APPENDIX A Landau Free-Energy Coefficients

APPENDIX A Landau Free-Energy Coefficients Long-Qing Chen Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 180 USA The thermodynamics of ferroelectrics is usually described by the phenomenological Landau Devonshire theory. Using the free energy for the unpolarized and unstrained crystal as the reference, the free energy of a ferroelectric crystal as a function of strain and polarization can be written as see, e.g., [1] F ε, P = 1 α ijp i P j + 1 3 β ijkp i P j P k + 1 γ ijklp i P j P k P l + 1 5 δ ijklmp i P j P k P l P m + 1 ω ijklmnp i P j P k P l P m P n + 1 c ijklε ij ε kl a ijk ε ij P k 1 q ijklε ij P k P l +, 1 where α ij, β ijk, γ ijkl, δ ijklm,andω ijklmn are the phenomenological Landau Devonshire coefficients, and c ijkl, a ijk,andq ijkl are the elastic, piezoelectric, and electrostrictive constant tensors, respectively. If the parent phase is centrosymmetrical, all odd terms are absent: F ε, P = 1 α ijp i P j + 1 γ ijklp i P j P k P l + 1 ω ijklmnp i P j P k P l P m P n + 1 c ijklε ij ɛ kl 1 q ijklε ij P k P l +. In, the set of coefficients, α, γ and ω, in the Helmholtz free energy correspond to those measured under a clamped boundary condition. Under the stress-free boundary condition, the macroscopic shape change of a crystal due to the ferroelectric phase transition is described by the spontaneous strain that can be obtained through the derivative of the Helmholtz free energy with respect to strain, i.e., σ ij = c ijkl ε 0 kl 1 q ijklp k P l =0. 3 Solving 3 for strain, we have ε 0 kl = 1 s ijklq klmn P m P n = Q ijmn P m P n, where s sijkl is the elastic compliance tensor and Q ijmn = 1 s ijklq klmn. 5 K. Rabe, C. H. Ahn, J.-M. Triscone Eds.: Physics of Ferroelectrics: A Modern Perspective, Topics Appl. Physics 105, 33 37 007 Springer-Verlag Berlin Heidelberg 007

3 Long-Qing Chen Substituting the spontaneous strain back into the free-energy expression, we have GP = 1 α ijp i P j + 1 γ ijkl 1 s mnorq mnij q orkl P i P j P k P l or + 1 ω ijklmnp i P j P k P l P m P n +, GP = 1 α ijp i P j + 1 γ ijkl c mnor Q mnij Q orkl P i P j P k P l + 1 ω ijklmnp i P j P k P l P m P n +. 7 The fourth-order coefficients are different for the clamped and stressfree 7 boundary conditions, and they are related by γ ijkl = γ ijkl 1 s mnorq mnij q orkl = γ ijkl c mnor Q mnij Q orkl, 8 where γ ijkl is the fourth-order coefficient for the stress-free boundary condition. In general, experimentally determined coefficients correspond to γ since it is usually easier to do measurements under stress-free boundary conditions. In the following, the Landau Devonshire coefficients are presented for a number of oxides, including the well-studied systems BaTiO 3, SrTiO 3 and PZT, collected from the open literature. All the data were provided for the stress-free boundary conditions unless noted otherwise. They are all in SI units with the temperature in K. 1 BaTiO 3 For BaTiO 3, a Landau Devonshire potential up to eighth order has been employed, GP x,p y,p z =α 1 P x y z + α11 P x y z +α 1 P x Py y Pz x Pz + α111 P x y z [ +α 11 P x P y z y P x z z P x y ] +α 13 Px Py Pz + α 1111 P 8 x y 8 z 8 [ +α 111 P x P y z y P x z z P x y ] +α 11 P x Py y Pz x Pz +α 113 P x Py Pz y Pz Px z Px Py. 9 Two sets of coefficients for 9 are given in Table 1. The elastic and electrostrictive coefficients are listed separately in Table. The free energy under

