Supplementary Materials to
“Local Probing of Mesoscopic Physics of Ferroelectric Domain Walls”
Vasudeva Rao Aravind,1,2,* A.N. Morozovska3, S. Bhattacharya, 1 D. Lee4, S. Jesse5, I.
Grinberg,6 Y. Li,1 S. Choudhury1, P. Wu,1 K. Seal,6 A.M. Rappe,6 S.V. Svechnikov,3
E.A.Eliseev,7 S.R. Phillpot,4 L.Q. Chen,1 Venkatraman Gopalan,1,† and S.V. Kalinin5,‡
1
Materials Research Institute and Department of Materials Science and Engineering,
Pennsylvania State University, University Park, PA 16802
2
Physics Department, Clarion University of Pennsylvania, Clarion, PA 16214
3
V. Lashkarev Institute of Semiconductor Physics,
National Academy of Science of Ukraine, 41, pr. Nauki, 03028 Kiev, Ukraine
4
Department of Materials Science and Engineering,
University of Florida, Gainesville FL 32611
5
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory,
Oak Ridge, TN 37831
6
Department of Chemistry, University of Pennsylvania, Philadelphia, PA
7
Institute for Problems of Materials Science,
National Academy of Science of Ukraine, 3, Krjijanovskogo, 03142 Kiev, Ukraine
*
This author was previously known by the name Aravind Vasudevarao
vgopalan@psu.edu
‡
sergei2@ornl.gov
†
Appendix A. Calculations within Landau-Ginzburg-Devonshire approach
I. Basic equations
Landau-Ginzburg-Devonshire free energy for the uniaxial ferroelectric is
2
d  
h 
 dz  P 2   P 4    P     P 2  P E e  E3    



 3
  2
4
2   z 
2
2   

  (A.1)
G P, E3e    dx  dy 0 
 




P 2 z  0 
P 2 z  h 
 

2 2
 2 1

where   0 and   0 are expansion coefficients of LGD free energy on polarization powers
for the second order ferroelectrics, electric field E3  E3  E3d is the sum of external and
depolarization fields. Corresponding LGD-equation:

  2 P3  2 P3
 2 P3
d

P3  P3  P33  


  x2   y2
dt
 z2


  E 3 ( x, y , z ) .


(A.2)
 in kinetic coefficient. The electric field E3 x, y, z     z can be expressed via electrostatic
potential (r ) .
The electrostatic potential distribution, (r ) , in ferroelectric obeys the equation
  2   2   1 P3
 2
,


 11  2 
 z2
 y 2   0  z
x
b
33
(A.3)
where b33 is the dielectric permittivity of background state1 and  0 is the universal dielectric
constant. The potential distribution induced by the probe yields boundary conditions
 ( x, y, z  0)  Ve ( x, y) . In the effective point charge approximation the distribution Ve(x,y) can
be approximated as Ve ( x, y, d )  V d
x 2  y 2  d 2 , where V is the applied bias and d is the
effective probe size.
For more realistic modeling of the tip shape the summation over the image charges
positions di (or integration over the line charge in order to account for the conic part of the probe
tip, see e.g. Refs. 2, 3, 4) should be performed, namely

 L  L  L  L 2  x 2  y 2  
2 e
1
1


V

ln 
2
 
2
2
2
2
2
2
ln ctg  2   e   
 i x  y  di
L  L  x  y



. (A.4)
Ve ( x , y ) 
2 e ln 1  L L 
1
i d i     ln ctg 2  2
e
Hereinafter    3311 is the effective dielectric constant,  e is the ambient dielectric constant.
The conical part potential is modeled by the linear charge of length L with a constant charge
density  L  40eV ln ctg 2  2 , where  is the cone apex angle. The distance between the
linear charge and the ferroelectric surface is L .
Allowing for the principle of the electric field superposition and linear electrostatic
equations below we could consider the single-charge component Ve(x,y,d) and the perform the
integration/averaging in the final results. The corresponding Fourier representation on transverse
~
~ z is the sum of external
coordinates {x,y} of electric field normal component E3 k, z    
(e) and depolarization (d) fields:
~
~
~
E3 k, z   E3e Ve , k, z   E3d P3 , k, z ,
(A.4a)
cosh k h  z   b  k
~
~
E3e k , z   Ve k 
,
sinh k h  b   b
(A.4b)
~
z

