A first principle study of the electronicstructure of the bismuthatesbyArash Khazraie ZamanpourB.Sc., The University of Toronto, 2010M.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2019c© Arash Khazraie Zamanpour 2019The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the dis-sertation entitled:A first principle study of the electronic structure of thebismuthatessubmitted by Arash Khazraie Zamanpour in partial fulfillment of therequirements for the degree of DOCTOR OF PHILOSOPHY in PHYSICS.Examining Committee:Supervisor: Prof. George SawatzkyCo-supervisor: Dr. Ilya ElfimovSupervisory Committee Member: Prof. Doug BonnSupervisory Committee Member: Prof. Joerg RottlerSupervisory Committee Member: Prof. Jeremy HeylUniversity Examiner: Prof. Philip StampUniversity Examiner: Prof. Roman KremsiiAbstractMotivated by the recently renewed interest in the High Tc superconductingbismuth perovskites, we investigate the electronic structure of the parentcompounds ABiO3 (A = Sr, Ba) using ab initio methods and tight-bindingmodelling. We use the density functional theory in the local density ap-proximation to understand the role of various contributions in shaping theABiO3 band structure. It is established that hybridization involving Bi-6sand O-2p orbitals plays the most important role. The opening of a gapwith the onset of the breathing distortion is associated with condensationof holes onto a1g-symmetric molecular orbitals formed by the O-2pσ orbitalson the collapsed BiO6 octahedra. The primary importance of oxygen pstates is thus revealed, in contrast to a popular picture of a purely ionicBi3+/Bi5+ charge-disproportionation. A single band model involving an ex-tended molecular orbital of both Bi-6s and a linear combination of six O-2porbitals is derived which provides a good description of the low energy scalebands straddling the chemical potential. In addition, a parameter-basedphase diagram associated with materials incorporating “skipped valence”ions is developed. A crossover from a bond disproportionated (BD) to acharge-disproportionated (CD) system in addition to the presence of a newmetallic phase is observed. We argue that three parameters determine theunderlying physics of the BD-CD crossover when electron correlation effectsare small: the hybridization between O-2pσ and s orbitals of the B cationin ABO3, their charge-transfer energy (∆), and the width of the oxygensub-lattice band (W ). In the BD system, we estimate an effective attractiveinteraction U between holes on the same O-a1g molecular orbital. Later,we show the possibility of surface electron doping of the bismuthates viaadatom. Finally, we propose a new class of materials, namely heterostruc-iiiAbstractture composed of LaLuO3 and SrBiO3, that can host coexisting electron andhole gases and potentially high-temperature superconductivity at their twoopposite interfaces. We argue that electronic reconstruction is the dominantmechanism for solving the diverging potential. The electronic structure ofthis system suggests the electron-hole gas interactions can be tuned withthe potential of obtaining excitonic insulating phases.ivLay SummaryThis thesis presents a computational study of the electronic structure ofbismuth oxides. We argue that the ideas of charge disproportionation of thebismuth atoms may not be a complete and true picture in describing theelectronic structure of the bismuthates. We argue in favour of formationof holes on oxygens. Surfaces of bismuthates are studied and it is shownthat surface electron doping is feasible by placing atoms on surfaces of bis-muthates. Finally, we propose a new class of materials composed of LaLuO3and SrBiO3, and predict the coexistence of electron and hole gases in thissystem with the possibility of inducing high temperature superconductivity.vPrefaceAll the work presented in this thesis was supervised by Prof. George Sawatzkyand Dr. Ilya Elfimov, who provided guidance in concept formulation andanalysis, as well as manuscript preparation. I was the main investigatorresponsible for the analysis and writing of the work in this thesis. Theopening chapters 1, 2 and chapters 6 and 7, along with the conclusion areoriginal and written specifically for this thesis. Chapter 3 was published inRef.[34] where Dr. Kateryna Foyevtsova is the first author and I am thesecond author. Chapters 4 and 5 are also based on original work that hasbeen published in Ref.[107] and Ref.[108] where I am the lead author. Thework presented in chapters 6 and 7 are currently under preparation to besubmitted for publication. In addition to my supervisors, Dr. KaterynaFoyevtsova contributed extensively to the manuscript preparation of thework in chapters 3, 4 and 5. I benefited from collaboration with these threethrough many useful discussions of the physics and technical details of ourresearch and in the analysis of the results.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Computational methods . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . 142.2.1 Hohenberg-Kohn theorems . . . . . . . . . . . . . . . 142.2.2 Kohn-Sham equations . . . . . . . . . . . . . . . . . . 172.2.3 Exchange-correlation functional . . . . . . . . . . . . 182.2.4 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.5 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . 222.2.6 Relaxation of the ionic system . . . . . . . . . . . . . 23viiTable of Contents2.2.7 Extensions of DFT: LDA+U . . . . . . . . . . . . . . 252.3 Tight-Binding model . . . . . . . . . . . . . . . . . . . . . . 262.4 Electron-phonon interaction . . . . . . . . . . . . . . . . . . 273 Hybridization effects and bond disproportionation . . . . . 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Electronic structure of SrBiO3 and molecular orbitals . . . . 303.3 Evolution of the breathing and tilting distortions . . . . . . . 333.4 Simplified tight-binding models . . . . . . . . . . . . . . . . . 363.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Oxygen holes and hybridization . . . . . . . . . . . . . . . . . 404.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Bi, BaBi, O3, and BaO3 sub-lattices of BaBiO3 . . . . . . . 414.3 Derivation of tight-binding models for BaO3, O3, and BaBiO3 444.4 Breathing and tilting distortions in two and three dimensions 474.5 Tight-binding models with a reduced number of orbitals . . . 514.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Bond versus charge disproportionation . . . . . . . . . . . . 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 BD-CD crossover and the Phase diagram . . . . . . . . . . . 615.3 Molecular orbitals of O-eg symmetry orbitals and insulator-to-metal transition . . . . . . . . . . . . . . . . . . . . . . . . 655.4 Why Bi is so special? . . . . . . . . . . . . . . . . . . . . . . 685.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 Surface electron doping via Cs adatom . . . . . . . . . . . . 736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Surface states of SrBiO3 and doping with adatom . . . . . . 746.3 Work function dependence on surface termination . . . . . . 786.4 Structural phase transition with doping . . . . . . . . . . . . 806.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84viiiTable of Contents7 Coexisting two dimensional electron and hole gases in LaLuO3/SrBiO3heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 The polar catastrophe . . . . . . . . . . . . . . . . . . . . . . 867.3 Electronic reconstruction in LLO/SBO films . . . . . . . . . 937.4 Role of oxygen vacancy in solving the polar problem . . . . . 967.5 Lattice response to potential divergence in thin films . . . . 1007.6 Metal-insulator transition . . . . . . . . . . . . . . . . . . . . 1027.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107AppendicesA Hybrid functionals . . . . . . . . . . . . . . . . . . . . . . . . . 118B Lindhard function . . . . . . . . . . . . . . . . . . . . . . . . . 119C Bulk structure parameters with doping . . . . . . . . . . . . 120D Electronic and structural properties of bulk LaLuO3 . . . 122E Electronic and structural properties of bulk SrBiO3 . . . . 124ixList of Tables4.1 The eigenstates and eigenvalues of an octahedron and a squareplaquette of O-pσ orbitals coupled via nearest-neighbour hop-ping integrals −tpp = (−tppσ + tpppi)/2. For oxygen site in-dexing and relative orbital phases, refer to Fig. 4.1 (d). Here,the O-pσ orbitals’ on-site energies are set to zero. . . . . . . . 434.2 On-site energies and hopping integrals in eV for BaBiO3 andits O3 and BaO3 sub-lattices. The values are obtained byfitting either the simplest or the extended tight-binding (TBor ETB) model or by using Wannier functions (WF) includingBi-6s and O-2p orbitals. This choice of WF orbitals results ina large next-nearest-neighbour hopping integral t′ppσ betweenthe O-pσ orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Variation of the nearest-neighbor hopping integrals in responseto the Bi-O bond-disproportionation of 0.1 A˚. . . . . . . . . . 494.4 The single-orbital TB model parameter values in eV for lat-tices with a varying degree of the breathing distortion b andno octahedra tilting. t, t′, and t′′ are the nearest, second-nearest, and fourth-nearest neighbour hopping integrals, re-spectively. cA1g and eA1gare the on-site energies of the A1g-like orbitals of the collapsed and expanded octahedron. Bonddisproportionation (B.D.) shows the number of Ac1g orbitalholes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1 Nearest-Neighbour TB model fit parameters of the undis-torted cubic perovskite crystal structure. . . . . . . . . . . . 70xList of Tables6.1 SBO lattice constant and breathing and tilting of the oxygenoctahedra. A comparison is made between the experimentaland the relaxed structures obtained with GGA, LDA andPBE-sol exchange and correlation functionals. An E-cut of410 eV and a k-point grid size of 8×8×8 is used in all thecalculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2 Lattice constant and structural distortions of SBO supercellat 0.5 electron or 0.5 hole doping per formula unit with GGAexchange-correlation functionals. An E-cut of 410 eV and ak-point grid size of 8×8×8 is used in all the calculations. . . 83C.1 Lattice constants and structural distortions of SrBiO3 bulkstructure upon 0.5 electron doping per formula unit. A com-parison is made between the experimental structure and therelaxed structures obtained with GGA, LDA and PBE-solexchange and correlation functionals. An E-cut of 410 eVand a k-point grid size of 8×8×8 is used in all the calculations.120C.2 Lattice constant and structural distortions of SrBiO3 bulkstructure upon 0.5 hole doping per formula unit. A com-parison is made between the experimental structure and therelaxed structures obtained with GGA, LDA and PBE-solexchange and correlation functionals. An E-cut of 410 eVand a k-point grid size of 8×8×8 is used in all the calculations.121D.1 Lattice constants and structural distortions of the four for-mula unit cell of bulk LaLuO3 is listed. An E-cut of 410 eVand a k-point grid size of 8×8×8 is used with GGA exchange-correlation functional. For the relaxed structure the volumeand all atomic positions are allowed to relax until all forcesare less than 0.01 eV/ A˚. . . . . . . . . . . . . . . . . . . . . 122D.2 LaLuO3 experimental structure atomic positions. . . . . . . . 123xiList of TablesE.1 Lattice constants and structural distortions of the four for-mula unit cell of bulk SrBiO3 is listed. An E-cut of 410 eVand a k-point grid size of 8×8×8 is used with GGA exchange-correlation functional. For the relaxed structure the volumeand all atoms are allowed to relax until the force on each atomis less than 0.01 eV/ A˚. . . . . . . . . . . . . . . . . . . . . . 124E.2 SrBiO3 experimental structure atomic positions. . . . . . . . 125E.3 Parameters of the 16 model structures of SrBiO3 with themonoclinic space group P21/n. For each structure, latticeconstants a, b, and c are given in the first row, fractionalcoordinates of oxygen atoms O1, O2, and O3 are given in,respectively, the second, third, and forth row, and relaxedfractional coordinates of Sr are given in the fifth row. Forthe t = 0 structures Sr is at high-symmetry position 0, 0,0.25. The two Bi atoms are at 0.5, 0, 0 and 0.5, 0, 0.5, themonoclinic angle equals 90o. . . . . . . . . . . . . . . . . . . . 126xiiList of Figures1.1 (a) Phase diagram for the BaPb1−xBixO3 [14, 15] system in-dicating the structural space groups as a function of doping(x) and temperature T in Kelvin. A CDW is present in theshaded region. Recently Ref. [16] has shown the existence of amuch richer phase diagram in this system. (b) Phase diagramfor the Ba1−xKxBiO3 adapted from Ref. [8]. . . . . . . . . . . 31.2 The crystal structure of (left) one formula unit cell of SrBiO3and (right) four formula unit cell of SrBiO3 showing the crys-tal structure distortions. The small balls at the corners ofthe octahedra represent oxygen atoms. The octahedra indark(light) colour represent the expanded (collapsed) Bi–Obond length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 LDA band structures of BaBiO3 and SrBiO3 calculated witha cubic unit cell of dimensions a × a × a and a supercell ofdimensions√2a×√2a× 2a. The Fermi energy EF is set tozero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Schematic illustration of all-electron(solid lines) and pseu-doelectron (dashed lines) potentials and their correspondingwave functions. The radius at which all-electron and pseudoelectron values match is designated rc. Figure adapted fromRef.[70] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23xiiiList of Figures3.1 (a) LDA electronic structure of SrBiO3 projected onto theBi-6s orbital and combinations of the O-pσ orbitals of a col-lapsed (top) and expanded (bottom) BiO6 octahedra. Forthe doublet Eg and the triplet T1u, only one projection isshown. The Fermi level is set to zero, and PDOS stands forprojected density of states and is given in states/eV/cell. (b)An octahedron of O-pσ orbitals coupled via nearest-neighbourhopping integrals −t and its eigenstates. . . . . . . . . . . . . 313.2 LDA characterization of SrBiO3 model structures with vary-ing degrees of the BiO6 octahedra’s tilting t and breathing b:(a) total energy per formula unit (f.u.) and (b) band gap. In(a), solid lines and solid circles (dashed lines and open circles)represent model structures with fixed (relaxed) Sr atoms. Thehorizontal dashed line marks the energy of the experimentalSrBiO3 structure. . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 LDA electronic structure of SrBiO3 as a function of breathingb and tilting t. Projections are made onto the Bi-6s orbitaland the A1g combination of the O-pσ orbitals of a collapsedBiO6 octahedron, as well as their bonding (“B”) and anti-bonding (“A”) combinations. . . . . . . . . . . . . . . . . . . 353.4 (a)LDA electronic structure of the oxygen sub-lattice of SrBiO3.Projections are made onto combinations of the O-pσ orbitalsof a collapsed oxygen octahedron. (b) Model density of statesas a function of breathing b and hybridization between s andp orbitals. CC (EC) stands for a collapsed (expanded) p-sitecage. The model states are 90% filled, i.e., there is one holeper s orbital; the Fermi energy is set to zero and marked withblack dashed vertical lines. . . . . . . . . . . . . . . . . . . . . 373.5 The effect of tilting on (a) the half-filled band and on (b) thestatic susceptibility χ(q, ω = 0), at zero breathing. In (b),solid (dashed) lines represent calculations where nonlinear ef-fects due to tilting are (are not) taken into account. . . . . . 38xivList of Figures4.1 The DFT (LDA) band structures of (a), (g), (h) BaBiO3 andits (b) BaBi, (c) Bi, (e) BaO3, and (f) O3 sub-lattices. In (a) -(c),the yellow-coloured fat bands represent the contribution ofthe Bi-s orbital, while in (e) - (h), the red-coloured fat bandsrepresent the contribution of the O-a1g molecular orbital. TheFermi level is marked with a horizontal dashed black line.Panel (d) shows the O-a1g molecular orbital combination ofoxygen-pσ orbitals in an octahedron. The oxygen sites 1 to6 are coupled via hopping integrals −tpp = (−tppσ + tpppi)/2.A nearest neighbour TB model fit of (e) BaO3 (f) O3 and(g)-(h) BaBiO3 is shown with dashed lines. In (h), Bi-6porbitals are added in an extended tight-binding model (ETB)for an improved fit. The parameter values resulting from thefits are listed in Table 4.2. . . . . . . . . . . . . . . . . . . . . 424.2 The band structures and projected DOS of the 3D (top pan-els) and 2D (bottom panels) TB models with varying strengthsof the breathing and tilting distortions. Here, breathing b ishalf the difference between the two disproportionated Bi-Obond lengths, and θ is the tilting angle. Molecular orbitalprojections are made for the compressed octahedron or squareplaquette following Table 4.1, while the Bi-s orbital projec-tion is made for the Bi atom located inside the compressedoctahedron or square plaquette. The red-coloured fat bandsrepresent the contribution of the O-a1g molecular orbital. . . 484.3 The charge gap as a function of the breathing distortion atvarious tilting distortions in (a) the 2D TB model, (b) the3D TB model, (c) SrBiO3 from DFT (LDA) calculations[34],and (d) the BiO3 sub-lattice from DFT (LDA) calculations. . 50xvList of Figures4.4 In (a) and (b), the full TB (solid line) and the four-orbitalTB (dashed line) models are compared for the b = 0.0 A˚ andb = 0.1 A˚ lattices, respectively. In (c) and (d), the DFT bandstructure (solid line) and the single-orbital TB model (dashedline) are compared for the b = 0.0 A˚ and b = 0.1 A˚ lattices,respectively. In (e), the single-orbital A1g coupling to nearest,second nearest, and fourth nearest neighbours are shown. . . 535.1 The band structures and projected density of states of thetight-binding model for (a) non-distorted lattice and (b) 0.1A˚ breathing distorted lattice. Oxygen molecular orbital pro-jections are made onto the compressed octahedron and theBi-6s orbital projection is for the Bi atom located inside thecompressed octahedron. The red-coloured fat bands repre-sent the contribution of the O-a1g molecular orbital of thecompressed octahedron. . . . . . . . . . . . . . . . . . . . . . 605.2 (a) The phase diagram representing the dominant character ofthe conduction band for the experimental breathing distortedlattice obtained from the tight-binding model as a functionof the charge-transfer energy, ∆ and tspσ. The character ofthe conduction band is obtained by integrating the partialdensity of states above the band gap. The boundary of thiscrossover is shown with a dashed line representing equal par-tial density of states between O-a1g and Bi-6s. Symbol xmarks the parameters relevant for ABiO3 [107] obtained fromtight-binding fit. (b) The charge gap in eV is calculated fromthe tight-binding model for the breathing distorted lattice.An insulator-to-metal transition is obtained at a critical valueof t∗spσ when holes transition into the O-eg molecular orbitals. 62xviList of Figures5.3 The energy level diagram of an oxygen octahedron calcu-lated within the three-band model at tspσ = 0(in black), andtspσ = 1 eV(in red) as a function of the charge-transfer en-ergy ∆. The character of the conduction band (dashed line)transitions from O-eg to anti-bonding Bi-6s–O-a1g band at acritical value of tspσ and charge-transfer energy (∆). . . . . . 645.4 (a) DFT band structure of the experimental structure of SrBiO3,(b) band structure of the experimental structure with a con-stant orbital potential of −11.4 eV on Bi-6s. An insulator-to-metal transition is obtained upon moving Bi-6s orbitals. Wenote that breathing distortions do not open a gap when holesoccupy molecular orbitals of the O-eg symmetry. . . . . . . . 665.5 DFT band structure of the relaxed structure of (a) MgPO3,(b) CaAsO3, (c) SrSbO3, (d)(h) BaBiO3, (e) BaPO3, (f)BaAsO3, (g) BaSbO3 assuming cubic perovskite crystal struc-ture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.6 TB model projected density of states with optimized latticeparameters of (a) BaPO3, (b) BaAsO3, (c) BaSbO3 and, (d)BaBiO3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.7 Partial density of states of (a) BaPO3, (b) BaAsO3, (c) BaSbO3and (d) BaBiO3 calculated using TB model with tsp set to zero. 705.8 Energy level diagram illustrating two cases before hybridiza-tion, (a) B-s band above O-a1g resulting in higher B-s con-tribution in the partial density of states at the conductionband, (b) B-s band below O-a1g resulting in higher O-a1gcontribution in the partial density of states at the conductionband. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1 Band structure and total density of states obtained usingGGA for slabs of (a) Sr-terminated and (b) Bi-terminatedSBO in the [001] direction. The atomic structure of slabsused in the calculations are shown on the right hand side. . . 75xviiList of Figures6.2 DFT band structure and total density of states obtained forCs adatom deposition of (a) 0.5Cs on Sr-terminated, (b) 0.5Cson Bi-terminated, (c) 0.25Cs on Sr-terminated, and (d) 0.25Cson Bi-terminated SBO [001] surface. . . . . . . . . . . . . . . 776.3 Plane averaged electrostatic potential calculated with GGAfor (a) SrO terminated (b) BiO2 terminated surfaces. Here,Z represents the distance along the slab in A˚. . . . . . . . . . 796.4 Band structure and total density of states obtained usingGGA for relaxed SBO supercell at (a) 0.5 hole doping and(b) 0.5 electron doping per formula unit. . . . . . . . . . . . . 816.5 Average Bi-O bond length (A˚) calculated in the collapsed andexpanded octahedra after electron(e)/hole(h) doping with GGA. 827.1 An artist’s view of the polar catastrophe is shown for LaAlO3film, as an example. (a) The electric field as a function of theposition normal to the layers alternates between zero and anon-zero value. (b) The electric potential obtained after inte-grating the electric field shows the existence of a net potentialdifference across the film which increases with the film thick-ness. This divergence of the potential by increasing the filmthickness is known as the polar catastrophe. Figure adaptedfrom Ref. [124] . . . . . . . . . . . . . . . . . . . . . . . . . . 867.2 An artist’s view of the polar catastrophe. The polar catastro-phe can be simply described if each layer is modelled as twowith half the charge of the original i.e. q/2. It can be seen [onthe right side] that the big capacitor (big dipole) produced bythe top (+q/2) and bottom (−q/2) planes is the origin of thepotential build up. The solution to the polar problem is toget rid of this big capacitor. . . . . . . . . . . . . . . . . . . . 887.3 An artist’s view of the Zener breakdown in which electronstunnel between the two interfaces of a semiconductor whenan electric field is applied. This breakdown is determined andlimited by the band gap of the system. . . . . . . . . . . . . . 89xviiiList of Figures7.4 (a)Total density of states of SrBiO3(LaLuO3) bulk experi-mental structure shown with projected density of states ontooxygen and La orbitals in GGA(GGA+U). Note that theband gap in SrBiO3 is mainly of oxygen character, (b) Anartist’s view of the unreconstructed interface which has neu-tral [001] planes in SrBiO3, but the [001] planes in LaLuO3have alternating net charges of +1, −1. This produces anelectric field, leading to an electric potential that divergeswith increasing LaLuO3 thickness. After electronic recon-struction a net charge of −|q| is transferred from the BiO2layer at the p-type(LuO2/SrO) interface into the BiO2 layerat the n-type(BiO2/LaO) interface resulting in the formationof 2DEG and 2DHG respectively. . . . . . . . . . . . . . . . 927.5 (a) The band structure of (n = 4, m = 4) LLO/SBO be-fore and (b) after atomic relaxation. The fat bands show theprojected density of states onto the collapsed Bi-6s octahe-dra (in blue) at the LaO/BiO2 interface and expanded Bi-6soctahedra (in red) at the LuO2/SrO interface. . . . . . . . . . 947.6 The Fermi surface of (n = 4, m = 4) LLO/SBO structureafter atomic relaxation. . . . . . . . . . . . . . . . . . . . . . 957.7 An artist’s view of formation of an oxygen vacancy in anLLO/SBO heterostructure. By removal of a neutral divalentoxygen atom, the system is left with two extra electrons in theconduction band. If the excess electrons move to the n-typeinterface (resulting in reduction of their energy), the electricpotential can in principle be compensated by the introductionof an equal an opposite potential in LLO. . . . . . . . . . . . 97xixList of Figures7.8 Relative formation energy of an oxygen vacancy as a func-tion of its position in (n = 4, m = 4) LLO/SBO supercellwith n-type and p-type interfaces indicated in red and bluedashed lines respectively. The concentration of oxygen vacan-cies is 1/8 per layer. All atomic positions are fixed to theircorresponding experimental structure in the bulk. We fix thein-plane lattice constant to the LLO experimental lattice con-stant of 4.18 A˚. . . . . . . . . . . . . . . . . . . . . . . . . . . 997.9 (a)(b)Layer resolved atomic displacements in the z-directionperpendicular to the interface in A˚. Atomic displacements(∆z) are averaged in each layer and are plotted against theposition of the layer in monolayers (ML) from the n-type in-terface(IF). (c) Dipole moments calculated from the net layerionic charges times their displacements following the calcu-lations of Ref. [136]. (d) Formation of ionic dipoles in bothLLO and SBO after atomic relaxation. The dipoles result inelectric fields shown by red arrows. . . . . . . . . . . . . . . . 