A Continuum Theory of Deformable, Semiconducting ... · PDF fileA Continuum Theory of Deformable, Semiconducting Ferroelectrics ... vacancies in barium titanate. ... A Continuum Theory

Digital Object Identifier (DOI) 10.1007/s00205-007-0096-yArch. Rational Mech. Anal. 189 (2008) 59–95

A Continuum Theory of Deformable,Semiconducting Ferroelectrics

Yu Xiao & Kaushik Bhattacharya

Communicated by R. D. James

Abstract

Ferroelectric solids, especially ferroelectric perovskites, are widely used assensors, actuators, filters, memory devices, and optical components. While thesehave traditionally been treated as insulators, they are in reality wide-band-gap semi-conductors. This semiconducting behavior affects the microstructures or domainpatterns of the ferroelectric material and the interaction of ferroelectrics with elec-trodes, and is affected significantly by defects and dopants. In this paper, we developa continuum theory of deformable, semiconducting ferroelectrics. A key idea is tointroduce space charges and dopant density as field (state) variables in additionto polarization and deformation. We demonstrate the theory by studying oxygenvacancies in barium titanate. We find the formation of depletion layers, regions ofdepleted electrons, and a large electric field at the ferroelectric–electrode bound-ary. We also find the formation of a charge double layer and a large electric fieldacross 90 domain walls but not across 180 domain walls. We show that theseinternal electric fields can give rise to a redistribution or forced diffusion of oxygenvacancies, which provides a mechanism for aging of ferroelectric materials.

1. Introduction

Ferroelectric perovskites are widely used in solid-state devices for their diverseproperties [14,25,34,49]. For example, lead zirconate titanate (PZT) is used inultrasonic transducers and actuators for its piezoelectric properties, lithium niobatein optical devices for its electro-optical properties, and barium strontium titanate(BST) in capacitors for its large dielectric constant. Until recently, many of theseapplications used only the linear response of ferroelectric materials. However, bycarefully controlling the domain patterns and switching processes in these materi-als, researchers have now begun to exploit their nonlinear properties as well [6,34].Thus, it has become important to understand the domain patterns and their switchingbehavior.

60 Yu Xiao & Kaushik Bhattacharya

A ferroelectric material is nonpolar (paraelectric) above its Curie temperature,but spontaneously polarized (ferroelectric) below it. Along with the spontaneouspolarization, there is usually a spontaneous distortion of the low-temperature phasecompared to the high-temperature phase. Further, the polarized state is less sym-metric than the nonpolar state and this gives rise to symmetry-related variants:crystallographically and energetically identical states that are oriented differentlywith respect to the parent nonpolar state. These variants can coexist as domainsseparated by domain walls. As an example, barium titanate (BaTiO3), an exten-sively investigated ferroelectric material, is cubic and nonpolar above its Curietemperature (120), and is tetragonal and spontaneously polarized along the 〈100〉direction at room temperature and therefore has six variants. These variants canform patterns involving 180 and 90 domain walls (see, for example, [25,33]).

Since symmetry-related variants are energy-equivalent, it is possible to switchone domain to another by the application of suitable electrical, mechanical or opti-cal loading. This gives rise to interesting applications. For example, nonvolatilememories are based on 180 domain switching [3,34]. The two opposite directionsof polarization represent the two logic states, and they can be switched from oneto another through the application of an electric field. Further, since both statesare stable, this information is unaltered even when the field is switched off. Largeactuation can be generated by non-180 domain switching, since such a switchis usually accompanied by a change of distortion. By applying a constant com-pressive stress and a cyclic electric field, Burcsu et al. [7,8] demonstrated 1%strain through 90 domain switching in barium titanate. Other examples of the roleof domain switching include poling, the extrinsic component of the piezoelectricresponse, and holographic storage.

These applications have motivated many theoretical investigations of domainpatterns and domain switching using a continuum (phase-field) model (see, forexample, [1,22,51,52]). These follow the theoretical framework going back toDevonshire [15–17] and Toupin [39], and treat the ferroelectric material as aninsulator. However, ferroelectric perovskites are wide-band-gap semiconductors[26,37]: the band-gap for BaTiO3 is about 3.0 eV, and that for PbZr0.40Ti0.60O3is about 3.4 eV. This has important consequences in practice, for example, it iswell known that the fatigue life and dielectric breakdown of a ferroelectric mate-rial are affected by the type of electrode one uses [13,31,34]. The semiconductingnature gives rise to a region of electron depletion close to the ferroelectric–electrodeinterface (the so-called depletion layer), and this depends on the particular materialcombination. Further, dopants including impurities and oxygen vacancies can affectthe switching behavior of these materials even in relatively dilute compositions. Inparticular, it has been recognized that defects often decorate domain walls [10,36].Furthermore, ferroelectric materials display an aging behavior where they developa memory of a domain pattern in which they have been held for a long time [32].Finally, the domain patterns of ferroelectric materials can be manipulated usinglight through the generation of photoelectrons [4]. We refer the reader to [46] fora detailed review of the experimental literature. There have been a few attemptsto study ferroelectrics as semiconductors. However, they either treat the polariza-tion distribution as frozen and calculate the space charges [41–43,45] or they treat

A Continuum Theory of Deformable, Semiconducting Ferroelectrics 61

the space charges as frozen and calculate the polarization [5,40]. Unfortunately,the polarization and space charges interact nontrivially through the electrostaticpotential. Therefore they require a uniform treatment, and this is the goal of thispaper.

In this paper, we revisit the continuum theory of ferroelectric materials. Thekey step—and departure from other work—is to treat the ferroelectric as a semi-conducting material in a unified manner by introducing the space charge densityand dopant density as field or state variables in addition to polarization and defor-mation. We provide the detailed kinematic description in Sections 2.1 and 2.2. Inthis presentation, we limit ourselves to a situation with a single species of donordopants for simplicity. This is appropriate for many ferroelectrics where oxygenvacancies are the predominant dopant. However, we note that the framework wedevelop can easily be generalized to multiple interacting dopants, some donorsand some acceptors, as suggested by recent observations [28]. The space chargesand polarization give rise to electric fields, and this requires some care in light ofthe finite deformation and possible jumps. This is described in Section 2.3. Wethen compute the rate of dissipation—defined as the difference between the rateof external working and the rate of change of stored energy—in a ferroelectric–conductor system in Section 2.4. We show that we can write this as a sum of theproducts between generalized forces (rate-independent quantities) and generalizedvelocities (rates). This allows us to derive the governing equations in Section 2.5.We discuss special cases in Section 2.6.

We examine the specific example of a single-crystal barium titanate slab coatedwith two platinum electrodes in Section 3. We assume that the barium titanateis doped with oxygen vacancies. In order to do so, we specialize to infinitesimaldisplacements and a particular additive assumption on the stored energy density.We conduct detailed two-dimensional numerical calculations using a finite-elementmodel of two problems, a slab with a 180 domain wall and a slab with a 90 domainwall in Section 3.2. We assume that the oxygen vacancies are distributed uniformlyand do not diffuse, and that the two electrodes are shorted. The calculations revealthe formation of depletion layers—layers with reduced charge and high electricfield—at the ferroelectric–electrode boundaries. They also show the formation ofcharge double layers and significant electric fields at the 90 domain wall, but notat the 180 domain wall. In short, the calculations reveal that there are significantinternal electric fields even with shorted electrodes. This is in marked contrast withthe classical continuum theories which would have predicted an almost uniformlyzero electric field in this situation. Using a simplified one-dimensional model, weexamine depletion layers and domain walls further in Sections 3.3 and 3.4, respec-tively. In these calculations, we allow the oxygen vacancies to diffuse within thespecimen while keeping their total number fixed. We find that they diffuse to theelectrodes, and argue that this can be a mechanism that promotes fatigue. We alsoshow that the oxygen vacancies diffuse to one side of 90 domain walls but havelong tails, and argue that this provides a mechanism for the observed aging.

The main results of this work were reported in Xiao et al. [47]. Further resultsand consequences of the one-dimensional problems will be presented elsewhere[48].


Fig. 1. A ferroelectric semiconducting system in an external field generated by conductorsCq and Cv. Cq , with fixed charge Q, is fixed in space by external forces, while Cv withfixed potential deforms with the ferroelectric body Ω

2. Continuum model

2.1. Kinematics

Consider a ferroelectric semiconducting crystal in an external field, as shownin Fig. 1. It occupies a region Ω ⊂ R

3 in the reference configuration. We find itconvenient to choose the undistorted nonpolar phase at the Curie temperature asour reference configuration. A deformation y : Ω → R

3 brings it to the proximityof electrodes Cv ⊂ R

3 with fixed potential φ and Cq ⊂ R3 with fixed charge Q

under the action of traction t. The deformation gradient is F = ∇xy, and we assumethat the deformation is invertible and that J = det F > 0 almost everywhere inΩ . We denote by p : y(Ω) → R

3 the polarization of ferroelectric material perunit deformed volume, and by p0 : Ω → R

3 the polarization per unit undeformedvolume. We have

p0(x) = (det ∇xy(x))p(y(x)). (1)

We shall make the following assumption for later use: the conductor Cq is fixedin space but the conductor Cv deforms with the body with negligible elastic energy.This is reasonable since electrodes are usually very thin compared to the body.

