PHASE-FIELD SIMULATION OF DOMAIN STRUCTURE EVOLUTION IN FERROELECTRIC THIN FILMS

Mat. Res. Soc. Symp. Proc. Vol. 652 2001 Materials Research Society PHASE-FIELD SIMULATION OF DOMAIN STRUCTURE EVOLUTION IN FERROELECTRIC THIN FILMS Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen Department of Materials Science and Engineering The Pennsylvania State University University Park, PA 16802, USA Abstract A phase-field model for predicting the domain structure evolution in constrained ferroelectric thin films is developed. It employs an analytical elastic solution derived for a constrained film with arbitrary eigenstrain distributions. In particular, the model is applied to the domain structure evolution during a cubic tetragonal proper ferroelectric phase transition. The effect of substrate constraint on the volume fractions of domain variants, domain-wall orientations, and domain shapes is studied. It is shown that the predicted results agree very well with existing experimental observations in ferroelectric thin films. Introduction A common feature for ferroelectric materials is the formation of domain structures when a paraelectric phase is cooled through the ferroelectric transition temperature or called the Curie temperature[1, 2]. The crystallography and thermodynamics of domain structures have been extensively studied and reasonably well understood[3]. For example, in a cubic tetragonal transformation, there are three possible orientation variants with the tetragonal axes along the [100], [010], and [001] directions of the cubic paraelectric phase. In the absence of any external field or constraint, all of them have the same probability to form in a parent cubic paraelectric phase below the ferroelectric transition temperature. The corresponding domain structure of the ferroelectric phase will contain all possible orientations of domains with equal volume fractions, separated by the so-called domain walls. Various domain configurations develop to minimize the sum of elastic energy, electrostatic energy, and domain wall energy. The main purpose of this paper is to describe a phase-field approach for predicting the domain structures in constrained ferroelectric thin films. It does not make any a priori assumptions with regard to the possible domain structures that might appear under a given temperature and substrate constraint. It is able to predict not only the effect of substrate constraint on phase transition temperatures and the volume fractions of orientation domains under different types of substrate constraint and film thickness, but also the detailed domain structures and their temporal evolution during a ferroelectric transition. Although similar approaches has been applied to ferroelectric domain evolution in bulk single crystals[8, 9, Y4.2.1

10], the presence of a stress-free surface and lattice constraint by the substrate requires an efficient elastic solution for a constrained three-dimensional film. In this paper, we briefly describe the phase-field model and the method to obtain the elastic solution for a constrained film. Preliminary results from our computer simulations of a cubic tetragonal ferroelectric transition will be presented. Phase field model of a ferroelectric film We consider a cubic thin film grown heteroepitaxially on a cubic substrate. The film undergoes a cubic-to-tetragonal ferroelectric phase transition when it is cooled below the Curie temperature. For a proper ferroelectric phase transition, the polarization vector P =(P 1,P 2,P 3 ) is the primary order parameter and its spatial distribution in the ferroelectric state describes the domain structure. The temporal evolution of the polarization field and thus the domain structure evolution is described by the Time-Dependent Ginzburg-Landau(TDGL) equations[8, 9, 10], P i (x,t) t = L δf, i =1, 2, 3, (1) δp i (x,t) where L is the kinetic coefficient. δf/δp i (x,t) forms the thermodynamic driving force for the spatial and temporal evolution of P i (x,t). F is the total free energy of the system. In this work, we ignore any possibe surface contributions[11, 12] and assume that the film s surface is compensated with free charge carriers so the depolarization energy is neglected. To solve equation (1), one has to formulate the total free energy functional F in terms of the polarization field variables, P 1, P 2 and P 3. We assume the transition is first-order and its bulk thermodynamics is characterized by the following Landau free energy expansion[13], f L (P i ) = α 1 (P1 2 + P2 2 + P3 2 )+α 11 (P1 4 + P2 4 + P3 4 )+α 12 (P1 2 P2 2 + P2 2 P3 2 + P1 2 P3 2 ) +α 111 (P1 6 + P2 6 + P3 6 ) (2) +α 112 [P1 4 (P2 2 + P3 2 )+P2 4 (P1 2 + P3 2 )+P3 4 (P1 2 + P2 2 )] + α 123 (P1 2 P2 2 P3 2 ) where α 1, α 11, α 12, α 111 α 112, α 123 are the expansion coefficients. The values of these coefficients determine the thermodynamic behavior of the bulk paraelectric and ferroelectric phases as well as the bulk ferroelectric properties such as the ferroelectric transition temperature, the stability and metastability of the parent paraelectric phase, the spontaneous polarization as a function of temperature, the susceptibility as a function of temperature, etc. For example, α 1 =1/2ɛ 0 χ,whereɛ 0 is the vacuum permittivity, and χ is the susceptibility of the material. A negative value of α 1 corresponds to an unstable parent paraelectric phase with respect to its transition to the ferroelectric state. A positive α 1 value indicates either a stable or metastable parent phase, depending on the relationships among α 1, α 11, and α 111 ;Ifα 2 11 > 3α 1 α 111, the parent phase is metastable, otherwise it is stable. In this work, we assume that α 1 is negative so the parent phase is unstable. Y4.2.2