1017.10 The band structure of (a) (n = 4, m = 2) (b) (n = 2, m = 4),and (c) (n = 4, m = 4) LLO/SBO super lattices after atomicrelaxation (with n representing the number of LLO and mnumber of SBO unit cells perpendicular to the interfaces).The fat bands show the projected density of states onto thecollapsed Bi-6s octahedra (in blue) at the LaO/BiO2 inter-face and expanded Bi-6s octahedra (in red) at the LuO2/SrOinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.1 The total and projected densities of states of SrBiO3 calcu-lated using (a) the LDA functional and (b) the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional. Bi1 and Bi2 denote thetwo inequivalent Bi atoms surrounded by, respectively, an ex-panded oxygen octahedron and a collapsed oxygen octahedron.118xxList of FiguresB.1 χ(q, ω = 0) for the t = texp structure: (a) a three-dimensionalscan through the Brillouin zone and (b) the dependence onhole doping, within a rigid-band approximation. The la-beling numbers x indicate concentrations of doped holes inSr1−xKxBiO3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119D.1 Band structure and density of states of the experimental andrelaxed structures. . . . . . . . . . . . . . . . . . . . . . . . . 123xxiGlossaryAL Atomic limitAPW Augmented plane waveBD Bond-disproportionationCD Charge-disproportionationCDW Charge-density waveDFT Density functional theoryDOS Density of statesETB Extended tight-bindingGGA Generalized gradient approximationHK Hohenberg-KohnIF Inter-faceKH Kohn ShamLAPW Linearized augmented plane waveLDA Local density approximationLO Local orbitalsLSDA Local spin density approximationMBE Molecular-beam epitaxyML Mono-layersPBE Perdew-Burke-ErnzerhofPDOS Partial density of statesPP PseudopotentialsSDW Spin-density waveTB Tight-binding modelWF Wannier function2D Two dimensional2DEG Two dimensional electron gasxxiiGlossary2DHG Two dimensional hole gas3D Three dimensionalxxiiiAcknowledgmentsI wish to thank Prof. George Sawatzky and Dr. Ilya Elfimov for their super-vision of this thesis. Without their continuous support this work would nothave been possible. I would also like to thank Dr. Kateryna Foyevtsova forher extensive help. The results presented in this thesis are due to the fruitfuldiscussions with the mentioned people. I am grateful to the members of mycommittee, Prof. Doug Bonn, Prof. Joerg Rottler and Prof. Jeremy Heylfor their feedback. Last but not least, I would like to thank my colleaguesDr. Reza Benam, Dr. Shadi Balandeh, Dr. Robert Green, Dr. Fengmiao Li,Dr. Oleksandr Foyevtsov, Dr. Matthias Hepting, Oliver Yam and EbrahimSajadi.xxivDedicationTo my family.xxvChapter 1IntroductionThe perovskite compounds, featuring BO6 octahedra (B = Bi, Cu, Mn,Ni,...) as building blocks, are exceptionally rich in physical properties. Theyexhibit a variety of structural, electronic, and magnetic phase transitionsand are hosts to interesting states of matter such as high-transition-temperature (Tc) superconductivity, pseudogap state and charge-density-wave (CDW), to name a few.In this thesis, we study the electronic structure of the bismuth perovskites,ABiO3(A = Sr, Ba)[1, 2], and attempt to better understand the individualrole played by its constituent elements in shaping their electronic struc-ture. The bismuthates are the parent compounds of the high-transition-temperature superconductors: BaPb1−xBixO3(BPBO)[3], Ba1−xKxBiO3(BKBO)[4]and Sr1−xKxBiO3(SKBO)[2]. Fig. 1.1 shows a schematic phase diagramof BPBO and BKBO for a range of temperatures T, and doping x, withthe superconducting region indicated with a red line. These materials areone of the few examples of transition-metal-free high-transition-temperaturesuperconductors. The highest superconducting transition temperature re-ported for any oxide superconductor not containing copper (i.e. cuprates)is found in the cubic structure of BKBO [4] with a Tc ≈ 34 K at x = 0.4.The electronic structure of the bismuthates is three dimensional comparedto the two-dimensional metal-oxygen planes in cuprates which are widelybelieved to be an essential factor in producing a high Tc in the copper-oxidesystems. In addition, local magnetic moments do not exist in the undopedbismuthates. They are diamagnetic and muon spin-resonance data [5] revealno evidence of static magnetic order and the magnetic-pairing mechanisms,as proposed for the cuprate superconductors, can be excluded for this ma-terial. Therefore, in order to understand the superconductivity mechanism1Chapter 1. Introductionin this material, it is essential to have a thorough understanding of the elec-tronic structure of the parent compounds and the interactions responsiblefor the superconductivity.The bismuthates also demonstrate a variety of temperature-driven electronicand structural phase transitions[6–8]. The crystal structure is monoclinicfrom 4.2 K to 430 K with the space group changing from P21/n to I2/m atabout 140 K. From 430 K, it transitions into a rhombohedral structure withthe R3 space group up until 820 K and finally becomes cubic in the Fm3mspace group shown in Fig. 1.1. In this work, we are mainly interested inthe low temperature monoclinic phase. In this phase, the oxygen octahedraaround the Bi ions exhibit alternating breathing-in and breathing-out dis-tortions along the three cubic crystallographic directions, resulting in thedisproportionated Bi–O bond lengths. Additionally, the oxygen octahedratilt and rotate, following the a−a−c0 pattern in BaBiO3 and the a−a−c+pattern in SrBiO3 in Glazer’s classification [9, 10]. The cubic one formulaunit cell and the four formula supercell of SrBiO3 is shown in Fig. 1.2.It is interesting to note that the tilting and rotation of the octahedra arefound to weaken with increasing temperature and vanish at around 800 K[8]. However, the bond-disproportionation is present at all temperatures.From the beginning of their discovery, much confusion has existed over theassignment of the oxidation states of Bi in BaBiO3. In general, it is assumedthat oxygen is closed shell with six 2p electrons resulting in an oxidationstate of 2−. For this reason, many assumed that this compound must con-tain a mixture of Bi3+ and Bi5+ [6, 7, 11–13] since an oxidation state of4+ for Bi has never been observed in any other compound and because Bain any oxide always is in a 2+ state. Therefore, an oxidation state of 4+for Bi is believed to be an energetically unstable oxidation state. However,if Bi4+ was possible, the ground state of BaBi4+O3 with a half-filled con-duction band should be a metal from band theory as can be seen in theband structure of the one formula unit cell in Fig. 1.3(a). Clearly, BaBiO3is a diamagnetic semiconductor and therefore Bi4+ can not be possible inthis compound. Therefore, it is argued that the breathing distortions of theoxygen octahedra form two different local environments around Bi ions and2Chapter 1. Introduction(b)(a)Figure 1.1: (a) Phase diagram for the BaPb1−xBixO3 [14, 15] system indicat-ing the structural space groups as a function of doping (x) and temperatureT in Kelvin. A CDW is present in the shaded region. Recently Ref. [16]has shown the existence of a much richer phase diagram in this system. (b)Phase diagram for the Ba1−xKxBiO3 adapted from Ref. [8].3Chapter 1. IntroductionFigure 1.2: The crystal structure of (left) one formula unit cell of SrBiO3and (right) four formula unit cell of SrBiO3 showing the crystal structuredistortions. The small balls at the corners of the octahedra represent oxy-gen atoms. The octahedra in dark(light) colour represent the expanded(collapsed) Bi–O bond length.lead to charge-disproportionation in this compound. This idea of charge-disproportionation into a Bi3+ and Bi5+ was later reinforced with neutrondiffraction data finding two different Bi–O bond lengths of 2.28 A˚ and 2.12A˚ at two distinct sites [6, 17]. The band structure of the distorted structureof BaBiO3 and SrBiO3 in the monoclinic phase obtained from density func-tional theory (DFT) in Fig. 1.3(b)(c) shows that indeed the bismuthatesare semiconductors with the existence of a larger gap in SrBiO3.In the charge-disproportionation picture, bismuth with the electronic con-figuration of [Xe]4f145d106s26p3, would donate its three 6p electrons to the4Chapter 1. Introductionsurrounding oxygens forming Bi3+ or donate the three 6p plus its two 6s elec-trons forming Bi5+. Compounds assumed to exhibit this type of behaviourare called the “skipped-valence” compounds. Among these elements, Tl1+in TlBr, TlCl and Tl3+ in TlCl3, TlF3; Pb2+ in PbBr2, PbCl2 and Pb4+in PbBr4, PbF4; Sn2+ in SnBr2, SnCl2 and Sn4+ in SnBr4, SnCl4; Sb3+ inSbBr3, SbCl3 and Sb5+ in SbBr5, SbCl5; Te4+ in TeF4 and Te6+ in TeF6; I5+in IF5 and I7+ in IF7 to name a few, are assumed to skip the intermediateoxidation state 6s1 or 5s1, but the 6s0 or 5s0 and the 6s2 or 5s2 states do oc-cur in nature. There also exist compounds that seem to have a mixture of s0and s2 configurations in the same compound. For example, SbO2 is assumedto have a mixture of Sb3+ and Sb5+. Thus, the formula is normally writtenas Sb2O4. Among other compounds that are assumed to have a mixture oftwo valences for the same atom are: CaFeO3 [18], CsTlF3 [19], CsAuCl3 [20],CsTe2O6 [21], SmNiO3 [22] and BaBiO3 [3]. In these compounds two differ-ent local environments for the same element (i.e. B-cation) is observed, aneffect generally referred to as bond or charge disproportionation. The basicphysics associated with this kind of disproportionation involves the creationof different B-X bond lengths, with X representing the anion i.e. O, F, Cl,Br. It is interesting to note that the bond-disproportionation in CaFeO3weakens with increasing temperature and becomes metallic at around 270K [8] in contrast to BaBiO3.In the bismuthates, such a valence skipping behaviour is often accounted forby the concept of an attractive Hubbard U [11, 13, 23] type model whichassumes attraction between alike charged particles. It is worth noting, how-ever, that the Bi–O bond lengths of the two sites do not differ by an amountthat is expected for two-electron charge difference between the two sites aspointed by Ref. [24]. This scenario of forming Bi3+ and Bi5+, is also notconsistent with the high binding energy of the Bi-6s states as discussed inRefs. [25] or the strongly covalent nature of the Bi–O bonding [25, 26] andis not supported by spectroscopic measurements finding little difference inthe Bi valence shell occupations [24, 27, 28]. This observation is similarto the well known case of the nickelates where the 3d7 configuration of Niwith one electron in the eg state is highly unstable. As was originally sug-5Chapter 1. Introduction-10-5 0 5 10Energy (eV)R Γ X M ΓA M Γ X N ΓA M Γ X N Γ(a)(b)a×a×a cubic √2a×√2a×2a supercellBaBiO3 BaBiO3 SrBiO3Figure 1.3: LDA band structures of BaBiO3 and SrBiO3 calculated witha cubic unit cell of dimensions a × a × a and a supercell of dimensions√2a×√2a× 2a. The Fermi energy EF is set to zero.gested by Mizokawa et al. [29] and later shown in Refs. [30–33], electroniccorrelations, hybridization between Ni-3d orbitals and oxygen molecular or-bitals of eg symmetry, and a negative charge-transfer gap results in holespreferring to occupy oxygen states rather than nickel states. The breathingdistortion of the NiO6 octahedra in the nickelates, gives a charge transferof type (d8L2)S=0(d8)S=1 in the two nickel sites, instead of the charge-disproportionated (d6)S=0 (d8)S=1 state previously assumed with L beingthe ligand hole, and S the spin of that particular configuration.The above body of experimental and theoretical evidence has led us to pro-pose in Ref. [34] and chapter 3, an alternative microscopic model for theinsulating state of ABiO3 using a “molecular orbitals” approach rather thanthe traditional atomic orbital centred derivation of the electronic structure.We use DFT methods and tight-binding model Hamiltonians to show thatthe metal-insulator transition with the breathing distortions of the oxygenoctahedra leads to spatial condensation of hole pairs into molecular-like or-6Chapter 1. Introductionbitals of the A1g symmetry on collapsed BiO6 octahedra and that all the Biions are close to being 6s2 and we have Bi3+. Note that the trivalent Bi leavesus with one missing electron or a positive charge (hole) per three oxygensaccording to the chemical formula. This will result in having on averagetwo holes per oxygen octahedra surrounding each Bi. After introductionof the breathing distortions, the formation of hole pairs on the collapsedoctahedra is favoured which pushes the molecular-like orbitals of oxygenoctahedra with the A1g symmetry above the Fermi level.Recently, a more advanced treatment of exchange and correlation effectshas been shown to result in an opening of a charge gap and enhance-ment of the electron-phonon coupling in BaBiO3 [19, 35–39]. However, weshow in chapters 3 and 4 that the states bridging the Fermi energy arebasically unchanged as compared to conventional DFT, indicating that theuse of LDA+U or hybrid functionals merely increases the gap value [referto Appendix. A]. On the other hand, as can be seen in Fig. 1.3, the localdensity approximation (LDA) gives very similar band structures of BaBiO3and SrBiO3. The few differences are (1) the appearance of the localizedBa-5p states below the Bi- 6s/O-2p band in BaBiO3 and (2) a larger chargegap and (3) a narrower band right below the Fermi level in SrBiO3, which isdue to a larger breathing distortion and a larger octahedra’s tilting presentin SrBiO3.An important and open question regarding the bismuthates is how large isthe electron-phonon coupling. This coupling naturally arises in the bonddisproportionation picture and it would be interesting to study how thecoupling strength evolves as a function of hole or electron doping of thebismuthates and better understand its role in the formation of Cooper pairs.In a crystalline solid, lattice vibrations of the atoms destroy the perfectperiodicity in the lattice. These vibrations of the lattice (phonons) will resultin a non-zero probability of electrons being scattered. The simplest approachto describe the electron-phonon scattering is to consider the electron-phononinteraction process induced by the annihilation or creation of a phonon withfinite momentum q accompanied with the excitation or de-excitation of anelectron from state |k, σ〉 to |k ± q, σ〉. In the following, we represent vectors7Chapter 1. Introductionin bold text. Here, k represents the wave vector of the electron and σ is thespin.In general, the Hamiltonian of the simplest electron-phonon interaction canbe written as:H =∑k,i,i′,qjg(k2, i,k1, i′,qj)c†k2i,σck1i′,σ(b†−qj + bqj) (1.1)where c†k2,i,σ and ck1,i′,σ are the creation and annihilation operators for therelevant quasiparticles i.e. electrons with wave vectors k1 = k and k2 = k+qin bands i′, i. The operators b†qj and bqj are the creation and annihilationoperators of the phonons which have an energy related to ωqj with wave vec-tor q for mode j. The displacement of the atomic sites can then be expressedas un =∑qj√~2Mωqjeiq.Rn(b†−q + bq). The pre-factor g(k2, i,k1, i′,qj) de-scribes the electron-phonon coupling parameter which in the most generalcase depends on both the electrons and phonons wave vectors k and q. Thiselectron-phonon Hamiltonian describes processes where an electron in bandi′ and with momentum k scatters to a state with momentum k + q in bandi. This process happens either by absorption of a phonon in mode j withmomentum q or emission of a phonon with momentum −q.Over the past few decades, there has been an extensive amount of researchon various models with electron-phonon coupling and I list the most highlystudied models here and show their relevant Hamiltonians and discuss theirrelation to the electron-phonon coupling in the bismuthates.The Frohlich Hamiltonian[40]: This model was written for the longitudinalacoustic phonons coupled to free electrons in example in the jellium modelwhere phonons are realized as motion of ions in a lattice. Here, for simplicitywe write the model Hamiltonian for a single phonon mode as:H =∑k,σ(k)c†k,σck,σ+∑q~ωqb†qbq+∑k,q√~2MNωqqV (q)(b†−q+bq)c†k+q,σck,σ(1.2)where the first term describes the energy of the free electron gas, the secondterm is the energy of the phonons and the last term is the coupling term.8Chapter 1. IntroductionNote that only the longitudinal acoustic phonon is present in this modelbecause in the term q.un all the the transverse modes vanish.The Holstein Hamiltonian[41]: In the Holstein model the electron-phononparameter is a q-independent coupling and a dispersionless phonon thatdescribes the interaction of an electron or hole with optical phonons. Thisis described by an Einstein model where ~ωq = Ω. Its Hamiltonian can bewritten as:H = −∑n,σnc†n,σcn,σ−∑n,m,σtn,mc†n,σcm,σ+Ω∑nb†nbn+g∑n,σc†n,σcn,σ(b†n+bn)(1.3)where the first term describes the on-site energy of the electron, the secondterm is the hopping of the electron between two different sites in the lattice,the third term is the energy of the phonons and the last term is the couplingbetween the electron and phonons. In this model c†n,σ, cn,σ represent theoperators that add or remove an electron at site n with spin σ. Looking atthe last term one can see that if g is positive, the interaction term says thatfor an electron at site n, the on-site energy of the electron must increaseor decrease depending on the direction of the movement of the site. Thisis due to the fact that for Einstein phonons in the harmonic approximationas the Hamiltonian suggests, un =√~2Mω (b†n + bn) is the displacement ofthe atomic site with mass M relative to its equilibrium position. Note thatthis model was originally written for a molecular crystal with more than oneatom per unit cell and consisting of optical phonons, whereas in the Frohlichmodel a monoatomic unit cell is considered and therefore we only have theacoustic phonons.The last model is the Peierls/SSH Hamiltonian[42]: The SSH model wasfirst introduced for the treatment of the half-filled case of a one-dimensionalpolymer chain polyacetylene. In this model the hopping tnm between twosites n and m depends on the distance between the sites and the closer theyare, the larger their overlap and bigger the hopping term. To first order thehopping integral can be written as tnm = t[1±α(un−um)]. In this expression9Chapter 1. Introductionthe nearest neighbour hopping is represented with t and we choose the ±sign such that the hopping integral increases if the sites move closer togetheror decreases if they move further apart. In this model one can think of thecoupling resulting from the Kinetic energy rather than potential energy. Thecoupling term for a one-dimensional model can be written as:He−ph = −g∑n,σ(c†n,σcn+1,σ + h.c.)(b†n + bn − b†n+1 − bn+1) (1.4)where the constant collects the constants, g = tα√~2MΩ with α being theTaylor expansion coefficient in tnm.The bismuthates are non magnetic and the electronic correlations are foundto be weak, therefore the most important degrees of freedom are the sin-gle electron degrees of freedom and their interaction with phonons. Thiselectron-phonon interaction can strongly contribute to the eventual electronpairing in a BCS like manner which results in superconductivity. Therefore,in order to study the electron-phonon coupling in the bismuthates one needsto choose the relevant model Hamiltonians for a system with two differentatoms in the unit cell. In the bismuthates the low energy scale electronicstructure is mainly described by the electrons of Bi and oxygen ions andthe hybridization between the states centred on Bi and those centred onoxygens. In chapter.3 and 4 by looking at the DFT band structures, wefind that as a function of lattice distortions the dominant change is in thehopping integrals between nearest neighbour Bi-s and O-2pσ orbitals ratherthan changes in their on-site energies. The Bi-O hopping integral thereforestrongly depends on the interatomic distances and the electron-phonon cou-pling can be realized as an extended SSH model with two atoms per unitcell. However, if one considers a molecular basis set with internal degrees offreedom (refer to chapter.4) rather than an atomic basis set for describingthe dispersion of the bands near the Fermi level, one finds from TB fitsthat the changes in the hopping integrals can be transformed into changesin local on-site energies of molecular orbitals which can then be describedwithin a Holstein type model.10Chapter 1. IntroductionSuperconductivity in the weak coupling, high density limit is described byBCS theory while in the strong coupling, low density limit could be mod-elled as a Bose-Einstein condensate (BEC). In the strong electron-phononcoupling regime, the lattice becomes locally unstable and electrons (holes)become heavily dressed particles (polarons). These polarons tend to attracteach other via electron-phonon coupling to lower their energy and form lo-cally bound pairs, called bipolarons that are real-space pairs [13] in contrastto Cooper pairs that are k-space pairs. Adding Fermionic carriers due tothe i.e. potassium (K) substitution as in Ba1−xKxBiO3 reduces the den-sity of bipolarons but also freeing them to move since they are no longerbound to an ordered bipolaronic lattice [43]. It would be, however, possibleto have a system which may be in a regime intermediate between thesetwo extreme liming cases, therefore one needs to study the crossover fromcollective Cooper pairing to the formation and condensation of independentbosons in BEC. In this regard, it has been proposed that two types of chargecarriers exist in the metallic BKBO: the local pairs (real-space bosons) andthe itinerant electrons. The real-space bosons might be responsible for thecharge transport in semiconducting BaBiO3 and for superconductivity inthe metallic BKBO [44]. Further theoretical work of Ref. [45], argued thatthe mechanism for the insulating behaviour of the lightly doped BKBO isthe formation of trapped states within the gap, which later was investigatedby optical reflectivity on BKBO samples in Ref. [46].In this thesis we study the electronic structure including the electron latticeinteractions with density functional theory and tight-binding model Hamil-tonians to obtain realistic estimates of the electron-phonon coupling andthe effective attractive interaction between the electrons (holes) formingthe bipolarons. I start in the next chapter, by giving an overview of thecomputational tools used to perform the calculations. In chapter 3 we in-troduce the bond disproportionation model of the bismuthates and discussthe interplay between distortions and localization of holes on oxygens in a“molecular-orbital” picture, that can be viewed as formation of bipolaronson the collapsed oxygen octahedron. In chapter 4 we show the importanceof hybridization and develop tight-binding models with various complexity11Chapter 1. Introductionin describing the electronic structure of the bismuthates. We show thatthe bands near the Fermi level are the anti-bonding combination of Bi-sand a1g symmetry of oxygen octahedra. Using our simplified model weestimate the electron-phonon coupling. It is found to correspond to thestrong coupling regime. In chapter 5, we explore the parameter space ofour tight-binding models and show that there exists a crossover betweenbond-disproportionation and charge-disproportionation regimes. It is foundthat bismuthates are on the boundary of this crossover. By varying theparameters in our model we find an attractive molecular orbital interactionbetween two holes on oxygens which results in formation of bipolarons in thebismuthates. In chapter 6, we propose an approach to electron dope surfacesof these materials using adatoms and show how the surface crystal structureand electronic structure change with doping. And finally in chapter 7, westudy a system that can host coexisting 2D electron and hole gases. Westudy interfaces formed between bismuthates and a polar terminated mate-rial and use the ideas of electronic reconstruction to argue that 2D electronand hole gases naturally form in the bismuthates to overcome the potentialbuild up in the polar terminated material.12Chapter 2Computational methods2.1 IntroductionNuclei of atoms are more than 1000 times heavier than electrons, thereforeone can use the Born-Oppenheimer approximation [47] which is the assump-tion that the motion of atomic nuclei and electrons can be separated. Thisapproximation greatly simplifies the solution of the interacting problem.We can now think of interacting electrons moving in the field created by thenuclei. Therefore the electronic Hamiltonian of the solid can be written as:H = T + Vee + Vei (2.1)where T , the kinetic energy of electrons of mass m is:T = − ~22m∑i∇2i (2.2)and Vee is the repulsive interaction between electrons written as:Vee =12∑i 6=je2|ri − rj | (2.3)and Vei, the attractive interaction between electrons and nuclei is:Vei =∑iVion(ri) = −∑iIZIe2|ri −RI | (2.4)where e (ZIe) and ri(RI) are the electric charge and spatial coordinates ofelectrons (nuclei) respectively.132.2. Density Functional TheoryUnfortunately, solving the electronic Schrodinger equation:HΨ = EΨ (2.5)still remains a very difficult man-body problem even within the Born-Oppenheimer approximation. However, density functional theory (DFT)offers an appealing alternative approach. The basic equations of DFT areformulated in terms of the electron density:n(r) = N∫dr2...drN |Ψ(r, r2...rN )|2 (2.6)where N is the number of electrons and Ψ(r, r2...rN ) is the eigenfunctionof H where r, r2...rN are the spatial and spin coordinates of the system’sN particles. The advantage of working with the electron