Phase diagram and domain splitting in thin ferroelectric films with incommensurate phase
ABSTRACT We studied the phase diagram of thin ferroelectric films with incommensurate phases and semiconductor properties within the framework of Landau-Ginzburg-Devonshire theory. We performed both analytical calculations and phase-field modeling of the temperature and thickness dependencies of the period of incommensurate 180°-domain structures appeared in thin films covered with perfect electrodes. It is found that the transition temperature from the paraelectric into the incommensurate phase as well as the period of incommensurate domain structure strongly depend on the film thickness, depolarization field contribution, surface and gradient energy. The results may provide insight on the temperature dependence of domain structures in nanosized ferroics with inherent incommensurate phases.
- A Y Borisevich, E A Eliseev, A N Morozovska, C-J Cheng, J-Y Lin, Y H Chu, D Kan, I Takeuchi, V Nagarajan, S V Kalinin[Show abstract] [Hide abstract]
ABSTRACT: Physical and structural origins of morphotropic phase boundaries (MPBs) in ferroics remain elusive despite decades of study. The leading competing theories employ either low-symmetry bridging phases or adaptive phases with nanoscale textures to describe different subsets of the macroscopic data, while the decisive atomic-scale information has so far been missing. Here we report direct atomically resolved mapping of polarization and structure order parameter fields in a Sm-doped BiFeO(3) system and their evolution as the system approaches a MPB. We further show that both the experimental phase diagram and the observed phase evolution can be explained by taking into account the flexoelectric interaction, which renders the effective domain wall energy negative, thus stabilizing modulated phases in the vicinity of the MPB. Our study highlights the importance of local order-parameter mapping at the atomic scale and establishes a hitherto unobserved physical origin of spatially modulated phases existing in the vicinity of the MPB.Nature Communications 01/2012; 3:775. · 10.02 Impact Factor - SourceAvailable from: Eugene A. Eliseev[Show abstract] [Hide abstract]
ABSTRACT: The role of elastic defects on the kinetics of 180-degree uncharged ferroelectric domain wall motion is explored using continuum time-dependent LGD equation with elastic dipole coupling. In one dimensional case, ripples, steps and oscillations of the domain wall velocity appear due to the wall-defect interactions. While the defects do not affect the limiting-wall velocity vs. field dependence, they result in the minimal threshold field required to activate the wall motions. The analytical expressions for the threshold field are derived and the latter is shown to be much smaller than the thermodynamic coercive field. The threshold field is linearly proportional to the concentration of defects and non-monotonically depends on the average distance between them. The obtained results provide the insight into the mesoscopic mechanism of the domain wall pinning by elastic defects in ferroelectrics.Journal of Applied Physics 01/2012; 113(18). · 2.21 Impact Factor - P. Ondrejkovic, M. Kempa, Y. Vysochanskii, P. Saint-Gregoire, P. Bourges, K. Z. Rushchanskii, J. Hlinka[Show abstract] [Hide abstract]
ABSTRACT: Ferroelectric phase transition in the semiconductor Sn2P2S6 single crystal has been studied by means of neutron scattering in the pressure-temperature range adjacent to the anticipated tricritical Lifshitz point (p=0.18GPa, T=296K). The observations reveal a direct ferroelectric-paraelectric phase transition in the whole investigated pressure range (0.18 - 0.6GPa). These results are in a clear disagreement with phase diagrams assumed in numerous earlier works, according to which a hypothetical intermediate incommensurate phase extends over several or even tens of degrees in the 0.5GPa pressure range. Temperature dependence of the anisotropic quasielastic diffuse scattering suggests that polarization fluctuations present above TC are strongly reduced in the ordered phase. Still, the temperature dependence of the (200) Bragg reflection intensity at p=0.18GPa can be remarkably well modeled assuming the order-parameter amplitude growth according to the power law with logarithmic corrections predicted for a uniaxial ferroelectric transition at the tricritical Lifshitz point.Physical Review B 12/2012; · 3.77 Impact Factor
Page 1
Phase diagram and domain splitting in thin ferroelectric films with incommensurate phase
A. N. Morozovska,1,*,†E. A. Eliseev,1,2JianJun Wang,3G. S. Svechnikov,1Yu. M. Vysochanskii,4Venkatraman Gopalan,3
and Long-Qing Chen3,*,‡
1V. Lashkarev Institute of Semiconductor Physics, NAS of Ukraine, prospect Nauki 41, 03028 Kiev, Ukraine
2Institute for Problems of Materials Science, NAS of Ukraine, Krjijanovskogo 3, 03142 Kiev, Ukraine
3Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA
4Institute for Solid State Physics and Chemistry, Uzhgorod University, 88000 Uzhgorod, Ukraine
?Received 23 December 2009; revised manuscript received 25 April 2010; published 27 May 2010?
We studied the phase diagram of thin ferroelectric films with incommensurate phases and semiconductor
properties within the framework of Landau-Ginzburg-Devonshire theory. We performed both analytical calcu-
lations and phase-field modeling of the temperature and thickness dependencies of the period of incommen-
surate 180°-domain structures appeared in thin films covered with perfect electrodes. It is found that the
transition temperature from the paraelectric into the incommensurate phase as well as the period of incom-
mensurate domain structure strongly depend on the film thickness, depolarization field contribution, surface
and gradient energy. The results may provide insight on the temperature dependence of domain structures in
nanosized ferroics with inherent incommensurate phases.
DOI: 10.1103/PhysRevB.81.195437PACS number?s?: 77.80.Dj, 64.70.Rh
I. INTRODUCTION
The incommensurate phase in bulk materials is the spa-
tially modulated state with period incommensurate with the
lattice constant.1The spontaneous modulation appears when
the homogeneous state is either unstable or less energetically
preferable ?metastable?. On the other hand, the initial ho-
mogenous states could become modulated in the spatially
confined systems. Typical examples are domain structures of
either ferroelectric or ferromagnetic films due to the depolar-
ization or demagnetization fields, respectively.
The influence of surfaces and interfaces on ferroic mate-
rials polar properties and their domain structure have been
attracting much attention since early seventies till the
present.2–8Laminar domain structure formation in thick films
with free surfaces was considered in the classic papers by
Kittel9for ferromagnetic media and Mitsui and Furuichi10for
ferroelectric media. The structure of a single boundary be-
tween two domains in the bulk ferroelectrics was considered
by Cao and Cross11and Zhirnov,12allowing for electrostric-
tion contribution. Formation and stability of ferroelastic do-
main structures were considered by different groups.13–15
The development of nonvolatile ferroelectric memory
technology has rekindled the interest in ferroelectric proper-
ties and polarization reversal mechanisms in ultrathin
films.16–19One of the key parameters controlling ferroic be-
havior is the structure and energetic of domain walls.