The contribution of domain walls to the total free energy, i.e., the domain wall energy, is introduced through gradients of the polarization field. For a cubic system[14], f G (P i,j ) = 1 2 G 11(P 2 1,1 + P 2 2,2 + P 2 3,3)+G 12 (P 1,1 P 2,2 + P 2,2 P 3,3 + P 1,1 P 3,3 ) + 1 2 G 44[(P 1,2 + P 2,1 ) 2 +(P 2,3 + P 3,2 ) 2 (P 1,3 + P 3,1 ) 2 ] + 1 2 G 44[(P 1,2 P 2,1 ) 2 +(P 2,3 P 3,2 ) 2 (P 1,3 P 3,1 ) 2 ] (3) where P i,j = P i / x j and G ij are gradient energy coefficients. In general, the domain wall energy is anisotropic. Since the proper ferroelectric phase transition involves the structural changes from cubic to tetragonal, strain appears as a secondary order parameter. The stress-free strain caused by the polarization field is given[13] ɛ o 11 = Q 11 P1 2 + Q 12 (P2 2 + P3 2 ) (4) ɛ o 22 = Q 11 P2 2 + Q 12 (P1 2 + P3 2 ) (5) ɛ o 33 = Q 11 P3 2 + Q 12 (P1 2 + P2 2 ) (6) ɛ o 23 = Q 44 P 2 P 3 (7) ɛ o 13 = Q 44 P 1 P 3 (8) ɛ o 12 = Q 44 P 1 P 2 (9) where Q ij are the electrostrictive coefficients. If we assume that the interfaces developed during a ferroelectric phase transition as well as the interface between the film and the substrate are coherent, elastic strains will be generated during the phase transition in order to accommodate the structural changes. They are given by e ij = ɛ ij ɛ o ij (10) where ɛ ij are the total strains. The corresponding elastic energy density is f E = 1 2 C 11(e 2 11 + e 2 22 + e 2 33)+C 12 (e 11 e 22 + e 22 e 33 + e 11 e 33 )+2C 44 (e 2 12 + e 2 23 + e 2 13) (11) where C ij is the elastic modulus constants using the Voigt s notation. The total free energy of film is then the sum of the Landau free energy F L, the domain wall energy F G and the elastic energy F E : F = F L (P i )+F G (P i,j )+F E (P i,ɛ ij )= [f L (P i )+f G (P i,j )+f E (P i,ɛ ij )] d 3 x (12) V Y4.2.3

In this model, although both the polarization field and the strain field appear as order parameters, one may assume that the mechanical relaxation of an elastic field is much faster than the relaxation of a polarization field. Consequently, during the process of ferroelectric transition, one can assume that the system reaches its mechanical equilibrium instantaneously for a given polarization field distribution. This assumption enables us to eliminate the strain field using the static condition of mechanical equilibrium. The calculation of the equilibrium elastic strain for the particular thin film boundary condition with substrate constraint is briefly described in the next section. Elastic field in constrained film We consider a thin film heteroepitaxially grown on a substrate. The top surface of the film is stress-free while the bottom surface is coherently constrained by the substrate (Fig.1). Within the film, there is an eigenstrain distribution developed due to a ferroelectric phase transition, ɛ o ij(x), x =(x 1,x 2,x 3 ) and (i, j = 1, 2, 3). The rectangular coordinates (x 1,x 2,x 3 ) are originated at the interface and x 3 is normal to the film. In Fig.1, h f is the thickness of the film. Because the thickness of the film is much smaller than its in-plane size, the x 1 x 2 plane is regarded as infinite. The elastic strain is related to stress σ ij by Hooke s law, σ ij = c ijkl e kl = c ijkl (ɛ ij ɛ o ij), where c ijkl are the components of elastic modulus tensor, and the summation convention for the repeated indices is employed. h f h s film substrate Figure 1: Schematic illustration of a thin film coherently constrained by a substrate. The mechanical equilibrium equation is given by σ ij =0, (i =1, 2, 3). (13) x j subject to the stress-free boundary condition at the top surface, σ 3j (x 1,x 2,x 3 = h f )=0, (14) For the substrate, stresses and displacements must occur due to the eigenstrain distribution, ɛ o ij in the film although ɛ o ij = 0 in the substrate. The stresses and displacements across the film-substrate interface are continuous. Generally, the thickness of a substrate is much larger than that of the film, so we consider the substrate to be an infinitely extended half-space body. Stresses in the substrate must disappear far away from the interface, i.e. Y4.2.4

