Lallart M. Ferroelectrics: Characterization and Modeling

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Intrinsic Interface Coupling in

Ferroelectric Heterostructures and Superlattices

K.-H. Chew

, L.-H. Ong

and M. Iwata

Department of Physics, University of Malaya, 50603 Kuala Lumpur,

School of Physics, Universiti Sains Malaysia, 11800 Minden, Penang,

Department of Engineering Physics, Electronics and Mechanics,

Graduate School of Engineering, Nagoya Institute of Technology, Nagoya,

1,2

Malaysia

Japan

1. Introduction

Ferroelectric superlattices comprising two or more different layers have received immense

attention due to their potential applications, as well as their striking new or enhanced

behaviors (Nakagawara et al., 2000; Dawber et al., 2005). In those structures, the coupling at

the interface between the two constituents has been demonstrated to play an important role

in governing their properties (Bousquet et al., 2005). Theoretical study of interface coupling

in ferroelectric superlattices was initially performed based on the Landau-like formulation

by taking the continuum limit of the transverse Ising model (Qu et al., 1997). In their model,

the extrapolation lengths describe the inhomogeneity of polarizations near the surfaces and

an interface-related parameter gives the strength of the coupling at the interface. We have

recently proposed a thermodynamic model with only one unknown parameter to study the

effect of interface on polarization behaviours at the interface region between two bulk

ferroelectrics (Chew et al., 2003; Tsang et al., 2004). The intrinsic ferroelectric coupling at the

interface leads to variation of polarization across the interface of the heterostructures.

In this contribution, we discuss some fundamental properties of the physics of interfaces in

ferroelectric heterostructures and superlattices using the Landau-Ginzburg theory. The key

issue that will be addressed is how the intrinsic ferroelectric coupling at the interface affects

the physical properties of the hybrid structures such as phase transitions, polarization

modulation profiles and dielectric susceptibilities. We begin with a discussion for

heterostructure of interfaces between a bulk ferroelectrics and dielectrics, in which the

influence of thickness is not significant. Explicit expressions describe the spatial profile of

polarization at the interface region of the heterostructure are obtained (Chew et al., 2003).

The influence of the intrinsic interface coupling on the inhomogeneity and discontinuity or

continuity of polarization at the interface is illustrated.

Since ferroelectric superlattice is an interesting system to study the interface effect, the

influence of thickness on the polarization profiles of the superlattice is investigated.

Analytical expressions of the polarization profile for superlattices are derived and discussed

in detailed (Ishibashi et al., 2007; Chew et al., 2009). Explicit expressions for dielectric

Ferroelectrics - Characterization and Modeling

374

susceptibilities in the paraelectric phase of the superlattice are also obtained (Chew et al.,

2008). Finally, we apply the model to epitaxial PbTiO

/SrTiO

by incorporating the

depolarization field and lattice strain in the free energy functional. Some calculated results

are discussed with experimental data. We conclude the chapter with some remarks.

2. Model of ferroelectric/dielectric heterostructure interfaces

In this section, the essential details for deriving the formalism of the ferroelectric/dielectric

heterostructure interface are presented (Chew et al., 2003). We assume a one-dimensional

problem in which the polarizations and related physical quantities vary along the x-

direction perpendicular to the interface of the heterostructure. The total energy associates

with the heterostructure can be expressed as

12i

FF F F=++, (1)

where

and

are the total free energy density of the ferroelectric constituent A and

dielectric constituent B, respectively.

is the coupling energy at the interface between the

two constituents.

The total free energy density of the ferroelectric constituent A and dielectric constituent B

are given by

()

111

222



=−





=−







(2)

which extend from x →−∞ to 0x

= and 0x = to x →∞, respectively.

denotes the

Landau-Ginzburg free energy densities of consituent layer

j , whereas

gives the energy

density in the single domain state of constituent

In the present study, the coupling energy

F between the polarizations at the interface is

described as

()

iii

Fpq

=−

, (3)

where

and

are the interface polarizations at 0x = .

is the coupling constant

describing the strength of the interaction.

The free energy density of the ferroelectric constituent

A with

11b

==− in the bulk

()x →−∞

is given by

'24 24

111 11

242 24

fpp pp

αβκ αβ



−= + + − +





. (4)

For the dielectric constituent

B, we have 0

qq== at x →∞, and the free energy density

contribution from the dielectric constituent (with the higher order

/4q

term truncated) is

ff q

ακ



−= +





, (5)

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

375

because

0f = for a non-polar dielectric. p and q are the order parameters of the

ferroelectric and dielectric consituents, respectively.

is a temperature-dependent

parameter

()

110 0

αα

=−, (6)

where

> is a temperarature-independent parameter.

> ,

> and

are all temperature-independent coefficients.

The equilibrium states of the heterostructures correspond to the minima of F with respect

to variations of p and q. These are given by solving the Euler-Lagrange equations for p and

pxp

qxq





∂∂∂

−=





′

∂∂∂









∂∂∂



−=





′

∂∂∂





(7)

with the boundary conditions







at 0x = , (8a)

and

and 0 at ,

pp x

qq x



==→−∞







==→+∞





(8b)

where

and

q are the bulk polarization of the ferroelectric constituent A (at x = -∞) and

the dielectric constituent B (at x =

∞ ), respectively.

