Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
The minimum writable domain size is not necessarily
related to the information limit in PFM and can be either larger
or smaller. This follows from the fact that while the signal
generation volume in PFM is independent of the tip bias, the
written domain size, and in particular, the critical size of the
nucleated domain, has a strong bias dependence, i.e. minimum
writable domain size can be smaller than the PFM resolution.
This suggests that in some cases the resolution is a limiting
factor precluding experimental observation of smaller domains
created by PFM. Clearly, this conclusion is non-universal and
strongly depends on the material, e.g. in polycrystalline films,
the grain-by-grain switching will result in minimal writable bit
sizes being larger than the resolution.
The resolution effect will clearly affect the analysis of the
parameters such as domain size distributions in the disordered
materials or geometry of the fractal and self-affine domain
walls. For example, the structure factor will be S(q) =
S ′ (q)R(q), where S ′ (q) is the intrinsic structure factor of the
interface and R(q) is the transfer function defining microscope
resolution. Practically, R(q) = 1 for q ≪ q c and R(q) =
q −n for q ≫ q c , where power law n is determined by the
image formation mechanism. For ultrathin interfaces such
as ferroelectric walls, q c ∼ 1/w c . Correspondingly, the
fractal properties h(x) for length scales below w c are likely
dominated by the scanner noise along the slow scan axis
(which can be established from the topographic imaging of
appropriate calibration standard, e.g. step edge of a cleaved
graphite surface), and by the tails of the transfer function for
the fast scan axis, rather than the intrinsic wall geometry [219].
3. Local polarization switching in ferroelectric
materials by PFM
3.1. Polarization dynamics at the nanoscale
The key characteristic of ferroelectric materials is that the
direction of spontaneous polarization can be reversed by the
application of sufficient electric field. Not surprisingly, SPM
of ferroelectrics has attracted considerable attention due to its
potential to manipulate ferroelectric materials at the nanoscale
by creating ferroelectric domains, studying their dynamics
during growth and relaxation and mapping their interaction
with structural and morphological defects [107, 108, 110]. In
this section, we summarize the existing results on the kinetics
of domain formation and relaxation, as well as theoretical
models for the description of the domain growth process in
the rigid ferroelectric approximation.
3.2. Experimental aspects of tip-induced polarization
switching
3.2.1. Domain growth kinetics. PFM allows a
straightforward approach to study the kinetics of domain
formation and relaxation by combining the writing step of
applying a bias pulse of preselected duration and magnitude,
and the reading stage at which the size of the resulting domain
is imaged. These studies have received a significant impetus
in the context of ferroelectric data storage [25], and have been
S V Kalinin et al
stimulated by the availability of high-quality epitaxial thin
films that have low (below 10 V) switching voltages.
A broad range of studies of domain wall growth on sol–
gel [220–223] ferroelectric films have been reported. The
implementation of high-voltage PFM [224] has allowed studies
of domain dynamics in single crystals as well, and particularly
has enabled the kinetic studies as a function of pulse parameters
[225–232]. As an example, shown in figure 22 is the
morphology of ferroelectric domains in LNO single crystal.
While for low bias pulses the domains are nearly round, large
domains adopt well-defined crystallographic shapes, mirroring
surface tension driven rounding of nanoparticles.
The radii of domains fabricated in lithium niobate
single crystals were found to scale linearly with applied
field and approximately logarithmically with time [226].
Similar scaling was found for other materials, suggesting the
universality of this rate law.
The time dependence of domain-wall velocity was studied
by several groups [128, 233, 234] in an attempt to relate domain
growth kinetics to the dominant wall pinning mechanisms. The
first link between wall velocity and disorder was established
in the seminal papers by Tybell et al [64] and Paruch et al
[234] and since then was actively studied by several groups
[233, 235]. Domain growth in epitaxial films has also
been compared with macroscopic measurements on capacitor
structures [380].
Significant efforts have been directed at fabricating ultrasmall
domains and the determination of minimal stable domain
size and associated lifetimes. Recently, 8 nm domain arrays
have been fabricated and detected using scanning nonlinear
dielectric microscopy [237, 238] (figure 23(a)). The Cho
group has also demonstrated the formation of 7 nm arrays and
2.8 nm domains (see figure 23(b)) and has written images using
the technique (figure 23(c)) [236]. This impressive result
corresponds to a density of 160 Tb inch −2 (10 Tb inch −2 for
8 nm array), approaching molecular storage level [239].
One of the key uncertainties for the characterization of
domain growth kinetics is the lack of quantitative information
on the electrostatic field structure produced by the probe. In
most studies to date, the field is approximated as a single point
charge, a model well validated for tip apex at large separations
from the contact area. On larger length scales, the conical
part of the tip provides a slow-decaying field component,
which is, however, attenuated by the dielectric gap effect.
Furthermore, the point charge model is clearly inapplicable
for small tip–surface separations of the order of the tip size
(which in turn is related to the domain wall width, as discussed
in section 2.3, on which the field is uniform). The second factor
affecting domain growth kinetics is the effect of mobile surface
charges, screening and liquid bridge formation, as discussed
in section 3.3.
3.2.2. Domain relaxation and retention. The retention
behavior of ferroelectric domains following local polarization
reversal presents obvious interest for data storage and
ferroelectric lithography applications. Retention behavior in
epitaxial and polycrystalline PZT [240], the thermal stability
[228] and the retention behavior [229] of fabricated domains
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