Using the various exchangecorrelation functionals, such as LDA, GGAPBE, GGAPBEsol and GGAAM05 functionals, first principle studies were conducted to determine the structures of paraelectric and ferroelectric PbTiO
_{3}
. Based on the structures determined by the various functionals, the piezoelectric properties of PbTiO
_{3}
are predicted under the densityfunctional perturbation theory (DFPT). The present prediction with the various GGA functionals are closer to the experimental findings compared to the LDA values. The present DFT calculations using the GGAPBEsol functional estimate the experimental data more reasonably than the conventional LDA and GGA fucntionals. The GGAAM05 functional also predicts the experimental data as well as the GGAPBEsol. The piezoelectric tensor calculated with PBEsol is relatively insensitive to pressure.
1. INTRODUCTION
Piezoelectric materials have a very important role in modern industries and are widely used as key materials in transducers, resonators, filters, and sensors. Discovered in 1880 by Pierre and Jacques Curie, piezoelectricity is a phenomenon whereby material becomes electrically polarized upon the application of stress. Complex insulating perovskite alloys are currently of great interest for use based on the exceptional piezoelectric properties. Examples include the Pb(Zr
_{1−x}
Ti
_{x}
)O
_{3}
(PZT) solid solutions that are currently used in piezoelectric transducers and actuators, and most recently, the class of PbMg
_{1/3}
Nb
_{2/3}
O
_{3}
−PbTiO
_{3}
(PMNPT) and PbZn
_{1/3}
Nb
_{2/3}
O
_{3}
−PbTiO
_{3}
(PZNPT) materials which, when synthesized in singlecrystal form, exhibit remarkably large piezoelectric constants and maximum strain levels. These materials promise dramatic improvements in the resolution and range of ultrasonic and SONAR listening devices. Moreover, new piezoelectric materials with excellent piezoelectric response combined with high temperature operation capabilities are currently sought. A fundamental understanding of factors that influence the piezoelectric properties of materials are of high scientiflc and technological value, but remains difficult to forecast.
Since the 1970s, scientists have persisted with efforts to make theoretic effort forecasting feasible. Due to rapidly increasing calculation speed and memory capacity of supercomputer and establish of modern polarization theory, direct quantitative calculation of piezoelectric coefficient of some relatively simple piezoelectric systems and qualitative forecasting of piezoelectric coefficient of some ferroelectric system from first principles are now possible. Based on the KohnSham formulation
[1]
, the densityfunctional theory (DFT)
[2]
has been used to calculate large systems where
abinito
HartreeFock calculation is not possible. Also, the densityfunctional perturbation theory (DFPT)
[3]
has been shown to give successful descriptions of the dielectric and piezoelectric properties of a wide range of materials in which electronic correlations are not too strong. The use of DFPT methods is becoming increasingly popular because of the ability to directly compute response properties.
Most previous DFT studies have employed the local density approximation (LDA) and the generalized gradient approximation (GGA) functionals to explain the exchangecorrelation energy. The gradient corrected PedrewBurkeErnzerhof (PBE)
[4]
functional has been regarded as the standard GGA. For the lattice constant of solids, which is one of the most important factors in predicting mechanical and piezoelectric properties, LDA clearly underestimates the lattice constants whereas the GGAPBE overestimates lattice constants to a similar degree. This discrepancy between the experiments and the predictions is intensified for the solids having d and fvalence electrons, such as transition metals and their oxides. Recently, to improve the lattice constants of solids, new GGA functionals (PBE for solids, PBEsol
[5]
and AM05
[6]
) have been suggested, and were shown to improve over PBE for various solids including transition metals.
In the present work, under the DFT, we investigate the physical properties of PbTiO
_{3}
(PT), which is one of the simplest ferroelectric oxide. Furthermore, because PT has a high Curie temperature, T
_{c}
= 766 K, it is considered one of the most important endcomponent of the promising single crystal piezoelectric perovskites, such as PZT, PMNPT, PZNPT, and PMNPIN(PbIn
_{1/2}
Nb
_{1/2}
O
_{3}
)PT. By employing LDA, GGAPBE, GGAPBEsol, GGAAM05, we calculate static dielectric matrix, stiffness matrix and ionclamped piezoelectric tensor at zero temperature. By comparing the DFT predictions with the available experimental results, we suggest more reliable functionals for the further studies, i.e. for the predictions of physical properties of PMNPT, PZNPT, and PMNPINPT, which are being actively studied for the real applications.
