Supplementary Materials to
“Surface polar states and pyroelectricity in ferroelastics induced by flexo-roto field”
A.N. Morozovska 1,2 , E.A. Eliseev 2 , S.V. Kalinin 3 , Long Qing Chen 4 and Venkatraman Gopalan 4
1 Institute of Semiconductor Physics, National Academy of Science of Ukraine,
41, pr. Nauki, 03028 Kiev, Ukraine
2 Institute for Problems of Materials Science, National Academy of Science of Ukraine,
3, Krjijanovskogo, 03142 Kiev, Ukraine
3
Center for Nanophase Materials Science, Oak Ridge National Laboratory,
Oak Ridge, TN, 37831
4
Department of Materials Science and Engineering, Pennsylvania State University,
University Park, Pennsylvania 16802, USA
S.1. Equations of state
Free energy function has the form:
F

V
 d
3 rf r
 
S d
2 r
 a i
S
P i
2  b i
S  i
2

(S.1a) f
 f
P
 f

 f
GR
 f
ST
 f
U
 f
P

 f
FL
 f
E
(S.1b)
Polarization-dependent energy is f
P
 a i
 
P i
2  a ij u
P i
2
P j
2
, (S.2a) structural order parameter dependent energy is f

 b i
 i
2  b ij u  i
2  2 j
, (S.2b) gradient energy is f
GR
 g ijkl
2

P i
 x j

P k
 x l
 v ijkl
2
  i
 x j
  k
 x l
 w ijklmn
2


 u ij
 x k
 u lm
 x n

 , (S.2c) elastic strain energy is f
ST
 c ijkl
2 u ij u kl
, (S.2d) electro-striction and roto-striction energy is f
U
  q ijkl u ij
P k
P l
 r ijkl u ij
 k
 l
, (S.2e)
1

biquadratic coupling term is [
1
,
2
,
3
] f
P

   u ijkl
P i
P j
 k
 l
. (S.2f)
Summation is performed over all repeated indices.
It should be noted that the flexoelectric coupling energy could be written in the different forms, namely it is either the form of Lifshitz invariant [4, 5, 6] f
FL
 f ijkl
2 u ij

P k
 x l

P k
 u ij
 x l
, (S.2g) or “short” form f
FL
 f ijkl u ij

P k
 x l
. Despite both forms give the same equations of state, they
have crucial implications on the boundary conditions [4]. Moreover if the Lifshitz invariant is
used, one should include squares of the strain gradients in the free energy (S.2c), w ijklmn
2
 u ij
 x m
 u
 x n kl , where the tensor w ijklmn
is positively defined in order to provide the system stability at arbitrary stresses. Since the strength of the material tensor w ijklmn
is typically unknown and regarded very small, most of the authors (see e.g. [
7
,
8
]) avoided its inclusion and used two different coefficients before u ij

P k
 x l
and P k
 u ij
 x l
(recent review and relations between these variants could be found in [9]). We should stress that any of the form could be safely used till the bulk system is considered and boundary conditions (surface terms) are not important.
At fixed values of conductors potentials (e.g. for short-circuited system placed between conducting electrodes) the energy of electric field E
could be written as (see e.g. Ref.[9, 10,
11]):
V
 f
E d
3 r

V




PE


0
 b
E
2
2

 d
3 r (S.3a)
At fixed values of conductors charges (e.g. for open system without electrode(s)) the electric field energy (S.3a) should be modified by adding the term DE (see e.g. Ref.[12]):

V
 f
E

DE
 d
3 r
 
V f
EL d
3 r
 
V

D
 grad

 d
3 r

 
V f
E d
3 r
 
V
 div
  d
3 r
 
S
 
 d
2 r
 div D

0
Dn
  f
Q
E
 
S
 f d 2 r
 
V



PE


0
 b
E
2
2

 d
3 r
 
Q
E
(S.3b)
Here we introduced the electrostatic displacement D and the electrostatic field potential as
E
  grad . It is seen that the energy at fixed values of conductors charges (S.3b) differ from
2

