Domain Dynamics in a Ferroelastic Epilayer on a Paraelastic Substrate

Transcription

1 Y. F. Gao Z. Suo Mechanical and Aerosace Engineering Deartment and Princeton Materials Institute, Princeton University, Princeton, NJ Domain Dynamics in a Ferroelastic Eilayer on a Paraelastic Substrate This aer models the domain dynamics in a ferroelastic eilayer within the timedeendent Ginzburg-Landau (TDGL) framework. Constrained on a araelastic substrate of square symmetry, the eilayer has rectangular symmetry, and forms domains of two variants. The domain wall energy drives the domains to coarsen. The sontaneous strains induce an elastic field, which drives the domains to refine. The cometition between coarsening and refining selects an equilibrium domain size. We model the eilayersubstrate as a nonequilibrium thermodynamic system, evolving by the changes in the elastic dislacements and the order arameters. The free energy consists of two arts: the bulk elastic energy, and the excess surface energy. The elastic energy density is taken to be quadratic in the strains. The surface energy density is exanded into a olynomial of the order arameters, the gradients of the order arameters, and the strains. In this exansion, the surface stress is taken to be quadratic in the order arameters. The evolution equations are derived from the free energy variation with resect to the order arameters. The elastic field is determined by suerosing the Cerruti solution. Examles of comuter simulation are resented. DOI: / Introduction The thermodynamics and kinetics of bulk ferroelectric materials have been extensively studied 1. However, there is little systematic study on the ferroelectric thin films 2,3. This aer considers some imortant issues on the domain dynamics in a ferroelastic eilayer on a araelastic substrate. Consider a ferroelastic eilayer on a araelastic substrate. The eilayer is much thinner than the substrate. They are coherent, accommodating the lattice mismatch by elastic deformation. At high temeratures, the eilayer and the substrate surface both have square symmetry, but have different lattice constants. To comensate for the lattice mismatch, the eilayer is under a uniform, biaxial stress state, and the substrate is unstressed. At low temeratures, the eilayer has rectangular symmetry, but the substrate surface still has square symmetry. Because of the broken symmetry, the eilayer now has two variants, equivalent uon a 90-deg rotation Fig. 1. Within the Ginzburg-Landau framework 1, we characterize the ferroelastic state by two order arameters, ( 1, 2 ), taken to be the comonents of a vector lying in the lane of the layer. If the eilayer were unconstrained by the substrate, the two variants would have different sontaneous strain states, equivalent after the 90-deg rotation. This aer considers the eilayer constrained on the substrate. It is sometimes assumed in the literature that the energy ground state is a single-variant eilayer on the substrate. This state is illustrated in Fig. 2a. The stress state in the eilayer is uniform and biaxial, with unequal comonents in the two directions. The substrate is unstressed. Assuming that the eilayer is of a single variant, Pertsev et al. 2 showed that the constraint of the substrate could alter the Curie temerature and even induce new hases. However, the eilayer of a single variant on the substrate is not the energy ground state 3 7. This is readily understood as Contributed by the Alied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for ublication in the ASME JOURNAL OF APPLIED ME- CHANICS. Manuscrit received by the ASME Alied Mechanics Division, March 15, 2001; final revision, December 22, Associate Editor: D. A. Kouris. Discussion on the aer should be addressed to the Editor, Prof. Robert M. McMeeking, Deartment of Mechanical and Environmental Engineering University of California Santa Barbara, Santa Barbara, CA , and will be acceted until four months after final ublication of the aer itself in the ASME JOURNAL OF APPLIED MECHANICS. follows. In Fig. 2b, the two variants coexist and form domains. To aid the argument, imagine that suitable forces are alied on the domain walls, so that the stress state in the old variant is unchanged, the stress state in the new variant is equivalent to the old one after the 90-deg rotation, and the substrate remains unstressed. Consequently, the state in Fig. 2b has the same elastic energy as the state in Fig. 2a. Now gradually reduce the forces on the domain walls, and allow the eilayer and the substrate to deform. After the forces are comletely removed Fig. 2c, the dislacements are generally in directions oosite from the alied forces in Fig. 2b. Consequently, after elastic relaxation, the two-variant state in Fig. 2c has lower elastic energy than the single-variant state in Fig. 