The electronic structure of perfect and defective perovskite crystals: Ab initio hybrid functional calculations

The electronic structure of perfect and
defective perovskite crystals: Ab initio
hybrid functional calculations
Ph.D. Thesis
Presented to the Department of Physics
of the University of Osnabrück
by
Sergejs Piskunovs
Thesis Advisor: Prof. Dr. Gunnar Borstel
October 2003
Contents
1 Introduction
1
2 Basic perovskite crystals: Strontium, Barium, and
Introduction . . . . . . . . . . . . . . . . . . . . . .
2.1 Experimental results . . . . . . . . . . . . . . . . .
2.1.1 Bulk crystals . . . . . . . . . . . . . . . . .
2.1.2 Impurity defects in perovskites . . . . . . .
2.1.3 Surfaces . . . . . . . . . . . . . . . . . . . .
2.2 Previous theoretical results . . . . . . . . . . . . . .
2.2.1 Bulk perovskites . . . . . . . . . . . . . . .
2.2.2 Point defects: SrTiO3 :Fe . . . . . . . . . . .
2.2.3 Calculations on surfaces . . . . . . . . . . .
2.3 Motivation . . . . . . . . . . . . . . . . . . . . . . .
3 DFT/HF formalism and methodology
Introduction . . . . . . . . . . . . . . . . . . . . .
3.1 DFT formalism . . . . . . . . . . . . . . . . . . .
3.1.1 Schrödinger equation . . . . . . . . . . . .
3.1.2 Total energy through the density matrices
3.1.3 Hohenberg-Kohn theorems . . . . . . . . .
3.1.4 Energy functional . . . . . . . . . . . . . .
3.1.5 Local density approximation . . . . . . . .
3.1.6 Generalized gradient approximation . . . .
3.1.7 Hybrid exchange functionals . . . . . . . .
3.1.8 Spin-density functional theory . . . . . . .
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Lead Titanates
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CONTENTS
3.2
3.3
ii
Practical implementation of DFT/HF calculation scheme . . . . . .
3.2.1 Selection of basis set . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Auxiliary basis sets for the exchange-correlation functionals .
3.2.3 Evaluation of the integrals. The Coulomb problem . . . . .
3.2.4 Reciprocal space integration . . . . . . . . . . . . . . . . . .
3.2.5 SCF calculation scheme . . . . . . . . . . . . . . . . . . . .
One-electron properties . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Properties in a direct space; population analysis . . . . . . .
3.3.2 Properties in a reciprocal space; band-structure and density
of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Calculations on bulk perovskites
Introduction . . . . . . . . . . .
4.1 Computational details . . . . .
4.2 Bulk properties . . . . . . . . .
4.3 Electronic properties . . . . . .
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surfaces.
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5 Point defects in perovskites: The case study of SrTiO3 :Fe
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 A consistent approach for a modelling of defective solids . .
5.2 Results for perfect STO and supercell convergence . . . . . .
5.3 Results for a single Fe impurity . . . . . . . . . . . . . . . .
6 Two-dimensional defects in perovskites: (001) and (110)
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 The choice of a model for surface simulation . . . . . . . .
6.2 Calculations on the ABO3 (001) surfaces . . . . . . . . . .
6.2.1 Surface structures . . . . . . . . . . . . . . . . . . .
6.2.2 Electronic charge redistribution . . . . . . . . . . .
6.2.3 Density of states and band structures . . . . . . . .
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces . . . . . . . . . . . . . . . . . .
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CONTENTS
iii
7 Low-temperature compositional heterogeneity in Bax Sr1−x TiO3 solid
solutions
Introduction . . . . . . . . . .
7.1 Perovskite solid solutions . . .
7.2 Thermodynamic theory . . . .
7.3 Application to Bax Sr(1−x) TiO3
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solid
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solutions
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8 Conclusions
153
A Hay-Wadt eﬀective core pseudopotentials for Ti, Sr, Ba and Pb
156
B Calculation of the elastic constants
160
C List of Acronyms
166
Presentation of the results of the present study
167
Acknowledgments
169
Bibliography
170
List of Figures
2.1
2.2
2.3
2.4
2.5
A prototype cubic structure of a perovskite crystal with the formula
unit ABO3 , where A=Sr, Ba or Pb, and B=Ti. . . . . . . . . . . .
The BTO and PTO crystals. Schematic sketch of a ferroelectric transition into a tetragonal broken-symmetry structure, where the origin
has been kept at the Ti atom. The arrows indicate atomic displacements. In the structure shown, the polarization is along [001]. . . .
The photoelectron energy distribution curves for STO and BTO.
Taken from Battye, Höchst and Goldmann (1976). . . . . . . . . .
Schematic illustration of three possible surfaces of cubic ABO3 perovskites (upper row). Each surface can be terminated by two types of
crystalline planes (pointed by arrows) consistent of diﬀerent atomic
compounds. The lower row demonstrates the relevant 7-layered slabs
(thin ﬁlms). Black rectangles represent the surface unit cells. . . .
One of possible relaxations of the ABO3 (001) surfaces. Arrows show
the directions of atomic displacements. The surface rumpling s is
shown for surface layer. Interlayer distances d12 and d23 are based
on the positions of relaxed metal ions which are known to be much
stronger electron scatterers than oxygen ions (Bickel, Schmidt, Heinz
and Müller, 1989). . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF FIGURES
3.1
3.2
4.1
4.2
4.3
5.1
5.2
6.1
6.2
v
The individual GTFs (solid lines) are relatively poor representatives
of true one-electron wavefunctions: GTFs have wrong asymptotics in
the inﬁnity (fall down too fast) and wrong behavior near the nucleus.
Left ﬁgure (a) shows the ”optimum” GTF obtained for the 1s orbital
by least-square ﬁt, preserving the normalization. The performance
can be improved by using ”contracted” GTF. Right ﬁgure (b) shows
1s wavefunction approximated by a contracted 4-GTF set. . . . . . . 40
Flow chart of the CRYSTAL code. . . . . . . . . . . . . . . . . . . . 55
The band structure of three cubic perovskites for selected high-symmetry
directions in the BZ. a) STO, b) BTO, c) PTO. The energy scale is
in atomic units (Hartree, 1 Ha = 27.212 eV), the dashed line is the
top of valence band. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
The calculated total and projected density of states (DOS and PDOS)
for three perovskites. a) STO, b) BTO, c) PTO. . . . . . . . . . . . 69
The diﬀerence electron density plots for three perovskites calculated
using DFT B3PW: a) STO, b) BTO, c) PTO. The electron density
plots are for AO-(001) (left column), (110) (middle column), and
TiO2 -(001) (right column) cross sections. Isodensity curves are drawn
from -0.05 to +0.05 e a.u.−3 with an increment of 0.005 e a.u.−3 . . . 72
(a) Schematic view of the Fe impurity in STO with asymmetric eg relaxation of six nearest O atoms, (b) The relevant energy levels before
and after relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
(a) The electronic density plots for the (010) cross section of Fe
and nearest ions in STO as calculated by means of the DFT-B3PW
method for the cyclic cluster of 160 atoms. Isodensity curves are
drawn from 0.8 to 0.8 e a.u.3 with an increment of 0.0022 e a.u.3 ,
(b) the same as (a) for the (001) section, (c) the same for the (110)
section. Left panels are diﬀerence electron densities, right panels spin
densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Models for simulating surfaces starting from a perfect 3D crystal. . . 92
Schematic illustration of the slab unit cells for ABO3 (001) surfaces:
a) AO-terminated, b) TiO2 -terminated, c) asymmetrical termination. 93
LIST OF FIGURES
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
Schematic illustration of the SrTiO-terminated SrTiO3 (110) 9-layer
slab unit cells: a) slab without vacancies (unstable, cannot exist
due to inﬁnite dipole moment perpendicular to the surface), b) TiOterminated SrTiO3 (110) surface (unreconstructed surface, stable according to Heifets, Kotomin and Maier (2000), also named as “unreconstructed surface”, see last section), c) Ti-terminated SrTiO3 (110)
surface (reconstructed surface). Vacancies created on Sr and O sites
are shown as green spots. . . . . . . . . . . . . . . . . . . . . . . .
Schematic illustration of two outermost surface layers relaxation with
respect to perfect 3d crystal positions: a) STO, b) BTO, c) PTO.
View from [010] direction. Arrows show the directions of atom displacements. Upper panels - AO termination, lower panels - TiO2
termination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diﬀerence electron density maps in the cross section perpendicular
to the (001) surface ((110) plane) with AO-, TiO2 and asymmetrical
terminations. Isodensity curves are drawn from -0.05 to +0.05 e a.u.−3
with an increment of 0.0025 e a.u.−3 . a) STO, b) BTO, c) PTO. .
Calculated electronic band structures for STO bulk and surfaces. .
Calculated electronic band structures for BTO bulk and surfaces. .
Calculated electronic band structures for PTO bulk and surfaces. .
Total and projected DOS for the bulk STO. . . . . . . . . . . . . .
Total and projected DOS for the SrO-terminated surface. . . . . .
Total and projected DOS for the STO TiO2 -terminated surface. . .
Total and projected DOS for the bulk BTO. . . . . . . . . . . . . .
Total and projected DOS for the BaO-terminated surface. . . . . .
Total and projected DOS for the BTO TiO2 -terminated surface. .
Total and projected DOS for the bulk PTO. . . . . . . . . . . . . .
Total and projected DOS for the PbO-terminated surface. . . . . .
Total and projected DOS for the PTO TiO2 -terminated surface. .
MIES and ab initio DOS results for the clean unreconstructed and
heated STO(110) surfaces. See text and inserts for detailed description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
UPS and ab initio DOS results, the same as for Fig. 6.18. . . . . .
vi
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LIST OF FIGURES
7.1
7.2
7.3
vii
Superstructures in quasibinary Bax Sr(1−x) TiO3 solid solutions that
are stable with respect to the formation of anti-phase boundaries. . . 140
The phase diagram of the quasi-binary disordered solid solution BST. 144
Phase diagram, the same as for Fig. 7.2. . . . . . . . . . . . . . . . . 146
A.1 A schematic illustration of all-electron (red lines) and pseudo- (blue
lines) potentials and their corresponding wavefunctions. The radius
at which all-electron and pseudopotential values match is rc . Taken
from Payne, Teter, Allan, Arias et al. (1992). . . . . . . . . . . . . . 157
List of Tables
2.1
2.2
2.3
Main applications of perovskite materials. . . . . . . . . . . . . . . . 4
Experimentally observed lattice constant a0 (Å), bulk modulus B
(GPa), and elastic constants cij (in 1011 dyne/cm2 ) for the three basic
perovskite crystals in their high-symmetry cubic phase. . . . . . . . 7
Experimentally observed surface rumpling s, and relative displacements of three near-surface planes for SrO- and TiO2 -terminated
STO(001) surfaces ∆dij (in percent of lattice constant). The negative sign means the reducing of interlayer distances. . . . . . . . . . 13
3.1
The exponents α (bohr−2 ) and contraction coeﬃcients dj of individually normalized Gaussian-type basis functions (see Eq. 3.45 and
Eq. 3.48). All atoms are described using the Hay-Wadt small core
pseudopotentials (Hay and Wadt, 1984c,b,a). . . . . . . . . . . . . . . 46
4.1
The optimized lattice constant a 0 (Å), bulk modulus B (GPa) and
elastic constants cij (in 1011 dyne/cm2 ) for three ABO3 perovskites as
calculated using DFT and HF approaches. The results of calculations
for standard BS are given in the brackets. The two last columns
contain the experimental data and the data calculated using other
QM techniques. The penultimate row for each perovskite contains the
bulk modulus calculated using the standard relation B=(c11 +2c12 )/3;
it is done for Experiment and Theory columns, respectively. . . . . . 64
The calculated optical band gap (eV). The results of calculations with
standard BS are given in the brackets. . . . . . . . . . . . . . . . . . 67
4.2
LIST OF TABLES
ix
4.3
Eﬀective Mulliken charges, Q (e), and bond populations, P (mili e),
for three bulk perovskites, the results of calculations with standard
BS are given in brackets. OI means the oxygen nearest to the reference one, OII oxygen from the second sphere of neighbour oxygens.
Negative populations mean repulsion between atoms. . . . . . . . . . 71
5.1
Convergence of results for pure STO (a0 = 3.904 Å) obtained for
DFT-B3PW band calculations corresponding to cyclic clusters of an
increasing size. εv is the upper level of valence band and εc is the
bottom of conduction band. All energies in eV, total energies are
presented with respect to the reference point of 314 a.u. = 8544.59
eV. q are the Mulliken eﬀective atomic charges (in e). L, NA , are
the primitive unit cell extension, number of atoms in the cyclic cluster, whereas RM and M are deﬁned by Eq. (5.1) and Eq. (5.6),
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The width of the Fe impurity band EW (in eV) calculated for the
relevant supercells. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eﬀective charges q of ions obtained in the DFT-B3PW band structure
calculations with Pack-Monkhorst k set 8 × 8 × 8 and diﬀerent cyclic
clusters modelling perfect and defective STO. The lengths in the ﬁrst
column are lattice constants of the relevant supercells whereas the
distances R given above for the eﬀective charges are calculated with
respect to the supercell coordinate origin, where the Fe ion is placed.
Positions of one-electron Fe levels (in eV) with respect to the VB top
calculated by means of DFT-B3PW method for L = 16 and L = 32
cyclic cluster with and without lattice relaxation. . . . . . . . . . . .
The eﬀective Mulliken charges of atoms q and bond populations P (in
milli e) for the L32 cyclic cluster with unrelaxed and relaxed lattices.
5.2
5.3
5.4
5.5
6.1
6.2
82
85
86
87
87
Atomic relaxation relative to ideal atomic positions of cubic ABO3 (001)
surfaces (in percent of lattice constant). A means Sr, Ba, or Pb. . . . 95
Surface rumpling s and relative displacements of the three nearsurface planes for AO- and TiO2 -terminated surfaces ∆dij (in percent of lattice constant). Results for asymmetrical slabs are given in
brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
LIST OF TABLES
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
Calculated surface energies (in eV per surface cell). Results for previous ab initio calculations (Cheng, Kunc and Lee, 2000; Tinte and
Stachiotti, 2001; Meyer, Padilla and Vanderbilt, 1999) are averaged
over AO and TiO2 terminated surfaces. . . . . . . . . . . . . . . .
AO termination. Charges and dipole moments. Numbers in brackets
are deviations from bulk values. Bulk charges in e; STO: Sr = 1.871,
Ti = 2.35, O = -1.407, BTO: Ba = 1.795, Ti = 2.364, O = -1.386,
PTO: Pt = 1.343, Ti = 2.335, O = -1.226 (see Table 4.3). . . . . .
TiO2 termination. The same as for Table 6.4 . . . . . . . . . . . .
Asymmetrical termination. The same as for Table 6.4 . . . . . . .
Charge densities in the (001) crystalline planes of the bulk perovskites
(in e, per TiO2 or AO unit, data are taken from Table 4.3) and in
four top planes of the AO-, TiO2 -terminated and asymmetrical slabs.
Changes of charge density with respect to the bulk are given in brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AO termination. Bond populations (in e ·10−3 ). Negative population
means atomic repulsion. The corresponding bond populations for
bulk perovskites are: Ti-O bond: STO) 88, BTO) 100, PTO) 98;
Pb-O bond: 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TiO2 termination. The same as for Table 6.8. . . . . . . . . . . . .
Asymmetrical termination. The same as for Table 6.8. . . . . . . .
The calculated optical gap (in eV) for the bulk (Table 4.2) and
surface-terminated perovskites. The last row contains experimental
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eﬀective Mulliken charges, Q (e), for two diﬀerent STO(110) terminations. Bulk charges of ions (in e): Sr = 1.871, Ti = 2.350, and O
= -1.407. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ti orbitals population for two diﬀerent STO(110) terminations. Ti
orbital populations for a bulk crystal: Ti 3p = 6.014, Ti 3d = 1.233,
Ti 4s = 0.163. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
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. 131
. 131
LIST OF TABLES
7.1
7.2
xi
Occupation probabilities, n(r), stoichiometric compositions, xst , and
the energies of formation, ∆U , for the ordering phases in Bax Sr(1−x) TiO3
solid solutions. Ṽ1 , Ṽ2 and Ṽ3 are Fourier transforms of the mixing
potential in the kjs points that correspond to the stars 1, 2, and 3
from Eq. 7.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Total energies, Etot , stoichiometric compositions, and equilibrium lattice parameters, aeq for the structures (a–i) from Fig. 7.1. Here, the
BTO and STO are represented by a supercell (2 × 2 × 2) containing
40 atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.1 Eﬀective core potentials for Ti, Sr, Ba, and Pb. . . . . . . . . . . . . 158
Chapter 1
Introduction
Density Functional Theory (DFT) is widely used as an eﬃcient and reliable method
for computing the ground state energetics for a wide spectrum of solids. In recent
years the use of functionals based on the generalized gradient approximation (GGA)
has signiﬁcantly improved the accuracy of computed energies when compared to the
original local density approximation (LDA). Nevertheless, the eigenvalues in DFT
calculations do not formally correspond to excitation energies. This is particulary
apparent in their use to estimate band gaps where the non-analytic dependence of
the eﬀective potential on the density often leads to a crystalline gap which is less
than half of that observed. The poor estimation of the band gap has a number of
unfortunate consequences. For instance, in some systems the band gap collapses
completely, and a qualitatively incorrect description of the ground state, as a metal
rather than an insulator, is obtained. Consequently the utility and reliability of
DFT calculations is limited, especially when properties depending explicitly on the
excited state energies are of interest.
An alternative to DFT, Hartree-Fock (HF) approach gives both ground and exited states, but considerably overestimates the optical band gap. In order to study
the electronic and optical properties of complex materials an approach providing a
reliable estimate of band gaps coupled with the reasonable description of the ground
state is required. In the present study of pure and defective perovskite crystals,
the fulﬁlment of such requirements is clearly demonstrated using a simple hybrid
HF/DFT scheme containing an admixture of non-local Fock exchange.
2
In present theoretical investigations, a wide class of perovskite oxides is represented by three, the most attractive (from a scientiﬁc point of view) crystals of
SrTiO3 , BaTiO3 , and PbTiO3 in their high symmetry cubic phases. These perovskite crystals present a great technological and fundamental interest due to their
numerous applications related to ferroelectricity, non-linear and electro-optics, superconductivity, and catalysis. Although the above-mentioned perovskite-type materials have been intensively investigated theoretically and experimentally at least
in the last ﬁfteen years, a proper description of their electronic properties is still an
area of active research. In order to make a contribution to the explanation of various
electro-optical eﬀects observed in perovskite materials, their ground-state properties
have been calculated from ﬁrst principles and analyzed in the present study.
The present work is divided into three main parts. The ﬁrst part, which includes
Chapters 2– 4, is concerned with the introduction into materials and diﬀerent computational approaches. In Chapter 2 the selected perovskite crystals are described
in detail, and the experimental and theoretical results known in the literature are
presented. Chapter 3 serves to explain the DFT/HF formalism and methodology.
Chapter 4 presents a comparison of the eﬀect of various approximations on the bulk
properties of perovskites calculated on equal computational grounds. In the second
part, including Chapters 5 and 6, the defective perovskite materials are considered.
Chapter 5 is dedicated to the investigations on point defects in perovskites which
are presented as a single Fe4+ impurity substituting a host Ti atom in strontium
titanate. In Chapter 6, two-dimensional defects (surfaces) are considered. The attention is paid to the technologically important (001) surfaces of all three perovskites
as well as to the Ti-terminated SrTiO3 (110) polar surface. Theoretical results are
compared with existing experiments. Finally, the third part containing of Chapter
7 demonstrates the theoretical predictions for Bax Sr1−x TiO3 solid solutions, which
are promising materials for future non-volatile computer memory cells. Such predictions are based on the combination of thermodynamic considerations with ab initio
DFT calculations for a number of Bax Sr1−x TiO3 structures.
Chapter 2
Basic perovskite crystals:
Strontium, Barium, and Lead
Titanates
Introduction
Perovskite oxides are the most signiﬁcant class of ferroelectric materials. Nowadays,
ferroelectrics are of great importance for a number of technological applications.
Their multistability due to a switchable macroscopic polarization allows the realization of nonvolatile ferroelectric computer memories based on stacked dielectric
capacitors. Ferroelectric materials display very large piezoelectric, pyroelectric, and
non-linear optic response. As an example, the piezoelectric eﬀect is exploited in
transducers and actuators, designed to convert an electrical signal into a mechanical
one and vice versa (Lines and Glass, 1977). In our time, the presence of such devices
is practically unexpendable in everyday’s life, the car-crash detector responsible for
air-bags is one of most common examples.
Perovskite-type ferroelectrics are generally denoted by a general formula unit
ABO3 , where A is a mono- or divalent cation represented by an alkali or rare-earth
element, and B is correspondingly a penta- or tetravalent cation of the transition
metal series. The classic examples are KNbO3 and BaTiO3 . The perovskites containing trivalent cations, e.g. LaMnO3 , are interesting due to the colossal magne-
4
Table 2.1: Main applications of perovskite materials.
Multilayered Capacitor
Piezoelectric Transducer
Electrooptical Modulator
Dielectric Resonator
Thick Film Resistor
Elecrostrictive Actuator
Magnetic Bubble Memory
Laser Host
Ferromagnet
Refractory Electrode
Second Harmonic Generator
BaTiO3
Pb(Zr,Yi)O3
(Pb,La)(Zr,Ti)O3
BaZrO3
BaRuO3
Pb(Mg,Nb)O3
GdFeO3
YAlO3
(Ca,La)MnO3
LaCoO3
KNbO3
toresistance eﬀect which is associated with a ferromagnetic-to-paramagnetic phase
transition (Ramirez, 1997), and by presence of mixed ionic-electronic conductivity
which makes them very attractive for use as cathodes in solid oxide fuel cells (Skinner, 2001). Some additional examples of the existing applications of wide spectra of
perovskite materials are collected in Table 2.1.
The SrTiO3 (STO), BaTiO3 (BTO), and PbTiO3 (PTO) are classical representatives of perovskites belonging to the AII BIV O3 group. They demonstrate quite
diﬀerent physical behaviour that make them very attractive from both technological
and computational points of view. The presence of diﬀerent crystal phases allows
them to be ferro- and paraelectrics. BTO has three phase transitions accompanied
by lattice distortion, when STO and PTO have only one transition (but of diﬀerent
nature). Nevertheless, to provide a comparative study of three perovskites in their
high temperature cubic phases it is enough to substitute only one cation while the
other atoms remain unchanged. Considered perovskites, namely STO, are widely
used as lattice-matched substrates for the epitaxial growth of various perovskite-type
thin ﬁlms, e.g. high-Tc superconductors, and of special interest are Bax Sr1−x TiO3
solid solutions (Kotecki, Baniecki, Shen, Laibowitz et al., 1999; Liu, Cheng, Chen,
Liu et al., 1998), which are promising dielectric materials for future ceramic capacitors and thermistor applications (due to the absence of phase segregation), and
whose Curie temperature Tc can be varied by adjusting the Ba/Sr ratio. Prior to
start theoretical investigations on selected materials, the existing experimental data
related to ABO3 perovskite crystals are covered in the next Section.
2.1 Experimental results
2.1
2.1.1
5
Experimental results
Bulk crystals
The fascinating feature of the perovskite crystalline structure is the extreme ease
with which it undergoes one or more structural phase transitions while the temperature is varied. At suﬃciently high temperatures all perovskites are paraelectrics, i.e.
the macroscopic polarization is zero in their high-symmetry cubic structure. The
prototype paraelectric structure is shown in Fig. 2.1. It is a simple cubic structure,
O
Ti
A
Figure 2.1: A prototype cubic structure of a perovskite crystal with the formula
unit ABO3 , where A=Sr, Ba or Pb, and B=Ti.
where the cation A is at the cube corners, the cation B is at the center of cube,
and the oxygen atoms are placed at the center positions of the cube faces, thus
showing the pattern of a corner-sharing octahedra. At room temperature, only the
STO has the simple cubic symmetry. At low temperatures the STO behaves as an
incipient ferroelectric (Barrett, 1952) (similar to KTaO3 ) in the sense that it has
a very large static dielectric response and is only barely stabilized against the condensation of the ferroelectric soft mode at low temperature (Lines and Glass, 1977;
Mitsui and Westphal, 1961; Sawaguchi, Kikuchi and Kodera, 1962). The observed
softening of polar phonons with the reduction of temperature to the range of 50–100
2.1 Experimental results
6
K would appear to extrapolate to a ferroelectric transition with the Curie temperature close to 40 K, but instead the softening saturates and no such transition is
experimentally observed (Viana, Lunkenheimer, Hemberger, Böhmer et al., 1994).
In addition, at 105 K the STO undergoes a structural phase transition from cubic
high-temperature to a tetragonal low-temperature phase. This is an antiferrodistortive (AFD) nonpolar transition in which the oxygen octahedra rotate in opposite
directions in neighboring unit cells.
O
Ti
A
Figure 2.2: The BTO and PTO crystals. Schematic sketch of a ferroelectric transition into a tetragonal broken-symmetry structure, where the origin has been kept
at the Ti atom. The arrows indicate atomic displacements. In the structure shown,
the polarization is along [001].
In contraposition with STO, BTO and PTO are ferroelectrics with the Curie
temperatures of 393 K and 763 K, respectively. At these temperatures BTO and
PTO undergo a ﬁrst ferroelectric transition into a broken-symmetry structure, a
tetragonal one with the polarization along the cube axis [001]. This transition is
caused by a displacement of oxygen octahedra and A atoms, with respect to the
Ti atom as it is shown in Fig.2.2. In the PTO this is the only transition observed,
while in BTO further lowering of the temperature induces a series of further transitions to the structures of diﬀerent symmetries, ﬁrst orthorhombic with polarization
2.1 Experimental results
7
Table 2.2: Experimentally observed lattice constant a0 (Å), bulk modulus B (GPa),
and elastic constants cij (in 1011 dyne/cm2 ) for the three basic perovskite crystals in
their high-symmetry cubic phase.
a0
STO
BTO
PTO
3.89 (i),
4.00 (i)
3.97 (ii)
20.60 (i)
∼22.90 (v)
3.90 (iii)
C11
31.72 (iv)
∼18.70 (v)
C12
10.25 (iv)
14.00 (i)
∼10.10 (v)
∼10.70 (v)
C44
12.35 (iv)
12.60 (i)
∼10.00 (v)
∼11.20 (v)
B
179 (i),
162 (i),
179±4.6 (vi)
195±5 (vi)
∼144†
i – Hellwege and Hellwege (1969)
ii – Shirane and Repinsky (1956)
iii – Abramov, Tsirelson, Zavodnik, Ivanov et al. (1995)
iv – Bell and Rupprecht (1963)
v – Li, Grimsditch, Foster and Chan (1996)
vi – Fischer, Wang and Karato (1993)
†
– calculated from elastic constants using the standard
formula: B=(c11 +2c12 )/3
along [110] direction, and ﬁnally rhombohedral (polarization along [111]). In these
transitions the oxygen octahedral cage is slightly deformed, while the metal atoms
are displaced oﬀ-center with respect to it. Also, from a ferroelectric phase, a back
transformation can be induced by external pressure, instead of increasing the temperature (Samara and Peercy, 1981).
Since the main aim of this study is a qualitatively description of the inﬂuence of
a number of methods on ﬁrst-principles calculations on the properties of perovskitetype crystals and to predict the method allowing the best description of the physicalchemical nature of selected materials, in order to keep the computational conditions
on equal ground, all perovskites are considered in their high-symmetry cubic phases.
Such theoretical approach allows one to make calculations much more computationally economical due to the increase of a number of symmetry operations and to
provide a most qualitative comparison between all three perovskites as well. Other
assertion in favor of high symmetry cubic phase applying stems from the fact that
the defect formation energy is much larger than the energy of transition to low symmetry phase.
2.1 Experimental results
8
The bulk properties of crystals play an important role in solid state physics and
are the basic measure allowing to understand how well experiments and theory correlate. These bulk properties could be mainly characterized by the lattice constant
a0 , the bulk modulus B, and the elastic constants cij , the experimental values of
which are collected in Table 2.2. Of peculiar interest are the elastic constants. For
cubic systems only three independent elastic constants (c11 , c12 , and c44 ) allowed by
symmetry. Experimentally, the individual elastic constants are measured by determining the velocities of ultrasonic wave propagation along various directions in high
quality mono-domain single crystals (Bell and Rupprecht, 1963; Ishidate and Sasaki,
1989; Li, Grimsditch, Foster and Chan, 1996). To provide the most comprehensive
comparison between calculated and observed elastic properties, one should keep
in mind that the cubic phase of perovskites under consideration is quite unstable.
Thus, the measured elastic constants depend strongly on temperature or pressure.
For example, according to the results of Bell and Rupprecht (1963), c11 measured for
STO increases by about 4% as the temperature is lowered from room temperature
to 115 K, before dropping abruptly as the transition temperature is approached. It
can therefore be supposed that the agreement between experiment and theory for
elastic properties cannot be too high, and a discrepancy of 10% could be taken as
quite satisfactory, taking into account the fact that the ab initio calculated elastic
constants are obtained using the approximation that the temperature is equal 0 K.
A substantial understanding of many crystalline properties requires the knowledge of the electronic structure and optical properties of the bulk material. The
detailed comparison of X-ray photoelectron spectra of the STO and BTO valence
bands (Battye, Höchst and Goldmann, 1976; Nakamatsu, Adachi and Ikeda, 1981)
demonstrate their strong similarity. The O 2p valence band of both materials consists of a central peak with a shoulder to each side, although, for BTO, the shoulder
of higher binding energy is not strongly pronounced, and the observed peak spacing
is a little bit smaller. This is partly explained by the inﬂuence of the increasing of
Ti-O bond distance and thus growing covalency of BTO Ti-O bond (Nakamatsu,
Adachi and Ikeda, 1981). The raw spectra obtained from each material for the
binding energy range 0 to 40 eV are shown in Fig. 2.3. The most recent resonant
photoelectron spectroscopy study on the electronic structure of BTO perovskite performed by Robey, Hudson, Henrich, Eylem et al. (1996) indicates a contribution of
the Ti 3d states to the predominantly O 2p valence band, and thus produces infor-
2.1 Experimental results
9
Figure 2.3: The photoelectron energy distribution curves for STO and BTO. Taken
from Battye, Höchst and Goldmann (1976).
mation on Ti 3d – O 2p hybridization. The optical band gap observed by Wemple
(1970) for the high temperature cubic phase of BTO is 3.2 eV. The indirect band
gap energy 3.25 eV and direct band gap energy 3.75 eV have been observed experimentally for STO by means of spectroscopic ellipsometry (van Benthem, Elsässer
and French, 2001). Unfortunately, no photoelectron spectroscopic measurements on
bulk PTO have been found in literature. The experimentally observed optical band
gap for PTO crystal is 3.4 eV (Peng, Chang and Desu, 1992).
2.1.2
Impurity defects in perovskites
It is well known that optical and mechanical properties of crystals are strongly
aﬀected by defects and impurities unavoidably present in any real material. Such
defects play an important role in electro-optical and non-linear optical applications
of ABO3 ferroelectric materials (Günter and Huignard, 1988). In this study, the
investigation of point defects in ferroelectrics is limited to a case study of the Fe4+
impurity substituted for a host Ti atom in STO. Experimentally the Fe impurities
2.1 Experimental results
10
in perovskites have been studied in detail by Schirmer, Berlinger and Müller (1975);
Wasser, Bieger and Maier (1990); van Stevendaal, Buse, Kämper, Hesse et al. (1996).
These studies point out the major role that iron impurities play in photochromic
and photorefractive processes in perovskite crystals. The isolated energy levels of
the Fe impurity in BTO have been observed at 0.95±0.05 eV above the valence
band edge (van Stevendaal, Buse, Kämper, Hesse et al., 1996). Additionally, the
high spin (S=2) state has been determined for Fe4+ impurity in the photochromic
STO (Schirmer, Berlinger and Müller, 1975).
2.1.3
Surfaces
The surfaces of ABO3 perovskites are very important for many innovative technological applications (Lines and Glass, 1977; Noguera, 1996; Henrick and Cox, 1994;
Scott, 2000). Due to the miniaturization of the relevant electronic devices, investigations on electronic properties and structure of ABO3 perovskite thin ﬁlms are an
object of intense interest. The cubic perovskite oxides have three low-index surfaces
(001), (110), and (111) schematically presented in Fig. 2.4.
According to the
classiﬁcation given by Tasker (1979), only the (001) surface of perovskite crystals
given by formula unit AII BIV O3 corresponds to the “Type I” stable surface, since
it exhibits no dipole moment perpendicular to the surface because of neutral cumulative charge in each layer, AO and BO2 , parallel to the surface. Because of their
stability, the (001) perovskite surfaces are interesting from both technological and
computational point of view and are considered in detail in this study. ABO3 (110)
and (111) surfaces correspond to “Type III” unstable surfaces in the Tasker’s classiﬁcation and cannot exist without substantial charge redistribution, usually caused
by surface reconstruction, e.g. vacancy creation. Such surfaces are also of great
technological interest and are intensively studied during the last few years.
The (001) surfaces, quite stable at room temperature, in cubic STO perovskite
are widely studied by various experimental groups. The STO surface structure has
been analyzed by means of Low Energy Electron Diﬀraction (LEED) by Bickel,
Schmidt, Heinz and Müller (1989). The structure and the electronic states of SrOand TiO2 -terminated STO surfaces have been intensively studied by Hikita, Hanada,
Kudo and Kawai (1993) by means of Reﬂection High Energy Electron Diﬀraction
(RHEED), X-ray Photoelectron Spectroscopy (XPS) and Ultraviolet Photoelectron
IV
III
II (BO2)
I (AO)
O2
ABO
(110)
III
IV
I (O2)
II (ABO)
B
AO3
AO3
(111)
II (B)
III
IV
A
B
O
I (AO3)
Figure 2.4: Schematic illustration of three possible surfaces of cubic ABO3 perovskites (upper row). Each surface can be
terminated by two types of crystalline planes (pointed by arrows) consistent of diﬀerent atomic compounds. The lower
row demonstrates the relevant 7-layered slabs (thin ﬁlms). Black rectangles represent the surface unit cells.
BO2
AO
(001)
2.1 Experimental results
11
2.1 Experimental results
O
s
Ti
12
I (surface) layer
d12
A
II layer
d23
z
x
III layer
Figure 2.5: One of possible relaxations of the ABO3 (001) surfaces. Arrows show
the directions of atomic displacements. The surface rumpling s is shown for surface
layer. Interlayer distances d12 and d23 are based on the positions of relaxed metal
ions which are known to be much stronger electron scatterers than oxygen ions
(Bickel, Schmidt, Heinz and Müller, 1989).
Spectroscopy (UPS). Ikeda, Nishimura, Morishita and Kido (1999) explored the
surface relaxation and rumpling of TiO2 -terminated STO(001) surface by means of
Medium Energy Ion Scattering (MEIS). Charlton, Brennan, Muryn, McGrath et al.
(2000) published results of examination of STO structure with both SrO and TiO2
terminations by means of Surface X-ray Diﬀraction (SXRD). The most recent experimental studies on STO(001) have been obtained by van der Heide, Jiang, Kim
and Rabalais (2001) (XPS, LEED, Time-Of-Flight Scattering and Recoiling Spectrometry), and by Maus-Friedrichs, Frerichs, Gunhold, Krischok et al. (2002) (UPS
and Metastable Impact Electron Spectroscopy (MIES)). Unfortunately, the results
of similar experimental studies on BTO and PTO surfaces are absent in literature,
despite their high-necessity and technological importance.
During the surface formation, the bond-breaking process induces forces which
push the atoms out of their bulk positions. The surface relaxation takes place.
Atoms in (001) surfaces of simple cubic crystals are allowed to relax only along
the z-axis since there are no forces along x- and y-axes by symmetry. Because of
relaxation, the surface layer acquires rumpling s, as well as changes in interlayer
distances ∆dij , where i and j are the numbers of surface layers, take place (See
Fig. 2.5). The experimentally obtained magnitudes of s and ∆dij of STO(001) are
collected in Table 2.3. It is clearly seen in Table 2.3 that experiments do not all
2.1 Experimental results
13
Table 2.3: Experimentally observed surface rumpling s, and relative displacements
of three near-surface planes for SrO- and TiO2 -terminated STO(001) surfaces ∆dij
(in percent of lattice constant). The negative sign means the reducing of interlayer
distances.
AO-terminated
TiO2 -terminated
s
∆d12
∆d23
s
∆d12
∆d23
LEED (i)
4.1±2
-5±1
2±1
2.1±2
1±1
-1±1
RHEED (ii)
4.1
2.6
1.3
2.6
1.8
1.3
1.5±0.2
0.5±0.2
12.8±8.5
0.3±1
MEIS (iii)
SXRD (iv)
1.3±12.1
-0.3±3.6
-6.7±2.8
i – Bickel, Schmidt, Heinz and Müller (1989)
ii – Hikita, Hanada, Kudo and Kawai (1993)
iii – Ikeda, Nishimura, Morishita and Kido (1999)
iv – Charlton, Brennan, Muryn, McGrath et al. (2000)
agree in the sign of ∆d12 and ∆d23 for the SrO-terminated surface as well as for
∆d23 of TiO2 -terminated STO. The disagreement between these experiments suggests that the experimental data should not to be taken as deﬁnite. The expected
quality of experimental analysis is not well established yet for a complicated metal
oxide surface such as STO(001). In the study of Bickel, Schmidt, Heinz and Müller
(1989) the authors did not determine the proportions of the surface exhibiting the
SrO and TiO2 terminations independently; they assumed that they both appear in
equal proportions. Reﬁning the structural parameters for surfaces simultaneously,
the authors then obtained a R-factor (reliability factor which allows to estimate
the eﬃciency of extracting information from LEED curves (Pendry, 1980)) of 0.529.
While it was argued to be acceptable that time because of the complexity of the surface, this value nevertheless seems to be too large. In the study of Hikita, Hanada,
Kudo and Kawai (1993), the surfaces were prepared under diﬀerent conditions in
order to obtain SrO and TiO2 terminations separately; for these R-factors of 0.28
and 0.26 were obtained, respectively. A possible explanation of the problems with
experimental reﬁnements has been given by Padilla and Vanderbilt (1998). The
explanation is an assumption accompanied by a theoretical conﬁrmation that there
is a substantial buckling in the second metal-oxygen layer, especially in the case of
the subsurface SrO layer on the TiO2 -terminated surface.
2.2 Previous theoretical results
14
The photoelectron spectra of surface electronic states of SrO- and TiO2 -terminated
STO(001) surfaces have been carefully studied (Hikita, Hanada, Kudo and Kawai,
1993; Maus-Friedrichs, Frerichs, Gunhold, Krischok et al., 2002). The electronic
states near the Fermi level, formed mainly by the Ti 3d O 2p orbitals, were found
to be sensitive to the coordination symmetry around the Ti atom, i.e. for the TiO2 terminated surface. Low symmetry in the O coordination gives a broadening in the
band structure of Ti and O. For the SrO terminated surface with high symmetry
coordination, a narrow band is observed (Hikita, Hanada, Kudo and Kawai, 1993).
The STO(110) and (111) polar perovskite surfaces were also studied experimentally using several diﬀerent techniques (Lo and Somorjai, 1978; Brunen and Zegenhagen, 1997; Bando, Ochiai, Haruyama, Yasue et al., 2001; Szot and Speier, 1999).
LEED experiments (Brunen and Zegenhagen, 1997) show a number of surface reconstructions at high temperatures. Atomic force microscopy (Szot and Speier, 1999)
also supports surface modiﬁcations due to an applied extensive thermal treatment.
However, there are no experimental estimates of the surface relaxation or the spectra of the STO or BTO at low temperatures to which theoretical calculations could
be compared.
2.2
2.2.1
Previous theoretical results
Bulk perovskites
The ﬁrst ab-initio theoretical investigations of the ferroelectric transitions in BTO
and PTO perovskite crystals have been performed by Cohen and Krakauer (1990).
The authors used the all-electron Full-Potential Linearized Augmented Plane Wave
(FLAPW) method to study ferroelectricity in BTO within the LDA. They performed
a series of frozen phonon calculations and demonstrated that the phase with a full
cubic symmetry is unstable with respect to zone-center distortions, in agreement
with the experimentally observed ferroelectric transition in this material. Authors
went on to study the depth and shapes of the energy well with respect to soft-mode
displacement, to demonstrate that the strain strongly inﬂuences the form of the
total-energy surface, later they extended this approach to the case of PTO (Cohen
and Krakauer, 1992). Using experimental data as a guide, they were able to show
2.2 Previous theoretical results
15
that the observed tetragonal ferroelectric ground state of this material is stabilized
by the large strain which appears upon transition from the cubic structure. Cohen
(1992) emphasized that the hybridization between the titanium 3d and oxygen 2p
is necessary for ferroelectricity in BTO and PTO.
King-Smith and Vanderbilt (1994) performed a systematic study of structural
and dynamical properties for eight various perovskites using the ﬁrst-principles
ultrasoft-pseudopotential method and the LDA. For the ﬁrst time, these authors
demonstrated the possibility of devising a computationally tractable scheme to compute the soft-mode total-energy surface correct to fourth order in the soft-mode
displacement. They showed the zone-center instabilities in the cubic perovskite
structure are very common, and highlighted the importance of extreme accuracy in
the k-point set in ﬁrst-principles calculations.
A few years later Tinte, Stachiotti, Rodriguez, Novikov et al. (1998) reported the
results of Local Spin Density Approximation (LSDA) and Perdew-Burke-Erzernhof
(PBE) GGA calculations on structural and dynamic properties of four ABO3 perovskites including BTO and STO. They demonstrated the underestimation of the
equilibrium volume of all material under investigation (an average of approximately
3%) that is typical for LDA calculations. The authors discussed about an ad hoc
correction of the theoretical equilibrium volume by selecting the adequate value of
the coeﬃcient κ related to the localization of the exchange-correlation hole, for each
particular system, that, of course, would not lead to a fully ab initio, free-parameter,
calculation scheme.
Most recently, Cora and Catlow (1999) explored the electronic structure of a
wide range of perovskites using the ab initio HF method. They performed an analysis of the ﬁrst-principles solution for the bulk materials based on tight binding-like
examinations of the band structures in reciprocal space. This treatment allowed an
understanding of the trends in the properties of bulk perovskites as a function of
their chemical composition via the tight binding parameter that controls the extent
of covalence in the Ti-O interaction. The authors demonstrated that symmetry
breaking around either a Ti or an O ion of the structure is responsible for the
ferroelectric-like distortion from the cubic perovskite phase; the electronic perturbation is then transferred to the neighboring sites through a delocalization of the
Ti-O bonding in the valence band.
The detailed description of the ferroelectric phase transition in PTO can be
2.2 Previous theoretical results
16
found in studies performed by Ghosez, Cockyane, Waghmare and Rabe (1999) and
Waghmare and Rabe (1997), and the most recent plane-wave-pseudopotential study
of Veithen, Gonze and Ghosez (2002) demonstrates the existence of essential Pb-O
covalency in a PTO crystal while the A-O bonds in BTO and STO remain fully ionic.
All these studies show the considerable progress in calculations of various perovskite properties and understanding the origin of ferroelectricity. However, the
previously calculated optical band gaps and lattice structure parameters are in disagreement with the relevant experimental results. Indeed, the band gap calculated
using the Kohn-Sham Hamiltonian usually strongly underestimates the experimental results, but, on the other hand, the HF overestimates the gap severely. The one
and only attempt (as found in the literature) to obtain a better approximation to
the band gap of perovskite material have been done by Cappelini, Bouette-Russo,
Amadon, Noguera et al. (2000) who applied a GW perturbative theory, based on
a set of self-consistent equations for the one-electron Green’s function involving
a screened potential (Hedin, 1965), to the DFT-LDA calculations of ground-state
properties of STO and related binary ionic oxides, SrO and MgO. The calculated
optical band gap of 2.7 eV for STO is in much better agreement with experiment
(3.25 eV) than the gaps calculated previously using LDA or HF. It seems that the
GW -method works reasonably well for pure ionic crystals, such as MgO and SrO,
in which the electronic states of the valence band and the bottom of the conduction
band consist mainly of s- and p-states, than for partly covalent STO, where localized
Ti 3d states contribute to bands around the Fermi level, and thus the GW -method
produces an overestimation of the self-energy correction.
2.2.2
Point defects: SrTiO3 :Fe
The only few theoretical calculations for ion impurities substituting for B atoms in
ABO3 have been found in the literature. The Fe impurities in KNbO3 have been
investigated by Donnerberg (1994, 1999); Postnikov, Poteryaev and Borstel (1998).
The Fe doped STO was considered by Selme, Pecheur and Toussaint (1984); MichelCalendini and Müller (1981)], and transition metal impurities in BTO were described
by Moretti and Michel-Calendini (1986). Most of these studies were semi-empirical
and/or cluster calculations, but the results of ab initio calculations on Fe impurity
in perovskite are quite scarse in literature. The one and only ﬁrst-principles investi-
2.2 Previous theoretical results
17
gation was performed recently for Fe in KNbO3 (Postnikov, Poteryaev and Borstel,
1998) using the linear muﬃn tin orbital method (LMTO) in the atomic sphere approximation. However, no lattice relaxation around the impurity was calculated,
and the calculated density of states depends considerably on the parameters of the
so-called LDA+U scheme.
2.2.3
Calculations on surfaces
The ﬁrst ab initio investigations on ABO3 perovskite surfaces were presented by
Kimura, Yamauchi, Tsukada and Watanabe (1995). The authors performed a series
of plane-wave (PW) ﬁrst-principle pseudopotential calculations on TiO2 -terminated
STO(001) surfaces to understand the inﬂuence of oxygen vacancies on conduction
properties of surface layers. Cohen (1996) presented his pioneering Linearized Augmented PW (LAPW) calculations performed for the periodic (001) and (111) slabs
of ferroelectric BTO where eﬀects on surfaces are well described (see also Cohen
(1997)). Padilla and Vanderbilt (1997) published the results of a PW ultrasoftpseudopotential calculations on both BaO- and TiO2 -terminated (001) surfaces of
cubic (paraelectric) and tetragonal (ferroelectric) BTO with detailed treatment of
surfaces relaxation and electronic structure of band gap region. Hereafter, Vanderbilt and coworkers extended their investigations over surfaces of other perovskite
oxides. In year 1998, results for STO have been published (Padilla and Vanderbilt,
1998) and recently, a common study for (001) surfaces of STO, BTO and PTO has
been presented (Meyer, Padilla and Vanderbilt, 1999). Cora and Catlow (1999)
reported the results of ab initio HF calculations on a set of perovskite-structured
transition metal oxides (including STO and BTO) with a detailed examination of
their ferroelectric behavior. The most recent study on the STO(001) surface relaxation has been provided by Cheng, Kunc and Lee (2000) using the DFT PW method
and Tinte and Stachiotti (2000, 2001) presented results of Shell Model (SM) simulations for BTO thin ﬁlms with parameters obtained from the ﬁrst-principles LAPW
method. In spite of great physical importance of all ab initio studies mentioned
above it should be noted that they have been carried out mostly using the LDA
method (the exception is the HF study by Cora and Catlow (1999)). It is the wellknown fact that the eigenvalues of the highest occupied and the lowest unoccupied
states in DFT-LDA calculations do not formally correspond to the real electron
excitation energies. As a result the calculated band gap underestimates the exper-
2.3 Motivation
18
imental one by more than a half. This discrepancy limits the proper theoretical
description of surface properties depending on the excited electron states such as
optical adsorption, for example. On the other hand, the band gap obtained through
the pure-HF calculations overestimates the experimental results severely (Pisani,
1996). The most recent comparative study performed by Heifets, Eglitis, Kotomin,
Maier et al. (2002) for the STO(001) surface demonstrates that the best agreement
with experiment could be achieved using so-called “hybrid” functionals (they contain the hybrid of the non-local HF exchange, DFT exchange, and GGA correlation
functionals) such as B3LYP and B3PW.
The one and only ﬁrst-principles study of ABO3 (110) surfaces, namely STO, has
been published recently by Bottin, Finocchi and Noguera (2003). The authors used
the PW DFT-LDA method to calculate the electronic structure of two stoichiometric SrTiO and O2 terminations as well as three nonstoichiometric terminations with
TiO, Sr, and O compositions, respectively, in the outermost atomic layer, which
automatically allows the surface to be free from any macroscopic polarization. The
authors suggested an insulating bulk-like electronic structure for all three nonstoichiometric terminations.
2.3
Motivation
As it is shown above, the electronic structure of perovskite materials have been the
subject of many experimental and theoretical investigations. A considerable progress
has been achieved in understanding the lattice dynamics and the origin of ferroelectricity by means of various ﬁrst-principles total-energy computation techniques.
However, the previously calculated optical band gaps are not in good agreement
with those observed experimentally. Obtaining the reliable prediction of the band
gap in semiconductors and insulators is a well recognized problem in ﬁrst-principles
calculations, and still remains an obstacle in ab initio band-gap engineering. This
is even more important for defects, since the defects level positions with respect to
the band edges are of key importance for their applications.
In recent years, a number of methods for obtaining a better approximation to the
band gap have been put forward. The GW perturbation theory (Hedin, 1965) yields
highly accurate gaps in weakly interacting systems such as semiconductors and simple oxides (Hybertsen and Louie, 1986; Cappelini, Bouette-Russo, Amadon, Noguera
2.3 Motivation
19
et al., 2000). In the case of semiconductors, the screened exchange approximation
has also been successful (Seidl, Görling, Vogl, Majewski et al., 1996; Engel, 1997).
The quantum Monte Carlo method (Ceperley, Chester and Kalos, 1977) has also
been used to estimate excitation energies based on explicitly correlated wavefunctions (Mitáš and Martin, 1994; Towler, Hood and Needs, 2000). Excitation energies
can also be extracted from the frequency-dependent linear response which may be
computed within the time-dependent DFT (Petersilka, Gossmann and Gross, 1996;
Tozer and Handy, 2000). Only a few calculations have been reported using these
latter methods, due to the complexity and computational costs involved in their
implementation.
In order to study the electronic and optical properties of more complex materials
such as ABO3 perovskites, an approach providing a reliable estimate of band gaps
while retaining a reasonable ground state description of the GGA is required. In
the present study the fact that these requirements are satisﬁed by a simple hybrid
scheme which contains an admixture of non-local Fock exchange to the DFT scheme
is clearly demonstrated.
Hybrid functional were originally developed to improve the description of the
ground state energetics of small molecules (Becke, 1993a). Subsequently, they have
been demonstrated to be signiﬁcantly more reliable than the best GGA functional
for computing atomisation enthalpies (Curtiss, Raghavachari, Redfern and Pople,
1997), ionization potentials and electron aﬃnities (Curtiss, Redfern, Raghavachari
and Pople, 1998), geometries and vibrational frequencies (Adamo, Ernzerhof and
Scuseria, 2000). The application of these methods in periodic calculations of solids
has been inhibited by diﬃculties in computing the non-local Fock exchange since
this is not convenient within the commonly used plane-wave basis set. Nevertheless
it can be implemented readily and very eﬃciently within the Linear Combination
of Atomic Orbitals (LCAO) technique that uses localized Gaussian Type Functions
(GTF) localized at atoms as the basis for expansion of the crystalline orbitals. The
main numerical approximation is the selection of the local Gaussian Basis Set (BS)
for each of the studied materials. Recently, a LCAO-GTF study has been performed
for both bulk and surface phases of STO crystal (Heifets, Eglitis, Kotomin, Maier
et al., 2002). In this study, the electronic structure of the STO(100) surface has been
calculated using “standard” BS available at Homepage (b) by means of various approximations including ab initio HF method with electron correlation corrections
2.3 Motivation
20
and DFT with diﬀerent exchange-correlation functionals, including hybrid (B3PW,
B3LYP) exchange techniques. This work demonstrated a noticeable improvement
of the calculated lattice constant, bulk modulus and optical band gap of STO with
respect to those experimentally observed when the hybrid B3LYP functional was
applied. Thus, further consistent ab initio investigations on ABO3 perovskite material is of high interest to give the most reliable theoretical predictions for many
crystal properties. Therefore the main aims of present study are the following:
• Development of the Basis Sets well suitable for LCAO computation of various
electronic properties of perovskites.
• Critical choice between diﬀerent Hamiltonians for the ab initio calculations of
selected ABO3 perovskites.
• Using the selected method to obtain the electronic structure of the perfect and
defective perovskite crystals.
• Based on the acquired knowledge, to predict the properties of Bax Sr1−x TiO3
superlatices, which are promising materials for non-volatile computer memory
cells.
Additionally, it could be noted that comparative ﬁrst-principles calculations on
ABO3 crystals, especially on their surfaces, are quite scarce in literature. Thus,
this study is an attempt to compensate such lack of information.
Chapter 3
DFT/HF formalism and
methodology
Introduction
The only possibility to study complex crystalline systems containing many atoms is
to perform computer simulations. This can be performed with a variety of methods ranging from classical to quantum mechanical (QM) approaches. The former
are force ﬁeld or semi-empirical schemes, in which the forces that determine the
interactions between the atoms are parameterized in order to reproduce a series of
experimental data, such as equilibrium geometries, bulk muduli or lattice vibrational
frequencies (phonons). These schemes have reached a high level of sophistication
and are often useful within a given class of materials provided good parameters are
already known. If, however, such parameters are not available, or if a system shows
unusual phenomena that are not yet understood, one must rely only on ab initio
calculations. They are more demanding in terms of computer requirements and thus
allow only the treatment of much smaller unit cells than semi-empirical calculations
do. The advantage of ﬁrst-principle methods lies in the fact that they do not require
any experimental knowledge to carry out such calculations. The following presentation will be restricted to ab initio methods whose main characteristics will be brieﬂy
