Lallart M. Ferroelectrics: Characterization and Modeling

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Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

389

Fig. 9. Schematic illustration of a periodic superlattice composed of a ferroelectric and a

paraelectric layers. The thicknesses of PbTiO

(PT) and SrTiO

(ST) layers are L

and L

respectively.

L denotes the periodic thickness of the PT/ST superlattice.

and the free energy per unit area for the ST layer with thickness

L is

246

11, 12,

2462

mST

ST ST ST ST

ST ST

dST

Fqqq

dz s s

eqdz

αβγκ





=++++













−







(68)

where

and

corresponds to the polarization of PT and ST layers, respectively. For the

superlattices with the polarizations perpendicular to the layer’s surfaces/interfaces, the

inhomogeneity of polarization means that the depolarization field effect is essential. In eqs.

(67) and (68),

and

are expressed as

12,

11, 12,

12,

11, 12,

αα

ββ



=−









(69)

where

and

are the Landau coefficients of layer j ( j : PT or ST), as usual.

11,

s and

12,

s are the elastic compliance coefficients, whereas

12,

Q is the electrostrictive constant.

()

uaaa=− denotes the in-plane misfit strain induced by the substrate due to the

lattice mismatch.

a is the unconstrained equivalent cubic cell lattice constants of layer j and

a is the lattice parameter of the substrate.

is the gradient coefficient, determining the

energy cost due to the inhomogeneity of polarization.

SrTiO

PbTiO

-L

substrate

Ferroelectrics - Characterization and Modeling

390

With the assumption that the ferroelectric layers are insulators with no space charges, the

depolarization field

e in the PT and ST layers can be expressed by

() ()

()

() ()

()

dPT

dST



=− −







=− −





(70)

respectively. In eq. (70),

denotes the dielectric permittivity in vacuum. The second term

describes the mean polarization of one-period superlattice

−









(71)

with the periodic thickness

PT ST

LL L=+. It is important to note here that

e acts as the

depolarization field, if its direction is opposite to the direction of ferroelectric polarization. If

e inclines in the same direction of polarization, it cannot be regarded as the

depolarization field; thus, we denote

e as “the internal electric field”. Hence, the average

internal electric field of one-period superlattice is defined as

() ()

ddPT dST

Eezdzezdz

−















. (72)

The intrinsic coupling energy between the polarizations at the interfaces 0z = of the two

layers is described as

()

Iii

Fpq

=−

, (73)

where

and

are the interface polarizations at 0z = for the PT and ST layers,

respectively. In eq. (73), the parameter

describes the strength of intrinsic interface

coupling and it can be conveniently related to the dielectric permittivity in vacuum

= , (74)

where

denote the temperature-independent interface coupling constant. In this case, the

existence of the interface coupling 0

≠ leads to the inhomogeneity of polarization near the

interfaces, besides the effect of the depolarization field.

In the calculations, it is assumed that 1 unit cell (u.c.) ≈ 0.4 nm and the thickness of ST layer is

maintained at L

≈ 3 u.c. The lattice constants in the paraelectric state are

a = 3.969 Å and

a = 3.905 Å for PT and ST layers, respectively. Based on the lattice constants, the lattice

strains are obtained as

,mPT

u = −0.0164 and

,mST

u = 0.

In Fig. 10, we show the average polarization P and internal electric fields

E of PT/ST

superlattices as a function of thickness ratio L

for different strength of interface

coupling

. It is seen that P and

E decrease with increasing

. As

increases, the

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

391

critical thickness ratio (at which P vanishes) shifts to a higher value. It is seen that there is

a good agreement between the calculated and measured polarizations. The calculated

polarizations using

= (black line) agree reasonably well with most of the

experimental measurements for L

> 0.4, implying that the strength of interface

coupling at this regime is strong. At the L

≤ 0.4 region, the predicted polarizations

with

0.2

= (red line) and 0.05 (blue line) agree well with some of the experimental

measurements. The

E versus L

curves show a trend similar to P versus L

e.g.

E disappears at a critical thickness ratio. For each

, the critical thickness ratio of

E coincides with that of P. It is remarkable to see that for 0

E > , internal electric field is

parallel to the direction of the ferroelectric polarization in PT layer, which enhances the

polarization of the superlattice.

