Supplementary materials to
Surface Effect on Domain Wall Width in Ferroelectrics
Eugene A. Eliseev,1, *, ‡ Anna N. Morozovska,1, †, ‡ Sergei V. Kalinin,2 Yulan L. Li,3 Jie Shen,4
Maya D. Glinchuk,1 Long-Qing Chen,3 and Venkatraman Gopalan3
1
Institute for Problems of Materials Science, National Academy of Science of Ukraine,
3, Krjijanovskogo, 03142 Kiev, Ukraine
2
The Center for Nanophase Materials Sciences and Materials Science and Technology Division,
Oak Ridge National Laboratory, Oak Ridge Tennessee 37831, USA
3
Department of Materials Science and Engineering, Pennsylvania State University,
University Park, Pennsylvania 16802, USA
4
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA
Appendix A. Depolarization field calculations
Introducing the potential  of the static electric field, E g , f ( x, y, z )   g , f ( x, y, z ) , can
write the equations for potential distribution as follows:
 2g
 z2
 
b
0 33

 2g
 y2
 2 f
 z2

 2g
 x2
 0,
  2 f  2 f
  

  y2
 x2

b
0 11
for  H  z  0 ,
(A.1a)
 P3

, for 0  z  L
 z

(A.1b)
Potentials  g and  f correspond to the dead layer and ferroelectrics respectively. Eqs.(A.1)
should be supplemented with the boundary conditions of fixed top and bottom electrode
potentials, continuous potential and normal component of displacement on the boundaries
between dead layer and ferroelectric, namely
 g z   H   0 ,
*
†
 g z  0   f z  0 ,
(A.2a)
E-mail: eliseev@i.com.ua
E-mail: morozo@i.com.ua, permanent address: V.Lashkaryov Institute of Semiconductor Physics, National
Academy of Science of Ukraine, 41, pr. Nauki, 03028 Kiev, Ukraine
‡
 f z  L   0
These authors contributed equally to this work.
  b33
Using
 f ( z  0)
2D
 g , f ( x , y , z )  2  
z
Fourier
1





P3 ( z  0)
0
  g
transformation
 g ( z  0)
(A.2b)
z
of
the
potentials
and
polarization,
~
 dk1  dk 2 exp  i k1 x  i k 2 y    g , f (k1 , k 2 , z ) , one can rewrite the equation
(A.1) as follows:
 2
~
 2  k 2 
for  H  z  0 ,
g  0,
z

~
 2
k 2 ~
1 P3
 2  2 
, for 0  z  L
f 
b

z

z





0 33
(A.3a)
(A.3b)
b
Here    b33 11
is the dielectric anisotropy factor, k  k12  k 22 . Since boundary
conditions (A.2) are linear on potentials and the coefficients do not depend on lateral
~ (k , k , z ) can be obtained from Eqs. (A.2) by simple
coordinates, boundary conditions for 
g, f
1
2
substitution of functions in real space by their Fourier images. General solution of Eqs. (A.3)
~ (k , z ) ~ exp  k z 

consists
of
the
sum
of
exponential
functions,
and
g
~ (k , z ) ~ exp  k z    
~ k , z  , where 
~

f
part
part is the partial solution of inhomogeneous Eq.
(A.3b). Using the well-known Green’s function1 of the Eq.(A.3b) for homogeneous boundary
conditions the partial solution can be found as
~
 z P3 k ,   sinh k    sinh k L  z   
d


k sinh k L  
 0  b33 0
~
 L P3 k ,   sinh k z   sinh k  L     

d

k sinh k L  
 0  b33 z
~ k , z   

part
It is more convenient to perform integration in parts in Eq. (A.4a):
~
z
cosh k    sinh k  L  z    P3 k ,  
~
 part k , z    d

sinh k L  
 0  b33
0
~
L
sinh k z   cosh k  L      P3 k ,  
  d
sinh k L  
 0  b33
z
(A.4a)
(A.4b)
~
~
~ k , z  z corresponding to potential 
Electric field Ed k , z    
part and related to the ideal
part
case of absent dead layer is
~
P3 k , z 
~
E d k , z   

 0  b33
~
~
k  z cosh k    cosh k L  z    P3 k ,  L cosh k z   cosh k L      P3 k ,   

   d
  d
 0
sinh k L  
sinh k L  
 0  b33
 0  b33 
z
(A.4c)
Using the conditions of short circuit (A.2a), one can write
~ (k , z )  C (k ) sinh k  z  H  ,

g
(A.5a)
g
~ (k , z )  C (k ) sinh k  L  z     
~ k , z 

f
f
part
(A.5b)
Unknown functions C g , f should be determined from the other boundary conditions (A.2).
Namely, applying the conditions of potential and normal components continuity on the boundary
between the dead layer and ferroelectric, one can write the following system of equations for
Cg , f :
C g sinh k H   C f sinh k L  
(A.6a)
~ k , z  0, L   0 . Condition (A.2b) gives the following
Here we take into account that 
part
equation
~
~
  P3 k , z  b  part k , z  
(A.6b)
L   
  33
  g C g k cosh k H 

z
  0
 z 0
~
Using the expression for the field corresponding to potential 
(A.4c) it is easy to get
k
k
 C f cosh 


b
33
part

b
33
~

part
z

z 0
~
P3 k ,0
0
L
  d
~
cosh k  L      P3 k ,   k
sinh k L  
0
0

and finally
k
 cosh 

b
33
~
L
cosh k L     P3 k , 

L C f   g cosh k H C g   d
0
sinh k L  
0

0
The solution of system of two equations (A.6a) and (A.6c) has the form
~
L
cosh k L     P3 k , 
sinh k H 
C f    d
sinh k L  
0
k 
k
0
 b33 cosh  L  sinh k H     g cosh k H  sinh 
 

(A.6c)

L 

(A.7)
Finally, using Eqs. (A.5b) and (A.7), one could write normal component of electric field inside
~
~ (k , z )  z as
ferroelectric, E f 3 (k , z )   
f
~
~
~
E f 3 (k , z )  Eh k , z   Ed k , z 
(A.8a)
where new designation is introduced as
L
~
E h k , z     d
0
~
cosh k L     P3 k ,   sinh k H  cosh k L  z    k 

k
 0   b33 cosh 



k
L  sinh k H     g cosh k H  sinh 



L   sinh k L  

(A.8b)
The depolarization field (A.8) is a linear integral operator, acting on polarization. It would be
~
~ ~
convenient to use below the following designation Eh,d k , z   Eh,d P3 k , z  .