density n(r) isthat it is a function of 4 variables i.e. electron density is a function ofr and spin. This can be contrasted to the quantum chemistry methodswhere one deals with wave functions that are functions of 4N variables andexponentially divergent Hilbert spaces. The justification that the groundstate electron density n0(r) can be used as the basic variable is based on thetwo Hohenberg-Kohn Theorems[48].2.2 Density Functional Theory2.2.1 Hohenberg-Kohn theoremsThe Hohenberg-Kohn (HK) theorems read:Theorem I: “The external potential Vext(r) is a unique functional of n(r),apart from a trivial additive constant.”This statement simply means that different ionic potentials must have dif-ferent ground state electron densities.Proof: We prove this by contradiction. Let us first assume two externalpotentials Vext(r) and V′ext(r). The Hamiltonians are going to be differentbecause the potentials are different. Therefore we have two ground statewave functions Ψ0 and Ψ′0 that correspond to the ground state energies E0142.2. Density Functional Theoryand E′0 . However both of them lead to the same ground state electrondensity n0(r). By definition, we haveE0 = 〈Ψ0|H |Ψ0〉 (2.7)E′0 = 〈Ψ′0|H ′ |Ψ′0〉 (2.8)Since |Ψ′0〉 is not the ground state of H, we must have:E0 < 〈Ψ′0|H |Ψ′0〉 < 〈Ψ′0|H ′ |Ψ′0〉+ 〈Ψ′0|H −H ′ |Ψ′0〉< E′0 +∫drn0(r)(Vext(r)− V ′ext(r))(2.9)Similarly,E′0 < 〈Ψ0|H ′ |Ψ0〉 < 〈Ψ0|H |Ψ0〉+ 〈Ψ0|H ′ −H |Ψ0〉< E0 +∫drn0(r)(V′ext(r)− Vext(r))(2.10)Combining Eq. 2.9 and Eq. 2.10, we obtain:E0 + E′0 < E0 + E′0 (2.11)which leads to contradiction. Therefore, two different external potentialsVext(r) and V′ext(r) can not lead to the same ground state density n0(r), i.e.,the ground state density determines the external potential Vext(r) uniquely,up to a constant.We can now see that specifying the external potential uniquely determinesthe Hamiltonian and therefore all ground-state properties are functionals ofn(r).Theorem II: “There exists a universal functional of the density, F [n(r)], in-dependent of the external potential Vext(r), such that the expression E[n] ≡∫drn(r)Vext(r)+F [n(r)] has as its minimum value the correct ground-stateenergy associated with Vext(r).”152.2. Density Functional TheoryProof : The universal functional F [n(r)] is written as:F [n(r)] = T [n(r)] + Vee[n(r)] (2.12)where T [n(r)] and Vee[n(r)] are the kinetic energy and the electron potentialenergy. Using the variational principle, for any wave function Ψ′, the energyfunctional written as:E[Ψ′] = 〈Ψ′|T + Vee + Vext |Ψ′〉 (2.13)has its global minimum value at the ground state wave function Ψ0. Fromthe first theorem we know Ψ′ must correspond to a ground state with ex-ternal potential V ′ext(r) and density n′(r). Then from variational principlewe have:E[Ψ′] = 〈Ψ′|T + Vee + Vext |Ψ′〉= E[n′(r)] =∫drn′(r)V ′ext(r) + F [n′(r)]> E[Ψ0] =∫drn0(r)Vext(r) + F [n0(r)]= E[n0(r)](2.14)Thus the energy functional E[n0(r)] of the true ground state electron densityn0(r), is indeed lower than the value of this functional for any other density.Therefore, to find the true ground state electron density and energy, onemust minimize the energy functional with respect to variations in the densityn(r).Although the two HK theorems show that particle density n(r) can be con-sidered as the basic variable, however, it is still impossible to calculate anyproperty of a system since we do not know the universal functional F [n(r)].But in 1965, Kohn and Sham [49], proposed a solution to this problem whichhas become a basis for modern DFT.162.2. Density Functional Theory2.2.2 Kohn-Sham equationsIn the Kohn-Sham ansatz, one maps the system onto an auxiliary non-interacting one where the electrons move within an effective Kohn-Shamsingle-particle potential VKS(r). This non-interacting system is chosen suchthat its ground state electron density is equal to that of the original systemof interacting electrons. By virtue of HK theorem, the ground state energyof these two systems is the same. The KS electron density is defined as:n(r) =N∑i=1|ψi(r)|2 (2.15)Let us now first write out the energy functional explicitly as:E[n(r)] = − ~22m∑i〈ψi| ∇2 |ψi〉+ e22∫ ∫drdr′n(r)n(r′)|r− r′| + Exc[n(r)]+∫drn(r)Vion(r).(2.16)Here, ψ is the KS wave function and i refers to the single-particle statesand the sum is over all the occupied states. Exc[n(r)] is the exchange-correlation functional where one groups the two unknown functionals forthe “correlation energy” and the “exchange energy”.Taking functional variation δE[n(r)]/δψ∗i (r) with the constraint that ψi(r)be normalized to one we obtain:(− ~22m∇2 + VKS)ψi(r) = iψi(r) (2.17)where VKS(r) is defined as:VKS(r) =∫dr′n(r′)|r− r′| + Vion(r) + Vxc(r) (2.18)The first term above is the Hartree potential, the second term is the localionic potential and the last term is the exchange-correlation potential defined172.2. Density Functional Theoryas:Vxc(r) =δExc[n]δn(r)(2.19)Solving the set of eq.2.15–2.19 self-consistently, we obtain the ground stateelectron density and the ground state energy.The Kohn-Sham equations 2.15–2.19 are a set of non-linear equations thathave to be solved for the electron density n(r), self-consistently starting froma guessed electron density and then the Kohn-Sham equations are solvednumerically.2.2.3 Exchange-correlation functionalThe Kohn-Sham ansatz enables practical applications of DFT. However, theexact form of the exchange-correlation functional is unknown. There hasbeen various approximations for this functional and the most widely usedapproximation in the solid state calculations are the Local Density Approxi-mation (LDA)[49] and the Generalized Gradient Approximation (GGA)[50–52]. The central concept of the LDA is adapted from the theory of electrongas [53] where each small region of space around a point r is approximatedas a uniform electron gas with the electron density n(r). Thus in the LDAapproximation, the exchange-correlation functional is of the form:Exc[n] ≈ ELDAxc =∫drn(r)xc(n(r)) (2.20)where xc(n) is the exchange and correlation energy per particle of a uni-form electron density, n(r) at r. Since, there is no analytical expression forxc(n(r)), it is approximated by the sum of the exchange and correlationcontributions:xc(n(r)) = x(n(r)) + c(n(r)). (2.21)In electron gas theory, the exchange is given by:x(n(r)) = −Cxn(r)1/3, Cx = 34( 3pi)1/3, (2.22)182.2. Density Functional Theorywhile the correlation part is obtained numerically[52, 54]. There is a numberof parametrized expressions for the correlation part. The most commonlyused one was proposed by Ceperley and Alder[55] who fitted quantum MonteCarlo results obtained in various coupling regimes.The GGA is another popular approximation and an improvement to theLDA. In GGA, xc(r) depends on the gradient of the electron density ∇n(r),in addition to the electron densities:Exc[n] ≈ EGGAxc =∫drf(n(r),∇n(r)) (2.23)There exist a number of GGA functionals[50, 51, 56–58] and the most widelyused one in solid state physics is the Perdew-Burke-Ernzerhof (PBE) func-tional [51].2.2.4 Basis setsThe ionic potential Vion in a crystalline solid possesses translational symme-try,Vion(r + RT ) = Vion(r) : (2.24)where RT is the translational lattice vector. From the Bloch’s theorem[59],the single-electron wave functions that correspond to the periodic potentialare the Bloch wave functions ψnk(r):ψnk(r) = eik.runk(r), (2.25)with unk(r) having the same periodicity as Vion. In order to solve the single-electron Kohn-Sham equations:(− ~22m∇2 + VKS)ψnk(r) = n(k)ψnk(r) (2.26)one needs to expand ψnk(r) in terms of some given basis functions φik(r):ψnk(r) =P∑icinφik(r), (2.27)192.2. Density Functional Theorywhere cin are the expansion coefficients. Evidently, the basis sets need to becomplete. However, it is highly desirable to have the smallest set possiblein order to reduce the size of Hamiltonian and increase the efficiency ofcomputation.The choice of the basis set is determined by considerations of accuracy vscomputing speed and ease of implementation. There exists a number of basissets that have been developed over the years. In this thesis, we have usedtwo DFT packages, VASP and WIEN2k, that are based on the plane wavesand the linearized augmented plane wave (LAPW) basis sets respectively.Plane wave basis setFor periodic systems, the most natural choice of basis is plane waves whichcan easily be expanded in Fourier series:ψnk(r) = eik.runk(r) =eik.r√V∑Gcnk(G)eiG.r (2.28)where G is a reciprocal lattice vector and V is the unit cell volume. Insertingexpression 2.28 into the Kohn-Sham equation 2.26 one finds that cnk(G)satisfy:(|k + G|)22cnk(G) +∑G′V¯KS(G−G′)cnk(G′) = nkcnk(G) (2.29)where V¯KS is the Fourier transform of the Kohn-Sham potential VKS . Theplane waves basis can be chosen up to an arbitrary reciprocal lattice vector.In practice, the expansion is truncated at some kinetic energy called thecutoff energy:(|k + G|)22≤ Ecut (2.30)The higher the Ecut, the more complete the basis, and the more accuratethe results. For this matter we always have to ensure the cut-off energy ishigh enough to give accurate results. We repeat the calculations with highercut-off energies until the properties of interest have converged.202.2. Density Functional TheoryEven by including a cutoff energy an extremely large plane wave basis set isrequired to perform an all-electron calculation, and a lot of computationaltime is required to compute the electronic wave functions. This is becauseof the tightly bound core orbitals that require a very large number of planewaves to describe their wave function and to follow the rapid oscillations ofthe wave functions of the valence electrons in the core region. This presentssignificant challenge for any method based on the plane wave basis set.Earlier calculations using this basis set were able to study the electronicstructure of only small unit cells. There are several ways to address thisproblem. One of them is called the pseudopotential method [60, 61].LAPW and APW+loThis basis set is used in all electron full potential calculations [62–64]. Thebasis functions consist of plane waves that are augmented with a combinationof atomic-like radial functions inside non-overlapping muffin-tin spheres thatare centred at atomic positions. The LAPW basis is described as:φLAPWG,k (r) =1√Vei(k+G)r, r /∈ RIMT∑lm[AI,k+Glm uIl (r, EI1l) +BI,k+Glm u˙Il (r, EI1l)]Ylm(r−RI), r ∈ RIMT(2.31)where uIl (r, EI1l) and u˙Il (r, EI1l) are the solutions to the radial Schrodingerequation of an isolated atom I and their energy derivatives, evaluated atlinearization energies EI1l. The coefficients AI,k+Glm and BI,k+Glm are obtainedfrom the requirement that the basis wave functions have to be differentiableand continuous at the muffin-tin sphere boundaries.In addition, an improvement on the description of semicore states is achiev-able, if the LAPW basis is complemented by local orbitals (LO), that are212.2. Density Functional Theoryonly defined inside muffin-tin spheres:φLOI,lm(r) =0, r /∈ RIMT[AI,LOlm uIl (r, EI1l) +BI,LOlm u˙Il (r, EI1l) + CI,LOlm uIl (r, EI2l)]Ylm(r−RI), r ∈ RIMT(2.32)The last term, CI,LOlm uIl (r, EI2l), takes into account all of the semicore statesfor a given atom I. Here, the expansion coefficients AI,LOlm , BI,LOlm and CI,LOlmare obtained by normalizing the local orbital and setting their values andderivatives to zero at the boundary of the muffin-tin sphere.The above basis functions, have been implemented in the WIEN2k [65] pack-age and achieve optimal accuracy at computational costs.2.2.5 PseudopotentialsThe physical properties of materials are dependent on the valence electronsto a much greater degree than on the core electrons. The pseudopotentialapproximation uses this fact by replacing the divergent potential in theregion close to the nucleus by a smooth shallow pseudopotential. Fig.2.1 il-lustrates an atomic potential, valence wave function and the correspondingpseudopotential and pseudo wave functions. It can be seen that the valenceelectron wave functions oscillate rapidly close to the core region to maintainthe orthogonality between the core and valence wave functions. Describ-ing these rapid oscillation of wave functions in the core region needs manyplane wave basis, which is very expensive computationally. To resolve thisproblem, one separates the potential into two regions. The potential in theinner region is replaced by a smooth function which insures proper boundaryconditions. For the outside core region the two potentials are identical andtherefore the scattering from the two potentials are indistinguishable. Thereis no unique recipe to choose parameters in calculation of pseudopotentialsfor an atom or an ion. There are quite a few different algorithms concerningthe generation of a pseudopotential, which can be found in the literature[66–69]. These parameters are obtained by trial and error while compar-ing to all electron full potential calculations. Pseudopotentials used in this222.2. Density Functional TheoryFigure 2.1: Schematic illustration of all-electron(solid lines) and pseudoelec-tron (dashed lines) potentials and their corresponding wave functions. Theradius at which all-electron and pseudo electron values match is designatedrc. Figure adapted from Ref.[70]work are provided by VASP developers. We have, however, tested them bycomparing calculated lattice parameters to that obtained using the WIEN2kcode.2.2.6 Relaxation of the ionic systemUp until now we only considered the calculations in which the ionic positionsand the size and shape of the unit cell are fixed. One can calculate atomicpositions by minimizing the total energy with respect to atomic displace-232.2. Density Functional Theoryments. In order to do this, one needs to know the forces acting on each atomin the unit cell. In the simplest approach, atoms are moved in the directionof forces until the change in the total energy or force is reduced below acertain threshold.Hellmann-Feynman theoremThe force on the ion I as stated in the force theorem is given by:FI = − ∂∂RI(E0 + Eii), (2.33)where E0 is the ground state energy and Eii is the ion-ion or nuclear inter-action energy. Since the Kohn-Sham states {ψi} implicitly depend on theionic positions, RI , the first term of the force equation gives:∂E0∂RI=∂E0∂RI∣∣∣{ψi}+∑i∫dr{δE0δψ∗i (r)∂ψ∗i (r)∂RI+δE0δψi(r)∂ψi(r)∂RI}(2.34)Taking the functional variations δE0/δψ∗i (r), we arrive at the Kohn-Shamequations:δE0δψ∗i (r)=[− ~22m∇2+∫dr′n(r′)|r− r′|+Vxc(r)+Vion(r)]ψi(r) = iψi(r). (2.35)The second equality holds since {ψi} are solutions of the Kohn-Sham equa-tions. Taking the complex conjugate of this functional variation, we canfurther simplify the total energy derivative: The second term in the lastequality vanishes since {ψi} are normalized. Finally, we have a simpleexpression for the force on nuclei known as the Hellmann-Feynman forcetheorem[71, 72]:FI = − ∂∂RI(E0 + Eii) = −∫drn(r)∂Vion∂RI− ∂Eii∂RI(2.36)This follows from the fact that the only term in the energy functional, E0,that explicitly depends on nuclei positions is the external potential. Note242.2. Density Functional Theorythat this expression depends solely on the electron density n(r), externalpotential Vion(r) and the interaction between nuclei Eii.2.2.7 Extensions of DFT: LDA+UThe local approximations to the exchange correlation functional (LDA, LSDA,GGA, etc.) have been very successful in describing the materials in whichthe electron correlation effects are not important. However, they fail todescribe the electronic structure and magnetic ground state of systems thatcontain d or f electrons. A simple way to fix some of LDA shortcomingswas proposed by Ref. [73, 74] . It is called the LDA+U method in whicha local functional is extended by introducing additional terms for selected(correlated) orbitals that resemble Hubbard-like interaction [73, 74]:ELDA+U [n(r)] = ELDA[n(r)] +Hint − 〈Hint〉 (2.37)where Hint represents the interaction part of the many-body Hamiltonian inmean field approximation and the last term 〈Hint〉 comprises the Coulombrepulsion that is included in the ELDA term and must be subtracted to avoiddouble counting. The Coulomb and the Hund’s rule coupling are representedby matrixes Umm′ and JHmm′ , respectively in:Hint =12∑m,m′ ,σUmm′nmσnm′−σ+12∑m6=m′ ,σ(Umm′−JHmm′)nmσnm′σ (2.38)with σ and nmσ denoting the spin projection and occupation number re-spectively.Among the most popular LDA+U methods are “around mean-field (AMF)”,and the “atomic limit (AL)” version of the LDA+U[73, 74]. The AMFLDA+U functional as an extension of the LSDA functional [74] is defined252.3. Tight-Binding modelas:EAMF = ELSDA +12∑m,m′ ,σUmm′ (nmσ − n¯σ)(nm′−σ − n¯−σ)+12∑m6=m′ ,σ(Umm′ − JHmm′ )(nmσ − n¯σ)(nm′σ − n¯σ)(2.39)where n¯σ is the spin-resolved average occupation number.In the AL version of the LSDA+U method the double counting part of themany-body Hamiltonian is written as:〈Hint〉AL = 12UN(N − 1)− 12JHN↑(N↑ − 1)− 12JHN↓(N↓ − 1) (2.40)whereN =∑mσ nmσ, U =1(2l+1)2∑mm′ Umm′ and JH = U− 12l(2l+1)∑mm′ (Umm′−JHmm′ ) with the AL functional:EAL = ELSDA +Hint − 〈Hint〉AL (2.41)2.3 Tight-Binding modelThe kinetic energy term of the Hubbard Hamiltonian[75]:H = −tij∑〈i,j〉,σ(c†i,σcj,σ + c†j,σci,σ) + UN∑i=1ni↑ni↓ (2.42)is referred to as the tight-binding (TB) Hamiltonian. In the Hamiltonian,〈i, j〉 represents summation over nearest-neighbour lattice sites and ni↑ni↓ isthe spin-density operator for spin σ on i-th site. By choosing an atomic-likebasis set one can evaluate the TB Hamiltonian and the overlap matrix ele-ments tij between the atomic orbitals. In the TB model the on-site energiesare defined as = 〈ψi|H |ψi〉 and the hopping integral as tij = 〈ψi|H |ψj〉,where i and j correspond to the atomic orbitals on neighbouring sites. Inorder to arrive at the TB band structure that resembles the band structureof a lattice obtained from DFT calculations, the TB Hamiltonian is first262.4. Electron-phonon interactionFourier-transformed into the k-space and the hopping integrals and on-siteenergies of the orbitals included in the model are adjusted such as to fitthe TB model band structure. To limit the number of fitting parameters, itis normally assumed that the inter-atomic matrix elements only extend tofirst or second neighbours. The explicit form of these integrals is obtainedfrom the Slater and Koster[76] table of inter-atomic matrix elements. In thisapproach the basis functions do not appear explicitly and are used only tohelp justify the chosen forms of the Hamiltonian and overlap matrices.Let us now briefly introduce the electron-phonon interaction in DFT.2.4 Electron-phonon interactionAn electron on the Fermi level interacts with the vibrations of the atoms ofthe crystal lattice. This process is most conveniently understood within anelectron-phonon scattering exchange interaction. A key component of thisinteraction is the matrix element for the electron-phonon scattering definedas:M[ν]k,k+q = (~mωq,ν)1/2 〈k| δνV |k + q〉 , (2.43)where, ωq,ν is the phonon frequency for a particular phonon wave vectorq and mode ν, δνV is the phonon perturbation for that mode, and 〈k| isthe Kohn-Sham electronic eigenstate. This matrix essentially accounts forthe details of the interaction between electronic eigenstates and all latticevibrations. Calculating the matrix elements allows one to determine thewave vector specific coupling parameter λq,ν within the Migdal [77] approx-imation:λq,ν =2N(0)ωq,ν1N∑k|M [ν]k,k+q|2δ(k)δ(k+q) (2.44)where N(0) is the density of states at the Fermi level, and δ(k) is the energyconserving delta-function.Now, the Eliashberg spectral function can be obtained from a Brillouin zone272.4. Electron-phonon interactionintegral of λq,ν as:α2F (ω) =12∑qωq,νλq,νδ(ω − ωq,ν) (2.45)This term has all the information about scattering events for the electronicstates scattering from k to k + q with an energy transfer of ω coupled tothe crystal vibrations.The total electron-phonon coupling constant is then obtained from the fre-quency moments of the spectral function and the average coupling:λ = 2∫ω−1α2F (ω)dω = 1/N∑q,νλq,ν . (2.46)In the the McMillan equation [78] one can estimate the superconductingtransition temperature Tc from:Tc =ωlog1.20exp[− 1.40(1 + λ)λ− µ∗(1 + 0.62λ)](2.47)where ωlog is the logarithmic frequency moment of the Eliashberg spec-tral function α2F (ω) and µ∗, a dimensionless parameter characterizing thescreened Coulomb interaction strength.28Chapter 3Hybridization effects andbond disproportionation3.1 IntroductionIn this chapter, we present a theoretical description of the bond dispropor-tionated insulating state of the bismuth perovskites ABiO3 (A = Sr, Ba)introduced in chapter 1 that recognizes the bismuth-oxygen hybridizationas a dominant energy scale. We show by using electronic structure meth-ods, such as DFT and model Hamiltonians, that the breathing distortionis accompanied by spatial condensation of hole pairs into local, molecular-like orbitals of the A1g symmetry composed of O-2pσ and Bi-6s atomic or-bitals of collapsed BiO6 octahedra. This contradicts the popular view of thepurely ionic Bi3+/Bi5+ charge-disproportionation which is often interpretedin terms of a charge-density-wave. The role played by oxygen molecularorbitals in the bismuthates is very similar to that in the nickelates and neg-ative charge-transfer transition-metal compounds in general. Furthermore,we show that the formation of localized states upon breathing distortionis, to a large extent, a property of the oxygen sub-lattice and the stronghybridization with cation of the shallow core orbitals.Finally, we show that the tilting distortions are found to strongly enhancethe breathing instability through an electronic mechanism, as manifested bya q = (pi,pi,pi) peak in the static susceptibility χ(q, ω = 0).293.2. Electronic structure of SrBiO3 and molecular orbitals3.2 Electronic structure of SrBiO3 and molecularorbitalsWe investigate the nature of the insulating bond-disproportionated state inthe bismuthates from the perspective of strong Bi-6s/O-2p hybridizationusing DFT and LDA [54, 65, 79–82], thanks to the weakly correlated natureof the Bi-6s and O-2p electrons in ABiO3[83–85].We begin by highlighting the basic features of the electronic structure ofthe bismuthates[19, 26, 35–37, 86–94]. Since the LDA band structures ofBaBiO3 and SrBiO3 are almost identical, in the following we will concentrateon SrBiO3 since LDA correctly predicts a band gap in SrBiO3. The Bi-6sand O-2p orbitals strongly hybridize to create a broad band manifold nearthe Fermi level EF extending from −14 to 3 eV [Fig. 3.1(a)]. The upperband of this manifold is a mixture of the Bi-6s orbitals and the σ 2p orbitalsof the oxygens, i.e., the O-2p orbitals with lobes pointing towards the centralBi. This upper band is half filled as a result of self-doping: The 18 2p statesof three oxygen ions and the two 6s states of a Bi ion are short of oneelectron to be fully occupied. Upon breathing, this band splits, producinga charge gap. We note that even in the bond-disproportionated state, thebismuthates are very far from the ionic limit of pure Bi3+ and Bi5+. Theself-doped holes reside predominantly on the oxygen-pσ orbitals, a situationsimilar to that in the nickelates. We now infer the exact distribution of holesby analyzing the DFT results in more detail.In this analysis, we will be guided by the following considerations. First,a spherically symmetric Bi-6s orbital couples only to the A1g combinationof the pσ orbitals of the surrounding oxygen atoms [Fig. 3.1(b)] [95]. Hop-pings to any other combination of the O-p orbitals are forbidden by symme-try. Second, since the Bi-6s orbitals are quite extended, the hybridizationbetween the Bi-6s and O-A1g orbitals is very strong and even becomes adominant effect. Third, in the bond-disproportionated phase, hybridizationwithin a collapsed (expanded) BiO6 octahedron is strongly enhanced (sup-pressed), resulting in a formation of local, molecular-like orbitals on thecollapsed octahedra. We are able to observe all of these effects in our DFT303.2. Electronic structure of SrBiO3 and molecular orbitals-10-5 0 5 10 Energy (eV)(b)(a) Top Bottom Left Right Back Front EnergyA1g1/√61/√61/√61/√61/√61/√6-4tEg1/√3 1/√3 -1/√12-1/√12-1/√12-1/√122tEg00 1/2 1/2-1/2-1/22tT1u 1/√2-1/√200000T1u 00 1/√2-1/√2000T1u 0000 1/√2-1/√20Bi-sEgT1uA1gCollapsed octahedron-10-5 0 5 10 0 1 2Energy (eV)PDOSA M Γ X N ΓA M Γ X N ΓBi-sEgT1uA1gExpanded octahedronFigure 3.1: (a) LDA electronic structure of SrBiO3 projected onto the Bi-6s orbital and combinations of the O-pσ orbitals of a collapsed (top) andexpanded (bottom) BiO6 octahedra. For the doublet Eg and the triplet T1u,only one projection is shown. The Fermi level is set to zero, and PDOSstands for projected density of states and is given in states/eV/cell. (b) Anoctahedron of O-pσ orbitals coupled via nearest-neighbour hopping integrals−t and its eigenstates.313.2. Electronic structure of SrBiO3 and molecular orbitalsresults by projecting the LDA single-particle eigenstates onto the combina-tions of O-pσ orbitals. Let us first focus on the collapsed octahedra. In thetop panel of Fig. 3.1(a), one finds the Bi-6s and O-A1g characters at about2 eV and −10 eV. The difference, 12 eV, is mainly a result of the hybridiza-tion splitting between the bonding and anti-bonding combinations, whichindeed turns out to be the dominant energy scale for SrBiO3. The anti-bonding combination is strongly peaked in the lowest two conduction bandsthroughout the whole Brillouin zone. In contrast, the anti-bonding A1g com-binations of the expanded octahedra (bottom panel) spread in energy, withsome weight seen both below and above EF depending on the position in theBrillouin zone. This indicates that the metal-insulator transition with bonddisproportionation in the bismuthates should be understood as a pairwisespatial condensation of holes into the anti-bonding A1g molecular orbitals ofthe collapsed octahedra. The small charge-disproportionation between theBi ions (±0.15e inside the Bi muffin tin spheres) appears to be a marginalside effect of the change in Bi-O distances. This state, which resembles the(d8L2)S=0 singlet proposed for the nickelates, was also hypothesized in Ref.