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lable at ScienceDirect
Materials Chemistry and Physics 205 (2018) 315e324
Contents lists avai
Materials Chemistry and Physics
journal homepage: www.elsevier .com/locate/matchemphys
First e principles calculations on stability and mechanical
propertiesof various ABO3 and their alloys
B. Akgenc a, b, *, A. Kinaci c, C. Tasseven d, T. Cagin b
a Department of Physics, Kirklareli University, Kirklareli,
39960, Turkeyb Department of Material Science and Engineering,
Texas A&M University, College Station, TX 77845, USAc Argonne
Natl Lab, Nanoscale Mat, Lemont, IL 60439, USAd Department of
Physics, Yildiz Technical University, Istanbul, 34210, Turkey
h i g h l i g h t s
* Corresponding author. Department of Physics, Ki39960,
Turkey.
E-mail address: [email protected] (B. Akgen
https://doi.org/10.1016/j.matchemphys.2017.11.0260254-0584/©
2017 Elsevier B.V. All rights reserved.
g r a p h i c a l a b s t r a c t
� Structure, elastic and dielectricproperties of ABO3 type
ceramicswere calculated.
� Composition and atomic configura-tion are effected on
pizeoelectricproperties.
� Our computations highlighted thatdeveloped small polarization
even ifcubic form.
a r t i c l e i n f o
Article history:Received 31 May 2017Received in revised form5
November 2017Accepted 12 November 2017Available online 16 November
2017
Keywords:Piezoelectric ceramicsABO3 type perovskitesMechanical
propertiesStructural propertiesPiezoelectric propertiesDensity
functional theory
a b s t r a c t
In this study, we perform firsteprinciple calculations based on
density functional theory (DFT) to obtainthe ground state
structural, elastic and dielectric properties of various ABO3 type
ceramics and their{AxA0(1-x)}BO3 and A{BxB0(1-x)}O3 alloys. To
represent alloy perovskites, we employ supercells with speciesA, A’
¼ Ba, Sr, Pb; B, B’ ¼ Ti, Zr. The effects of composition and atomic
configuration/order on latticestructure, thermodynamics, elastic
constants and dielectric properties are evaluated. In calculations,
wehave used linear response and homogeneous field methods and we
have also provided an assessment ofthe performance of these
approaches in the determination of aforementioned properties.
We have computed dielectric and piezoelectric properties for the
cubic form of alloy perovskites. Eventhough cubic form of alloy
perovskites does not have any piezoelectric properties, owing to
crystallo-graphic site occupied by different type of atoms, the
inversion symmetry breaks down and the structuresdevelop a small
tetragonality, in turn a small polarization and non-zero but quite
small piezoelectriccoefficients emerge as expected. For instance
the observed maximum piezoelectric constant forBaZrð1�xÞTixO3 is
0:554x10�15C=N. The magnitudes are smaller than the feasible ranges
for actualapplication needs, but they may increase substantially
upon phase to lower symmetry tetragonal formstransformation.
© 2017 Elsevier B.V. All rights reserved.
rklareli University, Kirklareli,
c).
1. Introduction
ABO3 type pure perovskite crystals have been extensivelystudied;
however, the structural, elastic, dielectric and piezoelectric
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B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324316
properties of their alloys are still a subject of inquiry [1e6].
Thesolid solutions of perovskites, owing to their tunable
dielectric andpiezoelectric responses, are of great technological
interest for in-dustrial and commercial applications such as
high-dielectric con-stant capacitors, ferroelectric thin film
memory devices, sensors,switches, actuators, piezoelectric sonars,
atomic microscopes, ul-trasound generators, piezoelectric motors,
ink-jet printers and fuelinjectors [7e12]. There are two ideal ways
of tuning physicalproperties of these ABO3 type perovskite alloys
by adjusting theconcentration and configuration/order of doped
atoms [13e15].
The elastic constants are used to characterize the
mechanicalresponse of materials to external macroscopic stress.
These con-stants are closely related to interatomic bonding,
mechanical sta-bility, thermal relaxation and internal strain.
Studying elasticconstants is essential to understand the
electromechanicalresponse of perovskite oxide ferroelectrics [16].
Especially today'shighly efficient density functional theory (DFT)
calculations, being afast alternative to experiments, can be
utilized to scan throughmany materials for required properties.
Elastic constants andrelated mechanical properties i.e. bulk
modulus, shear modulus,Young's modulus, and Poisson's ratio of a
large number of complexceramic crystals with different crystal
symmetries have beencalculated by means of density functional
theory [17]. The mainchallenge in estimating elastic constants from
first principles re-quires an accurate method to calculate the
total energy [18e20].The results presented here are obtained within
the generalizedgradient approximation (GGA).
Density functional perturbation theory (DFPT) is anothermethod,
which provides the desired response properties in auto-mated,
systematic and reliable fashion [21]. Therefore, it has beenwidely
used in calculations of structural, mechanical and
electronicproperties of perovskite materials and their alloys. DFT
study ofATiO3 compounds, where A represents the atoms Ca, Sr, Ba,
Ra, Cd,Mg, Ge, Sn and Pb, have revealed that the geometrical size
and theconfiguration of outer electronic shell of A atoms are
responsiblefor determining the off-center positioning of the A
atoms [1].Gonzalez-Garcia et al. have investigated the effect of
substituentconcentration on structural parameter, band gap energy,
mixingenthalpy and phase diagram of In1-xBxP semiconductor alloys.
It hasbeen found that the lattice parameters of the In1-xBxP
alloysdecrease with x, B-concentration, showing a negative
deviationfrom Vegard's law, while the bulk modulus increases
withcomposition x, showing a large deviation from the linear
concen-tration dependence [22]. Theoretical analysis has been
carried outin InxAl1-xN alloys on the applicability of Vegard's
linear rule anddeviations from its linearity with respect to
composition, inpiezoelectric polarization, surface orientation and
degree of strain.To determine and estimate these properties
correctly is of vitalimportance in technological applications of
these materials [23]. Ithas been found that Bi(Zn1/2Ti1/2)O3 and
PZT solid solution displayextremely large cation displacements and
leading to tetragonality.The displacements on A site are more
pronounced than B-sitedisplacements [24]. J. Bennett et al. have
investigated ground statestructural properties of Sn(Al1/2Nb1/2)O3
(SAN) solid solution anddemonstrated that these alloys can be
synthesized due to theirfavorable thermodynamics, and they possess
enhanced ferroelec-tric, and piezoelectric properties [25]. Halilov
et al. have investi-gated the main instabilities in CdTiO3 and
Cd0.5Pb0.5TiO3 to showthe piezoelectric potential of this
perovskite alloy which mayprovide much larger energy density than
traditional composition[26]. Majority work in the field, which has
mainly focused on B-sitealloying, we have investigated on A-site
alloying in addition to B-site. We have constructed {AxA01-x}BO3
and A{BxB01-x}O3 alloys forwhich x is varied from 0 to 1 with the
species A, A’ ¼ Ba, Sr, Pb; B,B’ ¼ Ti, Zr. We have shown the
effects substitutions at A-atom site
and B-atom site by quantifying properties of these alloys
namelytheir structural properties, elastic constants, dielectric
constants,and Born effective charges.
In recent years, researchers have focused on lead-free ABO3
typeperovskite ceramics for technological applications [27e29].
Sub-stantial effort spent on elucidating the influence of alloying
onemerging structures and associated enhanced physical
properties.As an example, Yang et al. reported the effects of
B-site cations onthe structure, elastic and thermodynamic
properties ofKNa0.5Nb0.5O3 perovskite ceramic based on first
principle calcula-tions [30]. Pontes et al. have investigated
effect of A-site chemicaldoping on the different properties of
Pb(Ca,Ba)TiO3, Pb(Sr,Ba)TiO3and Pb(Sr,Ca)TiO3 perovskite ceramics
through combined experi-mental and theoretical studies [31].
Riborino et al. conducted acomputational mapping of structural,
electronic and dielectricproperties of lead-free SZT [27]. In the
light of these very recentstudies, there is still need for a
comprehensive DFT study onAA’BB’O3 perovskite structure. Hence, in
this work, we study theeffects of A-site and B-site cations of
AA’BB’O3 type perovskites forwide concentration/configurations. The
specific aim of this study isto investigate the effects of alloying
(concentration and structural/chemical anisotropy) on their
thermodynamic, structural, me-chanical, and electromechanical
properties for broad concentrationranges rather than providing
comprehensive data for pure ABO3compounds for various existing
polymorphs.