2. METHODOLOGY
In the present spinpolarized DFT calculations, the wellestablished Vienna ab initio simulation package (VASP)
[7
,
8]
was used. We used augmented projected augmented wave (PAW) pseudopotentials, and included the 5
d
^{10}
and 3
s
^{2}
3
p
^{6}
electrons as the valence electrons for Pb and Ti, i.e., Pb 5
d
, 6
s
, 6
p
, Ti, 3
s
, 3
p
, 3
d
, 4
s
, and O 2
s
and 2
p
orbitalelectrons were treated as valence electrons. The exchangecorrelation energy was considered by employing the various functionals such as, LDA, GGAPBE, GGAPBEsol and GGAAM05. A planewave cutoff energy of 700eV was used for static structure energy simulations and piezoelectric properties calculations. The Brillouin zone integration was performed using a 6 × 6 × 6 MonkhorstPack kpoint mesh, by testing the convergence with respect to kpoint mesh number. All atoms were relaxed using the conjugate gradient method until residual forces on constituent atoms became smaller than 5×10
^{−2}
eV/Å. The convergence of atomic configurations and relative energies were carefully checked with respect to planewave energy cutoff.
3. RESULTS AND DISCUSSION
Energyvolume behaviors of the ferroelectric and paraelectric phases of PbTiO
_{3}
are shown in
Fig. 1
, where the PBEsol functional is used. Energetics of the system supports that lowest symmetry structure is the structure with the lowest energy. We can determine the minimum energy and the corresponding most stable structures given in
Figs. 2(a)
and
2(b)
. The cubic structure of PbTiO
_{3}
is completely described by the lattice constant
a
. The calculated structures of paraelectic cubic PbTiO
_{3}
using the various functionals are compared with the existing experimental data in
Table 1
. In comparison between the theoretical results with the experimental data, the temperature effect between the experimental data which are obtained at the Curie temperature (T
_{c}
= 766 K) and the present calculation which corresponds to zero temperature should be considered. Because the ions in paraelectric (PE) phase keep their symmetric cubic positions, the bond lengths of TiO and PbO are
a
/2 and
, respectively.
The FE and the PE phases energies per the formula unit (f.u) of PbTiO_{3} with respect to unit cell volume (Å^{3}). GGAPBEsol fuctional is used. Lines are Murnaghan fit results which will be discussed later.
Optimized structures of (a) the paraelectice cubic phase and (b) the ferroelectric tetragonal phase of PbTiO_{3}. GGAPBEsol fuctional is used.
Structural parameter of cubic paraelectric PbTiO3
^{a}data at the Curie temperature (766K) given in Mabud and Glazer [9].^{b}data at the Curie temperature (766K) given in Jona and Shirane [10].
The tetragonal structure of PbTiO
_{3}
is completely described by the lattice constant a, the ratio c/a, and by the displacement of ion their symmetric positions along the vertical zdirection (
δz
). The present calculation results using different functionals are summarized in
Table 2
, where all structural parameters determined by fully unconstrained calculations are compared with experimental data at room temperature. As expected, the traditional LDA functional underestimates the lattice parameter and bond lengths, whereas the GGAPBE overestimates. However, the recently suggested GGAPBEsol gives a result that is comparable or is in better agreement with the experimental data. Considering the temperature effect shown in Mabud and Galzer
[9]
, the agreement between the present with GGAPBEsol and experiment is much improved. These configurations are used in further mechanical and piezoelectric properties calculations. The GGAAM05 functional also predicts the experimental data as well as the GGAPBEsol.
Structural parameters and bond lengths of tetragonal ferroelectric PbTiO3. The displacement of ion their symmetric positions along the vertical zdirection (δz) are given in terms of the lattice constantc.
^{a}room temperature data of Mabud and Glazer [9].^{b}room temperature data given in Jona and Shirane [10].
Generally, at constant temperature, the bulk modulus is defined by:
Murnaghan
[11]
assumed that the bulk modulus is a linear function of pressure as
By integrating Eq. (1) with Eq. (2), we can get the following Murnaghan equation:
Also, we can also express the volume depending on the pressure:
Because in the present study we obtained the energy as a function of the volume, the following equation which connect the energy with volume is more useful:
The above equation is obtained by integrating Eq. (3) according to the relationship
P
= −(
∂E
/
∂V
)
_{T}
. The regression results with Eq. (5) are already given in
Fig. 1
. For the ferroelectric tetragonal phase of PbTiO
_{3}
, the bulk modulus at the fully relaxed state is
K
_{0}
=104GPa which is comparable with the previous experimental results
[12

14]
. Based on the calculation data given in
Fig. 1
, volumepressure relation for the ferroelectric tetragonal phase of PbTiO
_{3}
is summarized in
Fig. 3
.
Pressurevolume relation of the tetragonal ferroelectic PbTiO_{3}.
