A Continuum Theory of Deformable, Semiconducting Ferroelectrics

Transcription

1 Arch. Rational Mech. Anal. 189 (8) Digital Object Identifier (DOI) 1.17/s y A Continuum Theory of Deformable, emiconducting Ferroelectrics Yu Xiao & Kaushik Bhattacharya Communicated by R. D. James Abstract Ferroelectric solids, especially ferroelectric perovskites, are widely used as sensors, actuators, filters, memory devices, and optical components. While these have traditionally been treated as insulators, they are in reality wide-band-gap semiconductors. This semiconducting behavior affects the microstructures or domain patterns of the ferroelectric material and the interaction of ferroelectrics with electrodes, and is affected significantly by defects and dopants. In this paper, we develop a continuum theory of deformable, semiconducting ferroelectrics. A key idea is to introduce space charges and dopant density as field (state) variables in addition to polarization and deformation. We demonstrate the theory by studying oxygen vacancies in barium titanate. We find the formation of depletion layers, regions of depleted electrons, and a large electric field at the ferroelectric electrode boundary. We also find the formation of a charge double layer and a large electric field across 9 domain walls but not across 18 domain walls. We show that these internal electric fields can give rise to a redistribution or forced diffusion of oxygen vacancies, which provides a mechanism for aging of ferroelectric materials. 1. Introduction Ferroelectric perovskites are widely used in solid-state devices for their diverse properties [14,5,34,49]. For example, lead zirconate titanate (PZT) is used in ultrasonic transducers and actuators for its piezoelectric properties, lithium niobate in optical devices for its electro-optical properties, and barium strontium titanate (BT) in capacitors for its large dielectric constant. Until recently, many of these applications used only the linear response of ferroelectric materials. However, by carefully controlling the domain patterns and switching processes in these materials, 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 switching behavior.

2 6 Yu Xiao & Kaushik Bhattacharya A ferroelectric material is nonpolar (paraelectric) above its Curie temperature, but spontaneously polarized (ferroelectric) below it. Along with the spontaneous polarization, there is usually a spontaneous distortion of the low-temperature phase compared to the high-temperature phase. Further, the polarized state is less symmetric than the nonpolar state and this gives rise to symmetry-related variants: crystallographically and energetically identical states that are oriented differently with respect to the parent nonpolar state. These variants can coexist as domains separated by domain walls. As an example, barium titanate (BaTiO 3 ), an extensively investigated ferroelectric material, is cubic and nonpolar above its Curie temperature (1 ), and is tetragonal and spontaneously polarized along the 1 direction at room temperature and therefore has six variants. These variants can form patterns involving 18 and 9 domain walls (see, for example, [5,33]). ince symmetry-related variants are energy-equivalent, it is possible to switch one domain to another by the application of suitable electrical, mechanical or optical loading. This gives rise to interesting applications. For example, nonvolatile memories are based on 18 domain switching [3,34]. The two opposite directions of polarization represent the two logic states, and they can be switched from one to another through the application of an electric field. Further, since both states are stable, this information is unaltered even when the field is switched off. Large actuation can be generated by non-18 domain switching, since such a switch is usually accompanied by a change of distortion. By applying a constant compressive stress and a cyclic electric field, Burcsu et al. [7,8] demonstrated 1% strain through 9 domain switching in barium titanate. Other examples of the role of domain switching include poling, the extrinsic component of the piezoelectric response, and holographic storage. These applications have motivated many theoretical investigations of domain patterns and domain switching using a continuum (phase-field) model (see, for example, [1,,51,5]). These follow the theoretical framework going back to Devonshire [15 17] and Toupin [39], and treat the ferroelectric material as an insulator. However, ferroelectric perovskites are wide-band-gap semiconductors [6,37]: the band-gap for BaTiO 3 is about 3. ev, and that for PbZr.4 Ti.6 O 3 is about 3.4 ev. This has important consequences in practice, for example, it is well known that the fatigue life and dielectric breakdown of a ferroelectric material are affected by the type of electrode one uses [13,31,34]. The semiconducting nature gives rise to a region of electron depletion close to the ferroelectric electrode interface (the so-called depletion layer), and this depends on the particular material combination. Further, dopants including impurities and oxygen vacancies can affect the switching behavior of these materials even in relatively dilute compositions. In particular, it has been recognized that defects often decorate domain walls [1,36]. Furthermore, ferroelectric materials display an aging behavior where they develop a memory of a domain pattern in which they have been held for a long time [3]. Finally, the domain patterns of ferroelectric materials can be manipulated using light through the generation of photoelectrons [4]. We refer the reader to [46] for a detailed review of the experimental literature. There have been a few attempts to study ferroelectrics as semiconductors. However, they either treat the polarization distribution as frozen and calculate the space charges [41 43,45] or they treat

3 A Continuum Theory of Deformable, emiconducting Ferroelectrics 61 the space charges as frozen and calculate the polarization [5,4]. Unfortunately, the polarization and space charges interact nontrivially through the electrostatic potential. Therefore they require a uniform treatment, and this is the goal of this paper. In this paper, we revisit the continuum theory of ferroelectric materials. The key step and departure from other work is to treat the ferroelectric as a semiconducting material in a unified manner by introducing the space charge density and dopant density as field or state variables in addition to polarization and deformation. We provide the detailed kinematic description in ections.1 and.. In this presentation, we limit ourselves to a situation with a single species of donor dopants for simplicity. This is appropriate for many ferroelectrics where oxygen vacancies are the predominant dopant. However, we note that the framework we develop can easily be generalized to multiple interacting dopants, some donors and some acceptors, as suggested by recent observations [8]. The space charges and polarization give rise to electric fields, and this requires some care in light of the finite deformation and possible jumps. This is described in ection.3. We then compute the rate of dissipation defined as the difference between the rate of external working and the rate of change of stored energy in a ferroelectric conductor system in ection.4. We show that we can write this as a sum of the products between generalized forces (rate-independent quantities) and generalized velocities (rates). This allows us to derive the governing equations in ection.5. We discuss special cases in ection.6. We examine the specific example of a single-crystal barium titanate slab coated with two platinum electrodes in ection 3. We assume that the barium titanate is doped with oxygen vacancies. In order to do so, we specialize to infinitesimal displacements and a particular additive assumption on the stored energy density. We conduct detailed two-dimensional numerical calculations using a finite-element model of two problems, a slab with a 18 domainwallandaslabwitha9 domain wall in ection 3.. We assume that the oxygen vacancies are distributed uniformly and do not diffuse, and that the two electrodes are shorted. The calculations reveal the formation of depletion layers layers with reduced charge and high electric field at the ferroelectric electrode boundaries. They also show the formation of charge double layers and significant electric fields at the 9 domain wall, but not at the 18 domain wall. In short, the calculations reveal that there are significant internal electric fields even with shorted electrodes. This is in marked contrast with the classical continuum theories which would have predicted an almost uniformly zero electric field in this situation. Using a simplified one-dimensional model, we examine depletion layers and domain walls further in ections 3.3 and 3.4, respectively. In these calculations, we allow the oxygen vacancies to diffuse within the specimen while keeping their total number fixed. We find that they diffuse to the electrodes, and argue that this can be a mechanism that promotes fatigue. We also show that the oxygen vacancies diffuse to one side of 9 domain walls but have long 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 results and consequences of the one-dimensional problems will be presented elsewhere [48].