2.2. Space charge density in semiconducting solids

The total charge density at any point in a semiconductor in the current config-uration is [2]

ρ = e(zNd − z′Na − nd − nc + pa + pv

), (2)

where Nd is the density of donors (number per unit deformed volume), Na thedensity of acceptors, nd the density of electrons in the donor band, nc the densityof electrons in the conduction band, pa the density of holes in the acceptor band,pv the density of holes in the valence band, z and z′ the valency of donors andacceptors, respectively, and e the coulomb charge per electron.


We assume that oxygen vacancies are the dominant impurities.1 Since oxy-gen vacancies act as donors, we may set Na = pa = 0 in Equation (2). Further,electrons in the conduction band and holes in the valence band have a much highermobility than defects, so we group defect-based charges and electronic chargesseparately. Thus,

ρ = e(zNd − nd − nc + pv) = ez f Nd + ρc (3)

with

f = zNd − nd

zNd(4)

and

ρc = e(pv − nc). (5)

Here, we call f the ionization ratio of defects since f represents the ratio of ionizeddefects to the total. Further, ez f Nd is exactly the amount of charges contributedby the defects: f = 0 means that no defects are ionized and all the extra electronsfrom defects/donors are bound in the donor’s level; f = 1 means that all defectsare ionized and all the electrons in the donors level are activated into the conductionband. We call ρc the free charge density since it is the charge contribution from theconduction and valence bands.

We define the counterparts of Nd, ρ, and ρc in the reference configuration asNd0, ρ0, and ρc0 (number per unit undeformed volume), respectively, in analogywith Equation (1), and

ρ0 = ez f Nd0 + ρc0. (6)

It is worth noting that f is independent of the choice of configuration.Assuming that no oxygen vacancies or charges are generated in the interior, we

have the following conservation principles:

Nd0 = −∇x · JNd0 , (7)

ρ0 = −∇x · Jρ0 , (8)

where JNd0 and Jρ0 are the flux of defects and charges in the reference configuration,respectively.

We point out here that the dot on Nd0 and ρ0 denotes the material time derivativeof Nd0 and ρ0, respectively. For any variable ξ defined on y(Ω),

ξ (y(x, t), t) = ∂ξ(y(x, t), t)

∂t

x= ∂ξ(y, t)

∂t

∣∣∣∣y+ v · ∇yξ(y, t), (9)

where v = ∂y(x,t)∂t |x is the particle velocity of the material point x, and we call

∂ξ(y,t)∂t |y the spatial time derivative of ξ and denote it by

oξ .

1 We could proceed analogously for any other dopant, or combination of dopants.


2.3. Electric field

The polarization and the space charges in the ferroelectric body as well asthe charges on the surfaces of conductors generate an electric field in all space.The electrostatic potential φ at any point in R

3 is obtained by solving Maxwell’sequation:

∇y · (−ε0∇yφ + pχ(y(Ω, t))) = ρχ(y(Ω, t)) in R3\(Cv ∪ Cq),

∇yφ = 0 on Cv ∪ Cq (10)

subject to

∫

∂Cq

∂φ

∂ndSy = − Q

ε0,

φ = φ on Cv,

φ → 0 as |y| → ∞, (11)

where ε0 is the permittivity coefficient of free space, and χ(D) is the characteristicfunction of domain D.

Precisely, φ ∈ H1(R3) satisfies the following:

−∫

R3

(−ε0∇yφ + pχ(y(Ω)) · ∇ψ dy =

∫

y(Ω)ρ ψ dy +

∫

∂Cv∪∂Cq

σ ψ dSy,

(12)∫

∂Cq

σ dSy = Q, (13)

φ = φ on Cv (14)

for eachψ ∈ H1(R3), where σ : ∂Cv ∪∂Cq → R measurable is the surface chargedensity on the interface, and it is defined as

σ = −ε0∇yφ + pχ (y(Ω)) · n. (15)

Here denotes the jump across an interface: ξ = ξ+ − ξ−, with ξ being somevariable defined in both domains; n is the unit norm of the interface, pointing fromD− to D+.

Although φ is continuous in R3, other quantities such as ∇yφ can be discon-

tinuous across some interfaces, as Equation (15) shows. Here we discuss somejump conditions in a more general setting for later use. In particular, we shall beinterested in time-dependent processes. So the polarization p and the deformationy could depend on time, and we solve Equations (12)–(14) at each time to find theelectric potential.

The jump condition across any interface separating D+ and D− is (Fig. 2),

−ε0∇yφ + p · n = σ. (16)


nv

D

D

y(x)

pt

nv

D

D

y(x)

p

t

Fig. 2. An interface separating D− from D+ in an electric field. D− is a dielectric or ferro-electric body with polarization p; D+ can be a conductor or vacuum; σ is the surface chargedensity on the interface; n is the unit norm of the interface, pointing from D− to D+; andv is the velocity of a material point on the interface

If we assume as shown in Fig. 2 that p = 0 in D+, and if p denotes the polarizationin D− (for example, D+ can be a conductor or vacuum and D− a dielectric orferroelectric body), then Equation (16) can be rewritten as

∇yφ · n = − 1

ε0p · n − σ

ε0. (17)

Now, let y(α) be a curve on the interface at time t0 parameterized by α. Wehave, from the continuity of φ,

φ−(y(α)) = φ+(y(α)). (18)

Differentiating with respect to α, we have

∇yφ · ∂ y∂α

= 0. (19)

Since this holds for any curve on the interface, we obtain continuity of ∇yφ alongthe tangent, that is,

∇yφ · t = 0 ∀ t · n = 0. (20)

Combining this with Equation (17), we obtain

∇yφ = −(

1

ε0p · n + σ

ε0

)n. (21)

Now consider a material point x on the interface. Let us assume that the inter-face does not propagate in the reference configuration, so that the particle velocityremains continuous across the interface. Since the electric potentialφ is continuous,we have

φ−(y(x, t), t) = φ+(y(x, t), t), (22)

so that ˙φ−(y(x, t), t) = ˙

φ+(y(x, t), t), (23)


or,o

φ− + ∇yφ− · v =

o

φ+ + ∇yφ+ · v. (24)

Recall that φ here denotes the material time derivative of φ, andoφ denotes the

spatial time derivative of φ. Hence,

oφ = −∇yφ · v, (25)

where v is the particle velocity of the material point x as defined in Section 2.2.Inserting Equation (21) into Equation (25) yields

oφ = 1

ε0(p · n)(v · n)+ σ

ε0(v · n). (26)

Another quantity we will use later is the Maxwell stress tensor, defined as

TM = E ⊗ D − ε0

2E · E I, (27)

where E = −∇yφ is the electric field and D = ε0E + pχ (y(Ω)) is the electricdisplacement.

The discontinuity of E or D across an interface leads to the discontinuity ofTM. By a direct calculation on an interface as above between a ferroelectric andvacuum or a conductor, we have

TMn = (E ⊗ D − ε0

2E · E I)n

= 〈E〉 D · n + E 〈D〉 · n − ε0(〈E〉 · E)n

= 〈E〉σ + E

(ε0〈E〉 · n + 1

2p · n

)− ε0(〈E〉 · E)n

=(

E− + E

2

)σ + ε0E〈E〉 · n + 1

2(p · n)E (28)

− ε0〈E〉 · ((E · n)

n)

n

=(

E− + E

2

)σ + 1

2(p · n)E

= E−σ + 1

2ε0(p · n + σ)2n. (29)

The second equality uses the identity

φ ψ = φ〈ψ〉 + 〈φ〉ψ, (30)

where

〈φ〉 = φ+ + φ−

2(31)

is the average of the limiting values of a discontinuous quantity φ. The third equal-ity recalls the definition of D and the assumption that p = 0 on D+, and the fourthequality uses the continuity of E = −∇yφ along the tangential direction. The lastequality is obtained by using Equation (21).


2.4. Rate of dissipation of the system

The rate of dissipation of the whole system D is defined as the differencebetween the rate of external working F and the rate of the change of the totalenergy dE/dt :

D = F − dEdt. (32)

2.4.1. Rate of external working The rate of external working F includes themechanical work done by external forces, the electric work done by the electrodes,and the chemical energy flux from Cv into Ω:

F = φd

dt

∫

y(∂Cv)

σ dSy +∫

y(∂sΩ)

t · v dSy

−∫

∂Ω

µNd0 JNd0 · m dSx −∫

∂Ω

µρc0 Jρc0 · m dSx ,

(33)

where µNd0 and µρc0 are, respectively, the chemical potential carried by the fluxof the oxygen vacancies Nd0 and that of charges ρc0,2 y(∂sΩ) is the part of theboundary in the current configuration on which traction t acts, dSy and dSx arethe differential area in the current and reference configuration, respectively, m isthe normal to the surface in the reference configuration, and n will denote its coun-terpart in the current configuration.

