Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

375
because
'
2
0f = for a non-polar dielectric. p and q are the order parameters of the
ferroelectric and dielectric consituents, respectively.
1
α
is a temperature-dependent
parameter

()
110 0
TT
αα
=−, (6)
where
10
0
α
> is a temperarature-independent parameter.
2
0
α
> ,
1
0




∂∂∂

−=


′
∂∂∂


(7)
with the boundary conditions

i
i
p
p
qq
=


=

at 0x = , (8a)
and

and 0 at ,
and 0 at ,

and
i
q
as order parameters. This gives F as a function of
i
p
and
i
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,

2
22 24
111
()()
242
bb
d
p
pp pp
dx
αβκ

−+ −=


, (9)
and


1
K
α
κ
=− . (12)
For the dielectric constituent B, the solution of eq. (10) gives

2
exp( ),
i
qq Kx=− (13)
with

2
2
2
.K
α
κ
= (14)
If
i
p
is determined,
i
x can be obtained from eq. (11). In eqs. (11) and (13), the magnitude of
the interface polarizations
i
p
and

F
pp pq
p
βκ
λ
∂
=−+−=
∂
, (16)
and

22
()0
iii
i
F
qpq
q
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
λ
= and
λ
→∞. The variation of
polarization across the interface can be examined by looking into the continuity or

ii
p
q= , implying that the polarization
is continuos across the interface. In order to find
ii
p
q= , it is convenient to write eq. (15) in
term of only
i
p
as

22
323 2
11
1
(3 2)
32 2
iibb i
F
pppp p
ακ
βκ
=−++, (20)
and by minimizing it, we obtain

22 22
11 11
11
1

0.5
1.0p and q
x

p and q
p and qFig. 1. Spatial dependence of polarization at the interface region of ferroelectric/dielectric
heterostructures with
1
10
λ
−
= (top), 1 (middle) and 0 (bottom). In the curves, the
parameters are:
1
1
α
=− ,
2
1
α
= ,

−
, which is
governed by
2
α
and
2
κ
.

Ferroelectrics - Characterization and Modeling

378
0 1020304050
0.0
0.2
0.4
0.6
0.8
1.0

p
i
- q
i
λ
−1

Fig. 2. Mismatch in the polarization at the interface of ferroelectric/dielectric
heterostructures as a function of


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
1
and L
2
, respectively. L = L
1
+ L
2
is the periodic thickness of the superlattice.

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

379
In eq. (22), the total free energy density of the ferroelectric layer
1
F is given by

1
2
/2
24
111
1
0
d
242d
L

22d
L
L
q
Fq qEx
x
ακ



=+−









, (24)
respectively. In eqs. (23) and (24), p and q are the order parameters of the ferroelectric layer
and paraelectric layer, respectively.
E
denotes the external electric field.
The coupling energy at the interface between the ferroelectric- and dielectric-layers is as
shown in eq. (3). In this case, the boundary conditions at the interface (x = L
1
/2) are
described by

=−


(25)
3.1 Polarization modulation profiles
We first look at the polarization modulation profiles of the ferroelectric/dielectric
superlattice under the absence of an external electric field 0E = (Chew et al., 2009). The
polarization profiles of p and q for the ferroelectric and dielectric layers, respectively, can be
obtained using the Euler-Lagrange equation. For the dielectric layer, the Euler-Lagrange
equation is

2
22
2
d
d
q
q
x
κα
=
, (26)
and
()qx can be obtained as

c2
() cosh
2
L
qx q K x

x
κα β

=−+−


, (29)

Ferroelectrics - Characterization and Modeling

380
where
c
p
is the p value at d/d 0px= . In this case,
c
p
is the maximum value of p at 0x = .
Using
()
c
() sin
p
x
p
x
θ
= and
2
b11

, (30)
where
(,)Fk
θ
and
i
(,)Fk
θ
are the elliptic integral of the first kind with the elliptic modulus
k given by

2
2
c
22
bc
2
p
k
pp
=
−
. (31) Fig. 4. Spatial dependence of polarization for a superlattice with
1
5L = and
2
3L = for

λ
−
are: 100 (dot), 16 (dash-dot-dot),
8 (dash-dot), 2 (dash), and 0 (solid). Dotted circles represent the interface polarizations
(Chew et al., 2009).
Let us discuss the polarization modulation profiles in a ferroelectric/dielectric superlattice
using the explicit expressions. The characteristic lengths of polarization modulations in the
ferroelectric layer near the transition point and the dielectric layer are given by
1
111
/K
κα
−
=− and
1
222
/K
κα
−
= , respectively. Figure 4 illustrates an example of
1
λ
−
dependence of polarization modulation profiles. It is seen that the modulation of the
polarization is obvious in the ferroelectric layer, but not in the dielectric layer. This is
because
111
/2 / 2L
κα
>− = and

