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Spin-orbit coupling in iridates Zwartsenberg, Berend 2019

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Spin-Orbit Coupling in IridatesbyBerend ZwartsenbergBSc. Physics and Astronomy, University of Amsterdam, 2011MSc. Physics, University of Amsterdam, 2013a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Physics)The University of British Columbia(Vancouver)December 2019c© Berend Zwartsenberg, 2019The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Spin-Orbit Coupling in Iridatessubmitted by Berend Zwartsenberg in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in Physics.Examining Committee:Andrea Damascelli, University of British Columbia (Physics and Astron-omy)SupervisorDouglas Bonn, University of British Columbia (Physics and Astronomy)Supervisory Committee MemberMona Berciu, University of British Columbia (Physics and Astronomy)University ExaminerRoman Krems, University of British Columbia (Chemistry)University ExaminerChangyoung Kim, Seoul National University (Physics and Astronomy)External ExaminerAdditional Supervisory Committee Members:Marcel Franz, University of British Columbia (Physics and Astronomy)Supervisory Committee MemberGary Hinshaw, University of British Columbia (Physics and Astronomy)Supervisory Committee MemberiiAbstractTransition-metal oxides (TMOs) are a widely studied class of materialswith fascinating electronic properties and a great potential for applications.Sr2IrO4 is such a TMO, with a partially filled 5d t2g shell. Given the re-duced Coulomb interactions in these extended 5d orbitals, the insulatingstate in Sr2IrO4 is quite unexpected. To explain this state, it has been pro-posed that spin-orbit coupling (SOC) entangles the t2g states into a filledjeff = 3/2 state and a half-filled jeff = 1/2 state, in which a smaller Coulombinteraction can open a gap. This new scheme extends filling and bandwidth,the canonical control parameters for metal-insulator transitions, to the rel-ativistic domain. Naturally the question arises whether in this case, SOCcan in fact drive such a transition. In order to address this question, wehave studied the behaviour of Sr2IrO4 when substituting Ir for Ru or Rh.Both of these elements change the electronic structure and drive the systeminto a metallic state. A careful analysis of filling, bandwidth, and SOC,demonstrates that only SOC can satisfactorily explain the transition. Thisestablishes the importance of SOC in the description of metal-insulator tran-sitions and stabilizing the insulating state in Sr2IrO4.It has furthermore been proposed that the jeff = 1/2 model in Sr2IrO4is an analogue to the superconducting cuprates, realizing a two-dimensionalpseudo-spin 1/2 model. We test this directly by measuring the spin-orbitalentanglement using circularly polarized spin-ARPES. Our results indicatethat there is a drastic change in the spin-orbital entanglement throughoutthe Brillouin zone, implying that Sr2IrO4 can not simply be described asa pseudo-spin 1/2 insulator, casting doubt on direct comparisons to theiiicuprate superconductors. We thus find that the insulating ground state inSr2IrO4 is mediated by SOC, however, SOC is not strong enough to fullydisentangle the jeff = 1/2 state, requiring that Sr2IrO4 is described as amulti-orbital relativistic Mott insulator.ivLay SummaryThis thesis studies the behaviour of electrons in the crystalline compoundSr2IrO4. Generally, crystals can subdivided into two classes; those that doconduct electricity (metals), and those that do not (insulators); Sr2IrO4 be-longs to the latter of these two classes. In this work, we observe a transitioninto a metallic state, by changing the coupling between spin and momentum.This proves that the insulating properties in Sr2IrO4 derive from this cou-pling and is the first demonstration of a transition into a metallic state bychanging this parameter. It has further been proposed that Sr2IrO4 is a sys-tem that models cuprate superconductors. A superconductor is a materialthat conducts electricity without any loss. Sr2IrO4 does not superconduct,but comparing to it can help identify requirements for superconductivity.The experiments in this work show that Sr2IrO4 is in fact different on twocrucial aspects, highlighting their importance to superconductivity.vPrefaceThe work in this thesis is a representation of my scientific activity duringmy time as a graduate student at UBC. For all the work presented in thisthesis I was the primary responsible investigator. However, the nature ofexperimental physics dictates that none of the work presented here wasdone alone, in particular there has been incidental involvement from all themembers of our research group and technical staff. For the experimentalchapters, I will detail the contributions I and others have made below.Chapter 3 – Spin-orbit Controlled Metal-Insulator Transitionin Sr2IrO4This chapter investigates the metal-insulator transition in Sr2IrO4 upon sub-stitution of Rh and Ru. The work is a combination of experimental work(ARPES) and theoretical modelling (DFT, TB, matrix element analysis).The experiment was conceived by A. Damascelli and me. Preliminary stud-ies (that are not presented in this thesis) were done in the lab at UBC, forwhich all of our research group is responsible. The experimental work wasdone at multiple synchrotron radiation facilities. Planning and experimentdesign was done by me. A first set of data was taken at the Swiss LightSource in Villegen, Switzerland, together with E. Razzoli, M. Michiardi,with assistance from M.Shi and N. Xu. A second set of data was taken atthe Advanced Light Source in Berkeley, California together with E. Razzoliwith support from J. D. Denlinger. Sr2IrO4 samples were provided by thegroup of H. Takagi. The Rh doped samples were provided by B.J. Kim atMax Planck Institute in Stuttgart, samples that were grown by K. Ueda andviJ. Bertinshaw. Ru doped samples were provided by S. Calder and G. Cao atOakridge National Laboratory. Data analysis was done by me, with inputfrom E. Razzoli, R. P. Day and A. Damascelli. The numerical modelling wasdone by me, where the DFT calculations were supervised by I.S. Elfimov,the TB calculations and matrix element analysis had extensive inputs fromR. P. Day. The interpretation of the work has involved inputs from manypeople, but is mostly done by me, R. P. Day, I. S. Elfimov and A. Dama-scelli. Andrea Damascelli supervised the project. Andrea Damascelli wasresponsible for overall project direction, planning and management. Largeparts of the chapter form the basis for a manuscript which was written byme, R. P. Day and A. Damascelli. This manuscript has been accepted forpublication in Nature Physics.Chapter 4 – Spin and kz Resolved ARPES on Sr2IrO4This chapter discusses the spin-orbital entanglement of the states in Sr2IrO4.The work presented in this chapter is a combination of experimental (spin-ARPES) and numerical work. The experiment was conceived by A. Dama-scelli and me. The experiments were performed at the Elettra Syncrotronein Trieste, Italy. These experiments were done over multiple beam times.While I was the primary responsible for the planning and schedule, the prac-tical execution involved the help of R. P. Day, M. Michiardi, E. Razzoli, M.Schneider, S. Zhdanovic, M. X. Na, and G. Levy. Our experiments weresupported by C. Bigi, J. Fuji and I. Vobornik. The samples for these exper-iments came from the group of H. Takagi. The data analysis and numericalsimulations were done by me, with input from R. P. Day. Andrea Damas-celli supervised the project. Andrea Damascelli was responsible for overallproject direction, planning and management. These results are currentlybeing prepared for publication.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . xvList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . .xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The jeff = 1/2 state . . . . . . . . . . . . . . . . . . . 31.1.2 A pseudo-spin 1/2 model . . . . . . . . . . . . . . . . 51.1.3 This work . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Electronic structure . . . . . . . . . . . . . . . . . . . 81.2.2 Mott insulators . . . . . . . . . . . . . . . . . . . . . . 10viii1.2.3 The metal-insulator transition . . . . . . . . . . . . . 111.3 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 A simple description of spin-orbit coupling . . . . . . 131.3.2 The Dirac equation . . . . . . . . . . . . . . . . . . . . 141.4 SOC and angular momentum in solids . . . . . . . . . . . . . 151.4.1 SOC in atoms . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 SOC in the presence of a crystal field . . . . . . . . . 171.4.3 The jeff = 1/2 State . . . . . . . . . . . . . . . . . . . 181.4.4 Tetragonal splitting . . . . . . . . . . . . . . . . . . . 192 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Ab initio calculations . . . . . . . . . . . . . . . . . . . . . . . 222.1.1 Density functional theory . . . . . . . . . . . . . . . . 222.1.2 Wannier functions . . . . . . . . . . . . . . . . . . . . 262.1.3 The tight binding approach . . . . . . . . . . . . . . . 302.2 Band unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.1 A simple example . . . . . . . . . . . . . . . . . . . . 322.2.2 Projecting Bloch phases . . . . . . . . . . . . . . . . . 362.2.3 Impurity distributions . . . . . . . . . . . . . . . . . . 382.3 Angle-resolved photoelectron spectroscopy . . . . . . . . . . . 402.3.1 Theory of photoemission . . . . . . . . . . . . . . . . . 402.3.2 A photoemission experiment in practice . . . . . . . . 442.3.3 The photoemission dipole matrix element . . . . . . . 492.4 Spin-ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.4.1 A practical spin-ARPES experiment . . . . . . . . . . 512.4.2 Circularly polarized spin ARPES . . . . . . . . . . . . 523 Spin-orbit Controlled Metal-Insulator Transition in Sr2IrO4 583.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 The MIT in Rh and Ru substituted Sr2IrO4 . . . . . . . . . . 603.3 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . 643.3.1 Spin-orbit mixing and the impurity potential . . . . . 663.3.2 Supercell calculations and band unfolding . . . . . . . 69ix3.4 Experimental observation of SOC . . . . . . . . . . . . . . . . 733.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.A.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 783.A.2 Observation of SOC through the dipole transition ma-trix element . . . . . . . . . . . . . . . . . . . . . . . . 793.A.3 Tight binding and matrix element modelling . . . . . 813.A.4 Orbital weight for a two-site model with impurity po-tential . . . . . . . . . . . . . . . . . . . . . . . . . . . 834 Spin and kz Resolved ARPES on Sr2IrO4 . . . . . . . . . . 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 kz dispersion in Sr2IrO4 . . . . . . . . . . . . . . . . . . . . . 874.2.1 Body centered tetragonal structure . . . . . . . . . . . 874.2.2 Constant energy maps . . . . . . . . . . . . . . . . . . 894.2.3 Γ and X state dispersions. . . . . . . . . . . . . . . . . 924.3 Circularly polarized spin-ARPES . . . . . . . . . . . . . . . . 954.3.1 Interpretation of CPSA results for Sr2IrO4 . . . . . . 964.3.2 k dependent CPSA . . . . . . . . . . . . . . . . . . . . 994.3.3 kz dependent CPSA . . . . . . . . . . . . . . . . . . . 994.3.4 Slab Simulation of Sr2IrO4 . . . . . . . . . . . . . . . 1034.3.5 Magnitude of the kz hopping terms. . . . . . . . . . . 1084.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1 Spin-orbit controlled metal insulator transition in Sr2IrO4 . . 1115.2 Spin- and kz-resolved photoemission on Sr2IrO4 . . . . . . . . 1135.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A Overview of expectation values spin-orbit entangled states 134xList of TablesTable A.1 Table of expectation values for the spin-orbit coupled ` =2 states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Table A.2 Table of expectation values for the spin-orbit coupled ` =1 states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Table A.3 Table of expectation values for the spin-orbit coupled t2g(`eff = 1) states . . . . . . . . . . . . . . . . . . . . . . . . 137Table A.4 Construction of the jeff states in terms of the t2g orbitals . 138Table A.5 Table of expectation values for the spin-orbit coupled dxzand dyz states . . . . . . . . . . . . . . . . . . . . . . . . . 139xiList of FiguresFigure 1.1 Overview of different kinds of insulators . . . . . . . . . . 2Figure 1.2 Schematic representation of the jeff = 1/2 states . . . . . 5Figure 1.3 Overview of the physical and electronic structure of tran-sition metal oxides . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.4 Overview of the spin-orbital entanglement as function ofλ, 10Dq and δtet . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.1 The correspondence between a Wannier and a Bloch func-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.2 Example Wannier calculation with generated real-spaceorbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.3 Hybridization in a diatomic molecule . . . . . . . . . . . . 33Figure 2.4 Band unfolding for a one-dimensional chain of diatomicmolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.5 Example band unfolding calculation for a 20-site one-dimensional chain of atoms . . . . . . . . . . . . . . . . . 39Figure 2.6 Probability distribution for finding a number of impuritiesgiven system size . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.7 Schematic representation of an ARPES experiment . . . . 47Figure 2.8 Example ARPES measurement of Sr2RuO4 . . . . . . . . 48Figure 2.9 Illustration of the geometry used to calculate the photoe-mission matrixelement . . . . . . . . . . . . . . . . . . . . 50Figure 3.1 Dependence of the MIT on Rh and Ru substitution . . . 61xiiFigure 3.2 ARPES linewidth evolution with substitution of Ru andRh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.3 Observation of an impurity potential for Ru and Rh inSr2IrO4 from DFT . . . . . . . . . . . . . . . . . . . . . . 66Figure 3.4 Influence of the impurity potential on spin-orbit couplingin the valence band demonstrated in a 2-atom toy model 67Figure 3.5 Overview of the supercell calculations describing spin-orbit coupling dilution . . . . . . . . . . . . . . . . . . . . 71Figure 3.6 Reduction of SOC through supercell analysis for Sr2IrO4 72Figure 3.7 Observation of the reduction of SOC via the ARPESdipole matrix element . . . . . . . . . . . . . . . . . . . . 74Figure 3.8 Inference of the value of spin-orbit coupling through mod-elling of the matrix element . . . . . . . . . . . . . . . . . 75Figure 3.9 Orbital mixing between t2g orbitals and signatures ob-served in the matrix element . . . . . . . . . . . . . . . . 80Figure 3.10 Overview of the construction of Wannier models . . . . . 82Figure 4.1 Constant energy maps for a two-atom model in the bodycentred tetragonal structure . . . . . . . . . . . . . . . . . 88Figure 4.2 Overview of photon energy dependent results in Sr2IrO4 . 90Figure 4.3 Fits to kz dispersion along the Γ and X point for Sr2IrO4 93Figure 4.4 EDCs for both polarizations at Γ and X . . . . . . . . . . 94Figure 4.5 kz dependent tight binding model of Sr2IrO4 with 〈L · S〉projections . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 4.6 High binding energy CPSA measurements . . . . . . . . . 97Figure 4.7 k-dependent CPSA measurements . . . . . . . . . . . . . 100Figure 4.8 Photon energy dependent CPSA measurements . . . . . . 102Figure 4.9 CPSA repeatability . . . . . . . . . . . . . . . . . . . . . 103Figure 4.10 Slab model for Sr2IrO4 including antiferromagnetic (AFM)order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 4.11 CPSA simulations for different values of λSOC . . . . . . 107Figure 4.12 CPSA simulations for different values of the photoelectronmean free path . . . . . . . . . . . . . . . . . . . . . . . . 108xiiiFigure 4.13 CPSA simulations with enlarged kz hopping elements . . 109xivList of AbbreviationsARPES angle-resolved photoelectron spectroscopyAFM antiferromagneticBC bandwidth controlledBCT body centred tetragonalBZ Brillouin zoneCE constant energyCLS Canadian Light SourceCPSA circularly polarized spin-ARPESDFT density functional theoryDOS density of statesEDC energy distribution curveFC filling controlledGGA generalized gradient approximationHK Hohenberg-KohnKS Kohn-ShamLAPW linear augmented plane wavexvLDA local density approximationMDC momentum distribution curveMIT metal-insulator transitionQMSC Quantum Materials Spectroscopy CenterREXS resonant elastic x-ray scatteringRIXS resonant inelastic x-ray scatteringSOC spin-orbit couplingSTEM scanning transmission electron microscopySTM scanning tunnelling microscopySTS scanning tunnelling microscopyTB tight bindingTM transition metalTMO transition-metal oxideUV ultravioletVLEED very low energy electron diffraction (used to describe a type ofspin-detector)xviList of Symbols10Dq the crystal field splittingEF the Fermi energyε impurity potential strengtheg subset of the d orbitals (dz2 , dx2−y2)Γ the high symmetry point k = (0, 0, 0)Ipi pi-polarization, in the same plane as the slit (xˆ+ zˆ)Iσ σ-polarization, perpendicular to the slit direction (yˆ)jeff effective coupled angular momentumλ spin-orbit coupling strengthM the high symmetry point k = (pi/2a, pi/2a, 0), see XN the high symmetry point k = (pi/a, pi/a, 0), see Xt hopping strength (kinetic energy)t2g subset of the d orbitals (dxy, dxz, dyz)U Coulomb repulsionx concentration of impuritiesX the high symmetry point k = (pi/a, 0, 0), with xˆ along the TM-Obond, and a the TM-TM spacingZ the high symmetry point k = (0, 0, pi/c), see XxviiAcknowledgmentsAlthough this work has only one author, it would not have been possibleto write this thesis without a lot of help from the people around me. First,all the people in our research group (in approximate order of their time atUBC): Ilya Elfimov, Giorgio Levy, Sergey Zhdanovic, Art Mills, LudivineChauviere, Michael Schneider, Eduardo da Silva Netto, Elia Rampi, MartaZonno, Pascal Nigge, Alex Sheyerman, Fabio Boschini, Amy Qu, Ryan Day,Elia Razzoli, Matteo Michiardi, Ketty Na, Sean Kung, Christopher Gui-tierez, Danilo Ku¨hn, Sydney Dufresne and Cissy Suen. You have made mystay in Vancouver great, and you have all supported me countless times,both experimentally and intellectually.Thanks to technical staff Pinder Dosanjh and Doug Wong for givingtechnical design advice and helping us to fix all the things we break.An extra thank you goes to Ryan Day for helping me write up the SOCdilution paper to a level I could have never achieved by myself. You havebeen a large part of all of the projects during my PhD and I have reallyenjoyed your presence in it.Charlotte, Gijs, Sophie, thank you for being there for me, helping mekeep my confidence and listening to all my stories. Thank you Gabo, forbeing such a great and patient girlfriend and supporting me through thisproject the whole way. I would also like to say a big thank you to all myfriends (Amsterdam, Vancouver, Seattle) for their support. I feel incrediblylucky to have all of you in my life.Thank you to all the beamline staff that have helped me over the years,Nan Xu, Jonathan Denlinger, Chiara Bigi, Jun Fuji, Ivana Vobornik, TorxviiiPedersen and Sergey Gorovikov.Without samples to measure, none of the work here would have beenpossible and for that I would like to thank Hidenori Takagi, B.J. Kim, JoelBertinshaw, Kentaro Ueda, Guixin Cao and Stuart Calder.Thanks to Gabo Ailstock and Danika Wheeler for showing me the rightway to use the comma key, and patiently proofreading this thesis.A big thank you to my committee (supervisory and examining) for takingthe time and effort to read and help improve my thesis.I would like to thank my MSc supervisor Mark Golden for introducingme to Andrea, and quickly breaking the ice by letting Andrea know I waslooking for a PhD position.Finally I would like to thank my supervisor Andrea Damascelli for givingme the opportunity to move to this beautiful place and allow me to do aPhD. You have an amazing vision when it comes to understanding physicsand I have learned a lot from you over the last five years.xixChapter 1Introduction1.1 MaterialsThe compounds studied in this thesis are most commonly referred to astransition-metal oxides (TMOs). The physics describing these materialsarises from the interplay between the electronic states belonging to thetransition-metal and oxygen ions. Transition-metal oxides are materials thathave been studied intensively in contemporary physics for a multitude of rea-sons. This includes that their constituents are abundant in the earth’s crustand therefore inexpensive and accessible. Furthermore, these materials havegreat potential for possible applications [1, 2], as they host a plethora of in-teresting physics such as magnetism [3], superconductivity [4, 5], Mott andcharge transfer insulating behaviour [6, 7], charge and orbital ordering [8–10], and the colossal magnetoresistance effect [11–13]. Due to a combinationof a partially filled d shell and strong electron-electron interactions, TMOshost a large variety of physical phases, in a class of compounds that appearssimilar based on their constituents and structure. However, the partiallyfilled d shell and strong electron-electron interactions is also what makesthese compounds incredibly challenging to understand. A key challenge inthe field is the exploration of correlated insulating phases (referred to as“Mott-insulating” phases), where these electron-electron interactions causethe localization of charge carriers. More so, the transition from insulating1Figure 1.1: Overview of different kinds of insulators. A one-dimensional material is schematically depicted as a series of lat-tice sites (black horizontal lines) with a spin occupying each siteindicated as an arrow (a,b) Trivial corner cases of insulators, in(a) the lattice is empty, so no spins can be transported. In (b)the lattice is completely filled and the Pauli exclusion principleprevents movement of the spins. (c) In the case the system ishalf filled, charges can move if the transition probability t ishigh enough to overcome the Coulomb potential for double oc-cupation U . If U outweighs the effects of t the system is a Mottinsulator.to metallic phase is an important topic, as the aforementioned phenomenaoften arise in close vicinity to these transitions. The mechanisms and con-trol parameters describing these phases and transitions are therefore of greatinterest to the field.A description of different insulating phase is depicted in Fig. 1.1. Twotrivial corner cases of a completely empty and filled system are shown inFig. 