Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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570

19.1 Introduction

Understanding the physical properties in ferroelectric nanostructures (FEN),

which is a major requirement for the future progress in ferroelectric-based

technologies (Auciello et al., 1998; Scott, 1998, 2000; Ramesh et al., 2001;

Setter et al., 2006), has become an issue of interest over the past decade (see,

e.g., Dawber et al., 2005; Kornev et al., 2006; Ponomareva et al., 2005a;

Rabe, 2005; Yacoby et al., 2006). An intriguing problem in FEN concerns

their domain structures which are often non-equilibrium dipole patterns that

depend on sample history, temperature, boundary conditions, and other factors.

For instance, the various following patterns have recently been predicted or

observed in thin films: out-of-plane monodomains (Tybell et al., 1999; Ghosez

and Rabe, 2000; Junquera and Ghosez, 2003; Streiffer et al., 2002), 180°

out-of-plane stripe domains (Kopal et al., 1997; Tinte and Stachiotti, 2001;

Streiffer et al., 2002), 90° multidomains that are oriented parallel to the film

(Li et al., 2002), and microscopically paraelectric phases (Junquera and

Ghosez, 2003). The fact that different patterns have been reported for similar

mechanical boundary conditions supports a concept discussed in (Junquera

et al. (2004), Mehta et al. (1973) and Junquera and Ghosez (2003), namely

that they arise from different electrical boundary conditions. A reactive

atmosphere can indeed lead to a partial compensation of surface charges in

films with nominal ideal open-circuit conditions (Streiffer et al., 2002),

while metallic or semiconductor electrodes ‘sandwiching’ films do not always

provide ideal short-circuit conditions – resulting in a non-zero internal field

(Junquera and Ghosez, 2003). The degree of surface charges’ screening in

thin films can thus vary from one experimental set-up to another, possibly

generating different dipole patterns (Streiffer et al., 2002; Junquera and Ghosez,

2003).

The main theoretical methods used/developed so far in the study of dipole

patterns in low-dimensional ferroelectrics fall into two categories, namely:

(1) approaches based on the Ginzburg–Landau theory (Landau and Lifshitz,

Domains in ferroelectric nanostructures

from first principles

I A KORNEV, University of Arkansas, USA and

Madas Clausen Institute, University of Southern Denmark,

Denmark and B-K LAI, I NAUMOV,

I PONOMAREVA, H FU and L BELLAICHE,

University of Arkansas, USA

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Domains in ferroelectric nanostructures from first principles 571

1935, 1984) versus (2) first-principles-based schemes (Meyer and Vanderbilt,

2002; Ponomareva et al., 2005a; Rabe, 2005; Kornev et al., 2006).

The simplest description of the domain structure and a domain wall which

is an interface between uniformly polarized regions (domains) with different

polarization directions can be given within the Ginzburg–Landau theory for

a uniaxial ferroelectric. Within this oversimplified theory, below T

, two

uniform solutions exist, which correspond to two different orientations of

the spontaneous polarization. Under appropriate boundary conditions such

that the polarization points ‘up’ on the left, but ‘down’ on the right, the 180°

domain wall (phase boundary) will form separating regions corresponding to

the two different phases (Hubert and Schafer, 1998). Therefore, the domains

of alternating spontaneous polarization can be viewed as the regions of

different phases in local thermodynamic equilibrium separated by phase

boundaries. Generally, the phase boundaries will have a higher free energy

than in either phase and, naturally, cost energy. Although introduction of

180° domain walls raises the wall energy, at the same time it reduces high

stray field (or dipolar) energy (Hubert and Schafer, 1998). In order to minimize

the large depolarizing energy costs further, closure domains oriented at 90°

to the main domains may form if the wall energy advantage overwhelms the

anisotropy energy cost. The 90° closure domains eliminate electrostatic energy

but increase anisotropy energy in uniaxial material. Since 90° domain walls

are usually ferroelastic and since domains are constrained by their neighbors,

an additional complication appears due to a coupling between polarization

and strain (Roitburd, 1976) which is known to be strong in the technologically

and fundamentally important perovskite class of materials. All in all, the

domain structure in zero field results from a subtle competition between

different interactions, and depends considerably on the size, shape, and type

of ferroelectric material. Within the framework of the Ginzburg–Landau

theory, many of the characteristics of nanoscale ferroelectrics can be faithfully

reproduced (see, e.g., Tinte and Stachiotti, 2001; Li et al., 2002; Glinchuk et

al., 2003; Pertsev et al., 2003, and references therein). However, microscopic

foundations of phenomenological approaches are somewhat arbitrary and

atomic-scale mechanisms are difficult to assess.

