# 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.

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**ABSTRACT:**This work presents the study of domain structure in uniaxial semiconductor ferroelectrics Sn2P2S6, performed by multi-scale theoretical simulations and piezoresponse force microscopy. We reveal that in spite of the second-order ferroelectric phase transition, domain structure contains metastable paraelectric regions between ferroelectric domains after cooling from paraelectric phase to the polar one. Theoretical model shows that this feature is a result of the three-well local potential, which enables metastable non-polar regions in polar environment. We have investigated temperature evolution of the width of non-polar regions in heating-cooling cycle, which is in a good agreement with theoretical results.Ferroelectrics 01/2012; 438:55 - 67. · 0.38 Impact Factor - SourceAvailable from: Albina Y BorisevichEugene A. Eliseev, Sergei V. Kalinin, Yijia Gu, Maya D. Glinchuk, Victoria Khist, Albina Borisevich, Venkatraman Gopalan, Long-Qing Chen, Anna N. Morozovska[Show abstract] [Hide abstract]

**ABSTRACT:**We proved the existence of a universal flexo-antiferrodistortive coupling as a necessary complement to the well-known flexoelectric coupling. The coupling is universal for all antiferrodistortive systems and can lead to the formation of incommensurate, spatially-modulated phases in multiferroics. Our analysis can provide a self-consistent mesoscopic explanation for a broad range of modulated domain structures observed experimentally in multiferroics.Physical Review B 11/2013; 88(22). · 3.66 Impact Factor -
##### Article: On importance of higher non-linear interactions in the theory of type II incommensurate systems

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**ABSTRACT:**We reveal that the role of the higher non-linear local interactions in the conventional theoretical models developed to describe phase transitions in type II incommensurate systems is underestimated. Their consistent consideration in the thermodynamic potential expansion allows one to remove key contradictions in explanation of the experimental data for ferroelectric Sn2P2Se6 in the vicinity of the modulated-commensurate phase transition point.Physica B Condensed Matter 09/2013; 425:31–33. · 1.28 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?

1098-0121/2010/81?19?/195437?10?

©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

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