1.1a and b. In the former case no excitations can be made because noelectrons are available, the latter case is insulating because such excitationsare impossible due to the Pauli exclusion principle; both cases are insulat-ing. In the intermediate case, where the system is exactly half filled, aninteresting scenario arises. Electrons can move from one site to the next, asindicated by the hopping transition rate t, but whether or not the system2is insulating depends on the Coulomb repulsion U related to two electronsoccupying the same site. This system is referred to as the Hubbard model,and if the effect of U is large enough to suppress electron movement (U > t),the system is a Mott insulator.The cuprate superconductors are believed to be well described by a halffilled Hubbard model on a square, two-dimensional lattice. A great effort hastherefore been made to discover new materials that can be described withthe same basic Hamiltonian. This search has been primarily focussed onmaterials with 3d transition metals, although 4d and 5d compounds have alsobeen studied. In these latter categories, the discovery of superconductivityin Sr2RuO4 was made [14], but also Sr2RhO4 [15] and Sr2IrO4 [16, 17] gainedattention. Although these compounds are close to their 3d counterparts inthe periodic table, properties of 4d and 5d materials are markedly differentfrom those with 3d elements. The 4d and 5d orbitals are much more extendedthan 3d’s, which has a few effects. Firstly, since the electrons are more spreadout, the integrals determining the Coulomb and exchange interactions aresignificantly reduced. Secondly, the bandwidth is larger due to the increasedorbital size, which increases hopping between orbitals [16, 17].Because the itinerancy of the electrons increases and the Coulomb in-teraction decreases in the 4d and 5d orbitals, these systems are expected toshow more metallic behaviour. A metallic state is indeed found for the 4dmaterials Sr2RhO4 and Sr2RuO4 [14, 15]. Sr2IrO4 on the other hand, wasfound to be insulating [16], which was an unexpected result [16, 17], as theelectronic repulsion should not be sufficient to open a Mott gap.1.1.1 The jeff = 1/2 stateAn explanation for the insulating state was ultimately found by consideringthe effects of spin-orbit coupling (SOC). This interaction is a further im-portant deviation from the 3d properties as SOC is significantly increasedin 4d and 5d compounds [18]. SOC strength (as derived more thoroughly inSection 1.3) is given by the gradient of the potential and therefore increaseswith atomic number Z. If this derivation is done including the effects of3screening, one finds the magnitude scales as Z2 [19]. This Z2 behaviourgives a good agreement with experimental values found for materials inthese groups of the periodic table: ∼ 0.4 eV for Ir [20], ∼ 0.19 eV for Rhand Ru [18, 21, 22], ∼ 0.02− 0.1 eV [23, 24] for Fe.In their seminal work published in 2008 [25], B.J. Kim and colleaguespointed out that the insulating state in Sr2IrO4 could be interpreted byconsidering spin-orbit coupling. As stated, the insulating state in Sr2IrO4had been somewhat of a puzzle, since the increased bandwidth and reducedCoulomb interactions associated with the 5d compounds could not explainwhy the partially filled d band could result in insulating behaviour. Theelectrons in Sr2IrO4 occupy the t2g bands, a subset of the transition metald bands. Their reasoning was that SOC is large enough to couple the t2gorbitals into spin-orbitally entangled states. They suggest the bands cou-ple into a so-called jeff = 1/2 and 3/2 manifold (explained further in Sec-tion 1.4). This splits the bands, reducing the bandwidth, and creates asingly occupied jeff = 1/2 orbital. This half-filled band gives the smaller U(∼ 2 eV [26]) an opportunity to open a gap creating a “jeff = 1/2 Mott in-sulator”. A diagram for this process, schematically indicating the densitiesof states as a function of energy upon including the various energy terms, ispresented in Fig. 1.2a.Convincing evidence for the existence of this state was provided byresonant elastic x-ray scattering (REXS) measurements performed on the Iredge [27]. In particular, the branching ratio of the L2 and L3 at a magneticreflection is taken as indication that the ground state is well described bya jeff = 1/2 state. This result was later questioned, however, when it wasdemonstrated that for a system with magnetic moments lying in the abplane, as is the case for Sr2IrO4, the branching ratio is identically zero,regardless of the amount of spin-orbital entanglement [28]. The authors of[27] later acknowledge this in [29], where they analyze the matrix elementsfor REXS in more detail. Moreover, it was demonstrated in [30], that thejeff = 1/2 is quite close to a collapse. While spin-orbit coupling in Sr2IrO4 isundoubtedly very strong, it seems that a schematic of the density of statesfor Sr2IrO4 may in fact look more like Fig. 1.2b and therefore the precise4Figure 1.2: Schematic overview of the energy diagram of the jeffstates. (a) The idealized jeff = 1/2 model, where the spin-orbitsplitting is large enough that the jeff = 1/2 and the jeff = 3/2manifold no longer overlap. (b) A situation in which the band-width is large enough to cause significant overlap between thejeff = 1/2 and 3/2 states, which results in a ground state inwhich the jeff = 3/2 cannot simply be projected out.nature of the insulating state is still unknown.Alternative scenarios could still hold true, for example a Slater insulator,in which the insulating state is mediated by a band folding causing anti-ferromagnetism has been suggested by Arita and coworkers [26]. This hasbeen ruled out for Na2IrO3 [31] but the Ne´el temperature in Sr2IrO4 istoo high to allow for a similar analysis. Other suggestions that still fitwithin current observations are a multi-orbital Mott system in the presenceof strong spin orbit coupling, such as in Ca2RuO4 [32, 33], or a non-jeff = 1/2relativistic Mott insulator such as CaIrO3 [34].1.1.2 A pseudo-spin 1/2 modelWhile the true ground state may still be elusive, many consequences havebeen proposed if the jeff = 1/2 state were to hold true. It was quickly realizedthat the system would have a large number of properties that are commonly5found in cuprate superconductors [35]. The structure is identical to thatof La2CuO4. Moreover the compound is an anti-ferromagnetic (pseudo-)spin 1/2 model, in which the Coulomb interaction causes an insulating statein the parent compound. Resonant inelastic x-ray scattering (RIXS) datashow a remarkable similarity to inelastic neutron scattering data taken onLa2CuO4 [36]. This data is in good agreement with a Heisenberg model [37],which describes the ground state of the cuprates [38, 39]. More detailed anal-yses indeed predict a superconducting ground state [40, 41], with one keydifference from the cuprates; since the model Hamiltonian for the iridates isequivalent to that of the cuprates with t → −t, the superconductivity ap-pears in the phase diagram on the electron doped side. Attempts were madeto electron dope the system, using La (Sr2−xLaxIrO4 [42]) and oxygen [43].Promising signs were observed on La doped compounds, which showed apseudogap like state in angle-resolved photoelectron spectroscopy (ARPES)[44]. The spatial behaviour of the charge gap observed using STM, displayssimilarities to related cuprate compounds [45]. Moreover, using potassiumto surface dope electrons, signatures of a d-wave gap were seen in bothscanning tunnelling microscopy (STS) [46] and ARPES [47]. Unfortunately,so far no reports of bulk superconductivity have been made. It is possiblethat La is not able to dope enough electrons for the system to become bulksuperconductive. However, the models presented in [40, 41] assume the spin-orbit coupling is strong enough to cause full separation of the jeff = 1/2 andjeff = 3/2 states, an assumption which is explored in more detail in Chap-ter 4. If it turns out that electron doped Sr2IrO4 is in fact not superconduc-tive, this may lead to an interesting perspective on necessary ingredients forsuperconductivity.1.1.3 This workTo provide an overview of the field of relativistic correlated oxides, we canidentify two important open questions. The first question is whether spin-orbit-coupling should be considered as a parameter in the canonical phasediagram of Mott insulators and if Sr2IrO4 is a material that can be placed6on and tuned along this axis. Secondly, whether the jeff = 1/2 model is avalid description of the ground state of Sr2IrO4 and what that means forparallels identified with the superconducting cuprates. We will address boththese points in detail, in Chapter 3 and Chapter 4 respectively. The bodyof this thesis thus considers and attends to key questions in the field ofrelativistic correlated materials. In the remainder of this chapter, some ofthe fundamental physics of these materials will be considered, in particularthe crystal and electronic structure and the effects of spin-orbit coupling.1.2 StructureThe basic building block for all TMOs is the TM-O6 octahedron, depictedin Fig. 1.3a. This is a transition metal ion, surrounded by six oxygen atomsthat can be corner-shared between octahedra. The oxygen atoms provide apotential background, referred to as the crystal field, that breaks the spher-ical symmetry of the transition metal ion and thereby lifts the degeneracyof the d band. Moreover, the oxygen atoms mediate the electron bondingbetween octahedra through hybridization.A large fraction of these materials follow the basic formula of the Per-ovskite ABO3, in which A is an alkali, earth-alkali, or lanthanoid ion and Bis a transition metal. Another important element of the compounds stud-ied here is they are layered. These layered materials are members of theRuddlesden-Popper series [48] having alternating layers of AO and BO2,following the series AN+1BNO3N+1 (schematically shown in Fig. 1.3b). TheA2BO4 (N = 1) series of oxides, studied in this thesis, has a repeating unitof a stack of AO - BO2 - AO, isolating the BO2 layers from each other, toform a mostly two-dimensional compound (Fig. 1.3c for Sr2IrO4). It shouldbe clarified that these materials are not truly two-dimensional, but rathera layered three-dimensional structure. Two-dimensionality in this case ischaracterized by the fact that the electrons move mostly within one of thelayers, and the interaction between layers is relatively small. The A-site ion(Sr in this thesis) can be safely ignored, since it has a full shell and its elec-trons are all strongly bound to the core. The focus with these compounds is7thus on the transition metal (TM) ion seated on the B site of the perovskitestructure (Ir, Rh or Ru in this thesis), which is octahedrally coordinatedwith oxygen atoms. While oxygen hybridizes with the TM ion, which isparticularly true for the 4d and 5d compounds, the physics related to theTM ion can effectively be expressed in terms of TM-d O-p hybrids, whichfollow the symmetry of the original d orbitals.The structure of these materials can be seen in Fig. 1.3c. While the crys-tallographic space group for Sr2RuO4 is I4/mmm, the structure of Sr2RhO4and Sr2IrO4 is I4/acd. The key difference is that the TMO6 octahedra arerotated in a checkerboard pattern in Sr2RhO4 and Sr2IrO4 throughout thecrystal. A similar rotation occurs in Sr2RuO4, but only on the surface [49].1.2.1 Electronic structureThe degeneracy lifting, caused by the ligand oxygen atoms, induces welldefined shifts to particular d orbitals. Representing these orbitals in thebasis of cubic harmonics, it is the orbitals with lobes pointing towards theoxygen ions that gain energy (dx2−y2 and dz2−r2), also called the eg orbitals.The orbitals with lobes pointing in-between the oxygens, the t2g orbitals(dxy, dxz and dyz) are lowered in energy. The energy splitting related to theoxygen is referred to as 10Dq. A further degeneracy breaking can be inducedby a tetragonal distortion, in which octahedra are stretched or compressedalong one axis, further splitting the eg and t2g bands. In the case of alayered 3D material like Sr2IrO4, this symmetry is already broken in theglobal structure and therefore, a degeneracy would be purely accidental.A schematic representation of the induced splitting is plotted in Fig. 1.3d.While a tetragonal splitting indeed exists for all compounds studied in thisthesis, the splitting is much smaller than the bandwidth and therefore thet2g manifold is, for practical purposes, often assumed to be degenerate.We can gain some further insight into the electronic structure of Sr2IrO4,Sr2RhO4 and Sr2RuO4 by counting electrons. We can make the assumptionthat O and Sr assume completely filled shells as O2− and Sr2+. This dictatesthat the charge on Ir, Ru and Rh should be 4+, which implies that Ir and8Figure 1.3: Overview of the structure and electronic basics of thetransition metal oxides studied in this thesis. (a) A single oxy-gen octahedron, the building block that is the foundation forthe physics of the materials in this thesis. (b) A schematicoverview of the Ruddlesden-Popper series. Alternating units ofAO (SrO for Sr2IrO4) and BO2 (IrO2 for Sr2IrO4) are stackedwith varying layer thicknesses. The compound studied in thisthesis is the compound with N = 1, consisting of only singlyconnected layers of IrO2 between two layers of SrO. (c) The unitcell structure for Sr2IrO4. Alternating layers of SrO and IrO2can be seen stacked along the c-axis [50]. (d) Energy diagramexplaining the effect of the crystal field on the d orbitals, thatsplit into different subsets under influence of the octahedral field(10Dq) and the tetragonal splitting (δtet).9Rh are d5, while Ru is d4.For most 3d compounds Hund’s first rule is valid (U − U ′ > 10Dq) andions are high spin. However, 4d and 5d TMOs tend to have 10Dq > U −U ′since the orbitals are further extended and the effect of the crystal field islarge, and it is therefore more favourable for two electrons to occupy thesame site in the t2g manifold, in a low spin configuration [51]. Therefore,to the most basic extent, the physics of Sr2IrO4 can be considered as fiveelectrons in the (spin-degenerate) t2g manifold. This means a partially filledt2g shell for all materials, implying a metallic ground state in the absence ofelectron-electron interactions. Spin-orbit coupling then proceeds to modifythe structure of these (nominally) degenerate bands, as will be discussed inSection Mott insulatorsThe introduction of band theory has been very successful in the explanationof metallic and insulating properties. In the case of a partially filled band,gapless excitations are possible and the system is metallic. In the oppositecase, where a band is empty (or completely filled), a finite amount of en-ergy (the gap), has to be overcome for an excitation to be made, causingthese systems to instead be insulating. Band theory has been successful formany materials and has helped shape the world by explaining the behaviourof silicon and aiding in the creation of the transistor. However, already in1937, de Boer and Verwey pointed out that many transition-metal oxideshad partially filled d-bands and were insulators [52]. Commenting on thepaper of de Boer and Verwey, Peierls pointed out that this behaviour maybe due to the electrostatic interaction of electrons [53]. It was Mott wholater laid the ground-work for what is now known as a Mott insulator [54];a system in which electrons are localized and excitations are gapped dueto the strong electron-electron repulsion. Since then, an extensive amountof attention has been paid to these types of compounds. Not in the leastbecause a Mott insulating phase is often found in close proximity to many ofthe exotic states mentioned in Section 1.1. Because the nature of the insu-10lating states can be found in electron-electron interactions, Mott insulatorsare very challenging to study. As correlations require the exact treatmentof an exponentially growing number of possible interactions in a many-bodysystem, there remain many questions still unanswered.1.2.3 The metal-insulator transitionIn the case of Mott insulators, a frequently studied physical phenomenon isthe transition into a metallic state, called a metal-insulator transition (MIT).An excellent in depth review of both theoretical and experimental aspectsof this field of physics can be found in Ref. 6. A MIT can in general bedriven through the tuning of two individual parameters; the filling, n, andthe bandwidth, often denoted U/t. Tuning through the former of theseparameters is called a filling controlled (FC) MIT, whereas the latter isreferred to as a bandwidth controlled (BC) MIT. It is instructive to considerthe physical mechanism behind these two. In the BC MIT, the kinetic energyterms t are modified, to an extent to which it can be found more favourablefor electrons to be itinerant, regardless the cost of U that an electron paysto occupy the same site as another. Meanwhile, an FC MIT can be easilyunderstood from a system at integer filling. While for a system exactly athalf filling, excitations are energetically costly (U for one electron to hopto a site where another electron is already present), for a system that isnot exactly half filled, sites that are already doubly occupied can hop at noadditional cost, as that U was already expended.Experimentally, there are various tuning knobs that control these param-eters. For bandwidth control, pressure is effective, either applied chemicallyor externally. In the case of perovskite structures, changing the radius ofthe A site ion is an effective way to change the bandwidth and such MITscan be found for example in RNiO3 with R is a rare-earth ion [55]. Anotherexample is the substitution of Ca in Ca2RuO4 for Sr, which has a largerionic radius and drives the system from insulating to metallic [56]. Theadvantage of substituting an A site ion is that they generally do not par-take in the low-energy physics (as discussed in Section 1.2) and only provide11sideline support such as adjusting charge and chemical pressure. For thecase of filling control, modifications can be made chemically or by electronicgating. A classic example is to substitute an A site for one with a differentvalence, for example, substitute part of trivalent La in La2CuO4 for diva-lent Sr, to create SrxLa2−xCuO4, which changes n to n − x. This strategyhas been successfully applied to Sr2IrO4 to electron dope it, by creatingLaxSr2−xIrO4 [42, 44]. Another possibility is to change the oxygen content,a strategy that has seen a large success in the field of the superconductingcuprates and has also been demonstrated in Sr2IrO4 [43]. Lastly, a typeof transition that is mentioned here for completeness, is the control of di-mensionality. For example, of the Ruddlesden-Popper series of manganites,the single layer compound La1−xSr1+xMnO4 is insulating, while the bi-layercompound, La2−xSr1+xMn2O7, is metallic. Since all the compounds studiedin this thesis are single-layer compounds, dimensionality will not be consid-ered here. In Chapter 3, we will study a new variety of MIT: the one that iscontrolled by spin-orbit coupling. While it may not be immediately obviouswhy spin-orbit coupling is fundamentally different from a FC MIT, the sim-ulations in [57, 58] should provide a compelling answer: where U/t changesthe itinerancy, λ/t