Eq.(S.3a) only in the surface energy term, proportional to the surface free charges. Note that since f
E

DE
 
PE


0
 b
E
2
2

DE

D

P
 
0
 b
E


0
 b
E
2
2
, expression (S.3b), i.e.
V


0
 b
E
2
2 d
3 r
, looks consistent with corresponding expressions in [9, 13]. As anticipated, the
variation of both Eqs.(S.3a) and (S.3b) gives the same contribution into the equation of state for polarization.
The conditions of the free energy (S.1) extremum are given by zero variation:

F
  i


F
0 ,

P i

0 ,

F
 u ij
  ij
. (S.4a)
Here

is the variational derivative. Equations (S.4a) can be written in the evident form:
 f
  i


 x j


 
 f i
 x j
 
0 ,
 f

P i


 x j

 f


P i
 x j
 
0 , (S.4b)
 f
 u ij


 x k


 u
 f ij
 x k
   ij
. (S.4c)
Here
 ij
are the stress tensor components. In the case of inhomogeneous distributions of strains one should not regard them as independent ones, since compatibility relations should be satisfied.
The explicit form of equations (S.4) is
2 b i
 i

4 b u ij
 j
2  i
 v ijkl 
 2 x
 k j
 x l

2 r mjki u mj
 k

2
 u klij
P k
P l
 j

0 , (S.5a)
2 a i
P i

4 u a ijkl
P j
P k
P l
 g ijkl
 2
P k
 x j
 x l

2 q mjki u mj
P k
 ij
 c ijkl u kl
 r ijkl
 k
 l
 f ijkl


P k
 f mnil
 u mn
 x l

2
 u ijkl
P j
 k
 l
 x l

 q ijkl
P k
P l
 w ijklmn

 2 u lm

E i d
(S.5b)
 x k
 x n

(S.5c)
Mechanical equilibrium equation
  ij
 x j

0 , with
 ij
determined by Eq.(S.4c) gives:

 x j

 c ijkl u kl
 r ijkl
 k
 l
 f ijkl

P k
 x l
 q ijkl
P k
P l
 w ijklmn
 2 u lm
 x k
 x n



0 . (S.6)
In the dielectric limit depolarization electric field components E i d
(if any exist) are determined self-consistently from Maxwell equation

0
 b

E i d
 x i
 

P i
 x i
, (S.7)
3

where

0
=8.85

10
−12
F/m is the universal dielectric constant,
 b
is the “background” isotropic lattice permittivity, different from the ferroelectric soft mode permittivity [
14
,
15
,
16
].
S.2. Boundary conditions
Equations (S.5) should be supplemented with the boundary conditions (BC) at x
3

0 for the structural order parameter, polarization and stress components. As we mentioned above, the form of boundary conditions essentially depends on whether the higher order displacement derivatives are included in the free energy. Below we consider the cases of “short” form and Lifshitz invariant form for flexoelectric coupling energy.
For the structural order parameter BC is the same for both forms of the flexoelectric coupling. They were obtained by the variation of the surface energy and the gradient energy on components

: j

 2 b i
S  i
 n j v ijkl
  k
 x l

 x
3

0

0 , (S.8a)
Here n j
are the components of outer normal to the surface, in the considered case only n
3
 
1 is relevant. No summation over “ i
” is performed in the term
2 b
S  i i
. The first term in Eq.(S.8a) comes from the variation of the surface energy

S b i
S  i
2 d
2 r , while the second is obtained after integration in parts of the bulk energy gradients variations.
BC for elastic stress on the mechanically free flat
surface has the form [5]:



3 i
 f j 3 i 3
2

P j
 x
3
 f j 3 il
2

P j
 x l

 x
3

0

0 (S.8b)
BC for polarization were obtained by the variation of the surface energy

