Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
field. It involves a complex coupling of long-range
electromechanical interactions in a highly inhomogeneous
and nonequilibrium system. Analytical solutions for the
spatial and temporal distributions of polarization, electric field
and stress during switching under PFM are generally not
possible although semi-analytical solutions for the polarization
distributions corresponding to the critical states or stable states
have been attempted, as summarized in sections 3 and 4. To
better understand the polarization switching mechanisms in
PFM, a number of efforts to model the polarization evolution
process using the phase-field method [65, 263, 295, 350, 388,
389] have been attempted. One of the main advantages for
the phase-field method is the fact that one is able to model the
temporal/spatial evolution of arbitrary domain morphologies
under an applied electric field without explicitly tracking the
positions of domain walls. Secondly, the inhomogeneous
electric field and stress field distributions accompanying the
polarization switching are readily available. Furthermore, the
effect of structural defects such as grain boundaries, surfaces,
dislocations and random defects can be incorporated without
significantly increasing the computational time. Phase-field
models have previously been applied to domain evolution
during ferroelectric phase transitions and domain switching,
effect of random defects and dislocations, as well as strain
effect on transition temperatures and domain structures in thin
films [390].
5.1. Phase-field method
During polarization switching the polarization distribution
is always inhomogeneous, i.e. it depends on the spatial
positions. In the phase-field approach, one employs the
spatial distribution of local spontaneous polarization P (x) =
(P 1 (x), P 2 (x), P 3 (x)) to describe a domain structure. Using
the free energy for the unpolarized and unstrained crystal as
the reference, the local free energy density as a function of
strain and polarization using the Landau–Devonshire theory
of ferroelectrics is
f bulk (ε(x), P (x)) = 1 2 α ij P i (x) P j (x)
+ 1 4 γ ij klP i (x) P j (x) P k (x)P l (x)
+ 1 6 ω ij klmnP i (x)P j (x) P k (x)P l (x)P m (x)P n (x) + ···
1
2 c ij klε ij (x) ε kl (x) − 1 2 q ij klε ij (x)P k (x)P l (x), (5.1)
where α ij , γ ij kl and ω ij klmn are the phenomenological Landau
expansion coefficients and c ij kl and q ij kl are the elastic
and electrostrictive constant tensors, respectively. All the
coefficients are generally assumed to be constant except
α ij which is linearly proportional to temperature, i.e. α ij =
αij o (T − T o), where T o is the Curie temperature.
It should be noted that the coefficients in equation
(5.1) correspond to zero strain while experiments are usually
conducted at zero stress. In order to use the materials
constants and Landau coefficients from stress-free conditions,
we rewrite the free energy for zero stress. One first obtains
the spontaneous strain, i.e. the strain or crystal deformation at
zero stress,
ε o ij (P k) = 1 2 s ij mnq mnkl P k P l = Q ij kl P k P l , (5.2)
S V Kalinin et al
where Q ij kl are the electrostrictive coefficients measured
experimentally. Substituting the spontaneous strain from
equation (5.2) for the strain in equation (5.1), we have the
free energy at zero stress as
g bulk (P(x)) = 1 2 α ij P i (x)P j (x)
+ 1 4 γ ij ′
kl P i(x)P j (x)P k (x) P l (x)
+ 1 6 ω ij klmnP i (x)P j (x) P k (x)P l (x)P m (x)P n (x) + ···, (5.3)
where the α ij and ω ij klmn remain the same for zero stress as
for zero strain, but γ ij kl at constant strain is changed to γ
ij ′
kl at
zero stress with
γ ′
ij kl = γ ij kl − 2c mnop Q mnij Q opkl . (5.4)
The free energy at zero strain and that at zero stress are
related by
f bulk
(
εij (x), P (x) ) = g bulk (P(x)) + f elast
(
Pi (x) ,ε ij (x) ) ,
where
(
f elast P(x), εij (x) ) = 1 2 c (
ij kl εij (x) − εij o (x))
× ( ε kl (x) − εkl o (x)) . (5.5)
For a domain structure, the electrostatic energy contains
contributions from an external applied field E ex , the energy due
to inhomogeneous polarization distribution δP i (x) = P i (x)−
P¯
i and the depolarization energy F dep if the crystal is finite and
the surface polarization charge is not fully compensated:
∫
∫
F elec =
V
− 1 2
f elec (P i (x), E i (x)) dV =− P i (x) Ei
ex (x)dV
V
∫
( )
E i (x)δP j (x)dV + F dep P ¯i
, (5.6)
V
where P¯
i is the average polarization and E i is the ith
component of the electric field generated by the heterogeneous
polarization distribution δP i (x).
The total free energy of an inhomogeneous domain
structure is given by Ginzburg–Landau free energy functional:
∫
F = [f bulk (P i ) + f grad (∂P i /∂x j ) + f elast (P i ,ε ij )
V
+ f elec (P i ,E i )]d 3 x (5.7)
in which f bulk is the bulk free energy density, f grad is the
gradient energy that is only nonzero around domain walls and
other interfaces where the polarization is inhomogeneous,
f grad = 1 2 G ij klP i,j P k,l , (5.8)
where P i,j = ∂P i /∂x j and G ij kl is the gradient energy
coefficient.
To obtain the elastic strain energy density f elast in equation
(5.5), one needs to solve the mechanical equilibrium equation
for a given domain structure. For a bulk single crystal with
periodic boundary conditions, one can use Khachaturyan’s
elasticity theory [391, 392]. For thin films, the mechanical
boundary conditions become more complicated. The top
surface is stress-free and the bottom surface is constrained
by the substrate. As it has been shown in [393, 394],
the solution to the mechanical equilibrium equations for
56