0.0 0.5 1.0 1.5 2.0

0.2 0.4

0.0

0.1

0.2

0.2 0.4

Polarization

[

μC/cm

]

Internal Electric Field

[MV/cm]

Fig. 10. Polarization and internal electric field as a function of thickness ratio L

PT/ST superlattices at T = 300K. The values of

are: 10 (▬), 0.2 (▬) and 0.05 (▬). Solid

dots (●) represent experimental results from Dawber et al (Dawber et al., 2007). The insets

in each figure show the corresponding curves in smaller scale (Chew et al., unpublished).

4. Conclusion

We have proposed a model to study the intrinsic interface coupling in ferroelectric

heterostructure and superlattices. The layered structure is described using the Landau-

Ginzburg theory by incorporating the effect of coupling at the interface between the two

constituents. Explicit analytical expressions describing the polarization at the interface

Ferroelectrics - Characterization and Modeling

392

between bulk ferroelectrics and bulk dielectrics were derived and discussed. Here, we

mainly discussed only cases where the transition of the ferroelectric constituent is of second

order (Chew et al., 2003), though cases of heterostructure at the interfaces involving first-

order phase transition were also reported (Tsang et al., 2004).

We further extend the model to investigate the ferroelectricity of superlattice by

incorporating the thickness effect. Using the explicit expressions derived from the model,

the polarization modulation profiles, phase transitions and dielectric susceptibilities of a

superlattice are presented and discussed in detail (Ishibashi & Iwata, 2007; Chew et al., 2008;

Chew et al., 2009). The effort to obtain the explicit analytical solutions using the continuum

model of Landau-Ginzburg theory is worthwhile. This is because those expressions allow us

to gain general insight on how the intrinsic polarization coupling at the interface influences

the physical properties of those hybrid structures. Note that the effect of an applied electric

field on the polarization behaviors of heterostructure at the interfaces (Chew et al., 2005;

Chew et al., 2006) and superlattices (Chew et al., 2011; Chew et al., unpublished) is also very

important. However, those studies were not discussed. We have also constructed a one-

dimensional model on the basis of the Landau-Ginzburg theory to investigate the

polarization and dielectric behaviors (Chew et al., 2006; Chew et al., 2007), as well as the

switching characteristics (Chew et al., unpublished).

At the end of the discussion, we show how the present model can be applied to study the

ferroelectric polarization of epitaxial PT/ST superlattices with the polarizations

perpendicular to the surfaces/interfaces of the constituent layers (Chew et al., unpublished).

The effects of interface, depolarization field and substrate-induced strain are required to

include in the model. Our calculated polarizations (Chew et al., unpublished) agree

reasonably well with recent experimental measurements (Dawber et al., 2007). From our

study, it suggests that the recent experimental observation on the unusual recovery of

ferroelectricity at thickness ratio of L

< 0.5 (Dawber et al., 2005) may be related to a

weakening of ferroelectric coupling at the interface. It is certainly interesting to look at the

dielectric susceptibilities and polarization reversals of the superlattices, which will be

reported elsewhere.

5. Acknowlegement

This work is supported by the University of Malaya Research Grant (No: RG170-11AFR ).

L. H. Ong acknowledges the support from FRGS Grant (No: 203/PFIZIK/6711144) by the

Ministry of Higher Education, Malaysia.

6. References

Nakagawara, O.; Shimuta, T.; Makino, T.; Arai, S.; Tabata, H. & Kawai, T. (2000). Epitaxial

Growth and Dielectric Properties of (111) Oriented BaTiO

/SrTiO

Superlattices by

Pulsed-laser Deposition, Applied Physics Letter, Vol. 77, No. 20, (November 2000),

pp. 3257-3259, ISSN 0003-6951

Dawber, M.; Lichtensteiger, C.; Cantoni, M.; Veithen, M.; Ghosez, P.; Johnston, K.; Rabe, K.

