Partial polarization switching in ferroelectrics-semiconductors with charged defects
We propose the phenomenological description of ferroelectric disordering caused by charged defects in ferroelectric-semiconductors. The good agreement between the obtained experimental results for PZT films and theoretical calculations has been shown. We suppose that proportional to the averaged ch...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2004
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Partial polarization switching in ferroelectrics-semiconductors with charged defects / A.N. Morozovska, E.A. Eliseev, E. Cattan, D. Remiens // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 251-262. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1191202017-06-05T03:02:50Z Partial polarization switching in ferroelectrics-semiconductors with charged defects Morozovska, A.N. Eliseev, E.A. Cattan, E. Remiens, D. We propose the phenomenological description of ferroelectric disordering caused by charged defects in ferroelectric-semiconductors. The good agreement between the obtained experimental results for PZT films and theoretical calculations has been shown. We suppose that proportional to the averaged charge density of defects improper conductivity is sufficiently high to provide the screening of charge density random fluctuations drs in the absence of external field. When external electric field is applied, inner field fluctuations and induction fluctuations dD appear in the inhomogeneously polarized system “charged fluctuation + screening cloud”. We show that the macroscopic state of ferroelectric-semiconductor with random charged defects and sufficiently high improper conductivity can be described by three coupled equations for three order parameters. Averaged over sample volume induction determines the ferroelectric ordering in the system, its square fluctuation determines disordering caused by electric field fluctuations appeared around charged fluctuations drs, and reflects the correlations between the free carriers screening cloud and charged defects drs. For the first time, we derive the following system of three coupled equations: Also the obtained system of coupled equations qualitatively describes the peculiarities of polarization switching (footprint and minor hysteresis loops) in such ferroelectric materials with charged defects as PZT films with growth imperfections, PLZT ceramics and SBN single crystals doped with cerium. 2004 Article Partial polarization switching in ferroelectrics-semiconductors with charged defects / A.N. Morozovska, E.A. Eliseev, E. Cattan, D. Remiens // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 251-262. — Бібліогр.: 25 назв. — англ. 1560-8034 PACS: 77.80.-e, 77.80.Dj, 61.43.-j http://dspace.nbuv.gov.ua/handle/123456789/119120 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
We propose the phenomenological description of ferroelectric disordering caused by charged defects in ferroelectric-semiconductors. The good agreement between the obtained experimental results for PZT films and theoretical calculations has been shown.
We suppose that proportional to the averaged charge density of defects improper conductivity is sufficiently high to provide the screening of charge density random fluctuations drs in the absence of external field. When external electric field is applied, inner field fluctuations and induction fluctuations dD appear in the inhomogeneously polarized system “charged fluctuation + screening cloud”.
We show that the macroscopic state of ferroelectric-semiconductor with random charged defects and sufficiently high improper conductivity can be described by three coupled equations for three order parameters. Averaged over sample volume induction determines the ferroelectric ordering in the system, its square fluctuation determines disordering caused by electric field fluctuations appeared around charged fluctuations drs, and reflects the correlations between the free carriers screening cloud and charged defects drs. For the first time, we derive the following system of three coupled equations:
Also the obtained system of coupled equations qualitatively describes the peculiarities of polarization switching (footprint and minor hysteresis loops) in such ferroelectric materials with charged defects as PZT films with growth imperfections, PLZT ceramics and SBN single crystals doped with cerium. |
| format |
Article |
| author |
Morozovska, A.N. Eliseev, E.A. Cattan, E. Remiens, D. |
| spellingShingle |
Morozovska, A.N. Eliseev, E.A. Cattan, E. Remiens, D. Partial polarization switching in ferroelectrics-semiconductors with charged defects Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Morozovska, A.N. Eliseev, E.A. Cattan, E. Remiens, D. |
| author_sort |
Morozovska, A.N. |
| title |
Partial polarization switching in ferroelectrics-semiconductors with charged defects |
| title_short |
Partial polarization switching in ferroelectrics-semiconductors with charged defects |
| title_full |
Partial polarization switching in ferroelectrics-semiconductors with charged defects |
| title_fullStr |
Partial polarization switching in ferroelectrics-semiconductors with charged defects |
| title_full_unstemmed |
Partial polarization switching in ferroelectrics-semiconductors with charged defects |
| title_sort |
partial polarization switching in ferroelectrics-semiconductors with charged defects |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119120 |
| citation_txt |
Partial polarization switching in ferroelectrics-semiconductors with charged defects / A.N. Morozovska, E.A. Eliseev, E. Cattan, D. Remiens // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 251-262. — Бібліогр.: 25 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT morozovskaan partialpolarizationswitchinginferroelectricssemiconductorswithchargeddefects AT eliseevea partialpolarizationswitchinginferroelectricssemiconductorswithchargeddefects AT cattane partialpolarizationswitchinginferroelectricssemiconductorswithchargeddefects AT remiensd partialpolarizationswitchinginferroelectricssemiconductorswithchargeddefects |
| first_indexed |
2025-07-08T15:15:31Z |
| last_indexed |
2025-07-08T15:15:31Z |
| _version_ |
1837092289116110848 |
| fulltext |
251© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 3. P. 251-262.
PACS: 77.80.-e, 77.80.Dj, 61.43.-j
Partial polarization switching
in ferroelectrics-semiconductors with charged defects
A.N. Morozovska, Eu.A. Eliseev*, E. Cattan**, D. Remiens**,
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, pr. Nauky, 03028 Kyiv, Ukraine
E-mail: morozo@mail.i.com.ua
*Institute for Problems of Materials Science, National Academy of Sciences of Ukraine,
3, Krjijanovskogo str., 03142 Kyiv, Ukraine,
**IEMN, UMR 8520 OAE-dept/ MIMM, Universite de Valenciennes et du Hainaut-Cambresis,
Le Mont Houy, 59313 Valenciennes Cedex 9, France
Abstract. We propose the phenomenological description of ferroelectric disordering caused
by charged defects in ferroelectric-semiconductors. The good agreement between the obtained
experimental results for PZT films and theoretical calculations has been shown.
We suppose that proportional to the averaged charge density sρ of defects improper conduc-
tivity is sufficiently high to provide the screening of charge density random fluctuations δρs
in the absence of external field. When external electric field
)(
0 tE is applied, inner field
fluctuations and induction fluctuations δD appear in the inhomogeneously polarized system
�charged fluctuation + screening cloud�.