In light of Equation (6), and the fact that the activation/deactivation of donorshappens much faster than the defect diffusion, by further assuming that the electronsin the donor’s level diffuse with the defects with the same rate,3 we have

Jρ0 = ez f JNd0 + Jρc0 . (34)

Using the divergence theorem, Equation (7) and (8), and also noting that

ρ0 = ez f Nd0 + ez f Nd0 + ρc0 (35)

we can rewrite F as

F = φd

dt

∫

y(∂Cv)

σ dSy +∫

y(∂sΩ)

t · v dSy

−∫

∂Ω

(µNd0 − ez f µρc0

)JNd0 · m dSx −

∫

∂Ω

µρc0 Jρ0 · m dSx

= φd

dt

∫

y(∂Cv)

σ dSy +∫

y(∂sΩ)

t · v dSy

−∫

Ω

(∇x(µNd0 − ez f µρc0

)+ ez f ∇xµρc0

) · JNd0 + ∇xµρc0 · Jρc0

dx

+∫

Ω

µNd0 Nd0 + ezNd0µρc0 f + µρc0 ρc0

dx .

(36)

2 The exact meanings of µNd0 and µρc0 will be examined in Section 2.5.3 This is indeed a strong but reasonable assumption.


Fig. 3. A simplified version of Fig 1: Sv is the part of ∂y(Ω) with fixed potential

2.4.2. Total energy of the system The total energy of the system consists oftwo parts: the energy stored in the ferroelectric material and the electrostatic fieldenergy generated by external and internal sources, that is,

E =∫

Ω

W0 dx + ε0

2

∫

R3|∇φ|2 dx (37)

where W0 is the stored energy per unit reference volume in the ferroelectric material.We make the constitutive assumption that it depends on defect density Nd0, ioniza-tion ratio of defects f , free charge density ρc0, polarization p0, polarization gradient∇xp0, and deformation gradient∇xy, that is, W0 = W0(Nd0, f, ρc0,p0,∇xp0,∇xy).We require the stored energy density W0 to satisfy frame indifference and materialsymmetry.

Recall that the electrical potential φ is obtained by solving Equation (10) andits boundary conditions (Equation (11)). Recall also the assumption that Cq is fixedand that Cv is deformable with zero elastic energy. A better way to picture this isshown in Fig. 3, where we idealize Cv as an interface Sv = y(Sv0) between thevacuum and semiconductor y(Ω) on which the potential is fixed. We also denoteS f = ∂y(Ω)\Sv as the interface where y(Ω) has direct contact with the vacuum.

2.4.3. Rate of change of total energy The rate of change of the total energy inEquation (32) is,

dEdt

=∫

Ω

W0 dx+ d

dt

[1

2

∫

R3ε0|∇yφ|2 dy

]. (38)

We can directly calculate the first term on the right-hand side of Equation (38),

∫

Ω

W0(Nd0, f, ρc0,p0,∇xp0,∇xy) dx

=∫

Ω

(∂W0

∂Nd0Nd0 + ∂W0

∂ ff + ∂W0

∂ρc0ρc0

)dx

+∫

Ω

∂W0

∂p0· p0 −

[∇x ·

(∂W0

∂∇xp0

)]· p0

dx


+∫

∂Ω

(∂W0

∂∇xp0m)

· p0 dSx −∫

y(Ω)

[∇y ·

(1

J

∂W0

∂FFT)]

· v dy

+∫

∂y(Ω)

(1

J

∂W0

∂FFT n

)· v dSy. (39)

We have used the divergence theorem in the reference configuration to obtain thesecond and third integrals on the right, and the divergence theorem in the currentconfiguration to obtain the last two integrals. We clarify using indicial notation that

[∇x ·

(∂W0

∂∇xp0

)]· p0 =

(∂W0

∂p0I,J

)

,Jp0I ,

(∂W0

∂∇xp0m)

· p0 = ∂W0

∂p0I,Jm J p0I ,

[∇y ·

(1

J

∂W0

∂FFT)]

· v =(

1

J

(∂W0

∂Fi I

)F j I

)

, jvi ,

(1

J

∂W0

∂FFT n

)· v = 1

J

(∂W0

∂Fi I

)F j I vi n j

in Equation (39). Here, we use the summation convention where repeated indicesare summed.

The calculation of the second term in Equation (38), the change of electrostaticfield energy, needs some manipulation. The difficulty arises from the fact that theelectric energy exists in all space. So we follow a procedure similar to that used by[24] and divide the calculation into three steps in Sections 2.4.4–2.4.6. The finalresult is shown in Equation (49).

2.4.4. Rate of change of field energy: Step 1 First, by setting ψ = φ in Equa-tion (12), we have∫

R3ε0∇yφ · ∇yφ dy =

∫

y(Ω)∇yφ · p dy+

∫

y(Ω)φ ρ dy+

∫

Sv

φ σ dSy +∫

∂Cq

φ σ dSy .

(40)

Therefore,

d

dt

[∫

R3ε0|∇yφ|2 dy

]

= d

dt

∫

y(Ω)∇yφ · p dy +

∫

y(Ω)φ ρ dy

+ φ

d

dt

∫

Sv

σ dSy +∫

∂Cq

φ σ dSy

= d

dt

∫

Ω

∇yφ · p0 dx +∫

Ω

φ ρ0 dx

+ φ

d

dt

∫

Sv

σ dSy +∫

∂Cq

φ σ dSy

=∫

Ω

d

dt(∇yφ) · p0 dx +

∫

Ω

∇yφ · p0 dx +∫

Ω

φ ρ0 dx +∫

Ω

φ ρ0 dx

+φ d

dt

∫

Sv

σ dSy +∫

∂Cq

φ σ dSy


=∫

Ω

(∇yoφ + v(∇y∇yφ)) · p0 dx +

∫

Ω

∇yφ · p0 dx

+∫

Ω

(oφ + v · ∇yφ) ρ0 dx +

∫

Ω

φ ρ0 dx + φd

dt

∫

Sv

σ dSy +∫

∂Cq

φ σ dSy

=∫

y(Ω)(∇y

oφ + v(∇y∇yφ)) · p dy +

∫

y(Ω)(

oφ + v · ∇yφ) ρ dy

+∫

Ω

∇yφ · p0 dx +∫

Ω

φ ρ0 dx + φd

dt

∫

Sv

σ dSy +∫

∂Cq

φ σ dSy . (41)

Notice that we use the fact that Cq is the conductor with fixed charge Q in derivingthe second equality.

2.4.5. Rate of change of field energy: Step 2 Second, we multiplyoφ on both

sides of Maxwell equation (10), and integrate over R3 to obtain

∫

R3∇y · (−ε0∇yφ + pχ(y(Ω))

) oφ dy =

∫

R3ρχ(y(Ω))

oφ dy. (42)

We clarify that these integrals should be interpreted in the classical sense ratherthan in the sense of distributions. The left side of Equation (42) can therefore besplit into three parts on which the divergence theorem can be applied:

∫

R3∇y · (−ε0∇yφ + pχ(y(Ω))

) oφ dy

=∫

y(Ω)∇y · (−ε0∇yφ + p

) oφ dy +

∫

Cq

∇y · (−ε0∇yφ) oφ dy

+∫

R3\(y(Ω)∪Cq )

∇y · (−ε0∇yφ) oφ dy

= −∫

y(Ω)∇y

oφ · (−ε0∇yφ + p

)dy

+∫

S−v

oφ (−ε0∇yφ + p) · n dSy +

∫

S−f

oφ (−ε0∇yφ + p) · n dSy

−∫

Cq

∇yoφ · (−ε0∇yφ) dy +

∫

∂C−q

oφ (−ε0∇yφ) · nq dSy

−∫

R3\(y(Ω)∪Cq )

∇yoφ · (−ε0∇yφ) dy +

∫

∂C+q

oφ (−ε0∇yφ) · (−nq) dSy

+∫

S+v

oφ (−ε0∇yφ) · (−n) dSy +

∫

S+f

oφ (−ε0∇yφ) · (−n) dSy

=∫

R3ε0∇y

oφ · ∇yφ dy −

∫

y(Ω)∇y

oφ · p dy −

∫

∂Cq

oφ(−ε0∇yφ) · nq dSy

−∫

Sv

oφ(−ε0∇yφ + p) · n dSy −

∫

S f

oφ(−ε0∇yφ + p) · n dSy . (43)

Here, n, nq are the outward unit norms of y(Ω) and Cq , respectively, S−v and S−

f

are the inner surface of y(Ω), S−v and S+

f are the outer surfaces of y(Ω), and C−q

and C+q are the inner and outer surfaces of Cq , respectively.


