J. Y. Jo et al.
Domain Switching Kinetics in Disordered Ferroelectric Thin Films
J. Y. Jo,1 H. S. Han,1 J.-G. Yoon,2 T. K. Song,3 S.-H. Kim,4 and T. W. Noh1,∗
ReCOE&FPRD, Department of Physics andAstronomy, Seoul National University, Seoul 151-747, Korea
Department of Physics,University of Suwon, Suwon, Gyeonggi-do 445-743, Korea
School of Nano Advanced Materials, Changwon National University, Changwon, Gyeongnam 641-773, Korea
R&D center, Inostek Inc., Ansan, Gyeonggi-do 426-901, Korea
(Dated: October 23, 2018)
We investigated domain kinetics by measuring the polarization switching behaviors of polycrys-
talline Pb(Zr,Ti)O3 films, which are widely used in ferroelectric memory devices. Their switching
behaviors at various electric fields and temperatures could be explained by assuming the Lorentzian
distribution of domain switching times. We viewed the switching process under an electric field as
a motion of the ferroelectric domain through a random medium, and we showed that the local field
variation due to dipole defects at domain pinning sites could explain the intriguing distribution.
PACS numbers: 77.80.Fm, 77.80.Dj, 77.84.Dy
Domain switching kinetics in ferroelectrics (FEs) un-
der an external electric field Eext have been extensively
investigated for several decades [1, 2, 3, 4, 5, 6, 7, 8, 9].
The traditional approach to explain the FE switching
kinetics, often called the Kolmogorov-Avrami-Ishibashi
(KAI) model, is based on the classical statistical theory
of nucleation and unrestricted domain growth [10, 11].
For a uniformly polarized FE sample under Eext, the
KAI model gives the time (t)-dependent change in polar-
ization ∆P (t) as
∆P (t) = 2Ps[1− exp{−(t/t0)
n}], (1)
where n and t0 are the effective dimension and character-
istic switching time for the domain growth, respectively,
and Ps is spontaneous polarization. When the nuclei are
appearing in time with the same probability, n = 3 for
bulk samples and n = 2 for thin films [12]. In addition,
t0 is proportional to the average distance between the
nuclei, divided by the domain wall speed. Several stud-
ies have used the KAI model successfully to explain the
∆P (t) behaviors of FE single crystals and epitaxial thin
films [2].
Recently, FE thin films have been intensively investi-
gated for FE random access memory (FeRAM) [1]. Most
commercial FeRAM use polycrystalline Pb(Zr,Ti)O3
(poly-PZT) films, and their ∆P (t) behaviors determine
the reading and writing speeds of the FeRAM. In such
non-epitaxial FE films, a domain cannot propagate in-
definitely due to pinning caused by numerous defects, so
the KAI model cannot be applied. Therefore, it is impor-
tant both scientifically and technologically to clarify the
domain switching kinetics of polycrystalline FE films.
Numerous studies have examined the ∆P (t) behaviors
of polycrystalline FE films, and the reported results vary
markedly [3, 4, 5, 6, 7]. Lohse et al. measured the polar-
ization switching currents of poly-PZT films, and showed
that ∆P (t) slowed significantly compared to Eq. (1) [3].
Tagantsev et al. observed similar phenomena for poly-
PZT films. To explain these behaviors, they developed
the nucleation-limited-switching (NLS) model. They as-
sumed that films consist of several areas that have inde-
pendent switching kinetics:
∆P (t) = 2Ps
[1− exp{−(t/t0)
n}]F (log t0)d(log t0),
where F (log t0) is the distribution function for log t0
[4]. They assumed a very broad mesa-like function for
F (log t0), and could explain their ∆P (t) data. The same
500 ns
@ 1.7 V
1.7 V
1.4 V
1.2 V
1.1 V
1.0 V
0.9 V
0.8 V
-7 -6 -5 -4 -3 -2
300 K
150 K
  80 K
  25 K
  15 K
log t (s)
@ 300 K
Pole Read P
Write
A1 A2
Pole Read P
500 ns
A1 A2(a)
500 ns
FIG. 1: (color online). Schematic diagrams of the pulse trains
used to measure (a) non-switching polarization (Pns) and (b)
switching polarization (Psw). Time (t)-dependent switched
polarization ∆P (t) (c) under various external voltages (Vext)
at room temperature and (d) under 1.7 V at various temper-
atures. The dotted and solid lines correspond to fitted results
using the KAI model and the Lorentzian distribution function
in log t0, respectively.
http://arxiv.org/abs/0704.1053v1
0 100 200 300
  15 K
  25 K
  80 K
 (kV/cm)
150 K
300 K
-7 -6 -5 -4 -3
log t (s)
@ 80 K, 1.7 V 
 Experimental results
 Lorentzian distribution
 Gaussian distribution
 KAI model
-6 -4 -2
log t(s)
FIG. 2: (color online). (a) Values of n for various T and
Eext. (b) ∆P (t) results for (solid symbols) experimental data
and fitted results using the Lorentzian (solid line), Gaussian
(dashed line), and delta (dotted line) distributions for log t0.
