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 Osnabr¨uck
by
Sergejs Piskunovs
Thesis Advisor: Prof. Dr. Gunnar Borstel
October 2003

Contents
1 Introduction 1
2 Basic perovskite crystals: Strontium, Barium, and Lead Titanates 3
Introduction................................ 3
2.1 Experimentalresults ........................... 5
2.1.1 Bulkcrystals ........................... 5
2.1.2 Impuritydefectsinperovskites ................. 9
2.1.3 Surfaces .............................. 10
2.2 Previoustheoreticalresults........................ 14
2.2.1 Bulkperovskites ......................... 14
2.2.2 Point defects: SrTiO3:Fe..................... 16
2.2.3 Calculationsonsurfaces ..................... 17
2.3 Motivation................................. 18
3 DFT/HF formalism and methodology 21
Introduction................................ 21
3.1 DFTformalism .............................. 22
3.1.1 Schr¨odingerequation....................... 22
3.1.2 Total energy through the density matrices . . . . . . . . . . . 24
3.1.3 Hohenberg-Kohntheorems.................... 26
3.1.4 Energyfunctional......................... 27
3.1.5 Localdensityapproximation................... 29
3.1.6 Generalizedgradientapproximation............... 33
3.1.7 Hybridexchangefunctionals................... 33
3.1.8 Spin-density functional theory . . . . . . . . . . . . . . . . . . 35

CONTENTS ii
3.2 Practical implementation of DFT/HF calculation scheme . . . . . . . 35
3.2.1 Selectionofbasisset ....................... 35
3.2.2 Auxiliary basis sets for the exchange-correlation functionals . . 47
3.2.3 Evaluation of the integrals. The Coulomb problem . . . . . . 49
3.2.4 Reciprocalspaceintegration................... 52
3.2.5 SCFcalculationscheme ..................... 54
3.3 One-electronproperties.......................... 56
3.3.1 Properties in a direct space; population analysis . . . . . . . . 56
3.3.2 Properties in a reciprocal space; band-structure and density
ofstates .............................. 58
4 Calculations on bulk perovskites 61
Introduction................................ 61
4.1 Computationaldetails .......................... 62
4.2 Bulkproperties .............................. 63
4.3 Electronicproperties ........................... 67
5 Point defects in perovskites: The case study of SrTiO3:Fe 75
Introduction................................ 75
5.1 A consistent approach for a modelling of defective solids . . . . . . . 76
5.2 Results for perfect STO and supercell convergence . . . . . . . . . . . 83
5.3 ResultsforasingleFeimpurity ..................... 84
6 Two-dimensional defects in perovskites: (001) and (110) surfaces. 90
Introduction................................ 90
6.1 The choice of a model for surface simulation . . . . . . . . . . . . . . 91
6.2 Calculations on the ABO3(001) surfaces . . . . . . . . . . . . . . . . 94
6.2.1 Surfacestructures......................... 94
6.2.2 Electronicchargeredistribution .................100
6.2.3 Density of states and band structures . . . . . . . . . . . . . . 110
6.3 Calculations on TiO- and Ti-terminated
SrTiO3(110) polar surfaces . . . . . . . . . . . . . . . . . . . . . . . . 126

CONTENTS iii
7 Low-temperature compositional heterogeneity in BaxSr1−xTiO3solid
solutions 133
Introduction................................133
7.1 Perovskitesolidsolutions.........................133
7.2 Thermodynamictheory..........................137
7.3 Application to BaxSr(1−x)TiO3solidsolutions .............142
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 A prototype cubic structure of a perovskite crystal with the formula
unit ABO3,whereA=Sr,BaorPb,andB=Ti. ............ 5
2.2 The BTO and PTO crystals. Schematic sketch of a ferroelectric tran-
sition into a tetragonal broken-symmetry structure, where the origin
has been kept at the Ti atom. The arrows indicate atomic displace-
ments. In the structure shown, the polarization is along [001]. . . . . 6
2.3 The photoelectron energy distribution curves for STO and BTO.
Taken from Battye, H¨ochst and Goldmann (1976). . . . . . . . . . . 9
2.4 Schematic illustration of three possible surfaces of cubic ABO3per-
ovskites (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. . . . . 11
2.5 One of possible relaxations of the ABO3(001) surfaces. Arrows show
the directions of atomic displacements. The surface rumpling sis
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 M¨uller, 1989). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12