Since Cq is fixed in space, andoφ = φ on Cq , Equation (42) and (43) lead to

∫

R3ε0∇yφ · ∇y

oφ dy =

∫

y(Ω)∇y

oφ · p dy +

∫

y(Ω)

oφ ρ dy +

∫

∂Cq

φ σ dSy

+∫

Sv

oφ(−ε0∇yφ + p) · n dSy

+∫

S f

oφ(−ε0∇yφ + p) · n dSy . (44)

Therefore, by using Reynolds’ transport theorem, we have

d

dt

[1

2

∫

R3ε0|∇yφ|2 dy

]

= ε0

2

∫

y(Ω)

∂

∂t

∣∣∇yφ∣∣2 dy + ε0

2

∫

R3\y(Ω)

∂

∂t

∣∣∇yφ∣∣2 dy

−ε0

2

∫

∂y(Ω)∣∣∇yφ

∣∣2 v · n dSy

=∫

R3ε0∇yφ · ∇y

oφ dy − ε0

2

∫

Sv+S f

∣∣∇yφ

∣∣2 v · n dSy

=∫

y(Ω)∇y

oφ · p dy +

∫

y(Ω)

oφ ρ dy +

∫

∂Cq

φ σ dSy

+∫

Sv

oφ(−ε0∇yφ + p) · n dSy − ε0

2

∫

Sv

∣∣∇yφ

∣∣2 v · n dSy

+∫

S f

oφ(−ε0∇yφ + p) · n dSy − ε0

2

∫

S f

∣∣∇yφ

∣∣2 v · n dSy . (45)

Let S = Sv⋃

S f = ∂y(Ω). Using the jump conditions (16), (21), (25), (26),and (29), we can simplify the last four terms in Expression (45):

∫

S

oφ (−ε0∇yφ + p) · n dSy − ε0

2

∫

S∣∣∇yφ

∣∣2 v · n dSy

=∫

S〈 oφ〉−ε0∇yφ + p · n dSy +

∫

S

oφ〈−ε0∇yφ + p〉 · n dSy

−ε0

∫

S

(〈∇yφ〉 · ∇yφ) (

v · n)

dSy

=∫

S〈 oφ〉 σ dSy +

∫

S

oφ〈p〉 · n dSy

−ε0

∫

S

oφ〈∇yφ〉 · n dSy − ε0

∫

S

(〈∇yφ〉 · ∇yφ) (

v · n)

dSy

=∫

S

[φ − v · 〈∇yφ〉] σ dSy + 1

2ε0

∫

S(p · n)2(v · n) dSy

+ 1

2ε0

∫

Sσ (p · n)(v · n) dSy


=∫

Sφ σ dSy −

∫

Sv ·[∇yφ

− + 1

2∇yφ

]σ dSy

+ 1

2ε0

∫

S(p · n)2(v · n) dSy + 1

2ε0

∫


=∫

Sφ σ dSy −

∫

Sσ v · ∇yφ

− dSy

+ 1

2ε0

∫

Sσ(p · n)(v · n) dSy + 1

2ε0

∫

Sσ 2(v · n) dSy

+ 1

2ε0

∫

S(p · n)2(v · n) dSy + 1

2ε0

∫


=∫

Sφ σ dSy −

∫

Sσ v · ∇yφ

− dSy + 1

2ε0

∫

S

(p · n + σ

)2(v · n) dSy

=∫

Sφ σ dSy +

∫

STMn · v dSy

=∫

STMn · v dSy . (46)

The last equality comes from the fact that φ = φ, thus φ = 0 on Sv, and σ = 0on S f .

Substituting Equation (46) into Equation (45), we have

d

dt

[1

2

∫

R3ε0|∇yφ|2 dy

]=∫

y(Ω)∇y

oφ · p dy +

∫

y(Ω)

oφ ρ dy

+∫

∂Cq

φ σ dSy +∫

Sv⋃

S f

TMn · v dSy .

(47)

2.4.6. Rate of change of field energy: Step 3 Now, subtracting Equation (47)from Equation (41), we obtain,

d

dt

[1

2

∫

R3ε0|∇yφ|2 dy

]

=∫

Ω

∇yφ · p0 dx +∫

Ω

φ ρ0 dx +∫

y(Ω)v · ((∇y∇yφ

)p)

dy (48)

+∫

y(Ω)v · ∇yφ ρ dy + φ

d

dt

∫

Sv

σ dSy −∫

∂y(Ω)TMn · v dSy .

2.4.7. Rate of dissipation: The final expression Putting together Equations (32),(35), (36), (38), (39), and (49), we now have the final expression for the rate of dis-sipation of the whole system:

D = F − dEdt

= −∫

Ω

(∇x(µNd0 − ez f µρc0

)+ ez f ∇xµρc0

) · JNd0 + ∇xµρc0 · Jρc0

dx


+∫

Ω

(µNd0 − ∂W0

∂Nd0− ez f φ

)Nd0 dx +

∫

Ω

(µρc0 − ∂W0

∂ρc0− φ

)ρc0 dx

+∫

Ω

(ezNd0µρc0 − ∂W0

∂ f− ezNd0φ

)f dx

+∫

Ω

[∇x ·

(∂W0

∂∇xp0

)− ∂W0

∂p0− F−T ∇xφ

]· p0 dx

−∫

∂Ω

(∂W0

∂∇xp0m)

· p0 dSx

+∫

y(Ω)

[∇y ·

(1

J

∂W0

∂FFT)

− (∇y∇yφ)

p − ρ∇yφ

]· v dy

−∫

∂y(Ω)

(1

J

∂W0

∂FFT n

)· v dSy

+∫

∂y(Ω)TM n · v dSy +

∫

y(∂sΩ)

t · v dSy . (49)

From Equation (49), we can see that the dissipation of the system has threecontributions: the first four integrals are the dissipation caused by the diffusionof vacancies and charges and the activation (deactivation) of electrons from (to)the donor band, the next two terms are the dissipation caused by the polarizationevolution, and the remaining terms are the contribution from the deformation ofthe ferroelectric body.

2.5. Governing equations

According to the second law of thermodynamics specialized to isothermal pro-cesses, which we are currently considering, the rate of dissipation D should alwaysbe greater or equal to zero. Notice that in the expression (49) for the rate of dis-sipation each term is a product of conjugate pairs: generalized velocity (time rateof change of some quantity or flux of some quantity) multiplied by a generalizedforce (a quantity that depends on the state and not the rate of change of the state).Therefore we may argue as [11] to obtain the governing equations. Specifically, byconsidering various processes that have the same state at some instant of time butdifferent rates, and insisting that D ≥ 0 for all these processes, we conclude that

∇x

(∂W0

∂∇xp0

)− ∂W0

∂p0− F−T ∇xφ = 0 in Ω, (50)

∂W0

∂∇xp0m = 0 on ∂Ω, (51)

∇y ·(

1

J

∂W0

∂FFT)

− (∇y∇yφ)

p − ρ∇yφ = 0 in y(Ω), (52)

1

J

∂W0

∂FFT n − TMn − tχ(y(∂sΩ)) = 0 on ∂y(Ω) (53)


and

µNd0 − ∂W0

∂Nd0− ez f φ = 0 in Ω, (54)

ezNd0µρc0 − ∂W0

∂ f− ezNd0φ = 0 in Ω, (55)

µρc0 − ∂W0

∂ρc0− φ = 0 in Ω, (56)

JNd0 · (∇x(µNd0 − ez f µρc0

)+ ez f ∇xµρc0

) ≤ 0 in Ω, (57)

Jρc0 · ∇xµρc0 ≤ 0 in Ω. (58)

Equations (50) and (51) are, respectively, the equilibrium equation of polari-zation and its boundary condition. Equation (52) is the force equilibrium equationwith boundary condition (53).

If we define the Cauchy stress tensor as

σ = 1

J

(∂W0

∂F

)FT , (59)

and also notice that

− (∇y∇yφ)

p − ρ∇yφ = −φ,i j p j − ρφ,i

= −φ,i j (D j + ε0φ, j )− φ,i D j, j

= −(φ,i j D j + φ,i D j, j )− ε0φ,i jφ, j

= (−φ,i D j ), j −(ε0

2φ,kφ,kδi j

)

, j

=(

Ei D j − ε0

2

∣∣∇yφ∣∣2 δi j

)

, j

= ∇y ·(

E ⊗ D − ε0

2E · E I

)

= ∇y · TM,

Equations (52) and (53) can then be rewritten as

∇y · (σ + TM) = 0 in y(Ω), (60)

σ n − TMn − tχ (y(∂sΩ)) = 0 on ∂y(Ω). (61)

From Equations (54), (55), and (56), we have

µNd0 = ∂W0

∂Nd0+ ez f φ, (62)

µρc0 = ∂W0

∂ρc0+ φ, (63)

∂W0

∂ f= ezNd0

∂W0

∂ρc0. (64)

Equation (62) tells us that the chemical potential of defects consists of two parts:a compositional contribution ∂W0

∂Nd0and a electrostatic contribution ez f φ. Similarly,


Equation (63) indicates that the chemical potential of free charges also has twoparts: a compositional contribution ∂W0

∂ρc0and an electrostatic contribution φ. Fur-

ther, Equation (64) implies that the ionization ratio of defects f and the free chargedensity ρc0 are indeed dependent on each other.

In order to satisfy Equations (57) and (58), we make the additional constitutiveassumption that

JNd0 = −K1[∇x

(µNd0 − ez f µρc0

)+ ez f ∇xµρc0

],

= −K1[∇xµNd0 − ezµρc0∇x f

](65)

Jρc0 = −K2∇xµρc0 (66)

for some positive-definite symmetric tensors K1, K2. Physically, K2 is the conduc-tivity of the solid and K1 is the diffusivity of defects.