An alternative approach to continuum models is first-principles-based

calculations that require no experimental input. Some phenomena in low-

dimensional ferroelectrics can be understood by using direct first-principles

methods. However, their large computational cost currently prevents them

from being used to study some other rather complex phenomena and/or

systems with large number of atoms. On the other hand, first-principles-

derived computational approaches have proved to be powerful methods in

recent years for understanding complex materials. Moreover, a precise

correlation between the amount of screening of surface charges and the

morphology of the dipole pattern, and how this correlation depends on

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Handbook of dielectric, piezoelectric and ferroelectric materials572

mechanical boundary conditions, starts to emerge thanks to these latter

approaches (Ghosez and Rabe, 2000; Almahmoud et al., 2004; Kornev et al.,

2004; Wu et al., 2004; Ponomareva and Bellaiche, 2006).

In this chapter we review progress in first-principles-based calculations

that have addressed the morphology of the dipole patterns in FEN. Section

19.2 describes the theoretical method that was developed and used to tackle

domains in FEN. Section 19.3 discusses the morphology of domains in

ferroelectric thin films, and how this morphology (i) depends on the considered

material, film thickness, boundary conditions or film orientation and (ii)

evolves under an applied electric field. Section 19.4 discusses domain structures

never observed before in ferroelectrics, and that are predicted to occur in

ferroelectric nanodots and nanowires. Finally, Section 19.5 provides a summary.

19.2 Methods

In this section, we briefly describe a first-principles-derived approach allowing

predictions of finite-temperature properties of ferroelectric nanostructures –

under any possible electrical and mechanical boundary condition. More

precisely, the total energy for such low-dimensional systems is written as:

tot

({u

},{σ

},{v

},η) =

Heff

({u

},{σ

},{v

},η)

∑

< E

dep

>·Z*u

– ∑

E·Z*u

19.1

where u

is the local soft mode in the unit cell i of the film – whose product

with the effective charge Z* yields the local electrical dipole in this cell;

while η and {v

} are the homogeneous strain tensor and inhomogeneous

strain-related variables in unit cell i, respectively (Zhong et al., 1994, 1995).

The {σ

} arrangement characterizes the atomic configuration of the alloy.

Heff

represents the intrinsic (effective Hamiltonian) energy of the ferroelectric

nanostructure. Its analytical expression is the one from Zhong et al., (1994,

1995) for bulks made of simple perovskites (such as BaTiO

bulk) and from

Bellaiche et al. (2000, 2002) for bulks made of solid solutions (such as

Pb(Zr,Ti)O

), except for two modifications. The first modification applies

when the nanostructure is surrounded by vacuum, and consists of adding

energetic terms associated with the direct interaction between the vacuum

surrounding the dot and both the surface dipoles and inhomogeneous strain

near the surface (Almahmoud et al., 2004). The second modification consists

of replacing the (3D reciprocal-space-based) matrix associated with long-

range dipole–dipole interactions in the bulk (Zhong et al., 1994, 1995) by

the corresponding matrix characterizing dipole–dipole interactions in the

investigated ferroelectric nanostructure. Such a matrix is given in Ponomareva

et al. (2005c) for thin films, wires, and dots, and corresponds to ideal open-

circuit (OC) conditions. Such electrical boundary conditions naturally lead

to the possible existence of a maximum depolarizing field (denoted by 〈E

dep

〉)

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Domains in ferroelectric nanostructures from first principles 573

inside the nanostructure. The second term of Eq. (19.1) mimics a screening

of 〈E

dep

〉 via the β parameter. More precisely, the residual depolarizing field

resulting from the combination of the first and second term of Eq. (19.1) has

a magnitude equal to (1 – β)|〈E

dep

〉|. β = 0 thus corresponds to ideal OC

conditions, while an increase in β lowers the magnitude of the resulting

depolarizing field, and β = 1 corresponds to ideal short-circuit (SC) conditions

for which the depolarizing field has vanished. Technically, 〈E

dep

〉 is calculated

at an atomistic level via the proposed approach of Ponomareva et al. (2005c),

and depends on the dimensionality, size, and dipole pattern of the investigated

nanostructure. Note that real thin films are likely neither in ideal open-

circuit conditions − for which unscreened polarization-induced surface charges

can generate a large depolarizing electric field along the film orientation

(Meyer and Vanderbilt, 2001) − nor in ideal short-circuit conditions − that

are associated with a vanishing internal field resulting from the full screening

of surface charges − but rather experience a situation in-between.