The wall behavior at surfaces and interfaces will deter-
mine polarization switching and pinning mechanisms. Under
the absence of external fields in the bulk, the 180°-domain
wall is not associated with any depolarization effects. How-
ever, the symmetry breaking on the wall-surface or wall-
interface junction can give rise to a variety of unusual effects
due to the depolarization fields across the wall, as deter-
mined by screening mechanisms and strain boundary
conditions.20For instance, density-functional theory results
predicted the stabilization of vortex structure in ferroelectric
nanodots under the transverse inhomogeneous static electric
field.21,22This prediction has resulted in extensive experi-
mental efforts to discover toroidal polarization states in
ferroelectrics.23,24
However, despite numerous studies, size and surface ef-
fect on domain walls behavior in ferroics is still not clear.
Much remained to be done to clarify the peculiarities of the
order parameter redistribution in the wall vicinity and corre-
sponding wall energy in confined systems like thin films and
nanoparticles. For instance, simple analytical models typi-
cally face with “Kittel paradox:” 180-degree “rigid” domain
structure with ultra-sharp walls produces extra-high depolar-
ization fields near unscreened surface.25Possible formation
of closure domains in rigid ferroelectrics with infinitely thin
domain walls does not solve the problem. Relevant analytical
treatment of multiaxial polarization switching allowing for
domain walls intrinsic widths is still underway due to the
numerous obstacles. At the same time both first principle
calculations and phenomenological modeling revealed un-
usual domain structures in different ferroelectrics,26,27resem-
bling domain structures typical for ferromagnetics.
The axial next-nearest-neighbor Ising ?ANNNI? model
was successfully applied to study of the size effect on the
properties of magnetic films by W. Selke et al.28,29The se-
quence of the ordered phases for different film thickness and
the couplings in the surface layers obtained from mean-field
theory, Monte Carlo simulations, and low-temperature ex-
pansions was analyzed.
However the applicability of ANNNI model to the thin
ferroelectric films with out-of-plane polarization is question-
able, since the effect of strong depolarization field ?dipole-
dipole interactions? not considered within the model, while it
isofgreatimportancefor
geometry.30On the other hand continuum Landau-Ginzburg-
Devonshire ?LGD? models naturally involve depolarization
field and allow analytical calculations for spatially confined
ferroelectrics. Despite the remark, results obtained within the
ANNI model ?as well as in others discrete and atomistic
models? should call for care when using continuum theory in
the thin films.
out-of-planepolarization
PHYSICAL REVIEW B 81, 195437 ?2010?
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©2010 The American Physical Society195437-1
Page 2
Since the exchange forces are prevailing ones for the fer-
romagnetic systems, the magnetic dipole-dipole interaction is
often omitted under the theoretical consideration of these
systems. However dipole—dipole interaction strongly affects
the anisotropy energy of magnetic nanoparticles ?see, e.g.,
Ref. 31?. Unfortunately in the discrete model framework the
efforts to compute the dipole-dipole interaction energy dras-
tically increase with the increase of spins quantity ?see, e.g.,
Ref. 32?. In this case the continuum limit model could
present the simple picture of the system behavior without the
necessity to make huge efforts and involve complicated cal-
culation schemes.
Thus, the elaboration of the continuum LGD model seems
especially important for considered case of the thin ferroelec-
tric films with incommensurate phase, since the consistent
calculations with all necessary details included was virtually
absent. Possibly it is due to the fact the situation with the
theoretical description of the incommensurate ferroelectrics
is much more complex in comparison with the commensu-
rate ones. In particular within Landau phenomenological ap-
proach of the II-type incommensurate materials, the charac-
teristics of modulated phase should be found from forth
order nonlinear Euler-Lagrange differential equations ?see,
e.g., Refs. 33–35?; for commensurate ferroelectrics the equa-
tions are of the second order. Owing to the problem com-
plexity only few papers were published within oversimplified
LGD-model. Namely, Charnaya et al.36obtained the order
parameter distribution over the film thickness using the as-
sumption of slowly-varying amplitude and considered the
effect of size on the temperature of the phase transition into
the incommensurate phase. The direction of incommensurate
phase modulation was normal to the film plane, but the de-
polarization field was not considered, while the depolariza-
tion field becomes inevitable present in the system.
The most intriguing feature is the mechanism of
commensurate-incommensurate phase transitions. The transi-
tion in three-dimensional solids was considered as lock-in
transition from the incommensurate phase with negative en-
ergy of domain walls into the commensurate phase with
positive energy of domain walls.37Levanyuk et al. pointed
out that electrostriction coupling between polarization and
strains significantly changes the phase equilibrium.
The link between the phenomenological model of incom-
mensurate crystals and quasimicroscopic discrete lattice
model was established in Ref. 38. The temperature depen-
dence of the polarization wave number in ferroelectric
Sn2P2Se6as well as the anomalous heat capacity in the in-
commensurate phase were explained in the framework of
II-type phenomenological theory using the nonharmonic dis-
tribution of the order
calculations39may pour light of the local structure of the
incommensurate ferroelectrics, however their realization for
confined systems like thin films are almost not evolved to
date.
Possible pitfall of LGD theory application for the spatially
confined incommensurate ferroelectrics is the applicability of
the continuum media theory to nanosized systems. For nano-
sized ferroics the applicability of LGD theory is corroborated
by the fact that the critical sizes of the long-range order
appearance calculated from discrete atomistic40–43and
parameter.35
First-principle
phenomenological5,44,45theories are in a good agreement
with each other as well as with experimental results for
ferromagnetic46and ferroelectric16,47–50systems.
The paper is devoted to the size effects in thin ferroelec-
tric films with incommensurate phase and organized as fol-
lows. Section II is the problem statement. Here we listed
expressions for the depolarization field and the free energy
functional of ferroelectric thin films with II-type incommen-
surate phase and semiconductor properties. Approximate
analytical solution of the Euler-Lagrange equations is pre-
sented in Sec. III. Results of the analytical calculations of the
size effect on phase equilibrium and domain structure tem-
perature evolution are discussed in Secs. IV and I. Sections
IV and II contains results obtained by phase-field modeling.
Last section is a brief summary.
II. PHENOMENOLOGICAL DESCRIPTION OF THE
FERROELECTRIC THIN FILMS WITH THE
II-TYPE INCOMMENSURATE PHASE
Let us consider a film of incommensurate ferroelectric
with semiconductor properties. The spontaneous polarization
P3is directed along the polar axis z. The sample is dielectri-
cally isotropic in transverse directions, i.e., permittivity ?11
=?22at zero external field.
Further we assume that the dependence of in-plane polar-
ization components on E1,2 can be linearized as P1,2
??0??11−1?E1,2??0is the universal dielectric constant?.