In Eq. (A.8a) the first term is related to the dead layer influence, second term corresponds
to the case of homogenous space without screening (electrodes, dead layers etc.) and the third
term is related to the screening of free charges at the electrodes. Two latter terms in Eq. (A.8) are
absent in the case when polarization is independent on z.
For the case of semi-space filled with ferroelectric (neglecting the bottom electrode), one can
rewrite Eq. (A.8) as follows:
~
k  exp  k  z     P3 k ,   sinh k H 
~ ~
E f 3 P3 k , z     d

0
 0  b33 sinh k H     g cosh k H 
~
~
P3 k , z  k  exp  k z      exp  k   z    P3 k ,  

  d
0
2
 0  b33
 0  b33


(A.9)
In particular case, when the system is transversally uniform (polarization is independent
on lateral coordinates and thus only k=0 is relevant), one could obtain from Eq. (A.8) at k=0, the
following expression for electric field distribution:
E3  z   

P3  z  1 L P3  
H
1 L P3  
d



d

L 0
 0  b33 H   g L  0  b33 L 0  0  b33
P3  z 
 0  b33

g
L
 b33 H   g L 0
d
(A.10)
P3  
 0  b33
The terms, independent on H in Eq.(A.9) is similar to the expression for depolarization field
obtained by Kretschmer and Binder2 for the case of ideal electrodes and absence of background
polarizability (  b33  1 ).
It should be noted, that this problem could be considered for the case of non-ideal
electrodes (e.g., with finite screening length). For the transversally uniform system
corresponding expressions were presented by Tilley.3 Overall structure of depolarization field is
the same in this case, namely, terms related to homogeneous space, ideal screening and imperfect
screening, the latter involving additional length-scale, screening length of electrode.
For further consideration we will need to apply operator (A.8) to several specific distribution, for
~
instance to the constant polarization P0
 
~ ~
E h P0  
~
cosh k  L  z    sinh k H P0

k
 0   b33 cosh 


distributed as cosh si L  z 

k
L  sinh k H     g cosh k H  sinh 



L  

 
~ ~
, Ed P0  0 ;
~
E h cosh si  L  z  

k 
 k 
k cosh k  L  z    sinh k H  k cosh si L  sinh  L    si sinh si L  cosh  L  
 
  



k 
 k 
k 
 0   b33 cosh  L  sinh k H     g cosh k H  sinh  L   sinh  L k 2   2 si2 
 
  
 

~
E d cosh si  L  z  


k 
 si   si cosh si  L  z  sinh  L   k cosh k  L  z    sinh si L 
 



k 
 0  b33 sinh  L k 2   2 s i2 
 
and cosh si z 
~
E h cosh si z  


k 
k cosh k  L  z    sinh k H  k sinh  L    si sinh si L 
 




k 
 k 
k 
 0   b33 cosh  L  sinh k H     g cosh k H  sinh  L   sinh  L k 2   2 si2 
 
  
 

~
E d cosh si z  


k 
 si   si cosh si z  sinh  L   k cosh k z   sinh si L 
 



k 
 0  b33 sinh  L k 2   2 si2 
 
with constants si.
In the limit of semi-infinite media (L)
~
exp  k z   sinh k H P0 k 
~ ~
E h P0 k   
 0  b33 sinh k H     g cosh k H 


 exp  k z  k H ~
P0 k 




0
g
~ ~

E h P0 k   
b
 exp  k z  1  exp  2k H  1  exp  2k H   33    g

 b   

 0  b33    g 
g
 33




 ~
  P0 k 


Appendix B. Solution of linearized equation:
B.1. Dead layer influence
The free energy functional acquires the following form for Fourier images:
2
2
2

 ~
 ~
 ~


 P3 k , z   P32 k , z   P33 k , z  

4
6

2

2
L 
~

2
~
~
~ *e
 dz    P3 k , z    k 2 P



3 k , z   P3 k , z E 3 k , z  
  2


z
2
G   dk1  dk 2  0 
  (B.1)



 1~
sijkl ~
~ *d
~2
~*
~ * k , z  
 ij k , z 

  P3 k , z E3 k , z   Qij 33 ij k , z P3 k , z  

kl
2
 2


  ~

2
2
~
 P k ,0  P



 3
3 k , h  

 2 

~
~
Here we used Parseval theorem, identity E3d ,e  k, z   E3*d ,e k, z  (“e” is external, “d” is
depolarization field). Note, that Bogolubov approximation for Fourier images of the terms P34
2
4
2
6
~
~
~
~
and P36 leads to P32 k , z   P3 k , z  and P33 k , z   P3 k , z  . Variation of Eq.(B.1)
leads to:




 2
~
~
~
~
~ ~
~ ~
 S P3 k , z    S P33 k , z    P35 k , z     2   k 2  P3 k , z   Eh P3 k , z   Ed P3 k , z 
 z

(B.2a)
along with the boundary conditions
~
~
 
 P3 k , z    P3 k , z 
 0,


z

 z 0
~
~
 
 P3 k , z    P3 k , z 
 0. (B.2b)


z

 zL
~
Below we consider the perturbation ~
pk, z  of initial profile P0 (k ) , so that resulted profile will
~
~
have the form P3 k, z   P0 (k )  ~
pk, z  .
Since elastic degrees of freedom could be eliminated by the renormalization of free energy
~
coefficients (see main text and Appendix D), we could use Eq. (12) for the P3 k, z 
determination. Linearization with respect to ~
pk, z  gives the following.
 2
~
 S  3 S P  5 P  pk , z     2   k 2  ~pk , z  
 z

~ ~
~ ~
~
~
 E P k   p k , z   E P k   p k , z 
2
S
h

0
4
S

d


0
(B.2c)
Here we introduced the following designations  S    2q 2 PS2 ,  S    2q 2 .
Let us apply operator d 2 d z 2  k 2  2 to linearized Eq. (B.2c). As it follows directly from
(A.3b)
d
2
d z2  k 2
one
could
obtain
~
~
1
 2 E3 k , z    d 2 P3 k , z  d z 2  0  b33 
the
following
independently
on
relation
the
boundary
~
conditions and presence of dead layer. Since P0 is independent on z, we obtained from Eq.
(B.2c) the following equation
 d2
 d2
k 2 
1 d2~
pk , z 
2  ~
 2  2  S  3 S PS2  5 PS4    



k
p

k
,
z




2
b

 
 0  33 d z 2
d z
 dz

(B.3)
Looking for the solution of Eq. (B.3) in the form pk, z  ~ exp s z  , one can find characteristic
equation for s in the form:
 2 k2 
s2
 s  2  S  3 S PS2  5 PS4    s 2   k 2   
 
 0  b33

The
roots
of
this
biquadratic
(B.4)
equation

2 
  k 2  S  3 S PS2  5 PS4   k 2   0
 2  s 4  s 2   2  S  3 S PS2  5 PS4   k 2   k 2  
b 



0 33 
are
s12, 2 

1
2


1 
1   S  3 S PS2  5 PS4   k 2
k2




2 

 2  0  b33 

  S  3 S P  5 P   k




2
S
4
S
2
 k

2
2

2

4k
  2  S  3 S PS2  5 PS4   k 2

  
 
