Effect of substrate-induced strains on the spontaneous polarization ...

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114105-4 Zhang et al. J. Appl. Phys. 101, 114105 2007 would be a 001 c oriented BiFeO 3 film constrained by a 001 o orthorhombic <strong>substrate</strong> with an in-plane orientation BiFeO relationship <strong>of</strong> 100 3 s BiFeO c 110 o and 010 3 c 110 s o . The shear strain is 0 12 = 1 2cos , where is the angle between 110 o and 110 o crystallographic axes <strong>of</strong> the <strong>substrate</strong>. For simplicity, we assume 0 11 = 0 22 =0 in this work. Unlike previously studied cases, the energies <strong>of</strong> the eight polarization variants are no longer degenerate under the shear <strong>substrate</strong><strong>induced</strong> strain as shown in Fig. 2b. Under a negative shear <strong>substrate</strong>-<strong>induced</strong> strain, four variants r + 2 , r − 2 , r + 4 , and r − 4 have lower energy, while under a positive shear <strong>substrate</strong><strong>induced</strong> strain the energies <strong>of</strong> the other four variants r + 1 , r − 1 , r + 3 , and r − 3 are lower. From Fig. 2b, one can see that P s changes with the shear <strong>substrate</strong>-<strong>induced</strong> strain. For example, P s for r + 1 , r − 1 , r + 3 , and r − 3 increases about 3.6% for a strain 0 12 =1.6%, which is significant compared to the effect <strong>of</strong> normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. However, the magnitude <strong>of</strong> change is still dramatically smaller than other traditional ferroelectric systems such as BiFeO 3 . For comparison, we also performed calculations by assuming that the symmetry <strong>of</strong> BiFeO 3 was fixed to be rhombohedral P 1 =P 2 =P 3 = P s / 3 or tetragonal P1 = P 2 =0, P 3 = P s . By reducing Eq. 5 with such symmetry relations, the corresponding spontaneous polarization can be obtained by solving F/P i =0. For rhombohedral symmetry, P 2 s = 6c 1212−6a 1 c 1111 + c 1111 − c 1122 q 1111 +2q 1122 0 11 + 0 22 ±4c 1111 q 1212 12 c 1212 24a 11 + a 12 c 1111 − q 1111 +2q 1122 2 −8c 1111 q 1212 0 2 , 8 where “+” for the variants with P 1 = P 2 and “−” for the variants with P 1 =−P 2 . For tetragonal symmetry, P 2 s = −4a 1c 1111 −2c 1122 q 1111 − c 1111 q 1122 0 11 + 22 8a 11 c 1111 − q 1111 0 2 . From Eq. 8, one can see that, when BiFeO 3 has the rhombohedral symmetry, the effect <strong>of</strong> normal <strong>substrate</strong><strong>induced</strong> <strong>strains</strong> on P s is highly related to the magnitude <strong>of</strong> q 1111 +2q 1122 , which is given by q 1111 +2q 1122 =2c 1111 +2c 1122 Q 1111 +2Q 1122 . 9 10 For BiFeO 3 also true for many ferroelectrics, the magnitude <strong>of</strong> Q 1111 +2Q 1122 is quite small. Therefore, the dependence <strong>of</strong> P s on normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> is rather weak, and only a shear <strong>substrate</strong>-<strong>induced</strong> strain could affect the polarization effectively. However, P s is only a function <strong>of</strong> normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> when BiFeO 3 has a tetragonal symmetry as shown in Eq. 9. It may explain the strong dependence <strong>of</strong> spontaneous polarizations on normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> film thickness shown in Ref. 10, where the symmetry <strong>of</strong> BiFeO 3 was assumed to be tetragonal. It should be noted that the BiFeO 3 phases with tetragonal symmetry and rhombohedral symmetry have higher energy than the phase with monoclinic symmetry as shown in Fig. 1a; therefore, they are unstable in the range <strong>of</strong> <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> we studied. We now discuss 111 c oriented BiFeO 3 thin films grown on a dissimilar <strong>substrate</strong>. Following Ref. 13, a new coordinate system, x=x 1 ,x 2 ,x 3 , is set up with the x 1 , x 2 , and x 3 axes along the 011 c , 211 c , and 111 c crystallographic directions <strong>of</strong> the BiFeO 3 film, respectively. The free energy <strong>of</strong> BiFeO 3 in the coordinate system x is given by FP, = 1 t i1 P i 2 + t i2 P i 2 + t i3 P i 2 + 11 t i1 P i 4 + t i2 P i 4 + t i3 P i 4 + 12 t i1 P i 2 t i2 P i 2 + t i1 P i 2 t i3 P i 2 + t i2 P i 2 t i3 P i 2 + 1 2 c ijkl kl − 1 2 q ijkl ij P k P l , 11 where t ij is the transformation matrix from the coordinate system x to the coordinate system x. P i , , ij c ijkl , and q ijkl are the polarizations, <strong>strains</strong>, elastic constants, and electrostrictive constants in the coordinate system x, which are given by P i = t ij P j , ij = t im t jn mn , c ijkl = t im t jn t ko t lp c mnop , q ijkl = t im t jn t ko t lp q mnop , 12 with the transformation matrix 1 −1 0 2 2 −2 1 1 13 6 6 6 = 111 t c ij 1 1 1 3 3 3. For a 111 c oriented film deposited on a 111 c oriented cubic crystal <strong>substrate</strong> with an in-plane orientation relationship <strong>of</strong> 110 3 c 110 s c , the elastic boundary condition is BiFeO given by 11 = 0 11 , Downloaded 23 Oct 2007 to 128.118.53.4. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
114105-5 Zhang et al. J. Appl. Phys. 101, 114105 2007 FIG. 3. a Free energy F and P s ; b polarization components <strong>of</strong> various polarization variants for 111 c oriented BiFeO 3 thin films as a function <strong>of</strong> normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. FIG. 4. a Free energy F and P s ; b polarization components <strong>of</strong> various polarization variants for 101 c oriented BiFeO 3 thin films as a function <strong>of</strong> normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. 22 = 0 22 , 12 = 21 =0, 13 = 31 = 23 = 32 = 33 =0, 14 where 0 11 = 0 22 =a s −a BiFeO3 /a BiFeO3 , a s and a BiFeO3 are the lattice parameters <strong>of</strong> the <strong>substrate</strong> and film, respectively. Unlike the 001 c oriented film, under a biaxial <strong>substrate</strong><strong>induced</strong> strain the polarization variants do not have equal energy except for a critical point at 0 11 = 0 22 0% as shown in Fig. 3a. For a large compressive strain, the two polarization variants perpendicular to the film surface r + 1 and r − 1 have lower energy than the others, and the compressive <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> increase the P s <strong>of</strong> the variants, which is consistent with prior first-principles calculations. 8,9 It should be noted that in such a case, BiFeO 3 retains the rhombohedral R symmetry P 1 =P 2 =P 3 as shown in Fig. 3b, and indeed rhombohedral symmetry is found experimentally in 111 c epitaxial BiFeO 3 film deposited on a 111 c surface <strong>of</strong> SrTiO 3 , which has smaller lattice parameters than BiFeO 3 . 4,5,24 At the right side <strong>of</strong> the critical point in Fig. 3a, the other six polarization variants become more stable energetically. One can see that the <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> also increase the P s <strong>of</strong> these polarization variants. However, the symmetry <strong>of</strong> BiFeO 3 is no longer rhombohedral, and the stable phases have been marked in Fig. 3b. We also study 101 c oriented BiFeO 3 thin films. The coordinate system x was set up with the x 1 , x 2 , and x 3 axes along the010 c , 101 c , and 101 c crystallographic directions <strong>of</strong> the BiFeO 3 film, and the free energy <strong>of</strong> 101 c oriented BiFeO 3 thin films can be given by Eq. 11 with the transformation matrix from the coordinate system x to the coordinate system x, 0 1 0 − 1 1 0 2 2 15 = 101 t c ij 1 2 0 1 2. For a 101 c oriented BiFeO 3 film deposited on a 101 c oriented cubic crystal <strong>substrate</strong> with an in-plane orientation relationship <strong>of</strong> 010 c BiFeO 3 010 c s , the elastic boundary condition is given by 11 = 0 11 , 22 = 0 22 , 12 = 21 =0, Downloaded 23 Oct 2007 to 128.118.53.4. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
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