4 6 Yu Xiao & Kaushik Bhattacharya Fig. 1. A ferroelectric semiconducting system in an external field generated by conductors C q and C v. C q, with fixed charge Q, isfixedinspacebyexternalforces,whilec v with fixed potential deforms with the ferroelectric body. Continuum model.1. Kinematics Consider a ferroelectric semiconducting crystal in an external field, as shown in Fig. 1. It occupies a region R 3 in the reference configuration. We find it convenient to choose the undistorted nonpolar phase at the Curie temperature as our reference configuration. A deformation y : R 3 brings it to the proximity of electrodes C v R 3 with fixed potential ˆφ and C q R 3 with fixed charge Q under the action of traction t. The deformation gradient is F = x y, and we assume that the deformation is invertible and that J = det F > almost everywhere in. We denote by p : y() R 3 the polarization of ferroelectric material per unit deformed volume, and by p : R 3 the polarization per unit undeformed volume. We have p (x) = (det x y(x))p(y(x)). (1) We shall make the following assumption for later use: the conductor C q is fixed in space but the conductor C v deforms with the body with negligible elastic energy. This is reasonable since electrodes are usually very thin compared to the body... pace charge density in semiconducting solids The total charge density at any point in a semiconductor in the current configuration is [] ρ = e ( zn d z ) N a n d n c + p a + p v, () where N d is the density of donors (number per unit deformed volume), N a the density of acceptors, n d the density of electrons in the donor band, n c the density of electrons in the conduction band, p a the density of holes in the acceptor band, p v the density of holes in the valence band, z and z the valency of donors and acceptors, respectively, and e the coulomb charge per electron.

5 A Continuum Theory of Deformable, emiconducting Ferroelectrics 63 We assume that oxygen vacancies are the dominant impurities. 1 ince oxygen vacancies act as donors, we may set N a = p a = in Equation (). Further, electrons in the conduction band and holes in the valence band have a much higher mobility than defects, so we group defect-based charges and electronic charges separately. Thus, with and ρ = e(zn d n d n c + p v ) = ezf N d + ρ c (3) f = zn d n d zn d (4) ρ c = e(p v n c ). (5) Here, we call f the ionization ratio of defects since f represents the ratio of ionized defects to the total. Further, ezf N d is exactly the amount of charges contributed by the defects: f = means that no defects are ionized and all the extra electrons from defects/donors are bound in the donor s level; f = 1 means that all defects are ionized and all the electrons in the donors level are activated into the conduction band. We call ρ c the free charge density since it is the charge contribution from the conduction and valence bands. We define the counterparts of N d, ρ, and ρ c in the reference configuration as N d, ρ, and ρ c (number per unit undeformed volume), respectively, in analogy with Equation (1), and ρ = ezf N d + ρ c. (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: Ṅ d = x J Nd, (7) ρ = x J ρ, (8) where J Nd and J ρ are the flux of defects and charges in the reference configuration, respectively. We point out here that the dot on N d and ρ denotes the material time derivative of N d and ρ, respectively. For any variable ξ defined on y(), ξ(y(x, t), t) = ξ(y(x, t), t) t = x ξ(y, t) t + v y ξ(y, t), (9) y where v = y(x,t) x is the particle velocity of the material point x, and we call t ξ(y,t) t y the spatial time derivative of ξ and denote it by o ξ. 1 We could proceed analogously for any other dopant, or combination of dopants.

6 64 Yu Xiao & Kaushik Bhattacharya.3. Electric field The polarization and the space charges in the ferroelectric body as well as the 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 s equation: y ( ɛ y φ + p χ(y(, t))) = ρχ(y(, t)) in R 3 \(C v C q ), y φ = on C v C q (1) subject to φ C q ˆn d y = Q, ɛ φ = ˆφ on C v, φ as y, (11) where ɛ is the permittivity coefficient of free space, and χ(d) is the characteristic function of domain D. Precisely, φ H 1 (R 3 ) satisfies the following: ( ɛ y φ + p χ(y() ) ψ dy = ρψdy + σψd y, R 3 y() C v C q (1) σ d y = Q, (13) C q φ = ˆφ on C v (14) for each ψ H 1 (R 3 ), where σ : C v C q R measurable is the surface charge density on the interface, and it is defined as σ = ɛ y φ + p χ (y()) ˆn. (15) Here denotes the jump across an interface: ξ = ξ + ξ, with ξ being some variable defined in both domains; ˆn is the unit norm of the interface, pointing from D to D +. Although φ is continuous in R 3, other quantities such as y φ can be discontinuous across some interfaces, as Equation (15) shows. Here we discuss some jump conditions in a more general setting for later use. In particular, we shall be interested in time-dependent processes. o the polarization p and the deformation y could depend on time, and we solve Equations (1) (14) at each time to find the electric potential. The jump condition across any interface separating D + and D is (Fig. ), ɛ y φ + p ˆn = σ. (16)

7 A Continuum Theory of Deformable, emiconducting Ferroelectrics 65 v y(x) p nˆ tˆ D D Fig.. An interface separating D from D + in an electric field. D is a dielectric or ferroelectric body with polarization p; D + can be a conductor or vacuum; σ is the surface charge density on the interface; ˆn is the unit norm of the interface, pointing from D to D + ;and v is the velocity of a material point on the interface If we assume as shown in Fig. that p = ind +, and if p denotes the polarization in D (for example, D + can be a conductor or vacuum and D a dielectric or ferroelectric body), then Equation (16) can be rewritten as y φ ˆn = 1 ɛ p ˆn σ ɛ. (17) Now, let ŷ(α) be a curve on the interface at time t parameterized by α. We have, from the continuity of φ, Differentiating with respect to α, wehave φ (ŷ(α)) = φ + (ŷ(α)). (18) y φ ŷ =. (19) α ince this holds for any curve on the interface, we obtain continuity of y φ along the tangent, that is, y φ ˆt = ˆt ˆn =. () Combining this with Equation (17), we obtain ( 1 y φ = p ˆn + σ ) ˆn. (1) ɛ ɛ Now consider a material point x on the interface. Let us assume that the interface does not propagate in the reference configuration, so that the particle velocity remains continuous across the interface. ince the electric potential φ is continuous, we have φ (y(x, t), t) = φ + (y(x, t), t), () so that φ (y(x, t), t) = φ + (y(x, t), t), (3)