changes the entanglement of the multi-orbital Mott sys-tem and the two control-parameters lead to surprisingly different phases. Inthis work, we will show for the first time an experimental demonstration ofsuch a SOC controlled MIT, as is explained in Chapter 3.1.3 Spin-orbit couplingA central theme in this thesis is SOC, a term in the Hamiltonian that arisesfrom relativistic corrections to the Schro¨dinger equation, which entanglesspin and orbital angular momentum degrees of freedom. This entanglementcauses a wavefunction in which spin and orbital angular momentum degreesof freedom can no longer be factored out. I will first describe spin-orbitcoupling from the historical point of view, which is a treatment which isoften presented in introductory textbooks. After this, a description derivedfrom the relativistic Dirac equation will be given.121.3.1 A simple description of spin-orbit couplingThe initial description of spin-orbit coupling came after Goudsmit and Uh-lenbeck suggested that a new quantum number, related to intrinsic angularmomentum of the electron could better explain the spectral lines of Hy-drogen [59, 60]. They proposed the existence of such a “spin”, ad hoc andderived the associated energies following from electrostatics. Due to theeffects of relativity, an electron moving in the presence of the potential ofthe nucleus will experience a magnetic field in its rest frame. This magneticfield is:Bel =vc2×E = vec2×∇V. (1.1)The moment of the electron will align with this field, where the energy ofthe spin magnetic moment is given by:Vµs = µs ·Bel. (1.2)Writing the electron magnetic moment as µs = −gsµB S~ and substitutinggs = 2 we arrive (up to a constant) at the energy for the spin-orbit coupling:Vµs = − 1m2ec2S · p×∇V. (1.3)In a spherically symmetric potential we can write the gradient of the poten-tial as ∇V = −1r dVdr r which leads to the more familiar form:Vµs =1m2ec21rdVdrL · S. (1.4)This is the correct result up to a factor 2. The error arises from the fact thatan electron in a spherical potential is not moving with a constant velocitybut is continuously accelerated by the centripetal force. This was resolved byaccounting for the appropriate Lorentz factor by Thomas [61], which leadsto the correct expression. Although this derivation of spin-orbit coupling assome external perturbation is quite insightful, since it provides an intuitiveorigin for the term, it is not at all rigorous from a physics point of view.13Although this description gives the correct result, it is a complicated mix ofclassical, quantum and relativistic physics. Most importantly, electron spinis added ad hoc, justified by intrinsic angular momentum. It was pointed outby Slater that the picture of a spinning electron would violate relativity asthe periphery of the electron would spin significantly faster than the speedof light [62].1.3.2 The Dirac equationShortly after the description by Uhlenbeck and Goudsmit with the correctionfrom Thomas, the same solution was found more rigorously through theDirac equation[63]: (cα · p+ βmec2)ψ = Eψ. (1.5)Here, α and β are both 4× 4 matrices, given by:α =(0 σσ 0), β =(I 00 −I), (1.6)with all elements here being 2× 2 matrices, σ are the Pauli spin-matricesand I is the identity matrix. This derivation of spin-orbit coupling, mainlyfollows [64]. We start by writing the four-component wavefunction as twotwo-component vectors, ψ = (ψA, ψB). In this representation, we get twocoupled equations for ψA and ψB, from which we can eliminate ψB and weobtain for ψA (in the presence of an electronic potential V ):p · σ c2E − V +mec2p · σψA = (E − V −mec2)ψA. (1.7)We can substitute E = mec2 +  and in the non-relativistic limit, where−V << mc2, we can make an expansion in the energy around mc2 to findfor the middle term:c2E − V +mec2 =12me(1− − V2mec2+O((− V2mec2)2)). (1.8)14Here, the zeroth order gives a term containing (p · σ)2 = p2,1 which resultsin the ordinary, scalar Schro¨dinger equation, degenerate in the spin degreeof freedom: (p22me+ V)ψ = ψ. (1.9)Taking the first order expansion, we get an additional set of terms, arisingfrom the fact that V = V (r) and that generally [V (r),p] 6= 0. The result asstated in [64] is:(p22me+ V +p48m3ec2+~4m2ec2σ ·∇V × p+ ~28m2ec2∇2V)ψ = ψ. (1.10)The first two terms describe the non-relativistic Hamiltonian from the zerothorder. The third term is the relativistic correction to the kinetic energy ofthe electron, the fourth term is the spin-orbit coupling, and the fifth termis the Darwin term. Setting S = ~2σ, we can finally cast the spin-orbitcoupling in its usual form:HSOC = − 12m2ec2p× (∇V ) · S. (1.11)Or when the potential V is spherically symmetric:HSOC =12m2ec21rdVdrL · S. (1.12)This is the correct result including the factor of 2 that needed to be addedto the electrostatic picture.1.4 SOC and angular momentum in solidsAll of the work in this thesis is focussed on coupling of spin and orbitalangular momentum in solids. It therefore seems apt to give a thoroughdescription of some of the generally used bases and coupling terms. Thissection serves to expand the well known textbook treatment of spin-orbitcoupling and states of coupled angular momentum to the realm of solids in1This holds since all the squares of the Pauli matrices give the identity: (σi)2 = I.15which the crystal field breaks certain symmetries.1.4.1 SOC in atomsThe treatment of spin-orbit coupling here will deal with the inclusion ofthe Hamiltonian L · S as derived above. This is technically incorrect, asthe substitution ∇V = −dVdr rˆ made to obtain this form is only valid fora spherically symmetric potential. However, the largest influence on everyindividual orbital comes mostly from the region very close to the nucleus.Therefore it is still a reasonable approximation to neglect the influence fromneighbouring nuclei, as their contributions to the gradient are much smallerthan that from the centre atom.To start, we briefly revisit the states of coupled orbital angular momen-tum. Our challenge is to find an operator representation that commuteswith the spin-orbit coupling Hamiltonian derived in Section 1.3.2. It turnsout that while Sz and Lz no longer commute individually with the Hamil-tonian, their sum does. The Hamiltonian can be written as a combinationof the absolute value of the spin, orbital, and coupled angular momentum:HSOC = λSOCL · S = λSOC 12(J · J− S · S− L · L). (1.13)The eigenstates for this Hamiltonian are given in Table A.2 and Table A.1 for` = 1 and ` = 2 respectively, sorted by their angular momentum projectionon the z-axis. The values for L · S and their components are given for eachstate. Note that in the absence of a term that breaks the symmetry and liftsthe degeneracy between the different mj states, the different values for LiSiare artificial and arise from writing the states as separate mj states. Rather,these expectation values should be taken as an average over the degeneratestates, which gives a more satisfying result that the spin-angular momentumcoupling is independent of direction, as one would expect for a sphericallysymmetric system. These considerations are important for later, where weuse circularly polarized spin-ARPES (CPSA) to measure entanglement ofspin and orbital momentum, this technique essentially directly probes LiSialong a particular axis i.161.4.2 SOC in the presence of a crystal fieldWe now turn to the effect of spin-orbit coupling in the presence of a crystalfield potential. As mentioned, the effect the surrounding lattice has on anatom, in changing the gradient of the potential, is quite small. However, thecrystal field imbues big changes on orbital energies. If it is comparable toor larger than the energy of spin-orbit coupling, it influences the ability ofSOC to entangle spin and orbital degrees of freedom. The potential arisingfrom the crystal field will break certain symmetries and lift degeneracies.If the energy splittings related to these degeneracy liftings are larger thanspin-orbit coupling, that means the full ` manifold is no longer available andeigenstates of coupled orbital angular momentum like the ones in Table A.1may no longer form. The energy scale associated with the crystal field isgenerally around∼ 3 eV, making it between one and two orders of magnitudelarger than the relevant values for SOC (Section 1.1.1). This means thatin a first approximation, we can simply project out the crystal field splitstates and transform the SOC Hamiltonian into the new reduced basis. Weconsider the basis of the cubic harmonics, which are the eigenstates of theHamiltonian in an octahedral crystal field. These states all have quenchedorbital angular momentum, making each individual state insensitive to SOC.Nevertheless, for higher symmetries (e.g. cubic rather than tetragonal),degeneracies persist and spin-orbit interaction can couple degenerate subsetsof these states.We consider here the d states, since they are most appropriate for thiswork. We first consider an octahedral crystal field, which splits the degen-erate d band into a manifold of eg and t2g states. Although the eg states arein general above the Fermi level for the compounds studied in this thesis, forcompleteness we discuss their spin-orbit coupling here. Since the eg orbitalsconsist of spherical harmonics with ml = {−2, 0, 2}, which the L+ and L−terms arising from L · S are unable to couple, the Hamiltonian reduces tozero:HSOC,eg = B†egHSOCBeg = 0, (1.14)where HSOC is the Hamiltonian in Eq. 1.13 and Beg is the operator that17projects the eg states onto the spherical harmonics, the basis of HSOC1.4.3 The jeff = 1/2 StateThe same is not true for the t2g states, where the projected SOC Hamiltoniandoes entangle the states. Taking as a basis:bt2g = {dxy,↑, dxz,↑, dyz,↑, dxy,↓, dxz,↓, dyz,↓} , (1.15)we get for the HSOC :HSOC,t2g =λ20 0 0 0 −i 10 0 −i i 0 00 i 0 −1 0 00 −i −1 0 0 0i 0 0 0 0 i1 0 0 0 −i 0. (1.16)This gives rise to coupling of these degenerate substates as was the case forSOC without a crystal field. The t2g states have the special property thatthey mimic the behaviour of p states as has been noted in [25]. To show thederivation of these states, we start by writing down a new basis (the namingof which will become clear after the derivation), b`eff :|1eff〉 = 1√2(|dyz〉+ i |dxz〉) = i∣∣Y −12 〉 , (1.17)|0eff〉 = − |dxy〉 = − i√2(∣∣Y −22 〉− ∣∣Y 22 〉) , (1.18)|−1eff〉 = 1√2(− |dyz〉+ i |dxz〉) = −i∣∣Y 12 〉 . (1.19)18Note that this basis transformation reintroduces orbital angular momentum,having ml = {−1, 0, 1}. Within this basis, the L+ and Lz operators become:L+leff = B−1leffL+Bleff =√20 −1 00 0 −10 0 0 , (1.20)Lz,leff = B−1leffL+Bleff =−1 0 00 0 00 0 1 . (1.21)These are identical to the respective matrices for the p orbitals, except with` → −1. We can now construct coupled states of angular momentum ina similar way as was done for Eq. 1.13, which are given in Table A.3 andconstructed explicitly from t2g orbitals in Table A.4. Spin-orbit couplingthus entangles the manifold of t2g states into the so-called jeff states, called“effective” because their expectations are inconsistent with ordinary j val-ues. Instead, following the observation that the `eff states behave like stateswith ` = −1, the expectation values of L · S and their components are pre-cisely negative that of what the equivalent j state would yield. This is ofimportance for our investigation of the jeff states using CPSA later. Thebasis of jeff diagonalizes the SOC Hamiltonian:HSOC,jeff =1 0 0 0 0 00 1 0 0 0 00 0 −12 0 0 00 0 0 −12 0 00 0 0 0 −12 00 0 0 0 0 −12. (1.22)1.4.4 Tetragonal splittingIn Sr2IrO4, more than just the spherical symmetry is broken. An expansionor compression along the z-axis or even the dimensionality of the solid may19give rise to a tetragonal splitting, in which the eg and t2g orbitals split intonon-degenerate dx2−y2 , dz2−r2 , dxy and degenerate dxz and dyz states. Thissplitting is smaller than the spin-orbit coupling for Sr2IrO4 [65], but playsan important role in 4d materials with strong structural distortions, such asCa2RuO4 [66, 67] and 3d systems such as Fe superconductors [23]. More-over, spin-orbit coupling only couples states at one particular k-point. Thisimplies that the bandwidth plays an important role in determining how spin-orbit coupling is able to entangle certain states. For example in materialslike Sr2IrO4, Sr2RuO4 and Sr2RhO4, the dxy orbital has a two-dimensionalcharacter, while the dxz and dyz bands are one-dimensional. This meansthat the dxy has approximately double the bandwidth compared to the oth-ers, creating a k-dependent effective tetragonal splitting on the order of thebandwidth, which is in turn much larger than SOC. It is therefore of inter-est, to also study the states that form upon coupling only the dxz and dyz.The Hamiltonian in the reduced basis ({dxz,↑, dyz,↑, dxz,↓, dyz,↓}) becomes:Lz,leff = B−1leffL+Bleff =0 − i2 0 0i2 0 0 00 0 0 i20 0 − i2 0 . (1.23)This gives rise to a basis of two Kramers degenerate states, parallel andanti parallel, labelled here as |j+〉 and |j−〉, the relevant expectation valuesfor which are given in Table A.5. The tetragonal splitting thus causes adifferent splitting of the states and hence a different expectation value foreach component of L ·S. The remaining states are all singly degenerate andtherefore do not couple under spin-orbit coupling.Fig. 1.4 shows the splitting of states as a function of the discussed pa-rameters, with the colour encoding for the expectation value of LiSi. The di-agram shows that these expectation values can change significantly throughthe Brillouin zone, if a k-dependent energy splitting between the dxy anddxz/dyz is taken into account. Furthermore it shows that although spin-orbitcoupling itself does not change, its effects on the band-structure change sig-20Figure 1.4: Expectation values of the d bands subject to the introduc-tion of various of splittings (λSOC , 10Dq, δtet) for L ·S operator(a) and its components LxSx (b) and LzSz (c). Due to thefourfold rotational symmetry that is conserved, the expectationvalue for LySy is identical to LxSx and therefore not shown.nificantly depending on other crystal parameters.21Chapter 2MethodsThe work in this thesis makes use of a combination of ab initio, numerical,and experimental methods. A combination of these two approaches can bean extremely useful tool in understanding real-world systems. This chapteris meant to provide the reader with information about the frameworks andmodels used in later chapters.2.1 Ab initio calculationsThe description of many-body electron systems, such as transition-metaloxides (TMOs), is highly complex and requires the use of appropriate ap-proximations to avoid exponential scaling of computational resources. Thisis particularly true for a system like Sr2IrO4, in which the system is noteasily described simply by a reduced effective model. In general there isno silver bullet, and this work makes use of a combination of techniques todescribe the systems studied.2.1.1 Density functional theoryA very successful approach in dealing with the many-particle problem hascome from density functional theory (DFT), a technique that relies on thefact that a system can be uniquely described by its charge density, whichcan itself be described as a function of just three variables n(r), rather than223N variables Ψ(ri). To give an introduction to DFT, it is instructive tostart by considering the Hohenberg-Kohn (HK) theorems [68], the first ofwhich states:Theorem 1 The ground state particle density of a system with a potentialV , uniquely determines the potential of that system up to a constant.The consequence of this is that the ground state particle density must con-tain all the relevant physical properties of the system, as it uniquely describesthe potential of the system. This means any physical quantity should beable to be expressed in terms of the particle density n(r) alone. The secondtheorem states that:Theorem 2 There exists a functional F [n], that is independent of the exter-nal potential Vext, that has the property that when E0[n] = F [n]+∫drVext(r)n(r)has a minimum E0 which is the ground state energy of the system, and n0,the ground state particle density, is the functional that minimizes it.This implies that the system is completely described by E0[n], since it de-scribes n0. Meanwhile, F [n] is the functional that contains a description ofall kinetic and interaction energy terms in the system which is, remarkably,completely independent of the external potential. Unfortunately, there isno known form for F [n], meaning that we cannot use this result directly.However, Kohn and Sham later developed a strategy [69] that maps thisproblem onto a problem which we can solve using some assumptions. Westart by writing the particle density as a sum over occupied single particlestates:n(r) =∑i|ψ∗i ψi|2 . (2.1)We then write the equation from the second HK theorem, adding a Lagrangemultiplier to conserve particle number:δ{F [n] +∫drn(r)Vextr− µ(∫drn(r)−N)}. (2.2)23This yields the equation:δTs[n]δn+ VKS = µ, (2.3)where Ts[n] is the kinetic energy functional, and VKS is a potential, whichnot only includes the external potential but also the exchange and correlationpotentials. It is equal to:VKS(r) = Vext(r) +∫dr′n(r′)|r− r′| + Vxc(r), (2.4)where Vxc(r) is the exchange correlation term and the integral is the Hartreeterm, both arising from F [n]. This equation can now be solved by consid-ering the set of N coupled equations:HˆKSψi = εiψi, (2.5)with HˆKS =∑i~2m∇2i + VKX(r). Although the problem is now much moretractable, there is unfortunately, no simple description of what Vxc shouldbe. There have been many approximations to this problem. Among the mostfamous of these are the so-called local density approximation (LDA) and thegeneralized gradient approximation (GGA). The former only considers thelocal density, and treats the exchange potential like that of a homogeneouselectron gas, for which the exchange potential was calculated by Dirac [70].The correlation potential can be fit to Monte-Carlo simulations [71]. TheGGA also considers gradient terms, as implemented in Ref. 72 for example.Throughout this work, we rely mostly on the GGA exchange correlationpotential described by Perdew, Burke and Ernzerhof [73].Since VKS depends on the particle density, this equation has to be solvedself-consistently. The way this is done in practice is by making an assump-tion about the particle density, from which the potential is calculated andthe Kohn-Sham (KS) equations are solved, which leads to a new particledensity. This is repeated until a convergence criterion is reached. This canbe taken to be the total energy or charge density differing by less than a24user-set amount from one iteration to the next.There are various sets of basis functions one can choose for the ψi. An ob-vious choice is the atomic orbitals (Gaussian or Slater type orbitals), whichis a basis set often used for molecular calculations. For periodic systems,an appropriate choice is a set of plane waves on a k-point grid. A practicalproblem that arises in that case is that the 1/r like potentials of nuclei causea very sharply spiked wave function at the centre of the nuclei, for which avery large number of small wavelength plane waves is needed to accuratelydescribe the charge density. This means that the basis set becomes large,causing larger computational times for the matrix diagonalization step insolving the set of coupled equations in Eq. 2.5. These scale as O(S3), withS the size of the basis, so it is advantageous to reduce the number of basisfunctions as much as possible, without losing accuracy. A solution to thisproblem is to change the atomic potentials with so-called pseudopotentials,which give wavefunctions that are equal to the ones found from real poten-tials at r > rA. An example of a code that uses this scheme is QuantumEspresso [74]. Another solution to this problem is the linear augmentedplane wave (LAPW) basis set, in which the basis functions are defined to beatomic orbitals within a sphere around each atom, augmented with planewaves in the interstitial region. An example of a code that uses this schemeis Wien2k [75], which is the program that has been primarily used for thiswork.Although the KS approach allows us to find the ground state densityof the system, and therefore find its energy, there is no guarantee that theKS eigenenergies εi carry any physical meaning. Hartree Fock theory hasKoopmans’ theorem that states the ionization energy of the system is equalto the energy of the highest occupied molecular orbital [76]. This descrip-tion would in principle map onto the Kohn-Sham equations, however thisturns out to be difficult in practice and is very dependent on the form of thefunctional [77–79]. Unfortunately, this still does not give any meaning toother εi, and direct comparison to techniques that measure the one-particlespectral function, like angle-resolved photoelectron spectroscopy (ARPES),should only be done with this caveat in mind. Nevertheless, the comparison25is frequently