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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 . . . . . . . . . . . . . . . . . Lead Titanates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 5 9 10 14 14 16 17 18 . . . . . . . . . . 21 21 22 22 24 26 27 29 33 33 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 47 49 52 54 56 56 . 58 . . . . 61 61 62 63 67 . . . . 75 75 76 83 84 surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 90 91 94 94 100 110 . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 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 . . . . . . . . . solid . . . . . . . . . . . . . . . . . . solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 . 133 . 133 . 137 . 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 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . 6 . 9 . 11 . 12 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 . 94 . 96 . . . . . . . . . . . . . 109 111 112 113 117 118 119 120 121 122 123 124 125 . 128 . 129 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 . 99 . 101 . 101 . 102 . 104 . 106 . 106 . 107 . 114 . 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. Bibliography Abe, K. and S. Komatsu (1995). Ferroelectric properties in epitaxially grown Bax Sr1x TiO3 thin ﬁlms. J. Appl. Phys. 77(12), 6461. Abramov, Y. A., V. G. Tsirelson, V. E. Zavodnik, S. A. Ivanov et al. (1995). The chemical bond and atomic displacements in SrTiO3 from X-ray diﬀraction analysis. Acta Cryst. B51, 942. Adamo, C., M. Ernzerhof and G. E. Scuseria (2000). The meta-GGA functional: Thermochemistry with a kinetic energy density dependent exchange-correlation functional. J. Chem. Phys. 112(6), 2643. Akbas, M. A. and P. K. Davies (1998). Ordering-induced microstructures and microwave dielectric properties of the Ba(Mg1/3 Nb2/3 )O3 -BaZrO3 system. J. Am. Ceram. Soc. 81(3), 670. Akhtar, M. J., Z. N. Akhtar, R. A. Jackson and C. R. A. Catlow (1995). Computer simulation of strontium titanate. J. Am. Ceram. Soc. 78, 421. Auciello, O., J. F. Scott and R. Ramesh (1998). The physics of ferroelectric memories. Physics Today 51(7), 22. Baldereschi, A. (1973). Mean-value point in the Brillouin zone. Phys. Rev. B 7(12), 5212. Bando, H., Y. Ochiai, Y. Haruyama, T. Yasue et al. (2001). Metallic electronic states on SrTiO3 (110) surface: An in situ conduction measurement. J. Vac. Sci. Technol. A 19(4), 1938. BIBLIOGRAPHY 171 Baranek, P., G. Pinarello, C. Pisani and R. Dovesi (2000). Ab initio study of the cation vacancy at the surface and in bulk MgO. Phys. Chem. Chem. Phys. 2(17), 3893. Barrett, J. H. (1952). Dielectric constant in perovskite type crystals. Phys. Rev. 86(1), 118. Battye, F. L., H. Höchst and A. Goldmann (1976). Photoelectron studies of the BaTiO3 and SrTiO3 valence states. Solid State Commun. 19(3), 260. Becke, A. D. (1988a). Correlation energy of an inhomogeneous electron gas: A coordinate-space model. J. Chem. Phys. 88(2), 1053. Becke, A. D. (1988b). Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A. 38(6), 3098. Becke, A. D. (1993a). Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 98(7), 5648. Becke, A. D. (1993b). A new mixing of HartreeFock and local density-functional theories. J. Chem. Phys. 98(2), 1372. Bell, R. O. and G. Rupprecht (1963). Elastic constants of strontium titanate. Phys. Rev. 129(1), 90. Bellaiche, L. (2001). Piezoelectricity of ferroelectric perovskites from ﬁrst principles. Current Opinion in Solid State and Materials Science 6(1), 19. Bellaiche, L. and D. Vanderbilt (1998). Electrostatic model of atomic ordering in complex perovskite alloys. Phys. Rev. Lett. 81(6), 1318. Bickel, N., G. Schmidt, K. Heinz and K. Müller (1989). Ferroelectric relaxation of the srtio3 (100) surface. Phys. Rev. Lett. 62(17), 2009. Boon, M. H., M. S. Methfessel and F. M. Müller (1986). Singular integrals over the Brillouin zone: the analytic-quadratic method for the density of states. J. Phys. C: Solid State Phys. 19, 5337. Bottin, F., F. Finocchi and C. Noguera (2003). Stability and