Local polarization dynamics in ferroelectric materials

Rep. Prog. Phys. 73 (2010) 056502
S V Kalinin et al
(a) 1 µm (b) (c)
10
x 2 , ξ 2
L
E 3
P S
DW
P S
x 1 , ξ 1
H
Width/H
1
3
2
x 3 , ξ 3
0.1
1
(d)
(e)
10 -2 0.1 1 10
L/H
Figure 17. (a) Surface topography and (b) amplitude and (c) phase PFM images of an etched capacitor structure. (d) Coordinate systems in
a global excitation PFM experiment. (e) Domain wall width at level η = 0.76 as a function of the top electrode thickness for ν = 0.3 and
−d 33 /d 31 = 2, 3, 4 (curves 1, 2, 3). Solid curves represent the exact and dashed the approximate (equations (2.28a)–(2.28c)) solutions.
(a)–(c) Reproduced from [111]. Copyright 2008, Microscopy Society of America. (d), (e) Reproduced from [155] and copyright 2008 by
the American Institute of Physics.
2.3.4.5. Imaging of ferroelectric capacitors. Applications of
ferroelectric materials as non-volatile memory components
have resulted in significant interest in theoretical and
experimental studies of polarization dynamics in capacitor
structures. In PFM of capacitor structures, a periodic uniform
electric field is applied between the top and bottom electrodes,
and the resulting surface displacement is measured locally by a
SPM tip [215]. This global excitation method is different from
standard (local excitation) PFM, in which both the electric field
and the local strain are localized.
The domain wall width in PFM of capacitor structures
with uniform (in the vertical direction) domain structure, i.e.
the resolution limit in PFM with global excitation, can be
analyzed similarly to tip–electrode experiments. The vertical
PFM signal, i.e. the surface displacement u 3i (x) at location
(x 1 ,x 2 , 0) is given by equation (2.6) where the field is now
uniform. Considering a domain structure with two opposite
domains divided by a flat wall perpendicular to the film surface,
we obtain that for the uniform excitation, equation (2.6) can
be rewritten as follows:
u 3 (x 1 ,x 2 , 0) =−U(W 33 (x 1 ,L,H)d 33
+(1+2ν)W 31 (x 1 ,L,H,ν)d 31 ). (2.25)
Here, U = HE 3 is the potential difference between the top
and the bottom electrodes. L is the thickness of the top
electrode and H is the film thickness (see figure 17(d)). After
cumbersome integrations, functions W 33 and W 31 are derived
analytically as [155]
W 33 (x 1 ,L,H) = f 1 (x 1 ,L,H) + f 2 (x 1 ,L,H) , (2.26)
ν
W 31 (x 1 ,L,H,ν) = f 1 (x 1 ,L,H) +
2 (1+2ν) f 2 (x 1 ,L,H) ,
(2.27)
where f 1 and f 2 are simple algebraic functions as analyzed
in [155]. The piezoresponse signal in a uniformly poled
capacitor is u 3 (x 1 ,x 2 , 0)/U = −d 33 − (1+2ν)d 31 (due to
clamping). The non-zero contribution of d 31 is directly related
to the mechanical conditions at the boundary between the
film and the bottom electrode, which are assumed to have
identical mechanical properties. As an example, shown in
figure 17(a)–(c) are topography, PFM amplitude and PFM
phase images of a poled capacitor. Note that details within the
capacitor are much coarser than on a free surface, indicative
of lower spatial resolution.
The generalized domain wall width is defined as the
distance between the points at which the response is saturated
to the fraction, η, of the signal at infinity. Hence, the standard
edge resolution is defined as η = 0.5. The width can be
approximated as a bilinear function, w(L, H) ∼ = aH + bL,
of the film and top electrode thicknesses:
(
w (L, H) ∼ = a η, ν, d ) (
31
H + b η, ν, d )
31
L, (2.28a)
d 33 d 33
where coefficient, a, is the solution of the transcendental
equation:
( a
)
arctan +
2
and coefficient b is
(
d 33 + νd 31 /2
ln 1+ 4 ) a
d 33 + (1+2ν) d 31 a 2 2 = π 2 η, (2.28b)
b = η π 2
d 33 + (1+2ν) d 31
. (2.28c)
d 33 + νd 31 /2
Typical values for coefficient a vary from 0.08 to 0.4 at
η = 0.76 and −d 33 /d 31 = 2–4. The dependence of the
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