Different mechanical boundary conditions can also be simulated thanks

to the homogeneous strain η (Li et al., 2002; Pertsev et al., 2003). For

example during the simulation associated with a substrate-induced epitaxial

strain in the (x, y)-plane, three components of η – in Voigt notation – are kept

fixed (namely, η

= 0, and η

= η

= δ, with δ characterizing the lattice

mismatch between the material forming the nanostructure and the chosen

substrate) while the other three components relax.

On the other hand, all the

components of strain tensor are allowed to relax during the simulation of

freestanding (i.e. stress-free) phase. The third term of Eq. (19.1) mimics the

effect of an applied electric field on properties of ferroelectric nanostructures.

The parameters of Eq. (19.1) are determined from first-principles calculations

performed on relatively small supercells (typically, up to 40 atom-per-cell).

Once the effective Hamiltonian is fully specified, its total energy is used in

Monte-Carlo (MC) simulations on supercells mimicking the FEN structure

under investigation. Typical outputs of the MC procedure are the local mode

vectors − whose supercell average is directly proportional to the macroscopic

spontaneous polarization − and the homogeneous strain tensor − which provides

information about the crystallographic system.

19.3 Domains in ferroelectric thin films

19.3.1 Domains in Pb(Zr,Ti)O

ultra-thin films

Let’s first focus on the results of Ponomareva and Bellaiche, (2006) for thin

films made of Pb(Zr

0.4

0.6

(PZT), as predicted by the use of the effective

Here, we assumed homogeneously strained nanostructures, which should be realistic for

the ultra-small nanostructures we considered here. On the other hand, the strain may

become inhomogeneous and significantly relax in thicker systems.

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Handbook of dielectric, piezoelectric and ferroelectric materials574

Hamiltonian technique described above. The corresponding bulk (i.e., having

the same composition) possesses a polarization pointing along 〈001〉 and lies

outside the morphotropic phase boundary (Bellaiche et al., 2000; Noheda et

al., 2001) (and thus does not have dipoles that easily rotate). As a result, the

qualitative predictions for Pb(Zr

0.4

0.6

films should also apply to PbTiO

films. Three common growth orientations (Maruyama et al., 1998; Zybill et

al., 2000; Kelman et al., 2001; Kanno et al., 2003; Fong et al., 2004; Liang

and Wu, 2005) are investigated, namely [001], [110], and [111]. The films

all have Pb-O terminated surfaces, and are represented by supercells that are

finite along the film orientation, to be denoted as the z′-direction, and repeated

periodically along two in-plane x′- and y′-directions. The investigated thin

films are either stress-free or experience an epitaxial strain in the (x′, y′)-

planes. The atomic sites’ coordinates, the matrices appearing in the dipole–

dipole interactions (Ponomareva et al., 2005c) and the strain tensor are all

first expressed in the x′y′z′ coordinate system and then transformed into the

xyz system (in which the x-, y- and z-axes are along the [100], [010], and

[001] directions, respectively) before using the total energy in Monte-Carlo

simulation.

Figure 19.1 show the x-, y-, and z-components of the spontaneous

polarization (and the resulting dipole patterns) for the PZT thin films at

10K as a function of the β screening coefficient. Parts (a), (b) and (c) correspond

to [001] films under stress-free conditions, a tensile strain of 2.65% in the

(x′, y′)-planes, and under a compressive strain of –2.65% in the (x′, y′)-

planes, respectively. Parts (d)–(f) and (g)–(i) show similar data but for a film

orientation along [110] and [111], respectively. The thicknesses of the [001],

[110], and [111] films are equal to 48.0, 48.1, and 46.2 Å, respectively. A

broad variety of dipole pattern is clearly seen. In particular, one can enunciate

the following general rule: the films adopt patterns for which the out-of-

plane component of the spontaneous polarization vanishes for OC-like

conditions (i.e. below a critical value of β). Such vanishing occurs to ‘kill’

the large depolarizing field (associated with such boundary conditions) that

would occur if the polarization would have a non-zero component along the

out-of-plane direction. For instance, and as indicated by Fig. 19.1(a), the

stress-free [001] Pb(Zr

0.4

0.6

film exhibits dipoles that all point along

[100] for small β. In order to satisfy the general rule, this film thus adopts a

polarization lying along a direction that is a possible minimum-energy direction

of the polarization in the bulk and that is perpendicular to the film orientation.