electronic structure of the (1×1) SrTiO3 (110) polar surfaces by ﬁrst-principles calculations. Phys. Rev. B 68, 035418. BIBLIOGRAPHY 172 Brause, M., B. Braun, D. Ochs, W. Maus-Friedrichs et al. (1998). Surface electronic structure of pure and oxidized non-epitaxial Mg2 Si layers on Si(111). Surf. Sci. 398(1-2), 184. Bredow, T., R. A. Evarestov and K. Jug (2000). Implementation of the Cyclic Cluster Model in Hartree-Fock LCAO calculations of crystalline systems. Phys. Stat. Solidi. (b) 222, 495. Bredow, T., G. Geudtner and K. Jug (2001). Development of the cyclic cluster approach for ionic systems. J. Comput. Chem. 22(1), 89. Brunen, J. and J. Zegenhagen (1997). Investigation of the SrTiO3 (110) surface by means of LEED, scanning tunneling microscopy and Auger spectroscopy. Surf. Sci. 389, 349. Burke, J. P. P. K. and M. Ernzerhof (1996). Generalized gradient approximation made simple. Phys. Rev. Lett. 77(18), 3865. Burton, B. P. and E. Cockayne (1999). Why Pb(B,B’)O3 perovskites disorder at lower temperatures than Ba(B,B’)O3 perovskites. Phys. Rev. B 60(18), R12542. Cahn, J. W. (1965). Phase separation by spinodal decomposition in isotropic systems. J. Chem. Phys. 42(1), 93. Cappelini, G., S. Bouette-Russo, B. Amadon, C. Noguera et al. (2000). Structural properties and quasiparticle energies of cubic SrO, MgO and SrTiO3 . J. Phys.: Condens. Matter 12, 3671. Ceperley, D., G. V. Chester and M. H. Kalos (1977). Monte Carlo simulation of a many-fermion study. Phys. Rev. B 16(7), 3081. Ceperley, D. M. and B. J. Alder (1980). Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45(7), 566. Chadi, D. J. (1977). Special points for Brillouin-zone integrations. Phys. Rev. B 16(4), 1746. Chadi, D. J. and M. L. Cohen (1973). Special points in the brillouin zone. Phys. Rev. B 8(12), 5747. BIBLIOGRAPHY 173 Challacombe, M. and J. Cioslowski (1994). Eﬃcient implementation of the HillerSucher-Feinberg identity for the accurate determination of the electron density. J. Chem. Phys. 100(1), 464. Charlton, G., S. Brennan, C. A. Muryn, R. McGrath et al. (2000). Surface relaxation of SrTiO3 (001). Surf. Sci. 457, L376. Chen, Z. X., Y. Chen and Y. S. Jiang (2002). Comparative study of abo3 perovskite compounds. 1. ATiO3 (A = Ca, Sr, Ba, and Pb) perovskites. J. Phys. Chem. B 106(39), 9986. Cheng, C., K. Kunc and M. H. Lee (2000). Structural relaxation and longitudinal dipole moment of SrTiO3 (001) (1×1) surface. Phys. Rev. B 62(15), 10409. C.Lee, W. Yang and R. G. Parr (1988). Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 37(2), 785. Cohen, R. E. (1992). Origin of ferroelectricity in perovskite oxides. Nature 358, 136. Cohen, R. E. (1996). Periodic slab lapw computations for ferroelectric batio3 . J. Phys. Chem. Solids 57(10), 1393. Cohen, R. E. (1997). Surface eﬀects in ferroelectrics: periodic slab computations for batio3 . Ferroelectrics 194, 323. Cohen, R. E. and H. Krakauer (1990). Lattice dynamics and origin of ferroelectricity in BaTiO3 : Linearized-augmented-plane-wave total-energy calculations. Phys. Rev. B 42(10), 6416. Cohen, R. E. and H. Krakauer (1992). Ferroelectrics 136, 65. Cora, F. and C. R. A. Catlow (1999). Faraday Discussions 114: The Surface Science of Metal Oxides, chapter QM investigations on perovskite-structured transition metal oxides: bulk, surfaces and interfaces, 421–442. Royal Society of Chemistry, London. Cross, L. E. (1987). Relaxor ferroelectrics. Ferroelectrics 76, 241. BIBLIOGRAPHY 174 Cross, L. E. (1994). Relaxor ferroelectrics: An overview. Ferroelectrics 151, 305. Curtiss, L. A., K. Raghavachari, P. C. Redfern and J. A. Pople (1997). Assessment of Gaussian-2 and density functional theories for the computation of entalpies of formation. J. Chem. Phys. 106(3), 1063. Curtiss, L. A., P. C. Redfern, K. Raghavachari and J. A. Pople (1998). Assessment of Gaussian-2 and density functional theories for the computation of ionization potentials and electron aﬃnities. J. Chem. Phys. 109(1), 42. Deak, P. (2000). Choosing models for solids. Phys. Status Solidi B 