Thus thepolarizationvector
P?r?=??0??11−1?E1,?0??11−1?E2,P3?E,r?+?0??33−1?E3?.51
Maxwell’s equations for the inner electric field Ei=−??i?r?,
expressed via electrostatic potential ?i?r? and polarization
P?r?
reduces to the equation
=
?0?33
permittivity ?33is regarded much smaller than ferroelectric
contribution to temperature-dependent permittivity ?33
The boundaryconditions
?i?x,y,h?=0 used hereinafter correspond to the full screen-
ing of depolarization field outside the sample that is realized
by the ambient charges or perfect electrodes; h is the film
thickness.
In Debye approximation the Fourier representation on
transverse coordinates ?x,y? for the depolarization field E3
has the form ?see Appendix A in Ref. 52 for details?
acquirestheform:
?2?i
?z2+
?11
?33?
?2?i
?x2+
?2?i
?y2?−
?i
?33Rd
2
1
?P3
?z, Rdis the Debye screening radius. A background
f.
?i?x,y,0?=?e?x,y,0?,
d
E˜3
d?P˜3?k,z?? =?−P˜3?k,z?
?0?33
+?
+?
0
z
dz?P˜3?k,z??cosh?Kz??cosh?K?h − z??
?0?33· sinh?Kh?K
z
h
dz?P˜3?k,z??cosh?K?h
?0?33· sinh?Kh??.
− z???
cosh?Kz?K?k?
?1?
Here vector k=?k1,k2?, its absolute value k=?k1
2+k2
2, func-
MOROZOVSKA et al.
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-2
Page 3
tion K?k?=???11k2+Rd
polarization P3?z? independent on x,y coordinates and Rd
→? expression ?1? reduces to the expression for depolariza-
tion field obtained by Kretschmer and Binder.2
Figure 1 schematically illustrates the origin of depolariza-
tion fields in thin films with 180°-domain structure and in-
homogeneous polarization component P3?x,z?. Depolariza-
tion field Ei
polarization charges by the surrounding, inhomogeneous po-
larization distribution and/or its breaks at interfaces. Math-
ematically it arises once div P?0.
Correct phenomenological description of a nanosized sys-
tem requires the consideration of its surface energy FS. In-
cluding the surface energy term, LGD free-energy F=FS
+FVdepends on the chosen order parameter—spontaneous
polarization component P3.
Within the LGD theory for the II-type incommensurate
materials the spatial distribution of the spontaneous polariza-
tion component P3inside the film of thickness h could be
found by the minimization of the free-energy functional ?see
e.g., Refs. 45 and 53?
F ??
0
−?
2??P3
?xi
+?S
−?
−2?/?33. For the dielectric media with
dis caused by imperfect screening of the bound
h
dz?
?
dxdy???
?xi?
2?
2P3
2??2P3
2+??
4P3
4+?
6P3
2??P3
6− P3?E3
?xi?
d
2
+ E3
e?
+gi
2
+wi
2?
2
+vi
2· P3
2?
?
dxdy?P3
2?z=0+ P3
2?z=h?.
?2?
Coefficient ??T?=?T?T−TC? explicitly depends on tempera-
ture T, TCis the Curie temperature of a bulk material.
For thin films coefficients ???T? and ?? should be renor-
malized due to the electrostriction coupling as shown by
Cao and Cross12and Zhirnov11and misfit strain originated
from the film and substrate lattice mismatch,54,55as ??
=??T?−
s11+s12
−2
s11
=?+2
s11
?Q11+Q12?um
?
?Q11
2+Q12
2?s11−2s12Q11Q12
2−s12
, where sijis the elastic compliances
2
P¯0
2
and
??
?Q11
2+Q12
2?s11−2s12Q11Q12
2−s12
2
tensor at constant polarization; Qijis the electrostriction ten-
sor, ? and ? are stress-free expansion coefficients, the aver-
age spontaneous polarization is P¯0,56um
strain, at that um
L?hd?hdis the critical thickness of dislocations appearance?
in accordance with the model proposed by Speck and
Pompe.57The epitaxial strain um=?a/c?−1 originated from
the thin film ?a? and substrate ?c? lattice constants mismatch.
The gradient coefficient giand seeding coefficient ? could
be negative, other higher coefficients are positive. Last inte-
gral term in Eq. ?2? is the surface contribution to the system
free energy. Expansion coefficients of the polarization-
dependent surface energy may be different for different sur-
face regions. Below we approximate the coordinate depen-
dence by effective value ?Sand neglect higher order terms in
the surface energy. The depolarization field E3
Eq. ?1?.
The necessary condition of the incommensurate phase ap-
pearance, gi?0, should be satisfied at least in one spatial
direction ?i.e., for one value of i?, while for other directions
the homogeneous state would be stable if gj?0. Since most
of ferroelectrics with incommensurate phase are uniaxial or
biaxial ones, the coefficients giare not necessarily equal. In
order to simplify our consideration and obtain close-form
analytical results, we suppose that g3?0 and either g1=g2
?0 ?symmetric biaxial x,y-incommensurate case? or g1?0
& g2?0 ?uniaxial x-incommensurate case?. In this case one
could neglect the higher order derivatives on z and put w3
=0 and v3=0 in Eq. ?2?. Coefficients w1and v1should be
nonzero positive values for the x-incommensurate modula-
tion existence; or coefficients w1=w2and v1=v2should be
nonzero positive values for the x,y-incommensurate modu-
lation existence. This simplified model allows analytical con-
sideration of the influence of size effects, depolarization field
and surface energy on the incommensurate phase features.
So, under the conditions g1=g2?0, w1=w2and v1=v2,
g3?0 and w3=0 and v3=0, minimization of the free-energy
?2? results into the relaxation equation for polarization distri-
bution
?is the effective misfit
??L??umhd/L,
??L??um, L?hd and um
dis given by
??P3
?t
+ ???T?P3+ ??P3
3+ ?P3
5− g3
?y2?+ w1??4P3
+??P3
?2P3
?z2
− ?g1+ v1P3
− v1P3???P3
2???2P3
?x?
?x2+?2P3
?y?
?x4+?4P3
?y4?
22?= E3
d?x,y,z? + E0
eexp?i?t?,
?3?
where ? is a positive relaxation coefficient, ? is the fre-
quency of external field E0
?P3/?y=0 and consider P3?x,z? in Eq. ?3?. The important
feature of the Eq. ?3? is depolarization field E3
cally depends on P3?x,y,z? as given by the linear integral
operator in Eq. ?1?.
