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

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
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