In order to demonstrate the explicit form of the matrix elements of the Hamiltonians shown in Eq. (2.10) and Eq. (2.21) it is more convenient to discuss a simplified model Hamiltonian for bulk STO, with fewer hopping terms. Within this model the matrix elements of Eq. (2.12) can be separated into three terms ↵↵SO0, ↵↵T 0 andt˜↵↵k 0 that describe the atomic spin-orbit coupling, tetragonal distortion and hopping up to next nearest neighbour lattice sites respectively. This Hamiltonian is written as
Hbulk=HSO+HT +Hk (B.1)
Hbulk= X
k↵↵0
h ↵↵0
SO + ↵↵0 ↵
T + ↵↵0˜t↵ki
c†k↵ck↵0 (B.2) where all of thekdependence is contained in the hopping term and the delta functions ↵↵0further simplify the Hamiltonian by prohibiting hoppings that change the electron orbital character. Thus the only offdiagonal matrix elements come from the atomic spin-orbit matrix. For the basis
{↵}={xy", xy#, yz", yz#, xz", xz#} (B.3)
the matrixHSO / ·L, where is the Pauli matrix vector and Lis the
orbital angular momentum vector, takes the form
where the energy SO controls the strength of spin-orbit coupling.
Parametrization of the band structure shown in Fig. 3.17 gives SO = 19.95 meV. The effect of the tetragonal distortion of the STO crystal structure can be modelled by
where T describes the increase in energy of the dxyorbitals due to the cubic-to-tetragonal reduction of symmetry. The DFT calculation for STO used in this thesis uses a cubic crystal structure such that parametrisation of the band structure shown in Fig. 3.17 results in T = 0. Forab initio calculations that include the tetragonal distortion, T <5meV [29, 96].
The kinetic energy contribution toH is contained in the hopping Hamil-tonian given by elements is given explicitly by the Fourier transform of effective hopping parameters t↵↵R 0 such that
˜txy"k = ˜txy#k =t0+ 2t1cos(kx) + 2t1cos(ky) + 2t2cos(kz) + 4t3cos(kx)cos(ky) (B.7)
˜tyzk"= ˜tyzk #=t0+ 2t2cos(kx) + 2t1cos(ky) + 2t1cos(kz) + 4t3cos(ky)cos(kz) (B.8)
˜txzk "= ˜txzk #=t0+ 2t1cos(kx) + 2t2cos(ky) + 2t1cos(kz) + 4t3cos(kx)cos(kz) (B.9) Here the subscript of the effective hopping parameterst0, t1, t2 andt3 no longer refers to the vectorRsince the cubic symmetry of the model means that many t↵↵R0 in Eq. (2.12) are identical. Only terms where ↵=↵0 are included. The energy constants found by parametrizing theab initio DFT calculations used in this thesis aret1= 0.277eV, t2 = 0.031eV, t3 = 0.076eV, andt0= 1.4811eV to fix the conduction band minimum at 0 eV.
Writing this model Hamiltonian in the quasi 2D supercell form of Eq. (2.21) allows us to include the term HV that simulates an on-site potentialV(z)in the surface region of the crystal. The Hamiltonian is now given by matrix that must be diagonalized at every kk point in the Brillouin zone.