Landau Free-Energy Coefficients 35 Table 1. Landau Devonshire potential coefficients for BaTiO 3 T in K SI units and Coefficients Sixth order [, 3] Eight-order [] α 110 5 C m N 3.3T 381.1T 388 α 1110 C m N.9T 393 0 09.7 α 110 8 C m N 3.3 7.97 α 11110 7 C m 10 N 5.5T 10 + 7 19. α 1110 9 C m 10 N.7 1.950 α 1310 9 C m 10 N.919.5009 α 111110 10 C 8 m 1 N 0.0 3.83 α 11110 10 C 8 m 1 N 0.0.59 α 1110 10 C 8 m 1 N 0.0 1.37 α 11310 10 C 8 m 1 N 0.0 1.37 Table. Elastic and electrostrictive coefficients of BaTiO 3 [1, 3, 5 9]. Note that additional data on the elastic constants of BaTiO 3 can be found in [1] and[8] c 1110 11 N m 1.78 c 110 11 N m 0.9 c 10 11 N m 1. Q 11C m 0.10, 0.11 Q 1C m 0.03, 0.05 Q C m 0.09, 0.059 a constant strain, ε, can be easily obtained from the above stress-free free energy through F P, ε =GP, σ =0+ 1 c ijkl εij ε 0 ij εkl ε 0 kl, 10 where ε 0 ij is given by. SrTiO 3 To describe both the proper ferroelectric and the antiferroelastic distortion AFD structural transition in SrTiO 3 requires both the spontaneous polarization P =P x,p y,p z and the structural order parameter q =q 1,q,q 3 as the order parameters. The structural order parameter represents the linear oxygen displacement that corresponds to simultaneous out-of-phase rotations of oxygen octahedra around one of their four-fold symmetry axes. A fourth-order Landau polyno-

3 Long-Qing Chen mial as a function of the polarization and structural order parameter is given by GP i,q i =A ij P i P j + A ijkl P i P j P k P l + B ij q i q j +B ijkl q i q j q k q l + C ijkl P i P j q k q l, 11 where i, j = x, y, z, A ijkl, B ijkl and C ijkl are constants and A ij and B ij are functions of temperature. Keeping only the terms allowed by the cubic symmetry of the SrTiO 3 crystal, one has G = α 1 P x y z + α11 P x y z +α 1 P x Py y Pz x Pz + β1 q x + qy + qz +β 11 q x + qy + qz + β1 q x qy + qyq z + qxq z t 11 P x qx y qy z qz t 1 [Px q y + qz y q x + qz z q x + qy ] t P x P y q x q y y P z q y q z z P x q z q x, 1 where α ij, β ij,andt ij are assumed to be constants and α 1 and β 1 depend on temperature. ε 0 ij is the stress-free strain or the transformation strain as a result of the structural and/or ferroelectric transitions, ε 0 ij = Q ijkl p k p l + Λ ijkl q k q l, 13 in which Q ijkl and Λ ijkl represent, respectively, the electrostrictive coefficient and the linear quadratic coupling coefficient between the strain and structural order parameter. 3 PbZr 1 x Ti x O 3 PZT Existing experimental measurements in PZT have been fitted to a sixth-order polynomial: GP x,p y,p z =α 1 P x y z + α11 P x y z +α 1 P x Py y Pz x Pz + α111 P x y z +α 11 [Px P y z y P x z z P x y ] +α 13 Px Py Pz. 1