Equations (62), (63), (64), (65), and (66), together with the continuity equa-tions (7) and (8), describe the two diffusion processes. The first one is the diffusionof oxygen vacancies,

Nd0 = ∇x ·

K1

[∇x

(∂W0

∂Nd0− ez f

∂W0

∂ρc0

)+ ez f ∇x

(∂W0

∂ρc0+ φ

)], (67)

and the second one is the diffusion of charges,

ρ0 = −∇x · (Jρc0 + ez f JNd0

)

= ∇x ·

K2∇x

(∂W0

∂ρc0+ φ

)+ ez f K1

[∇x

(∂W0

∂Nd0− ez f

∂W0

∂ρc0

)

+ ez f ∇x

(∂W0

∂ρc0+ φ

)]. (68)

Notice that f in Equations (67) and (68) is determined by Equation (64).In summary, Equations (50), (59), (60), (64), (67), and (68), plus Maxwell’s

equation (10) are the governing equations subject to the boundary conditions (11),(51), and (61), plus suitable boundary and initial conditions for Nd0 and ρ0.

2.6. Special cases

We now discuss some special cases.

2.6.1. Transient conduction The diffusion of free charges is usually much fasterthan the diffusion of defects (min λ(K2) max λ(K1), where λ(K) is an eigen-value of K). By choice, we can consider phenomena on timescales on which thereis no diffusion of defects but only transport of free charges. We do so by settingK1 = 0, so that Nd0 = 0 and Nd0 is fixed and given. Thus,

ρ0 = ∇x ·(

K2∇x

(∂W0

∂ρc0+ φ

)). (69)

In summary, in this case, we have Equations (50), (59), (60), (69), and (10) as thegoverning equations, plus Equations (51), (61), and (11) as their boundary condi-tions.


We point out that the real transportation of free carriers in semiconductors isusually more complicated than that described by Equation (69). In Equation (69),electrons and holes are assumed to have the same mobility, which is usually notthe case [2]. Further, the recombination dynamics of holes and electrons are notconsidered here. However, our derivation can be modified to account for thosephenomena.

2.6.2. Defect diffusion We now work on a time scale appropriate for the diffu-sion of defects. We do so by assuming K2 = ∞, so that the free charges ρc arealways in equilibrium, or Jρc0 = 0 and f = 0.

The assumption that K2 = ∞, Jρc0 = 0, together with Equation (66), leads to

µρc0 = ∂W0

∂ρc0+ φ = 0, (70)

which may be interpreted as a constitutive relation between the free charge densityand electric potential. If W0 is convex in ρc0, we can invert Equation (70) to obtainρc0 = ρc0(φ, Nd0, f,p0,∇xp0,∇xy). If we further assume that

W0 = Wd0(Nd0, f, ρc0)+ We0(p0,∇xp0,∇xy), (71)

we haveρ0 = ρ0(φ, Nd0) (72)

by using Equation (64).As an example we shall use later, for a typical semiconductor, the charge density

in the thermal equilibrium state is [38]:

ρ(φ, Nd) = −eNc F12

(Efm − Ec + eφ

KbT

)+ eNv F1

2

(Ev − eφ − Efm

KbT

)

+zeNd

⎛

⎝1 − 1

1 + 1g exp

(Ed−eφ−Efm

KbT

)

⎞

⎠

−z′eNa

⎛

⎝1 − 1

1 + 1g′ exp

(Efm+eφ−Ea

KbT

)

⎞

⎠ , (73)

where Nc and Nv are the effective density of states in the conduction and valenceband, respectively, Ec is the energy at the bottom of the conduction band, Ev isthe energy at the top of the valence band, Ed and Ea are the donor and acceptorlevels, respectively, Kb is the Boltzmann constant, T is the absolute temperature,F1

2is the Dirac-Fermi integral, Efm is the Fermi level of the semiconductor, and

g (g′) is the ground-state degeneracy of the donor (acceptor) level and equals 2for z(z′) = 1. The first two terms in Equation (73) calculate the electrons in theconduction band and the holes in valence band, respectively, while the last twoterms are the contributions from donors and acceptors. The relation (73) we givehere is a general relation including acceptors in the current configuration. For ourcase, Na = 0. We notice that Equation (64) is implicitly embodied in this relation:


the ionization ratio of defects f is a function of φ and equals the expression in thelarge brackets in the third term on the right-hand side, that is,

f (φ) = 1 −(

1 + 1

gexp

(Ed − eφ − Efm

KbT

))−1

. (74)

Finally, we point out that Efm of the semiconductor varies with its doping typeand dopant/defect level, and if the semiconduction is in contact with a metal, Efmshould be equal to the metal’s Fermi level.

Therefore, instead of solving the diffusion equation for ρ0, we only need tosolve the Maxwell equation (10) with ρ determined by Equation (73).

Furthermore, if we assume that the total number of defects is conserved in Ωand that the diffusivity of oxygen vacancies is isotropic (K1 = k1I), then the finaldiffusion equation we need to solve is

Nd0 − ∇x ·(βNd0∇x

(∂Wd0

∂Nd0+ ez f (φ)φ

))= 0 in Ω, (75)

where β = k1/Nd0 is the defect’s mobility, with an integral constraint

d

dt

∫

Ω

Nd0 dx = 0. (76)

We point out that, by splitting the stored energy functional W0 into two partsas in Equation (71), we ignore the effect of strain/stress on the concentration ofdefects Nd0. Consequently, the stress-induced diffusion of Nd0 is not considered.

In summary, we now need to solve Equations (10), (50), (59), (60), (73), (74),and (75), subject to the boundary conditions (11), (51), and (61), and an integralconstraint (76).

2.6.3. Steady state In this case, we assume K1 = 0 and K2 = ∞, that is, weassume that the defects are immobile and that free charges adjust themselves intothermal equilibrium in no time. Consequently, we only need to solve Equations (10),(50), (59), (60), and (73) subject to the boundary conditions (11), (51), and (61).

2.6.4. Polarization evolution The polarization equation (50) that we derived isfor the equilibrium state. One way of quickly deriving equations for polarizationevolution is to state that it has to overcome a dissipation of

∫Ωµp2

0 dx . Hence,

D = F − dEdt

−∫

Ω

µp20 dx ≥ 0. (77)

By recourse to Equation (49) and the argument by [11], we obtain,

( f [p0] − µp0) · p0 ≥ 0, (78)

where

f [p0] = ∇x ·(∂W0

∂∇xp0

)− ∂W0

∂p0− F−T ∇xφ. (79)


Consider any process p0(t) such that p0(t0) = p0, p0(t0) = q0 = 0. We havefor Equation (78) that ( f [p0] − µq0) · q0 ≥ 0. Now let us consider another processp0(t) with p0(t0) = p0, ˙p0(t0) = −q0. Then

− ( f [p0] + µq0) · q0 = − ( f [p0] − µq0) · q0 − 2µq20 ≥ 0,

which means( f [p0] − µq0) · q0 ≤ 0. (80)

In view of Equation (78) and (80), we therefore conclude:

f [p0] − µq0 = 0. (81)

Since p0 was an arbitrary process, we conclude that

µp0 = ∇x ·(∂W0

∂∇xp0

)− ∂W0

∂p0− F−T ∇xφ. (82)

Equation (82) is of gradient flow type, which has been widely used in simulatingthe phase evolution problems [1,12,20,21,51,52].

2.6.5. Infinitesimal displacement We have derived our force equilibrium equa-tion (60) from a general setting with finite deformations. However, searching for asuitable constitutive relation like Equation (59) and working with finite deforma-tion is a demanding task, even if we assume that the stored energy functional W hasa simple form such as the one in Equation (71). For some materials, like BaTiO3,a small-strain description often suffices.

The Devonshire-Ginzburg-Landau (DGL) energy [15–17], widely used inthe ferroelectric community, is such an example, which assumes that thepolarization- and deformation-related energy We

4 in Equation (71) depends onthe infinitesimal strain ε instead of the deformation gradient F, that is,

We = We(∇p,p, ε) (83)

with

ε = 1

2

(∇u + (∇u)T

)(84)

being the infinitesimal strain and u the displacement. The Maxwell stress is usuallyomitted in this setting, and the Cauchy stress is defined as

σ = ∂We(∇p,p, ε)∂ε

. (85)

Consequently, the force equilibrium equation (60) becomes

∇ · σ = 0 in Ω. (86)

4 Since we now consider small strain, there is no need to differentiate between the refer-ence and current configuration; the subscript 0 is therefore omitted.


BaTiO3 single crystal

Pt

Pt

y

xz

BaTiO3 single crystal

Fig. 4. Computation domains

However, the DGL energy We(∇p,p, ε) is not a genuine linearization ofWe(∇p,p,F). When an energy functional W is a function of F only, it is wellknown that frame indifference leads to W (F) = W (C), where C = FT F .= 1 + ε.However, when W = W (p,F), in general, we do not have W (p,F) = W (p,C),which means, when we linearize W using infinitesimal displacement gradient whilekeeping p finite, we do not obtain

W (p,F) .= W (p, ε). (87)

in general. Instead, W also depends on the antisymmetric part of ∇u. Only a com-plete linearization of both displacement gradient and polarization can lead to theusual piezoelectric constitutive relation:

σi j = Ci jklεkl + αi jk pk . (88)

Unfortunately, this is not suitable for ferroelectric materials. Fortunately, the DGLenergy is sufficient for most ferroelectric materials. For a detailed discussion onthis matter, we refer to [39].

