# Phase Diagrams and Domain Splitting in Thin Ferroelectric Films with Incommensurate Phases

**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 modelling of the temperature and thickness dependencies of the
period of incommensurate 180 degree 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 film thickness, and surface
and gradient energy contributions. The results may provide insight on the
temperature dependence of domain structures in nanosized ferroics with inherent
incommensurate phases.

# Figures

Phase diagram and domain splitting in thin ferroelectric ﬁlms with incommensurate phase

A. N. Morozovska,

1,

*

,†

E. A. Eliseev,

1,2

JianJun Wang,

3

G. S. Svechnikov,

1

Yu. M. Vysochanskii,

4

Venkatraman Gopalan,

3

and Long-Qing Chen

3,

*

,‡

1

V. Lashkarev Institute of Semiconductor Physics, NAS of Ukraine, prospect Nauki 41, 03028 Kiev, Ukraine

2

Institute for Problems of Materials Science, NAS of Ukraine, Krjijanovskogo 3, 03142 Kiev, Ukraine

3

Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA

4

Institute for Solid State Physics and Chemistry, Uzhgorod University, 88000 Uzhgorod, Ukraine

共Received 23 December 2009; revised manuscript received 25 April 2010; published 27 May 2010

兲

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.81.195437 PACS number共s兲: 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.

1

The spontaneous modulation appears when

the homogeneous state is either unstable or less energetically

preferable 共metastable兲. 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.

2–8

Laminar domain structure formation in thick ﬁlms

with free surfaces was considered in the classic papers by

Kittel

9

for ferromagnetic media and Mitsui and Furuichi

10

for

ferroelectric media. The structure of a single boundary be-

tween two domains in the bulk ferroelectrics was considered

by Cao and Cross

11

and Zhirnov,

12

allowing 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.

16–19

One 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.

20

For instance, density-functional theory results

predicted the stabilization of vortex structure in ferroelectric

nanodots under the transverse inhomogeneous static electric

ﬁeld.

21,22

This 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.

25

Possible 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,

26,27

resem-

bling domain structures typical for ferromagnetics.

The axial next-nearest-neighbor Ising 共ANNNI兲 model

was successfully applied to study of the size effect on the

properties of magnetic ﬁlms by W. Selke et al.

28,29

The 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 共dipole-

dipole interactions兲 not considered within the model, while it

is of great importance for out-of-plane polarization

geometry.

30

On the other hand continuum Landau-Ginzburg-

Devonshire 共LGD兲 models naturally involve depolarization

ﬁeld and allow analytical calculations for spatially conﬁned

ferroelectrics. Despite the remark, results obtained within the

ANNI model 共as well as in others discrete and atomistic

models兲 should call for care when using continuum theory in

the thin ﬁlms.

PHYSICAL REVIEW B 81, 195437 共2010兲

1098-0121/2010/81共19兲/195437共10兲 ©2010 The American Physical Society195437-1

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 共see, e.g.,

Ref. 31兲. Unfortunately in the discrete model framework the

efforts to compute the dipole-dipole interaction energy dras-

tically increase with the increase of spins quantity 共see, e.g.,

Ref. 32兲. 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 共see,

e.g., Refs. 33–35兲; 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.

36

obtained 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.

37

Levanyuk 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

Sn

2

P

2

Se

6

as 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 parameter.

35

First-principle

calculations

39

may 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 atomistic

40–43

and

phenomenological

5,44,45

theories are in a good agreement

with each other as well as with experimental results for

ferromagnetic

46

and ferroelectric

16,47–50

systems.

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

P

3

is directed along the polar axis z. The sample is dielectri-

cally isotropic in transverse directions, i.e., permittivity

11

=

22

at zero external ﬁeld.

Further we assume that the dependence of in-plane polar-

ization components on E

1,2

can be linearized as P

1,2

⬇

0

共

11

−1兲E

1,2

共

0

is the universal dielectric constant兲.

Thus the polarization vector acquires the form:

P共r兲=关

0

共

11

−1兲E

1

,

0

共

11

−1兲E

2

, P

3

共E , r兲+

0

共

33

−1兲E

3

兴.