M.; & Triscone, J. M. (2005). Unusual Behavior of the Ferroelectric Polarization in

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

393

PbTiO

/SrTiO

Superlattices, Physical Review Letters, Vol. 95, No.17, (October 2005),

pp. 177601, ISSN 0031-9007

Bousquet, E.; Dawber, M.; Stucki, N.; Lichtensteiger, C.; Hermet, P.; Gariglio, S.; Triscone,

J. M. & Ghosez, P. (2008). Improper Ferroelectricity in Perovskite Oxide Artificial

Superlattices, Nature, Vol. 452, No. 7188, (April 2008), pp. 732-U4, ISSN 0028 -

0836

Qu, B. D.; Zhong, W. L.; & Prince, R. H. (1997) Interfacial Coupling in Ferroelectric

Superlattices, Physical Review B, Vol. 55, No. 17, (May 1997), pp. 11218-11224, ISSN

0163-1829

Chew, K.-H.; Ishibashi, Y. ; Shin F. G. & Chan, H. L. W. (2003). Theory of Interface Structure

in Double-Layer Ferroelectrics, Journal of the Physical Society of Japan, Vol. 72, No.9,

(September 2003), pp. 3158-3165, ISSN 0031-9015

Tsang, C. H.; Chew, K.-H.; Ishibashi, Y. & Shin F. G. (2004). Structure of Interfaces in

Layered Ferroelectrics of First and/or Second Order Transition, Journal of the

Physical Society of Japan, Vol. 73, No.11, (November 2004), pp. 3158-3165, ISSN 0031-

9015

Ishibashi, Y. & Iwata, M. (2007). Landau-Ginzburg Theory of Phase Transition of

Ferroelectric Superlattices, Ferroelectrics, Vol. 354, (2007), pp. 8-12, ISSN 0015 -

0193

Chew, K.-H.; Iwata, M.; Ishibashi, Y. & Shin F. G. (2009). Polarization Modulation Profiles in

Ferroelectric Superlattices, Ferroelectrics Letters Section, Vol. 36, No.1-2, (2009), pp.

12-19, ISSN 0731-5171

Chew, K.-H.; Iwata, M.; & Shin F. G. & Ishibashi Y. (2008). Exact Expressions for Dielectric

Susceptibilities in the Paraelectric Phase of Ferroelectric Superlattices Based on

Ginzburg-Landau Theory, Integrated Ferroelectrics, Vol. 100, No.1, (2008), pp. 79-87,

ISSN 1058-4587

Dawber, M.; Stucki, N.; Lichtensteiger C.; Gariglio, S.; Ghosez, P. & Triscone J. M. (2007).

Tailoring the Properties of Artificially Layered Ferroelectric Superlattices, Advanced

Materials, Vol. 19, No. 23, (December 2007), pp. 4153-4159, ISSN 0935-9648

Chew, K.-H.; Ong, L.-H. & Iwata, M. Interface-induced Sign Change of Local Internal

Electric Field in Nanoscale Ferroelectric Superlattices, (unpublished)

Chew, K.-H.; Ishibashi Y. & Shin F. G. (2005). Ferroelectric Hysteresis Loops as the

Manifestation of Interfacial-aided Polarization Reversals in Ferroelectric

Heterostructures, J. Phys. Soc. Jpn. Vol. 74, No. 8, (August 2005), pp. 2338-2346, ISSN

0031-9015

Chew, K.-H.; Ishibashi Y. & Shin F. G. (2006). Intrinsic Ferroelectric Hysteresis Behaviors for

Heterostructures, Physica Status Solidi A–Applications and Materials Science. Vol. 203,

No. 9, (July 2006) pp. 2205-2208, ISSN 0031-8965

Chew, K.-H.; Ong, L.-H. & Iwata, M. (2011). Switching Dynamics in Ferroelectric

Superlattices, Current Applied Physics, Vol. 11, No.3, (May 2011), pp. 755-761, ISSN

15671739

Chew, K.-H.; Ong, L.-H. & Iwata, M. Influence of Dielectric Stiffness, Interface and Layer

Thickness on Hysteresis Loops of Ferroelectric Superlattices, (unpublished)

Ferroelectrics - Characterization and Modeling

394

Chew, K.-H.; Ishibashi, Y. & Shin F. G. (2006). A Lattice Model for Ferroelectric

Superlattices, Journal of the Physical Society of Japan, Vol. 75, No.6, (June 2006), pp.