3. Oxygen-vacancy doped barium titanate with platinum electrodes

We illustrate the theory above by considering a slab of barium titanate withplatinum electrodes in a parallel-plate capacitor geometry, as shown in Fig. 4. Thisis a common geometry in many applications. We assume that the barium titanate isa single crystal in the tetragonally polarized room-temperature phase, and that thecrystallographic [001]c direction coincides with the normal to the slab. We assumethat the material, polarization distribution, and other quantities are invariant alongthe z direction, and restrict ourselves to two dimensions. We also restrict ourselvesto infinitesimal displacements, as described in Section 2.6.5.

We assume that the stored energy function W takes a simple additive form:

W = Wd(Nd, f, ρc)+ Wg(∇p)+ Wp(p, ε). (89)

The first term is the energy of defects and charges. The second term, associ-ated with the polarization gradient, penalizes rapid changes of polarization. The


last term is the energy of deformation and polarization; it contains importantcrystallographic information. Finally, we assume that K2 = ∞ and K1 = k1I.Under these assumptions, the governing equations are now:

Nd − ∇ ·(βNd∇

(∂Wd

∂Nd+ ez f (φ)φ

))= 0 in Ω, (90)

µp − ∇ ·(

dWg

d∇p

)+ ∂Wp

∂p+ ∇φ = 0 in Ω, (91)

∇ · σ = 0 in Ω, (92)

∇ · (−ε0∇φ + pχ(Ω)) = ρ(φ, Nd)χ(Ω) in R3. (93)

with the constitutive Equations (59), (73), and (74), and subject to appropriateboundary conditions, initial conditions, and the constraint Equation (76).

3.1. Normalization and parameter selection

Following [50], we choose the stored energy of polarization gradient, polariza-tion, and deformation to be the Devonshire-Ginzburg-Landau energy [15–17]with slight modification. Specifically, we choose Wg to be

Wg(∇p) = a0

2|∇p|2 , (94)

where

|∇p|2 := ∇p · ∇p := Trace(∇p∇pT

)= p2

x,x + p2x,y + p2

y,x + p2y,y, (95)

and

Wp(p, ε) = a1

2(p2

x + p2y)+ a2

4(p4

x + p4y)+ a3

2p2

x p2y + a4

6(p6

x + p6y)

+a5

4(p4

x p4y)− b1

2(εxx p2

x + εyy p2y)− b2

2(εxx p2

y + εyy p2x )

−b3εxy px py + c1

2(ε2

xx + ε2yy)+ c2εxxεyy + c3

2ε2

xy . (96)

For computational purpose and for better interpretation of domain structures,the variables are normalized as follows:

x′ = xL0, t ′ = t

T0, W ′

g = Wg

c0, W ′

p = Wp

c0,

p′ = pp0, φ′ = φ

φ0, ρ′ = ρ/ρ0, σ ′ = σ

c0

(97)

with characteristic constants,

c0 = 1 GPa, p0 = 0.26 C/m2,

L0 = p0

√a0

c0, φ0 = √

a0c0, ρ0 =√

c0

a0, T0 = µp2

0

c0.

(98)


Here, p0 = 0.26 C/m2 is chosen to be the spontaneous polarization of BaTiO3at room temperature, so that the normalized spontaneous polarization is 1. Theresultant W ′

g and W ′p have the same forms as Wg and Wp, with ai , b j , and c j being

replaced, respectively, by a′i , b′

j and c′j , for i = 0, . . . , 5 and j = 1, 2, 3. Specif-

ically, a′0 = 1, a′

1 = a1 p02/c0, a′

2 = a2 p04/c0, a′

3 = a3 p04/c0, a′

4 = a4 p06/c0,

a′5 = a5 p0

8/c0, b′j = b j p0

2/c0, and c′j = c j/c0. We point out that specific choices

of c0 and L0 are made so that a′0 = 1, and both the domain wall thickness5 and the

normalized elastic moduli are of moderate range (10–100). This choice also hasthe feature that the normalized solution for the classical perfect crystal case doesnot depend on a0.

However, a0 does play a very important role in defected crystals, and thereforeit is important to decide the range of a0. From experimental data [36] and first-principle calculations [30], it is believed that the domain wall thickness is usu-ally about 1–10 nm, although thicknesses as large as 150 nm have been reported inLiNbO3 [44]. Here, we will work on two cases: a0 = 10−9 Vm3C−1, correspond-ing to a domain wall thickness of a few nanometers, and a0 = 10−7 Vm3C−1 fordomain walls one order thicker.

Other material constants we choose are [50]: c′1 = 185, c′

2 = 111, c′3 = 54,

b′1 = 1.4282, b′

2 = −0.185, b′3 = 0.5886, a′

1 = −0.007, a′2 = −0.009, a′

3 =0.003, a′

4 = 0.0261, and a′5 = 5. Notice that a′

1 and a′2 are both negative since

the cubic-to-tetragonal phase transition of BaTiO3 is a first order phase transition[30,35].

It is advantageous to rewrite W ′p as

W ′p(p

′, ε) = a′1

2(p′

x2 + p′

y2)+

(a′

2

4− d ′

)(p′

x4 + p′

y4)

+(

a′3

2− f ′

)p′

x2 p′

y2 + a′

4

6(p′

x6 + p′

y6)+ a′

5

4(p′

x4 p′

y4)

+1

2(ε − εs) · C′ (ε − εs), (99)

so that the normalized stress σ ′ = σ/c0 can be easily written as

σ ′ = C′(ε − εs), (100)

where C′ is the normalized stiffness matrix,

C′ =⎛

⎝c′

1 c′2 0

c′2 c′

1 00 0 c′

3

⎞

⎠ (101)

in Voigt notation, εs the eigenstrain caused by spontaneous polarization,

εs =⎛

⎜⎝

a′ p′x

2 − b′ p′y

2

b′ p′x

2 − a′ p′y

2

c′ p′x p′

y,

⎞

⎟⎠ (102)

5 The domain wall thickness is proportional to√

a0/|a1| [18].


and

a′ = b′1c′

1 − b′2c′

2

2(c′1

2 − c′2

2)

= 0.0065, (103)

b′ = b′2c′

1 − b′1c′

2

2(c′1

2 − c′2

2)

= −0.0044, (104)

c′ = b′3

c′3

= 0.0109, (105)

d ′ = −2b′1b′

2c′2 + (b′

12 + b′

22)c′

1

8(c′1

2 − c′2

2)

= 0.0025, (106)

f ′ = 2b′1b′

2c′1 − (b′

12 + b′

22)c′

2

4(c′1

2 − c′2

2)

− b′3

2

2c′3

= −0.0005. (107)

With Equation (3.1), we can easily see the meaning of each term in the energyfunctional W ′

p. The last term is the strain energy. The remaining part of W ′p is a

polynomial of polarization with a multiwell structure. The minima correspond tothe four spontaneous states: p′

x = ±1, p′y = 0 or p′

x = 0, p′y = ±1. The energy

barrier between different wells is E ′b = 3.924 × 10−3, or Eb = 3.924 MPa, which

is about the right range for BaTiO3 [53]. From Equation (102), we notice that thec/a ratio of the tetragonal phase of BaTiO3 is (1 + a′)/(1 + b′) = 1.0109, whichis consistent with experimental data.

Finally, the normalized versions of Equations (91)–(93) are:

µ′ ∂p′

∂t ′− ∇′ ·

(dW ′

g

d∇′p′

)

+ ∂W ′p

∂p′ + ∇′φ′ = 0, (108)

∇′ · σ ′ = 0, (109)

∇′ · (−ε′∇′φ + p′ χ(Ω)) = ρ′(φ′, Nd) (110)

with material constants

µ′ = 1, ε′ = ε0c0

p20

= 0.131. (111)

We point out that, although a0 does not explicitly appear in the normalizedequations, it is implicitly included in ρ′(φ′) in Equation (110) since

ρ′(φ′) = 1

ρ0

−eNc F1

2

(Efm − Ec + eφ0φ

′

KbT

)+ eNv F1

2

(Ev − eφ0φ

′ − Efm

KbT

)

+zeNd

⎛

⎝1 − 1

1 + 12 exp

(Ed−eφ0φ′−Efm

KbT

)

⎞

⎠

−z′eNa

⎛

⎝1 − 1

1 + 12 exp

(Efm+eφ0φ′−Ea

KbT

)

⎞

⎠

⎫⎬

⎭(112)


and ρ0, φ0 defined by Equation (98) depend on a0. For example, for a0 =10−9 Vm3C−1, ρ0 = 109 Cm−3 and φ0 = 1 V, while for a0 = 10−7 Vm3C−1,ρ0 = 108 Cm−3 and φ0 = 10 V. Therefore, the solution of the normalized equa-tions still depends on a0, and thus depends on the physical thickness of domainwalls. The only exception is when ρ′(φ′) ≡ 0. However, even in this case, a0 stillenters since the size of the computational specimen depends on a0.

We now specify the material constants we use for BaTiO3 in Equation (112).Nc(Nv) is the effective density of states in the conduction (valence) band, and isapproximately 1.0 × 1024 m−3 [2,38]. Kb is the Boltzmann constant, 1.3807 ×10−23 JK−1. T , the absolute temperature, is set to be 300 K. The Fermi level Efmof platinum electrodes is −5.3 eV. The constants of the band structure of BaTiO3are chosen to be [34]: Ec = −3.6 eV, Ed = −4.0 eV, Ea = −6.2 eV, and Ev =−6.6 eV.