51

Maxwell’s equations for the inner electric ﬁeld E

i

=−ⵜ

i

共r兲,

expressed via electrostatic potential

i

共r兲 and polarization

P共r兲 reduces to the equation

2

i

z

2

+

11

33

共

2

i

x

2

+

2

i

y

2

兲−

i

33

R

d

2

=

1

0

33

P

3

z

, R

d

is the Debye screening radius. A background

permittivity

33

is regarded much smaller than ferroelectric

contribution to temperature-dependent permittivity

33

f

.

The boundary conditions

i

共x , y ,0兲=

e

共x , y ,0兲,

i

共x , y , h兲= 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 兵x, y其 for the depolarization ﬁeld E

3

d

has the form 共see Appendix A in Ref. 52 for details兲

E

˜

3

d

关P

˜

3

共k,z兲兴 =

再

−

P

˜

3

共k,z兲

0

33

+

冕

0

z

dz

⬘

P

˜

3

共k,z

⬘

兲cosh共Kz

⬘

兲

cosh关K共h − z兲兴

0

33

· sinh共Kh兲

K

+

冕

z

h

dz

⬘

P

˜

3

共k,z

⬘

兲cosh关K共h

− z

⬘

兲兴

cosh共Kz兲K共k兲

0

33

· sinh共Kh兲

冎

. 共1兲

Here vector k = 兵k

1

,k

2

其, its absolute value k=

冑

k

1

2

+k

2

2

, func-

MOROZOVSKA et al. PHYSICAL REVIEW B 81, 195437 共2010兲

195437-2

tion K共k兲=

冑

共

11

k

2

+R

d

−2

兲/

33

. For the dielectric media with

polarization P

3

共z兲 independent on x , y coordinates and R

d

→ ⬁ expression 共1兲 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 P

3

共x , z兲. Depolariza-

tion ﬁeld E

i

d

is caused by imperfect screening of the bound

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 F

S

. In-

cluding the surface energy term, LGD free-energy F=F

S

+F

V

depends on the chosen order parameter—spontaneous

polarization component P

3

.

Within the LGD theory for the II-type incommensurate

materials the spatial distribution of the spontaneous polariza-

tion component P

3

inside the ﬁlm of thickness h could be

found by the minimization of the free-energy functional 共see

e.g., Refs. 45 and 53兲

F ⬇

冕

0

h

dz

冕

−⬁

⬁

dxdy

冋

␣

⬘

2

P

3

2

+

⬘

4

P

3

4

+

␥

6

P

3

6

− P

3

冉

E

3

d

2

+ E

3

e

冊

+

g

i

2

冉

P

3

x

i

冊

2

+

w

i

2

冉

2

P

3

x

i

2

冊

2

+

v

i

2

· P

3

2

冉

P

3

x

i

冊

2

册

+

␣

S

2

冕

−⬁

⬁

dxdy共P

3

2

兩

z=0

+ P

3

2

兩

z=h

兲. 共2兲

Coefﬁcient

␣

共T兲=

␣

T

共T − T

C

兲 explicitly depends on tempera-

ture T, T

C

is the Curie temperature of a bulk material.

For thin ﬁlms coefﬁcients

␣

⬘

共T兲 and

⬘

should be renor-

malized due to the electrostriction coupling as shown by

Cao and Cross

12

and Zhirnov

11

and misﬁt strain originated

from the ﬁlm and substrate lattice mismatch,

54,55

as

␣

⬘

=

␣

共T兲−

共Q

11

+Q

12

兲u

m

ⴱ

s

11

+s

12

−2

共Q

11

2

+Q

12

2

兲s

11

−2s

12

Q

11

Q

12

s

11

2

−s

12

2

P

¯

0

2

and

⬘

=

+2

共Q

11

2

+Q

12

2

兲s

11

−2s

12

Q

11

Q

12

s

11

2

−s

12

2

, where s

ij

is the elastic compliances

tensor at constant polarization; Q

ij

is the electrostriction ten-

sor,

␣

and

are stress-free expansion coefﬁcients, the aver-

age spontaneous polarization is P

¯

0

,

56

u

m

ⴱ

is the effective misﬁt

strain, at that u

m

ⴱ

共L兲⬇u

m

, L ⱕh

d

and u

m

ⴱ

共L兲⬇u

m

h

d

/ L,

L⬎h

d

共h

d

is the critical thickness of dislocations appearance兲

in accordance with the model proposed by Speck and

Pompe.