σ ij (x 1,x 2,x 3 ) 0, x 3. (15) To obtain the elastic solution, we separate the total strain of the system into a sum of an average macroscopic strain and a heterogeneous strain, ɛ ij (x) = ɛ ij + η ij (x). (16) We assume that the average macroscopic in-plane strain of the film is totally controlled by the sufficiently thick substrate. For example, for a [001] oriented cubic substrate lattice parameter a s and a film with cubic lattice parameter a f, the mean in-plane strains of the film are ɛ 11 = ɛ 22 = a s a f, ɛ 12 =0. (17) a s The heterogeneous strain along the (x 1,x 2 ) plane is given by η αβ (x) d 3 x =0, (α, β =1, 2) (18) V where V is the volume of the film. The actual macroscopic shape deformation of the film along x 3 is determined by the elastic solution for the whole system which satisfies the boundary conditions. We choose the quantity ɛ 3j,(j =1, 2, 3) of eq.(16) in such a way that it satisfies σ 3j = c 3jkl ɛ kl =0, (j =1, 2, 3). (19) It should be emphasized that, with the definition (19), the heterogeneous strain η 3j do not satisfy equation (18) and the average strain ɛ 3j do not represent the actual macroscopic shape deformation along x 3 direction since it does not contain the contributions from the eigenstrain distribution within the film and the stress-free boundary condition. The latter two contributions are obtained by solving the elastic equations for the heterogeneous strains. To solve the heterogeneous strain, η ij, we introduce a new set of displacements u i (x), η ij = 1 ( ui + u ) j, (20) 2 x j x i The equations of equilibrium can be rewritten as The associated stresses are calculated by c ijkl u k,lj = c ijkl ɛ o kl,j (21) σ ij = c ijkl (u k,l ɛ o kl) (22) and meet the requirement of the stress-free boundary condition (14). In the following, the condition (15) is replaced by u i (x 1,x 2,x 3 = h s )=0. (23) where h s is the distance from the film-substrate interface into the substrate, beyond which the elastic deformation can be ignored (Fig.1). For the sake of simplicity for deriving an analytical solution, the elastic properties of the film and the substrate are taken to be homogeneous and isotropic. The case of elastically anisotropic modulus tensor will be discussed elsewhere. For an isotropic material, Y4.2.5

c ijkl = λδ ij δ kl + µδ ik δ jl + µδ il δ jk, λ =2µν/(1 2ν), where µ is the shear modulus, and ν =1/m is the Poisson ratio, m is the Poisson constant. Because of the limitation on the space for this paper, we will simply outline procedures to obtain the elastic solution. We take two steps. First, we use a three-dimensional (3D) Fourier transform to solve equation(21) in an infinite 3D space with eigenstrain distribution ɛ o ij within 0 <x 3 <h f [15, 16]. From the full 3D solution, the stress at h f, σ3j(x A 1,x 2,h f ), (j =1, 2, 3), can be calculated. Similarly, we can calculate the displacements at x 3 = h s : u A i (x 1,x 2, h s ), (i =1, 2, 3). The next step is to find an elastic solution denoted with superscript B in an infinite plate of thickness h f + h s with no body force, but subject to the following boundary conditions: σ B 3j(x 1,x 2,h f )= σ A 3j(x 1,x 2,h f ), (24) u B j (x 1,x 2, h s )= u A j (x 1,x 2, h s ), (25) j =1, 2, 3, <x 1,x 2 <. The sum of solutions A and B yields the solution for the boundary value problem of equations(14,20-23). With the elastic solution, the total free energy of the system is only a functional of the polarization field. Simulation results and discussions The temporal evolution of the polarization vector fields, and thus the domain structures, is obtained by numerically solving the TDGL equations (1). In particular, we used the semiimplicit Fourier-spectral method[17] for the time-stepping and spatial discretization. Since, experimentally, lead titanate (PbTiO 3 ) thin film is one of the most extensively studied systems, we use it as an example for the numerical simulation. The corresponding material constants are from the literature[18] and [19] (in SI units and temperature in degree C: α 1 =3.8(T 479) 10 5, α 11 = 7.3 10 7, α 12 =7.5 10 8, α 111 =2.6 10 8, α 112 =6.1 10 8, α 123 = 3.7 10 9, Q 11 =0.089, Q 12 = 0.026, Q 44 =0.03375, µ =4.762 10 10, ν =0.312. In the computer simulations, we defined reduced units following reference [9]. In the reduced units, the gradient energy coefficients are G 11 =0.6, G 12 =0,G 44 = G 44 =0.3. We employed a 128 128 36 discrete grid points and the grid spacing spacing in real space is chosen to be x = y =1.0 and z =0.5. Periodic boundary conditions are applied along the x and y axes. The step for integration is t =0.06. We first studied the effect of h s on the volume fractions of different orientation domains and the domain morphology. The value of h s represents the region of the substrate that is allowed to deform, and thus to a certain degree, it represents the rigidity of the substrate. For example, h s = 0 represents a complete rigid substrate. If the substrate is deformable with the same elastic moduli as the film, the displacements should be zero at a distance sufficiently far away from the interface. Therefore, as h s increases, the results should converge. We considered a particular case of ɛ = 0.002, T = 25 C, and film thickness h f = 20 z. The formation and evolution of ferroelectric domains in the film are simulated for h s = 0, 2 z, 4 z,..., 12 z. The initial values for the polarization field are assigned zero everywhere with small random perturbations, corresponding to a paraelectric state. During the simulation,polarization field in the substrate is kept equal to zero. To characterize a domain structure, the three tetragonal variants of the ferroelectric tetragonal phase are denoted as a 1, a 2 and c domains according to their c axis orientation. The c axis of a c domain is normal to film surface. An a domains has its c axis parallel to the 1 Y4.2.6

