Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
S V Kalinin et al
(a)
d 3
Q 1
Q 2
Q 3
V Q
(b)
(c)
0.8 µm
Piezoresponse, a.u.
4
3
2
1
0
-1
-2
(d)
single data set
one charge model
0 1000 2000 3000 4000 5000
Distance, nm
Figure 19. (a) Representation of a realistic tip by a set of image charges. (b) Surface topography and (c) PFM image of domain wall in
LNO. (d) Fit of the extrapolated data set with equal weighting for all points. Reproduced from [218]. Copyright 2007, American Institute of
Physics.
Table 2. Effective image charge parameters for different ferroelectrics.
Wall R 0 for R 0 for
Material width (nm) Q d (nm) R d (nm) ε e = 1(µm) ε e = 80 (nm)
LiNbO 3 96 1000 92 58.6 4.8 60
Epitaxial PZT 107 2550 125 79.6 5 62.5
PZT in air 58 723 86.5 55.1 44 541
PZT in liquid 6 104 11.8 7.5 75
charges can be deconvoluted from the experimental domain
wall profile by minimizing the functional
∫ ( 1
F [u 3 ] = PR(a) −
2πε 0 (ε e + κ)
N∑
m=0
) 2
Q m
ũ 3 (s m ) da
d m
(2.29)
with respect to the number, N, magnitude and charge–surface
separation of a set of image charges {Q i ,d i } N representing the
tip. Here PR(a) is the measured piezoresponse and integration
is performed over all available a values. The dielectric constant
of the medium can be fixed to the value of free air (ε e = 1)
or water in the tip–surface junction or imaging in liquid
(ε e = 80). The output of the fitting process is the set of reduced
charges q i = Q i /2πε 0 and the charge surface separation, d i .
Note that the charges and the dielectric constants cannot be
determined independently, since only Q m /(ε e + κ)ratios enter
equation (2.29).
Shown in figures 19(b)–(d) is an example of a domain
wall profile and the corresponding fit by equation (2.29) with
N = 1 for LNO. The corresponding image charge parameters
are summarized in table 2. To improve the fit quality, more
complex fitting functions with N = 2 and N = 3 were
attempted. However, independent of the choice of the initial
values of the image charge, the fit converged to a single image
charge, i.e. d i = d and ∑ Q i = Q. Similar behavior was
observed for other domain walls.
This analysis suggests that the electrostatic field produced
by the tip is consistent with a single point charge positioned
at large separation from the surface, contrary to the behavior
anticipated in contact mode imaging. The analysis was
extended to the case of the sphere–plane (radius of curvature
R 0 ) and disk–plane (radius R d ) models. The sphere
parameters are calculated both from ambient and water
environment to account for possible capillary condensation
effects. From the data, it is clear that the use of the
sphere/air model leads to implausibly large radii. Hence,
experimental data are consistent either with the presence of
a capillary water film in the sphere model, or conductive
disk model [218].
2.4.2. Resolution for non-zero contact area. Extensive
quantitative analysis of domain wall width, taking into account
the finite contact tip–surface contact area, was reported by
the Gopalan group [183]. The geometric structure of more
than 100 SPM tips was ascertained using scanning electron
microscopy to yield effective tip sizes [212], see figures 20(a)
and (b). The domain wall width in periodically poled lithium
niobate was measured as a function of tip radius. In parallel,
the domain wall width was calculated using a numerical finite
element analysis package, figure 20.
The comparison of the experimentally measured PFM
signal and domain wall width as a function of tip radius
with numerical and analytical theory predictions is shown in
figure 21. The data clearly suggest that the PFM signal is
independent of contact area, in agreement with the theory
of Karapetian [175, 176]. At the same time, the domain
wall width saturates at ∼100 nm, corresponding to a tipindependent
domain wall width. This can be attributed both
to the effect of an ambient environment on imaging, and to
wall broadening, necessitating further studies in an ultra-high
vacuum environment.
21