2. Computational method
The first-principles calculations are performed by
employinggeneralized gradient approximation (GGA) based on density
func-tional theory (DFT) which allows ground-state and
mechanicalproperty calculations and density functional perturbation
theory(DFPT) that lets to calculate of Born-effective charges,
dielectric andpiezoelectric tensors as implemented in Vienna
abinitio SimulationPackage (VASP) program [32]. The 5s5p and 6s of
Ba, 3s;3p;3dand4s of Ti, 2sand 2p of O, 5d;6sand 6pof Pb, 4s;4pand
5s of Sr and4s;4p;5s and 4d of Zr are considered as valence states
in the con-struction of the pseudopotentials. In any accurate first
principlesimulation, it is necessary to test the accuracy of
exchange-correlation potential forms, and sufficiently large
k-point meshesand energy cut-off value [33]. Energy convergence
tests have beencarried out to determine the cut-off energy and the
number ofk-points for all structures. The total energy is converged
at the cut-off energy of 520 eV for all structures. The
Brillouin-zone integra-tion is done using special k-points sampled
within the Monkhorst-Pack scheme. It is found that a mesh of 8x8x8
k-points is required todescribe well stability and mechanical
properties in the cubicstructures. In order to optimize
computational cost, a (2 � 2 � 2)supercell, which contains 40 atoms
(eight formula units) isemployed. As the system size doubled in
each direction the k-pointmesh is reduced to 4x4x4in reciprocal
space. In order to makecomparisons standard all results for
extensive variables such asvolume, energy, etc. are reported for
one formula unit. All calcu-lations are performed on charge neutral
systems and with thefollowing common parameters: (1) a high
accuracy for electronicground-state convergence (10�4eV); (2) a
small tolerance for ionicrelaxation convergence (�10�3eV); and (3)
maximum ionic steps of100 for all cases. While using energy
minimization, the unit cellparameters and atomic coordinates are
allowed to relax.
2.1. Construction of alloys
The prototype of ABO3 type perovskite structure has a spacegroup
of Pm30m space group (No: 225). Ba, Sr, Pb are considered foratoms
of A-sites; Ti and Zr are considered for atoms of B-sites. We
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B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324 317
choose 2 � 2 � 2 super-cells with periodic boundary
conditionscontaining 40 atoms for the creation of {AxA0(1-x)}BO3
and A{BxB0(1-x)}O3 alloys in which some equivalent crystallographic
sites areoccupied by different atoms. We studied the alloys varying
con-centration from 01 for A-site and B-site in steps of 0.125. The
eight Aand A0 atoms, eight B and B0 atoms are situated on a simple
cubicsublattice. {AxA0(1-x)}BO3 and A{BxB0(1-x)}O3 alloys can be
classifiedas homovalent and heterovalent alloys from the
chemicalperspective. Homovalent alloys are defined as A atoms
belonging tothe same column of periodic table, while heterovalent
alloys belongto different column of periodic table. {BaPb}TiO3,
{SrPb}TiO3, {SrBa}TiO3, Pb{ZrTi}O3, Ba{ZrTi}O3, and Sr{ZrTi}O3
alloys are studied inthis work. We model our supercell with
different concentrations ofx ¼ 0.875, 0.75, 0.625, 0.50, 0.375,
0.25 and 0.125 as all the valuesallowed for a 2x2x2 supercell.
To the best of our knowledge, there are no reports on
structur-eeproperty relationship in A-site and B-site mixed
perovskites forthe compounds we considered. All possible local
chemical orderingof A-ions, B-ions and their influence on the
mixing energy areinvestigated by first-principles calculations
based on densityfunctional theory. The number of crystallographic
sites is altered
Fig. 1. All possible arrangements for 2x2x2 supercell used in
different concentration.The dopant and host atoms are represented
by red and green atoms, respectively. (Forinterpretation of the
references to colour in this figure legend, the reader is referred
tothe web version of this article.)
Fig. 2. 40 atoms in Pb8ZrxTi8-xO24 cell with (a) 0.125:0.875 and
(b) 0.250:0.750 stoichiometspheres represent Pb atoms (A-site),
green spheres represent Zr atoms (B-site), blue spheresalloys is
visualized using the program VESTA. (For interpretation of the
references to colou
from one to eight with different stoichiometric ratio. All
possiblearrangements for 2x2x2 supercells are shown in Fig. 1.
The0.125:0.875 stoichiometric ratio of A and A0 in {AxA0(1-x)}BO3
has aminimum energy without any position limit, as the symmetry
ofthe structure does not change. The 0.25:0.75 ratio that two
crys-tallographic site are occupied by different atoms, can break
thesymmetry. We have shown that minimum energy changed
withdifferent configurations when atoms are placed at corners
andsurface positions, due to a slight change in symmetry. We
havescanned all possible configurations for each stoichiometric
ratiostudied; and used symmetry equivalent ones in Fig. 1 and
incalculations.
The initial lattice parameters of 40-atom super cells are
chosenaccording to Vegard's law at different contents for
structural opti-mization and energy minimization. For instance, the
lattice pa-rameters of PbZrxTi1-xO3 alloys can be calculated by the
followingformula:
aPbZrxTi1�xO3 ¼ xaPbZrO3 þ ð1� xÞaPbTiO3 (1)
where aPbZrO3 and aPbTiO3 are the lattice constants of PbZrO3
andPbTiO3 compounds, respectively. Different types of cations
arereplaced by the element from the base component at the
corre-sponding crystallographic site. Replacement of different
typescations influences the chemical ordering, hence may induce
abreaking of centro-symmetry and furthermore at some
concen-trations different configurationmay also break the cubic
symmetry.In turn, the structure influences energetics, mechanical,
dielectric,though probably small piezoelectric properties. The
symmetricallyunique structural motifs shown in Fig. 1 are analyzed
for energetics,structure, mechanical and electrical properties.
Furthermore, themixing energies of the alloys for all
concentrations are determined.The detailed all atom structures for
two stoichiometric ratios0.125:0.875 and 0.250:0.750 are shown in
2x2x2 supercells as anexample of Pb8ZrxTi8-xO24 in Fig. 2.
3. Results and discussions
3.1. Equation of state
The calculated total energies as a function of a primitive
cellvolume (5 atoms) for ABO3 are used to determine structural
andmechanical properties of the pure systems by a least-squares fit
tothe third-order Birch-Murnaghan equation of state:
ric ratio with three different configurations of Zr-Ti
placement, respectively. The blackrepresent Ti atoms (B0-site) and
red spheres represent O atoms. The crystal structure ofr in this
figure legend, the reader is referred to the web version of this
article.)
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B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324318
EðVÞ ¼ E0 þ9V0B016
("�V0V
�2=3� 1#3
B00 þ
"�V0V
�2=3� 1
#2"6
� 4�V0V
�2=3#);
(2)
where, E0 is the total energy, V0 is the equilibriumvolume, B0
is thebulk modulus at 0 GPa pressure and B
00 is the first derivative of bulk
modulus with respect to pressure. All of these are treated as
freeparameters during the least-square fitting. The equilibrium
latticeparameter a0 is determined as from the value of V0, the
volume atthe minimum total energy.
To assess the experimental accuracy of our DFT calculations,
wehave selected the cubic phase of six pure compounds and
comparedtheir calculated lattice parameters and bulk modulus with
theavailable experimental data. In Table 1, we present the
parametersobtained after the fitting procedure along with the
experimentallattice parameters and bulkmodulus. As it can be seen,
the errors inlattice parameters and bulk modulus are less than 2%
and 6%respectively.
DFT calculations on pristine ABO3 perovskites have been
re-ported in literature earlier, eg. Ref. [42-45] On the other
hand, theeffects of alloying on the mechanical and structural
properties havenot been fully explored. Total energy versus volume
behavior(Equations of State) of different types of {AxA0(1-x)}BO3
and A{BxB0(1-x)}O3 are shown in Fig. 3, where volume and energy are
per singleABO3 formula unit is used in all cases. Dashed line Fig.
3 representsthe fit of data to Birch-Murnaghan equation of state.
For each alloyconcentration, it is sought to find the energetically
favorablearrangement. Doped perovskites {AxA0(1-x)}BO3 and
A{BxB0(1-x)}O3have intermediate formation energies between those of
the pureforms in accordance to the degree of doping. This behavior
supportsthe idea that the difference between the groups depends on
theoxidation states. The calculated equation of states
parameters,namely the equilibrium lattice constants, the
equilibrium volume,bulk modulus, the pressure derivative of bulk
modulus and cohe-sive energy for the PZT are listed in Table 2. We
have generated theresults for all different concentrations of the
other perovskitescompounds as well. The rest of them have been
given inSupplementary Material.
