. Comm.122, 49.

Morrison, FD, JF Scott, M Alexe, TJ Leedham, T Tatsuta,and O Tsuji, 2003, Microelectron. Eng. 66, 591.

Mott, N.F., 1938, Proc. Cambridge Philos. Soc. 34, 568.Munkholm, A., S. K. Streiffer, M. V. R. Murty, J. A. East-

man, C. Thompson, O. Auciello, L. Thompson, J. F.

45

Moore, and G. B. Stephenson, 2002, Phys. Rev. Lett. 88,016101.

Muralt, P., 2000, J. Micromech. Microeng 10, 136.Murphy, E.L., R.H. Good, 1956, Phys. Rev. 102, 1464.Nagarajan, V., A. Roytburd, A. Stanishevsky, S. Praserth-

choung, T. Zhao, L. Chen, J. Melngailis, O. Auciello, andR. Ramesh, 2003, Nature Materials 2, 43.

Nakagawara, O., T. Shimuta, T. Makino, T. Makino, andT. Makino, 2000, Appl. Phys. Lett. 77, 3257.

Neaton, J. B., and K. M. Rabe, 2003, Appl. Phys. Lett. 82,1586.

Nielsen, O. H., and R. M. Martin, 1985, Phys. Rev. B 32,3780.

Nowotny, J., M. Rekas, 1994, Ceram. Int. 20, 251.Nuffer, J., D.C. Lupascu, J. Rdel, and M. Schroeder, 2001,

Appl. Phys. Lett. 79, 3675.Ogawa, Y., H. Yamada, T. Ogasawara, H. Yamada, and

H. Yamada, 2003, Phys. Rev. Lett. 90, 217403.Okuyama, M., H. Sugiyama, T. Nakaiso, and M. Noda, 2001,

Integ. Ferroelec. 34, 1477.Orihara, H., S. Hashimoto, and Y. Ishibashi, 1994, J. Phys.

Soc. Japan 63, 1031.Ottow, S., V. Lehmann, and H. Fo , 1996, Appl. Phys. A 63,

153.Outzourhit, A., A. Naziripour, J.U. Trefny, T. Kito, B. Yarar,

R. Yandrofski, J.D. Cuchiaro, and A.M. Hermanna 1995,Integ. Ferroelec. 8, 227.

Padilla, J., and D. Vanderbilt, 1997, Phys. Rev. B 56, 1625.Padilla, J., and D. Vanderbilt, 1998, Surf. Sci. 418, 64.Pan, M.J., S.F. Park, C.W. Park, K.A. Markowski., S.

Yoshikawa, and C.A. Randall, 1996, J. Amer. Ceram. Soc.79, 2971.

Park, J.D., and T.S. Oh, 2001, Integ. Ferroelec. 34, 1561.Park, C. H., 2003, J. Kor. Phys. Soc. 42, 1420.Park, C. H., and D. J. Chadi, 1998, Phys. Rev. B 57, 13961.Park, S. E., and T. R. Shrout, 1997, J. Appl. Phys. 82, 1804.Paruch, P., T. Tybell, and J.-M. Triscone, 2001, Appl. Phys.

Lett. 79, 530.Patzke, G.R., F. Krumeich, and R. Nesper, 2002, Angew.

Chem. Int. Ed. 41, 2446.Peacock, P.W., and J. Robertson, 2002, J. Appl. Phys. 92,

4712.Pertsev, N.A., J. Rodriguez Contreras, V.G. Kukhar, B. Her-

manns, H. Kohlstedt, R. Waser, Appl. Phys. Lett. 83, 3356.Pertsev, N. A., V. G. Kukhar, H. Kohlstedt, and R. Waser,

2003, Phys. Rev. B 67, 054107.Pertsev, N. A., A. K. Tagantsev, and N. Setter, 2000a, Phys.

Rev. B 61, 825.Pertsev, N. A., A. K. Tagantsev, and N. Setter, 2000b, Phys.