8 66 Yu Xiao & Kaushik Bhattacharya or, o φ + y φ o v = φ + + y φ + v. (4) Recall that φ here denotes the material time derivative of φ, and o φ denotes the spatial time derivative of φ. Hence, o φ = y φ v, (5) where v is the particle velocity of the material point x as defined in ection.. Inserting Equation (1) into Equation (5) yields o φ = 1 ɛ (p ˆn)(v ˆn) + σ ɛ (v ˆn). (6) Another quantity we will use later is the Maxwell stress tensor, defined as T M = E D ɛ E EI, (7) where E = y φ is the electric field and D = ɛ E + p χ (y()) is the electric displacement. The discontinuity of E or D across an interface leads to the discontinuity of T M. By a direct calculation on an interface as above between a ferroelectric and vacuum or a conductor, we have T M ˆn = (E D ɛ E EI)ˆn = E D ˆn + E D ˆn ɛ ( E E )ˆn = E σ + E (ɛ E ˆn + 1 ) p ˆn ɛ ( E E )ˆn ( = E + E ) σ + ɛ E E ˆn + 1 (p ˆn) E (8) ɛ E (( E ˆn ) ˆn ) ˆn ( = E + E ) σ + 1 (p ˆn) E = E σ + 1 (p ˆn + σ) ˆn. (9) ɛ The second equality uses the identity φψ = φ ψ + φ ψ, (3) where φ = φ+ + φ (31) is the average of the limiting values of a discontinuous quantity φ. The third equality recalls the definition of D and the assumption that p = ond +, and the fourth equality uses the continuity of E = y φ along the tangential direction. The last equality is obtained by using Equation (1).

9 A Continuum Theory of Deformable, emiconducting Ferroelectrics Rate of dissipation of the system The rate of dissipation of the whole system D is defined as the difference between the rate of external working F and the rate of the change of the total energy de/dt: D = F de dt. (3).4.1. Rate of external working The rate of external working F includes the mechanical work done by external forces, the electric work done by the electrodes, and the chemical energy flux from C v into : F = ˆφ d σ d y + t v d y dt y( C v ) y( s ) (33) µ Nd J Nd ˆm d x µ ρc J ρc ˆm d x, where µ Nd and µ ρc are, respectively, the chemical potential carried by the flux of the oxygen vacancies N d and that of charges ρ c, y( s ) is the part of the boundary in the current configuration on which traction t acts, d y and d x are the differential area in the current and reference configuration, respectively, ˆm is the normal to the surface in the reference configuration, and ˆn will denote its counterpart in the current configuration. In light of Equation (6), and the fact that the activation/deactivation of donors happens much faster than the defect diffusion, by further assuming that the electrons in the donor s level diffuse with the defects with the same rate, 3 we have J ρ = ezf J Nd + J ρc. (34) Using the divergence theorem, Equation (7) and (8), and also noting that ρ = ez fn d + ezf Ṅ d + ρ c (35) we can rewrite F as F = ˆφ d σ d y + t v d y dt y( C v ) y( s ) ( ) µnd ezf µ ρc JNd ˆm d x µ ρc J ρ ˆm d x = ˆφ d σ d y + t v d y dt y( C v ) y( s ) {( ( ) ) } x µnd ezf µ ρc + ezf x µ ρc JNd + x µ ρc J ρc dx { } + µnd Ṅ d + ezn d µ ρc f + µ ρc ρ c dx. (36) The exact meanings of µ Nd and µ ρc will be examined in ection.5. 3 This is indeed a strong but reasonable assumption.

10 68 Yu Xiao & Kaushik Bhattacharya Fig. 3. A simplified version of Fig 1: v is the part of y() with fixed potential.4.. Total energy of the system The total energy of the system consists of two parts: the energy stored in the ferroelectric material and the electrostatic field energy generated by external and internal sources, that is, E = W dx + ɛ R 3 φ dx (37) where W is the stored energy per unit reference volume in the ferroelectric material. We make the constitutive assumption that it depends on defect density N d, ionization ratio of defects f, free charge density ρ c, polarization p, polarization gradient x p, and deformation gradient x y, that is, W = W (N d, f,ρ c, p, x p, x y). We require the stored energy density W to satisfy frame indifference and material symmetry. Recall that the electrical potential φ is obtained by solving Equation (1) and its boundary conditions (Equation (11)). Recall also the assumption that C q is fixed and that C v is deformable with zero elastic energy. A better way to picture this is shown in Fig. 3, where we idealize C v as an interface v = y( v ) between the vacuum and semiconductor y() on which the potential is fixed. We also denote f = y()\ v as the interface where y() has direct contact with the vacuum Rate of change of total energy The rate of change of the total energy in Equation (3) is, de dt = W dx + d dt [ ] 1 ɛ y φ dy. (38) R 3 We can directly calculate the first term on the right-hand side of Equation (38), Ẇ (N d, f,ρ c, p, x p, x y) dx ( W = Ṅ d + W N d f { W + ṗ p [ x ) ρ c dx ρ c f + W ( W x p )] ṗ } dx

11 A Continuum Theory of Deformable, emiconducting Ferroelectrics 69 ( W + ˆm x p + y() ) ( 1 W J F FT ˆn [ ( 1 ṗ d x y y() J ) )] W F FT v dy v dy. (39) We have used the divergence theorem in the reference configuration to obtain the second and third integrals on the right, and the divergence theorem in the current configuration to obtain the last two integrals. We clarify using indicial notation that ( ( ) W W [ x [ y )] ṗ = ṗ I, x p p I,J,J ( ) W ˆm ṗ = W ˆm J ṗ I, x p p I,J ( ( ( ) ) 1 1 W F ji W F FT J ( 1 W J F FT ˆn )] v = ) v = 1 J J ( W F ii F ii, j ) F ji v i ˆn j in Equation (39). Here, we use the summation convention where repeated indices are summed. The calculation of the second term in Equation (38), the change of electrostatic field energy, needs some manipulation. The difficulty arises from the fact that the electric energy exists in all space. o we follow a procedure similar to that used by [4] and divide the calculation into three steps in ections The final result is shown in Equation (49) Rate of change of field energy: tep 1 First, by setting ψ = φ in Equation (1), we have ɛ y φ y φ dy = y φ p dy+ φρdy+ R 3 y() y() ˆφσd y + v φσd y. C q (4) Therefore, [ ] d ɛ y φ dy dt R3 = d { } y φ p dy + φρdy + ˆφ d σ d y + φσd y dt y() y() dt v C q = d { } y φ p dx + φρ dx + ˆφ d σ d y + φσd y dt dt v C q d = dt ( yφ) p dx + y φ ṗ dx + φρ dx + φ ρ dx + ˆφ d σ d y + φσd y dt v C q v i,