We show that the macroscopic state of ferroelectric-semiconductor with random charged
defects and sufficiently high improper conductivity can be described by three coupled equa-
tions for three order parameters. Averaged over sample volume induction D determines the
ferroelectric ordering in the system, its square fluctuation 2Dδ determines disordering caused
by electric field fluctuations appeared around charged fluctuations δρs, and sDδρδ reflects
the correlations between the free carriers screening cloud and charged defects δρs. For the first
time, we derive the following system of three coupled equations:
)(3 0
32 tEDDD
t
D
=+
++
∂
∂
Γ βδβα , ( ) ( )
s
s D
tEDDDD
t ρ
δδρ
δβδβαδ )(3
2 0
22222 −=
+++
∂
∂Γ
,
s
s
ss tEDDDD
t ρ
ρδ
δρδδββαρδδ
2
0
22 )(3 −=
+++
∂
∂
Γ .
Also the obtained system of coupled equations qualitatively describes the peculiarities of
polarization switching (footprint and minor hysteresis loops) in such ferroelectric materials
with charged defects as PZT films with growth imperfections, PLZT ceramics and SBN single
crystals doped with cerium.
Keywords: ferroelectrics-semiconductors, charged defects, partial switching, footprint and
minor hysteresis loop.
Paper received 24.04.04; accepted for publication
1. Introduction
In most cases, stable partial switching of the spontane-
ous polarization can be achieved in imperfect ferro-
electric materials [1]. Sometimes, the polarization rever-
sal process is strongly asymmetric even in thick ferro-
electric films and bulk samples, and in particular the
minor hysteresis loops appear in imperfect ferroelectrics
(see e.g. [2�5]). For the doped ferroelectrics, this phe-
nomena is strongly dependent on the type and concentra-
tion of dopants [3] and the external field frequency [2, 3].
For the system of sandwich type metal �ferroelectrics �
semiconductor [5], this type of switching is defined by the
depolarization field and built-in charge layer in the
ferroelectrics � semiconductor interface [6]. The similar
system was theoretically studied in [7] allowing for the
semiconductor properties of ferroelectrics. The model of
21.10.04.
252
SQO, 7(3), 2004
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
ref. [7] predicts that only one direction of spontaneous
polarization is stable with the ferroelectric layer thick-
ness decrease.
Another interesting feature of the polarization switch-
ing in ferroelectrics with non-isovalent dopants are the
clamped or pinched hysteresis loops observed in La-doped
lead zirconate titanate sol-gel films after annealing them
in hydrogen atmosphere [8]. This type of hysteresis loops
are sometimes called as footprint loops, their appearance
was attributed to the influence of the sample conductivity
[9]. Constricted hysteresis loops similar to those observed
by authors of ref. [8] were found in the La-doped lead
zirconate titanate ceramics with composition near the
morphotropic boundaries between ferroelectric tetrago-
nal, rhombohedral and antiferroelectric phases [10]. This
effect was attributed to the structural changes from cubic
matrix with embedded microdomains to orthorhombic
macrodomain state twice during one switching cycle of
the external field. However, we do not know the adequate
quantitative model describing the hysteresis loops with
constrictions.
We would like to underline that in all the aforemen-
tioned materials where footprint and minor loops exist
either non-isovalent impurities or some unavoidable im-
perfections manifest themselves as charged defects.
Moreover, these �dusty� materials would be rather con-
sidered as improper semiconductor than ideal dielectrics
with random electric fields [11]. These two facts give the
basis of our model.
We propose the new phenomenological model that
can give both the simple qualitative explanation and ana-
lytical description of partial polarization switching phe-
nomena in bulk ferroelectric materials with charged de-
fects and sufficiently high improper conductivity. We try
to involve the minimum number of hypothesis into our
model. Moreover, we have not used the detailed descrip-
tion of the chemical nature, concentration and sizes dis-
tribution of randomly situated immovable charged de-
fects that are the sources of movable charge carriers, in-
ner electric field and induction fluctuations. Also our
model admits continuous transformation from the ordered
ferroelectric to the disordered material under increasing
the charged defects concentration fluctuations.
The main goal of our paper is to demonstrate that
macroscopic state of the thick sample with random charged
defects can be described by the system of three coupled
equations, which is similar to such well-known nonlinear
systems of first order differential equations as the Lorenz
one [12]. Such dynamical systems of equations can re-
veal chaotic regimes, strange attractors as well as strongly
non-ergodic behaviour and continuous relaxation time
spectrum. Up to date, we have studied only the stability
of the system stationary states by means of the reduced
free energy and simulated quasi-equilibrium ferroelectric
hysteresis loops. The good agreement between the theo-
retical calculations and obtained experimental results for
thick PZT films has been shown. Certainly, the dynami-
cal dielectric response of this first obtained system re-
quires further investigations.
2. The problem
We assume that unavoidable charged defects or non-
isovalent impurity atoms are embedded into hypotheti-
cal �pure� uniaxial ferroelectric. We suppose also that
even in the absence of proper conductivity, imperfections
provide a rather high improper conductivity in the bulk
of the sample. The concentration of these atoms sρ fluc-
tuates due to the great variety of misfit effects (different
ionic radii, local symmetry breaking, clusterization).
These fluctuations δρs are considered in the continuous
medium framework, i.e., the discrete charge density
(point charges in Scheme 1) is approximated by the smooth
function sss δρρρ += . In this approach, the short-range
fluctuations will be neglected, and the smallest period d
in δρs spatial spectrum is much greater than the average
distance h between real point defect atoms.
Hereinafter, we regard embedded defects almost im-
movable and charged with density )(rsρ . The sample as
a whole is electro-neutral. In this case, the movable free
charges ),( tn r surround each charged impurity center (see
Scheme 1). The characteristic size of these screening
clouds is of the same order as the Debye screening radius
RD. For the large enough average defect concentration
sρ , radius sDR ρ1~ is much smaller than the average
distance h between defects. It is obvious that in the ab-
sence of the external electric field E0 the inner field is
close to zero outside the screening clouds (see Sche-
me 1a, c). But when one applies the external field E0, the
screening clouds of free charges are deformed, and nano-
system �defect center + screening cloud� becomes polari-
zed (Scheme 1b, d). If the defect charge density fluctua-
tions δρs(r) are absent, the short-range electric fields
caused by homogeneously distributed induced dipoles
are canceled, and no long-range electric field arises in
the bulk of the sample (Scheme 1b). Moreover, the fluc-
tuations δρs(r) do not reveal themselves in the absence of
the external filed E0 due to the complete screening by
movable space-charge fluctuations δn (Scheme 1 c). Con-
trary to this at 00 ≠E polarized regions �δρs(r) + δn(r, t)�
cause the long-range inner electric field fluctuations
δE(r, t) (Scheme 1d). According to the equations of state,
the fluctuations of the inner electric field δE cause induc-
tion fluctuations δD. The rigorous conditions that deter-
mine the existence of these induction fluctuations will be
given below.