Rev. Lett. 84, 3722.Pertsev, N. A., A. G. Zembilgotov, and A. K. Tagantsev, 1998,

Phys. Rev. Lett. 80, 1988.Pertsev, N. A., A. G. Zembilgotov, and A. K. Tagantsev, 1999,

Ferroelectrics 223, 79.Petroff, P.M., A. Lorke, and A. Imamoglu, 2001, Physics To-

day 54, 46.Piskunov, S., E. Heifets, and R. I. E. et al., 2004, Comp. Mat.

Sci. 29, 165.Pokropivny, V.V., 2001, Physica C 351, 71.Poykko, S., and D. J. Chadi, 2000, J. Chem. Phys. Solids 61,

291.Rabe, K. M., UnpublishedRabe, K. M., and U. V. Waghmare, 1995, Phys. Rev. B 52,

13236.

Rabe, K. M., and U. V. Waghmare, 2002, Phys. Rev. B 65,214111.

Rabson, T.A., T.A. Rost, and H. Lin, 1995, Integ. Ferro. 6,15.

Ramesh, R., T. Sands, and V.G. Keramidas, 1993, Appl.Phys. Lett 63, 731.

Randoshkin, V.V., 1995, Fiz. Tverd. Tela. 37, 3056.Rao, F., M. Kim, A. J. Freeman, S. Tang, and M. Anthony,

1997, Phys. Rev. B 55, 13953.Reiner, J. W., F. J. Walker, R. A. McKee, C. A. Billman,

J. Junquera, K. M. Rabe, and C. H. Ahn, UnpublishedResta, R., 1994, Rev. Mod. Phys. 66, 899.Resta, R., 2003, Mod. Sim. Mat. Sci. Eng. 11, R69.Rhoderick, E.H., and R.H. Williams 1988, Metal-

semiconductor contacts, 2nd Ed. ( Oxford,Clarendon)Rios, S., A. Ruediger, J. Q. Jiang, J. F. Scott, H. Lu, and

Z. Chen, 2003, J. Phys. Cond. Matt. 15, 305.Robblee, L.S and S.F. Cogan 1986, in Enc. Mat. Sci. and

Eng., Suppl. Vol 1, edited by R.W. Cahn (Pergamon, Ox-ford), pp. 276.

Robels, U., G. Arlt, 1993, J. Appl. Phys. 73, 3454.Robertson, J., 2003, J. Appl. Phys. 93, 1054.Robertson, J., and C. W. Chen, 1999, Appl. Phys. Lett. 74,

1168.Robertson, J., C. W. Chen, and W. L. W. et al., 1996, Appl.

Phys. Lett. 69, 1704.Rodriguez Contreras, J., H. Kohlstedt, U. Poppe, and R.

Waser, 2003, Appl. Phys. Lett 83, 4595.Roelofs, A., N.A. Pertsev, R. Waser, F. Schlaphof, L.M. Eng,

C. Ganpule V. Nagarajan, and R. Ramesh, 2002, Appl.Phys. Lett. 80, 1424.

Roelofs, A., T. Schneller, K. Szot, and R. Waser, 2003, Nan-otechnology 14, 250.

Rossetti, G.A., L.E. Cross, and K. Kushida, 1991, Appl. Phys.Lett. 59, 2524.

Roytburd, A. L., S. P. Alpay, L. A. Bendersky, V. Nagarajan,and R. Ramesh, 2001, J. Appl. Phys. 89, 553.

Ruediger, A., T. Schneller, A. Roelofs, S. Tiedke, T. Schmitz,and R. Waser, Appl. Phys. A (In Press), Nanosize ferro-electric oxides-tracking down the superparaelectric limit.

Rudd, R.E., G.A.D. Briggs, A.P. Sutton, G. Medeiros-Ribeiro, and R. Stanley Williams, 2003, Phys. Rev. Lett.90, 146101.

Ruini, A., R. Resta, and S. Baroni, 1998, Phys. Rev. B 57,5742.