064712, ISSN 0031-9015

Chew, K.-H.; Ishibashi, Y. & Shin F.G. (2007). Competition between the Thickness Effects of

Each Constituent Layer in Ferroelectric Superlattices, Ferroelectrics, Vol. 357, No.6,

(2007), pp. 697-701, ISSN 0015-0193

Chew, K.-H., Ong L.-H. & Iwata M. A One-Dimensional Lattice Model of Switching

Characteristics in Ferroelectric Superlattices, (unpublished)

First-principles Study of ABO

: Role of the B–O

Coulomb Repulsions for Ferroelectricity and

Piezoelectricity

Kaoru Miura

Corporate R&D Headquarters, Canon Inc., Shimomaruko, Ohta, Tokyo

Japan

1. Introduction

Since Cohen (Cohen & Krakauer, 1990; Cohen, 1992) proposed an origin for ferroelectricity in

perovskite oxides, investigations of ferroelectric materials using ﬁrst-principles calculations

have been extensively studied (Ahart et al., 2008; Bévillon et al., 2007; Bousquet et al., 2006;

Chen et al., 2004; Diéguez et al., 2005; Furuta & Miura, 2010; Khenata et al., 2005; Kornev et

al., 2005; Miura & Tanaka, 1998; Miura, 2002; Miura et al., 2009; 2010a;b; Miura & Furuta, 2010;

Miura et al., 2011; Oguchi et al., 2009; Ricinschi et al., 2006; Uratani et al., 2008; Vanderbilt,

2000; Z. Wu et al., 2005). Currently, using the pseudopotential (PP) methods, most of the

crystal structures in ferroelectric perovskite oxides (ABO

) as well as perovskite-relatedoxides

can be precisely predicted. However, it is also known that the most stable structures of

ABO

optimized by the ﬁrst-principles PP methods are sometimes inconsistent with the

experimental results.

BaTiO

is a well-known ferroelectric ABO

, and shows the tetragonal structure at room

temperature. However, even in this well-known material, the optimized structure by the

PP methods of ﬁrst-principles calculations is strongly dependent on the choice of the Ti PPs,

i.e., preparation for Ti 3s and 3p semicore states in addition to Ti 3d and 4s valence states

is essential to the appearance of the tetragonal structure. This is an important problem for

ferroelectricity, but it has been generally recognized for a long time that this problem is within

an empirical framework of the calculational technics (Gonze et al., 2005).

It is known that ferroelectric state appears when the long-range forces due to the dipole-dipole

interaction overcome the short-range forces due to the Coulomb repulsions. Cohen (Cohen

& Krakauer, 1990; Cohen, 1992) proposed that the hybridization between Ti 3d state and

O 2p state (Ti 3d–O 2p) in BaTiO

and PbTiO

, which weakens the short-range force of

the Coulomb repulsions between Ti and O ions, is origin of ferroelectricity. However,

it seems to be difﬁcult to consider explicitly whether the long-range force due to the

dipole-dipole interaction can or cannot overcome the short-range force only with the

Ti 3d–O 2p hybridization. Investigations about the relationship between the Ti–O Coulomb

repulsions and the appearance of ferroelectricity were separately reported. Theoretically, we

previouly investigated (Miura & Tanaka, 1998) the inﬂuence of the Ti–O

Coulomb repulsions

on Ti ion displacement in tetragonal BaTiO

and PbT iO

,whereO

denotes the O atom to the

z-axis (Ti is displaced to the z-axis). Whereas the hybridization between Ti 3d state and O

state stabilize Ti ion displacement, the strong Coulomb repulsions between Ti 3s and 3p

First-Principles

2 Ferroelectrics

states and O 2p

states do not favourably cause Ti ion displacement. Experimentally, on the

other hand, Kuroiwa et al. (Kuroiwa et al., 2001) showed that the appearance of ferroelectric

state is closely related to the total charge density of Ti–O bondings in BaT iO

. As discussed

above, investigation about a role of Ti 3s and 3p states is important in the appearance of the

ferroelectric state in tetragonal BaTiO

It has been generally known (Miura & Furuta, 2010) that the most stable structure of ABO

closely related to the tolerance factor t,

≡

+ r

√

2 ( r

+ r

)

,(1)

where r

, r

,andr

denote the ionic radii of A, B , and O ions, respectively. Generally,

the most stable structure is tetragonal for t

 1, cubic for t ≈ 1, and rhombohedral or

orthorhombic for t

 1. In fact, BaTiO

(t = 1.062) and SrTiO

(t = 1.002) show tetragonal

and cubic structures in room temperature, respectively. However, under external pressure,

e.g., hydrostatic or in-plane pressure (Ahart et al., 2008; Fujii et al., 1987; Haeni et al., 2004),

the most stable structures of ABO

generally change; e.g., SrTiO

shows the tetragonal and

ferroelectric structure even in room temperature when the a lattice parameter along the [100]

axis (and also the [010] axis) is smaller than the bulk lattice parameter with compressive

stress (Haeni et al., 2004). Theoretical investigations of ferroelectric ABO

under hydrostatic

or in-plane pressure by ﬁrst-principles calculations have been reported (Bévillon et al., 2007;