2a. That is, when the symmetry is broken, due to the substrate constraint, the single-variant eilayer is no longer the energy ground state. This suggests that Pertsev et al. s assumtion of a single variant is excessively simlistic and questions the validity of their results. Figure 3 illustrates a domain attern of a eriodic array of alternating variants. The smaller the domain size, the more elastic energy is relaxed. Consequently, elasticity drives domains to refine. On the other hand, variants coexist at the cost of adding the domain wall energy. The smaller the domain size, the longer the collective domain walls. That is, the domain wall energy drives the domain to coarsen. The cometition between the elastic energy and the domain wall energy selects an equilibrium domain size, which minimizes the combined energy It is also exected that the minimization of the combined energy can select an equilibrium domain attern. This aer introduces a model within the time-deendent Ginzburg-Landau TDGL framework to simulate domain dynamics in the eilayer. The eilayer may have different roerties from the bulk, articularly if the eilayer is only a few monolayers thick. It may also be ossible that a marginally araelastic bulk crystal is ferroelastic within a few surface layers. Motivated by these considerations, we will model the eilayer-substrate as a single system. The model arallels that for an eilayer with comosition modulation In the resent model, the free energy of the system consists of two arts: the bulk elastic energy and the excess surface energy. The elastic energy density is taken to be quadratic in the strains. The excess surface energy density is exanded into a olynomial of the order arameters, the gradients Journal of Alied Mechanics Coyright 2002 by ASME JULY 2002, Vol. 69 Õ 419

2 Fig. 1 Broken symmetry and lattice mismatch. The substrate has square symmetry. The eilayer has rectangular symmetry, with two variants. of the order arameters, and the strains. In this exansion, the surface stress is taken to be quadratic in the order arameters. We derive the evolution equations from the free-energy variation associated with variation in the order arameters, and determine the elastic field by suerosing the Cerruti solution. To illustrate the model, we resent reliminary results from comuter simulations. The TDGL framework has been alied to bulk ferroelastic crystals In a bulk ferroelastic single crystal, however, elasticity does not refine domains. In fact, the single-domain crystal is the energy ground state. The bulk elasticity roblem lacks an intrinsic length scale. For an infinite, olydomain crystal, the total elastic energy in the crystal is indeendent of the domain size so long as the domain attern is self-similar as domains coarsen. In this case, domains coarsen to reduce the domain wall energy. The situation is different if the system is not an infinite single crystal. For a thin ferroelastic film on a substrate 4 9, the film thickness rovides a length scale. In a olycrystalline crystal, the grain size rovides a length scale 18,19. In both cases, elasticity does limit the size of the domains. In the model introduced in this aer, the length scale is rovided by the introduction of the surface stress, as will be identified in a later section. 2 Free Energy and Kinetic Law Even for a single crystal, atoms in a few surface layers have different energy from atoms in the bulk. The deosition of the eilayer further changes the energy state. We model the substrateeilayer system as a bulk solid couled with a suerficial object. The total free energy consists of two arts: the bulk elastic energy and the excess surface energy, written as G WdV da, (1) where W is the elastic energy density, and is the excess surface energy density. The first integral is over the volume of the entire system, and the second integral is over the surface area covered by the eilayer. Following 14, we interret the surface energy as the excess free energy relative to bulk elastic energy. Thus, includes the effects of the mismatch between the two materials, as well as the resence of the emty sace above. In the rectangular coordinate system, the surface coincides with the (x 1,x 2 ) lane, and the material occuies the half-sace x 3 0. Reference the dislacement vector in the system u i from an infinite, unstressed crystal of the same comosition as the substrate. The strain tensor ij relates to the dislacement gradient, namely, ij 1 2 u i, j u j,i. (2) The Latin subscrits run from 1 to 3. Fig. 2 Schematic illustration of elastic refining; a a singlevariant eilayer, b a two-variant eilayer with aroriate forces alied around the new domain, c the external forces removed We assume that the substrate is elastically isotroic, being the shear modulus, and Poisson s ratio. The elastic energy density W is quadratic in the strain tensor: W ij ij 12 kk 2. (3) Fig. 3 A domain attern in the eilayer constrained on the substrate 420 Õ Vol. 69, JULY 2002 Transactions of the ASME