There is a second way to satisfy the general rule, as followed by the

stress-free Pb(Zr

0.4

0.6

films grown along [110] and [111] for small β:

the supercell average of the out-of-plane polarization component in these

two cases is still equal to zero as in [001] Pb(Zr

0.4

0.6

, but (as indicated

in the left schematizations above Figs 19.1d and e) this annihilation occurs

by developing ≈90° periodic nanostripe domains [note that the average angle

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Domains in ferroelectric nanostructures from first principles 575

[001] [110]

[111]

17 22

[001]

[100]

[110]

[11 0]

[111]

[11 0]

[001]

[100]

[110]

[11 0]

[111]

[11 0]

[001]

[100]

[110]

[11 0]

[111]

[11 0]

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Polarization (C/m

)

1.2

0.6

0.0

–0.6

–1.2

1.2

0.6

0.0

–0.6

–1.2

1.2

0.6

0.0

–0.6

–1.2

0.85 0.90 0.95 1.00

19.1

Cartesian components of the ground-state spontaneous polarization in Pb(Zr

0.4

0.6

(≈ 46–48Å-thick) films as a

function of the screening coefficient β and expressed in the

xyz

coordinate system, when choosing 16 and 24 unit cells

for the periodicity of the (in-plane)

′- and

′-directions for [110] and [111] films under compressive strain and 12 unit

cells for the periodicity of the

′ - and

′-directions for all other films (see text for the definition of the

xyz

and

′

coordinate systems). (a), (b) and (c): [001] films under stress free, 2.65% tensile strain and

–2.65% compressive strain,

respectively; (d)–(f): same as (a)–(c) but for a [110] film; (g) and same as (a)–(c) but for a [111] film. The vertical lines

characterize the transition of the dipoles pattern from SC-like conditions to OC-like conditions. The schematization of

these two different patterns is given above each part. The width of the stripe domains is given in

Å. Open circles,

diamonds and stars display

and

, respectively. (Reprinted with permission from Ponomareva and Bellaiche,

2006).

δ = 0% δ = + 2.65% δ = – 2.65%

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Handbook of dielectric, piezoelectric and ferroelectric materials576

made by the dipoles belonging to different stripes is in fact close to 70°]. The

dipoles inside each domain are aligned along directions that are close to

[100] or [010] (that are possible minimum-energy directions for the bulk

polarization). The reason behind the dramatic difference in homogeneity of

dipole patterns between the [111] and [001] films is that all the possible

〈001〉 minimum-energy directions of the bulk polarization are far away from

the planes perpendicular to the film orientation in the case of the [111] film.

An homogeneously in-plane polarized [111] film would thus require a

significant deviation of the direction of the dipoles with respect to the minimum-

energy directions of the bulk polarization. Such significant deviation does

not occur in stress-free [111] Pb(Zr

0.4

0.6

because of its large anisotropy.

On the other hand, the dipoles in the [110] film do have the possibility to all

lie along an in-plane direction that is also a minimum-energy direction for

the bulk, namely [001]. However, the analysis of our computations reveals

that the stress-free [110] film prefers to rather form ≈90° nanostripe domains

because of a gain in short-range interactions. Interestingly, 90° periodic

stripe domains were previously observed in [111] PZT films (Zybill et al.,

2000), but not yet in [110] ferroelectric films − to the best of our knowledge.

A tensile strain and OC-like conditions both disfavor out-of-plane

components of dipoles, because of the strain–dipole coupling and the existence

of a large depolarizing field along the film orientation, respectively. As a

result, most of the films under tensile strain develop homogeneous in-plane

polarization pattern for small β (see, left schematizations above Fig. 19.1b

and e). The only exception from this trend is the [111] Pb(Zr

0.4

0.6

film,

as seen in Fig. 19.1(h). Such exception is once again caused by the material

desire to have dipoles close to the possible minimum-energy polarization

directions of the bulk.

