Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
‘threshold’ domain nucleation is similar to the well-known where
first order phase transition (compare figure 31(e) with the
well-known dependence of the free energy profile on the order
parameter for first order ferroelectrics).
In the thermally induced nucleation limit, the domain
nucleation process is analyzed as a thermally activated motion
in the phase space of the system along the minimum energy
path connecting the origin and one of the local minima. The
relaxation time necessary for the stable domain formation
at U cr is maximal and the critical slowing down appears
in accordance with the general theory of phase transitions.
Within the framework of the activation rate theory, the domain
nucleation takes place at a higher activation voltage U a
determined from the condition (U a ) = E a , corresponding
to the activation time τ = τ 0 exp(E a /k B T). For instance,
the activation energy E a = 20k B T corresponds to a relatively
fast nucleation time τ ∼ 10 −3 s for phonon relaxation time
τ 0 = 10 −12 s, while the condition E a 2k B T corresponds to
‘instant’ or thermal nucleation.
×
The difference between the voltages corresponding to
the formation of a saddle point and a stable domain, U S −
U cr , determines the width of the thermodynamic hysteresis
loop (see figures 31(e)–(g)). More realistic models of
piezoresponse hysteresis loop formation consider domain wall
pinning effects. In the weak pinning limit, the domain
growth in the forward direction is assumed to follow the
thermodynamic energy minimum, while with decreasing bias,
the domain remains stationary due to domain wall pinning
by lattice and atomic defects. For stronger pinning, the
domain size is limited by the wall mobility in the tip field
and ferroelectric (as, e.g., studied for domain wall dynamic
by Molotskii in [311]), and the full analysis of this problem
remains the matter of future research.
3.3.2. Analytical treatment for Landauer geometry. The
integral expressions for the free energy components in
equation (3.1b) are extremely complex, and can be evaluated
analytically only for simple domain configurations. Here we
summarize the Pade approximations for the individual terms in
the free energy (r, l, U) = S (r, l)+ p (r,l,U)+ D (r, l) of
the semi-ellipsoidal domain for ferroelectric semiconductors
allowing for Debye screening and uncompensated surface
charges.
The domain wall energy has the form
(
r
S (r, l) = πψ S lr
l + arcsin √ )
1 − r 2 /l
√ 2
1 − r2 /l 2
(
)
≈ π 2 ψ S lr
1+ 2 (r/l)2 , (3.2)
2 4+π (r/l)
where r is the semi-ellipsoid domain radius, l is the domain
length.
The Pade approximation for the depolarization energy
of a semi-ellipsoidal domain including the effects of Debye
screening in the material is
DL (r, l) = 4πP2 S r2 R d n D (r, l)
ε 0 κ 4n D (r, l) +3R d (γ/l) , (3.3a) ×
32
n D (r, l) =
⎛ (√ )
(rγ/l)2 ⎝ arcth 1 − (rγ/l)
2
1 − (rγ/l) 2 √
1 − (rγ/l)
2
S V Kalinin et al
⎞
− 1⎠
(3.3b)
is the depolarization factor, κ = √ ε a ε c is the effective
dielectric constant of the medium, and, γ = √ ε c /ε a is the
anisotropy factor. For an infinite Debye length (i.e. a rigid
dielectric), equation (3.3a) becomes the well-known Landauer
depolarization energy.
The energy of the depolarization field created by the
surface charges (σ S − P S ) located on the domain face has the
form
4πr 3 R d
DS (r, l) ≈
ε 0 (16κr +3πR d (κ + ε e ))
(
(σ S − P S ) 2 + 2P S (σ S − P S )
1+(l/rγ)
)
. (3.4)
The surface charge density σ S is −P S σ S P S , while
σ S =−P S without screening charges.
The driving force for the switching process is the
interaction energy determined by the electrostatic field
structure produced by the probe. The Pade approximation for
the interaction energy between a spherical tip and the surface
based on image charge series is
p (r, l, U) ≈ 4πε 0 ε e UR 0
∞ ∑
m=0
q m (3.5)
× R d ((σ S − P S ) F W (r, 0,d − r m ) +2P S F W (r, l, d − r m ))
ε 0
(
(κ + ε e ) R d +2κ √ (d − r m ) 2 + r 2 ) .
The image charges q m are located at distances r m from the
spherical tip center, where
( ) κ − m εe sinh (θ)
q 0 = 1, q m =
κ + ε e sinh ((m +1) θ) , (3.6a)
r 0 = 0, r m = R 0
sinh (mθ)
sinh ((m +1) θ) ,
cosh (θ) = d R 0
. (3.6b)
Here R 0 is the tip radius of curvature, R is the distance
between the tip apex and the sample surface, so d = R 0 + R.
The function
F W (r, l, z) ≈
r 2
√
r2 + z 2 + z + (l/γ)
(3.7)
is the Pade approximation of a cumbersome exact expression
obtained originally by Molotskii [309].
Under the typical condition R ≪ R 0 , i.e. when
the tip is in contact with the surface, equation (3.5) can be
approximated as
R d UC t /ε 0
p (r, l) ≈
(κ + ε e ) R d +2κ √ d 2 + r
(
2
(σ S − P S ) r 2
√
r2 + d 2 + d + 2P S r 2
√
r2 + d 2 + d + (l/γ)
)
, (3.8)