3 elasticity theory by adding a strain-deendent surface energy to the free energy In this aer, we exand the surface stress f in terms of the order arameters: f 11 f f 22 f (8) Fig. 4 The Landau exansion as a function of the order arameters. The four wells corresond to sontaneous states. The stresses ij in the bulk are the differential coefficients of the elastic energy density: W ij ij. (4) The reeated subscrits imly the summation convention. We characterize the ferroelastic state of the eilayer by two order arameters, 1 and 2, which form the comonents of a vector lying in the lane of the eilayer. For examle, when the layer is also ferroelectric, this vector coincides with the olarization. The excess surface energy density is a function of the order arameters, the gradients of the order arameters, and the strains. Following the Ginzburg-Landau formalism, we exand into a Taylor series to the lowest terms in, and : f L f G, f. (5) The Greek letters run from 1 to 2. The hysical content of every term in 5 is interreted as follows. The first term in 5 is the Landau exansion. We assume that the eilayer has square symmetry in the high-temerature hase. The Landau exansion takes the following form 15 17: f L a 0 a a a a a (6) Here a 0 is the surface energy density of the eilayer on the substrate when the eilayer is in the araelastic hase ( 0) and the substrate is unstrained ( 0). The coefficient a 1 is roortional to TT C, i.e., the temerature difference from critical oint, T C. When TT C, a 1 0, and the function has four wells Fig. 4. Each well corresonds to an unconstrained ferroelastic variant. There is no reason to regard the coefficients in 6 to be the same as those of bulk materials. The second term in 5 stands for the gradient energy, which is taken to be quadratic in the gradients of the order arameters 15 17: f G, 1 2 g ,1 2 2,2 g 12 1,1 2,2 1 2 g 44 1,2 2, g 44 1,2 2,1 2. (7) The g arameters are ositive constants. This is a continuum descrition of the domain wall energy. The third term in 5 gives the strain deendence. We have only included the linear terms of the strains, neglecting the excess elastic constants of the eilayer relative to the substrate. The latter could also be included as a refinement of the theory. The coefficients f are the comonents of the surface stress tensor. The surface stress tensor has been incororated into the continuum f Here f 0 is the surface stress when the eilayer is in the araelastic state. In the ferroelastic state, the surface stress is taken to be quadratic in the order arameters. The surface stress coules the domain attern in the eilayer to the elastic deformation in the substrate. The eilayer-substrate is a nonequilibrium thermodynamic system. The system can vary by two means: the elastic dislacements and the order arameters. Characterize a virtual change of the system by u i and. The free-energy variation associated with this virtual change is G f L x f G, f da 3 f x u 33 u 3dA ij, j u i dv. (9) We assume that elastic relaxation is much faster than domain switching. At a given time the system reaches elastic equilibrium instantaneously. Consequently, the energy variation associated with the elastic dislacements vanishes, leading to the familiar equilibrium equations in elasticity: ij, j 0, (10) and the boundary conditions: 3 f, x (11) Associated with the virtual change of the order arameters, the variation of the free energy defines a thermodynamic force, F, namely, F dag. (12) A comarison of 9 and 12 gives that F f L x f G, f. (13) Following 12 14, we adot a linear kinetic law that the rate of the change of the order arameters is roortional to the driving force, namely, LF t, (14) where L is the kinetic coefficient. Combining 13 and 14, we obtain the evolution equations for the order arameters: t L f L x f G, f. (15) Assuming elastic equilibrium and using the divergence theorem, we obtain the free energy as follows: G f L f G u da. (16) The integral extends over the surface area. Journal of Alied Mechanics JULY 2002, Vol. 69 Õ 421