2
2
sin ,
2422 2
1
LD
FJppppCpqq
L
k
ακ α β λ
θ


−

=+++−+



+





(32)
where

()
i





=+





=−




(33)
with
()
1
iic
sin /
p
p
θ
−
= . By utilizing
()
22 2
b
/2

22
KL
AKL
ακ
λ
−
=− +
, (35)
and O(p
c
4
) indicates the higher order terms of p
c
4
.
From the equilibrium condition for q
c
, dF/dq
c
= 0, the condition of the transition point can
be obtained as A - C
2
/D = 0, i.e.,

11
2
11
11
sin cos 0
22

−
for different dielectric stiffness
2
α
.
For a superlattice with a soft dielectric layer
2
0.1
α
=
and 1,
c
p
remains almost the same as
the bulk polarization
cb
~
p
p
for all
1
λ
−
. For the case with
2
5
α
=
,
c
Fig. 5. p
c
and q
c
as a function of
1
λ
−
for various
2
α
, where
2
α
is 0.1, 1, 5, 10, and 50. The
other parameters are the same as Fig. 4 (Chew et al., 2009).
As the temperature increases, the ferroelectric layer can be in the ferroelectric state or in the
paraelectric state. Phase transition may or may not take place, depending on the model
parameters. Let us examine the stability of superlattice in the paraelectric state by taking
into account the polarization profile to appear in the ferroelectric state. Instead of the exact
solutions obtained from the Euler-Lagrange equations, which are in term of the Jacobi
Elliptic Functions, we use (Ishibashi & Iwata, 2007)

1
cos
c
p
pKx= , (38)

(40)
where

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

383

()
2
22
111 11
11111 11
1
1 1 11 11
1
11
2
222 22 22
2
2
11 22
1
sin cos ,
42
3sin sin2
,
44 8
sinh cosh cosh ,
22 2
cos cosh .




=+ +





=+



=


(41)
Similarly, from the equilibrium condition for q
c
, dF/dq
c
= 0, we find eq. (40) can be reduced
to a more simple form as

*
24
11
cc
2
24





(43)
where
(,)Rr
λ
is given by eq. (37). r is a function of
2
α
,
2
κ
and
2
L . The transitions of the
superlattice from a paraelectric phase to a ferroelectric state occurs when
*
1
0a =
. Note here
that
*
1
a
consists of the physical parameters from both the ferroelectric and dielectric layers.
It is seen that the influence of the dielectric layer via
λ
becomes stronger with increasing

, we obtain the wave number k. It is qualitatively
Fig. 6. The dependence of the wave number k for various R/L
1
when κ
1
= 1 and L
1
= 1/2. The
curves show the cases 1) R/L
1
= 0, 2) R/L
1
= 2, 3) R/L
1
= 20, 4) R/L
1
= 200 and 5) R/L
1
=∞.
Dotted lines denote the transition point of each case (Ishibashi & Iwata, 2007).

Ferroelectrics - Characterization and Modeling

384
obvious that k is small, implying a flat polarization profile, when the contribution from the
dielectric layer R, is small, while
2

,
d
p
p
E
x
q
qE
x
ακ
ακ

−=




−=


(44)
with the condition that F (eq. (22)) including the interface energy (eq. (3)) takes the
minimum value. Note that in the present system, the ferroelectric transition point
c
α
is
negative. Thus, one must consider both cases
1
0
α

. (45)
3.3.1 Case
1
0
α
≥

For the case of
1
0
α
≥ , the exact solutions are

c1
1
c2
2
cosh ,
cosh ,
2
E
ppE Kx
LE
qqE Kx
α
α

=+






=+


(47)
In this case,
111
/K
ακ
= and
2
K is given by eq. (14). By utilizing eqs. (46) and (47), we can
express F in terms of
c
p
and
c
q as

Intrinsic Interface Coupling in Ferroelectric Heterostructures and Superlattices

385

22 2
12
cccc1c2c
2
22

22
2
12
sinh cosh ,
22
2
sinh cosh ,
22
cosh cosh ,
22
11
cosh ,
2
11
cosh .
2
KL
aKL
K
KL
aKL
K
KL KL
c
KL
d
KL
d
α
λ