P k , z '
cosh k h  z   b  k
 dz ' 3

cosh

k
z
'



b

b


sinh

k
h





~
b
b
0 33
0
E3d P3 , k , z   

~
~
h
P3 k , z '
cosh k z  b  k P3 k , z  

  dz '   b cosh k h  z '  b  sinh k h       b 
z
b
b
0 33
0 33 
(A.4c)
Here  b   b33 11 is the “bare” dielectric anisotropy factor, k  k1,k2  is a spatial wavevector, its absolute value k  k12  k22 . For typical ferroelectric material parameters the
inequality 2 0  b33   1 is valid.
II. Perturbation theory
To obtain the spatial distribution of the polarization at small positive biases, V, Eq. (A.1a)
was linearized as P3 r    P0 x   pr  , where P0  x  is the initial flat domain wall profile
positioned at x=x0:


P0  x   PS tanh x  x0  2 L .
(A.5)
where the correlation length is L    2 , and the spontaneous polarization is PS2     .
Polarization p(r) is the induced due to materials response to a biased probe. The condition
pr   0 is valid far from the probe at an arbitrary applied bias. Here, we derive the solution
within a perturbation approach.
Under the condition of a thick sample, h  d , the approximate closed form expression
for the linearized stationary solution of Eq. (A.2) is derived as (see Supplement in Ref.[5] for
details):


d  z  d 2


3
/
2
 L d  z    d  z  2   2

*
V

.
p , z  
d  L  S  d 2 d 2   2  3d 4
 z 
  
L
exp

z
 L 
2
2 5/ 2


d


z 








(A.6)

Here   x 2  y 2 is the radial coordinate. The correlation length Lz  0b33 is extremely
small for typical values of gradient term . The effective dielectric anisotropy factor
   b2  1 11 0  S  and the “bare” dielectric anisotropy factor  b   b33 11 are introduced.
When deriving expression (A.6), we utilized the inequalities 20b33 S  1 , b33  33 ,
L  0.5...5 nm
and
Lz  1 Å,
valid
for
typical
ferroelectric
material
parameter
~ 108…1010 J m3/C2 and the background permittivity b33  5. Assuming the validity of
additional inequalities Lz  L  d , the approximate solution was derived as:
d  z  d 2
V
p, z  
 S d d  z  2   2


3/ 2
,
at z  Lz ,
(A.7a)
E3 (, z ) 

V d  z   d
 d  z     2
2

3 2
.
(A.7b)
The linear approximation for the polarization distribution given by Eq.(A.7) is quantitatively
valid until p  PS or, alternatively, V d  PS , i.e. at biases V much smaller than the coercive
bias, at which polarization reversal is absent. This means that the probe induced domain
formation cannot be considered quantitatively within the linearized LGD-equation.
Below we take into account the ferroelectric material nonlinearity within direct variation
method. Using trial function with variational parameter PV
P3 x, y,0  P0 x  
11 0  2 d 2  PV (V )
d
  2   d 2  x 2  y 2
d
2
 x2  y2
.
(A.8)
one can obtain renormalized free energy.
Under the reasonable assumption L  d , polarization distribution (A.8) produces the
following depolarization field:
d 2 PV 
d  z  
E , z   2
2
b 
   b  0  33   d  z  2   2

d
3
 b2

Polarization distribution is shown in Fig. A.1.

3/ 2


d  z
b 
 b d  z  b    2
2

3/ 2


.