The diagonal blocks forHSO andHT are given by Eq. (B.4) and Eq. (B.5) respectively. The diagonal block ofHV iszdependent and has the form
HV =
The explicit inclusion of thez position index in the Hamiltonian now means that ˜t↵kkzz0 must be split into diagonal terms for z = z0 and off-digonal terms forz6=z0. In this simplified model, which only includes next nearest neighbour hopping there are two cases
t˜↵kkzz0 =
(˜t↵kk0 forz z0= 0
˜t↵kk1 forz z0= 1 (B.13)
The diagonal6⇥6blocks of the hopping Hamiltonian Hkk have the form
with the matrix elements are found from Eq. (2.20) and are explicitly
˜txyk " which defines the Hamiltonian of a single atomic plane of STO. The off-diagonal6⇥6blocks of Hkk have the form
with the matrix elements found from Eq. (2.20) given by
t˜xy"kk1= ˜txy#kk1=t2 (B.19) The6L⇥6Lsupercell matrix for eachkpoint inH is represented visually in Fig. B.1. Each matrix is divided into6⇥6blocks represented by the smaller (coloured) squares. The diagonal (red) blocks containHSO,HT, the diagonal terms ofHkk given by Eq. (B.14) andHV whosezdependent behaviour is indicated by the red colour scale. V(z) has a high magnitude at near the surface (z= 0, bright red) and a low magnitude towards the bulk side of the
Z 0 1 2 … … L 0
1
2
……
L
0 1 2 … … L
(b) (a)
Figure B.1:Visual representation of the6L⇥6Lmatrices that are the building blocks of the supercell HamiltonianH at eachk. Each small square represents a 6⇥6 matrix and the row/column index encode a lattice vectorz z0. The red diagonal squares are defined byHSO+HT andHV (whosezdependent magnitude is represented by the red-colour scale) and the diagonal blocks ofHkk (Eq. (B.14)).
The green squares contain the off-diagonal blocks ofHkk (Eq. (B.18)). The inclusion of higher-order hoppiung is represented by the blue squares in (b).
supercell (z!L, light red). The green offdiagonal squares are given by the blocks of Eq. (B.18) and can be understood as inter-layer coupling terms.
The truncation of the interlayer hopping described by Eq. (2.24) is obvious in the asymmetry of the green off-diagonal squares forz= 0andz=L. In the case that higher order hopping terms are included in˜t↵k
kzz0, as for the realistic effective tight binding model used in Sect. 3.1.3, the Hamiltonian contains additional offdiagonal terms which are represented by blue squares in Fig. B.1(b).
In contrast to the results of the self consistent calculations for a realistic effective tight-binding Hamiltonian, the results of self consistent calculations (for the same dielectric constant and boundary conditions as Fig. 3.18 and Fig. 3.25) using this model Hamiltonian do not produce a Rashba-like spin splitting of the subband structure. This can be seem in Fig. B.2 and is due to the absence of inter-orbital hopping in the simplified model that are sensitive to the broken inversion symmetry at the surface. In STO, the inter-orbital hopping amplitudes change sign with thez-axis hopping direction and are thus not mirror symmetric[40, 107, 110]. Their inclusion in the realistic tight-binding parametrization of the STO bulk band structure means that Rashba-like spin splitting is a natural consequence of confinement.
-0.3 -0.2 -0.1 0.0
E -EF (eV)
1.0 0.8
0.6 0.4
0.2 0.0
kx (π/a)
Figure B.2:Subband structure resulting from the self consistent solution of coupled Poisson-Schrödinger equations for the realistic tight binding supercell Hamiltonian Eq. (3.5) (red dashed lines) using hopping parameters found by downfoldingab initioDFT calculations and for the simplified model supercell Hamiltonian Eq. (B.11) (black lines) usingt1= 0.277eV,t2= 0.031eV,t3= 0.076eV,t0= 1.4811eV and SO= 0.0193eV which are found by parametrization of the same DFT calcual-tion.
Acknowledgements
I am very grateful to Prof. Felix Baumberger for having offered me the opportunity to do a PhD in Geneva and for his guidance throughout. I would particularly like to thank Anna Tamai and Flavio Bruno for their support across all aspects of my project and Alberto de la Torre, Sara Riccò and Irène Cucchi for always offering their help when I got stuck. I have enjoyed working with you all very much as a part of "Team ARPES".