We have shown that E0 does not change with respect to the
Table 1The calculated equilibrium total energies (E0 in eV), the
lattice parameters (a in Å), the bulhere with other theoretical and
available experimental data for pure perovskites.
Equation of State Parameters
Eo(eV) a(Å)
This work Expt. Theory
BaTiO3 �40.21 4.03 4.00a 4.00bPbTiO3 �38.01 3.96 3.97c
3.96dSrTiO3 �40.29 3.94 3.91f 3.95dBaZrO3 �41.67 4.25 4.19h 4.21
hPbZrO3 �39.11 4.20 4.13 gSrZrO3 �41.23 4.19 4.10 h 4.17ia Ref.
[33].b Ref. [34].c Ref. [35].d Ref. [36].e Ref. [37].f Ref. [38].g
Ref. [39].h Ref. [40].i Ref. [41].
symmetry equivalent positions, the symmetry non-equivalent
po-sitions generate a very small change in total energy. Due to
theatom size difference in alloys, the structures deviate from
cubicform to tetragonal form, which is an indication that alloying
leadsto emergence of permanent dipole moments/polarization in
ABO3systems. Therefore, we may provide the first step for an avenue
toimprove polarization properties: concentration specific,
orderspecific alloy-compounds. Hence, assessing the
observedenhancement in electromechanical response in these well
knownwidely used ABO3 materials. The calculated magnitude of
dielectricand piezoelectric coefficients and the details on
dielectric andpiezoelectric properties will be described in section
3.4.
In order to assess the influence on the structural parameters
inequilibrium using different possible equilibrium
configurationswith different ordering for various alloys, the
structural parametersare computed for different values of
concentration x of AxA0(1-x)BO3alloys, specific results for PZT
alloys are given in Table 2 and shownin Fig. 3 as lattice
parameters and c/a ratio.
3.2. Calculation of elastic constants
Two closely related approaches can be used to calculate
theelastic constants from the first principles calculations. The
first isrelated to analysis of the calculated total energy of a
crystal as afunction of applied strain, and the second is based on
the analysis ofthe changes in the calculated stress values arising
from the varia-tion in the strain [8]. In our calculation, we have
chosen the firstmethod to determine the elastic constants. At the
same time, theelastic constants are calculated with stress-strain
relationship is touse IBRION ¼ 6 and ISIF ¼ 3 that is implemented
in the VASP andthe results are compared with the former. The
elastic tensor isdetermined by performing six finite distortions of
the lattice andderiving the elastic constants from the
stress-strain relationship. Inthis way, calculated elastic
constants include both the contributionsfor distortions with rigid
ions and the contributions from the ionicrelaxation [21].
To obtain the elastic constants, we calculate the total energy
forseveral values of lattice parameters of the cubic cell that
corre-sponds to the isotropic dilatational strain -expanding and
con-tracting up to %10% with respect to zero stress state for all
the pureand alloyed structures. For ABO3 alloys, each structure
which cor-responds to a different concentration, we deformed the
lattice byvarying the strain parameter d from -0.07 to 0.07 in
steps of 0.005 to
k modulus (B in GPa), and the pressure derivative of bulk
modulus (B0) are presented
B(GPa) B00
This work Expt. Theory This work
163.64 162a 172b 4.55176.68 144e 140d 4.51171.51 174g 170d
4.42149.47 - 157i 4.45156.94 4.39152.74 - 160i 4.35
-
Fig. 3. Total energies as a function of primitive cell volume
for cubic (PbxBa8-x)Ti8O24, (SrxPb8-x)Ti8O24, (SrxBa8-x)Ti8O24,
Ba8(ZrxTi(8-x))O24, Pb8(ZrxTi(8-x))O24 and Sr8(ZrxTi(8-x))O24from
ab-initio calculations for 2x2x2 unit cell.
B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324 319
obtain the total minimum energies as a function of strain.
Thedeformation energy of a crystal can be expressed as a Taylor
seriesin strain:
E�V ; εij
� ¼ EðV0;0Þ þ V0 X3i;j¼1
sijεij þ12!V0
X6i;j;k;l¼1
Cijklεijεkl þ…
(3)
where, ε and s are the strain and stress tensors, respectively.
V0 isthe volume of unstrained state, EðV0;0Þ is the corresponding
un-strained total energy, and Cij are the second order elastic
constants.The second order elastic constants are defined as
Cijkl ¼vsijvεij
¼ 1V0
"d2EðV0; εÞdεijdεkl
#z¼0
(4)
The Voight notation is used in Equation (4) that replaces xx,
yy,zz, yz, xz and xy components with indices: 1, 2, 3, 4, 5 and
6,respectively. Eq. (3) becomes
EðV ; εÞ ¼ EðV0;0Þ þ V0X6i¼1
siεi þ12!V0X6i;j¼1
Cijεiεj þ… (5)
The elasticity of the cubic crystal is completely described
bythree independent constants C11, C12 and C44 and they are
obtainedby the three strains (C1, C2, C3) given in Table 3. The
elastic
-
Table 2Calculated equilibrium lattice parameters (a in Å), bulk
modulus (B in GPa), and the pressure derivative of the bulk modulus
(B0) together with the theoretical and availableexperimental values
for PZT perovskites. (a, b, c corresponds to different positions
shown in Fig. 1).
Eo(eV) V0 (Å3) B(GPa) B0 Lattice parameter(Å) c=a
PZT Pb8Zr7Ti1O24 �38.94 72.60 157.81 4.47 4.170
1.0000Pb8Zr6Ti2O24a �38.77 71.22 158.13 4.51 a ¼ c ¼ 4.136, b ¼
4.157 1.0052Pb8Zr6Ti2O24b �38.78 71.01 160.53 4.45 4.139
1.0000Pb8Zr6Ti2O24c �38.79 71.04 159.25 4.53 a ¼ b ¼ 4.140, c ¼
4.138 1.0004Pb8Zr5Ti3O24a �38.62 69.76 159.25 4.59 a ¼ b ¼ 4.122, c
¼ 4.099 1.0056Pb8Zr5Ti3O24b �38.63 69.54 161.01 4.58 a ¼ c ¼ 4.103,
b ¼ 4.125 1.0054Pb8Zr5Ti3O24c �38.67 69.40 162.46 4.50 4.108
1.0000Pb8Zr4Ti4O24a �38.46 68.40 160.37 4.61 a ¼ b ¼ 4.104, c ¼
4.054 1.0123Pb8Zr4Ti4O24b �38.49 67.96 165.34 4.47 a ¼ c ¼ 4.063, b
¼ 4.112 1.0121Pb8Zr4Ti4O24c �38.57 67.67 165.02 4.48 4.073
1.0000Pb8Zr4Ti4O24d �38.48 68.19 161.17 4.66 a ¼ c ¼ 4.083, b ¼
4.084 1.0003Pb8Zr4Ti4O24e �38.50 68.00 163.42 4.53 b ¼ c ¼ 4.088, a
¼ 4.063 1.0061Pb8Zr4Ti4O24f �38.49 68.22 161.65 4.60 4.084
1.0000Pb8Zr3Ti5O24a �38.34 66.75 163.58 4.63 a ¼ b ¼ 4.064, c ¼
4.037 1.0065Pb8Zr3Ti5O24b �38.36 66.52 165.02 4.66 a ¼ c ¼ 4.041, b
¼ 4.068 1.0068Pb8Zr3Ti5O24c �38.40 66.37 167.26 4.54 4.047
1.0000Pb8Zr2Ti6O24a �38.22 65.20 167.43 4.60 a ¼ c ¼ 4.013, b ¼
4.042 1.0073Pb8Zr2Ti6O24b �38.24 64.96 170.63 4.49 4.018
1.0000Pb8Zr2Ti6O24a �38.25 65.00 168.06 4.65 a ¼ b ¼ 4.019, c ¼
4.020 1.0002Pb8Zr1Ti7O24 �38.11 63.56 171.43 4.56 3.989 1.0000
Table 3Applied strains εi (or strain parameter d) and
corresponding energy densitiesEðV ; εÞ � EðV0 ;0ÞV0 (or
corresponding mechanical response constant) for the
cubicperovskites.
Strain Parameters EðV ; εÞ � EðV0 ;0ÞV0C1 ε1 ¼ ε2 ¼ ε3 ¼ d
3=2ðC11 þ 2C12ÞC2
ε1 ¼ d; ε2 ¼ �d; ε3 ¼ d2=1� d2 ðC11 � C12Þd2C3
ε1 ¼ d2=1� d2; ε6 ¼ d ð2C44Þd2
Table 4The calculated elastic constants (Cij in GPa) for pure
perovskites.