The boundary conditions for polarization acquire the form
e. For 1D case one should put
dthat nonlo-
x
z
P3
P3
boundcharge
free screening charge
z=h
Bottom electrode
Top electrode
free screening charge
FIG. 1. ?Color online?. 180°-domain structure of thin film cov-
ered with perfect electrodes. Break of double electric layers at the
film-surface junction ?marked by a circle? and polarization inhomo-
geneity ?arrows of different length? cause depolarization field.
PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN…
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-3
Page 4
???SP3− g3
?P3
?x3??
x3=0
= 0, ???SP3+ g3
?P3
?x3??
x3=h
= 0.
?4?
Similarly to the case of commensurate ferroelectrics one
could introduce extrapolation length ?=g3/?Sthat is usually
positive. Infinite extrapolation length corresponds to an ideal
surface ??S→0? and so-called natural boundary conditions,
while zero extrapolation length ??S→?? corresponds to
P3?x3=0?=0 at a strongly damaged surface without long-
rangeorder.Reported
?=2–50 nm.58,59
experimental valuesare
III. APPROXIMATE ANALYTICAL SOLUTION OF THE
EULER-LAGRANGE EQUATIONS
Then one could find the solution of Eq. ?3? linearized for
the small polarization modulation p?k,z? in the form of se-
ries on the eigen functions fn?k,z?
n?An?k?fn?k,z?exp?− ?n?k?t
p?k,z,t? =?
??
+ En?k,??fn?k,z?exp?i?t?
?n?k? + i???.
k=?kx,ky?
or
?5?
Hereinafter
x,y-incommensurate
x-incommensurate modulation.
The first term in Eq. ?5? is related to the relaxation of
initial conditions while the second one is the series expan-
sion external stimulus E˜0
eigen functions fn?k,z? and eigen values ?n?k? should be
found from the nontrivial solutions of the following problem:
k=?k?
and
for the
the
modulation
k=?kx,0?
for
evia the eigen functions fn?k,z?. The
???− g3
= ?n?k?fn?k,z?,
???Sfn− g3
?d2
dz2+ g1
?k2+ w1k4?fn?k,z? − E3
d?fn?k,z??
?6a?
?fn
?z??
z=0
= 0, ???Sfn+ g3
?fn
?z??
z=h
= 0,
?6b?
where ??=??+3?P¯0
erage polarization ?for a bulk single domain sample the spon-
taneous polarization P0
tion field E3
involved into the problem ?6? determines the solution form.
Note, that the linear harmonic approximation is valid in
the paraelectric phase as well as in it’s the immediate vicin-
ity, where ferroelectric nonlinearity can be neglected.
The solution of Eq. ?6? was derived as ?see Appendix A in
Ref. 52 for details?
fn?k,z? ? cosh?qn1?z
qn1sinh?qn1/2?
2+5?P¯0
4, g1
?=g1+v1P¯0
2and P¯0is the av-
2=???2−4??−??/2??. Depolariza-
d?fn?k,z?? given by integral operator ?1? and
h−1
qn2sinh?qn2/2?cosh?qn2?z
2??
−qn2
qn1
2− h2K2
2− h2K2
h−1
2??.
?7?
Here qn1,2are expressed via the eigen value ?nas the solu-
tions of the biquadratic equation
??+ g1
?k2+ w1k4− g3
qn
h2+
2
qn
2− h2K2?= ?n?k?. ?8?
2
?0?33?qn
The equation for the eigen spectrum ?n?k? is
qn1sinh?qn1/2?
qn1
2− h2K2
=qn2sinh?qn2/2?
qn2
2− h2K2 ?cosh?qn1/2???S/g3? + ?qn1/h?sinh?qn1/2?
cosh?qn2/2???S/g3? + ?qn2/h?sinh?qn2/2??.
?9?
Note, that similar equations could be found for “sinh”-eigen functions. Since the smallest ?first? eigenvalue should correspond
to eigen function of constant sign, we restrict our consideration for the first symmetric “cosh”-eigen functions ?7?.
The equilibrium dependence of the transverse modulation wave vector k on the temperature T and film thickness h should
be found from Eqs. ?8? and ?9? under the conditions ?min=0.
IV. SIZE EFFECT ON THE PHASE EQUILIBRIUM AND DOMAIN STRUCTURE TEMPERATURE EVOLUTION
A. Harmonic approximation
Transcendental Eq. ?9? was essentially simplified at the domain structure onset ?see Appendix B in Ref. 52? so that the
approximate expression for the highest and lowest roots was derived in the form
k?
2?h,T? ? −
g1
2w1
?
??g1
?2
4w1
2−
1
w1????T? +
2?Sg3
??S?g3?0?33+ g3?h + ?Sg3?0h2/4Rd
2?.
?10?
MOROZOVSKA et al.
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-4
Page 5
Note,that
2?Sg3
thesize-dependent terminEq.
?10?,
??S?g3?0?33+g3?h+?Sg3?0h2/4Rd
polarization field given by Eq. ?1? and the finite surface en-
ergy ??S?0?. The finite-size effect is absent only for the
unrealistic case without surface energy, when ?S=0. Under
the typical conditions Rd?50 nm and g3?10−10J·m3/C2,
the term ?Sg3?0h2/4Rd
of Eq. ?10? without any noticeable loss of precision. For the
particular case the depolarization term is proportional to 1/h
as anticipated for ferroelectrics dielectrics.2,20,53,54The depo-
larization effect vanishes with the film thickness increase
?h→??.
The root k−?h,T? is always stable only in the incommen-
surate phase of the bulk material. The root k+?h,T? can be
?meta?stable in thin films even in the temperature range cor-
responding to the bulk commensurate phase, since domain
stripes with definite period correspond to smaller depolariza-
tion field in comparison with a single domain distribution.
The direct comparison of the corresponding free energies ?2?
should be performed in order to determine the film thickness
range, where the roots k??h,T? are stable ?and thus single
domain distribution is unstable?.
The comparison of the free energies ?2? was performed in
harmonic approximation. It was demonstrated that the root
k+?h,T? can be stable in the wide temperature range starting
from the low temperatures ?much smaller than TC? the up to
the vicinity of the transition temperature into the paraelectric
phase. This striking result can be explained in the following
way. A single domain state should be energetically preferable
in the commensurate ferroelectric defect-free film placed be-
tween perfect conducting electrodes, but only for the case of
zero surface energy ?coefficient ?S=0, ?→??. Even under
the absence of defects domain stripes originate from imper-
fect screening of depolarization field outside the film ?either
imperfect electrodes, dielectric gap? and/or the spatial con-
finement ?i.e., surface energy contribution determined by
nonzero ?S?. Zero value of ?Smeans natural boundary con-
ditions and the absence of finite size and depolarization ef-
fects, since the polarization is homogeneous along the polar
axis and the depolarization field is absent for the case. The
nonzero values ?Slead to appearance of polarization inho-
mogeneity along the polar axis, localized near the surfaces.