The inset shows the distribution functions corresponding to
the fitted results.
authors also studied La-doped poly-PZT films and found
that ∆P (t) at room temperature is limited mainly by
nucleation, while at a low temperature (T ), the switching
kinetics are governed by domain wall motion, implying
the validity of the KAI model [5].
In this Letter, we investigate the polarization switch-
ing behaviors of poly-PZT films. We can explain the
measured ∆P (t) in terms of the Lorentzian distribution
function for F (log t0), irrespective of T . We show that
such distribution arises from local field variation in a dis-
ordered system with dipole-dipole interactions.
Note that (111)-oriented poly-PZT films with a Ti
concentration near 0.7 are the most widely used mate-
rial in FeRAM applications. We prepared our polycrys-
talline PbZr0.3Ti0.7O3 thin film on Pt/Ti/SiO2/Si sub-
strates using the sol-gel method. The poly-PZT film
had a thickness of 150 nm. X-ray diffraction studies
showed that it has the (111)-preferred orientation, and
scanning electron microscopy studies indicated that our
poly-PZT film consists of grains with a size of about 200
nm. We deposited Pt top electrodes using sputtering
with a shadow mask. The areas of the top electrodes
were about 7.9×10−9 m2.
We obtain the ∆P (t) values of our Pt/PZT/Pt capac-
itors using pulse measurements [2, 4, 13, 14]. Figure
1(a) shows the pulse trains used to measure the non-
switching polarization change (Pns). Using pulse A1, we
poled all the FE domains in one direction. Then, we ap-
plied pulse A2 with the same polarity, and measure the
current passing a sensing resistor. By integrating the cur-
rent, we could obtain the Pns values. Figure 1(b) shows
the pulse trains used to measure the switching polariza-
tion (Psw). Inserting pulse B with the opposite polarity
between pulses A1 and A2, we could reverse some portion
of the FE domains, so the difference between the values
of Psw and Pns represents the polarization change due
to domain switching, namely ∆P (t). We varied t from
200 ns to 1 ms, and Vext from 0.8 to 4 V. The value of
Eext can be estimated easily by dividing Vext by the film
thickness. At T of 80∼300 K, we used pulses A1 and A2
with a height of 4 V, which was larger than the coercive
voltage. Below 80 K, the coercive voltage increases, so
we increased the pulse height to 6 V [15].
Figure 1(c) shows the values of ∆P (t)/2Ps at room
temperature with numerous values of Vext. Figure 1(d)
shows the values of ∆P (t)/2Ps at various T with Vext
= 1.7 V. The dotted lines in both figures are the curves
best fitting Eq. (1). The KAI model predictions deviated
markedly from the experimental ∆P (t) values in the late
switching stage, in agreement with Gruverman et al. [6].
In addition, the best fitting results with the KAI model
gave unreasonable values of n. As shown in Fig. 2(a),
the values of n varied markedly with T and Eext. In ad-
dition, in the low Eext region, we obtained n values much
smaller than 1, which are not proper as an effective di-
mension of domain growth. Therefore, Eq. (1) fails to
describe the polarization switching behaviors of our PZT
films.
To explain the measured ∆P (t), we tried simple func-
tions for F (log t0) in Eq. (2). The opposite domain,
once nucleated, will propagate inside the film, so we fixed
n=2. The solid circles in Fig. 2(b) show the experimen-
tal ∆P (t) at 80 K with Vext = 1.7 V. For F (log t0),
we tried the delta, Gaussian, and Lorentzian distribu-
tion functions, as shown in the inset. The dotted line
indicates the fitting results using Eq. (2) with a delta
function. Note that this curve corresponds to a fit with
the KAI model, and thus the classical theory cannot ex-
plain our experimental data. The dashed line shows the
Gaussian fitting results. Although this fitting seems rea-
sonable, some discrepancies occur. The solid black shows
the fitting results with the Lorentzian distribution:
F (log t0) =
(log t0−log t1)2+w2
, (3)
where A is a normalization constant, and w (log t1) is
the half-width at half-maximum (a central value) [16].
The Lorentzian fit can account for our observed ∆P (t)
behaviors quite well.
We applied the Lorentzian fit to all of the other exper-
imental ∆P (t) data. As shown by the solid lines in Figs.