1
2
b
0 33

(B.5)

It is seen that for any real values of k values of s1, 2 are real. So that the general solution of Eq.
(B.3) acquires the form
pk, z   A1 coshs1 z   B1 coshs1 L  z   A2 coshs2 z   B2 coshs2 L  z 
(B.6)
After substitution of this solution into Eq.(B.2c) one could obtain that terms proportional to
exp  si z  are cancelled out, and the only remained part coming from depolarization field is
linear is linear combination of cosh k z   and cosh k L  z   . Since these functions are
linearly independent at k0, one should equate coefficients near these functions to zero.
k
Coefficients near cosh k z    0  b33 sinh 



L  gives the equation

 s1 k sinh s1 L 
k
2
  2 s12 
A1 
Coefficient near cosh k L  z    gives the equation
 s 2 k sinh s 2 L 
k
2
  2 s 22 
A2  0
(B.7a)

~
sinh k H P0 k 

k
 0   b33 cosh 



k
L  sinh k H     g cosh k H  sinh 



L  


 si k sinh si L Bi
k
 0  b33 sinh 


L k 2   2 si2 



k 
 k 
k sinh k H  k cosh si L  sinh  L    si sinh si L  cosh  L   Bi
 
  



 b
k 
 k 
k  2
2 2
 0   33 cosh  L  sinh k H     g cosh k H  sinh  L   sinh  L k   si 
 
  
 



k
 0   b33 cosh 




k 
k sinh k H  k sinh  L    si sinh si L  Ai
 



 k 
k
L  sinh k H     g cosh k H  sinh  L   sinh 

  


L k 2   2 si2 

0
(B.7b)
Boundary conditions (B.2b) at infinite extrapolation length (natural boundary conditions) give
two equations
A1 s1 sinh s1 L  A2 s2 sinh s2 L  0
(B.8a)
B1 s1 sinh s1 L  B2 s2 sinh s2 L  0
(B.8b)
It is seen that subsystem (B.7a) and (B.7a) has the solution A1=A2=0. Thus Eq. (B.7b) is reduced
to
~
k 
sinh  L  sinh k H P0 k  
 
 s1 k sinh s1 L   b
k 
k
  cosh  L  sinh k H     g cosh k H  sinh 
b
2
2 2  33
 33 k   s1  
 


k 
k
k sinh k H  k cosh s1 L  sinh  L    s1 sinh s1 L  cosh 
 



2
2 2
k   s1 

L   B1 


L   B1


(B.9a)
 s 2 k sinh s 2 L   b
k 
 k 


cosh
L
sinh

k
H




cosh

k
H

sinh


 L   B2 
33
g
 
 b33 k 2   2 s 22  


  

k 
 k 
k sinh k H  k cosh s 2 L  sinh  L    s 2 sinh s 2 L  cosh  L   B2
 
  


0
2
2 2
k   s 2 
B2   B1
And finally one equation for B1 has the view:
s1 sinh s1 L 
s 2 sinh s 2 L 
(B.9b)
k
sinh 

1 ~
L  P0 k  
k


1
1
 k   g
 k  
B 
 s1 sinh s1 L  cosh  L   b coth k H  sinh  L   2

2 2
2
2 2  1



k


s
k


s






33
1
2 

k 
 k 
(B.9c)
 k cosh s1 L  sinh  L    s1 sinh s1 L  cosh  L   B1
 
  



k 2   2 s12 

k 
k
 k cosh s 2 L  sinh  L   s 2 sinh s 2 L  cosh 
 



2
2 2
k   s 2 s2 sinh s2 L 

L  

s1 sinh s1 L B1  0
At the limit of high film thickness it has the following solution:
~
2 exp  s1 L  b33 k 2   2 s12 k 2   2 s 22 P0 k 
s2
B1 
k s 2  s1  b33    g coth k H s 2 s1  2 s 2  s1    b33 k   s1 k  s 2 k   s1  s 2 
(B.10a)
and
B2   B1

s1 exp s1 L 
s 2 exp s 2 L 
s1

~
2 exp  s 2 L  b33 k 2   2 s12 k 2   2 s 22 P0 k 
k s 2  s1  b33    g coth k H s 2 s1  2 s 2  s1    b33 k   s1 k  s 2 k   s1  s 2 
(B.10b)
So, perturbation distribution is
pk , z  
B1
2
exp s1  L  z  
B2
exp s 2  L  z  
2
~
s 2 exp  s1 z   s1 exp  s 2 z  b33 k 2   2 s12 k 2   2 s 22 P0 k 
1

k s 2  s1  b33    g coth k H  s 2 s1  3 s 2  s1    b33 k   s1 k  s 2 k   s1  s 2 
(B.11)
B.2. No dead layer: Surface energy influence.
~
Here we also can use solution in the form P0 (k )  ~
pk, z  with evident dependence (B.9) on
coordinate z with the same characteristic factors si, since Eq. (B.3) is independent on the
~
boundary conditions. After substitution of this solution into Eq.(B.2) and dropping Eh part
(since we put dead layer thickness to zero) one could obtain that terms proportional to exp  si z 
are cancelled out, and the only remained part coming from depolarization field is linear
combination of cosh k z   and cosh k L  z   . Since these functions are linearly
independent at k0, one should equate coefficients near these functions to zero.
Coefficients near cosh k z   gives equation:

 s1 k sinh s1 L 
k
  sinh 

b
0 33

L  k 2   2 s12



A1 
 s 2 k sinh s 2 L 
k
  sinh 

b
0 33

L  k 2   2 s 22



A2  0
(B.12a)
Coefficient near cosh k L  z    gives equation:
:

 s1 k sinh s1 L B1
k
  sinh 

b
0 33

L  k 2   2 s12




 s 2 k sinh s 2 L B2
k
  sinh 

b
0 33

L  k 2   2 s 22



0
(B.12b)
Next we recall the boundary conditions (B.2b) which the following equations for constants Ai
and Bi:
 A1  B1 cosh s1 L   A2  B2 cosh s 2 L   

   P0 (k )
  B1 s1 sinh s1 L   B2 s 2 sinh s 2 L 

(B.13c)
 A1 cosh s1 L   B1  A2 cosh s 2 L   B2

  A1 s1 sinh s1 L   A2 s 2 sinh s 2 L 
(B.13d)

   P0 (k )