4 l 4g (22) 11 s This length scales the size of an individual domain in an equilibrium attern. As stated in the Introduction, the equilibrium domain attern is an outcome of the cometition between refining due to elasticity and coarsening due to domain wall energy. The time scale can be derived from the evolution Eqs. 15, giving 1 2a 1 L. (23) We will reort time in the unit of. In terms of the dimensionless quantities introduced above, the evolution equations become 1 t 1,11 1 2,21 2 1,22 b l a 11 1 Fig. 5 of b. Domain wall shae. The domain width is on the order a a a t 2,22 1 1,12 2 2,11 (24) 3 Dimensionless Equations, Length Scales, and Time Scale The four wells in the function f L Fig. 4 corresond to four states of sontaneous olarization. The value of the sontaneous olarization s satisfies a 1 2a a , (17) giving s a 1 a 11 a 11 1/2 2 3a 111 a 1. (18) We use s to normalize the order arameters 1 and 2. Introduce dimensionless arameters a 11 a 2 11 s, a a 1 12 a 2 12 s, a a a s, a a a s, a 1 (19) 1 g 12g 44 g 44, g 2 g 44g 44. (20) 11 g 11 The driving force F in 13 contains three terms. The comarison of the terms involving f L and f G defines a length scale: b g 11. (21) 2a 1 This length reresents the width of the domain wall in the Ginzburg-Landau framework. We use b to normalize the satial coordinates. Figure 5 shows the shae of a 90-deg domain wall. In 15, a boundary value roblem was solved to obtain the analytical result for the 90-deg domain wall. In our work, the simulation from evolution equations shows similar shae, and the domain wall width is comarable to b. In 13 a comarison between the gradient energy term and the surface stress term defines another length scale: b l a 11 2 a a a In the above the strains have to be determined by solving the elastic field in the semi-infinite solid subject to the following boundary conditions on the surface: x x x x x x x 1 1 x 1 (25) 33 0 where the stress is normalized by 11 s 2 /b. The elastic field in a half-sace due to a tangential oint force on the surface is known as the Cerruti field 20. A suerosition gives the needed strain field: x x 2 y 2 3/2 6x 5/2 3 x 2 y 2 dd 32 2y x 2 y 2 3/2 5/2 6yx2 x 2 y 2 dd 422 Õ Vol. 69, JULY 2002 Transactions of the ASME

5 x x 2 y 2 3/2 5/2 6xy2 x 2 y 2 dd y x 2 y 2 3/2 6y 5/2 3 x 2 y 2 dd y x 2 y 2 3/2 12x2 y 5/2 x 2 y 2 dd x x 2 y 2 3/2 5/2 12xy2 x 2 y 2 dd. (26) The strains are normalized by 11 2 s /4b. The dislacements, normalized by 11 2 s /4, are u x 2 y 2 1/2 2x 3/2 2 x 2 y 2 dd 32 u xy 3/2 x 2 y 2 dd 2xy 3/2 x 2 y 2 dd 21 x 2 y 2 1/2 2y 3/2 2 x 2 y 2 dd. 4 Comuter Simulation (27) To illustrate the model, we now resent several reliminary comuter simulations. Referring to 16,17, we adot the following arameters: a , a , a , a g 12 /g , g 44 /g , g 44 /g b/l1.0, 12 / , 44 / , 0.3. (28) We have not carried out a comrehensive arametric study. The evolution equations have both satial and temoral derivatives. We discretize the infinite surface into an array of squares of size NN. All fields are assumed to be eriodically relicated from one square to another. Each square is the comutation unit cell, which is subdivided into grids of sacing x. Satial derivatives are aroximated with finite difference. At every time ste, the elastic field is calculated by evaluating the double integrals, and the order arameter field is udated by using the Euler method. In our simulation, we use N128, x1.0, and adative time stes. The ratio of two length scales is b/l1.0. The singular integrals in 26 and 27 are evaluated by adoting a technique in the Aendix. Since the double integrals must be calculated for Fig. 6 Free energy versus domain sacing. Without substrate constraint, the energy decreases as the domain size increases. With substrate constraint, the energy reaches a minimum at a secific domain size. every grid oint, the simulation is rather slow. The evolution can also be simulated in recirocal sace 12 17, which we will imlement for this model in the future. For a given domain attern, i.e., a rescribed field of the order arameters, the free energy can be comuted from Eqs. 16, 25, and 27. In the first simulation, we secify a arallel domain attern, fix the domain sacing, but allow the domain walls to relax locally. We calculate the free energy as a function of the domain sacing. Figure 6 illustrates the effect of substrate constraint. The domain size in this figure is the width of the domain along x,y coordinates of Fig. 5. As discussed in the Introduction, without the substrate, the total free energy of the eilayer decreases with the increase of the domain size, and the single domain has the lowest energy. With the substrate constraint, since the cometition of elasticity and domain wall energy selects an equilibrium domain size, a valley exists on the curve of energy versus domain sacing. For the domains smaller than the equilibrium size, the relaxation of the elastic energy cannot accommodate the raid increase of the domain wall length, so the more domains, the higher the free energy. Figure 7 comares the domain evolution with substrate and without substrate constraint. Through every grid oint we draw a short line segment, reresenting the magnitude and the direction of the order arameter. The left column shows the result with the substrate, and the right one without substrate. In both cases, we start with the initial condition of a small random erturbation of the order arameter field