57

The epitaxial strain u

m

=共a / c兲− 1 originated from

the thin ﬁlm 共a兲 and substrate 共c兲 lattice constants mismatch.

The gradient coefﬁcient g

i

and seeding coefﬁcient

could

be negative, other higher coefﬁcients are positive. Last inte-

gral term in Eq. 共2兲 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

␣

S

and neglect higher order terms in

the surface energy. The depolarization ﬁeld E

3

d

is given by

Eq. 共1兲.

The necessary condition of the incommensurate phase ap-

pearance, g

i

⬍0, should be satisﬁed at least in one spatial

direction 共i.e., for one value of i兲, while for other directions

the homogeneous state would be stable if g

j

⬎0. Since most

of ferroelectrics with incommensurate phase are uniaxial or

biaxial ones, the coefﬁcients g

i

are not necessarily equal. In

order to simplify our consideration and obtain close-form

analytical results, we suppose that g

3

⬎0 and either g

1

=g

2

⬍0 共symmetric biaxial x, y-incommensurate case兲 or g

1

⬍0

& g

2

⬎0 共uniaxial x-incommensurate case兲. In this case one

could neglect the higher order derivatives on z and put w

3

=0 and

v

3

=0 in Eq. 共2兲. Coefﬁcients w

1

and

v

1

should be

nonzero positive values for the x-incommensurate modula-

tion existence; or coefﬁcients w

1

=w

2

and

v

1

=

v

2

should 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 g

1

=g

2

⬍0, w

1

=w

2

and

v

1

=

v

2

,

g

3

⬎0 and w

3

=0 and

v

3

=0, minimization of the free-energy

共2兲 results into the relaxation equation for polarization distri-

bution

⌫

P

3

t

+

␣

⬘

共T兲P

3

+

⬘

P

3

3

+

␥

P

3

5

− g

3

2

P

3

z

2

− 共g

1

+

v

1

P

3

2

兲

冉

2

P

3

x

2

+

2

P

3

y

2

冊

+ w

1

冉

4

P

3

x

4

+

4

P

3

y

4

冊

−

v

1

P

3

冋

冉

P

3

x

冊

2

+

冉

P

3

y

冊

2

册

= E

3

d

共x,y ,z兲 + E

0

e

exp共i

t兲,

共3兲

where ⌫ is a positive relaxation coefﬁcient,

is the fre-

quency of external ﬁeld E

0

e

. For 1D case one should put

P

3

/

y = 0 and consider P

3

共x , z兲 in Eq. 共3兲. The important

feature of the Eq. 共3兲 is depolarization ﬁeld E

3

d

that nonlo-

cally depends on P

3

共x , y , z兲 as given by the linear integral

operator in Eq. 共1兲.

The boundary conditions for polarization acquire the form

x

z

P

3

P

3

bound charge

free screening charge

z=

h

Bottom electrode

Top electrode

free screening charge

FIG. 1. 共Color online兲. 180°-domain structure of thin ﬁlm cov-

ered with perfect electrodes. Break of double electric layers at the

ﬁlm-surface junction 共marked by a circle兲 and polarization inhomo-

geneity 共arrows of different length兲 cause depolarization ﬁeld.

PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN… PHYSICAL REVIEW B 81, 195437 共2010兲

195437-3

冏

冉

␣

S

P

3

− g

3

P

3

x

3

冊

冏

x

3

=0

=0,

冏

冉

␣

S

P

3

+ g

3

P

3

x

3

冊

冏

x

3

=h

=0.