The solution has form:
A1 s1 , s 2   B1 s1 , s 2  
A2 s1 , s 2   B2 s1 , s 2   
P0 k   sinh s 2 L 2M s 2 
cosh s1 L 2Det I s1 , s 2 , L 
P0 k sinh s1 L 2M s1 
cosh s 2 L 2Det I s1 , s 2 , L 
,
,

qh
 sh
 sh
 qh 
 M s  cosh 
 sinh    M q  cosh   sinh 
 
 2 
 2 
 2 
 2  

Det I s, q, h   2


 sh
 qh
  q M s   s M q  sinh   sinh 


 2 
 2 


Where M s  
s
k2
 s2
2
(B.14a)
(B.14b)
(B.14c)
. It is clear that A1 s1, s2    A2 s2 , s1 .
At a given extrapolation length , linearized solution of the system diverges at several k
values determined from the condition Det I s1 (k ), s2 (k ), , L  0 . Corresponding solution cr(k)
or kcr() indicates the instability point of bulk domain structure P0  x  with period 2/kcr()
induced by the surface influence. Dependence cr(k) is shown in Fig.B1 for typical ferroelectrics
material parameters and different thickness h.
0
0
2
/L
0.5

3
1

1
2

(a)
h=50L
2

3 
2 
1 0

1
2
3
(b)
h=10L
3 
2 
1 0

1
2
3
0
2
0.5

/L
4
k1L
0
2, 3
2

3
1

1
4

1
1.5

2.5

1
1.5

k1L
2

3
1

4
1.5

2
0.5

6

h=5L
4
3 
2 
1 0

1
k1L
(c)
2
3
8

h=L
4
3 
2 
1 0

(d)
1
2
3
k1L
Figure S2. (Color online) Dependence cr(k1) calculated from Eq.(B.12c) in LiNbO3 at
Rz/R=1.5 (curves 1), LiTaO3 at Rz/R=1 (curves 2), PbZr0.5Ti0.5O3 at Rz/R=1 (curves 3) and
BaTiO3 at RL/R=2 (curves 4) for different film thickness h/R=50, 10, 5, 1 (parts a, b, c, d).
It is clear that zero determinant DetI given by Eq.(B.12c) is possible only at negative 
values, at that two maximums cr(k1) exist in semi-infinite sample and thick films as shown in
Figs.B1a-c; they split into the single maxima cr(0) with film thickness decrease as shown in Fig.
B1d. Note, that thickness-induced paraelectric phase transition at h<hcr takes place only at 0.
The considered spontaneous stripe domain splitting near the film surface appeared at
negative  values could not be treated in terms of linearized approach (13), however the
condition Det I s1 (k ), s2 (k ), , h  0 determine the most probable structure period at a given .
Then, in order to determine the polarization amplitude direct variational method should be used.
Below we consider the range of extrapolation length values where the bulk domain
structure P0  x  is stable and so one may suspect PV  1 to be a good approximation (e.g >0 and
<-2 for thickness h>10L).
For particular case k  0 (transversally homogeneous film) one can obtain characteristic

33 
k 2  2 11
11k 2
2
 L 
33  1L2z    1 and determinant
increments as s 
 0 , s20  2 1 
Lz  33 
33
33
   0
2
10
 1
h

s h
s h
 s h 
Det I s10 , s 20 , h   2
sinh  20  
 cosh  20    s 20 sinh  20    . Under the
 2  2 33  1 
 2 
 2 
 s 20
typical condition s20 h  1 , the solution is

~
233  1 tanh s20 h 2 

P3 k  0, z   PS 1 




s
h
1


s
tanh
s
h
2
20
20
20


1


cosh s20 h 2  z 
1 
 ,




cosh
s
h
2


s
sinh
s
h
2
20
20
20


(B.15)
For
particular
Det I s, q, L    
As, q  
L 
case
one
obtains
determinant
q  s  L 
1
M s   M q   q M s   s M q exp 
,
2
2


~
P0 k   2 exp  s L M q 
M s   M q   q M s   s M q 
coefficient
, polarization Fourier image

exp  s1 z M s2   exp  s2 z M s1  
~
~
P3 k , z   P0 (k )1 

 M s1   M s 2   s 2 M s1   s1 M s 2  
explicit form is:
2
2
 

  exp  s1 z  k  s12  s 2  exp  s 2 z  k  s 22  s1  
2
2





 
~
~



 
P3 k , z   P0 (k )1 

k2



s1  s 2  2  s1 s 2  s1 s 2 s1  s 2 






(B.16a)
At k tending to zero and neglecting  2 S  b33  0 with respect to unity:


z


exp


 b33 0 
~
~


P3 k , z   P0 (k )1 


1

 b33 0 



 11
 k
 2 S  b33



1



~
P0 (k ) exp   2 S  0 11 k z

 b33 0 

 11
 2 S  b33

(B.16b)
k
Approximate analytical results can be derived from (B.16b) for a single domain wall profile in
iL Delta( k 2 )  PS
~
the second order ferroelectrics, since P0 (k ) 
, where Delta( k ) is Diracsinh  k1 L 

delta function. For odd functions f 0 ( x) 
~
f
0
(k1 ) exp ik1 x dk1 , so


k1 exp ik 1 x  k1 zC 
~
~
f ( x, z )   f 0 ( k 1 )
dk1   f 0 (k1 ) k1 exp ik 1 x   zC  q  k1 dk1 
2
1  q k1  Q k1  ...



d ~

f 0 (k1 ) exp ik 1 x  k1  zC  q dk1 
Cdz 


 dyf 0 ( x  y)




2 zC  q 
2  zC  q   y 2
d

dyf
(
x

y
)

0
2 2
2
Cdz y 2   zC  q  2 
y   zC  q 

2

4 zC  q 
2

  dyf 0 ( x  y ) 2

2
 y   zC  q  2

y 2   zC  q 




2
2

.




(B.17)
Appendix C. The case of finite screening length of imperfect top electrode
Introducing the potential  of the static electric field, E g , f ( x, y, z )   g , f ( x, y, z ) , can
write the equations for potential distribution as follows:
 2g
z
2

 2g
y
2
 0  b33

 2 g
x
 2 f
z
2
2

g
Rd2
for    z  0 ,
(C.1a)
 P3

, for 0  z  L
 z

(C.1b)
 0,
  2 f  2 f
b 
  0 11

  y2
 x2

Potentials  g and  f correspond to the top “semi-conducting” media and ferroelectrics
respectively. Here Rd is the screening length of semiconductor. Eqs.(C.1) should be
supplemented with the boundary conditions of fixed potentials at top and bottom electrodes,
continuous potential and normal component of displacement on the boundaries between
electrode and ferroelectric, namely
 g z  0   f z  0 ,
 g  z     0 ,
  b33
 f ( z  0)
z