sketched. The fact that electrons are indistinguishable and are Fermions requires
that their wave functions must be antisymmetric when two electrons are permuted.
3.1 DFT formalism
22
This leads to the phenomenon of exchange and correlation. There are two types of
approaches for a full quantum mechanical treatment: HF and DFT. The traditional
scheme is the HF method which is based on a wave function in a form of one Slater
determinant. Exchange is treated exactly but correlation eﬀects are neglected. The
latter can be included by more sophisticated approaches such as conﬁguration interaction but they progressively require much more computer time. As a consequence,
it is only feasible to study small systems which contain a few atoms. An alternative
scheme is DFT which is commonly used to calculate the electronic structure of complex systems containing many atoms such as large molecules or solids. It is based
on the electron density rather than on the wave functions and treats both exchange
and correlation, but both approximately. The ideal crystal is deﬁned by the unit
cell which may contain several atoms (up to about 100, in practical state-of-the-art
calculations) and is repeated inﬁnitely according to the translational symmetry. Periodic boundary conditions are used to describe an inﬁnite crystal. The additional
symmetry operations (inversion, rotation, mirror planes, etc.) that retain the ideal
crystal invariant allow to simplify the calculations, which always correspond to the
absolute zero temperature.
3.1
3.1.1
DFT formalism
Schrödinger equation
The ground state energy of an ensemble of atoms may be computed by solving the
Schrödinger equation which in the time-independent nonrelativistic case reads:
ĤΨk (r1 , r2 , . . . rN ) = Ek Ψk (r1 , r2 , . . . rN ).
(3.1)
The Hamiltonian operator (in atomic units m = = e2 = 1), Ĥ, consists of a sum
of three terms; the kinetic energy, the interaction with the external potential (V̂ext )
and the electron-electron interaction (V̂ee ). That is:
1 2
1
Ĥ = −
∇i + V̂ext +
.
2 i
|ri − rj |
i<j
N
N
(3.2)
3.1 DFT formalism
23
In materials simulation the external potential of interest is simply the interaction of
the electrons with the atomic nuclei1 :
V̂ext = −
Nat
α
Zα
,
|ri − Rα |
(3.3)
where ri is the coordinate of electron i and the charge on the nucleus at Rα is Zα .
Equation 3.2 is solved for a set of wavefunctions Ψ subject to the constraint that the
Ψ are antisymmetric, i.e. they change sign if the coordinates of any two electrons
are interchanged. The lowest energy eigenvalue, E0 , is the ground state energy, and
the probability density of ﬁnding an electron with any particular set of coordinates
{ri } is |Ψ0 |2 . The average total energy for a state speciﬁed by a particular Ψ, not
necessarily one of the eigenfunctions of Equation 3.2, is the expectation value of Ĥ,
that is:
(3.4)
E[Ψ] = Ψ∗ ĤΨdr ≡ Ψ|Ĥ|Ψ.
The notation [Ψ] emphasizes the fact that the energy is a functional of the wavefunction. The energy is higher than that of the ground state unless Ψ corresponds
to Ψ0 which is the variational theorem:
E[Ψ] E0 .
(3.5)
The ground state wavefunction and energy may be found by searching all possible
wavefunctions for the one that minimizes the total energy. HF theory consists of
an ansatz for the structure of Ψ - it is assumed to be an antisymmetric product of
functions φi each of which depends on the coordinates of a single electron, that is:
1
ΨHF = √ det[φ1 φ2 φ3 . . . φN ],
N!
(3.6)
where det indicates a matrix determinant (Szabo and Ostlund, 1982). Substitution
of this ansatz for Ψ into the Schrödinger equation results in an expression for the
1
In order to simplify the notation and to focus on the main feature of DFT the spin coordinate
is omitted here.
3.1 DFT formalism
24
HF energy:
EHF =
1
+
2 i,j
N
N
1 2
−
∇i + V̂ext φi (r)dr+
2 i
φ∗i (r)
N
φ∗i (r1 )φi (r1 )φ∗j (r2 )φj (r2 ) 1 −
|ri − rj |
2 i,j
φ∗i (r1 )φj (r1 )φi (r2 )φ∗j (r2 )
. (3.7)
|ri − rj |
The second term is simply the classical Coulomb energy written in terms of the
orbitals and the third term is the exchange energy. The ground state orbitals are
determined by applying the variation theorem to this energy expression under the
constraint that the orbitals are orthonormal. This leads to the HF equations:
ρ(r )
1 2
dr φi (r) + νX (r, r )φi (r )dr = εi φi (r),
− ∇ + νext (r) +
2
|r − r |
(3.8)
where the non-local exchange potential, νX , is such that:
N φj (r)φ∗j (r )
νX (r, r )φi (r )dr = −
φi (r )dr .
|
|r
−
r
j
(3.9)
The Hartree-Fock equations describe non-interacting electrons in the mean ﬁeld
potential consisting of the classical Coulomb potential and a non-local exchange
potential. From this starting point on, better approximations (additional correlation
methods) for Ψ and E0 are readily obtained but the computational cost of such
improvements is very high and scales prohibitively with the number of electrons
(Szabo and Ostlund, 1982). In addition, accurate solutions require a very ﬂexible
description of the wavefunction’s spatial variation, i.e. a large basis set is needed
which also adds to the expense for practical calculations.
3.1.2
Total energy through the density matrices
The Hamiltonian operator (Equation 3.2) consists of single electron- and bi-electronic
interactions, i.e. operators that depend on the coordinates of one or two electrons
only. In order to compute the total energy, we do not need to know the 3N dimensional wavefunction. A knowledge of the two-particle probability density - that is,
3.1 DFT formalism
25
the probability of ﬁnding an electron at r1 and another at r2 is suﬃcient. A quantity
of great use in analyzing the energy expression is the second-order density matrix,
which is deﬁned as:
P2 (r1 , r2 ; r1 , r2 )
N (N − 1)
=
2
Ψ∗ (r1 , r2 , . . . , rN )Ψ(r1 , r2 , . . . , rN )dr3 dr4 . . .drN .
(3.10)
The diagonal elements of P2 , often referred to as the two-particle density matrix or
pair density, are:
P2 (r1 , r2 ) = P2 (r1 , r2 ; r1 , r2 ).
(3.11)
This is the required two electron probability function that completely determines
all two particle operators as, e.g., Coulomb interaction in the many-electron system.
The ﬁrst-order density matrix is deﬁned in a similar manner and may be written in
terms of P2 as:
2
(3.12)
P2 (r1 , r2 ; r1 , r2 )dr2 .
P1 (r1 , r1 ) =
N −1
Given P1 and P2 , the total energy is determined exactly:
Nat
Zα
1 2 P1 (r1 , r1 )
E=
− ∇1 −
2
|r
−
R
|
1
α
α
r1 =r1
dr1 +
1
P2 (r1 , r2 )dr1 dr2 .
|r1 − r2 |
(3.13)
It is well seen that the diagonal elements of the ﬁrst- and second-order density
matrices completely determine the total energy. This appears to vastly simplify the
task in hand. The solution of the full Schrödinger equation for Ψ is not required - it
is suﬃcient to determine P1 and P2 , and the problem in a space of 3N coordinates
has been reduced to a problem in a six-dimensional space. Approaches based on the
direct minimization of E(P1 , P2 ) suﬀer from the speciﬁc problem of ensuring that the
density matrices are legal, that is, they must be constructible from an antisymmetric
Ψ. Imposing this constraint is non-trivial and is currently an unsolved problem. In
view of this, the Equation (3.13) does not lead immediately to a reliable method
for computing the total energy without calculating the many body wavefunction.
The observation which underpins the density functional theory is that: the P2 is
not required to ﬁnd E in the ground state. The ground state energy is completely
determined by the diagonal elements of the ﬁrst-order density matrix, that is, the
charge density.
3.1 DFT formalism
3.1.3
26
Hohenberg-Kohn theorems
In 1964 Hohenberg and Kohn proved two theorems (Hohenberg and Kohn, 1964).
The ﬁrst theorem may be stated as follows: The electron density determines
the external potential (to within an additive constant) which gives rise
to it. If this statement is true then it immediately follows that the electron density
uniquely determines the Hamiltonian operator (Eq. 3.2). This follows as the Hamiltonian is speciﬁed by the external potential and the total number of electrons, N ,
which can be computed from the density simply by integration over all space. Thus,
in principle, given the charge density, the Hamiltonian operator could be uniquely
determined and thus the wave functions Ψ (of all states) and all material properties computed. Hohenberg and Kohn (1964) gave a straightforward proof of this
theorem, which was generalized to include systems with degenerate states by Levy
(1979, 1982). It is said that the theoretical spectroscopist E. B. Wilson put forward
a very straightforward proof of this theorem during a meeting in 1965 at which it
was being introduced. Wilson’s observation is that the electron density uniquely
determines the positions and charges of the nuclei and thus trivially determines the
Hamiltonian. This proof is both transparent and elegant, it is based on the fact that
the electron density has a cusp at the nucleus, such that:
−1 ∂ ρ̄(rα )
,
Zα =
2ρ̄(0)
∂rα rα =0
(3.14)
where ρ̄(r) is the spherical average of ρ and so a suﬃciently careful examination of the
charge density uniquely determines the external potential and thus the Hamiltonian.
Although less general than the Levy’s proof, this observation establishes the theorem
for the case of interest - electrons interacting with nuclei. The ﬁrst theorem may be
summarized by saying that the energy is a functional of the density i.e. E[ρ]. The
second theorem establishes a variational principle: For any positive definite trial
density, ρt , such that ρt (r)dr = N , E[ρt ] E0 . The proof of this theorem is
straightforward. From the ﬁrst theorem we know that the trial density determines a
unique trial Hamiltonian - Ht and thus wavefunction - Ψt ; E[ρt ] = Ψt |Ht |Ψt E0
follows immediately from the variational theorem of the Schrödinger equation (Eq.
3.5). This theorem restricts density functional theory to studies of the ground state.
A slight extension allows variation to excited states that can be guaranteed to be
orthogonal to the ground state, but in order to achieve this knowledge the exact
3.1 DFT formalism
27
ground state wavefunction is required. The two theorems lead to the fundamental
statement of density functional theory:
δ E[ρ] − µ( ρ(r)dr − N ) = 0.
(3.15)
The ground state energy and density correspond to the minimum of some functional E[ρ] subject to the constraint that the density contains the correct number
of electrons. The Lagrange multiplier of this constraint is the electronic chemical
potential µ. The above discussion establishes the remarkable fact that there is a
universal functional E[ρ] (i.e. it does not depend on the external potential which
represents the particular system of interest) which, be its form known, could be
inserted into the above equation and minimized to obtain the exact ground state
density and energy.
3.1.4
Energy functional
From the form of the Schrödinger equation (Eq. 3.2) it is clearly seen that the
energy functional contains three terms: the kinetic energy, the interaction with the
external potential and the electron-electron interaction, and thus the functional may
be written as:
(3.16)
E[ρ] = T [ρ] + Vext [ρ] + Vee [ρ].
The interaction with the external potential is trivial:
Vext [ρ] =
V̂ext ρ(r)dr.
(3.17)
The kinetic and electron-electron functionals are unknown. If good approximations
to these functionals could be found, direct minimization of the energy would be
possible; this possibility is the subject of much current researches, see for instance
Foley and Madden (1996). Kohn and Sham (1965) proposed the following approach
to approximate the kinetic and electron-electron functionals. They introduced a ﬁctitious system of N noninteracting electrons to be described by a single determinant
wavefunction in N “orbitals” φi . In this system the kinetic energy and electron
3.1 DFT formalism
28
density are known exactly from the orbitals:
1
φi |∇2 |φi ,
Ts [ρ] = −
2 i
N
(3.18)
here the suﬃx s emphasises that this is not the true kinetic energy but is that of a
system of non-interacting electrons, whose charge density however, by construction,
equals that of true (interacting) system:
ρ(r) =
N
|φi |2 .
(3.19)
i
The construction of the density explicitly from a set of orbitals ensures that it is
legal, i.e., it can be constructed from an asymmetric wavefunction. Taking into
account that a signiﬁcant component of the electron-electron interaction is the classical Coulomb interaction or Hartree energy (this is simply the second term of Eq.
3.7 written in terms of the density),
1
VH [ρ] =
2
ρ(r1 )ρ(r2 )
dr1 dr2 ,
|r1 − r2 |
(3.20)
the energy functional can be rearranged as:
E[ρ] = Ts [ρ] + Vext [ρ] + VH [ρ] + Exc [ρ],
(3.21)
where the exchange-correlation functional is:
Exc [ρ] = (T [ρ] − Ts [ρ]) + (Vee [ρ] − VH [ρ]).
(3.22)
Exc is simply the sum of the error made in using a non-interacting kinetic energy
and the error made in treating the electron-electron interaction classically. Writing
the functional (Eq. 3.21) explicitly in terms of the density built from noninteracting
orbitals (Eq. 3.19) and applying the variational theorem (Eq. 3.15) we ﬁnd that the
3.1 DFT formalism
29
orbitals which minimize the energy satisfy the following set of equations:
1 2
ρ(r )
− ∇ + νext (r) +
dr + νxc (r) φi (r) = εi φi (r),
2
|r − r |
(3.23)
where a local multiplicative potential is introduced, which is the functional derivative
of the exchange correlation energy with respect to the density:
νxc (r) =
∂Exc [ρ]
.
∂ρ
(3.24)
This set of non-linear equations, the Kohn-Sham (KS) equations, describes the behavior of non-interacting “electrons” in an eﬀective local potential. For the exact
functional, and thus exact local potential, the “orbitals” yield the exact ground
state density via Eq. 3.19 and exact ground state energy via Eq. 3.21. These KS
equations have the same structure as the HF equations (Eq. 3.8) with the nonlocal exchange potential replaced by the local exchange-correlation potential νxc . It
should be noted at this point that the notations in general use are often misleading.
As stated above, Exc contains an element of the kinetic energy and is not the sum of
the exchange and correlation energies as they are understood in HF and correlated
wavefunction theories. The KS approach achieves an exact correspondence of the
density and ground state energy of a system consisting of non-interacting electrons
and the “real” many body system described by the Schrödinger equation.
The computational cost of solving the KS equations (Eq. 3.23) scales formally
as N 3 (due to the need to maintain the orthogonality of N orbitals) but in practice
can be in principle redused to ∼ N , for large systems, through the exploitation of
the locality of the orbitals.
3.1.5
Local density approximation
The generation of approximations for Exc has lead to a large and still rapidly expanding ﬁeld of research. There are now many diﬀerent ﬂavours of density functional
available which may be more or less appropriate for study in question. Ultimately
such judgments must be made in terms of results, but knowledge of the derivation
and structure of functionals is very valuable when selecting which to use in any particular study. The early thinking that lead to practical implementations of density
3.1 DFT formalism
30
functional theory was dominated by one particular system for which near exact results could be obtained, the homogeneous electron gas. In this system the electrons
are subject to a constant external potential and thus the charge density is constant.
The system is thus speciﬁed by a single number - the value of the constant electron
density ρ = N/V . Thomas (1927) and Fermi (1928) independently studied the homogeneous electron gas in the early 1920s. For the sake of the simplicity, the one
electron functions can be taken in the form of plane waves. If the electron-electron
interaction is approximated by the classical Hartree potential (that is, exchange and
correlation eﬀects are neglected) then the total energy functional can be readily
computed (Thomas, 1927; Fermi, 1928). Under these conditions the dependence of
the kinetic and exchange energy (Eq. 3.7) on the density of the electron gas can
be extracted (Dirac, 1930a; Lieb, 1981) and expressed in terms of a local functions
of the density. This suggests that in an inhomogeneous system it is possible to approximate the functional as an integral over a local function of the charge density.
Using the kinetic and exchange energy densities of the non-interacting homogeneous
electron gas leads to
T [ρ] = 2.87
ρ5/3 (r)dr
(3.25)
ρ4/3 (r)dr.
(3.26)
and
Exc [ρ] = 0.74
These results are highly suggestive of a representation for Exc in an inhomogeneous
system. The local exchange correlation energy per electron might be approximated
as a simple function of the local charge density, εxc (ρ), yielding an approximation
of the form:
Exc [ρ] = ρ(r)εxc (ρ(r))dr.
(3.27)
An obvious choice for the exchange and correlation energy density εxc (ρ) is that of
the uniform electron gas – this is the Local Density Approximation. Within the
LDA εxc (ρ) is a function of only the local value of the density. It can be separated
into exchange and correlation contributions:
εxc [ρ] = εx [ρ] + εc [ρ].
(3.28)
3.1 DFT formalism
31
The Dirac form can be used for εx (Eq. 3.26):
εx [ρ] = −Cρ1/3 ,
(3.29)
where for generality a free constant C has been introduced rather than that determined for the homogeneous electron gas. This functional form is much more widely
applicable than is implied from its derivation, or could be established from scaling
arguments (Parr and Yang, 1989). The functional form for the correlation energy
density εc is unknown and has been simulated for the homogeneous electron gas in
numerical quantum Monte Carlo calculations which yield essentially exact results
(Ceperley and Alder, 1980). The resultant exchange-correlation energy has been
ﬁtted by a number of analytic forms (Vosko, Wilk and Nusair, 1980; Perdew and
Zunger, 1981; von Barth and Hedin, 1972), all of which yield similar results in practice and are collectively referred to as LDA functionals.
The LDA has proven to be a remarkably fruitful approximation. Properties such
as structure, vibrational frequencies, elastic moduli and phase stability (of similar
structures) are described reliably for many systems. However, in computing energy
diﬀerences between rather diﬀerent structures the LDA can have signiﬁcant errors.
For instance, the binding energy of many systems is overestimated (typically by
20-30%) and energy barriers in diﬀusion or chemical reactions may be too small or
absent. Nevertheless, the remarkable fact is that the LDA works as well as it does,
given the reduction of the energy functional to a simple local function of the density.
The magnitude of the errors in the LDA energy densities has recently been estimated by computing the energy density of bulk silicon with variational quantum
Monte Carlo calculations (Hood, Chou, Williamson, Rajagopal et al., 1997). There
are very signiﬁcant errors in the exchange and correlation energies but, as the exchange energy is generally underestimated and the correlation energy overestimated,
these errors tend to cancel. The success of the LDA appears to be in part due to
this cancellation of errors. As demonstrated above (Eq. 3.13) the ﬁrst and second
order density matrices are suﬃcient to determine the exact total energy. An insight
into the behavior of functionals can be obtained by examining how well they approximate P2 . A commonly used device is to convert P2 , the probability of ﬁnding
an electron at r1 and an electron at r2 , into the conditional probability of ﬁnding
an electron at r2 given that there is an electron at r1 . This quantity is called the
3.1 DFT formalism
32
exchange-correlation hole:
Pxc (r1 , r2 ) =
P2 (r1 , r2 )
− ρ(r2 ).
ρ(r1 )
(3.30)
The comparison of the exact variation of the exchange hole with that computed
within LDA (Gunnarsson, Jonson and Lundqvist, 1979) demonstrates that the LDA
is a very poor approximation to P2 . The LDA is able to produce the reasonable
energetics while the pair correlation function is so poorly described, only because
of the structure of the Coulomb operator. From Eq. 3.13 the electron-electron
interaction can be written in terms of P2 as:
1
Vee =
2
1
P2 (r1 , r2 )dr1 dr2 .
|r1 − r2 |
(3.31)
From this it seems apparent that a poor approximation to P2 leads directly to a
poor estimate of the electron-electron interaction. However, the Coulomb operator
depends only on the magnitude of the separation of r1 and r2 . A substitution
u = r1 − r2 yields:
P2 (r1 , r1 + u)dr1 dΩu
1
1
2
P2 (r1 , r1 + u)dr1 du =
4πu
du.
u
2
u
4π
(3.32)
Therefore the electron-electron interaction depends only on the spherical average of
the pair density P (u):
1
Vee =
2
P (u) =
P2 (r1 , r1 + u)dr1
dΩu
.
4π
(3.33)
Thus, the remarkable performance of the LDA is a consequence of its reasonable
description of the spherically averaged exchange correlation hole coupled with the
tendency for errors in the exchange energy density to be cancelled by errors in the
correlation energy density.
3.1 DFT formalism
3.1.6
33
Generalized gradient approximation
The local density approximation can be considered to be the zero’th order approximation to the semi-classical expansion of the density matrix in terms of the density
and its derivatives (Dreizler and Gross, 1990). A natural progression beyond the
LDA is thus to the gradient expansion approximation in which ﬁrst order gradient
terms in the expansion are included. This results in an approximation for the exchange hole (Dreizler and Gross, 1990) which has a number of unphysical properties;
it does not normalize to −1, it is not negative deﬁnite and it contains oscillations at
large u (Wang, Perdew, Chevary, Macdonald et al., 1990). In the GGA a functional
form is adopted which ensures the normalization condition and that the exchange
hole is negative deﬁnite (Perdew and Wang, 1986). This leads to an energy functional that depends on both the density and its gradient but retains the analytic
properties of the exchange correlation hole inherent in the LDA. The typical form
for a GGA functional is:
Exc ≈
ρ(r)εxc (ρ, ∇ρ)dr.
(3.34)
The GGA improves signiﬁcantly on the LDA’s description of the binding energy
of molecules. It was this feature which lead to the very widespread acceptance of
DFT in the chemistry community during the early 1990’s. A number of functionals
within the GGA family have been developed and were discussed by Perdew and
Wang (1986); Langreth and Mehl (1983); Becke (1988b); C.Lee, Yang and Parr
(1988); Burke and Ernzerhof (1996).
3.1.7
Hybrid exchange functionals
There is an exact connection between the non-interacting density functional system
and the fully interacting many body system via the integration of the work done in
gradually turning on the electron-electron interactions. This “adiabatic connection”
(Pines and Nozières, 1966; Harris and Jones, 1974; Gunnarsson and Lundqvist, 1976)
allows the exact functional to be formally written as:
1
Exc [ρ] =
2
drdr
1
dλ
λ=0
λe2
[ρ(r)ρ(r )ρ,λ − ρ(r)δ(r − r )],
|r − r |
(3.35)
3.1 DFT formalism
34
where the expectation value ρ(r)ρ(r )ρ,λ is the density-density correlation function
and is computed at density ρ(r) for a system described by the eﬀective potential:
Vef f
1 λe2
= Ven +
2 i=j |r − r |
(3.36)
Thus the exact energy could be computed if one knew the variation of the densitydensity correlation function with the coupling constant λ. The LDA is recovered
by replacing the pair correlation function with that for the homogeneous electron
gas. The adiabatic integration approach suggests a diﬀerent approximation for the
exchange-correlation functional. At λ = 0 the non-interacting system corresponds
identically to the HF ansatz, while the LDA and GGA functionals are constructed
to be excellent approximations for the fully interacting homogeneous electron gas,
that is, a system with λ = 1. It is therefore not unreasonable to approximate the
integral over the coupling constant as a weighted sum of the end points, that is, it
might be set:
GGA
(3.37)
Exc ≈ aEF ock + bExc
with the coeﬃcients are to be determined by reference to a system for which the exact result is known. Becke adopted this approach (Becke, 1993b,a) in the deﬁnition
of a new functional with coeﬃcients determined by a ﬁt to the observed atomization energies, ionization potentials, proton aﬃnities and total atomic energies for a
number of small molecules (Becke, 1993b,a). The resultant three parameter energy
functional is:
LDA
F ock
LDA
B88
Exc = Exc
+ 0.2(EX
− EX
) + 0.72EX
+ 0.81EcP W 91 ,
(3.38)
B88
and EcP W 91 are widely used GGA corrections (Becke, 1988a; Perdew
where EX
and Wang, 1992) to the LDA exchange and correlation energies respectively. Hybrid
functionals of this type are now very widely used in chemical applications with the
B3LYP or B3PW functionals (in which the parameterisation is as given above but
with a diﬀerent GGA treatment of correlation (C.Lee, Yang and Parr, 1988; Perdew
and Wang, 1992) being the most notable). Computed binding energies, geometries
and frequencies are systematically more reliable than the best GGA functionals.
3.2 Practical implementation of DFT/HF calculation scheme
3.1.8
35
Spin-density functional theory
Up to now, only closed-shell systems have been considered, i.e. the spin coordinate
was omitted. The extension of density functional theory to spin polarized systems
is quite natural (Perdew and Zunger, 1981; Gunnarsson and Lundqvist, 1976). One
must deﬁne an electron density for each spin state: ρ↑ , ρ↓ or equivalently, a total
density, ρ = ρ↑ + ρ↓ and a spin electron density m = ρ↑ − ρ↓ . All functionals, for
instance εxc , will depend on both densities: εxc ≡ εxc (r; [ρ, m]). The spin-density
functional theory allows the investigations of magnetic systems which exhibit spontaneous magnetic phenomena. A signiﬁcant improvement of DFT calculations could
be obtained by applying its spin-density extension to systems in the absence of the
magnetic ﬁeld. The underlying reason for this lies in the fact that in this approach
electrons with diﬀerent spin quantum numbers feel a diﬀerent spin-dependent potentials, which is clearly a better approximation for systems with an odd number of
electrons.
3.2
Practical implementation of DFT/HF calculation scheme
3.2.1
Selection of basis set
In order to calculate crystalline structures and properties, it is necessary to determine the eigenfunctions Ψ and eigenvalues E of the Schrödinger equation (Eq. 3.1).
For systems of interest in chemistry, analytic solutions to this problem are impossible
to ﬁnd, so one normally resorts to the variational approach involving the introduction of a trial wave function Ψ̄(α) that depends on a set of variable parameters {α}.
If the functional
Ψ̄|Ĥ|Ψ̄
Ē(α) =
(3.39)
Ψ̄|Ψ̄
is minimized with respect to variations in {α}, the energy converges from above
towards the true energy in Eq. 3.1, and the wave function converges to the true
wave function as the parameter set {α} is expanded to completeness.
The most obvious way of implementing this approach is to make the trial wave
function depend linearly on the parameters {α}. The resulting linear expansion may
3.2 Practical implementation of DFT/HF calculation scheme
be written most generally as:
Ψ̄ =
Φµ cµ ,
36
(3.40)
µ
i.e. the parameters {α} are represented by {Φµ , cµ }. A set of equations for the linear
coeﬃcients cµ in this expansion may be derived on substitution of Eq. 3.40 into Eq.
3.39 by making the energy stationary with respect to variations in the coeﬃcients.
The N -particle basis functions {Φ} are a set of ﬁxed analytic functions that depend
on coordinates of all electrons in the system. They can conveniently be taken to be
orthonormal, in which case the variational equations correspond to the eigenvalue
problem. If the N -particle basis were a complete set of N -electron functions, the
use of the variational approach would introduce no error, because the true wave
function could be expanded exactly in such a basis. However, the basis would then
be of inﬁnite dimension, and in practice, the fact that one must work with an
incomplete set of N -particle functions is one of our major practical approximations.
In most practical quantum chemistry techniques, the basis is constructed using
linear combinations of products of one-electron wave functions or orbitals. These
are usually antisymmetrized to account for the permutational symmetry of the wave
function and may also be spin- and symmetry-adapted:
N
Φµ = Â
φµi (xi ),
(3.41)
i
where Â is an antisymmetrization operator and the xi are the space and spin coordinates of a single electron. The unknown one-electron functions {φ} in Eq. 3.41 are
referred to as atomic, molecular or crystalline orbitals, depending on the physical
nature of the problem. To ﬁnd the unknown orbitals, one generally expands them
as an orthonormal linear combination of known one-electron basis functions χa :
φµi =
χa Ca, µi .
(3.42)
a
The coeﬃcients Ca, µi then span the variational space {α} mentioned above. One
then obtains a set of algebraic equations for the optimum orbitals which may be
solved by standard matrix techniques. The set of functions χ in Eq. 3.42 constitute
the one-particle basis set given as input to most quantum chemistry calculations.
3.2 Practical implementation of DFT/HF calculation scheme
37
The simplest truncation of the N -particle space is that in which only one N electron basis function is used - a single conﬁguration which is the best variational
approximation to the exact ground-state wave function. In this case all coeﬃcients
in the Eq. 3.40 are zero with the exception of that of the groundstate conﬁguration.
Minimizing the expectation value of the Hamiltonian with respect to the one-electron
orbitals allows one to derive the self-consistent Hartree-Fock or Kohn-Sham equations (Eq. 3.8 and 3.23), which may be solved to ﬁnd the optimum orbitals (i.e. the
best coeﬃcients of Eq. 3.42).
The nature of the one-particle basis functions used in the expansion of the orbitals
depends on the periodicity of the system. In the periodic structure, the one-particle
basis must be made up of Bloch functions (BFs) Φik (r) i.e. products of a function
periodic in the primitive lattice and a phase factor whose frequency and direction
of oscillation is dependent on the wave vector k. These BFs might be, for example,
simple plane-waves exp[i(k + G)r], where G is a vector in the reciprocal lattice, or
a combination of a localized function χj and all its periodic images, modulated by
a phase factor:
1 g
χj (r − rj − g) exp(ik · g),
(3.43)
Φik (r) = √
N g
where χgj refers to the jth localized atomic function (lying at the position rj relative
to the origin of the unit cell, which is translated by the lattice vector g). Because the
wave vector k is a continuous variable, the basis set of BFs is in principle inﬁnite;
in practice however, the problem is solved at a ﬁnite set of k points, and the results
interpolated. Thus the one-particle basis of atomic functions (BFs) determines the
one-particle orbitals. If the one-particle basis is complete, it would in principle be
possible to form a complete N -particle basis, and hence to obtain an exact wave
function variationally. However, such a complete one-particle basis would be of inﬁnite dimension, and thus the basis must be truncated in practical applications. One
must therefore use truncated N -particle spaces that are constructed from truncated
one-particle bases. These truncations are the most important sources of uncertainty
in quantum chemical calculations. It should be noted that the ultimate accuracy of
any calculation, in correlated calculations over many N -particle basis functions as
well as at the self consistent ﬁeld (SCF) level, is determined by the one-particle BS.
Historically, basis functions with exponential asymptotic behaviour, Slater-type
orbitals, were the ﬁrst to be used. These are characterized by an exponential factor
3.2 Practical implementation of DFT/HF calculation scheme
38
in the radial part (similar to that in the analytical solution for the hydrogen atom):
χSlater = rn−1 exp(−ζr)Ylm (Θ, φ),
(3.44)
where ζ is called exponent, the Ylm (Θ, φ) is the spherical harmonic or angular momentum part, and the n, l, m are quantum numbers. Unfortunately such functions
are not eﬃcient for fast calculations of multicentre integrals, so GTFs were introduced to calculate the integrals (2- to 4-centered in the construction of Hamiltonian,
as well as in many cases matrix elements of other operators of interest) analytically,
these can be written:
(3.45)
χGT F = exp(−αr2 )xl y m z n ,
where α is again the exponent, and the l, m, n are not quantum numbers, but simply
integer powers of Cartesian coordinates. In this form, called Gaussian primitives,
the GTFs can be factorized into their Cartesian components i.e.:
F GT F GT F
χGT F = χGT
χy χz ,
x
(3.46)
where each Cartesian component has the form (introducing an origin such that the
Gaussian is located at position xa ),
F
= (x − xa )l exp[−α(x − xa )2 ].
χGT
x
(3.47)
Factorization simpliﬁes considerably the calculation of integrals. In GTF, only functions with l = n − 1 are explicitly used, i.e. 1s, 2p, 3d etc., but not 2s, 3p, 4d etc.
However, combinations of Gaussians are able to approximate correct nodal properties of atomic orbitals if the primitives are included with diﬀerent signs. The sum of
exponents of Cartesian coordinates L = l + m + n is used analogously to the angular momentum quantum number for atoms to mark Gaussian primitives as s-type
(L = 0), p-type (L = 1), d-type (L = 2), f-type (L = 3) etc.
The present success of GTFs as the BS of choice in virtually all molecular quantum chemistry calculations originally was far from obvious. In particular, it is clear
that the behavior of a Gaussian is qualitatively wrong both at the nuclei and in the
long-distance limit for a Hamiltonian with point-charge nuclei and Coulomb interaction. It has therefore been a commonly held belief that Slater-type orbitals would
3.2 Practical implementation of DFT/HF calculation scheme
39
be the preferred basis if only the integral evaluation problem could be solved. It has
been claimed (Roos, 1994) that this is not necessarily the case and that the “cusp”
behavior represents an idealized point nucleus, and for more realistic nuclei of ﬁnite
extension the Gaussian shape may actually be more realistic. If accurate solutions
for a point-charge model Hamiltonian are desired, they can be obtained to any desired accuracy in practice by expanding the “core” basis functions in a suﬃciently
large number of Gaussians to ensure their correct behavior. Furthermore, properties
related to the behavior of the wave function at or near nuclei can often be predicted
correctly, even without an accurately “cusped” wave function (Challacombe and
Cioslowski, 1994). In most molecular applications the asymptotic behavior of the
density far from the nuclei is considered much more important than the nuclear
cusp. The wave function for a bound state must fall oﬀ exponentially with distance,
whenever the Hamiltonian contains Coulomb electrostatic interaction between particles.
More important are limitations arising from the convergence of results with the
size of the BS. Both the number of integrals over basis functions to be stored on
disk and the total CPU time nominally scale rather unpleasantly with the number
of functions in the BS. Thus it usually pays to consider the issue of BS compactness,
that is, the ability to expand the orbitals as accurately as possible using the minimum number of basis functions. In most applications therefore, Gaussian-type basis
functions are expanded as a linear combination (or “contraction”) of individually
normalized Gaussian primitives gj (r) characterized by the same center and angular
quantum numbers, but with diﬀerent exponents:
χi (r) =
S
dj gj (r),
(3.48)
j=1
where S is the length of the contraction, the dj contraction coeﬃcients. By proper
choice of these quantities, the “contracted Gaussians” may be made to assume any
functional form consistent with the primitive functions used. One may therefore
choose the exponents of the primitives and the contraction coeﬃcients so as to lead
to basis functions with desired properties, such as reasonable cusp-like behavior at
the nucleus (see Fig. 3.1). Integrals involving such basis functions reduce to sums
of integrals involving the Gaussian primitives. Even though many primitive inte-
3.2 Practical implementation of DFT/HF calculation scheme
a
40
b
Figure 3.1: The individual GTFs (solid lines) are relatively poor representatives
of true one-electron wavefunctions: GTFs have wrong asymptotics in the inﬁnity
(fall down too fast) and wrong behavior near the nucleus. Left ﬁgure (a) shows
the ”optimum” GTF obtained for the 1s orbital by least-square ﬁt, preserving the
normalization. The performance can be improved by using ”contracted” GTF. Right
ﬁgure (b) shows 1s wavefunction approximated by a contracted 4-GTF set.
grals may need to be calculated for each basis function integral, the basis function
integrals will be rapidly calculated provided the method of calculating primitive integrals is fast, and the number of orbital coeﬃcients in the wavefunction will have
been considerably reduced. The exponents and contraction coeﬃcients are normally
chosen on the basis of relatively cheap atomic SCF calculations so as to give basis
functions suitable for describing exact HF atomic orbitals. An approximate atomic
basis function, whose shape is suitable for physical and chemical reasons, is thus expanded in a set of primitive Gaussians, whose mathematical properties are attractive
from a computational point of view (i.e. the calculations of multicentre integrals are
much easier). The physical motivation for this procedure is that, while many primitive Gaussian functions may be required to provide an acceptable representation of
an atomic orbital, the relative weights of these primitives are unchanged when the
atoms are formed into molecules or crystals. The relative weights of the primitives
can therefore be ﬁxed from a previous calculation and only the overall scale factor
for this contracted Gaussian function need be determined in the extended calculation. It is clear that contraction will in general signiﬁcantly reduce the number of
basis functions.
3.2 Practical implementation of DFT/HF calculation scheme
41
The separability of GTFs into Cartesian components (Eq. 3.46) allows a computationally eﬃcient transition from the spherical symmetry of the atom, naturally
represented in a polar coordinate system, to a more general Cartesian representation
which is useful for describing crystalline geometries. Another equally important reason for the usefulness of a Gaussian BS is embodied in the Gaussian product theorem
(GPT), which in its simplest form states that the product of two simple Gaussian
functions with exponents α and β, located at centres A and B, is itself a simple
Gaussian with exponent γ, multiplied by a constant factor F , located at a point C
along the line segment A − B, where:
γ = α + β,
C=
αA + βB
γ
F = exp −
(3.49)
,
αβ
(A − B)2 .
γ
(3.50)
(3.51)
The product of two polynomial GTFs, of degree µ and ν and located at points A
and B is therefore another polynomial GTF located at C, of the degree µ + ν in xC ,
yC and zC , which can be expressed as a short expansion of one-centre Gaussians:
χax (x)χbx (x) =
µ+ν
Ciµ+ν ϕci(x−xc ) ,
(3.52)
i=0
2
where ϕci (x) = xi e−αp(x−xc ) and xc = (αxa + βxb )/(α + β). The product of two
Gaussians which are functions of the coordinates of the same electron is referred to
as an overlap distribution, and all the integrals which must be calculated involve
at least one such overlap distribution. The most important consequence of the
GPT is that all four-center two-electron integrals can be expressed in terms of twocenter quantities. However, the cost of integral evaluating still scales nominally
as N 4 , where N is the number of functions in the expansion. This scaling is far
from satisfactory and this must be reduced in order to treat large systems. One
way of doing this is the method of pre-screening where, rather than attempting
to calculate the integrals more eﬃciently, one seeks where possible to avoid their
evaluation altogether. Since the expression for an integral over primitive Gaussians
3.2 Practical implementation of DFT/HF calculation scheme
42
can be formally written as:
ab|cd = Aab Scd Tabcd ,
(3.53)
where Sab is a radial overlap between functions χa and χb , and Tabcd is a slowly
varying angular factor. In many situations the product Sa Sb thus constitutes a
good estimate of the magnitude of the integral, and it may seem attractive to use
that product as an estimate in screening out small integrals. In order to estimate
these overlaps quickly, a single, normalized s-type Gaussian is associated with each
shell, whose exponent α is the smallest of the exponents in the shell contraction.
This function thus reproduces approximately the absolute value of the corresponding atomic orbitals at intermediate and long range. The normalized s-type Gaussian
is used in fast algorithms for estimating overlaps on the basis of which integrals are
either evaluated exactly, approximately, or not at all.
The simplest alternative to Gaussians in solid-state calculations are PW:
1 i(k+G)r
e
,
χG
k = √
Ω
(3.54)
where k stands for a point in the Brillouin zone (BZ) labelling a certain irreducible
representation of the translation group and G labels the sites of the reciprocal lattice. Plane waves are an orthonormal complete set; any function belonging to the
class of continuous normalizable functions (which are those of interest in quantum
mechanics) can be expanded with arbitrary precision in such a BS. The set is universal, in the sense that it does not depend on the positions of the atoms in the
unit cell, nor on their nature. One thus does not have to invent a new BS for every
atom in the periodic table nor modify them in diﬀerent materials as is the case with
Gaussian functions, and the basis can be made better (and more expensive) or worse
(and cheaper) by varying a single parameter (planewave cutoﬀ i.e. the largest vector
used in planewave expansion |k + G| ≤ Gmax ). This characteristic is particularly
valuable in ab initio molecular dynamics calculations, where nuclear positions are
constantly changing. The algorithms mainly (involving fast Fourier transforms) are
easier to program since the algebraic manipulation of PW is very simple. Using PW
it is relatively easy to compute forces on atoms. Finally, PW calculations do not
suﬀer from basis set superposition error (BSSE). In practice, one must use a ﬁnite
3.2 Practical implementation of DFT/HF calculation scheme
43
set of PWs, and this restricts the detail that can be revealed in real space to such
an extent that core electrons cannot be described in this manner. One must either
augment the BS with additional functions, or use pseudopotentials to imitate the
eﬀect of the core states.
In comparison with PWs, the use of all-electron Gaussian calculations allows
one to describe accurately electronic distributions both in the valence and the core
region with a limited number of basis functions. The local nature of the basis
allows a treatment of both ﬁnite systems and systems with periodic boundary conditions in one, two or three dimensions. This has advantages over PW calculations
of molecules, polymers or surfaces which work by imposing artiﬁcial periodicity:
the calculation must be done on e.g. a three-dimensional array of molecules with
a suﬃciently large distance between them. Gaussian total energies can be made
very precise (i.e. reliable to many places of decimals) since all integrals can be
done analytically. Having an “atomic-like” basis facilitates population analysis, the
computation of properties such as projected densities of states, and “pre-SCF” alteration of orbital occupation. Many PW programs cannot compute exact non-local
exchange which is required not only for Hartree-Fock calculations, but also in the
“hybrid” DFT exchange-correlation functionals and in overcoming self-interaction
problems in DFT calculations of Mott insulators. The cost that is paid for using
Gaussians is the loss of orthogonality, of universality, the need for more sophisticated algorithms for the calculations of the integrals, the diﬃculty of computing
forces, and an overly heavy reliance on the presence of lots of space group symmetry
operators for eﬃcient calculations.
In choosing a BS the paramount but conﬂicting issues are accuracy and computational cost which are obviously inversely related. However, computational cost alone
should not determine what BS is used. Selecting a smaller set purely on the basis of
a lack of suﬃciently powerful computers will often prove unsuitable for describing
the system in question, which rather defeats the object of performing the calculation
in the ﬁrst place. The minimum BS requirements of all properties to be computed
should always be considered. The following general principles are suggested to be
taken into account when a BS for a periodic problem is constructed:
• Diﬀuse functions
The pre-screening procedure is based on overlaps between Gaussian s functions
associated with each shell whose exponents are set equal to the lowest exponent
3.2 Practical implementation of DFT/HF calculation scheme
44
of all the primitive Gaussians in the contraction. Thus one should keep in
mind that the number of integrals to be calculated increases very rapidly with
decreasing exponents of the primitive Gaussians.
• Number of primitives
A typical BS will have “core functions” with higher exponents and a relatively
large number of primitives. These will have a large weight in the expansion
of the core states. The “valence functions” with a large weight in the outer
orbitals will have lower exponents and contractions of only a very few primitives. One can get away with putting a lot of primitives in the core since core
states have very little overlap with neighboring atoms and thus the use of a
large number of primitives in the GTF contraction is of limited cost in CPU
time. The use of many primitives in the valence shells would add signiﬁcantly
to the cost of a calculation.
• Numerical catastrophes
Under certain conditions calculations may fall into a non-physical state during
the SCF part characterized by an oscillating total energy signiﬁcantly higher
than the true energy. Such calculations will not, in general, converge. It is
observed that the risks of numerical problems like this increases rapidly with
decreasing value of the most diﬀuse Gaussian exponent in the BS.
• Basis Set Superposition Error
A rather serious problem associated with Gaussian BS is basis set superposition
error. A common response to this problem is to ignore it, since it will go away
in the limit of a complete basis. Sometimes this approach is justiﬁed, but this
requires investigation that is seldom performed, and some understanding of
BSSE is indispensable in order to perform accurate and reliable calculations.
The problem of BSSE is a simple one: in a system comprising interacting
fragments A and B, the fact that in practice the BSs on A and B are incomplete
means that the fragment energy of A will necessarily be improved by the basis
functions on B, irrespective of whether there is any genuine binding interaction
in the compound system or not. The improvement in the fragment energies
will lower the energy of the combined system giving a spurious increase in the
binding energy.
3.2 Practical implementation of DFT/HF calculation scheme
45
• Pseudopotentials
It is well known that core states are not in general aﬀected by changes in chemical bonding. The idea behind pseudopotentials is therefore to treat the core
electrons not as particles occupying KS (or atomic) orbitals, but only by their
eﬀect on the potential ﬁlled by electrons in valence shells. Pseudopotential are
thus not orbitals but modiﬁcations to the Hamiltonian and are used because
they can introduce signiﬁcant computational eﬃciencies. The most important
characteristic of a pseudopotential designed for such calculations is that it is
as smooth as possible in the core region. It is quite easy to incorporate relativistic eﬀects into pseudopotentials which is increasingly important for heavy
atoms. The use of pseudopotentials will decrease the number of coeﬃcients in
the wave function and give signiﬁcant savings in the SCF computations.
In the present study, according to these rules, the BSs for Sr, Ba, Pb, Ti and
O have been carefully selected and optimized. The Hay-Wadt small-core Eﬀective
Core Pseudopotentials (ECPs) have been adopted for Ti, Sr, and Ba atoms, and the
large-core version of Hay-Wadt ECP for Pb atoms (Hay and Wadt, 1984c,b,a) (see
Appendix A for more details). The small-core ECPs replace only inner core orbitals,
but orbitals for outer core electrons as well as for valence electrons are calculated
self-consistently. The full electron BS has been adopted for the light oxygen atom.
The BSs have been constructed and optimized in the following forms: for oxygen
has been adopted a basis of contracted GTFs of the form s(8)sp(4)sp(1)sp(1)d(1),
where the letters give the shell type and the numbers in brackets give the number of
Gaussian primitives in each shell contraction, for Ti - sp(4)sp(1)sp(1)d(3)d(1)d(1),
for Sr and Ba - sp(3)sp(1)sp(1)d(1), and for Pb - sp(2)sp(1)sp(1)d(1). The exponents and contraction coeﬃcients are reported in Table 3.1. These BSs is expected
to be reasonably good – there are three valence sp shells on the anion and three
on the cations. The d electrons of Ti are described by three shells, a contraction of
three Gaussians for the inner part, and two single Gaussians for the outer part. The
calculation with this basis is cheap, taking only a few minutes on a medium-sized
workstation. This is because the unit cell contains only ﬁve atoms, the system has
high symmetry and the external Gaussians of the two cations have reasonably large
exponents (0.36, 0.26, 0.20 and 0.14 for Ti, Sr, Ba and Pb respectively) and that of
the anion is not too diﬀuse (0.17). In a comparison with the standard BSs (Homepage, a) the polarization d -function has been added on O, the inner core orbitals
3.2 Practical implementation of DFT/HF calculation scheme
46
Table 3.1: The exponents α (bohr−2 ) and contraction coeﬃcients dj of individually
normalized Gaussian-type basis functions (see Eq. 3.45 and Eq. 3.48). All atoms
are described using the Hay-Wadt small core pseudopotentials (Hay and Wadt,
1984c,b,a).
Shell
Ti
3sp
4sp
5sp
3d
Sr
4d
5d
4sp
Ba
5sp
6sp
4d
5sp
Pb
6sp
7sp
5d
6sp
7sp
8sp
6d
Exponents
16.66
3.823
3.767
1.334
0.773
0.437
21.43
6.087
2.079
0.831
0.356
16.73
2.232
1.985
0.654
0.261
0.470
8.552
2.114
1.872
0.509
0.204
0.332
1.335
0.752
0.554
0.142
0.193
s
0.00529
0.349
0.2
-0.847
1.0
1.0
Coeﬃcients
p
-0.00247
-0.491
0.5
0.0475
1.0
1.0
d
0.0881
0.417
1.0
1.0
1.0
-0.0409
1.0
9.261
1.0
1.0
0.00643
1.0
-0.964
1.0
1.0
0.00445
-0.761
1.0
1.0
1.0
0.0109
-0.598
1.0
1.0
1.0
1.0
1.0
-0.145
1.0
1.0
1.0
-0.107
1.0
1.0
1.0
1.0
3.2 Practical implementation of DFT/HF calculation scheme
47
of Ti have been replaced by small-core Hay-Wadt ECP, and two most diﬀuse s and
p Gaussians have been used consistently as separate basis functions on Ti, Ba, Sr,
Pb. The BS optimization procedure has been divided into several stages. In the
ﬁrst stage, the optimization of Gaussian exponents and contraction coeﬃcients have
been done through the energy minimization of the free-ion state of metal atoms
(Ti4+ , Sr2+ , Ba2+ , Pb2+ ), using the small computer code that implements Conjugated Gradients optimization (Press, Teukolsky, Vetterling and Flannery, 1997)
with a numerical computation of derivatives. In the second stage, using the same
optimization code, the outer Gaussian exponents have been optimized in bulk crystals through the minimization of the total energy per unit cell. This brings the
BS into its ﬁnal shape. The only exception is BS of an oxygen atom taken from
(Homepage, a). The optimization of the outermost diﬀused exponents of oxygen
atoms (αsp = 0.5 and 0.191 bohr−2 ) in STO leads to 0.452 and 0.1679, respectively.
The oxygen d polarization orbital (αd = 0.451 bohr−2 ) has been added, which provides the ﬂexibility to BS and reﬂects the oxygen additional polarizability in the
crystalline environment. The same BS for oxygen was then employed for BTO and
PTO. Thus, the Gaussian BS have been successfully generated for all three perovskite oxides composed of 18 atomic orbitals for O, 27 for Ti and 17 for Sr, Ba and
Pb. The adequacy of selected BSs is carefully tested in the next chapter.
3.2.2
Auxiliary basis sets for the exchange-correlation functionals
The matrix elements of the exchange-correlation potential operator νxc (Eq. 3.24)
in a basis of GTFs χj (r − sj − g), where g labels the cell containing the j-th basis
function at atomic position sj , take the form:
i, jg
νxc
=
dr χi (r − si )νxc (r)χj (r − sj − g).
(3.55)
In general the νxc potentials have an exceedingly complex analytic form, even at
a local level, and so for periodic systems the matrix elements must be evaluated
numerically. Performing the numerical integration directly over each pair of real
space basis functions is expensive in crystalline calculations, and in the present
study a more eﬃcient procedure has been adopted. In this method, the exchange-
3.2 Practical implementation of DFT/HF calculation scheme
48
correlation potential is expanded in an auxiliary BS of contracted GTFs {G(r)}:
νxc (r) =
N
ca Ga (r).
(3.56)
a
This expansion allows the evaluation of the DFT potential integrals in Equation 3.55
as a linear combination of integrals gai, jg over the auxiliary basis functions. These
primitive integrals need only be calculated once and stored:
i, jg
νxc
=
N
ca gai, jg ,
(3.57)
dr χi (r)Ga (r)χ(r − s − g).
(3.58)
a=1
where
gai, jg
=
At each SCF iteration the auxiliary BS is ﬁtted to the actual analytic form of the
exchange-correlation potential, which changes with the evolving charge density. The
best-ﬁt coeﬃcients ca are evaluated by solving the linear least-squares equation:
Ac = b,
(3.59)
where A is the overlap matrix between auxiliary basis function:
Aab =
dr Ga (r)Gb (r).
(3.60)
unit cell
c is the vector of unknown coeﬃcients ca , and b is a density-dependent vector of
overlap integrals between the ﬁtting functions and the exchange-correlation potential,
ba =
dr Ga (r)νxc (r).
(3.61)
unit cell
Both the ba integrals and the elements of the A matrix are evaluated numerically to
ensure a consistent level of accuracy. The ba integrals are much simpler than direct
i, jg
numerical integration of the matrix elements νxc
since they are restricted to the
unit cell. Furthermore, since the integrand is totally symmetric, the evaluation of
the integrand may be restricted to an irreducible set of sampling points.
3.2 Practical implementation of DFT/HF calculation scheme
49
Since νxc is a basis for the totally-symmetric irreducible representation of the
space group, the chosen auxiliary basis functions must be totally symmetric with
respect to all operations of the space group. In this study, a set of atom-independent
Gaussian functions has been employed, with even-tempered exponents chosen to
span the range from 200 down to 0.05. Functions of angular symmetry s, p, d, f
and g are available in the present implementation. An input parameter has been
deﬁned to allow a straightforward variation in the number of ﬁtting functions and
the addition of further functions at the bond midpoints. Preliminary calculations
have indicated that accurate results can be obtained using only s-type functions,
although it is desirable to use functions of higher l to reduce the number of grid
points per atom in the numerical integration while maintaining a given level of
accuracy.
3.2.3
Evaluation of the integrals. The Coulomb problem
One of the main strengths of the present calculation scheme is the accuracy of the
treatment of the Coulomb interactions described by the Coulomb and nonlocal exchange contribution to the Fock and KS operators. No “cutoﬀ” is introduced in
the evaluation of these interactions in a periodic system, and all the charge is correctly introduced in the summation of the whole Coulomb series up to inﬁnity. The
sole approximation appears in the transformation of the inﬁnite series of long-range
bielectronic integrals into an inﬁnite series of monoelectronic integrals, which is evaluated using Ewald techniques. This transformation is performed via a multipolar
analysis of the charge density, in essentially the same way as in the method that in
recent molecular applications has been referred to as distributed multipolar analysis
or fast multipolar analysis. The use of this technique in the periodic LCAO scheme
is documented and analyzed in (Pisani, 1996), and only a brief summary is given
here.
Inﬁnite summations of Coulomb terms appear in the electrostatic energy contribution to the real space Fock and KS matrices. Each matrix element refers to
g
the interaction of a charge distribution ρO,
12 (r) with the charge density of the whole
system ρ(r):
1
g
g
drdr ρO,
(3.62)
(F12 )Coulomb =
ρ(r ),
12 (r)
|r − r |
3.2 Practical implementation of DFT/HF calculation scheme
50
where
g
g
O
ρO,
12 (r) = χ1 (r)χ2 (r).
(3.63)
The lattice vector O refers to the reference cell of the crystal. Similar terms also
appear in the evaluation of the Coulomb contribution to the total energy:
ECoulomb
1
=
2N
drdr ρ(r)
1
ρ(r ),
|r − r |
(3.64)
which corresponds to the interaction of the whole charge density with itself. The total charge density ρ(r) can be partitioned into electronic and nuclear contributions,
and the various inﬁnite series involving the diﬀerent partitions may then be evaluated independently. The nuclear-electron and nuclear-nuclear terms are evaluated
without approximation, using Ewald summation. The electron-electron Coulomb
terms are evaluated using a more complex approximate scheme. The general contribution to the matrix element may be written as:
g e− e
)Coulomb
(F12
=
3, 4, l
l
P34
h
g
dr dr ρO,
12 (r)
1
ρh, h+l (r ),
|r − r | 34
(3.65)
where two new lattice vector labels h and l have been introduced to identify the cells
containing the two components of the ρ34 distributions, and P34 is an element of the