(A.9)
Depth z (nm)
0
0
Probe
Probe
L
increase
2
4
3
2
(a)
L
increase
2
(b)
1
4
1
4
-5
0
5
-5
0
x (nm)
0
1
2
5
Probe
2 1
5
3
15
4
L
increase
10
(c)
15
-5
Probe
3
4
10
5
x (nm)
0
Depth z (nm)
4
3
2
0
L
increase
(d)
5
-5
x (nm)
0
5
x (nm)
Fig. A.1. Domain wall vertical cross-section for different distances from initial flat wall x 0=,
15, 5, 0 nm (panels (a), (b), (c) and (d) respectively). Curves 1, 2, 3, 4 corresponds to different
values of L=0, 0.5, 1, 2 nm. Other parameters: effective distance d=5 nm, 11=500, = -1.66108
m/F, = -1.44108 m5/(C2F).
After the substitution of Eq.(A.8-9) into the free energy functional (S.1a) and integration,
renormalized free energy was derived as
PV   d
w1  1, w2 ( x0 ) 
 0 11 
w
w
w
2
3
4
  PV V  1 PV  2 PV  3 PV ,
 2 
2
3
4

 2110
 3PS x0
L  d  x 4 L  d 


2
2
0
2
, w3 
110
2
4 L  d 
2
(A.10a)
(A.10b)
it is easy to find the equation of state from Eq.(S.8a). So, in the presence of lattice pinning of
viscous friction type, the amplitude PV should be found from Landau-Khalatnikov equations as:

d
2
3
PV  w1PV  w2 ( x0 ) PV  w3 PV  V (t )
dt
(A.11)
The parameter PV serves as effective variational parameter describing domain geometry, and
allows reducing complex problem of domain dynamics in the non-uniform field to an algebraic
equation Eq. (A.11).
Critical points of polarization bias dependence (inflection points, coercive biases) could
be found from the static equation d V dPV  0 , namely we derived expressions for coercive
biases Vc, loop halfwidth Vc and imprint bias VI as
Vc 
Vc 
Vc  Vc

2

 
w2 2w22  9w3 w1  2 w22  3w3 w1

2 w22  3w3 w1

3/ 2
,
27 w32

3/ 2
, VI 
27 w32
Vc  Vc
2


w2 2w22  9w3 w1
27 w32
.
(A.12)
III. Effective piezoresponse calculations
In decoupled approximation and object transfer functions approach (see Refs.[6, 7, 8]),
analytical V dependence of effective piezoelectric response PRV  were found as:
PRV , x0   d 0eff ( x0 ) 
 0 11