12 7 Yu Xiao & Kaushik Bhattacharya o = ( yφ + v( y y φ)) p dx + y φ ṗ dx + ( o φ + v y φ)ρ dx + φ ρ dx + ˆφ d σ d y + φσd y dt v C q o = ( yφ + v( y y φ)) p dy + ( φ o + v y φ)ρ dy y() y() + y φ ṗ dx + φ ρ dx + ˆφ d σ d y + φσd y. (41) dt v C q Notice that we use the fact that C q is the conductor with fixed charge Q in deriving the second equality Rate of change of field energy: tep econd, we multiply φ o on both sides of Maxwell equation (1), and integrate over R 3 to obtain y ( ɛ y φ + p χ(y()) ) o φ dy = ρχ(y()) φ o dy. (4) R 3 R 3 We clarify that these integrals should be interpreted in the classical sense rather than in the sense of distributions. The left side of Equation (4) can therefore be split into three parts on which the divergence theorem can be applied: y ( ɛ y φ + p χ(y()) ) φ o dy R 3 = y ( ɛ y φ + p ) o φ dy + y ( ɛ y φ ) φ o dy y() C q + y ( ɛ y φ ) φ o dy R 3 \(y() C q ) o ( = yφ ɛ y φ + p ) dy y() o o + φ( ɛ y φ + p) ˆn d y + φ( ɛ y φ + p) ˆn d y v f o o y φ ( ɛ y φ)dy + φ( ɛ y φ) ˆn q d y C q Cq o o yφ ( ɛ y φ)dy + φ( ɛ y φ) ( ˆn q ) d y R 3 \(y() C q ) C q + o o + φ( ɛ y φ) ( ˆn) d y + φ( ɛ y φ) ( ˆn) d y v + + f o o = ɛ y φ y φ dy yφ p dy φ( ɛ R 3 y() C o y φ) ˆn q d y q o φ( ɛ y φ + p) ˆn d y φ( ɛ o y φ + p) ˆn d y. (43) v f Here, ˆn, ˆn q are the outward unit norms of y() and C q, respectively, v and f are the inner surface of y(), v and + f are the outer surfaces of y(), and Cq and C q + are the inner and outer surfaces of C q, respectively.

13 A Continuum Theory of Deformable, emiconducting Ferroelectrics 71 ince C q is fixed in space, and φ o = φ on C q, Equation (4) and (43) lead to o o o ɛ y φ y φ dy = yφ p dy + φρdy + φσd y R 3 y() y() C q + φ( ɛ o y φ + p) ˆn d y v + φ( ɛ o y φ + p) ˆn d y. (44) f Therefore, by using Reynolds transport theorem, we have [ ] d 1 ɛ y φ dy dt R 3 = ɛ y φ dy + ɛ y φ dy y() t R 3 \y() t ɛ y φ v ˆn d y y() o = ɛ y φ y φ dy ɛ y φ v ˆn d y R 3 v + f o o = yφ p dy + φρdy + φσd y y() y() C q + φ( ɛ o y φ + p) ˆn d y ɛ y φ v ˆn d y v v + φ( ɛ o y φ + p) ˆn d y ɛ y φ v ˆn d y. (45) f f Let = v f = y(). Using the jump conditions (16), (1), (5), (6), and (9), we can simplify the last four terms in Expression (45): φ( ɛ o y φ + p) ˆn d y ɛ y φ v ˆn d y = o φ ɛ y φ + p ˆn d y + φ ɛ o y φ + p ˆn d y ( ɛ y φ y φ )( v ˆn ) d y = o φ σ d y + φ p ˆn o d y ɛ o ( φ y φ ˆn d y ɛ y φ y φ )( v ˆn ) d y [ = φ v y φ ] σ d y + 1 (p ˆn) (v ˆn) d y ɛ + 1 σ(p ˆn)(v ˆn) d y ɛ

14 7 Yu Xiao & Kaushik Bhattacharya = + 1 ɛ = + 1 ɛ + 1 [ φσd y v y φ + 1 ] yφ σ d y (p ˆn) (v ˆn) d y + 1 σ(p ˆn)(v ˆn) d y ɛ φσd y σ v y φ d y σ(p ˆn)(v ˆn) d y + 1 σ (v ˆn) d y ɛ (p ˆn) (v ˆn) d y + 1 σ(p ˆn)(v ˆn) d y ɛ ɛ = φσd y σ v y φ d y + 1 ( ) p ˆn + σ (v ˆn) dy ɛ = φσd y + T M ˆn v d y = T M ˆn v d y. (46) The last equality comes from the fact that φ = ˆφ, thus φ = on v, and σ = on f. ubstituting Equation (46) into Equation (45), we have [ ] d 1 ɛ y φ o o dy = yφ p dy + φρdy dt R 3 y() y() (47) + φσd y + T M ˆn v d y. C q v f.4.6. Rate of change of field energy: tep 3 Now, subtracting Equation (47) from Equation (41), we obtain, [ ] d 1 ɛ y φ dy dt R 3 = y φ ṗ dx + φ ρ dx + v (( y y φ ) p ) dy (48) y() + v y φρdy + ˆφ d σ d y T M ˆn v d y. y() dt v y().4.7. Rate of dissipation: The final expression Putting together Equations (3), (35), (36), (38), (39), and (49), we now have the final expression for the rate of dissipation of the whole system: D = F de dt {( ( ) ) } = x µnd ezf µ ρc + ezf x µ ρc JNd + x µ ρc J ρc dx