P3 ( z  0)
0
  g
 f z  L   0
 g ( z  0)
z
(C.2a)
(C.2b)
Here we consider semiconducting media as sufficiently thick in order to neglect the effects of
finite thickness of top electrode.
Using
2D
 g , f ( x , y , z )  2  
Fourier
1




transformation
of
the
potentials
and
polarization,
~
 dk1  dk 2 exp  i k1 x  i k 2 y    g , f (k1 , k 2 , z ) , one can rewrite the equation
(A.1) as follows:
 2
1 ~
 2  k 2  2 
for  H  z  0 ,
g  0,

z
R

d 
~
 2
k 2 ~
1 P3
 2  2  f 
, for 0  z  L
 
 0  b33 z
z
(C.3a)
(C.3b)
b
Here    b33 11
is the dielectric anisotropy factor, k  k12  k 22 . Since boundary
conditions (C.2) are linear on potentials and the coefficients do not depend on lateral
~ (k , k , z ) can be obtained from Eqs. (C.2) by simple
coordinates, boundary conditions for 
g, f
1
2
substitution of functions in real space by their Fourier images. General solution of Eqs. (C.3)
consists
of
the
sum
of
exponential
functions,

~ (k , z ) ~ exp  k 2   2 z

g

and
~ (k , z ) ~ exp  k z    
~ k , z  , where =1/R and 
~

is the partial solution of
d
f
part
part
inhomogeneous Eq. (C.3b). Using the well-known Green’s function4 of the Eq.(C.3b) for
homogeneous boundary conditions the partial solution can be found as
~
 z P3 k ,   sinh k    sinh k L  z   
~
 part k , z   
d


k sinh k L  
 0  b33 0
~
 L P3 k ,   sinh k z   sinh k  L     

d

k sinh k L  
 0  b33 z
(C.4a)
It is more convenient to perform integration in parts in Eq. (C.4a):
~
z
cosh k    sinh k  L  z    P3 k ,  
~
 part k , z    d

sinh k L  
 0  b33
0
~
L
sinh k z   cosh k  L      P3 k ,  
  d
sinh k L  
 0  b33
z
(C.4b)
~
~
~ k , z  z corresponding to potential 
Electric field Ed k , z    
part and related to the ideal
part
case of absent dead layer is
~
P3 k , z 
~
E d k , z   

 0  b33
~
~
k  z cosh k    cosh k L  z    P3 k ,  L cosh k z   cosh k L      P3 k ,   

   d
  d
b

 0
sinh k L  
sinh

k
L


 0  b33


0 33 
z
(C.4c)
Using the conditions of short circuit (C.2a), one can write the solution as


~ (k, z )  C (k ) exp k 2   2 z ,

g
g
(C.5a)
~ (k , z )  C (k ) sinh k  L  z     
~ k , z 

f
f
part
(C.5b)
~ k , z  z  C (k ) cosh k  z  L    k   
~ k , z  z
 
f
f
part
Unknown functions C g , f should be determined from the other boundary conditions (C.2).
Namely, applying the conditions of potential and normal components continuity on the boundary
between the semiconsuctor and ferroelectric, one can write the following system of equations for
Cg , f :
C g  C f sinh k L  
(C.6a)
~ k , z  0, L   0 . Condition (C.2b) gives the following
Here we take into account that 
part
equation
~
~
  P3 k , z  b  part k , z  
L   
  33
  g k 2   2 C g (C.6b)

z
  0
 z 0
~
Using the expression for the field corresponding to potential 
(C.4c) it is easy to get
k
k
 C f cosh 


b
33
part

b
33
~

part
z

z 0
~
P3 k ,0
0
L
  d
~
cosh k  L      P3 k ,   k
sinh k L  
0
0

and finally
k
 k cosh 

b
33
~
L
cosh k L      P3 k ,  

2
2
L C f   g k   C g  k  d
0
sinh k L  
0

0
The solution of system of two equations (C.6a) and (C.6c) has the form
~
L
k P3 k ,  
cosh k  L     
C f    d
sinh k L  

k 
k
0
 0   b33 k cosh  L     g k 2   2 sinh 
 


(C.6c)

L  

(C.7)
Finally, using Eqs. (C.5b) and (C.7), one could write normal component of electric field inside
~
~ (k , z )  z as
ferroelectric, E f 3 (k , z )   
f
~
~
~
E f 3 (k , z )  Eh k , z   Ed k , z 
(C.8a)
where new designation is introduced as
L
~
E h k , z     d
0
~
cosh k L     P3 k ,   cosh k L  z    k 2 

k
 0   b33 k cosh 



 k 
L     g k 2   2 sinh  L   sinh k L  

  
(C.8b)
The depolarization field (C.8) is a linear integral operator, acting on polarization. It would be
~
~ ~
convenient to use below the following designation Eh,d k , z   Eh,d P3 k , z  .


For the case of semi-space filled with ferroelectric (neglecting the bottom electrode), one
can rewrite Eq. (C.8) as follows:
~
k  exp  k  z     P3 k ,  k
~ ~
E f 3 P3 k , z     d

0
 0  b33 k    g k 2   2
~
~
P3 k , z  k  exp  k z      exp  k   z    P3 k ,  

  d
0
2
 0  b33
 0  b33




(C.9)
In particular case, when the system is transversally uniform (polarization is independent
on lateral coordinates and thus only k=0 is relevant), one could obtain from Eq. (C.8) at k=0, the
following expression for electric field distribution:
E3  z   

Rd
P3  z  1 L P3  
1 L P3  
d



d

L 0
 0  b33 Rd   g L  0  b33 L 0
 0  b33
P3  z 
 0  b33

g
L
 b33 Rd   g L 0
d
P3  
(C.10)
 0  b33
Similar expression was presented by Tilley5 for the case of non-ideal electrodes with finite
thickness. The terms, independent on Rd n Eq.(C.10) is similar to the expression for
depolarization field obtained by Kretschmer and Binder6 for the case of ideal electrodes and
absence of background polarizability (  b33  1 ).
Appendix D. Elastic sub-problem solution.
The equations of state for elastic fields (2b),  qijkl Pk Pl  cijkl u kl  ij , could be also transformed
to the following Qijkl Pk Pl  sijkl  kl  uij by the convolution with compliance tensor, s ijkl . The
latter equation is more suitable for our purposes, since we will consider mechanically free
system. For cubic symmetry ferroelectrics with the considering only P3 component of
polarization equation Qijkl Pk Pl  sijkl  kl  uij can be rewritten as:
u11  s1111  s12  22   33   Q12 P32 ,
(D.1a)
u 22  s11 22  s12 11   33   Q12 P32 ,
(D.1b)
u 33  s11 33  s12  22  11   Q11 P32
(D.1c)
u12 
s 44
12
2
(D.1d)
u13 
s 44
13
2
(D.1e)
u 23 
s 44
 23
2
(D.1f)
Note, that denominator “2” appearance in Eqs. (D.1d)-(D.1f) is related to the fact, that we use
tensor notation for strain and stress components, and matrix notations for compliances.
Since we are interested in solution in terms of stresses, let us write the conditions, which
strain and stress distribution should satisfy, namely compatibility condition7