At the same time, inhomogeneity along the polar axis should
induce the depolarization field that affects periodically
modulated domain structures. This effect is a general feature
of all ferroics ?see e.g., Refs. 53 and 60?, it is not related with
a bulk incommensurate phase.
The value P¯0→0 in the vicinity of the transition from the
paraelectric into modulated ferroelectric phase, while P¯0
?0 in the paraelectric phase under the absence of external
field. So, the transition temperature from the paraelectric into
theincommensurateferroelectric
−
thicknessintotheincommensurate
=
??T?T−TC?−g1
exact, since linear harmonic approximation is valid in the
paraelectric phase ?T?TIC? as well as in the immediate vi-
cinity of TIC, where ferroelectric nonlinearity can be ne-
glected.
2, originates from the influence of de-
2can be neglected in the denominator
phase is
TIC?h?=TC
2?Sg3
?Th??S?g3?0?33+g3?+
g1
2
4w1?T. At fixed temperature T the transition
phaseis
hIC?T?
−2?Sg3
2/4w1???S?g3?0?33+g3?. These expressions are almost
The transition temperature into commensurate ferroelec-
tric phase is TCF?h?=TC−
=TCF?h?+
At fixed temperature T the transition thickness into the
commensurate phase is hCF?T?=
surprise numerical simulations showed that these expressions
also appeared almost exact, most probably due to the same
reasons as in commensurate ferroelectrics ?see also Ref. 53
and next section?.
Note, that the size-dependent term in TCF?h? and TIC?h?,
namely,
polarization field given by Eq. ?1? and the finite surface en-
ergy ??S?0?. It is absent only for the unrealistic case with-
out surface energy, when ?S=0. Naturally, the depolarization
field influence and finite surface energy determine the value
of the transition thicknesses hCF?T? and hIC?T?. The depolar-
ization effect vanishes with the film thickness increase, i.e.,
at h?hCF?T?.
As anticipated, Eq. ?10? reduces to the well-known bulk
solution kB
4w1
h increase. So, for the bulk sample the incommensurate
modulation exists in the temperature range TC?T?TIC,
where TIC=TC+
analytical solution differs from the bulk solution in renormal-
ization of ? by depolarization field and surface effects, both
contribute into finite-size effects.
Phase diagram in coordinates temperature—film thickness
with paraelectric ?PE?, incommensurate ?IC?, and commen-
surate ?CF? ferroelectric phases, and corresponding domain
structure profiles are shown in Fig. 2.
It is seen from Fig. 2 that the transition temperatures into
incommensurate and commensurate phases strongly depend
2?Sg3
?Th??S?g3?0?33+g3?. Note that TIC?h?
g1
2
4w1?Tas anticipated.
−2?Sg3
?T?T−TC???S?g3?0?33+g3?. To our
2?Sg3
?Th??S?g3?0?33+g3?, originates from the influence of de-
2?T?=−
g1
2w1??
g1
2
2−
?T?T−TC?
w1
with the film thickness
g1
2
4w1?Tand kB
2?TIC?=−g1/2w1. Thus obtained
z
x
P3
z
x
P3
Temperature T (K)
110102
103
0
100
200
PE
IC
CF
hIC
hCF
Film thickness h (nm)
I
II
I
II
FIG. 2. ?Color online?. Phase diagram in the coordinates
temperature-thickness of the film deposed on the matched substrate
?um
PE is a paraelectric phase, IC is an incommensurate phase and
CF is a commensurate ferroelectric phase. Solid and dashed curves
correspond to the surface energy coefficient ?S=1 and 10 m2/F,
respectively. Insets schematically show the polarization ?x,z? pro-
files in the points I and II of the phase diagram. Material parameters
ofS2P2Se6:
?T=1.6·106J·m/?C2·K?,
=−4.8·108J·m5/C4,
?=8.5·1010J·m9/C6,
=−5.7·10−10J·m3/C2,
w1=1.8·10−27J·m5/C2,
=1.2·10−8J·m7/C4, g3=5·10−10J·m3/C2, and positive g2?g3
were taken from Refs. 61 and 62, ?11=?33=10 ?reference medium is
isotropic dielectric?.
??0.003? and placed between perfect conducting electrodes.
TC=193 K,
?
g1
v1
PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN…
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-5
Page 6
on the film thickness due to the surface energy, depolariza-
tion field and polarization gradient, which contributions in-
creases with the film thickness decrease. Thus “soft” incom-
mensurate modulation appears at thickness hIC?T? and
becomes “harder” with the thickness increase in CF phase
?compare insets I and II plotted for thicknesses hI?hII?.
Calculated modulation period q??h,T?=2?k?
presented in Fig. 3 for different values of the surface energy
coefficient ?S?compare curves 1–4?. It is seen that in the
most cases approximate Eq. ?10? ?dotted curves? and numeri-
cal calculations from Eqs. ?7?–?9? ?solid curves? give almost
the same results.
Summarizing results obtained in this section, we would
like to underline that transition temperatures TCF, TICand the
maximal period q−=2?k−
structure strongly depend on the film thickness, depolariza-
tion field and surface energy contributions, while the mini-
mal period q+=2?k+
The correlation effects, which strength is in turn determined
from the value of gradient coefficient g1
of both periods. The dependence of all polar properties on
the Debye screening radius Rdis rather weak under the typi-
cal conditions Rd?50 nm.
−1?h,T? is
−1of the incommensurate domain
−1weakly depends on the film thickness.
?, determine the scale
B. Phase-field modeling
In order to check the validity of the analytical calculations
performed in harmonic approximation, we study the problem
by the phase-field modeling.63–65Phase-field method allows
rigorous numerical calculations of the spontaneous polariza-
tion spatial distribution and temporal evolution. The distribu-
tion of electric field is obtained by solving the electrostatic
equations supplemented by the boundary conditions at the
top and bottom electrodes. All-important energetic contribu-
tions ?including depolarization field energy, electrostriction
contribution, elastic energy, and surface energy? are incorpo-
ratedintothetotalLGD
F?P1,P2,P3,uij?. The temporal evolution of the polarization
vector field, and thus the domain structure, is then described
by the time-dependent LGD equations
the kinetic coefficient related to the domain-wall mobility.
For a given initial distributions, numerical solution of the
time-dependent LGD equations yields the temporal and spa-
tial evolution of the polarization. We use periodic boundary
conditions along both the x and y directions.
Approximate analytical results are compared with numeri-
cal phase-field calculations of 2D ?x,y?-modulated domain
structures in Figs. 4 for a thin film with dimensions 100
?100?40 nm at different temperatures. A positive surface
energy coefficient ?Swas employed. It is seen from the plots
?b?–?e? that domain structure originated at low temperatures.