15 A Continuum Theory of Deformable, emiconducting Ferroelectrics ( µ Nd W ) ( ezf φ Ṅ d dx + µ ρc W ) φ ρ c dx N d ρ c ( ezn d µ ρc W ) ezn d φ f dx f [ ( ) W x W ] F T x φ ṗ dx x p p ( ) W ˆm ṗ d x x p [ ( ) 1 W y J F FT ( y y φ ) ] p ρ y φ v dy ( ) 1 W J F FT ˆn v d y T M ˆn v d y + t v d y. (49) y() y() y() y( s ) From Equation (49), we can see that the dissipation of the system has three contributions: the first four integrals are the dissipation caused by the diffusion of vacancies and charges and the activation (deactivation) of electrons from (to) the donor band, the next two terms are the dissipation caused by the polarization evolution, and the remaining terms are the contribution from the deformation of the ferroelectric body..5. Governing equations According to the second law of thermodynamics specialized to isothermal processes, which we are currently considering, the rate of dissipation D should always be greater or equal to zero. Notice that in the expression (49) for the rate of dissipation each term is a product of conjugate pairs: generalized velocity (time rate of change of some quantity or flux of some quantity) multiplied by a generalized force (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. pecifically, by considering various processes that have the same state at some instant of time but different rates, and insisting that D for all these processes, we conclude that ( 1 y J 1 J ( ) W x W F T x φ = x p p in, (5) W ˆm = x p on, (51) ) W F FT ( y y φ ) p ρ y φ = in y(), (5) W F FT ˆn T M ˆn tχ(y( s )) = on y() (53)

16 74 Yu Xiao & Kaushik Bhattacharya and µ Nd W ezf φ = N d in, (54) ezn d µ ρc W ezn d φ = f in, (55) µ ρc W φ = in, (56) ρ c J Nd ( ( ) ) x µnd ezf µ ρc + ezf x µ ρc in, (57) J ρc x µ ρc in. (58) Equations (5) and (51) are, respectively, the equilibrium equation of polarization and its boundary condition. Equation (5) is the force equilibrium equation with boundary condition (53). If we define the Cauchy stress tensor as and also notice that σ = 1 J ( W F ) F T, (59) ( y y φ ) p ρ y φ = φ,ij p j ρφ,i = φ,ij (D j + ɛ φ, j ) φ,i D j, j = (φ,ij D j + φ,i D j, j ) ɛ φ,ij φ, j = ( φ,i D j ), j ( = E i D j ɛ = y = y T M, Equations (5) and (53) can then be rewritten as ( ɛ φ,kφ,k δ ij ), j y φ ) δ ij, j ( E D ɛ E EI ) y (σ + T M ) = in y(), (6) σ ˆn T M ˆn tχ (y( s )) = on y(). (61) From Equations (54), (55), and (56), we have µ Nd = W + ezf φ, N d (6) µ ρc = W + φ, ρ c (63) W f = ezn d W ρ c. (64) Equation (6) tells us that the chemical potential of defects consists of two parts: a compositional contribution W N d and a electrostatic contribution ezf φ. imilarly,

17 A Continuum Theory of Deformable, emiconducting Ferroelectrics 75 Equation (63) indicates that the chemical potential of free charges also has two parts: a compositional contribution W ρ c and an electrostatic contribution φ. Further, Equation (64) implies that the ionization ratio of defects f and the free charge density ρ c are indeed dependent on each other. In order to satisfy Equations (57) and (58), we make the additional constitutive assumption that [ ( ) ] J Nd = K 1 x µnd ezf µ ρc + ezf x µ ρc, [ = K 1 x µ Nd ezµ ρc x f ] (65) J ρc = K x µ ρc (66) for some positive-definite symmetric tensors K 1, K. Physically, K is the conductivity of the solid and K 1 is the diffusivity of defects. Equations (6), (63), (64), (65), and (66), together with the continuity equations (7) and (8), describe the two diffusion processes. The first one is the diffusion of oxygen vacancies, Ṅ d = x {K 1 [ x ( W N d ezf W ρ c ) ( )]} W + ezf x + φ, (67) ρ c and the second one is the diffusion of charges, ρ = x (J ) ρc + ezf J Nd { ( ) [ ( W W = x K x + φ + ezf K 1 x ezf W ) ρ c N d ρ c ( )]} W + ezf x + φ. (68) ρ c Notice that f in Equations (67) and (68) is determined by Equation (64). In summary, Equations (5), (59), (6), (64), (67), and (68), plus Maxwell s equation (1) are the governing equations subject to the boundary conditions (11), (51), and (61), plus suitable boundary and initial conditions for N d and ρ. We now discuss some special cases..6. pecial cases.6.1. Transient conduction The diffusion of free charges is usually much faster than the diffusion of defects (min λ(k ) max λ(k 1 ), where λ(k) is an eigenvalue of K). By choice, we can consider phenomena on timescales on which there is no diffusion of defects but only transport of free charges. We do so by setting K 1 =, so that Ṅ d = and N d is fixed and given. Thus, ( ( )) W ρ = x K x + φ. (69) ρ c In summary, in this case, we have Equations (5), (59), (6), (69), and (1) asthe governing equations, plus Equations (51), (61), and (11) as their boundary conditions.

18 76 Yu Xiao & Kaushik Bhattacharya We point out that the real transportation of free carriers in semiconductors is usually more complicated than that described by Equation (69). In Equation (69), electrons and holes are assumed to have the same mobility, which is usually not the case []. Further, the recombination dynamics of holes and electrons are not considered here. However, our derivation can be modified to account for those phenomena..6.. Defect diffusion We now work on a time scale appropriate for the diffusion of defects. We do so by assuming K =, so that the free charges ρ c are always in equilibrium, or J ρc = and f =. TheassumptionthatK =, J ρc =, together with Equation (66), leads to µ ρc = W + φ =, (7) ρ c which may be interpreted as a constitutive relation between the free charge density and electric potential. If W is convex in ρ c, we can invert Equation (7) to obtain ρ c = ρ c (φ, N d, f, p, x p, x y). If we further assume that W = W d (N d, f,ρ c ) + W e (p, x p, x y), (71) we have ρ = ρ (φ, N d ) (7) 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]: ( ) ( ) Efm E c + eφ Ev eφ E fm ρ(φ, N d ) = en c F1 + en v F1 K b T K b T 1 +zen d 1 ( ) 1 + g 1 exp Ed eφ E fm K b T z 1 en a 1 ( ), (73) g exp Efm +eφ E a K b T where N c and N v are the effective density of states in the conduction and valence band, respectively, E c is the energy at the bottom of the conduction band, E v is the energy at the top of the valence band, E d and E a are the donor and acceptor levels, respectively, K b is the Boltzmann constant, T is the absolute temperature, F1 is the Dirac-Fermi integral, E fm is the Fermi level of the semiconductor, and g (g ) is the ground-state degeneracy of the donor (acceptor) level and equals for z(z ) = 1. The first two terms in Equation (73) calculate the electrons in the conduction band and the holes in valence band, respectively, while the last two terms are the contributions from donors and acceptors. The relation (73) wegive here is a general relation including acceptors in the current configuration. For our case, N a =. We notice that Equation (64) is implicitly embodied in this relation:

19 A Continuum Theory of Deformable, emiconducting Ferroelectrics 77 the ionization ratio of defects f is a function of φ and equals the expression in the large brackets in the third term on the right-hand side, that is, ( f (φ) = ( )) g exp Ed eφ E 1 fm. (74) K b T Finally, we point out that E fm of the semiconductor varies with its doping type and dopant/defect level, and if the semiconduction is in contact with a metal, E fm should be equal to the metal s Fermi level. Therefore, instead of solving the diffusion equation for ρ, we only need to solve the Maxwell equation (1) with ρ determined by Equation (73). Furthermore, if we assume that the total number of defects is conserved in and that the diffusivity of oxygen vacancies is isotropic (K 1 = k 1 I), then the final diffusion equation we need to solve is ( ( )) Wd Ṅ d x β N d x + ezf (φ)φ = in, (75) N d where β = k 1 /N d is the defect s mobility, with an integral constraint d N d dx =. (76) dt We point out that, by splitting the stored energy functional W into two parts as in Equation (71), we ignore the effect of strain/stress on the concentration of defects N d. Consequently, the stress-induced diffusion of N d is not considered. In summary, we now need to solve Equations (1), (5), (59), (6), (73), (74), and (75), subject to the boundary conditions (11), (51), and (61), and an integral constraint (76) teady state In this case, we assume K 1 = and K =, that is, we assume that the defects are immobile and that free charges adjust themselves into thermal equilibrium in no time. Consequently, we only need to solve Equations (1), (5), (59), (6), and (73) subject to the boundary conditions (11), (51), and (61) Polarization evolution The polarization equation (5) that we derived is for the equilibrium state. One way of quickly deriving equations for polarization evolution is to state that it has to overcome a dissipation of µṗ dx. Hence, D = F de µṗ dx. (77) dt By recourse to Equation (49) and the argument by [11], we obtain, ( f [p ] µṗ ) ṗ, (78) where ( ) W f [p ]= x W F T x φ. (79) x p p

20 78 Yu Xiao & Kaushik Bhattacharya Consider any process p (t) such that p (t ) = p, ṗ (t ) = q =. We have for Equation (78) that ( f [ p ] µ q ) q. Now let us consider another process ˆp (t) with ˆp (t ) = p, ˆp (t ) = q. Then which means ( f [ p ]+µ q ) q = ( f [ p ] µ q ) q µ q, ( f [p ] µ q ) q. (8) In view of Equation (78) and (8), we therefore conclude: f [ p ] µ q =. (81) ince p was an arbitrary process, we conclude that ( ) W µṗ = x W F T x φ. (8) x p p Equation (8) is of gradient flow type, which has been widely used in simulating the phase evolution problems [1,1,,1,51,5] Infinitesimal displacement We have derived our force equilibrium equation (6) from a general setting with finite deformations. However, searching for a suitable constitutive relation like Equation (59) and working with finite deformation is a demanding task, even if we assume that the stored energy functional W has a simple form such as the one in Equation (71). For some materials, like BaTiO 3, a small-strain description often suffices. The Devonshire-Ginzburg-Landau (DGL) energy [15 17], widely used in the ferroelectric community, is such an example, which assumes that the polarization- and deformation-related energy W e 4 in Equation (71) depends on the infinitesimal strain ε instead of the deformation gradient F, that is, W e = W e ( p, p, ε) (83) with ε = 1 ( u + ( u) T ) (84) being the infinitesimal strain and u the displacement. The Maxwell stress is usually omitted in this setting, and the Cauchy stress is defined as σ = W e( p, p, ε). (85) ε Consequently, the force equilibrium equation (6) becomes σ = in. (86) 4 ince we now consider small strain, there is no need to differentiate between the reference and current configuration; the subscript is therefore omitted.

21 A Continuum Theory of Deformable, emiconducting Ferroelectrics 79 Pt y Pt BaTiO3 single crystal z x Fig. 4. Computation domains However, the DGL energy W e ( p, p, ε) is not a genuine linearization of W e ( p, p, F). When an energy functional W is a function of F only, it is well known that frame indifference leads to W (F) = Ŵ (C), where C = F T F. = 1 + ε. However, when W = W (p, F), in general, we do not have W (p, F) = Ŵ (p, C), which means, when we linearize W using infinitesimal displacement gradient while keeping 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 complete linearization of both displacement gradient and polarization can lead to the usual piezoelectric constitutive relation: σ ij = C ijkl ε kl + α ijk p k. (88) Unfortunately, this is not suitable for ferroelectric materials. Fortunately, the DGL energy is sufficient for most ferroelectric materials. For a detailed discussion on this 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 with platinum electrodes in a parallel-plate capacitor geometry, as shown in Fig. 4. This is a common geometry in many applications. We assume that the barium titanate is a single crystal in the tetragonally polarized room-temperature phase, and that the crystallographic [1] c direction coincides with the normal to the slab. We assume that the material, polarization distribution, and other quantities are invariant along the z direction, and restrict ourselves to two dimensions. We also restrict ourselves to infinitesimal displacements, as described in ection.6.5. We assume that the stored energy function W takes a simple additive form: W = W d (N d, f,ρ c ) + W g ( p) + W p (p, ε). (89) The first term is the energy of defects and charges. The second term, associated with the polarization gradient, penalizes rapid changes of polarization. The

22 8 Yu Xiao & Kaushik Bhattacharya last term is the energy of deformation and polarization; it contains important crystallographic information. Finally, we assume that K = and K 1 = k 1 I. Under these assumptions, the governing equations are now: ( ( )) Wd Ṅ d β N d + ezf (φ)φ = in, (9) N d ( ) dwg µṗ + W p + φ = in, (91) d p p σ = in, (9) ( ɛ φ + p χ()) = ρ(φ, N d )χ() in R 3. (93) with the constitutive Equations (59), (73), and (74), and subject to appropriate boundary conditions, initial conditions, and the constraint Equation (76) Normalization and parameter selection Following [5], we choose the stored energy of polarization gradient, polarization, and deformation to be the Devonshire-Ginzburg-Landau energy [15 17] with slight modification. pecifically, we choose W g to be where and p := p p := Trace W g ( p) = a p, (94) ( p p T ) = p x,x + p x,y + p y,x + p y,y, (95) W p (p, ε) = a 1 (p x + p y ) + a 4 (p4 x + p4 y ) + a 3 p x p y + a 4 6 (p6 x + p6 y ) + a 5 4 (p4 x p4 y ) b 1 (ε xx px + ε yy p y ) b (ε xx p y + ε yy px ) b 3 ε xy p x p y + c 1 (ε xx + ε yy ) + c ε xx ε yy + c 3 ε xy. (96) For computational purpose and for better interpretation of domain structures, the variables are normalized as follows: with characteristic constants, x = x, t = t, W g L T = W g, W p c = W p, c p = p, φ = φ, ρ = ρ/ρ, σ = σ (97) p φ c c = 1GPa, p =.6 C/m, L = p a c, φ = a c, ρ = c a, T = µp c. (98)