Evidently non-homogeneous mechanical stresses ap-
pearing near the defects should be taken into account
[13]. But the consideration of non-homogeneous mechani-
cal stresses significantly complicates the problem, and
we hope that the system behavior will not change qualita-
tively under the influence of non-homogeneous mechani-
cal stresses. It is also known (see e.g. [14]) that homoge-
neous elastic stresses due to the electrostriction coupling
with the polar order parameter can be taken into account
by the renormalization of the free energy expansion co-
efficients.
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
253SQO, 7(3), 2004
3. General equations
Maxwell�s equations for the electric induction D, field Å
and equation of continuity have the form:
0,0,4 =+
∂
∂== cdiv
t
rotdiv jED
ρπρ . (1)
They have to be supplemented by the equations of
state:
zzSz EDDD )(, εε +== ⊥⊥⊥ ED , (2)
( ) ., ∑∑ +=−=
m
sm
m
mmmmc grad ρρρρκρµ Ej (3)
The sample is regarded as linear dielectric in trans-
verse x,y-directions and as nonlinear polar material in
longitudinal z-direction. Here ε⊥ is transversal compo-
nent of dielectric permittivity, ρm, µm and κm are the m-
type movable charge volume density (m = n, p), mobility
and diffusion coefficient, respectively, jc is the macro-
scopic free-carriers current, ρs(r) is the given charge den-
sity of static defects.
The spatial-temporal distribution of the induction z-
component Dz ≡ D can be obtained from the Landau-
Ginsburg-Khalatnikov equation:
Scheme 1. The system of charged defects with the charge den-
sity ρs (dots) screened by the free charges with density n (circles
or ellipses) and screening radius RD. Parts a, b represent charges
homogeneous distributions (δρs = 0, δn = 0). Parts c, d represent
the distribution with the long-range fluctuations (δρs ≠ 0, δn ≠ 0)
and the space period d much greater than the average distance
h between defects. The parts a, c and b, d show the system with
the zero and nonzero external field E0, respectively.
n
h
a
c d
d
b
r
dr = 0
dr ¹ 0
d » rn �d
d ~ r / )E E ( � d r
S
S
S
dE+
+
+
+
+
+
+
+
+
+ �
�
�
�
�
�
�
�
�
�
�
E = 0 E 0¹0
s
s sz 0
0
D~R
zE
D
DD
t
D =
∂
∂−++
∂
∂Γ
2
2
3
r
γβα . (4)
Here à > 0 is the kinetic coefficient, α(T) = αT(T �
� T*), Ò is the absolute temperature, Ò* is the Curie tem-
perature of the hypothetical pure (free of defects) sample,
b>0, g>0. Equations (1), (2) can be rewritten as:
( ) ( ) .0
4
1
,4
,
=
+
∂
∂
+−
−=
∂
∂
⊥⊥
=
⊥⊥
∑ EeE
E
ε
π
ρκρµ
επρ
z
hem
mmmm D
t
graddiv
divD
z
Here ez is the unit vector directed along z-axis.
Hereinafter, we suppose that homogeneous external
field )(0 tE is applied along polar z-axis. The sample oc-
cupies the region ll <<− z , i.e., it is infinite in the
transverse directions. Let us consider that the electrodes
potential difference )(2 0 tEl=ϕ is independent on trans-
verse coordinates. So, the inner field satisfies the condi-
tions:
.0),(),(),(
2
1
0 == ∫∫ ⊥⊥
−
rrEr dttEdztEz
l
l
l
(6)
Boundary conditions depend on the mechanism of the
spontaneous induction screening, which associates with
the formation of oppositely charged space-charge layers
with thickness cl [15]. We can assume that the induction
distribution is symmetrical for a rather thick sample
( cll >> ) with equivalent boundaries l±=z , i.e.:
∫∫ ⊥⊥⊥⊥ −=≈=
SS
dtzD
S
dtzD
S
rrrr ),,(
1
),,(
1
ll . (7)
S is the sample cross-section. We also introduce the aver-
aging over the sample volume:
∫=+=
V
dtf
V
tftftftf rrrr ),(
1
)(),,()(),( δ . (8)
Hereinafter, the dash designates the averaging over the
sample volume V, { }...,,,,,, jDEf sm ρρρ= , 0),( =tf rδ .
All the functions δf(r, t) consist of the regular part cau-
sed by spontaneous induction screening [15] and the
random one caused by fluctuations. Since the contribu-
tion from the screening region cz ll −> to the integrals
∫
V
n dtf rr ),(δ is negligibly small for the rather thick sam-
ple cll >> , and δ f is the fast oscillating function in the
remainder of the sample cz ll−≤ , one can conclude that:
...2;1,),(~),( 22 =
ntftf
n
n rr δδ (9)
...2;1,0),(12 =≈+ ntf n rδ (10)
(5)
254
SQO, 7(3), 2004
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
Also, we suppose that the correlation between the dif-
ferent δf-functions is equal to zero, if the total power of
the functions is an odd number.
It follows from (5) and (4) that:
),,()(),( 0 tEtEtE zz rr δ+=
),,(),( tt rErE ⊥⊥ = δ (11)
i.e., E is the applied uniform field )(0 tE and 0=⊥E .
Notice that the average values E, D are determined ex-
perimentally [15, 16] in most cases. Having substituted
(6)�(8) into (5) and averaged, one can obtain the expres-
sions for the average quantities, namely:
.
4
)(
)()(
,0
,
π
ρρρ
tD
t
tt zc
s
hem
m
∂
∂
+=
−=⇒= ∑
=
ejj
(12)
The absence of the space charge average density ρ
follows from the sample electro-neutrality and corre-
sponds to the result [15], [17]. Here )(tj is the total mac-
roscopic current. Using (5)-(7) one can obtain:
⊥⊥
=
−
+=
∂
∂ ∑ Er δεδρδρπδ divD
z s
hem
m )(4
,
, (13)
( )[ ]( )
( ) .0
4
1
,
0
=+
∂
∂
+
+−−++
⊥⊥
=
∑
Ee
EEe
δεδ
π
δρκδδρδδρρδρµ
D
t
gradE
z
hem
mmmmmzmm
(14)
Using the nonlinear equation (4) and formulae (8)-
(11), it is easy to obtain the following system of equa-
tions:
),(3 0
32 tEDDD
t
D =+
++
∂
∂Γ βδβα (15)
.