[44].323.3. Evolution of the breathing and tilting distortions-0.8-0.6-0.4-0.2 0 0.2 0.4 0 0.5 1 1.5E-E cubic (eV/f. u.)b/bexp 0 0.4 0.8 1.2 0 0.5 1 1.5Gap (eV)b/bexp(a) (b)t=00.5texptexp1.5texpFigure 3.2: LDA characterization of SrBiO3 model structures with varyingdegrees of the BiO6 octahedra’s tilting t and breathing b: (a) total energyper formula unit (f.u.) and (b) band gap. In (a), solid lines and solidcircles (dashed lines and open circles) represent model structures with fixed(relaxed) Sr atoms. The horizontal dashed line marks the energy of theexperimental SrBiO3 structure.3.3 Evolution of the breathing and tiltingdistortionsIt is interesting to trace the evolution of the LDA projected density of statesas a function of breathing b and tilting t of the octahedra. For this, weprepare a set of SrBiO3 model structures with varying degrees of b and t [seeAppendix. E for experimental bulk structure]. We characterize the breathingdistortion b by the Bi-O bond length disproportionation, bexp =∆dBi-Od¯Bi-O=9%, and the tilting distortion t by the mismatch between the Bi-Bi andthe Bi-O bond lengths, texp =2d¯Bi-O−d¯Bi-Bid¯Bi-Bi= 16.5o. In Figs. 3.2(a) and3.2(b), the model structures are characterized in terms of, respectively, thetotal energy and the charge gap calculated within LDA. Parameters b andt are given compared to the experimentally observed distortions bexp andtexp. We find that a finite t is required to stabilize the breathing distortion.Although this result is sensitive to details of DFT calculations [19, 38, 39, 91],the absence of the breathing instability at zero tilting can be qualitatively333.3. Evolution of the breathing and tilting distortionsunderstood in terms of a missing linear component in the elastic energy ofthe breathing distortion when Bi-O-Bi angle is 180 degrees. With increasingt, the equilibrium b shifts to higher values [Fig. 3.2(a)], while the chargegap opens sooner as a function of b [Fig. 3.2(b)]. We conclude, in agreementwith previous DFT studies[36, 90, 91], that tilting enhances the breathinginstability, through a mechanism which will be discussed later. Interestingly,letting Sr atoms relax from their high-symmetry positions (while keeping band t fixed) can significantly lower the total energy for structures with hight [open circles in Fig. 3.2(a)]. For the (bexp, texp ) structure, for instance, theenergy drop is 0.15 eV per formula unit. The lattice constants, the atomicpositions of oxygen atoms O1, O2, and O3, as well as the relaxed atomicpositions of Sr of the 16 model structures are given in Appendix. E. We notethat in the experimental structure the Sr positions are slightly away fromthe high symmetry positions.Figure 3.3 shows projections onto collapsed octahedra for the (0, 0), (bexp,0), (0, texp), and (bexp, texp) structures. Here, the bonding (“B”) and anti-bonding (“A”) combinations of the Bi-6s and O-A1g orbitals are shownexplicitly. An important observation is that, even with no breathing, amajor portion of the anti-bonding orbital weight is above EF . Thus, thesystem is predisposed to hole condensation. The breathing distortion justmakes collapsed octahedra more preferable for the holes to go to. On theother hand, the primary role of the tilting distortion is to reduce the band-width. This band narrowing, however, is very nonuniform, leading to certainimportant consequences, as will be discussed later.343.3.Evolutionofthebreathingandtiltingdistortions-10-5 0 5 10Energy (eV)Bi-sA1gt=0, b=0"A""B"-10-5 0 5 10 0 1Energy (eV)PDOSA M Γ X N ΓA M Γ X N ΓBi-sA1gt=texp, b=0 0 1 2"A""B"Bi-sA1gt=0, b=bexp"A""B" 0 1PDOSA M Γ X N ΓA M Γ X N ΓBi-sA1gt=texp, b=bexp 0 1 2"A""B"Figure 3.3: LDA electronic structure of SrBiO3 as a function of breathing b and tilting t. Projections are madeonto the Bi-6s orbital and the A1g combination of the O-pσ orbitals of a collapsed BiO6 octahedron, as well astheir bonding (“B”) and anti-bonding (“A”) combinations.353.4. Simplified tight-binding models3.4 Simplified tight-binding modelsThe formation of well-defined molecular O-A1g states on collapsed octahedrais a property of the oxygen sub-lattice. This is illustrated in Fig. 3.4(a),which shows LDA calculations for an artificial system consisting of onlythe oxygen sub-lattice of SrBiO3. Here, however, the A1g states are thelowest in energy and fully occupied. It is the strong hybridization with theBi-6s states that pushes them above EF . Upon hybridization, the 2 eVbroad oxygen band and the essentially flat Bi-6s band form bonding/anti-bonding combinations with splitting of about −15 eV. To demonstrate thissplitting, we consider for simplicity a tight-binding (TB) model for a two-dimensional perovskite like lattice of pσ, ppi-orbital sites and s-orbital sites,with nearest-neighbour s−p and p−p hoppings as found in SrBiO3: |tspσ| ≈2.3, |tppσ| ≈ 0.64, and |tpppi| ≈ 0.03 eV as calculated using the Wannierfunctions technique in DFT for a cubic unit cell. Switching on the breathingdistortion helps to form molecular A1g states associated with collapsed pσ-orbital cages at the bottom of the p band [compare the left and central panelsof Fig. 3.4(b)]. Subsequent hybridization with the flat s bands, located rightbelow the A1g states, has a much stronger effect (since |tsp| >> |tpp|) andpushes the anti-bonding Bi-6s-A1g combination above EF [right panel ofFig. 3.4(b)].Finally, we show the effects of the tilting-induced band narrowing [also seeAppendix. B]. For this purpose, we calculate the static non interacting sus-ceptibility χ(q, ω = 0) with the help of a single-band TB model for struc-tures with zero breathing and varying tilting. For structures with tilting,(b = 0, t 6= 0), calculations in a cubic unit cell are desirable in view oftheir straightforward interpretation but cannot be done directly owing tothe broken cubic symmetry. For these structures, we therefore adopt thefollowing TB parametrization scheme. First, we use the Wannier functionstechnique to describe the four LDA bands near the Fermi level calculatedwith a√2×√2× 2 supercell. Then, the TB parameters thus obtained aresymmetrized according to a cubic symmetry and the TB model is rewrittenfor a small cubic unit cell. Symmetrization turns out to have very little effect363.4. Simplified tight-binding models-2-1 0 0 5 10Energy (eV)PDOSA M Γ X N Γ A M Γ X N Γ(a)(b)EgT1uA1gJust oxygen sublatticet=texp, b=bexpt=0, b=0 0 1CCb=0, h=0Model PDOS TotA1g 0 1-4 -2 0ECEnergy (eV)sb≠0, h=0-4 -2 0Energy (eV)b≠0, h≠0-12 -8 -4 0Energy (eV)Figure 3.4: (a)LDA electronic structure of the oxygen sub-lattice of SrBiO3.Projections are made onto combinations of the O-pσ orbitals of a collapsedoxygen octahedron. (b) Model density of states as a function of breathing band hybridization between s and p orbitals. CC (EC) stands for a collapsed(expanded) p-site cage. The model states are 90% filled, i.e., there is onehole per s orbital; the Fermi energy is set to zero and marked with blackdashed vertical lines.373.4. Simplified tight-binding models-6-4-2 0 2 4 R Γ X M Γ Energy (eV) 0.5 1 1.5 (0,0,0) (π,π,π) -χ(q) (a. u.)q(b) t=00.5texptexp1.5texp(a)Figure 3.5: The effect of tilting on (a) the half-filled band and on (b) thestatic susceptibility χ(q, ω = 0), at zero breathing. In (b), solid (dashed)lines represent calculations where nonlinear effects due to tilting are (arenot) taken into account.on the bands, which validates our approach. For our analysis, two kinds ofTB models, describing the band crossing the Fermi level, are considered:(1) realistic t 6= 0 models closely following the LDA bands and (2) t 6= 0models uniformly rescaled with respect to the t = 0 case, such as to matchthe LDA bandwidth. In Fig. 3.5(a), the band dispersions of the realisticmodels are plotted in the Brillouin zone of a small cubic cell so that thenonuniform nature of the t-induced band narrowing is particularly clear:The changes at R, for example, are much smaller than at Γ. Using therealistic models and considering all the energies spanned by the band, wefind a dominating susceptibility peak at q = (pi,pi,pi) indicating the exis-tence of breathing instability that quickly grows with increasing tilting [Fig.3.5(b), solid lines]. This growth is much quicker than it would be in thecase of a uniform band narrowing (compare with the dashed lines). Thisindicates that nonlinear effects due to tilting, possibly, approaching perfectnesting conditions [96–98], are very important for stabilizing the breathingdistortion.383.5. Summary3.5 SummaryIn summary, we have studied hybridization effects in the bismuth perovskitesand their interplay with structural distortions, such as breathing and tilting.It is shown that strong hybridization between the Bi-6s and O-2p orbitalsprecludes purely ionic charge-disproportionation of a Bi3+/Bi5+ form. In-stead, the (self-doped) holes spatially condense into molecular-orbital-likeA1g combinations of the Bi-6s and O-2pσ orbitals of collapsed BiO6 octa-hedra, with predominantly O-2pσ molecular orbital character. The tiltingdistortion is found to strongly enhance the breathing instability through (atleast in part) an electronic mechanism, as manifested by a q = (pi,pi,pi) peakin the static susceptibility χ(q,ω = 0). It is expected that similar processestake place in other perovskites as well and can involve localized molecularorbitals of symmetries other than A1g. Thus, in the rare-earth nickelates,the orbitals of relevance would be the Eg combinations of the O-2pσ orbitalshybridized with the Ni-eg orbitals[30–33].39Chapter 4Oxygen holes andhybridization4.1 IntroductionIn chapter 3 and Ref. [34], we used DFT methods to validate an alternativemicroscopic model[34, 99] for the insulating state of ABiO3, in which the holepairs condense spatially onto collapsed O6 octahedra occupying molecularorbitals of the a1g symmetry while all the Bi ions are close to being 6s2,i.e., the following process is taking place: Bi3+L+ Bi3+L→ Bi3+L2 + Bi3+,where L represents a ligand hole in an a1g-symmetric molecular orbital ona collapsed O6 octahedron.In this chapter, we intend to better understand the relevance of variousinteractions in determining the electronic structure of ABiO3 and derive itsminimal tight-binding (TB) model. Such a minimal model describing thelow energy scale states is especially useful in constructing model Hamilto-nians. The models can be used to include the electron-phonon interactionsand study bipolaron formation and superconductivity in hole or electrondoped systems. We first use DFT calculations to study the hybridizationstrengths between the constituent elements of ABiO3, demonstrating theextreme effects of interatomic hybridization involving the Bi-6s and O-2porbitals. Having determined the most relevant atomic orbitals and interac-tions, we then study the properties of our derived TB model as a function ofstructural distortions in two and three dimensions. Finally, we explore pos-sible simplifications of the TB model with a focus on describing low-energyelectronic excitations.404.2. Bi, BaBi, O3, and BaO3 sub-lattices of BaBiO34.2 Bi, BaBi, O3, and BaO3 sub-lattices ofBaBiO3In order to study the hybridization strengths between the constituent ele-ments of ABiO3, we calculate and compare the electronic band structures ofBaBiO3 and of its Bi, BaBi, O3, and BaO3 sub-lattices. It is to be expectedthat all the conclusions in this section will be equally applicable to SrBiO3because of its close similarity to BaBiO3.The electronic structure calculations are performed with the full-potentiallinearized-augmented-plane wave code WIEN2k[65]. Exchange and correla-tion effects are treated within the generalized gradient density approxima-tion (GGA)[100]. For now, we will neglect the effects of lattice distortionsand consider an idealized cubic unit cell containing one formula unit. Avalue of 4.34 A˚ is used for the lattice constant, taken as the average over thenearest-neighbour Bi-Bi distances in the experimentally measured distortedstructure[6], and a 7 × 7 × 7 k-point grid is used for the Brillouin-zoneintegration.As discussed earlier, the band structure of BaBiO3 near the Fermi level isfeatured by a collection of strongly dispersive states with predominant Bi-6sand O-2p orbital characters. Most of the Bi-6s character is concentrated inthe 8 eV broad lowest band centred at −10 eV [Fig. 4.1(a)]. In contrast,in either the BaBi [Fig. 4.1(b)] or the Bi [Fig. 4.1(c)] sub-lattices, the Bi-6s band width is less than 1 eV. This indicates that interatomic hoppingintegrals involving only Ba and Bi atomic orbitals are rather small.Let us now consider the electronic band structures of the BaO3 and O3 sub-lattices, shown in Figs. 4.1 (e) and (f), respectively. As we demonstratedin Ref. [34], it is helpful to analyze the dispersion of the O-2p states in aperovskite structure in terms of molecular orbital combinations of the O-2pσatomic orbitals in an isolated octahedron. There are six such combinationslisted in Table 4.1 with the oxygen sites indexed in Fig. 4.1 (d). For futurereference, Table 4.1 also contains molecular orbital combinations of the O-2pσ orbitals in an isolated square plaquette. The O-a1g-symmetric molecularorbital combination is of particular interest as it is the only combination that414.2. Bi, BaBi, O3, and BaO3 sub-lattices of BaBiO3-15-10-5 0 5 R Γ X M Γ Energy (eV)(a) BaBiO3Bi-s-15-10-5 0 5 R Γ X M Γ (b) BaBi-15-10-5 0 5 R Γ X M Γ (c) Bi-5 0 R Γ X M Γ Energy (eV)(e) BaO3O-a1gTB fit-5 0 R Γ X M Γ (f) O3-15-10-5 0 5 R Γ X M Γ (g) BaBiO3-15-10-5 0 5 R Γ X M Γ (h) BaBiO3(d) O-pσ octahedronFigure 4.1: The DFT (LDA) band structures of (a), (g), (h) BaBiO3 and its(b) BaBi, (c) Bi, (e) BaO3, and (f) O3 sub-lattices. In (a) - (c),the yellow-coloured fat bands represent the contribution of the Bi-s orbital, while in(e) - (h), the red-coloured fat bands represent the contribution of the O-a1gmolecular orbital. The Fermi level is marked with a horizontal dashed blackline. Panel (d) shows the O-a1g molecular orbital combination of oxygen-pσorbitals in an octahedron. The oxygen sites 1 to 6 are coupled via hoppingintegrals −tpp = (−tppσ + tpppi)/2. A nearest neighbour TB model fit of(e) BaO3 (f) O3 and (g)-(h) BaBiO3 is shown with dashed lines. In (h),Bi-6p orbitals are added in an extended tight-binding model (ETB) for animproved fit. The parameter values resulting from the fits are listed inTable 4.2.424.2. Bi, BaBi, O3, and BaO3 sub-lattices of BaBiO3Octahedron Square plaquetteO site a1g t1u t1u t1u eg eg a1g eu eu b1g1 1√61√20 0 1√30 1√41√20 1√42 1√6−1√20 0 1√30 1√4−1√20 1√43 1√60 1√20 −1√121√41√40 1√2−1√44 1√60 −1√20 −1√121√41√40 −1√2−1√45 1√60 0 1√2−1√12−1√46 1√60 0 −1√2−1√12−1√4Energy −4tpp 0 0 0 2tpp 2tpp −2tpp 0 0 2tppTable 4.1: The eigenstates and eigenvalues of an octahedron and a squareplaquette of O-pσ orbitals coupled via nearest-neighbour hopping integrals−tpp = (−tppσ+tpppi)/2. For oxygen site indexing and relative orbital phases,refer to Fig. 4.1 (d). Here, the O-pσ orbitals’ on-site energies are set to zero.is allowed by symmetry to hybridize with the Bi-6s orbital. In the calculatedband structures of BaO3 and O3, its character is concentrated at the bottomof the O-2p band, mirroring the situation in an isolated octahedron (see thebottom of Table 4.1). As expected, the intensity of the O-a1g character isstrongly k-point dependent in the Brillouin zone of a cubic unit cell, van-ishing at the Γ-point and reaching maximum at the R-point. We find littledifference between the widths (1.89 eV versus 2.28 eV) and the dispersionsof the O-2p bands in the BaO3 and O3 sub-lattices, which indicates thatthe hybridization between the Ba and O-2p orbitals is much weaker thanthat between the O-2p orbitals themselves. This is due to a large separationbetween the O-2p and Ba-5p atomic energy levels as well as a large distanceof 3.04 A˚ between the O and Ba atoms.We can now appreciate the enormous effect that the Bi-6s–O-2p orbitalhybridization has on the electronic structure of BaBiO3, whereby the valenceband width increases from 2 eV or less in the isolated sub-lattices to 15 eVin the full BaBiO3 structure. After the hybridization, the O-a1g molecularorbital in an anti-bonding combination with the 6s orbital lands above theFermi level [see Figs. 4.1 (g) or (h)]. Such behaviour of the O-a1g molecularorbital paves the path for the bipolaronic condensation of oxygen holes upon434.3. Derivation of tight-binding models for BaO3, O3, and BaBiO3breathing distortion, as will be discussed later.Apart from being strongly coupled via spσ-type overlap integrals, the hy-bridizing Bi-6s atomic orbital and the O-a1g molecular orbital also takeadvantage of their energetic proximity. To demonstrate this, the position ofthe Bi-6s band in, e.g. the Bi sub-lattice in Fig. 4.1 (c) has been aligned withthat in BaBiO3 at the Γ point where the Bi-6s–O-2p hybridization vanishesby symmetry, marking the Bi-6s on-site energy at s=−6.1 eV. Similarly,the top of the O-2p band in the BaO3 sub-lattice in Fig. 4.1 (e) has beenaligned with the top of the O-2p non-bonding states in BaBiO3. After suchalignments, one clearly sees that the Bi-6s orbital is only about 2 eV belowthe O-a1g molecular orbital which is much smaller than the hopping integralbetween the O-a1g and the Bi-6s orbitals.4.3 Derivation of tight-binding models for BaO3,O3, and BaBiO3In order to quantify the hybridization effects discussed above, we will nowderive minimal tight-binding models for BaBiO3 and its BaO3 and O3 sub-lattices by fitting their DFT band structures. In all our nearest-neighbourTB models, there are two inter-site oxygen hopping integrals tppσ and tpppi.We additionally include an spσ hopping integral between the Ba-6s and O-2porbitals, tBa-Ospσ , for the BaO3 sub-lattice and an spσ hopping integral betweenthe Bi-6s and O-2p orbitals, tspσ, for BaBiO3. These simple models cannevertheless provide an overall good description of the DFT band structure,see Figs. 4.1 (e)-(g). The parameter values resulting from the fits are listedin Table 4.2. The Bi-6s–O-2p hybridization parameter tspσ = 2.10 eV isindeed found to be by far the dominant hopping integral in the system.Surprisingly, the ratios |tppσ|/|tpppi| = 5 in O3, and |tppσ|/|tpppi| = 10 inBaBiO3 are considerably larger than the empirical ratio of 3 typically as-sumed in cuprates [101, 102]. The origin of such an enhancement of the|tppσ|/|tpppi| ratio is not clear, although we have ruled out a possible sen-sitivity of this parameter to the system’s dimensionality and to the O-O444.3. Derivation of tight-binding models for BaO3, O3, and BaBiO3bond length variation by comparing calculations with accordingly modifiedstructural parameters.For BaBiO3, we also find that the TB model parameter values are in goodagreement with hopping integrals calculated using Wannier function (WF)projections[81, 82], where we only included Bi-6s and O-2p orbitals, sincethe low energy states spanning the Fermi energy are primarily of O-2p andBi-6s character (see the fourth column of Table 4.2). However, this tech-nique gives a nonphysically large next-nearest-neighbour hopping betweenthe O-pσ orbitals t′ppσ. This result is a consequence of the rather strong hy-bridization of the O-2p orbitals with the empty Bi-6p orbitals, which havenot been included in the Wannier basis. It has motivated us to also consideran extended tight-binding (ETB) model with added hybridization betweenthe O-2p and Bi-6p orbitals (see the fifth column of Table 4.2). The ETBmodel indeed provides an improved description of the DFT band structure[see Fig. 4.1 (h)], but also gives a more realistic value for the Bi-6s orbitalon-site energy s = −6.2 eV, which is very close to the value of −6.1 eVcorresponding to the position of the strongest Bi-6s character band at Γ inFig. 4.1 (a).454.3. Derivation of tight-binding models for BaO3, O3, and BaBiO3O3 BaO3 BaBiO3TB TB TB WF ETBs −4.73 −6.93 −6.2pσ −2.57 −2.92 −5.13 −4.84 −3.11ppi −2.49 −2.82 −3.06 −3.73 −2.78tppσ 0.26 0.30 0.63 0.64 0.4tpppi −0.05 −0.01 −0.04 −0.03 −0.03tspσ 2.10 2.31 2.09Bas 3.47tBa-Ospσ 0.9t′ppσ −1.0Bip 2.1tBi-Oppσ 2.34tBi-Opppi −0.53Table 4.2: On-site energies and hopping integrals in eV for BaBiO3 andits O3 and BaO3 sub-lattices. The values are obtained by fitting eitherthe simplest or the extended tight-binding (TB or ETB) model or by usingWannier functions (WF) including Bi-6s and O-2p orbitals. This choice ofWF orbitals results in a large next-nearest-neighbour hopping integral t′ppσbetween the O-pσ orbitals.464.4. Breathing and tilting distortions in two and three dimensions4.4 Breathing and tilting distortions in two andthree dimensionsIn this section, we will consider lattice distortions present in the real ABiO3structure. We are interested in whether our TB model can capture thechanges in the electronic structure due to the lattice distortions as observedin DFT calculations[34], such as the opening of the charge gap with theonset of the breathing distortion. Behaviours of the TB model in three andtwo dimensions will be compared to study the role of dimensionality in theproblem.Since our focus is mainly on the top valence band crossing the Fermi level,which is of the O-a1g symmetry and does not mix with the Bi-6p orbital, wewill use here the simpler TB model from the third column of Table 4.2 withonly Bi-6s and O-2p orbitals in the basis. The coupling of electrons to latticedistortions is modelled through a 1/d2 dependence of hopping integrals onthe interatomic separation d[101].In order to study the individual roles of the breathing and tilting distortions,let us consider four model structures of ABiO3 with the following charac-teristics: (i) b = 0 A˚, θ = 0◦, (ii) b = 0.1 A˚, θ = 0◦, (iii) b = 0 A˚, θ = 16.5◦,and (iv) b = 0.1 A˚, θ = 16.5◦. Here, b is half the difference between thetwo disproportionated Bi-O bond lengths and θ is the tilting angle of theoctahedron in three-dimensions, or the rotation of the square plaquette intwo-dimensions. The values of b = 0.1 A˚ and θ = 16.5◦ in structure (iv)correspond to the respective strengths of the breathing and tilting distor-tions in the experimental SrBiO3 structure. Because the distortions breaktranslational symmetry, there are four formula units in the three-dimensional(3D) unit cell and two formula units in the two-dimensional (2D) unit cell.474.4.Breathingandtiltingdistortionsintwoandthreedimensions-10-5 0A M Γ X N Γ Energy (eV)O-a1g 0 1 2 3 4States/eV3D: θ = 0, b = 0Sa1gegt1u-10-5 0A M Γ X N Γ 0 1 2 3 4States/eV3D: θ = 0, b = 0.1 A-10-5 0A M Γ X N Γ 0 1 2 3 4States/eV3D: θ = 16o, b = 0-10-5 0A M Γ X N Γ 0 1 2 3 4States/eV3D: θ = 16o, b = 0.1 A-10-5 0Γ X M Γ Energy (eV) 0 0.5 1 1.5 2States/eV2D: θ = 0, b = 0Sa1geub1g -10-5 0Γ X M Γ 0 0.5 1 1.5 2States/eV2D: θ = 0, b = 0.1 A-10-5 0Γ X M Γ 0 0.5 1 1.5 2States/eV2D: θ = 16o, b = 0-10-5 0Γ X M Γ 0 0.5 1 1.5 2States/eV2D: θ = 16o, b = 0.1 AFigure 4.2: The band structures and projected DOS of the 3D (top panels) and 2D (bottom panels) TB modelswith varying strengths of the breathing and tilting distortions. Here, breathing b is half the difference between thetwo disproportionated Bi-O bond lengths, and θ is the tilting angle. Molecular orbital projections are made forthe compressed octahedron or square plaquette following Table 4.1, while the Bi-s orbital projection is made forthe Bi atom located inside the compressed octahedron or square plaquette. The red-coloured fat bands representthe contribution of the O-a1g molecular orbital.484.4. Breathing and tilting distortions in two and three dimensionstspσ (eV) tppσ (eV) tpppi (eV)b = 0 A˚, θ = 0◦ 2.1 0.63 −0.04b = 0.1 A˚, θ = 0◦ 2.37 1.96 0.71 0.59 −0.045−0.03Table 4.3: Variation of the nearest-neighbor hopping integrals in responseto the Bi-O bond-disproportionation of 0.1 A˚.Figure 4.2 presents the band structures and the projected densities of states(DOS) of our 3D (top panels) and 2D (bottom panels) model structures.Molecular orbital projections are made for the compressed octahedron orsquare plaquette following Table 4.1, and the red-coloured fat bands rep-resent the contribution of the O-a1g molecular orbital. We find that themodels’ electronic structures exhibit similar characteristics irrespective ofthe dimensionality. Close to the Fermi level, the tilting distortion opensa gap at around −1.5 eV and causes an overall band narrowing while thebreathing distortion opens a charge gap transforming the system into asemiconductor. This metal-to-semiconductor transition is accompanied bya shift of the O-a1g molecular orbital character into the empty states. Itsintensity becomes k-point independent meaning that in real space holes spa-tially condense into well-defined molecular orbitals on the collapsed octahe-dra. The observed strong tendency towards formation of molecular orbitalscan be due to the fact that oxygen hopping integrals are rather sensitiveto the Bi-O bond-disproportionation. As one can see in Table 4.3, bond-disproportionation of 0.1 A˚ results in 0.4 eV difference in the tspσ hoppingintegrals for the collapsed and the expanded octahedron.Let us also have a closer look at the behaviour of the charge gap as a functionof tilting θ and breathing, b. It is depicted in Figs. 4.3 (a) and (b) for the2D and 3D TB