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

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 ,
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