S a i
S
P i
2 d
2 r , the gradient energy and the flexoelectric energy on P . In the case of Lifshitz invariant form of j flexoelectric coupling (S.2g) the higher order derivatives of displacement should be included into the free energy w ijklmn
2


 u ij
 x k
 u lm
 x n

 and the BC for polarization are

 2 a i
S
P i
 n j g ijkl

P k
 x l
 n j f lmij
2 u lm

 x
3

0

0 (S.8c)
No summation over “ i ” is performed in the term 2 a i
S
P i
.
For the
“short”
form of the flexoelectric energy and w ijklmn

0 variation gives:
4


 2 a i
S
P i
 n j g ijkl

P k
 x l
 n j f lmij u lm

 x
3

0

0 , (S.8d)
Note that Eq.(S.8c) and (S.8d) differ in factor ½ in the flexoelectric term (see also [4]).
Also additional boundary conditions should be considered for the case w ijklmn

0
[4]:
n k w ijklmn
 u lm
 x n
 f ijkl
P l
2 x
3

0

0 (S.9)
The latter comes from the facts that the variation of the free energy on displacement vector u i besides (S.4d) leads to the following BC n j 

 2 u i
 f
 x j
 x k

S

0 and n j



 f


 u i
 x j
 

 x k


 2 u i
 f
 x j
 x k



S

0 and Lame type equation
 f
 u i


 x j

 f

 u i
 x j
 
 2
 x j
 x k


 2 u i
 f
 x j
 x k
 
0 . Note that Eq.(S.7d) loses its sense in the limit w ijklmn

0 , if one regard the surface energy independent on the stress and strain.
Otherwise it simply relates the unknown surface energy coefficients and w ijklmn
.
S.3. Evident form of equations of state and elastic solution
For the parent phase of m3m symmetry Eqs.(S.5) have the form:
2 b
1

1

4 b u
11
 3
1

2 b u
12

1

 2
2
  2
3


2

 u
11
P
1
2   u
12

P
2
2 
P
3
2
 

1
  u
44

P
1
P
2

2

P
1
P
3

3

 v
44
 2 
1
 x
3
 x
3

2
 r
11 u
1
 r
12
 u
2
 u
3
 

1
 r
44 u
6

2
 r
44 u
5

3

0
2 b
1

2
 v
44

4 b u
11
 3
 2 
2
 x
3
 x
3

2
 u
2 b
12

2


1
2
2
 r
11 u
2

  2
3


2

 u
11
P
2
2 r
12
 u
1
 u
3
 

2

 r
44 u
6

1
 u
12

P
1
2 
 r
44 u
4

3
P
3
2
 

2

0
2 b
1

3

4 b u
11
 3
3
 u
2 b
12

3


1
2   2
2


2

 u
11
P
3
2   u
12

P
1
2 
P
2
2
 

3
  u
44

P
1
P
2

1
  u
44

P
1
P
3

1

P
2
P
3

3


P
2
P
3

2

(S.10a)
(S.10b)
 v
11
 2 
3
 x
3
 x
3

2
 r
11 u
3
 r
12
 u
2
 u
1
 

3
 r
44 u
5

1
 r
44 u
4

2

0
(S.10c)
Equations of state for polarization components could be written as
2 a
1
P
1