Diéguez et al., 2005; Furuta & Miura, 2010; Khenata et al., 2005; Kornev et al., 2005; Miura et

al., 2010a; Ricinschi et al., 2006; Uratani et al., 2008; Z. Wu et al., 2005), and their calculated

results are consistent with the experimental results. However, even in BaTiO

,whicharea

well-known lead-free ferroelectric and piezoelectric ABO

, few theoretical papers about the

piezoelectric properties with in-plane compressive stress have been reported.

Recently, we investigated the roles of the Ti–O Coulombrepulsions in the appearance of a

ferroelectric states in tetragonal BaTiO

by the analysis of a ﬁrst-principles PP method (Miura

et al., 2010a). We investigated the structural properties of tetragonal and rhombohedral

BaTiO

with two kind of Ti PPs, and propose the role of Ti 3s and 3p states for ferroelectricity.

Moreover, we also investigated the structural, ferroelectric, and piezoelectric properties of

tetragonal BaTiO

and SrTiO

with in-plane compressive structures (Furuta & Miura, 2010).

We discussed the difference in the piezoelectric mechanisms between BaTiO

and SrTiO

with

in-plane compressive structures, which would be important for piezoelectric material design.

In this chapter, based on our previous reports (Furuta & Miura, 2010; Miura et al., 2010a),

the author discusses a general role of B–O Coulomb repulsions for ferroelectricity and

piezoelectricity in ABO

, especially in BaTiO

and SrTiO

2. Calculations

Calculations of BaTiO

and SrTiO

were performed using the ABINIT packagecode(Gonze

et al., 2002), which is one of the norm-conserving PP (NCPP) methods. Electron-electron

interaction was treated in the local-density approximation (LDA) (Perdew & Wang, 1992).

Pseudopotentials were generated using the

OPIUM code (Rappe, 2004):

(i) In order to investigate the role of Ti 3s and 3p states for BaTiO

, two kinds of Ti PPs were

prepared: one is the Ti PP with 3s, 3p, 3d and 4s electrons treated as semicore or valence

electrons (Ti3spd4s PP), and the other is the Ti PP with only 3d and 4s electrons treated as

valence electrons (Ti3d4s PP). The above seudopotentials were generated using the

OPIUM

396

Ferroelectrics - Characterization and Modeling

First-principles Study of ABO

: Role of the B–O Coulomb Repulsions for Ferroelectricity and Piezoelectricity 3

code (Rappe, 2004), and the differences between the calculated result and the experimental

one are within 1.5 % of the lattice parameter and within 10 % of the bulk modulus in the

optimized calculation of bulk Ti in both PPs. Moreover, Ba PP with 5s, 5p and 6s electrons

treated as semicore or valence electrons, and O PP with 2s and 2p electrons treated as semicore

or valence electrons, were also prepared. The cutoff energy for plane-wave basis functions was

settobe50Hartree(Hr).A6

×6 ×6 M onkhorst-Pack k-point mesh was set in the Brillouin zone

of the unit cell. The number of atoms in the unit cell was set to be ﬁve, and positions of all

the atoms were optimized within the framework of the tetragonal (P4mm)orrhombohedral

( R3m) structure.

(ii) The ferroelectric and piezoelectric properties of SrTiO

and BaTiO

with compressive

tetragonal structures are investigated. Pseudopotentials were generated using the

OPIUM

code (Rappe, 2004); 4s (5s), 4p (5p) and 5s (6s) electrons for Sr (Ba), 3s, 3p, 3d and 4s electrons

for Ti, and 2s and 2p electrons for O were treated as semicore or valence electrons. The cutoff

energy fo the plane wave basis functions was set to be 50 Hr. A 6

×6 ×6 Monkhorst-Pack

k-point mesh was set in the Brillouin zone of the unit cell. The number of atoms in the ABO

unit cell was set to be ﬁve, and the coordinations of all the atoms were optimized within a

framework of the tetragonal (P4mm) structure. An 6

×6×6 Monkhorst-Pack k-point sampling

was s et in Brillouin zone of the unit cell.