film/substrate interface and aligned with one of the cube directions (x 1 axis) of the substrate. Similarly, an a 2 domain has its c axis parallel to the film/substrate interface but rotated 90 from the a 1 domain (along x 2 axis). We calculated the volume fraction of c domains, V c, from the simulated domain structures as a function of h s and present the results in Table 1. Since we assume the cubic substrate has a [001] orientation, the volume fractions of a 1 and a 2 domains are approximately the same, i.e. V a1. = Va2 =(1 V c )/2. It can be seen from Table 1 that the volume fraction V c varies with h s and approaches to a constant value as h s increases. Table 1. Volume fraction of c-domains, V c, as a function of h s h s 0 z 2 z 4 z 6 z 8 z 10 z 12 z V c 0.800 0.787 0.775 0.767 0.762 0.756 0.753 It is also interesting to examine the h s=0.0 effect of h s on the domain wall orientations and the domain shapes. Shown h s=0.2h f in Fig.2 are the 2D sections of domain structures cut at the same position for h s=0.4h different values of h s. It is shown that f for h s =0.0, the width of a 2 domains become smaller and sharpen close to h s=0.6h f the film-substrate interface. As h s increases, the a 2 domain shapes show showing the dependence of the a 2 -domain shapes on h s. Figure 2: Some 2D cross-sections of 3D domain structures some changes near the film-substrate interface. However, even with a relatively large value of h s, the width of a 2 domains is slightly smaller at the substrate interface than that close to the surface. It is shown that the domain shapes practically do not change when h s exceeds half of the film thickness. We studied the effect of substrate constraint on the volume fractions of domains and domain morphology at T =25 C. The effect of substrate is reflected by the macroscopic average strain. We consider the simple case of a cubic substrate with the [001] orientation, and hence ɛ 11 = ɛ 22 = ɛ. We allow elastic deformation in the substrate by choosing a value of h s = 12 z =0.6h f beyond which our simulations showed little changes in the results (Table 1 and Fig.2). The volume fraction of c domains, V c, as a function of mismatch strain is given in Table 2. It is shown that the substrate constraint can dramatically alter the volume fractions of different orientation domains. The volume fraction of c domains decreases as the magnitude of ɛ increases and there are no c domains when ɛ =0.02 or larger. Table 2. Volume fraction of c-domains, V c, as a function of ɛ ɛ -0.012-0.008-0.006-0.004-0.002 0.0 0.002 0.006 0.020 V c 1.0 0.97 0.901 0.832 0.751 0.646 0.539 0.261 0.0 Y4.2.7