33
2
2
3
222
zE
D
D
DDDDDD
t
δ
δ
γβδ
δδβδβαδ
=
∂
∂
−+
+
−+
++
∂
∂
Γ
r
(16)
The system of equations (13)-(16) is complete, because
the quantities Eδδρ ,m can be expressed via the fluctua-
tions of induction δD and )(rsδρ allowing for (13), (14).
It determines the spatial-temporal evolution of the induc-
tion in the bulk sample and has to be supplemented by the
initial distributions of all variables.
The system (13)�(16) can be used to study the mecha-
nisms of domain wall pinning by the given distribution of
charged defects fluctuations )(rsδρ , domain nucleation
during spontaneous induction reversal in the ferroelectric
semiconductors with non-isovalent impurities. These prob-
lems for the ferroelectric ideal insulators were consid-
ered earlier in detail (see e.g. [18], [19]). Hereinafter, we
consider only the average characteristics of the system.
4. Coupled equations
In order to simplify the nonlinear system (13)�(16), the
following hypotheses have been used.
a) The sample is the improper semiconductor with
rather high n-type conductivity:
.0,0,0
,,
,
><=
−≈+−≈∑
=
s
ss
hem
m
n
nn
ρµδ
ρδρρ
(17)
Hereinafter, we neglect the proper conductivity and omit
the subscript �m�.
b) The equations (13)�(14) can be linearized with
respect to δn in the bulk of the sample, where δE ~ δn
and so:
EEE δρδδδδ ⋅<<− snn . (18)
c) The characteristic time of δn, δE, δD and E0 chang-
ing is the same order as the maxwellian time, which is
much smaller than the Landau-Khalatnikov relaxation
time:
αρπµ
Γ<<
s4
1
. (19)
d) We suppose also that the smallest period d of
the inhomogeneities distribution (see Scheme 1) is
much greater than the Debye screening radius RD =
sρµπεκ ⊥−= 4 and correlation length (the thick-
ness of the neutral domain wall) αγ=cl , namely
122 <<dRD , 122 <<dcl . (20)
Note that for the typical defects concentration ~1�10%
that provides sufficiently high improper conductivity at
the room temperature RD ~ 5 nm [15], [17], d ~ 50 nm,
cl ~ 1 nm [13], [15], i.e. inequalities (20) is valid.
After neglecting the temporal derivatives of δE and
δD (compare with [17]), linearization over δn and el-
ementary transformations, the equations (13)-(14) acquire
the form:
( )ngrad
n
tE
ss
z δ
ρµ
κ
ρ
δδ −≈ )(0eE , (21a)
s
s
D
z
nn δρδ
π
δ
ρπµ
κεδ −
∂
∂
≈∆− ⊥⊥
4
1
4
. (21b)
The gradient terms in (21b) can be neglected in accord-
ance with (20), because ( )sDR ρµπεκ ⊥−= 42 is trans-
verse Debye screening radius (see Scheme 1). Moreo-
ver, if only the concentration of free carriers is high
enough to provide the good screening of the charged
inhomogeneities δρs, the gradient of the induction fluc-
tuations is small in the bulk of a sample (see Scheme 2),
( ) ( )nzD s δδρδπ +≅∂∂41 namely:
sD
z
δρδ
π
<<
∂
∂
4
1
. (22)
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
255SQO, 7(3), 2004
Thus, the field variation δEz from (21a) can be ex-
pressed via δD and δρs (see Appendix A and [20]):
s
s
s
s
z tE
z
E
ρ
δρ
ρ
δρ
µ
κδ )(0−
∂
∂
≈ . (23)
Having substituted solution (23) into (16), we obtain
from (15)�(16) the self-consistent system of the nonlinear
integral-differential equations for D and δD . Its non-
homogeneity is proportional to charge fluctuations δρs
and external field E0.
The approximate system of first-order differential
equations for average induction D , its square fluctua-
tion 2Dδ and correlation
sDδρδ can be derived after
some elementary transformations (see Appendix A). Thus,
we obtain three coupled equations:
)(3 0
32 tEDDD
t
D =+
++
∂
∂Γ βδβα , (24a)
( )
( )
s
s D
tED
DDD
t
ρ
δδρ
δβ
δβαδ
)(
3
2
0
22
222
−=
+
+++
∂
∂Γ
,
(24b)
s
s
s
s
tED
DDD
t
ρ
δρ
δρδ
δββαδρδ
2
0
22
)(
3
−=×
×
+++
∂
∂
Γ
(24c)
The system (24) determines the temporal evolution of
the bulk sample dielectric response and have to be sup-
plemented by the initial values of D, 2Dδ and sDδρδ at
t = 0.
Coupled equations (24) have the following physical
interpretation (compare with modified approach [20]).
The macroscopic state of the bulk sample with charged
defects can be described by three parameters: D , 2Dδ
and sDδρδ . The long-range order parameter D descri-
bes the ferroelectric ordering in the system, and the dis-
order parameter 2Dδ describes disordering caused by
inner electric fields arising near charged non-homo-
geneities δρs. The correlation sDδρδ determines the cor-
relations between the movable screening cloud δn and
static charged defects δρs.
We will show that equations (24) admit the continu-
ous transformation from the perfect ferroelectric (δρs → 0
and so 2Dδ (t) → 0) to the local disordering ( 02 ≠sδρ and
so 2Dδ (t) ≠ 0) and then to the completely disordered ma-
terial ( 2Dδ (t) > |α|/3β) under the increase of fluctua-
tions δρs.
As a resume to this section, we would like to stress
that the derived non-Hamiltonian system of coupled equa-
tions (24) is similar to other well-known nonlinear sys-
tems of first order differential equations (e.g., the Lorenz
system). Such dynamical systems possess chaotic regions,
strange attractors as well as strongly non-ergodic behav-
iour and continuous relaxation time spectrum [12]. Any
new system of such type demands a separate detailed
mathematical study that was not the aim of this paper.
Hereinafter, we discuss only the system behaviour in the
vicinity of the equilibrium states far from the possible
chaotic regions.
5. Quasi-stationary states
Let us consider the stationary solution of (24), which cor-
responds to changing the quasi-static external field. It is
easy to check that the system (24) is not the Hamiltonian
one, i.e., it can not be directly obtained by varying of
some free energy functional. But in order to study the
stability of the stationary points under the external field
changing, we try to obtain the reduced free energy.