Using the fullyrelaxed structures, the piezoelectric tensor is determined by the densityfunctional perturbation theory (DFPT). In the present study, we calculated the piezoelectric tensor
e
_{ij}
which connects the induced polarization
P
_{i}
and strain tensor element
ε
_{j}
as
The tetragonal phase PbTiO
_{3}
is polarized along the (001) axis. For a tetragonal lattice the three piezoelectric constants
e
_{3,1}
(=
e
_{3,2}
),
e
_{3,3}
and
e
_{1,5}
(=
e
_{2,4}
) are sufficient to describe the piezoelectric effects. Additionally,
e
_{3,1}
(=
e
_{3,2}
) and
e
_{3,3}
describe the zero field polarization induced along the zaxis when crystal is uniformly strained in the basal xyplane or along the zaxis, respectively. Yet,
e
_{1,5}
(=
e
_{2,4}
) represents the change of polarization perpendicular to the zaxis induced by shear strain. In a tetragonal crystal, the induced polarizations are related as
In
Table 3
we summarized all the values calculated with the various functionals, also the available experimental data were given
[12

14]
. As shown in this table, the calculated results strongly depend on the choice of the exchangecorrelation functional. Our
e
_{1,5}
,
e
_{3,1}
and
e
_{3,3}
values calculated with GGA functionals are in good agreement with the experimental data of Li et al.
[12]
. However, there is a significant spread between various experiments and the LDA values are consistently larger than those with GGA.
Piezoelectric stress tensor elements (C/m2) of ferroelectric tetragonal PbTiO3.
Piezoelectric stress tensor elements (C/m^{2}) of ferroelectric tetragonal PbTiO_{3}.
The pressuredependency of the piezoelectric tensor is given in
Table 4
. Here, we calculate the piezoelectric constants for the structure optimized at fixed crystal volume, and estimate pressure by using Eq. (3). As shown in
Table 4
, the piezoelectric coefficient increases with pressure.
Predicted piezoelectric stress tensor elements (C/m2) of ferroelectric tetragonal PbTiO3. PBEsol functional is used for the exchangecorrelation.
Predicted piezoelectric stress tensor elements (C/m^{2}) of ferroelectric tetragonal PbTiO_{3}. PBEsol functional is used for the exchangecorrelation.
4. CONCLUSIONS
The structures of the PE and FE phases of PbTiO
_{3}
were investigated using the first principle quantum calculations. In the present study, under the density functional theory (DFT), various exchangecorrelation functionals, such as LDA, GGAPBE, GGAPBEsol, GGAAM05 were tested. All functionals predict the experimental results correctly to a certain degree, however the modified functionals for solid materials, i.e. GGAPBEsol, GGAAM05, yielded the structures and physical properties of PbTiO
_{3}
more reasonably than the conventional LDA and GGAPBE functionals. In future studies, we will use GGAPBEsol and GGAAM05 functionals to investigate the structures and physical properties of the various piezoelectric materials such as PbMg
_{1/3}
Nb
_{2/3}
O
_{3}
(PMN), PbMg
_{1/3}
Nb
_{2/3}
O
_{3}
PbTiO
_{3}
(PMNPT) and PbIn
_{1/2}
Nb 1/2O
_{3}
 PbMg
_{1/3}
Nb
_{2/3}
O
_{3}
PbTiO
_{3}
(PINPMNPT).
Acknowledgements
This work was supported in part by Korean R&D program (10043193) funded by the Ministry of Trade, Industry & Energy (MOTIE), Korea and ONRNICOP Research Grant (Award No. N6290914N029). This work was also in part by International Collaborative R&D Program funded by the Defense Acquisition Program Administration (DAPA), Korea.
Kohn W.
,
Sham L. J.
(1965)
Phys. Rev.
140
A1133 
1133
Parr R. G.
,
Yang W.
1989
DensityFunctional Theory of Atoms and Molecules
Oxford University Press
Baroni S.
,
Giannozzi P.
,
Testa A.
(1987)
Phys. Rev. Lett.
58
1861 
Perdew J. P.
,
Burke K.
,
Ernzerhof M.
(1996)
Phys. Rev. Lett.
77
3865 
Perdew J. P.
,
Ruzsinszky A.
,
Csonka G. I.
,
Vydrov O. A.
,
Scuseria G. E.
,
Constantin L. A.
,
Zhou X.
,
Burke K.
(2008)
Phys. Rev. Lett.
100
136406 
Mattsson A. E.
,
Armiento R.
,
Paier R, J.
,
Kresse G.
,
Wills J. M.
,
Mattsson T. R.
(2008)
J. Chem. Phys.
128
084714 
Kresse G.
,
Furthmüller J.
(1996)
Comput. Mat. Sci.
6
15 
Kresse G.
,
Furthmüller J.
(1996)
Phys. Rev. B
54
11169 
Mabud S. A.
,
Glazer A.M.
(1979)
J. App. Cryst.
12
49 
Joan F.
,
Shirane G.
1962
Ferroelectric Crystals
Pergamon Press Inc.
New York
Murnaghan F. D.
(1944)
Proc. Natl. Acad. Sci. U.S.A.
30
244 
Li Z.
,
Crimsditch M.
,
Xu X.
,
Chan S. K.
(1993)
Ferroelectrics
141
313 
Kalinichev A. G.
,
Bass J. D.
,
Sun B. N.
,
Payne D. A.
(1997)
J. Mater. Res.
12
2623 
Ikegami S.
,
Ueda I.
,
Nagata T.
(1971)
J. Acoust. Soc. Am.
50
1060 