Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
S V Kalinin et al
halfwidth qd
10
1
0.1 1 10 10 2
anisotropy γ
(a)
halfwidth qR0
50
10
5
10
5
1
0.5 0.2
1
0.1 0.2 0.5 1 2 5 10
2
anisotropy γ
(b)
halfwidth qR0
10 2
10
(c)
2
1
0.75
0.5
0.25
1
5
0.01 0.1 1 10 100
permittivity κ/ε e
Figure 13. The half-width of OTF ˜W 333 (a) for the point charge model of the tip versus the dielectric anisotropy, γ . Shown are results of
exact calculations on the basis of equation (2.16) (solid curves) and Pade approximations (2.17) (dotted curves). (b), (c) The halfwidth
calculated in the sphere–plane model of the tip versus the dielectric anisotropy, γ (b) and relative permittivity, κ/ε e (c). Figures near the
curves correspond to the values of the ratio κ/ε e (b) and γ (c). Adapted from [186].
The dependence of dimensionless half-width q FWHM of the
˜W 333 (i.e. Rayleigh resolution) on dielectric anisotropy, γ ,
and relative permittivity, κ/ε e , is illustrated in figure 13. For
the point charge model, the half-width, qd,scales linearly
with dielectric anisotropy, γ , for small γ , while it saturates
for γ ≫ 1. Hence, γ ≪ 1 favors high-resolution imaging.
For the sphere–plane model, the half-width, qR 0 , decreases
with the dielectric anisotropy factor, γ , and increases with the
dielectric ratio κ/ε e .
The Rayleigh resolution r min in PFM experiments (i.e. in
r-space) is determined as r min
∼ = 1/qFWHM . Using d = ε e R 0 /κ
in the point charge model, the relationship between resolution,
tip geometry and materials’ parameters can be derived as
γε e R 0 1+2γ
⎧⎪ ⎨
r min
∼
2κ (1+γ ) 2 ,
=
⎪γε e R 0 1+2γ
⎩
ε e + κ
effective point charge model,
(1+γ ) 2 , sphere–plane model. (2.18)
Thus, the functional dependence, r min ∼ ε e R 0 /ε 11 , is valid at
ε e ≪ κ for both the point charge and sphere–plane models of
the tip. Hence, it is desirable to decrease external permittivity,
ε e (e.g. by imaging in dry air), and decrease tip radius, R 0
(sharp tips), in order to increase the lateral resolution of PFM,
while maintaining good contact. Furthermore, the formation
of liquid necks in the tip–surface junction will decrease the
resolution due to an increase in ε e (for dielectric liquid) or
effective radius R 0 (conductive liquid). Note that higher lateral
resolution is possible in materials with high ε 11 values.
The resolution function and OTF approach allows
approximate calculation of the piezoresponse from those
domain structures for which the Fourier image, ˜d klj (q), exists
in a usual (e.g. single, multiple or periodic domain stripes,
cylindrical domains, rings, etc) or generalized (infinite plane
domain wall) sense.
2.3.4.3. The response near the flat domain wall. A natural
experimental observable in PFM is a domain wall between
antiparallel domains. Here, we apply the resolution function
equation (2.16) to determine analytically the domain wall
profile in vertical and lateral PFM, and establish the
relationship between domain wall width, tip radius and
materials properties.
The surface displacement vector below the tip located at
distance a from the infinitely thin planar domain wall located
at y 1 = a 0 (figure 14(a)) is given by equation (2.15)as
∫ ∞ ∫ ∞
u i (0,a) = dξ 1 dξ 2 W ij kl (−ξ 1 , −ξ 2 ) d lkj
−∞
−∞
×sign (a − a 0 − ξ 1 ) . (2.19)
The resolution function W ij kl is given by equation (2.16). For
both the point charge and the sphere–plane models of the tip
of curvature R 0 that touches the sample, the displacement
components can be derived in the analytical form as
⎧
( )
Q
a −
2πε 0 (ε e + κ) d g a0
ij k ,γ,ν d kj ,
d
⎪⎨
point charge model,
u i (a − a 0 ) = ( )
(2.20)
a − a0
Ug ij k ,γ,ν d kj ,
fR ⎪⎩
0
sphere–plane model,
where i = 1, 3 (since u 2 ≡ 0) and f =
(2ε e /(κ − ε e )) ln((ε e + κ)/2ε e ). Exact expressions of g ij k and
Pade-exponential approximations gij Pade
k
are derived in [186].
In the case of weak dielectric anisotropy γ ∼ = 1 the signal
components are the following:
g333 Pade
1+2γ s
(s, γ, ν) =−
(1+γ ) 2 |s| +1/4 ,
(2.21a)
(
−1
g133 Pade (s, γ, ν) = 8 (1+γ )3
+ |s|)
, (2.21b)
3 γ
g351 Pade
2
γ
(s, γ) =−
(1+γ ) 2 · s
|s| +3/4 ,
(
) −1
g151 Pade (s, γ) = 1 2 (1+γ )3
+
2/π − 3/8 (3+γ ) γ |s| ,
2
( 1+2γ
g313 Pade (s, γ, ν) = (1+γ ) 2 − 2 1+ν
1+γ
(
g113 Pade (s, γ, ν) =− 8 (1+γ )3
+
3 γ
)
|s|) −1
+
s
|s| +1/4 ,
(2.21c)
(2.21d)
(2.21e)
(1+ν)
1+(1+γ ) 2 |s| .
(2.21f)
16