Bi    ln  e  bi   C PV (V )
4
 b    ln  e  b   C L
i 1
i

i
 d ln  e  bi   C 
.
(A.13)
Where d 0eff  x 0  is the bias-independent PFM profile of the flat 180o-domain wall located at
distance x from the tip apex. Response d 0eff  x 0  was calculated in Ref.[9]; dielectric anisotropy
factor is    33 11 . Constants e2.71828… is the natural logarithm base and C0.577216…
2
is Euler's constant. Constants b1     2 1    , b2     1    , b3    1  2  21    ,
2
b4     16  15 2  41   
2
B3  2 0  33Q11 1  2  1    ,
2
and
B1  2 0  33Q12  1    ,
B4  2 0 11Q44  2 1   
electrostriction tensor for cubic symmetry).
B2  21  2   0  33Q12 1    ,
2
2
(
is
Poisson
ratio,
Qij
is
The expected behavior of the hysteresis loops as a function of tip surface separation is
illustrated in Fig.A.2. Directly at the wall, the loop is closed and the local response originates
from the bias-induced bending of the domain wall. It is clear from the figure, that the loop
halfwidth, determined as the difference of coercive biases Vc  Vc  Vc  2 , appears and
monotonically increases with the distance x increase. The bistability is possible and Vc is
defined only for x02  2L  d  . Far from wall (x0 >> d) corresponding coercive biases are
2
symmetric, Vc  2PS L  d  3  611 0 . The inclusion of viscous friction leads to the loop
broadening and smearing far from the wall, while near the wall the minor loop opening is
observed (compare solid and dotted curves). Note that the observed evolution of the loop shape
PR loop width V/Vc
2
(a)
=0
0
1
2
3
4
1
1
10
10
x0=0
0
-10
-3
2
Distance from the wall x0/d
10
0
(c)
x0<d
-2 -1 0
x0=d
3
1
1
V/Vc
2
3
(d)
-3 -2 -1 0 1
V/Vc
2 3
2
1
4
2
0
(b)
-1
0
Vс
1
3
2
Distance x0/d
3
x0=2d
10
x0=2.5d
0
0
-10
Vс+
4 3
2
0
0
PR (pm/V)
Coercive bias Vc/Vc
and switching parameters agrees with the experimental observations.
-10
(e)
-3
-2 -1 0
1
V/Vc
2 3
-10
(f)
-3 -2 -1 0
1
2
V/Vc
3
Fig. A2. (a) Piezoresponse (PR) loop relative width Vc Vc ( Vc is the static coercive bias far
from the wall) vs. the distance from the wall x0/d. (b) Left (bottom curves) and right (top curves)
coercive biases of the PR loops. Curves 1, 2, 3, 4 correspond to the different relaxation
coefficients =0, 10-8, 10-7, 10-6 SI units. Plots (c-f) show piezoresponse loops vs. applied bias
(V) calculated for increasing distance x0 from domain wall (labels at the plots). Material
parameters for LiNbO3 are 11=84,  = 1.95109 m/F,  = 3.61109 m5/(C2F), PS=0.75 C/m2,
Poisson ratio is =0.3, parameter d=60 nm, frequency 104 rad sec-1 and maximal bias
Umax=15 V.
IV. The influence of the probe tip conical part on the domain nucleation
The tip of the probe induces strong but localized electric field, while the conical part of
the probe produces weaker but more diffused field distribution. The influence of the probe tip
conical part on the domain nucleation is shown in Fig. A3. This effect is evident from
Figs. A3a,b, since the nascent domain is more diffuse for the case with conical part included. It is
also seen from Figs. A3c, d that the flat domain wall is practically unaffected by the field of
probe tip for distances x0>10 d between them, while the conical part field induces wall bending
even in this region.
y (nm)
20
20
(a)
(b)
0
0
0
0.3
0.6
0.3
0.6
0
-0.3
0.6
-0.3
-0.6
-0.6
-20
-20
-20
0
-20
0
x (nm)
x (nm)
200
200
0.6
y (nm)
0.6
0.3
0.3
0
-0.3
0
0
0
-0.6
-0.6
(c)
-200
99
-0.3
100
x (nm)
101
(d
-200
99
100
101
)
x (nm)
Fig. A3. Contour maps of the bound charge distribution (values near the curves) on the surface
(z=0) for nucleation near the flat wall at x0=10 nm (a, b) and far from the flat wall at x0=100 nm
(c, d); for two different probe models, effective point charge alone (a, c) and effective point and
line charges (b, d). For the tip far from the wall only the near wall region is shown. Effective
distance between the charge and surface d=10 nm, applied voltage V=30 V, line charge length is
1 m.
Thus, the long-range influence of the probe conical part on the initial domain wall
behavior could explain logarithmically slow saturation of nucleation bias shown in Fig.A4.
Bias Vc (V)
30
20
10
0
1
102
10
103
104
Distance x0 (nm)
Fig.A4. (a) PFM hysteresis loop halfwidth Vc  Vc  Vc  2 vs. the distance from the wall.
Material parameters for LiNbO3 are 11=84, =  1.95109 SI units, PS=0.75 C/m2, Poisson ratio
is =0.3; domain wall intrinsic width L=0.5 nm. Filled boxes are experimental points. Red solid
curve is the fitting for the model that takes into account point charge with d=30 nm and line
charge L spanning from 100 nm to 1000 nm. Blue dashed curve is the fitting with the equation
Vc  7.6  lg 2.2  x0 / 2.6 (with x0 in nm). Theoretical curves are calculated for threshold bias
Vth = 3 V originated from the lattice pinning.
Appendix B. Calculations of nucleation bias within MW and BC approaches
Excess free energy for nascent nucleus at the domain under the external field after Miller
and Weinreich [10] is
a2
F  2 PS W0  2W c a  l  2 p c
l
2
2
(B.1)
First term is the energy of nucleus interaction with external field, second term is the excess wall
energy and the third one is the depolarization field energy. Here PS is the spontaneous
polarization, W0 is the external field E0 and integrated on the nucleus volume, c is the nucleus
width normal to the wall, a is the nucleus half-width on the surface along the wall, l is the size of
nucleus along the wall and normal to the surface (see Fig. B1). The surface energy of the domain
wall W is regarded independent on the wall orientation. p is an effective surface density of the
depolarization field energy,  p  ln 0.74a c  PS2 c 0 11  in SI units. Here width c was
regarded of lattice constant order and considered much smaller than other sizes of nucleus.
(b)
(a)
y
2a
l
PS
PS
Fig. B1. Schematics of calculations: (a) triangular prism nucleus from Miller and Weinreich, (b)
Burtsev-Chervonobrodov smooth nucleus.
For the case of homogeneous external field, considered by Miller and Weinreich, W0 is
simply the product of nucleus volume and the electric field value W0  E0 c0 al . For considered
case of inhomogeneous electric field of SPM probe W0 is
W0