Let us exclude one of the order parameters from the
system (24). Indeed we can easy obtain from (24c) that
++
⋅
−=
22
2
0
)(3 DtD
E
D
s
s
s
δββαρ
δρ
δρδ . (25a)
Substituting (25) into (24b) and extracting square root
one obtains
22
0
222 3 ssEDDD ρδρδββαδ ±=
++⋅ . (25b)
In the stationary case (24a), it acquires the form:
0
323 EDDD =+
+ βδβα . (25c)
It is easy to see that system of equations (25b), (25c) can
be obtained from the variational principle. Really, the
equations (25b), (25c) can be integrated over averaged
induction fluctuations 2Dδ and induction D , and so
the reduced free energy functional determining the sta-
bility of the stationary points has the following form:
a b
d
s
s
div D /4 = ( ) d p d dr n +
E = 0 E 0≠ 0 0
dr
� nd
Scheme 2. The screening of the charged defects δρs by free
charges δn.
256
SQO, 7(3), 2004
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
.
2
3
42
,
2
22
0
22
2
24222
⋅
+−⋅+
+
++
+=
s
s D
DEDD
DDDDDDG
ρ
δρδ
δ
β
δ
β
δ
α
δ
(26a)
Hereinafter, we suppose that the root 2Dδ can be both
negative and positive depending on the external condi-
tions. It is easy to verify that the equations of state
0, 2 =
∂
∂
DDG
D
δ , 0, 2
2
=
∂
∂
DDG
D
δ
δ
(26b)
coincide with the equations (25c) and (25b). Let us re-
write energy functional (26a) in dimensionless variables.
( ) ( ) ( )
( ).
2
3
4
1
2
1
,
22
4422
DmmDm
DmDmDmm
RDED
DDDG
∆⋅+−∆⋅+
+∆++∆+−=∆
(27)
Here α < 0, Sm DDD /= , βα−=SD , Em = E0/(�αDS),
SD DD /2δ=∆ , 22
ssR ρδρ= .
The contour plots of negative values of the free en-
ergy (27) at different Em amplitudes and small R value
are depicted in Figs 1, 2.
The free energy minimums (crosses and stars) prove
the existence of two sorts of stable states: the first one
with non-zero averaged induction values 0≠D and
2Dδ ≈ 0 (ordered state), and the second one with its zero
value 0≈D and 2Dδ ≠ 0 (disordered state). It is obvious
that at zero external electric field all these four states
have the same energy, and therefore the system can be in
any of these states (see Fig. 1a, c). However, when one
applies the external field E0 above the definite threshold
value, the states with induction direction opposite the
field one vanish, and the system has to switch to the states
with induction directed along the external field. In con-
trast to the pure ferroelectrics, where the switching takes
place between two ordered states 1±=D , the considered
system from the ordered state 1=D (Fig. 1c) switches
first to the disordered one 0=D (Fig. 1d) and vice versa
(Fig. 1a, b). The plots in Fig. 1 correspond to the small
amplitudes of the external field Em, which can switch the
system only between two adjacent minima denoted by the
asterisk.
If the external field Em is increased, full switching
between two ordered states 1±=D through disordered
ones 0=D is possible (see Fig. 2). In such case, the field
Em can successively switch the system between four
minima (e.g., from Fig. 2a through Fig. 2b to Fig. 2c,
Fig. 2d and then to Fig. 2e, f).
d c
ba
�1
�1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
�1
�1
�1
�1
�1
�1
D
D
m
∆
Fig. 1. The contour plots of negative values of the free energy
(27) at R2 = 0.1 and different values of Em = 0, 0.3, 0, �0.3 (parts
a, b, c, d, respectively). Crosses denote the positions of minima;
asterisk is the position of the current minimum.
Fig. 2. The contour plots of negative values of the free energy (27)
at R2 = 0.1 and different values of Em = 0, 0.3, 0.6, 0, �0.3, �0.6
(parts a, b, c, d, e, f, respectively). Crosses denote the positions of
minima; asterisk is the position of the current minimum.
de
ba
f c
�1
�1
�1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
�1
�1
�1
�1
�1
�1
�1
�1
�1
D
m
D∆
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
257SQO, 7(3), 2004
The contour plots of the free energy (27) at fixed Em
and different R values are depicted in Fig. 3. Again, free
energy minima testify in favour of existence of the stable
ordered and disordered states. The disordered state
2Dδ ≠ 0 is absent in pure ferroelectrics.
The cross-sections of the free energy (27) at 2Dδ = 0
(completely ordered states) and 0=D (completely disor-
dered states) at fixed Em and different R values are de-
picted in Fig. 4. As it should be expected, the ordered
state is more energetically preferable than the disordered
one at 12 <<R even in the infinitely small external field
(see Figs 3a, b and 4a, b). One can see that the probabili-
ties of the system existence in ordered and disordered
states become very close at 5.02 >R (see Fig. 4c). Moreo-
ver, at 1→R ordered and disordered states become ener-
getically indistinguishable (see Fig. 4d).
Our calculations proved that at R > 1 disordered state
with 0=D , 2Dδ ≠ 0 becomes the most energetically pref-
erable even in the infinitely small external field. Actu-
ally, this means that the phase transition into the disor-
dered state takes place at the critical value of fluctua-
tions 2
sδρ = 2
sρ , i.e. the ferroelectrics sample with charged
defects splits into the polar regions (domains or Cross
regions [21]) with different induction orientations.
6. Quasi-equilibrium dielectric hysteresis
In this section, we demonstrate how the dielectric quasi-
equilibrium hysteresis loop )( 0ED changes its shape un-
der the presence of charged defects. First of all, let us
rewrite equation (26) in dimensionless variables: .)31(
,)31(
2
1
,)31(
222
422
2
32
REKD
d
dK
KED
d
d
EDD
d
dD
mDDm
D
DmDDm
D
mmmD
m
−=∆−−−
−=∆+∆−−
∆
=+∆−−
ρ
ρ
ρ
τ
τ
τ
(28)
Here α < 0, Sm DDD /= , βα−=SD , Em = E0/(�αDS),
SD DD /2δ=∆ , 2/ SsD DDK ρδδρ = , ( )Γ−= ατ /t ,
222
ssR ρδρ= . The dependence of the dimensionless in-
duction Dm over the external field Em is represented in
Figs 5�9 for the case of harmonic modulation of the ex-
ternal field Em = EmA sin(wt). Hereinafter, we use the
dimensionless frequency ωα ⋅Γ−=w .