made and we can find some comfort in symmetry considera-tions that constrain a large part of the problem. On a more intuitive level,the Kohn-Sham equations appear to describe exactly the problem of a sin-gle electron moving through the background of all other electrons and theexternal potential of the system. This justifies the interpretation of the εi asthe single particle energies but does not guarantee the same interpretationfor many-body systems.Although band structure comparisons only hold in the single particlelimit, there are other properties that DFT is able to capture more robustly.This includes, for example, ion valencies and charge densities, as well asatomic forces and the derived results using structure optimization. Recog-nizing the weaknesses and strengths of DFT makes it an incredibly powerfuland useful tool for modelling and understanding experimental data. DFT isused throughout this thesis as the basis of many models and calculations.2.1.2 Wannier functionsWannier functions are the real space counterpart of Bloch functions and canbe useful if an interpretation in real space is appropriate. In a simple de-scription, Wannier functions are a non-unique set of functions that form thebuilding blocks for the periodic un,k(r) part of the Bloch waves. Conversely,the Fourier transform of a Bloch function should yield a Wannier functioncentred at all real-space lattice sites. Wannier defined a set of real-spacefunctions as the building blocks for Bloch functions as[80–82]:|Rn〉 = V2pi∫BZdke−ik·R |ψnk〉 , (2.6)where |Rn〉 is the Wannier function with label n at lattice position R, |ψnk〉is the Bloch state with label n at k, and the integral is over the entireBrillouin zone. In the case of a discrete k grid, as is often the case inelectronic structure calculations, this integral would be replaced by a sum26Figure 2.1: The relationship between a Bloch wave and a Wannierfunction. (a) A Bloch wave consisting of a phase changing withwavevector k, that multiply the repeating Wannier function de-picted in (b).over all k-points. The inverse of the transformation is given by:|ψnk〉 =∑Reik·R |Rn〉 . (2.7)From this equation it can easily be seen that Wannier functions form build-ing blocks of Bloch waves, which are modulated only by a phase factorthroughout a crystal. Fig. 2.1a shows a Bloch wave, consisting of Wannierfunctions in Fig. 2.1b. Both form a complete orthonormal basis set, and theFourier transform is the relation between the two.There is, however, one difficulty arising from the calculation of Wannierfunctions, which is that Bloch waves have a gauge freedom that propagatesinto the shape of the Wannier functions. In other words, the Wannier func-tions belonging to a set of Bloch waves are not unique. The gauge invarianceis [82]: ∣∣∣ψ˜nk〉 = eiφ(k) |ψnk〉 , (2.8)with φ (k) being any smooth function that follows the periodicity of the lat-tice in k-space. Both∣∣∣ψ˜nk〉 and |ψnk〉 give physically accurate descriptionsof the system in question, but their calculated Wannier functions differ. Thecase gets more complex in the multi band case, as any unitary transforma-tion applied to the set of Bloch waves results in the same physical description27of the system: ∣∣∣ψ˜mk〉 = ∑mUnm (k) |ψnk〉 , (2.9)where Unm (k) is now an arbitrary unitary transformation that is periodicand smooth in k.To fix the gauge and set the matrix of phases, a widely used criterionis the one of maximal localization, the gauge that yields Wannier functionsthat are the most localized. The method for calculating maximally localizedWannier functions (MLWFs) was developed by Marzari and Vanderbilt [83],and relies on minimizing the functional that describes the spread in thehome unit cell:Ω =∑n(〈0n|r2|0n〉 − 〈0n|r|0n〉2). (2.10)The minimization happens with respect to Unm (k), to find the correct gaugethat describes a set of Wannier functions that are maximally localized withinthe home cell.An example of a Wannier calculation performed on Sr2IrO4 is presentedin Fig. 2.2. The calculated DFT band structure is plotted in black withthe corresponding Wannier band structure displayed in red. The real spaceshape of the Wannier orbitals is plotted too as an iso-surface for each in-dividual orbital. The shapes of the Wannier orbitals reflect a previouslymentioned point (Section 1.2.1): the physics that is often only discussed interms of Ir d orbitals should rather be seen as Ir-O hybrids, here visiblein the Wannier functions as additional lobes to the t2g orbitals. Note thatthese hybrids still follow the symmetry of the original Ir orbitals.Wannier functions in this work are mainly used to derive a model Hamil-tonian from first principles calculations, which is more versatile for furthercalculations than the results from DFT. In particular, when the right choicefor Unm (k) is found, this defines a transformation from k-space into realspace that can be applied to the KS Hamiltonian Hk. We obtain the real28Figure 2.2: An example of a Wannier calculation: the black thin linesrepresent a DFT calculation of Sr2IrO4, with the red curve aWannier band structure generated from the t2g orbitals. The in-ferred Wannier model projects onto the low energy physics andis more compact and interpretable than the original DFT cal-culation. Iso-surface plots of the generated orbitals are shown,with red and blue colours representing positive and negativeamplitudes. The iso-surface for each orbital is taken at 10% ofthe maximum value of the wavefunction.29space Hamiltonian by applying the transformation:HR−R′ =∫BZdke−ik·(R−R′)U †nm (k)HkUnm (k) , (2.11)which can be interpreted as a tight-binding Hamiltonian with the obtainedWannier functions as an orbital basis. This can now be used as a strat-egy to extract a tight-binding like Hamiltonian from ab initio calculationsfor orbitals with particularly chosen symmetries, for example, the Ir-t2g or-bitals in Fig. 2.2. A projected tight binding Hamiltonian, in which onlyorbitals of particular character are kept, is often preferable to an electrondensity obtained from an ab initio code for two reasons. Firstly, simplifyingthe Hamiltonian makes the physics that is relevant to the problem muchmore interpretable. Secondly, because calculations are simpler and there-fore faster. This makes it easier to define relevant quantities and calculateexpectation values.2.1.3 The tight binding approachThe tight binding approach has been used extensively throughout this thesis.The following section briefly highlights important aspects of the technique,its strengths, and why it is convenient and appropriate to use for this work.The rudimentary idea is that we treat the electrons as independent particleswhich move independently in the crystal potential. If we use a basis set{|i〉}, the Hamiltonian can be expressed in terms of its matrix elements:Hij = 〈i|H|j〉 , (2.12)where {|i〉} are some set of localized atomic like wavefunctions, limited toa particular set of orbital angular momenta. In the case of this thesis, thisis often the iridium d-orbitals. The Hamiltonian matrix elements can inprinciple be calculated from the chosen basis and the Hamiltonian by directintegration. In practice they can be derived from Wannier calculations orfitting a model with orbitals of interest to either ab initio calculations orexperimental data. When fitting such models, symmetry restricts many of30the possible matrix elements to particular values. After collecting all matrixelements, the wavefunctions can be simply found by diagonalizing Hij :Hvn = εnvn. (2.13)These wavefunctions include phases, which means we can easily calcu-late expectation values of operators, band projections, (partial) densities ofstates, as well as more complex quantities like the photoemission matrix ele-ment. The approach is computationally cheap, the calculation time requiredis the diagonalization of an N ×N matrix, where N is the size of the basis.This allows us to study large clusters of atoms (up to 12× 12 with 10 basisstates per site for this work), without computational time becoming a lim-itation. Moreover, for a system like Sr2IrO4, in which all d-orbitals on twoseparate Ir sites are important, it becomes computationally very costly totreat electron-electron interactions fully. Furthermore, the generated mod-els are quite interpretable and therefore aid in explaining the underlyingphysics of the problem.The downside of this is that the only way electronic interactions areincluded is as a background potential term arising from the original KSequations. Hence, this method cannot make accurate predictions about cor-related physics, like the existence of a Mott gap. Nevertheless, the preva-lence of orbital symmetry in solid state physics makes this a very predictivetechnique, particularly when it comes to calculating dipole matrix elements,on which both experimental chapters put a significant amount of emphasis.Therefore, tight binding built on ab intio calculations is one of the preferredmethods for calculating quantities in this thesis.2.2 Band unfoldingIn physical systems, one of the useful control parameters is often chemicaldoping or substitution. To describe systems with impurities or dopants, themost accurate approach would be to work in the position basis, studyinglarge clusters of atoms. However, this has the disadvantage that the basisset becomes very large and secondly, because of working in the position ba-31sis, calculations lose their interpretation with regards to k-dependence. Aworkaround to this problem is to take a combined approach and to make su-percells of atoms. This consists of repeating the unit cell (Nx, Ny, Nz) timesand adding impurities, vacancies, or making substitutions in the originallattice. A new synthetic periodic system is hereby effectively created, whichno longer has the periodicity of the original crystal but models a regular dis-tribution of impurities. Although this can, at best, be an approximation tothe real system, the advantage is that it fits within the previously discussedmodels and momentum is kept as a good quantum number. A complicationthat arises, is that since the unit cell becomes larger, the reciprocal latticevectors become Ni times shorter, with Ni times more bands in the first Bril-louin zone. One can easily plot this band structure in the unfolded zone butthat still leaves NxNyNz folded replica’s that hinder a direct comparison tothe original band structure. To interpret the folded band structure, we makeuse of band unfolding, a technique that attempts to recover the characterof bands in the original Brillouin zone. A projection onto the Bloch wavesthat follow the periodicity of the original unit cell can be made, a techniquediscussed in [84, 85].2.2.1 A simple exampleWe first consider a simple example, the case of a 1D chain of atoms with asingle s orbital, with nearest neighbour hopping t. We will then consider asupercell consisting of two atoms, that makes up a new representation of anidentical chain (see Fig. 2.4(a)). Before considering these band structures,it is insightful to highlight that the units of this supercell chain can beconsidered as diatomic molecules. The energy spectrum and wavefunctionsfor each of these molecules can be easily calculated. When two atoms areallowed to interact via some kinetic term 2t, we can write the Hamiltonian:H =(0 2t2t 0). (2.14)32Figure 2.3: The energy levels and wavefunctions of a diatomicmolecule. The separate atoms are depicted on the outside ofthe figure, with the symmetric and anti-symmetric wavefunc-tions and energies in the centre.This Hamiltonian has eigen-energies E = ±2t, with corresponding wave-functions 1√2(1,±1). The wavefunctions and energies are also plotted inFig. 2.3, for the uncoupled (a) and coupled case (b). The coupling of thesestates causes a symmetric (low energy) and anti-symmetric state (often re-ferred to as a bonding and anti-bonding orbital). We now construct a chainof these atoms and compare it to a chain of molecules. Since these sys-tems are physically identical (they only differ in their description), theyshould present the same observable results. A schematic of the comparisonis sketched in Fig. 2.4a, with the description of the same system in terms ofa size a and 2a unit cell. We start by considering the chain of single atoms,with unit cell size a. The dispersion for this system is shown in Fig. 2.4b, inblack, from Γ(k = 0) to X(k = pi/a) (with a the spacing between two singleatoms). To obtain the wavefunction at the Γ and X points, we use Bloch’stheorem, which dictates that the wavefunction is only allowed to change upto a phase from unit cell and that phase should progress as eikr. We fixthe phase of the first atom to one and plot the wavefunction for both the Γ(|ψΓ(r)〉) and the X (|ψX(r)〉) point, in Fig. 2.4(c) and (d) respectively.33Figure 2.4: Band unfolding in a chain of diatomic molecules. (a)Schematic overview of the model. The atoms are spaced aapart, and a supercell of two atoms (sized 2a) is considered.(b) Dispersion relation for a 1D chain of atoms (blue) and a 1Dchain of diatomic molecules with the same hopping parameters(red). (c) The wavefunction for the 1D chain of atoms (black)and the progression of the phase (blue) at Γ. (d) The same atX. (e). Wavefunction from the symmetric state for the chainof molecules (black) with the phase indicated at Γ (blue) andΓ′ (red). (f) The same as (e) for the anti-symmetric state. Inthis case the phase at Γ is indicated in red, while Γ′ is in blue.The blue coloured phase lines correspond to the correct phasefor the original unfolded cell.34Now we turn to the band structure of the two atom unit cell, indicatedwith the red dashed lines in Fig. 2.4a, which adds the folded replica of theoriginal band structure plotted in red in panel (b). Note that these foldedbands are effectively plotted in the first and second Brillouin zone, since theunit cell size doubled and the Brilluoin zone halved in size. The two foldedbands have a simple interpretation at the Γ (and Γ′ = pi/a) point: they arethe band structure of the symmetric (low energy) and anti-symmetric (highenergy) hybrids that form in the previously discussed diatomic molecule.To identify which of these bands holds physical meaning at which k-points,we look at the phase progression of the wavefunction and require the sameprogression as in the single-atom case. Using the hybrid wavefunctions asthe basis for the Bloch wave, we plot the wavefunctions (|ψsym(r)〉 and|ψasym(r)〉) of the folded band structure at the Γ point (identical to Γ′),including their phase progression for both Γ and Γ′ in Fig. 2.4 (e) and (f).In this supercell model, the states displayed in (c) and (e) are reproducedas expected. However, the wavefunction that occurs at Γ′ (X) can now befound at Γ and similarly for the wavefunction at Γ can now also be foundat Γ′ (X). As stated, since these models are identical, they should yieldthe same observable results. Comparing the wavefunctions in (e) and (f)to the plotted Bloch phases of the unreconstructed cell, it is clear whichwavefunction is the correct one for which k-point. The wavefunction in (e)belongs to Γ while the wavefunction in (f) belongs to Γ′. We therefore findthat both models yield the same results as expected, as long as we enforcethe correct progression of Bloch phases.For this example, it is easy to see which state belongs to the originalcell. However, for unit cells with more than two atoms, this quickly getsrather complex. We can extend the requirement of a physically correctband beyond our simple model; we need to require that the phases of thewavefunction in the supercell follow the phase progression of the Bloch wavesin the primitive cell. Quantitatively, we project the supercell wavefunctiononto the Bloch phases of the original cell. By doing this, it is easy to see that|ψsym(r)〉 in Fig. 2.4 has the correct phase for Γ and would project to unitybut for X it would project to zero. This method implicitly assumes that the35impurities are similar to the original species, as it enforces the symmetriesof the original crystal onto the supercell. Practically, it also requires thatthe number and type of orbitals are the same.2.2.2 Projecting Bloch phasesA more rigorous derivation calculates the weight of the spectral function forBloch functions in the original primitive cell from the spectral function inthe supercell. This is demonstrated in Ref. 85, we present that derivationhere, including some intermediate steps. The spectral function for a Wannierfunction |kn〉 reads:Akn(ω) =∑K,J|〈kn|KJ〉|2AKJ(ω), (2.15)where |kn〉 = 1√Nr∑r eik·r |rn〉, with n an index that labels the Wannierorbital. |KJ〉 are the eigenstates of the supercell, with K the momentumin the folded Brillouin zone, and J a band index. We can express |KJ〉 interms of its Wannier constituents as:|KJ〉 =∑N〈KN |KJ〉 |KN〉 . (2.16)Where |KN〉 is a Bloch wave constructed of Wannier functions in the sameway the |kn〉 was defined:|KN〉 =∑ReiK·R |K,R〉 . (2.17)We can then write the projection as:〈kn|KJ〉 =∑R,N〈kn|RN〉 〈R|KN〉 〈KN |KJ〉=∑R,R′,N,re−ik·r 〈rn|RN〉 eiK·R′ 〈RN ∣∣R′N〉 〈KN |KJ〉 . (2.18)36To evaluate 〈rn|RN〉, we write the normal cell position in terms of thesupercell position and set r = R + ρ(N), a sum of the supercell position,and a vector within the supercell pointing to the particular atom ρ(N). Wecan now eliminate most projections and write the expression as:〈kn|KJ〉 =∑R,Nei(K−k)·Re−ik·ρNδn,n′(N) 〈KN |KJ〉=∑Ne−ik·ρNδn,n′(N)δK′(k),K 〈KN |KJ〉 . (2.19)Here, n′ (N) is the function that recovers the band index in the original cell.This means that the sum should only run over orbitals that are identical inthe primitive cell. Meanwhile, K′ (k) describes the momentum of k in thefolded unit cell. In the case when the supercell model is constructed as atight binding model from Wannier orbitals, the terms 〈KN |KJ〉 are simplythe coefficients of the eigenvectors cJ,N (K), we recover the expected resultfor our weighted spectral function:AJk,n (ω) =∑N in SC∣∣∣e−ik·ρ(N)cJ,N (k)∣∣∣2ANk,J (ω) . (2.20)This shows the intuition from the two-atom unit cell was correct. We needto require the phase within the supercell to progress the same way thatthe state in the original cell would. This is mathematically enforced by theproduct of coefficient and exponential; any solution that does not have theright phase relation will interfere out.As an example, we extend the chain diatomic molecules presented inFig. 2.4a to a supercell size of 20 atoms. The band structure for this system isplotted in thin black lines in Fig. 2.5a. To illustrate the unfolded bands, theband structure is also plotted in blue markers, with their transparency set bythe unfolded weight. We see that this method effectively recovers the originalexpected band (red curve) from the many folds of bands. As an example of away in which this technique can be used for a non-trivial calculation, we showthe calculated unfolded band structure for the same system, where one atom37has an added impurity potential of 0.4t in Fig. 2.5b. The unfolded bandsare shown in green and it can be seen that the single impurity introducesdisorder into the band structure, in the form of intensity on the other foldedbands, particularly around the Γ point. The interpretation of results that areobtained using this technique will be discussed in more detail in Chapter 3,in particular sections Section 3.3 and Section Impurity distributionsTo represent a real substituted system and model a material that properlycaptures the observations as seen by ARPES, the system size would needto be on the order of the probe size (150 µm) and the computation wouldbe intractable (∼ 1011 atoms). Therefore, simulations are limited to smallersystem sizes, in the case of this thesis up to 12 × 12. To reflect the differentpossibilities of distributions of impurities, many configurations are averageduntil no further change is seen in the resulting spectrum.A point to note here is the amount of impurities; the number of impuritiesfor a given doping follows a binomial distribution. For a system with size nand doping p, the probability that we find k impurities is:P (n, p, k) =(nk)pk (1− p)n−k . (2.21)The binomial distribution has the property that its width increases as σk ∝√n. Since the actual observed doping can be calculated as p¯ = kn , thespread in concentrations follows σp¯ ∝ 1√n and thus reduces as the modelledsystem size increases. This effect can be clearly seen in Fig. 2.6, where thedistribution of impurities, Pn (p¯), is plotted for various system sizes at afixed p. To circumvent this issue, we find that more sensible results areobtained if the number of dopants for a particular configuration is fixed toa set number, obtained by rounding 〈k〉 = pn to the nearest integer. Wethen average over different configurations of impurities to simulate a systemthat reflects the impurity distribution of a large system size. This makes theresults independent of the modelled system size. The approach equates toartificially setting the width of the distribution to zero, which is reflective38Figure 2.5: 20-site supercell model for a one-dimensional chain ofatoms with a single s-orbital. (a). The band structure forthe 20-site supercell (thin black curves) and the unfolded bandstructure (blue markers, transparency set by unfolded weight).