models, respectively. In both cases, the gap is linear in bbut its θ-dependence is stronger in the 2D case. Qualitatively our modelcalculations can reproduce the DFT results for SrBiO3[34] [Fig. 4.3 (c)],but quantitatively the effects of both the breathing and tilting distortionsare rather underestimated, especially in the 3D case. This is due to theapproximations we used in the model calculations, such as neglecting the494.4. Breathing and tilting distortions in two and three dimensions 0 0.5 1 1.5Charge gap (eV) (a) 2D TBθ=0oθ=11.5oθ=16.5oθ=20o(b) 3D TB 0 0.5 1 1.5 0 0.05 0.1 0.15Charge gap (eV)b (angstoms)(c) SrBiO3 0 0.05 0.1 0.15b (angstoms)(d) BiO3Figure 4.3: The charge gap as a function of the breathing distortion atvarious tilting distortions in (a) the 2D TB model, (b) the 3D TB model,(c) SrBiO3 from DFT (LDA) calculations[34], and (d) the BiO3 sub-latticefrom DFT (LDA) calculations.variation of on-site energies or limiting the number of orbitals. To exemplifythe effect of the latter approximation, we compare the DFT gap in SrBiO3with that in BiO3 [Fig. 4.3 (d)]. Here, the θ-dependence of the gap isnoticeably reduced, similarly to what we find in the model calculations wherethe A cation orbitals are also neglected. This suggests that hybridizationwith the A cation orbitals plays a role in determining the size of the chargegap as a function of θ.504.5. Tight-binding models with a reduced number of orbitals4.5 Tight-binding models with a reduced numberof orbitalsFinally, we discuss possible simplifications of the ABiO3 TB model thatwould still allow an accurate description of low-energy electronic excita-tions. As was shown previously, the bands straddling the Fermi level aredominantly of the Bi-s and O-pσ orbital character. Therefore, a naturalsimplification of the TB model could be to eliminate the O-ppi orbitalsfrom the basis. This reduces the basis size from ten to four orbitals performula unit. As one can see in Figs. 4.4 (a) and (b), the four-orbital TBmodel gives a good agreement with the full ten-orbital model near the Fermilevel even without adjustment of model parameters. Here, the calculationsare done for a face-centred cubic unit cell with two Bi sites, and the non-distorted lattice [panel (a)] is compared with a lattice featuring an 0.1 A˚breathing distortion and no octahedra tilting [panel (b)]. The comparisonillustrates, in particular, that the four-orbital model is capable of describingthe distortion-induced metal-to-semiconductor transition in ABiO3.Despite its reduced basis size, the four-orbital model, however, containsredundant degrees of freedom, as far as low-energy physics is concerned.They give rise to the bonding Bi-s and O-pσ states at −10 eV and theoxygen non-bonding states at −3 eV, i.e. in the energy regions deep belowthe Fermi level. One can take a step further and write down a single-orbitalTB model with an A1g-symmetric orbital at each octahedron site. For thismodel, which could represent only the low energy scale bands, the basisconsists of anti-bonding combinations of Bi-s and O-a1g orbitals:|ψA1g〉 = 1√α2 + β2(α |ψBi−s〉 − β |ψO−a1g〉)(4.1)where ψO−a1g orbital is a symmetric linear combination of O-pσ[Fig. 4.1 (d)].Neglecting spin, the effective Hamiltonian in this basis can be written as:H =∑icA1g cˆ†i cˆi +∑jeA1g dˆ†j dˆj +Hc−e +Hc−c +He−e. (4.2)514.5. Tight-binding models with a reduced number of orbitalsb cA1g eA1gt t′ t′′ B.D.0.00 A˚ −0.13 −0.13 −0.45 −0.09 0.10 1.000.05 A˚ 0.35 −0.51 −0.47 −0.10 0.11 1.460.10 A˚ 0.99 −0.65 −0.48 −0.11 0.115 1.680.15 A˚ 1.86 −0.78 −0.50 −0.12 0.125 1.78Table 4.4: The single-orbital TB model parameter values in eV for latticeswith a varying degree of the breathing distortion b and no octahedra tilting.t, t′, and t′′ are the nearest, second-nearest, and fourth-nearest neighbourhopping integrals, respectively. cA1g and eA1gare the on-site energies of theA1g-like orbitals of the collapsed and expanded octahedron. Bond dispro-portionation (B.D.) shows the number of Ac1g orbital holes.Here, indices i and j run over collapsed and expanded octahedron sites,respectively; cˆ†i (cˆi) create (annihilate) a hole on site i and dˆ†j (dˆj) create(annihilate) a hole on site j; cA1g (eA1g) is the on-site energy of the A1gorbital on a collapsed (expanded) octahedron site. The hybridization termscan be written as:Hc−e =n.n.∑<ij>tcˆ†i dˆj + h.c.Hc−c =∑i∑i′∈{i}t′cˆ†i cˆi′ +∑i∑i′′∈{i}t′′cˆ†i cˆi′′ + h.c.He−e =∑j∑j′∈{j}t′dˆ†j dˆj′ +∑j∑j′′∈{j}t′′dˆ†j dˆj′′ + h.c.(4.3)where < ij > represents sum over nearest-neighbour sites, i′ and i′′ (j′ andj′′) are sites at distances√2a and 2a from site i(j), respectively [Fig. 4.4(e)].The model parameter values are obtained by fitting to the DFT states closestto the Fermi level. The parameter values are given in Table 4.4 for latticeswith a varying degree of the breathing distortion, and the fits for the b = 0 A˚and b = 0.1 A˚ lattices are shown in Figs. 4.4 (c) and (d), respectively.524.5.Tight-bindingmodelswithareducednumberoforbitals-15-10-5 0 5 W L Γ X WEnergy (eV)(a) Full TBFour-orbital-15-10-5 0 5 W L Γ X W(b) Full TBFour-orbital-15-10-5 0 5 W L Γ X WEnergy (eV)(c) DFTSingle-orbital-15-10-5 0 5 W L Γ X W(d) DFTSingle-orbital(e)Figure 4.4: In (a) and (b), the full TB (solid line) and the four-orbital TB (dashed line) models are compared forthe b = 0.0 A˚ and b = 0.1 A˚ lattices, respectively. In (c) and (d), the DFT band structure (solid line) and thesingle-orbital TB model (dashed line) are compared for the b = 0.0 A˚ and b = 0.1 A˚ lattices, respectively. In (e),the single-orbital A1g coupling to nearest, second nearest, and fourth nearest neighbours are shown.534.5. Tight-binding models with a reduced number of orbitalsWe find that within the single-orbital TB model the appearance and growthof the charge gap with an increasing breathing distortion can be well de-scribed by a splitting of the two A1g orbital on-site energies, with essentiallyno need of modifying the hopping integrals [see Table 4.4]. This modelcan be interpreted as an effective low energy model of bismuthates thatcan well describe the bands near the Fermi level and can now be used forexample to include electron-phonon coupling keeping in mind the origin ofthese wave functions and the effects of electron or hole doping looking forpossible superconductivity.In order to clarify the physics involved in these effective hopping integrals,it is instructive to obtain estimates for hopping integrals, t, t′, and t′′ byconsidering the composition of the A1g orbitals described above. Thesehoppings can be directly related to our full ten orbital model parameters,tppσ, tpppi, and tspσ, with taking into account O-O and Bi-O hoppings onlyup to nearest-neighbour. In the following, we consider for simplicity a non-distorted case, approximate the coefficients α and β to be α = β = 1,and also neglect the nonorthogonality that occurs in the O-a1g orbitals onnearest-neighbours in estimating t, since the overlap is only 16 . The A1gnearest-neighbour hopping t [see Fig. 4.4(e)] can be written as:t = 〈ψA1gi |H |ψA1gj 〉 = 〈1√2(ψs − 1√6(−p1 + p2 − p3 + p4 − p5 + p6))i|H| 1√2(ψs − 1√6(−p′1 + p′2 − p′3 + p′4 − p′5 + p′6))j〉= − 12√6〈ψsi |H |(p′2)j〉+12√6〈p1i |H |ψsj 〉+112〈−p1i |H |(−p′3 + p′4 − p′5 + p′6)j〉+112〈(−p3 + p4 − p5 + p6)i|H |(p′2)j〉 = −1√6tspσ +23tpp =− 1√6tspσ +13(tppσ − tpppi) ≈ −0.64eV,(4.4)The next-nearest neighbour hopping term can be written:544.5. Tight-binding models with a reduced number of orbitalst′ = 〈ψA1gi |H |ψA1gi′ 〉 = 〈1√2(ψs − 1√6(−p1 + p2 − p3 + p4 − p5 + p6))i|H| 1√2(ψs − 1√6(−p′′1 + p′′2 − p′′3 + p′′4 − p′′5 + p′′6))i′〉=12× 16〈(p1 + p3)i|H |(−p′′2 − p′′4)i′〉= − 212tpp =112(−tppσ + tpppi) ≈ −0.05 eV,(4.5)and the fourth nearest-neighbour term:t′′ = 〈ψA1gi |H |ψA1gi′′ 〉 = 〈ψA1gi |ψA1gj 〉 〈ψA1gj |H |ψA1gi′′ 〉 = 〈ψA1gi |ψA1gj 〉 × t= 〈 1√2(ψs − 1√6(−p1 + p2 − p3 + p4 − p5 + p6))i|1√2(ψs − 1√6(−p′1 + p′2 − p′3 + p′4 − p′5 + p′6))j〉 × t=12× 16〈−p1| p′2〉 × t ≈ +0.05 eV(4.6)where 〈ψA1gi |ψA1gj 〉 = − 112 is the overlap integral between site i and j. Theabove estimates are within the order of magnitude of the single-orbital pa-rameters in Table 4.4 obtained from the fit. For the on-site energies wehave:A1g = 〈ψA1gi |H |ψA1gi 〉 =12〈(ψs − ψa1g)i|H |(ψs − ψa1g)i〉=12(s + a1g)− 6√6tspσ(4.7)where on-site energy of O-a1g is a1g = σ − 4tpp and 6√6 tspσ is the couplingenergy of Bi-s and O-a1g orbitals. The change in the on-site energies due to554.5. Tight-binding models with a reduced number of orbitalsbreathing can now be written as:∆ = cA1g − eA1g = (−2tcpp −6√6tcspσ)− (−2tepp −6√6tespσ). (4.8)Here, tcpp(tepp) represent O-O hoppings [see Fig.4.1(d)] of the collapsed (ex-panded) A1g orbitals, and tcspσ(tespσ) are the corresponding Bi-O hoppings,and we have assumed that Bi-s on-site energies are unchanged. Using hop-ping parameters of a distorted lattice of 0.1 A˚ from Table 4.3, a direct gapvalue of ∆ ≈ 1.1 eV is estimated. This value of the gap is comparable to∆ in Table 4.4 obtained from the fit.In Table 4.4 we show an effective orbital occupation corresponding to thenumber of holes in the collapsed Ac1g orbital calculated by integrating theprojected density of states above the chemical potential as a function of thebreathing distortion. Since we have two Bi and two holes per unit cell thenumber of holes in the expanded Ae1g orbital is also 1.0 for no breathing. Asthe distortion increases we see that the number of holes in the collapsed Ac1gmolecular orbital gradually moves towards 2.0 at which point there wouldbe no holes anymore in the expanded Ae1g molecular orbital. This looks verymuch like charge-disproportionation. However, in the structure all the oxy-gens are identical which means that this cannot be charge-disproportionationinvolving the oxygen. Each oxygen indeed participates in both the collapsedand expanded A1g molecular orbitals however it participates more in thecollapsed Ac1g molecular orbital than in the expanded one. So again, thishas to do with bond-disproportionation. We note however that there is nosymmetry change in moving from a bond to a charge-disproportionationpicture and so there is no clear boundary but rather a gradual cross-over. Areal charge-disproportionation would have to imply an attractive Coulombinteraction while a bond-disproportionation results from the electron densitychanges in the bonds driven by an electron-phonon coupling involving thehopping integral changes.We can infer the electron-phonon coupling strength from our single-orbitalmodel as the A1g molecular orbital on-site energy lowering in Table 4.4,which is about 1 eV in the presence of the experimental bond modulation564.5. Tight-binding models with a reduced number of orbitalsof 0.1 A˚ or equivalently ddx = 10 eV/A˚. Given a Raman breathing modephonon frequency of ωph ≈ 70 meV[103, 104] at q = (pi, pi, pi) for BaBiO3the electron-phonon coupling g can be estimated as:g =∂∂x√~2Moωph≈ 10 eV/A˚× 0.04321 A˚ ≈ 0.4321 eV. (4.9)where Mo is the oxygen mass. The electron-phonon coupling g translatedinto the dimensionless coupling λ is:λ =2g2~ωphW=2× 0.432120.07× 6 ≈ 0.89, (4.10)where W ≈ 6 eV is the single-orbital A1g bandwidth read from Fig. 4.4 (d).Following the same argument we can estimate a value of λ = 1.21 forBa0.6K0.4BiO3 given a breathing mode phonon frequency of ωph ≈ 60 meV[105]for Ba0.6K0.4BiO3 and assuming that the bandwidthW is unchanged upon Ksubstitution. These electron-phonon couplings are much higher than whathas been obtained from previous LDA calculations on Ba0.6K0.4BiO3[88–90, 92, 93, 106]. Of course, to make a real comparison with LDA estimatesone would have to determine the q dependence of this coupling and averagethis over an assumed Fermi surface.574.6. Summary4.6 SummaryIn summary, we have studied the electronic structure of the bismuth per-ovskites ABiO3 (A = Sr, Ba) using ab initio calculations and tight-bindingmodelling. We found that the hopping integrals involving the Bi-6s and O-2porbitals play a leading role in shaping the electronic band structure of ABiO3near the Fermi level. A minimal TB model with ten orbitals per formulaunit (one Bi-6s and nine O-2p orbitals) was derived and shown to be ableto describe the changes in the electronic structure due to lattice distortionsthat had been observed in previous DFT studies, such as the opening ofthe charge gap due to the breathing distortion and the associated formationof molecular orbitals on collapsed octahedra. We also showed that for thepurpose of exploring low-energy excitations in ABiO3 this TB model canbe further reduced to a four-orbital one with one Bi-6s and three O-2pσorbitals in the basis and even further down to a single-orbital one with asingle A1g-like orbital at each octahedron site. Within this model, we furtherestimated electron-phonon couplings of λ = 0.89 and λ = 1.21 for BaBiO3and Ba0.6K0.4BiO3 respectively. This single band model is a good represen-tation of the band structure close to the chemical potential and can be usedin a more detailed study including the influence of electron-phonon couplingand possible mechanisms for superconductivity in the doped materials, butone has to keep in mind the rather extended molecular orbital character ofthe basis states in such a model.58Chapter 5Bond versus chargedisproportionation5.1 IntroductionThe two starting points in describing the electronic structure of the bis-muthates: charge-disproportionation (CD) and bond-disproportionation (BD)[34, 107, 108] both yield the same symmetry and lattice deformation. How-ever, we can expect that the nature of the states involved in the low energyscale properties are quite different. For example in the BD picture it is quitenatural to obtain solutions in which two holes in the collapsed oxygen octa-hedra cooperatively bond with the Bi in that octahedra forming a lattice ofbipolarons with an attractive interaction resulting from the combined effectof the electron-phonon coupling. This is important for the potential descrip-tion of superconductivity in the hole doped systems as in Ba1−xKxBiO3[4].In this chapter, we intend to better understand the nature of the two start-ing points and demonstrate that there is quite a natural crossover [108]between a bond and a charge-disproportionated system having on averagetwo holes per collapsed BiO6 octahedron. The difference being of the twoholes wave functions i.e. mostly delocalized on the O-2p molecular orbitalof a1g symmetry in the bond disproportionated system or mostly localizedon the Bi-6s atomic orbital in the charge-disproportionated system. Weargue that only three parameters determine the underlying physics of thiscrossover: tspσ, the strength of the hybridization between O-2pσ and Bi-6sorbitals, ∆ = (Bi-6s) − (O-a1g), the charge-transfer energy, and W, thewidth of the oxygen sub-lattice band. In addition, we show that a new595.1. Introduction-10-5 0A M Γ X N Γ Energy (eV)(a) non-distortedO-a1g 0 1 2 3 4States/eVSa1gegt1u-10-5 0A M Γ X N Γ (b) 0.1 A breathing distortion 0 1 2 3 4States/eVFigure 5.1: The band structures and projected density of states of the tight-binding model for (a) non-distorted lattice and (b) 0.1 A˚ breathing distortedlattice. Oxygen molecular orbital projections are made onto the compressedoctahedron and the Bi-6s orbital projection is for the Bi atom located in-side the compressed octahedron. The red-coloured fat bands represent thecontribution of the O-a1g molecular orbital of the compressed octahedron.metallic phase appears when holes occupy oxygen molecular orbitals of egsymmetry in which Bi is 3+ with band-like delocalized O-2p states.605.2. BD-CD crossover and the Phase diagram5.2 BD-CD crossover and the Phase diagramTo study the BD-CD crossover in the bismuthates, we use the tight-binding(TB) model derived in Ref. [107] which consists of one Bi-6s and nine O-2porbitals per formula unit with two nearest-neighbour inter-site O-2p hoppingintegrals, tppσ and tpppi, and the Bi-6s–O-2pσ hopping integral, tspσ. Thissimple TB model can describe well the changes in the electronic structuresuch as the opening of the charge gap due to the breathing distortion ob-served in previous DFT studies [34, 86, 87], as well as the formation ofmolecular orbitals on the collapsed octahedra as shown in Fig. 5.1. Here forsimplicity, we have neglected the tilting distortions of the oxygen octahe-dra, and consider a four formula unit supercell lattice with an experimentalbreathing distortion of 0.1 A˚[2]. From the projected density of states [seeFig. 5.1(a)] we can see that the Bi-6s band is very broad, extending from+1 eV above the Fermi level down to −12 eV below. This is due to thestrong hybridization between Bi-6s and O-a1g states. The breathing dis-tortion splits the bonding and anti-bonding bands in sub-bands associatedwith collapsed and expanded octahedra. The collapsed BiO6 octahedra arerepresented by the red-coloured fat bands in Fig. 5.1(b).A BD-CD phase diagram can be studied by scanning regions of strong andweak hybridization and varying the charge-transfer energy (i.e. by varyingBi-6s on-site energy) in a breathing distorted lattice. The results are shownin Fig. 5.2(a). Here, x marks the parameters relevant for ABiO3 [107]. Wefind that the holes’ character changes from O-a1g to Bi-6s as we go froma negative to positive charge-transfer energy. However, it is important tonote that since there is no symmetry change in moving from a BD to a CDsystem, there is no clear boundary but rather a gradual crossover that canbe defined by equal contribution of Bi-6s and O-a1g in the wave function ofthe collapsed octahedra. Note that since each octahedron has on averagetwo holes, hole density per octahedron does not change as we go from CDto a BD state. In addition to the BD-CD crossover we find that at a criticalhybridization t∗spσ for a given ∆, at which orbital energies interchange andmolecular orbitals of eg symmetry become lowest electron addition states.615.2. BD-CD crossover and the Phase diagram(b)(a)Figure 5.2: (a) The phase diagram representing the dominant character ofthe conduction band for the experimental breathing distorted lattice ob-tained from the tight-binding model as a function of the charge-transferenergy, ∆ and tspσ. The character of the conduction band is obtained by in-tegrating the partial density of states above the band gap. The boundary ofthis crossover is shown with a dashed line representing equal partial densityof states between O-a1g and Bi-6s. Symbol x marks the parameters relevantfor ABiO3 [107] obtained from tight-binding fit. (b) The charge gap in eV iscalculated from the tight-binding model for the breathing distorted lattice.An insulator-to-metal transition is obtained at a critical value of t∗spσ whenholes transition into the O-eg molecular orbitals.625.2. BD-CD crossover and the Phase diagramIn the BD regime two hole states on a collapsed oxygen octahedron stronglybond with the central Bi-6s orbital, which results in a lowering of the sys-tem’s total energy relative to when the holes are well separated in differentoctahedra. This quite naturally results in tendencies to bipolaron formationand leads to an effective “molecular” [107] rather than on-site attractive in-teraction U which can result in superconductivity. To estimate the value ofthis attractive U , we calculate the energy of a system with two spatially-well-separated holes in A1g molecular orbitals (we define A1g as a combinationof O-a1g molecular orbitals and Bi-6s,) by including the polaronic effects[42, 109, 110] resulting from the increase of Tspσ =√6tspσ molecular orbitalhopping integral due to electron-phonon interaction and compare that withthe energy of two holes of opposite spin on the same A1g molecular orbital.The polaronic lowering of the single hole energy due to the decrease in theBi−O bond length can be estimated as δTspσ =√6δtspσ and the total energylowering for two of these well separated holes is twice this i.e. 2δTspσ. Re-calling that tsp ≈ 1/(d0+δd)2 [101], where d0 is the Bi-O bond length and δdis the breathing distortion, for small δd, δtsp is proportional to −δd. Now, ifthere are two holes on the same A1g molecular orbital the change in the bondlength will be twice as large in linear response theory and so δtsp will be twiceas large and the total energy lowering will be 4δTsp. So the energy differencebetween two holes on the same A1g orbital and two times one hole per A1gorbital will be −2δTsp. Given the breathing distortion of 0.1 A˚ we calculatean effective attractive interaction Ueff = −2δTspσ = −2√6(2.37 − 2.15) ≈−1.1 eV between two holes on an A1g molecular orbital.Let us now calculate the band gap in our TB model for the breathing dis-torted lattice and above considered ranges of tspσ and ∆. We find thatthe distorted structure has a gap for all values of ∆ and hopping integralslarger than t∗spσ[see Fig. 5.2(b)]. However, for weaker hybridization whenholes cross over into O-eg orbitals the system transitions into a metallicstate. This insulator-to-metal transition can be understood by noting thatthe Peierls-like breathing distortion of the oxygen octahedra does not opena gap in the O-eg states, because this distortion is not coupled to the O-egstates due to symmetry. As a result, the system can not lower its energy635.2. BD-CD crossover and the Phase diagramW<latexit sha1_base64="tdhIGO5jxxDhu1ILMwqKhNEqcDg=">AAAB6nicdZDLSgMxFIbP1Futt6pLN8FWcFUy1VqXBUFcVrQXaIeSSTNtaOZC khHKUPAF3LhQxK1P5M63MTOtoKI/BD7+c8I553cjwZXG+MPKLS2vrK7l1wsbm1vbO8XdvbYKY0lZi4YilF2XKCZ4wFqaa8G6kWTEdwXruJOLtN65Y1LxMLjV04g5PhkF3OOUaGPdlDvlQbGEKxjb1ZM6ygDbtQzOTmsY2QZSlWCh5qD43h+GNPZZoKkgSvVsHGknIVJzKtis0I8ViwidkBHrGQyIz5 STZKvO0JFxhsgLpXmBRpn7/UdCfKWmvms6faLH6nctNf+q9WLtnTsJD6JYs4DOB3mxQDpE6d1oyCWjWkwNECq52RXRMZGEapNOwYTwdSn6H9rVio0r9nW11Li8n8eRhwM4hGOwoQ4NuIImtIDCCB7gCZ4tYT1aL9brvDVnLSLchx+y3j4BxMyN3A==</latexit><latexit sha1_base64="tdhIGO5jxxDhu1ILMwqKhNEqcDg=">AAAB6nicdZDLSgMxFIbP1Futt6pLN8FWcFUy1VqXBUFcVrQXaIeSSTNtaOZC khHKUPAF3LhQxK1P5M63MTOtoKI/BD7+c8I553cjwZXG+MPKLS2vrK7l1wsbm1vbO8XdvbYKY0lZi4YilF2XKCZ4wFqaa8G6kWTEdwXruJOLtN65Y1LxMLjV04g5PhkF3OOUaGPdlDvlQbGEKxjb1ZM6ygDbtQzOTmsY2QZSlWCh5qD43h+GNPZZoKkgSvVsHGknIVJzKtis0I8ViwidkBHrGQyIz5 STZKvO0JFxhsgLpXmBRpn7/UdCfKWmvms6faLH6nctNf+q9WLtnTsJD6JYs4DOB3mxQDpE6d1oyCWjWkwNECq52RXRMZGEapNOwYTwdSn6H9rVio0r9nW11Li8n8eRhwM4hGOwoQ4NuIImtIDCCB7gCZ4tYT1aL9brvDVnLSLchx+y3j4BxMyN3A==</latexit><latexit sha1_base64="tdhIGO5jxxDhu1ILMwqKhNEqcDg=">AAAB6nicdZDLSgMxFIbP1Futt6pLN8FWcFUy1VqXBUFcVrQXaIeSSTNtaOZC khHKUPAF3LhQxK1P5M63MTOtoKI/BD7+c8I553cjwZXG+MPKLS2vrK7l1wsbm1vbO8XdvbYKY0lZi4YilF2XKCZ4wFqaa8G6kWTEdwXruJOLtN65Y1LxMLjV04g5PhkF3OOUaGPdlDvlQbGEKxjb1ZM6ygDbtQzOTmsY2QZSlWCh5qD43h+GNPZZoKkgSvVsHGknIVJzKtis0I8ViwidkBHrGQyIz5 STZKvO0JFxhsgLpXmBRpn7/UdCfKWmvms6faLH6nctNf+q9WLtnTsJD6JYs4DOB3mxQDpE6d1oyCWjWkwNECq52RXRMZGEapNOwYTwdSn6H9rVio0r9nW11Li8n8eRhwM4hGOwoQ4NuIImtIDCCB7gCZ4tYT1aL9brvDVnLSLchx+y3j4BxMyN3A==</latexit><latexit sha1_base64="tdhIGO5jxxDhu1ILMwqKhNEqcDg=">AAAB6nicdZDLSgMxFIbP1Futt6pLN8FWcFUy1VqXBUFcVrQXaIeSSTNtaOZC khHKUPAF3LhQxK1P5M63MTOtoKI/BD7+c8I553cjwZXG+MPKLS2vrK7l1wsbm1vbO8XdvbYKY0lZi4YilF2XKCZ4wFqaa8G6kWTEdwXruJOLtN65Y1LxMLjV04g5PhkF3OOUaGPdlDvlQbGEKxjb1ZM6ygDbtQzOTmsY2QZSlWCh5qD43h+GNPZZoKkgSvVsHGknIVJzKtis0I8ViwidkBHrGQyIz5 STZKvO0JFxhsgLpXmBRpn7/UdCfKWmvms6faLH6nctNf+q9WLtnTsJD6JYs4DOB3mxQDpE6d1oyCWjWkwNECq52RXRMZGEapNOwYTwdSn6H9rVio0r9nW11Li8n8eRhwM4hGOwoQ4NuIImtIDCCB7gCZ4tYT1aL9brvDVnLSLchx+y3j4BxMyN3A==</latexit>BondingAnti-bondingFigure 5.3: The energy level diagram of an oxygen octahedron calculatedwithin the three-band model at tspσ = 0(in black), and tspσ = 1 eV(in red) asa function of the charge-transfer energy ∆. The character of the conductionband (dashed line) transitions from O-eg to anti-bonding Bi-6s–O-a1g bandat a critical value of tspσ and charge-transfer energy (∆).by adopting the breathing distortion and therefore will stay in the metallicnon-disproportionated state.Finally, we show that this phase transition can simply be described withina three-band model consisting of only O-a1g, O-eg, and Bi-6s orbitals withthe hybridization Tspσ =√6tspσ. This hybridization couples only the O-a1gand Bi-6s orbitals, since the O-eg orbital does not hybridize with eitherBi-6s or O-a1g due to symmetry. We find that at zero hybridization, O-eg(O-a1g) states are at the top(bottom) of the oxygen band which resultsin the conduction band (represented by a dashed line) being mainly of O-egcharacter[see Fig. 5.3]. However, after Bi-6s–O-a1g hybridization is turnedon, at a certain value of the charge transfer-energy, the anti-bonding Bi-6s–645.3. Molecular orbitals of O-eg symmetry orbitals and insulator-to-metal transitionO-a1g band is pushed above the O-eg states. We can realize this transitionfor a fixed hybridization (tspσ = 1 eV) by going from a negative to positivecharge-transfer energy as shown in Fig. 5.3. It can be seen that the con-duction band changes character from O-eg to anti-bonding Bi-6s–O-a1g at∆ ≈ 1 eV. We find that the boundary of this transition is described by afunction of ∆ and W :t∗spσ =√W∆ +W 2√6. (5.1)where W=(O-eg)−(O-a1g) is the width of the oxygen band.5.3 Molecular orbitals of O-eg symmetry orbitalsand insulator-to-metal transitionIn order to check the validity of this insulator-to-metal transition in a self-consistent calculation we use the constant orbital dependent potential fea-ture in WIEN2k [65] within the GGA approximation [51]. Here, one caneffectively change the on-site energy of Bi-6s orbitals. We find that for theinsulating state in the experimental structure of SrBiO3 [refer to Fig. 5.4(a)],the transition of the conduction band from O-a1g to O-eg is achieved self-consistently at an orbital potential value of −11.4 eV on Bi-6s as shown inFig. 