The quasi-equilibrium hysteresis loops were obtained
at the low frequency of external field (see Figs 5�6). It is
seen from Figure 5 that a constriction on hysteresis loops
appears for the nonzero R values, and the loop area in-
creases with the increase of R parameter. The so-called
�footprint� loops were observed in PLZT ceramics [10].
But for a given external field amplitude there is a critical
value of R parameter. For the R value above critical one
instead of �full� loops, we obtained only minor loops
(see Fig. 6) or their absence (see Fig. 5d) depending on
the ordered (Dm(τ = 0) = 1) or disordered (Dm(τ = 0) = 0)
initial state of the sample. The similar minor loops were
observed in PZT [1] and SBN: Ce single crystals [2].
On the other hand, there is a critical value of the ex-
ternal electric field amplitude for the given value of R
parameter. One can also see minor loops only for the
amplitude of electrical field smaller than the critical one
Fig. 3. The contour plots of negative values of the free energy
(27) at Em = 0.15 and different values of R2 = 0, 0.3, 0.5, 1 (parts
a, b, c, d, respectively). Crosses denote the positions of minima;
marked crosses are the positions of the absolute minima.
c d
a b
�1
�1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
�1
�1
�1
�1
�1
�1
D
m
D∆
Fig. 4. The cross-sections of the free energy (27) at 2Dδ = 0
(solid curves) and 0=D (dashed curves) for Em = 0.15 and dif-
ferent values of R2 = 0, 0.3, 0.5, 1 (parts a, b, c, d, respectively).
�1
�1
�1
�1
0
0
0
0
0
0
0
0
1
1
1
1
�0.4
�0.4
�0.2
�0.2
0.2
0.2
0.4
0.4
G
D
a b
dc
m
m
258
SQO, 7(3), 2004
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
(compare Fig. 5 with Fig. 6). Even for the field amp-
litude below the thermodynamic coercive field
( 385.0332 ≈=mCE ), there is partial switching, when
the hysteresis loop is absent in �pure� ferroelectrics (com-
pare dotted and solid curves). It should be noted that for
R = 1 one can observe only minor loops (see Fig. 6d) for
the finite value of the external electric field.
Note that two minor loops in Fig. 6a�d are obtained
with several cycles of external field and switching be-
tween two loops is caused by the fluctuations. Upper mi-
nor loops represent solutions with the positive initial val-
ues of the induction, while the lower ones correspond to
its negative initial values.
The hysteresis loops represented in Figs 7 and 8 are
obtained with the same parameters as in Fig. 5, but with
the higher frequency values. It is seen that Fig. 7 is quali-
tatively similar to the Fig. 5: there are constrictions on
hysteresis loops for R2 ≤ 0.5, but for R = 1 only minor
loop exists and paraelectric-like dependence D(E) (see
Fig. 5d) cannot be achieved. With the frequency increase,
the constrictions are smeared and loops tend to those of
pure ferroelectrics (see dotted lines in Fig. 8). Also under
the frequency increase, minor loops arise for the smaller
values of R parameter at the same external field ampli-
tude (compare Figs 7c and 8c).
Switching between ordered and disordered states in
ferroelectrics with charged defects causes both the minor
hysteresis loops for the smaller external field and the con-
Fig. 5. The dependence of dimensionless induction on dimen-
sionless field (hysteresis loops) for frequency w = 0.001, external
field amplitude EmA = 2 and different values of R2 = 0.1, 0.3, 0.5,
1 (parts a, b, c, d, respectively). Initial conditions for the different
solid curves are 0)0(2 ==∆ τD , 0)0( ==τρDK , Dm(t = 0) = 1 (parts
a, b, c) and Dm(τ = 0) = 0 (part d). The dotted curve corresponds
to the �classic� loop in the pure ferroelectric (R = 0) at the same
frequency value.
�1 �1
�1 �1
�1
�1
�2
�2
�2
�2
�1
�1
0 0
0 0
0
0
0
0
1 1
1 1
1
1
2
2
2
2
1
1
E
D
a b
dc
m
m
Fig. 6. Hysteresis loops for the frequency w = 0.001, external
field amplitude EmA = 0.4 and different values of R2 = 0.1, 0.3,
0.5, 1 (parts a, b, c, d, respectively). Initial conditions are
0)0(2 ==∆ τD , 0)0( ==τρDK , Dm(τ = 0) = �1. The dotted curve cor-
responds to the �classic� loop in the pure ferroelectric (R = 0) at
the same frequency value.
�0.4
�0.4
�1 �1
�1 �1
�0.4
�0.4
0
0
0 0
0 0
0
0
0.4
0.4
1 1
1 1
0.4
0.4
E
D
a b
dc
m
m
striction on loops for the larger external field amplitude.
These phenomena can be easily explained on the basis of
simple pictures of the free energy map evolution for dif-
ferent values of external field (see comments to Figs 1, 2).
Therefore, our theory predicts footprint and minor po-
larization hysteresis loops in ferroelectric materials with
charged impurities and relatively high improper conduc-
tivity (see e.g., [1�3, 8, 10]). It should be noted that the
origin of the constricted or double loops in aged
ferroelectric ceramics BaTiO3 and (Pb,Ca)TiO3 without
charged impurities is caused by the mechanical clamp-
ing of spontaneous polarization switching [22], and so it
lies outside of our theoretical consideration.
7. The experimental results
Let us compare the theoretically calculated dielectric
response with our experimental results obtained for thick
PZT films on Si substrate (see Scheme 3). The studied
PZT-films with Zr/Ti ratio 54/46 are near to the morpho-
tropic phase boundary, which corresponds to best per-
formances for bulk PZT ceramics. Investigated Pt-PZT-
Pt/Ti-SiO2/Si structures with oriented PZT layer were
manufactured by RF magnetron sputtering in the system
and under conditions described previously [25]. The sput-
tering target obtained by uniaxial cold pressing includes
the mixture of PbO, TiO2 and ZrO2 in a stochiometric
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
259SQO, 7(3), 2004
composition. The structure includes the top 150 nm Pt
electrode, 1.9 µm layer of (111)-oriented PZT, bottom
Pt/Ti-bilayer (150 nm of Pt, 10 nm of Ti) deposited onto
the oxidized (350 nm of SiO2) (100) n-type Si 350 µm
substrate.
For PZT � Si-substrate structure, it is necessary to
design the bottom electrode, which possesses not only a
stable and high enough electrical conductivity but also
simultaneously prevents the interfacial reactions between
electrode, PZT and SiO2 components in PZT-film and
Si-substrate surroundings under rather high temperature.