(49)
Using the equilibrium conditions
cc
//0Fp Fq∂∂=∂∂=
, we find

211
c22
22 12
11
cosh sinh ,
22
KL
pKL
aA K
λα
αα

−
=−


(50)
and

122

22
sinh sinh
22
pq
KL L KL L
KL L KL L
χ
αα
=+++. (53)
3.3.2 Case
1
0
α
<

In this case, the exact solutions of eq. (44) are

c1
1
c2
2
cos ,
cosh ,
2
E
ppE Kx
LE
qqE Kx
α
α

2
cosh .
2
KL E
ppE
KL E
qqE
α
α

=+




=+


(55)
Similarly, we find

22 2
12
cccc1c2c
2
22
aa
F
pq
c

22
sinh cosh ,
22
cos cosh ,
22
11
cos ,
2
11
cosh ,
2
KL
aKL
K
KL
aKL
K
KL KL
c
KL
d
KL
d
α
λ
α
λ
λ
λ
αα

αα

−
=−


(58)
and

122
c11
21 12
11
cosh sin ,
22
KL
qKL
aA K
λα
αα

=−


(59)
with

2
1
2

where the phase transition point is given by
2
12
/0Aa c a=− =. Using
2
12
/0Aa c a=− =,
the condition of the transition point is

2
22
2
1112
11
2
222
1
22
2
sin
2
sin cos 0.
22
sin cosh
22
KL
KLK
KL
KL
K

,
2
1
α
=
, for cases of: (1) 0
λ
= , (2) 0.3
λ
= , (3) 3
λ
=
(Chew et al., 2008). Fig. 8. Spatial dependence of polarization for a superlattice with
12
3LL==. The parameters
adopted for the calculation are:
12
1
κκ
==,
2
1
α
= ,3
λ
= , for cases of (1)
1

8
-1 0 1 2 3 4
x
p
q
1
2
3
Polarizatio
n

Ferroelectrics - Characterization and Modeling

388
The result indicates that the second-order phase transition is possible in our model of the
superlattice structure. It is seen that the susceptibility is continuous at
1
0
α
= , though the
susceptibility is divided into two different functions at
1
0
α
= . Taking the limit of
1
0
α
=±
from both the positive and negative sides, the explicit expression for the susceptibility at



, (63)
implying that the susceptibility is always continuous at
1
0
α
= . It is worthwhile to look at
the field-induced polarization profile at
1
0
α
= because
1
K becomes zero at
1
0
α
= . By
taking the limit of
1
0
α
=± from both the positive and negative sides for the polarization p,
the expressions for the polarization profiles in
()
p
x and ()qx can be explicitly expressed as

()

KL
EK L
LE
qx K x
KL
αα


=−+






(65)

Equation (64) depicts the polarization profile
()
p
x that exhibits a parabolic modulation at
1
0
α
= , as shown in Fig. 8. The polarization profile obtained near the transition point may
coincide with the polarization modulation pattern of the ferroelectric soft mode in the
paraelectric phase.
3.4 Application of model to epitaxial PbTiO
3
/SrTiO

246

11, 12,
,
2462
1
,
2
PT
mPT
PT PT PT PT
PT
L
PT PT
dPT
u
dp
Fppp
dz s s
epdz
αβγκ
−


=++++


+



**

,
246
0
11, 12,
,
2462
1
.
2
ST
L
mST
ST ST ST ST
ST
ST ST
dST
u
dq
Fqqq
dz s s
eqdz
αβγκ


=++++


+

2
12,
*
11, 12,
4
,
4
,
j
jj
m
j
jj
j
jj
jj
Q
u
ss
Q
ss
αα
ββ

=−

+




j
S
uaaa=− denotes the in-plane misfit strain induced by the substrate due to the
lattice mismatch.
j
a is the unconstrained equivalent cubic cell lattice constants of layer j and
S
a is the lattice parameter of the substrate.
j
κ
is the gradient coefficient, determining the
energy cost due to the inhomogeneity of polarization.
SrTiO
3

SrTiO
3
PbTiO
3
PbTiO
3
0
-L
PT

L
ST

z
L

ez
q
zP
ε
ε

=− −




=− −


(70)
respectively. In eq. (70),
0
ε
denotes the dielectric permittivity in vacuum. The second term
describes the mean polarization of one-period superlattice

0
0
1
,
ST
PT
L
L
P

,,
0
1
() ()
ST
PT
L
ddPT dST
L
Eezdzezdz
L
−


=+





. (72)
The intrinsic coupling energy between the polarizations at the interfaces 0z = of the two
layers is described as

()
2
2
Iii
Fpq
λ

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
ST
≈ 3 u.c. The lattice constants in the paraelectric state are
A
a = 3.969 Å and
B
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
d
E of PT/ST
superlattices as a function of thickness ratio L
PT
/L
ST
for different strength of interface
coupling
0
λ
. It is seen that P and
d
E decrease with increasing
0
λ

measurements. The
d
E versus L
PT
/L
ST
curves show a trend similar to P versus L
PT
/L
ST
,
e.g.
d
E disappears at a critical thickness ratio. For each
0
λ
, the critical thickness ratio of
d
E coincides with that of P. It is remarkable to see that for 0
d
E > , internal electric field is
parallel to the direction of the ferroelectric polarization in PT layer, which enhances the
polarization of the superlattice.