共4兲

Similarly to the case of commensurate ferroelectrics one

could introduce extrapolation length ⌳=g

3

/

␣

S

that is usually

positive. Inﬁnite extrapolation length corresponds to an ideal

surface 共

␣

S

→ 0兲 and so-called natural boundary conditions,

while zero extrapolation length 共

␣

S

→ ⬁兲 corresponds to

P

3

共x

3

=0兲= 0 at a strongly damaged surface without long-

range order. Reported experimental values are

⌳=2–50 nm.

58,59

III. APPROXIMATE ANALYTICAL SOLUTION OF THE

EULER-LAGRANGE EQUATIONS

Then one could ﬁnd the solution of Eq. 共3兲 linearized for

the small polarization modulation p共k , z兲 in the form of se-

ries on the eigen functions f

n

共k , z兲

p共k,z,t兲 =

兺

n

冋

A

n

共k兲f

n

共k,z兲exp

冉

−

n

共k兲

t

⌫

冊

+ E

n

共k,

兲

f

n

共k,z兲exp共i

t兲

n

共k兲 + i

⌫

册

. 共5兲

Hereinafter k=兩k兩 and k=兵k

x

,k

y

其 for the

x, y-incommensurate modulation or k=兵k

x

,0其 for the

x-incommensurate modulation.

The ﬁrst term in Eq. 共5兲 is related to the relaxation of

initial conditions while the second one is the series expan-

sion external stimulus E

˜

0

e

via the eigen functions f

n

共k , z兲. The

eigen functions f

n

共k , z兲 and eigen values

n

共k兲 should be

found from the nontrivial solutions of the following problem:

冋

␣

ⴱ

− g

3

ⴱ

d

2

dz

2

+ g

1

ⴱ

k

2

+ w

1

k

4

册

f

n

共k,z兲 − E

3

d

关f

n

共k,z兲兴

=

n

共k兲f

n

共k,z兲, 共6a兲

冏

冉

␣

S

f

n

− g

3

f

n

z

冊

冏

z=0

=0,

冏

冉

␣

S

f

n

+ g

3

f

n

z

冊

冏

z=h

=0,

共6b兲

where

␣

ⴱ

=

␣

⬘

+3

P

¯

0

2

+5

␥

P

¯

0

4

, g

1

ⴱ

=g

1

+

v

1

P

¯

0

2

and P

¯

0

is the av-

erage polarization 共for a bulk single domain sample the spon-

taneous polarization P

0

2

=共

冑

2

−4

␣␥

−

兲/ 2

␥

兲. Depolariza-

tion ﬁeld E

3

d

关f

n

共k , z兲兴 given by integral operator 共1兲 and

involved into the problem 共6兲 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. 共6兲 was derived as 共see Appendix A in

Ref. 52 for details兲

f

n

共k,z兲⬃cosh

冋

q

n1

冉

z

h

−

1

2

冊

册

−

q

n2

2

− h

2

K

2

q

n1

2

− h

2

K

2

q

n1

sinh共q

n1

/2兲

q

n2

sinh共q

n2

/2兲

cosh

冋

q

n2

冉

z

h

−

1

2

冊

册

.

共7兲

Here q

n1,2

are expressed via the eigen value

n

as the solu-

tions of the biquadratic equation

␣

ⴱ

+ g

1

ⴱ

k

2

+ w

1

k

4

− g

3

q

n

2

h

2

+

q

n

2

0

33

共q

n

2

− h

2

K

2

兲

=

n

共k兲. 共8兲

The equation for the eigen spectrum

n

共k兲 is

q

n1

sinh共q

n1

/2兲

q

n1

2

− h

2

K

2

=

q

n2

sinh共q

n2

/2兲

q

n2

2

− h

2

K

2

冉

cosh共q

n1

/2兲共

␣

S

/g

3

兲 + 共q

n1

/h兲sinh共q

n1

/2兲

cosh共q

n2

/2兲共

␣

S

/g

3

兲 + 共q

n2

/h兲sinh共q

n2

/2兲

冊

. 共9兲

Note, that similar equations could be found for “sinh”-eigen functions. Since the smallest 共ﬁrst兲 eigenvalue should correspond

to eigen function of constant sign, we restrict our consideration for the ﬁrst symmetric “cosh”-eigen functions 共7兲.