So, the single domain state appeared unstable starting from
much lower temperatures than TCF. This supports the as-
sumption made in Secs. V and I that domain stripes in ferro-
electric phase possibly originate from finite surface energy
value determined by nonzero ?S. It should be emphasized
that periodic boundary conditions along both the x and y
directions should affect the periodicity of the incommensu-
rate structures.
Let us underline that phase field modeling results mimics
labyrinth domain structures ?see especially Figs. 4?b?–4?g??.
Possible qualitative explanation of the striking fact can be
found in the Bjelis et al. papers,66,67where it was shown that
the nonlinear Euler-Lagrange problem ?3? without depolar-
ization field and surface energy contribution represents an
example of nonintegrable problem with chaotic phase por-
trait. Dananic et al. revealed that periodic solutions are iso-
lated trajectories at the phase portrait in the uniaxial case
and are physically trains of commensurate and incommensu-
rate domains of various periods. Unfortunately the general
results of Dananic and Bjelis cannot be applied quantita-
tively to the ferroelectric films on substrate considered in
our paper, since they did not consider neither surface energy
contribution nor depolarization field and strain effects ?com-
pare Eq. ?2.2? in Ref. 66 with considered free energy ?2??.
However it is well known that exactly depolarization field
and surface energy contribution rule finite-size effects and
size-induced phase transitions
ferroelectrics.2,5,6,8,15,20,25,30,45,53,54,59
Additional numerical simulations proved that depolariza-
tion field decrease ?reached by artificial increase of ?33in Eq.
?1? up to 102and higher? and the second kind boundary
conditions
sence of surface energy, ?S=0? strongly facilitate the spon-
taneous splitting on random domains in the considered sys-
tem,while thedepolarization
polarization distribution more ordered and eventually laby-
free-energyfunctional
?Pi
?t=−??F
?Pi, where ? is
in spatiallyconfined
?P3
?x3?x3=0,h=0 ?that exactly corresponds to the ab-
fieldincreasemakes
150170190210
10
102
103
180200220
10
102
103
110102
103
104
12
14
16
18
20
22
110102
103
104
0
20
40
60
80
Film thickness h (nm)
Modulation period (nm)
(a)
T=195 K
(b)
T=215 K
Film thickness h (nm)
Modulation period (nm)
Temperature T (K)
Modulation period (nm)
(c) h=30 nm
(d)
h=100 nm
Temperature T (K)
Modulation period (nm)
PE
IC
CF
TIC
TCF
T (K)
1
2
3
4
hIC
paraelectric phase
1
2
3
4
hIC
paraelectric phase
1
2
3
4
TIC
TCF
1
2
3
4
TIC
TCF
q? ?
q+
q? ?
q+
q? ?
q+
q? ?
q+
FIG. 3. ?Color online?. Thickness dependences of modulation
periods q−=2?k−
q+=2?k+
different values of temperature ?a? T=195 K and?b? 215 K. Tem-
perature dependences of q−=2?k−
?bottom curves? for different values of the film thickness ?c? h
=30 nm and ?d? 100 nm. Curves 1, 2, 3, and 4 correspond to the
surface energy coefficient ?S=0.03, 0.1, 0.3, and 1 m2/F, respec-
tively. Solid and dotted curves represent exact numerical calcula-
tions from Eqs. ?7?–?9? and approximate analytical dependences
?10?, respectively. Material parameters are the same as in Fig. 2 and
Rd?500 nm.
−1?top curves above the dashed horizontal line? and
−1?bottom curves below the dashed horizontal line? for
−1?top curves? and q+=2?k+
−1
MOROZOVSKA et al.
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-6
Page 7
rinthlike domains disappear. We also lead to the conclusion
that the structures with mainly rectangular corners ?shown in
Figs. 4?b?–4?g?? possibly originate from the spatial confine-
ment of the simulation volume in ?x,y?-directions. However
the structures only mimic true chaotic labyrinths with
smooth random shapes. True labyrinthic domain structures
appear near the surfaces of relaxor ferroelectrics ?compare
Fig. 1 from Ref. 68 and Fig. 2 from Ref. 69 with Figs.
4?b?–4?g??. Relaxor ferroelectrics have anomalously high ?33
and accordingly very small depolarization fields are ex-
pected, therefore Dananic and Bjelis formalism may quanti-
tatively describe their chaotic incommensurate domain struc-
ture.
We also performed 1D-phase field simulations to calculate
the incommensurate x-modulated structures in thin plates
with sizes hx=250 nm, hy=2 nm, hz=40 nm. We calculated
the polarization distribution at different temperatures ?Fig.
5?. From Fig. 5 it can be seen that the P3distribution across
6080100 120 140 160 180 200 220
10
102
103
Temperature T (K)
Phase-field modeling, h=40 nm,
x-y cross-sections for T = 0, 20, 40, 60, 80, 100, 140, 180 K
Modulation period (nm)
1
2
3
4
TIC
TCF
q? ?
q+
5
(a)
Analytical
calculations,
h=40 nm
FIG. 4. ?Color online?. ?a? Temperature dependence of the modulation period q−?top curves above the dashed horizontal line? and q+
?bottom curves below the dashed horizontal line? calculated analytically for the film thickness h=40 nm. Curves 1, 2, 3, 4, and 5 correspond
to the surface energy coefficient ?S=0.03, 0.1, 0.3, 1, and 10 m2/F respectively. Solid and dotted curves represent numerical calculations
from Eqs. ?7?–?9? and approximate analytical dependences ?10?, respectively. ?b?–?j? Temperature evolution of the ?x,y? modulated domain
structure calculated by the phase-field modeling for the film with sizes 100?100?40 nm, ?S=10 m2/F, and temperatures T=0, 20, 40, 60,
80, 100, 140, and 180 K. Other parameters are the same as in Fig. 2, but g2=g1=−5.7·10−10J·m3/C2and Rd?500 nm.
PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN…
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-7
Page 8
FIG. 5. ?Color online?. ?a? Sketch of the 1D x-modulated domain structure. ?b? Phase field 1D simulation of the polarization component
P3variation for different z positions at T=160 K and x=160. ?c?–?h? Polarization component P3morphologies in thin plates with sizes
hx=250 nm, hy=2 nm, and hz=40 nm at temperatures T=0, 40, 80, 120, 160, and 180 K. The pink solid circles, navy rhombus and dark
cyan squares represent P3at z=20, 33, and 1, respectively. Other parameters are the same as in Fig. 4, but g1=−5.7·10−10J·m3/C2and
g2?0.
MOROZOVSKA et al.