Landau Free-Energy Coefficients 37 Table 3. Landau Devonshire coefficients for SrTiO 3 SI units and T in K [10 15] α 110 5 C m N 05 [coth5/t coth5/30] 3.5 [coth/t 0.907],T <50 K 7.37T 8,T >70 K 7.0T 35.5,T > 100 K α 1110 9 C m N 1.70 α 110 9 C m N 1.37 Q 11 C m 0.057, 0.08, 0.0 Q 1 C m 0.0135, 0.015, 0.013 Q C m 0.00957 c 11 10 11 N m 3.15, 3.181, 3.3, 3.8 c 1 10 11 N m 1.01, 1.05, 1.07, 1.07 c 10 11 N m 1.19, 1.15, 1.3, 1.7 β 1 10 9 N m 1.3 [coth15/t coth15/105] β 11 10 50 N m 1.9, 1.58, 1.0, 0.99 β 1 10 50 N m 3.88, 3.78,.88,.73 Λ 1110 18 m 8.7, 1.7, 9.3, 8.35 Λ 110 18 m 7.8, 7.3,., 5.5 Λ 10 18 m 9., 9.88,.93, 7.5 t 11 10 9 C N 1.7,.10 t 1 10 9 C N 0.755, 0.85 t 10 9 C N 5.85 The corresponding Landau Devonshire and electrostrictive coefficients are given by [1 0] α 1 =T T 0 /ε 0 C 0,ε=8.85 10 1, α 11 = 10.1.55x +10.955x 10 13 /C 0, α 111 = 1.0 17.9x +9.179x 10 13 /C 0, α 11 =.90 3.375x +58.80e 9.397x 10 1 /C 0, α 1 = η 1 /3 α 11,α 13 = η 3α 111 α 11, [ η 1 =.13 + 0.73x 9.+0.01501x e 1.x] 10 1 /C 0, [ η = 0.887 0.7973x +1.5 0.08851x e 1.55x] 10 15 /C 0, T 0 =.3 + 83.x 105.5x + 01.8x 3 388.3x + 1337.8x 5,.171 C 0 = +0.131x +.01 10 5, when 0.0 x 0.5, 1+ 500.05x 0.5.8339 C 0 = 1+ 1.5x 0.5 +1.13 10 5, when 0.5 x 1.0, 0.09578 Q 11 = +0.079x +0.05, 1+ 00x 0.5 0.058 Q 1 = +0.01093x 0.01338, 1+ 00x 0.5 Q = 1 0.0535 +0.00857x +0.017, 1 + 00x 0.5

38 Long-Qing Chen Table. The compliance tensor was estimated for a number of compositions [1] Ti content x 0. 0.5 0. 0.7 0.8 0.9 s 11 10 1 m /N 8.8 10.5 8. 8. 8. 8.1 s 1 10 1 m /N.9 3.7.8.7..5 s 10 1 m /N. 8.7 1. 17.5 1. 1 Table 5. Landau Devonshire potential coefficients for PbTiO 3 T in K [1, ] SI units and α 110 5 C m N 3.8T 75 Q 11 C m 0.089 α 1110 8 C m N 0.73 Q 1 C m 0.0 α 110 8 C m N 7.5 Q C m 0.075 α 11110 8 C m 10 N. s 11 C 1 m /N 8.0 α 1110 8 C m 10 N.1 s 1 C 1 m /N.5 α 1310 8 C m 10 N 37 s C 1 m /N 9.0 where x is the mole fraction of PbTiO 3 in PZT. The units are SI with the temperature in K. The elastic compliance values for a number of selected compositions were provided in [1] Table. PbTiO 3 For pure PbTiO 3, the free-energy coefficients are given in Table 5. 5LiTaO 3 and LiNbO 3 LiNbO 3 and LiTaO 3 belong to the 3m point group. Denoting the crystallographic uniaxial directions as the z-axis, the free-energy expansion is given by [3] P x y + β1 ε 3 + β ε 1 + ε F = α 1 P z + α P z + α 3 ] +β 3 [ε 1 ε + ε + β ε 3 ε 1 + ε +β 5 ε + ε 5 +β [ε 1 ε ε + ε 5 ε ]+γ 1 ε 1 + ε Pz + γ ε 3 Pz +γ 3 [ε 1 ε P y P z + ε P x P z ]+γ ε 5 P x P z + ε P y P z, 15 where electrostriction terms that do not involve the primary z component of polarization have been ignored. The corresponding coefficients are given in Table noted that in this example, α i are determined at constant zero strain rather than constant zero stress.