C11 C12 C44 Methods
BaTiO3 281.67 103.75 119.78 Stress- Strain281.38 104.23 119.68
Energy-Strain
PbTiO3 288.21 122.52 96.64 Stress- Strain287.67 119.05 96.94
Energy-Strain
SrTiO3 316.69 103.79 111.93 Stress- Strain316.47 99.33 108.44
Energy-Strain
BaZrO3 293.18 76.08 84.44 Stress- Strain293.18 77.61 82.06
Energy-Strain
SrZrO3 317.18 70.52 71.92 Stress- Strain315.37 71.57 70.37
Energy-Strain
PbZrO3 301.27 84.91 58.94 Stress- Strain303.12 85.13 61.48
Energy-Strain
B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324320
constants are identified by the constants in the quadratic term
d (inthe polynomial fit to the total energy as a function of the
strainparameter) in Table 4.
The elastic constants are calculated with two different
ap-proaches: i) energy-strain and ii) stress-strain relationship.
Theyobey the following mechanical stability conditions for instance
inthe cubic structure, they are given as;
C11 � jC12j>0
C44 >0 (6)
C11 þ 2C12 >0More relevant to the objective of the work here,
we have
calculated elastic constants tensor for all alloys, all
concentrations
and all unique symmetry configurations corresponding these
con-centrations. In Table 5, we present the elastic constants for
cubicand non-cubic cases (due to emergence of a small
tetragonalstructure in some structures where 6 independent elastic
constantsarise) for PZT. The results for complete set of alloys
with uniquesymmetry configurations for all concentration were given
indetailed tables (in Supplementary material).
It is a point of interest to investigate the behavior of
elasticconstants as a function concentration and different
symmetryunique configuration is to apply Vegard's rule for the
property andcalculate the deviation from this value to note if the
property in-crease or decreases as a result of mixing. In Fig. 4 we
have plottedthe deviation of the calculated elastic constants from
the Vegard'srule (dot-dash line is 0) for PZT, BZT and SZTas
examples. The y-axison each constant is the amount of difference in
GPa. We willcomment not only on magnitude but on fractional or
percentagedeviation from the Vegard's rule. For C44 for all
compounds and allconfigurations show a negative deviation, hence
the alloys PZT, BZTand SZT all weakens for pure shear response of
the resulting ma-terial which in many cases this represents a
weakening ca. 10%. Thevariation in C12 values display similar
behavior in almost all con-centrations as decrease with respect to
Vegard's Rule result rangingfrom 2% to 6%, except in 50-50 case in
only one configuration eachBZT and SZT shows around 1% increase.
However for C11 values fordifferent concentrations and
corresponding unique configurationsshow both negative and positive
deviations, distribution tending tobe heavily positive. In these
calculations, for slightly tetragonalsupercells, we assumed systems
are pseudo cubic. This assumptionis reasonable, since the largest
c/a ratio is 1.015.
3.3. Calculation of Born effective charges
In order to obtain Born effective charges, we perform the
cor-responding linear response computations. The technical details
ofcomputation of responses to atomic displacements,
homogeneouselectric fields, and strains are based on density
functional pertur-bation theory (DFPT). The Born effective charge
(BEC) tensorsrepresent the coupling of a macroscopic field to
relative sub-latticedisplacements in the crystal structure. They
relate the macroscopicpolarization to the atomic scale
displacements by the followingequation
-
Table 5The calculated elastic constants (Cij in GPa) for
PZT.
C11 C12 C44 C13 C33 C66
PZT Pb8Zr7Ti1O24 300.44 86.23 63.01 e e ePb8Zr6Ti2O24a 300.65
88.79 65.61 88.94 297.69 65.73Pb8Zr6Ti2O24b 299.86 91.45 66.62 e e
ePb8Zr6Ti2O24c 298.64 90.71 67.11 90.22 300.51 65.92Pb8Zr5Ti3O24a
296.42 92.04 69.44 91.38 300.55 68.46Pb8Zr5Ti3O24b 299.06 94.27
69.51 95.48 296.39 70.09Pb8Zr5Ti3O24c 298.29 96.07 70.65 e e
ePb8Zr4Ti4O24a 293.59 94.13 73.23 93.32 302.50 72.99Pb8Zr4Ti4O24b
301.35 98.51 73.45 102.76 291.01 76.03Pb8Zr4Ti4O24c 296.44 101.84
76.53 e e ePb8Zr4Ti4O24d 294.90 95.47 73.47 94.18 295.73
70.84Pb8Zr4Ti4O24e 299.67 98.68 74.39 97.80 294.77
74.10Pb8Zr4Ti4O24f 297.43 96.52 73.11 e e ePb8Zr3Ti5O24a 288.23
93.15 74.09 92.97 291.74 50.61Pb8Zr3Ti5O24b 296.53 100.72 75.93
103.12 290.09 73.66Pb8Zr3Ti5O24c 294.03 102.97 51.56 e e
ePb8Zr2Ti6O24a 289.44 103.58 82.78 104.30 295.47 69.69Pb8Zr2Ti6O24b
295.29 108.40 84.23 e e ePb8Zr2Ti6O24c 293.62 106.91 75.32 105.62
292.01 69.30Pb8Zr1Ti7O24 292.29 110.67 87.34 e e e
Fig. 4. The variation of elastic constants as a function of
concentration and dependence on symmetry inequivalent
configurations at each concentration in PZT, BZT and SZTdisplayed
through deviation from the values that are obtained via application
of Vegard's rule of combination.
B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324 321
-
Fig. 5. Dielectric constants of Pb1�xBaxTiO3 according to Ba
concentration at 0 K.
B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324322
Pf ¼ eU
XZiabduib: (7)
where, i denotes the ith atom and a and b are the components
ofthe polarization and atomic displacement vectors, respectively.
Instrongly ferroelectric materials the BEC's are significantly
differentfrom the anticipated nominal electric charge ossn the ion.
BECsemphasize why some charges are larger than nominal charges,
andis also an indicator of long-range Coulomb interactions that
help toclarify origin of polarization in the material. Resta et al.
[46] havesuggested that strong variations of BECs relative to
nominal ionicvalues are due to the large in-equivalence of the O
ions.
The present calculations of BECs values for all pure
perovskitesare given in Table 6. It is seen that the BECs of these
materials aremuch larger than formal charges as expected [47e49].
For proto-type of cubic perovskite structure, the metal atoms A or
B sit at thecenter of cubic symmetry that results in the isotropic
effectivecharges along the diagonal of charge tensor. Oxygen atoms
that sitat the face centers have two independent components of
effectivecharge, which are arising from displacement of oxygen ion
along B-O direction or perpendicular. It is well understood that
the largevalue of B charges (Ti and Zr) resulting from the effect
of covalentbonding with O atoms. The acoustic sum rule of Born
effectivecharges is well satisfied in our calculations:Xi
Zi ¼ ZA þ ZB þ ZOk þ 2ZO⊥ (8)
3.4. Calculation of dielectric and piezoelectric constants
The dielectric properties of the alloys for various
concentrationsare determined using linear response theory as
implemented inVASP. This method computes the components of
ion-clamped staticdielectric tensor from the change in the
electronic dipole momentdue to electric field.
εij ¼ dij þ4pε0
vPivEj
ði; j ¼ x; y; zÞ (9)
For some concentrations where structure slightly deviates into
atetragonal form show this character in dielectric constants, as
adeviation from cubic symmetry ε11 ¼ ε22 ¼ ε33. The deviation
islower than 1%, which is comparable with the deviations in
latticeparameters. As seen from Fig. 5, the variation in dielectric
constantsis linear with respect to concentration. We made pseudo
cubicassumption for slightly tetragonal phases, and plotted only
one(average of nonzero components of dielectric tensor). Fig. 5
displaysquite a close agreement with Vegard's law (εalloy¼ xεAþ
(1-x)εB) forall concentrations. The variation of dielectric
constants for the otheralloys are given in Supplementary Materials
and they exhibit prettyclose agreement with Vegard's law as
well.
Dielectrics can have polarization contribution due to the
elec-trons, ions and molecular dipoles contribution. Each of
these
Table 6Comparative Born effective charges for different ABO3
perovskites.
Perovskites Z*A Z*B Z
*O;k Z
*O;⊥
BaTiO3 2.745 7.488 �5.943 �2.144PbTiO3 3.852 7.338 �6.076
�2.555SrTiO3 2.545 7.448 �5.928 �2.033BaZrO3 2.741 6.213 �4.905
�2.024PbZrO3 3.913 6.045 �4.957 �2.499SrZrO3 2.577 6.107 �4.935
�1.874
polarization contributions causes piezoelectricity. In this
study, wehave calculated the dielectric constants under zero stress
at 0 K. Wehave also investigated piezoelectric properties of alloys
withdifferent ratios of mixing within the density-functional
perturba-tion theory (DFPT). Basically, the piezoelectric effect
stems fromchanging of polarization upon a deformation.