In the present calculations, spontaneous polarizations and piezoelectric constants were also

evaluated, due to the Born effective charges (Resta, 1994) and density-functional perturbation

theory (Hamann et al., 2005; X. Wu et al., 2005). The spontaneous polarization of tetragonal

structures along the [001] axis, P

,isdeﬁnedas,

≡

∑

∗

(k)u

(k) ,(2)

where e, c,andΩ denote the charge unit, the lattice parameter of the unit cell along the [001]

axis, and the volume of the unit cell, respectively. u

(k) denotes the displacement along the

[001] axis of the kth atom, and Z

∗

(k) denotes the Born effective charges (Resta, 1994) which

contributes to the P

from the u

(k).Thepiezoelectrice constants, on the other hand, are

deﬁned as

αβ

≡



∂P

∂η



∑



∂P

∂u

(k)



∂u

(k)

∂η

,(3)

where P, η,andu

(k) denote the spontaneous polarization, the strain, and the displacement of

the kth atom, respectively. α and β denote the direction-indexes of the axis, i.e., 1 along the

[100] axis, 2 along the [010] axis, and 3 along the [001] axis, respectively. In eq. (3), the ﬁrst

term of the right hand denotes the clamped term evaluated at vanishing internal strain, and

the second term denotes the relaxed term that is due to the relative displacements. According

to the eqs. (2) and (3), therefore, e

or e

can be especially written as,

3β



∂P

∂η



∑

∗

(k)

∂u

(k)

∂η

(β = 3, 1) .(4)

397

First-Principles Study of ABO

: Role of the

B–O Coulomb Repulsions for

Ferroelectricity and Piezoelectricity

4 Ferroelectrics

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㻜

㻠㻜

㻤㻜

㻝㻞㻜

㻙㻞㻚㻠㻙㻞㻚㻞㻙㻞㻙㻝㻚㻤㻙㻝㻚㻢㻙㻝㻚㻠㻙㻝㻚㻞㻙㻝

㻰㻻㻿㻌㻔㻛㻴

㼞

㻙㻝㻙㻜㻚㻤㻙㻜㻚㻢㻙㻜㻚㻠㻙㻜㻚㻞㻜㻜㻚㻞㻜㻚㻠

㻳㻔㻕

㻙㻝㻌㻌㻌㻙㻜㻚㻤㻌㻌㻙㻜㻚㻢㻌㻌㻙㻜㻚㻠㻌㻌㻙㻜㻚㻞㻌㻌㻌㻌㻜㻌㻌㻌㻌㻌㻜㻚㻞㻌㻌㻌㻌㻜㻚㻠㻙㻞㻚㻠㻌㻌㻙㻞㻚㻞㻌㻌㻌㻙㻞㻌㻌㻌㻙㻝㻚㻤㻌㻌㻙㻝㻚㻢㻌㻌㻙㻝㻚㻠㻌㻌㻙㻝㻚㻞㻌㻌㻌㻙㻝

㻱㻺㻱㻾㻳㼅㻌㻔㻴㼞㻕

㻱㻺㻱㻾

㻳

㼅㻌

㻔

㻴㼞

㻕

㻱㻺㻱㻾㻳㼅㻌㻔㻴㼞㻕

Fig. 2. Total density of states (DOS) of fully optimized tetragonal BaTiO

with the

Ti3spd4s PP (solid line) and cubic BaTiO

with the Ti3d4s PP (red dashed line) (Miura et al.,

2010a).

3. Results and discussion

3.1 BaTiO

: Role of Ti 3s and 3p states for ferroelectricity

In this subsection, the author discusses the role of Ti 3s and 3p states for ferroelectricity for

ferroelectricity in tetragonal BaTiO

Figures 1(a) and 1(b) show the optimized results for the ratio c/a of the lattice parameters and

the value of the Ti ion displacement (δ

)asafunctionofthea lattice parameters in tetragonal

BaTiO

, respectively. Results with arrows are the fully optimized results, and the others results

are those with the c lattice parameters and all the inner coordinations optimized for ﬁxed

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398

Ferroelectrics - Characterization and Modeling