The layer of Ti plays an important role in limiting the
diffusion of Ti in Pt/Ti intermediate bilayer through Pt-
layer into the PZT-layer and directly into SiO2-layer,
and also in correction of poor adhesion of Pt-layer. The
annealing treatment of the Pt/TiOx/SiO2/Si-substrate
structure just before of PZT deposition was performed
for substrate stabilization and post-annealing treatment
of PZT-film was performed for crystallizing the film in
the polar perovskite phase. The top Pt-electrodes have
1 mm2 area.
The ferroelectric hysteresis loops for the PZT films
were obtained using the conventional Sawyer-Tower cir-
cuit (see, e.g., [16]), and the bipolar triangular voltage U
(symmetrical saw-tooth) with the frequency about 1 KHz
was applied (see Scheme 3).
For the most of the films, the hysteresis loops had
rather �slim� than �square� shape and were strongly
asymmetric. The typical minor loop is represented in
Fig. 9. Notice that mm-thick semiconductor films are
thick enough to neglect the size-driven effects of dielec-
tric properties changing [23], and partial switching phe-
nomena are caused by other reasons. In our model, the
high fluctuations of the charged defects concentration
can lead to the partial switching of the induction. In the
considered material, the source of this charged defects
can be related to the numerical randomly distributed Pb
vacancies (so R ~ 1) appeared during the high tempera-
ture annealing due to the high PbO volatility.
One can see from Fig. 9 that proposed model quanti-
tatively well describes experimental data. Some discrep-
ancy between the theory and experiment in the vicinity of
the coercive field values can be related to the mobility of
charged defects under the external field high values, which
are treated as static in our model, as well as to the dead
layer appearance [1].
Fig. 7. Hysteresis loops for the frequency w=0.1, external field
amplitude EmA = 2 and different values of R2 = 0.1, 0.3, 0.5, 1
(parts a, b, c, d, respectively). Initial conditions are )0(2 ==∆ τD 1,
0)0( ==τρDK , Dm(τ = 0) = 0. The dotted curves corresponds to
the �classic� loops in the pure ferroelectric (R = 0) at the same
frequency value.
Fig. 8. Hysteresis loops for the frequency w=0.5, external field
amplitude EmA=2 and different values of R2=0.1, 0.3, 0.5, 1 (parts
a, b, c, d, respectively). Initial conditions are )0(2 ==∆ τD 1,
0)0( ==τρDK , Dm(τ = 0)=0. The dotted curves correspond to the
�classic� loop in the pure ferroelectric (R = 0) at the same fre-
quency value.
�1
�1 �1
�1
�1
�1
�1
�1
�2
�2
�2
�2
0
0 0
0
0
0
0
0
1
1 1
1
1
1
1
1
2
2
2
2
E
D
a b
dc
m
m
�1
�1 �1
�1
�1
�1
�1
�1
�2
�2
�2
�2
0
0 0
0
0
0
0
0
1
1 1
1
1
1
1
1
2
2
2
2
E
D
a b
dc
m
m
Scheme 3. The PZT film on Si substrate, U is the applied voltage.
Pt
PZT
Voltage U
U
0
T = 2 /p w
Dead layer
Pt/Ti
SiO2
260
SQO, 7(3), 2004
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
The dielectric permittivity hysteresis calculated by
means of the data in Fig. 9 is shown in Fig. 10. The di-
electric permittivity dUDd⋅= lε was calculated numeri-
cally by the simple central difference without smoothing.
It is seen from the figure that slight variation of the ex-
perimental data in Fig. 9 led to the anomalous oscilla-
tions of permittivity, which have no physical meaning.
The capacity of a 1.9 µm-thick ferroelectric PZT film
for triangular applied voltage has been measured inde-
pendently at the lower frequencies and higher applied
voltage amplitudes. The typical capacity hysteresis is
depicted in Fig. 11. It is seen that the dead layer influ-
ence increases under external voltage increasing. The
dead layer has been taken into account as the effective
series capacity Cd (see Scheme 3), thus the resulting meas-
ured capacity C = CF ⋅Cd/(CF + Cd) differs from the film
capacity CF.
We can conclude that coupled equations (24) describe
the polarization switching and ferroelectric disordering
caused by charged defects in the thick ferroelectric PZT
films with Pb vacancies.
8. Dicussion
We have proposed the phenomenological description of
polarization switching peculiarities in some ferroelectric
semiconductor materials with charged defects. The com-
parison with our experimental results obtained for thick
PZT films has been performed.
It is shown that the impurity concentration fluctua-
tions δρs(r) result in to the ferroelectric disordering of the
considered system. The quantitative degree of this disor-
der is the parameter 2Dδ characterizing the inhomo-
geneity of the induction distribution. The mean induc-
tion D is the order parameter. For the first time, the sys-
tem of coupled equations (24) that determines the evolu-
tion of these parameters has been derived.
Solving the system of coupled equations one can get
the information about system ordering as a whole, with-
out defining concrete space distribution of the appeared
inhomogeneities, domain walls characteristics, correla-
tion radius of Cross regions or sizes of originated
microdomains. In order to obtain this kind of informa-
tion, one has to solve the system of equations (13)-(16)
with the specified distribution of impurity concentration
fluctuations δρs(r), but the consideration of this problem
was not the purpose of the present paper.
We would like to underline that in contrast to the sys-
tem (13)�(16) the averaged system of the coupled equa-
tions (24) does not contain any information about induc-
tion gradient across the sample. This happened rather
due to the local compensation of the strong inhomogene-
ous electric field in the vicinity of charged defects by the
movable charge carriers, then due to enough sample thick-
ness in order to neglect the size effects and the depolari-
zation field influence [23], [24]. We also suppose that
inhomogeneous mechanical stresses arisen near defects
are rather small or compensated by the sample treatment.
Fig. 9. Minor loop observed in a thick ferroelectric PZT film
(l = 1.9 µm) for the triangular applied voltage. Triangles are
experimental data measured at U0 = 20V, solid curve is our fit-
ting for the material with charged defects (R = 0.95) at w = 0.35,
DS = 40 µC/cm2, UC = 4V. The dotted curve corresponds to the
loop in pure ferroelectric (R = 0) at the same other parameters.