4 a u
11
P
1
3  u
2 a
12
P
1

P
3
2 
P
2
2


2

 u
11

1
2   u
12

 2
3
  2
2
 
P
1
  u
44


1

3
P
3
 
2

1
P
2

 g
44
 2 P
1
 x
3
 x
3

2
 q
11 u
1
 q
12
 u
2
 u
3
 
P
1
 q
44 u
6
P
2
 q
44 u
5
P
3
 f
44
 u
5
 x
3

0
(S.10d)
5

2 a
1
P
2

4 a u
11
P
2
3  u
2 a
12
P
2

P
1
2 
P
3
2


2

 u
11
 2
2
  u
12


1
2  
3
2
 
P
2
  u
44


2

3
P
3
 
2

1
P
1

 g
44
2 a
1
P
3
 g
11
 2
P
2
 x
1
 x
1

 2
P
3
 x
3
 x
3

4 u a
11
P
3
3

2
 q
11 u
2

2
 q
11 u
3

2 u a
12
P
3
 q
12
 u
1 q
12
 u
2
 u
3

P
1
2 
P
2
2


 u
1
 
P
2
 
P
3

2

 u
11

 q
44 u
6
P
1
2
3
  u
12 q
44 u
5
P
1


 q
44 u
4
P
3

1
2   2
2 q
44 u
4
P
2
 f
44
 u
4
 x
5

0

 
P
3
  u
44

 f
11
 u
3
 x
3

1

3

P
1
 u
1


 u
2
2

 f
12  x
3

3
P
2
E d

(S.10e)
(S.10f)
For considered geometry (

||
 
 
1
 

0 ,


 
 
3
 

0 , P
||
 

P
1
 

0 ) only three equations remained:
2 b
1

1

4 b u
11
 3
1
 u
2 b
12

1
  
3
 u
11
P
1
2


1
 v
44
 2 
1
 x
3
 x
3
2 b
1

3

4 b u
11
 3
3
 u
2 b
12

3
 
1

2
 u
12
P
1
2 
3
 v
11
 2 
3
 x
3
 x
3

2
 r
11 u
1
 r
12
 u
2
 u
3
 

1
 r
44 u
5

3

0
(S.11a)

2
 r
11 u
3
 r
12
 u
2
 u
1
 

3
 r
44 u
5

1

0
2 a
1
P
1
 u
4 a
11
P
1
3 
2

 u
11

1
2   u
12

3
2

P
1
 g
44
 2
P
1
 x
3
 x
3

2
 q
11 u
1
 q
12
 u
2
 u
3
 
P
1
 f
44
 u
5
 x
3

0
(S.11b)
(S.11c)
Note that for considered case of 1D distribution of order parameters and strains along x
3 axis, the second and the third terms in Eq.(S.8b) are cancelled out so that Eq.(S.8b) reduces to the conventional BC in the form

3 i x
3

0

0 . w ijklmn

Decoupling on polarization vector in Eqs.(S.11) and neglecting the strain gradient
 2 u lm
 x k
 x n

in (S.5c) leads to the following equations:
2
2 b
1

1 b
1

3


4 b u
11
 3
1
4 b u
11
 3
3


2 b u
12

1
 2
3
2 b u
12

3
 2
1

 v
44 v
11
 2 
1
 x
3
 x
3
 2 
3
 x
3
 x
3


2
 r
11 u
1
2
 r
11 u
3

 r
12
 u
2
 u
3
 

1 r
12
 u
2
 u
1
 

3
 r
44 u
5

3
 r
44 u
5

1


0 , (S.12a)
0 , (S.12b)
2 a
1
P
1

4 a u
11
P
1
3 
2

 u
11

1
2   u
12

3
2

P
1
 g
44
 2
P
1
 x
3
 x
3

2
 q
11 u
1
 q
12
 u
2
 u
3
 
P
1
 f
44
 u
5
 x
3
. (S.12c)
For considered geometry (see Fig. 1a ) elastic solution for strain tensor in Voigt notations
(11=1, 22=2, 33=3, 23=4, 13=5, 12=6) has the form: u
1
 u
3 u
4