We have calculated piezoelectric constants tensor for all
alloys,all concentrations and all unique symmetry configurations
corre-sponding concentrations. We present the results of
piezoelectriccoefficient for BaZrð1�xÞTixO3 cubic form of
perovskite structures inTable 7. According to symmetries of
tensors, there are maximum 18dij components for lowest symmetry
case. dij is a 3x6matrix. As it iswell known, the cubic form of
perovskite structures does not showany piezoelectric effect because
of high symmetry. But in this sce-nario, owing to the fact that
crystallographic site is occupied bydifferent type of atoms, the
inversion symmetry breaks down andthe structures developed a small
tetragonality. Our result showsthat these alloys have non-zero
piezoelectric coefficients on theorder of 10�15C=N magnitude. The
chemical/configuration inducedpiezoelectricity exists in pseudo
cubic structures as well. Theirmagnitude is smaller than the
feasible ranges for actual applicationneeds, but it can be
significantly improved upon phase trans-forming to tetragonal
symmetry at various temperatures.
4. Concluding remarks
We studied structural and mechanical properties of some
pureperovskites and their alloys using first principle DFT
calculations.The elastic constants were obtained using the
energy-strain andstress-strain relations.We conclude that in
linewith our theoreticalpredictions on the considered properties
some perovskites andtheir alloys would be serving as a reliable
reference for the future.
The results on prototype alloy structures indicate the
potentialfor improvement of piezoelectric response through alloying
as itmanifests itself as changes in structural and energetic form
in thepreliminary results we provided in the previous section.
Beside the effects of alloying/doping and deformation in
thecrystal lattice, another intrinsic mechanism that may lead to
po-larization is through the emergence of oxygen vacancies. This
isquite common for most oxides due to volatility of
Oxygen.Furthermore there are other mechanisms one can develop
-
Table 7Calculated piezoelectric
dijin 10
�15CN
!coefficients for BaZrð1�xÞTixO3cubic form of perovskite
structures.
d11 d12 d13 d21 d22 d23 d31 d32 d33
Ba8Zr7Ti1O24 �0.157 0.176 0.176 0.201 �0.210 0.201 0.180 0.107
�0.164Ba8Zr6Ti2O24a 0.080 0.554 0.330 0.119 �0.200 0.119 0.326
0.546 0.083Ba8Zr6Ti2O24b �0.173 �0.037 �0.037 �0.018 �0.110 �0.018
�0.021 �0.021 �0.158Ba8Zr6Ti2O24c �0.171 0.127 0.158 0.136 �0.171
0.151 0.327 0.327 �0.280Ba8Zr5Ti3O24a �0.144 0.186 0.175 0.170
�0.132 0.152 0.155 0.155 �0.043Ba8Zr5Ti3O24b �0.131 0.167 0.086
0.167 �0.181 0.167 0.095 0.164 �0.151Ba8Zr5Ti3O24c �0.469 0.294
0.294 0.298 �0.472 0.298 0.293 0.293 �0.466Ba8Zr4Ti4O24a �0.228
0.199 0.109 0.189 �0.213 0.098 0.049 0.049 �0.218Ba8Zr4Ti4O24b
�0.503 �0.409 �0.561 0.150 �0.215 0.150 �0.560 �0.394
�0.492Ba8Zr4Ti4O24c �0.571 0.256 0.256 0.250 �0.524 0.250 0.266
0.266 �0.569Ba8Zr4Ti4O24d �0.226 0.358 0.038 0.090 �0.460 0.090
0.036 0.348 �0.204Ba8Zr4Ti4O24e �0.370 0.124 0.124 0.111 �0.195
0.115 0.116 0.091 �0.181Ba8Zr4Ti4O24f �0.278 0.237 0.237 0.225
�0.262 0.225 0.228 0.228 �0.282Ba8Zr3Ti5O24a �0.479 0.3844 0.147
0.4232 �0.451 0.197 0.094 0.094 �0.249Ba8Zr3Ti5O24b �0.163 0.267
0.382 0.101 �0.269 0.101 0.378 0.274 �0.139Ba8Zr3Ti5O24c �0.196
0.124 0.124 0.110 �0.210 0.110 0.125 0.125 �0.232Ba8Zr2Ti6O24a
0.509 0.252 0.365 0.204 �0.276 0.204 0.387 0.233
�0.518Pb8Zr2Ti6O24b �0.552 �0.158 �0.158 0.106 �0.289 0.106 �0.149
�0.149 �0.541Pb8Zr2Ti6O24c �0.323 0.210 0.176 0.222 �0.323 0.164
0.073 0.073 �0.261Ba8 Zr1Ti8O24 �0.451 0.266 0.266 0.269 �0.443
0.269 0.252 0.252 �0.495
B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324 323
materials systems with Oxygen vacancies (for instance through
theintroduction of aliovalent elements to base oxide). These
vacancies,both charged and neutral, may have substantial effect on
the values ofpolarization and piezoelectric response of the
materials as well as onthermal, mechanical and dynamic properties.
Hence, understand-ing the influence of vacancies is essential for
engineering perfor-mance of ABO3 type perovskites. Systematic
theoretical studiesalong these lines are quite recent. Some recent
calculations worthto mention are on orbital reconstruction [50], on
carrier density[51] and on ferromagnetism [52] induced by oxygen
vacancies. Liet al. have shown that Nb-doped SrTiO3 can deform the
crystallattice and increased deformation observed in the presence
of ox-ygen vacancies. They have also investigated that Ti 3d
electrons inSrTiO3 originating from oxygen vacancies are naturally
more inac-tive and localized [52]. Yildiz et al. have shown that
lattice strainaffects the vacancy formation energy of SrTiO3 in
Ref. [53]. Inanother theoretical work they have investigated the
relationshipbetween defect chemistry and vacancy concentration
[54].
Acknowledgement
This work was partially supported by The Scientific and
Tech-nological Research Council of Turkey (TUBITAK) and Council
ofHigher Education of Turkey (YOK) for fellowship to
conductresearch in Texas A&M University. T.C acknowledges the
supportfrom the International Institute of Materials for Energy
Conversion(IIMEC) at Texas A&M University, an NSF International
MaterialsInstitute (DMR 0844082) and also support provided by
Scientificand Technological Research Council of Turkey
(TUBITAK-BIDEB) forhis visit to initiate the project in 2013. All
calculations are per-formed at the facilities of Supercomputing
Center of Texas A&MUniversity, and computing facilities of
Laboratory of ComputationalEngineering of Nanomaterials directed by
TC.
Appendix A. Supplementary data
Supplementary data related to this article can be found
athttps://doi.org/10.1016/j.matchemphys.2017.11.026.
References
[1] A. Lebedev, Ab initio calculations of phonon spectra in
ATiO3 perovskitecrystals (A¼ Ca, Sr, Ba, Ra, Cd, Zn, Mg, Ge, Sn,
Pb), Phys. Solid State 51 (2009)362e372.
[2] Ph Ghosez, E. Cockayne, U.V. Waghmare, K.M. Rabe, Lattice
dynamics ofBaTiO3, PbTiO3 and PbZrO3: a comparative
first-principles study, Phys. Rev. B60 (1999) 836.
[3] Y. Duan, G. Tang, C. Chen, T. Lu, Z. Wu, First-principles
investigations offerroelectricity and piezoelectricity in
BaTiO3/PbTiO3 superlattices, Phys. Rev.B 85 (2012) 054108.
[4] A. Erba, K.E. El-Kelany, M. Ferrero, I. Baraille, M. Rerat,
Piezoelectricity ofSrTiO3: an ab initio description, Phys. Rev. B
88 (2013) 035102.
[5] P. Ghosez, X. Gonze, J.P. Michenaud, Lattice dynamics of
BaTiO3, PbTiO3, andPbZrO3: a comparative first-principles study,
Phys. Rev. B 60 (1999) 836.
[6] M. Uludogan, P.D. Guarin, Z.E. Gomez, T. Cagin, W.A.
Goddard, CMES-ComptModel. Eng. Sci. 24 (2008) 215e238.
[7] W. Guyton Candy, Piezoelectricty, McGraw-Hill Book Company,
New York,1946.
[8] J.F. Nye, Physical Properties of Crystals, Clarendon,
Oxford, 1957.[9] A.W. Hewat, Structure of rhombohedral
ferroelectric barium titanate, Ferro-
electrics 6 (1973) 215e218.[10] M.E. Lines, A.M. Glass,
Principles and Applications of Ferroelectrics and
Related Materials, Oxford university press, 1977.[11] J.F.