The equilibrium dependence of the transverse modulation wave vector k on the temperature T and ﬁlm thickness h should

be found from Eqs. 共8兲 and 共9兲 under the conditions

min

=0.

IV. SIZE EFFECT ON THE PHASE EQUILIBRIUM AND DOMAIN STRUCTURE TEMPERATURE EVOLUTION

A. Harmonic approximation

Transcendental Eq. 共9兲 was essentially simpliﬁed at the domain structure onset 共see Appendix B in Ref. 52兲 so that the

approximate expression for the highest and lowest roots was derived in the form

k

⫾

2

共h,T兲⬇−

g

1

ⴱ

2w

1

⫾

冑

g

1

ⴱ2

4w

1

2

−

1

w

1

冉

␣

ⴱ

共T兲 +

2

␣

S

g

3

共

␣

S

冑

g

3

0

33

+ g

3

兲h +

␣

S

g

3

0

h

2

/4R

d

2

冊

. 共10兲

MOROZOVSKA et al. PHYSICAL REVIEW B 81, 195437 共2010兲

195437-4

Note, that the size-dependent term in Eq. 共10兲,

2

␣

S

g

3

共

␣

S

冑

g

3

0

33

+g

3

兲h+

␣

S

g

3

0

h

2

/4R

d

2

, originates from the inﬂuence of de-

polarization ﬁeld given by Eq. 共1兲 and the ﬁnite surface en-

ergy 共

␣

S

⫽ 0兲. The ﬁnite-size effect is absent only for the

unrealistic case without surface energy, when

␣

S

=0. Under

the typical conditions R

d

Ⰷ50 nm and g

3

⬃10

−10

J·m

3

/ C

2

,

the term

␣

S

g

3

0

h

2

/ 4R

d

2

can be neglected in the denominator

of Eq. 共10兲 without any noticeable loss of precision. For the

particular case the depolarization term is proportional to 1 / h

as anticipated for ferroelectrics dielectrics.

2,20,53,54

The depo-

larization effect vanishes with the ﬁlm thickness increase

共h → ⬁兲.

The root k

−

共h , T兲 is always stable only in the incommen-

surate phase of the bulk material. The root k

+

共h , T兲 can be

共meta兲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 共2兲

should be performed in order to determine the ﬁlm thickness

range, where the roots k

⫾

共h , T兲 are stable 共and thus single

domain distribution is unstable兲.

The comparison of the free energies 共2兲 was performed in

harmonic approximation. It was demonstrated that the root

k

+

共h , T兲 can be stable in the wide temperature range starting

from the low temperatures 共much smaller than T

C

兲 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 共coefﬁcient

␣

S

=0, ⌳→ ⬁兲. Even under

the absence of defects domain stripes originate from imper-

fect screening of depolarization ﬁeld outside the ﬁlm 共either

imperfect electrodes, dielectric gap兲 and/or the spatial con-

ﬁnement 共i.e., surface energy contribution determined by

nonzero

␣

S

兲. Zero value of

␣

S

means 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

␣

S

lead 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 共see e.g., Refs. 53 and 60兲, 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

the incommensurate ferroelectric phase is T

IC

共h兲= T

C

−

2

␣

S

g

3

␣

T

h共

␣

S

冑

g

3

0

33

+g

3

兲

+

g

1

2

4w

1

␣

T

. At ﬁxed temperature T the transition

thickness into the incommensurate phase is h

IC

共T兲

=

−2

␣

S

g

3

共

␣

T

共T−T

C

兲−g

1

2

/4w

1

兲共

␣

S

冑

g

3

0

33

+g

3

兲

. These expressions are almost

exact, since linear harmonic approximation is valid in the

paraelectric phase 共T ⬎ T

IC

兲 as well as in the immediate vi-

cinity of T

IC

, where ferroelectric nonlinearity can be ne-

glected.