0
30
60
90
0.0 0.5 1.0 1.5 2.0
0
Internal Electric Field
[MV/cm]

Fig. 10. Polarization and internal electric field as a function of thickness ratio L
PT
/L
ST
of
PT/ST superlattices at T = 300K. The values of
0
λ
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

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
3
/SrTiO
3
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
3
/SrTiO
3
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

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

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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)
0
First-principles Study of ABO
3
: Role of the B–O
Coulomb Repulsions for Ferr oelectricity 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

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 hy bridization. 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
z
Coulomb repulsions
on Ti ion displacement in tetragonal BaTiO
3
and PbTiO
3
,whereO
z
denotes t he O atom to the
z-axis (Ti is displaced to the z-axis). Whereas the hybridization between Ti 3d state and O
z
2p
z
state stabilize Ti ion displacement, the strong Coulomb repulsions between Ti 3s and 3p
z
First-Principles
20
2 Ferroelectrics
states and O 2p
z
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 BaTiO
3

 1, cubic for t ≈ 1, and rhombohedral or
orthorhombic for t
 1. In fact, BaTiO
3
(t = 1.062) and SrTiO
3
(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
3
generally change; e.g., SrTiO
3
shows the tetragonal and
ferroelectric s tructure even in room temperature when the a lattice parameter along the [100]
axis (and al so the [010] axis) is smaller than the bulk lattice parameter with compressive
stress (Haeni et al., 2004). Theoretical investigations of ferroelectric ABO
3
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
3
,whicharea
well-known lead-free ferroelectric and piezoelectric ABO
3
, 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

3
and SrTiO
3
were performed u sing 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 u sing the
OPIUM code (Rappe, 2004):
(i) In order to investigate the role of Ti 3s and 3p states for BaTiO
3
, two kinds of Ti PPs were
prepared: one is the T i 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
3
: 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 s emicore 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 Monkhorst-Pack k-point mesh was set in the Brillouin zone
of the unit cell. The n umber 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.

Z
∗
33
(k)u
3
(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
3
(k) denotes the displacement along the
[001] ax is of the kth atom, and Z
∗
33
(k) d enotes the Born effective charges (Resta, 1994) which
contributes to the P
3
from the u
3
(k).Thepiezoelectrice constants, on the other hand, are
defined as
e
αβ
≡

∂P
α
∂η
β

u

=

∂P
3
∂η
β

u
+
∑
k
ec
Ω
Z
∗
33
(k)
∂u
3
(k)
∂η
β
(β = 3,1) .(4)
397
First-Principles Study of ABO
3
: Role of the
B–O Coulomb Repulsions for

Ferroelectricity and Piezoelectricity

㼀㼕
㻔䊅㻕
㻟㻚㻡 㻟㻚㻢 㻟㻚㻣 㻟㻚㻤 㻟㻚㻥 㻠
㼍㻌㻔䊅㻕
㻟㻚㻡 㻟㻚㻢 㻟㻚㻣 㻟㻚㻤 㻟㻚㻥 㻠
㼍㻌㻔䊅㻕
Fig. 1. Optimized calculated results as a function of a lattice parameters in tetragonal BaTiO
3
:
(a) c/a ratio and ( b) δ
Ti
to the [001] axis. Blue lines correspond to the results with the
Ti3spd4s PP, and re d lines correspond to those with the Ti3d4s PP. Results with arrows are
the fully optimized results, and the o ther results are those with c and all the inner
coordinations optimized for fixed a (Miura et al., 2010a).
㻮㼍㼀㼕㻻㻟
㻝㻢㻜
㻞㻜㻜
㼞
㻕
㻮㼍㼀㼕㻻㻟
㼀㼕㻌㻟㼟㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㼀㼕㻌㻟㼜㻌㻌㻌㻌㻌㻌㻌㻌㻮㼍㻌㻡㼟㻌㻌㻌㻻㻌㻞㼟㻌㻌㻮㼍㻌㻡㼜㻌㻌㻻㻌㻞㼜㻌㻌㼀㼕㻌㻟㼐
㻼㻠㼙㼙㻌㻮㼍㼀㼕㻻
㻟
㻦㻌㻰㼑㼚㼟㼕㼠㼥㻌㼛㼒㻌㻿㼠㼍㼠㼑㼟
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

Ti
)asafunctionofthea lattice parameters in tetragonal
BaTiO
3
, 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
㻔㼍㻕 䠄㼎䠅
398
Ferroelectrics - Characterization and Modeling