As indicated by the left schematizations above Figs 19.1(c)–(i), the thin

films all adopt flux-closure periodic nanostripe domains (which bear

resemblance with those observed in magnetic films; Patel et al., 1968; Aitlamine

et al., 1991; Hubert and Schafer, 1998) for compressive strains and large

residual depolarizing fields − because of the competition between the

compressive strain that favors out-of-plane components of the dipoles and

the large depolarizing field that desires to kill the out-of-plane component of

the polarization (Fong et al., 2004; Kornev et al., 2004). Interestingly, the

morphology of the stripe domains depends on the film orientation: [001]

Pb(Zr

0.4

0.6

exhibits the out-of-plane 180° nanostripes that were observed

(Fong et al., 2004) in [001] PbTiO

ultra-thin films under similar boundary

conditions. On the other hand, [110] and [111] Pb(Zr

0.4

0.6

films are

more reluctant to have dipoles along the film orientation and prefer to adopt

≈ 90° nanodomains (note that the average angle made by the dipoles belonging

to different stripes is in fact close to 105°). The existence of these latter

domains for compressive strain can be traced back to the difficulty of rotating

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Domains in ferroelectric nanostructures from first principles 577

the polarization away from a 〈100〉 direction in the Pb(Zr

0.4

0.6

bulk − as

demonstrated by the fact that it is possible to transform these ≈90° domains

into 180° out-of-plane nanostripe domains in thin films grown along [110] or

[111] and having compositions associated with a tetragonal phase in the bulk

via two different processes: (1) increase the magnitude of the compressive

strain – as we numerically found out in Pb(Zr

0.4

0.6

; or (2) decrease the

composition towards the morphotropic phase boundary of PZT bulk − i.e.

making the polarization easier to rotate – as we numerically checked when

going from Pb(Zr

0.4

0.6

to Pb(Zr

0.5

The equilibrium width (which is half the period) of the periodic stripe

domains is found by varying the in-plane supercell sizes, and is given in Å

in Fig. 19.1. The predicted equilibrium domain width is ≈24Å for the

compressed [001] film (having a thickness of 48Å). This is in excellent

agreement with the experimental data of Fong et al. (2004) and Streiffer et

al. (2002) for PbTiO

ultra-thin films having similar thickness and growth

orientation. Such width is unusually small with respect to the domain width

of around 1000−10 000 Å observed in bulks (Merz, 1954; Arlt and Sasko,

1980). Interestingly, Kornev et al. (2004) predicted that [001] PZT thin films

having a thickness of ≈20Å exhibit an equilibrium width of ≈16 Å (i.e. four

lattice constants) for the laminar domains, which implies that the equilibrium

width of domains in ultra-thin films has a square root dependency on the

film thickness − as consistent with the prediction of Landau and Lifshitz

(1935), Kittel (1946), Kopal et al. (1997) and Catalan et al. (2006).

The stripe domain sizes can thus be intriguingly close to the ultimate limit

of a single-molecule ferroelectric memory element. The domain width of

≈ 22Å we found for the stress-free and tensile [111] 46.2Å-thick films

agrees also very well with the interpolation, via this square root dependency

on the film thickness (Landau and Lifshitz, 1935; Kittel, 1946; Kopal et al.,

1997), of the experimental data of Zybill et al. (2000) down to the thickness

of ≈46Å. Specifically, such interpolation results in a domain width of

≈ 21Å. We are not aware of any measurements done in stress-free [110],

compressed [110], or compressed [111] ultra-thin films, for which we predict

a domain width of ≈ 17Å, 22Å, and 29Å, respectively, when these films

have a thickness around ≈ 46-48Å.

19.3.2 Domains in BaTiO

ultra-thin films

We now want to determine the morphology of stripe nanodomains in [001]

BaTiO

(BTO) ultra-thin films, and compare it with those in [001] Pb(Zr,Ti)O

ultra-thin films. We focus here on a 20 Å-thick BTO film, as mimicked by a

24 × 24 × 5 supercell, under a compressive strain of –2.2% and having a

realistic electrical boundary condition corresponding to β = 0.8 (Lai et al.,

2005). As consistent with Streiffer et al. (2002), Kornev et al. (2004) and Lai

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Handbook of dielectric, piezoelectric and ferroelectric materials578

et al. (2006), the combination of a significant compressive epitaxial strain

(which favors dipoles along the z direction) with such electrical boundary

condition (that will lead to a large enough residual depolarizing field if all

the dipoles would point in parallel along the z-axis) generates stripe domains

at low temperature. Figure 19.2(a) displays such domains at 10K (which

were obtained by slowly cooling down the system) and indicates that the

stripe domains in the BTO thin films consist of periodically alternating ‘up’

and ‘down’ domains (here, the up (down) domains refer to the domains with

the z-component of the local dipoles along the +z (–z) direction), as in

compressively strained [001] PZT and PbTiO

thin films (Streiffer et al.,

(a)

