The electronic structure of perfect and defective perovskite crystals [Elektronische Ressource] : ab initio hybrid functional calculations / by Sergejs Piskunovs

Sergejs Piskunovs

Sujets

• Physik

Informations

• Publié par: universitat_osnabruck
• Publié le: 01 janvier 2003
• Nombre de lectures: 34
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

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 Osnabruc¨ k
by
Sergejs Piskunovs
Thesis Advisor: Prof. Dr. Gunnar Borstel
October 2003Contents
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: SrTiO:Fe.....................163
2.2.3 Calculationsonsurfaces.....................17
2.3 Motivation.................................18
3 DFT/HF formalism and methodology 21
Introduction21
3.1 DFTformalism..............................2
3.1.1 Schr¨odingerequation.......................2
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...............3
3.1.7 Hybridexchangefunctionals.33
3.1.8 Spin-density functional theory . . . . . . . . . . . . . . . . . . 35CONTENTS 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 SrTiO :Fe 753
Introduction75
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 ABO (001) surfaces . . . . . . . . . . . . . . . . 943
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
SrTiO (110) polar surfaces . . . . . . . . . . . . . . . . . . . . . . . . 1263CONTENTS iii
7 Low-temperature compositional heterogeneity in Ba Sr TiO solidx 1−x 3
solutions 133
Introduction................................13
7.1 Perovskitesolidsolutions.........................13
7.2 Thermodynamictheory..........................137
7.3 Application to Ba Sr TiO solidsolutions .............142x (1−x) 3
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 170List of Figures
2.1 A prototype cubic structure of a perovskite crystal with the formula
unit ABO,whereA=Sr,BaorPb,andB=Ti. ............ 53
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, Hoc¨ hst and Goldmann (1976). . . . . . . . . . . 9
2.4 Schematic illustration of three possible surfaces of cubic ABO per-3
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 ABO (001) surfaces. Arrows show3
the directions of atomic displacements. The surface rumpling s is
shown for surface layer. Interlayer distances d and d are based12 23
on the positions of relaxed metal ions which are known to be much
stronger electron scatterers than oxygen ions (Bickel, Schmidt, Heinz
and Muller,¨ 1989). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12LIST OF FIGURES v
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. . . . . . . 40
3.2 FlowchartoftheCRYSTALcode. ...................5
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
topofvalenceband. ...........................6
4.2 The calculated total and projected density of states (DOS and PDOS)
for three perovskites. a) STO, b) BTO, c) PTO. . . . . . . . . . . . 69
4.3 The difference electron density plots for three perovskites calculated
usingDFTB3PW:a)STO,b)BTO,c)PTO.Theelectrondensity
plots are for AO-(001) (left column), (110) (middle column), and
TiO -(001) (right column) cross sections. Isodensity curves are drawn2
−3 −3from -0.05 to +0.05 e a.u. with an increment of 0.005 e a.u. ...72
5.1 (a) Schematic view of the Fe impurity in STO with asymmetric e re-g
laxation of six nearest O atoms, (b) The relevant energy levels before
andafterrelaxation............................84
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
3 3drawn from 0.8 to 0.8 e a.u. with an increment of 0.0022 e a.u. ,
(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. .................................8
6.1 Models for simulating surfaces starting from a perfect 3D crystal. . . 92
6.2 Schematic illustration of the slab unit cells for ABO (001) surfaces:3
a) AO-terminated, b) TiO -terminated, c) asymmetrical termination. 932LIST OF FIGURES vi
6.3 Schematic illustration of the SrTiO-terminated SrTiO (110) 9-layer3
slab unit cells: a) slab without vacancies (unstable, cannot exist
due to infinite dipole moment perpendicular to the surface), b) TiO-
terminated SrTiO (110) surface (unreconstructed surface, stable ac-3
cording to Heifets, Kotomin and Maier (2000), also named as “unre-
constructed surface”, see last section), c) Ti-terminated SrTiO (110)3
surface (reconstructed surface). Vacancies created on Sr and O sites
areshownasgrenspots. ........................94
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 dis-
placements. Upper panels - AO termination, lower panels - TiO2
termination. ...............................96
6.5 Difference electron density maps in the cross section perpendicular
to the (001) surface ((110) plane) with AO-, TiO and asymmetrical2
−3terminations. Isodensity curves are drawn from -0.05 to +0.05 e a.u.
−3with an increment of 0.0025 e a.u. . a) STO, b) BTO, c) PTO. . . 109
6.6 Calculated electronic band structures for STO bulk and surfaces. . . 111
6.7 e band s for BTO bulk and . . 112
6.8 Calculated electronic band structures for PTO bulk and surfaces. . . 113
6.9 Total and projected DOS for the bulk STO. . . . . . . . . . . . . . . 117
6.10 Total and projected DOS for the SrO-terminated surface. . . . . . . 118
6.11 Total and projected DOS for the STO TiO -terminated surface. . . . 1192
6.12 Total and projected DOS for the bulk BTO. . . . . . . . . . . . . . . 120
6.13 Total and projected DOS for the BaO-terminated surface. . . . . . . 121
6.14 Total and projected DOS for the BTO TiO -terminated surface. . . 1222
6.15 Total and projected DOS for the bulk PTO. . . . . . . . . . . . . . . 123
6.16 Total and projected DOS for the PbO-terminated surface. . . . . . . 124
6.17 Total and projected