Saghi-Szabo, G., R. E. Cohen, and H. Krakauer, 1998, Phys.Rev. Lett. 80, 4321.

Sai, N., B. Meyer, and D. Vanderbilt, 2000, Phys. Rev. Lett.84, 5636.

Sai, N., and A. M. Rappe, UnpublishedSai, N., and D. Vanderbilt, 2000, Phys. Rev. B 62, 13942.Sajeev, J, and K Busch, 20 June 2002, US Patent Number

US2002074537.Sakamaki, Set al. 2001, US Patent Number 20010412.Samara, G. A., 1987, Ferroelectrics 73, 145.Samara, G.A., 2003, J. Phys. Condens. Matter 15, R367.Sarin Kumar, A.K., P. Paruch, J.-M. Triscone W. Daniau, S.

Ballandras, L. Pellegrino, D. Marr, and T. Tybell, 2004,Appl. Phys. Lett 85, 1757.

Saunders, V. R., R. Dovesi, C. Roetti, M. Causa, N. M. Har-rison, R. Orlando, and C. M. Zicovich-Wilson, 1999 (Uni-versity of Torino).

Sawyer, C.B., and C.H. Tower, 1930, Phys. Rev. 35, 269.Sayer, M, A Mansingh, AK Arora, and A Lo, 1992, Integ

46

Ferroelec 1, 129.Schilling, J., F. Muller, S. Matthias, R.B. Wehrspohn, U.

Gosele, and K. Busch, 2001, Appl. Phys. Lett 78, 1180.Schloss, L.F., P.C. McIntyre, B.C. Hendrix, S.M. Bilodeau,

J.F. Roeder, and S.R. Gilbert, 2002, Appl. Phys. Lett. 81,3218.

Schloss, L.F., H. Kim, and P.C. McIntyre, 2004, J. Mat. Res.19, 1265.

Schluter, M., 1978, Phys. Rev. B 17, 5044.Schnupp, P., 1967, Phys. Stat. Sol. 21, 567.Schottky, W., 1938, Naturewissenschaften 26, 843.Scott, J.F., 1977, Phys. Rev. B 16, 2329.Scott, J.F., 1979, Rep. Prog. Phys. 42, 1055.Scott, J.F., L. Kammerdiner, M. Parris, S. Traynor, V. Ot-

tenbacher, A. Shawabkeh, and W.F. Oliver, 1988, J. Appl.Phys. 64, 787.

Scott, J.F., and B. Pouligny, 1988, J. Appl. Phys 64, 1547.Scott, J.F., C.A. Araujo, B.M. Melnick, L.D. McMillan, and

R.Zuleeg, 1991, J. Appl. Phys. 70, 382.Scott, J.F., M. Azuma, E. Fujii, T. Otsuki, G. Kano, M.C.

Scott, C.A. Paz de Araujo, L.D. McMillan, and T. Roberts,1992, Proc ISAF (IEEE, New York, 1992) 356.

Scott, J.F., B.M. Melnick, L.D. McMillan, and C.A. Paz deAraujo, 1993, Integ. Ferroelec. 3, 225.

Scott, J.F., 1995, Science and Technology of ElectroceramicThin Films, ed. O. Auciello and R. Waser NATO ASI Se-ries 284 (Kluwer, Dordrecht)

Scott, J.F., 1996, Integ. Ferro 12, 71.Scott, J.F., 1998, Ferroelectrics Rev. 1, 82.Scott, J.F., 2000, Ferroelectric Memories (Springer-Verlag,

Berlin)Scott, J.F, 2000, J. Appl. Phys 88, 6092.Scott, J.F., and M. Dawber, 2000, Appl. Phys. Lett. 76, 3801.Scott, J.F., A.Q. Jiang, S.A.T. Redfern, M. Zhang, and M.

Dawber, 2003, J. Appl. Phys. 94, 3333.Seifert, A., A. Vojta, J.S. Speck, and F.F. Lange, 1996, J.