My thesis also involved collaborations with many people beyond the group in Geneva. I would like to thank Phil King for his advice and insight, and Saeed Bahramy and Christophe Berthod for their help with the theoretical aspects of my thesis. I am grateful to Jaime Sánchez-Barriga for making our spin-ARPES collaboration possible and to Zhiming Wang for sharing his expertise in the projects we worked on together. My thesis is formed of data acquired at several synchrotron facilities and I would like to thank those who’s tireless efforts made this possible, in particular Moritz Hoesch and Timur Kim from I05 at the Diamond Lightsource; Nick Plumb, Milan Radovic and Ming Shi from SIS-HRPES at the Swiss Light Source; Andrei Varykhalov from PHOENEXS and One-Squared at Bessy II. I am also grateful to the many unmentioned people who made invaluable contributions to many successful beamtimes and to the publications related to my thesis.
I would also like to express my gratitude to Prof. Jean-Marc Triscone, Prof.
Ralph Claessen and Prof. Karsten Held for forming my thesis committee, and for taking the time to read my thesis.
I am fortunate that my parents Christine and Paul always encourage me to do what I enjoy and to take opportunities as they come, which lead me firstly to St-Andrews and eventually to Geneva. I am very grateful for their unwavering faith in me. I would also like to say thank you to my brother Michan for his advice and great sense of irony. I would like to thank Tiger, Rex and Jenny, the girls of 123, and the lads from Leeds for their invaluable friendship. I also want to thank the friends I have made over the last few
years in Geneva; Joe, Nacho, David, Bene, Gaetan and Sara, I have been lucky to have met all of you. Finally I would like to thank Théo for his support and encouragement.
I acknowledge financial support from the Swiss National Science Foun-dation (200021-146995) and the financial and academic support of the University of Geneva.
Bibliography
[1] K. v. Klitzing, G. Dorda, and M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance”, Physical Review Letters 45, 494–497 (1980).
[2] Y. A. Bychkov and E. I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, JETP Letters39, 78–81 (1984).
[3] J. Nitta et al., “Gate Control of Spin-Orbit Interaction in an Inverted In0.53Ga0.47As/In0.48Al0.48As heterostructure ”, Physical Review Let-ters 78, 1335–1338 (1997).
[4] S. Datta and B. Das, “Electronic analog of the electro-optic modula-tor”, Applied Physics Letters56, 665 (1990).
[5] H. C. Koo et al., “Control of Spin Precession in a Spin-Injected Field Effect Transistor”, Science325, 1515–1518 (2009).
[6] P. D. C. King et al., “Quasiparticle dynamics and spin-orbital texture of the SrTiO3two-dimensional electron gas.”, Nature Communications 5, 3414 (2014).
[7] P. Zubko et al., “Interface Physics in Complex Oxide Heterostructures”, Annual Review of Condensed Matter Physics2, 141–165 (2011).
[8] J. Mannhart and D. G. Schlom, “Oxide Interfaces–An Opportunity for Electronics”, Science327, 1607–1611 (2010).
[9] Y. Tokura et al., “Filling dependence of electronic properties on the verge of metal–Mott-insulator transition in Sr1 xLaxTiO3”, Physical Review Letters70, 2126–2129 (1993).
[10] A. Ohtomo et al., “Artificial charge-modulation in atomic-scale per-ovskite titanate superlattices”, Nature419, 378–380 (2002).
[11] A. Ohtomo and H. Y. Hwang, “A high-mobility electron gas at the LaAlO3/SrTiO3heterointerface.”, Nature427, 423–426 (2004).
[12] N. Ogawa et al., “Nonlinear optical detection of a ferromagnetic state at the single interface of an antiferromagnetic LaMnO3/SrMnO3
double layer”, Physical Review B78, 212409 (2008).
[13] N. Reyren et al., “Superconducting interfaces between insulating oxides.”, Science317, 1196–1199 (2007).
[14] A. Brinkman et al., “Magnetic effects at the interface between non-magnetic oxides.”, Nature Materials6, 493–496 (2007).
[15] A. D. Caviglia et al., “Tunable Rashba Spin-Orbit Interaction at Oxide Interfaces”, Physical Review Letters104, 126803 (2010).