inc i, j, u   eikl e jmnuln,km  0
(D.2)
and equilibrium conditions
 ij, j  0
(D.3)
here comma separated subscript means the derivative on corresponding coordinate, e.g.
ij xk  ij,k .
For the considered case when the system is homogeneous along x2 direction one could
solve only quasi-2D problem, i.e. all functions depend on x1 and x3 only.
Compatibility conditions (D.2) for i=1, j=2 and i=3, j=2 along with equations of state
(D.1e) and (D.1f) lead to 12,33   23,13  0 and 12,13  23,11  0 . Mechanical equilibrium (D.3)
conditions for i=2 lead to 12,1  23,3  0 , so 12,11  23,31  0 and 12,13  23,33  0 after
differentiating on x1 and x3. So one can write two independent equations as 12,33  12,11  0 and
23,33  23,11  0 . These 2D Laplace equations has only zero conditions  23  0 and 12  0
allowing for the boundary conditions on free surfaces and absence of stress on infinity.
The remained mechanical equilibrium (D.3) conditions for i=1 and i=3 11,1  13,3  0
and 13,1  33,3  0 can be fulfilled by introducing of stress function ( x1 , x3 ) as follows:8
11   ,33 ( x1 , x3 ) , 13   ,13 ( x1 , x3 ) ,
33   ,11 ( x1 , x3 ) .
(D.4)



Compatibility conditions (D.2) for the components inc 1,1, u  , inc 1,3, u  and inc 3,3, u 
lead to the conditions u22,33  u22,11  u22,13  0 , which gives u22  const owing to the finite strain
conditions at infinity. The constant strain u22 should be determined from the corresponding
equation of state as u22  Q12 PS2 since stress vanishes and P3   PS at infinity. Then
corresponding equation u22  Q12 P32  s12 11  33   s1122 immediately gives:




s12 11  33  s11 22  Q12 PS2  P32 ( x, z ) .
Eq. (D.5) allows one to determine 22 from known 11 and 33 components.
(D.5)

Compatibility condition inc 2,2, u  is
u11,33  u 33,11  2u13,13  0
(D.6)
Using the equations of state (D.1a)-(D.1c) and definition (D.4), one can rewrite the condition
(D.6) as the equation for ( x1 , x3 ) :
 ,3333   ,1111 s112  s122   2s11  s12 s12  s44 s11  ,1133 
 Q12 s11  s12 P32 ,33  Q11s11  Q12 s12 P32 ,11
For
elastically
isotropic
media
with
s44  2s11  s12 
.
(D.7a)
(and
thus
2s11  s12 s12  s 44 s11  2s112  s122  ) one obtains well-known biharmonic equation for stress
function ( x1 , x3 ) . In general case
,3333  ,1111  2 ,1133 
2
S
Q12P32,33
s11  s12

Q11s11  Q12s12
P32,11 .
s112  s122
(D.7b)
here the elastic anisotropy factor  2S  s11  s12 s12  s 44 s11 2 s112  s122  and designation
P32 ( x, z )  PS2  P32 ( x, z ) are introduced.
Eqs. (D.7) should be supplemented with appropriate boundary conditions. For the case of
mechanically free slab it is
13 ( x3  0)  0 ,  33 ( x3  0)  0 , 13 ( x3  h)  0 and  33 ( x3  h)  0
(D.8)
Here h is the slab thickness.
In order to solve (D.7) along with (D.8) let us use Fourier transformations on coordinate x1 as
( x1 , x3 ) 
1

 dk
2  
1
~ (k , x )
exp  i k1 x1   
1
3
(D.9)
Hereinafter sigh “~” denotes the Fourier image of corresponding function.
Eq. (D.7b) gives the following expression for the Fourier image of stress function:
~
2~
Q12 d 2 P32 Q11s11  Q12 s12 2 ~ 2
d 4~

4~
2 2 d 
 k1   2 S k1


k1 P3 .
d x34
d x32 s11  s12 d x32
s112  s122
(D.10)
Using Eq. (D.8), (D.9) and definitions (D.4), it is easy to find the boundary conditions for stress
function as:
~ ( k , h)  0 .
~(k ,0)  0 , d ~
~(k , h)  0 , d 
(k1 ,0)  0 , 

1
1
1
dz
dz
(D.11)
Below we consider elastically isotropic material with  2S  1 and P32 ( x1 , x3 )  P02 ( x1 ) as
zero approximation for 1D domain structure. For this case solution of Eq.(D.10) was obtained in
the following form:
Q s Q s
~
~
(k1 , z )   112 11 2 12 212 P32 k1  
s11  s12 k
 1  k z e  k z  1  k h  k z e k  z  h   1  k z e  k  2 h  z   1  k h  k z e  k  h  z  

 1 
1  2k h e  k h  e  2 k h


, (D.12)
Here we introduced designation k1  k and x3z.
Using (D.12) and (D.4)-(D.5), one can easily obtain Fourier images of stress components as
Q11s11  Q12 s12 ~ 2
~ (k , z )  k 2 ~

P k  
33
1
1  
s112  s122  3 1

1  k z e k z  1  k h  k z e k  z  h   1  k z e k 2 h z   1  k h  k z e k h z  

   1 
1  2k h e  k h  e  2 k h


2~
~2
~ (k , z )  d    Q11 s11  Q12 s12 P

11
1
3 k 1 , k 2  
2
2
2
dz
s11  s12
 1  k z e  k z  1  k h  k z e  k  z  h   1  k z e  k  2 h  z   1  k h  k z e  k  h  z  

 
k h
2k h
1

2
k
h
e

e


(D.13a)
(D.13b)

e  k z  e  k  z h   e  k 2h z   e  k h z  
~2
~ (k , z )  
~ (k , z )  Q11s11  Q12 s12 P




k

1

2
33
1
11 1
s112  s122  3 1 
1  2k h e k h  e 2 k h

k z
k  z h 
 e  k 2 h z   e  k h z    ~ 2
~ (k , z )   Q12  s12 Q11s11  Q12 s12 1  2 e  e
 P3 k1  (D.13c)

22
1

s
s112  s122
1  2k h e k h  e 2 k h


 11 s11
~2
~ (k , z )  i k Q11 s11  Q12 s12 P
k , k 

13
1
1
2
2
s11  s12  3 1 2

z e  k z   h  z e  k  z  h   z e  k  2 h  z    h  z  e  k  h  z 
(D.13d)
1  2k h e  k h  e  2 k h
~ (k , z )  0

13
1
It should be noted that

,
does not contribute into the convolution

~ Q 
~
~
~
Qij33
ij
11 33  Q12 11  22 for cubic symmetry.
For the arbitrary slab thickness h and coordinate z, expression (13) could not be
transformed into real space. However, in the limit of half-space (h) for the points on the
surface (z=0) and far from surface (z) closed form expressions could be obtained. For
instance, far from the surface originals of Eqs.(D.13) have the form:
11( x, )  0 ,  22 ( x, ) 
33 ( x, ) 
Q12 s11  Q11s12 2