The transition temperature into commensurate ferroelec-

tric phase is T

CF

共h兲= T

C

−

2

␣

S

g

3

␣

T

h共

␣

S

冑

g

3

0

33

+g

3

兲

. Note that T

IC

共h兲

=T

CF

共h兲+

g

1

2

4w

1

␣

T

as anticipated.

At ﬁxed temperature T the transition thickness into the

commensurate phase is h

CF

共T兲=

−2

␣

S

g

3

␣

T

共T−T

C

兲共

␣

S

冑

g

3

0

33

+g

3

兲

. To our

surprise numerical simulations showed that these expressions

also appeared almost exact, most probably due to the same

reasons as in commensurate ferroelectrics 共see also Ref. 53

and next section兲.

Note, that the size-dependent term in T

CF

共h兲 and T

IC

共h兲,

namely,

2

␣

S

g

3

␣

T

h共

␣

S

冑

g

3

0

33

+g

3

兲

, originates from the inﬂuence of de-

polarization ﬁeld given by Eq. 共1兲 and the ﬁnite surface en-

ergy 共

␣

S

⫽ 0兲. 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 h

CF

共T兲 and h

IC

共T兲. The depolar-

ization effect vanishes with the ﬁlm thickness increase, i.e.,

at hⰇh

CF

共T兲.

As anticipated, Eq. 共10兲 reduces to the well-known bulk

solution k

B

2

共T兲=−

g

1

2w

1

⫾

冑

g

1

2

4w

1

2

−

␣

T

共T−T

C

兲

w

1

with the ﬁlm thickness

h increase. So, for the bulk sample the incommensurate

modulation exists in the temperature range T

C

⬍T⬍T

IC

,

where T

IC

=T

C

+

g

1

2

4w

1

␣

T

and k

B

2

共T

IC

兲=−g

1

/ 2w

1

. Thus obtained

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 共PE兲, incommensurate 共IC兲, and commen-

surate 共CF兲 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

z

x

P

3

z

x

P

3

Temperature T

(

K

)

1 10 10

2

10

3

0

100

200

PE

IC

CF

h

IC

h

CF

Film thickness

h

(nm)

I

II

I

II

FIG. 2. 共Color online兲. Phase diagram in the coordinates

temperature-thickness of the ﬁlm deposed on the matched substrate

共u

m

ⴱ

⬇0.003兲 and placed between perfect conducting electrodes.

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 m

2

/ F,

respectively. Insets schematically show the polarization 兵x ,z其 pro-

ﬁles in the points I and II of the phase diagram. Material parameters

of S

2

P

2

Se

6

:

␣

T

=1.6·10

6

J·m/ 共C

2

·K兲, T

C

=193 K,

=−4.8·10

8

J·m

5

/ C

4

,

␥

=8.5·10

10

J·m

9

/ C

6

, g

1

=−5.7·10

−10

J·m

3

/ C

2

, w

1

=1.8·10

−27

J·m

5

/ C

2

,

v

1

=1.2·10

−8

J·m

7

/ C

4

, g

3

=5·10

−10

J·m

3

/ C

2

, and positive g

2

⬃g

3

were taken from Refs. 61 and 62,

11

=

33

=10 共reference medium is

isotropic dielectric兲.

PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN… PHYSICAL REVIEW B 81, 195437 共2010兲

195437-5

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 h

IC

共T兲 and

becomes “harder” with the thickness increase in CF phase

共compare insets I and II plotted for thicknesses h

I

⬍h

II

兲.

Calculated modulation period q

⫾

共h , T兲=2

k

⫾

−1

共h , T兲 is

presented in Fig. 3 for different values of the surface energy

coefﬁcient

␣

S

共compare curves 1–4兲. It is seen that in the

most cases approximate Eq. 共10兲共dotted curves兲 and numeri-

cal calculations from Eqs. 共7兲–共9兲共solid curves兲 give almost

the same results.

Summarizing results obtained in this section, we would

like to underline that transition temperatures T

CF

, T

IC

and the

maximal period q

−

=2

k

−

−1

of the incommensurate domain

structure strongly depend on the ﬁlm thickness, depolariza-

tion ﬁeld and surface energy contributions, while the mini-

mal period q

+

=2

k

+

−1

weakly depends on the ﬁlm thickness.