density matrix (i.e. the direct space representation of the ﬁrst-order density operator). The vector sum over the l lattice vectors converges rapidly, since according to
h+l the Gaussian product theorem the overlap density ρh,
(r ) decays exponentially
34
functions. By contrast, the h vector
with increasing separation of the χh3 and χh+L
4
sum refers to a long-range interaction which decays only Coulomb-like. The resulting
series is conditionally and very slowly convergent and requires an extremely careful
analysis. The various inﬁnite series contributing to each matrix element are therefore limited and approximated in diﬀerent ways. The bielectronic Coulomb integrals
are disregarded completely when the space integral of either the overlap distribution
g
O l
χO
1 χ2 or that of the overlap distribution χ3 χ4 is less than a prespeciﬁed threshold
Tl . The conditionally-convergent Coulomb series over h vectors is not truncated,
but is approximated beyond a certain threshold using a distributed point multipole
model of the charge distribution, as follows. The local basis functions associated
with each nuclear site in the reference cell are partitioned into nonintersecting sets
3.2 Practical implementation of DFT/HF calculation scheme
51
q (’shells’) sharing similar asymptotic decay properties. For each shell, the charge
density of an associated shell distribution is then deﬁned according to a Mulliken
partition scheme as
ρhq (r) =
3∈ q
4
l
χh3 (r)χh+l
P34
4 (r) .
(3.66)
l
This deﬁnition saturates the {4, l} indices of the basis function χh+l
4 , and allows the
conversion of many four-centre bielectronic integrals between overlap distributions
into a single three-centre integral between an overlap distribution and a shell distribution. The list of h vectors is partitioned into a ﬁnite internal set {hbi } and
g
an inﬁnite external set {hmono } for χO
1 χ2 (r), using overlap criteria and a second
penetration threshold T2 . The electron-electron Coulomb contributions to the Fock
matrix may then be split into internal and external terms:
g e− e
g bi
g mono
)Coulomb = (F12
)Coulomb + (F12
)Coulomb .
(F12
(3.67)
The bielectronic part is calculated through explicit evaluation of the four-centre
bielectronic integrals I(12g; 34l), to give:
g bi
)Coulomb
(F12
=
l
P34
I(12g; 34l)
3, 4, l
=
3, 4, l
l
P34
{bi} h
g
drdr χO
1 χ2
1
χh χh+l .
|r − r | 3 4
(3.68)
The monoelectronic term, which involves an inﬁnite sum of Coulomb integrals involving electronic distributions that are “external” to each other, is calculated in
an approximate way. The potential at a ﬁeld point r due to each shell distribution
ρhq (r) is given to a good approximation by the spherical multipolar expansion:
ρhq (r )
=
l
l
max ρhq
dr
γql, m Φl, m (r − sq − h).
|r − r |
l=0 m=−l
(3.69)
Here γql, m is the l, m multipole of the shell charge distribution centered at sq (the
position of the function χ3 in cell h) and the ﬁeld term Φl, m (r) represents the
potential at the point r due to an inﬁnite array of unit point multipoles γ l, m . The
multipolar expansion is truncated at some maximum value lmax (which may be up
3.2 Practical implementation of DFT/HF calculation scheme
52
to six). With this approximation, the external Coulomb contribution to the Fock
matrix may be rewritten as:
g mono
(F12
)Coulomb
=
l
max
l
l=0 m=−l
{hEXT }
h
dr γql, m Φl, m (r − sq − h),
(3.70)
q
where the sums over h in the ﬁeld terms are treated using Ewald techniques. This
approximate scheme produces a speed up of around an order of magnitude and lowers
considerably the amount of disk space required, with little or no loss of accuracy
with respect to an “exact” scheme.
3.2.4
Reciprocal space integration
In periodic structure (i.e. when the periodic boundary conditions are imposed), BFs
(Eq. 3.43) associated with diﬀerent k points within the ﬁrst BZ belong to diﬀerent
irreducible representations of the group of one-electron Hamiltonians, Ĥ. It is then
possible to factorize the solution of HF or KS equations (Eq. 3.8 and 3.23) into
separate parts for each k:
1. Consider the p BFs, φµ (r, k), associated with k
k
k
= φkµ |Ĥ|φkν and Sµν
= φkµ |φkν 2. Calculate the matrix elements: Hµν
3. Solve the p × p matrix equation:
H kC k = S kC kE k,
(3.71)
where the diagonal matrix E k contains the eigenvalues εki , and the matrix C k
contains, columnwise, the coeﬃcients of the crystalline orbitals (COs):
ψi (r; k) =
p
ckµi φµ (r; k).
(3.72)
µ=1
The above procedure should be carried on for the complete set of k points in the ﬁrst
BZ, so as to determine the complete set of COs (that is, the KS or HF spin-orbitals)
with the precision granted by the BS adopted and by the accuracy of the algorithms.
3.2 Practical implementation of DFT/HF calculation scheme
53
From the knowledge of the eigenvalues, εki and the eigenfunctions, ψi (r; k), at a
few sample k points (to be indicated in the following by k ), it is possible to obtain
accurate estimates of quantities such as the number of states below a certain energy,
i(e) and the matrix of integrated densities of states, I(e), which imply a sum over
all k points:
−1
dk θ(e − εki ),
(3.73)
i(e) = 2VBZ
−1
T
Iµν
(e) = 2VBZ
i
i
BZ
k
k
dk ck∗
µi cνi exp(ık · T)θ(e − εi ).
(3.74)
BZ
The sum has been replaced with an integral over the BZ, due to the fact that k
points are uniformly distributed in reciprocal space; the condition that only the
orbitals of energy less than e are included in the sum is expressed by the presence in
the integrand of the step function θ(e−εki ), whose value is 1 if εki is less than e, and is
zero otherwise. The quantities i(e) and I(e) are very important. The Fermi energy,
eF , is determined by imposing the condition: i(eF ) = 2n, that is, by requiring
that there are exactly 2n spin-orbitals per cell with energy less than eF . The total
density of states (DOS): n(e) = di(e)/de and the integrated densities of states
derivative, N (e) = dI(e)/de, which is called the projected density of states (PDOS),
give rich information on the chemical structure of the system and allow all oneelectron properties to be obtained within the independent-electron approximation
(Pisani, Dovesi and Roetti, 1988). The value of I(e) at the Fermi energy eF is the
P matrix [P = I(eF )]. The determination of the Fermi energy, eF , is a delicate
problem only in the case of metals. For insulators, n bands are fully occupied, the
others are void. The θ(e − εki ) function can then be dropped from the integrand and
the sum over i is limited to the n lowest eigenvalues at each k point. In summary,
one must estimate integrals of the form:
Ξ(e) =
−1
VBZ
dk [ξ(k)θ(e − ε(k))] =
BZ
−1
VBZ
dk ξ(k),
(3.75)
BZ where ξ(k) and ε(k) are well-behaved, periodic functions in reciprocal space, and
BZ (e) is the portion of BZ where ε(k) < e: it coincides with the whole BZ, if ε(k)
is less than e everywhere and vanishes if ε(k) is always greater than e. In the general
case (BZ (e) = BZ), the linear (Jepson and Anderson, 1971) or quadratic (Boon,
Methfessel and Müller, 1986; Wiesenekker, te Velde and Baerends, 1988) tetrahedron
3.2 Practical implementation of DFT/HF calculation scheme
54
techniques are usually adopted. The BZ is subdivided into tetrahedral mini-cells:
the integral will be the sum of sub-integrals over each of them. After evaluating ξ
and ε at the vertices of the tetrahedra, k , a linear or quadratic approximation is
obtained for both ξ(k) and ε(k) inside the tetrahedron: the sub-integral is ﬁnally
evaluated analytically using these approximate expressions. If BZ (e) = BZ, special
points techniques may be used (Baldereschi, 1973; Chadi, 1977; Monkhorst and Pack,
1976). This means that one may select a special set of points {k } within the BZ,
with a weight, w(k ), associated with each of them, evaluate ξ(k) at each of them
and substitute the integral with a weighted sum: Ξ ≈ k w(k )ξ(k ).
3.2.5
SCF calculation scheme
Finally, to provide the calculations with a high accuracy the following points should
be taken into account:
• The problem of how many and which k points should be considered is an
extremely important one. In order to reconstruct the Ĥ operator and to
calculate the crystalline properties from the solution, we need all occupied spinorbitals, in principle. Due to the continuity of eigenvalues and eigenfunctions
with respect to k, it is, however, possible to derive the required information
from the results obtained at a few suitably sampled k points.
k
matrix elements, one must consider functions and
• In order to calculate Hµν
operators which are extended to the whole crystalline structure. Often there
is a problem of summations over one or more indices associated with diﬀerent
crystalline cells. The accurate and eﬃcient handling of such series determines
the ﬁnal quality of a computational scheme.
• The process must be repeated until self-consistency is achieved, that is, until
eigenvalues and eigenvectors coincide, within a given tolerance, with those used
for the reconstruction of Ĥ. This may be very diﬃcult to achieve at times.
The general structure of the programm that satisﬁes demands mentioned above
(CRYSTAL (Pisani, 1996; Saunders, Dovesi, Roetti, Causa et al., 1998; Homepage,
b,c)) is demonstrated in Fig. 3.2. CRYSTAL (in this study version CRYSTAL’98
has been adopted) is a “direct space” program, in the sense that all the relevant
3.2 Practical implementation of DFT/HF calculation scheme
Definition of geometry
Definition of translation symmetry
Specification of the basis set
Clasification and computation of
one- and two-electron integrals
Coulomb and exchange series
Calculation of Fermi
energy eF and
reconstruction
of density
matrix ||P||
Reconstruction
of matrix ||H||
SCF
Fourier transformation
of ||H|| to ||Hκ||
κ
Diagonalization of ||H ||
Properties
Figure 3.2: Flow chart of the CRYSTAL code.
55
3.3 One-electron properties
56
quantities (mono- and bi-electronic integrals, overlap and H matrices), are computed in the conﬁguration space. Just before the diagonalization step the H matrix
is Fourier transformed to reciprocal space, then the eigenvalues and eigenvectors
of the H k matrices are combined to generate the “direct space” density matrix for
the next SCF cycle. CRYSTAL’98 can solve the HF as well as the KS equations.
The most popular local and non local functionals are available, as well as hybrid
schemes. Schemes are also available, that permit to correct the HF total energy by
estimating the correlation energy a posteriori, integrating a correlation-only functional of the HF charge density. Many steps of the calculation are common to the
HF and DFT options (for example the treatment of the Coulomb series, which are
evaluated analytically); the main diﬀerence concerns, obviously, the exchange (and
correlation) contribution to the Hamiltonian matrix and total energy: in the HF case
the exchange bi-electronic integrals are evaluated analytically, and the exchange series is truncated after a certain number of terms, as discussed below. In the DFT
calculations, the exchange-correlation potential is expanded in an auxiliary basis
set of GTF, with even tempered exponents. At each SCF iteration the auxiliary
BS is ﬁtted to the actual analytic form of the exchange-correlation potential, which
changes with the evolving charge density. CRYSTAL can treat systems periodic in 0
(molecules), 1 (polymers), 2 (slabs) and 3 (crystals) directions with similar accuracy;
this permits the evaluation of energy diﬀerences such as bulk-minus-molecule (lattice
energy of a molecular crystal), bulk-minus-slab (surface energy), bulk-minus-chain
(inter-chain interactions) with high accuracy, as well as energy diﬀerences between
crystals with diﬀerent cell size, shape, and number of atoms.
3.3
3.3.1
One-electron properties
Properties in a direct space; population analysis
To characterize the electronic properties of the system studied in the coordinate
space, the key quantity to construct is the one-electron density, ρ(r), which describes
the probability of ﬁnding one electron in the position r. The density operator, ρ̂, is
3.3 One-electron properties
57
deﬁned from the occupied eigenstates (l) as follows:
ρ̂ =
nl |ψl ψl |.
(3.76)
l
Its matrix representation, for a solid and in the starting localized function basis set,
is given by
g
nl cpl (k)c∗ql (k)eik·g dk,
(3.77)
Ppq =
BZ l
where the l-th eigenstate is
|ψl (k) =
cpl (k)|Φp (k).
(3.78)
p
By analyzing the electronic density ρ(r) (or its representative matrix P ), it is possible to estimate the chemical features of the system under investigation, in particular
the type and strength of bonds between atoms. One of the widely used methods
of analysis is the Mulliken population analysis (Mulliken, 1955a,b,c,d). It uses the
matrix representation of P in the basis set of localized functions (see Eq. 3.77), and
exploits the localization of the starting basis functions, the atomic orbials χ, to partition the electronic density into atomic (ionic), bond and single orbital contributions.
A new matrix M is deﬁned, whose elements are given by
g
g
g
= Ppq
· Spq
.
Mpq
(3.79)
g
are attributed to the atoms on
The electrons described by the matrix element Ppq
which the basis functions p and q are centered, and are equally shared between the
two atoms which p and q belong to. Under this “partition rule”, the electronic
charge attributed to the pth basis function is simply
Qp =
g, q
g
Mpq
.
(3.80)
3.3 One-electron properties
58
The charge attributed to the Ath atom is obtained by summing the orbital charge
of all the atomic orbitals in its basis set:
QA =
Qp
(3.81)
p∈A
while a bond charge QAB between atoms A and B is obtained by summing the value
g
of all the out-of-diagonal elements Mpq
in which one of the atomic orbitals belongs
to atom A and the other to atom B:
QAB =
Mpq +
p∈A, q∈B
p∈B, q∈A
Mpq = 2
Mpq .
(3.82)
p∈A, q∈B
Comparing the value of QA with a formal electronic charge, one can deﬁne a net
charge for atom A, and relate the solution with the degree of ionicity of A in its
crystalline environment. The drawbacks of the Mulliken population analysis are
the arbitrariness of its “partition rule”, which always attributes half of the electrons
g
to each of the atomic orbitals p and q, and its basis set
described by the element Mpq
dependence. Nevertheless, qualitative indications can be obtained from a Mulliken
population analysis. As a general rule, the more a bond-population is positive, the
more the bond is covalent, while (slightly) negative values of QAB usually indicate
non-bonded interactions between atoms A and B. Despite the shortcomings, the
formal simplicity makes a Mulliken population analysis very easy to calculate, this
scheme is still widely used and is adopted in present study.
3.3.2
Properties in a reciprocal space; band-structure and
density of states
The most important observable of interest for examining the properties of a crystalline solid in reciprocal space is its spectrum of one-electron energy levels. Following the secular Equation 3.71, in principle the set of solutions is diﬀerent in
each point k of reciprocal space. The band-structure maps are the energy solutions
along a representative path of reciprocal space (usually including its high-symmetry
positions, where bands often have their minimum and maximum allowed energy levels). Examination of the band-structure can provide information not only on the
3.3 One-electron properties
59
conductivity of the system (linked to the value of the band-gap between occupied
and empty levels), but also on its chemistry, in a way alternative to the Mulliken
population analysis. Examining the atomic composition of eigenvalues and the energy dispersion of the eigenvectors in reciprocal space can be a very eﬀective way to
understand and quantify the nature of the interactions in the solid.
An eigenvalue of the solid can be considered describing a covalent bond between
a pair of atomic orbitals, p and q, on nearest neighbor atoms A and B and the element of the Hamiltonian matrix Hpq can be called as β. The value of β describes the
eﬀective energy of the bond examined. The weight of the p and q atomic orbitals in
the crystalline BFs is modulated by a wave factor ek·RA, B (Eq. 3.43). In the Bloch
function BS in point k, Φ(k), the value of the integral between p and q is therefore
modulated by a wave exponent ek·(RA −RB ) and will contribute to the Hamiltonian
H(k) a value β · ek·RAB . The corresponding eigenvalue will (in general) oscillate
as a function of k, with an amplitude proportional to β. The higher the energy
dispersion of one eigenvalue in reciprocal space, the more eﬀective is therefore the
covalent bond between the atomic orbitals p and q. As a general rule, relatively
ﬂat bands are indicative of ionic systems, while materials with wide bands usually
contain covalent bonds, whose strength is proportional to the band-width.
The number of bands (energy levels in each k point) equals the number of basis
functions in the unit cell; it therefore increases with the size and complexity of the
system studied. Only for materials with a relatively small unit cell is it possible
to follow the dispersion of individual energy levels in reciprocal space. When the
number of atoms in the unit cell increases, a convenient way to examine the solution
is via the DOS. The latter is obtained by integrating the band structure over the
whole of reciprocal space, and gives information on the number of energy levels
available for the system as a function of the energy. The DOS can also be divided
in atomic components according to a Mulliken partition scheme, by projecting the
contribution of a selected set of atoms or atomic orbitals to the eigenstates at each
energy. The result provides the partial or PDOS, for which one can deﬁne both
orbital Np (e) and atomic NA (e) values, as follows:
Np (e) =
−1
VBZ
q, l, g
BZ nl (κ)Spq (κ)cpl (κ)c∗ql (κ)eiκ·g δ[e − el (κ)]dκ
(3.83)
3.3 One-electron properties
60
NA (e) =
Np (e)
(3.84)
p∈A
Ntot (e) =
NA
(3.85)
A
By examining the projected density of states, one can indirectly investigate the
extent of hybridization of atomic orbitals on diﬀerent atoms (that is the covalence
in their bonding). Extensive overlap in the PDOS for two diﬀerent atoms usually
indicates covalent bonds between the two atoms.
Chapter 4
Calculations on bulk perovskites
Introduction
In this Chapter, in order to compare the various type Hamiltonians, the results of
detailed calculations for bulk properties and the electronic structure of the cubic
phase of SrTiO3 , BaTiO3 , and PbTiO3 perovskite crystals employing the basis set
described in Section 3.2.1 are documented and discussed. These are obtained using
both ab initio Hartree-Fock and Density Functional Theory applying a number of
diﬀerent exchange-correlation functionals including hybrid (B3PW and B3LYP) exchange techniques. Results, obtained for seven methods, are compared with previous
Quantum Mechanical calculations and available experimental data. Great attention
is paid to the calculated optical band gap which is responsible for a many observed
crystal properties (e.g. optical adsorption) and to the elastic constants Cij which
play an important role in the physics of materials as they characterize the behavior of the crystal in the ﬁeld of external forces, e.g. on the substrate. Cij can be
easily determined from the ﬁrst principles calculations done in the direct computation. Nevertheless, only a few of the ab initio studies (King-Smith and Vanderbilt,
1994; Waghmare and Rabe, 1997) mentioned in Chapter 2 discuss calculations of
the elastic constants for perovskite crystals. In this study, a detailed comparison of
calculated elastic constants using diﬀerent approximation, with existing theoretical
and experimental data is demonstrated.
4.1 Computational details
4.1
62
Computational details
Using newly generated basis sets, the total energies and the electronic structures
have been calculated for all three perovskite crystals by means of several, quite different methods: “pure” HF and DFT schemes accompanied with various exchange
correlation functionals. In DFT computations, the LDA scheme with the DiracSlater exchange (Dirac, 1930b) and the Vosko-Wilk-Nusair correlation (Vosko, Wilk
and Nusair, 1980) energy functionals have been used as well as a set of GGA exchange and correlation functionals as suggested by Perdew and Wang (PWGGA)
(Perdew and Wang, 1986, 1992), by Perdew, Burke and Ernzerhof (PBE) (Burke
and Ernzerhof, 1996), and lastly by the Becke exchange potential (Becke, 1988b)
combined with the correlation potential by Lee, Yang and Parr (BLYP) (C.Lee,
Yang and Parr, 1988). Also calculations using the hybrid functionals (Becke, 1993a)
mixing the Fock exchange and Becke’s gradient corrected exchange functional have
been performed. Two versions of the gradient corrected correlation potentials together with hybrid exchange potentials have been employed (see Eq. 3.38): by Lee,
Yang and Parr (B3LYP) or by Perdew and Wang (B3PW). For the DFT calculations the exchange and correlation potentials have been expanded in the auxiliary
BS according to that described in Section 3.2.2. The reciprocal space integration was
performed by sampling the BZ with the 8 × 8 × 8 Pack-Monkhorst net (Monkhorst
and Pack, 1976) which provides a balanced summation in the direct and reciprocal
lattices (Bredow, Evarestov and Jug, 2000). To achieve high accuracy, large enough
tolerances (N = 7, 8, 7, 7, 14) were employed in the evaluation of the overlap for inﬁnite Coulomb and exchange series (Saunders, Dovesi, Roetti, Causa et al., 1998) (i.e.
the calculation of integrals with an accuracy of 10−N were chosen for the Coulomb
overlap, Coulomb penetration, exchange overlap, the ﬁrst exchange pseudo-overlap,
and for the second exchange pseudo-overlap respectively). In particular, the energy
versus strain curves lose their smoothness if smaller tolerances are used. Thus, the
truncation parameters have been also selected to reproduce the smooth behavior of
the total energy versus the lattice strain. During the lattice constant optimizations,
all atoms were ﬁxed in the sites of perfect cubic perovskite structure.
The deﬁnition and calculation of the elastic constants for cubic crystals is described in detail in Appendix B. The bulk modulus could be calculated in two ways,
4.2 Bulk properties
63
ﬁrstly as:
2 ∂ 2 Eun.cell
,
9V0
∂V 2
(4.1)
B = (c11 + 2c12 )/3 .
(4.2)
B=
or using the elastic constants:
The results for both type of bulk modulus evaluation are presented below.
4.2
Bulk properties
To describe the three cubic perovskite crystals, the lattice constants have been optimized, independently for the HF and for DFT accompanied with each exchangecorrelation functional. The results are presented in Table 4.1. In Table 4.1 data
obtained on cubic STO using the proposed BSs are compared with the same values obtained employing the same methods of calculations, but using the standard
BS available on Web-site (Homepage, a) (results are given in brackets). It is clear
from Table 4.1 that the LDA calculations underestimate the lattice constant for
all three perovskites, whereas pure HF and GGA overestimate this. The diﬀerent
GGA schemes give quite good results only for the PTO crystal. The PTO lattice
constants computed using PWGGA and PBE functionals are close to the experimental values, whereas in other cases the DFT-GGA gives overestimated values.
The best agreement with experimental lattice constant was obtained for the hybrid
DFT B3PW and B3LYP methods. On the average, the disagreement between the
lattice constants computed using hybrid functionals and experimental values for all
three perovskites is less than 0.5%.
Table 4.1 also lists the computed bulk moduli and the static elastic constants
obtained by means of all methods. The presented results for both ways of bulk
moduli evaluation diﬀer usually no more than 10-15%. The calculations conﬁrm
the tendency, well known in the literature, that the HF calculations overestimate
the elastic constants. The overestimated elastic constants have been also obtained
here for all three perovskites, when the DFT-LDA scheme was used. In the case of
a cubic STO, which is experimentally well investigated, almost perfect coincidence
with the experimental data has been obtained for both the bulk modulus and elastic
4.2 Bulk properties
64
Table 4.1: The optimized lattice constant a 0 (Å), bulk modulus B (GPa) and elastic
constants cij (in 1011 dyne/cm2 ) for three ABO3 perovskites as calculated using DFT
and HF approaches. The results of calculations for standard BS are given in the
brackets. The two last columns contain the experimental data and the data calculated using other QM techniques. The penultimate row for each perovskite contains
the bulk modulus calculated using the standard relation B=(c11 +2c12 )/3; it is done
for Experiment and Theory columns, respectively.
STO
BTO
PTO
a0
LDA
3.86
(3.86)
PWGGA
3.95
(3.93)
PBE
3.94
(3.93)
BLYP
3.98
(3.98)
P3PW
3.90
(3.91)
B3LYP
3.94
(3.94)
HF
3.92
(3.93)
Exper.
3.89 (i)
3.90 (iii)
c11
42.10
31.29
31.93
29.07
31.60
32.83
41.68
31.72 (vii)
c12
12.21
9.80
9.75
9.39
9.27
10.57
7.11
10.25 (vii)
c44
13.32
11.34
11.30
11.09
12.01
12.46
10.50
12.35 (vii)
B
222
170
171
159
167
180
186
174 (vii)
B
a0
214
(215)
3.96
167
(195)
4.03
169
(195)
4.03
164
(165)
4.08
177
(186)
4.01
177
(187)
4.04
219
(211)
4.01
179 (i)
179 (ix)
4.00 (i)
c11
35.81
30.11
31.04
28.22
31.12
29.75
30.07
c12
11.52
10.35
10.72
10.78
11.87
11.57
13.46
c44
14.98
13.22
13.98
12.24
14.85
14.54
17.34
B
196
169
175
166
183
176
190
B
204
175
180
154
188
172
194
a0
3.93
3.96
3.96
4.02
3.93
3.96
3.94
20.60 (i)
18.70 (x)
14.00 (i)
10.70 (x)
12.60 (i)
11.20 (x)
162 (i)
134 (x)
195 (ix)
162 (i)
3.97 (xi)
c11
45.03
32.47
34.25
23.03
43.04
34.42
39.83
22.90 (x)
c12
26.14
15.81
15.52
9.93
24.95
18.08
16.90
10.10 (x)
c44
11.28
10.69
10.96
8.25
10.93
10.35
17.20
10.00 (x)
B
324
213
217
143
310
235
245
144 (x)
B
321
246
252
140
279
242
299
i – Hellwege and Hellwege (1969)
ii – Cora and Catlow (1999)
iii – Abramov et al. (1995)
iv – Cappelini et al. (2000)
v – King-Smith and Vanderbilt (1994)
vi – Tinte, Stachiotti, Rodriguez, Novikov et al. (1998)
vii – Bell and Rupprecht (1963)
Theory
3.93 (ii)
3.85 (iv)
3.86 (v)
3.86 (vi)
38.9 (v)
30.15 (viii)
10.5 (v)
13.74 (viii)
15.5 (v)
13.78 (viii)
200 (v)
192 (viii)
203 (iv)
204 (vi)
4.02 (ii)
3.94 (v)
4.03 (vi)
35.1 (v)
12.5 (v)
13.9 (v)
200 (v)
195 (vi)
3.88 (xii)
3.89 (xiii)
32.02 (xiii)
34.2 (v)
14.12 (xiii)
14.9 (v)
37.49 (xiii)
10.3 (v)
201 (xiii)
213 (v)
203 (xiii)
209 (v)
viii – Akhtar, Akhtar, Jackson and Catlow (1995)
ix – Fischer, Wang and Karato (1993)
x – Li, Grimsditch, Foster and Chan (1996)
xi – Shirane and Repinsky (1956)
xii – Ghosez, Cockyane, Waghmare and Rabe (1999)
xiii – Waghmare and Rabe (1997)
4.2 Bulk properties
65
constants calculated using B3PW and B3LYP hybrid schemes. The disagreement
of elastic constants is less than 5%, and the bulk moduli practically coincide with
the experimental magnitudes. The DFT-GGA calculations have tendency to underestimate slightly the bulk modulus, while the lattice constant is overestimated.
The elastic constants are underestimated by 5-10% in the GGA calculations. At
the same time, the improvement of bulk properties calculated using newly generated BSs, as compared to the values calculated using the standard BSs, is well seen
for STO. It is clearly seen that the hybrid DFT functionals give the best description of the STO perovskite, i.e. the best agreement with experiment was achieved
for both bulk modulus and lattice constant, as well for the elastic constants, and
lastly, as will be shown below, for the optical gap as well. Unfortunately, the experimental data for BTO and PTO are more limited. In the case of BTO, relying on
the literature data (see Table 4.1 for references), the DFT B3PW scheme has been
chosen for further calculations, since BTO has the same tendencies as STO in the
present ab initio calculations. Unlike the cases of BTO and STO, results of computations for PTO perovskite show better agreement with experimental data for DFT
B3LYP, PWGGA and PBE. The DFT B3LYP scheme is favored since, as will be
shown below, only B3PW and B3LYP give the optical gap close to the experimental one. The last column of Table 4.1 presents the data of recent QM calculations
performed by other theoretical groups (all references are given in Table 4.1). Most
of them have worse agreement with experimental values than the results obtained
using the proposed BSs. Nevertheless, our data correlate well with them, especially
with results obtained by King-Smith and Vanderbilt (1994) using the DFT-LDA
and Ultra-soft-pseudopotential Augmented-Plane-Wave method. Furthermore, it is
necessary to note that a cubic phase of perovskites is quite unstable, and thus the
measured elastic constants strongly depend on the temperature. For example, c11 of
STO increases by about 4% when the temperature decreases from 30 ◦ C to -145 ◦ C,
as reported by Bell and Rupprecht (1963), then c11 drops as the phase transition
temperature is achieved. The same is true for c44 and c12 . BTO and PTO elastic
constants as a function of temperature have been considered by Li, Grimsditch, Foster and Chan (1996). Thus, if disagreement for calculated elastic properties with
experimental results is about of 10%, it may be still considered as a good agreement.
4.2 Bulk properties
66
STO B3PW
E / a.u.
0.300
0.200
0.100
a)
Γ12
0.000
Γ25'
-0.100
Γ15
-0.200
Γ25
Γ15
-0.300
Γ
X
M
Γ
R
X
M'
R
BTO B3PW
E / a.u.
0.300
0.200
0.100
Γ12
b)
0.000
Γ25'
-0.100
Γ15
-0.200
Γ25
Γ15
-0.300
Γ
X
M
Γ
R
X
M'
R
PTO B3PW
E / a.u.
0.200
0.100
Γ12
0.000
c)
Γ25'
-0.100
Γ15
-0.200
Γ25
-0.300
Γ15
-0.400
Γ
X
M
Γ
R
X
M'
R
Figure 4.1: The band structure of three cubic perovskites for selected high-symmetry
directions in the BZ. a) STO, b) BTO, c) PTO. The energy scale is in atomic units
(Hartree, 1 Ha = 27.212 eV), the dashed line is the top of valence band.
4.3 Electronic properties
67
Table 4.2: The calculated optical band gap (eV). The results of calculations with
standard BS are given in the brackets.
STO
BTO
PTO
Optical gap
LDA
PWGGA
PBE
BLYP
P3PW
B3LYP
HF
Experiment
Γ-Γ
2.36
2.31
2.35
2.27
3.96(4.43)
3.89
12.33
3.75 -
X-X
2.94
2.79
2.84
2.72
4.53(5.08)
4.42
13.04
direct gap
M-M
4.12
3.69
3.74
3.56
5.70(6.45)
5.50
14.45
3.25 -
R-R
4.77
4.25
4.31
4.09
6.47(7.18)
6.23
15.72
indirect gap
X-Γ
2.78
2.69
2.73
2.63
4.39
4.31
12.86
(i)
M-Γ
2.15
2.06
2.08
2.03
3.71(4.23)
3.66
12.02
R-Γ
2.04
1.97
1.99
1.94
3.63(4.16)
3.57
11.97
Γ-Γ
1.98
1.97
1.99
1.91
3.55
3.49
11.73
X-X
2.85
2.73
2.74
2.57
4.39
4.26
12.83
M-M
3.81
3.47
3.50
3.24
5.39
5.19
14.11
R-R
4.45
4.03
4.07
3.76
6.12
5.89
15.22
X-Γ
2.64
2.55
2.57
2.44
4.20
4.10
12.57
M-Γ
2.01
1.93
1.95
1.84
3.60
3.51
11.95
R-Γ
1.92
1.84
1.86
1.76
3.50
3.42
11.85
Γ-Γ
2.65
2.61
2.65
2.48
4.32
4.15
12.74
X-X
1.54
1.68
1.70
1.77
3.02
3.05
10.24
M-M
3.78
3.58
3.61
3.33
5.55
5.33
13.76
R-R
4.16
3.91
3.94
3.65
5.98
5.78
15.07
X-Γ
1.40
1.56
1.58
1.67
2.87
2.92
10.01
M-Γ
2.01
1.98
2.00
1.88
3.66
3.53
11.43
R-Γ
2.03
1.98
2.00
1.89
3.66
3.52
12.03
3.2 (ii)
3.4 (iii)
i – van Benthem, Elsässer and French (2001)
ii – Wemple (1970)
iii – Peng, Chang and Desu (1992)
4.3
Electronic properties
All electronic properties have been calculated for the equilibrium geometry for each
calculation scheme, respectively. Data on the optical band gaps are collected in
Table 4.2. Table 4.3 lists the calculated Mulliken charges and bond populations between an oxygen ion and its neighbors. However, the band structures, the densities
of states, and the maps of electron densities for each crystal have been calculated
using the hybrid B3PW functional only (Fig. 4.1-4.3), because the same properties
obtained using other calculation schemes look quite similar to them.
The band structures of all three perovskites (Fig. 4.1) look very similar and
agree with band structures published previously in the literature using diﬀerent ab
initio methods and basis sets, including plane waves (see, e.g., Tinte, Stachiotti,
Rodriguez, Novikov et al. (1998); Veithen, Gonze and Ghosez (2002)). Nine valence
4.3 Electronic properties
68
bands derived from O 2p orbitals at the Γ point form the three 3-fold degenerate
levels (Γ15 , Γ25 and Γ15 ). The crystalline ﬁeld and the electrostatic interaction between O 2p orbitals split these bands. But the top of the valence band is displaced
from the Γ-point of the BZ to the R-points in STO and BTO, and to the X-points
in PTO. The highest valence electron states at the M point appear only about 0.1
eV below the highest states in R-points, for STO, BTO, and PTO (except HF case).
The dispersion of the top valence band is almost ﬂat between R and M points for all
three crystals. The highest valence states at the Γ-point stay very close to the top
of the valence band in BTO, only 0.1 eV below the R-point. In STO and PTO the
diﬀerence becomes 0.3 and 0.6 eV respectively. The additional s-orbitals on Pb ions
in PTO cause the appearance of an additional valence band below the other bands.
They cause also the highest states at the X-point to rise above all other valence
states and to make a new top of the valence band. The bottom of the conduction
band lies at the Γ-point in all three perovskite crystals. The bottom of the conduction band is presented by the 3-fold (Γ25 ) and 2-fold (Γ12 ) degenerate states, which
are built from the t2g and eg states of Ti 3d orbitals, respectively. The electron
energy in the lowest conduction band at the X-point is just 0.1-0.2 eV higher than
at the bottom of conduction bands. So, there is a little dispersion in the lowest
conduction band between the Γ and X points in the BZ.
The optical band gaps of three perovskites obtained using various functionals are
summarized in Table 4.2. This Table clearly demonstrates that pure HF calculations
overestimate the optical gap by several times for all three perovskites whereas LDA
and GGA calculations dramatically underestimate it. This tendency is well known
in solid-state physics. The most realistic band gaps have been obtained using the
hybrid B3LYP and B3PW functionals. The STO experimental band gaps are 3.25
eV (indirect gap) and 3.75 eV (direct gap), as determined by van Benthem, Elsässer
and French (2001) using spectroscopic ellipsometry; a 3.2 eV band gap has been
measured for BTO (Wemple, 1970) and 3.4 eV for PTO (Peng, Chang and Desu,
1992). In the present calculations using the B3LYP functional a STO indirect band
gap (R-Γ) has been obtained of 3.57 eV to be smaller than 3.89 eV for the direct
(Γ-Γ) band gap. Using the B3PW hybrid functional 3.96 eV and 3.63 eV have been
obtained for the STO direct and indirect band gap, respectively. Calculated band
gaps are very close to the experimental ones. The best agreement with the experimental results (in contrast to calculations with standard BS given in brackets in
-0.1
0.0
0.1
100
200
10
20
10
20
1
2
2
-0.3
-0.2
0.1
-0.2
-0.1
0.0
0.1
0.2
0.3
100
-0.2
-0.1
0.0
0.2
0.3
120
100
200
10
20
30
10
20
30
30
60
90
15
30
-0.4
-0.3
-0.2
-0.1
0.0
Total DOS
0.1
0.2
DOS projected to
pz AOs of O(z) atom
DOS projected to
px AOs of O(z) atom
DOS projected to p AOs of Pb atom
DOS projected to s AOs of Pb atom
0.3
100
200
10
20
30
10
20
30
30
60
90
15
30
45
60
45
60
60
60
30
DOS projected to O atom
DOS projected to Ti atom
30
60
60
120
0.1
120
DOS projected to Pb atom
-0.3
60
-0.4
60
120
c)
-0.3
100
200
Total DOS
200
20
20
10
40
30
15
2
30
60
10
DOS projected to pz AOs of O(z) atom
DOS projected to
px AOs of O(z) atom
DOS projected to p AOs of Ba atom
DOS projected to s AOs of Ba atom
DOS projected to O atom
10
20
20
10
40
30
15
2
30
60
60
120
DOS projected to Ti atom
120
60
100
200
0.3
100
0.2
b)
0.3
0.0
a)
0.2
-0.1
DOS projected to Ba atom
-0.2
Energy / a.u.
0.1
-0.3
200
Energy / a.u.
0.0
100
200
10
20
10
20
1
2
DOS for BaTiO 3 bulk (B3PW) / arb. units
Energy / a.u.
-0.1
Total DOS
DOS projected to
pz AOs of O(z) atom
DOS projected to
px AOs of O(z) atom
DOS projected to
p AOs of Sr atom
2
4
DOS projected to s AOs of Sr atom
4
60
30
DOS projected to O atom
60
30
60
60
120
DOS projected to Ti atom
200
0.3
120
0.2
100
DOS projected to Sr atom
-0.2
100
-0.3
DOS for PbTiO3 bulk (B3PW) / arb. units
Figure 4.2: The calculated total and projected density of states (DOS and PDOS) for three perovskites. a) STO, b) BTO,
c) PTO.
DOS for SrTiO3 bulk (B3PW) / arb. units
200
4.3 Electronic properties
69
4.3 Electronic properties
70
Table 4.2) were obtained due to the adding of d polarization orbital to the basis set
of the oxygen atom. The band gaps calculated for BTO and PTO crystals, 3.42 eV
and 2.87 eV, respectively, are also in a good agreement with the experiment, the
discrepancy is less than 7%. This is acceptable if one takes into account diﬃculties
in determining the band gap experimentally, including the optical absorption edge
tails which extend up to several tenths of eV (Lines and Glass, 1977).
As it is seen in Figure 4.2, oxygen p-orbitals give the primary contribution to
the valence band of all three studied perovskites. The additional valence band in
PTO contains contributions mostly from Pb 6s-orbitals. These orbitals contribute
to the valence bands through the entire set of the bands. But these contributions
are small, except in the vicinity of the top of valence spectra. The top of valence
bands in STO and BTO is created by O 2p-orbitals, which are perpendicular to
the Ti-O-Ti bridge and lie in the SrO- (BaO-) planes. In case of PTO, the top of
valence bands contains the same O 2p-orbitals with a signiﬁcant admixture of Pb
6s-orbitals. The bottom of conduction bands is formed by Ti 3d -orbitals. These
orbitals give the main contribution to conduction bands at about the lowest portion
(0.1-0.2 atomic units) of the spectrum. There is some small contribution from O 2porbitals to this part of the spectrum. Sr(Ba) valence s-orbitals and Pb 6p-orbitals
contribute to the conduction bands at higher energies. Ti 3d -orbitals also contribute
to the lower half of the valence state spectra. Such an admixture of Ti 3d -orbitals
to O 2p-orbitals demonstrates the weak covalency of the chemical bonds between
Ti and O.
The Mulliken net charges of Ti and O diﬀer quite from the formal ionic charges
of ABO3 perovskites: B4+ , and O2− (see Table 4.3). The reason for this is that,
despite the ABO3 perovskites often are treated as completely ionic, there is a large
overlap between the Ti 3d and oxygen 2p orbitals, resulting in a partly covalent O-Ti
chemical bonding. This is conﬁrmed by the O-Ti bond populations, which vary from
0.108 to 0.072 e, depending on the calculation method and material. In contrast,
there is practically no bonding of O with Sr and Ba atoms in STO and BTO. Sr
and Ba charges remain close to the formal +2 e. These results are very close for all
methods used. The atomic eﬀective charges increase in a series of DFT functionals
better accounting for the exchange eﬀect, i.e. LDA, GGA, hybrid functionals, and
lastly HF. The calculated optical band gaps (Table 4.2) also increase in the same
series (GGA, LDA, hybrid, HF). Since vacant orbitals in perovskites are localized on
4.3 Electronic properties
71
Table 4.3: Eﬀective Mulliken charges, Q (e), and bond populations, P (mili e), for
three bulk perovskites, the results of calculations with standard BS are given in
brackets. OI means the oxygen nearest to the reference one, OII oxygen from the
second sphere of neighbour oxygens. Negative populations mean repulsion between
atoms.
atom
Charge Q,
LDA
PWGGA
PBE
BLYP
P3PW
B3LYP
HF
Bond
populations P
STO
Sr2+
Q
1.854
1.853
1.852
1.848
1.871
1.869
1.924
(1.830)
(1.834)
(1.832)
(1.835)
(1.852)
(1.852)
(1.909)
Ti4+
Q
2.179
2.239
2.245
2.257
2.350
2.369
2.785
(2.126)
(2.212)
(2.206)
(2.266)
(2.272)
(2.325)
(2.584)
-1.344
-1.364
-1.365
-1.368
-1.407
-1.413
-1.570
(-1.319)
(-1.349)
(-1.346)
(-1.367)
(-1.375)
(-1.392)
(-1.497)
O2− (z)
O
Q
OI
Sr
Ti
OII
BTO
PTO
-52
-32
-32
-30
-44
-40
-58
(-48)
(-34)
(-32)
(-30)
(-36)
(-36)
(-40)
-10
-6
-4
-4
-10
-10
-22
(-10)
(-6)
(-6)
(-4)
(-10)
(-8)
(-10)
86
96
96
100
88
92
72
(52)
(70)
(74)
(66)
(82)
(74)
(112)
-8
-6
-6
-6
-8
-8
-12
(-2)
(-2)
(-8)
(-2)
(-4)
(-4)
(-8)
1.855
Ba2+
Q
1.783
1.769
1.766
1.772
1.795
1.796
Ti4+
Q
2.195
2.240
2.245
2.252
2.364
2.370
2.808
O2− (z)
Q
-1.326
-1.337
-1.337
-1.342
-1.386
-1.388
-1.554
O
OI
-44
-28
-28
-24
-36
-34
-46
Ba
-34
-26
-24
-22
-34
-32
-52
Ti
100
104
106
108
100
102
80
OII
-6
-4
-4
-4
-6
-6
-10
Q
1.312
1.257
1.231
1.292
1.343
1.407
1.612
Pb2+
Ti4+
Q
2.172
2.210
2.217
2.232
2.335
2.343
2.785
O2− (z)
Q
-1.161
-1.156
-1.149
-1.175
-1.226
-1.250
-1.466
O
OI
-52
-42
-40
-34
-50
-46
-60
Pb
24
30
32
24
16
14
-20
Ti
104
108
106
110
98
102
76
OII
-8
-6
-6
-4
-8
-6
-10
4.3 Electronic properties
0.045
0.035
0.015
0
-0.0050
72
O
0.030
0.020
0
Ti
O
0.030
0.030
0.0050
0
0.010
0.010
-0.0050
0
-0.0050
0.020
0.020
0
0.030
0
0
0.010
0.030
0.030
-0.0050
O
Sr
STO B3PW
Sr
0.020
-0.0050
-0.0050
a)
O
STO B3PW
STO B3PW
O
0
Ti
0.020
O
0.020
0.030
Ti
0.030
0.020
0.020
0
-0.0050
b)
Ti
0.020
-0.0050
0.030
-0.0050
-0.0050
0
0
0.010
-0.0050
0
0.020
0.020
0
0
0.010
-0.0050
0.030
O
O
BTO B3PW
Ba
-0.0050
0.030
0.010
0.020
0
O
0
Ti
0.020
O
-0.0050
0
0
0
0.010
0
0.030
0.030
0.020
-0.0050
0.010
0.020
0
Ti
0.030
0.020
-0.0050
-0.0050
c)
BTO B3PW
BTO B3PW
Ba
0.020
0.030
-0.0050
0.030
O
Pb
PTO B3PW
Pb
PTO B3PW
O
PTO B3PW
Figure 4.3: The diﬀerence electron density plots for three perovskites calculated
using DFT B3PW: a) STO, b) BTO, c) PTO. The electron density plots are for
AO-(001) (left column), (110) (middle column), and TiO2 -(001) (right column) cross
sections. Isodensity curves are drawn from -0.05 to +0.05 e a.u.−3 with an increment
of 0.005 e a.u.−3 .
4.3 Electronic properties
73
cations, an increase of the band gap leads to an additional transfer of the electron
density from cations to anions, accompanied by a growth of the crystal ionicity. In
contrast, Pb charges turn out to be much less than the formal +2 e charge. Also,
in contrast to the negative bond populations of STO and BTO, the positive O-Pb
bond populations are obtained in all DFT calculations, except the HF where it is
negative. This means the PTO has a weak covalent O-Pb bonding. The diﬀerent
sign of O-Pb bond population can be explained partly by the fact that “pure” HF
calculations do not include the electron correlation corrections. Because a “large
core” ECP was employed for Pb, there was no explicit treatment of 5d orbitals on
lead ions. We expect that inclusion of Pb 5d -orbitals could slightly increase the
covalency of the Pb-O bond. The O-O bond populations are always negative. This
is evidence that repulsion between oxygens in the perovskites has contributions from
both Coulomb interactions, and due to the antibonding interaction.
The diﬀerence electron density maps, calculated with respect to the superposition density for A2+ , B4+ , and O2− ions are presented in Figure 4.3. These maps
were plotted in the three most signiﬁcant crystallographic plains, (001) containing
Sr and O atoms, (001) containing Ti and O atoms, and (110) containing Ti, Sr and
O atoms. Analysis of the electron density maps fully conﬁrms the Ti-O covalent
bonding eﬀect discussed above. The positive solid isodensity curves easily distinguishable in Fig. 4.3, explicitly show the concentration of the electronic density
between Ti and O ions. This picture is essentially the same for all three perovskites
(see the middle and right columns in Fig. 4.3, which correspond to the (110) and
TiO2 -(001) cross sections, respectively). At the same time, the density maps drawn
for the AO-(001) cross section (the left column in Fig. 4.3) show no trace of covalent
bonding between the oxygen atom and Sr, Ba or Pb. Calculated electron density
maps fully conﬁrm the Mulliken population analysis presented in Table 5.
In conclusion of this Chapter it can be stressed that the re-optimized Gaussian
type basis sets for several key perovskite crystals permit to considerably improve
the quality of calculations of basic electronic properties based on the HF and DFT
SCF LCAO methods combined with six diﬀerent exchange-correlation functionals.
Careful comparison of the seven types of Hamiltonians shows that the best agreement
with the experimental results give the hybrid exchange techniques (B3LYP and
B3PW). On the other hand, a good agreement between the results computed using
identical Hamiltonians (e.g. LDA), but diﬀerent type basis sets (e.g. PW and
4.3 Electronic properties
74
Gaussian) is observed (see Table 4.1). The calculations demonstrate a considerable
Ti-O covalent bonding in all three ABO3 perovskites studied, and an additional weak
covalent Pb-O bond in PbTiO3 . Results of the present Chapter are quite useful
for further simulations of perovskite surfaces, multi-layered structures, interfaces
between perovskites and other materials, and defects in perovskite crystals.
Chapter 5
Point defects in perovskites: The
case study of SrTiO3:Fe
Introduction
In this Chapter, a consistent and economic approach for defective solids is presented
and applied to ab initio calculations of the iron impurity in STO, with a focus on
detailed treatment of lattice relaxation around a single defect. For defective crystal
the supercell model and hybrid density functional theory calculations based on the
linear combination of atomic localized Gaussian basis sets are used. Despite the
fact that the supercell approach is widely used already for two decades in defect
calculations, very little attention is paid to the supercell size optimization and the
eﬀect of periodically repeated defect interaction. A study of the convergence of
results to the limit of a single defect is one of main aims of this Chapter. All
results presented here are obtained in close cooperation with Prof. R. A. Evarestov
(St.Petersburg, Russia).
5.1 A consistent approach for a modelling of defective solids
5.1
76
A consistent approach for a modelling of defective solids
Usually defect concentrations in solids are so low that point defects could be treated
as single ones. The main problem is to understand changes induced by a single
point defect in the electronic and atomic structure of a host crystal (electronic density redistribution, additional local energy levels in the optical gap, lattice relaxation
around defects, etc.) This requires use of adequate models for both perfect and defective crystals. When a single point defect appears, perfect crystalline translation
symmetry is lost so that use of a k-mesh in the BZ and primitive unit cell, commonly
used in perfect crystal calculations, becomes formally impossible. The simplest and
direct approach in this case is a molecular cluster model of the defective crystal.
This is obtained by cutting out in the crystal some fraction of atoms consisting of
the point defect and several spheres of nearest neighbors, followed by an embedding
of this cluster into the ﬁeld of the surrounding crystal and (or) by saturating cluster
dangling bonds with pseudoatoms. There are well-known (Deak, 2000) diﬃculties
of the cluster model connected with changes of host crystal symmetry, pseudoatom
choice at the cluster boundaries and the necessity to consider nonstoichiometric
(charged) clusters. Nevertheless, a reasonable choice of cluster is possible when well
localized point defects are considered (Sousa, de Graaf and Illas, 2000).
In recent years, in view of the fact that powerful computers allow to carry out
calculations on solids with quite a large number of atoms in the primitive unit cell
(Pisani, 1996; Makov, Shah and Payne, 1996; Ordejon, 2000), two models alternative to the cluster model became popular. These use the translation symmetry not
only for a perfect but also for defective crystals: the supercell model (SCM) (Mallia,
Orlando, Roetti, Ugliengo et al., 2001; Lichanot, Baranek, Mérawa, Orlando et al.,
2000) and the cyclic cluster model (CCM). These two models have both similarities
and discrepancies (Deak, 2000; Bredow, Evarestov and Jug, 2000). Similarity is
that in both models not a standard primitive unit cell but an extended unit cell is
used (this is why it is called supercell or also large unit cell ). Another well-known
procedure to treat localized defects exists, which is based on the Green’s functions
method and called the embedded cluster model (Pisani, Dovesi, Roetti, Causa et al.,
2000). This model considers a ﬁnite cluster including defects embedded into the
rest of the host crystal, by assuming that the electronic structure in the external
5.1 A consistent approach for a modelling of defective solids
77
region remains the same as in the perfect crystal. The assumption of locality of
the perturbation is exploited diﬀerently by diﬀerent embedding techniques, starting
from the pioneering studies of Baraﬀ and Schluter for semiconductors with later extension to ionic systems (Pisani, Dovesi, Roetti, Causa et al., 2000). The embedding
approach is, in principle, more adequate than the SCM, but it is computationally
more demanding and also faces convergence problems of the self-consistent procedure (for more details and illustrations see Ref. Pisani, Birkenheuer, Cora, Nada
et al. (1996); Sulimov, Casassa, Pisani, Garapon et al. (2000); Baranek, Pinarello,
Pisani and Dovesi (2000)). In this work, the attention is focused on the periodic
defect models.
In defect calculations there are two criteria to be met: the model used for solving the quantum mechanical problem has to describe suﬃciently well both (i) the
extended crystalline states and (ii) the localized states of the isolated defect. The
CCM can be deﬁned from two points of view: it can be regarded either as the application of the Born-Karman cyclic boundary conditions directly to the large unit
cell (supercell), or as the band structure calculation on SCM with (a) applying the
k = 0 approximation and (b) neglecting interactions beyond the Wigner-Seitz cell
corresponding to the supercell. The SCM have no restrictions such as (a) and (b),
and thus CCM could be considered as a special approximation to the SCM.
From the viewpoint of criterion (i), approximation (a) is suﬃcient, provided
the primitive k vectors represented by the k = 0 point of the narrowed BZ of the
supercell form a special k-point set of suﬃcient quality. As for the criterion (ii),
the approximation (b) decouples eﬀectively the interactions between periodically repeated atoms of the supercell calculation, provided the change in the charge density,
caused by the defect, is negligible at the boundary of the corresponding Wigner-Seitz
cell. The problem is, however, that there is no direct way of checking how well these
two conditions are satisﬁed, except, of cause, a series of calculations for a trivial,
direct increase of the SCM size, which is very time consuming.
In practical self-consistent HF or DFT band structure calculations with the primitive unit cell in direct space the convergence of the bulk electronic properties (total
energy per unit cell, band gap, and one-electron energies of band edges, the density of states, and electronic charge distribution) can be obtained by increasing the
number of the used k points in the primitive BZ. The speed of this convergence
and the ﬁnal number of necessary k points depend on the particular system under
5.1 A consistent approach for a modelling of defective solids
78
consideration, basis set used, etc. When performing the BZ summation, the theory
of so-called special k points in the BZ is widely used (Chadi and Cohen, 1973).
The one-to-one correspondence was demonstrated (Evarestov and Smirnov, 1983;
Moreno and Soler, 1992; Evarestov and Smirnov, 1997b) between any ﬁxed k mesh
and the supercell in a real space, deﬁned by its translation vectors:
Aj =
3
lij ai ,
(5.1)
i=1
with the translation vectors of the primitive cell ai , and its volume Va = a1 [a2 × a3 ],
det l = L, i, j = 1, 2, 3.
Here L > 1 means a number of primitive unit cells in the corresponding supercell.
Introducing a supercell, deﬁned by Eq. (5.1), one receives from the equation below
the corresponding mesh {kt } of the k points in the BZ:
exp(−ikt · Aj ) = 1, j = 1, 2, 3 t = 1, 2, . . . , L.
(5.2)
The absolute value RM of the smallest Aj in Eq. (5.1) deﬁnes the accuracy of
the special points set chosen and might be called cutoﬀ length for any k mesh. Each
RM may be characterized by some number of spheres M of the lattice translation
vectors ordered in such a way that the sphere radii are not decreasing (Moreno and
Soler, 1992; Evarestov, Lovchikov and Typitsyn, 1983).
It is possible to choose the matrix l in Eq. (5.1) both diagonal and nondiagonal but maintaining the point symmetry of the crystalline lattice (Evarestov and
Smirnov, 1983). By increasing L, one can ensure the increase of k-mesh accuracy
and thus the accuracy of the corresponding supercell modelling of the perfect crystal.
Using Wannier functions in one-electron density matrix (DM) deﬁnition, it was
shown (Bredow, Evarestov and Jug, 2000; Evarestov and Tupitsyn, 2002; Evarestov
and Smirnov, 1997a) that the convergence of the self-consistent results with an increase of the k-mesh accuracy takes place when the diagonal DM elements (used
in DFT calculations) decay to zero at the cutoﬀ length RM . In the HF method
(due to its nonlocal exchange) the calculated oﬀ-diagonal DM elements (between
the reference primitive cell and that centered at the lattice site on the sphere of the
radius RM ) must decay to zero.
5.1 A consistent approach for a modelling of defective solids
79
Thus, one can say that an increase of k-mesh accuracy in self-consistent band
structure calculations with primitive cell means in fact that the perfect crystal is
modelled by a sequence of supercells of increasing size. The convergence of the results (size of the converged supercell) depends on the system under consideration
(for small or zero-band gap crystals convergence is very slow, but for ionic crystals
already relatively moderate k meshes are suﬃcient, as it is shown below for the case
of the wide-gap STO crystal).
Use of SCM means, in fact, consideration of a “new crystal” with artiﬁcially introduced point defect periodicity. The point defect period is deﬁned by the supercell
choice, the space group of a defective crystal in SCM is deﬁned by the local point
symmetry of a defect and the chosen lattice of supercells (Bredow, Evarestov and
Jug, 2000; Evarestov and Smirnov, 1999; Bredow, Geudtner and Jug, 2001). The
calculation is made in the same way as for a perfect crystal using the k sampling of
the BZ [a new, narrowed BZ is L-times smaller than the original (primitive) one and
may diﬀer, when the type of lattice is changed by the transformation, Eq. (5.1)]. In
practical calculations those k sets are used which allow to minimize the defect-defect
interaction (Makov, Shah and Payne, 1996) for a ﬁxed supercell size and shape.
Use of the k meshes in SCM allows one to estimate for each supercell chosen
the role of defect-defect interaction through the width of the defect energy bands:
the narrower these bands are, the closer the results obtained are to the single defect limit. When the convergence is reached, SCM gives the same results as CCM
for the same l-matrix choice in Eq. (5.1). Unfortunately, SCM faces the following
diﬃculties: the lattice relaxation around the defect is periodically repeated which
aﬀects the total energy per cell, for charged point defects this artiﬁcial periodicity
requires use of some charge compensation. These diﬃculties are absent in CCM.
The economic approach to a single point defect, as suggested in this study, consists of three stages. At stage 1 the band structure calculation of a perfect crystal
is performed, in order to ﬁx the shape and size of the supercell which reasonably
models the host crystal, i.e., when the above-described condition (i) is met. These
calculations are made using a primitive unit cell and k sampling in the usual (primitive) BZ. Due to the abovementioned one-to-one correspondence between k-point
sampling and the supercell size in real space, it is possible to ﬁnd such a k mesh
which ensures a compromise between its size and a reasonable reproduction of the
total and one-electron energies, as well as the electron density distribution in the
5.1 A consistent approach for a modelling of defective solids
80
host crystal. At this stage the k-point sets satisfying Eq. (5.2) are used.
At stage 2 the calculations are made for a defective crystal using SCM, in order
to check the above-described criterion (ii). It is reasonable to begin from the smallest supercell, chosen at stage 1, i.e., corresponding to the converged results of the
band calculations. In the particular case of the STO crystal a supercell of 80 atoms
may be used for a perfect crystal in the DFT-B3PW calculations. When estimating
at the second stage the defect-defect interaction from the calculated defect band
width, one makes a decision about the need of a further increase of the supercell.
As it will be shown below, the iron band width in the DFT-B3PW calculations still
changes when the supercell is increased from 80 to 160 atoms. It means that the
local states induced by the point defect are suﬃciently well localized only in the
larger, 160-atom super-cell. That is, at stage 2 the comparison of supercell results
for diﬀerent k meshes allows one to decide if it is necessary to further increase a
supercell, in order to surpass artiﬁcial defect-defect interaction. When energies at
k = 0 and nonzero k supercell calculations turn out to be close, this means that the
corresponding cyclic cluster is chosen for the isolated defect study.
At the most time-consuming stage 3 the CCM is used (i.e., performing band
structure calculations for the chosen super-cell only at k = 0) for the relaxation
of the crystalline lattice around the point defect and calculation of other defective
crystal properties. In particular, diﬀerent charge states of the point defect could
be also considered without diﬃculties, since in CCM the charge is not periodically
repeated over the lattice. The results of CCM calculations of SrTiO3 :Fe with lattice
relaxation are discussed below. This approach guarantees a correct study of the
convergence to the limit of a single defect.
The following serves to illustrate what was said above for the simple cubic (sc)
lattice of perovskite-type ABO3 structure. The 23 k-point Monkhorst-Pack mesh
(Monkhorst and Pack, 1976) consists of eight points in the BZ and corresponds to
the transformation (Eq. 5.1) with the following diagonal matrix:


2 0 0
l =  0 2 0 .
(5.3)
0 0 2
In the irreducible part of the BZ this k mesh consists of four points Γ(000),
M ( 12 21 0), X( 12 00) (in units of the reciprocal lattice basic translations, M
and X points form three-branch stars in the whole BZ). The corresponding supercell
R( 12 21 21 ),
5.1 A consistent approach for a modelling of defective solids
81
in the real space consists of eight primitive unit cells, for STO perovskite this results
in 5 × 8 = 40 atoms. Next k meshes based on the diagonal transformation matrix,
(Eq. 5.1) correspond to 4 × 4 × 4 = 64 and 6 × 6 × 6 = 216 k points in the BZ, with
the relevant supercells of 320 and 1080 atoms, respectively.
However, the transformation, (Eq.
matrices:

1 1
 1 0
0 1
and
5.1) could also be done for nondiagonal l

0
1 , L = 2
1
(5.4)


1
1 −1
 1 −1 1  , L = 4
−1 1
1
(5.5)
which result in the face-centered cubic (fcc) and body-centered cubic (bcc) lattices,
respectively. The corresponding k sets are the Γ, R and Γ, 3M , respectively. Further 2 × 2 × 2 increase of the unit cells for these two lattices gives the k meshes
corresponding to the super-cells of L = 16 and L = 32 primitive unit cells (80 and
160 atoms, respectively.) At last, the k mesh with L = 108 (3 × 3 × 3 extension of
the bcc lattice with L = 4) corresponds to the supercells of 540 atoms.
Thus, at the ﬁrst stage the band calculation of a perfect STO crystal is made for
the k meshes obtained for transformations (Eq. 5.1) and (Eq. 5.2), with L = 8, 16,
32, 64, 108 and the convergence of the results for host crystal is investigated. The
relevant sets of the k meshes in the BZ are as follows:
L8 (Γ, R, 3M, 3X), where Γ(000), R( 12
1 1
),
2 2
L16 (Γ, R, 3M, 3X, 8Λ), i.e., as L8 and Λ( 14
M ( 12 21 0), X( 12 00),
1 1
),
4 4
L32 (Γ, R, 3M, 3X, 12Σ, 12S), i.e., as L8 and Σ( 14 41 0), S( 14
1 1
),
4 2
L64 (Γ, R, 3M, 3X, 12Σ, 12S, 8Λ, 6∆, 6T, 12Z), i.e., as L32 and ∆( 14 00), T ( 14
Z( 14 21 0), Λ( 14 41 41 ),
L108(Γ, 3M, 6∆, 6T, 12Σ, 12Σ , 8Λ, 12Z, 24B, 24C), where ∆( 13 00), T ( 13
Σ ( 16 61 0), Λ( 13 31 31 ), Z( 16 21 0), B( 16 21 31 ), and ( 16 61 31 ).
1 1
),
2 2
1 1
),
2 2
Σ( 13 31 0),
The transformation matrices deﬁned by Eq. (5.1) for these k-point sets are given
5.1 A consistent approach for a modelling of defective solids
82
Table 5.1: Convergence of results for pure STO (a0 = 3.904 Å) obtained for DFTB3PW band calculations corresponding to cyclic clusters of an increasing size. εv
is the upper level of valence band and εc is the bottom of conduction band. All
energies in eV, total energies are presented with respect to the reference point of
314 a.u. = 8544.59 eV. q are the Mulliken eﬀective atomic charges (in e). L, NA ,
are the primitive unit cell extension, number of atoms in the cyclic cluster, whereas
RM and M are deﬁned by Eq. (5.1) and Eq. (5.6), respectively.
L, NA
M
RM , Å
Etot , eV
εv , eV
εc , eV
q(Ti)
q(O)
q(Sr)
8, 40 (sc)
A
4
7.81
-19.431
-3.317
-0.088
2.35
-1.41
1.88
16, 80 (fcc)
B
7
11.04
-20.031
-3.200
0.388
2.36
-1.41
1.88
32, 160 (bcc)
C
11
13.53
-20.459
-3.241
0.417
2.37
-1.42
1.88
64, 320 (sc)
D
14
15.62
-20.461
-3.241
0.418
2.37
-1.42
1.88
108, 540 (bcc)
E
24
20.29
-20.461
-3.241
0.418
2.37
-1.42
1.88
Transformation matrices, Eq. (5.1)