Scott, Ferroelectric Memories, Vol. 3, Springer Science &
Business Media,
2013.[12] R. Waser, H. Keller, U. Erb, Nanoelectronics and
Information Technology:
Advanced Electronic Materials and Novel Devices, MRS BULLETIN,
2004.[13] F. Calle- Vallejo, J. I.Martinez, J.M. Garcia e Lastra,
M. Mogensen, J. Rossmeisl,
Angew. Chem. Int. Ed. 49 (2010) 1e4.[14] L. Bellaiche, J.
Padilla, D. Vanderbilt, Phys. Rev. B 59 (1998).[15] D. Damjanovic,
Piezoelectric properties of perovskite ferroelectrics: unsolved
problems and future research, Annales de Chimie Science des
Materiaux. Vol.26. No. 1, No longer published by Elsevier,
2001.
[16] G. Rohrer, M. Affatigato, M. Backhaus, R. Bordia, K. Scott
Weil, Challenges inceramic science: a report from the workshop on
emerging research areas inceramic science, J. Am. Ceram. Soc. 95
(2012) 3699e3712.
[17] H. Yao, L. Ouyang, W.Y. Ching, Ab initio calculation of
elastic constants ofceramic crystals, J. Am. Ceram. Soc. 90 (2007)
3194e3204.
[18] P. Ravindran, L. Fast, P.A. Korzhavyi, B. Johansson, J.
Wills, O. Eriksson, J. Appl.Phys. 84.9 (1998) 4891e4904.
[19] M. Kalay, H.H. Kart, S. Ozdemir Kart, T. Cagin, Elastic
properties and pressureinduced transitions of ZnO polymorphs from
first-principle calculations,J. Alloys Compd. 484 (2009)
431e438.
[20] I. Erdem, H.H. Kart, T. Cagin, High pressure phase
transitions in SnO2 poly-morphs by first-principles calculations,
J. Alloys Compd. 587 (2014) 638e645.
[21] X. Wu, D. Vanderbilt, D.R. Hamaann, Systematic treatment of
displacements,strains, and electric fields in density-functional
perturbation theory, Phys.Rev. B 72 (2005) 035105.
[22] A. Gonzalez-Garcia, W. Lopez-Perez, R. Palacio-Mozo, R.
Gonzalez- Hernandez,J. Comp. Mat. Sci. 91 (2014) 279e284.
[23] M.-Y. Xie, F. Tasnadi, I.A. Abrikosov, L. Hultman, V.
Darakchieva, Elastic
https://doi.org/10.1016/j.matchemphys.2017.11.026http://refhub.elsevier.com/S0254-0584(17)30905-7/sref1http://refhub.elsevier.com/S0254-0584(17)30905-7/sref1http://refhub.elsevier.com/S0254-0584(17)30905-7/sref1http://refhub.elsevier.com/S0254-0584(17)30905-7/sref1http://refhub.elsevier.com/S0254-0584(17)30905-7/sref1http://refhub.elsevier.com/S0254-0584(17)30905-7/sref1http://refhub.elsevier.com/S0254-0584(17)30905-7/sref2http://refhub.elsevier.com/S0254-0584(17)30905-7/sref2http://refhub.elsevier.com/S0254-0584(17)30905-7/sref2http://refhub.elsevier.com/S0254-0584(17)30905-7/sref2http://refhub.elsevier.com/S0254-0584(17)30905-7/sref2http://refhub.elsevier.com/S0254-0584(17)30905-7/sref2http://refhub.elsevier.com/S0254-0584(17)30905-7/sref3http://refhub.elsevier.com/S0254-0584(17)30905-7/sref3http://refhub.elsevier.com/S0254-0584(17)30905-7/sref3http://refhub.elsevier.com/S0254-0584(17)30905-7/sref3http://refhub.elsevier.com/S0254-0584(17)30905-7/sref3http://refhub.elsevier.com/S0254-0584(17)30905-7/sref4http://refhub.elsevier.com/S0254-0584(17)30905-7/sref4http://refhub.elsevier.com/S0254-0584(17)30905-7/sref4http://refhub.elsevier.com/S0254-0584(17)30905-7/sref5http://refhub.elsevier.com/S0254-0584(17)30905-7/sref5http://refhub.elsevier.com/S0254-0584(17)30905-7/sref5http://refhub.elsevier.com/S0254-0584(17)30905-7/sref5http://refhub.elsevier.com/S0254-0584(17)30905-7/sref5http://refhub.elsevier.com/S0254-0584(17)30905-7/sref6http://refhub.elsevier.com/S0254-0584(17)30905-7/sref6http://refhub.elsevier.com/S0254-0584(17)30905-7/sref6http://refhub.elsevier.com/S0254-0584(17)30905-7/sref7http://refhub.elsevier.com/S0254-0584(17)30905-7/sref7http://refhub.elsevier.com/S0254-0584(17)30905-7/sref8http://refhub.elsevier.com/S0254-0584(17)30905-7/sref9http://refhub.elsevier.com/S0254-0584(17)30905-7/sref9http://refhub.elsevier.com/S0254-0584(17)30905-7/sref9http://refhub.elsevier.com/S0254-0584(17)30905-7/sref10http://refhub.elsevier.com/S0254-0584(17)30905-7/sref10http://refhub.elsevier.com/S0254-0584(17)30905-7/sref11http://refhub.elsevier.com/S0254-0584(17)30905-7/sref11http://refhub.elsevier.com/S0254-0584(17)30905-7/sref11http://refhub.elsevier.com/S0254-0584(17)30905-7/sref12http://refhub.elsevier.com/S0254-0584(17)30905-7/sref12http://refhub.elsevier.com/S0254-0584(17)30905-7/sref13http://refhub.elsevier.com/S0254-0584(17)30905-7/sref13http://refhub.elsevier.com/S0254-0584(17)30905-7/sref13http://refhub.elsevier.com/S0254-0584(17)30905-7/sref13http://refhub.elsevier.com/S0254-0584(17)30905-7/sref14http://refhub.elsevier.com/S0254-0584(17)30905-7/sref15http://refhub.elsevier.com/S0254-0584(17)30905-7/sref15http://refhub.elsevier.com/S0254-0584(17)30905-7/sref15http://refhub.elsevier.com/S0254-0584(17)30905-7/sref16http://refhub.elsevier.com/S0254-0584(17)30905-7/sref16http://refhub.elsevier.com/S0254-0584(17)30905-7/sref16http://refhub.elsevier.com/S0254-0584(17)30905-7/sref16http://refhub.elsevier.com/S0254-0584(17)30905-7/sref17http://refhub.elsevier.com/S0254-0584(17)30905-7/sref17http://refhub.elsevier.com/S0254-0584(17)30905-7/sref17http://refhub.elsevier.com/S0254-0584(17)30905-7/sref18http://refhub.elsevier.com/S0254-0584(17)30905-7/sref18http://refhub.elsevier.com/S0254-0584(17)30905-7/sref18http://refhub.elsevier.com/S0254-0584(17)30905-7/sref19http://refhub.elsevier.com/S0254-0584(17)30905-7/sref19http://refhub.elsevier.com/S0254-0584(17)30905-7/sref19http://refhub.elsevier.com/S0254-0584(17)30905-7/sref19http://refhub.elsevier.com/S0254-0584(17)30905-7/sref20http://refhub.elsevier.com/S0254-0584(17)30905-7/sref20http://refhub.elsevier.com/S0254-0584(17)30905-7/sref20http://refhub.elsevier.com/S0254-0584(17)30905-7/sref20http://refhub.elsevier.com/S0254-0584(17)30905-7/sref21http://refhub.elsevier.com/S0254-0584(17)30905-7/sref21http://refhub.elsevier.com/S0254-0584(17)30905-7/sref21http://refhub.elsevier.com/S0254-0584(17)30905-7/sref22http://refhub.elsevier.com/S0254-0584(17)30905-7/sref22http://refhub.elsevier.com/S0254-0584(17)30905-7/sref22http://refhub.elsevier.com/S0254-0584(17)30905-7/sref23
-
B. Akgenc et al. / Materials Chemistry and Physics 205 (2018)
315e324324
constants, composition, and piezolectric polarization in
InxAl1-xN : from abinitio calculations to experimental implications
for the applicability ofVegard's rule, Phys. Rev. B 86 (2012)
155310.
[24] T. Qi, I. Grinberg, M. Rappe, First-principles
investigation of the highlytetragonal ferroelectric material Bi
(Zn1/2Ti1/2)O3, Phys. Rev. B 79 (2009)094114.