5.4(b). This transition to a system where holes localize on O-eg statesis accompanied with an insulator-to-metal transition.655.3. Molecular orbitals of O-eg symmetry orbitals and insulator-to-metal transition-10-5 0A M Γ X N ΓEnergy (eV)(a)-10-5 0A M Γ X N ΓEnergy (eV)(b)Figure 5.4: (a) DFT band structure of the experimental structure of SrBiO3,(b) band structure of the experimental structure with a constant orbitalpotential of −11.4 eV on Bi-6s. An insulator-to-metal transition is obtainedupon moving Bi-6s orbitals. We note that breathing distortions do not opena gap when holes occupy molecular orbitals of the O-eg symmetry.665.3.MolecularorbitalsofO-egsymmetryorbitalsandinsulator-to-metaltransition-15-10-5 0 5 R Γ X M Γ Energy (eV)(a) MgPO3-15-10-5 0 5 R Γ X M Γ (b) CaAsO3-15-10-5 0 5 R Γ X M Γ (c) SrSbO3-15-10-5 0 5 R Γ X M Γ Energy (eV)(e) BaPO3-15-10-5 0 5 R Γ X M Γ (f) BaAsO3-15-10-5 0 5 R Γ X M Γ (g) BaSbO3-15-10-5 0 5 R Γ X M Γ (h) BaBiO3-15-10-5 0 5 R Γ X M Γ (d)BaBiO3Figure 5.5: DFT band structure of the relaxed structure of (a) MgPO3, (b) CaAsO3, (c) SrSbO3, (d)(h) BaBiO3,(e) BaPO3, (f) BaAsO3, (g) BaSbO3 assuming cubic perovskite crystal structure.675.4. Why Bi is so special?5.4 Why Bi is so special?We can now appreciate the dominant role played by the hybridization be-tween Bi-6s and O-a1g molecular orbitals and the charge-transfer energy ∆in determining the electronic structure of the bismuthates. One might won-der if there exists other materials also with strong O-2p–B-s hybridizations,that have similar electronic structure to the bismuthates and possibly canbecome superconductors. We are not aware of any existing compounds withthese properties, however, we can still study the electronic structure of somehypothetical compounds theoretically with DFT, by replacing the A and/orB-cations of BaBiO3. These elements are chosen from the correspondingcolumns of Ba and Bi respectively. Here, for simplicity, we only considerperfect cubic structures with no distortion. However, the real perovskitestructures if existed, might have breathing and tilting distortions as in thebismuthates.Substituting the A and B-cation in BaBiO3 with the elements of the corre-sponding column and row in the periodic table, we have: MgPO3, CaAsO3,SrSbO3, and by substitution of only the B-cation we have: BaPO3, BaAsO3,BaSbO3. Fig. 5.5(a)–(h) shows the band structure of the above perovskites.Here, the optimal lattice constants are obtained by minimizing the totalenergy as a function of the unit cell volume. Comparing the band structurescarefully we observe very similar characteristics, in particular in all casesone band crosses the Fermi-level, with the exception of MgPO3.In order to determine the low energy electronic structure of the proposedcompounds we fit their DFT band structures using our simplified nearest-neighbour TB model with nine O-2p and one s orbitals. [ Table.5.1 lists themodel parameters obtained from the fit] It can be seen that the B-cation-sand O-2p hopping parameter decreases as we go down the column from P toBi. This is a product of increased inter atomic distances due to increase inionic radii. Looking at the character of the conduction band by projectingthe TB density of states onto the molecular orbitals formed of O-pσ and theB-6s orbital [see Fig. 5.6], we notice that the conduction bands are againmostly of B-s and O-a1g orbital character. The dominate character in the685.4. Why Bi is so special?-15-10-5 0 5 0 1 2 3 4Energy (eV)States/eV(a) BaPO3SA1gEgT1u-15-10-5 0 5 0 1 2 3 4States/eV(b) BaAsO3-15-10-5 0 5 0 1 2 3 4States/eV(c) BaSbO3-15-10-5 0 5 0 1 2 3 4States/eV(d) BaBiO3Figure 5.6: TB model projected density of states with optimized latticeparameters of (a) BaPO3, (b) BaAsO3, (c) BaSbO3 and, (d) BaBiO3.anti-bonding state is determined by the difference in the on-site energies.We find that when the B-s band is above O-a1g band, the anti-bondingstate is of mostly B-s character and this would naturally lead to chargedisproportionation. Interestingly, BaBiO3 is the only compound that hassignificantly higher oxygen contribution in the DOS at the Fermi level ascompared to Bi-s.Charge-transfer energy ∆ = (B-s)−(O-a1g) is another important parame-ter in determining the strength of O-pσ–B-6s hybridization. One can obtainthe value of ∆ by setting the hybridization tspσ to zero, and read the on-siteenergy of the O-a1g molecular orbital from Fig. 5.7. The on-site energies canbe obtained by looking at the largest peak position. We observe that theonly compound that has a negative ∆ is BaBiO3. Also in agreement withour parameter phase diagram, there is indeed a change in the conductionband character from O-a1g to B-6s as we go from a negative to positive∆. This can be simply explained from an energy level diagram showingthe relative energy of the O-a1g and B-6s band before their hybridizationas shown in Fig. 5.8. We find that when B-6s band is above O-a1g the695.4. Why Bi is so special?ABO3 a=b=c A˚ s (eV) a1g (eV) tspσ (eV) tppσ (eV) tpppi (eV)BaPO3 3.9410 −5.65 −7.0 2.33 0.77 −0.01BaAsO3 4.0663 −6.30 −6.5 2.30 0.74 −0.03BaSbO3 4.2818 −5.25 −6.1 2.00 0.53 −0.03BaBiO3 4.4170 −6.58 −5.1 1.98 0.46 −0.03Table 5.1: Nearest-Neighbour TB model fit parameters of the undistortedcubic perovskite crystal structure.-15-10-5 0 5 0 1 2 3 4Energy (eV)States/eV(a) BaPO3SA1gEgT1u-15-10-5 0 5 0 1 2 3 4States/eV(b) BaAsO3-15-10-5 0 5 0 1 2 3 4States/eV(c) BaSbO3-15-10-5 0 5 0 1 2 3 4States/eV(d) BaBiO3Figure 5.7: Partial density of states of (a) BaPO3, (b) BaAsO3, (c) BaSbO3and (d) BaBiO3 calculated using TB model with tsp set to zero.character of the conduction band is dominated by B-6s and when belowO-a1g the conduction band character is mainly of O-a1g.705.5. SummaryB-sO-a1gA AB BB-sO-a1g(a) (b)Figure 5.8: Energy level diagram illustrating two cases before hybridization,(a) B-s band above O-a1g resulting in higher B-s contribution in the partialdensity of states at the conduction band, (b) B-s band below O-a1g resultingin higher O-a1g contribution in the partial density of states at the conductionband.5.5 SummaryIn summary, we have studied the orbital character of the holes in the bis-muthates and have shown that it depends strongly on the strength of Bi-6sand O-2pσ hybridization, tspσ, and the charge-transfer energy, ∆. We havedemonstrated that there exists a natural crossover between a bond and acharge-disproportionated system with holes preferring to occupy Bi-6s orO-a1g molecular orbitals. In the BD regime, an effective attractive “molec-ular” orbital interaction between two holes residing on the collapsed oxygenoctahedra is estimated as Ueff = −1.1 eV, via considering electron-phononcoupling effects through changes in tspσ. We have further shown that holescan transition into non-bonding O-eg molecular orbitals which is accom-panied by an insulator-to-metal transition. One might expect that similarcrossovers can occur in other multi-band systems in which the symmetryof bands crossing the Fermi energy could change depending on changes in715.5. Summaryhybridization between their respective molecular orbitals and atomic orbitalsat the centre.We expand our study by doing more realistic density functional calculationsof various model compounds which belong to the same category as SBOand we find that only Bi containing compounds are in the negative charge-transfer regime.72Chapter 6Surface electron doping viaCs adatom6.1 IntroductionTo date, superconductivity in the bismuth perovskites, ABiO3(A = Sr, Ba),have only been observed upon hole doping[2–4]. This is in part due to thealready large atomic radii of Bi, Sr and Ba, hence electron doping by substi-tution has not been feasible because we cannot find 3+ ions large enough tofit into the lattice. On the other hand, high temperature superconductivityin copper-oxides has been achieved upon both hole and electron dopingof the parent Mott insulating material by chemical substitution. In addi-tion, in situ doping control of the surface of cleaved YBa2Cu3O6+x havebeen attained through deposition of potassium [111, 112]. Among othersuperconductors, deposition of potassium onto FeSe films converted non-superconducting films with various thicknesses into superconductors withHigh Tc [113]. Therefore, it is interesting to study electron doping [114]of the bismuthates via adatom and explore their superconducting phasediagram.In this chapter, we study (theoretically) the possibility of surface electrondoping of the bismuthates by deposition of Cs. This is done by constructingsymmetric slabs by stacking the SBO bulk monoclinic unit cells along the[001] direction. A 15 A˚ vacuum is placed above each slab to minimizeinteractions between periodic copies of the slab and facilitate the study ofthe surface electronic structure. Due to the large unit cells, the use of allelectron (full potential) codes like WIEN2k are computationally prohibitive.736.2. Surface states of SrBiO3 and doping with adatomHence, we use VASP which is a plane-wave DFT package generally used forstudying large systems.6.2 Surface states of SrBiO3 and doping withadatomLooking at the band structure and total density of states of (a) SrO or (b)BiO2 terminated surfaces [see Fig. 6.1 ], the first thing that one notices is thelarge difference in the band gap of the two structures. The BiO2 terminatedsurface has a gap of 0.1 eV compared to a 0.5 eV gap in the SrO terminatedsurface. This strong dependence of the gap on surface termination can berelated to the large difference in the Madelung potentials at the two surfaces.Another important factor that can strongly influence the size of the bandgap is the strength of the hybridization between Bi-6s and O-2p orbitals ofa1g symmetry. In chapter 3 and 4 we found that the size of the band gaphas a strong dependence on the size of the breathing distortion. Therefore,a breaking of the oxygen octahedron at the BiO2 terminated surface canresult in smaller hybridization and a decrease in the surface band gap[114].Let us now turn to the electron doping study of SBO surfaces by Cs adatom.Here, the choice of Cs over other elements is due to the fact that Cs withan ionic radii of 170 pm [115] is unlikely to substitute for Sr or Bi ionsin the deeper layers of SBO. We simulate Cs adatom deposition at twoconcentrations: 0.5 Cs and 0.25 Cs per 2D surface unit cell (i.e.√2a ×√2a unit cell). First, we place Cs on top of sub-surface Sr in the BiO2terminated structure and on top of sub-surface Bi in the SrO terminatedstructure. After relaxing the structures, we notice that the surfaces turnmetallic with the Cs-s band donating its electron to the surfaces [see Fig.6.2].This doping happens at both levels of adatom concentration, but, lookingat the Cs-6s band character in Fig.6.2(a) we see the SrO surface is notcompletely electron doped at the Γ point where Cs-band crosses the Fermilevel. But, on the other hand, the BiO2 surface is completely doped with1e [see Fig.6.2(b)]. This is not surprising since we showed earlier that the746.2. Surface states of SrBiO3 and doping with adatom-1 0 1 2Γ X M ΓEnergy (eV)(a) Sr terminatedStates/eV-1 0 1 2Γ X M ΓEnergy (eV)(b) Bi terminatedStates/eVFigure 6.1: Band structure and total density of states obtained using GGAfor slabs of (a) Sr-terminated and (b) Bi-terminated SBO in the [001] direc-tion. The atomic structure of slabs used in the calculations are shown onthe right hand side.756.2. Surface states of SrBiO3 and doping with adatomBiO2 surface has a small band gap of only 0.1 eV, naturally, it would beenergetically more favourable for electrons to go to a surface with smallerband gap. Interestingly, as we decrease the Cs concentration to 0.25 Cs,complete electron doping of both terminated surfaces is achieved [refer to6.2(c)(d)]. The reason for this is that at 0.5 Cs concentrations, the adatomshybridize with each other and form a band. By reducing the concentration,Cs-Cs hybridization is now decreased and their electrons are transferred tothe SBO surface. To quantify and obtain the distance at which the couplingof Cs atoms can be negligible, we calculate the dispersion of Cs atoms in asimple cubic structure with varying lattice constants (i.e. Cs-Cs distance).We find that at a distance of 8.5 A˚(i.e. close to the Cs-Cs distance at 0.25 Cscoverage), the dispersion of Cs-6s band is reduced to less than 1 eV. Thislimits Cs states to an energy range well above bottom of the conductionband and therefore results in electron doping of the surfaces regardless oftermination.766.2. Surface states of SrBiO3 and doping with adatom-1 0 1 2Γ X M ΓEnergy (eV)(a) 0.5 Cs + SrBiO3 (Sr termination)Cs-6s 0 50 100States/eV-1 0 1 2Γ X M ΓEnergy (eV)(c) 0.25 Cs + SrBiO3 (Sr termination)Cs-6s 0 50 100States/eV-1 0 1 2 3Γ X M ΓEnergy (eV)(b) 0.5 Cs + SrBiO3 (Bi termination)Cs-6s 0 50 100States/eV-1 0 1 2 3Γ X M ΓEnergy (eV)(d) 0.25 Cs + SrBiO3 (Bi termination)Cs-6s 0 50 100States/eVFigure 6.2: DFT band structure and total density of states obtained forCs adatom deposition of (a) 0.5Cs on Sr-terminated, (b) 0.5Cs on Bi-terminated, (c) 0.25Cs on Sr-terminated, and (d) 0.25Cs on Bi-terminatedSBO [001] surface.776.3. Work function dependence on surface termination6.3 Work function dependence on surfaceterminationOne property that is of importance to the surface studies of bismuthates istheir work function. The work function is defined as the minimal energyrequired to remove an electron from inside the material across its surfaceinto the vacuum. In DFT calculations one typically defines work functionas the difference between the plane averaged electrostatic potential of theslab in the vacuum region and the Fermi level, Φ = Vvac− EF (EF is thematerials Fermi energy) [116–118].To the best of our knowledge, the work function of SBO has not been calcu-lated before. Fig. 6.2 shows the planar averaged electrostatic potential of theBiO2 and SrO terminated slabs. Using this potential and the correspondingEF we find the SrO and BiO2 terminated structures work functions of ΦSrO= 3.03 eV and ΦBiO2 = 4.75 eV, respectively. Clearly, SBO work functionsstrongly depend on the detail of the surfaces and show strong sensitivity tothe surface termination. This is quite unusual but can be easily understoodconsidering breaking of the a1g molecular orbitals due to missing one oxy-gen on BiO2 terminated surface and strongly changed hybridization withthe central Bi-6s of the octahedron.786.3. Work function dependence on surface termination-25-20-15-10-5 0 5 0 5 10 15 20 25 30 35Electrostatic potential (eV)Z (A)(a) SrO terminated -25-20-15-10-5 0 5 0 5 10 15 20 25 30 35 40Electrostatic potential (eV)Z (A)(b) BiO2 terminatedFigure 6.3: Plane averaged electrostatic potential calculated with GGA for(a) SrO terminated (b) BiO2 terminated surfaces. Here, Z represents thedistance along the slab in A˚.796.4. Structural phase transition with dopinga b c tiltingo breathing A˚Expt 5.95 6.10 8.48 16 0.10GGA 5.95 6.14 8.50 17 0.11PBE-sol 5.89 6.08 8.41 18 0.125LDA 5.84 6.04 8.34 19 0.115Table 6.1: SBO lattice constant and breathing and tilting of the oxygenoctahedra. A comparison is made between the experimental and the relaxedstructures obtained with GGA, LDA and PBE-sol exchange and correlationfunctionals. An E-cut of 410 eV and a k-point grid size of 8×8×8 is used inall the calculations.6.4 Structural phase transition with dopingPreviously it was observed that upon hole doping, the SBO and BBO[2,4, 119, 120] monoclinic supercells, turn cubic or tetragonal (depending onthe dopant, i.e. A or B cation substitution), and become superconductingwith Tc as high as 30 K [4]. One might expect to observe similar structuralphase transitions upon Cs adsorption and the subsequent electron dopingof bismuthates surfaces. Therefore, it is interesting to study the structuralproperties and possible phase transitions of SBO surfaces.From our calculations of electron doped SBO we found that the breath-ing distortions of the oxygen octahedra are reduced near the surface. Thisreduction occurs at the surface and subsurface layers, whereas, in deeperlayers the compressed and expanded octahedra remain without any signifi-cant changes, and maintain their insulating charge gap as in the bulk. Onthe other hand, we find no suppression of the oxygen octahedra’s tiltingswith electron doping. One can compare this apparent structural phase tran-sition (with electron doping) to the hole doped bulk structures where it wasfound experimentally that breathing distortions are suppressed. However,since surface calculations can be difficult to relate to actual materials dueto many approximations that are used in optimizing the calculations, weperform simple electron and hole doping calculations of the bulk to studythe possible structural transitions.Let’s start from the experimental bulk structure and relax the lattice volume806.4. Structural phase transition with doping-10-5 0A M Γ X N ΓEnergy (eV)(a) 0.5 hole + SrBiO3 0 5 10 15 20States/eV-10-5 0A M Γ X N ΓEnergy (eV)(b) 0.5 electron + SrBiO3 0 5 10 15 20States/eVFigure 6.4: Band structure and total density of states obtained using GGAfor relaxed SBO supercell at (a) 0.5 hole doping and (b) 0.5 electron dopingper formula unit.and all atoms. This gives an optimized structure that is used for electron orhole doping. The lattice constants and the degrees of distortions obtainedfrom the relaxation of the bulk with GGA, LDA and PBE-sol functionalsare listed in Table. 6.1. Comparing the structural parameters, (i.e. latticeconstants, atomic positions and distortions) we see good agreement betweenthe experimental values and GGA.Next, we add 0.5 electrons or holes per formula unit with fixing the atomicpositions. We observe a rigid band shift as expected. Relaxing the latticevolume and atomic positions again with extra 0.5 electrons or hols, onefinds the breathing distortions of the oxygen octahedra are washed out andconsequently the charge gap diminishes as shown in Fig. 6.4 similar to the ob-servations of the insulator-to-metal transition in hole doped Ba1−xKxBiO3.Looking more closely at the doped structures with GGA, we see that tilt-ing distortions and the average cubic lattice constants (Bi-Bi distances) areincreased from a value of t = 17o, aavg = 4.26 A˚ (undoped structure) to t= 20o, aavg = 4.39 A˚ in electron doped structure and decreased to t = 15o,aavg = 4.12 A˚ in the hole doped structure. The observed structural changes816.4. Structural phase transition with doping 1.9 2 2.1 2.2 2.3 2.4 2.5 0 0.1 0.2 0.3 0.4 0.5Bi-O bond(A)dopingSBO bulk Bi-O bonde-collapseh-collapsee-expandh-expandFigure 6.5: Average Bi-O bond length (A˚) calculated in the collapsed andexpanded octahedra after electron(e)/hole(h) doping with GGA.can be understood by noting that by adding electrons to the anti-bondingstates, the Bi-O bond lengths increase and by removing and electron (i.e.hole doping) from the anti-bonding states the Bi-O bond lengths decrease.For completion, we have performed all electron and hole doping calculationswith GGA, LDA and PBE-sol exchange and correlation functionals and havelisted the structural parameters obtained after relaxation in Table. C.1 andTable. C.2 in Appendix. C. We see that irrespective of the functional used,hole(electron) doping results in decrease(increase) of the lattice constantsand the average Bi-O bond length in the oxygen octahedra.In order to find the critical value of doping at which structural distortionsdisappear, we perform several calculations with various doping levels asshown in Fig. 6.5. It can be seen that the average Bi-O bond lengths of thecollapsed and expanded octahedra converge to the same value at a criticaldoping of about 0.4 (hole or electron) per formula unit and the structuretransitions to a cubic phase. This transition can be indicative of a Fermi-surface nesting although one still needs to calculate the static susceptibilityof an electron doped system.The above results agree qualitatively well with findings of Ref.[120] where it826.4. Structural phase transition with dopinga b c tiltingo breathing A˚Expt 5.95 6.10 8.48 16 0.10GGA 5.95 6.14 8.50 17 0.11GGA(electron) 6.10 6.34 8.79 20 0.0GGA(hole) 5.81 5.89 8.23 15 0.0Table 6.2: Lattice constant and structural distortions of SBO supercell at 0.5electron or 0.5 hole doping per formula unit with GGA exchange-correlationfunctionals. An E-cut of 410 eV and a k-point grid size of 8×8×8 is used inall the calculations.was shown that an insulator-metal transition occurs near a cubic perovskitephase which exists for x = 0.37 (at Tc = 10 K) in Ba1−xKxBiO3. At roomtemperature, with decreasing the potassium concentration, it was shownthat first the cubic structure distorts by the BiO6 octahedral tilting andthen by symmetric oxygen breathing-mode distortions.If superconductivity in the bismuthates can be associated with the disap-pearance of a charge-density-wave (CDW) in a way that is analogous tothe role played by anti ferromagnetism, or spin-density waves (SDW), inCu-based oxide superconducting systems, we can expect that electron dop-ing can also suppress CDW and therefore possibly induce superconductivitywith possible transition to a cubic phase near a metal-insulator transition.836.5. Summary6.5 SummaryIn summary, we have studied the electronic structure of SBO [001] thinfilms and have shown that electron doping of surfaces is achievable via Csadsorption. This electron doping is found to be strongly sensitive to surfacetermination (i.e. SrO or BiO2) and the adatom concentration. We alsoshowed that the band gap and surface work functions are very sensitive tosurface termination with a difference of upto 1.5 eV in work functions of thetwo terminations. We further investigated the structural phase transitionswith electron doping of SBO on its surface and in the bulk and found thatelectron doping of SBO leads to disappearance of breathing distortions andconsequently a transition from an insulating to a metallic state. One canexpect that electron doping can suppress the bond/charge modulations inthe bismuthates and therefore possibly induce electron-doped superconduc-tivity.84Chapter 7Coexisting two dimensionalelectron and hole gases inLaLuO3/SrBiO3heterostructures7.1 IntroductionIn this chapter, we propose a new class of materials, namely heterostructurescomposed of LaLuO3(LLO) and SrBiO3(SBO), that can host coexisting elec-tron and hole gases and potentially high temperature superconductivity attheir two opposite interfaces. We argue that electronic reconstruction result-ing from the polar termination of the LLO, is the dominant mechanism forsolving the diverging potential (“polar catastrophe”) in this system which inreturn results in simultaneous electron and hole doping of LLO/SBO/LLOheterostructure at its interfaces. We further show that the electronic struc-ture of this system suggests the electron-hole gas interactions can be tunedby changing the spatial separation with the potential of obtaining excitonicinsulating phases.857.2. The polar catastrophe7.2 The polar catastropheThe polar catastrophe is most simply described as an electrostatic effect thatemerges due to charge distribution and spatial termination of a crystal atits surface or interfaces [121–123]. Let us introduce the polar problem in thehighly studied LaAlO3/SrTiO3 system by first considering the crystal struc-ture of LaAlO3(LAO) and SrTiO3(STO) separately. The Sr and Ti cationsin STO have formal charges of 2+ and 4+ respectively with oxygen being2−. Calculating the net plane charges for example in the [001] direction,we arrive at a structure that consists of alternating charge neutral SrO andTiO2 layers which is of type 1 in Tasker’s classification of ionic crystals[123].La3+O2—Al3+O24—La3+O2—Al3+O24—+1-1+1-1La3+O2— +1E-1Al3+O24—V(a) (b)Figure 7.1: An artist’s view of the polar catastrophe is shown for LaAlO3film, as an example. (a) The electric field as a function of the position normalto the layers alternates between zero and a non-zero value. (b) The electricpotential obtained after integrating the electric field shows the existenceof a net potential difference across the film which increases with the filmthickness. This divergence of the potential by increasing the film thicknessis known as the polar catastrophe. Figure adapted from Ref. [124]867.2. The polar catastropheFor LAO however, the situation is different with La and Al both having avalency of 3+. In this case, the LaO plane has a net charge of +1 and theAlO2 a charge of −1 per unit cell and there exists a dipole in the repeat unit.However, the repeat unit is always charge neutral following Tasker’s type 3classification[123]. Thus, LAO in the [001] direction has a polar terminationwith alternating oppositely charged layers similar to a capacitor as shownin Fig. 7.1.In order to determine the electric potential of this system, we need to knowthe electric field produced by the charged planes. From elementary electro-statics, it is known that a charged plane produces a constant electric field of(σ/2)nˆ, with nˆ being the unit vector normal to the planes of charge density±σ and dielectric constant . Given this, we find that the electric fieldalternates between 0 and +(σ/)zˆ in the regions between the charged planes[see Fig. 7.1(a)]. Now, the electric potential profile within such a system isobtained by integrating the electric field as a function of the perpendiculardistance to the planes. It can be seen clearly [see Fig. 7.1(b)] that thepotential increases as more LAO layers are added. Therefore, such a systemis unstable and must change itself in some way to remove the divergingpotential with thickness. This is known as the “polar catastrophe”. It isinstructive to demonstrate the origin