[16] S. Thiel et al., “Tunable quasi-two-dimensional electron gases in oxide heterostructures.”, Science313, 1942–1945 (2006).
[17] Y. Segal et al., “X-ray photoemission studies of the metal-insulator transition in LaAlO3/SrTiO3 structures grown by molecular beam epitaxy”, Physical Review B80, 241107(R) (2009).
[18] E. Slooten et al., “Hard x-ray photoemission and density functional theory study of the internal electric field in SrTiO3/LaAlO3 oxide heterostructures”, Physical Review B87, 085128 (2013).
[19] G. Berner et al., “Direct k-Space Mapping of the Electronic Structure in an Oxide-Oxide Interface”, Physical Review Letters110, 247601 (2013).
[20] P. R. Willmott et al., “Structural Basis for the Conducting Interface betweenLaAlO3andSrTiO3”, Physical Review Letters99, 155502 (2007).
[21] G. Herranz et al., “High Mobility in LaAlO3/SrTiO3Heterostructures:
Origin, Dimensionality, and Perspectives”, Physical Review Letters 98, 216803 (2007).
[22] N. Nakagawa, H. Y. Hwang, and D. A. Muller, “Why some interfaces cannot be sharp”, Nature Materials5, 204–209 (2006).
[23] X. Lin et al., “Critical Doping for the Onset of a Two-Band Super-conducting Ground State in SrTiO3 ”, Physical Review Letters112, 207002 (2014).
[24] D. W. Reagor and V. Y. Butko, “Highly conductive nanolayers on strontium titanate produced by preferential ion-beam etching.”, Na-ture Materials4, 593–596 (2005).
BIBLIOGRAPHY
[25] K. Ueno et al., “Electric-field-induced superconductivity in an insula-tor.”, Nature materials7, 855–858 (2008).
[26] M. D’Angelo et al., “Hydrogen-Induced Surface Metallization of SrTiO3 (001)”, Physical Review Letters108, 116802 (2012).
[27] Y. Chen et al., “Metallic and insulating interfaces of amorphous SrTiO3-based oxide heterostructures.”, Nano Letters11, 3774–3778 (2011).
[28] Y. Z. Chen et al., “A high-mobility two-dimensional electron gas at the spinel/perovskite interface of -Al2O3/SrTiO3”, Nature Commu-nications 4, 1371 (2013).
[29] D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin, “Common Fermi-liquid origin of T2resistivity and superconductivity in n-type SrTiO3”, Physical Review B84, 205111 (2011).
[30] X. Lin et al., “Fermi Surface of the Most Dilute Superconductor”, Physical Review X 3, 021002 (2013).
[31] X. Lin, B. Fauque, and K. Behnia, “Scalable T2resistivity in a small single-component Fermi surface”, Science 349, 945–948 (2015).
[32] A. Spinelli et al., “Electronic transport in doped SrTiO3: Conduction mechanisms and potential applications”, Physical Review B81, 155110 (2010).
[33] J. Schooley et al., “Dependence of the Superconducting Transition Temperature on Carrier Concentration in Semiconducting SrTiO3”, Physical Review Letters 14, 305–307 (1965).
[34] R. C. Neville, “Permittivity of Strontium Titanate”, Journal of Applied Physics 43, 2124 (1972).
[35] O. Copie et al., “Towards Two-Dimensional Metallic Behavior at LaAlO3/SrTiO3 Interfaces”, Physical Review Letters 102, 216804 (2009).
[36] A. D. Caviglia et al., “Electric field control of the LaAlO3/SrTiO3
interface ground state.”, Nature 456, 624–627 (2008).
[37] A. Joshua et al., “A universal critical density underlying the physics of electrons at the LaAlO3/SrTiO3 interface.”, Nature Communications 3, 1129 (2012).
