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

More documents

Recommendations

Info

114105-2 Zhang et al. J. Appl. Phys. 101, 114105 2007 employed and i, j,k,l=1,2,3. The electrostrictive coefficients, q ijkl , defined here can be easily obtained from q ijkl =2c ijmn Q mnkl , where Q mnkl are the electrostrictive coefficients typically measured experimentally. Although sixth-order terms in polarization are required for a first-order ferroelectric transition in BiFeO 3 , 17 we only considered terms up to fourth order due to the lack <strong>of</strong> sufficient experimental data to obtain the numerical values <strong>of</strong> all the coefficients. Since our focus is on the effect <strong>of</strong> <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> on the spontaneous polarization <strong>of</strong> BiFeO 3 thin films at room temperature, such approximation will not impact the main results <strong>of</strong> this article. We also neglect the magnetoelectric coupling term in Eq. 1 due to fact that BiFeO 3 has a large polarization but a quite small saturated magnetization, 2 and the coupling term should have small effect on the spontaneous polarization. With these approximations, the free-energy expression <strong>of</strong> BiFeO 3 was reduced to FP, = 1 P 1 2 + P 2 2 + P 32 + 11 P 1 4 + P 2 4 + P 34 + 12 P 1 2 P 2 2 + P 1 2 P 3 2 + P 2 2 P 32 2 + 1 2 c ijkl ij kl − 1 2 q ijkl ij P k P l . 3 The elastic constants <strong>of</strong> cubic phase BiFeO 3 are used, c 1111 =3.0210 11 Nm −2 , c 1122 =1.6210 11 Nm −2 , and c 1212 =0.6810 11 Nm −2 , which were obtained from firstprinciples calculations. The Landau expansion coefficients and the electrostrictive coefficients were obtained by fitting them to experimental measurements, 18–21 i.e., 1 =4.9T −110310 5 C −2 m 2 N, 11 =6.510 8 C −4 m 6 N, 12 =1.010 8 C −4 m 6 N, Q 1111 =0.032 C −2 m 4 , Q 1122 = −0.016 C −2 m 4 , and Q 1212 =0.01 C −2 m 4 , where T is temperature in Kelvin. It should be noted that, due to the lack <strong>of</strong> reliable experimental data about the ferroelectric properties TABLE I. Properties <strong>of</strong> BiFeO 3 films at room temperature, T=300 K. Properties <strong>of</strong> BiFeO 3 films on SrTiO 3 <strong>substrate</strong>s This work a Experimental measurements Polarization P 3 <strong>of</strong> 0.63 0.55, b 0.64 c 001 c BiFeO 3 film C/m 2 Polarization P 3 <strong>of</strong> 0.80 0.80, b 0.81 c 101 c BiFeO 3 film C/m 2 Polarization P 3 <strong>of</strong> 0.96 1.00, b 0.96 c 111 c BiFeO 3 film C/m 2 Dielectric constant <strong>of</strong> 111 c BiFeO 3 film 62 45, b 60 c a The <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> were assumed to be 0 11 = 0 0 22 =−0.015, 12 =0 in the calculations. b Reference 20. c Reference 21. <strong>of</strong> bulk BiFeO 3 , we calculated the ferroelectric properties <strong>of</strong> different oriented BiFeO 3 films grown on SrTiO 3 <strong>substrate</strong>s, and fitted them with the corresponding experimental measurements <strong>of</strong> BiFeO 3 films 20,21 to obtain the needed coefficients. The calculated ferroelectric properties <strong>of</strong> BiFeO 3 films at room temperature are compared in Table I with the available experimentally measured values. For a general case, <strong>strains</strong> are imposed along the x 1 -x 2 directions, while all the stress components involving the x 3 direction are zero due to the existence <strong>of</strong> the free film surface, i.e., 11 = 0 11 , 22 = 0 22 , 12 = 21 = 0 12 , 13 = 31 = 23 = 32 = 33 =0, where the misfit <strong>strains</strong> 0 ij depend on the lattice parameters <strong>of</strong> the film and <strong>substrate</strong>. With such specified boundary conditions, the free energy <strong>of</strong> 001 c oriented BiFeO 3 thin film is given by 4 F = 1 − c 1111q 1111 − c 1122 q 1122 0 0 11 + c 1111 q 1122 − c 1122 q 1122 22 2 P 2c 1 1111 + 1 − c 1111q 1122 − c 1122 q 1122 0 0 11 + c 1111 q 1111 − c 1122 q 1122 22 2 P 2c 2 1111 + 1 − c 1111q 1122 − c 1122 q 1111 0 0 11 + c 1111 q 1122 − c 1122 q 1111 22 2 P 2c 3 1111 −2q 1212 0 12 P 1 P 2 + 12 − q 2 1122 4c 1 1111P 2 P 2 2 + 12 − q 1111q 1122 − q 2 1212 4c 1111 2c 1 1212P 2 P 2 3 + P 2 2 P 32 + 11 − q 2 1122 P 4 1 + P 24 + 11 − q 2 1111 8c 3 1111P 4 − c 2 1122 0 11 + 0 22 2 + 1 2c 1111 2 c 1111 11 8c 1111 0 2 + 0 22 2 + c 1122 0 11 0 22 +2c 1212 12 0 2 . 5 Downloaded 23 Oct 2007 to 128.118.53.4. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