23 A Continuum Theory of Deformable, emiconducting Ferroelectrics 81 Here, p =.6 C/m is chosen to be the spontaneous polarization of BaTiO 3 at room temperature, so that the normalized spontaneous polarization is 1. The resultant W g and W p have the same forms as W g and W p, with a i, b j, and c j being replaced, respectively, by a i, b j and c j,fori =,...,5 and j = 1,, 3. pecifically, a = 1, a 1 = a 1 p /c, a = a p 4 /c, a 3 = a 3 p 4 /c, a 4 = a 4 p 6 /c, a 5 = a 5 p 8 /c, b j = b j p /c, and c j = c j/c. We point out that specific choices of c and L are made so that a = 1, and both the domain wall thickness5 and the normalized elastic moduli are of moderate range (1 1). This choice also has the feature that the normalized solution for the classical perfect crystal case does not depend on a. However, a does play a very important role in defected crystals, and therefore it is important to decide the range of a. From experimental data [36] and firstprinciple calculations [3], it is believed that the domain wall thickness is usually about 1 1 nm, although thicknesses as large as 15 nm have been reported in LiNbO 3 [44]. Here, we will work on two cases: a = 1 9 Vm 3 C 1, corresponding to a domain wall thickness of a few nanometers, and a = 1 7 Vm 3 C 1 for domain walls one order thicker. Other material constants we choose are [5]: c 1 = 185, c = 111, c 3 = 54, b 1 = 1.48, b =.185, b 3 =.5886, a 1 =.7, a =.9, a 3 =.3, a 4 =.61, and a 5 = 5. Notice that a 1 and a are both negative since the cubic-to-tetragonal phase transition of BaTiO 3 is a first order phase transition [3,35]. It is advantageous to rewrite W p as ( W p (p, ε) = a 1 + p a ) y ) + (p x 4 + p 4 y ) (p x + ( a ) 3 f 4 d p x p a y (p 6 x + p 6 a y ) (p 4 x p 4 y ) + 1 (ε ε s) C (ε ε s ), (99) so that the normalized stress σ = σ /c can be easily written as σ = C (ε ε s ), (1) where C is the normalized stiffness matrix, c C 1 c = c c 1 (11) c 3 in Voigt notation, ε s the eigenstrain caused by spontaneous polarization, a p x b p y ε s = b p x a p y (1) c p x p y, 5 The domain wall thickness is proportional to a / a 1 [18].

24 8 Yu Xiao & Kaushik Bhattacharya and a = b 1 c 1 b c (c 1 c =.65, (13) ) b = b c 1 b 1 c (c 1 c =.44, (14) ) c = b 3 c 3 =.19, (15) d = b 1 b c + (b 1 + b )c 1 8(c 1 c ) f = b 1 b c 1 (b 1 + b )c 4(c 1 c ) =.5, (16) b 3 c 3 =.5. (17) With Equation (3.1), we can easily see the meaning of each term in the energy functional 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 to the four spontaneous states: p x =±1, p y = orp x =, p y =±1. The energy barrier between different wells is E b = ,orE b = 3.94 MPa, which is about the right range for BaTiO 3 [53]. From Equation (1), we notice that the c/a ratio of the tetragonal phase of BaTiO 3 is (1 + a )/(1 + b ) = 1.19, which is consistent with experimental data. Finally, the normalized versions of Equations (91) (93) are: ( ) µ p dw t g d p + W p p + φ =, (18) σ =, (19) ( ɛ φ + p χ()) = ρ (φ, N d ) (11) with material constants µ = 1, ɛ = ɛ c p =.131. (111) We point out that, although a does not explicitly appear in the normalized equations, it is implicitly included in ρ (φ ) in Equation (11) since ρ (φ ) = 1 { ( Efm E c + eφ φ ) ( Ev eφ φ ) E fm en c F1 + en v F1 ρ K b T K b T 1 +zen d 1 ( ) exp Ed eφ φ E fm K b T z 1 en a 1 ( ) (11) exp Efm +eφ φ E a K b T

25 A Continuum Theory of Deformable, emiconducting Ferroelectrics 83 and ρ, φ defined by Equation (98) depend on a. For example, for a = 1 9 Vm 3 C 1, ρ = 1 9 Cm 3 and φ = 1 V, while for a = 1 7 Vm 3 C 1, ρ = 1 8 Cm 3 and φ = 1 V. Therefore, the solution of the normalized equations still depends on a, and thus depends on the physical thickness of domain walls. The only exception is when ρ (φ ). However, even in this case, a still enters since the size of the computational specimen depends on a. We now specify the material constants we use for BaTiO 3 in Equation (11). N c (N v ) is the effective density of states in the conduction (valence) band, and is approximately m 3 [,38]. K b is the Boltzmann constant, JK 1. T, the absolute temperature, is set to be 3 K. The Fermi level E fm of platinum electrodes is 5.3 ev. The constants of the band structure of BaTiO 3 are chosen to be [34]: E c = 3.6eV, E d = 4.eV, E a = 6. ev, and E v = 6.6eV. As to the defects, we set N a = here since we are mainly interested in oxygen vacancies, which act like donors, and we estimate the oxygen vacancy density N d as follows. For BaTiO 3, a = 3.99 Å and c = Å [7], therefore the volume of a unit cell is about 6 Å 3. ince there are five atoms per unit cell, the volume per atom is approximately 1 Å 3, thus the total atoms sites per unit volume is N t = 1 9 m 3. According to [53], the oxygen vacancy density ranges from 1 to 1 ppm, corresponding to N d of m 3. The nominal valency of oxygen vacancy z is equal to, although the effective valency is usually less [53]. We choose z = 1 here. We now consider the diffusion equation (9), which can also be written as: Ṅ d = J, (113) ( ) Wd J = βn d + ezf (φ)φ, (114) N d where J is the flux of oxygen vacancies. The constraint Equation (76) is equivalent to the zero normal flux boundary condition, that is, J n = J ˆn =. (115) We assume that the part of W d that explicitly depends on N d is the usual free energy of mixing at small concentration [9]. This leads to: W d = µ v, (116) N d N a where N a is Avagadro s number, and µ v = G v + (1 C) + N a K b T ln C (117) is the partial molar free energy of vacancies. Here, C = N d /N t is the mole fraction of vacancies, G v the molar free energy of vacancies when C = 1, and = N a qɛ with q being the number of bonds per atom and ɛ the energy difference per bond with and without vacancies.