For the present study of ferroelectric/dielectric heterostructure of interface, it turns out that

the free energy F of eq. (1) can be rewritten in terms of the interface polarizations

and

as order parameters. This gives F as a function of

and

q without the usual integral form.

Solving eqs. (1) and (7) simultaneously with the boundary conditions (i.e. eqs. (8a) and (8b))

imposed, and integrating once, the Euler-Lagrange equations becomes,

22 24

111

()()

242

pp pp

αβκ



−+ −=





, (9)

and

ακ







. (10)

By solving eq. (9), the polarization of the ferroelectric constituent A becomes

Ferroelectrics - Characterization and Modeling

376

tanh ( )

pxx=−, (11)

where

=− . (12)

For the dielectric constituent B, the solution of eq. (10) gives

exp( ),

qq Kx=− (13)

with

= (14)

is determined,

x can be obtained from eq. (11). In eqs. (11) and (13), the magnitude of

the interface polarizations

and

q are determined by the interface coupling parameter

The total energy, eq. (1), of the heterostructure can be written in terms of

and

q as

323 2 2

(3 2) ( ).

32 2 2

iibb i ii

Fppppqpq

ακ

βκ λ

=−+++− (15)

The equilibrium structure can be found from

()()0

ib ii

pp pq

βκ

∂

=−+−=

∂

, (16)

and

()0

iii

qpq

ακ λ

∂

=−−=

∂

. (17)

Let us examine the variation of polarization across the interface and the total energy F of

the heterostructure for the particular conditions of 0

= and

→∞. The variation of

polarization across the interface can be examined by looking into the continuity or

discontinuity in interface polarizations

q− . Without interface coupling ( 0

= ), we find

that

p= and 0

q = . Thus, the mismatch of interface polarizations and the total energy of

the heterostructure are found to be

ii b

qp−=

, (18)

and

0F =

, (19)

respectively.

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

377

For a strong interface coupling, i.e.,

→∞, we have

q= , implying that the polarization

is continuos across the interface. In order to find

q= , it is convenient to write eq. (15) in

term of only

323 2

(3 2)

32 2

iibb i

pppp p

ακ

βκ

=−++, (20)

and by minimizing it, we obtain

22 22

11 11

iib

pqp

ακ ακ





 



== + − − −





 

 





, (21)

which clearly indicates that the polarizations at the interface are determined by the

intermixed properties of two constituents.

-10 -5 0 5 10

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

p and q

Fig. 1. Spatial dependence of polarization at the interface region of ferroelectric/dielectric

heterostructures with

−

= (top), 1 (middle) and 0 (bottom). In the curves, the

parameters are:

=− ,

= ,

= and

= . Solid circles denote the

polarization at interface.

Figure 1 shows a typical example of a ferroelectric/dielectric heterostrucutre of interface

with different strength of interface coupling

. It is seen that the mismatch in the

polarization across the interface is notable for a loose coupling at the interface

−

= . The

mismatch in the interface polarization becomes smaller with increasing coupling strength. It

is interesting to see that the coupling at the interface induces polarization in the dielectric

consituent. This may be called the interface-induced polarization, and it extends into the

bulk over a distance governed by the characteristic length of the material

−

, which is

governed by

and

Ferroelectrics - Characterization and Modeling

378

0 1020304050

0.0

0.2

0.4

0.6

0.8

1.0

- q

−1

Fig. 2. Mismatch in the polarization at the interface of ferroelectric/dielectric

heterostructures as a function of

−

. Other parameters are the same as for Fig. 1.

In Fig. 2, the mismatch in polarizations across the interface is examined under various

strengths of interfacial coupling. The results clearly show that the mismatch in the interface

polarizations is decreased with increasing interface coupling strength.

3. Model of ferroelectric/dielectric superlattices

We now consider a periodic superlattice composed of alternating ferroelectric layer and

dielectric layer (ferroelectric/dielectic suprelattices), as shown in Fig. 3. Some key points are

repeated here for clarity of discussion. Similarly, we assume that all spatial variation of

polarization takes place along the x-direction. The thickness of ferroelectric layer and

dielectric layer are L

and L

, respectively. L is the periodic thickness of the superlattice. The

two layers are coupled with each other across the interface. Periodic boudary conditions are

used for describing the superlattices.

By symmetry, the average energy density of the ferroelectric/dielectric superlattice F is

(Ishibashi & Iwata, 2007; Chew et al., 2008; Chew et al., 2009)

()

12i

FFFF

=++

. (22)

Fig. 3. Schematic illustration of a periodic ferroelectric superlattice composed of a

ferroelectric and dielectric layers. The thickness of ferroelectric layer A and dielectric layer

B are L

and L

, respectively. L = L

+ L

is the periodic thickness of the superlattice.