Supplementary Information
Sub-Critical Field Domain Reversal in Epitaxial
Ferroelectric Films
Jason Chen1, Alexei Gruverman2, Anna N. Morozovska3and Nagarajan Valanoor1*
1
School of Materials Science and Engineering, University of New South Wales, Sydney, New South Wales 2052,
Australia
2
Department of Physics and Astronomy University of Nebraska Lincoln,
Lincoln, NE 68588, United States
3
Institute of Physics, National Academy of Sciences of Ukraine, 03028 Kyiv, Ukraine
Figure S1. (a) Reciprocal space mapping (RSM) & (b) -2 scans for PZT(20nm)/LSMO/STO
showing fully c-oriented and highly strained PZT growth on STO.
* [email protected]
1
Figure S2 Domains produced by 0.5V, 1000s voltage pulses and visualized right after bias
application (a) and one hour later (b).
Figure S3 Typical hysteresis loops obtained in the (a) as-grown PZT film and (b) the same film
poled by a -3.5V applied by the probing tip.
2
S4. Mathematical details
Fitting of experimental data for the domain radius r (V ) can be done by using the
dependence1, 2:
r (V )  r0  a V Vc t   1

(1)
Here r0 is the minimal radius of the critical nuclei, a is the effective radius related with the tipsurface contact size a, power factor 2 3    2 depends on the system peculiarities of the
ferroelectric domain nucleation in the ferroelectric, Vc t  is the critical (activation) voltage that
can depend on the pulse duration t. We can fit the experimental data for r V  by using the
dependence (1) with using , a and Vc as fitting parameters. Effective radius a can be calculated
from the tip curvature a0 using the relation a   e a0
1133 for the spherical tip apex. Tip
nominal curvature is a0=25 nm. Effective ambient permittivity  e can reach the values from e.g.
8 to 80 due to the water meniscus under the tip. Relative effective dielectric permittivity of PZT,
11 33 , is about 111 at room temperature. So the effective parameter a can be treated as a fitting
parameter within the range of 2 to 20 nm. Experiment gives a=8-10 nm for the tip-surface
contact radius. Domain radius vs. pulse duration t was fitted using logarithmic law3
r (t )  R0 ln t  .
t
Instant domain radius r t    vt 'dt ' , trivially
0
R
dr
dr
 v and so the pulse duration t  
,
dt


v
r
0
where R is the fixed domain radius. Let us find the dependence t (V ) at fixed R as.

 1 R
 E 
32 
  exp    0 2 r 2  a 2  dr , d  a
  Va
 
 v0 0

t V   

  E 0 d 2

1 R
2 12 

 dr , d  a
 v  exp    Va r  a
 
 0 0
 



Approximation of the integral in Eq.(2) is
3

(2)
  11 

 v0 
t V , R   
  11 

 v0 
1  E 
 Va 2   1 

     , 0 2 R 2  a 2
   Ua
 E 0     


 1  E d
 Va   1 

     , 0
   Ua
 E 0 d    


32





 , d  a


(3)
 
R 2  a 2   , d  a
  

(z) is the Euler gamma function, (x,z) is the incomplete gamma function. For the case   1
and R>>a we get
t V  
1
v0

 E d   Va
 exp  0 R   1
.
 Va   E0 d

(4)
We can fit the experimental data for the pulse duration vs. applied voltage by using the
dependences (2)-(3). Figure S4 illustrates this.
1000
R=20 nm
10
1
0.1
(a)
1
1.5
2
2.5
3
R=35 nm
100
(b)
Voltage V (V)
10
1
1
1.5
2
2.5
3
3.5
4
Voltage V (V)
Figure S4. Pulse duration vs. applied voltage calculated from the dependence (3) for domain
radius R  20 nm (a, red solid curves) and R  35 nm (b, blue solid curves). Dotted curves are
calculated from the dependence (4). A large deviation for R=20nm indicates that the condition
R>>a is not satisfied in this case. Data points are experimental results showing the relationship
between the applied bias and the time required for domains to reach a radius of 20nm (a) and
35nm (b). Fitting parameters:   2, a = 10 nm, Vc  0.60 V, r0  15 nm, and =0.78  = 0.52,
v0  4104 nm/s and E0 =2 MV/cm.
4
Domain radius r (nm)
100
Pulse duration t (s)
Pulse duration t (s)
1000
S5. Subcoercive switching studies in 10-nm and 40-nm thick epitaxial PZT thin films
Figure S5.1. Typical hysteresis loops for (a) 10-nm-thick and (b) 40-nm-thick PZT films poled
by a -3.5V bias applied to the tip. 30 loops were measured for each sample and the average Ecloc
values obtained are 0.7± 0.2V and 1.5± 0.2V for the 10nm and 40nm thick film, respectively.
5
Figure S5.2. PFM phase images showing domain formation in (a) 10-nm-thick and (b) 40-nmthick PZT films as a function of bias time for a fixed subcoercive bias. Similar to the findings for
the 20-nm-thick film, only under very long bias application times we find that it is possible to
initiate domain nucleation. (c) and (d) PFM phase images (with amplitude images shown as
insets) after (0.5V, 1000s) and (1V, 1000s) bias application, in the 10-nm-thick and 40-nm-thick
films, respectively.
6
S6. Thickness dependence of pulse duration: Theoretical calculations based on Eqn 4 in S4
above
The plot below shows that the pulse required to initiate a critical domain size increases with
reducing applied bias and increasing thickness. Note that the model does not assume any size
effects or take into account contributions from defects. Particularly note that for thin samples the
system relaxation time to the "new" equilibrium state increases tending to infinity when the
film thickness approaches the critical thickness. This is observed experimentally above in S5.
V=0.5V
107
106
105
V=1V
104
V=1.5V
3
10
V=2V
2
10
10
10
(a)
15
20
25
30
35
Film thickness d (nm)
V=0.5V
108
107
V=1.5V
106
105
104
V=2V
3
10
102
40
(b)
V=1V
10
10
R=35 nm
15
20
25
30
35
40
Film thickness d (nm)
Figure S6. Pulse duration vs. film thickness calculated from the dependence (4) for different
applied voltages V= (0.5, 1, 1.5, 2) V and domain radii R  20 nm (a) and R  35 nm (b). Fitting
parameters:   2, a = 10 nm, Vc  0.60 V, r0  15 nm, and =0.78  = 0.52, v0  4104 nm/s
and E0 =2 MV/cm.
7
Domain radius r (nm)
Pulse duration t (s)
108
109
R=20 nm
Pulse duration t (s)
109
V=
Film
Puls
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