The correlation effects, which strength is in turn determined

from the value of gradient coefﬁcient g

1

ⴱ

, determine the scale

of both periods. The dependence of all polar properties on

the Debye screening radius R

d

is rather weak under the typi-

cal conditions R

d

Ⰷ50 nm.

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.

63–65

Phase-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 共including depolarization ﬁeld energy, electrostriction

contribution, elastic energy, and surface energy兲 are incorpo-

rated into the total LGD free-energy functional

F共P

1

, P

2

, P

3

,u

ij

兲. The temporal evolution of the polarization

vector ﬁeld, and thus the domain structure, is then described

by the time-dependent LGD equations

P

i

t

=−⌫

␦

F

␦

P

i

, where ⌫ is

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 兵x , y其-modulated domain

structures in Figs. 4 for a thin ﬁlm with dimensions 100

⫻100⫻ 40 nm at different temperatures. A positive surface

energy coefﬁcient

␣

S

was employed. It is seen from the plots

共b兲–共e兲 that domain structure originated at low temperatures.

So, the single domain state appeared unstable starting from

much lower temperatures than T

CF

. 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 关see especially Figs. 4共b兲–4共g兲兴.

Possible qualitative explanation of the striking fact can be

found in the Bjelis et al. papers,

66,67

where it was shown that

the nonlinear Euler-Lagrange problem 共3兲 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 共com-

pare Eq. 共2.2兲 in Ref. 66 with considered free energy 共2兲兴.

However it is well known that exactly depolarization ﬁeld

and surface energy contribution rule ﬁnite-size effects and

size-induced phase transitions in spatially conﬁned

ferroelectrics.

2,5,6,8,15,20,25,30,45,53,54,59

Additional numerical simulations proved that depolariza-

tion ﬁeld decrease 关reached by artiﬁcial increase of

33

in Eq.

共1兲 up to 10

2

and higher兴 and the second kind boundary

conditions

P

3

x

3

兩

x

3

=0,h

=0 共that exactly corresponds to the ab-

sence of surface energy,

␣

S

=0兲 strongly facilitate the spon-

taneous splitting on random domains in the considered sys-

tem, while the depolarization ﬁeld increase makes

polarization distribution more ordered and eventually laby-

150 170 190 210

10

10

2

10

3

180 200 220

10

10

2

10

3

1 10 10

2

10

3

10

4

12

14

16

18

20

22

1 10 10

2

10

3

10

4

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)

Tem

p

erature T

(

K

)

Modulation period (nm)

(c)

h

=30 nm

(d)

h

=100 nm

Temperature T (K)

Modulation period (nm)

PE

IC

CF

T

IC

T

CF

T

(

K

)

1

2

3

4

h

IC

paraelectric phase

1

2

3

4

h

IC

paraelectric phase

1

2

3

4

T

IC

T

CF

1

2

3

4

T

IC

T

CF

q

q

+

q

q

+

q

q

+

q

q

+

FIG. 3. 共Color online兲. Thickness dependences of modulation

periods q

−

=2

k

−

−1

共top curves above the dashed horizontal line兲 and

q

+

=2

k

+

−1

共bottom curves below the dashed horizontal line兲 for

different values of temperature 共a兲 T =195 K and共b兲 215 K. Tem-

perature dependences of q

−

=2

k

−

−1

共top curves兲 and q

+

=2

k

+

−1

共bottom curves兲 for different values of the ﬁlm thickness 共c兲 h

=30 nm and 共d兲 100 nm. Curves 1, 2, 3, and 4 correspond to the

surface energy coefﬁcient

␣

S

=0.03, 0.1, 0.3, and 1 m

2

/ F, respec-

tively. Solid and dotted curves represent exact numerical calcula-

tions from Eqs. 共7兲–共9兲 and approximate analytical dependences

共10兲, respectively. Material parameters are the same as in Fig. 2 and

R

d

⬃500 nm.