[38] M. Salluzzo et al., “Orbital Reconstruction and the Two-Dimensional Electron Gas at the LaAlO3/SrTiO3 Interface”, Physical Review Letters102, 166804 (2009).
[39] M. Stengel, “First-Principles Modeling of Electrostatically Doped Perovskite Systems”, Physical Review Letters106, 136803 (2011).
[40] Z. Zhong, A. Tóth, and K. Held, “Theory of spin-orbit coupling at LaAlO3/SrTiO3 interfaces and SrTiO3 surfaces”, Physical Review B 87, 161102 (2013).
[41] Z. Popović, S. Satpathy, and R. Martin, “Origin of the Two-Dimensional Electron Gas Carrier Density at the LaAlO3on SrTiO3
Interface”, Physical Review Letters101, 256801 (2008).
[42] G. Berner et al., “LaAlO3/SrTiO3oxide heterostructures studied by resonant inelastic x-ray scattering”, Physical Review B82, 241405 (2010).
[43] K. Zhou et al., “Localized and delocalized Ti 3d carriers in LaAlO3/SrTiO3 superlattices revealed by resonant inelastic x-ray scattering”, Physical Review B83, 201402(R) (2011).
[44] C. Cancellieri et al., “Interface Fermi States of LaAlO3/SrTiO3and Re-lated Heterostructures”, Physical Review Letters110, 137601 (2013).
[45] C. Cancellieri et al., “Doping-dependent band structure of LaAlO3/SrTiO3
interfaces by soft x-ray polarization-controlled resonant angle-resolved photoemission”, Physical Review B89, 121412 (2014).
[46] W. Meevasana et al., “Creation and control of a two-dimensional electron liquid at the bare SrTiO3 surface.”, Nature Materials 10, 114–118 (2011).
[47] A. F. Santander-Syro et al., “Two-dimensional electron gas with universal subbands at the surface of SrTiO3.”, Nature469, 189–193 (2011).
[48] Y. Kim, R. M. Lutchyn, and C. Nayak, “Origin and transport signa-tures of spin-orbit interactions in one- and two-dimensional SrTiO3 -based heterostructures”, Physical Review B87, 245121 (2013).
[49] A. Fête et al., “Growth-induced electron mobility enhancement at the LaAlO3/SrTiO3interface”, Applied Physics Letters106, 051604 (2015).
[50] P. Delugas et al., “Intrinsic origin of two-dimensional electron gas at the (001) surface of SrTiO3”, Physical Review B91, 115315 (2015).
[51] N. Plumb et al., “Mixed Dimensionality of Confined Conducting Electrons in the Surface Region of SrTiO3”, Physical Review Letters 113, 086801 (2014).
[52] Y Aiura et al., “Photoemission study of the metallic state of lightly electron-doped SrTiO3”, Surface Science515, 61–74 (2002).
[53] S. H. Wemple, “Some Transport Properties of Oxygen-Deficient Single-Crystal Potassium Tantalate (KTaO3)”, Physical Review137, A1575–
A1582 (1965).
BIBLIOGRAPHY
[54] K. Ueno et al., “Discovery of superconductivity in KTaO3by electro-static carrier doping”, Nature Nanotechnology6, 408–412 (2011).
[55] P. D. C. King et al., “Subband Structure of a Two-Dimensional Electron Gas Formed at the Polar Surface of the Strong Spin-Orbit Perovskite KTaO3”, Physical Review Letters108, 117602 (2012).
[56] A. F. Santander-Syro et al., “Orbital symmetry reconstruction and strong mass renormalization in the two-dimensional electron gas at the surface of KTaO3”, Physical Review B86, 121107(R) (2012).
[57] P. O. Gartland and B. J. Slagsvold, “Transitions conserving parallel momentum in photoemission from the (111) face of copper”, Physical Review B12, 4047–4058 (1975).
[58] R. Claessen et al., “Complete band-structure determination of the quasi-two-dimensional Fermi-liquid reference compound TiTe2”, Phys-ical Review B54, 2453–2465 (1996).