�60
�40
�20
�20 �10
20
2010
40
60
80
0
0
Voltage , V U
Fitting for the
model with
defects
Fitting
for pure
material
D
is
p
la
c
em
e
n
t,
C
/c
m
µ
2
Experimental
data for thick
PZT film
Fig. 10. The dielectric permittivity hysteresis of 1.9 µm-thick
ferroelectric PZT film for triangular applied voltage. Triangles
are experimental data calculated from the loop in Fig. 9, solid
curve is our fitting for the material with charged defects (R = 0.95)
at U0 = 20 V, w = 0.35, DS = 40 µC/cm2, UC = 4V. The dotted
curve corresponds to the hysteresis in pure ferroelectric (R = 0)
at the same other parameters.
�20 �10 20100
0
0
1
1
22
3
3
Voltage , V U
Fitting
for pure
material
Experimental
data for thick
PZT film
P
er
m
it
ti
v
it
y
,
1
0
�
3
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
261SQO, 7(3), 2004
In accord to our theory, random inhomogeneities in
the defect distribution throughout the sample lead to the
stabilization of the disordered state ( DD >>2δ ). For
sufficiently small external field amplitudes, this state re-
veals itself as switching from the ordered state to the dis-
ordered one (the so-called minor loop, see e.g. [1]). When
the external field increases this minor loop transforms
into the loop with constriction or footprint-type hyster-
esis loop [9]. In this case, the constriction corresponds to
the switching from the ordered state to disordered one
and then again to the ordered state with the opposite di-
rection of the induction. Ferroelectric hysteresis loops
with the constrictions or footprint loops are observable
in some ferroelectric materials. For example, footprint
loops exist in the plumbum zirconate-titanate ceramics
doped with La [10], namely in at õ = 0.35, y = 0.08, 0.084
and õ = 0.3, y = 0.076, 0.079, which is regarded as relaxor
material. In this material, La ions have excess charge
and can be regarded as charged defects. Notice that our
theory predicts transformation from footprint to the mi-
nor loop with the external field frequency increase.
The transformation from full loop to the minor one
was observed in SBN single crystals doped with cerium
under applied field frequency increasing (see Fig. 6 in
[2]). It is clear that our theory [20] describes qualitatively
minor loops observed in SBN:Ce [2], Pb(Zr,Ti)O3-
Pb(Sb,Mn)O3 ceramics [3], PZT thin films [8], thick PZT
films with Pb vacancies [25] and TGS [5], but not the
aging process seen as the loop degradation. This may be
related to the fact that neither finite domain wall thick-
ness cl , nor possible evolution of the charge fluctuations
δρs caused by the relaxation/origin of internal stresses
around defects was taken into account in our model. These
problems as well as the calculation of the system dielec-
tric response are in progress now.
We can conclude that coupled equations (24) qualita-
tively describe the polarization switching and
ferroelectric disordering caused by charged defects in
bulk ferroelectric-semiconductors.
Acknowledgments
The authors are greatly indebted to Profs. S.L. Bravina
and N.V. Morozovsky for frutfull discussions of the model
and useful remarks to the manuscript.
Appendix A
Let us express the field variation δEzcan be via δD and
δρs. In accordance with (21b) and (20), one obtains that
( ) szDn δρδπδ −∂∂≈ 41 . Having substituted this expres-
sion into (21a), one obtains (23) using the inequality (22):
.)(
4
1
)(
0
0
s
s
s
s
s
s
s
z
tE
z
D
zz
tEE
ρ
δρ
ρ
δρ
µ
κ
ρ
δρ
δ
ρπµ
κδ
−
∂
∂
≈
≈
−
∂
∂
∂
∂
−≈
(A.1)
The equations for 2Dδ and sD ρδδ obtained directly
from (16) have the form:
( )
zED
D
DD
DtDD
t
δδδδγδβ
δβαδ
+
∂
∂=+
+++
∂
∂Γ
2
2
4
222 )(3
2
r
,
(A.2)
( )
zsss
ss
E
D
D
DtDD
t
δρδδρδγρδδβ
ρδδβαδρδ
+
∂
∂=+
+++
∂
∂Γ
2
2
3
2 )(3
r
.
(A.3)
One can derive from (A.1) the following approxima-
tions for the correlations:
( )
( )
.)(
)(
0
0
s
s
s
s
s
s
z
D
tE
z
D
D
tEED
ρ
δδρ
ρ
δρ
δ
µ
κ
ρ
δδρ
δδ
−≈
≈
∂
∂
+−=
(A.4)
Fig. 11. The capacity hysteresis of a 1.9 µm-thick ferroelectric
PZT film for triangular applied voltage. Squares are experimental
data measured at U0 = 36V, solid curve is our fitting for the ma-
terial with charged defects (R = 0.95) at w = 0.1, DS = 40 µC/cm2,
UC = 4V. The dotted curve is our fitting for the material with
charged defects and dead layer with capacity Cd = 0.16nF at
the same other parameters.
�20�30 �10 20 30100
4
6
8
10
12
14
16
18
Voltage , V U
Experimental
data for thick
PZT film
C
a
p
a
ci
ty
,
n
F
C
262
SQO, 7(3), 2004
A.N. Morozovska et al.: Partial polarization switching in ferroelectrics-semiconductors ...
In (A.4) the term ( )D
dz
D s
ss
s δρδ
ρµ
κ
ρ
ρδδ
µ
κ
~ =
∂
∂
( )D
d
R
s
D δρδπε 24 ⊥= can be neglected under the assump-
tion that the screening of defects is rather strong to sat-
isfy the inequality ( )sD dEdR ρπε⊥<< 40
22 (see (20)).
For a thick sample with equivalent boundaries l±=z we
obtain from (A.1) that
s
s
s
s
s
s
zs tE
z
tEE
ρ
ρδ
ρ
ρδ
µ
κ
ρ
ρδ
δρδ
2
0
22
0 )(
2
)( −≡
∂
∂
+−= .
(A.5)
Taking into account (20) one obtains that
,
,
22
2
2
2
2
2
2
DD
d
D
d
DD
D
δαδ
γ
δγδγδδγ
<<
−≅
∂
∂
−=
∂
∂
rr (A.6a)
,
,~
2
22
2
DD
d
D
d
D
ss
ss
δρδαδρδ
γ
δδρ
γδ
δργ
<<
−
∂
∂
r , (A.6b)
and so gradient terms in (A.2)-(A.3) can be either ne-
glected at αγ <<2d or the coefficient a can be
renormalized as ( )2dR γααα +=→ . One obtains also
from (9)-(10) that
2
24
≈ DD δδ , ss DDD δρδδδρδ 23 ≈ . (A.7)
Using (A.3)�(A.6) we obtain the equations (26b) and
(26c) from the equations (A.1) and (A.2) if only αγ <<2d
(see (20)).
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