(d)

(b)

(e)

(c)

(f)

19.2

=10K three-dimensional polarization patterns in 20Å-thick BTO

(001) films under a compressive strain of –2.2% and a 80% screening

of the maximum depolarizing field, for different

: stripe domains

under (a)

= 0, (b)

= 30 × 10

and (c)

= 41 × 10

V/m; bubble

domains under (d)

= 51 × 10

and (e)

= 71×10

V/m; and

monodomain under (f)

= 104 × 107 V/m. Light (dark) arrows

characterize local dipoles having a positive (negative) component

along the

-axis. (Reprinted with permission from Lai

et al.

, 2007).

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Domains in ferroelectric nanostructures from first principles 579

2002; Kornev et al., 2004; Wu et al., 2004; Fong et al., 2005; Lai et al.,

2006; Ponomareva and Bellaiche, 2006). However, the stripe domains in the

BTO film run along [110] and alternate along

[110]

– as also previously

found in Tinte and Stachiotti (2001) – while the stripe domains in PZT and

PbTiO

thin films run along [100] and alternate along [010] (Streiffer et al.,

2002; Kornev et al., 2004; Wu et al., 2004; Fong et al., 2005; Lai et al.,

2006; Ponomareva and Bellaiche, 2006). Moreover, the periodicity of these

‘diagonal’ stripes in the 20 Å-thick BTO film is of ≈4.3 lattice constants

along

[110]

(which is consistent with Tinte and Stachiotti (2001) since Fig.

19.2(a) indicates an alternation of the stripe of six lattice constants along the

[100] and [010] directions. (In the following, we will refer to this ground-

state of BTO films as BTO-110-4.3.) On the other hand, as indicated above,

the periodicity of the laminar domains in a PZT film with the same thickness

(namely, 20 Å) is equal to eight lattice constants along the [010] direction.

We numerically found that the 〈110〉-oriented stripes are energetically preferred

over the 〈010〉-alternating stripes in BTO films mainly because of long-range

dipole–dipole interaction energy, E

dpl

, albeit at the cost of short-range energy,

short

. Interestingly, the lower E

dpl

in BTO-110-4.3 stripes can be understood

by solely focusing on the dipoles close to the domain walls: any of such

dipoles, say located at site i, will interact with the two (respectively, one)

antiparallel dipoles and with the two (respectively, three) parallel dipoles

located at the four sites that are nearest neighbor (in the (001) plane) of site

i, when the stripes alternate along

[110]

(respectively, [010]). This gain in

number of nearest-neighbor antiparallel dipoles when going from stripes

alternating along [010] to stripes alternating along

[110]

effectively lowers

dpl

− while raising E

short

at the same time. In the case of BTO, E

dpl

lowered more than E

short

is raised, while we numerically found that the

opposite occurs for PZT films because of the different parameters inherent to

that latter material – which explains the difference in morphology of stripe

domains in these two films.

Furthermore, and as can be seen in Fig. 19.3, which shows the real-space

distribution of the local dipoles in the ground-states, two other main differences

exist between the morphology of the stripe domains in BTO versus PZT

films. They are: (1) the dipoles of a given stripe domain are nearly constant

in direction (i.e. parallel or antiparallel to the z-axis) and magnitude inside

the BTO films, while such dipoles continuously rotate across the stripe

inside the PZT films; (2) the dipoles at the surfaces can ‘only’ deviate from

the z-axis by up to 45° in the BTO film, while in-plane surface dipoles ocurs

in PZT films in order to close the flux (Kornev et al., 2004; Wu et al., 2004;

Lai et al., 2006). To better understand such differences, we decided to construct

another domain pattern in a 20Å-thick BTO film (keeping the same boundary

conditions as above), to be denoted by BTO

PZT

-010-8. Such latter state exhibits

the same dipole configuration as the equilibrium domain pattern of a 20Å-

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