of the polar catastrophe by splittingeach layer into two, each with half the charge (i.e. q/2). From Fig. 7.2one can see that the potential build-up is produced by uncompensated ±q/2 surface charges and in order to solve the polar catastrophe one needs toeliminate this big capacitor.There are many interesting phenomena including 2DEG [124, 125], 2DHG[126], magnetism [127] and superconductivity [128, 129] that have been ob-served at the LAO/STO thin film interfaces. The formation of 2DEG atsemiconductor interfaces of AlGaAs/GaAs heterostructures[130–132] havealso been the subject of extensive research. The mechanism responsible forthe formation of these interesting phenomena is still an open area of re-search, however there is a general consensus that the polar catastrophe andits compensation mechanism are of critical importance in these systems.One of the mechanisms suggested to compensate for this electrostatic po-877.2. The polar catastrophe+V +q/2=V+q/2-q/2-q/2+q/2-q/2-q/2+q/2-q/2-q/2+q/2+q/2+q/2V-q/22E E EFigure 7.2: An artist’s view of the polar catastrophe. The polar catastrophecan be simply described if each layer is modelled as two with half the chargeof the original i.e. q/2. It can be seen [on the right side] that the bigcapacitor (big dipole) produced by the top (+q/2) and bottom (−q/2) planesis the origin of the potential build up. The solution to the polar problem isto get rid of this big capacitor.887.2. The polar catastropheConduction bandValence bandElectron tunnelingHole formationp-type n-typeFigure 7.3: An artist’s view of the Zener breakdown in which electronstunnel between the two interfaces of a semiconductor when an electric fieldis applied. This breakdown is determined and limited by the band gap ofthe system.897.2. The polar catastrophetential build-up is the “electronic reconstruction” [133–135](the phrase wasfirst coined by Hesper et al. [135] in the study of K3C60 surfaces). In thismechanism, the polar interfaces electronically reconstruct by transferringhalf an electron per unit cell from its p-type interface(surface) to the n-typeinterface. In other words a Zener breakdown occurs resulting in transfer ofelectrons as shown in Fig. 7.3. This transfer of electrons is simply to dis-charge the big capacitor. After the reconstruction, the electric potential nolonger diverges with film thickness and the consequent electrons transferredto the interface form the 2DEG[124, 125].Over the years there have been several other proposed compensation mech-anisms for solving the polar catastrophe. Among the possible mechanismsbeside electronic reconstruction are, atomic reconstruction [123, 136–138],interface defects [139, 140], and O-vacancy formation [140–147]. Many havesuggested that there is an interplay between these mechanisms which de-pends on the details of the crystal growth conditions.Now, one might wonder if it would be possible to use the polar catastrophe toinduce 2DEG and 2DHG and possibly superconductivity in the bismuthates.To answer this question, first we need to find a polar terminated perovskitethat has a small lattice mismatch with SBO so it can be easily used forepitaxial film growth. To the best of our knowledge, LaLuO3(LLO) is theonly material which is currently available and has a very good lattice matchto bismuthates. It is a non-magnetic band insulator that is polar in the[001] direction similar to LAO with the formal charges of La3+ and Lu3+,and has a small lattice mismatch of only 2% with SBO(lattice constant ofLLO and SBO are 4.18 A˚ [148] and 4.25 A˚ [2] respectively). Bulk LLOadopts the perovskite crystal structure in the monoclinic space group [referto Appendix D]. The oxygen octahedra are tilted and rotated. The Bulkelectronic structure has a band gap of about 5 eV as shown in Fig. 7.4(a)and one does not expect the [001] surface of LLO to be electronically recon-structed. However, since SBO has a band gap of only about 1 eV it providesthe potential for pure electronic reconstruction due to the low energy costsassociated with it.In the following, we study LLO/SBO supercells with√2a × √2a lateral907.2. The polar catastrophelattice constants (where a is the cubic lattice constant), containing (n,m)monolayers (ML) of experimental structure of LLO and SBO. Since samplesare grown on LLO substrates, we fix the in-plane lattice constant of the het-erostructure to the LLO experimental lattice constant [refer to Appendix. D].Thus, SBO with a cubic lattice constant of 4.25 A˚ is subject to compression.We perform all the calculations using the generalized gradient approxima-tions to the exchange correlation functional and the projector augmentedplane wave method as implemented in the Vienna ab initio simulation pack-age VASP. A kinetic energy cutoff of 410 eV was used for the basis set.Integration in reciprocal space is performed using 11×11×1 k-point grid.We use U = 6 eV on La-f orbitals within LSDA+U method to correct LDAerrors in determining energies of empty La-4f orbitals. This inclusion doesnot change our main conclusion which is dominated by the polarity of thesystem. In the relaxed structures all atoms are allowed to relax until theforce on each atom is less than 0.01 eV/ A˚.917.2. The polar catastrophe-5 -4 -3 -2 -1 0 1 2 3 4 5 6Energy (eV)24681012141618Density (states/eV)TotalLa O LaLuO3-6 -5 -4 -3 -2 -1 0 1 2 3Energy (eV)0510152025303540Density (states/eV)TotalOSrBiO3(a)BiO2SrOLa3+O2—Lu3+O24—BiO2+12DEG2DHG+1+1-1-1-1-e/2(b)La3+O2—Lu3+O24—La3+O2—Lu3+O24—SrOFigure 7.4: (a)Total density of states of SrBiO3(LaLuO3) bulk experimen-tal structure shown with projected density of states onto oxygen and Laorbitals in GGA(GGA+U). Note that the band gap in SrBiO3 is mainlyof oxygen character, (b) An artist’s view of the unreconstructed interfacewhich has neutral [001] planes in SrBiO3, but the [001] planes in LaLuO3have alternating net charges of +1, −1. This produces an electric field, lead-ing to an electric potential that diverges with increasing LaLuO3 thickness.After electronic reconstruction a net charge of −|q| is transferred from theBiO2 layer at the p-type(LuO2/SrO) interface into the BiO2 layer at then-type(BiO2/LaO) interface resulting in the formation of 2DEG and 2DHGrespectively.927.3. Electronic reconstruction in LLO/SBO films7.3 Electronic reconstruction in LLO/SBO filmsLet us first focus on pure electronic reconstruction in LLO/SBO super-lattices where all atomic degrees of freedom are frozen out. We considerstoichiometric superlattices which contain both interfaces i.e. LaO/BiO2and LuO2/SrO as shown in Fig. 7.4(b). Note that the number of atomiclayers in the figure are only for illustration purposes and do not representthe actual calculated structures.Now, in order to dope SBO at its two interfaces, we need to transfer electronsbetween its valence and conduction band i.e. we desire a Zener breakdownto occur in which electrons tunnel between the two interfaces as shown inFig. 7.3. The electric field in the system enables tunnelling of electronsfrom the valence to the conduction band, leading to the formation of freecarriers at the interfaces. This breakdown can naturally happen at a criticalthickness of LLO when the electric potential across the interfaces exceedsthe band gap of the system. From simple electrostatics, for planes of op-posite charge Q and area A with distance d between them, we estimate theelectrostatic potential for one unit cell of LLO at ∆V = Qd2oA ≈ 11 V, andgiven that SBO has a small band gap of only 1 eV it can be expected thatthe system will be electronically reconstructable.After electronic convergence of the LLO/SBO supercell with no atomic re-laxation [see Fig. 7.5(a)], we find that indeed the LLO/SBO system becomesmetallic and electronically reconstructs for n = 4, m = 4 unit cells of LLOand SBO. However, we find that the system crosses into the metallic state atn = m = 3 unit cells of LLO and SBO, with a minimum of m = 3 unit cellsof SBO required for the insulator-metal transition. After relaxing all atomicpositions we find that the hole pocket at the Γ point [see Fig. 7.5(a)] dis-appears and the corresponding bands are pushed away from the Fermi leveland only two bands corresponding to the BiO6 octahedra at the two oppositeinterfaces cross the Fermi level as shown in Fig. 7.5(b). The projections ofBi-6s is a convenient marker which is possible due to its strong hybridiza-tion with the O-a1g molecular orbitals. From the Bi-6s projected densityof states at the two interfaces it can be seen that electrons are transferred937.3. Electronic reconstruction in LLO/SBO filmsΓ X M Y Γ-101Energy (eV)EFΓ X M Y Γ-101Energy (eV)EF -|q|(b)(a)Figure 7.5: (a) The band structure of (n = 4, m = 4) LLO/SBO beforeand (b) after atomic relaxation. The fat bands show the projected densityof states onto the collapsed Bi-6s octahedra (in blue) at the LaO/BiO2interface and expanded Bi-6s octahedra (in red) at the LuO2/SrO interface.from the expanded BiO6 octahedra at the p-type (LuO2/SrO) interface intothe collapsed BiO6 octahedra at the n-type interface (LaO/BiO2). Then-type interface develops electron pockets at the X and Y symmetry pointswhile the p-type interface has a hole pocket at the M point. In Fig. 7.5 |q|represents the amount of charge transferred between the two interfaces. Theelectronic reconstruction in this system benefits from the electron transferbetween orbitals of the same character i.e. O-2p in SBO. This is becausethe charge-transfer gap in SBO is mainly of oxygen character which is deter-mined by the occupied and unoccupied O-2p bands of SBO [see Fig. 7.4(a)]while in LLO, the gap originates from charge-transfer between occupiedO-2p and unoccupied La-d. Therefore, as a result of band alignment, theempty oxygen band of SBO will be located below the energy level of theempty La-d band in LLO, and the electron transfer would favourably occurin SBO between the empty and full O-2p orbitals.We expect that electrons and holes formation to be robust because the holeand electron pockets are formed at different points in the Brillouin zone947.3. Electronic reconstruction in LLO/SBO filmsY MXGFigure 7.6: The Fermi surface of (n = 4, m = 4) LLO/SBO structure afteratomic relaxation.as can be seen in the Fermi surface of the relaxed LLO/SBO structure inFig. 7.6. In addition, the existence of a large electrostatic potential in LLOprevents the recombination of holes and electrons. From the above findings,it is plausible to expect the possibility of forming coexisting 2DEG and2DHG with potential for inducing High-Tc superconductivity at the twointerfaces of the LLO/SBO heterostructures.The purely electronic solution to the polar problem discussed above is veryattractive in its simplicity and can result in forming 2D systems (with pos-sibility of superconductivity) that are free of defects or vacancies. However,to date many attempts to achieve this goal have been unsuccessful, mainlydue to the large effective band gaps of materials involved in such interfaces.Therefore, the LLO/SBO system due to the small band gap of SBO can bea candidate for inducing both electron and hole gases at its two interfaces.957.4. Role of oxygen vacancy in solving the polar problem7.4 Role of oxygen vacancy in solving the polarproblemIf the electric discontinuity at the interfaces can be largely screened by theelectrons arising from oxygen vacancies, there may be an interplay betweenelectronic reconstruction and oxygen vacancy formation. By removal of aneutral divalent oxygen from the p-type interface, the system is left withtwo extra electrons in the conduction band [see Fig. 7.7]. If the excesselectrons move to the n-type interface (resulting in the reduction of theirenergy), the electric potential can in principle be compensated by the in-troduction of an equal an opposite field in LLO. A key difference with pureelectronic reconstruction is that the oxygen vacancies are expected to beimmobile at the SrO/LuO2 interface and therefore it is unlikely that theylead to any conductivity at that interface. There is indeed a competitionbetween vacancy formation and pure electronic reconstruction. This compe-tition depends strongly on the size of the effective band gap of the system,and materials with smaller band gaps are more prone to pure electronicreconstruction[149]. Therefore, oxygen vacancies can in principle contributeto interface reconstruction and produce non conducting hole doped interfacefor example similar to the surface of LAO films grown on STO substrate[150].In order to gain more insight into the role of oxygen vacancies in solvingthe polar instability in the LLO/SBO system, we calculate the relative for-mation energy of an oxygen vacancy as a function of its location in thesupercell with 2a × 2a lateral constant and n = 4, m = 4 unit cells ofLLO/SBO without atomic relaxation. This is done by calculating the totalenergy difference between various configurations in which one oxygen atomis removed from one atomic layer in the supercell. The concentration of theoxygen vacancies is 1/8 which is exactly what is needed for compensationof the polar catastrophe.Lets first define the energy for vacancy formation as ΩVac = EVacSC −ESC+µO,where µO is the chemical potential of oxygen. The energies ESC can becalculated from first principles and in the present case where we use periodic967.4. Role of oxygen vacancy in solving the polar problemBiO2SrOLa3+O2—Lu3+O24—BiO2+1+1+1-1-1-12eLa3+O2—Lu3+O24—La3+O2—Lu3+O24—SrOVoFigure 7.7: An artist’s view of formation of an oxygen vacancy in anLLO/SBO heterostructure. By removal of a neutral divalent oxygen atom,the system is left with two extra electrons in the conduction band. If theexcess electrons move to the n-type interface (resulting in reduction of theirenergy), the electric potential can in principle be compensated by the intro-duction of an equal an opposite potential in LLO.977.4. Role of oxygen vacancy in solving the polar problemsupercells, EVacSC and ESC are the energies of supercells with and without anoxygen vacancy. Therefore, the relative energy of an oxygen vacancy in anLLO/SBO supercell with respect to that in the bulk of SBO can be writtenas, ∆ΩVac = ΩVac-LLO/SBO − ΩVac-SBO. In this case, assuming that theoxygen chemical potential is the same in LLO/SBO and SBO supercells, µOwill cancel in the expression for ∆ΩVac. Now, calculating ∆ΩVac by lookingat the energy dependence of the oxygen vacancies, we find that vacanciesare more favourably formed in SBO as shown in Fig. 7.8. We note, due tothe small band gap of SBO, the relative energies of vacancies are essentiallyposition independent within the SBO block, although, there is some positiondependence in LLO. This is in contrast to large band gap materials i.e. LAOand STO supercells [147] where it was found that the relative formationenergy is lowest for vacancies at the p-type interface. Our findings suggestthat the reconstruction mechanism is of pure electronic nature.987.4. Role of oxygen vacancy in solving the polar problem-2-1 0 1 2 3 4 5Relative formation energy (eV)Position of oxygen vacancyn pLLOSBO SBOFigure 7.8: Relative formation energy of an oxygen vacancy as a functionof its position in (n = 4, m = 4) LLO/SBO supercell with n-type andp-type interfaces indicated in red and blue dashed lines respectively. Theconcentration of oxygen vacancies is 1/8 per layer. All atomic positions arefixed to their corresponding experimental structure in the bulk. We fix thein-plane lattice constant to the LLO experimental lattice constant of 4.18A˚.997.5. Lattice response to potential divergence in thin films7.5 Lattice response to potential divergence inthin filmsAtomic reconstruction can also play an important role in compensating thepotential build-up in polar-non-polar terminated heterostructures at smallfilm thicknesses. To demonstrate the lattice response to the diverging po-tential, we compare all atomic positions in the relaxed supercell to the un-relaxed structure while fixing the lattice constant to the LLO in both cases.We find that atoms are displaced along the direction normal to the interface.Fig. 7.9(a)(b) shows the average atomic displacements (∆z) in each layer.Note, displacements away from the n-type interface are defined positive inSBO and negative in LLO. It can be seen that the LaO layers show displace-ments of La (0.08-0.15 A˚) and O (0.0-0.05 A˚), while the LuO2 layers moveby Lu (0.0-0.1 A˚) and O (0.0-0.2 A˚), resulting in formation of dipoles with anet dipole moment calculated in Fig. 7.9(c). Similarly, SrO and BiO2 layersin SBO show relaxations of Sr, Bi and oxygen atoms. Note that the ionicdipoles in LLO and SBO are found to be in opposite directions to each other,and are formed throughout the supercell. These dipoles can be viewed assplitting of each layer into a cation and an anion (oxygen) planes as shownin Fig. 7.9(d) and have a significant impact on the electronic behaviour ofthe system and becomes a dominant effect at smaller thicknesses. Similarresults were also reported by Ref. [136] for LAO over layers on STO.The observed partial atomic reconstruction suppresses the electronic recon-struction by reducing the electron transfer between the two interfaces, lead-ing to an electron doping of approximately 0.1e per formula unit for n = 4,m = 4 LLO/SBO structure obtained from the Fermi surface integration[seeFig. 7.6] by taking the ratio of the states above and below the Fermi sur-face for each band crossing it. The doping level obtained in the LLO/SBOsystem is less than 0.5e per formula unit expected from pure electronicreconstruction, however it is comparable to the doping levels found in thesuperconducting states of BKBO.1007.5. Lattice response to potential divergence in thin filmsn-type. . .. . .La+1OO2 -1Lu+1OLa-1O2LuO2BiOSrO2Bi EEOSr-0.2-0.1 0 0.1 0.2IF IF + 1 IF + 2 IF + 3 IF + 4Δz (A)LaOLa(a)-0.2-0.1 0 0.1 0.2IF IF + 1 IF + 2 IF + 3 IF + 4Δz (A)(b)LuOLu-1-0.8-0.6-0.4-0.2 0 0.2 0.4IF IF + 1 IF + 2 IF + 3 IF + 4d(eA)distance from n-type IF (ML)(c)-0.2-0.1 0 0.1 0.2IF IF + 1 IF + 2 IF + 3 IF + 4Δz (A)SrOSr(a)-0.2-0.1 0 0.1 0.2IF IF + 1 IF + 2 IF + 3 IF + 4Δz (A)(b)BiOBi-0.4-0.2 0 0.2 0.4 0.6 0.8 1IF IF + 1 IF + 2 IF + 3 IF + 4d(eA)distance from n-type IF (ML)(c)(d)Figure 7.9: (a)(b)Layer resolved atomic displacements in the z-directionperpendicular to the interface in A˚. Atomic displacements (∆z) are averagedin each layer and are plotted against the position of the layer in monolayers(ML) from the n-type interface(IF). (c) Dipole moments calculated from thenet layer ionic charges times their displacements following the calculationsof Ref. [136]. (d) Formation of ionic dipoles in both LLO and SBO afteratomic relaxation. The dipoles result in electric fields shown by red arrows.1017.6. Metal-insulator transitionΓ X M Y Γ-101Energy (eV)EF(a)Γ X M Y Γ-101Energy (eV)EF(b)Γ X M Y Γ-101Energy (eV)EF(c)Figure 7.10: The band structure of (a) (n = 4, m = 2) (b) (n = 2, m =4), and (c) (n = 4, m = 4) LLO/SBO super lattices after atomic relaxation(with n representing the number of LLO and m number of SBO unit cellsperpendicular to the interfaces). The fat bands show the projected densityof states onto the collapsed Bi-6s octahedra (in blue) at the LaO/BiO2interface and expanded Bi-6s octahedra (in red) at the LuO2/SrO interface.7.6 Metal-insulator transitionThe energy cost associated with the transfer of 0.5 an electron per unit cell,which is essentially the band gap of SBO, predicts the onset of electronicreconstruction to occur at an LLO thickness of 1-2 unit cell. Therefore, itwould be interesting to study the LLO/SBO system as a possible candidatefor an excitonic insulator in the proximity of this thickness.Now, one might wonder if the SBO thickness would also have some effect onthe degree of electronic reconstruction. To find out, we construct a systemwith four unit cells of LLO and two unit cells of SBO i.e. (n = 4, m = 2)as shown in Fig. 7.10(a). Upon atomic relaxation we find that the systemdoes not electronically reconstruct and as a result stays insulating with aband gap of about 0.4 eV. This insulating behaviour can be explained dueto the small thickness of SBO. In this case the holes and electrons are inclose proximity and therefore can not overcome the attractive Coulomb forcebetween them and recombine to lower their energy. At this thickness, atomicreconstruction and formation of ionic dipoles must be the dominant compen-1027.6. Metal-insulator transitionsation mechanism. However, as we increase the SBO thickness to four unitcells [refer to Fig. 7.10(b)(c)], electrons and holes separate and overcome theCoulomb attraction between them at the two interfaces resulting in electronand hole doping.Constructing the structure formed of two unit cells of LLO and four unit cellsof SBO i.e. (n = 2, m = 4) with the band structure shown in Fig. 7.10(b),we find that after atomic relaxation, some electronic reconstruction occursin this system as can be seen from the band structure where a hole and anelectron pocket is present at the M and Y symmetry points respectively. Theelectrostatic potential build up in LLO in this case is just enough to form2D gases at the two interfaces, however, naturally, as the LLO thicknessis increased to four unit cells [see Fig. 7.10(c)] an increase in the potentialbuild up results in a larger transfer of electrons and therefore, larger carrierdensity at the two interfaces.Finally, we find that the band gap can be tuned as a function of SBOthickness. As shown previously bulk SBO has a band gap of about 1 eV.After growing four unit cells of LLO on two unit cells of SBO in Fig. 7.10(a)the band gap decreases to approximately 0.4 eV. We find that the bandgap of the LLO/SBO system can be further increased to about 0.6 eV byreducing the LLO thickness to two unit cells. We expect that tuning theLLO and SBO thickness as well as introducing lateral stress and strain onthe system can also result in changing its band gap.1037.7. Summary7.7 SummaryIn summary, we have shown that a new class of materials composed of polarterminated LaLuO3(LLO) and non-polar terminated SBO, can be simulta-neously doped by electrons and holes at its two opposite interfaces. Weargue that electronic reconstruction is the dominant mechanism for solvingthe diverging potential (“polar catastrophe”) in this system which in returnresults in the doping of the LLO/SBO system. We further showed that mate-rials of this type can possibly exhibit excitonic insulator properties with theexistence of an insulator-metal transition as a function of the LLO and SBOfilm thickness. Such heterostructures can open the door to exploring possible2D electron and hole gases, and possible electron-doped superconductivitywhich has been relatively unexplored in the bismuthates, and could lead tonew exciting physics that has not been realized in the corresponding bulkmaterials.104Chapter 8Summary and OutlookWe studied the electronic structure of the bismuthates showing that hy-bridization effects play a dominant role in determining their electronic struc-ture. We argued that strong hybridization between the Bi-6s and O-2p or-bitals precludes purely ionic charge-disproportionation of a Bi3+/Bi5+ form.Instead, the (self-doped) holes spatially condense into molecular-orbital-likeA1g combinations of the Bi-6s and O-2pσ orbitals of collapsed BiO6 oc-tahedra, with predominantly O-2pσ molecular orbital character. Later weshowed that the holes character strongly depends on the strength of Bi-6s and O-2pσ hybridization, tspσ, and the charge-transfer energy, ∆. Wedemonstrated that there exists a natural crossover between a bond and acharge-disproportionated system with holes preferring to occupy Bi-6s orO-a1g molecular orbitals. In the BD regime, an effective attractive “molec-ular” orbital interaction between two holes residing on the collapsed oxygenoctahedra was estimated via considering electron-phonon coupling effectsthrough changes in tspσ. A thorough study of the phonon dispersion in theundoped and doped bismuthates accompanied with detailed calculations isneeded in order to fully study the strength of electron-phonon coupling inthis system.Moreover, we studied the electronic structure of SrBiO3 thin films in the[001] direction and showed that electron doping of their surfaces are achiev-able via Cs adsorption. This electron doping is found to be strongly sensitiveto surface termination (i.e. SrO or BiO2) and adatom concentration. Wealso showed that the band gap and surface work functions are tunable viasurface termination with a difference of upto 1.5 eV in work functions of thetwo surfaces. To prove or refute the possibility of surface electron dopingof the bismuthates it is essential to make clean samples with well defined105Chapter 8. Summary and Outlookterminations for adatom depositions and explore the possible electron dopingin this system that might lead to superconductivity.Finally, we proposed and studied a new class of materials composed of po-lar terminated LaLuO3 and non-polar terminated SrBiO3, that could besimultaneously doped by electrons and holes at its two opposite interfaces.We argued that electronic reconstruction is the dominant mechanism forsolving the diverging potential in this system due to the small band gapof the bismuthates. If electronic reconstruction is achieved experimentally,this can be an interesting and new field of research with the potential forthe existence of an excitonic insulator. Next challenge is to grow the het-erostructure system with MBE and characterize their interfaces with newand improved X-ray reflectometry techniques.106Bibliography[1] R. Scholder, K. Ganter, H. Glaser, G. Merz, Z. Anorg. Allgem. Chem.,319 (1963).