[59] Z.-X. Shen et al., “Anomalously large gap anisotropy in the a-b plane of Bi2Sr2CaCu2O8+ ”, Physical Review Letters70, 1553–1556 (1993).
[60] H. Ding et al., “Momentum Dependence of the Superconducting Gap in Bi2Sr2CaCu2O8”, Physical Review Letters74, 2784–2787 (1995).
[61] A. Damascelli, Z. Hussain, and Z.-X. Shen, “Angle-resolved photoe-mission studies of the cuprate superconductors”, Reviews of Modern Physics 75, 473–541 (2003).
[62] Y. L. Chen et al., “Experimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3”, Science325, 178–181 (2009).
[63] D. Hsieh et al., “A topological Dirac insulator in a quantum spin Hall phase”, Nature452, 970–974 (2008).
[64] A. A. Soluyanov et al., “Type-II Weyl semimetals”, Nature527, 495–
498 (2015).
[65] L. Yang et al., “Discovery of a Weyl Semimetal in non-Centrosymmetric Compound TaAs”, arXiv:1507.00521 (2015).
[66] A. Tamai et al., “Fermi arcs and their topological character in the candidate type-II Weyl semimetal MoTe2”, arXiv:1604.08228 (2016).
[67] G. D. Mahan, “Theory of Photoemission in Simple Metals”, Physical Review B2, 4334–4350 (1970).
[68] C. Berthod,Applications of the Many-Body Formalism in Condensed-Matter Physics, Lecture Notes (University of Geneva, 2012).
[69] S. Hüfner, Photoelectron Spectroscopy, Advanced Texts in Physics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2003).
[70] M. P. Seah and W. A. Dench, “Quantitative electron spectroscopy of surfaces: A standard data base for electron inelastic mean free paths in solids”, Surface and Interface Analysis1, 2–11 (1979).
[71] T. Okuda and A. Kimura, “Spin- and Angle-Resolved Photoemission of Strongly Spin–Orbit Coupled Systems”, Journal of the Physical Society of Japan82, 021002 (2013).
[72] S. G. Louie et al., “Periodic Oscillations of the Frequency-Dependent Photoelectric Cross Sections of Surface States: Theory and Experi-ment”, Physical Review Letters44, 549–553 (1980).
[73] J. J. Paggel, T. Miller, and T.-C. Chiang, “Angle-resolved photoemis-sion from atomic-layer-resolved quantum well states in Ag/Fe(100)”, Journal of Electron Spectroscopy and Related Phenomena101-103, 271–275 (1999).
[74] W. Meevasana et al., “Strong energy-momentum dispersion of phonon-dressed carriers in the lightly doped band insulator SrTiO3”, New Journal of Physics12, 023004 (2010).
[75] A. D. Caviglia et al., “Two-Dimensional Quantum Oscillations of the Conductance at LaAlO3/SrTiO3 Interfaces”, Physical Review Letters 105, 236802 (2010).
[76] H. Boschker et al., “Electron-phonon Coupling and the Supercon-ducting Phase Diagram of the LaAlO3-SrTiO3Interface”, Scientific Reports5, 12309 (2015).
[77] Z. Wang et al., “Anisotropic two-dimensional electron gas at SrTiO3
(110).”, Proceedings of the National Academy of Sciences of the United States of America111, 3933–3937 (2014).
[78] S. McKeown Walker et al., “Carrier-Density Control of the SrTiO3
(001) Surface 2D Electron Gas studied by ARPES”, Advanced Mate-rials27, 3894–3899 (2015).
[79] H. O. Jeschke, J. Shen, and R. Valenti, “Localized versus itinerant states created by multiple oxygen vacancies in SrTiO3”, New Journal of Physics17, 023034 (2015).
[80] C. Lin and A. A. Demkov, “Electron Correlation in Oxygen Vacancy in SrTiO3”, Physical Review Letters111, 217601 (2013).