As to the defects, we set Na = 0 here since we are mainly interested in oxy-gen vacancies, which act like donors, and we estimate the oxygen vacancy densityNd as follows. For BaTiO3, a = 3.9920 Å and c = 4.0361 Å [27], therefore thevolume of a unit cell is about 60 Å

3. Since there are five atoms per unit cell, the

volume per atom is approximately 10 Å3, thus the total atoms sites per unit volume

is Nt = 1029 m−3. According to [53], the oxygen vacancy density ranges from10 to 1000 ppm, corresponding to Nd of 1024–1026 m−3. The nominal valency ofoxygen vacancy z is equal to 2, although the effective valency is usually less [53].We choose z = 1 here.

We now consider the diffusion equation (90), which can also be written as:

Nd = −∇ · J, (113)

J = −βNd∇(∂Wd

∂Nd+ ez f (φ)φ

), (114)

where J is the flux of oxygen vacancies. The constraint Equation (76) is equivalentto the zero normal flux boundary condition, that is,

Jn = J · n = 0. (115)

We assume that the part of Wd that explicitly depends on Nd is the usual freeenergy of mixing at small concentration [29]. This leads to:

∂Wd

∂Nd= µv

Na, (116)

where Na is Avagadro’s number, and

µv = Gv +Ω(1 − C)2 + Na KbT ln C (117)

is the partial molar free energy of vacancies. Here, C = Nd/Nt is the mole fractionof vacancies, Gv the molar free energy of vacancies when C = 1, and Ω = Naqεwith q being the number of bonds per atom and ε the energy difference per bondwith and without vacancies.


Substituting Equations (116) and (117) into Equation (114), we have

J = −d∇Nd − βezNd∇ ( f (φ)φ) , (118)

whered = βKbT F (119)

is the diffusion coefficient, and F is the thermodynamic factor, defined as

F = 1 − 2Ω(1 − C)C

Na KbT. (120)

For the dilute case, C 1, F is approximately equal to 1. For simplicity, we onlywork with the dilute case here.

Equations (113) and (118) can be normalized as

∂C

∂ t ′= −∇′ · J′, (121)

J′ = −α∇′C − αγC∇′ ( f (φ′)φ′) , (122)

where

J′ = T0

L0 NtJ, α = T0

L20

d, γ = ezφ0

KbT, (123)

and t ′, x′, and φ′ are defined by Equation (97), and L0, φ0, and T0 by Equation (98).In a steady state, ∂C/∂ t ′ = 0, and J′ = const. In the case of one dimension,

if we assume the defect population is conserved in the computed domain [−L , L],we have

dC

dx ′ + γCd f (φ′)φ′

dx ′ = 0 (124)

with ∫ +L

−LC dx ′ = 2LC0 (125)

where C0 is the average defect concentration, or

C = Q exp(−γ f (φ′)φ′) (126)

with

Q = 2LC0∫ L−L exp(−γ f (φ′)φ′) dx ′ . (127)

From Equation (126), we can see that the steady defect concentration has anexponential relation with f (φ)φ. γ , defined in Equation (123), is approximatelyequal to 40φ0 at room temperature with z chosen to be 1. Therefore, a slight dif-ference of potential φ will result in a huge difference in defect concentration. Thisposes a computational challenge, and it is physically unlikely since Equation (126)is based on the assumption that we are working in the dilute case and the thermo-dynamic factor F can be approximated by 1. Indeed, in most diffusion processesin semiconductors, the diffusion coefficient d, defined in Equation (119), is nota constant, and it approaches zero when the diffused species reaches a saturatedvalue [38]. Nevertheless, the potential difference is indeed a crucial driving forcefor defect redistribution, and a small potential difference does result in a big dif-ference in defect concentration. Therefore, we choose γ = 4φ0 for our numericalcalculation here.


3.2. Results of numerical computation

We use the finite-element method to obtain numerical solutions to Equations(91)–(93) on a 400 × 200 rectangular domain. In this section we take the defectdensity Nd to be uniform and constant at a value of 1.0 × 1024 m−3, and hencewe do not consider equation (90). This is because the diffusion of vacancies ismuch slower than the revolution of polarization, or, α 1 in Equation (122).So the results of this section correspond to a time scale that is too short for thediffusion of defects but long enough for the formation of domains. We choosea0 = 1.0 × 10−7Vm3C−1 here, which means the slab size is 1,040 nm × 520 nm.a0 is high and corresponds to a domain wall width of 50 nm, but is convenient forvisualization of the qualitative features. The boundary conditions we choose are:upper or lower boundary electrically shorted, no flux at the left and right sides;no rotation/reflection allowed, no displacement at the lower left corner point; no xdisplacement along the left side; stress free on other sides.

We consider two initial conditions, one with a 180 and another with a 90domain wall inside the computational domain. We integrate the equations forwardin time until equilibrium is reached. The resulting space charge distribution andelectrostatic potential are shown in Fig. 5. For comparison, we have also computedthe results in a classical theory (perfectly insulating crystal with no dopants orspace charge density) for a double-sized slab, and Fig. 6 shows this electrostaticpotential. A prominent difference between the two calculations is a layer of charges(or depletion of electrons) close to the electrodes in the current theory (Fig. 5). Thisis the so-called depletion layer that is a typical feature of a metal–semiconductorinterface. This depletion layer is accompanied by a large electric field, and this canaid the injection of charges from the electrodes, which is a mechanism of electri-cal fatigue. Such a depletion layer is completely absent in the classical calculation(Fig. 6).

In the case of the 180 domain walls (Fig. 5a), the depletion layer overwhelmsany contribution from the domain wall and we barely notice the domain wall ineither the distribution of potential or charges. Thus we conclude that 180 domainwalls have very little interaction with oxygen vacancies and effect on their diffu-sion. In contrast, there is a significant interaction between the 90 domain wallsand oxygen vacancies, as shown in Fig. 5b. We see the depletion layers as before,but we also see that very large amounts of negative charges are accumulated alongthe domain wall. In other words, electrons are injected from the electrodes andtrapped at the domain wall at equilibrium. The reason for this can be understoodby going back to the electric potential in Fig. 6 without defects and noting the largeelectric field at the domain wall; this drives the injection of charges into the domainwall. Despite this, an electric field remains at the domain wall, as shown in Fig. 5,and this can in turn force the diffusion of oxygen vacancies and lead to pinning ofdomain walls.

An electrostatic feature is visible where the 180 meets the electrode in theclassical calculation (Fig. 6); such features are completely masked by the depletionlayer in the current theory. (Note that the two figures are plotted with differentranges of potential.)


Fig. 5. The electric potential (V) and charge densities (C m−3) near a 180 and b 90 domainwalls. The location of domain walls is marked by white dashed lines, and the polarizationdirections are indicated by white arrows. The density of oxygen vacancies Nd = 1024 m−3

and a0 = 10−7 Vm3C−1. The last is high, corresponding to a domain wall width of 50 nm,but is convenient for visualization of the qualitative features

Figures 7 and 8 show the stress and the strain distributions computed using thecurrent theory. These are raised near the domain walls, and have a concentrationnear where the domain wall meets the electrodes. These strain/stress concentrationsites may likely serve as starting points for microcracking or domain wall pinning.It is also interesting to notice that there is not much difference in terms of the mag-nitude of the stress concentration between the 180 and 90 domain walls. This isa little surprising, since conventional wisdom states that 90 domain walls undergomuch more distortion than 180 domain walls. The results obtained by the classicaltheory are similar and are not displayed here. We refer the reader to [46] for theseand further calculations where the deformation is completely ignored.


Fig. 6. Electric potential near 180 (upper) and 90 (lower) domain walls in perfect crys-tals. The real value of φ depends on a0, the number (V) shown here is obtained by choosinga0 = 1.0 × 10−7 Vm3C−1

3.3. Depletion layers

We now seek to examine the depletion layers closely, and also to understand anypossible diffusion of oxygen vacancies. To do so, we notice from Fig. 5 that, awayfrom any domain wall, depletion layers are essentially one dimensional. So we seekto study the one-dimensional problem along the section marked A–A′ in Fig. 5.To this end, we assume that p = 0, p′(y) and that all quantities are independentof x . We also ignore the deformation for simplicity. The governing equations nowreduce to:

∂C

∂ t ′= α

∂

∂y′

(∂C

∂y′ − γC∂

∂y′(

f φ′)), (128)

µ′ ∂p′

∂t ′= d2 p′

dy′2 − dW ′p

d p′ − dφ′

dy′ , (129)

−ε′ d2φ′

dy′2 + d p′

dy′ = ρ′(φ′, Nd). (130)

with

W ′p = a′

1

2p′2 +

(a′

2

4− d ′

)p′4 + a′

4

6p′6. (131)

The results are shown in Fig. 9 for the cases where the defect diffusion is small(α 1, shown by the blue curves). The electrostatic potential rises quickly fromzero at the electrode to a value of approximately 1.4 V. This value is known asthe build-in potential φbi, and is approximately equal to the difference between theFermi levels of the film and the electrode:

φbi = EBTfm − EPt

fm

e≈ 1

2e(Ec + Ed − 2EPt

fm) = 1.5 V. (132)


0 50 100 150 200 250 300 350 4000

50

100

150

200

εxx

−5

−4

−3

−2x 10

−3

0 50 100 150 200 250 300 350 4000

50

100

150

200

εyy

34567

x 10−3

0 50 100 150 200 250 300 350 4000

50

100

150

200

εxy

−2

0

2

x 10−3

0 50 100 150 200 250 300 350 4000

50

100

150

200

εxx

−5

0

5

x 10−3

0 50 100 150 200 250 300 350 4000

50

100

150

200

εyy

−4−20246

x 10−3

0 50 100 150 200 250 300 350 4000

50

100

150

200

εxy

−2

−1

0

1x 10

−3

(a)

(b)

Fig. 7. Strain profile near a 180 and b 90 domain walls in oxygen-vacancy doped crystals

This acts as a barrier against charge injection; metal electrodes with higher workfunction or smaller Fermi level would be less susceptible to fatigue. This is consis-tent with observations [34]. Further, we can show that the width of the depletion

layer is given by d ≈√

2εr ε0φbiezNd

, where εr =(∂2W∂p2

∣∣p0ε0

)−1can be viewed as

the effective relative dielectric constant [46]. Since smaller depletion layers makeit more susceptible to charge injection through tunneling, it follows that failurebecomes worse with increasing doping density and better with large effective dielec-tric modulus. In [48], we will present the detailed influence of the doping level ondepletion layers, including a tail-to-tail domain wall with an interior depletion layerin films above a critical doping level.