PS  P02 ( x)  ,
2
2
s11  s12
Q11s11  Q12 s12 2
PS  P02 ( x) .
s112  s122
On the surface (z=0) original of Eqs.(D.13) have the form:
(D.14a)
(D.14b)
Q11s11  Q12 s12 2
PS  P02 ( x) ,
2
2
s11  s12


33 ( x,0)  0 ,
(D.15a)
Q
s Q s  Q12s12  2
 PS  P02 ( x) ,
22 ( x,0)   12  12 11 112
2
s11  s12 
 s11 s11
(D.15b)
11 ( x,0)  


Finally stress convolution with electrostriction Qij33ij  Q1133  Q12 11   22  , induced by the
unperturbed solution P0 x  , for elastically isotropic semi-infinite ferroelectric material has the
form:


 2

2
 Q11  Q12 s11  2Q12Q11s12


2
2


s11  s12


Q11s11  Q12 s12

 ~2
~
Q11  Q12  
Qij 33ij (k1 , z )    k1 z exp  k1 z 
(D.16a)
2
2
P3 k1 
s11  s12




Q s  Q12 s12 
s12   

 
  exp  k1 z  11 112
Q

Q
1

2
12 
2
 11


s

s
s
11
12
11

  


~
here Voigt notations are used; P32 k1  is the Fourier image of the difference




P32  x   PS2  P02 ( x) , PS is bulk spontaneous polarization. For the second-order ferroelectrics
PS2     , while PS2 
   4   2
2
for the first order ones. At the bulk and surface
(D.16a) is simplified to
Qij33ij ( x, ) 
Q
2
11
Qij33ij ( x,0)  Q12

 Q122 s11  2Q12Q11s12 2
PS  P02 ( x)
2
2
s11  s12



Q12 s11  2s12  Q11s11 2
PS  P02 ( x)
s11 s11  s12





(D.16b)
(D.16c)
Substitution of inhomogeneous stresses (D.16) into equation of state for polarization
leads to the following renormalization of coefficients  and  near the surface:



Q s  2s12  Q11s11 2 
 S ( z  0)  1  2Q12 12 11
PS  ,
s
s

s

11 11
12







Q s  2s12  Q11s11 
 ,
S ( z  0)  1  2Q12 12 11
s11 s11  s12 




(D.17a)
(D.17b)
While in the bulk



Q 2  Q122 s11  2Q12Q11s12 2 
 S ( z  Lbz )  1  2 11
PS  ,
s112  s122 







Q 2  Q122 s11  2Q12Q11s12 
 .
S ( z  Lbz )  1  2 11
2
2
s

s

11
12




(D.18a)
(D.18b)
This immediately leads to different transverse correlation radius near the domain wall. Namely,
at the surface
R , z  z  0   R
b
, z

4Q12 Q12 s11  2s12   Q11 s11  2 
1 
PS 
2
4
 s11 s11  s12   3PS  5 PS  
1 2
.
(D.19a)
While far from the surface:
R , z ( z   )  R
b
, z

Q 2  Q 2 s  2Q12Q11s12 P 2 
1  4 2 11 2 12 11
s11  s12   3PS2  5 PS4  S 

1 2
.
(D.19b)
Here Rb     3PS2  5 PS4  and R zb     3PS2  5 PS4  are stress-free transverse
and longitudinal correlation length. The stress-free correlation lengths are typically from several
to tens of lattice constants for ~3-10 K below phase transition temperature. However, they
strongly depend on temperature and tend to infinity at Curie temperature for the second order
ferroelectrics.9
Estimations of correlation length in typical ferroelectrics are summarized in Table 1.
Striction and free energy expansion coefficients were taken from Refs. [10,
11
]. Note that for
diffraction methods the observable quantity is R,z, not R,zb. Similar mechanism of elastic stress
influence on domain wall width should exist in all ferroic materials.
Table S1. Dielectric permittivity ii and correlation radii ratio for typical ferroelectrics.
Material
11
33
R,z(0)/R,zb
R,z()/R,zb
R,z(0)/R,z()
PbZr0.6Ti0.4O3
529
295
0.63
0.54
1.17
PbZr0.5Ti0.5O3
1721 382
0.30
0.27
1.11
PbZr0.4Ti0.6O3
498
197
0.36
0.28
1.27
PbTiO3
140
105
0.66
0.58
1.14
BaTiO3
2920 168
0.80
0.74
1.09
LiNbO3
85
29
0.996
0.986
1.01
LiTaO3
54
44
0.994
0.988
1.01
The ratios R,z(0)/R,z() can be closer to 1 allowing for the stress relaxation on the
defects typically concentrated in the vicinity of domain walls. Qualitatively, inhomogeneous
elastic stress leads to clamping and decrease of domain wall width in perovskites with high
striction coefficients, since the wall width w(z)~R(z), and it slightly increases when approaching
the surface because R(0)/R()>1. However, for materials from Tab.1 we obtained the typical
ratio 1<R(0)/R()<1.3 and so 1<w(0)/w()<1.3 assuming that the wall broadening is ascribed
to inhomogeneous elastic stress effect only.
Next we consider the surface displacement caused by the spontaneous strain and effects
of stress concentration near the domain wall on displacement.
Using Eqs. (D.1) and (D.13), it is easy to get the Fourier images of strain distribution. However,
since far from wall the dilatational strain components tend to constant value (spontaneous
internal strain), one should consider the Fourier images of inhomogeneous part only, namely
quantities u11  u11  Q12 PS2 , u 22  u 22  Q12 PS2 , u 33  u 33  Q11 PS2 . Since some of the
expressions are very cumbersome, we listed below the images on the surface (z=0):
~

2 k h e  k h Q11  Q12 s11  1  e 2 k h Q11  Q12 s11  2Q12 s12 P32 k1 
~
u11 (k1 , z  0) 
s11 1  2k h e k h  e 2 k h 
u~22 (k1 , z)  0
(D.20b)
2 k h e k h s11  1  e 2k h s11  2s12 
~
u~33 (k1 , z  0)  Q11s11  Q12 s12 P32 k1 
s11 s11  s12 1  2k h e k h  e 2 k h 
Q s  Q12 s12 ~ 2
u~13 (k1 , z )  i k1 11 11
P3 k1 , k 2 
s11  s12 

(D.20a)
z e  k z   h  z e  k  z  h   z e  k  2 h  z    h  z  e  k  h  z 
(D.20c)
(D.20d)
1  2k h e  k h  e  2 k h
Also we need z-derivative for the displacement calculations:
s12  ~ 2

2k 1  e  k h 
~
P3 k1 
u11,3 (k1 , z  0)   Q11  Q12
s11 
1  2k h e  k h  e  2 k h