Mater. Res. 11, 1470.Sepliarsky, M., S. R. Phillpot, M. G. Stachiotti, and R. L.

Migoni, 2002, J. Appl. Phys. 91, 3165.Sepliarsky, M., S. R. Phillpot, D. Wolf, D. Wolf, and D. Wolf,

2001, J. Appl. Phys. 90, 4509.Sepliarsky, M., Z. Wu, and A. A. and, 2004, Ferroelectrics

301, 55.Seshadri R. and N.A. Hill, 2001, Chem. Mater. 13, 2892.Shchukin, V.A., and D. Bimberg, 1999 , Rev. Mod. Phys. 71,

1125.Shimada, Y., K. Arita, Y. Kato, K. Uchiyama, V. Joshi, and

M. Lim, 2001, Integ. Ferroelec. 34, 1467.Shimakawa, Y., Y. Kubo, Y. Nakagawa, T. Kamiyama, H.

Asano, and F. Izumi, 1999, Appl. Phys. Lett. 74, 1904.Shimizu, M., H. Fujisawa, H. Nonomura, and H. Niu, 2003,

ISIF 2003 Colorado SpringsShimuta, T., O. Nakagawara, T. Makino, T. Makino, and

T. Makino, 2002, Jap. J. Appl. Phys. 91, 2290.Shin, C.-H., S.Y. Cha, H.C. Lee, W.-J. Lee, B.-G. Yu, and

D.-H. Kwak, 2001, Integ. Ferroelec. 34, 1553.Shur, V. Ya., A. Gruverman, V.P. Kuminov, and N.A.

Tonkachyova, 1990, Ferroelectrics 111, 197.Shur, V.Ya, A.L. Gruverman, N.Yu. Ponomarev, E.L.

Rumyantsev, and N.A. Tonkacheva, 1991, JETP Lett. 53,615.

Shur, V. Ya., 1996, in Ferroelectric Thin Films: Synthesis andBasic Properties, edited by C. Paz de Araujo,G.W. Tay-lor,J.F. Scott (Gordon and Breach, Amsterdam), pp. 153 -

192.Sigman, J., D. P. Norton, H. M. Christen, P. H. Fleming, and

L. A. Boatner, 2002, Phys. Rev. Lett. 88, 079601.Simmons, J.G., 1965, Phys. Rev. Lett 15, 967.Singh, D. J., 1995, Phys. Rev. B 52, 12559.Sinharoy, S., H. Buhay, M.H. Francombe, W.J. Takei, N.J.

Doyle, J.H. Rieger, D.R. Lampe, and E. Stepke, 1991, J.Vac. Sci. Technol. A 9, 409.

Sinharoy, S., D.R. Lampe, H. Buhay, and M.H. Francombe,1992, Integ. Ferroelec. 2, 377.

Sinharoy, S., H. Buhay, M.H. Francombe, and D.R. Lampe,1993, Integ. Ferroelec. 3, 217.

Smith, R.L. and S.D. Collins, 1992, J. Appl. Phys. 71, R1.Smyth, D.M, 1984, Prog. Sol. Stat. Chem. 15, 145.Soler, J. M., E. Artacho, J. D. Gale, A. Garcia, J. Junquera,

P. Ordejon, and D. Sanchez-Portal, 2002, J. Phys. Cond.Matt. 14, 2745.

Souza, I., J. Iniguez, and D. Vanderbilt, 2002, Phys. Rev.Lett. 89, 117602.

Speck, J. S., and W. Pompe, 1994, J. Appl. Phys. 76, 466.Stachiotti, M. G., C. O. Rodriguez, C. Ambrosch-Draxl, and

N. E. Christensen, 2000, Phys. Rev. B 61, 14434.Stachiotti, M. G., 2004, Appl. Phys. Lett 84, 251.Steinhart, M. , Z. Jia, A.K. Schaper, R.B. Wehrspohn, U.