from zero, corresonding to the araelastic state. When the layer is constrained, arallel domains form, and the domain size aroaches to what has been shown in Fig. 5. When the layer is unconstrained, the domains coarsen, being limited only by the calculating cell size. 5 Summary This aer resents a formalism to simulate the evolution of domain atterns in ferroelastic eilayers. The surface stress due to the sontaneous strains can be relaxed by the substrate mediated elastic interaction. Free energy is divided into two arts. One is the elastic energy of the semi-infinite substrate, and the other is the excess surface energy, which has the ingredients of the Landau exansion, the gradient energy, and the surface stress. Evolution equations are derived from energy variation. Comuter simulation ascertains that the cometition of coarsening and refining is resonsible for equilibrium domain atterns. The formalism resented in this aer is flexible, and can be used to study other henomena involving surface hase transition and attern formation. Journal of Alied Mechanics JULY 2002, Vol. 69 Õ 423

6 Aendix The integrals 26 and 27 extend over the entire surface. To save comutation time, we only extend the integrals to a finite square of size 16b16b in our simulation. The integrals are singular when x and y. Let be a small number, say the grid sacing x. We treat the singularity as follows: x y x y x 31,x x 2 y 2 3/2 dd x y y 31, 31 x,yx x 2 y 2 3/2 dd x y x y 31 x 31 x x 31 x x 31 y y x x y 1 1 x 2 y 2 3/2 y x ln&1 31 x. x 2 y 2 3/2 dd 1 s 2 s 2 t 2 3/2 dsdt dd Fig. 7 Comarison of domain evolution in simulations with substrate constraint and without. The left column is the result with substrate, and arallel domains can be seen. The right column is the result without constraint, and the structure kees coarsening. Acknowledgments This work is suorted by the National Science Foundation through grant CMS , and by Princeton University through the Sir Gordon Wu Fellowshi for Y. F. G. We would like to thank Dr. Wei Lu for helful discussions. References 1 Lines, M. E., and Glass, A. M., 1977, Princiles and Alications of Ferroelectrics and Related Materials, Clarendon Press, Oxford. 2 Pertsev, N. A., Zembilgotov, A. G., and Tagantsev, A. K., 1998, Effect of Mechanical Boundary Conditions on Phase Diagrams of Eitaxial Ferroelectric Thin Films, Phys. Rev. Lett., 80, Suo, Z., 1998, Stress and Strain in Ferroelectrics, Curr. Oin. Solid State Mater. Sci., 3, Pome, W., Gong, X., Suo, Z., and Seck, J. S., 1993, Elastic Energy Release due to Domain Formation in the Strained Eitaxy of Ferroelectric and Ferroelastic Films, J. Al. Phys., 74, Seck, J. S., and Pome, W., 1994, Domain Configurations due to Multile Misfit Relaxation Mechanisms in Eitaxial Ferroelastic Thin Films: I Theory, J. Al. Phys., 76, Kwak, B. S., and Erbil, A., 1992, Strain Relaxation by Domain Formation in Eitaxial Ferroelectric Thin Films, Phys. Rev. Lett., 68, Sridhar, N., Rickman, J. M., and Srolovitz, D. J., 1996, Twinning in Thin Films I. Elastic Analysis, Acta Mater., 44, Sridhar, N., Rickman, J. M., and Srolovitz, D. J., 1996, Twinning in Thin Films II, Equilibrium Microstructures, Acta Mater., 44, Seul, M., and Andelman, D., 1995, Domain Shaes and Patterns The Phenomenology of Modulated Phases, Science, 267, Ibach, H., 1997, The Role of Surface Stress in Reconstruction, Eitaxial Growth and Stabilization of Mesoscoic Structures, Surf. Sci. Re., 29, Alerhand, O. L., Vanderbilt, D., Meade, R. D., and Joannooulos, J. D., 1988, Sontaneous Formation of Stress Domains on Crystal Surfaces, Phys. Rev. Lett., 61, Lu, W., and Suo, Z., 1999, Coarsening, Refining, and Pattern Emergence in Binary Eilayers, Z. Metallkd., 90, Lu, W., and Suo, Z., 2001, Dynamics of Nanoscale Pattern Formation of an Eitaxial Monolayer, J. Mech. Phys. Solids, 49, Suo, Z., and Lu, W., 2000, Comosition Modulation and Nanohase Searation in a Binary Eilayer, J. Mech. Phys. Solids, 48, Cao, W., and Cross, L. E., 1991, Theory of Tetragonal Twin Structures in Ferroelectric Perovskites With a First-Order Phase Transition, Phys. Rev. B, 44, Hu, H.-L., and Chen, L.-Q., 1998, Three-Dimensional Comuter Simulation of Ferroelectric Domain Formation, J. Am. Ceram. Soc., 81, Nambu, S., and Sagala, D. A., 1994, Domain Formation and Elastic Long- Range Interaction in Ferroelectric Perovkites, Phys. Rev. B, 50, Cao, W. W., and Randall, C. A., 1996, Grain Size and Domain Size Relations in Bulk Ceramic Ferroelectric Materials, J. Phys. Chem. Solids, 57, Randall, C. A., Kim, N., Kucera, J. P., Cao, W. W., and Shrout, T. R., 1998, Intrinsic and Extrinsic Size Effects in Fine-Grained Morhotoroic-Phase Boundary Lead Zirconate Titanate Ceramics, J. Am. Ceram. Soc., 81, Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK, Õ Vol. 69, JULY 2002 Transactions of the ASME