[25] J. Bennett, I. Grinberg, P. Davies, A.M. Rappe, Pb-free
ferroelectrics investi-gated with density functional theory:
SnAl1/2Nb1/2O3 perovskites, Phys. Rev.B 83 (2011) 144112.
[26] S.V. Halilov, M. Fornari, D.J. Singh, Lattice instabilities
in (Pb,Cd)TiO3 alloys,Appl. Phys. Lett. 81 (2002) 3443e3445.
[27] R.A.P. Ribeir, S.R. de Lazaro, DFT/PBE0 study on
structural, electronic anddielectric properties of SnZr0.5Ti0.5O3
lead-free ferroelectric material, J. AlloysCompd. 714 (2017)
553e559.
[28] J. Zhou, Z. Xia, M.S. Molokeev, X. Zhang, D. Peng, Q. Liu,
Composition design,optical gap and stability investigations of
lead-free halide double perovskiteCs2AgInCl6, J. Mater. Chem. A 5
(2017) 15031e15037.
[29] S. Mehmood, Z. Ali, I. Khan, I. Ahmad, Effects of cobalt
substitution on thephysical properties of the perovskite strontium
ferrite, Mater. Chem. Phys.196 (2017) 222e228.
[30] W. Yang, J. Han, L. Wang, Y. Shen, L. Li, Y. Yang, H. Li,
L. Chen, Effect of orderedB-site cations on the structure, elastic
and thermodynamic properties ofKTa0.5Nb0.5O3 crystal, Appl. Phys. A
123.7 (461) (2017).
[31] D.S.L. Pontes, W.B. Bastos, A.J. Chiquito, E. Longo, F.M.
Pontes, Combinedtheoretical and nanoscale experimental study of
Pb(Ca,Ba)TiO3, Pb(Sr,Ba)TiO3,and Pb(Sr,Ca)TiO3 complex perovskite
structures: an investigation of theferroelectric and electronic
properties, J. Alloys Compd. 702 (2017) 327e337.
[32] G. Kresse, M. Marsman, J. Furtmuller, Vienna Ab-initio
Simulation PackageWien, Austria, 2001.
[33] K.H. Hellwege, A.M. Hellwege, Ferroelectrics and Related
Substances, Landolt-B€ornstein, New Series, 1969.
[34] J.J. Wang, F.Y. Meng, X.Q. Ma, X. Xu, L.Q. Chen, Lattice,
elastic, polarization, andelectrostrictive properties of BaTiO3
from first-principles, J. Appl. Phys. 108(2010) 034107.
[35] H. Landolt, R. Bornstein, Numerical Data and Functional
Relationship in Sci-ence and Technology, Springer, 1987.
[36] S. Piskonov, E. Heifets, R.I. Eglitis, G. Bornstel, J.
Comp. Mat. Sci. 29.2 (2004)165e178.
[37] Z. Li, M. Grimsditch, C.M. Foster, S.K. Chan, Dielectric
and elastic properties offerroelectric materials at elevated
temperature, J. Phys. Chem. Solids 57(1996) 1433e1438.
[38] L. Cao, E. Sozontov, J. Zegenhagen, Cubic to tetragonal
phase transition ofSrTiO3 under epitaxial stress: an X-ray
backscattering study, Phys. status solidi(a) 181.2 (2000)
387e404.
[39] S. Yamanaka, H. Fujikane, T. Hamaguchi, H. Muta, T. Oyama,
T. Matsuda,S. Kobayashi, K. Kurosaki, Thermophysical properties of
BaZrO3 and BaCeO3,
J. Alloys Compd 359 (2003) 109.[40] R. Terki, H. Feraoun, G.
Bertrand, H. Aourag, Full potential calculation of
structural, elastic and electronic properties of BaZrO3 and
SrZrO3, Phys. StatusSolidi B 242.5 (2005) 1054e1062.
[41] B.J. Kennedy, C.J. Howard, B.C. Chakooumakos,
High-temperature phasetransitions in SrZrO3, Phys. Rev. B 59 (1999)
4023.
[42] M.D. Fontana, G. Metrat, J.L. Servoin, F. Gervais, Infrared
spectroscopy inKNbO3 through the successive ferroelectric phase
transitions, J. Phys. C SolidState Phys. 17 (1984) 483.
[43] Ph Sciau, A. Kania, B. Dkhil, E. Suard, A. Ratuzna,
Structural investigation ofAgNbO3 phases using x-ray and neutron
diffraction, J. Phys. Condens. Matter16 (2004) 2795.
[44] J.C. Chervin, J.P. Itie, D. Gourdain, Ph Pruzan, Energy
dispersive X-raydiffraction study of KNbO3 up to 19GPa at room
temperature, Solid stateCommun. 110 (1999) 247e251.
[45] A. Mahmood, S.M. Ramay, H.M. Rafique, Y. Al-Zaghayer, U.
Khan, First-prin-ciples study of electronic, optical and
thermoelectric properties in cubicperovskite materials AgMO3 (M¼ V,
Nb, Ta), Mod. Phys. Lett. B 28 (2014)1450077.
[46] R. Resta, M. Posternak, A. Baldereschi, Towards a quantum
theory of polari-zation in ferroelectrics: the case of KNbO3, Phys.
Rev. Lett. 70 (1993) 1010.
[47] W. Zhong, R.D. King-Smith, D. Vanderbilt, Giant LO-TO
splittings in perovskiteferroelectrics, Phys. Rev. Lett. 72 (1994)
3618.
[48] M. Posternak, R.Resta, A. Baldereschi, Role of covalent
bonding in the polari-zation of perovskite oxides: the case of
KNbO3, Phys. Rev. B 50 (1994) 8911.
[49] G. Henkelman, A. Arnaldsson, H. Jonsson, A fast and robust
algorithm forBader decomposition of charge density, Comput. Mater.
Sci. 36 (2006)354e360.
[50] Z.Q. Liu, W. Lu, S.W. Zeng, J.W. Deng, Z. Huang, C.J. Li,
M. Motapothula,W.M. Lu, L. Sun, K. Han, J.Q. Zhong, P. Yang, N.N.
Bao, W. Chen, J.S. Chen,Y.P. Feng, J.M.D. Coey, T. Venkatesan,
Ariando, Bandgap control of the oxygen-vacancy-induced two
dimensional electron gas in SrTiO3, Adv. Mater. In-terfaces 1
(2014) 1400155.
[51] Z.Q. Liu, D.P. Leusink, W.M. Lu, X. Wang, X.P. Wang, K.
Gopinadhan, Y.T. Lin,A. Annadi, Y.L. Zhao, A.R. Barman, S. Dhar,
Y.P. Feng, H.B. Su, G. Xiong,T. Venkatesan, Ariando, Reversible
metal-insulator transition in LaAlO3 thinfilms mediated by intragap
defects: an alternative mechanism for resistiveswitching, Phys.
Rev. B 84 (2011) 165106.
[52] Z.Q. Liu, W.M. Lu, S.L. Lim, X.P. Qiu, N.N. Bao, M.
Motapothula, J.B. Yi, M. Yang,S. Dhar, T. Venkatesa, Ariando,
Reversible room-temperature ferromagnetismin Nb-doped SrTiO3 single
crystals, Phys. Rev. B 87 (2013) 220405.
[53] D. Marrocchelli, L. Sun, B. Yildiz, Dislocations in SrTiO3
: easy to reduce but notso fast for oxygen transport, J. Am. Chem.
Soc. 137 (2015) 4735e4748.