1(c) and (d), the Lorentzian fit provides excellent expla-
nations. Figure 3(a) presents the Lorentzian distribution
-4 -2 0 2 4
0.8 V
1.0 V
1.2 V
1.4 V
1.7 V
log t - log t
-8 -7 -6 -5 -4 -3 -2
log t
 0.8 V
 1.0 V
 1.2 V
 1.4 V
 1.7 V
@ 300 K
@ 300 K
FIG. 3: (color online). (a) The Eext-dependent Lorentzian
distribution functions at room temperature. (b) Rescaled
∆P (t) using fitting parameters for the Lorentzian distribu-
tion function.
functions used for the 300 K data. As Vext increases,
log t1 and w decrease. We rescaled the experimental
∆P (t)/2Ps data using (log t - log t1)/w. All the data
merge into a single line, an arctangent function [16], as
shown in Fig. 3(b). Although not indicated in this fig-
ure, the experimental data for all other T also merged
with this line. This scaling behavior suggests that the
Lorentzian distribution function for log t0 is intrinsic.
Note that F (log t0) follows not the Gaussian distribu-
tion, but the Lorentzian distribution. For a statistically
independent random process, it is a basic statistical rule
that the resulting distribution should become Gaussian,
regardless of the process details [17]. For example, im-
purities (or crystal defects) inside a real crystal result in
inhomogeneous broadening of the light absorption line,
which has a Gaussian line shape.
However, some studies have observed that magnetic
resonance line broadening of randomly distributed dipole
impurities follows the Lorentzian distribution [18]. The
first rigorous theoretical result for this problem is that of
Anderson, who showed that the distribution of any in-
teraction field component in the system of dilute aligned
dipoles should be Lorentzian [19, 20]. Polycrystalline FE
films should contain many dipole defects that will act
as pinning sites for the domain wall motion. To explain
our observed Lorentzian distribution of log t0, we assume
that a local field E exists at the FE domain pinning sites
and that it has a Lorentzian distribution:
F (E) = A
, (4)
where ∆ is the half-width at half-maximum of the E dis-
tribution function, related to the concentration of pin-
ning sites.
In the low Eext region, the domain wall motion should
be governed by thermal activation process at the pin-
ning sites. Without E effects, thermal activation results
in a domain wall speed in the form v ∝ 1/t0 ∝ exp[-
(U/kBT )(E0/Eext)], where U is the energy barrier and
E0 is the threshold electric field for pinned domains [21].
Since E results in a change in the effective electric field
at pinning sites, the associated t0 can be expressed as
t0 ∼ exp
)( E0
Eext+E
. (5)
Then, the distribution of E results in a distribu-
tion in log t0, using the relation F (log t0) = F (E) ·
|dE(log t0)/d(log t0)|. With
log t1 ≈
w ≈ UE0∆
, (7)
we can obtain the desired Lorentzian distribution for
F (log t0), i.e., Eq. (3), from Eqs. (4) and (5).
Our experimental values for log t1 and w agree with
the analytical forms. Figures 4(a) and (b) plot log t1 vs.
1/Eext and w vs. 1/E
ext at various T , respectively. Both
log t1 and w follow the expected Eext-dependence in the
low Eext region. Note that Eq. (6) is consistent with
Merz’s law [22], which states that the current coming
from FE polarization switching should have a character-
istic time of exp(α/Eext), where α is the activation field.
Using this empirical law, several studies have measured α
values. For example, So et al. reported α ≈1700 kV/cm
for 100-nm-thick epitaxial PZT films [2], and Scott et
al. reported α ≈ 270 kV/cm for 350-nm-thick poly-PZT
films [23]. These values are consistent with our room
temperature value of UE0/kBT , i.e., 1400 kV/cm.
Our model viewed the FE domain switching kinet-
ics as domain wall motion driven by Eext with a ran-
dom pinning potential. In the low Eext region, thermal
activation at the pinning sites can be important, result-
ing in the so-called domain wall creep motion. Applying
atomic force microscopy, Tybell et al. [21] and Paruch
et al. [24] demonstrated that the domain-switching ki-
netics in epitaxial PZT films are governed by the domain
wall creep motion. Some theoreticians studied the do-
main wall creep motion of an elastic string in a random
potential. They found a linear increase in U with an in-
crease in T [25]. The insets in Fig. 4(a) show that the
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005
0 150 300
0 150 300
0.000 0.005 0.010 0.015 0.020 0.025 0.030
  15 K
  25 K
  80 K
150 K
300 K
T (K)
(cm/kV)
  15 K
  25 K
  80 K
150 K
300 K
 (cm/kV)
FIG. 4: (color online). Eext-dependent (a) log t1 and (b) w at
various T . Note that log t1 and w are proportional to 1/Eext
and to 1/E2extin the low Eext region, respectively. The insets
show UE0/kB and ∆. The solid lines are guidelines for eyes.
value of UE0/kB obtained from the linear fits in Fig. 4(a)
increase linearly with T , consistent with the theoretical
prediction for U [25]. The inset in Fig. 4(b) shows ∆
obtained from the fits to Fig. 4(b). Similar exponential
decay behavior was predicted in a magnetic resonance
study of randomly distributed dipoles [26].