217(1), 9. DeHoﬀ, R. T. (1993). Thermodynamics in materials science. McGraw-Hill, New York. Dirac, P. A. M. (1930a). Exchange phenomena in the Thomas atom. Proc. Camb. Phil. Soc. 26, 376. Dirac, P. A. M. (1930b). Note on exchange phenomena in the Thomas-Fermi atom. Proc. Camb. Phil. Soc. 26, 376. Donnerberg, H. (1994). Geometrical microstructure of Fe3+ N b -VO defects in KNbO3 . Phys. Rev. B 50(13), 9053. Donnerberg, H. (1999). Atomic simulation of electrooptic and magnetooptic oxide materials. Springer, Berlin. Dreizler, R. M. and E. K. U. Gross (1990). Density functional theory : an approach to the quantum many-body problem. Springer, Berlin. Engel, G. E. (1997). Linear response and the exchange-correlation hole within a screened-exchange density functional theory. Phys. Rev. Lett. 78(18), 3515. Evarestov, R. A., V. A. Lovchikov and I. I. Typitsyn (1983). Hartree-Fock exchange and LCAO approximation in the band structure calculations of solids. Phys. Status Solidi B 117(1), 417. Evarestov, R. A. and V. P. Smirnov (1983). Special points of the Brillouin zone and their use in the solid state theory. Phys. Status Solidi B 119(1), 9. BIBLIOGRAPHY 175 Evarestov, R. A. and V. P. Smirnov (1997a). Site Symmetry in Crystals: Theory and Applications, volume 108 of Springer Series in Solid State Sciences. SpringerVerlag, Berlin, 2 edition. Evarestov, R. A. and V. P. Smirnov (1997b). Symmetrical transformation of basic translation vectors in the supercell model of imperfect crystals and in the theory of special points of the Brillouin zone. J. Phys. C 9(14), 3023. Evarestov, R. A. and V. P. Smirnov (1999). Supercell model of V-doped TiO2 : Unrestricted Hartree-Fock calculations. Phys. Status Solidi B 215(2), 949. Evarestov, R. A., V. P. Smirnov and D. E. Usvyat (2003). Local properties of the electronic structure of cubic SrTiO3 , BaTiO3 and PbTiO3 crystals in Wanniertype function approach. Submitted to Solid State Communications. Evarestov, R. A. and I. I. Tupitsyn (2002). The HartreeFock method and densityfunctional theory as applied to an inﬁnite crystal and to a cyclic cluster. Phys. Solid State 44(9), 1656. Fermi, E. (1928). A statistical method for the determination of some atomic properties and the application of this method to the theory of the periodic system of elements. Z. Phys. 48, 73. Fischer, G. J., Z. Wang and S. Karato (1993). Elasticity of CaTiO3 , SrTiO3 and BaTiO3 perovskites up to 3.0 GPa: the eﬀect of crystallographic structure. Phys. Chem. Minerals 20, 97. Foley, M. and P. A. Madden (1996). Further orbital-free kinetic-energy functionals for ab initio molecular dynamics. Phys. Rev. B 53(16), 10589. George, A. M., J. Íñiguez and L. Bellaiche (2001). Anomalous properties in ferroelectrics induced by atomic ordering. Nature 413, 54. Ghosez, P., E. Cockyane, U. V. Waghmare and K. M. Rabe (1999). Lattice dynamics of BaTiO3 , PbTiO3 , and PbZrO3 : A comparative ﬁrst-principles study. Phys. Rev. B 60(2), 836. Goniakowski, J. and C. Noguera (1996). The concept of weak polarity: an application to the SrTiO3 (001) surface. Surf. Sci. 365, L657. BIBLIOGRAPHY 176 Gonis, A., X. G. Zhang, J. M. MacLaren and S. Crampin (1990). Multiple-scattering Green-function method for electronic-structure calculations of surfaces and coherent interfaces. Phys. Rev. B 42(7), 3798. Gunhold, A., L. Beuermann, M. Frerichs, V. Kempter et al. (2003a). Island formation on 0.1 at.% La-doped SrTiO3 (100) at elevated temperatures under reducing conditions. Surf. Sci. 523(1), 80. Gunhold, A., L. Beuermann, K. Gömann, G. Borchardt et al. (2003b). Electronic and atomic structure of thermally treated SrTiO3 (110) surfaces. Submitted to Surf. Sci. Gunhold, A., K. Gömann, L. Beuermann, M. Frerichs et al. (2002). Geometric structure and chemical composition of SrTiO3 surfaces heated under oxidizing and reducing conditions. Surf. Sci. 507-510, 447. Gunnarsson, O., M. Jonson and B. I. Lundqvist (1979). Descriptions of exchange and correlation eﬀects in inhomogeneous electron systems. Phys. Rev. B 20(8), 3136. Gunnarsson, O. and B. I. Lundqvist (1976). Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B 13(10), 4274. Günter, P. and J. P. Huignard, editors (1988). Photorefractive Materials and Their Application: Topics in Applied Physics, volume 61, 62. Springer Verlag, Berlin. Harada, Y., S. Masuda and H. Ozaki (1997). Electron spectroscopy using metastable atoms as probes for solid surfaces. Chem. Rev. 97(6), 1897. Harris, J. and R. O. Jones (1974). The surface energy of a bounded electron gas. J. Phys. F 4(8), 1170. Hay, P. J. and W. R. Wadt (1984a). Ab initio eﬀective core potentials for molecular calculations. Potentials for K to Au including the outermost core orbitals. J. Chem. Phys. 82(1), 299. Hay, P. J. and W. R. Wadt (1984b). Ab initio eﬀective core potentials for molecular calculations. Potentials for main group elements Na to Bi. J. Chem. Phys. 82(1), 284. BIBLIOGRAPHY 177 Hay, P. J. and W. R. Wadt (1984c). Ab initio eﬀective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg. J. Chem. Phys. 82(1), 270. Hedin, L. (1965). New method for calculating the one-electron Green’s function with application to the electron-gas problem. Phys. Rev. 139(3A), A796. Heifets, E., R. I. Eglitis, E. A. Kotomin, J. Maier et al. (2002). First-principles calculations for SrTiO3 (100) surface structure. Surf. Sci. 513(1), 211. Heifets, E., W. A. Goddard III, E. A. Kotomin, R. I. Eglitis et al. (2003). Ab initio calculations of the SrTiO3 (110) polar surface. Submitted to Phys. Rev. B. Heifets, E., E. A. Kotomin and J. Maier (2000). Semi-empirical simulations of surface relaxation for perovskite titanates. Surf. Sci. 462(1–3), 19. Hellwege, K. H. and A. M. Hellwege, editors (1969). Ferroelectrics and Related Substances, volume 3 of New Series. Landolt-Bornstein, Springer Verlag, Berlin. Group III. Hemmen, R. and H. Conrad (1991). New interpretation of Penning spectra from alkali-metal atoms chemisorbed on metal surfaces. Phys. Rev. Lett. 67(10), 1314. Henrick, V. E. and P. A. Cox (1994). The Surface Science of Metal Oxides. Cambridge University Press, New-York. Hikita, T., T. Hanada, M. Kudo and M. Kawai (1993). Structure and electronic state of the tio2 and sro terminated srtio3 (100) surfaces. Surf. Sci. 287/288, 377. Hohenberg, P. and W. Kohn (1964). Inhomogeneous electron gas. Phys. Rev. 136(3B), B864. Homepage (a). http://www.chimifm.unito.it/teorica/crystal/basis sets/mendel.html. Homepage (b). http://www.chimifm.unito.it/teorica/crystal/crystal.html. Homepage (c). http://www.cse.clrc.ac.uk/cmg/crystal. Hood, R. Q., M. Y. Chou, A. J. Williamson, G. Rajagopal et al. (1997). Quantum monte carlo investigation of exchange and correlation in silicon. Phys. Rev. Lett. 78(17), 3350. BIBLIOGRAPHY 178 Hybertsen, M. S. and S. G. Louie (1986). Spin-orbit splitting in semiconductors and insulators from the ab initio pseudopotential. Phys. Rev. B 34(4), 2920. Ikeda, A., T. Nishimura, T. Morishita and Y. Kido (1999). Surface relaxation and rumpling of tio2 -terminated srtio3 (001) determined by medium energy ion scattering. Surf. Sci. 433-435, 520. Ishidate, T. and S. Sasaki (1989). Elastic anomaly and phase transitio of BaTiO3 . Phys. Rev. Lett. 62(1), 67. Jepson, O. and O. K. Anderson (1971). The electronic structure of h.c.p. Ytterbium. Solid State Comm. 9(20), 1763. Kalkstein, D. and P. Soven (1971). A Green’s function theory of surface states. Surf. Sci. 85(1), 85. Kaufman, L. and H. Bernstein (1970). Computer calculation of phase diagrams: with special reference to refractory metals. Acad. Press, New York. Kawashima, S., M. Nishida, I. Ueda and H. Ouchi (1983). Ba(Zn1/3 Ta2/3 )O3 ceramics with low dielectric loss at microwave frequencies. J. Am. Ceram. Soc. 66(6), 421. Khachaturyan, A. G. (1983). Theory of structural transformations in solids. Wiley, New York. Kiat, C. M. J. M., B. Dkhil, M. Dunlop, H. Dammak et al. (2002). Structural evolution and polar order in Sr1−x Bax TiO3 . Phys. Rev. B 65, 224104. Kimura, S., J. Yamauchi, M. Tsukada and S. Watanabe (1995). First-principles