MOROZOVSKA et al. PHYSICAL REVIEW B 81, 195437 共2010兲

195437-6

rinthlike domains disappear. We also lead to the conclusion

that the structures with mainly rectangular corners 关shown in

Figs. 4共b兲–4共g兲兴 possibly originate from the spatial conﬁne-

ment of the simulation volume in 兵 x , y其-directions. However

the structures only mimic true chaotic labyrinths with

smooth random shapes. True labyrinthic domain structures

appear near the surfaces of relaxor ferroelectrics 关compare

Fig. 1 from Ref. 68 and Fig. 2 from Ref. 69 with Figs.

4共b兲–4共g兲兴. 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 h

x

=250 nm, h

y

=2 nm, h

z

=40 nm. We calculated

the polarization distribution at different temperatures 关Fig.

5兴. From Fig. 5 it can be seen that the P

3

distribution across

60 80 100 120 140 160 180 200 220

10

10

2

10

3

Temperature T (K)

Modulation period (nm)

1

2

3

4

T

IC

T

CF

q

q

+

5

(a)

Analytical

calculations,

h=40 nm

Phase-field modeling, h=40 nm,

x-y cross-sections for T = 0, 20, 40, 60, 80, 100, 140, 180 K

FIG. 4. 共Color online兲. 共a兲 Temperature dependence of the modulation period q

−

共top curves above the dashed horizontal line兲 and q

+

共bottom curves below the dashed horizontal line兲 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 m

2

/ F respectively. Solid and dotted curves represent numerical calculations

from Eqs. 共7兲–共9兲 and approximate analytical dependences 共10兲, respectively. 共b兲–共j兲 Temperature evolution of the 兵x , y其 modulated domain

structure calculated by the phase-ﬁeld modeling for the ﬁlm with sizes 100⫻ 100⫻ 40 nm,

␣

S

=10 m

2

/ F, and temperatures T =0, 20, 40, 60,

80, 100, 140, and 180 K. Other parameters are the same as in Fig. 2, but g

2

=g

1

=−5.7·10

−10

J·m

3

/ C

2

and R

d

⬃500 nm.

PHASE DIAGRAM AND DOMAIN SPLITTING IN THIN… PHYSICAL REVIEW B 81, 195437 共2010兲

195437-7

FIG. 5. 共Color online兲. 共a兲 Sketch of the 1D x-modulated domain structure. 共b兲 Phase ﬁeld 1D simulation of the polarization component

P

3

variation for different z positions at T=160 K and x =160. 共c兲–共h兲 Polarization component P

3

morphologies in thin plates with sizes

h

x

=250 nm, h

y

=2 nm, and h

z

=40 nm at temperatures T =0, 40, 80, 120, 160, and 180 K. The pink solid circles, navy rhombus and dark

cyan squares represent P

3

at z= 20, 33, and 1, respectively. Other parameters are the same as in Fig. 4, but g

1

=−5.7·10

−10

J·m

3

/ C

2

and

g

2

⬎0.

MOROZOVSKA et al. PHYSICAL REVIEW B 81, 195437 共2010兲

195437-8

the ﬁlm depth z looks like a dome at ﬁxed x position. P

3

is

maximal in the middle of the ﬁlm, and its module decreases

from the middle to edges. Transversal x distribution of P

3

is

periodic and it looks like sine wave with a period about 17.6

nm 共18 grids兲关Figs. 5共c兲–5共h兲兴. 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 T

IC

.

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 共as well as the period of incommensurate

domain structures兲 strongly depend on the ﬁlm thickness,

depolarization ﬁeld contribution, surface energy and gradient

coefﬁcients. At the same time their dependences on Debye

screening radius R

d

are rather weak for the typical values

R

d

Ⰷ50 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 共much smaller

than the bulk Curie temperature兲 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 共e.g., spontaneous polariza-

tion or magnetization兲 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 共Materials

World Network, Grants No. DMR-0820404 and No. DMR-

0908718兲.

*

Corresponding author.

†

morozo@i.com.ua

‡

lqc3@psu.edu

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- CitationsCitations11
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