Gsele, and J.H. Wendorff, 2003, Adv. Mat. 15, 706.Stemmer, S., S.K. Streiffer, F. Ernst, and M. Ruhle, 1995,

Phil. Mag. A 71, 713.Stolichnov, I., A.K. Tagantsev, E.L. Colla, and N. Setter,

1998, Appl. Phys. Lett. 73, 1361.Stolichnov, I., and A.K. Tagantsev, 1998, J. Appl. Phys. 84,

3216.Stolichnov, I., A.K. Tagantsev, E.L. Colla, and N. Setter,

1999, Ferroelectrics. 225, 125.Stolichnov, I., A. Tagantsev, N. Setter, S. Okhonin, P. Fazan,

J.S. Cross, M. Tsukada, A. Bartic, and D. Wouters, 2001,Integrated Ferroelectrics 32, 737.

Streiffer, S. K., J. A. Eastman, D. D. Fong, C. Thompson,A. Munkholm, M. V. R. Murty, O. Auciello, G. R. Bai,and G. B. Stephenson, 2002, Phys. Rev. Lett. 89, 067601.

Sugiyama, H., K. Kodama, T. Nakaiso, M. Noda, and M.Okuyama, 2001, Integ. Ferroelec. 34, 1521.

Sun, H. P., W. Tian, X. Q. Pan, J. H. Haeni, and D. G.Schlom, 2004, Appl. Phys. Lett. 84, 3298.

Suzuki, T., Y. Nishi, and M. Fujimoto, 1999, Phil. Mag. A79, 2461.

Suzuki, T., and M. Fujimoto, 2001, J. Appl. Phys. 89, 5622.Switzer, J.A., M.G. Shumsky, and E.W. Bohannan, 1999, Sci-

ence 284, 293.Szafraniak, I., C. Harnagea, R. Scholz, S. Bhattacharyya, D.

Hesse, and M. Alexe, 2003, Appl. Phys. Lett. 83, .Sze, S.M., D.J. Coleman, and A. Loya, 1971, Solid State Elec-

tron. 16, 85.Tabata, H., H. Tanaka, and T. Kawai, 1994, Appl. Phys. Lett.

65, 1970.Tagantsev, A.K., I. Stolichnov, E.L. Colla, and N. Setter,

2001, J. Appl. Phys 90, 1387.Tagantsev, A.K., I. Stolichnov, N. Setter, J.S. Cross, and M.

Tsukada, 2002, Phys. Rev. B 66, 214109.Takahashi, K. S., M. Kawasaki, and Y. Tokura, 2001, Appl.

Phys. Lett. 79, 1324.Taylor, D.V., and Damjanovic D., 1997, J. Appl. Phys 82,

1973.Taylor, D.V., and D. Damjanovic, 1998, Appl. Phys. Lett 73,

2045.

47

Tian, W., J. C. Jiang, X. Q. Pan, J. H. Haeni, D. G. Schlom,J. B. Neaton, and K. M. Rabe, Unpublished, Structural ev-idence for enhanced polarization in a commensurate short-period batio3 / srtio3 superlattice.

Tinte, S., and M. G. Stachiotti, 2000, Amer. Inst. Phys. Conf.Proc. 535, 273.

Tinte, S., M. G. Stachiotti, M. Sepliarsky, R. L. Migoni, andC. O. Rodriguez, 1999, J. Phys. Cond. Matt. 11, 9679.

Tinte, S., and M. G. Stachiotti, 2001, Phys. Rev. B 64,235403.

Troeger, G, UnpublsihedTsai, F., V. Khiznishenko and J.M. Cowley, 1992, Ultrami-

croscopy 4, 5.Tsai, M. H., Y. H. Tang, and S. K. Dey, 2003, J. Phys. Cond.

Matt. 15, 7901.Tsurumi, T., S-M Num, Y-B Kil, S. Wada, Ferroelectrics 259,

43.Tybell, T., P. Paruch, T. Giamarchi, and J.-M. Triscone,

2002, Phys. Rev. Lett. 89, 097601.Ueda, K., H. Tabata, and T. Kawai, 1998, Science 280, 1064.Ueda, K., H. Tabata, and T. Kawai, 1999a, Jap. J. Appl.