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-8
Page 9
the film depth z looks like a dome at fixed x position. P3is
maximal in the middle of the film, and its module decreases
from the middle to edges. Transversal x distribution of P3is
periodic and it looks like sine wave with a period about 17.6
nm ?18 grids? ?Figs. 5?c?–5?h??. This result provides neces-
sary background for harmonic approximation used in Secs.
IV and I in the deep enough incommensurate phase, i.e., at
temperatures essentially lower than TIC.
V. SUMMARY
We proposed the theoretical description of finite size, de-
polarization field effect, surface, and correlation energy in-
fluence on the phase diagram of thin ferroelectric films with
II-type incommensurate phases and semiconductor proper-
ties.
Within the framework of Landau-Ginzburg-Devonshire
theory we performed analytical calculations and phase-field
modeling of the temperature evolution and thickness depen-
dence of the period of incommensurate 180° domains ap-
peared in thin films covered with perfect electrodes. Despite
numerous efforts, the problem has not been solved previ-
ously.
It was shown analytically that the transition temperature
between paraelectric, incommensurate, and commensurate
ferroelectric phases ?as well as the period of incommensurate
domain structures? strongly depend on the film thickness,
depolarization field contribution, surface energy and gradient
coefficients. At the same time their dependences on Debye
screening radius Rdare rather weak for the typical values
Rd?50 nm.
Unexpectedly, both the analytical theory and phase-field
modeling results demonstrate that the incommensurate
modulation can be stable in thin films in the wide tempera-
ture range starting from the low temperatures ?much smaller
than the bulk Curie temperature? up to the temperature of
paraelectric phase transition. Phase field modeling results
mimics labyrinth domain structures.
These domain stripes possibly originate even at low tem-
peratures from the spatial confinement and finite surface en-
ergy contribution. Nonzero values of the surface energy lead
to appearance of polarization inhomogeneity along the polar
axis and depolarization field, localized near the surfaces.
Similar effects could be the common feature of various con-
fined ferroelectrics and ferromagnetics. Thus, we expect that
the long-range order parameter ?e.g., spontaneous polariza-
tion or magnetization? subjected to either spatial confinement
or imperfect screening could reveal incommensurate modu-
lation in nanosized ferroics. The result can be important for
applications of the nanosized materials in nanoelectronics
and memory devices.
We hope that our results would stimulate the experimental
study of the size-induced phase transitions in thin ferroelec-
tric films with incommensurate phase.
ACKNOWLEDGMENTS
Research sponsored by Ministry of Science and Education
of Ukraine and National Science Foundation ?Materials
World Network, Grants No. DMR-0820404 and No. DMR-
0908718?.
*Corresponding author.
†morozo@i.com.ua
‡lqc3@psu.edu
1P. Bak, Rep. Prog. Phys. 45, 587 ?1982?.
2R. Kretschmer and K. Binder, Phys. Rev. B 20, 1065 ?1979?.
3E. V. Chensky and V. V. Tarasenko, Sov. Phys. JETP 56, 618
?1982? ?Zh. Eksp. Teor. Fiz. 83, 1089 ?1982??.
4J. Junquera and Ph. Ghosez, Nature ?London? 422, 506 ?2003?.
5A. N. Morozovska, M. D. Glinchuk, and E. A. Eliseev, Phys.
Rev. B 76, 014102 ?2007?.
6Q. Y. Qiu, V. Nagarajan, and S. P. Alpay, Phys. Rev. B 78,
064117 ?2008?.
7C. W. Huang, Lang Chen, J. Wang, Q. He, S. Y. Yang, Y. H. Chu,
and R. Ramesh, Phys. Rev. B 80, 140101?R? ?2009?.
8Yue Zheng, C. H. Woo, and Biao Wang, J. Phys.: Condens. Mat-
ter 20, 135216 ?2008?.
9C. Kittel, Phys. Rev. 70, 965 ?1946?.
10T. Mitsui and J. Furuichi, Phys. Rev. 90, 193 ?1953?.
11V. A. Zhirnov, Zh. Eksp. Teor. Fiz. 35, 1175 ?1959? ?Sov. Phys.
JETP 8, 822 ?1959??.
12W. Cao and L. E. Cross, Phys. Rev. B 44, 5 ?1991?.
13A. M. Bratkovsky and A. P. Levanyuk, Phys. Rev. Lett. 86, 3642
?2001?.
14W. T. Lee, E. K. H. Salje, and U. Bismayer, Phys. Rev. B 72,
104116 ?2005?.
15I. A. Luk’yanchuk, A. Schilling, J. M. Gregg, G. Catalan, and J.
F. Scott, Phys. Rev. B 79, 144111 ?2009?.
16D. D. Fong, G. B. Stephenson, S. K. Streiffer, J. A. Eastman, O.
Auciello, P. H. Fuoss, and C. Thompson, Science 304, 1650
?2004?.
17C. Lichtensteiger, J.-M. Triscone, J. Junquera, and P. Ghosez,
Phys. Rev. Lett. 94, 047603 ?2005?.
18G. B. Stephenson and K. R. Elder, J. Appl. Phys. 100, 051601
?2006?.
19V. Nagarajan, J. Junquera, J. Q. He, C. L. Jia, R. Waser, K. Lee,
Y. K. Kim, S. Baik, T. Zhao, R. Ramesh, Ph. Ghosez, and K. M.
Rabe, J. Appl. Phys. 100, 051609 ?2006?.
20E. A. Eliseev, A. N. Morozovska, C V. Kalinin, Y. L. Li, Jie
Shen, M. D. Glinchuk, L. Q. Chen, and V. Gopalan, J. Appl.
Phys. 106, 084102 ?2009?.
21S. Prosandeev, I. Ponomareva, I. Kornev, I. Naumov, and L.
Bellaiche, Phys. Rev. Lett. 96, 237601 ?2006?.
22P. Aguado-Puente and J. Junquera, Phys. Rev. Lett. 100, 177601
?2008?.
23G. Catalan, A. Schilling, J. F. Scott, and J. M. Gregg, J. Phys.:
Condens. Matter 19, 132201 ?2007?.
24A. Gruverman, D. Wu, H.-J. Fan, I. Vrejoiu, M. Alexe, R. J.
Harrison, and J. F. Scott, J. Phys.: Condens. Matter 20, 342201
?2008?.
25I. A. Luk’yanchuk, L. Lahoche, and A. Sené, Phys. Rev. Lett.
PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN…
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-9
Page 10
102, 147601 ?2009?.
26D. A. Scrymgeour, V. Gopalan, A. Itagi, A. Saxena, and P. J.
Swart, Phys. Rev. B 71, 184110 ?2005?.