It is noted that a positive value of ɛ indicates a tensile constraint by the substrate. Therefore, under a large tensile stress, a 1 and a 2 domains dominate. An example of domain structure for ɛ =0.02 is shown in Fig.3 in which only a 1 and a 2 domains exist. In Fig.3, only the domain-walls are shown and the a 1 -domains and a 2 domains are not distinguished. Essentially all the domain walls are perpendicular to the film surface and along the [110] or [1 10] directions parallel to Figure 3: A predicted domain structure consisting of a 1 and a the surface. The domain structure 2 domains is very similar to those predicted for two-dimensional square to rectangular structural transformations[20]. The domain structures with a compressive or very small tensile stress are dramatically different from that in Fig.3. An example of a domain structure obtained for the case of ɛ =0.0 is shown in Fig.4. The volume fraction of c domainsisroughly 0.65. The a 1 and a 2 -domains are plates aligning about 45 degrees from the substrate (the c-domains are not shown in the figure). The shapes of a 1 and a 2 domains under different degrees of compressive substrate constraints are similar to those in Fig. 4 although their volume fraction will Figure 4: Three dimensional representation of a 1 and a 2 domains become less and less as the compressive substrate misfit strain increases (Table 2). The a domain orientation and shapes predicted from the simulations under a compressive substrate constraint agree very well with existing experimental observations [21, 22]. The volume fractions of domains and domain morphology also depend strongly on the temperature and results of our simulations on temperature dependence of domain structures will be reported elsewhere. Y4.2.8

Conclusions A phase-field model for predicting the domain structure evolution in constrained ferroelectric thin films is proposed for the first time. An analytical elastic solution was derived for a constrained film with arbitrary eigenstrain distributions. We studied the domain structure evolution during a cubic-tetragonal proper ferroelectric phase transition in a film constrained by a cubic substrate with the [001] orientation. It is shown that the rigidity of the substrate affects the domain morphology close to the coherent film-substrate interface. The volume fractions of domains and the domain structures strongly depend on the degree of lattice mismatch between the film and the substrate. Depending on the misfit strain, the domain structures range from a single c domain to a two-domain state with only a 1 and a 2 domains. The domain-wall orientations under a compressive stress are along the [101] and [110] directions and 45 degrees away from the substrate-film interface. A large tensile stress produces domain walls perpendicular to the film-substrate interface separating the a 1 and a 2 domains. The predicted volume fractions of domain variants, domain-wall orientations, and domain shapes under a given substrate constraint agree very well with existing experimental observations of domain structures in ferroelectric thin films. References [1] M.E.Lines and A.M.Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 1977. [2] L.E.Cross, Ferroelectric Ceramics: Tailoring Properties for Specific Applications, pp.1-85 in Ferroelectric Ceramics, Birkhauser Verlag, Basel, Switzerland, 1993. [3] G. Arlt, Ferroelectrics 104, 217 (1990). [4] J. S. Speck and W. Pompe, J. of Appl. Phys. 76, 466 (1994). [5] A. L. Roytburd, J. of Appl. Phys. 83, 228 (1998) [6] A. L. Roytburd, J. of Appl. Phys. 83, 239 (1998) [7] N. A. Pertsev and V. G. Koukhar, Phys. Rev. Lett. 84, 3722(2000). [8] S. Nambu and D. A. Sagala, Phys. Rev. B 50, 5838(1994) [9] H. L. Hu and L. Q. Chen, J. Am. Ceram. Soc. 81, 492 (1998). [10] S. Semenovskaya and A. G. Khachaturyan, J. Appl. Phys. 83, 5125 (1998). [11] K. Binder, Ferroelectrics 35, 99 (1981). [12] D. R. Tilley and B. Zeks, Solid State Commu. 49, 823 (1984). [13] A. F. Devonshire, Phil. Mag. Suppl. 3, 85 (1954). [14] W.Cao and L.E.Cross, Phys.Rev.B 44 5(1991). [15] A.G.Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983. Y4.2.9

[16] T. Mura, Micromechanics in Solids, Kluwer Academic Publishers, 1982. [17] L. Q. Chen and J. Shen, Comp. Phys. Comm. 108, 147(1998). [18] M. J. Haun, E. Furman, S. J. Jang, H. A. Mckinstry and L. E. Cross, J. Appl. Phys. 62, 3331(1987). [19] N. A. Pertsev, A. G. Zembilgotov, and A. K. Tabantev, Phys. Rev. Lett. 80, 1988(1998). [20] L. Q. Chen and A. G. Khachaturyan, Phil. Mag. Lett. 65, 15(1992). [21] A. Seifert, F. F. Lange, and J. S. Speck, J. Mater. Res. 10, 680(1995). [22] S. P. Alpay, V. Nagarajan, L. A. Bendersky, M. D. Vaudin, S. Aggarwal, R. Ramesh and A. L. Roytburd, J. of Appl. Phys. 85, 3271(1999). Y4.2.10