study on electronic structure of the (001) surface of srtio3 . Phys. Rev. B 51(16), 11049. King-Smith, R. D. and D. Vanderbilt (1994). First-principles investigation of ferroelectricity in perovskite compounds. Phys. Rev. B 49(9), 5828. Kohn, W. and L. J. Sham (1965). Self-consiestent equations including exchange and correlation eﬀects. Phys. Rev. 140(4A), A1133. BIBLIOGRAPHY 179 Kornev, I. A. and L. Bellaiche (2002). Planar defects and incommensurate phases in highly ordered perovskite solid solutions. Phys. Rev. Lett. 89, 115502. Kotecki, D. E., J. D. Baniecki, H. Shen, R. B. Laibowitz et al. (1999). (Ba,Sr)TiO3 dielectrics for future stacked-capacitor DRAM. IBM J. Res. Develop. 43(3), 367. Landau, L. D. and E. M. Lifshitz (1980). Statistical Physics, volume 5 of Course of theoretical physics. Pergamon Press, Oxford, 3 edition. Langreth, D. C. and M. J. Mehl (1983). Beyond the local-density approximation in calculations of ground-state electronic properties. Phys. Rev. B 28(4), 1809. Lemanov, V. V., E. P. Smirnova, P. P. Syrnikov and E. A. Tarakanov (1996). Phase transitions and glasslike behavior in Sr1−x Bax TiO3 . Phys. Rev. B 54(5), 3151. Levy, M. (1979). Universal variational functionals of electron densities, ﬁrst-order density matrices, and natural spin-orbitals and solution of the ν-representability problem. Proc. Natl. Acad. Sci. USA 76(12), 6062. Levy, M. (1982). Electron densities in search of Hamiltonians. Phys. Rev. A 26(3), 1200. Li, Z., M. Grimsditch, C. M. Foster and S. K. Chan (1996). Dielectric and elastic properties of ferroelectric materials at elevated temperature. J. Phys. Chem. Solids 57(10), 1433. Lichanot, A., P. Baranek, M. Mérawa, R. Orlando et al. (2000). VOH and VOD centers in alkaline-earth oxides: An ab initio supercell study. Phys. Rev. B 62(19), 12812. Lieb, E. H. (1981). Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53(4), 603. Lines, M. E. and A. M. Glass (1977). Principles and Applications of Ferroelectrics and Related Materials. Clarendon Press, Oxford. Liu, R. S., Y. C. Cheng, J. M. Chen, R. G. Liu et al. (1998). Crystal and electronic structures of (Ba,Sr)TiO3 . Mater. Lett. 37, 285. BIBLIOGRAPHY 180 Lo, W. J. and G. A. Somorjai (1978). Temperature-dependent surface structure, composition, and electronic properties of the clean SrTiO3 (111) crystal face: Lowenergy-electron diﬀraction, Auger-electron spectroscopy, electron energy loss, and ultraviolet-photoelectron spectroscopy studies. Phys. rev. B 17(12), 4942. Makov, G., R. Shah and M. C. Payne (1996). Periodic boundary conditions in ab initio calculations. II. Brillouin-zone sampling for aperiodic systems. Phys. Rev. B 53(23), 15513. Mallia, G., R. Orlando, C. Roetti, P. Ugliengo et al. (2001). F center in LiF: A quantum mechanical ab initio investigation of the hyperﬁne interaction between the unpaired electron at the vacancy and its ﬁrst seven neighbors. Phys. Rev. B 63, 235102. Matsumoto, K., T. Hiuga, K. Takada and H. Ichimura (1986). Ba(Mg1/3 Ta2/3 )O3 ceramics with ultra-low loss at microwave frequencies. In Proceedings of the 8th IEEE International Symposium on Application of Ferroelectrics, 118. Institute of Electrical and Electronic Engineers, New York. Maus-Friedrichs, W., M. Frerichs, A. Gunhold, S. Krischok et al. (2002). The characterization of SrTiO3 (001) with MIES, UPS(HeI) and ﬁrst-principles calculations. Surf. Sci. 515, 499. McQuarrie, M. (1955). J. Am. Ceram. Soc. 38, 444. Meyer, B., J. Padilla and D. Vanderbilt (1999). Faraday Discussions 114: The Surface Science of Metal Oxides, chapter Theory of PbTiO3 , BaTiO3 , and SrTiO3 surfaces, 395–405. Royal Society of Chemistry, London. Michel-Calendini, F. M. and K. A. Müller (1981). Interpretation of charge transfer bands in Fe doped SrTiO3 crystals. Solid State Commun. 