26 84 Yu Xiao & Kaushik Bhattacharya ubstituting Equations (116) and (117) into Equation (114), we have J = d N d βezn d ( f (φ)φ), (118) where d = β K b TF (119) is the diffusion coefficient, and F is the thermodynamic factor, defined as (1 C)C F = 1. (1) N a K b T For the dilute case, C 1, F is approximately equal to 1. For simplicity, we only work with the dilute case here. Equations (113) and (118) can be normalized as C t = J, (11) J = α C αγc ( f (φ )φ ), (1) where J = T J, α = T L N t L d, γ = ezφ K b T, (13) and t, x, and φ are defined by Equation (97), and L, φ, and T by Equation (98). In a steady state, C/ t =, 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 + γ C d f (φ )φ dx = (14) with +L C dx = LC (15) L where C is the average defect concentration, or C = Q exp( γ f (φ )φ ) (16) with LC Q = L L exp( γ f (φ )φ ) dx. (17) From Equation (16), we can see that the steady defect concentration has an exponential relation with f (φ)φ. γ, defined in Equation (13), is approximately equal to 4 φ at room temperature with z chosen to be 1. Therefore, a slight difference of potential φ will result in a huge difference in defect concentration. This poses a computational challenge, and it is physically unlikely since Equation (16) is based on the assumption that we are working in the dilute case and the thermodynamic factor F can be approximated by 1. Indeed, in most diffusion processes in semiconductors, the diffusion coefficient d, defined in Equation (119), is not a constant, and it approaches zero when the diffused species reaches a saturated value [38]. Nevertheless, the potential difference is indeed a crucial driving force for defect redistribution, and a small potential difference does result in a big difference in defect concentration. Therefore, we choose γ = 4φ for our numerical calculation here.

27 A Continuum Theory of Deformable, emiconducting Ferroelectrics Results of numerical computation We use the finite-element method to obtain numerical solutions to Equations (91) (93) on a 4 rectangular domain. In this section we take the defect density N d to be uniform and constant at a value of m 3, and hence we do not consider equation (9). This is because the diffusion of vacancies is much slower than the revolution of polarization, or, α 1 in Equation(1). o the results of this section correspond to a time scale that is too short for the diffusion of defects but long enough for the formation of domains. We choose a = Vm 3 C 1 here, which means the slab size is 1,4 nm 5 nm. a is high and corresponds to a domain wall width of 5 nm, but is convenient for visualization 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 x displacement along the left side; stress free on other sides. We consider two initial conditions, one with a 18 and another with a 9 domain wall inside the computational domain. We integrate the equations forward in time until equilibrium is reached. The resulting space charge distribution and electrostatic potential are shown in Fig. 5. For comparison, we have also computed the results in a classical theory (perfectly insulating crystal with no dopants or space charge density) for a double-sized slab, and Fig. 6 shows this electrostatic potential. 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). This is the so-called depletion layer that is a typical feature of a metal semiconductor interface. This depletion layer is accompanied by a large electric field, and this can aid the injection of charges from the electrodes, which is a mechanism of electrical fatigue. uch a depletion layer is completely absent in the classical calculation (Fig. 6). In the case of the 18 domain walls (Fig. 5a), the depletion layer overwhelms any contribution from the domain wall and we barely notice the domain wall in either the distribution of potential or charges. Thus we conclude that 18 domain walls have very little interaction with oxygen vacancies and effect on their diffusion. In contrast, there is a significant interaction between the 9 domain walls and 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 along the domain wall. In other words, electrons are injected from the electrodes and trapped at the domain wall at equilibrium. The reason for this can be understood by going back to the electric potential in Fig. 6 without defects and noting the large electric field at the domain wall; this drives the injection of charges into the domain wall. 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 of domain walls. An electrostatic feature is visible where the 18 meets the electrode in the classical calculation (Fig. 6); such features are completely masked by the depletion layer in the current theory. (Note that the two figures are plotted with different ranges of potential.)

28 86 Yu Xiao & Kaushik Bhattacharya Fig. 5. The electric potential (V) and charge densities (C m 3 )neara 18 and b 9 domain walls. The location of domain walls is marked by white dashed lines, and the polarization directions are indicated by white arrows. The density of oxygen vacancies N d = 1 4 m 3 and a = 1 7 Vm 3 C 1. The last is high, corresponding to a domain wall width of 5 nm, but is convenient for visualization of the qualitative features Figures 7 and 8 show the stress and the strain distributions computed using the current theory. These are raised near the domain walls, and have a concentration near where the domain wall meets the electrodes. These strain/stress concentration sites 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 magnitude of the stress concentration between the 18 and 9 domain walls. This is a little surprising, since conventional wisdom states that 9 domain walls undergo much more distortion than 18 domain walls. The results obtained by the classical theory are similar and are not displayed here. We refer the reader to [46] for these and further calculations where the deformation is completely ignored.

29 A Continuum Theory of Deformable, emiconducting Ferroelectrics 87 Fig. 6. Electric potential near 18 (upper) and 9 (lower) domain walls in perfect crystals. The real value of φ depends on a, the number (V) shown here is obtained by choosing a = Vm 3 C Depletion layers We now seek to examine the depletion layers closely, and also to understand any possible diffusion of oxygen vacancies. To do so, we notice from Fig. 5 that, away from any domain wall, depletion layers are essentially one dimensional. o we seek to study the one-dimensional problem along the section marked A A in Fig. 5. To this end, we assume that p ={, p (y)} and that all quantities are independent of x. We also ignore the deformation for simplicity. The governing equations now reduce to: C t = α ( C y y γ C ( f φ ) ) y, (18) µ p t = d p dy dw p d p dφ dy, (19) ɛ d φ d p + dy dy = ρ (φ, N d ). (13) with ( W p = a 1 p a ) + 4 d p 4 a p 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 from zero at the electrode to a value of approximately 1.4 V. This value is known as the build-in potential φ bi, and is approximately equal to the difference between the Fermi levels of the film and the electrode: φ bi = EBT fm EPt fm e 1 e (E c + E d Efm Pt ) = 1.5V. (13)