2
(D.20e)
Using the definition of strain components u ij  u i , j  u j ,i , it is easily to get the following
relations for the considered case, when all functions depend on x1 and x3 only:7
u 3,11  2u13,1  u11,3 ,
u 3,3  u 33
(D.21a)
u1,33  2u13,3  u 33,1 ,
u1,1  u11
(D.21b)
Since on the free surface (x3=0) 13=0 and u13=0, the first relation of Eq. (D.21a) could rewritten
for the Fourier images as  k 2 u~3 (k1 , x3  0)  u~11,3 (k1 , x3  0) and thus Fourier image of the
vertical component of the surface displacement is
2 ~
21  e  k h  P32 k1  
s 
~
 Q  Q12 12 
u3 (k1 , x3  0) 
(D.22)
k h
 2 k h  11

k 1  2k h e  e
s11 
~
Using the Fourier image of polarization square P32 k1   PS2 2 2 L2 k1 sinh  k1 L  , it is easy
to obtain the distribution of vertical displacement on the surface as
s 
8L2 1  exp  k h  cosk x1 d k

u3 ( x1 , x3  0)  PS2  Q11  Q12 12  
s11  0 1  2k h exp  k h   exp  2 k h  sinh  k L 

2
(D.23a)
Note, that while writing Eqs.(D.22), (D.23) we assumed that the displacement far from wall
tends to zero. In this case maximal displacement is achieved at the wall. For the latter
approximate expression were obtained from (D.23a) as follows
s  8  h

u3 ( x1  0, x3  0)  PS2  Q11  Q12 12  L  ln 
s11      L



  0.571 .


(D.23b)
The comparison of exact and approximate dependences is presented in Fig. S1.
Displacement u3/L
0.5
0.5
0
0
0.1
1
10
102
103
slab thickness h/L
0
100
200
slab thickness h/L
Fig. S1. The dependence of maximal value of vertical surface displacement on the slab thickness
for PS2 Q11  Q12 s12 s11  =0.04. Solid and dotted curves represent the exact and approximate
dependences (D.23a) and (D.23b) respectively.
 10 2 )
It is seen, that for the typical values of spontaneous strain (for most ferroelectrics PS2 Qij ~
surface displacement does not exceed several L in the wide interval of slab thickness.
Appendix E. Phase field modeling
Below we study numerically the effect of finite extrapolation length on periodic c-domain
structure near the surfaces of a thin film by using phase field method. The spontaneous
polarization, P=(P1, P2, P3), is taken as the order parameter. For the considered uniaxial
ferroelectrics LiTaO3 and LiNbO3, P1=P2=0 are assumed. The spatial-temporal evolution for P3
is calculated from the Landau-Khalatnikov equation
P3 (r, t )
t
 
G
,
P3 (r, t )
(E.1)
where  is the kinetic coefficient, related to the domain wall mobility, radius-vector r ( x, y, z ) ,
G (or F depending on the mechanical boundary conditions) is the free energy of the system given
by Eq.(5). Variational derivative G / P3 (r, t ) represents the thermodynamic driving force for
the spatial and temporal evolution of the simulated system.
P 
P 


Corresponding boundary conditions are  P3 1 3  0,  P3 2 3  0 .
z  z 0 
z  z h

The free energy bulk density g includes polarization (or Landau) energy, domain wall (or
correlation) energy, electrostatic and elastic energy. So the free energy density is written as
g  f Lan ( P3 )  f grad ( P3, j )  f elec ( E3 , P3 )  f elastic (ij , P3 ) ,
where
f Lan 
(E.2)
 2  4
P3  P3 . The expansion coefficients in SI units are =-1.256109,
2
4
=5.043109 for LiTaO3 and =-2.012109, =3.608109 for LiNbO3, respectively.
The correlation energy density is f grad
2
2
2
1  P3 
1  P3   P3  
  , where 
 
  
 
2  z 
2  x   y  
1
and  are the gradient energy coefficients. In the simulations, we take    * L2 and
2
1
    * H 2 , where * and  * are dimensionless parameters, H and L represent the real
2
simulation cell size of 2L2H in a 2D model, and  
*
4 L2z
H2
, 
*
4 L2
L2
.
1 

The electrostatic energy density, which can be expressed as f elec  E30  E3d  P3 , where
2 

E 3d is the component of the depolarization electric field. Without any applied electric field E 30 ,
depolarization field is induced only by the inhomogeneous spontaneous polarizations allowing
for screening charges on the electrodes. Depolarization field potential  satisfy electrostatic
  2   2   1 P3
 2

equilibrium equation (2), namely
(where  0  8.85 10 12 Fm 11  2 
2
2 
z
 y  0  z
x
1
and 11 =54 for LiTaO3 and 11 =85 for LiNbO3) and short-circuit boundary condition
 | z 0   | z  h  0 .
Eq. (E.1) was solved by using a mixed Chebyshev-collocation Fourier-Galerkin
method.12,
13
The simulations started from a 180o periodic domain structure with sharp interface
and uniform polarization at each domain. For 180o-domain wall in LiTaO3 or LiNbO3 the elastic
energy contribution appeared relatively small allowing for small striction coefficients. We
assumed that electric equilibrium is established instantaneously for a given polarization
distribution. The polarization profiles of Fig. 7 are the stable profiles that existed at the end of
each simulation at times t much longer that Khalatnikov relaxation time.
1
G.A. Korn, and T.M. Korn. Mathematical handbook for scientists and engineers (McGraw-Hill,
New-York, 1961) p.262.
2
R. Kretschmer, and K. Binder. Phys. Rev. B 20, 1065 (1979).
3
Tilley (1996).
4
G.A. Korn, and T.M. Korn. Mathematical handbook for scientists and engineers (McGraw-Hill,
New-York, 1961) p.262.
5
Tilley (1996).
6
R. Kretschmer, and K. Binder. Phys. Rev. B 20, 1065 (1979).
7
S.P. Timoshenko, and J.N. Goodier, Theory of Elasticity (McGraw-Hill, New-York 1970).
8
L.D. Landau and E.M. Lifshitz, Theory of Elasticity. Theoretical Physics, Vol. 7 (Butterworth-
Heinemann, Oxford, 1976).
9
M. E. Lines and A. M. Glass, Principles and Application of Ferroelectrics and Related
Materials (Clarendon Press, Oxford, 1977)
10
D.A. Scrymgeour, V. Gopalan, A. Itagi, A. Saxena, and P.J. Swart, Phys. Rev. B 71, 184110
(2005).
11
12
M.J. Haun, E. Furman, S.J. Jang, and L.E. Cross, Ferroelectrics 99, 63 (1989).
D. Gottlieb, and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and
Applications (SIAM-CBMS, Philadelphia, 1977).
13
C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral methods. Scientific
Computation (Springer-Verlag, Berlin, 2006).