40, 255. Mitáš, L. and R. M. Martin (1994). Quantum Monte Carlo of nitrogen: Atom, dimer, atomic, and molecular solids. Phys. Rev. Lett. 72(15), 2438. Mitsui, T., S. Nomura, M. Adachi, J. Harada et al. (1981). Landolt-Bornstein, New Series, Group III: Crystal and Solid State Physics, Ferroelectrics and Related Substances, subvolume a: Oxides, volume 16, chapter Numerical Data and Functional Relationships in Science and Technology, 400–449. Springer-Verlag, Berlin. BIBLIOGRAPHY 181 Mitsui, T. and W. B. Westphal (1961). Dielectric and X-Ray studies of Cax Ba1−x TiO3 and Cax Sr1−x TiO3 . Phys. Rev. 124(5), 1354. Monkhorst, H. J. and J. D. Pack (1976). Special points for Brillouin-zone integrations. Phys. Rev. B 13(12), 5188. Moreno, J. and J. M. Soler (1992). Optimal meshes for integrals in real- and reciprocal-space unit cells. Phys. Rev. B 45(24), 13891. Moretti, P. and F. M. Michel-Calendini (1986). Mn(IV) and Cr(III) impurities in cubic BaTiO3 : Theoretical study through a molecular-orbital model. Phys. rev. B 34(12), 8538. Mulliken, R. S. (1955a). Electronic population analysis on LCAO-MO molecular wave functions. I. J. Chem. Phys. 23(10), 1833. Mulliken, R. S. (1955b). Electronic population analysis on LCAO-MO molecular wave functions. II. Overlap populations, bond orders, and covalent bond energies. J. Chem. Phys. 23(10), 1841. Mulliken, R. S. (1955c). Electronic population analysis on LCAO-MO molecular wave functions. III. Eﬀects of hybridization on overlap and gross AO populations. J. Chem. Phys. 23(12), 2338. Mulliken, R. S. (1955d). Electronic population analysis on LCAO-MO molecular wave functions. IV. Bonding and antibonding in LCAO and Valence-Bond theories. J. Chem. Phys. 23(12), 2343. Nakamatsu, H., H. Adachi and S. Ikeda (1981). Electronic structure of the valence band for perovskite-type titanium double oxides studied by XPS and DV-Xα cluster calculations. Journal of Electron Spectroscopy and Related Phenomena 24, 149. Noguera, C. (1996). Physics and Chemistry at Oxide Surfaces. Cambridge University Press, New-York. Northrup, J. E. and M. L. Cohen (1984). Total energy of the adatom and pyramidalcluster models for si(111). Phys. Rev. B 29(4), 1966. BIBLIOGRAPHY 182 Ochs, D., W. Maus-Friedrichs, M. Brause, J. Günster et al. (1996). Study of the surface electronic structure of mgo bulk crystals and thin ﬁlms. Surf. Sci. 365(2), 557. Ordejon, P. (2000). Linear scaling ab initio calculations in nanoscale materials with SIESTA. Phys. Status Solidi B 217(1), 335. Padilla, J. and D. Vanderbilt (1997). Ab initio study of batio3 surfaces. Phys. Rev. B 56(3), 1625. Padilla, J. and D. Vanderbilt (1998). Ab initio study of srtio3 surfaces. Surf. Sci. 418, 64. Park, S.-E. and T. R. Shrout (1997). Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals. J. Appl. Phys. 82(4), 1804. Parr, R. G. and W. Yang (1989). Density-functional theory of atoms and molecules. Oxford Univ. Press, Oxford. Payne, M. C., M. P. Teter, D. C. Allan, T. A. Arias et al. (1992). Iterative minimisation techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev. Mod. Phys. 64(4), 1045. Pendry, J. B. (1980). Reliability factor for LEED calculations. J. Phys. C.: Solid. St. Phys. 13, 937. Peng, C. H., J. F. Chang and S. Desu (1992). Mater. Res. Soc. Symp. Proc. 243, 21. Perdew, J. P. and Y. Wang (1986). Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. Phys. Rev. B 33(12), 8800. Perdew, J. P. and Y. Wang (1992). Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45(23), 13244. Perdew, J. P. and A. Zunger (1981). Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23(10), 5048. BIBLIOGRAPHY 183 Petersilka, M., U. J. Gossmann and E. K. U. Gross (1996). Excitation energies from Time-Dependent Density-Functional Theory. Phys. Rev. Lett. 76(8), 1212. Pines, D. and P. Nozières (1966). The theory of quantum liquids. Benjamin, New York. Pisani, C., editor (1996). Quantum-Mechanical Ab-initio Calculations of the Properties of Crystalline Materials, volume 67 of Lecture Notes in Chemistry. Springer. Pisani, C., U. Birkenheuer, F. Cora, R. Nada et al. (1996). EMBED-96 Users Manual . Universita di Torino, Torino. Pisani, C., R. Dovesi and C. Roetti (1988). Hartree-Fock ab initio treatment of crystalline systems, volume 