(b) Band structure for the same model where one atom has animpurity potential of 0.4t added (green markers, transparencyencodes for the unfolded weight).39Figure 2.6: Probability to find a number of impurities as a functionof the system size. While the number of impurities that areobserved k grows linearly with system size n, the width of thedistribution only grows as√n. This implies that the effectiverange of concentrations (k/n) reduces as the system size goesup.of a very large system size as 1√n→ 0.2.3 Angle-resolved photoelectron spectroscopyAngle resolved photoelectron spectroscopy (ARPES) is a powerful techniqueused for studying the electronic structure of electrons in solids experimen-tally. It is a direct probe for both electron momentum and energy, andmoreover provides a direct measure of the interactions between electronsand other particles. The information overlaid in this section mostly followsRef. [86].2.3.1 Theory of photoemissionThe technique is based on the photoelectric effect, which is the liberation ofan electron from a material after excitation by a photon. If the energy ofthe incident photon is known by using a monochromatized source of light,the kinetic energy Ek of the emitted photoelectron can be measured and thebinding energy Eb can be calculated. We start by invoking conservation of40energy:Eb = hν − Ek − Φ. (2.22)Here, hν is the photon energy and Φ is the work-function, a sample depen-dent potential that describes the energy difference between the Fermi energyand the vacuum energy. If the photoelectron is emitted into vacuum, thekinetic energy also gives the absolute value of the momentum of the emittedelectron, through the dispersion relation of a free particle:~k =√2meEk. (2.23)By using conservation of momentum and the take-off angles of the photo-electron, we can extract the in-plane momentum of the electron before itwas emitted:k‖ = k sin θ (cos (φ)xˆ+ sin (φ)yˆ) . (2.24)Unfortunately such a simple form does not exist for k⊥. Since transla-tional symmetry perpendicular to the surface is broken, this momentum isnot conserved. The perpendicular momentum can however often be approx-imated by:~k⊥ =√2me (Ek cos2 θ + V0), (2.25)where V0 is the inner potential, describing the bottom of the valence band.The inner potential, a parameter on the order of 10 eV is often inferredfrom experiment. The perpendicular momentum is only of importance forthree-dimensional materials however. In general the preferred materials foran ARPES experiment are (quasi-) two-dimensional, in part because herekz is irrelevant, but also because two-dimensional materials often have anatural cleavage plane, which is important for sample preparation. Thedefinition of “two-dimensional” in this case means the electron dispersionis fully (or mostly) in a plane, embedded as a stack of layers in a three-dimensional material. A useful criterion for determining two-dimensionalityis the requirement that no bands cross, and no electron or hole pockets closeas a function of kz.We consider the theory behind photoemission, by treating the interaction41with light perturbatively. We make the canonical substitution p→ p− eActo account for the light. Dropping the quadratic term in A, we obtain forHint:Hint = − e2mc(p ·A+A · p) (2.26)Using the commutator [p,A] = −i~∇ ·A and the fact that the wavelengthof ultraviolet (UV) light is large enough that we can set ∇ ·A = 0, we getfor Hint:Hint = − emcA · p (2.27)With the interaction Hamiltonian, we can describe the transition probabilityfrom the many body grounds state ΨNi into a particular final state ΨNf usingFermi’s golden rule:wf,i =2pi~∣∣ 〈ΨNf ∣∣Hint∣∣ΨNi 〉∣∣2 δ(ENf − ENi − hν). (2.28)Here, ENf and ENi are the energies of the initial and final state many bodyelectron wavefunctions.To calculate this, one would need full knowledge of the many body wave-function, properly taking into account the sample surface and related effects.This method is therefore quite cumbersome and often the process of photoe-mission is modelled by the so-called three-step model. In this case, the pho-toemission process is phenomenologically split up into separate processes:excitation into the bulk, travel to the surface, and emission from the sur-face. This allows one to approximate the photoemission intensity by tran-sition probabilities that ignore the effect of the surface on the Hamiltonian.Another assumption that is made is the so-called sudden approximation,which considers the process of photoemission instantaneous and makes itpossible to neglect any interactions between the photoelectron and core holethat is left by the photoemission process. The many body final state canthen be written as a product of a single particle ψf and an N − 1 particlemany body state ΨN−1f :ΨNf = AφfΨN−1f , (2.29)42where A is an operator that anti-symmetrizes the wavefunction. If we as-sume that the initial state can be written as a single Slater determinant, wecan write a similar factorization:ΨNi = AφiΨN−1i , (2.30)where the subscript i now denotes the initial state wavefunction. As aconvenient choice for the possible final states, we can take the eigenstates ofthe N−1 electron Hamiltonian, labelled by m as ΨN−1m with energies EN−1m .The matrix element for the scattering amplitude in to one such eigenstatethen becomes:〈ΨNf,m∣∣Hint∣∣ΨNi 〉 = 〈φf |Hint|φi〉〈ΨN−1m ∣∣∣ΨN−1i 〉 , (2.31)in which the first term is the dipole matrix element and the second termdescribes the many-body spectrum. To gain insight into the second term, wenote that for systems in which electron correlations are important, ΨN−1i =ckΨNi is not an eigenstate of the N − 1 electron Hamiltonian. Equivalently,if electron-electron interactions are ignored, the N particle Hamiltonian isidentical to the N − 1 Hamiltonian and ΨN−1i will be an eigenstate. In theformer case, ΨN−1i may have many projections onto the N − 1 eigenstatescm,i =〈ΨN−1i∣∣∣ΨN−1m 〉 that are non-zero. To obtain the full transition prob-ability, we sum over all initial and final single particle states, as well as allN − 1 eigenstates:I (k, Ekin) =∑i,f,m∣∣∣Mkf,i∣∣∣2 | ci,m|2 δ (Ekin + EN−1m − ENi − hν) . (2.32)The quantity〈ΨN−1i∣∣∣ΨN−1m 〉 can be related to the one-particle removalGreen’s function:G− (k, ω) =∑m∣∣ 〈ΨN−1m ∣∣c−k ∣∣ΨNi 〉∣∣2ω − EN−1m + ENi − iη(2.33)Taking the limit η → 0+, this corresponds to the one-particle removal spec-43tral function as:A− (k, ω) = − 1piImG− (k, ω)=∑m∣∣ 〈ΨN−1m ∣∣c−k ∣∣ΨNi 〉∣∣2δ (ω − EN−1m + ENi )=∑m|ci,m|2δ(ω − EN−1m + ENi), (2.34)in which we can see the right-hand side of Eq. 2.32 reappear. Note that inthe case where electron-electron interactions are ignored, we can describeΨN−1m using a single eigenstate with m = m0, and all other terms vanish.This means that the spectrum looks like a single delta function peak atenergy m. When electron-electron interactions are taken into account, thespectrum will have many peaks or a continuum of peaks as a broad hump.A famous example is the case of photoemission from molecular hydrogen,which produces an excited state of H+2 under photoemission. One observes apeak for each possible vibrational mode that overlaps with the ground state,minus one electron [87].2.3.2 A photoemission experiment in practiceIn this section we will discuss the parameters and considerations of a typicalphotoemission experiment. Experiments are normally performed using pho-ton energies between 20 and 200 eV. In principle, the lower limit is set by thesample work function Φ, with an additional amount for the photoelectron tohave non-zero velocity, so it can reach the analyzer. Although in practice,at such low energies, stray magnetic and electric fields tend to influence themeasurements. Additionally, the aforementioned sudden approximation islikely invalid and the final state can no longer be modelled by a free electron.Although in theory there is no upper limit, the disadvantage of using higherphoton energies is that the absolute energy and momentum resolution wors-ens linearly and quadratically, respectively. All data presented in this thesisare taken between 44 and 120 eV.A challenge arising with these particular energies is that electron scat-44tering cross sections are very high and therefore the escape depth is onlyon the order of a few atomic spacings. This requires measurements to bedone at ultra-high vacuum pressures (< 5 10−10 mbar) and samples to beprepared in-situ. This limits the technique to samples than can be preparedappropriately. This implies either cleaving of bulk crystals, in-situ growthlike molecular beam epitaxy, evaporation or pulsed laser deposition, or somecleaning of a polished surface by sputtering and annealing. Meanwhile, inorder to measure momentum, single crystals are needed so that the systemis translationally symmetric. Samples studied in this thesis are all singlecrystalline, prepared by cleaving in-situ.Since ARPES attempts to measure energies precisely, experiments arecarried out at low temperatures to combat the broadening arising from kBT .In practice, a cryostat is integrated into a movable sample stage, with liq-uid He providing cooling (displayed in gold in Fig. 2.7). Base temperaturedepends on the design of the system but is ordinarily between 4.2 and 20 K.While lower temperatures are better for optimizing energy resolution, thedata presented in this thesis are taken at higher temperatures because of theinsulating nature of Sr2IrO4. Since electrons are removed from the materialby the UV light, these need to be replenished or the sample will gain an elec-trostatic charge. This changes the energy of the photoemitted electrons andin some cases completely prevents photoemission. To mitigate this problemin Sr2IrO4, most of the measurements in this thesis were performed at 150K, at which temperature the resistivity is lower [17].The UV light required for ARPES experiments can be generated in amultitude of ways. One strategy is to use a discharge lamp in conjunctionwith an inert gas, often He, which produces spectral lines at 21.1 and 40.8eV. The disadvantage of using a discharge lamp is that the spot-size istypically quite large (∼ 1 mm2) and the energies are limited to the spectrallines of the gas used. Another strategy is to use a synchrotron light source,in which electrons are stored at highly relativistic energies in an acceleratorring. Devices with periodic arrays of magnets are inserted into this ringand the electrons generate light as they fly by. The advantage of using asynchrotron light source is that a large range of photon energies is available,45with a spot-size that is on the order of ∼ 100× 100 µm, with controllable,arbitrary polarization. However, in this case the user relies on allocatedperiods of “beamtime”, which are highly competitive and limited. All of thework presented in this thesis was done at synchrotron light sources, althoughadditional work using He-lamp ARPES was done for sample characterizationand other initial experiments.The electron analyzer used for ARPES experiments consists of two con-centric hemispheres. A constant electric field is applied to an inner andouter shell, causing electrons with different energies to follow circular tra-jectories with different radii. Electrons enter the analyzer through an en-trance slit, which allows a one-dimensional range of angles to be detectedsimultaneously. The energy separation in the hemisphere expands this one-dimensional cut into a two-dimensional detector image. The electrons areincident on a phosphor screen and are subsequently detected by a camera.The full process of emission and dispersion in the hemisphere is depicted inFig. 2.7, as well as the geometry of sample, analyzer and detector. CurrentARPES chambers have an energy resolution on the order of 1 meV and anangular resolution better than 0.1◦. Although such resolutions of ∼ 1 meVare technically attainable at most of the experimental setups used in this the-sis, the broad nature of the spectrum of Sr2IrO4 means that measurementspresented here have been optimized towards counts, and are performed atan energy resolution of approximately 20 meV.In Fig. 2.8 we show an example of a high quality ARPES dataset. Thisdata was taken as a part of the commissioning of the Quantum Materi-als Spectroscopy Center (QMSC) endstation at the Canadian Light Source(CLS). The data shown were collected on Sr2RuO4, using 48 eV photonswith horizontal polarization (indicated with a black arrow marked ~ε), at asample temperature of 20 K. Sr2RuO4 has been studied using ARPES nu-merous times before [49, 88] and it is an excellent demonstration of what therequirements are to obtain a high quality dataset. This involves a high qual-ity sample, with a freshly cleaved surface, a low vacuum pressure (6 10−11mbar), a small spot-size (150 µm), and a high quality analyzer with a goodangular resolution.46Figure 2.7: Schematic representation of an ARPES experiment. A(cutaway) hemispherical analyzer is shown with the trajec-tory of the electrons indicated. The one-dimensional entranceslit combined with the energy dispersion results in a two-dimensional image. The electrons are incident on a phosphorscreen and are subsequently detected by a camera. The geome-try is variable to access all possible take-off angles. Rather thanmoving the entire analyzer, the sample is rotated with the aid ofa 6-axis manipulator. The manipulator can be rotated aroundits axis, the other two axes of rotation are internal. Light isincident from an angle next to the detector.47Figure 2.8: Example measurement of Sr2RuO4, a dataset that wastaken during commissioning time of the QMSC beamline atCLS. The data was taken at 20K, using 48 eV photons withlinear horizontal polarization (indicated with a black arrow inpanel (a)). The colour encodes the intensity of the photocur-rent, with a linear scale indicated. (a) “Fermi surface map”, arepresentation that plots the intensity recorded at many anglesat the Fermi energy. The data here is integrated over 2 meVto improve statistics. (b) A single slice as measured. The datapresented in (a) is recorded by measuring many of these indi-vidual slices at different manipulator angles. The curve tracedout by the slit through k-space to record (b) is indicated in (a)as a thin dashed line.The data is collected as a so-called Fermi-Surface mapping, in whichthe manipulator angles are changed sequentially and a spectrum I(E, θ)(Fig. 2.8(b)) is collected at each angle. These spectra can be put togetherto form a complete solid angle. After transforming the data into k-spaceusing the expression in Eq. 2.24, the data can be plotted at constant energyin Fig. Fig. 2.8(a).The many sharp bands that are visible correspond to the α, β and γpockets of Sr2RuO4 and their folded replicas (α′, β′ and γ′) [49]. Further-more, the two different surface octahedral rotations give rise to the doubled48bands, as explained in [88].2.3.3 The photoemission dipole matrix elementThe work in this thesis emphasizes on the simulation and modelling of thedipole matrix element introduced in Section 2.3.1. The purpose of thissection is to explain the approach that is taken to calculate this quantityfrom tight binding and first principles calculations.The goal is to calculate the photoemission matrix element arising fromthe dipole term in Fermi’s golden rule in Eq. 2.28, which is given by [89]:∣∣∣Mki,f ∣∣∣2 ∝ ∣∣∣〈ψkf |r · ε|φki 〉∣∣∣2 , (2.35)where φki is the initial state, ε is the light polarization vector, and ψkf is thefinal state wavefunction. For the final state, a plane wave is used, which isan approximation that is valid if the kinetic energy is sufficiently high. Theinitial state is taken to be an eigenstate of a tight-binding Hamiltonian. Itis insightful to first look at the symmetries of particular states and try toinfer some information about the value of the matrix element in Eq. 2.35.Considering the illustration in Fig. 2.9, we can directly see that the dipolematrix element carries information about the symmetry of the initial state.The integrand, consisting of polarization, orbital, and plane wave like finalstate, needs to be an even function or it will vanish. Considering the planespanned by the incoming light and the outgoing photoelectron, the freeelectron state is even, while for σ-polarization (as shown), the term r · ε isodd, which requires the initial state to be odd.We can improve this simple picture by including the phases of the tightbinding wavefunction and calculating the actual value of the matrix element.To do so for an arbitrary initial state φki , we expand it in terms of sphericalharmonics:φki = Rn,`∑m,`Ck`,mYm` , (2.36)where the phase and amplitude information is contained in Ck`,m. Tightbinding models in this thesis are in terms of cubic harmonics, which are49Figure 2.9: Illustration of the geometry used to calculate the photoe-mission matrixelement.easily converted to spherical harmonics, making this a convenient basis towork in. Similarly, we write the final state in terms of spherical harmonicsusing the plane wave expansion:ψkf ∝ eik·r = 4pi∞∑`=0m=∑`m=−`i`j`(kr)Ym` (θk, φk) (Ym` (θr, φr))∗ , (2.37)where j`(kr) is the spherical Bessel function and the spherical harmonics arefunctions of the angles describing the r and k vectors. We can also writethe polarization vector in terms of the ` = 1 spherical harmonics, to obtain:M,σ =∑`i,`f ,mimε,mfBni,`i,`f(∫dΩrYmi`iY mε1 Ymf`f)Ymf`f(θk, φk) . (2.38)The product of three spherical harmonics can be found using the Clebsch-Gordan coefficients. The radial integrals are written as :Bn` =∫drr3Rni,`i(r)j`f (r). (2.39)These integrals can be calculated using some assumption for the initial stateradial wavefunction. In this work, Slater type orbitals are most often usedas implemented in the chinook package [90]. The argument of j`(kr) in50Eq. 2.39 depends on the photon energy through k, as it increases with thefree electron kinetic energy Ek, which causes the photoemission final stateratio to be photon energy dependent. Since selection rules, based on conser-vation of orbital angular momentum, only allow the final states of `f = `i±1like character, this implies that a model that only describes a single mani-fold of `i states, all but two terms in Eq. 2.38 vanish. A single parameter(i.e. the ratio Bn`i−1/Bn`i+1), in addition to an overall normalization factorsuffices in such cases. The ratio can be kept as a model parameter, whichcan be adjusted to best match experimental data.2.4 Spin-ARPESThe experimental results presented in Chapter 4 make use of an extension ofARPES: spin-resolved ARPES, in which, aside from energy and momentum,the spin of the photoelectron is also measured. The spin is measured aftertravel through the hemispherical analyzer. The electrons are redirected byelectron lenses, away from the two-dimensional detector, to a separate spindetector.2.4.1 A practical spin-ARPES experimentVarious strategies have been used for detecting the spin of electrons. Theone used for this thesis is very low energy electron diffraction (VLEED) offmagnetic targets [91–94]. This technique uses a magnetized film to scatterelectrons, which follow different trajectories for the spin up and spin downchannels. The targets for the spin-detectors used in this thesis are made ofFeO films, which are magnetized prior to the experiment, using electromag-nets. The magnetic easy axis for the films is in plane [91], therefore, one filmis able to detect a total of four spin-directions (both up and down spin for twoquantization axes). To measure the full three-dimensional spin-structure,two of such detectors are needed. The geometry of the experiment is suchthat a total of 8 spin channels can be measured (x↑, x↓, z↑, z↓, y↑, y↓, z′↑, z′↓),of which two are redundant and can be used to check the consistency be-tween the two detectors. Relevant parameters for these detectors are the51efficiency in which electrons are detected, and efficiency in which electronspins are separated. The former is captured in the “figure of merit” (FOM)[94]:FOM = S2I/I0, (2.40)where I/I0 is the fraction of detected photoelectrons and S is the Shermanfunction, which describes the efficiency with which the spin is separated.The Sherman function can be determined from:I↑ − I↓I↑ + I↓= PS, (2.41)in which Iσ are the measured spin up and down currents, P is the actualpolarization of the photoemitted states, and S is the Sherman function.Spin-resolved photoemission experiments are often complicated by low val-ues for these parameters. As an example, a different strategy to observespin is the so-called Mott detector, for which a typical value for the FoM isbetween 10−3 and 10−4 with a Sherman function of 0.068 [95]. For VLEED,these parameters are drastically improved to 10−2 and 0.5 respectively [94].Nevertheless, the challenge of high statistics data persists and often spin-resolved ARPES is limited to a single k-point and long acquisition