[54] A. Kushima, B. Yildiz, Role of lattice strain and defect
chemistry on the oxygenvacancy migration at the (8.3%
Y2O3-ZrO2)/SrTiO3 hetero-interface: a firstprinciples study, ECS
Trans. 25 (2009) 159e1609.
http://refhub.elsevier.com/S0254-0584(17)30905-7/sref23http://refhub.elsevier.com/S0254-0584(17)30905-7/sref23http://refhub.elsevier.com/S0254-0584(17)30905-7/sref23http://refhub.elsevier.com/S0254-0584(17)30905-7/sref23http://refhub.elsevier.com/S0254-0584(17)30905-7/sref23http://refhub.elsevier.com/S0254-0584(17)30905-7/sref24http://refhub.elsevier.com/S0254-0584(17)30905-7/sref24http://refhub.elsevier.com/S0254-0584(17)30905-7/sref24http://refhub.elsevier.com/S0254-0584(17)30905-7/sref24http://refhub.elsevier.com/S0254-0584(17)30905-7/sref24http://refhub.elsevier.com/S0254-0584(17)30905-7/sref24http://refhub.elsevier.com/S0254-0584(17)30905-7/sref25http://refhub.elsevier.com/S0254-0584(17)30905-7/sref25http://refhub.elsevier.com/S0254-0584(17)30905-7/sref25http://refhub.elsevier.com/S0254-0584(17)30905-7/sref26http://refhub.elsevier.com/S0254-0584(17)30905-7/sref26http://refhub.elsevier.com/S0254-0584(17)30905-7/sref26http://refhub.elsevier.com/S0254-0584(17)30905-7/sref26http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref27http://refhub.elsevier.com/S0254-0584(17)30905-7/sref28http://refhub.elsevier.com/S0254-0584(17)30905-7/sref28http://refhub.elsevier.com/S0254-0584(17)30905-7/sref28http://refhub.elsevier.com/S0254-0584(17)30905-7/sref28http://refhub.elsevier.com/S0254-0584(17)30905-7/sref28http://refhub.elsevier.com/S0254-0584(17)30905-7/sref28http://refhub.elsevier.com/S0254-0584(17)30905-7/sref29http://refhub.elsevier.com/S0254-0584(17)30905-7/sref29http://refhub.elsevier.com/S0254-0584(17)30905-7/sref29http://refhub.elsevier.com/S0254-0584(17)30905-7/sref29http://refhub.elsevier.com/S0254-0584(17)30905-7/sref30http://refhub.elsevier.com/S0254-0584(17)30905-7/sref30http://refhub.elsevier.com/S0254-0584(17)30905-7/sref30http://refhub.elsevier.com/S0254-0584(17)30905-7/sref30http://refhub.elsevier.com/S0254-0584(17)30905-7/sref30http://refhub.elsevier.com/S0254-0584(17)30905-7/sref30http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref31http://refhub.elsevier.com/S0254-0584(17)30905-7/sref32http://refhub.elsevier.com/S0254-0584(17)30905-7/sref32http://refhub.elsevier.com/S0254-0584(17)30905-7/sref33http://refhub.elsevier.com/S0254-0584(17)30905-7/sref33http://refhub.elsevier.com/S0254-0584(17)30905-7/sref33http://refhub.elsevier.com/S0254-0584(17)30905-7/sref34http://refhub.elsevier.com/S0254-0584(17)30905-7/sref34http://refhub.elsevier.com/S0254-0584(17)30905-7/sref34http://refhub.elsevier.com/S0254-0584(17)30905-7/sref34http://refhub.elsevier.com/S0254-0584(17)30905-7/sref35http://refhub.elsevier.com/S0254-0584(17)30905-7/sref35http://refhub.elsevier.com/S0254-0584(17)30905-7/sref36http://refhub.elsevier.com/S0254-0584(17)30905-7/sref36http://refhub.elsevier.com/S0254-0584(17)30905-7/sref36http://refhub.elsevier.com/S0254-0584(17)30905-7/sref37http://refhub.elsevier.com/S0254-0584(17)30905-7/sref37http://refhub.elsevier.com/S0254-0584(17)30905-7/sref37http://refhub.elsevier.com/S0254-0584(17)30905-7/sref37http://refhub.elsevier.com/S0254-0584(17)30905-7/sref38http://refhub.elsevier.com/S0254-0584(17)30905-7/sref38http://refhub.elsevier.com/S0254-0584(17)30905-7/sref38http://refhub.elsevier.com/S0254-0584(17)30905-7/sref38http://refhub.elsevier.com/S0254-0584(17)30905-7/sref38http://refhub.elsevier.com/S0254-0584(17)30905-7/sref39http://refhub.elsevier.com/S0254-0584(17)30905-7/sref39http://refhub.elsevier.com/S0254-0584(17)30905-7/sref39http://refhub.elsevier.com/S0254-0584(17)30905-7/sref39http://refhub.elsevier.com/S0254-0584(17)30905-7/sref39http://refhub.elsevier.com/S0254-0584(17)30905-7/sref40http://refhub.elsevier.com/S0254-0584(17)30905-7/sref40http://refhub.elsevier.com/S0254-0584(17)30905-7/sref40http://refhub.elsevier.com/S0254-0584(17)30905-7/sref40http://refhub.elsevier.com/S0254-0584(17)30905-7/sref40http://refhub.elsevier.com/S0254-0584(17)30905-7/sref40http://refhub.elsevier.com/S0254-0584(17)30905-7/sref41http://refhub.elsevier.com/S0254-0584(17)30905-7/sref41http://refhub.elsevier.com/S0254-0584(17)30905-7/sref41http://refhub.elsevier.com/S0254-0584(17)30905-7/sref42http://refhub.elsevier.com/S0254-0584(17)30905-7/sref42http://refhub.elsevier.com/S0254-0584(17)30905-7/sref42http://refhub.elsevier.com/S0254-0584(17)30905-7/sref42http://refhub.elsevier.com/S0254-0584(17)30905-7/sref43http://refhub.elsevier.com/S0254-0584(17)30905-7/sref43http://refhub.elsevier.com/S0254-0584(17)30905-7/sref43http://refhub.elsevier.com/S0254-0584(17)30905-7/sref43http://refhub.elsevier.com/S0254-0584(17)30905-7/sref44http://refhub.elsevier.com/S0254-0584(17)30905-7/sref44http://refhub.elsevier.com/S0254-0584(17)30905-7/sref44http://refhub.elsevier.com/S0254-0584(17)30905-7/sref44http://refhub.elsevier.com/S0254-0584(17)30905-7/sref44http://refhub.elsevier.com/S0254-0584(17)30905-7/sref45http://refhub.elsevier.com/S0254-0584(17)30905-7/sref45http://refhub.elsevier.com/S0254-0584(17)30905-7/sref45http://refhub.elsevier.com/S0254-0584(17)30905-7/sref45http://refhub.elsevier.com/S0254-0584(17)30905-7/sref45http://refhub.elsevier.com/S0254-0584(17)30905-7/sref45http://refhub.elsevier.com/S0254-0584(17)30905-7/sref46http://refhub.elsevier.com/S0254-0584(17)30905-7/sref46http://refhub.elsevier.com/S0254-0584(17)30905-7/sref46http://refhub.elsevier.com/S0254-0584(17)30905-7/sref47http://refhub.elsevier.com/S0254-0584(17)30905-7/sref47http://refhub.elsevier.com/S0254-0584(17)30905-7/sref48http://refhub.elsevier.com/S0254-0584(17)30905-7/sref48http://refhub.elsevier.com/S0254-0584(17)30905-7/sref48http://refhub.elsevier.com/S0254-0584(17)30905-7/sref49http://refhub.elsevier.com/S0254-0584(17)30905-7/sref49http://refhub.elsevier.com/S0254-0584(17)30905-7/sref49http://refhub.elsevier.com/S0254-0584(17)30905-7/sref49http://refhub.elsevier.com/S0254-0584(17)30905-7/sref50http://refhub.elsevier.com/S0254-0584(17)30905-7/sref50http://refhub.elsevier.com/S0254-0584(17)30905-7/sref50http://refhub.elsevier.com/S0254-0584(17)30905-7/sref50http://refhub.elsevier.com/S0254-0584(17)30905-7/sref50http://refhub.elsevier.com/S0254-0584(17)30905-7/sref50http://refhub.elsevier.com/S0254-0584(17)30905-7/sref51http://refhub.elsevier.com/S0254-0584(17)30905-7/sref51http://refhub.elsevier.com/S0254-0584(17)30905-7/sref51http://refhub.elsevier.com/S0254-0584(17)30905-7/sref51http://refhub.elsevier.com/S0254-0584(17)30905-7/sref51http://refhub.elsevier.com/S0254-0584(17)30905-7/sref51http://refhub.elsevier.com/S0254-0584(17)30905-7/sref52http://refhub.elsevier.com/S0254-0584(17)30905-7/sref52http://refhub.elsevier.com/S0254-0584(17)30905-7/sref52http://refhub.elsevier.com/S0254-0584(17)30905-7/sref52http://refhub.elsevier.com/S0254-0584(17)30905-7/sref53http://refhub.elsevier.com/S0254-0584(17)30905-7/sref53http://refhub.elsevier.com/S0254-0584(17)30905-7/sref53http://refhub.elsevier.com/S0254-0584(17)30905-7/sref53http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54http://refhub.elsevier.com/S0254-0584(17)30905-7/sref54
First – principles calculations on stability and mechanical
properties of various ABO3 and their alloys1. Introduction2.
Computational method2.1. Construction of alloys
3. Results and discussions3.1. Equation of state3.2. Calculation
of elastic constants3.3. Calculation of Born effective charges3.4.
Calculation of dielectric and piezoelectric constants
4. Concluding remarksAcknowledgementAppendix A. Supplementary
dataReferences