[81] F. Lechermann et al., “Electron correlation and magnetism at the LaAlO3/SrTiO3 interface: A DFT+DMFT investigation”, Physical Review B90, 085125 (2014).
[82] M. Sing et al., “Profiling the interface electron gas of LaAlO3/SrTiO3
heterostructures with hard x-ray photoelectron spectroscopy”, Physi-cal Review Letters102, 176805 (2009).
BIBLIOGRAPHY
[83] Y. Z. Chen et al., “Room Temperature Formation of High-Mobility Two-Dimensional Electron Gases at Crystalline Complex Oxide In-terfaces”, Advanced Materials 26, 1462–1467 (2014).
[84] M. L. Knotek and P. Feibelman, “Ion Desorption by Core-Hole Auger Decay”, Physical Review Letters40, 964–967 (1978).
[85] M. L. Knotek, “Stimulated desorption from surfaces”, Physics Today 37, 24 (1984).
[86] O. Dulub et al., “Electron-induced oxygen desorption from the TiO2(011) 2⇥1 surface leads to self-organized vacancies.”, Science 317, 1052–1056 (2007).
[87] S. Moser et al., “Tunable Polaronic Conduction in Anatase TiO2”, Physical Review Letters 110, 196403 (2013).
[88] R. Hesse et al., “Comparative study of the modelling of the spectral background of photoelectron spectra with the Shirley and improved Tougaard methods”, Journal of Electron Spectroscopy and Related Phenomena186, 44–53 (2013).
[89] M. Altmeyer et al., “Magnetism, Spin Texture, and In-Gap States:
Atomic Specialization at the Surface of Oxygen-Deficient SrTiO3”, Physical Review Letters 116, 157203 (2016).
[90] S. S. Ghosh and E. Manousakis, “Structure and ferromagnetic in-stability of the oxygen-deficient SrTiO3 surface”, arXiv:1511.07495 (2015).
[91] A. C. Garcia-Castro et al., “Spin texture induced by oxygen vacancies in strontium perovskite (001) surfaces: A theoretical comparison between SrTiO3 and SrHfO3”, Physical Review B93, 045405 (2016).
[92] J. Hemberger et al., “Electric-field-dependent dielectric constant and nonlinear susceptibility in SrTiO3”, Physical Review B 52, 13159–
13162 (1995).
[93] F. Stern, “Iteration methods for calculating self-consistent fields in semiconductor inversion layers”, Journal of Computational Physics6, 56–67 (1970).
[94] P. Blaha et al.,WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties(Karlheinz Schwarz, Techn.
Universität Wien, Austria, 2001).
[95] Z. Zhong, Q. Zhang, and K. Held, “Quantum confinement in perovskite oxide heterostructures: Tight binding instead of a nearly free electron picture”, Physical Review B88, 125401 (2013).
[96] G. Khalsa and A. MacDonald, “Theory of the SrTiO3 surface state two-dimensional electron gas”, Physical Review B 86, 125121 (2012).
[97] P. Moetakef et al., “Quantum oscillations from a two-dimensional electron gas at a Mott/band insulator interface”, Applied Physics Letters101, 151604 (2012).
[98] A. McCollam et al., “Quantum oscillations and subband properties of the two-dimensional electron gas at the LaAlO3/SrTiO3interface”, APL Materials2, 022102 (2014).
[99] M. Kim et al., “Fermi surface and superconductivity in low-density high-mobility -doped SrTiO3”, Physical Review Letters107, 106801 (2011).
[100] S Gariglio, A Fête, and J.-M. Triscone, “Electron confinement at the LaAlO3/SrTiO3interface”, Journal of Physics: Condensed Matter27, 283201 (2015).
[101] S. McKeown Walker et al., “Absence of Giant Spin Splitting in the
[101] S. McKeown Walker et al., “Absence of Giant Spin Splitting in the