times.2.4.2 Circularly polarized spin ARPESNaturally, spin-resolved ARPES is limited to those compounds that havespin-polarized electron states to photoemit from. This means that spin-ARPES is frequently used to study the spin-polarized surface states of topo-logical insulators [96, 97] or Rashba states that arise on the surface due tostrong spin-orbit coupling and the absence of inversion symmetry [98, 99].Contrary to that, in the case where spin-orbit coupling is strong and inver-sion symmetry is not broken, Kramers degeneracy dictates that althoughstates might be highly entangled with the spin degree of freedom, thosestates are not in fact spin-polarized. When there are no such spin-polarizedstates available to probe, it is possible to make use of the spin-orbital entan-glement and circularly polarized light to preferentially excite one particular52spin direction. In this section, we will discuss the theoretical and practicalfoundations of circularly polarized spin-ARPES (CPSA). The technique wasfirst described in [100], after which it was successfully applied to Ca2RuO4([101]), Sr2RuO4 [21] and the iron superconductors ([23]). As this techniquerelies on the coupling of spin and orbital angular momentum to measurespin-polarization, it is particularly suited to use on systems with significantspin-orbit coupling, as for example the iridates. We will use the techniqueto understand spin-orbital entanglement in Sr2IrO4 in Chapter 4.The Spin-Orbital EntanglementMeasuring spin, while limiting the orbital angular momentum states probed,captures the correlation between orbital and spin angular momentum. Morequantitatively, performing this experiment corresponds precisely to measur-ing the z component of the 〈L · S〉 operator: LzSz.We start by considering the photoemission matrix element as discussedin Section 2.3.3. Using circularly polarized light with positive helicity givesε⊕ · r = ε0 (x+ iy) = ε0Y 11 . The matrix element then becomes:Mki,f =〈ψkf∣∣∣r · ε∣∣∣φi〉 =ε0∑`fmf ,miBni,`i,`f〈Ymf`f∣∣∣Y 11 ∣∣∣cmi`i Y mi`i 〉Y mflf (θk, φk) , (2.42)where Bni,`i,`f specifies the radial integral as defined in Eq. 2.39. At theΓ point, we can simplify this equation by using the fact that the sphericalharmonic Ymflf(θk, φk)) has nodes for all mf except mf = 0, where its valueis 1. With the spherical harmonic arising from the polarization vector setto Y 11 , we only emit from a single initial state spherical harmonic. We cantherefore simplify the expression in Eq. 2.42 to:Mkσi,f = ε0∑`fBni,`i,`f〈Y 0`f∣∣∣Y 11 ∣∣∣cmi=−1,σ`i Y −1`i 〉 . (2.43)The effect of the spherical harmonic, arising from the polarization vec-53tor, is to raise the angular momentum of the initial state, meaning we canwrite the product as: Y 11 Y−1`i=∑`fa`i,`f c−1`iL+Y −1`f , where a`i,`f is a co-efficient that describes the possible resulting angular momenta, determinedby the Clebsch-Gordan coefficients arising from the product of two sphericalharmonics. This allows one to write the matrix element as:Mkσi,f = ε0∑`fBni,`i,`fa`i,`f〈Y 0`f∣∣∣L+∣∣∣cmi=−1,σ`i Y −1`f 〉= ε0∑`fBni,`i,`fa`i,`f〈Y −1`f∣∣∣L−L+∣∣∣cmi=−1,σ`i Y −1`f 〉= ε0Bni,`i,`fa`i,`f cmi=−1,σ`i. (2.44)This means that the measured photoemission intensity is:I⊕σ = ε20∑`fBni,`i,`fa`i,`f2 ∣∣∣cmi=−1,σ`i ∣∣∣2 = A∣∣∣c−1,σ`i ∣∣∣2. (2.45)We can thus measure using σ =↑, ↓ and ε = ⊕,	 and construct:I	↑ − I⊕↑ − I	↓ + I⊕↓ =A(∣∣∣c1,↑∣∣∣2 − ∣∣∣c−1,↑∣∣∣2 − ∣∣∣c1,↓∣∣∣2 + ∣∣∣c−1,↓∣∣∣2) (2.46)Noting that in the basis of |ml = 1, ↑〉 , |−1, ↑〉 , |1, ↓〉 , |−1, ↓〉, we have:LzSz =~221 0 0 00 −1 0 00 0 −1 00 0 0 1 , (2.47)we get for 〈LzSz〉:〈LzSz〉 = ~22(∣∣∣c1,↑∣∣∣2 − ∣∣∣c−1,↑∣∣∣2 − ∣∣∣c1,↓∣∣∣2 + ∣∣∣c−1,↓∣∣∣2) , (2.48)54which is precisely the expression found in Eq. 2.46, aside from the prefactor.Note that the expression derived above is independent (up to the prefactorA) of the values for Bni,`i,`f . Since there is only a single term of mli foreach configuration, there are no interference terms and the sum in Eq. 2.45can be evaluated separately. The only limiting case is when Bni,`i,`i+1 =−Bni,`i,`i−1, in which case the photoemission signal is zero.The Geometric MeanThis formulation of 〈LzSz〉 in terms of Iε,σ is unfortunately only valid ifall factors Bni,`i,`f are the identical for both both polarizations ε⊕ and ε	,which may not be the case in a system where there is circular dichroism.Moreover, if the sensitivity of the spin-detectors is not equal for up and downchannels, the description also breaks down. By denoting the sensitivity ofthe detector of each spin detector as ησ, and the factor related to the circulardichroism as αε, we can write the measured photoemission signal as:I˜εσ = αεησIε,σ = αεησA∣∣∣cmi,σ`i ∣∣∣2, (2.49)where mi = −1 for ε⊕ and 1 for ε	. Substituting the I˜ into Eq. 2.46, theexpectation value 〈LzSz〉 is no longer recovered as a result of the prefactors.We can instead take the geometric mean P which divides out the prefactors:P =√I˜	↑I˜⊕↓ −√I˜⊕↑I˜	↓√I˜	↑I˜⊕↓ +√I˜⊕↑I˜	↓=√Aα	η↑|c1,↑|2Aα⊕η↓|c−1,↓|2 −√Aα⊕η↑|c−1,↑|2Aα	η↓|c1,↓|2√Aα	η↑|c1,↑|2Aα⊕η↓|c−1,↓|2 +√Aα⊕η↑|c−1,↑|2Aα	η↓|c1,↓|2=√|c1,↑|2|c−1,↓|2 −√|c−1,↑|2|c1,↓|2√|c1,↑|2|c−1,↓|2 +√|c−1,↑|2|c1,↓|2. (2.50)55In the case of Kramers degeneracy, we should have |cm,σ|2 = |c−m,σ¯|2, andusing the fact that the states are normalized (∑ |cm,σ|2 = 1,) we obtain:P =∣∣c1,↑∣∣2 − ∣∣c1,↓∣∣2|c1,↑|2 + |c1,↓|2=∣∣∣c1,↑∣∣∣2− ∣∣∣c1,↓∣∣∣2− ∣∣∣c−1,↑∣∣∣2 + ∣∣∣c−1,↓∣∣∣2 = 2~2 〈LzSz〉 .(2.51)Using the geometric mean, we can thus extract the expectation value for〈LzSz〉 without the need to know the exact detector sensitivities or circulardichroism effects. This method is therefore the sole method used in thisthesis to present spin-polarized data.So far, the only expectation value discussed is the one along the z direc-tion. In principle, other components can be measured, such as 〈LzSx〉, asthe detectors used have three separate spin-axes available to measure. Whilethis is possible, it is used in this thesis only to demonstrate the robustnessof the technique. As no terms in the spin-orbit coupling hamiltonian appearto actually couple momentum perpendicularly, the expectation value andmeasured spin-polarization should always be zero.Furthermore, the calculated expectation values are only valid at theΓ-point. Despite this, the technique has been successfully applied awayfrom Γ [23]. Later in this thesis, data away from the Γ-point will also bepresented. The equations hold true as long as not too much weight comesfrom final states with ml 6= 0. Following the k-dependent spherical harmonicin Eq. 2.38, these other components have a dependence ∝ (1− cos2 θk),where θk is the angle of the photoemitted electron and the surface normal.In particular, if the photon energy is large, this angle is small, even at largermomentum values at the edges of the Brillouin zone. However, in such casesit is recommended to properly model the matrix elements as implementedfor example in chinook [90].While it is possible to measure in-plane components like 〈LxSx〉, in prac-tice, this is difficult as it would require measuring ml = 2, 0,−2 coefficients,which have matrix elements identical zero for photoemission at normal emis-sion. An experiment would therefore have to be off Γ, as close to the pre-56ferred spin-orbital axis to measure as possible.Light PolarizationUntil now, the calculations presented have assumed that the incident lightis perfectly perpendicular to the surface. In the geometry of a realisticARPES experiment (Fig. 2.7), the electron analyzer would be in the lightpath. Therefore, the incidence angle of the light is usually approximately45◦. Taking the direction of the sample surface normal to be zˆ, we can writefor the incoming light:ε⊕ = ε0(1√4(xˆ− zˆ) + iyˆ). (2.52)This can be converted into spherical harmonics that are used for the calcu-lation as:ε⊕ =1√4Y 01 + (1√4+1√8)Y 11 + (1√4− 1√8)Y −11 . (2.53)This deviates from the ideal case where we only make excitations with Y 11 .However, at Γ, there are no available final state channels for Y 01 to scat-ter into, so that term can be safely be ignored. The Y −11 term meanwhilecreates excitations of the opposite spin-orbital entanglement. Taking thesquares of these coefficients, we get 0.73, and for 0.02 for Y 11 and Y−11 re-spectively. This means that this configuration leads to an opposite signalof just 3%, generating a net 6% of additional, unpolarized signal. This isfar less than the Sherman function of 0.5 of the (high-efficiency) VLEEDdetectors discussed in Section 2.4, and can therefore be safely ignored.57Chapter 3Spin-orbit ControlledMetal-Insulator Transition inSr2IrO4The transition of Sr2IrO4 into a metallic state is here reported, by mak-ing careful substitutions using Rh and Ru. Throughout this chapter wewill argue that the transition is mediated by a reduction of the spin-orbitinteraction. This is not only the first demonstration of a metal insulatortransition driven by spin-orbit coupling, but also showcases the pivotal rolespin-orbit plays in stabilizing the insulating phase of Sr2IrO4. For correlatedinsulators, where electron-electron interactions (U) drive the localization ofcharge carriers, the metal-insulator transition (MIT) is described as eitherbandwidth controlled (BC) or filling controlled (FC) [6]. Where the formerdescribes the modification of the kinetic energy terms versus the interactionterms, the latter brings a Mott system away from integer filling to induce ametallic state. Spin-orbit coupling is in this regard fundamentally different,as it does not change the respective scales of kinetic and interaction termsdirectly, but rather the entanglement of different orbitals in a multi-orbitalsystem.583.1 IntroductionAs discussed in Chapter 1, the insulating phase in Sr2IrO4 was quite puz-zling and it was proposed to be part of a new class of correlated insulators,in which spin-orbit coupling (SOC) is believed to fully entangle the Ir t2gorbitals into a jeff = 3/2 and 1/2 state. The bandwidth of the half-filledjeff = 1/2 doublet is then significantly reduced, allowing a modest U toinduce a charge-localized phase [25, 27]. The insulating state has been be-lieved to be stabilized by spin-orbit coupling for a while, but the evidencethat was put forward has been debated. The existence of the jeff = 1/2 statewas demonstrated by resonant elastic x-ray scattering (REXS) [27], whichshows a complete quenching of the ratio of the L2 and L3 absorption edge,indicative of a complete spin-orbital entanglement of the unoccupied state.It was later however shown that for systems with in-plane magnetic mo-ments like Sr2IrO4, the branching ratio is identically zero regardless of thespin-orbital entanglement [28]. This was later acknowledged by the authorsof [27] in [29], where they perform a detailed study of the matrix elements forx-ray scattering. Meanwhile, different suggestions to explain the insulatingstate have emerged, such as a Slater insulator [26], in which the antiferro-magnetic order breaks the translational symmetry of the lattice and opensa gap through band folding. Moreover, the results in [30] show that Sr2IrO4is quite close to a quenching of the spin-orbital entanglement, further ques-tioning the validity of the jeff = 1/2 description. Here we provide evidencefor the central role of spin-orbit coupling, by directly modifying its strengthand showing that this causes a collapse of the insulating state.The effect of spin-orbit coupling in the valence band is modified by mak-ing substitutions that replace Ir with Ru and Rh, both lighter, 4d elements,with substantially lower SOC. Rh substituted Sr2IrO4 has been studied pre-viously and is known to drive the MIT. Previous studies have consideredthe role of SOC [102, 103], but a clear interpretation has been hindered byconcurrently occurring changes to the filling [104–106]. We overcome thischallenge by employing multiple substituents that introduce well definedchanges to the signatures of SOC and carrier concentration in the electronic59structure, as well as a new methodology that allows us to monitor SOC di-rectly. Specifically, we study Sr2Ir1−xTxO4 (T = Ru, Rh) by angle-resolvedphotoelectron spectroscopy (ARPES) combined with ab-initio and super-cell tight-binding calculations. This allows us to distinguish relativistic andfilling effects, thereby establishing conclusively the central role of SOC instabilizing the insulating state of Sr2IrO4. Most importantly, we estimatethe critical value for spin-orbit coupling in this system to be λc = 0.42 eVand provide the first demonstration of a spin-orbit-controlled MIT.3.2 The MIT in Rh and Ru substituted Sr2IrO4The familiar tools of chemical doping and pressure have provided straightfor-ward access to both FC and BC MIT in conventional correlated insulators.In an effort to unveil the role of SOC in the insulating behaviour of Sr2IrO4and whether it can indeed drive a MIT, we have attempted to controllablydilute SOC in the valence electronic structure by substituting Ir (λSOC ∼0.4 eV [20, 37, 107]) with Ru and Rh (λSOC ∼ 0.19 eV [18, 21, 22]). Whilethese substituents have similar values of λSOC and are both 4d ions withcomparable values for U [108, 109] and ionic radii [110], they are other-wise distinct: Ru has one less electron than Rh and is therefore associatedwith a markedly larger impurity potential. We will show through supercelltight-binding model calculations that this leads to a pronounced contrast inthe consequences of Rh and Ru substitution: the larger impurity potentialassociated with Ru precludes a significant reduction of the valence SOC.By comparison, Rh is electronically more compatible with Ir, facilitating asuccessful dilution of SOC. We measure this evolution directly, through or-bital mixing imbued by SOC, manifest experimentally in the photoemissiondipole matrix elements. To comprehend all aspects of the MIT observedhere for both Rh and Ru substitution, we consider individually the effectsof filling (Fig. 3.1), correlations/bandwidth (Fig. 3.2), and spin-orbit cou-pling (Fig. 3.6 and Fig. 3.7), ultimately concluding that the transition inSr2Ir1−xTxO4 is a spin-orbit controlled MIT.Having highlighted the three relevant aspects of the MIT, we begin our60Figure 3.1: Dependence of the MIT on Rh and Ru substitu-tion. a-d ARPES spectra along Γ−X for the pristine sample,xRh = 0.22, xRu = 0.40 and xRu = 0.20, respectively. e and fshow Fermi surface maps for x = 0.22 Rh and x = 0.40 Ru. Thesizes of the pockets are indicated with white lines. Fermi surfacemaps are integrated over 50 meV. All data taken at hν = 64 eVwith temperatures between 120 K and 150 K for x ≤ 0.10, andbelow 40 K otherwise.disquisition by showcasing the changes that each substituent introduces tothe electronic structure of Sr2IrO4 as measured by ARPES. We investigatesamples with various nominal concentrations of Ru and Rh, for which detailsof growth and consistency are given in Section 3.A.1. Fig. 3.1a-d summarizeARPES spectra for x = 0, xRh = 0.22, and xRu = 0.20, 0.40. As reportedpreviously [25], the pristine sample supports an energy gap, with a bandmaximum at X at a binding energy of around Eb = 0.25 eV. When sub-stituting Rh, spectral weight appears at the Fermi level for concentrations61x & 0.13, signalling the formation of a metallic state. This observation isin line with previously reported ARPES [104–106]. This is exemplified byour xRh = 0.22 data, shown in Fig. 3.1b,e. At comparable values of xRu,the system remains insulating (cf. xRu = 0.20 in Fig. 3.1d), and only bygoing as high as xRu = 0.40 (Fig. 3.1c,f) do we find that the MIT has beentraversed [111–113], consistent with transport measurements [112].Within the metallic phase, the Fermi surface volume provides a directmeasure of the hole doping introduced by the impurity atoms. We report aBrillouin zone coverage of (16±2)% and (46±5)% for Rh and Ru respectively,with the uncertainty arising from the ambiguity caused by the broad natureof the states. These surface areas correspond to a nominal doping of 0.16±0.02 holes (at xRh = 0.22) and 0.46±0.05 holes (at xRu = 0.40), per formulaunit. Note that there is no apparent transition from a small to a large Fermisurface as is observed in the cuprate superconductors [114–116] for the widedoping range we study, and the Fermi surface size n progresses with thedoping p as n = p (due to there being two iridium atoms per unit cell). Apossible reason for this may be found in the fact that whereas in the cupratesspecific order such as antiferromagnetism is required to fold the bands, thisdistortion is already present in Sr2IrO4 through the staggered rotation ofthe octahedra. To within our level of certainty, each impurity atom thencontributes approximately one hole carrier, with Ru contributing a slightlylarger number than Rh. This observation runs contrary to the expectationsfor a FC transition: despite contributing at least as many holes as Rh,the MIT critical concentration required for Ru is roughly double that ofRh. This precludes a transition described in terms of filling, despite earlierreports to the contrary [104–106].As the concentration of Ru increases, the octahedral distortions reduceslowly until a structural transition is observed at x > 0.5. The octahedraldistortions in Sr2IrO4 cause hybridization between the dxy and dx2−y2 or-bitals, which influences the overal bandwidth of the system. An explanationin terms of modification to the crystal structure however can be equallyexcluded, since changes to the TM-O bond length and octahedral distor-tions are minimal (a reduction from 12◦ to 10◦) up to the concentrations62Figure 3.2: ARPES linewidth evolution with substitution. En-ergy distribution curves (EDC’s) for Ru a and Rh b substitutedsamples, taken at the momentum with the leading edge closestto the Fermi energy. Photon energies and temperatures for theEDCs are the same as in Fig. 3.1 c Momentum distributioncurve (MDC) curves for xRu = 0.40 and xRh = 0.22. d MDCfits for xRu = 0.40 and xRh = 0.22. MDC data shown in c,dwere taken using hν = 92 eV at a temperature of 20 K.used in our study [112]. The changes to bandwidth that are associated withsuch distortions are expected to be negligible, as the effect of the full 12◦is to reduce the bandwidth by at most 4% [117]. Furthermore, a reductionof distortions would increase the bandwidth, opposite to our observations.Equally, recent studies regarding disorder in Mott systems point out thatsuch effects could push the critical concentration to lower values [113, 118].Both disorder and the progression of octahedral rotations would thus predictthe opposite trend (xc,Ru < xc,Rh) to what we observe.Looking beyond the disparate critical concentrations associated with Ruand Rh substitution, analysis of the ARPES spectral features allows for a63more thorough comparison of these materials to be made. The selectedenergy distribution curves (EDCs) cut through the valence band maximumfor each doping in Fig. 3.2a (Ru) and b (Rh), reflecting the evolution of eachmaterial across the MIT. This coincides with a definitive Fermi level cross-ing in the EDCs of Fig. 3.2a and b. By observing which samples are metallicand insulating in our experiments, we can infer xc to be within the range setby highest insulating and lowest metallic concentration. This yields a rangeof possible critical concentrations as xRh = 0.13±0.03 and xRu = 0.30±0.10.These values are limited by the available samples, but match previous pho-toemission work on the Rh substituted compound [104–106]. Ru-substitutedsamples have not previously been studied by photoemission, as such no di-rect comparison to literature can be made. The values for the Ru dopedsamples do however agree with available transport data [112, 119]. As theinterpretation of EDCs lineshape is non-trivial [120], we turn to an analysisof momentum distribution curves (MDCs) for a more quantitative analysisof the evolution of correlation effects. The MDC linewidth is directly relatedto the state lifetime, and by extension to both electronic interactions anddisorder [86, 121, 122]. Two representative MDCs are shown in Fig. 3.2cfor xRh = 0.22 and xRu = 0.40. Widths from these, and other MDCs alongthe dispersion, are summarized in Fig. 3.2d. As can be inferred by the com-parison of data from 20 K and 150 K, correlations – rather than thermalbroadening – are the limiting factor in determining the MDC linewidth.Consideration of both xRu = 0.40 and xRh = 0.22 reveal remarkably similarinteraction effects in the two compounds, despite their significant differencesin composition and disorder. In addition, while spectral broadening at highbinding energies precludes a precise evaluation of the bandwidth, we es-timate the latter to be constant to within 10% over the range of Rh/Ruconcentrations considered.3.3 Spin-orbit couplingThrough study of the Fermi surface sizes, and MDC widths, we have thusdetermined that while doping effects are comparable for Ru and Rh, similar64correlated metallic phases are observed at very different concentrations. Torectify this apparent contradiction, one must consider the context of