At this point, we wish to compare our model with the
NLS model. Although both models use Eq. (2), the ori-
gins and forms for F (log t0) are quite different. In the
NLS model, the FE film consists of numerous areas, each
with its own and independent t0. Subsequently, it was
suggested that the individually switched regions corre-
spond to single grains or clusters of grains in which the
grain boundaries act as frontiers limiting the propaga-
tion of the switched region [8]. Consequently, the NLS
model can be applied for polycrystalline films only, and
the form of F (log t0) should depend on their microstruc-
ture. Conversely, in our model, the interaction between
dipole defects inside the FE film induces a distribution
in the local field, which results in F (log t0). Therefore,
both point defects and the grain boundaries could act
as pinning sites. Using the Lorentzian distribution for
F (log t0), our model can be used for both epitaxial and
polycrystalline FE films [2]. Using Eqs. (2) and (3) with
small w values, we could successfully explain the ∆P (t)
for FE single crystals or epitaxial thin films [2]. We also
found that our model can explain the ∆P (t) data for
poly-PZT films with Ti concentrations of 0.48 and 0.65.
Note that our model for thermally activated domain
switching kinetics can be viewed as the famous prob-
lem that treats the propagation of elastic objects driven
by an external force in presence of a pinning potential
[21, 24, 25]. It can be applied to many FE films, since
the domain wall motion with a disordered pinning po-
tential should be the dominant mechanism for ∆P (t).
Therefore, the ∆P (t) studies can be used to investi-
gate numerous intriguing issues concerning nonlinear sys-
tems, such as creep motion, avalanche phenomenon, pin-
ning/depinning transition, and so on.
In summary, we investigated the polarization switch-
ing behaviors of (111)-oriented poly-PZT films and
found that the characteristic switching time obeyed the
Lorentzian distribution. We explained this intriguing
phenomenon by introducing the local electric field due
to the defect dipole.
The authors thank D. Kim for fruitful discussions.
This study was financially supported by Creative Re-
search Initiatives (Functionally Integrated Oxide Het-
erostructure) of MOST/KOSEF.
∗ Electronic address: twnoh@snu.ac.kr
[1] Ferroelectric Memories, edited by J. F. Scott (Springer-
Verlag, Berlin, 2000).
[2] Y. W. So et al., Appl. Phys. Lett. 86, 92905 (2005) and
references therein.
[3] O. Lohse et al., J. Appl. Phys. 89, 2332 (2001).
[4] A. K. Tagantsev et al., Phys. Rev. B 66, 214109 (2002).
[5] I. Stolichnov et al., Appl. Phys. Lett. 83, 3362 (2003).
[6] A. Gruverman et al., Appl. Phys. Lett. 87, 082902 (2005).
[7] V. Shur et al., J. Appl. Phys. 84, 445 (1998).
[8] I. Stolichnov et al., Appl. Phys. Lett. 86, 012902 (2005).
[9] B. H. Park et al., Nature 401, 682 (1999).
[10] N. Kolmogorov, Izv. Akad. Nauk. Ser. Math. 3, 355
(1937).
[11] M. Avrami, J. Chem. Phys. 8, 212 (1940).
[12] If all nuclei of opposite polarization arise through whole
process, n could be larger than the actual dimension.
[13] Y. S. Kim et al., Appl. Phys. Lett. 86, 102907 (2005).
[14] J. Y. Jo et al., Phys. Rev. Lett. 97, 247602 (2006).
[15] Complications can occur due to charge trapping or do-
main pinning, called the imprint effect. Refer to Ref. [1].
To prevent the imprint effect, we applied a pulse with
the opposite polarity at the end of each pulse train mea-
surement (i.e., after pulse A2).
[16] A double exponential function exp[-{10log t/10log t0}n]
with n >1 can be approximated as a step function cen-
tered at log t0=log t. As a result, Eq. (2) can be approx-
imated as 2PsA/π · [arctan{(log t− log t1)/w}+ π/2].
[17] F. Reif, Fundamentals of Statistics and Thermal Physics
(McGraw-Hill, Singapore, 1985).
[18] J. H. V. Vleck, Phys. Rev. 74, 1168 (1948).
[19] P. W. Anderson, Phys. Rev. 82, 342 (1951).
[20] J. R. Klauder and P. W. Anderson, Phys. Rev. 125, 912
(1962).
[21] T. Tybell et al., Phys. Rev. Lett. 89, 097601 (2002).
[22] W. J. Merz, Phys. Rev. 95, 690 (1954).
[23] J. F. Scott et al., J. Appl. Phys. 64, 787 (1998).
[24] P. Paruch et al., Phys. Rev. Lett. 94, 197601 (2005).
[25] A. B. Kolton et al., Phys. Rev. Lett. 94, 047002 (2005).
mailto:twnoh@snu.ac.kr
[26] M. W. Klein, Phys. Rev. 173, 552 (1968).

  We investigated domain kinetics by measuring the polarization switching
behaviors of polycrystalline Pb(Zr,Ti)O$_{3}$ films, which are widely used in
ferroelectric memory devices. Their switching behaviors at various electric
fields and temperatures could be explained by assuming the Lorentzian
distribution of domain switching times. We viewed the switching process under
an electric field as a motion of the ferroelectric domain through a random
medium, and we showed that the local field variation due to dipole defects at
domain pinning sites could explain the intriguing distribution.