The large electric field in the depletion layers also drives the diffusion of defects,including oxygen vacancies, especially during annealing at high temperatures, whenα ≈ 1. To understand this, we solve diffusion equation (128), (129) and (130) until


0 50 100 150 200 250 300 350 4000

50

100

150

200

σxx

−0.6

−0.4

−0.2

0

0 50 100 150 200 250 300 350 4000

50

100

150

200

σyy

0

0.2

0.4

0.6

0 50 100 150 200 250 300 350 4000

50

100

150

200

σxy

−0.1

0

0.1

0 50 100 150 200 250 300 350 4000

50

100

150

200

σxx

−0.1

0

0.1

0 50 100 150 200 250 300 350 4000

50

100

150

200

σyy

−0.05

0

0.05

0.1

0 50 100 150 200 250 300 350 4000

50

100

150

200

σxy

0

0.05

0.1

(a)

(b)

Fig. 8. Stress (GPa) profile near a 180 and b 90 domain walls in oxygen-vacancy dopedcrystals

steady state is reached in both Nd/C and p. The results given by dashed red curvesin Fig. 9 indicate that the oxygen vacancies are almost fully depleted from the inte-rior and tend to accumulate near the electrodes. The increased donor compositionnear the electrodes may accelerate failure for the reasons described earlier. Finally,this calculation shows that oxygen vacancies can be rearranged even when theirtotal number is conserved.

3.4. Domain walls

We now turn to domain walls and examine a region of the ferroelectric contain-ing a domain wall that is far away from the electrodes. Thus we isolate the effectsof the domain wall from the depletion layer. Further, since the domain wall areplanar defects, it suffices to study the problem in one dimension transverse to thedomain wall, that is, along the sections B-B′ and C-C′ in Fig. 5.


0 0.5 10.2

0.25

0.3

0.35

p (C

/m2 )

y/L0 0.5 1

0

0.5

1

1.5

φ (V

)

y/L

0 0.5 10

0.5

1

1.5

2x 10

5

ρ (C

m−

3 )

y/L0 0.5 1

0

1

2

3

4x 10

6

0 0.5 1

1

x 1024

Nd (

m−

3 )

y/L0 0.5 1

0

0.5

1

1.5

2x 10

25

Fig. 9. The polarization, electric potential, space charge density, and defect density alongthe line A–A′ of the BaTiO3 slab in Fig. 5. Solid blue lines without diffusion of defects;dashed red lines with diffusion of defects

In order to understand the essentials, here, instead of scaling Wg and Wp by c0in Equation (97), we rescale them by Eb, the depth (difference between maximaand minima) of the wells of Wp. Consequently, all the scalings in Equation (97)and (98) follow with c0 being replaced by Eb. We first consider the simplest casewhere no defects are present in the crystal. The governing equations then reduce to

d2 p′r

dr ′2 − ∂W ′p

∂p′r

− dφ′

dr ′ = 0, (133)

d2 p′s

dr ′2 − ∂W ′p

∂p′s

= 0, (134)

−ε′ d2φ′

dr ′2 + d p′r

dr ′ = 0, (135)

where ε′ = ε0 Eb/p20, and r ′ and s′ are coordinates normal and parallel to the

domain wall, respectively. Specifically, r ′ = x ′, s′ = y′ for a 180 domain wall,and r ′ = (x ′ − y′)/

√2, s′ = (x ′ + y′)/

√2 for a 90 domain wall. These equa-

tions differ from those used in the classical studies of domain walls [9], whereit is assumed a priori that the normal component of the polarization p′

r and theelectrostatic potential φ′ are constant, and therefore Equations (133) and (135) areomitted. Finally, to solve these equations in closed form, we choose

W ′p = a′

2

(p′

x2 + p′

y2)

+ b′

2p′

x2 p′

y2 + c′

4

(p′

x4 + p′

y4), (136)

where a′ = −4, c′ = −a′ = 4, and b′ = 20. We have verified that the resultsobtained with this choice hold qualitatively for any generic form of W ′

p.For a 180 domain wall, the classical kink solution [18], p′

r = 0, p′s =

tanh(√

2 r ′)

and φ′ = const. also solves the more general system of equations

(133)–(135).


−10 −8 −6 −4 −2 0 2 4 6 8 100.706

0.7062

0.7064

0.7066

0.7068

0.707

0.7072

p r′

r′

NumericalPerturbative

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

p s′

r′


−10 −8 −6 −4 −2 0 2 4 6 8 10−2

−1.5

−1

−0.5

0

0.5

φ′

r′


Fig. 10. Structure of a 90 domain wall without defects

The situation is very different for a 90 domain wall. It is easy to verify thatthe classical kink solution [18] with p′

r ≡ √2/2, and φ′ ≡ 0 does not solve these

equations. While we are unable to obtain an exact solution to Equations (133)–(135) in closed form, we are able to obtain a perturbative solution using the factthat ε′ ≈ 10−4 for BaTiO3 [46]:

p′r =

√2

2(1 − 2ε′sech(

√3r ′)), (137)

p′s = −

√2

2tanh(

√3r ′), (138)

φ′ = −√

6

3(tanh(

√3r ′)+ 1). (139)

This approximates well the exact numerical solution (Fig. 10).The key feature of this solution is that the transverse component of the polar-

ization (p′r) shows a dip at the domain wall, and consequently one has a drop in

voltage of 0.115 V across the 90 domain wall, which is consistent with ab initiocalculation [23].


ρ (C

m−

3 )r (nm)

−150 −100 −50 0 50 100 150−400

−200

0

200

400

Nd (

m−

3 )

r (nm)−150 −100 −50 0 50 100 150

9.9

9.95

10

x 1023

Fig. 11. The distribution of space charge and donors (average density Nd = 1024 m−3) for aperiodic array of 90 domain walls in BaTiO3 (Eb = 5 × 106 Jm−3, a0 = 10−9 Vm3C−1).In this calculation, the defects are allowed to diffuse and achieve a steady state. Noticethe formation of charge double layers and asymmetric redistribution of defects near thedomain walls. The colors represent different domains, with polarization direction indicatedby arrows

The variation in the electrostatic potential across the 90 walls provides a driv-ing force for the rearrangement of space charges and defects (Fig. 5b, refer to [47]for more detail). In distinct contrast, there is no interaction between the 180 wallsand space charges. If diffusion of defects is allowed, we expect them to decorate thedomain walls as in the case of the depletion layer. This is indeed what is observedfor a periodic array of domain walls, as shown in Fig. 11, where in a steady statethe dopants segregate along a domain wall in an asymmetric manner, creating adefect dipole. This result is consistent with experimental observations [19,36] andprovides a mechanism for the domain wall to have a memory of its location duringannealing. To understand this mechanism, note that the domains that disappear onthe application of an electric field will not in general reappear at their old locationswhen the electric field is switched off. However, the inhomogeneous charge distri-bution allows the walls to have a memory of their location (by providing favorablenucleation sites) even after the field is switched off. Thus, it provides a mechanismfor imprinting the domain wall and for the large electrostriction through agingrecently observed by [32]. Finally, the presence of these charges and defect lay-ers at the 90 domain walls means that such walls promote electrical failure byproviding a high-conductivity pathway from electrode to electrode.

Acknowledgements. This work draws from the doctoral thesis of Yu Xiao at the CaliforniaInstitute of Technology. We thank Vivek Shenoy and Wei Zhang for useful discussions. Weare also glad to acknowledge the financial support of the US Army Research Office throughthe MURI grant no. DAAD 19-01-1-0517.


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Division of Engineering,Brown University, Providence, RI 02912, USA.

e-mail: [email protected]

and

Division of Engineering and Applied Science,California Institute of Technology,

Pasadena, CA 91125, USA.e-mail: [email protected]

(Received August 14, 2006 / Revised July 31, 2007 / Accepted August 3, 2007)Published online April 29, 2008 – © Springer-Verlag (2008)

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