 c
11 c
11


 c
12 c
12

 r
11 c
11


2 c
12 r
12
2 c
12
 q
12
P
1
2  r
11
 2
3 c
11

 2
B
, u
2 r
12

1
2
 c
12 c
11

0 , u
5
  f
44 c
44

P
1
 x
3
 r
44 c
44



 c
11 c
11
 c
11 r
11

 c
11 r
12



 c
12 c
11 c
12

 c
11 c
12 c
11 r
11


2 c
12
2 c
12
 r
12
2 c
12


1

3
, u
6

0

 2
B
, (S.13a)


 2
B
, (S.13b)
. (S.13c)
6

Stress tensor components in Voigt notations has the form

1

2



 r
11
 r
12


1
 r
12 c
12 c
11 c
12 c
11



 2
B



 2
B

3
 
1
2
 
1
2
 
4








 r
11 r
11

5 c c
12
11 c
12  c
11
 
6
 r
12



3
2 

 q
12 c c
12
11
 q
11

 P
1
2
, (S.13d) r
12


 2
3

0
 q
12

 c c
12
11


1

P
1
2
, (S.13e)
. (S.13f)
Substituting Eqs.(S.10, 11) in Eq.(S.5) we get coupled system of equations for structural order parameters and polarization. Polarization satisfies the following equation:
2
 a
1



 u
11



 u
4 a
11

2
2 q
12 c
11 q
12 r
12

 P
1
3 c
11



1
2


 g
44




 
11
2 f
44 c
44
  u
11


 2
P
1
 x
3
2

 q
12 r
12 c
11 r
44 f
44 c
44


 2
B



 u
12



1

3

 x
3
 q
12 r
11 c
11


 2
3
P
1
(S.14)
Here we introduced the expansion coefficients at given stress. Namely, for coupling coefficients
 
11
  u
11

2
 q
11

3
 q
12 c
11

 r
11 c
12

 r
12


 q
11

3

2 q
12 c
11

 r
11
2 c
12


2 r
12

, (S.15a)
 
12
  u
12

 q
11

3
 q
12 c
11

 r
11 c
12

 r
12


 q
11

3

2 q
12 c
11

 r
11
2 c
12


2 r
12

, (S.15b)
 
44
  u
44
 q
44 r
44 c
44
. (S.15c)
And for nonlinearity coefficients b

11
 u b
11

 r
11
3
 c
11

 r
12 c
12

2


 r
11
6
 c
11


2 r
12

2
2 c
12
 , b

12
 u b
12

3

 r
11 c
11

 r
12 c
12

2


3

 r
11 c
11


2 r
12
2 c
12

2


2 r
44
2 c
44
(S.16a)
 a
11
 u a
11

 q
11
3
 c
11

 q
12 c
12

2


 q
11
6
 c
11


2 q
12
2 c
12

2
 ,
 a
12
 u a
12

 q
11
3
 c
11

 q
12 c
12

2


 q
11
3
 c
11


2 q
12
2 c
12

2

 q 2
44
2 c
44
(S.16b)
7

S.4. SrTiO
3
material parameters
Table S.I
. SrTiO
3
material parameters included into LGD free energy (S.1)
Parameter SI units Value Source and notes
 b

T
( )
T
0
E dimensionless
10
6
K
43
 m/(F K) 0.75
30
[ a, b
[ c, d
]
] ibidem
T
( E ) q
K 54 ibidem a q ij ij
10
9  m
5
/(C
2
F) u a
11
=2.025, u a
12
=1.215,
10
10 
m/F a

=1.724,
11 a

=1.396
12 q
11
=1.251, q
12
=

0.108, q
44
=0.243
10 -11 
V
 m 3 /C g
11
= g
44
=1, g
12
=0.5
[ ibidem calculated from c
] a ij u g
 ijkl
T
T
S
T
(

) q
10
K
K
26 
J/(m
5
K) 9.1
105
145
[
[
Estimation based on
Ref. [ e
]
[ c ] c c
]
] b r
 v c s f ij ij ij ij ijkl ijkl ijkl
10
10
10
10
10
10
V
50
30
29
10
11
-12