114105-3 Zhang et al. J. Appl. Phys. 101, 114105 2007 FIG. 1. a Free energy F and P s ; b polarization components <strong>of</strong> 001 c oriented BiFeO 3 thin films as a function <strong>of</strong> normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. The free energy and P s <strong>of</strong> rhombohedral phase R and tetragonal phase T were also given for comparison. It should be noted that Eq. 5 is essentially the same as that reported in Ref. 14 except for the sixth-order terms and the different notation <strong>of</strong> coefficients. The spontaneous polarization can be derived from Eq. 5 by solving F/P i =0 i =1,2,3. For the case <strong>of</strong> a 001 c oriented BiFeO 3 film epitaxially grown on a 001 c oriented cubic single-crystal <strong>substrate</strong> with an in-plane orientation relationship <strong>of</strong> BiFeO 100 3 c 100 s c , a constant dilatational plane strain, socalled biaxial strain, is imposed along the x 1 -x 2 directions, 0 11 = 0 22 , 0 12 =0, 6 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. The dependence <strong>of</strong> the polarizations on the <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> is shown as Fig. 1. We can see that the in-plane components <strong>of</strong> polarization P 1 =P 2 are not equal to the out<strong>of</strong>-plane component P 3 due to the constraint <strong>of</strong> the <strong>substrate</strong>. As expected, compressive <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> lead to an increase in P 3 and decrease in P 1 and P 2 , while tensile <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> have the opposite effect. Therefore, the direction <strong>of</strong> the polarization vector deviates from its equilibrium 111 c directions toward 001 c /001 c directions for compressive <strong>strains</strong> and 110 c /110 c /110 c /110 c directions for tensile <strong>strains</strong>. As shown in Fig. 1b, BiFeO 3 becomes the monoclinic M A FIG. 2. Free energy F and P s <strong>of</strong> various polarization variants for 001 c oriented BiFeO 3 thin films as a function <strong>of</strong> a anisotropic normal <strong>substrate</strong><strong>induced</strong> <strong>strains</strong> and b shear <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. phase and M B phase, 22 respectively, which is consistent with prior experimental observations. 4,23 However, the change <strong>of</strong> the absolution value <strong>of</strong> the spontaneous polarization P s =P with the <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong> is rather small as shown in Fig. 1a. For example, a rather large compressive strain <strong>of</strong> 1.6% leads to an increase <strong>of</strong> P s only about 0.3%. It should be noted that all eight polarization variants are degenerate in energy and hence should exist with the same probability in the 001 c oriented BiFeO 3 thin film. For the case <strong>of</strong> a 001 c oriented BiFeO 3 film epitaxially grown on a 110 o oriented orthorhombic <strong>substrate</strong> for example 110 o oriented DyScO 3 with an in-plane orientation BiFeO relationship <strong>of</strong> 100 3 c 110 s BiFeO o and 010 3 c 001 s o ,an anisotropic in-plane strain is imposed along the x 1 -x 2 directions, while there is no shear strain. 0 11 0 22 , 0 12 =0, 7 where 0 11 = 2 as +b 2 s −a BiFeO3 /a BiFeO3 , 0 22 =c s −a BiFeO3 / a BiFeO3 , and a s , b s , and c s are the lattice parameters <strong>of</strong> the orthorhombic <strong>substrate</strong>. By fixing the ratio <strong>of</strong> 0 11 / 0 22 ,we plot the dependence <strong>of</strong> free energy and P s on the <strong>substrate</strong><strong>induced</strong> <strong>strains</strong> in Fig. 2a. It was interesting to note that all eight polarization variants have equal energy even under the anisotropic <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. The absolution value <strong>of</strong> the spontaneous polarization P s is insensitive to the normal <strong>substrate</strong>-<strong>induced</strong> <strong>strains</strong>. Shear <strong>substrate</strong> strain may also exist for 001 c oriented BiFeO 3 films, i.e., 0 11 = 0 22 0, 0 12 0. A potential example Downloaded 23 Oct 2007 to 128.118.53.4. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
Page 1: JOURNAL OF APPLIED PHYSICS 101, 114
Page 5 and 6: 114105-5 Zhang et al. J. Appl. Phys

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

Create successful ePaper yourself

Delete template?

Save as template?