J. Y. Jo et al.
Domain Switching Kinetics in Disordered Ferroelectric Thin Films
J. Y. Jo,1 H. S. Han,1 J.-G. Yoon,2 T. K. Song,3 S.-H. Kim,4 and T. W. Noh1,∗
ReCOE&FPRD, Department of Physics andAstronomy, Seoul National University, Seoul 151-747, Korea
Department of Physics,University of Suwon, Suwon, Gyeonggi-do 445-743, Korea
School of Nano Advanced Materials, Changwon National University, Changwon, Gyeongnam 641-773, Korea
R&D center, Inostek Inc., Ansan, Gyeonggi-do 426-901, Korea
(Dated: October 23, 2018)
We investigated domain kinetics by measuring the polarization switching behaviors of polycrys-
talline Pb(Zr,Ti)O3 films, which are widely used in ferroelectric memory devices. Their switching
behaviors at various electric fields and temperatures could be explained by assuming the Lorentzian
distribution of domain switching times. We viewed the switching process under an electric field as
a motion of the ferroelectric domain through a random medium, and we showed that the local field
variation due to dipole defects at domain pinning sites could explain the intriguing distribution.
PACS numbers: 77.80.Fm, 77.80.Dj, 77.84.Dy
Domain switching kinetics in ferroelectrics (FEs) un-
der an external electric field Eext have been extensively
investigated for several decades [1, 2, 3, 4, 5, 6, 7, 8, 9].
The traditional approach to explain the FE switching
kinetics, often called the Kolmogorov-Avrami-Ishibashi
(KAI) model, is based on the classical statistical theory
of nucleation and unrestricted domain growth [10, 11].
For a uniformly polarized FE sample under Eext, the
KAI model gives the time (t)-dependent change in polar-
ization ∆P (t) as
∆P (t) = 2Ps[1− exp{−(t/t0)
n}], (1)
where n and t0 are the effective dimension and character-
istic switching time for the domain growth, respectively,
and Ps is spontaneous polarization. When the nuclei are
appearing in time with the same probability, n = 3 for
bulk samples and n = 2 for thin films [12]. In addition,
t0 is proportional to the average distance between the
nuclei, divided by the domain wall speed. Several stud-
ies have used the KAI model successfully to explain the
∆P (t) behaviors of FE single crystals and epitaxial thin
films [2].
Recently, FE thin films have been intensively investi-
gated for FE random access memory (FeRAM) [1]. Most
commercial FeRAM use polycrystalline Pb(Zr,Ti)O3
(poly-PZT) films, and their ∆P (t) behaviors determine
the reading and writing speeds of the FeRAM. In such
non-epitaxial FE films, a domain cannot propagate in-
definitely due to pinning caused by numerous defects, so
the KAI model cannot be applied. Therefore, it is impor-
tant both scientifically and technologically to clarify the
domain switching kinetics of polycrystalline FE films.
Numerous studies have examined the ∆P (t) behaviors
of polycrystalline FE films, and the reported results vary
markedly [3, 4, 5, 6, 7]. Lohse et al. measured the polar-
ization switching currents of poly-PZT films, and showed
that ∆P (t) slowed significantly compared to Eq. (1) [3].
Tagantsev et al. observed similar phenomena for poly-
PZT films. To explain these behaviors, they developed
the nucleation-limited-switching (NLS) model. They as-
sumed that films consist of several areas that have inde-
pendent switching kinetics:
∆P (t) = 2Ps
[1− exp{−(t/t0)
n}]F (log t0)d(log t0),
where F (log t0) is the distribution function for log t0
[4]. They assumed a very broad mesa-like function for
F (log t0), and could explain their ∆P (t) data. The same
500 ns
@ 1.7 V
1.7 V
1.4 V
1.2 V
1.1 V
1.0 V
0.9 V
0.8 V
-7 -6 -5 -4 -3 -2
300 K
150 K
  80 K
  25 K
  15 K
log t (s)
@ 300 K
Pole Read P
Write
A1 A2
Pole Read P
500 ns
A1 A2(a)
500 ns
FIG. 1: (color online). Schematic diagrams of the pulse trains
used to measure (a) non-switching polarization (Pns) and (b)
switching polarization (Psw). Time (t)-dependent switched
polarization ∆P (t) (c) under various external voltages (Vext)
at room temperature and (d) under 1.7 V at various temper-
atures. The dotted and solid lines correspond to fitted results
using the KAI model and the Lorentzian distribution function
in log t0, respectively.
http://arxiv.org/abs/0704.1053v1
0 100 200 300
  15 K
  25 K
  80 K
 (kV/cm)
150 K
300 K
-7 -6 -5 -4 -3
log t (s)
@ 80 K, 1.7 V 
 Experimental results
 Lorentzian distribution
 Gaussian distribution
 KAI model
-6 -4 -2
log t(s)
FIG. 2: (color online). (a) Values of n for various T and
Eext. (b) ∆P (t) results for (solid symbols) experimental data
and fitted results using the Lorentzian (solid line), Gaussian
(dashed line), and delta (dotted line) distributions for log t0.