48 of Lecture Notes in Chemistry. Springer, Berlin. Pisani, C., R. Dovesi, C. Roetti, M. Causa et al. (2000). CRYSTAL and EMBED, two computational tools for the ab initio study of electronic properties of crystals. Int. J. Quantum Chem. 77(6), 1032. Postnikov, A. V., A. I. Poteryaev and G. Borstel (1998). First-principles calculations for Fe impurities in KNbO3 . Ferroelectrics 206–207, 69. Press, W. H., S. A. Teukolsky, W. T. Vetterling and B. P. Flannery (1997). Numerical Recipies in Fortran77 . Cambridge Univ. Press, Cambridge, MA, 2 edition. Puchin, V. E., J. D. Gale, A. L. Shluger, E. A. Kotomin et al. (1997). Atomic and electronic structure of the corundum (0001) surface: comparison with surface spectroscopies. Surf. Sci. 370(2), 190. Ramirez, A. P. (1997). Colossal magnetoresistance. J. Phys.: Condens. Matter 9, 8171. Rao, C. N. R. and K. J. Rao (1978). Phase transitions in solids: An approach to the study of the chemistry and physics of solids. McGraw-Hill, New York. Robey, S. W., L. T. Hudson, V. E. Henrich, C. Eylem et al. (1996). Resonant photoelectron spectroscopy studies of BaTiO3 and related mixed oxides. J. Phys. Chem. Solids 57(10), 1385. BIBLIOGRAPHY 184 Roos, B. O., editor (1994). European Summer School in Quantum Chemistry, volume 64 of Lecture Notes in Chemistry. Springer, Berlin. Samara, G. A. and P. S. Peercy (1981). The study of soft-mode transitions at high pressure. In H. Ehrenreich, editor, Solid State Physics, volume 36, 1–118. New York: Academic. Saunders, V. R., R. Dovesi, C. Roetti, M. Causa et al. (1998). CRYSTAL’98 User’s Manual . Universita di Torino, Torino. Sawaguchi, E., A. Kikuchi and Y. Kodera (1962). Dielectric constant of strontium titanat at low temperature. J. Phys. Soc. Japan 17, 1666. Schirmer, O. F., W. Berlinger and K. A. Müller (1975). Electron spin resonance and optical identiﬁcation of Fe4+ -V0 in SrTiO3 . Solid State Commun. 16(12), 1289. Scott, J. F. (2000). Ferroelectric Memories. Advanced Microelectronics 3. Springer, Berlin. Seidl, A., A. Görling, P. Vogl, J. A. Majewski et al. (1996). Generalized kohn-sham schemes and the band-gap problem. Phys. Rev. B 53(7), 3764. Selme, M. O., P. Pecheur and G. Toussaint (1984). A tight-binding study of energy levels of iron in SrTiO3 . J. Phys. C: Solid State Phys. 17, 5185. Semenovskaya, S. and A. G. Khachaturyan (1995). Pseudotetragonal and orthorhombic ordered structures in substoichiometric YBa2 Cu3 O6+x oxides at x < 0.4. Phys. Rev. B 51(13), 8409. Semenovskaya, S. and A. G. Khachaturyan (1996). Low-temperature ordering in YBa2 Cu3 O6+x oxides at x > 0.5: Computer simulation. Phys. Rev. B 54(10), 7545. Semenovskaya, S. V. (1974a). Application of X-ray diﬀuse scattering to the calculation of the iron-aluminum equilibrium diagram. Phys. Status Solidi (b) 64(1), 291. Semenovskaya, S. V. (1974b). Use of X-ray diﬀuse scattering data for the construction of the Fe-Si equilibrium diagram. Phys. Status Solidi (b) 64, 627. BIBLIOGRAPHY 185 Setter, N. and L. E. Cross (1980). The role of B-site cation disorder in diﬀuse phase transition behavior of perovskite ferroelectrics. J. Appl. Phys. 51(8), 4356. Shirane, B. G. and R. Repinsky (1956). X-ray and neutron diﬀraction study of ferroelectric PbTiO3 . Acta. Cryst. 9, 131. Singh, N., A. P. Singh, C. D. Prasad and D. Pandey (1996). Diﬀuse ferroelectric transition and relaxational dipolar freezing in (Ba,Sr)TiO3 : III. Role of order parameter ﬂuctuations. J. Phys.: Condens. Matter 8, 7813. Skinner, S. J. (2001). Recent advances in Perovskite-type materials for solid oxide fuel cell cathodes. International Journal Of Inorganic Materials 3, 113. Sousa, C., C. de Graaf and F. Illas (2000). Core exciton energies of bulk MgO, Al2 O3 , and SiO2 from explicitly correlated ab initio cluster model calculations. Phys. Rev. B 62(15), 10013. Sulimov, V., S. Casassa, C. Pisani, J. Garapon et al. (2000). Embedded cluster ab initio study of the neutral oxygen vacancy in quartz and cristobalite. Modell. Simul. Mater. Sci. Eng. 8, 763. Szabo, A. and