27D. Lee, R. K. Behera, P. Wu, H. Xu, S. B. Sinnott, S. R. Phillpot,
L. Q. Chen, and V. Gopalan. Phys. Rev. B 80, 060102?R?
?2009?.
28W. Selke, D. Catrein, and M. Pleimling, J. Phys. A 33, L459
?2000?.
29W. Selke, M. Pleimling, and D. Catrein, Eur. Phys. J. B 27, 321
?2002?.
30D. R. Tilley, Finite-size Effects on Phase Transitions in Ferro-
electrics in Ferroelectic Thin Films, edited by C. Paz de Araujo,
J. F. Scott, and G. W. Teylor ?Gordon and Breach, New York,
1996?.
31J. d’Albuquerque e Castro, D. Altbir, J. C. Retamal, and P. Var-
gas, Phys. Rev. Lett. 88, 237202 ?2002?.
32W. Zhang and S. Haas, J. Magn. Magn. Mater. 321, 3687 ?2009?.
33Yu. V. Sereda, Phys. Rev. B 66, 054109 ?2002?.
34A. V. Babich, S. V. Berezovsky, V. F. Klepikov, and Yu. V.
Sereda, Condens. Matter Phys. 9, 121 ?2006?.
35S. V. Berezovsky, V. Yu. Korda, and V. F. Klepikov, Phys. Rev.
B 64, 064103 ?2001?.
36E. V. Charnaya, S. A. Ktitorov, and O. S. Pogorelova, Ferroelec-
trics 297, 29 ?2003?.
37A. P. Levanyuk, S. A. Minyukov, and A. Cano, Phys. Rev. B 66,
014111 ?2002?.
38G. H. F. van Raaij, K. J. H. van Bemmel, and T. Janssen, Phys.
Rev. B 62, 3751 ?2000?.
39K. Z. Rushchanskii, Y. M. Vysochanskii, and D. Strauch, Phys.
Rev. Lett. 99, 207601 ?2007?.
40C.-G. Duan, S. S. Jaswal, and E. Y. Tsymbal, Phys. Rev. Lett.
97, 047201 ?2006?.
41G. Geneste, E. Bousquest, J. Junquera, and P. Ghosez, Appl.
Phys. Lett. 88, 112906 ?2006?.
42M. Q. Cai, Y. Zheng, B. Wang, and G. W. Yang, Appl. Phys.
Lett. 95, 232901 ?2009?.
43J. W. Hong, G. Catalan, D. N. Fang, Emilio Artacho, and J. F.
Scott, arXiv:0908.3617 ?unpublished?.
44C. L. Wang and S. R. P. Smith, J. Phys.: Condens. Matter 7,
7163 ?1995?.
45A. N. Morozovska, E. A. Eliseev, and M. D. Glinchuk, Phys.
Rev. B 73, 214106 ?2006?.
46A. Sundaresan, R. Bhargavi, N. Rangarajan, U. Siddesh, and C.
N. R. Rao, Phys. Rev. B 74, 161306?R? ?2006?.
47D. Yadlovker and Sh. Berger, Phys. Rev. B 71, 184112 ?2005?.
48D. Yadlovker and Sh. Berger, Appl. Phys. Lett. 91, 173104
?2007?.
49D. Yadlovker and Sh. Berger, J.Appl. Phys. 101, 034304 ?2007?.
50E. Erdem, H.-Ch. Semmelhack, R. Bottcher, H. Rumpf, J.
Banys, A. Matthes, H.-J. Glasel, D. Hirsch, and E. Hartmann, J.
Phys.: Condens. Matter 18, 3861 ?2006?.
51The dependence of ?11on E3is absent for uniaxial ferroelectrics.
It may be essential for perovskites with high coupling constant.
52A. N. Morozovska, E. A. Eliseev, JianJun Wang, G. S. Svechni-
kov, Yu. M. Vysochanskii, Venkatraman Gopalan, and Long-
Qing Chen, arXiv:0912.4423 ?unpublished?.
53M. D. Glinchuk, E. A. Eliseev, V. A. Stephanovich, and R. Farhi,
J. Appl. Phys. 93, 1150 ?2003?.
54M. D. Glinchuk, A. N. Morozovska, and E. A. Eliseev, J. Appl.
Phys. 99, 114102 ?2006?.
55N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Phys.
Rev. Lett. 80, 1988 ?1998?.
56Note that the renormalization does not contribute into the equa-
tions of state in a single-domain sample with constant polariza-
tion P0?x,y,z?=P¯0, since ??P¯0+??P¯0
57J. S. Speck and W. Pompe, J. Appl. Phys. 76, 466 ?1994?.
58C.-L. Jia and Valanoor Nagarajan, J.-Q. He, L. Houben, T. Zhao,
R. Ramesh, K. Urban, and R. Waser, Nat. Mater. 6, 64 ?2007?.
59M. D. Glinchuk, E. A. Eliseev, A. Deineka, L. Jastrabik, G.
Suchaneck, T. Sandner, G. Gerlach, and M. Hrabovsky, Integr.
Ferroelectr. 38, 101 ?2001?.
60M. I. Kaganov and A. N. Omelyanchouk, Zh. Eksp. Teor. Fiz.
61, 1679 ?1971? ?Sov. Phys. JETP 34, 895 ?1972??.
61Yu. M. Vysochanskii, M. M. Mayor, V. M. Rizak, V. Yu. Slivka,
and M. M. Khoma, Zh. Eksp. Teor. Fiz. 95, 1355 ?1989? ?Sov.
Phys. JETP 68, 782 ?1989??.
62M. M. Khoma, A. A. Molnar, and Yu. M. Vysochanskii, J. Phys.
Stud. 2, 524 ?1998?.
63Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen, Appl. Phys. Lett.
78, 3878 ?2001?.
64Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen, Acta Mater. 50,
395 ?2002?.
65S. V. Kalinin, A. N. Morozovska, L. Q. Chen, and B. J. Rod-
riguez, Rep. Prog. Phys. 73, 056502 ?2010?.
66V. Dananic and A. Bjelis, Phys. Rev. E 50, 3900 ?1994?.
67V. Dananic, A. Bjelis, M. Rogina, and E. Coffou, Phys. Rev. A
46, 3551 ?1992?.
68S. V. Kalinin, B. J. Rodriguez, S. Jesse, A. N. Morozovska, A. A.
Bokov, and Z.-G. Ye, Appl. Phys. Lett. 95, 142902 ?2009?.
69S. V. Kalinin, B. J. Rodriguez, J. D. Budai, S. Jesse, A. N.
Morozovska, A. A. Bokov, and Z.-G. Ye, Phys. Rev. B 81,
064107 ?2010?.
3??P¯0+?P¯0
3.
MOROZOVSKA et al.
PHYSICAL REVIEW B 81, 195437 ?2010?
195437-10