The inset shows the distribution functions corresponding to
the fitted results.
authors also studied La-doped poly-PZT films and found
that ∆P (t) at room temperature is limited mainly by
nucleation, while at a low temperature (T ), the switching
kinetics are governed by domain wall motion, implying
the validity of the KAI model [5].
In this Letter, we investigate the polarization switch-
ing behaviors of poly-PZT films. We can explain the
measured ∆P (t) in terms of the Lorentzian distribution
function for F (log t0), irrespective of T . We show that
such distribution arises from local field variation in a dis-
ordered system with dipole-dipole interactions.
Note that (111)-oriented poly-PZT films with a Ti
concentration near 0.7 are the most widely used mate-
rial in FeRAM applications. We prepared our polycrys-
talline PbZr0.3Ti0.7O3 thin film on Pt/Ti/SiO2/Si sub-
strates using the sol-gel method. The poly-PZT film
had a thickness of 150 nm. X-ray diffraction studies
showed that it has the (111)-preferred orientation, and
scanning electron microscopy studies indicated that our
poly-PZT film consists of grains with a size of about 200
nm. We deposited Pt top electrodes using sputtering
with a shadow mask. The areas of the top electrodes
were about 7.9×10−9 m2.
We obtain the ∆P (t) values of our Pt/PZT/Pt capac-
itors using pulse measurements [2, 4, 13, 14]. Figure
1(a) shows the pulse trains used to measure the non-
switching polarization change (Pns). Using pulse A1, we
poled all the FE domains in one direction. Then, we ap-
plied pulse A2 with the same polarity, and measure the
current passing a sensing resistor. By integrating the cur-
rent, we could obtain the Pns values. Figure 1(b) shows
the pulse trains used to measure the switching polariza-
tion (Psw). Inserting pulse B with the opposite polarity
between pulses A1 and A2, we could reverse some portion
of the FE domains, so the difference between the values
of Psw and Pns represents the polarization change due
to domain switching, namely ∆P (t). We varied t from
200 ns to 1 ms, and Vext from 0.8 to 4 V. The value of
Eext can be estimated easily by dividing Vext by the film
thickness. At T of 80∼300 K, we used pulses A1 and A2
with a height of 4 V, which was larger than the coercive
voltage. Below 80 K, the coercive voltage increases, so
we increased the pulse height to 6 V [15].
Figure 1(c) shows the values of ∆P (t)/2Ps at room
temperature with numerous values of Vext. Figure 1(d)
shows the values of ∆P (t)/2Ps at various T with Vext
= 1.7 V. The dotted lines in both figures are the curves
best fitting Eq. (1). The KAI model predictions deviated
markedly from the experimental ∆P (t) values in the late
switching stage, in agreement with Gruverman et al. [6].
In addition, the best fitting results with the KAI model
gave unreasonable values of n. As shown in Fig. 2(a),
the values of n varied markedly with T and Eext. In ad-
dition, in the low Eext region, we obtained n values much
smaller than 1, which are not proper as an effective di-
mension of domain growth. Therefore, Eq. (1) fails to
describe the polarization switching behaviors of our PZT
films.
To explain the measured ∆P (t), we tried simple func-
tions for F (log t0) in Eq. (2). The opposite domain,
once nucleated, will propagate inside the film, so we fixed
n=2. The solid circles in Fig. 2(b) show the experimen-
tal ∆P (t) at 80 K with Vext = 1.7 V. For F (log t0),
we tried the delta, Gaussian, and Lorentzian distribu-
tion functions, as shown in the inset. The dotted line
indicates the fitting results using Eq. (2) with a delta
function. Note that this curve corresponds to a fit with
the KAI model, and thus the classical theory cannot ex-
plain our experimental data. The dashed line shows the
Gaussian fitting results. Although this fitting seems rea-
sonable, some discrepancies occur. The solid black shows
the fitting results with the Lorentzian distribution:
F (log t0) =
(log t0−log t1)2+w2
, (3)
where A is a normalization constant, and w (log t1) is
the half-width at half-maximum (a central value) [16].
The Lorentzian fit can account for our observed ∆P (t)
behaviors quite well.
We applied the Lorentzian fit to all of the other exper-
imental ∆P (t) data. As shown by the solid lines in Figs.