Phys. 38, 6690.Ueda, K., H. Tabata, and T. Kawai, 1999b, Phys. Rev. B 60,

12561.Ullmann, M., H. Goebel, H. Hoenigschmid, and T. Haneder,

2001, Integ. Ferroelec. 34, 1595.Van Aken, B.B., Palstra T.T.M, Filippetti A., and Spaldin

N.A., 2004, Nature Materials 3, 164.Vasco, E., R. Dittmann, S. Karthauser, and R. Waser, 2003,

Appl. Phys. Lett. 82, 2497.Vendik, O.G., and S.P. Zubko, 2000, J. Appl. Phys. 88, 5343.Von Hippel, A., 1935, Ergeb. Exact Naturwiss 14, 118.Waghmare, U. V., and K. M. Rabe, 1997a, Phys. Rev. B 55,

6161.Waghmare, U. V., and K. M. Rabe, 1997b, Ferroelectrics 194,

135.Walker, L.R., 1963, in Magnetism III, edited by G.T. Rado

and H. Suhl (New York), p. 450.

Wang, J et al. 2003, Science 299, 1719.Warusawithana, M. P., E. V. Colla, J. N. Eckstein, and M. B.

Weissman, 2003, Phys. Rev. Lett. 90, 036802.Waser, R., T. Baiatu, and K-H. Hartdl, 1990, J. Am. Ceram.

Soc. 73, 1654.Waser, R., and D.M. Smyth, 1996, in Ferroelectric Thin

Films: Synthesis and Basic Properties, edited by C. Paz deAraujo,G.W. Taylor,J.F. Scott (Gordon and Breach, Ams-terdam), pp. 47-92.

Watanabe, K., A.J. Hartmann, R.N. Lamb, and J.F. Scott,1998, J. Appl. Phys. 84, 2170.

Wieder, H.H., 1956, J. Appl. Phys. 27, 413.Wieder, H.H., 1957, J. Appl. Phys. 28, 367.Williams, R. Stanley, G. Medeiros-Ribeiro, T.I. Kamins,

D.A.A. Ohlberg, 2000, Annual Rev. Phys. Chem. 51, 527.Woodward, D.I., I.M. Reaney, G.Y. Yang, E.C. Dickey, C.A.

Randall, 2004, Appl. Phys. Lett. 84, 4650.Wu, Z., R. E. Cohen, and D. J. Singh, 2004, Phys. Rev. B

70, 104112.Wu, Z., and H. Krakauer 2003, Phys. Rev. B. 68, 014112.Xiaohua, L., J. Yin, Z.G. Liu, L. Wang, J. Li, X.H. Zhu, K.J.

Chen, 2001, Integ. Ferroelec. 34, 1571.Yamada, H., M. Kawasaki, Y. Ogawa, Y. Ogawa, and

Y. Ogawa, 2002, Appl. Phys. Lett. 81, 4793.Yoo, I.K., and S.B. Desu, 1992, Phys. Status Solidi A 133,

565.

Yoon, S.-M., and H. Ishiwara, 2001, IEEE Transactions onElectron Devices 48, 2002.

Yu, B., C. Zhu, and F. Gan, 1997, J. Appl. Phys 82, 4532.Zafar, S., R.E. Jones, B. Jiang, B. White, V. Kaushik, and S.

Gillespie, 1998, Appl. Phys. Lett 73, 3533.Zafar, S., B. Hradsky, D. Gentile, P. Chu, R.E. Jones, and S.

Gillespie, 1999, J. Appl. Phys. 86, 3890.Zhong, W., D. Vanderbilt, and K. M. Rabe, 1994, Phys. Rev.

Lett. 73, 1861.Zhou, W., 1992, J. Sol. St. Chem. 101, 1.