N. S. Ostlund (1982). Modern Quantum Chemistry. Macmillan, New York. Szot, K. and W. Speier (1999). Surfaces of reduced and oxidized SrTiO3 from atomic force microscopy. Phys. Rev. B 60(8), 5909. Tanaka, H., H. T. K. Ota and T. Kawai (1996). Molecular-dynamics prediction of structural anomalies in ferroelectric and dielectric BaTiO3 -SrTiO3 -CaTiO3 solid solutions. Phys. Rev. B 53(21), 14112. Tasker, P. W. (1979). The stability of ionic crystal surfaces. J. Phys. C: Solid State Phys. 12, 4977. Tenne, D. A., A. Soukiassian, M. H. Zhu, A. M. Clark et al. (2003). Raman study of Bax Sr1x TiO3 ﬁlms: Evidence for the existence of polar nanoregions. Phys. Rev. B (67), 012302. Thomas, L. H. (1927). The calculation of atomic ﬁelds. Proc. Camb. Phil. Soc. 23, 542. BIBLIOGRAPHY 186 Tinte, S. and M. D. Stachiotti (2000). Atomistic simulation of surface eﬀects in batio3 . AIP. conf. proc. 535, 273. Tinte, S. and M. D. Stachiotti (2001). Surface eﬀects and ferroelectric phase transitions in batio3 ultrathin ﬁlms. Phys. Rev. B 64, 235403. Tinte, S., M. G. Stachiotti, C. O. Rodriguez, D. L. Novikov et al. (1998). Applications of the generalized gradient approximation to ferroelectric perovskites. Phys. Rev. B 58(18), 11959. Towler, M. D., R. Q. Hood and R. J. Needs (2000). Minimum principles and level splitting in quantum Monte Carlo excitation energies: Application to diamond. Phys. Rev. B 62(4), 2330. Tozer, D. J. and N. C. Handy (2000). On the determination of excitation energies using density functional theory. Phys. Chem. Chem. Phys. 2, 2117. van Benthem, K., C. Elsässer and R. H. French (2001). Bulk electronic structure of SrTiO3 : experiment and theory. J. Appl. Phys. 90(12), 6156. van der Heide, P. A. W., Q. D. Jiang, Y. S. Kim and J. W. Rabalais (2001). X-ray photoelectron spectroscopic and ion scattering study of the srtio3 (001) surface. Surf. Sci. 473, 59. van Stevendaal, U., K. Buse, S. Kämper, H. Hesse et al. (1996). Light-induced charge transport processes in photorefractife barium titanate doped with rhodium and iron. Appl. Phys. B 63, 315. Veithen, M., X. Gonze and P. Ghosez (2002). Electron localization: Band-by-band decomposition and application to oxides. Phys. Rev. B 66, 235113. Viana, R., P. Lunkenheimer, J. Hemberger, R. Böhmer et al. (1994). Dielectric spectroscopy in SrTiO3 . Phys. Rev. B 50(1), 601. Vikhnin, V. S., R. I. Eglitis, S. E. Kapphan, E. A. Kotomin et al. (2001). A new phase in ferroelectric oxides: The phase of charge transfer vibronic excitons. Europhys. Lett. 56(5), 702. von Barth, U. and L. Hedin (1972). A local exchange-correlation potential for the spin polarized case. J. Phys. C: Solid State Phys. 5, 1629. BIBLIOGRAPHY 187 Vosko, S. H., L. Wilk and M. Nusair (1980). Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 58, 1200. V.S.Vikhnin, R.I.Eglitis, S.E.Kapphan, G.Borstel et al. (2002). Polaronic-type excitons in ferroelectric oxides: Microscopic calculations and experimental manifestation. Phys. Rev. B 65, 104304. Waghmare, U. V. and K. M. Rabe (1997). Ab initio statistical mechanics of the ferroelectric phase transition in PbTiO3 . Phys. Rev. B 55(10), 6161. Wang, Y., J. P. Perdew, J. A. Chevary, L. D. Macdonald et al. (1990). Exchange potentials in density-functional theory. Phys. Rev. A 41(1), 78. Wasser, R., T. Bieger and J. Maier (1990). Determination of acceptor concentrations and energy levels in oxides using an optoelectrochemical technique. Solid State Commun. 76(8), 1077. Wemple, S. H. (1970). Polarization ﬂuctations and the optical-absorption edge in BaTiO3 . Phys. Rev. B 2(7), 2679. Wiesenekker, G., G. te Velde and E. J. Baerends (1988). Analytic quadratic integration over the two-dimensional Brillouin zone. J. Phys. C: Solid State Phys. 21, 4263. Wu, Z. and H. Krakauer (2001). Charge-transfer electrostatic model of compositional order in perovskite alloys. Phys. Rev. B 63, 184113. Ziman, J. M. (1979). Models of disorder. The theoretical physics of homogeneously disordered systems. Cambridge University Press, Cambridge.