[2] S. M. Kazakov, C. Chaillout, P. Bordet, J. J. Capponi, M. Nunez-Regueiro, A. Rysak, J. L. Tholence, P. G. Radaelli, S. N. Putilin, andE. V. Antipov, Nature 390, 148 (1997).[3] A. Sleight, J. Gillson, and P. Bierstedt, Solid State Communications 17,27 (1975).[4] R. J. Cava, B. Batlogg, J. J. Krajewski, R. Farrow, L. W. R. Jr., A. E.White, K. Short, W. F. Peck, and T. Kometani, Nature (London) 332,814 (1988).[5] Y. J. Uemura, B. J. Sternlieb, D. E. Cox, J. H. Brewer, R. Kadono, J.R. Kempton, R. F. Kiefl, S. R. Kreitzman, G. M. Luke, P. Mulhern, T.Riseman, D. L. Williams, W. J. Kossler, X. H. Yu, C. E. Stronach, M.A. Subramanian, J. Gopalakrishnan and A. W. Sleight, Nature 335, 151(1988).[6] D. Cox and A. W. Sleight, Solid State Communications 19, 969 (1976).[7] D. E. Cox and A. W. Sleight, Acta Crystallographica Section B 35, 1(1979).[8] A. Sleight, Physica C 514 , 152 (2015).[9] A. M. Glazer, Acta Crystallographica Section B 28, 3384 (1972).[10] A. M. Glazer, Acta Crystallographica Section A 31, 756 (1975).107Bibliography[11] C. M. Varma, Phys. Rev. Lett. 61, 2713 (1988).[12] I. Hase and T. Yanagisawa, Phys. Rev. B 76, 174103 (2007).[13] T. M. Rice and L. Sneddon, Phys. Rev. Lett. 47, 689 (1981).[14] D.T. Marx, P.G. Radaelli, J.D. Jorgensen, R.L. Hitterman, D.G. Hinks,S. Pei, and B. Dabrowski, Phys. Rev. B 46, 1144 (1992).[15] E. Climent-Pascual, N. Ni, S. Jia, Q. Huang, and R.J. Cava, Phys. Rev.B 83, 174512 (2011).[16] D. Nicoletti, E. Casandruc, D. Fu, P. Giraldo-Gallo, I. R. Fisher, andA. Cavalleri, PNAS 114, 9020 (2017).[17] G. Thornton and A. J. Jacobson, Acta Crystallographica Section B 34,351, (1978).[18] J. Matsuno, T. Mizokawa, A. Fujimori, Y. Takeda, S. Kawasaki, andM. Takano, Phys. Rev. B 66, 193103 (2002).[19] Z. P. Yin, A. Kutepov, and G. Kotliar, Phys. Rev. X 3, 021011 (2013).[20] N. Matsushita, H. Ahsbahs, S. S. Hafner, and N. Kojima, J. Solid StateChem. 180, 1353 (2007).[21] T. Siritanon, J. Li, J. K. Stalick, R. T. Macaluso, A. W. Sleight, andM. A. Subramanian, Inorg. Chem. 50, 8494 (2011).[22] P. F. Henry, M. T. Weller, and C. C. Wilson, Chem. Mater. 14, 4104(2002).[23] G. Vielsack and W. Weber, Phys. Rev. B 54, 6614 (1996).[24] G. K. Wertheim, J. P. Remeika, and D. N. E. Buchanan, Phys. Rev. B26, 2120 (1982).[25] W. A. Harrison, Phys. Rev. B 74, 245128 (2006).[26] L. F. Mattheiss and D. R. Hamann, Phys. Rev. B 28, 4227 (1983).108Bibliography[27] J. de Hair and G. Blasse, Solid State Communications 12, 727 (1973).[28] A. F. Orchard and G. Thornton, J. Chem. Soc. Dalton Trans. p. 1238(1977).[29] T. Mizokawa, D. Khomskii and G. A. Sawatzky, Phys. Rev. B 61, 11263(2000).[30] H. Park, A. J. Millis and C. A. Marianetti, Phys. Rev. Lett. 109, 156402(2012).[31] B. Lau, A. J. Millis, Phys. Rev. Lett. 110, 126404 (2013).[32] S. Johnston, A. Mukherjee, I. Elfimov, M. Berciu and G. Sawatzky,Phys. Rev. Lett. 112, 106404 (2014).[33] R.J. Green, M.W. Haverkort, G.A. Sawatzky, Phys. Rev. B 94, 195127(2016).[34] K. Foyevtsova, A. Khazraie, I. Elfimov and G. A. Sawatzky, Phys. Rev.B 91, 121114(R) (2015).[35] R. Nourafkan, F. Marsiglio, and G. Kotliar, Phys. Rev. Lett. 109,017001 (2012).[36] D. Korotin, V. Kukolev, A. V. Kozhevnikov, D. Novoselov, and V. I.Anisimov, J. Phys.: Condens. Matter 24, 415603 (2012).[37] D. M. Korotin, D. Novoselov, and V. I. Anisimov, J. Phys.: Condens.Matter 26, 195602 (2014).[38] C. Franchini, G. Kresse, and R. Podloucky, Phys. Rev. Lett. 102,256402 (2009).[39] C. Franchini, A. Sanna, M. Marsman, and G. Kresse, Phys. Rev. B 81,085213 (2010).[40] H. Frohlich, Adv. Phys. 3, 325-361 (1954).[41] T. Holstein, Ann. Phys. 8, 325-342 (1959).109Bibliography[42] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698(1979).[43] J. F. Federici, B. I. Greene, E. H. Hartford, and E. S. Hellman, Phys.Rev. B 42, 923 (1990).[44] A. P. Menushenkov, K. V. Klementev, A. V. Kuznetsov, M. Y. Kagan,Journal of Experimental and Theoretical Physics, 93, 3, 615( 2001).[45] I. B. Bischofs, V. N. Kostur, and P. B. Allen, Phys. Rev. B 65, 115112(2002).[46] J. Ahmad and H. Uwe, Phys. Rev. B 72, 125103 (2005).[47] M. Born and R. Oppenheimer, Annalen der Physik 84, 457(1927).[48] P. Hohenberg and W. Kohn, Phys. Rev. B.136, 864(1964).[49] W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133(1965).[50] J. P. Perdew and Y. Wang, Phys. Rev. B, 33, 8800(1986).[51] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett., 77,3865(1996).[52] J. P. Perdew and A. Zunger, Phys. Rev. B, 23, 5048(1981).[53] P. A. M. Dirac, Proc. Cambridge Phil. Soc. 26, 376(1930).[54] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244(1992).[55] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566(1980).[56] D. C. Langreth and M. J. Mehl, Phys. Rev. B 28, 1809(1983).[57] Z. Wu and R. E. Cohen, Phys. Rev. B 73, 235116 (2006).[58] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria,L. A. Constantin, X. Zhou and K. Burke, Phys. Rev. Lett. 100, 136406(2008).110Bibliography[59] F. Bloch. Zeitschrift fur Physik A Hadrons and Nuclei 52, 555(1929).[60] J. C. Phillips, Phys. Rev. 112, 685 (1958).[61] L. Kleinman and J. C. Phillips, Phys. Rev. 116, 880 (1959).[62] O. K. Andersen, Phys. Rev. B 12, 3060(1975).[63] E. Wimmer, H. Krakauer, M. Weinert and A. J. Freeman, Phys. Rev.B 24, 864(1981).[64] P. Blaha, K. Schwarz, P. Sorantin and S.B. Trickey, Computer PhysicsCommunications 59, 399(1990).[65] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz,WIEN2K, An Augmented Plane Wave Local Orbitals Program for Calcu-lating Crystal Properties (Karlheinz Schwarz, Techn. Universitat Wien,Austria, 2001.)[66] D. R. Hamann, M. Schluter, and C. Chiang, Phys. Rev. Lett. 43, 1494(1979).[67] J. D. Joannopoulos, Th. Starkloff, and Marc Kastner. Theory of pres-sure dependence of the density of states and reflectivity of selenium. Phys.Rev. Lett., 38, 660 (1977).[68] T. Starkloff and J. D. Joannopoulos, Phys. Rev. B, 16, 5212 (1977).[69] D. Vanderbilt, Phys. Rev. B, 41, 7892(R) (1990).[70] M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D.Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992).[71] H. Hellmann. Einfuhrung in die Quantenchemie. Leipzig: FranzDeuticke (1937).[72] R. P. Feynman, Phys. Rev. 56, 340(1939).[73] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B, 44, 943(1991).111Bibliography[74] M. T. Czyzyk and G. A. Sawatzky, Phys. Rev. B 49, 14211 (1994).[75] J. Hubbard, Proceedings of the Royal Society of London 276 238 (1963).[76] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498(1954).[77] A. B. Migdal, Qualitative methods in quantum theory (1977).[78] W. L. McMillan, Phys. Rev., 167, 331(1968).[79] G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15 (1996).[80] J. Paier, R. Hirschl, M. Marsman, and G. Kresse, J. Chem. Phys. 122,234102 (2005).[81] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, andN. Marzari, Computer Physics Communications 178, 685 (2008), ISSN0010-4655.[82] J. Kunes, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and K. Held,Comp. Phys. Commun. 181, 1888 (2010).[83] J. Zannen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418(1985).[84] N. C. Plumb, D. J. Gawryluk, Y. Wang, Z. Ristic, J. Park, B. Q. Lv, Z.Wang, C. E. Matt, N. Xu, T. Shang, K. Conder, J. Mesot, S. Johnston,M. Shi and M. Radovic, Phys. Rev. Lett. 117, 037002 (2016).[85] S. Balandeh, R. J. Green, K. Foyevtsova, S. Chi, O. Foyevtsov, F. Li,and G. A. Sawatzky, Phys. Rev. B 96, 165127 (2017).[86] L. F. Mattheiss and D. R. Hamann, Phys. Rev. B 26, 2686 (1982).[87] L. F. Mattheiss, Phys. Rev. B 28, 6629 (1983).[88] M. Shirai, N. Suzuki, and K. Motizuki, J. Phys.: Condens. Matter 2,3553 (1990).112Bibliography[89] N. Hamada, S. Massidda, A. J. Freeman, and J. Redinger, Phys. Rev.B 40, 4442 (1989).[90] A. I. Liechtenstein, I. I. Mazin, C. O. Rodriguez, O. Jepsen, O. K.Andersen and M. Methfessel, Phys. Rev. B 44, 5388 (1991).[91] K. Kunc, R. Zeyher, A. Liechtenstein, M. Methfessel, and O. Andersen,Solid State Commun. 80, 325 (1991).[92] K. Kunc and R. Zeyher, Phys. Rev. B 49, 12216 (1994).[93] V. Meregalli and S. Y. Savrasov, Phys. Rev. B 57, 14453 (1998).[94] T. Thonhauser and K. M. Rabe, Phys. Rev. B 73, 212106 (2006).[95] F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 (1988).[96] N. Hiraoka, T. Buslaps, V. Honkimaki, H. Minami, and H. Uwe, Phys.Rev. B 71, 205106 (2005).[97] N. Hiraoka, T. Buslaps, V. Honkimaki, J. Ahmad, and H. Uwe, Phys.Rev. B 75, 121101 (2007).[98] S. Sahrakorpi, B. Barbiellini, R. S. Markiewicz, S. Kaprzyk, M. Lin-droos, and A. Bansil, Phys. Rev. B 61, 7388 (2000).[99] A. Ignatov, Nuclear Instruments and Methods in Physics Research A448 332(2000).[100] J. Perdew, K. Burke and Y.Wang, Phys. Rev. B 54, 16533, (1996).[101] S. Froyen and W. A. Harrison, Phys. Rev. B 20, 2420 (1979).[102] A. K. McMahan and R. M. Martin, Phys. Rev. B 38, 6650 (1988).[103] S. Sugai, S. Uchida, K. Kitazawa, S. Tanaka, and A. Katsui, Phys.Rev. Lett. 55, 426 (1985).[104] S. Tajima, M. Yoshida, N. Koshizuka, H. Sato, and S. Uchida, Phys.Rev. B 46, 1232 (1992).113Bibliography[105] M. Braden, W. Reichardt, A. S. Ivanov, and A. Yu. Rumiantsev, Eu-rophys. Lett. B 34, 531 (1996).[106] T. Bazhirov, S. Coh, S. G. Louie, and M. L. Cohen, Phys. Rev. B 88,224509 (2013).[107] A. Khazraie, K. Foyevtsova, I. Elfimov, and G. A. Sawatzky, Phys.Rev. B 97, 075103 (2018).[108] A. Khazraie, K. Foyevtsova, I. Elfimov, and G. A. Sawatzky, Phys.Rev. B 98, 205104 (2018).[109] M. Moeller, G. A. Sawatzky, M. Franz and M. Berciu, Nature Com-munications 8, 2267 (2017).[110] M. Moeller and M. Berciu, Phys. Rev. B 93, 035130 (2016).[111] D. Fournier, G. Levy, Y. Pennec, J. L. McChesney, A. Bostwick, E.Rotenberg, R. Liang, W. N. Hardy, D. A. Bonn, I. S. Elfimov, and A.Damascelli, Nature Phys. 6, 905 (2010).[112] M. A. Hossain, J. D. F. Mottershead, D. Fournier, A. Bostwick, J. L.Mcchesney, E. Rotenberg, R. Liang, W. N. Hary, G. A. Sawatzky, I. S.Elfimov, D. A. Bonn, and A. Damascelli, Nature Phys. 4, 527 (2008).[113] Y. Miyata, K. Nakayama, K. Sugawara, T.Sato and T.Takahashi, Na-ture Mater. 14, 775 (2015).[114] V.Vildosola, F. Guller, and A.M. Llois, Phys. Rev. Lett. 110, 206805(2013).[115] CRC Handbook of Chemistry and Physics, 93rd ed., edited by W.Haynes (Taylor & Francis, London 2012).[116] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehartand Winston, Philadelphia, 1976).[117] Z. Zhong and P. Hansmann, Phys. Rev. B 93 , 235116 (2006).114Bibliography[118] R. Jacobs, J. Booske, and D. Morgan, Adv. Funct. Mater.26, 5471(2016).[119] S. Pei, N. J. Zaluzec, J. D. Jorgensen, B. Dabrowski, D. G. Hinks, A.W. Mitchell, and D. R. Richards, Phys. Rev. B 39, 811(R) (1989).[120] S. Pei, J.D. Jorgensen, B.Dabrowski, D. G. Hinks, D. R. Richards, A.W. Mitchell, J. M. Newsam, S. K. Sinha, D. Vaknin, and A. J. Jacobson,Phys. Rev. B 41, 4126(1990).[121] G. A. Baraff, J. A. Appelbaum, and D. R. Hamann, Phys. Rev. Lett.38, 237 (1977).[122] W. A. Harrison, E. A. Kraut, J. R. Waldrop, and R. W. Grant, Phys.Rev. B 18, 4402 (1978).[123] P. W. Tasker, J. Phys. C 12, 4977 (1979).[124] A. Ohtomo and H. Y. Hwang, Nature London 427, 423 (2004).[125] S. Thiel, G. Hammerl, A. Schmehl, C. W. Schneider, and J. Mannhart,Science 313, 1942 (2006).[126] H. Lee, N. Campbell, J. Lee, T. J. Asel, T. R. Paudel, H. Zhou, J. W.Lee, B. Noesges, J. Seo, B. Park, L. J. Brillson, S. H. Oh, E. Y. Tsymbal,M. S. Rzchowski and C. B. Eom, Nature Mater. 17, 231 (2018).[127] A. Brinkman, M. Huijben, M. V. Zalk, J. Huijben, U. Zeitler, J.C. Maan, W. G. V. der Wiel, G. Rijnders, D. H. A. Blank, and H.Hilgenkamp, Nature Mater. 6, 493 (2007).[128] N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G. Ham-merl, C. Richter, C. W. Schneider, T. Kopp, A. S. Ruetschi, D. Jaccard,M. Gabay, D. A. Muller, J.M. Triscone, and J. Mannhart, Science 317,1196 (2007).[129] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M.Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, Nature(London) 456, 624 (2008).115Bibliography[130] R. Dingle, W. Wiegmann, and C. H. Henry, Phys. Rev. Lett, 33,827(1974).[131] L. L. Chang, L. Esaki, and R. Tsu,, Appl. Phys Lett, 24, 593 (1974).[132] S. Mori and T. Ando, Surface Science, 98, 101 (1980).[133] N. Nakagawa, H. Y. Hwang, and D. A. Muller, Nature Mater. 5, 204(2006).[134] A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. MilettoGranozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L.Marrucci, Phys. Rev. B 80, 075110 (2009).[135] R. Hesper, L. H. Tjeng, A. Heeres, and G. A. Sawatzky, Phys. Rev. B62, 16046 (2000).[136] R. Pentcheva and W. E. Pickett, Phys. Rev. Lett. 102, 107602 (2009).[137] V. Vonk, M. Huijben, K. J. I. Driessen, P. Tinnemans, A. Brinkman,S. Harkema, and H. Graafsma, Phys. Rev. B 75, 235417 (2007).[138] D. R. Hamann, D. A. Muller, and H. Y. Hwang, Phys. Rev. B 73,195403 (2006).[139] A. S. Kalabukhov, Y. A. Boikov, I. T. Serenkov, V. I. Sakharov, V. N.Popok, R. Gunnarsson, J. Borjesson, N. Ljustina, E. Olsson, D. Winkler,and T. Claeson, Phys. Rev. Lett. 103, 146101 (2009).[140] L. Yu and A. Zunger, Nat. Commun. 5, 5118 (2014).[141] B. G. Levi, Phys. Today 60, 23 (2007).[142] J. N. Eckstein, Nat. Mater. 6, 473 (2007).[143] C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C.S. Hellberg, J. Mannhart, and J. Levy, Nature Mater. 7, 298 (2008).116[144] W. Siemons, G. Koster, H. Yamamoto, W. A. Harrison, G. Lucovsky,T. H. Geballe, D. H. A. Blank, and M. R. Beasley, Phys. Rev. Lett. 98,196802 (2007).[145] S. Pauli and P. Willmott, J. Phys.: Condens. Matter 20, 264012(2008).[146] H. Chen, A. Kolpak, and S. Ismail-Beigi, Phys. Rev. B 82, 085430(2010).[147] Z. Zhong, P. X. Xu, and P. J. Kelly, Phys. Rev. B 82, 165127 (2010).[148] J. Varghese, T. Joseph, and M. T. Sebastian, N. Reeves-McLaren, andA. Feteira, J. Am. Ceram. Soc., 93, 2960 (2010).[149] X. Gu, I. Elfimov, G. A. Sawatzky, arXiv:0911.4145[150] J. Zhou, T. C. Asmara, M. Yang, G. A. Sawatzky, Y. P. Feng, and A.Rusydi, Phys. Rev. B 92, 125423 (2015).117Appendix AHybrid functionalsThe effect of HSE hybrid functional is shown on the band gap of the exper-imental structure of SrBiO3. A comparison is made with the correspondingcalculations in LDA. 0 2 4 6DOS (states/eV/spin)(a) LDA TotalBi1-6sBi2-6sO-2p 0 2 4 6-12 -10 -8 -6 -4 -2 0 2DOS (states/eV/spin)Energy (eV)(b) HSEFigure A.1: The total and projected densities of states of SrBiO3 calculatedusing (a) the LDA functional and (b) the Heyd-Scuseria-Ernzerhof (HSE)hybrid functional. Bi1 and Bi2 denote the two inequivalent Bi atoms sur-rounded by, respectively, an expanded oxygen octahedron and a collapsedoxygen octahedron.118Appendix BLindhard functionFigure. B.1 shows the Sr1−xKxBiO3 Lindhard function for various q in theBrillouin zone as a function of x, the degree of doped holes. It can be seenthat there exists a pick at (pi, pi, pi) with a maximum at zero doping x = 0in Fig. B.1(b). 0.5 1(0,0,qz) (π,0,qz) (π,π,qz) (0,0,qz)-χ(q) (a. u.)qqz=0qz=πt=texp 0.5 1 (0,0,0) (π,π,π) -χ(q) (a. u.)q(b)(a) 0.00.10.20.30.40.50.54t=texpFigure B.1: χ(q, ω = 0) for the t = texp structure: (a) a three-dimensionalscan through the Brillouin zone and (b) the dependence on hole doping,within a rigid-band approximation. The labeling numbers x indicate con-centrations of doped holes in Sr1−xKxBiO3.119Appendix CBulk structure parameterswith dopingLattice constants and degrees of breathing and tilting distortions of theBiO6 octahedra are listed for the electron (Table. C.1) and hole (Table.D.1) doped bulk structure of SrBiO3. For comparison we have listed thelattice parameters for GGA, LDA, and PBE-sol exchange and correlationfunctionals. We observe that upon 0.5 electron and 0.5 hole doping, thebreathing distortions are eliminated and the structure transitions into ametallic state.0.5 electron doping per SrBiO3Structure a b c tiltingo breathing A˚Expt (initial structure) 5.95 6.10 8.48 16 0.1GGA (electron) 6.10 6.34 8.79 20 0.0PBE-sol (electron) 6.02 6.28 8.67 15 0.0LDA (electron) 5.96 6.23 8.60 21 0.0Table C.1: Lattice constants and structural distortions of SrBiO3 bulk struc-ture upon 0.5 electron doping per formula unit. A comparison is made be-tween the experimental structure and the relaxed structures obtained withGGA, LDA and PBE-sol exchange and correlation functionals. An E-cut of410 eV and a k-point grid size of 8×8×8 is used in all the calculations.120Appendix C. Bulk structure parameters with doping0.5 hole doping per SrBiO3Structure a b c tiltingo breathing A˚Expt (initial structure) 5.95 6.10 8.48 16 0.1GGA (hole) 5.81 5.89 8.23 15 0.0PBE-so (hole) 5.74 5.82 8.13 15 0.0LDA (hole) 5.69 5.78 8.06 15 0.0Table C.2: Lattice constant and structural distortions of SrBiO3 bulk struc-ture upon 0.5 hole doping per formula unit. A comparison is made betweenthe experimental structure and the relaxed structures obtained with GGA,LDA and PBE-sol exchange and correlation functionals. An E-cut of 410eV and a k-point grid size of 8×8×8 is used in all the calculations.121Appendix DElectronic and structuralproperties of bulk LaLuO3The Lu-O bond lengths in the Pnma [148] structure range from 2.179 to2.241 A˚ and, after atomic and volume relaxation bond lengths converge to2.20 A˚. In the experimental and relaxed structure the LuO6 octahedra showtiltings of about 16o around the Lu ions. The tilting of the LuO6 octahedrareduces the coordination number of the La from 12 to 8, and the structurecan be described in terms of Glazer’s notation as a−b+a− with a calculatedtolerance factor of 0.852. A charge gap of approximately 4.5 eV is obtainedwith GGA compared to the experimental gap of 5.3 eV. The lattice constantsare listed in Table. D.1.LaLuO3Structurea b c tiltingoExperimental 6.018 88 8.374 89 5.818 41 16Relaxed 6.087 32 8.447 83 5.916 29 16Table D.1: Lattice constants and structural distortions of the four formulaunit cell of bulk LaLuO3 is listed. An E-cut of 410 eV and a k-point gridsize of 8×8×8 is used with GGA exchange-correlation functional. For therelaxed structure the volume and all atomic positions are allowed to relaxuntil all forces are less than 0.01 eV/ A˚.The atomic positions of bulk experimental structure of LaLuO3 are listed inTable. D.2.122Appendix D. Electronic and structural properties of bulk LaLuO3-5-4-3-2-1 0 1 2 3 4 5 6A M Γ X N ΓEnergy (eV)(a) Experimental structure LLO 0 5 10 15 20States/eVTotalLa-dO-p-5 0 5A M Γ X N ΓEnergy (eV)(b) Relaxed structure 0 5 10 15 20States/eVFigure D.1: Band structure and density of states of the experimental andrelaxed structures.Space group Atom x y zPnma La 0.0542 0.2500 −0.0135Lu 0.5000 0.0000 0.0000O(1) 0.4460 0.2500 0.1000O(2) 0.3092 0.0541 0.6784Table D.2: LaLuO3 experimental structure atomic positions.123Appendix EElectronic and structuralproperties of bulk SrBiO3The Bi-O bond lengths in the P21/n [2] structure of SrBiO3 ranges from 2.09to 2.321 A˚ . The structure shows breathing distortions of about 0.1 A˚. Inthe experimental and relaxed structures the BiO6 octahedra show tiltings ofabout 16o around the Bi ions and the structure can be described in terms ofGlazer’s notation as a−a−c+ with a calculated tolerance factor of 0.838. Acharge gap of approximately 0.8 eV is obtained with GGA compared to theexperimental gap of 1.2 eV. The lattice constants are listed in Table. E.1.SrBiO3Structurea b c tiltingoExperimental 5.948 02 6.095 12 8.485 43 16Relaxed 5.956 67 6.138 65 8.506 39 16Table E.1: Lattice constants and structural distortions of the four formulaunit cell of bulk SrBiO3 is listed. An E-cut of 410 eV and a k-point gridsize of 8×8×8 is used with GGA exchange-correlation functional. For therelaxed structure the volume and all atoms are allowed to relax until theforce on each atom is less than 0.01 eV/ A˚.The atomic positions of bulk experimental structure of SrBiO3 are listed inTable. E.2.124Appendix E. Electronic and structural properties of bulk SrBiO3Space group Atom x y zP21/n Sr −0.01527 0.54554 0.2511Bi1 0.0000 0.0000 0.0000Bi2 0.0000 0.0000 0.5000O(1) 0.40567 0.46177 0.2411O(2) 0.2841 0.1901 0.5461O(3) 0.1871 0.7141 0.5561Table E.2: SrBiO3 experimental structure atomic positions.125AppendixE.ElectronicandstructuralpropertiesofbulkSrBiO3Table E.3: Parameters of the 16 model structures of SrBiO3with the monoclinic space group P21/n. For each structure,lattice constants a, b, and c are given in the first row, frac-tional coordinates of oxygen atoms O1, O2, and O3 are givenin, respectively, the second, third, and forth row, and relaxedfractional coordinates of Sr are given in the fifth row. Forthe t = 0 structures Sr is at high-symmetry position 0, 0,0.25. The two Bi atoms are at 0.5, 0, 0 and 0.5, 0, 0.5, themonoclinic angle equals 90o.6.0147, 6.0147, 8.5061 A˚ 6.0147, 6.0147, 8.5061 A˚ 6.0147, 6.0147, 8.5061 A˚ 6.0147, 6.0147, 8.5061 A˚O1: 0.0000, 0.5000, 0.2500 O1: 0.0000, 0.5000, 0.2444 O1: 0.0000, 0.5000, 0.2391 O1: 0.0000, 0.5000, 0.2339O2: 0.7500, 0.2500, 0.0000 O2: 0.7556, 0.2556, 0.0000 O2: 0.7609, 0.2609, 0.0000 O2: 0.7661, 0.2661, 0.0000O3: 0.2500, 0.2500, 0.0000 O3: 0.2444, 0.2556, 0.0000 O3: 0.2391, 0.2609, 0.0000 O3: 0.2339, 0.2661, 0.00005.9559, 6.0730, 8.5061 A˚ 5.9559, 6.0730, 8.5061 A˚ 5.9559, 6.0730, 8.5061 A˚ 5.9559, 6.0730, 8.5061 A˚O1: 0.0704, 0.4903, 0.2500 O1: 0.0704, 0.4903, 0.2442 O1: 0.0704, 0.4903, 0.2386 O1: 0.0703, 0.4903, 0.2333O2: 0.7141, 0.2845, 0.0352 O2: 0.7200, 0.2902, 0.0353 O2: 0.7257, 0.2957, 0.0354 O2: 0.7312, 0.3009, 0.0355O3: 0.2141, 0.2155, 0.9648 O3: 0.2082, 0.2212, 0.9649 O3: 0.2025, 0.2267, 0.9650 O3: 0.1971, 0.2319, 0.9652Sr: 0.9933, 0.0314, 0.2500 Sr: 0.9930, 0.0312, 0.2501 Sr: 0.9943, 0.0276, 0.2501 Sr: 0.9918, 0.0325, 0.25065.8976, 6.1295, 8.5061 A˚ 5.8976, 6.1295, 8.5061 A˚ 5.8976, 6.1295, 8.5061 A˚ 5.8976, 6.1295, 8.5061 A˚O1: 0.1001, 0.4807, 0.2500 O1: 0.1001, 0.4807, 0.2440 O1: 0.1000, 0.4807, 0.2382 O1: 0.0999, 0.4807, 0.2326O2: 0.6980, 0.2982, 0.0501 O2: 0.7043, 0.3040, 0.0503 O2: 0.7103, 0.3095, 0.0505 O2: 0.7161, 0.3148, 0.0506126AppendixE.ElectronicandstructuralpropertiesofbulkSrBiO3O3: 0.1980, 0.2018, 0.9499 O3: 0.1917, 0.2077, 0.9502 O3: 0.1857, 0.2133, 0.9505 O3: 0.1800, 0.2187, 0.9507Sr: 0.9862, 0.0536, 0.2500 Sr: 0.9863, 0.0514, 0.2498 Sr: 0.9861, 0.0522, 0.2493 Sr: 0.9859, 0.0524, 0.24845.8400, 6.1845, 8.5061 A˚ 5.8400, 6.1845, 8.5061 A˚ 5.8400, 6.1845, 8.5061 A˚ 5.8400, 6.1845, 8.5061 A˚O1: 0.1232, 0.4713, 0.2500 O1: 0.1232, 0.4713, 0.2437 O1: 0.1231, 0.4713, 0.2377 O1: 0.1229, 0.4713, 0.2320O2: 0.6848, 0.3081, 0.0616 O2: 0.6914, 0.3141, 0.0619 O2: 0.6979, 0.3197, 0.0622 O2: 0.7040, 0.3251, 0.0625O3: 0.1848, 0.1918, 0.9384 O3: 0.1781, 0.1978, 0.9388 O3: 0.1718, 0.2035, 0.9392 O3: 0.1658, 0.2090, 0.9396Sr: 0.9804, 0.0681, 0.2500 Sr: 0.9804, 0.0679, 0.2493 Sr: 0.9807, 0.0674, 0.2483 Sr: 0.9812, 0.0664, 0.2474127
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A first principle study of the electronic structure of the bismuthates Khazraie Zamanpour, Arash 2019
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Title | A first principle study of the electronic structure of the bismuthates |
Creator |
Khazraie Zamanpour, Arash |
Publisher | University of British Columbia |
Date Issued | 2019 |
Description | Motivated by the recently renewed interest in the High Tc superconducting bismuth perovskites, we investigate the electronic structure of the parent compounds ABiO₃ (A = Sr, Ba) using ab initio methods and tight-binding modelling. We use the density functional theory in the local density approximation to understand the role of various contributions in shaping the ABiO₃ band structure. It is established that hybridization involving Bi-6s and O-2p orbitals plays the most important role. The opening of a gap with the onset of the breathing distortion is associated with condensation of holes onto a₁g-symmetric molecular orbitals formed by the O-2pσ orbitals on the collapsed BiO₆ octahedra. The primary importance of oxygen p states is thus revealed, in contrast to a popular picture of a purely ionic Bi³⁺/Bi⁵⁺ charge-disproportionation. A single band model involving an extended molecular orbital of both Bi-6s and a linear combination of six O-2p orbitals is derived which provides a good description of the low energy scale bands straddling the chemical potential. In addition, a parameter-based phase diagram associated with materials incorporating “skipped valence” ions is developed. A crossover from a bond disproportionated (BD) to a charge-disproportionated (CD) system in addition to the presence of a new metallic phase is observed. We argue that three parameters determine the underlying physics of the BD-CD crossover when electron correlation effects are small: the hybridization between O-2pσ and s orbitals of the B cation in ABO₃, their charge-transfer energy (∆), and the width of the oxygen sub-lattice band (W ). In the BD system, we estimate an effective attractive interaction U between holes on the same O-a₁g molecular orbital. Later, we show the possibility of surface electron doping of the bismuthates via adatom. Finally, we propose a new class of materials, namely heterostructure composed of LaLuO₃ and SrBiO₃, that can host coexisting electron and hole gases and potentially high-temperature superconductivity at their two opposite interfaces. We argue that electronic reconstruction is the dominant mechanism for solving the diverging potential. The electronic structure of this system suggests the electron-hole gas interactions can be tuned with the potential of obtaining excitonic insulating phases. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2019-01-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0376122 |
URI | http://hdl.handle.net/2429/68279 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2019-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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