Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
i.e. as a convolution of a function describing the spatial
distribution of material properties, d mnk (x), and a function
related to probe parameters (integral in parentheses).
Note that equation (2.11) belongs to the linear model
(equations (2.9a) and (2.9b)), with the term in parentheses
being the resolution function and d mnk (y − ξ) an ideal
image.
Note that this analysis is rigorously valid in the point
mechanical contact approximation, i.e. the displacement
transferred to the tip is equal to the displacement at the tip–
surface contact junction. In this case, the non-uniformity of
strains within the contact area is neglected. This assumption is
expected to be valid when the electric contact area significantly
exceeds the mechanical contact area (e.g. due to the formation
of a liquid droplet at the tip–surface junction) and to provide a
good approximation when the response changes only weakly
within the contact area. Experimentally, the two can be
compared by comparing topographic resolution (e.g. width
of step edges) with the PFM resolution (domain wall width).
Approximately, this effect can be represented as a convolution
of the calculated point-contact responses with an additional
smoothing function with the half-width of the order of contact
radius.
2.3.4.1. Electrostatic field structure. Determination of
the resolution function in equation (2.11) necessitates the
knowledge of the electrostatic field structure in the material.
The detailed analysis of electric field structure for the uniform
half-plane of arbitrary symmetry of dielectric properties is
givenin[185]. In particular, for symmetries of isotropic
and transversally isotropic materials the electrostatic problems
for spherical tip geometry can be solved using the image
charge method, in which the tip is represented by a set of
image charges chosen so that the corresponding isopotential
contour represents the tip geometry [207–209]. For lower
material symmetries and complex tip geometries, the image
charge method, while not being rigorous, provides a good
approximation (e.g. the line charge model for the conical part
of the tip [210]).
For the case of transversally isotropic symmetry of
dielectric properties, the potential V Q in the point charge-based
models of the tip has the form
V(ρ,z)=
1
2πε 0 (ε e + κ)
∞∑
m=0
Q m
√
ρ2 + (z/γ + d m ) 2 , (2.12)
√
where x1 2 + x2 2
= ρ and ξ 3 = z are the radial and vertical
coordinates, respectively, ε e is the dielectric constant of the
ambient, κ = √ ε 33 ε 11 is the effective dielectric constant of
the material, γ = √ ε 33 /ε 11 is the dielectric anisotropy factor,
−d m is the z-coordinate of the point charge Q m and summation
is performed over the set of image charges representing the tip.
The potential in the sphere–plane model can be obtained from
equations (2.12)–(2.14), where the summation is performed
over image charges. In the case of the rigorous sphere–plane
model of a tip of curvature R 0 located at a distance R from
S V Kalinin et al
the sample surface, the image charges are given by recurrent
relations
R0
2 d m+1 = R 0 + R −
R 0 + R + d m
(2.13)
and
Q m+1 = κ − ε e R 0
·
Q m ,
κ + ε e R 0 + R + d m
(2.14)
where Q 0 = 4πε 0 ε e R 0 U, d 0 = R 0 + R and U is the tip bias.
An alternative approach to describe electric fields in
the immediate vicinity of the tip–surface junction is the use
of the effective point charge model, in which the charge
magnitude and charge–surface separation are selected such that
the corresponding isopotential surface reproduces the tip radius
of curvature and tip potential. In this case, the tip is represented
by a single charge Q = 2πε 0 ε e R 0 U(κ + ε e )/κ located at
d = ε e R 0 /κ [211]. Note that the intrinsic limitation of the
point-charge model is that it can approximate only two out
of three characteristic parameters—potential at the tip–surface
junction, curvature of the isopotential surface and effective
decay length of the field. However, its simplicity allows it to
be used as a zero-order approximation.
Effective point charge parameters used to describe the
spherical [211] or flattened [212] tip in the immediate vicinity
or at the contact with the sample surface, corresponding to
the typical geometry of PFM experiments, are summarized in
table 1.
In the framework of the effective point charge model the
isopotential surface curvature reproduces the tip curvature at
the point of contact. This model is appropriate for electric
field description in the immediate vicinity of the tip–surface
junction, relevant for, e.g., modeling nucleation processes. For
the film thickness h d, the effective point charge model gives
d ≈ ε e R 0 /κ for the spherical tip with curvature R 0 .
The field structure in most part of the piezoresponse
volume can be represented by the point charge model
in which the effective charge value Q is equal to the
product of tip capacitance on applied voltage. For the
film thickness h d, it gives the effective separation as
d ≈ 2ε e R 0 ln((ε e + κ)/2ε e )/(κ − ε e ) for the spherical tip with
curvature R 0 .
Finally, for the conductive disk of radius R 0 representing
contact area, d ≈ 2R 0 /π. The proposed model should
be clearly distinguished from the conventional capacitance
approximation (with d = R 0 ) that describes electric field far
from the tip only.
2.3.4.2. Resolution function and materials properties effect.
In the case when x 1 = x 2 = 0 (the displacement directly
below the tip) and the strain piezoelectric coefficient d klj (ξ) is
independent of ξ 3 (system is uniform in the z-direction), the
Fourier transformation of the surface displacement (i.e. VPFM
signal) is [186]
ũ i (q) = ˜d klj (q) ˜W ij kl (−q) + N (q) , (2.15)
where ũ i (q) = ∫ u i (x)e iqx dx, ˜d klj (q) and N(q) are the
Fourier transforms of the measured image, ideal image and
14