Landau Free-Energy Coefficients 39 Table. Landau-Devonshire potential coefficients for LiTaO 3 and LiNbO 3 [3] Coefficients LiTaO 3 LiNbO 3 α 110 9 C N m 1.5.01 α 10 9 C N m 5.03 3.08 α 310 9 C N m. 1.35 β 110 10 N m 13.55 1.5 β 10 10 N m.75. β 310 10 N m.95 3.75 β 10 10 N m 7. 7.5 β 510 10 N m.8 3 β 10 10 N m 1. 0.9 γ 110 9 C N m 0.0 0.1 γ 10 9 C N m 1.317 1.88 γ 310 9 C N m.8 0.33 γ 10 9 C N m.99 3.9 Table 7. Landau-Devonshire potential coefficients for SrBi Nb O 9 E is Young s modulus and ν is Poisson s ratio [] α 110 C m N 1.03T 3 Q 1110 3 C m 0.385 α 1110 8 C m N 0.9 Q 110 3 C m 0.0 α 110 8 C m N 9.38 Q 10 3 C m 0.05 α 11110 8 C m 10 N 11.8 E10 1 m /N 0.9 α 1110 8 C m 10 N 3. ν10 1 m /N 0.31 Sr 0.8 Bi. Ta O 9 For Sr 0.8 Bi. Ta O 9, the only existing Landau free-energy description is a single double-well potential [5], F =.03 10 5 T 0P +3.75 10 9 P, 1 where T is in K and F and P are in SI units. 7 SrBi Nb O 9 The thermodynamics of SrBi Nb O 9 ferroelectrics was modeled using the following free-energy function, GP x,p y =α 1 P x y + α11 P x y + α1 Px Py +α 111 P x y + α11 P x P y y P x. 17 Due to the lack of experimental data in this system, many of the coefficients were estimated Table 7.

370 Long-Qing Chen References [1] F. Jona, G. Shirane: Ferroelectric Crystals Macmillan, New York 19 33, 35 [] W. R. Buessem, L. E. Cross, A. K. Goswami: Phenomenological theory of high permittivity in fine-grained barium titanate, J. Am. Ceram. Soc. 9, 33 19 35 [3] J. Bell, L. E. Cross: A phenomenological Gibbs function for BaTiO 3 giving correct E-field dependence of all ferroelectric phase-changes, Ferroelectrics 59, 197 03 198 35 [] Y. L. Li, L. E. Cross, L. Q. Chen: A phenomenological thermodynamic potential for BaTiO 3 single crystals, J. Appl. Phys. 98, 0101 005 35 [5] A. F. Devonshire: Theory of barium titanate, Philos. Mag., 105 1079 1951 35 [] D. Berlincourt, H. Jaffe: Elastic and piezoelectric coefficients of single-crystal barium titanate, Phys. Rev. 111, 13 18 1958 35 [7] T. Yamada: Electromechanical properties of oxygen-octahedra ferroelectric crystals, J. Appl. Phys. 3, 38 197 35 [8] Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology, Group III, vol. 3a, New Series Springer, Heidelberg 00 pp. 3 35 [9] A. Yamanaka, M. Kataoka, Y. Inaba, K. Inoue, B. Hehlen, E. Courtens: Evidence for competing orderings in strontium titanate from hyper-raman scattering spectroscopy, Europhys. Lett. 50, 88 9 000 35 [10] N. A. Pertsev, A. K. Tagantsev, N. Setter: Phase transitions and strain-induced ferroelectricity in SrTiO 3 epitaxial thin films, Phys. Rev. B 1, R85 89 000 37 [11] N. A. Pertsev, A. K. Tagantsev, N. Setter: Erratum: Phase transitions and strain-induced ferroelectricity in SrTiO 3 epitaxial thin films, Phys. Rev. B 5, 19901 00 37 [1] H. Uwe, T. Sakudo: Stress-induced ferroelectricity and soft phonon modes in SrTiO 3,Phys.Rev.B13, 71 197 37 [13] A. K. Tagantsev, E. Courtens, L. Arzel: Prediction of a low-temperature ferroelectric instability in antiphase domain boundaries of strontium titanate, Phys. Rev. B, 107 001 37 [1] J. C. Slonczewski, H. Thomas: Interaction of elastic strain with the structural transition of strontium titanate, Phys. Rev. B 1, 3599 1970 37 [15] R. O. Bell, G. Rupprecht: Elastic constants of strontium titanate, Phys. Rev. 19, 90 193 37 [1] M. J. Haun, E. Furman, S. J. Jang, H. A. McKinstry, L. E. Cross: Thermodynamic theory of PbTiO 3,J.Appl.Phys., 3331 3338 1987 37, 38 [17] M. J. Haun: Ph.D. thesis, The Pennsylvania State University 1988 37 [18] M. J. Haun, E. Furman, S. J. Jang, L. E. Cross: Thermodynamic theory of the lead zirconate-titanate solid-solution system: 1. Phenomenology, Ferroelectrics 99, 13 5 1989 37 [19] M. J. Haun, E. Furman, H. A. McKinstry, L. E. Cross: Thermodynamic theory of the lead zirconate-titanate solid-solution system:. Tricritical behavior, Ferroelectrics 99, 7 1989 37

