Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
Figure 30. (a) Antipoling effect observed on epitaxial PZT. (b)
Schematic representation of the effect: (left) poling at high voltage
and (right) collection of surface charges and formation of opposite
field, leading to polarization switching. The domain wall between
the upper and lower regions reduces the electrostatic energy of the
head-to-head configuration. In the center schematic the contributing
potentials from the spherical model are presented. (c) Illustrates the
surface charge distribution at the beginning and at the end of a
biasing experiment, and after the bias is turned off. (a), (b)
Reproduced with permission from [304], copyright 2005, American
Physical Society.
known by now [305–307]. While the electrochemistry
of classical ferroelectrics such as PZT and BTO in
ultrathin films and in the presence of water vapor is not
studied, experimental observations suggest that tip-induced
electrochemical processes are extremely common. Typically,
their onset is associated with the formation of ‘switched’
patterns with low contrast and highly disordered and broad
domains walls. Often the process is associated with the
formation of topographic features following the bias pattern.
For high biases and humidities, the complete breakdown of the
film and the formation of particulates are observed. Recent
measurements suggest that the onset of the electrochemical
processes is related to the leakage current flow. Note that
these processes are closely related to the transport properties
of ferroelectrics, actively studied in the context of resistive
switching [42, 267, 268, 308], and hence must be considered
in this context.
3.3. Theory of tip-induced polarization switching
The analysis of the tip-induced switching behavior by
PFM necessitates the comparison between macroscopic and
microscopic switching as a first step of theoretical analysis.
The seminal work by Landauer has demonstrated that
the experimentally observed switching fields correspond to
S V Kalinin et al
impossibly large (∼10 3 –10 5 kT) values for the nucleation
activation energy [79]. This discrepancy is resolved by
postulating the presence of discrete switching centers initiating
low-field nucleation. The spatial and energy distribution of
the nucleation center has become the central component of all
statistical theories for polarization switching in macroscopic
ferroelectrics [66, 95]. Notably, this behavior is universal
to all first order phase transitions including solidification,
martensitic transformations and phenomena such as plastic
deformation and electrochemical reactions, in which the
formation of a new phase is always initiated at a specific
defect site or is controlled by defect dynamics. Given that
typical spacings between extended defects are on the order
of 10–100 nm, even high-quality epitaxial films will have
multiple defects within a typical device structure such as a
micrometer-sized capacitor.
In comparison, in an SPM experiment, the field is
concentrated to a small (10–30 nm) volume within a material
that may possess small number of defects or be (extended)
defect free. This opens the pathway for probing defectmediated
switching in ferroelectrics on a single defect
level. The original work by Landauer [79] utilized the
rigid ferroelectric approximation (polarization is P or −P ,
domain walls are infinitely thin) to describe thermodynamics
of domain formation in a uniform field. This approach was
extended in a series of groundbreaking works by Molotskii
et al [309–313] to the analysis of the domain formation
induced by the inhomogeneous electric field of a biased SPM
probe. This thermodynamic analysis was further developed
by Morozovska et al to account for the finite electric field
below the probe, surface and bulk screening and presence of
defects [211, 314]. Below, we develop a general description
of switching thermodynamics in the rigid approximation.
3.3.1. General formalism of the problem. The free energy of
a nucleating ferroelectric domain is [314]
(r,U) = d (r) + Dom (r,U) ,
(3.1a)
where the domain geometry is described by the N-dimensional
parameter vector, r (which can be, e.g., coordinates of all
points on a domain wall, or the radius and length in the semiellipsoidal
approximation) and U is the electric bias applied to
the local probe in the proximity of the surface. The first term
in equation (3.1a) is the contribution from the frozen disorder
(i.e. random electric fields in the material), d (r), within the
volume of the domain. The second term in equation (3.1a) is
the thermodynamic work of switching
Dom (r,U) = S (r) + p (r,U) + D (r) .
(3.1b)
It comprises the contributions of positive domain wall energy,
S (r) = ψ S S, where S is the domain surface and ψ S is the
domain wall energy density, p (r,U) is the energy density
due to the probe field and D (r) is the depolarization energy.
The analysis of the switching process can be simplified
for a rigid piezoelectric, for which effective material constants
are independent of the electric field. In this case, the
interaction energy with the probe tip electric field, E p 3 (x),
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