1(c) and (d), the Lorentzian fit provides excellent expla-
nations. Figure 3(a) presents the Lorentzian distribution
-4 -2 0 2 4
0.8 V
1.0 V
1.2 V
1.4 V
1.7 V
log t - log t
-8 -7 -6 -5 -4 -3 -2
log t
 0.8 V
 1.0 V
 1.2 V
 1.4 V
 1.7 V
@ 300 K
@ 300 K
FIG. 3: (color online). (a) The Eext-dependent Lorentzian
distribution functions at room temperature. (b) Rescaled
∆P (t) using fitting parameters for the Lorentzian distribu-
tion function.
functions used for the 300 K data. As Vext increases,
log t1 and w decrease. We rescaled the experimental
∆P (t)/2Ps data using (log t - log t1)/w. All the data
merge into a single line, an arctangent function [16], as
shown in Fig. 3(b). Although not indicated in this fig-
ure, the experimental data for all other T also merged
with this line. This scaling behavior suggests that the
Lorentzian distribution function for log t0 is intrinsic.
Note that F (log t0) follows not the Gaussian distribu-
tion, but the Lorentzian distribution. For a statistically
independent random process, it is a basic statistical rule
that the resulting distribution should become Gaussian,
regardless of the process details [17]. For example, im-
purities (or crystal defects) inside a real crystal result in
inhomogeneous broadening of the light absorption line,
which has a Gaussian line shape.
However, some studies have observed that magnetic
resonance line broadening of randomly distributed dipole
impurities follows the Lorentzian distribution [18]. The
first rigorous theoretical result for this problem is that of
Anderson, who showed that the distribution of any in-
teraction field component in the system of dilute aligned
dipoles should be Lorentzian [19, 20]. Polycrystalline FE
films should contain many dipole defects that will act
as pinning sites for the domain wall motion. To explain
our observed Lorentzian distribution of log t0, we assume
that a local field E exists at the FE domain pinning sites
and that it has a Lorentzian distribution:
F (E) = A
, (4)
where ∆ is the half-width at half-maximum of the E dis-
tribution function, related to the concentration of pin-
ning sites.
In the low Eext region, the domain wall motion should
be governed by thermal activation process at the pin-
ning sites. Without E effects, thermal activation results
in a domain wall speed in the form v ∝ 1/t0 ∝ exp[-
(U/kBT )(E0/Eext)], where U is the energy barrier and
E0 is the threshold electric field for pinned domains [21].
Since E results in a change in the effective electric field
at pinning sites, the associated t0 can be expressed as
t0 ∼ exp
)( E0
Eext+E
. (5)
Then, the distribution of E results in a distribu-
tion in log t0, using the relation F (log t0) = F (E) ·
|dE(log t0)/d(log t0)|. With
log t1 ≈
w ≈ UE0∆
, (7)
we can obtain the desired Lorentzian distribution for
F (log t0), i.e., Eq. (3), from Eqs. (4) and (5).
Our experimental values for log t1 and w agree with
the analytical forms. Figures 4(a) and (b) plot log t1 vs.
1/Eext and w vs. 1/E
ext at various T , respectively. Both
log t1 and w follow the expected Eext-dependence in the
low Eext region. Note that Eq. (6) is consistent with
Merz’s law [22], which states that the current coming
from FE polarization switching should have a character-
istic time of exp(α/Eext), where α is the activation field.
Using this empirical law, several studies have measured α
values. For example, So et al. reported α ≈1700 kV/cm
for 100-nm-thick epitaxial PZT films [2], and Scott et
al. reported α ≈ 270 kV/cm for 350-nm-thick poly-PZT
films [23]. These values are consistent with our room
temperature value of UE0/kBT , i.e., 1400 kV/cm.
Our model viewed the FE domain switching kinet-
ics as domain wall motion driven by Eext with a ran-
dom pinning potential. In the low Eext region, thermal
activation at the pinning sites can be important, result-
ing in the so-called domain wall creep motion. Applying
atomic force microscopy, Tybell et al. [21] and Paruch
et al. [24] demonstrated that the domain-switching ki-
netics in epitaxial PZT films are governed by the domain
wall creep motion. Some theoreticians studied the do-
main wall creep motion of an elastic string in a random
potential. They found a linear increase in U with an in-
crease in T [25]. The insets in Fig. 4(a) show that the
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005
0 150 300
0 150 300
0.000 0.005 0.010 0.015 0.020 0.025 0.030
  15 K
  25 K
  80 K
150 K
300 K
T (K)
(cm/kV)
  15 K
  25 K
  80 K
150 K
300 K
 (cm/kV)
FIG. 4: (color online). Eext-dependent (a) log t1 and (b) w at
various T . Note that log t1 and w are proportional to 1/Eext
and to 1/E2extin the low Eext region, respectively. The insets
show UE0/kB and ∆. The solid lines are guidelines for eyes.
value of UE0/kB obtained from the linear fits in Fig. 4(a)
increase linearly with T , consistent with the theoretical
prediction for U [25]. The inset in Fig. 4(b) shows ∆
obtained from the fits to Fig. 4(b). Similar exponential
decay behavior was predicted in a magnetic resonance
study of randomly distributed dipoles [26].