thepresent MIT: it has been proposed that the correlated insulating phase inSr2IrO4 is stabilized by the strong spin-orbit coupling. This motivates con-sideration of the role SOC plays in the MIT for both Ru- and Rh- substitutedcompounds. The low-energy influence of SOC can be characterized by aneffective value in the valence band, determined by the hybridization betweenatomic species as demonstrated in Refs. 123, 124. This effect could causea reduction of SOC effects in the valence band as a function of (Ru,Rh)substitution. We find the reduction of SOC to be strongly dependent onthe presence of an impurity potential, which limits hybridization of hostand impurity states, ultimately curtailing the dilution of SOC effects (seeSection 3.3.1). Various works report some form of electronic phase sepa-ration in the Ru doped compounds [125–127]. In particular in Ref. 126,Raman scattering shows the coexistence of multiple electronic states. Suchelectronic phase separation cannot be caused by structural phase separation,as Z-contrast (Z being the atomic number) scanning transmission electronmicroscopy (STEM) shows a homogeneous distribution of Ru dopants [119].This suggests that electronically, the Ru ions form a separate manifold, awayfrom the Ir bands. A SOC may therefore be more effective for Rh, providinga natural explanation for their disparate critical concentration in substitutedSr2IrO4 compounds. In the following sections we will quantify the influenceof an impurity potential on the mixing of spin-orbit coupling in the valenceband.Firstly we will provide more quantitative evidence for the existence ofthe impurity potential. Using density functional theory (DFT), at x = 0.25substitution, in Fig. 3.3a we observe good overlap between the Rh and Irprojected density of states (DOS). This can be compared against the samescenario for Ru in Fig. 3.3b, where the substituent DOS is found to alignpoorly with Ir. Such an offset, observed most clearly through considerationof the centre of mass of the Ru-projected DOS, has been reported previouslyfor similar substitutions (Co into Fe superconductors) [128, 129]. Calculat-ing the band’s centre of mass in terms of the projected densities of states for65Figure 3.3: In a and b, an analysis of the impurity potential of Rhand Ru in Sr2IrO4 is plotted, as calculated by density-functionaltheory. The grey background represents the total DOS, normal-ized by the number of TM sites. The black curves show the Irprojected DOS per Ir ion in the 25% substituted calculation,while the orange and green colored curves reflect the projectedDOS per substituent ion for Rh and Ru respectively. The arrowsindicate the centre of mass for the projected bands.both, we find an impurity potential for Ru of 0.3 eV, which is close to thenumber found in [128, 129] (0.25 eV), and agrees with Wannier calculations(0.2 eV) performed on the same supercells. This establishes a reasonablestarting point from which we can explore the influence of doping on SOCeffects in more detail.3.3.1 Spin-orbit mixing and the impurity potentialTo illustrate how the impurity potential and spin-orbit coupling on twoseparate sites combine, we have distilled the phenomenon of mixing into asimple and more insightful model. We will demonstrate the sensitivity ofhybridization to the impurity potential strength: strong hybridization pro-duces mixed states that host an effective spin-orbit coupling derived fromthe atomic composition. The model used in this simple case contains onlytwo atoms, with two orbitals (dxz and dyz) each, and only one-dimensionalhopping along the z-direction. The atoms have different spin-orbit coupling(λ1 = 0.45, λ2 = 0.19), and atom 2 has an impurity potential ε, representedschematically in Fig. 3.4f. Since the Hamiltonian for spin-orbit coupling inthe basis of dxz and dyz orbitals is degenerate in the spin degree of free-66Figure 3.4: Influence of the impurity potential on spin-orbit coupling in the valence band demonstratedin a 2-atom toy model. a Bandstructure for the system that is schematically depicted in f. Thebands (1,2) and (3,4) are split by spin-orbit coupling, the splitting corresponds to the average valuebetween the two atoms λ¯. The dashed black line illustrates the bands in the absence of spin-orbitcoupling. An arrow at Γ indicates the position at which the band energies are plotted in c. b Thesame bandstructure with an impurity potential of  = 0.3. The splittings are no longer the samesize, and are k-dependent. c The band energies at Γ as a function of the impurity potential. Thered and purple arrows correspond to the impurity potentials of panels a and b, the splitting for theupper and lower manifold are indicated. d The progression of the splittings ∆A and ∆B as a functionof the impurity potential. e The orbital weight projected onto atom 1 (solid) and atom 2 (dashed)for each of the four bands. g The splitting of the lower manifold ∆A (green) plotted together withλeff , calculated from the orbital weights (see text). h The splitting of the lower manifold ∆A (green)plotted together with the analytic form discussed in the text.67dom, we only consider the spin up part of the Hamiltonian. This yields theHamiltonian:H =0 iλ12 −2t cos k 0−iλ12 0 0 −2t cos k−2t cos k 0 ε iλ220 −2t cos k −iλ22 ε . (3.1)The band structure along from k = 0 to k = pi is presented in thepresence (Fig. 3.4a) and absence (Fig. 3.4b) of an impurity potential ofε = 0.3. The band structure without spin-orbit coupling (black dashed line,both panels) is plotted for reference in Fig. 3.4a,b. As indicated by thepurple contours, upon including SOC a 4-fold degeneracy is lifted, with aconstant splitting along the dispersion. This splitting consistent with thatexpected for a pure crystal with λ¯ = (λ1 + λ2)/2.Introducing a relative impurity potential of 0.3 eV between the two sites(Fig. 3.4b), the SOC-splitting persists, but is now momentum-dependent.Furthermore, the splitting between the lower (1,2) and upper (3,4) bandsbecomes inequivalent.To investigate the evolution of this system as a function of impuritypotential, we plot the eigenvalues at k = 0 for a range of ε in Fig. 3.4c.As indicated by the black dashed lines, the eigenvalues at k = 0 are de-fined largely by the onsite energies, with an avoided crossing due to kineticterms in the Hamiltonian. Labelling the lower and upper bands as A andB, respectively, we plot their splittings, ∆A,B vs ε in Fig. 3.4d. These split-tings derive from spin-orbit coupling, but only in the limit of |ε| >> t do∆A,B → λ1,2. In the opposite limit (|ε| >> 0), both splittings converge to λ¯introduced previously. This asymptotic behaviour reflects the essential roleof hybridization in mediating a mixing of spin-orbit coupling strengths inimpurity-substituted systems.To make this point more definitive, we include a comparison of orbitalcompositions on each of the four bands at k = 0 in Fig. 3.4e. In each case,for ε ∼ 0, the eigenvectors are an equal mix of atom 1 (solid) and atom 268(dashed). This even mixing changes rapidly with finite ε, saturating in thelimit of large impurity potential. We attempt to relate the state compositionfor the lower two states to an effective spin-orbit coupling by calculating:λeff (k) = λ1|ψ1,A(k)|2 + λ2|ψ2,A(k)|2. (3.2)This quantity is the average spin-orbit coupling on the lower band (A)weighted by the atom projected wavefunctions. It is plotted in Fig. 3.4g(white dashed line), and exactly traces the splitting (green curve) of thelower (A) state. This observation provides evidence for the idea that spin-orbit coupling in the valence band can be regarded as arising from a mixturebetween states of different atomic character, the mixture being influenced byhybridization, controlled by hopping and the impurity potential. A relevantscale for the process can be found in Fig. 3.4h, where the splitting is plottedas a function of the normalized ε/4t, as well as the function x√1+x2(whitedashed). The latter is the functional form of the orbital weight for a two-site model with an impurity potential and hopping, which is derived inSection 3.A.4.We have thus shown that the effects of spin-orbit coupling as they occurin the valence band, are sensitive to the presence of different atomic species,subject to hybridization and impurity potential strength. This shows that itis possible to mix and tune spin-orbit coupling in the valence band by usingthe appropriate atomic species and controlling the hybridization through theimpurity potential. It gives a direct handle on influencing spin-orbit couplingas a continuous parameter in an experimental scenario. It should be pointedout that the results of this model apply directly to other systems too, becausethe Hamiltonian presented does not assume any material specific properties.3.3.2 Supercell calculations and band unfoldingThe model presented in Section 3.3.1 to illustrate the mechanism of spin-orbit mixing, can be made quantitative for the Ru/Rh iridates throughconsideration of impurity-substituted models. This is carried out throughdevelopment of a supercell tight binding (TB) model. We expand the single69iridium TB Hamiltonian derived in Section 3.A.3 to a 64 site supercell, ran-domly substituting a fraction x of sites with an impurity atom. A schematicof the generated model is given in Fig. 3.5aFor the sake of simplicity, the impurities are assumed to differ from Irin only their λSOC (0.19 eV for both Ru and Rh, 0.45 for Ir), and onsitepotential (0.0 eV for Rh and Ir, 0.25 ± 0.05 eV for Ru). Similarly, octahe-dral distortions and electron correlations are neglected to better illustratethe energy shift of the jeff states. We have used the unfolding method[84, 85, 130, 131] (described in detail in Section 2.2) to project bands intothe original Brillouin zone. This method makes use of the phase proper-ties of Bloch waves to define a weight for each band representative of theband’s projection onto the extended Brillouin zone. We average 200 pos-sible permutations to reconstruct a stochastic representation of the dopedsystem. The spectral function is found to have converged after averagingover 200 configurations, with no further appreciable changes observed forlarger samplings. We observe a smooth evolution of effective SOC in thissystem, which depends strongly on the impurity potential.Results of the calculations for the Rh substituted case are summarizedin Fig. 3.5b-e. The splitting at k =(3pi4a , 0)(indicated with a red arrow)allows for direct estimation of the effective SOC. This k-point is chosenbecause later experimental results will be presented at this Brillouin zonelocation. Also indicated are the nominal values for 100% Ir (purple) and100% Rh/Ru (yellow). Experimentally, such a straightforward extraction ofSOC is not possible due to the dominant effects of linewidth which precludedetailed characterization of the level spacing. Furthermore, direct inferencewould require a more detailed analysis of the effect of correlations, as corre-lations enhance the splittings created by spin-orbit coupling, dependent onthe crystal momentum [132, 133]. The calculated spectral functions indicatea monotonic reduction of this splitting as the substituent concentration isincreased. Aside from the reduction of spin-orbit coupling, spectral broad-ening is observed at several points in the Brillouin zone. This broadening iscaused by the disorder introduced by the Rh atoms, and is predominantlyfound in the places where the spin-orbit coupling alters the dispersion most.70Figure 3.5: Overview of the supercell calculations describing spin-orbit coupling dilution. a Schematic overview of the supercellgeneration, indicating the extension of the orbital basis set overthe larger unit cell. As indicated by the image colouring, afraction of atoms is assigned reduced spin-orbit coupling andan impurity potential of εi = 0.3 (Ru) or εi = 0 (Rh). b-eCalculated averaged unfolded spectral function for Rh concen-trations of x = 0.0, 0.1, 0.2 and 0.3. The high symmetry pointsare indicated in Fig. 3.9d. The arrows indicate the SOC in-duced splitting (red), with the associated splittings of the endlimit compounds x = 0 (purple) and x = 1 (yellow) added forreference.71Figure 3.6: a Cuts (EDCs) for different concentrations of dopants atthe position of the red arrow in Fig. 3.5 [k =(3pi4a , 0)]. The sub-stituted atoms have different SOC and on-site energy. We useλSOC = 0.19 eV and εi = 0.0 eV for Rh (black), while we useλSOC = 0.19 eV and εi = 0.25 ± 0.05 eV for Ru (red). b Pro-gression of the splitting between the outermost peaks for simu-lations in a for Rh (black markers) and Ru (red markers). ForRh, a linear interpolation is plotted between the end membersof the phase diagram. For Ru, the resulting range of splittingfor εi = 0.25± 0.05 is indicated by red shaded rectangles. Thecritical concentrations obtained from our measurements are in-dicated by the black (Rh) and red (Ru) vertical lines. The blueshaded area indicates the inferred λc = 0.42± 0.01.Such disorder is much stronger in the calculated spectra for Ru doping.We extract the effective value of SOC by fitting EDCs for each impu-rity concentration. The results are summarized in Fig. 3.6. The changein splitting is seen clearly in Fig. 3.6a, where we present a series of EDCsat k =(3pi4a , 0), for models with a non-zero on-site impurity potential (Ru,red), and those without (Rh, black). This doping dependence is summa-rized in Fig. 3.6b. The right vertical axis reflects the splitting observed atk =(3pi4a , 0), and the left the value of λSOC that would produce the cor-responding splitting in a model without substitutions (i.e. for an overalluniform value of λSOC). This second axis serves to illustrate the effectivespin-orbit coupling caused by substitution of Ir with Rh and Ru. Fromthe progression in Fig. 3.6b it is evident that Rh should dilute SOC more72efficiently than Ru: the black markers trace the interpolation between thevalues of Ir and Rh, indicated by the grey line. Meanwhile the modelled im-purity potential for Ru (0.25±0.05 eV) prevents successful dilution of SOC.The results in Fig. 3.6b suggest that the different critical concentrations forthe two substituents can be attributed to a common parameter: a value forspin-orbit coupling of λc ∼ 0.42 ± 0.01 (indicated as a blue shaded area inFig. 3.6b) yields critical concentrations (xRh ∼ 0.15 and xRu ∼ 0.3) thatfit well with our experimental observations. Theoretical results presented inRef. 134 suggest that SOC in Sr2IrO4 is only marginally above the thresh-old for the insulating state, and that such a small change could drive thetransition. The dilution of spin-orbit coupling is therefore found to providea compelling theoretical picture of the transition.3.4 Experimental observation of SOCHaving demonstrated this evolution of SOC via substitution and its abilityto provide a natural explanation for the transition in Sr2Ir1−xTxO4, we aimto substantiate these predictions experimentally. To establish a convenientmetric for SOC, we leverage the symmetry constraints of the photoemissionmatrix element. Dipole selection rules allow transitions from only certainorbitals: since dxz (dyz, dxy) is even (odd) in the experimental scatteringplane, states composed of this cubic harmonic are only observable with pi-(σ)-polarization. As SOC mixes these orbitals into linear combinations pre-scribed by the jeff construction [25], we quantify SOC by comparing theratio of even/odd states at strategically chosen points in the Brillouin zonewhere these symmetry-based selection rules are most well defined. In theabsence of SOC, the state along Γ − Xx (defined in Fig. 3.7) in Sr2IrO4would be of pure dxz character: any photoemission from this state usingσ-polarization must be due to the admixture of dyz and dxy introduced bySOC. More quantitatively, of interest here is the value of Mσx , the matrixelement at the Xx point, which we normalize in our results through divisionby Mσy . A simulation of this quantity based on an ab-initio tight bindingmodel for Sr2IrO4 with variable spin-orbit coupling is shown as a black solid73Figure 3.7: Observation of the reduction of SOC via theARPES dipole matrix element. a-d Constant energy mapsfor different concentrations of xRh, using σ-polarized light. Theconstant energy maps are integrated over 150 meV to improvenumerical accuracy, and taken at an energy such that the sizeof the pocket around X is the same for all concentrations. Therelevant states used for the analysis are indicated using the redboxes, and their integrated values are shown within. All dataare taken at 64 eV, with temperatures at 120 K for x = 0 andxRu = 0.1, 70 K for xRh = 0.1, and 20 K xRh = 0.16, all chosento mitigate the effects of charging.74Figure 3.8: Inference of the value of spin-orbit couplingthrough modelling of the matrix element. Calculated ra-tio of matrix elements for a model of Sr2IrO4 (details in Sec-tion 3.A.3), plotted as a function of spin-orbit coupling (blackcurve). The coloured markers indicate the ratio of the experi-mental values shown in panels a-d of Fig. 3.7. The error barsare calculated from the standard deviation over the integratedrange in energy. The top axis indicates the substitution re-quired to produce the spin-orbit coupling value on the bottomaxis, predicted by the supercell calculations in Fig. 3.6e.line in Fig. 3.8. The model takes into account effects of experimental geom-etry as well as photon energy and polarization; for further details refer toSection 3.A.3. The curve shows a clear decrease of Mσx /Mσy as a functionof spin-orbit coupling, demonstrating the possibility for a direct measure ofλSOC via ARPES. The model omits electron-electron interactions, whichcould in principle change the mixing of orbitals, and hence the observedSOC. However, since we measure the relative change to the pristine sample,our conclusions are robust against such an overall change.Motivated by the supercell calculations, we investigate the progressionof Mσx /Mσy experimentally in a series of Rh and Ru substituted samples.75In Fig. 3.7a-d we plot constant-energy contours for each of the concentra-tions, as recorded with σ-polarized light. To compare the different samples,we consider constant energy maps at the energy which places the state ofinterest at kx =(3pi4a , 0). We subtract a uniform background based on the av-erage of the intensity in areas where no states are present. Then, integratingand dividing the ARPES intensity within the indicated regions of Fig. 3.7a-dyields the ratio Mσx /Mσy . We can proceed to make a quantitative connectionwith an effective spin-orbit coupling strength by plotting the experimentaldata points alongside the simulated curve in Fig. 3.8. The latter has beennormalized to the experimental data for pristine Sr2IrO4, allowing for aneffective λSOC strength to be extracted for the Rh/Ru substituted samples.This analysis yields λSOC values of 0.443, 0.424, and 0.408 respectively. Aconnection to the supercell calculations can be made through these λSOCvalues: the associated impurity concentrations in Fig. 3.6e agree remarkablywell with the actual experimental values, made explicit in the case of Rhwith the top horizontal axis of Fig. 3.8. This confirms the premise of oursupercell model and the sensitivity to the impurity potential for successfuldilution of λSOC . In connection to the MIT, the λc = 0.42 ± 0.01 eV atxRh = 0.15 obtained from Fig. 3.6e is overlain in Fig. ConclusionGenerally speaking, λc is a function of filling, U , bandwidth, disorder, amongothers, and SOC represents but a single axis within a higher dimensionalphase space. As filling, distortions, and disorder may be anticipated to ex-pedite the metal-insulator transition in Ru-substituted samples, SOC seemsalone capable of explaining the dichotomy in xc observed for Ru and Rh.This indicates the critical role of SOC in the MIT of Sr2Ir1−xTxO4 for bothRh and Ru substitution.The combination of SOC-sensitive techniques and the comparison of Ruand Rh substituted samples has put us in a unique position to comment onthe role of SOC in the metal-insulator transition of Sr2IrO4, demonstratingfor the first time an SOC controlled-collapse of a correlated insulating phase.76Through doing so, as an important corollary to these results, our workconclusively establishes Sr2IrO4 as a relativistic Mott insulator.Additionally, we note that the investigation into mixing spin-orbit cou-pling discussed in Section 3.3.1 was calculated for a generic two-site Hamil-tonian. As such this mechanism pertains to other systems in which thistype of physics appears, and has broad ranging implications for attempts totune and tailor spin-orbit coupling. Controlling spin-orbit coupling is a ma-terial properties design challenge at which many attempts have been made,with varying success. This result not only sets the boundary conditions forsuccessful dilution of spin-orbit coupling, it also explains what the control-ling parameters are that influence the resultant spin-orbit coupling in mixedsystems. It gives