2 0 0
2 2 0
2
2 −2





A=
0 2 0 , B= 2 0 2 , C=
2 −2 2  ,
0 0 2
0 2 2
−2 2
2




4 0 0
3
3 −3
D =  0 4 0  , E =  3 −3 3 
0 0 4
−3 3
3
5.2 Results for perfect STO and supercell convergence
83
in Table 5.1. These special Kq points sets satisfy the well known Chadi-Cohen
condition (Chadi and Cohen, 1973):
Kq
wq
exp(iKq · Rj ) = 0, m = 0, 1, 2, 3, . . . ,
(5.6)
|Rj |=dm
where the second lattice sum is over lattice vectors of the same length equal with the
mth neighbor distance dm , the ﬁrst sum is over a set of these special Kq points, and
wq are weighting factors equal to the number of branches in their stars. The larger
the number m, the better is the electronic density approximation for the perfect
crystal. The numbers M (m = 0, 1, 2, . . . , M − 1), deﬁning according to Eq. (5.6)
the accuracy of the corresponding Kq sets, are given in Table 5.1.
5.2
Results for perfect STO and supercell convergence
Table 5.1 demonstrates the eﬀect of the cyclic cluster increase for DFT-B3PW calculations. The main calculated properties are: the total energy (per primitive unit
cell), one-electron band edge energies of the valence band top and conduction band
bottom εv and εc and Mulliken eﬀective atomic charges q. As is well seen, the results converge, as the supercell size increases. The convergence of local properties
of the electronic structure (atomic charges) is faster than that for the total and
one-electron energies. Based on results of Table 5.1, the conclusion can be drawn
that in the DFT-B3PW calculations of the perfect crystal, the electronic structure
is reasonably well reproduced by the supercell of 80 atoms (L = 16). This is conﬁrmed by the band structure analysis. The results of the standard band structure
calculations for the STO primitive unit cell with Pack-Monkhorst k set 8 × 8 × 8
and the cyclic cluster of 80 atoms (k = 0) are very similar. It appears that the most
important features of the electronic structure of a perfect crystal (valence and conduction band widths, local properties of electronic structure) are well reproduced at
the Γ point of the cyclic cluster. The corresponding one-electron energies practically
do not change along all symmetry directions in the narrowed BZ. Analysis of the
diﬀerence electron density plots, calculated for the primitive unit cell with the k set
8 × 8 × 8, and for the 80 atom cyclic cluster conﬁrms that the latter reproduces well
5.3 Results for a single Fe impurity
84
the electron density distribution in a perfect crystal. Lastly, the total and projected
density of states for a perfect crystal shows that the upper valence band consists
of O 2p atomic orbitals with admixture of Ti 4d orbitals, whereas the Sr states
contribute mainly to the energies close to the conduction band bottom. Thus, the
calculations for Fe impurity in STO have been performed using 80 and 160 atom
cyclic clusters (L = 16 and 32).
5.3
Results for a single Fe impurity
a
b
z
y
2
E / eV
b1g
1
x
eg
eg
t2g
Fe
4+
2-
O
2+
b
a1g 2g
0
Valence band
Sr
Figure 5.1: (a) Schematic view of the Fe impurity in STO with asymmetric eg
relaxation of six nearest O atoms, (b) The relevant energy levels before and after
relaxation.
Figure 5.1(a) shows schematically the iron atom substituting for a host Ti atom
and surrounded by six nearest neighbor O ions in face-centered positions. Table 5.1
has shown that an increase of the cyclic cluster from L16 to L32 does not change
the calculated top of the valence band. However, the calculated width of the defect
impurity band EW found using our standard Monkhorst-Pack set 8 × 8 × 8 for three
5.3 Results for a single Fe impurity
85
Table 5.2: The width of the Fe impurity band EW (in eV) calculated for the relevant
supercells.
Supercell
No. atoms
Fe-Fe distance, Å
EW , eV
L8
40
7.81
1.53
L16
80
11.04
0.25
L32
160
13.53
0.13
diﬀerent supercells (Table 5.2), demonstrates clearly a considerable dispersion of
defect energies across the BZ. Indeed, the EW decreases rapidly, from 1.53 eV (L8)
down to 0.25 eV (L16), and further down to 0.13 eV (L32), when the Fe-Fe distance
increases only by a factor of about 2, from 7.81 to 13.53 Å, since the overlap of
the impurity atomic functions decreases exponentially. This is why only a L32 (160
atom) cyclic cluster is suitable for a careful modelling of the single Fe impurity
and lattice relaxation around it. This is in contrast with many previous supercell
calculations of defects where L8 supercells were often used without any convergence
analysis.
Mulliken eﬀective charges calculated for ions at diﬀerent positions in supercells
modelling pure and Fe-doped STO are summarized in Table 5.3. Its ﬁrst two lines
demonstrate that the standard band structure calculation and the L64 cyclic cluster give essentially identical charges. The more so, charges of the same ions in 320
atom supercell are the same, irrespective of the ion position inside the cyclic cluster.
Next, in the defective crystal calculations, say, for the L32 cyclic cluster, the eﬀective charges of atoms close to its boundary are the same as in the perfect crystal.
This conﬁrms that the chosen cyclic cluster is large enough.
The L32-DFT-B3PW supercell calculations for the zero-spin and high-spin (S =
2) states show that the latter is much lower in energy (by 5.4 eV) (after lattice
relaxation). In the perovskite crystalline ﬁeld a ﬁve-fold degenerate Fe 3d state
splits into eg and t2g states [Fig. 5.1(b)] separated by 0.75 eV (for an undistorted
lattice). In the high spin state with S = 2 the upper eg level is occupied by one α
(up-spin) electron and three other α electrons occupy t2g states. As is well known
(Donnerberg, 1994; Postnikov, Poteryaev and Borstel, 1998) in this case an Eg ⊗ eg
Jahn-Teller eﬀect takes place. This means that an orbital degeneracy is lifted by an
asymmetrical eg displacement of six O ions, as shown in Fig. 5.1(a): four equatorial
5.3 Results for a single Fe impurity
86
Table 5.3: Eﬀective charges q of ions obtained in the DFT-B3PW band structure calculations with Pack-Monkhorst k set 8 × 8 × 8 and diﬀerent cyclic clusters modelling
perfect and defective STO. The lengths in the ﬁrst column are lattice constants of
the relevant supercells whereas the distances R given above for the eﬀective charges
are calculated with respect to the supercell coordinate origin, where the Fe ion is
placed.
R, Å
0.00
1.95
3.38
3.90
4.37
5.52
5.86
6.48
6.76
7.04
7.81
Latt. const., Å
q(Fe)
q(O)
q(Sr)
q(Ti)
q(O)
q(Ti)
q(O)
q(Sr)
q(Ti)
q(O)
q(Ti)
-1.407
1.871
2.350
-1.417
1.877
2.373
-1.417
2.373
-1.417
1.877
2.373
-1.417
2.373
-1.417
2.373
3.90
band structure,
perfect crystal
15.62 (L64),
perfect crystal
7.81 (L8)
2.899
-1.512
1.873
2.365
11.04 (L16)
2.893
-1.512
1.881
2.369
-1.420
2.380
13.53 (L32)
2.890
-1.512
1.878
2.370
-1.418
2.373
-1.417
1.877
2.373
15.62 (L64)
2.890
-1.512
1.878
2.370
-1.418
2.373
-1.417
1.877
2.373
O atoms lying in the x − y plane relax towards the impurity, whereas the two other
O ions relax outwards along the z axis. This results in two nondegenerate levels
close to the valence band top: eg level at 0.3 eV above the band, and a virtual
nondegenerate b1g level lying much higher (Table 5.4).
If one assumes that x, y, z displacements have equal magnitudes, calculations
give practically the same magnitude of the six O displacements σ = 0.04 Å, a quite
ﬂat minimum and an energy gain of 1.33 eV. However the magnitudes of the O atom
displacements along the x, y and the z axes could be diﬀerent and indeed, in the
latter case a small additional energy decrease, down to 1.42 eV has been found for
the following asymmetrical dispacements: 0.028 Å along the x, y axis and -0.052 Å
along the z axis, i.e., outward displacements of two O atoms are twice larger than
those for four equatorial O atoms. This means that there is a combination of the
Jahn-Teller and breathing modes of surrounding O atom displacements.
The eﬀective charges q of atoms collected in Table 5.4 demonstrate considerable
covalency eﬀects, well known for ABO3 perovskites. In particular, in pure STO
the eﬀective charges are q(Ti) = 2.373 e, q(Sr)= 1.877 e, and q(O) = -1.417 e.
5.3 Results for a single Fe impurity
87
Table 5.4: Positions of one-electron Fe levels (in eV) with respect to the VB top
calculated by means of DFT-B3PW method for L = 16 and L = 32 cyclic cluster
with and without lattice relaxation.
Before relaxation
After relaxation
Cluster
t2g
eg
a1g
b2g
eg
b1g
80 atoms
0.2
0.85
0.007
0.02
0.3
2.0
160 atoms
0.15
0.9
0.005
0.01
0.3
1.7
Cyclic
Table 5.5: The eﬀective Mulliken charges of atoms q and bond populations P (in
milli e) for the L32 cyclic cluster with unrelaxed and relaxed lattices.
Pure STO
q(Ti)
q(Ox,y )
q(Oz )
P (Ti-Ox )
P (Ti-Oz )
2.373
-1.417
-1.417
86
86
STO:Fe
q(Fe)
q(Ox,y )
q(Oz )
P (Fe-Ox )
P (Fe-Oz )
unrelaxed
2.890
-1.418
-1.418
92
92
relaxed
2.914
-1.394
-1.488
118
74
The Ti-O bond population in a pure crystal is 86 milli e. When the two O atoms
are displaced outwards from the Fe impurity along the z axis and thus approach
the nearest Ti atoms, the Ti-O bond population increases up to 118 milli e. The
combination of a large lattice relaxation energy and relatively small O displacements
is not surprising in the light of a considerable covalent bonding between the unpaired
iron electrons occupying Fe 3d orbitals and the 2p orbitals of four equatorial oxygen
ions: the Fe-Ox,y bond populations (Table 5.5) increase upon mutual approach of
these atoms from 92 milli e to 118 milli e. Analysis of the total electron density
and spin density distribution (Fig. 5.2) shows that in DFT-B3PW calculations four
unpaired electrons are well localized on the Fe ion.
To summarize this Chapter, it can be noted that a regular method to check on the
convergence of periodic defect calculations to the limit of the single defect has been
suggested. This method could be very eﬃcient for many impurities in insulators
characterized by high symmetry and when calculating forces is computationally
expensive. It has been demonstrated that the size of the cyclic cluster, large enough
5.3 Results for a single Fe impurity
88
O
O
Sr
a)
Ti
Fe
Fe
Ti
O
Sr
O
b)
Ti
O
Fe
O
Fe
Ti
c)
Figure 5.2: (a) The electronic density plots for the (010) cross section of Fe and
nearest ions in STO as calculated by means of the DFT-B3PW method for the
cyclic cluster of 160 atoms. Isodensity curves are drawn from 0.8 to 0.8 e a.u.3 with
an increment of 0.0022 e a.u.3 , (b) the same as (a) for the (001) section, (c) the
same for the (110) section. Left panels are diﬀerence electron densities, right panels
spin densities.
5.3 Results for a single Fe impurity
89
for a correct reproduction of the single Fe4+ impurity, should be not smaller than
160 atoms. This is in contrast with many previous supercell calculations, where
as small as L8 supercells were used without convergence analysis. It should be
mentioned here that the correct estimation of the optical band gap provided by the
DFT-B3PW scheme, allows to reproduce the defect level positions within the gap,
even determined with respect to the valence band top. The present calculations
have demonstrated the strong covalent bonding between unpaired electrons of Fe
impurity and four nearest O ions relaxed towards an impurity. Positions of Fe
energy levels in a STO gap are very sensitive to the lattice relaxation which was
neglected in previous studies. Based on this, a considerable dependence of the optical
absorption bands of transition metals in perovskites can be predicted on the external
or local stresses (e.g., in solid solutions, like Srx Ba1−x TiO3 ). This is important for
the interpretation of experimental data. The positions of the Fe energy levels with
respect to the valence band top could be checked by means of the UPS spectroscopy
whereas the local lattice relaxation around iron and its spin state by means of the
EXAFS. This is the more important since the single Fe4+ ions so far are not detected
by ESR (only Fe4+ -V0 complexes were studied by Schirmer, Berlinger and Müller
(1975)) and their optical absorption bands at 2.1 and 2.8 eV (Wasser, Bieger and
Maier, 1990) are tentative.
Chapter 6
Two-dimensional defects in
perovskites: (001) and (110)
surfaces.
Introduction
Many of ABO3 applications, including substrates for growth of high-Tc superconductors, are closely related to the surface properties of perovskites (Noguera, 1996;
Henrick and Cox, 1994). This is true, ﬁrst of all, for catalysis when small molecule
absorption on active sites and defects, and related surface diﬀusion-controlled reactions take place. Moreover, further miniaturization and development of desired
nonvolatile computer memories of the next generation could be also achieved using
multi-layer ABO3 thin ﬁlms and related four-component structures (e.g. A=Sr, Ba,
K or Pb, and B=Ti or Nb) (Auciello, Scott and Ramesh, 1998), where the surface
properties are of high importance. In this Chapter the results of ﬁrst-principles
calculations on two possible terminations of the (001) surfaces of SrTiO3 , BaTiO3 ,
and PbTiO3 perovskites are presented. Surface structures and their electronic conﬁgurations have been calculated using ab initio Density Functional Theory with
hybrid (B3PW) exchange-correlation technique. Results are compared with previous quantum mechanical calculations and available experimental data. Surface
relaxations and electronic states near valence band gap are discussed in detail for all
6.1 The choice of a model for surface simulation
91
three perovskites. In addition, the ab initio calculations have been carried out on
TiO- and Ti-terminated SrTiO3 (110) polar surfaces (i.e. SrTiO-terminated (110)
surface with created vacancies on Sr and O sites) to be compared with MIES and
UPS experiments performed by Prof. Kempter’s group in the Technische Universität Clausthal (Gunhold, Beuermann, Gömann, Borchardt et al., 2003b). Such
comparison is discussed in detail in the last Section.
6.1
The choice of a model for surface simulation
By cutting a 3D crystal through a crystalline plane (hkl), two ideal semi-inﬁnite
crystals are generated each limited by an ideal surface (see Fig. 6.1). Each semicrystal preserves 2-D periodicity parallel to the selected face but loses all symmetry
elements which involve displacements in a perpendicular direction (conventionally,
the z-direction). The ideal surface may undergo relaxation, without loss of translational symmetry, or exhibit partial reconstruction, whereby the 2-D unit cell becomes
larger. This is a typical ﬁnite+inﬁnite problem: a subsystem consisting of a few layers close to the surface, whose properties are interesting, is connected to the rest
of the semi-crystal, an inﬁnite system whose electronic structure is known (Pisani,
1996). There are various techniques (based on the Green’s functions) which are suitable for studying similar problems and have been applied to study the semi-crystals
(Kalkstein and Soven, 1971; Gonis, Zhang, MacLaren and Crampin, 1990). In this
study, a diﬀerent technique, the slab model, is considered. A slab (also called a thin
ﬁlm or isolated slab) is created, formed by a few atomic layers, parallel to the (hkl)
surface; relaxation or partial reconstruction can be taken into account. The unit cell
of this periodic 2D structure comprises a ﬁnite number of atoms and can, therefore,
be studied using the same techniques as for the perfect crystal. At variance with the
semi-crystal, the slab may possess symmetry elements (a mirror plane, a glide plane,
a 2-axis parallel to the surface, an inversion center) which involve displacements in
the z-direction. In the multi-slab approach, an inﬁnite number of identical slabs
is considered, regularly spaced along the z-direction: a typical separation is 10 a.u.
Thus, one has a 3D crystal, whose unit cell comprises a ﬁnite number of atoms across
the slab, and a portion of the vacuum region separating two neighboring slabs. The
multi-slab model is particularly advantageous for computational schemes based on
the use of PWs and soft-core pseudo-potentials (Pisani, 1996; Northrup and Cohen,
6.1 The choice of a model for surface simulation
92
sin
gl
double
cut
Semi-infinite crystal
Slab
t
cu g
le in
tip ac
ul sp
m nd
a
ec
ut
Infinite crystal
Multi-slab
Figure 6.1: Models for simulating surfaces starting from a perfect 3D crystal.
6.1 The choice of a model for surface simulation
93
1984). Figure 6.1 represents schematically all three diﬀerent models.
O
I
I
I
Ti
II
II
II
A
III
III
III
IV
IV
IV
V
VI
VII
VIII
a)
b)
c)
Figure 6.2: Schematic illustration of the slab unit cells for ABO3 (001) surfaces: a)
AO-terminated, b) TiO2 -terminated, c) asymmetrical termination.
In the present simulations, surfaces of perovskite materials have been modelled
using a slab model, where an artiﬁcial periodicity of supercells separated by vacuum
regions (multi-slab model) is eliminated. It allows to exclude any spurious interaction
between periodically repeated slabs and assumes the proper boundary condition of
an electric ﬁeld in the slab surface region. Both (001) and (110) ABO3 surfaces have
been modelled considering the crystal as a set of crystalline planes perpendicular to
the given surface, and cutting out 2D slab of a ﬁnite thickness, periodic in x − y
plane. Figure 6.2 demonstrates AO- and TiO2 - symmetrically terminated slab unit
cell (Fig. 6.2a and Fig. 6.2b respectively), and asymmetrically AO- and TiO2 terminated ABO3 (001) slab unit cell is shown in Fig. 6.2c. The slabs containing
seven (symmetric) and eight (asymmetric) monolayers can be treated thick enough
since the convergence of the calculated slab total energy per ABO3 unit is achieved.
This energy diﬀers less than 0.0005 Hartree for 7- and 9-layer (or 8- and 10-layer for
asymmetrical termination) slabs for all three perovskites. Using the same approach,
the STO(110) Ti- TiO- and SrTiO-terminated surface has been modelled by a 9-layer
6.2 Calculations on the ABO3 (001) surfaces
94
slab. This slab (with and without vacancies) is shown in Fig. 6.3.
O
Ti
Sr
a)
b)
c)
Figure 6.3: Schematic illustration of the SrTiO-terminated SrTiO3 (110) 9-layer slab
unit cells: a) slab without vacancies (unstable, cannot exist due to inﬁnite dipole
moment perpendicular to the surface), b) TiO-terminated SrTiO3 (110) surface (unreconstructed surface, stable according to Heifets, Kotomin and Maier (2000), also
named as “unreconstructed surface”, see last section), c) Ti-terminated SrTiO3 (110)
surface (reconstructed surface). Vacancies created on Sr and O sites are shown as
green spots.
6.2
6.2.1
Calculations on the ABO3(001) surfaces
Surface structures
In present simulations the z-position of the two outermost surface layers have been
allowed to relax for each structure. By symmetry, surfaces of perfect cubic crystals
have no forces along x- and y-axes. Displacements of third layer atoms were negligible in the calculations and are not treated in present simulations. The optimization of atomic coordinates has been done through the slab total energy minimization using the small computer code implements Conjugated Gradients optimization
technique (Press, Teukolsky, Vetterling and Flannery, 1997) with numerical computation of derivatives. The calculated atomic displacements are presented in Table
6.1 and are schematically illustrated in Fig. 6.4. Comparison with surface atomic
-3.74
O
A
7
0.15
2.27
O2
Ti
...
-0.5
-0.7
-0.5
2.5
-1.6
-3.4
-0.1
-1.2
0.0
1.2
0.1
-5.7
iii
0.26
-0.26
0.77
4.61
-0.26
-1.79
0.26
-1.54
0.26
1.79
1.02
-6.66
iv
4.72
...
...
...
-1.98
-1.8
-0.72
-0.81
-0.50
2.36
-2.74
-4.14
-0.28
-0.69
2.75
1.70
-1.09
-0.72
vi
3.03
-0.35
-0.75
-0.62
1.31
-1.63
-3.89
0.26
0.53
0.48
0.92
-1.40
-2.79
v
3.52
-0.01
-0.33
-0.17
2.19
-0.94
-2.72
0.16
-0.51
0.76
1.25
1.00
-3.72
ii
0.34
-2.54
-0.37
...
...
...
1.50
1.70
-0.69
-2.09
0.38
2.51
-0.35
-3.08
1.40
1.74
-0.63
-1.99
study
vi – Tinte and Stachiotti (2000)
vii – Cohen (1997)
viii – Meyer, Padilla and Vanderbilt (1999)
iii – Padilla and Vanderbilt (1998)
iv – Cheng, Kunc and Lee (2000)
4.79
2.68
...
...
...
-3.26
-4.28
vii
ii – Heifets, Kotomin and Maier (2000)
v – Padilla and Vanderbilt (1997)
8
-0.64
...
...
...
...
...
...
...
0.61
O2
0.39
1.55
O
Ti
-5.22
-0.29
A
-0.6
-0.21
3.46
-1.73
-2.96
O2
0.01
2.18
-0.93
-2.19
Ti
3.55
0.57
O
-0.13
O2
A
-2.25
Ti
0.7
0.87
O
0.85
1.57
1.15
-7.10
-1.42
0.77
O2
1.25
0.61
-4.29
A
...
2
1
3
2
1
3
1.75
Ti
-4.84
0.84
O
ii
This
i
This
study
BTO
STO
A
At.
i – Heifets, Eglitis, Kotomin, Maier et al. (2002)
asymmetrical
TiO2
1
AO
2
N
Termination
2.84
-0.74
-5.44
-1.22
...
...
...
1.95
3.01
-0.24
-4.02
1.28
5.32
0.31
-2.81
2.30
3.07
-0.31
-3.82
Study
This
PTO
-0.27
-0.92
0.43
4.53
-0.34
-3.40
-0.20
-1.37
1.21
2.39
-0.46
-4.36
viii
Table 6.1: Atomic relaxation relative to ideal atomic positions of cubic ABO3 (001) surfaces (in percent of lattice constant).
A means Sr, Ba, or Pb.
6.2 Calculations on the ABO3 (001) surfaces
95
6.2 Calculations on the ABO3 (001) surfaces
96
displacements obtained by other QM calculations is also presented in Table 6.1. In
agreement with results obtained from previous ab initio studies the metal and the
oxygen atoms, generally, move in the same direction. The relaxation of metal atoms
is much larger than oxygen relaxation, which in turn leads to considerable rumpling
of relaxed layers. The outward relaxation of all atoms from second layer relative to
I (surface) layer
O
Ti
A
II layer
III layer
AO-term.
TiO2-term.
I (surface) layer
II layer
z
x
III layer
a
b
c
Figure 6.4: Schematic illustration of two outermost surface layers relaxation with
respect to perfect 3d crystal positions: a) STO, b) BTO, c) PTO. View from [010]
direction. Arrows show the directions of atom displacements. Upper panels - AO
termination, lower panels - TiO2 termination.
their original positions is found for surfaces of all three perovskites for both possible
terminations. Atoms of ﬁrst surface layer, mostly, relax inward toward the bulk. The
exceptions are the top oxygens of STO SrO-terminated and PTO TiO2 -terminated
surfaces. Moreover, the displacement of oxygen of TiO2 -terminated PTO is in disagreement with result obtained by Meyer, Padilla and Vanderbilt (1999) with PW
pseudopotential method, but the magnitudes of these displacements are relative
small, 0.31 and -0.34 percents of lattice constant, respectively. The displacements
obtained for asymmetrically terminated slabs are practically the same as for symmetrically terminated, which is normal when thick enough slabs are considered and
convergency on calculated properties is achieved.
6.2 Calculations on the ABO3 (001) surfaces
97
Table 6.2: Surface rumpling s and relative displacements of the three near-surface
planes for AO- and TiO2 -terminated surfaces ∆dij (in percent of lattice constant).
Results for asymmetrical slabs are given in brackets.
AO-terminated
STO
TiO2 -terminated
s
∆d12
∆d23
s
∆d12
∆d23
This study
5.66
-6.58
1.75
2.12
-5.79
3.55
(5.61)
(-6.79)
(1.55)
(2.43)
(-6.02)
(3.74)
ab initio (i)
4.9
-5.5
1.3
-4.4
ab initio (ii)
5.8
-6.9
2.4
1.8
-5.9
3.2
ab initio (iii)
7.7
-8.6
3.3
1.5
-6.4
4.9
Shell model (iv)
8.2
-8.6
3.0
1.2
-6.4
4.0
LEED expt. (v)
4.1±2
-5±1
2±1
2.1±2
1±1
-1±1
RHEED expt. (vi)
4.1
2.6
1.3
2.6
1.8
1.3
1.5±0.2
0.5±0.2
SXRD expt. (viii)
1.3±12.1
-0.3±3.6
-6.7±2.8
12.8±8.5
0.3±1
This study
1.37
-3.74
1.74
2.73
-5.59
2.51
(1.40)
(-3.79)
(1.70)
(2.69)
(-5.57)
2.54
1.39
-3.71
0.39
2.26
-5.2
2.06
MEIS expt. (vii)
BTO
ab initio (ix)
PTO
Shell model (x)
0.37
-2.42
2.39
1.4
-6.5
3.17
Shell model (xi)
4.72
-4.97
1.76
1.78
-4.91
2.52
This study
3.51
-6.89
3.07
3.12
-8.13
5.32
(3.78)
(-7.03)
(3.01)
(3.58)
(-8.28)
(5.44)
3.9
-6.75
3.76
3.06
-7.93
5.45
ab initio (xii)
i – Heifets, Eglitis, Kotomin, Maier et al. (2002)
ii – Padilla and Vanderbilt (1998)
iii – Cheng, Kunc and Lee (2000)
iv – Heifets, Kotomin and Maier (2000)
v – Bickel, Schmidt, Heinz and Müller (1989)
vi – Hikita, Hanada, Kudo and Kawai (1993)
vii – Ikeda, Nishimura, Morishita and Kido (1999)
viii – Charlton, Brennan, Muryn, McGrath et al. (2000)
ix – Padilla and Vanderbilt (1997)
x – Tinte and Stachiotti (2000)
xi – Heifets, Kotomin and Maier (2000)
xii – Meyer, Padilla and Vanderbilt (1999)
6.2 Calculations on the ABO3 (001) surfaces
98
In order to compare the calculated surface structures with available results obtained experimentally, the amplitudes of surface rumpling s (the relative displacement of oxygen with respect to the metal atom in the surface layer) and the changes
in interlayer distances ∆dij (i and j are the numbers of layers) are presented in
Table 6.2. The calculations of the interlayer distances are based on the positions
of relaxed metal ions (see Fig. 2.6), which are known to be much stronger electron
scatterers than oxygen ions (Bickel, Schmidt, Heinz and Müller, 1989). The agreement is quite good for all theoretical methods, which give the same sign for both
the rumpling and change of interlayer distances. The amplitude of surface rumpling
of SrO-terminated STO is predicted much larger in comparison to that for TiO2 terminated STO surface, when the rumpling of BTO TiO2 -terminated surface is
two times larger than the same for BaO-terminated surface and PTO demonstrates
practically equal rumpling for both terminations. From the Table 6.2 one can see
all surfaces display the reduction of interlayer distance d12 and expansion of d23 .
The calculated surface rumpling amplitudes agree quite well with LEED, RHEED
and MEIS experiments (Bickel, Schmidt, Heinz and Müller, 1989; Hikita, Hanada,
Kudo and Kawai, 1993; Ikeda, Nishimura, Morishita and Kido, 1999) which are
available for STO surfaces only. Nevertheless, the calculated changes in interlayer
distances are mostly in disagreement with experimental data. As an example, the
experiments show the expansion of d12 for TiO2 terminated STO, but all calculations on the contrary demonstrate the reduction of this magnitude. Moreover, from
Table 6.2 is clearly seen that experiments contradict each other in the sign of ∆d12
and ∆d23 for SrO-terminated surface as well as for ∆d23 of TiO2 -terminated STO.
Another problem is that LEED, RHEED and MEIS experiments demonstrate that
the topmost oxygen always move outward from the surfaces whereas all calculations predict for the TiO2 -terminated STO surface that oxygen goes inwards. Even
more important is the contradiction between three above mentioned experiments
and most recent SXRD data (Charlton, Brennan, Muryn, McGrath et al., 2000)
where oxygen atoms are predicted to move inwards for both surface terminations,
reaching rumpling amplitude up to 12.8% for the TiO2 terminated surface. Up to
now the reason for such discrepancies between the diﬀerent experimental data is
not clear and was discussed in Section 2.1.3. Thus, the disagreement between data
obtained theoretically and in experiment can not be taken seriously till the conﬂict
between experimental results will be resolved.
6.2 Calculations on the ABO3 (001) surfaces
99
Table 6.3: Calculated surface energies (in eV per surface cell). Results for previous ab initio calculations (Cheng, Kunc and Lee, 2000; Tinte and Stachiotti, 2001;
Meyer, Padilla and Vanderbilt, 1999) are averaged over AO and TiO2 terminated
surfaces.
STO
This study
BTO
TiO2
asymm.
BaO
TiO2
asymm.
PbO
TiO2
asymm.
1.15
1.23
1.19
1.19
1.07
1.13
0.83
0.74
0.85
1.45
1.40
i
1.18
1.22
ii
1.32
1.36
iii
1.21
1.19
iv
v
PTO
SrO
1.17
1.26
1.24
0.97
i – Heifets, Eglitis, Kotomin, Maier et al. (2002)
ii – Heifets, Kotomin and Maier (2000)
iii – Cheng, Kunc and Lee (2000)
iv – Tinte and Stachiotti (2001)
v – Meyer, Padilla and Vanderbilt (1999)
In order to calculate the surface energy, one can start with the cleavage energy
for unrelaxed AO- and TiO2 -terminated surfaces. For example, the two 7-layer
AO- and TiO2 -terminated slabs represent together 7 bulk unit cells (one 8-layer
symmetrical slab represents 4 bulk unit cells). Surfaces with both terminations
arise simultaneously under cleavage of the crystal and the relevant cleavage energy
is distributed equally between created surfaces. Therefore, one can assume that the
cleavage energy is the same for both terminations:
Esunrel =
1 unrel
unrel
(Eslab, AO + Eslab,
T iO2 − 7Ebulk ),
4
(6.1)
unrel
unrel
where Eslab,
AO and Eslab, T iO2 are unrelaxed AO- and TiO2 -terminated slab energies,
Ebulk the energy per bulk unit cell, and a factor of four comes from the fact that four
surfaces upon cleavage procedure are created. Next, one can calculate the (negative)
relaxation energies for each of AO and TiO2 terminations, when both sides of slabs
relax:
1 rel
unrel
(6.2)
Exrel = (Eslab,
x − Eslab, x ),
2
rel
where Eslab,
x is the slab energy after relaxation, x = AO or TiO2 . Lastly, the surface
6.2 Calculations on the ABO3 (001) surfaces
100
energy sought for is just a sum of the cleavage and relaxation energies:
Es, x = Exrel + Esunrel .
(6.3)
The calculated surface energies of the relaxed surfaces are presented in Table 6.3.
In this Table, the agreement with surface energies obtained in previous calculations is clearly demonstrated. The energies calculated for AO- and TiO2 -terminated
surfaces demonstrate only small diﬀerences, that means the both terminations are
quite stable and energetically favorable. Nevertheless, the energy computed for
TiO2 -terminated STO surface is a little bit larger than for SrO-termination and the
opposite situation is observed for BTO and PTO crystals, where TiO2 -terminated
surface is a little bit more energetically favorable. The predicted stability of (001)
surfaces of AII BIV O3 perovskite materials (which are all three perovskite crystals
under consideration) is in good agreement with Tasker’s theoretical classiﬁcation
(Tasker, 1979) who classiﬁed ABO3 (001) as “ type I ” stable surfaces without inﬁnite dipole moment perpendicular to surface (but only when the formal ionic plane
charges are taken into account; the weak-polarity which exists in AII BIV O3 (001)
surfaces due to the partly covalent nature of perovskites is discussed in the next
section).
6.2.2
Electronic charge redistribution
It is preferable to start a discussion of the electronic structure of surfaces with an
analysis of charge redistribution in surface planes. The eﬀective atomic charges (calculated using standard Mulliken population analysis) and dipole atomic moments
are presented for all AO-, TiO2 - and asymmetrical-terminated surfaces in Tables
6.4, 6.5 and 6.6, respectively. The diﬀerences in charge densities at (001) planes in
ABO3 bulk crystals and on (001) surfaces are analyzed in Table 6.7.
In case of AO-terminated surfaces STO and BTO show a similar behavior. The
charges of cations of top layer become smaller with respect to the bulk charges
when the oxygens acquires additional electron charge and become more negative.
The titanium ions in second layer demonstrate the slight increasing of their charges
when oxygens again become more negative due to addition electron charge transfer.
Changes in atomic charges in deeper layers become very small and practically equal
6.2 Calculations on the ABO3 (001) surfaces
101
Table 6.4: AO termination. Charges and dipole moments. Numbers in brackets are
deviations from bulk values. Bulk charges in e; STO: Sr = 1.871, Ti = 2.35, O =
-1.407, BTO: Ba = 1.795, Ti = 2.364, O = -1.386, PTO: Pt = 1.343, Ti = 2.335, O
= -1.226 (see Table 4.3).
N
Ion
STO
Q, e
BTO
d,
Q, e
e a.u.
1
A
1.845
-0.2202
(-0.026)
O
-1.524
-0.0336
Ti
2.363
O2
-1.449
0.0106
A
1.875
-0.0191
-1.429
Ti
2.336
0.0008
-1.411
2.377
-1.417
1.801
-1.415
0
2.386
0.0070
0
-1.392
-0.4804
-1.131
0.0248
2.333
-0.0211
(-0.002)
0.0182
-1.257
-0.0062
(-0.031)
-0.0433
1.354
-0.0484
(+0.011)
-0.0084
-1.258
-0.0155
(-0.032)
0
(+0.004)
(-0.004)
1.277
(+0.095)
(-0.029)
(-0.014)
O2
-0.0532
(+0.006)
(-0.022)
4
-1.473
d,
e a.u.
(-0.066)
(-0.031)
-0.0232
(+0.004)
O
-0.4634
(+0.013)
(-0.042)
3
1.751
(-0.087)
(+0.013)
Q, e
e a.u.
(-0.044)
(-0.117)
2
PTO
d,
2.342
0
(+0.007)
0
(-0.006)
-1.232
0
(-0.006)
Table 6.5: TiO2 termination. The same as for Table 6.4
N
Ion
STO
Q, e
BTO
d,
Q, e
e a.u.
1
Ti
2.314
0.0801
(-0.036)
O2
-1.324
0.0418
A
1.851
O
-1.361
0.0423
Ti
2.386
-0.0436
-1.389
A
1.871
-0.0167
-1.399
(+0.008)
1.765
-1.343
2.362
-1.369
0
1.794
0.0949
-1.381
(+0.005)
0.0962
-1.182
-0.0307
1.270
0.0990
(-0.073)
-0.0303
-1.166
-0.0060
(+0.060)
0.0079
2.332
0.0183
(-0.003)
-0.0108
-1.205
-0.0169
(+0.021)
0
(-0.001)
0
2.279
(+0.044)
(+0.017)
(0.000)
O
0.0207
(-0.002)
(+0.018)
4
-1.278
d,
e a.u.
(-0.056)
(+0.043)
0.0139
(+0.036)
O2
0.0816
(-0.030)
(+0.046)
3
2.304
(+0.108)
(-0.020)
Q, e
e a.u.
(-0.060)
(+0.083)
2
PTO
d,
1.337
0
(-0.006)
0
-1.219
(+0.007)
0
6.2 Calculations on the ABO3 (001) surfaces
102
Table 6.6: Asymmetrical termination. The same as for Table 6.4
N
1
Ion
A
O
2
Ti
O2
3
A
O
4
Ti
O2
5
A
O
6
Ti
O2
7
A
O
8
Ti
O2
STO
Q, e
1.845
(-0.026)
-1.520
(-0.113)
2.361
(+0.011)
-1.450
(-0.043)
1.874
(+0.003)
-1.425
(-0.018)
2.352
(+0.002)
-1.408
(-0.001)
1.870
(-0.001)
-1.404
(+0.003)
2.348
(-0.002)
-1.383
(+0.024)
1.846
(-0.022)
-1.365
(+0.042)
2.291
(-0.059)
-1.297
(+0.110)
d,
e a.u.
-0.2238
-0.0428
0.0117
-0.0156
-0.0225
0.0067
0.0026
-0.0006
-0.0042
0.0102
-0.0089
0.0057
-0.0633
0.0288
-0.0748
-0.0493
BTO
Q, e
1.751
(-0.044)
-1.470
(-0.084)
2.376
(+0.012)
-1.417
(-0.031)
1.800
(+0.005)
-1.414
(-0.028)
2.366
(+0.002)
-1.389
(-0.003)
1.795
(0.000)
-1.384
(+0.002)
2.362
(-0.002)
-1.370
(+0.016)
1.765
(-0.030)
-1.343
(+0.043)
2.305
(-0.059)
-1.278
(+0.108)
d,
e a.u.
-0.4653
-0.0548
0.0054
0.0265
-0.0461
-0.0082
0.0006
-0.0024
-0.0074
0.0089
-0.0084
0.0103
-0.0950
0.0318
-0.0823
-0.0189
PTO
Q, e
1.249
(-0.094)
-1.124
(+0.102)
2.318
(-0.017)
-1.242
(-0.016)
1.345
(+0.002)
-1.252
(-0.026)
2.325
(-0.010)
-1.223
(+0.003)
1.334
(-0.009)
-1.222
(+0.004)
2.325
(-0.010)
-1.200
(+0.026)
1.260
(-0.083)
-1.149
(+0.077)
2.272
(-0.063)
-1.176
(+0.050)
d,
e a.u.
-0.3055
0.0305
-0.0350
-0.0493
0.0660
0.0062
0.0025
-0.0323
0.1315
0.0425
-0.0115
0.0042
0.0019
0.0254
-0.1508
-0.0048
6.2 Calculations on the ABO3 (001) surfaces
103
zero in the center of the slabs. The PbO-terminated surface demonstrates a diﬀerent
electron charge redistribution. The charge of Pb ion of the surface layer becomes
smaller with the largest deviation from the bulk charge. The surface oxygen of AOterminated PTO(001), unlike the STO and BTO surfaces, becomes more positive.
Ti cations in second layer of PbO-terminated surface show practically no changes
in their eﬀective atomic charges when oxygen becomes more negative as well as in
case of STO and BTO. That charge redistribution is in good agreement with the
surface relaxation discussed in the section above and caused by shortening or even
dangling of crystal bonds due to surface boundary conditions.
The charge redistributions in TiO2 -terminated surfaces of all perovskites under
consideration demonstrate a quite similar behavior. For both topmost layers all
cations demonstrate charge reduction with cations becoming more positive. The
charge reduction for surface Ti cations is a little bit stronger pronounced then for A
ions in a subsurface layer. Changes of charges for ions of asymmetrically terminated
slabs (Table 6.6) are practically the same as in symmetrically terminated AO- and
TiO2 - slab as it should be, when convergency is achieved.
The real charges of ions in partly-covalent materials usually are far from from
their formal ionic charges due to charge electron density redistribution caused by
covalency eﬀect, what is conﬁrmed by present calculations. As a result, (001) planes
in the bulk perovskites turn out to be charged with charge density per unit cell:
σB (AO) = −σB (T iO2 ), where B means “Bulk” (Table 6.7). Half of this charge
density comes from TiO2 planes to each of the two neighboring AO planes. If the
formal ionic charges are considered the calculated charge densities of (001) planes
would be equal zero and the pure “type I surfaces” (according to generally-accepted
classiﬁcation given by Tasker (1979)) would be constructed. In reality, the charge
redistribution makes the ABO3 (001) surfaces to be polar with a non-zero dipole
moment perpendicular to the surface, caused by the macroscopic electric ﬁeld in
the slab, or produces “type III polar surfaces”, which are unstable. In case of symmetrically terminated slabs this moment disappears due to symmetry of planes, but
the case of asymmetrical slabs is quite of interest. One of possibilities to stabilize
the surface is to create a defect, what is neglected in this study. The second possibility, which is realized here, is to add the compensating charge density to the
topmost layers of surfaces. The charge density changes for all perovskite surfaces
with respect to the bulk magnitudes are summarized in Table 6.7. It is clearly seen
6.2 Calculations on the ABO3 (001) surfaces
104
Table 6.7: Charge densities in the (001) crystalline planes of the bulk perovskites
(in e, per TiO2 or AO unit, data are taken from Table 4.3) and in four top planes of
the AO-, TiO2 -terminated and asymmetrical slabs. Changes of charge density with
respect to the bulk are given in brackets.
Termination
N
Bulk
AO
1
2
3
4
TiO2
1
2
3
4
Asymmetrical
1
Unit
0.464
0.409
0.117
-0.464
-0.409
-0.117
AO
TiO2
AO
TiO2
TiO2
AO
TiO2
AO
AO
3
AO
6
7
8
PTO
AO
TiO2
5
BTO
TiO2
2
4
STO
TiO2
AO
TiO2
AO
TiO2
0.321
0.278
0.146
(-0.143)
(-0.131)
(0.029)
-0.535
-0.457
-0.181
(-0.071)
(-0.049)
(-0.064)
0.446
0.386
0.096
(-0.018)
(-0.023)
(-0.021)
-0.486
-0.398
-0.122
(-0.022)
(0.010)
(-0.005)
-0.334
-0.252
-0.085
(0.130)
(0.156)
(0.032)
0.490
-0.422
0.104
(0.026)
(0.013)
(-0.013)
-0.392
-0.376
-0.078
(0.072)
(0.032)
(0.039)
0.472
0.413
0.118
(0.008)
(0.004)
(0.001)
0.325
0.281
0.125
(-0.139)
(-0.128)
(0.008)
-0.539
-0.458
-0.166
(-0.075)
(-0.050)
(-0.049)
0.449
0.386
0.093
(-0.015)
(-0.023)
(-0.024)
-0.464
-0.421
-0.121
(0.000)
(-0.004)
(-0.004)
0.466
0.411
0.112
(0.002)
(0.002)
(-0.005)
-0.418
-0.378
-0.075
(0.046)
(0.030)
(0.042)
0.481
0.422
0.111
(0.017)
(0.013)
(-0.006)
-0.303
-0.251
-0.080
(0.161)
(0.157)
(0.037)
6.2 Calculations on the ABO3 (001) surfaces
105
that the additional charge densities are mostly localized on two upper layers while
the central layers practically remain the bulk charge density. Such electron charge
density redistribution, accompanied by atomic displacement in the surfaces, allows
to avoid the non-zero dipole moment even for asymmetrical terminated slabs. As
one can see, the sum of changes in charge densities for three topmost layers of asymmetrically terminated slab (for both sides) of any ABO3 perovskite approximately
equals half of the charge density of the corresponding bulk plane. This is necessary
condition to turn the dipole moment into zero. Weak ion polarization in perovskite
surfaces, demonstrated here, is in a line with ideas of “weak polarity” described by
Goniakowski and Noguera (1996).
The dipole moments of surface atoms, characterizing atomic polarization and
deformations, are presented in Tables 6.4, 6.5, 6.6 for AO-, TiO2 - and asymmetrical terminated surfaces respectively. In particular, the dipole d characterize atomic
deformation and polarization along the z-axis perpendicular to the surface (Saunders, Dovesi, Roetti, Causa et al., 1998). On the AO-terminated surfaces of all
perovskites the cations have the negative dipole moments. This means that their
dipole moments are directed inwards to the surface (direction outwards is chosen as
a positive). The dipole moments of Sr, Ba and Pb in AO-terminated surfaces are
surprisingly large, a few times larger than those of other ions, including the case
of TiO2 -terminated surfaces. Oxygens of AO-terminated surfaces of STO and BTO
demonstrate negative dipole moments whereas in case of PTO the dipole moment
is positive. The polarization of second and third layers is quite small.
On the TiO2 terminated surfaces the polarization of cations has a positive sign,
as well as for oxygens of STO and BTO. Oxygen on PTO surface has negative dipole
moment, opposite to the case of AO-terminated surface. Cations of subsurface layers for all perovskites have positive dipole moments, whereas the dipole moments of
oxygens are negative. The asymmetrically terminated slabs, actually, reproduce the
polarization picture obtained for symmetrically terminated slabs.
The bond populations between atoms in surface layers (positive bond populations correspond to increasing of covalency) are presented in Tables 6.8, 6.9 and 6.10
for AO-, TiO2 - and asymmetrically terminated slabs, respectively. Considering the
AO-terminated surfaces, the main eﬀect is observed for the PTO crystal Pb-O bond
of the top layer whose population, with respect to the bulk, is increased by a factor
of three. The partly covalent nature of the Pb-O bond in lead titanate crystal due
6.2 Calculations on the ABO3 (001) surfaces
106
Table 6.8: AO termination. Bond populations (in e ·10−3 ). Negative population
means atomic repulsion. The corresponding bond populations for bulk perovskites
are: Ti-O bond: STO) 88, BTO) 100, PTO) 98; Pb-O bond: 16.
STO
BTO
PTO
Atom A
Atom B
Atom B
Atom B
O(1)
O(1)
O(2)
O(3)
O(4)
4
O(1)
2
O(1)
0
Sr(1)
-6
Ba(1)
-30
Pb(1)
54
Ti(2)
72
Ti(2)
80
Ti(2)
102
-74
O(2)
-54
O(2)
-58
O(2)
Sr(1)
-30
Ba(1)
-56
Pb(1)
52
O(2)
-46
O(2)
-38
O(2)
-60
80
Ti(2)
78
Ti(2)
88
Ti(2)
Sr(3)
-10
Ba(3)
-30
Pb(3)
6
O(3)
-48
O(3)
-34
O(3)
-42
Ti(2)
86
Ti(2)
90
Ti(2)
72
O(3)
-8
O(3)
-6
O(3)
-8
Sr(3)
-12
Ba(3)
-36
Pb(3)
24
Ti(4)
84
Ti(4)
98
Ti(4)
96
O(4)
-46
O(4)
-38
O(4)
-54
Sr(3)
-10
Ba(3)
-34
Pb(3)
24
O(4)
-8
O(4)
-6
O(4)
-8
Ti(4)
86
Ti(4)
98
Ti(4)
94
Table 6.9: TiO2 termination. The same as for Table 6.8.
Atom A
O(1)
O(2)
O(3)
O(4)
STO
BTO
PTO
Atom B
Atom B
Atom B
O(1)
-30
O(1)
-24
O(1)
-34
Ti(1)
114
Ti(1)
126
Ti(1)
114
Sr(2)
-14
Ba(2)
-38
Pb(2)
42
O(2)
-28
O(2)
-20
O(2)
-42
Ti(1)
142
Ti(2)
140
Ti(2)
162
0
O(2)
2
O(2)
2
O(2)
Sr(2)
-8
Ba(2)
-30
Pb(2)
8
Ti(3)
72
Ti(3)
90
Ti(3)
80
-36
O(3)
-36
O(3)
-32
O(3)
Sr(2)
-4
Ba(2)
-24
Pb(2)
14
O(3)
-42
O(3)
-36
O(3)
-44
110
Ti(3)
94
Ti(3)
106
Ti(3)
Sr(4)
-10
Ba(4)
-34
Pb(4)
18
O(4)
-42
O(4)
-36
O(4)
-44
Ti(3)
92
Ti(3)
102
Ti(3)
106
O(4)
2
O(4)
2
O(4)
2
Sr(4)
-10
Ba(4)
-34
Pb(4)
14
6.2 Calculations on the ABO3 (001) surfaces
107
Table 6.10: Asymmetrical termination. The same as for Table 6.8.
Atom A
O(1)
O(2)
O(3)
O(4)
O(5)
O(6)
O(7)
O(8)
STO
BTO
PTO
Atom B
Atom B
Atom B
O(1)
4
O(1)
2
O(1)
0
Sr(1)
-6
Ba(1)
-30
Pb(1)
58
Ti(2)
70
Ti(2)
82
Ti(2)
104
O(2)
-56
O(2)
-60
O(2)
-70
Sr(1)
-32
Ba(1)
-58
Pb(1)
48
O(2)
-46
O(2)
-38
O(2)
-58
Ti(2)
76
Ti(2)
88
Ti(2)
88
Sr(3)
-10
Ba(3)
-30
Pb(3)
6
O(3)
-48
O(3)
-32
O(3)
-44
Ti(2)
86
Ti(2)
90
Ti(2)
82
O(3)
2
O(3)
-6
O(3)
0
Sr(3)
-12
Ba(3)
-36
Pb(3)
26
Ti(4)
86
Ti(4)
98
Ti(4)
98
O(4)
-46
O(4)
-38
O(4)
-52
Sr(3)
-10
Ba(3)
-34
Pb(3)
14
O(4)
-8
O(4)
2
O(4)
-8
Ti(4)
88
Ti(4)
98
Ti(4)
100
Sr(5)
-10
Ba(5)
-34
Pb(5)
14
O(5)
-44
O(5)
-38
O(5)
-50
Ti(4)
88
Ti(4)
102
Ti(4)
102
O(5)
2
O(5)
-6
O(5)
0
Sr(5)
-10
Ba(5)
-34
Pb(5)
18
Ti(6)
88
Ti(6)
102
Ti(6)
102
O(6)
-44
O(6)
-36
O(6)
-50
Sr(5)
-10
Ba(5)
-34
Pb(5)
24
O(6)
-42
O(6)
-36
O(6)
-44
Ti(6)
96
Ti(6)
104
Ti(6)
110
Sr(7)
-4
Ba(7)
-24
Pb(7)
14
O(7)
-36
O(7)
-32
O(7)
-34
Ti(6)
80
Ti(6)
90
Ti(6)
84
O(7)
2
O(7)
2
O(7)
0
Sr(7)
-8
Ba(7)
-30
Pb(7)
8
Ti(8)
132
Ti(8)
142
Ti(8)
160
-40
O(8)
-28
O(8)
-20
O(8)
Sr(7)
-14
Ba(7)
-38
Pb(7)
42
Ti(8)
118
Ti(8)
126
Ti(8)
116
O(8)
-32
O(8)
-24
O(8)
-32
6.2 Calculations on the ABO3 (001) surfaces
108
to hybridization of 6s states of Pb with the 2p states of the oxygen atoms is already
pronounced in the bulk (Table 4.3), but due to surface relaxation its covalency is
increased. Further analysis of Table 6.8 shows no signiﬁcant increasing of Ti-O
bond covalency for all perovskites, whereas repulsion (negative bond population)
between oxygen and Sr (or Ba) is multiplied. The Ti-O bonds of all perovskites in
TiO2 terminated surfaces increase their covalency due to bond shortening, caused
by surface relaxation and breaking bonds on the surface, as it is seen from Table
6.9. The asymmetrically terminated slabs demonstrate practically the same bond
populations as for AO- and TiO2 -terminated slabs discussed above.
The diﬀerence electron density maps calculated with respect to the superposition
density for A2+ , Ti4+ and O2− ions for surfaces of STO, BTO and PTO perovskites
are presented on Fig. 6.5. These maps demonstrate considerable electron charge
density redistribution for all perovskites surfaces and are entirely consistent with
Mulliken charges and bond population analysis. For all three perovskites the excess
of electron density (the solid isodensity curves), what corresponds to covalency,
is observed in Ti-O bonds. In all terminations the nearest to surface Ti-O bond
becomes stronger, but the next nearest bond becomes weaker. The A atoms on
AO-terminated surfaces demonstrate an intense polarization as it was predicted by