At this point, we wish to compare our model with the
NLS model. Although both models use Eq. (2), the ori-
gins and forms for F (log t0) are quite different. In the
NLS model, the FE film consists of numerous areas, each
with its own and independent t0. Subsequently, it was
suggested that the individually switched regions corre-
spond to single grains or clusters of grains in which the
grain boundaries act as frontiers limiting the propaga-
tion of the switched region [8]. Consequently, the NLS
model can be applied for polycrystalline films only, and
the form of F (log t0) should depend on their microstruc-
ture. Conversely, in our model, the interaction between
dipole defects inside the FE film induces a distribution
in the local field, which results in F (log t0). Therefore,
both point defects and the grain boundaries could act
as pinning sites. Using the Lorentzian distribution for
F (log t0), our model can be used for both epitaxial and
polycrystalline FE films [2]. Using Eqs. (2) and (3) with
small w values, we could successfully explain the ∆P (t)
for FE single crystals or epitaxial thin films [2]. We also
found that our model can explain the ∆P (t) data for
poly-PZT films with Ti concentrations of 0.48 and 0.65.
Note that our model for thermally activated domain
switching kinetics can be viewed as the famous prob-
lem that treats the propagation of elastic objects driven
by an external force in presence of a pinning potential
[21, 24, 25]. It can be applied to many FE films, since
the domain wall motion with a disordered pinning po-
tential should be the dominant mechanism for ∆P (t).
Therefore, the ∆P (t) studies can be used to investi-
gate numerous intriguing issues concerning nonlinear sys-
tems, such as creep motion, avalanche phenomenon, pin-
ning/depinning transition, and so on.
In summary, we investigated the polarization switch-
ing behaviors of (111)-oriented poly-PZT films and
found that the characteristic switching time obeyed the
Lorentzian distribution. We explained this intriguing
phenomenon by introducing the local electric field due
to the defect dipole.
The authors thank D. Kim for fruitful discussions.
This study was financially supported by Creative Re-
search Initiatives (Functionally Integrated Oxide Het-
erostructure) of MOST/KOSEF.
∗ Electronic address: twnoh@snu.ac.kr
[1] Ferroelectric Memories, edited by J. F. Scott (Springer-
Verlag, Berlin, 2000).
[2] Y. W. So et al., Appl. Phys. Lett. 86, 92905 (2005) and
references therein.
[3] O. Lohse et al., J. Appl. Phys. 89, 2332 (2001).
[4] A. K. Tagantsev et al., Phys. Rev. B 66, 214109 (2002).
[5] I. Stolichnov et al., Appl. Phys. Lett. 83, 3362 (2003).
[6] A. Gruverman et al., Appl. Phys. Lett. 87, 082902 (2005).
[7] V. Shur et al., J. Appl. Phys. 84, 445 (1998).
[8] I. Stolichnov et al., Appl. Phys. Lett. 86, 012902 (2005).
[9] B. H. Park et al., Nature 401, 682 (1999).
[10] N. Kolmogorov, Izv. Akad. Nauk. Ser. Math. 3, 355
(1937).
[11] M. Avrami, J. Chem. Phys. 8, 212 (1940).
[12] If all nuclei of opposite polarization arise through whole
process, n could be larger than the actual dimension.
[13] Y. S. Kim et al., Appl. Phys. Lett. 86, 102907 (2005).
[14] J. Y. Jo et al., Phys. Rev. Lett. 97, 247602 (2006).
[15] Complications can occur due to charge trapping or do-
main pinning, called the imprint effect. Refer to Ref. [1].
To prevent the imprint effect, we applied a pulse with
the opposite polarity at the end of each pulse train mea-
surement (i.e., after pulse A2).
[16] A double exponential function exp[-{10log t/10log t0}n]
with n >1 can be approximated as a step function cen-
tered at log t0=log t. As a result, Eq. (2) can be approx-
imated as 2PsA/π · [arctan{(log t− log t1)/w}+ π/2].
[17] F. Reif, Fundamentals of Statistics and Thermal Physics
(McGraw-Hill, Singapore, 1985).
[18] J. H. V. Vleck, Phys. Rev. 74, 1168 (1948).
[19] P. W. Anderson, Phys. Rev. 82, 342 (1951).
[20] J. R. Klauder and P. W. Anderson, Phys. Rev. 125, 912
(1962).
[21] T. Tybell et al., Phys. Rev. Lett. 89, 097601 (2002).
[22] W. J. Merz, Phys. Rev. 95, 690 (1954).
[23] J. F. Scott et al., J. Appl. Phys. 64, 787 (1998).
[24] P. Paruch et al., Phys. Rev. Lett. 94, 197601 (2005).
[25] A. B. Kolton et al., Phys. Rev. Lett. 94, 047002 (2005).
mailto:twnoh@snu.ac.kr
[26] M. W. Klein, Phys. Rev. 173, 552 (1968).