J/m
J/(m
7
(F m)
J/m
J/m m
3
3
3
5
/J
)
-1 b u
11
=1.94, b u
12
=3.96, b

11
=1.69, b

12
=3.88 r
11
=1.3, r
12
=

2.5, r
44
=-2.3
 u
11
=-3.366,
 u
12
= 0.135,
 u
44
=6.3
 
11
=-2.095,
 
12
=

0.849,
 
44 v
11
=0.28, v
12
=

7.34, v
44
=7.11
=5.860
[
[
[
[ c calculated from c c
]
]
] calculated from d, f ] b u ij
 u ij c
11
=3.36, c
12
=1.07, c
44
=1.27 s
11
=3.52, s
12
=

0.85, s
44
=7.87 f
11 e
=

3.24 , f e
12
= 1.44 , e f
44
[ c, d
] calculated from c ij
= 1.08 Recalculated from
Ref.[ g ] at given stress a
G. Rupprecht and R.O. Bell, Phys. Rev. 135 , A748 (1964).
b
G.A. Smolenskii, V.A. Bokov, V.A. Isupov, N.N Krainik, R.E. Pasynkov, A.I. Sokolov, Ferroelectrics and Related Materials (Gordon and Breach, New York, 1984). P. 421 c
N. A. Pertsev, A. K. Tagantsev, and N. Setter, Phys. Rev. B 61 , R825 (2000). d A.K. Tagantsev, E. Courtens and L. Arzel, Phys. Rev. B, 64 , 224107 (2001). e J. Hlinka and P. Marton, Phys. Rev. B 74 , 104104 (2006).
f W. Cao and R. Barsch, Phys. Rev. B, 41 , 4334 (1990) g
P. Zubko, G. Catalan, A. Buckley, P.R. L. Welche, J. F. Scott. Phys. Rev. Lett. 99 , 167601 (2007).
1
M. J. Haun, E. Furman, T. R. Halemane and L. E. Cross, Ferroelectrics, 99 , 55 (1989), ibidem p.13.
2
A.K. Tagantsev, E. Courtens and L. Arzel, Phys. Rev. B, 64 , 224107 (2001).
3
. B. Houchmanzadeh, J. Lajzerowicz and E Salje, J. Phys.: Condens. Matter 3 , 5163 (1991)
4
E.A. Eliseev, A.N. Morozovska, M.D. Glinchuk, and R. Blinc. Phys. Rev. B. 79 , 165433,
(2009).
8

5
A.S. Yurkov, JETP Letters, 94 , 455 (2011).
6
A. K. Tagantsev, A. S. Yurkov, Flexoelectric effect in finite samples, arXiv:1110.0380v1
[cond-mat.mtrl-sci] 3 Oct 2011
7
M. S. Majdoub, R. Maranganti, and P. Sharma, Phys. Rev. B 79 , 115412 (2009).
8
Yue Zheng, Biao Wang, C. H. Woo. Appl. Phys. Lett. 89 , 062904 (2006).
9
N.D. Sharma, C.M. Landis, and P. Sharma, J. Appl. Phys. 108 , 024304 (2010).
10
M. Marvan and J. Fousek, Phys. Stat. Sol. (b) 208 , 523 (1998).
11
A.N. Morozovska, E.A. Eliseev, G.S. Svechnikov, and S.V. Kalinin. Phys. Rev. B 84 , 045402
(2011).
12 L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of Continuous Media , 2nd ed. (Butterworth-Heinenann, Oxford, 1984) Vol. 8.
13 A.K. Tagantsev, L.E. Cross, and J. Fousek: "Domains in Ferroic Crystals and Thin Films"
Springer, 2010; Section 2.3 Thermodynamic approach, pgs. 81-84; and section 8.2 Basics of
Domain State Reorientation, pgs. 354-360.
14
A.K. Tagantsev, and G. Gerra, J. Appl. Phys. 100 , 051607 (2006).
15
C.H. Woo and Yue Zheng, Appl. Phys. A 91 , 59 (2007)
16
G. Rupprecht and R.O. Bell, Phys. Rev. 135 , A748 (1964).
9