 a 2  d l   a 2  l  2 a 2  d 2  y 2 
2 c aVd

ln 


2
2
2
 a 2  l  
 d  l   l   a 2  l   d  l    y 2




2
2
2
  2 c Vd ln  a  a  d  y 




d 2  y2












(B.2)
Here V is the bias, applied to the probe, d is the effective charge –surface distance, y is the
distance between the probe axis and the domain wall,    33 11 is the dielectric anisotropy
factor.
When
F  2 PS

the
d 2Vc a l
 d 2  y2
nucleus

3
sizes
are
 2 W c a 2  l 2  2 p c
W0 
small

d 2Vc a l
 d 2  y2

3
and
a2
. It is seen, that in this case the free energy
l
F in the inhomogeneous field is the same as in homogeneous one, but with substitution of E0
with E P V  

d 2V
 d y
2
2

3
. Thus, Miller-Weinreich activation energy of domain wall step
nucleation, obtained with respect to probe tip electric field inhomogeneity, is

   d 2  x2

0
Fa (W ,V , x0 ) 
ln  W
2
2cPS d V

3 3

8
  c
3
W

3  d 2  x02
  
 0 11

d 2V
.
3
(B.3)
Directly at the wall (x0=0)
FaMW (W ,V ) 
  d
ln  W
3 3
 2cPS V
16
 cW 3  d

 4  V .
0 11

(B.4)
It should be noted, that Miller and Weinreich considered lattice discreteness in very rough
model and do not take the possibility of wall to bent into account. Burtsev and
Chervonobrodov11 considered a more realistic model with continuous lattice potential and
diffuse domain walls, at that the nucleus shape and domain wall width are calculated selfconsistently. Using their approach we obtained expression:
BC
a
F

 d  min   c  min 

(W , V )  ln 
 2cPS V  40 11



3
d
.
V
(B.5)
Using dependence of activation energy on applied bias, one could find activation voltage
from the equality of activation energies (B.4-5) to some relevant level.
Below
we
used
the
following
values
of
lattice
potential:
minimal
value
min  0.160 J m2 and modulation depth min  0.150 J m2 calculated for domain walls in
LiNbO3. Other parameters were c=0.5 nm, PS=0.75 C/m2, 11  84  33  30 , effective distance d
(tip size) was determined from the expression d  Vc
2711 0
2 2
, where Vc is coercive bias far
from the wall.
In Table 1 we presented results of activation voltage calculations for the models of
Miller-Weinreich and modified Burtsev-Chervonobrodov.
Table 1. Values of activation voltage for domains wall in LiNbO3
Model
Barrier level Fa (W ,V , x0  0)
25 kB T
kB T
d values (nm)
d values (nm)
21
61
86
21
61
86
MW
3.9 V
11.4 V
16.1 V
5.2 V
15.1 V
21.2 V
BC
0.9 V
2.6 V
3.6 V
2.6 V
7.4 V
10.5 V
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