Landau Free-Energy Coefficients 371 [0] M. J. Haun, Z. Q. Zhuang, E. Furman, S. J. Jang, L. E. Cross: Thermodynamic theory of the lead zirconate-titanate solid-solution system: 3. Curie constant and th-order polarization interaction dielectric stiffness coefficients, Ferroelectrics 99, 5 5 1989 37 [1] N. A. Pertsev, V. G. Kukhar, H. Kohlstedt, R. Waser: Phase diagrams and physical properties of single-domain epitaxial PbZr 1 xti xo 3 thin films, Phys. Rev. B 7, 05107 003 38 [] N. A. Pertsev, A. G. Zembilgotov, A. K. Tagantsev: Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films, Phys. Rev. Lett. 80, 1988 1991 1998 38 [3] D. A. Scrymgeour, V. Gopalan, A. Itagi, A. Saxena, P. J. Swart: Phenomenological theory of a single domain wall in uniaxial trigonal ferroelectrics: Lithium niobate and lithium tantalate, Phys. Rev. B 71, 18110 005 38, 39 [] Y. L. Li, L. Q. Chen, G. Asayama, D. G. Schlom, M. A. Zurbuchen, S. K. Streiffer: Ferroelectric domain structures in SrBi Nb O 9 epitaxial thin films: Electron microscopy and phase-field simulations, J. Appl. Phys. 95, 33 30 00 39 [5] M. Tanaka, K. Hironaka, A. Onodera: Thermal behavior of ferroelectric switching properties of SBT thin films, Ferroelectrics, 103 110 00 39 Index BaTiO 3, 3, 35 coefficient, 33 absent, 33 corresponding, 38 elastic, 33 35 electric compliance, 33 electrostrictive, 33 35, 37 free energy, 38 Landau free-energy, 33 Landau Devonshire, 33, 3, 37 piezoelectric, 33 description Landau free-energy, 39 Landau Devonshire, 33 potential, 3 potential coefficient, 35, 38 LiNbO 3, 38, 39 LiTaO 3, 38, 39 PbTiO 3, 38 PbZr 1 xti xo 3 PZT, 3, 3, 38 Sr 0.8Bi.Ta O 9, 39 SrBi Nb O 9, 39 SrTiO 3, 3 37