dipole moment calculations (see Tables 6.4, 6.5 and 6.6). Nevertheless, density maps
demonstrate no trace of covalent bonding (zero dot-dashed curves in area between
A cations and Ti-O pairs) between A cations and oxygen even for PTO, despite the
Pb-O bond population calculated for atoms on PbO-terminated surfaces (see Table
6.8) reaches the quite large magnitude of 54 me. This means, in reality the covalency
of the Pb-O bond in PTO is quite weak and plays no important role in terms of
covalency of the whole PTO crystal. Such behavior of Pb-O bond population can be
explained by the dependence of calculated local properties (on the basis of Mulliken
analysis) on the atomic BS chosen for Pb (Table 3.1), whose most diﬀuse exponent
(0.142 bohr−2 ) contributes to an increasing covalency. The recently suggested, based
on a minimal valence basis of Wannier Type Atomic Orbitals (WTAO) (Evarestov,
Smirnov and Usvyat, 2003), approach for population analysis is directly connected
with the electronic band structures. Such analysis predicts, that fully ionic charge
+2e remains practically the same for bulk PTO crystal, with small charge decreasing
down to +1.99e. Thus, the large magnitude of the Pb-O bond population on the
surface obtained through the Mulliken population analysis can be treated as an
0
0
STO TiO 2
0
O (4)
Ti (3)
O (2)
Ti (1)
Ti (4)
O (3)
a
Sr (7)
Sr (5)
Sr (3)
Ti (2) Sr (1)
O (1)
0
0
0
0
0
STO
asymmetrical
termination
Ti (8)
O (7)
Ti (6)
O (5)
Ti (4)
O (3)
Ti (2)
O (1)
Ba (4)
Ba (2)
Ba (3)
Ba (1)
0
0
0
BTO TiO 2
0
0
BTO BaO
O (4)
Ti (3)
O (2)
Ti (1)
Ti (4)
O (3)
Ti (2)
O (1)
b
Ba (7)
Ba (5)
Ba (3)
Ba (1)
0
0
0
0
0
BTO
asymmetrical
termination
Ti (8)
O (7)
Ti (6)
O (5)
Ti (4)
O (3)
Ti (2)
O (1)
Pb (4)
Pb (2)
Pb (3)
Pb (1)
0
0
0
PTO TiO 2
0
0
0
PTO PbO
O (4)
Ti (3)
O (2)
Ti (1)
Ti (4)
O (3)
Ti (2)
O (1)
c
Pb (7)
Pb (5)
Pb (3)
Pb (1)
0
0
0
0
0
Ti (8)
O (7)
Ti (6)
O (5)
Ti (4)
O (3)
Ti (2)
O (1)
PTO
asymmetrical
termination
Figure 6.5: Diﬀerence electron density maps in the cross section perpendicular to the (001) surface ((110) plane) with
AO-, TiO2 and asymmetrical terminations. Isodensity curves are drawn from -0.05 to +0.05 e a.u.−3 with an increment
of 0.0025 e a.u.−3 . a) STO, b) BTO, c) PTO.
Sr (4)
Sr (2)
Sr (3)
Sr (1)
0
STO SrO
6.2 Calculations on the ABO3 (001) surfaces
109
6.2 Calculations on the ABO3 (001) surfaces
110
increasing of electron attraction between Pb and O ions with presence of weak
covalency and not as a presence of fully covalent or ion-covalent Pb-O chemical
bond.
6.2.3
Density of states and band structures
The calculated band structures for STO and BTO surfaces and bulk (see Fig. 6.6
and 6.7) are quite similar to each other. The band structure for bulk perovskites
(also for PTO) have been calculated considering the bulk as a supercell, with a
crystal unit cell extended four time along z-axis. Such supercell allows to model
the eight-layered “slab” periodically repeated in 3D space and to provide the most
natural comparison with the surface band structures. In bulk band structures bands
are plotted using the Γ-X-M-Γ path in the typical “surface” BZ, which is simply the
square for cubic crystals with high symmetry points: Γ in the center, M in the corner
and X on the center of the square edge. The upper valence bands (VB) for STO
and BTO bulks are quite ﬂat with the top in M point and contain a perfectly ﬂat
fragment between M and X points. The main contribution into the upper VB make
the 2px and 2py oxygen orbitals as it is well seen from calculated density of states
(DOS) projected to corresponding atomic orbitals (see Fig. 6.9 and 6.12). The bottom of lowest conduction band (CB) lies in Γ, with quite ﬂat fragment between Γ-X
points, and consists of Ti 3d threefold degenerated T2g level. The optical band gaps
for surfaces and bulk of all three perovskites, calculated by means of DFT-B3PW
technique, are presented in Table 6.11. It is clearly seen that a good agreement with
experiment is achieved. The band structure for SrO-terminated surface demonstrates practically the same ﬂatness of upper valence bands as the bulk STO, with
the top of VB in M point and the bottom of CB in Γ. The optical band gap for SrOterminated surfaces becomes smaller with respect to the band gap of bulk STO. The
most narrow place located between Γ and M points (indirect gap) is equal to 3.3 eV
whereas the most narrow gap obtained for bulk is 3.63 eV (see Table 6.11 for details).
Analysis of DOS plots calculated for SrO-terminated surface (see Fig. 6.10)
demonstrates that there is no contribution of surface O 2p electronic states into the
top of VB, which consists mainly of 2p electrons of the central oxygens. The main
contribution into the bottom of CB make the 3d electronic states of Ti atom from the
second layer. The band structure calculated for the TiO2 -terminated surface of STO
6.2 Calculations on the ABO3 (001) surfaces
111
STO Bulk
E / a.u.
0.100
0.000
-0.100
-0.200
Γ
X
M
E / a.u.
Γ
STO SrO-terminated
0.100
0.000
Ti 3d
(subsurface layer)
O 2p
(central layer)
-0.100
-0.200
-0.300
O 2p
(surface layer)
Γ
X
Γ
M
STO TiO2-terminated
E / a.u.
0.000
-0.100
O 2p
(surface layer)
Ti 3d
(3rd layer)
Ti 3d
(surface layer)
-0.200
-0.300
-0.400
Γ
X
Γ
M
STO asymmetrical terminated
E / a.u.
0.000
Ti 3d
(subsurface layer,
SrO-termination)
Ti 3d
(surface layer,
TiO2-termination)
-0.100
O 2p
(surface layer,
TiO2-termination)
-0.200
-0.300
-0.400
Γ
O 2p
(surface layer,
SrO-termination)
X
M
Γ
Figure 6.6: Calculated electronic band structures for STO bulk and surfaces.
6.2 Calculations on the ABO3 (001) surfaces
112
BTO Bulk
E / a.u.
0.100
0.000
-0.100
-0.200
-0.300
Γ
X
M
E / a.u.
Γ
BTO BaO-terminated
0.100
0.000
Ti 3d
(subsurface layer)
O 2p
(central layer)
-0.100
-0.200
O 2p
(surface layer)
-0.300
Γ
X
Γ
M
BTO TiO2-terminated
E / a.u.
0.000
-0.100
Ti 3d
(3rd layer)
Ti 3d
(surface layer)
-0.200
O 2p
(surface layer)
-0.300
-0.400
Γ
X
Γ
M
BTO asymmetrical terminated
E / a.u.
0.000
Ti 3d
(subsurface layer,
BaO-termination)
Ti 3d
(surface layer,
TiO2-termination)
-0.100
O 2p
(surface layer,
TiO2-termination)
-0.200
-0.300
Γ
O 2p
(surface layer,
BaO-termination)
X
M
Γ
Figure 6.7: Calculated electronic band structures for BTO bulk and surfaces.
6.2 Calculations on the ABO3 (001) surfaces
113
PTO Bulk
E / a.u.
0.100
0.000
-0.100
-0.200
-0.300
-0.400
Γ
X
M
E / a.u.
Γ
PTO PbO-terminated
0.000
-0.100
Pb 6s and O 2p
(3rd layer)
Ti 3d
(subsurface layer)
-0.200
-0.300
-0.400
O 2p
(surface layer)
-0.500
Pb 6s (surface layer)
Γ
X
Γ
M
PTO TiO2-terminated
E / a.u.
-0.100
-0.200
Ti 3d
(3rd layer) Ti 3d
(surface layer)
O 2p
(surface layer)
Pb 6s and O 2p
(central layers)
-0.300
-0.400
-0.500
Γ
X
-0.100
Γ
M
PTO asymmetrical terminated
E / a.u.
Ti 3d
(subsurface layer,
PbO-termination)
Ti 3d
(surface layer,
TiO2-termination)
Pb 6s (central layers)
O 2p
(surface layer,
TiO2-termination)
-0.200
-0.300
-0.400
-0.500
Γ
X
M
O 2p
(surface layer,
PbO-termination)
Pb 6s
(surface layer,
PbO-termination)
Γ
Figure 6.8: Calculated electronic band structures for PTO bulk and surfaces.
3.17
3.92
5.17
(i)
3.25 - indirect gap
3.75 - direct gap
3.30
3.55
5.62
2.31
3.41
4.66
iii – Peng, Chang and Desu (1992)
ii – Wemple (1970)
i – van Benthem, Elsässer and French (2001)
Experiment
3.63
Γ-R
4.20
4.39
3.71
Γ-X
Γ-M
6.12
6.47
3.50
3.60
5.39
4.39
5.70
4.09
R-R
4.04
3.55
bulk
M-M
4.37
3.03
asymm.
4.53
3.95
TiO2
STO
X-X
3.72
SrO
3.96
bulk
Γ-Γ
Optical gap
3.32
3.49
5.40
4.22
3.49
BaO
(ii)
3.2
2.33
3.41
4.17
3.63
2.96
TiO2
BTO
2.10
3.18
4.17
3.72
2.73
asymm.
3.66
3.66
2.87
5.98
5.55
3.02
4.32
bulk
3.55
2.96
5.37
3.79
3.58
PbO
(iii)
3.4
3.19
2.98
5.01
3.10
3.18
TiO2
PTO
2.96
2.78
4.88
3.28
3.08
asymm.
Table 6.11: The calculated optical gap (in eV) for the bulk (Table 4.2) and surface-terminated perovskites. The last row
contains experimental data.
6.2 Calculations on the ABO3 (001) surfaces
114
6.2 Calculations on the ABO3 (001) surfaces
115
demonstrates a less ﬂat top of the VB in comparison with SrO-termination. The
indirect optical band gap (Γ-M) becomes even more narrow with a value of 3.17 eV.
For the TiO2 -terminated STO surface the main contribution into the top of VB make
the electrons from the 2px and 2py orbitals, which are perpendicular to the Ti-O-Ti
bridge (see Fig. 6.11). The main contribution to the bottom of the CB make the 3d
states of Ti from third layer and the electronic states of surface Ti atom lie a little
bit higher in energy range. The calculated STO DOS are in good agreement with
MIES and UPS, spectra recently obtained on TiO2 -terminated STO(001) surface by
Maus-Friedrichs, Frerichs, Gunhold, Krischok et al. (2002). Moreover, the calculated
position of the top of VB for TiO2 -terminated STO with respect to the vacuum (5.9
eV) practically coincides with the experimentally observed 5.7 eV (Maus-Friedrichs,
Frerichs, Gunhold, Krischok et al., 2002), if the error margins of the experiment
(±0.2 eV) are taken into account. The band structure calculated for asymmetrical
terminated slab demonstrates the mixture of band structures obtained for both
symmetrical terminated slabs. The band gap becomes more narrow (2.31 eV), the
top of VB mainly consists of O 2p electronic states from TiO2 -terminated slab
surface, and in the bottom of the CB the main contribution make the 3d states
of Ti from the subsurface layer. The split of the upper VB (around 0.8 eV) for
the asymmetrical STO slab is well pronounced. The band structures calculated for
cubic BTO(001) surfaces demonstrate practically the same behavior as in case of
STO (see Fig. 6.7, 6.12, 6.13, 6.14, and Table 6.11). Nevertheless, the split of
the upper band in the VB region is more pronounced for TiO2 -terminated BTO
in comparison with the STO surface. Due to presence of hybridization between
Pb 6s and O 2p orbitals in PTO, the calculated band structures and DOS of this
perovskite are diﬀer slightly from those calculated for STO and BTO (see Fig. 6.8,
6.15, 6.16, 6.17, and Table 6.11). The narrowest gap distance of all bulk and surface
band structures lies between Γ and X points of The BZ. In bulk, the top of the
VB is formed signiﬁcantly by 6s Pb orbitals which also make the main contribution
into the bottom of the VB. The bottom of the CB for PTO bulk is formed by
Ti 3d orbitals, the same as it is for other perovskites. The optical band gap for
PbO-terminated surfaces does not become smaller as it was in the case of BTO and
STO, but it increases slightly up to 2.96 eV with respect 2.87 for bulk. The top of
the VB for PbO-terminated PTO consists of a mixture of Pb 6s and O 2p orbitals
from the third layer while the bottom of the CB is formed by Ti 3d orbitals from
6.2 Calculations on the ABO3 (001) surfaces
116
the subsurface layer. The top of VB for the TiO2 -terminated PTO(001) surface in
the X point surprisingly consists of a mixture of O 2p and Pb 6s electronic states
from surface and central layers. Moreover, the main contribution make the orbitals
of the central atoms. The bottom of the CB for TiO2 -terminated PTO consists
mainly of Ti 3d orbitals from the third layer. The calculated band gaps of PbOand TiO2 -terminated PTO(001) surfaces are practically equal, 2.96 and 2.98 eV,
respectively. The band structure calculated for asymmetrical PTO slab represents
a mixture of band structures calculated for symmetrical slabs, as well as in the case
of STO and BTO. The diﬀerent behavior of electronic properties of (001) surfaces
of PTO and other perovskite under investigation can be explained by the presence
of hybridization between Pb 6s and O 2p, and the presence of weak covalency in
Pb-O bond.
In conclusion of this Section it can be stressed that the data obtained for surface
structure DFT-B3PW calculations are in good agreement with theoretical results
published previously and partly with data obtained in experiments. The computed
relaxed surface energies show that the surfaces with both termination are quite stable in agreement with Tasker’s classiﬁcation and existing experiments. Also, the
calculations on charge densities for all perovskite surfaces demonstrate the presence
of a weak polarity predicted earlier in literature for surfaces of ionic-covalent crystals. The analysis of the dipole moments shows that the cations on AO-terminated
surfaces demonstrating a strong polarization along the z-axis. The calculated diﬀerence electron density maps demonstrate an increasing of covalency in Ti-O bonds for
atoms near the surfaces for all surfaces and only weak covalency for the Pb-O bond
on PbO terminated surface. The illustrated absence of surface electronic states in
the upper valence bands for AO-terminated (001) surfaces of all perovskites and the
presence of Pb 6s orbitals in the top of the VB region of PTO could be important
for the treatment of electronic structure of surface defects on perovskite surfaces, as
well as for adsorption and surfaces diﬀusion of atoms and small molecules relevant
for catalysis.
6.2 Calculations on the ABO3 (001) surfaces
117
Density of states for STO bulk
-0.3
1000
-0.2
Top of VB
500
DOS /arb. units
0
1200
-0.1
0.0
0.1
0.2
1000
DOS projected
to Ti atoms
500
DOS projected to px and py
AOs of O(z) atoms
600
0
0
1200
600
0
DOS projected to pz
AOs of O(z) atoms
500
0
6000
500
DOS projected to O atoms
0
6000
3000
3000
0
6000
0
6000
Total DOS
3000
3000
0
0
-0.3
-0.2
-0.1
0.0
0.1
0.2
Energy / at. un.
Figure 6.9: Total and projected DOS for the bulk STO.
6.2 Calculations on the ABO3 (001) surfaces
118
STO, SrO terminated (100) surface
-0.3
150
-0.2
-0.1
Top of VB
0.0
0.1
0.2
DOS projected to O atom
in the 1st layer
150
0
200
0
DOS projected to Ti
atom in the 2nd layer
200
0
DOS / arb. un.
0
DOS projected to O atom
in the 2nd layer
150
150
0
0
DOS projected to O atom
in the 3rd layer
150
150
0
0
200
DOS projected to Ti
atom in the 4th layer
200
0
0
DOS projected to O
atom in the 4th layer
150
150
0
0
Total DOS
1600
0
-0.3
-0.2
-0.1
0.0
0.1
1600
0
0.2
Energy / at. un.
Figure 6.10: Total and projected DOS for the SrO-terminated surface.
6.2 Calculations on the ABO3 (001) surfaces
119
STO, TiO2 terminated (100) surface
-0.4
300
-0.3
-0.2
DOS projected to Ti atom
in the 1st layer
0
150
-0.1
0.0
300
Top of VB
DOS projected to px AOs of
O atom in the 1st layer
0
0
DOS projected to py AOs
of O atom in the 1st layer
40
0
DOS / arb. un.
0
250
DOS projected to O atom
in the 1st layer
0
0
250
300
0
DOS projected to Ti atom
in the 3rd layer
0
200
300
DOS projected to O atom
in the 3rd layer
0
200
0
DOS projected to O atom 300
in the 4th layer
0
3000
Total DOS
0
300
0
3000
0
80
0
DOS projected to O atom
in the 2nd layer
300
300
40
0
DOS projected to pz AOs
of O atom in the 1st layer
80
0
0
150
-0.4
-0.3
-0.2
-0.1
0.0
0
Energy / at. un.
Figure 6.11: Total and projected DOS for the STO TiO2 -terminated surface.
6.2 Calculations on the ABO3 (001) surfaces
120
Density of states for BTO bulk
-0.3
2000
-0.2
Top of VB
1000
DOS / arb. units
0
1000
-0.1
0.0
0.1
0.2
2000
DOS projected
to Ti atoms
1000
DOS projected to px and py
AOs of O(z) atoms
500
0
300
0
1000
500
0
300
DOS projected to pz
AOs of O(z) atoms
150
DOS projected to O atoms
0
3000
150
0
3000
1500
1500
0
3000
0
3000
Total DOS
1500
1500
0
0
-0.3
-0.2
-0.1
0.0
0.1
0.2
Energy / at. un.
Figure 6.12: Total and projected DOS for the bulk BTO.
6.2 Calculations on the ABO3 (001) surfaces
121
BTO, BaO terminated (100) surface
-0.3
175
-0.2
-0.1
DOS projected to O
atom in the 1st layer
0.0
0.1
0.2
175
Top of VB
0
0
DOS projected to Ti atom
150 in the 2nd layer
150
DOS / arb. un.
0
0
DOS projected to O atom
in the 2nd layer
100
100
0
0
DOS projected to O atom
in the 3rd layer
100
100
0
0
200
DOS projected to Ti
atom in the 4th layer
200
0
0
DOS projected to O atom
in the 4th layer
100
0
1600
100
0
1600
Total DOS
800
800
0
-0.3
-0.2
-0.1
0.0
0.1
0
0.2
Energy / a.u.
Figure 6.13: Total and projected DOS for the BaO-terminated surface.
6.2 Calculations on the ABO3 (001) surfaces
122
BTO, TiO2 terminated (100) surface
-0.5
300
-0.4
-0.3
DOS projected to Ti atom
in the 1st layer
-0.2
-0.1
0.0
300
Top of VB
0
0
DOS projected to px AOs of
O atom in the 1st layer
80
0
60
DOS projected to py AOs
of O atom in the 1st layer
DOS / arb. un.
0
150
DOS projected to pz AOs
of O atom in the 1st layer
0
150
DOS projected to O atom
in the 1st layer
0
100
DOS projected to O atom
in the 2nd layer
DOS projected to Ti atom
in the 3rd layer
0
100
0
150
0
150
0
100
500
DOS projected to O atom
in the 3rd layer
0
150
0
2000
Total DOS
-0.4
-0.3
-0.2
-0.1
0
100
0
150
DOS projected to O
atom in the 4th layer
0
2000
0
-0.5
0
60
0
0
500
80
0.0
0
Energy / at. un.
Figure 6.14: Total and projected DOS for the BTO TiO2 -terminated surface.
6.2 Calculations on the ABO3 (001) surfaces
123
Density of states for PTO bulk
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
800
800
DOS projected
to Pb atoms
Top of VB
400
400
0
2000
DOS / arb. units
1000
0
DOS projected
to Ti atoms
0
1000
2000
1000
DOS projected to px and py
AOs of O(z) atoms
500
0
0
1000
500
0
500
DOS projected to pz
AOs of O(z) atoms
250
0
4000
500
250
0
4000
DOS projected to O atoms
2000
2000
0
0
4000
4000
Total DOS
2000
2000
0
-0.4
-0.3
-0.2
-0.1
0.0
0.1
Energy / at. un.
Figure 6.15: Total and projected DOS for the bulk PTO.
0
0.2
6.2 Calculations on the ABO3 (001) surfaces
124
Density of States for PTO
PbO-terminated (100) slab
-0.5
100
-0.4
-0.3
-0.2
-0.1
0.0
0.1
DOS projected to Pb atom in the 1st layer
100
Top of valence
band
0
0
DOS projected to O atom
in the 1st layer
100
0
350
0
0
DOS projected to O atom
in the 2nd layer
DOS / arb. units
100
0
200
DOS projected to O atom
in the 3rd layer
0
0
150
0
DOS projected to Ti
atom in the 4th layer
400
0
100
DOS projected to O atom
in the 4th layer
0
0
100
0
Total DOS
1500
0
100
0
DOS projected to Pb atom
in the 3rd layer
0
150
400
100
0
DOS projected to Ti atom
in the 2nd layer
350
200
0.2
-0.5
-0.4
-0.3
-0.2
-0.1
1500
0.0
0.1
0
0.2
Energy / at. un.
Figure 6.16: Total and projected DOS for the PbO-terminated surface.
6.2 Calculations on the ABO3 (001) surfaces
125
Density of States for PTO
TiO2-terminated (100) slab
-0.5
-0.4
-0.3
-0.2
-0.1
Top of valence
band
400
0.1
DOS projected to pxAOs of
O atom in the 1st layer
0
30
DOS projected to pz
AOs of O atom in the 1st layer
0
40
0
DOS projected to O atom in the 1st layer 150
150
0
70
DOS projected to Pb atom
in the 2nd layer
0
400
DOS projected to O atom
in the 2nd layer
DOS projected to O atom
in the 3rd layer
0
100
0
250
0
200
0
100
DOS projected to Pb atom
in the 4th layer
DOS projected to O atom
in the 4th layer
0
2000
0
400
0
400
DOS projected to Ti atom
in the 3rd layer
0
200
0
0
150
0
0
0
400
400
DOS projected to pyAOs 30
of O atom in the 1st layer
0
40
0
70
0.2
DOS projected to Ti
atom in the 1st layer
0
150
DOS / arb. units
0.0
0
250
0
Total DOS
-0.5
-0.4
-0.3
-0.2
-0.1
2000
0.0
0.1
0
0.2
Energy / at. un.
Figure 6.17: Total and projected DOS for the PTO TiO2 -terminated surface.
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
6.3
126
Calculations on TiO- and Ti-terminated
SrTiO3(110) polar surfaces
The STO(110) surface consists of a sequence of alternating charged SrTiO and O2
planes (see Fig. 6.3). It is well known that such a surface is unstable due to an
inﬁnite dipole moment produced by the charged planes perpendicularly to the surface. This is why cleavage of this surface should result in formation of two stable
surfaces: Sr-terminated and TiO-terminated (Heifets, Kotomin and Maier, 2000).
In present calculations, TiO-terminated surface and its reduction to Ti-terminated
surface, when all O atoms are removed from the ﬁrst plane, have been simulated.
The relaxed positions of atoms in surface and subsurface layers have been taken
from Heifets, Goddard III, Kotomin, Eglitis et al. (2003). In order to compare calculations with MIES experiments, the PDOS, projected on the ﬁrst plane (TiO- or
Ti-terminated), and the eﬀective charges of atoms in the slab, as well as the electron
populations of Ti atoms in diﬀerent situations have been compared. The calculated
DOS is convoluted with a Gaussian function, in order to account for the electronphonon broadening of the electron emission spectra and the experimental resolution
(Ochs, Maus-Friedrichs, Brause, Günster et al., 1996). A value of 1 eV was chosen
for the width of the Gaussian because it gives good agreement between DOS and
UPS spectra.
The measurements on surface electronic structure of the TiO2 rows on heated
STO(110) surfaces were performed by Prof. Kempter’s group (Gunhold, Beuermann, Gömann, Borchardt et al., 2003b) using two diﬀerent set-ups. MIES and
UPS(HeI) measurements were performed in an ultrahigh vacuum (for more details
see Ochs, Maus-Friedrichs, Brause, Günster et al. (1996); Brause, Braun, Ochs,
Maus-Friedrichs et al. (1998); Maus-Friedrichs, Frerichs, Gunhold, Krischok et al.
(2002)). A time-of-ﬂight technique was used to separate the electrons emitted in the
interaction of He* atoms and HeI photons with the surface. The electron spectra
were recorded with a resolution of 250 meV under normal emission within 100 s.
The angle of incidence for the mixed He*/HeI beam is 45◦ . The MIES spectra are
displayed as a function of the electron binding energy with respect to the Fermi level.
Studied crystals (Crystec, Berlin) have been heated in vacuum and in synthetic air,
respectively. Preparation under vacuum was done by heating to 700 ◦ C for 20 h at
pressures below 10-8 mbar, in order to clean its surface completely. The cleanliness
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
127
of the surface was checked by MIES and XPS. After the MIES measurements this
crystal was transferred to the AFM/STM apparatus ex-situ where it was cleaned
by heating to 700 C for 20 min. This procedure does not inﬂuence STO crystals
composition. The crystals heated in synthetic air (consisting of 80% N2 and 20%
O2 ) were cleaned by heating to 700 ◦ C for 20 min after introduction into vacuum, respectively. The heating procedures in the AFM/STM and the MIES apparatus were
reproduced with a precision of better than 10 K, respectively. The target temperatures were controlled by a commercial optical pyrometer (Impac IGA 120) through
viewports in both apparatus. All experiments were performed at room temperature.
The MIES and UPS results are presented together with those of the ab initio
DFT-B3PW calculations. Before discussing the results, some remarks concerning
the interaction of He* with surfaces might be useful: Metastable He*(23 S) atoms
interact with the surfaces via various processes. Three diﬀerent processes may occur
(for details see Harada, Masuda and Ozaki (1997)):
a) On pure STO surfaces the impinging He* atoms are ionized by a resonant electron
transfer into localized Ti3+ 3d surface orbitals. Subsequently, the remaining
He+ is neutralized in front of the surface by Auger Capture (AC). Hereby a
surface electron ﬁlls the He 1s orbital emitting a Ti3+ 3d electron. Due to
the fact that the Ti 3d orbital possesses a rather small Full Width at Half
Maximum (FHWM) the resulting MIES spectrum looks quite similar to the
Auger Deexcitation (AD) spectrum (see below), but shifted to lower kinetic
energies (i.e. to higher binding energies) by 1.2 eV (Maus-Friedrichs, Frerichs,
Gunhold, Krischok et al., 2002).
b) For work functions below about 3.5 eV AD becomes the dominating process. In
this process a surface electron ﬁlls the He 1s orbital and the He 2s electron is
emitted, carrying away the excess energy. The energy balance is similar to UPS
with the exception of the diﬀerent excitation energy (19.8 eV for He*(23 S)).
c) For work functions below about 2.2 eV and high electron density just below
the Fermi level EF , the probability for resonant electron transfer from the
surface to He* becomes sizeable forming He− *1s2s2 ions in front of the surface
(Hemmen and Conrad, 1991). These species decay rapidly in an intraatomic
autodetachment process.
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
128
MIES Spectra and Projected DOS of SrTiO3 (110) surface / arb. units
PDOS projected on Ti 3d orbitals (upper
layer of SrTiO3 (110) Ti-terminated)
MIES spectra of:
STO(110) clean, unreconstructed surface
STO(110) heated in air
STO(110) heated in vacuum
O(2p)
Ti(3d)
20
10
0
binding energy / eV
Figure 6.18: MIES and ab initio DOS results for the clean unreconstructed and
heated STO(110) surfaces. See text and inserts for detailed description.
MIES / UPS Spectra and Projected DOS of SrTiO3 (110) surface / arb. units
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
129
PDOS projected on 1st plane of SrTiO3 (110)
TiO-terminated slab (Ti and O)
PDOS projected on Ti 3p orbitals (upper
layer of SrTiO3 (110) Ti-terminated)
PDOS projected on Ti 3d orbitals (upper
layer of SrTiO3 (110) Ti-terminated) (5x)
UPS spectrum of STO(110) heated in vacuum
O(2p)
Ti(3d)
5x
20
10
0
binding energy / eV
Figure 6.19: UPS and ab initio DOS results, the same as for Fig. 6.18.
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
130
The results are presented as a function of the binding energy EB , with respect to EF ,
of the electrons emitted in the AD process. Only for the clean STO(110) surface the
interaction in MIES is due to the defect-modiﬁed AC process, producing however
spectra similar to those from AD (Maus-Friedrichs, Frerichs, Gunhold, Krischok
et al., 2002). MIES spectra of the clean and unreconstructed surface are shown in
Fig. 6.18. This surface was studied after heating in air at 900 ◦ C for 1 h in vacuum
at 1000 ◦ C for 1 h.
The MIES spectra are compared with the calculated PDOS projected on the Ti
3d orbitals of the reconstructed surface (corresponding to Fig. 6.3(c)). The UPS
spectrum of the clean STO(110) heated in vacuum at 1000 ◦ C for 1 h is shown in
Fig. 6.19. In this ﬁgure the PDOS projected to the 1st plane of the TiO-terminated
slab of the unreconstructed surface (corresponding to Fig. 6.3(b)) is also plotted,
the PDOS of the Ti 3p orbitals and the Ti 3d orbitals belong to the ﬁrst layer
atom. The MIES and UPS spectra of the unreconstructed (110) surface are similar
to those for STO(100) (Gunhold, Gömann, Beuermann, Frerichs et al., 2002; MausFriedrichs, Frerichs, Gunhold, Krischok et al., 2002; Gunhold, Beuermann, Frerichs,
Kempter et al., 2003a): A dominant structure denoted by O(2p) appears around
EB =7 eV from the ionization of O 2p orbitals. In contrast to MIES, the UPS(HeI)
spectra display a double-peak structure in this region. As it was shown previously
in studies on MgO (Ochs, Maus-Friedrichs, Brause, Günster et al., 1996), STO(100)
(Harada, Masuda and Ozaki, 1997) and Al2 O3 (Puchin, Gale, Shluger, Kotomin
et al., 1997), MIES is particularly sensitive to orbitals protruding out of the surface.
Here, this results in a higher probability for the detection of O2p-orbitals directed
perpendicular to the surface. The MIES and UPS spectra of the surface heated in
vacuum show additional peaks near zero binding energy. The same surface heated in
synthetic air shows a similar feature. The emission detected beyond about EB =10
eV are attributed to secondary electrons and are not discussed here.
The DOS projected onto the topmost plane is also shown in Fig. 6.19. For the
Ti-terminated surface, contributions of Ti 3p and 3d states are shown separately.
Tables 6.12 and 6.13 give atomic charges and electron occupancy of the Ti atomic
orbitals in diﬀerent planes for both surface terminations. The theory is able to reproduce the double-peak structure from the O 2p states seen with UPS. Positions
and shape of the O 2p emission in the DOS and UPS spectra agree well. The peak
at lower binding energies (EB = 6.8eV) involves the contributions of 2p orbitals
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
131
Table 6.12: Eﬀective Mulliken charges, Q (e), for two diﬀerent STO(110) terminations. Bulk charges of ions (in e): Sr = 1.871, Ti = 2.350, and O = -1.407.
Layer
Atoms
Q (TiO-term.)
Q (Ti-term.)
1
Ti
2.496
1.399
O
-1.315
2
O
-1.162
-1.145
3
Sr
1.856
1.855
Ti
2.335
2.364
O
-1.370
-1.324
O
-1.503
-1.411
4
Table 6.13: Ti orbitals population for two diﬀerent STO(110) terminations. Ti
orbital populations for a bulk crystal: Ti 3p = 6.014, Ti 3d = 1.233, Ti 4s = 0.163.
Layer
Orbitals
TiO-term.
Ti-term.
1
Ti 3p
6.031
5.980
Ti 3d
1.441
2.606
Ti 4s
0.482
0.408
Ti 3p
6.056
6.054
Ti 3d
1.574
1.548
Ti 4s
0.506
0.509
3
directed perpendicularly to the surfaces while that at larger binding energies stems
from orbitals parallel to the surface. The reproduction of the double-peak structure
shows that UPS(HeI), in the present case, mainly images the DOS of the initial
states. MIES possesses a higher sensitivity for the detection of initial states at the
STO surface with py, z character. These partial DOS have projections directed perpendicular to the (110) surface. The superposition of the DOS contributions from
diﬀerently oriented O 2p orbitals combined with the diﬀerent sensitivity for the detection of py, z and px orbitals produces a broad single peak in MIES, rather than
the double-peak structure seen in UPS(HeI). The reconstructed (110) surface shows
a very similar O 2p contribution in the spectra.
In addition, a novel peak is observed in the DOS near zero binding energy for
6.3 Calculations on TiO- and Ti-terminated
SrTiO3 (110) polar surfaces
132
the reconstructed surface. The calculations establish that Ti 3d states become populated during the reconstruction of the surface, and are responsible for the DOS
contribution near zero binding energy; 3p orbitals give only a small contribution
around 7 eV. The good agreement in shape and position between DOS and MIES
spectra suggests that MIES directly images the DOS of the initially populated Ti
3d states of the surface via the AD process.
Tables 6.12 and 6.13 give a conﬁrmation to these conclusions. Due to the Ti-O
chemical bond covalency, the eﬀective charges of Ti and O atoms in the bulk diﬀer
considerably from the ionic model (+4e and -2e, respectively), unlike the charge of
the Sr ions (1.9 e instead of 2e). On the TiO terminated surface, the Ti ion charge
(2.5 e) does not considerably diﬀer from that in the central 3rd plane of a slab simulating the bulk (2.33 e). The O ion charges on the TiO surface and in the slab center
are also close (-1.31 e vs -1.37 e). In contrast, on the Ti-terminated surface, the
Ti eﬀective charge is decreased by ≈ 1 e, as compared to that on the TiO surface;
1.4 e vs 2.5 e. This is accompanied by a strong increase of the population of Ti 3d
orbitals, from 1.44 e to 2.61 e, respectively. Indeed, these Ti 3d orbitals give the
main contribution to the MIES peak around 1.2 eV below the EF .
Summarizing this Section, one can stress that the electronic structure of the
STO(110) surfaces studied by means of MIES and UPS(HeI) are compared with
ab initio DFT-B3PW calculations. Besides giving good overall agreement with
the observed O 2p emission, the calculations identify an additional peak close to
zero binding energy for the vacuum heated, Ti-terminated surface as due to Ti 3d
occupied states, giving direct evidence for the termination of the reconstructed,
microfaceted surface by reduced Ti3+ -ions.
Chapter 7
Low-temperature compositional
heterogeneity in BaxSr1−xTiO3
solid solutions
Introduction
In this last Chapter it is demonstrated how a thermodynamic formalism based on ab
initio DFT-B3PW calculations can be adopted for a consistent study of the eﬀect of
external conditions on the ferroelectric phase transformation in Bax Sr1−x TiO3 solid
solutions. All results discussed in this Chapter (especially the thermodynamic part)
have been obtained in close cooperation with Prof. S. Dorfman from the Technion
Institute of technology (Haifa, Israel).
7.1
Perovskite solid solutions
Complex perovskite solid solutions with common formula (A,A’,A”,...)(B,B’,B”,...)O3
attract a growing attention during the last decade because of numerous unusual and
sometimes unexpected properties. These properties open new important ﬁelds of
application of such materials, stimulating future eﬀorts in the study of their behavior under diﬀerent conditions. It is well recognized nowadays that the dielectric
7.1 Perovskite solid solutions
134
and piezoelectric properties, response on external excitations, etc. in these alloys are
entirely linked to the structural properties including compositional ordering and formation of complicated heterostructures. This concerns, for example, several groups
of materials: a) so-called “super-Q” mixed metal perovskites, Ba(Zn1/3 Ta2/3 )O3
(BZT) or Ba(Mg1/3 Nb2/3 )O3 -BaZrO3 (BMN-BZ); b) “new” relaxor ferroelectric alloys, such as Pb(Mg1/3 Nb2/3 )O3 -PbTiO3 (PMN-PT) or Pb(Zn1/3 Nb2/3 )O3 -PbTiO3
(PZN-PT); c) Bax Sr1−x TiO3 (BST) or Srx Ca1−x TiO3 (SCT). BZT and BMN-BZ
alloys exhibit ultra low losses at microwave frequencies (Akbas and Davies, 1998),
while PMN-PT and PZN-PT have extremely high values of the piezoelectric constants (Park and Shrout, 1997). BST is considered as the most promising candidate
for memory cell capacitors in dynamic random access memories with extremely high
scale integration (Abe and Komatsu, 1995). Experimental results show that for the
ﬁrst group of materials the microwave loss properties may be very sensitive to the
B-site cation order (Kawashima, Nishida, Ueda and Ouchi, 1983; Matsumoto, Hiuga, Takada and Ichimura, 1986). For the second group compositional ﬂuctuations
play an important role in the “relaxor” behavior (Setter and Cross, 1980). For BST
solid solutions in the Ba-rich region the dielectric anomalies were associated with the
ﬂuctuations of the order parameter (Singh, Singh, Prasad and Pandey, 1996). The
dielectric and ultrasonic study in Sr-rich BST was reported by Lemanov, Smirnova,
Syrnikov and Tarakanov (1996), where it was shown that a small addition of Ba
to STO leads to formation of a glassy state at very low Ba concentrations and
complicates signiﬁcantly the sequence of phase transitions near x=0.15. Structural
evolution and polar order in BST was reported by Kiat, Dkhil, Dunlop, Dammak
et al. (2002) on the basis of combination of diﬀraction and diﬀusion of neutron and
high-resolution x-ray experiments as well as dielectric susceptibility and polarization
measurements. It is shown that the STO-type antiferrodistortive phase exists up
to a concentration of Ba xcr ≈0.094, the progressive substitution of Sr by Ba leads
to a monotonic decrease and to a vanishing of the oxygen octahedral tilting. The
critical concentration xcr separates the phase diagram in two regions, one with a
sole antiferrodistortive phase transition (x < xcr ) and one with a succession of three
BTO-type ferroelectric phase transitions (x > xcr ). Moreover, inside the nonferroelectric antiferrodistortive phase a local polarization is observed, with a magnitude
that is comparable to the values of spontaneous polarization in the ferroelectric
phases of the rich in Ba compounds. Tenne, Soukiassian, Zhu, Clark et al. (2003)
7.1 Perovskite solid solutions
135
have published the results of a Raman study of BST ﬁlms with thickness ∼1 µm
and with Ba atomic fraction x=0.05, 0.1, 0.2, 0.35, and 0.5. They show the striking
similarity with the behavior of relaxor ferroelectrics which is explained by the existence of polar nanoregions in the BST thin ﬁlms.
To describe and to explain the ties of the structural and dielectric properties in these materials signiﬁcant eﬀorts were employed. A simple purely ionic
model that accounts for the electrostatic interaction was presented by Bellaiche
and Vanderbilt (1998) to reproduce the compositional long-range order observed
in a large class of perovskite alloys, including Pb(Sc1/2 Ta1/2 )O3 , Pb(In1/2 Nb1/2 )O3 ,
Pb(Mg1/2 W1/2 )O3 , Ba(Mg1/3 Nb2/3 )O3 , Ba(Zn1/3 Ta2/3 )O3 , Ba(Zn1/3 Nb2/3 )O3 , etc.
To go beyond the ground state behavior and to make conclusions on the thermodynamic behavior as a function of temperature, Metropolis Monte Carlo simulations
were further applied with the energy deﬁned as the excess electrostatic energy in
heterovalent binaries. This model automatically does not allow homovalent binary
alloys to order. To describe the weak order in PMN, for example, it was necessary
to account for the multivalent nature of Pb atoms. Account for charge transfer
may be performed by direct modelling in the framework of an electrostatic model
as was reported by Wu and Krakauer (2001) or may be carried out by ab initio calculations. A comparative study of Pb(B,B’)O3 and Ba(B,B’)O3 perovskites
was performed by Burton and Cockayne (1999) on the basis of PW pseudopotential calculations and the trends of lower temperature disordering in Pb(B,B’)O3 ,
as compared with Ba(B,B’)O3 , was associated with the enhanced Pb-O bonding to
less-bonded oxygens in B2+ -O-B2− environments. This indicates that the long-range
Coulomb interactions that drive B-site ordering in Ba systems do not dominate in
Pb systems. The results of ab initio calculations obtained by George, Íñiguez and
Bellaiche (2001) show that a certain class of atomic rearrengement should lead simultaneously to large electromechanical responses and to unusual structural phases in
a given class of perovskite alloys. The simulations of George, Íñiguez and Bellaiche
(2001) reveal also the microscopic mechanism responsible for these anomalies. Evidence, that planar defects aﬀect strongly the structural properties of highly ordered
perovskite solid solutions, and a mechanism, that involves these defects, may be
responsible for the existence and anomalous features of the incommensurate phases
in these alloys, were provided by Kornev and Bellaiche (2002). A rather short but
comprehensive review of the very recent use of ﬁrst-principle-derived approaches to
7.1 Perovskite solid solutions
136
investigate piezoelectricity in simple and complex ferroelectric perovskites may be
found in Bellaiche (2001). It is interesting to note that most of these investigations
were performed for heterovalent solid solutions and the case of homovalent alloys,
such as BST, is much less studied. In this sense it worth to mention the study of
Tanaka, Ota and Kawai (1996) where the Molecular Dynamics calculations have
been performed. The interatomic pairwise potential used in these calculations included a Coulomb interaction, Born-Mayer-type repulsive interaction, and a Van
der Waals attractive interaction. Although the giant dielectric constant in BST for
x=0.7 was explained, other ﬁne features of the phase transformations in this system
described above were not found. This clearly demonstrates the importance of ab
initio calculations for homovalent perovskite alloys, on one hand, and, on the other
hand, raises the question how to explain the peculiarities of phase transformations
in complex BST alloys where compositional long-range order is absent. Additional
questions of interest are what may be the reason for ferroelectric phase transformations in a BST alloy when the atomic fraction of Ba in BST is very small and why
ferrodistortive transformations occur when the atomic fraction of Sr in BST is very
small.
Summing up, there exists an interest for a study of this system in the 0 < x < 1
range of substitution for low temperatures, where phase transformations occur. Unfortunately, the experimental phase diagrams for BST solid solutions are usually
given for high temperatures as, for example, by McQuarrie (1955), where the temperatures are in the interval 1538–1703 K. The complete picture of the whole phase
diagram is lacking, although numerous experimental data specify the diﬀerences in
the low temperature phase transformations in BST, when composition is varied in
a wide range.
In this Chapter it is shown for the ﬁrst time that a statistical thermodynamic
approach combined with ab initio DFT-P3PW calculations allows to predict the
main features of the quasi-binary phase diagram for BST alloys in a wide range of
concentrations and to shed some light on the complicate picture of the sequence
of phase transformations in this system. The results clearly demonstrate that ﬁne
peculiarities of these transformations are decorated to a considerable extent by the
spinodal decomposition that occurs in this system at relatively low temperatures.
Formation of speciﬁc morphology of alloys, that is essentially diﬀerent between the
binodal and spinodal and below the spinodal phase enables to “explain” the unusual
7.2 Thermodynamic theory
137
behavior of both STO slightly doped by Ba and Ba-enriched solid solutions.
7.2
Thermodynamic theory
Here a statistical thermodynamic approach in modelling the formation of BST solid
solutions will be discussed. This will be combined with ab initio atomistic calculations. DFT-B3PW calculations that are going to be used here are applicable only
for the absolutely ordered structures. This enforces to formulate the problem in a
way that allows to extract the necessary energy parameters from the calculations
for these phases and than to apply these parameters to a study of the disordered
or partly ordered solid solutions to get the information on the thermodynamic behavior of the BST solid solution. From this microscopic study and from a survey of
experimental data (Mitsui, Nomura, Adachi, Harada et al., 1981), it follows that in
perovskite alloy BST Ba substitutes for Sr for all atomic fractions x. This allows to
focus only on the alloying sub-lattice, see e.g. Bellaiche and Vanderbilt (1998), and
to consider the solid solutions between these components on the sites of a simple
cubic lattice immersed in the external ﬁeld of the rest Ti and O ions and in the
ﬁeld of the electronic charge distribution created by these atoms. When the atomic
fraction of Ba changes, this may inﬂuence the external ﬁeld and change the charge
distribution that will be accounted by further DFT-B3PW calculations. The thermodynamics of such solid solution that one can call quasi-binary solid solution, may
be formulated with the help of the eﬀective mixing interatomic potential :
Ṽ (r, r ) = VAA (r, r ) + VBB (r, r ) − 2VAB (r, r ),
(7.1)
where VAA (r, r ), VBB (r, r ) and VAB (r, r ) are the eﬀective interatomic potentials
between Ba atoms (A), between Sr atoms (B), and between Ba and Sr atoms, respectively; r and r are the positions of the sites in a simple cubic lattice. The
eﬀective mixing interatomic potential (Eq. 7.1) describes the interactions of A and
B components in such a system in the ﬁeld of the rest atoms in the perovskite alloy.
The atomic fractions of Ba atoms or of Sr in this simple cubic solid solution can
be determined in the usual way. The total number of particles in this system is
conserved, being equal to the number of simple cubic lattice sites. This simpliﬁes
the application of the traditional thermodynamic theory of substitutional solid solu-
7.2 Thermodynamic theory
138
tions. The analysis of thermodynamic stability of this solid solution becomes a study
of the ordering and/or decomposition tendencies in such a binary system, and the
stability may be considered in terms of the phase diagram of the BST alloy. In the
present study the Concentration Wave (CW) approach, developed by Khachaturyan
(1983), is used. This theory has several advantages over other statistical theories
of alloys. One such advantage is that CW theory is formulated so as to use the
Fourier transforms of interatomic interaction potentials. Thus, it accounts formally
for the interactions in all coordination shells, and does not make the usually used
approximation of the ﬁrst, or ﬁrst and second, etc. nearest neighbor interactions.
In this theory the distribution of atoms A in a binary A-B alloy is described by
a single occupation probability function n(r). This function gives the probability to
ﬁnd the atom A (Ba) at the site r of the crystal lattice. The conﬁgurational part of
the free energy of formation of the solid solution per atom is given by:
F =
1
2N
Ṽ (r, r )n(r)n(r ) + kT
[n(r) · ln n(r) + (1 − n(r)) · ln(1 − n(r))].
r
r, r, r=r
(7.2)
The summation in Eq. 7.2 is performed over the sites of the Ising lattice that is
a simple cubic lattice in the present case, with atoms Ba and Sr distributed on it.
The function n(r), that determines a distribution of dissolved atoms in the ordering
phase, may be expanded in a Fourier series. It is presented as a superposition of
CWs:
1 Q(kjs )eikjs r + Q∗ (kjs )e−ikjs r ,
(7.3)
n(r) = cA +
2 j ,s
s
where cA is the concentration of particles A, eikjs r is a CW, kjs is a nonzero wave
vector deﬁned in the ﬁrst BZ of the disordered binary alloy, the index {js } numerates
the wave vectors in the BZ, that belong to the star s, and Q(kjs ) is the amplitude
of the CW. As it was shown by Khachaturyan (1983), all Q(kjs ) are linear functions
of the long-range-order (LRO) parameters of the superlattices that may be formed
on the basis of the Ising lattice of the disordered solid solution:
Q(kjs ) = ηs γs (js ),
(7.4)
where the ηs are the LRO parameters, and the γs (js ) are coeﬃcients that determine
the symmetry of the occupation probabilities n(r) (the symmetry of the super-
7.2 Thermodynamic theory
139
structure) with respect to rotation and reﬂection symmetry operations. The LRO
parameters are deﬁned in such a way that they should be equal to unity in a completely ordered state, where the occupation probabilities, n(r), are either unity or
zero on all the lattice sites {r}. This requirement completely deﬁnes the constants
γs (js ). This deﬁnition of the LRO parameters coincides with the conventional definition in terms of the occupation probabilities of sites in the diﬀerent sublattices.
Substituting of Eqs. 7.3 and 7.4 in the ﬁrst term of Eq. 7.2 gives the internal energy
of formation per atom for the ordering superstructure in a form:
1
1 2
∆U = Ṽ (0) · c2A +
γ (js )ηs2 Ṽ (kjs ),
2
2 s, j s
(7.5)
s
where Ṽ (kjs ) is the Fourier transform of the mixing interatomic potential and Ṽ (0)
is the same for the vector equal to zero. It may be shown that the value V (kjs ) is
the same for diﬀerent vectors kj belonging to the same star of vectors. Eqs. 7.2 and
7.5 deﬁne the Helmholtz free energy and internal energy of the ordering phases with
respect to the reference state.
In simulating the superstructures in quasi-binary solid solution the structures
given in Fig. 7.1 have been used. Occupation probabilities for these structures are
presented in the form of Eq. 7.3. They are found with the determination of vectors
kjs . Superstructure vectors kjs deﬁne the positions of the additional X-ray reﬂections
that appear when the binary system changes from a disordered state on the Ising
lattice to an ordered or partly ordered state. kjs determines new unit translations in
the reciprocal lattice arising from the reduction of the translation symmetry caused
by the ordering. These vectors describe the structures, which have the minimum
of V (k) from the symmetry considerations. To choose these vectors, the Lifshitz
criterion (Landau and Lifshitz, 1980) is used. According to this criterion the point
group of the vector contains the intersecting elements of symmetry. The stars of
vectors kjs for the simple cubic lattice are:
1) ( 12 00), (0 12 0), (00 12 );
2) ( 12 21 0), ( 12 0 12 ), (0 12 12 );
3) ( 12 21 12 ).
(7.6)
7.2 Thermodynamic theory
140
Ba
Ba
Sr
Sr
a)
b)
Ba
Sr
Ba
Sr
c)
d)
Sr
Ba
Ba
Sr
e)
f)
Ba
Sr
Sr
Ba
g)
h)
Sr
Ba
i)
Figure 7.1: Superstructures in quasibinary Bax Sr(1−x) TiO3 solid solutions that are
stable with respect to the formation of anti-phase boundaries.
7.2 Thermodynamic theory
141
Table 7.1: Occupation probabilities, n(r), stoichiometric compositions, xst , and the
energies of formation, ∆U , for the ordering phases in Bax Sr(1−x) TiO3 solid solutions.
Ṽ1 , Ṽ2 and Ṽ3 are Fourier transforms of the mixing potential in the kjs points that
correspond to the stars 1, 2, and 3 from Eq. 7.6.
n(r)
xst
γ
c + γ · η1 · eiπz
1
2
1
2
2
1
2 Ṽ (0)c
+ 18 Ṽ1 η12
c + γ · η2 · eiπ(x+y)
1
2
1
2
2
1
2 Ṽ (0)c
+ 18 Ṽ2 η22
c + γ · η3 · eiπ(x+y+z)
1
2
1
2
2
1
2 Ṽ (0)c
+ 18 Ṽ3 η32
1
4
1
4
3
4
− 14
2
1
2 Ṽ (0)c
+
iπ(x+y)
iπ(x+z)
c + γ · η2 · e
iπx
c + γ1 · η1 · e
+e
iπy
+e
iπ(y+z)
+e
iπ(x+y)
+ γ2 · η2 · e
c + γ1 · η1 · eiπx + eiπy + eiπz +
γ2 · η2 · eiπ(x+y) + eiπ(x+z) + eiπ(y+z) +
iπ(x+y+z)
γ3 · η3 · e
1
4
γ1 = γ2 =
∆U
3
4
γ1 = γ2 = − 14
1
8
γ1 = γ2 = γ3 =
7
8
2
1
2 Ṽ (0)c
1
4
+
2
3
32 Ṽ2 η2
2
1
16 Ṽ1 η1
+
2
1
32 Ṽ2 η2
1
8
γ1 = γ2 = γ3 = − 18
2
1
2 Ṽ (0)c
+
2
3
128 Ṽ2 η2
2
3
128 Ṽ1 η1 +
+
2
1
128 Ṽ3 η3
These vectors are given in units (2π/a), where a is the lattice parameter.
The structures displayed in Fig. 7.1(a–c), being ordered from the disordered
solid solution on the simple cubic lattice, are described by only one vector kjs . The
superstructure vector k31 = 2π
(00 12 ) deﬁnes the structure displayed in Fig. 7.1a.
a
The structure that is shown in Fig. 7.1b is deﬁned by the vector k12 = 2π
( 1 1 0),
a 22
( 1 1 1 ). The
while the structure shown in Fig. 7.1c is deﬁned by the vector k12 = 2π
a 222
superstructure from Fig. 7.1e is described, for example, by the combination of three
CWs with vectors kjs : k11 = 2π
( 1 00), k21 = 2π
(0 12 0), and k12 = 2π
( 1 1 0). The
a 2
a
a 22
occupation probabilities for the structures shown in Fig. 7.1 are presented in Table
7.1 together with the stoichiometric compositions and the energies of formation for
these phases, ∆U , with respect to the heterophase mixture xBTO+(1-x)STO. This
mixture is assumed to be considered as a standard case in present study. Table
7.1 contains a comprehensive list of the binary superstructures that may be formed
on the simple cubic lattice and are stable with respect to the formation of anti-
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
142
Table 7.2: Total energies, Etot , stoichiometric compositions, and equilibrium lattice
parameters, aeq for the structures (a–i) from Fig. 7.1. Here, the BTO and STO are
represented by a supercell (2 × 2 × 2) containing 40 atoms.
Structure xst
Etot , a.u.
aeq , Å
a
1/2
-2497.06046
3.9631
b
1/2
-2497.06005
3.9655
c
1/2
-2497.05988
3.9505
d
1/4
-2507.49056
3.9445
e
3/4
-2486.63722
3.9772
f
1/4
-2507.49030
3.9394
g
3/4
-2486.63705
3.9756
h
1/8
-2512.70707
3.9262
i
7/8
-2481.42596
3.9917
BTO
-2476.21745
4.0045
STO
-2571.92863
3.9030
phase boundaries according to the Lifshitz criteria. In Table 7.1 x and y are the
coordinates of the lattice sites of the Ising lattice and should be substituted in the
lattice parameter units. It is easy to check by direct substitution of coordinates of
the simple cubic lattice sites, that for the displayed structures in absolutely ordered
states and stoichiometric compositions the occupation probabilities are equal to
unity in the sites where Ba atoms are placed and are equal to zero in the sites with
Sr.
7.3
Application to BaxSr(1−x)TiO3 solid solutions
The reference state energy has been chosen in the conventional way (Kaufman and
Bernstein, 1970) as the energy of heterogeneous mixture xBTO+(1-x)STO. In the
present case it is calculated as the sum of weighted (according to atomic fraction)
total energies per lattice site for BTO and for STO. From the DFT-B3PW calculations the total energies and the equilibrium lattice parameter have been obtained
for all structures from Fig. 7.1. These data are collected in Table 7.2.
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
143
Table 7.2 demonstrates also the results of analogous calculations for BTO and
for STO, that are necessary for the further analysis. With the data from Table 7.2,
using the deﬁnition:
BT O
ST O
+ (1 − x) · Etot
)
∆U = Etot − (x · Etot
(7.7)
the energies of formation ∆U have been calculated for all absolutely ordered phases
from Fig. 7.1. These energies are positive. The states represented by the phases
considered in Fig. 7.1 and Table 7.1 have a higher energy than the reference state
and formation of the considered phases is unfavorable at T = 0 K with respect to
the heterophase mixture xBTO+(1-x)STO. A total solubility or decomposition in
absolutely disordered BST solid solution should occur. The obtained data allow to
calculate the energy parameter needed to describe the situation at T = 0. Solving
the system of equations:
1
∆Ua = Ṽ (0)c2 +
2
1
∆Ub = Ṽ (0)c2 +
2
1
∆Uc = Ṽ (0)c2 +
2
1
∆Uh = Ṽ (0)c2 +
2
1
Ṽ1 η12
8
1
Ṽ2 η22
8
1
Ṽ3 η32
8
3
3
1
Ṽ1 η12 +
Ṽ2 η22 +
Ṽ3 η32
128
128
128
(7.8)
for the parameters Ṽ (0), Ṽ1 , Ṽ2 and Ṽ3 , it was got Ṽ (0) = −0.149 eV per atom
in the quasi-binary solid solution BST. Here the indexes a–c and h correspond to
the phases from Fig. 7.1, c was taken equal to the stoichiometric composition of
the corresponding phases, and all LRO parameters were equal to unity. ∆Ua , ∆Ub ,
∆Uc , and ∆Uh were obtained from the data given in Table 7.2 with Eq. 7.7, and
they are equal to 0.3422 eV, 0.3534 eV, 0.3581 eV, and 0.2087 eV, respectively (per
cell of the BST solid solution). It has been assumed also that Ṽ (0), Ṽ1 , Ṽ2 and Ṽ3
are concentration-independent. This assumption is based mainly on the results of
the diﬀuse X-ray scattering data for the alloys (see Semenovskaya (1974a,b); Semenovskaya and Khachaturyan (1995, 1996)).
The condition n(r) = cA = const corresponds to the case of the disordered quasi-
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
144
binary solid solution when all LRO parameters in Eqs. 7.8 or in Eq. 7.5 are equal
to zero. Substitution of n(r) = cA into Eq. 7.2 gives the free energy of this solution:
1
F (c) = Ṽ (0) · c2 + kT [c ln c + (1 − c) ln(1 − c)] ,
2
(7.9)
where the index A is omitted. From simple thermodynamic considerations, it follows
that an equilibrium phase diagram remains unaﬀected if the free energy given by
Eq. 7.9 is replaced by:
1
F (c) = Ṽ (0) · c(1 − c) + kT [c ln c + (1 − c) ln(1 − c)] .
2
(7.10)
This expression includes the chemical potential term, and is more convenient be-
spinodal
solvus
500
Temperature, in K
400
T0
1
o
300
T"
o
o
o
200
T' o
o
o
2
100
C1
C0
C2
0
0.0
0.2
0.4
0.6
0.8
1.0
Atomic fraction of Ba in (BaxSr1-x)TiO3
Figure 7.2: The phase diagram of the quasi-binary disordered solid solution BST.
cause of its symmetry with respect to c = 1/2. The phase diagram of the quasibinary disordered solid solution BST calculated with Eq. 7.10 is given in Fig. 7.2.
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
145
It has the miscibility gap, and a decomposition reaction takes place because the
obtained value Ṽ (0) < 0. Although this result seems to be in contradiction with
the phase diagram presented by McQuarrie (1955), actually it does not, because the
data from McQuarrie (1955) correspond to much higher temperatures (∼1500–1700
K). At these temperatures the total solubility in quasi-binary solid solution has been
obtained with no formation of ordering phases, that coincides with data obtained by
McQuarrie (1955). Decomposition occurs at relatively low temperatures where the
thermodynamic measurements that are necessary to reproduce phase equilibrium in
an alloy are very diﬃcult because of extremely slow kinetics of the evolution of the
system towards the equilibrium state. The kinetics of the single-phase decomposition when the temperature lowers down in this case may be controlled by thermal
ﬂuctuations in the system or by some speciﬁc features of the Jahn-Teller-type interaction connected with charge transfer vibronic excitons (Vikhnin, Eglitis, Kapphan,
Kotomin et al., 2001; V.S.Vikhnin, R.I.Eglitis, S.E.Kapphan, G.Borstel et al., 2002).
The calculated phase diagram represents the case of the limited solid solubility
in this alloy. The solubility curve (bimodal) is shown in Fig. 7.2 by the bold line,
and the dashed line describes the spinodal. The solubility curve is determined by
the necessary minimum condition dF (c)/dc = 0. The spinodal curve is given by the
equation d2 F (c)/dc2 = 0. According to the suggested model, the two-phase region is
symmetric with respect to concentration c = 0.5. This follows from the assumption
that the energy parameter Ṽ (0) is concentration-independent.
To analyze the decomposition in the solid solution, let us start from the point 1
on Fig. 7.2. This point represents the high-temperature state of a perovskite alloy
with an equilibrium concentration of Ba atoms c0 at the temperature T0 . This is a
single-phase state, corresponding to a disordered solid solution in the alloy, when Ba
and Sr atoms randomly occupy the sites of the simple cubic lattice immersed in the
ﬁeld of the rest lattice with Ti and O atoms. Cooling of the system to temperature
T brings the system to the state shown by point 2, below the spinodal.
After annealing at this new temperature T the equilibrium two-phase state of
the solid solution on this simple cubic lattice is obtained. The thermodynamic
mechanism of the formation of this state is a decomposition of single-phase state
into two-phase state. This two-phase state is a mixture of two random solid solutions in the Ba-Sr sub-system. One phase is an extremely dilute solid solution of Ba
atoms, randomly distributed on the lattice sites with the equilibrium concentration
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
146
c1 (phase 1). The second phase is also a random solid solution of the same type as
the ﬁrst one, but with extremely high concentration of Ba atoms, c2 (phase 2). Thus,
the two-phase state represents the mixture of the phases: one is highly enriched with
Ba, whereas the second one is depleted of Ba atoms. The relative fraction of the
phase 2 in a two-phase mixture is deﬁned by the “lever rule” (DeHoﬀ, 1993), and
is equal to (c0 − c1 )/(c2 − c1 ), whereas the fraction of the phase 1, with its smaller
concentration of Ba atoms, is much higher, and is equal to (c2 − c0 )/(c2 − c1 ). If
the solubility regions are narrow, we have only a very small fraction of phase 2.
Nevertheless, it has to exist.
spinodal
solvus
500
Temperature, in K
400
T01
1
3
o
o
300
4
2
o
T' 1
o
o o
200
100
C3
0
0.0
0.2
0.4
0.6
0.8
1.0
Atomic fraction of Ba in (BaxSr1-x)TiO3
Figure 7.3: Phase diagram, the same as for Fig. 7.2.
The two-phase state that corresponds to the temperature T and atomic fraction c0 is characterized therefore as Ba-rich regions (with Ba atomic fraction c2 )
that are immersed in a Sr-enriched lattice with few Ba atoms randomly distributed
on its sites. These small Ba-rich regions are also random solid solutions, but the
concentration of Ba in them is very large and the number of sites occupied by Sr
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
147
correspondingly small. For the temperature T > T the atomic fraction of Ba atoms
in Ba-rich regions decreases, while the fraction of Sr on the sites increases in these
regions.
Let us consider now the case when, after cooling from the temperature T01 (points
1 or 3 in Fig. 7.3) to the temperature T1 , the system comes to the region of the
phase diagram between the binodal and the spinodal (points 2 or 4 in Fig. 7.3). It
is easy to see from Eq. 7.10 that the condition d2 F (c)/dc2 > 0 is satisﬁed in this
region of the phase diagram. For all points c inside this interval the curve F (c )
is concave, and this condition means that the homogeneous solid solution is stable
with respect to inﬁnitesimal heterogeneity. Indeed, if d2 F (c)/dc2 > 0 it is always
possible to choose an inﬁnitesimal region of concentrations c1 < c < c2 in the vicinity of the point c , where d2 F (c)/dc2 > 0, i.e. where the curve F (c) is concave. This
curve lies below the straight line connecting the points (c1 , F (c1 )) and (c2 , F (c2 )).
Therefore the homogeneous single-phase alloy is more stable than a mixture of two
phases having inﬁnitesimally diﬀerent compositions.
If a homogeneous alloy characterized by the condition d2 F (c)/dc2 > 0 at the
point c is unstable with respect to the formation of a two-phase mixture with cα
and cβ phase compositions that are substantially diﬀerent from the alloy composition, the alloy is nevertheless stable with respect to inﬁnitesimally small composition
heterogeneity. This is a metastable alloy, and the described situation corresponds to
the points 2 and 4 in Fig. 7.3. The decomposition reaction in this case should involve
the formation of a ﬁnite composition heterogeneity and follow the nucleation-andgrowth mechanism. A small increase of the atomic fraction of Ba beyond the value
c3 (see Fig. 7.3) to the right from the binodal curve will leave the quasi-binary solid
solution in single-phase state. The system will also remain in single-phase state if
the temperature T is changed to bring the “alloy” to the state above the binodal.
Thus, following these discussions, one can formulate a simple thermodynamic rule to
get nanoparticles of BTO in STO even if the Ba atomic fraction in BST is very small
(Sr-rich side of the phase diagram). The Ba-rich clusters will be obtained at low Ba
composition if the cooling process is such that, at the end, the BST system exists in
the region of the phase diagram between the binodal and spinodal with subsequent
decomposition into a two-phase state. A more complicated wave-like or percolation
structure will be obtained if, at the end of cooling, the system ﬁnds itself in the region below the spinodal on the phase diagram. At very high temperatures, alloying
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
148
by Ba atoms will leave the system in a one-phase state, namely a disordered Ba-Sr
quasi-binary solid solution immersed as a simple cubic Ising lattice in the lattice of
the rest, i.e. Ti and O, atoms. It is diﬃcult to reach the thermodynamic equilibrium
in this single-phase state at low temperature, because the solubility region at rather
low temperature T1 in Fig. 7.3 is extremely narrow. The decomposition reaction
for low concentration of Ba in BST involves the formation of a ﬁnite composition
heterogeneity and follows the nucleation-and-growth mechanism. Particles of the
Ba-rich phase that are formed in this region of phase diagram are well separated.
They have low connectivity and may be considered as isolated BTO clusters. The
number of Sr atoms in these clusters is extremely small. This situation is typical for
decomposition of a binary dilute solid solution with limited solubility (Cahn, 1965;
Rao and Rao, 1978). The analogous decomposition occurs at the Ba-rich side of
the phase diagram. In the thermodynamic analysis of a cluster of given size one is
interested neither in the history of its appearance nor in its future evolution. Here,
the cluster is considered as a static formation that is in internal thermodynamic
equilibrium. The cluster is treated as being in partial or complete thermodynamic
equilibrium with the ambient old phase. The obtained decomposition in BST solid
solution means that at low temperatures and small atomic fraction of Ba there exist
clusters in the old phase which consist of large number of Ba atoms, i.e. there are
the clusters of “almost pure” BTO in “almost pure” STO. When atomic fraction
of Ba in BST is large, one may obtain at low temperatures the clusters of “almost
pure” STO in “almost pure” BTO. The existence of such clusters is one of the basic
assumptions of the kinetic theory of nucleation that allow to develop a mathematical
formalism for a detailed description of the evolution of the process. The fact of the
existence of clusters reﬂects one of the most outstanding features of the nucleation
process – the initial localization of the new phase in nanoscopically small spatial
regions. It is in line with the fact that ﬁrst-order phase transitions occur along the
path of non-uniform transformations of the density of the old phase into the density
of the new phase.
Keeping in mind the proved changes of the morphology in the BST system when
the temperature decreases or the atomic fraction is varied, one may link the experimental facts on ferroelectric phase transformations with the spinodal decomposition
in this perovskite alloy.
As it was shown by Kiat, Dkhil, Dunlop, Dammak et al. (2002), the STO-type
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
149
antiferrodistortive phase exists up to the concentration of Ba ccr ∼ 0.094. The
critical concentration ccr separates the phase diagram in two regions, one with an
antiferrodistortive phase transition (c < ccr ) and one with a succession of three
BTO-type ferroelectric phase transitions (c > ccr ). It may be seen from our Fig. 7.2
that at the corresponding temperature, that is ∼100K, there is the transformation
from the structure with Ba-enriched BTO clusters in the STO matrix in the region
between the binodal and spinodal to the percolated, snake-like structure below the
spinodal, where these ferroelectric transformations may occur. At the same time
the local polarization inside the nonferroelectric antiferrodistortive phase observed
by Kiat, Dkhil, Dunlop, Dammak et al. (2002) may be associated according to this
approach with the formation of small Ba-rich separated clusters, that are formed between the solvus and spinodal. This explains the assumption made by Kiat, Dkhil,
Dunlop, Dammak et al. (2002) who measured a magnitude of spontaneous polarization in dilute BST, which is comparable to the value of spontaneous polarization
observed in the ferroelectric phases of the Ba-rich BST compounds. It is worth to
note in this context that the number of Ba-rich clusters in the left side of the phase
diagram and their ﬁne structure strongly depends on the regime of the temperature
decrease and may change the total picture of phase transformations in this region.
For example, if these clusters are very small (several nanometers) and their number is large, one may consider such state as a state when the ferroelectric phase
transformations are suﬃciently damped by the external pressure applied to these
clusters from the rest Sr-rich matrix. This pressure is caused by the mismatch of
the lattice parameters of BTO and STO. The glassy state reported by Lemanov,
Smirnova, Syrnikov and Tarakanov (1996), that is formed at very low temperatures
(below 20 K) and very small Ba concentration (c less then 0.035) in BST perovskite
solid solutions, may be associated with the cluster-type morphology of the solid solution between the spinodal and bimodal in the very bottom at the left corner of
the phase diagram in Fig. 7.2. In this context the question formulated by Lemanov,
Smirnova, Syrnikov and Tarakanov (1996) “is it really important to have oﬀ-center
impurity ions for glasslike behavior in these systems” has a special sense. Actually
one may have a speciﬁc morphology of the BST solid solution analogous to dilute
Ising ferromagnets (see, for example, Ziman (1979)), that may be interpreted as
the glassy state with suﬃciently suppressed (or even vanished) by the surrounding
matrix pressure ferroelectric properties, which are induced by the dipole moments
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
150
of oﬀ-center impurities. On the contrary, the mechanism of formation of quadrupole
moments due to elastic strains induced by the lattice mismatch between the Ba-rich
regions and STO matrix may be dominant in BST. The present approach, combining
thermodynamics with ab initio calculations is also supported by the experimental
data obtained by Tenne, Soukiassian, Zhu, Clark et al. (2003) where the existence
of polar Ba-rich nanoregions in dilute BST thin ﬁlms was evidently proved. It was
shown that the features of lattice dynamics in BST ﬁlms are remarkably similar to
those in the relaxor ferroelectric PMN. In relaxors the formation of polar nanoregions is caused by the compositional heterogeneity (Cross, 1987, 1994). As follows
from the present results, analogous heterogeneity may be obtained in BST, although
this happens at relatively low temperatures, and formation of Ba-rich clusters is just
the result of the spinodal decomposition in BST. Formation of polar nanoregions
in BST thus is not necessarily associated with the presence of oxygen vacancies to
cause the TO phonon hardening. This phonon hardening also may be stimulated
by the external pressure eﬀect on BTO clusters immersed in a STO matrix. The
fact that the analogous eﬀect was not observed in the bulk single crystal BST, may
be explained by the freezing down of the kinetics of the formation of such clusters
in the bulk in the corresponding measurements, or the number of clusters was not
suﬃcient to provide an interaction of Ba-rich nanoregions. At the same time, from
the microscopic point of view, the BST ﬁlm of the thickness about 1 µm studied
in Tenne, Soukiassian, Zhu, Clark et al. (2003) contains about 2.5 × 103 cubic cells
in the height and is thick enough to be considered as a medium where the spinodal
decomposition may occur.
Actually in the present non-empirical study of the phase diagram in quasi-binary
BST perovskite alloy, the ground state energies of the competing cubic phases are
considered, and a question that arises is to what extend the neglect of the ferroelectric transformations to low-temperature non-cubic phases is signiﬁcant and may
inﬂuence the predictions. To answer this question it should be reminded that the
energies of formation, ∆U from Eq. 7.7 deﬁne the competition of the ordering and
decomposition processes. These values are ∼0.2-0.35 eV, while the values of the
total energy diﬀerences between cubic, tetragonal, and rhombohedral structures in
BTO are ∼0.035-0.008 eV (i.e. at least smaller by one order of magnitude) as follows
from recent ab initio calculations (Chen, Chen and Jiang, 2002). Analysis of the potential energy surfaces of atomic displacements for Ba and Ti in BTO performed by
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
151
Chen, Chen and Jiang (2002) shows also that the corresponding potential barriers
for Ti<001>, and Ti<111> displacements do not exceed 0.02 eV and for Ba<001>
displacements the potential energy surface is a single well. For STO the potential
energy surface is a single-well potential for Ti<001>, Ti<111>, and Sr<111> displacements. These data justify the present consideration.
The model used in this study to calculate the phase diagram for BST is suﬃciently simpliﬁed. First of all, the mean-ﬁeld approximation used in present theory,
does not account for some ﬁne peculiarities of phase formation, including the correlation eﬀects in solid solutions. The approximation of pairwise interactions is
implicit in our study, even though many-body interactions may play an important
role changing the relative energies of the phases that compete in the phase diagram.
Several additional factors may also aﬀect the conditions of the phase formation in
the above-studied quasi-binary solid solution. Among them is small misﬁt in the lattice parameters of the “pure” BTO and “pure” STO. This means that it is necessary
to account the elastic part of the energy in the minimization of the free energy of the
perovskite alloy. The structure of the alloy in this case may become more complicated. It may include some analog of the Guinier-Preston zones that may be formed,
as in the case of small misﬁts in metallic alloys (Rao and Rao, 1978). Lattice strains
may lead to the formation of strain-modulated structures complicating the morphology of the structures formed in the alloy. Each of the above-mentioned factors needs
to be the subject of a special investigation. Nevertheless, the approach in terms of
the phase diagrams may be still extremely fruitful in understanding the main trends
in the ferroelectric or ferrodistortive phase transformations in perovskite solid solutions where they may be accompanied by the spinodal decomposition. From the
analysis of the phase diagram obtained in the present calculation it follows also that
in the case of BST perovskite solid solution with concentration close to unity the
analogous decomposition should occur with the formation of Sr-rich clusters immersed in Ba-rich BTO matrix. In this case ferroelectric phase transformatios of
the matrix would be accompanied by the ferrodistortive transformations in small
Sr-rich clusters between the binodal and spinodal. Also the glasslike state at these
compositions for low temperatures should be expected. For all compositions at low
temperatures the decomposition changes the morphology of solid solution and complicates the common picture of ferroelectric properties.
In conclusion of this Chapter, it can be noted that the distinguishing feature of
7.3 Application to Bax Sr(1−x) TiO3 solid solutions
152
the present approach is that the absolutely ordered super-structures in the Ba-Sr
simple cubic sublattice are considered. Although these structures are unstable with
respect to the decomposition, the results of total energy calculations allow to extract
the necessary energy parameters and thus to predict the phase diagram for this system. A novel approach applied to the BST system, enables to predict the conditions
when the Ba and Sr atom distribution are random, or when Ba atoms aggregate into
clusters on this simple cubic sublattice, leading to the formation of Ba-rich complexes
of “almost pure” BTO. As follows from this study, such nanoregions may be formed
in extremely dilute BST, when the temperature is lowered down in such a way that
the spinodal decomposition of the perovskite alloy occurs. This decomposition is
dictated by the general thermodynamic properties of the considered system. Similar
decomposition in Ba-rich region of BST allows to predict the formation of “almost
pure” STO nanoregions when the temperature decreases. The eﬀects of changing
the morphology of solid solution as the temperature and/or composition in the alloy
is varied, control the total pattern of ferroelectric or ferrodistortive phase transformations in BST. A novel theory could be applied to many perovskite systems, which
would permit the prediction of the conditions for a random A-B atom distribution,
or for A (or B) atoms to aggregate into clusters to form A/B-rich complexes with
corresponding ferroelectric properties even if the atomic fraction of these atoms in
the alloy is small. Summing up, in this Chapter a new physical mechanism explaining the eﬀect of external conditions on the ferroelectric phase transformations is
suggested. For a particular BST perovskite alloy it is deﬁnitely shown that this is
a spinodal decomposition. This demonstrates a nontrivial situation observed experimentally. Depending on the temperature and concentration of Ba and Sr atoms,
the picture of ferroelectric or ferrodistortive phase transformations is complicated
by the spinodal phase separation and by the formation of speciﬁc alloy morphologies
above or below the spinodal line.
Chapter 8
Conclusions
In this Thesis, reasonably good Gaussian-type basis sets for the ab initio simulation
of several key perovskite crystals have been developed and their adequacy has been
carefully tested. These basis sets contain three valence sp shells on the anion and
three on the cations. The Ti d electrons are described by three shells, a contraction
of three Gaussians for the inner part, and two single Gaussians for the outer part.
In comparison with the widely used standard basis sets, a polarization d -function
has been added on O, the inner core orbitals of Ti have been replaced by small-core
Hay-Wadt eﬀective core pseudopotential, and two most diﬀuse s and p Gaussians
have been used consistently as the separate basis functions on Ti, Ba, Sr, Pb. The
calculation with this basis is cheap, taking only a few minutes on a medium-sized
workstation. In order to understand the relationship between basis set ﬂexibility
and the selection of Hamiltonian, the elastic and electronic properties of bulk perovskite crystals have been carefully considered. The comparison of seven types of
Hamiltonians shows that the best agreement with the experimental results is obtained by the hybrid exchange techniques (B3LYP and B3PW). The polarization
orbital added to the basis set of oxygen atom allows to get the optical band gaps
of 3.57 eV, 3.42 eV and 2.87 eV for STO, BTO and PTO, respectively, which are
very close to those experimentally observed. The best representation of the bulk
properties has been obtained using the B3PW hybrid functional. Thus, it gives the
ground to recommend this computation scheme for further calculations on defective
perovskite structures (i.e. surfaces, interfaces, solid solutions, etc.).
The DFT/B3PW calculations on point defects in perovskite materials (the case
154
study of Fe impurity in SrTiO3 ) demonstrate the applicability of the suggested
method to get the convergence of periodic defect calculations to the limit of a single defect. This method could be very eﬃcient for many impurities in insulators
characterized by a high symmetry and when calculating forces is computationally
expensive. It has been demonstrated that the size of the cyclic cluster large enough
for a correct reproduction of the single Fe4+ impurity should be not smaller than
160 atoms. This is in contrast with many previous supercell calculations, where as
small as 2 × 2 × 2 extended supercells were used without any convergence analysis.
It should be mentioned here that the correct estimation of the optical band gap provided by DFT-B3PW scheme accompanied with a newly developed basis set allows
to reproduce reliably defect level positions within the optical age gap. The present
calculations have demonstrated the strong covalent bonding between unpaired electrons of Fe impurity and four nearest O ions relaxed towards an impurity. The
positions of Fe energy levels in a STO gap are very sensitive to the lattice relaxation
which was neglected in previous studies. Based on this, a considerable dependence
of the optical absorption bands of transition metals in perovskites on the external
or local stresses (e.g., in solid solutions, like Srx Ba1−x TiO3 ) can be predicted. This
is important for the interpretation of experimental data and device development.
The positions of the Fe energy levels with respect to the valence band top could be
checked by means of UPS spectroscopy whereas the local lattice relaxation around
iron and its high spin state by means of EXAFS.
The data obtained in surface structure DFT-B3PW calculations are in a good
agreement with ab initio theoretical results published previously and partly with
data obtained in experiments. The computed relaxed surface energies for AO and
BO2 terminations argue that the surfaces are quite stable, in agreement with Tasker’s
classiﬁcation and existing experiments. Also, the calculations on charge densities for
all perovskite surfaces demonstrate the presence of a weak polarity, predicted earlier
in literature for surfaces of ion-covalent crystals. The analysis of atomic dipole moments shows the cations on AO-terminated surfaces are strongly polarized along the
z-axis. The calculated diﬀerence electron density maps demonstrate an increasing
of covalency in Ti-O bonds for atoms near the surfaces for all perovskite surfaces,
and only weak covalency for Pb-O bond on PbO terminated surface. The absence
of surface electronic states in the upper valence bands for AO-terminated (001) surfaces of all perovskites and the presence of Pb 6s orbitals in the top of valence band
155
region of PTO could be important for the treatment of the electronic structure of
surface defects on perovskite surfaces as well as for adsorption and surfaces diﬀusion
of atoms and small molecules, relevant for catalysis.
The electronic structures of the STO(110) polar surfaces calculated using the ab
initio DFT-B3PW method have been compared with MIES and UPS(HeI) experiments performed by Prof. Kempter’s group in the Technische Universität Clausthal.
Besides giving good overall agreement with the observed O 2p emission, the calculations identify an additional peak close to zero binding energy for the vacuum heated,
Ti-terminated surface as due to Ti3+ 3d occupied states, giving direct evidence for
the termination of the reconstructed, microfaceted surface by reduced Ti3+ -ions.
In order to give a theoretical prediction for technologically important Bax Sr1−x TiO3
perovskite solid solutions, a thermodynamic formalism based on ab initio DFTB3PW calculations has been developed in collaboration with Prof. S. Dorfman
(Technion Institute of technology, Haifa, Israel). This approach is based on the ordered super-structures on the Ba-Sr simple cubic sublattice. Although these structures are unstable with respect to the decomposition, the results of total energy
calculations allow to extract the necessary energy parameters and to calculate the
phase diagram for the solid solutions (alloys). A novel approach applied to the
Bax Sr1−x TiO3 system enables to predict the conditions when the Ba and Sr atom
distribution is random, or when Ba atoms aggregate into clusters on this simple
cubic sublattice, leading to the formation of Ba-rich complexes of “almost pure”
BaTiO3 . As follows from this study, such nanoregions may be formed in an extremely dilute Bax Sr1−x TiO3 , when the temperature is lowered down in such a way
that the spinodal decomposition of the perovskite alloy occurs. This decomposition is dictated by the thermodynamic properties of the considered system. Similar
decomposition in Ba-rich region of Bax Sr1−x TiO3 allows to predict the formation
of “almost pure” SrTiO3 nanoregions when the temperature decreases. The varied
morphology of perovskite solid solution with the temperature and/or composition
aﬀects the ferroelectric or ferrodistortive phase transitions in Bax Sr1−x TiO3 . The
novel theory could be applied to many perovskite systems, which would permit the
prediction of the conditions for a random A-B atom distribution, or for A (or B)
atoms to aggregate into clusters to form A/B-rich complexes with the wanted ferroelectric properties, even if the atomic fraction of these atoms in the alloy is relatively
small.
Appendix A
Hay-Wadt eﬀective core
pseudopotentials for Ti, Sr, Ba
and Pb
In many respects, most physical properties of solids are dependent on the valence
electrons to a much greater extent than on the core electrons. The pseudopotential
approximation exploits this by removing the core electrons and by replacing them
and the strong ionic potential by a weaker pseudopotential that acts on a set of
pseudo wave functions rather than the true valence wave functions. An ionic potential, valence wave function and the corresponding pseudopotential and pseudo
wave function are illustrated schematically in Fig. A.1. The valence wave functions
oscillate rapidly in the region occupied by the core electrons due to the strong ionic
potential in this region. These oscillations maintain the orthogonality between the
core wave functions and the valence wave functions, which is required by the Pauli
exclusion principle. The pseudopotential is constructed, ideally, so that its scattering properties or phase shifts for the pseudo wave functions are identical to the
scattering properties of the ion and the core electrons for the valence wave functions,
but in such a way that the pseudo wave functions have no radial nodes in the core region. Thus, to generate an appropriate eﬀective core pseudopotential (ECP) from ab
initio atomic wave functions, the following basic rules should be taken into account.
First of all, the atomic orbitals should be partitioned into valence and core orbitals.
157
Ψpseudo
ΨV
rc
r
Vpseudo
Z/r
Figure A.1: A schematic illustration of all-electron (red lines) and pseudo- (blue
lines) potentials and their corresponding wavefunctions. The radius at which allelectron and pseudopotential values match is rc . Taken from Payne, Teter, Allan,
Arias et al. (1992).
In the present study, the valence electrons of Pb were taken to be the outermost s
and p orbitals, i.e., 6s and 6p, that means a large core ECP containing [Xe]4f 14 5d10
core orbitals. For Ti, Sr, and Ba atoms small core ECPs have been adopted, i.e. the
outermost core orbitals of these atoms corresponding to the ns2 np6 conﬁguration are
not replaced by ECP, but are treated on an equal ground with the valence orbitals:
3s2 3p6 3d2 4s2 for Ti, 4s2 4p6 5s2 for Sr, and 5s2 5p6 6s2 for Ba, respectively. After this
partitioning, atomic wave functions should be generated to provide valence orbitals
(φl ) of all angular momentum, 0 < l < L, where L is typically by one greater than
the highest l of any core orbital. For heavy atoms (Z > 36) relativistic eﬀect should
be taken into account. Then, the all-electron valence orbitals should be transformed
into smooth, nodeless pseudo-orbitals (φ̃l ) that match the all-electron orbitals in the
valence region. After that, numerical eﬀective potentials (Ul ) are derived for each
l by inverting the one-electron Schrödinger equations for φl using φ̃l . This requires
that φ̃l in the ﬁeld of Ul , gives the same orbital energy, l , as φl . The total potential
0
1
2
2
2
0
1
2
2
2
-10.0000000
-51.8427816
-9.1429145
3.0000000
19.4825579
207.3349279
235.6744501
-166.8784387
50.2966943
63.5089754
26.0996084
5.6022573
5.2171069
5.0000000
5.5348822
177.8419384
107.4207153
-71.9065902
p − d potential
81.4730696
72.6496724
31.8128213
6.1664468
5.8268347
s − d potential
265.3263909
47.7687815
11.8903334
1
2
2
dk
d potential
ζk
Ti
nk
0
1
2
2
2
0
1
2
2
2
0
1
2
2
0
1
2
2
2
Sr
nk
2.9989022
25.6552669
183.1818533
58.4384739
4.9551189
25.4472367
203.8002780
155.0518740
39.3605192
65.8291301
32.7282621
21.1146030
9.1071292
2.8110754
3.0056451
26.7064119
74.5756901
63.1742121
20.2961162
d − f potential
92.1201991
46.8132559
48.6566432
14.9503238
3.4268785
p − f potential
59.3240631
55.2038472
20.4692092
3.9588141
s − f potential
-0.384323
-20.6174271
-101.1737744
-38.7743603
-4.6479243
dk
f potential
782.3804631
124.6542338
36.9874966
9.8828819
3.2829588
ζk
0
1
2
2
2
0
1
2
2
2
0
1
2
2
0
1
2
2
2
2
Ba
nk
2.8131160
55.3050626
149.5513402
50.5078553
4.9191812
39.3872075
335.07533584
131.2535153
36.3175025
95.2752536
34.6604608
15.4891040
5.0015895
1.3236266
2.9764140
46.0571141
117.1658588
54.0130815
15.5784906
d − f potential
99.4922261
68.2711309
36.1518181
10.1051537
2.0232648
p − f potential
140.3669200
39.1436547
12.9553493
2.4308751
s − f potential
-0.0834652
-33.3257671
-190.8607232
-55.1984172
-18.0236340
-2.1978281
dk
f potential
620.9690488
146.3648826
42.3207114
11.2135151
3.6963891
1.3169502
ζk
0
1
2
2
2
2
0
1
2
2
2
2
0
1
2
2
2
2
0
1
2
2
2
2
0
1
2
2
2
2
Pb
nk
Table A.1: Eﬀective core potentials for Ti, Sr, Ba, and Pb.
2.8115386
65.0367205
212.7868545
72.1053175
33.0140940
-5.7708461
4.8754911
63.9148102
148.1064358
47.3106301
21.0306702
-7.0930772
3.2161388
55.7386086
121.4168351
19.3456064
15.3675168
6.1298724
128.1021322
54.8029154
24.5529308
8.1144792
1.6931290
0.7670500
4.1353628
67.5128446
258.7373107
113.2478264
34.1680201
-6.5531956
f − g potential
68.8336005
24.2815874
9.4532762
2.4788185
2.4789161
0.5551738
d − g potential
67.8966454
24.9898225
10.7052939
3.2792568
0.8452522
0.6416245
p − g potential
132.4248796
47.2376044
17.6312727
5.4744712
1.2634856
0.7651447
s − g potential
-0.1789605
-54.3972337
-199.7061759
-79.1223941
-24.9869020
-4.4397939
dk
g potential
376.5803786
86.4840014
26.6784276
9.4261986
2.7101719
0.8792031
ζk
158
159
is then given in terms of projection operators Pl = |ll| by
U (r) = UL (r) +
L−1
[Ul (r) − UL (r)]Pl .
(A.1)
l=0
For computational convenience, an analytic form for U (r) is obtained by ﬁtting (Nc
is the number of valence electrons)
r2 [Ul (r) − UL (r)],
r [Ul (r) − Nc /r],
2
l = 0, 1, . . . , L − 1,
(A.2)
l=L
to Gaussian functions of the form
dk rnk exp(−ζk r2 ),
(A.3)
k
where nk = 0, 1, or 2. The linear (dk ) and nonlinear (ζk ) parameters have been
optimized for many elements in the periodic table by Hay and Wadt (1984c,b,a).
The analytic ﬁts to the numerical ECP for the Ti, Sr, Ba, and Pb are given in Table
A.1. In this Table the diﬀerence potentials (e.g., s − f potential) refer to r2 (Ul − UL )
and the potentials UL with no angular momentum projectors in Eq. A.1 refer to
r2 (UL − Nc /r) as in Eq. A.2.
Appendix B
Calculation of the elastic constants
When a body is experiencing some forces from its surrounding, it is said to be
stressed. In fact it can also exert force on neighboring parts. These forces can be
divided into body-forces such as gravity acting on each elements of the body and
proportional to the volume and forces exerted on its surface by the surrounding
material. These forces are proportional to the area of the surface. The stress is
homogeneous if the forces acting on the surface of a non-moving element of ﬁxed
shape are independent of the position in the stressed body. Consider a unit cube
with edges parallel to the reference axes X, Y, Z, also noted 1, 2 and 3, respectively.
The basis is necessary to describe the forces, that are vectors. This cube experiences
homogeneous stress exerted by the surrounding. Then these forces are described by
three terms for every face: one perpendicular to and two within the given face.
These forces can be labelled according their orientation (X, Y, Z or 1, 2, 3) and
that of the face, in fact the normal to the face (X, Y, Z or 1, 2, 3). So, the stress
component perpendicular to the face perpendicular to X is labelled σ11 , where the
ﬁrst 1 indicates the orientation of the force (X) and the second the face or more
exactly its normal (X). So σ31 describes the component parallel to Z on the face
perpendicular to the X direction. The σii are the normal components and the σij, j=i
are the shear components. The σij deﬁne a tensor because the stress is independent
of the basis, only its representation is dependent on it. The tensor is deﬁned with
reference to directions. The stress tensor is a second rank tensor and needs two
161
indices. The tensor is represented as:


σ11 σ12 σ13
σ21 σ22 σ23  .
σ31 σ32 σ33
(B.1)
The unit volume is static, so there is no moment about any axes, this yields σij = σji .
So the stress tensor is symmetrical for homogeneous stress:


σ11 σ12 σ13
σ12 σ22 σ23  .
(B.2)
σ13 σ23 σ33
Now it contains 6 independents components instead of 9. In the case of nonhomogeneous stress, it can be shown that in the absence of any body-torque (a
torque proportional to volume as in an anisotropic polarized material) the stress
tensor remains symmetrical.
The deformation of a solid is described by strain. This deformation is expressed
on the basis of the relative displacement of points. It is not expressed with respect
to some origin. In the one-dimensional case the strain for the arbitrary section AB
is deﬁned as the ratio of the increase of length (∆u) to the original length (∆x).
The strain at some point P is deﬁned as:
du
∆u
=
.
∆x→0 ∆x
dx
e = lim
(B.3)
The strain e is without dimension. The change of distance is given by u = u0 + ex.
The displacement can be homogeneous (e is constant), that is proportional to the
distance, or inhomogeneous (e is not constant). The change of distance between two
points along one direction (i.e. ∆u2 ) is dependent on the projection of the relative
position vector supported by the points along the reference directions (δx, δy and
δz). The strain is deﬁned by nine terms in three-dimension cases:


∂u1
∂u1
1
e11 = ∂u
e
=
e
=
12
13
∂x
∂y
∂z
e = ∂u
∂u2
∂u2 
2
e
=
e
=
(B.4)
 21
22
23
∂x
∂y
∂z  .
∂u3
∂u3
∂u3
e31 = ∂x e32 = ∂y e33 = ∂z
The change of the relative position vector is:
∆ui =
∂ui
∂ui
∂ui
∆x +
∆y +
∆z, i = 1, 2, 3.
∂x
∂y
∂z
(B.5)
162
The strain can be expressed as the sum of two tensors: a symmetrical one and an
anti-symmetrical one:
(B.6)
eij = ij + ωij ,
where ij is symmetrical:
1
ij = (eij + eji ) = ij
2
(B.7)
1
ωij = − (eij − eji ) = −ωij .
2
(B.8)
and ωij is anti-symmetrical:
The symmetrical part [ij ], is deﬁned as the strain tensor.
A material is stretched if it experiences stresses. So strain in any direction
depends on the diﬀerent stress components. Strain parallel to Y is related to stress
parallel to Z and perpendicular to the surface normal to Z. The exact reaction of the
material to stress is determined by its structure, and therefore symmetry. Within the
elastic limit, strain and stress are linearly related by some constants characteristic
of the material:
σ = c,
(B.9)
where c is called the elastic or stiﬀness constant. From this, c is a tensor because
it is a feature of the material. Its rank is four because any strain component ij can
contribute to any stress component kl. For example:
σ11 = c1111 σ11 + c1112 σ12 + c1113 σ13 + c1121 σ21 +
+ c1122 σ22 + c1123 σ23 + c1131 σ31 + c1132 σ32 + c1133 σ33 . (B.10)
The set of elastic constants form a tensor, this tensor contains 81 elements. Due
to the symmetry of stress and strain tensors, the elastic constant tensor must be
symmetrical:
(B.11)
cijkl = cjikl = cijlk = cjilk .
This reduces the number of independent elements from 81 to 21.
Because of the symmetry of the ﬁrst two, and the last two indices, it is possible
163
to use a 2 × 2 matrix-like notation for strain, stress and therefore the elastic tensor.
Pairs of indices are contracted according to the rules:
11
1
22
2
33
3
32 or 23
4
31 or 13
5
21 or 12
6
In order to obtain a convenient and compact expression the tensors transformed as:




σ11 σ12 σ13
σ1 σ6 σ5
σ21 σ22 σ23  −→ σ6 σ2 σ4 
(B.12)
σ31 σ32 σ33
σ5 σ4 σ3




11 12 13
1 21 6 12 5
21 22 23  −→  1 6 2 1 4 
(B.13)
2
2
1
1
31 32 33
3
2 5 2 4
and the elastic tensor is

c1111 c1122 c1133
c2211 c2222 c2233

c3311 c3322 c3333

c2311 c2322 c2333

c1311 c1322 c1333
c1211 c1222 c1233
written:
c1123
c2223
c3323
c2323
c1323
c1223
c1113
c2213
c3313
c2313
c1313
c1213


c1112
c11
c21
c2212 



c3312 
 −→ c31
c41
c2312 


c51
c1312 
c1212
c61
c12
c22
c32
c42
c52
c62
c13
c23
c33
c43
c53
c63
c14
c24
c34
c44
c54
c64
c15
c25
c35
c45
c55
c65

c16
c26 

c36 

c46 

c56 
c66
(B.14)
Using these representation, it becomes:
σi =
cij j .
(B.15)
j
In order to calculate elastic constants one needs a relation between them and
the energy. Strain is equivalent to a displacement, stress to a force. The work done
by the stress component σi acting on the cube faces, and moving the faces by di is:
dW = σi di .
From it:
dW =
cij di ,
(B.16)
(B.17)
j
but also:
∂W
= cij j
∂i
(B.18)
164
and
∂
∂j
∂W
∂i
= cij .
(B.19)
If the deformation process is isothermal and reversible, the work done is equal to
the change of the free energy dE. But since free energy is a state function here,
speciﬁed in terms of strain components, the order of diﬀerentiation is not relevant
and cij = cji . Integrating the work equation, the strain energy per unit volume then
is:
1 cij i j .
(B.20)
2 ij, 1...6
So, ﬁtting the change of energy between relaxed and strained states as function of
the strains by polynomials should yield the elastic constants as the coeﬃcients of
the polynomials. Unfortunately, the stressed state has a low symmetry with respect
to the relaxed one, and numerous distortions have to be applied to extract every
elastic modulus.
For an unspeciﬁed material, 21 components deﬁne the elastic constant tensor.
However, symmetry will reduce this number to some extent. For cubic systems, only
3 elements are independent(c11 , c12 and c44 ), as the symmetry analysis shows that:
c11 = c22 = c33
c44 = c55 = c66
c12 = c13 = c23
cij = 0 i = 1, 6 j = 4, 6 i = j
Then, according to Eq. B.20 the elastic energy ∆Ee for unit volume can be deﬁned
as:
2δEe = c11 1 1 + c12 1 2 + c12 1 3 + c12 2 1 + c11 2 2 + c12 2 3 +
+ c12 3 1 + c12 3 2 + c11 3 3 + c44 4 4 + c44 5 5 + c44 6 6 (B.21)
By choosing the strain elements, the elastic components can be estimated. For
strain:


δ 0 0
0 0 0
(B.22)
0 0 0
165
it becomes:
1
δEe = c11 1 1
2
(B.23)
If the elastic energy is ﬁtted with a polynomial, at least of order 2 (bδ 2 + cδ 3 . . .),
c11 = 2b.
(B.24)
Thus, using the strain elements:


δ 0 0
0 −δ 0 ,
0 0 0
c11 − c12 = b
and c44 can be deﬁned as:
c44 =
using the strain elements:

0
0
0
0
0
1
δ
2
1
b
2

0
1 
δ .
2
0
(B.25)
(B.26)
(B.27)
(B.28)
Appendix C
List of Acronyms
AC – Auger Capture
LCAO – Linear Combination of Atomic Or-
AD – Auger Deexcitation
BF – Bloch Function
BMN-BZ – Ba(Mg1/3 Nb2/3 )O3 -BaZrO3
bitals
LDA – Local Density Approximation
LEED – Low Energy Electron Diﬀraction
BS – Basis Set
BSSE – Basis Set Superposition Error
LRO – Long-Range-Order
LSDA – Local Spin Density Approximation
BST – Bax Sr(1−x) TiO3
BTO – BaTiO3 (Barium Titanate)
MEIS – Medium Energy Ion Scattering
MIES – Metastable Impact Electron Spec-
BZ – Brillouin Zone
BZT – Ba(Zn1/3 Ta2/3 )O3
troscopy
PDOS – Projected Density of States
CB – Conduction Band
CO – Crystalline Orbital
CPU – Central Processing Unit
PMN-PT – Pb(Mg1/3 Nb2/3 )O3 -PbTiO3
PTO – PbTiO3 (Lead Titanate)
PW – Plane Waves
CW – Concentration Wave
DFT – Density Functional Theory
PZN-PT – Pb(Zn1/3 Nb2/3 )O3 -PbTiO3
QM – Quantum Mechanical
DM – Density Matrix
DOS – Density of States (total)
RHEED – Reﬂection High Energy Electron
Diﬀraction
ECP – Eﬀective Core Pseudopotential
SCF – Self Consistent Field
FLAPW – Full-Potential Linearized Aug- SCM – SuperCell Model
mented Plane Wave
FWHM – Full Width Half Maximum
GGA – Generalized Gradient Approximation
SCT – Srx Ca(1−x) TiO3
SM – Shell Model
STO – SrTiO3 (Strontium Titanate)
GPT – Gaussian Product Theorem
GTF – Gaussian Type Function
SXRD – Surface X-Ray Diﬀraction
UPS – Ultraviolet Photoelectron Spectroscopy
HF – Hartree-Fock
KS – Kohn and Sham
VB – Valence Band
XPS – X-ray Photoelectron Spectroscopy
LAPW – Linearized Augmented Plane Waves
Presentation of the results of the
present study
Main publications
I. R. A. Evarestov, R. I. Eglitis, S. Piskunov, E. A.Kotomin, and G. Borstel,
Large scale ab initio simulations of Fe-doped SrTiO3 perovskites, Mat. Res.
Soc. Symp. Proc., Vol. 731 (2002), W3.12.1, p. 1-6.
II. R. A. Evarestov, S. Piskunov, E. A. Kotomin, and G. Borstel, Single impurities
in insulators: Ab initio study of Fe-doped SrTiO3 , Phys. Rev. B. 67 (2003),
064101.
III. S. Piskunov, E. Heifets, R. I. Eglitis, and G. Borstel, Bulk properties and
electronic structure of SrTiO3 , BaTiO3 , and PbTiO3 perovskites: an ab initio
HF/DFT study, Comp. Mat. Sci. (2003), in press.
IV. S. Piskunov, E. Heifets, E. A. Kotomin, R. I. Eglitis, and G. Borstel, Atomic
and electronic structures of SrTiO3 , BaTiO3 , and PbTiO3 (001) surfaces: a
ﬁrst-principles DFT-B3PW study, Surf. Sci. (2003), submitted.
V. A. Gunhold, L. Beuermann, K. Gömann, G. Borchardt, V. Kempter, W.
Maus-Friedrichs, S. Piskunov, E. A. Kotomin, and S. Dorfman, Electronic
and atomic structure of thermally treated SrTiO3 (110) surfaces, Surface and
Interface Analysis (2003), in press.
VI. S. Dorfman, S. Piskunov, E. A. Kotomin, and D. Fuks, Low-temperature compositional heterogeneity in Bax Sr1−x TiO3 solid solutions from ab initio study,
Phys. Rev. B. (2003), submitted.
168
Presentations at workshops and conferences
I. A poster at European Summer school “Ab initio Modelling in Solid State
Chemistry”, Torino, Italy, September 11 – 15, 2001.
Poster’s title: “Hartree-Fock calculations on the SrTiO3 crystal in a cubic
phase”
S. Piskunov, Yu. F. Zhukovskii, E. A. Kotomin, and Yu. N. Shunin.
II. A talk at 3rd international conference “Advanced optical materials and devices”, Riga, Latvia, August 19 – 22, 2002.
http://www.fpd.lu.lv/AOMD3/index.html
Talk’s title: “Large scale ﬁrst-principles calculations of Fe-doped SrTiO3 ”
S. Piskunov, R. A. Evarestov, R. I. Eglitis, E. A. Kotomin, and G. Borstel.
III. A poster at International conference “Information technologies and management”, Riga, Latvia, April 16 – 17, 2003.
Poster’s title: “B3PW and B3LYP exchange-correlation techniques in CRYSTAL computer code: the case of ABO3 perovskites”
S. Piskunov, E. Heifets, E. A. Kotomin, and Yu. N. Shunin.
IV. A talk in the Block Seminar, Quakenbrück, Germany, July 10 – 11, 2003.
http://www.physik.uni-osnabrueck.de/pp/
Talk’s title: “Ab initio DFT-B3PW study on SrTiO3 , BaTiO3 , and PbTiO3
(001) surfaces”. Presented by S. Piskunov.
Acknowledgments
I would like to express my deep gratitude to my research superviser Prof. Dr. Gunnar Borstel for many fruitful scientiﬁc discussions and his constant support in solving
various administrative problems.
I am greatly indebted to Prof. Dr. Eugene Kotomin for his invaluable support,
encouragement, and patience. Without his help this work would have never been
done.
My special thanks are due to Dr. Eugene Heifets for many explanations and
helpful discussions in the beginning of my study.
I am indebted to Prof. Dr. Robert A. Evarestov, who’s criticism, comments, and
arguments allowed me to gain a lot of knowledge in fundamental solid state physics.
I would like to say my thanks to Prof. Dr. Simon Dorfman, Privatdozent Dr.
Andrei Postnikov, Prof. Dr. David Fuks, Dr. Roberts Eglitis, Dr. Yuri Zhukovskii
and Stefan Bartkowski for many useful and stimulating discussions.
I am very thankful to my Latvian friends Oleg and Marina Sychev for their essential moral support, as well as I am much obliged to my German friend Amalia
Ament for giving me the possibility to practice my German language and learn more
about Germany and the Germans beyond oﬃcial relationships.
I am very grateful to my parents and my sister for their omnifarious support
during my Ph.D. study, and last, but of course not least, I thank my own family –
my wife and son, simply for what they are.
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