a direct explanation of the effects seen in systems suchas Ga1−xBixAs, where Bi readily enhances the effects of SOC [135]. It alsosheds light on attempts to drive a topological to trivial transition in topo-logical insulators by reducing spin-orbit coupling through substitution [136–140]. Moreover, the sensitivity of this phenomenon to an impurity potentialhas implications for ongoing efforts to enhance SOC effects in grapheneand related systems through adatom deposition and other proximity-relatedtechniques [123, 124, 141–143]. In particular, experimental results have beenunable to observe the predicted enhancement of spin-orbit coupling so far.It is possible that an impurity potential, or small hopping parameter limitsthe hybridization between the adatom species and the graphene lattice, thusnot effectively facilitating enhanced effects of spin-orbit coupling. In sum-mary, this work has important implications for experiments that attempt tomodify spin-orbit coupling, and should serve as a practical guide for futureendeavours.773.A Appendices3.A.1 MethodsSingle crystals of Sr2Ir1−xRhxO4 were grown with nominal concentrations ofxRh = 0.0, 0.10, 0.16, 0.22 and measured with electron probe microanalysisto be within 0.01 of their nominal concentration. Crystals of Sr2Ir1−xRuxO4were grown with nominal concentrations of xRu = 0.10, 0.20, 0.40. Qualityof the Ru doped samples was assured by comparing magnetization measure-ments to available literature [112, 119]. Chemical homogeneity of the Rudoped samples was ensured using Z-contrast STEM. Measurements werecarried out at the SIS beamline at the Swiss Lightsource (Rh substitutedsamples) and at the Merlin beamline at the Advanced Lightsource (Rh andRu substituted samples). All measurements were done on freshly cleavedsurfaces, where the pressure during measurement and cleaving was alwayslower than 3.3 · 10−10 mbar. Measurements used for inference of spin-orbitcoupling values were performed with 64 eV photons, using light polarizedperpendicular to the analyzer slit direction (σ-polarization). The rotationaxis of the manipulator for the acquisition of the Fermi surface was parallelto the slit direction. The sample was mounted such that the Ir-O bonds(Γ − X) were aligned to this axis of rotation. Temperatures were chosenas low as possible while mitigating the effects of charging and are reportedin the figure captions. A tight-binding model was constructed from a Wan-nier orbital calculation using the Wannier90 package [144]. The Wannier90calculations were performed on results from density functional theory cal-culations done with the Wien2k package [75, 145]. The DOS calculationspresented in Fig. 3.6 were performed with the Wien2k package. The super-cell configuration assumed a single layer with 8 TM ions per unit cell. Thepresented results at x = 0.25 are similar to those found for x = 0.125 andx = 0.5.783.A.2 Observation of SOC through the dipole transitionmatrix elementIn the Section 3.4, we have demonstrated the ability to quantify spin-orbitcoupling by taking advantage of the dipole selection rules associated withthe photoemission process. To illustrate how the effects of SOC are manifestin polarization-dependent ARPES on Sr2IrO4, it is instructive to consider aminimal tight-binding model consisting of a t2g basis set with nearest neigh-bour hopping and spin-orbit coupling. Although too simple to capture thefull detail of Sr2IrO4, this model serves to highlight the general concept ofthis technique and its broad applicability beyond this particular set of ma-terials. The calculations presented in the Fig. 3.8 use the more elaboratemodel described in Section 3.A.3. Suitable points in k-space are fixed by thesymmetry of the lattice, as can be illustrated by the orbital-projected bandstructure plotted in Fig. 3.9 as a function of spin-orbit coupling. The disper-sion of the model system is plotted for several values of λSOC in Fig. 3.9a-c.The colour scale encodes the expectation value of 〈L · S〉. SOC can be seento cause both an increase in the entanglement of spin and orbital angularmomentum, as well as a splitting between the bands originating from dxzand dyz orbitals that eventually form the jeff = 3/2 and 1/2 states. Thefully entangled jeff = 3/2 and 1/2 have 〈L · S〉 = −1/2 (red) and 1 (blue),respectively. Note that these values are the negative of the expectation valuefor ` = 1, where j = 3/2 (1/2) yields 〈L · S〉 = 1/2 (−1), since the jeff statesare derived from ` = 2 states. Any deviations from these numbers reflectthe competition between SOC and kinetic terms.The orbital mixing resulting from the entanglement of spin and orbitalangular momentum can be seen in Fig. 3.9e-g, where the colour scale nowreflects the orbital projection along the same k-space path as before. Neg-ligible mixing is observed for λSOC = 0.01, becoming quite prominent byλSOC = 0.4 eV. As with the spin-orbital entanglement, except for extremelylarge values of SOC, the mixing is k-dependent as orbital angular momen-tum remains partially quenched. Directing our attention towards the (pi, 0)point on the edge of the Brillouin zone, we can further explore the evolutionof the orbital character of the valence band. Although in the absence of79Figure 3.9: Orbital mixing between t2g orbitals. a-c Bandstructures for λSOC 0.01, 0.1 and 0.4 eV along thek-path indicated in d; colour encodes the expectation value of 〈L · S〉 for each band. In all calculationsthe Fermi level was set by a total occupation of 5 electrons per site. d Fermi surface plots for λSOC0.01 (left) and 0.4 eV (right), with band character colour encoded. e-g The same band structure as ina-c, where the colour now indicates the weight of dxy, dxz and dyz for each state (the colour mappingis defined in the triangle in the top left). h The orbital composition of the state and matrix elementratio as a function of SOC, at the k-point indicated by a grey box in d and g. i The SOC inducedsplitting at Γ denoted by the grey double arrow in g.80SOC this state is of pure dxz character, substantial admixture of dyz anddxy weight is observed in Fig. 3.9h for large SOC. Providing a connectionto the experiment, the ratio of photoemission intensity between the (pi, 0)and (0, pi) points (Mσx /Mσy ) is also plotted on the same axes. To withina global renormalization factor, this quantity is found to follow the ratioof dxz/dyz weight precisely, demonstrating the correspondence between thisexperimental signature and the orbital structure of the underlying electronicstates. Consequently, Mσx /Mσy provides an experimental measure of SOC inthese states. This is particularly useful for the study of Sr2Ir1−xTxO4 (T =Ru,Rh) where we might expect the effective SOC to reduce with increasingsubstituent content.We stress that this technique is not limited to this particular material,and could be used in other systems that are subject to spin-orbit couplingthat causes mixing between even and odd orbitals, for example in otheriridates [146, 147], ruthenates [14, 21] or the iron-based superconductors[23, 24]. Although the general principle of the idea is fully captured by thissimple model, a better effort in predicting the intricacies of this ratio can bemade by considering the full nature of the system, including its octahedraldistortions and higher order hopping elements, as is done in the next section.3.A.3 Tight binding and matrix element modellingQualitatively, the extent to which the matrix elements can be relied uponto convey information regarding orbital mixing is found to be independentof the details of a given model. For example, the Hamiltonians describedin [148] as well as [44] have been tested, and found to be consistent withthe intensity variations observed experimentally. To provide a quantitativeconnection to the experimental Mσx /Mσy values, a more sophisticated modelthan that presented in Section 3.A.2 is required. We describe here the modelused for matrix element calculations in Fig. 3.8, which relies on DFT andmaximally localized Wannier functions to generate a tight-binding Hamilto-81Figure 3.10: Overview of the model used in this work. The high sym-metry points are indicated in Fig. 3.9d. a Initial DFT calcula-tion in the I4/mmm space group (black), overlaid on the calcu-lated tight binding band structure as extracted from Wannierorbitals (grey). b Calculated band structure for the formerHamiltonian after it has been doubled, rotated, and strippedof out-of-plane hopping matrix elements (grey) c Calculatedband structure for the rotated model to which atomic spin-orbit coupling has been added (coloured). The experimentalconfiguration is represented schematically in panel d.nian [75, 144, 145]. DFT calculations were performed with the Wien2Kpackage [75], under the generalized gradient approximation (GGA). Tominimize the size of the basis set for Wannier down folding, octahedraldistortions of Sr2IrO4 were suppressed at this stage, allowing for use of theI4/mmm space group. We used the lattice parameters reported in Ref. 16,and kept the Ir−O distance fixed to conserve overall bandwidth. Max-imally localized Wannier functions for five Ir 5d orbitals were calculatedusing the Wien2Wannier [145] and Wannier90 packages [144]. The resultingtight-binding model was truncated beyond fifth nearest neighbour hoppingintegrals, with matrix elements smaller than 0.5 meV suppressed. The as-sociated bandstructure is plotted in grey in Fig. 3.10a, and agrees well withthe full DFT band structure (black curve). The quasi-two-dimensionalityof the electronic structure has been strictly imposed through suppressionof out-of-plane hopping terms. Staggered octahedral rotations of T=11.5degrees can then be introduced as outlined in Ref. 148, recovering the true82I41/acd of both the pristine and substituted Sr2IrO4 lattice [16, 112]. Theband structure in Fig. 3.10b reflects the effects of these distortions. Finally,spin-orbit coupling for the Ir d-orbitals has been added as:HˆSOC = λ Lˆ · Sˆ, (3.3)resulting in the band structure plotted in Fig. 3.10c with the projection of〈Lˆ · Sˆ〉 indicated by the colourscale. While a jeff = 1/2 (〈Lˆ · Sˆ〉 = 1) andjeff = 3/2 (〈Lˆ ·Sˆ〉 = −1/2) manifold can be defined, significant hybridizationbetween the two persists due to the comparable energy scales of bandwidthand λSOC . To best match our experimental data we have set λSOC = 0.45,consistent with other reports [37, 107].The tight-binding model defined here was used for the simulation ofthe ARPES matrix elements over a range of λSOC values, producing theblack curve plotted in Fig. 3.8. The method uses the transition probabilitydescribed by Fermi’s golden rule [86, 121]:∣∣∣Mki,f ∣∣∣2 ∝ ∣∣∣〈φkf |r · ε|ψkf〉∣∣∣2 , (3.4)where φki is the initial state, derived from our tight binding model, ε is thelight polarization vector, and φkf is the final state wavefunction. The finalstate is assumed to be free-electron like, well justified by the high photonenergies hν ≈ 50 eV [86]. Fig. 3.10d illustrates the various components ofthe process. The matrix element can then be further calculated similar tothe procedure outlined in 149. Matrix element calculations were performedusing the chinook package [90].3.A.4 Orbital weight for a two-site model with impuritypotentialIn Section 3.3.1 we discuss control of spin-orbit coupling dilution through theimpurity potential. Here we derive the orbital weight for an eigenstate of a83two-site model, as controlled by hopping parameter t and impurity potential. We take for the Hamiltonian:H =( tt −), (3.5)which we normalize by the hopping strength t and write:H =(x 11 −x), (3.6)with x = /t. The eigenvalues of this matrix are given by:λ± = ±√1 + x2, (3.7)and eigenvectors:v± =1√A(1λ± − x), (3.8)with A the norm of the vector. We can now find the orbital weight as afunction of the normalized impurity potential by taking the absolute valuesquared of one of the elements in the vector:|v+,0|2 = 1A=12 + 2x2 − 2x√1 + x2 . (3.9)We can rewrite this expression as:|v+,0|2 = x+√1 + x22√1 + x2=12(x√1 + x2+ 1), (3.10)which corresponds to the functional form in the four-orbital model discussedin Section 4Spin and kz ResolvedARPES on Sr2IrO4This chapter describes experiments studying the dispersion perpendicular(kz) to the atomic layers and its effects on the spin-orbital entanglement.With these experiments we test validity of the jeff = 1/2 model, and attemptto compare Sr2IrO4 to the superconducting cuprates, in particular La2CuO4.The two-dimensionality of the system will be investigated using photon en-ergy dependent angle-resolved photoelectron spectroscopy (ARPES). Spin-ARPES will be used to investigate the spin-orbital entanglement in Sr2IrO4,which should give a clear signature in light of the jeff = 1/2 model. We ulti-mately find that the complex electronic structure in Sr2IrO4 cannot simplybe explained as a two-dimensional pseudo-spin 1/2 insulator.4.1 IntroductionSince the discovery of the cuprate superconductors [150], the physics com-munity has put a lot of effort into finding other superconducting transition-metal oxides (TMOs). The first successful attempt was the discovery ofSr2RuO4 [14], but many other compounds with similar properties had beensuggested. Among them Sr2IrO4 [17], with the same structure as La2CuO4and an antiferromagnetic ground state in the pristine “parent” compound.85A difference is that this system has a single hole in the t2g manifold, ratherthan the eg hole in the cuprates. Spin-orbit coupling has been suggestedto play an important role in the insulating ground state of these materi-als, with t2g orbitals possibly entangling into a filled jeff = 3/2, and a halffilled jeff = 1/2 manifold [25]. It was quickly realized that this scenariobrings Sr2IrO4 even closer to the quintessential cuprate superconductor: a(pseudo-)spin 1/2 Mott insulator on a square 2D lattice. Theoretical calcu-lations predicted a superconducting state may exist in such a pseudo-spin1/2 system when the system is electron doped [35], with more sophisticatedanalysis including all t2g orbitals and strong spin-orbit coupling painting asimilar picture [40, 41]. It was moreover found that the excitations of thepseudospins probed by resonant inelastic x-ray scattering (RIXS) are remi-niscent of a Heisenberg model [37, 151], the expected low energy behaviourfor a spin 1/2 Mott insulator [38, 39]. Promising observations were madein experiments: features reminiscent of doped Mott insulators, such as aparticular gap shape and spatial distribution were found in scanning tun-nelling microscopy (STM) [45], and a pseudogap was found in ARPES [44].Stronger evidence was found using surface doped samples: using STM, a gapvery reminiscent of those found in superconductors was observed [46], whilein ARPES a d-wave gap was observed, a classic signature of the cuprate su-perconductors [47]. However, so far no signatures of bulk superconductingbehaviour have been reported in the literature.A possible explanation for the lack of superconductivity may be found inthe multi-band nature of Sr2IrO4: the theoretical models predicting super-conductivity are either done on a pseudo-spin 1/2 model [35], or a system inthe strong spin-orbit coupling limit in which the jeff = 3/2 states can be ef-fectively projected out [40, 41]. Although spin-orbit coupling is large in thissystem (∼ 0.45 eV [20, 37, 107]), it is still modest compared to the overallbandwidth of the t2g bands [104–106]. Furthermore, it can be anticipatedthat the Ir t2g bands have a more significant out-of-plane dispersion: notonly are the 5d orbitals more extended than their 3 and 4d counterparts, thedxz and dyz bands have stronger pi-like bonds between layers (as opposed toδ for the dx2−y2 orbitals in cuprates). In this chapter, we will address these86two arguments: the out-of-plane dispersion will be measured using photonenergy dependent ARPES, and the spin-orbital entanglement will be mea-sured using spin-ARPES. The observations indicate that the kz dispersion inthis compound is indeed significant. Moreover the highest unoccupied statecan be found to be either of jeff = 1/2 or jeff = 3/2 character dependingon kz. We therefore conclude that the occupied states of Sr2IrO4 should bedescribed by a three-dimensional multi-orbital Mott system, rather than a2D, pseudo-spin 1/2 Mott insulator.4.2 kz dispersion in Sr2IrO4In this section we will discuss the out-of-plane (kz-axis) dispersion of Sr2IrO4,considering results from photon energy dependent ARPES experiments. Wefind that Sr2IrO4 has a significant inter-layer coupling, which will help to ex-plain the results using spin-ARPES in Section 4.3, and has consequences forthe description of Sr2IrO4 as a two-dimensional Mott insulator. It helps hereto define exactly what is meant in this work by two-dimensionality. For sucha multi-orbital system we require that no bands significantly change theircharacter, or cross in energy as a function of the perpendicular momentumkz. We will furthermore demonstrate that the electron removal spectrumof Sr2IrO4 is not the single-band spectrum that should be expected from apure jeff = 1/2 model.4.2.1 Body centered tetragonal structureIn order to guide the reader through the various effects arising from thestructure of Sr2IrO4, this section presents a simple tight binding model inthe same structure, to facilitate easy interpretation of the different effects.The structure of Sr2IrO4 is body centred tetragonal, which leads to theappearance of a few characteristic effects that will be discussed here.In Sr2IrO4, the IrO4 octahedra are rotated around their z-axis by 11.5 degin a checkerboard pattern. This implies that two separate iridium ions areneeded to describe the unit cell of Sr2IrO4. The symmetry reduces fromthe I4/mmm to the I41/acd space group, with a c-axis stacking order ex-87Figure 4.1: Constant energy maps for a two-atom model in the bodycentred tetragonal structure. (a) Dispersion of the states alongselected high symmetry points. (b,c) Constant energy contoursat indicated energies. The locations of the cuts are indicatedusing coloured lines in (a).panding the unit cell to four iridium atoms. To demonstrate the effects offolding, we calculate the dispersion for a structure with two iridium atoms.Although we describe two atoms per unit cell, no changes to the hoppingintegrals are made. The result is a dispersion that is mirrored along thezone anti-diagonal ((pi, 0), (0, pi)), folding the bands at the N (pi, pi) point toΓ and vice-versa.The resulting dispersion is reminiscent of the band structure that is88frequently presented for Sr2IrO4. Although this model neglects any changesto the band structure from the symmetry breaking, there is an importanteffect that can be clearly seen in Fig. 4.1(a). The bands originating fromthe N point (pi, pi, 0), disperse with a different period due to the staggerednature of the reciprocal space BZ cells. This causes folded bands at Γ thatdisperse with this doubled periodicity, whereas in the unfolded band modelno such bands were present. The dispersion from the X point (pi, 0, 0) tothe X + Z point (pi, 0, pi) is completely flat in this model only consideringpi hopping. Fig. 4.1(b-c) display cuts at constant energy (CE), effectivelyplotting contours at constant ky = 0 and varying values of kx and kz, in thesame way that data will be presented in the next section. The CE maps areshown for an identical k-range, with locations of relevant high-symmetrypoints indicated. The bands folded from the N point (b) show differentperiodicity from the original band at Γ (c). The reason is that whereas theoriginal bands cut through the centres of the Brillouin zone (BZ) cells, thefolded bands cut in between the staggered BZ cells, causing an apparentdouble periodicity. This is relevant since the interpretation of the electronicstructure of Sr2IrO4 is such that the state at Γ arises from a folded band,which shows such double periodicity. Another feature arising due to thebody centred tetragonal structure are the bands that “wave” in between theBZ cells visible in (c). Both these features are observed and will be discussedin Section Constant energy mapsAs discussed in Section 2.3, assuming a free-electron-like final state, we canobtain the out-of-plane momentum using:~k2⊥2m= Ek + V0 −~k2‖2m. (4.1)The dependence of k⊥ on Ek implies we can measure the dispersion alongkz by changing the photon energy. This section will show the results fromphoton energy dependent measurements to highlight the kz dispersion in89Figure 4.2: Overview of photon energy dependent results in Sr2IrO4.(a-d) Constant energy maps at E = −0.35 (a,b) and E = −0.55(c,d), using σ-polarized light (a,c) and pi-polarized light. In (d)the faint elliptical pockets around Γ are highlighted by thin redlines on the right side of the centre. (e) Spectrum at hν = 100eV (5.29 A˚−1). The red lines indicate the locations of the otherdata in this figure. (f) A series of MDCs from hν = 60 eV(4.18 A˚−1) to hν = 100 eV (5.29 A˚−1), with σ-polarization atE = −0.55 eV. The area encompassed is indicated by a red boxin (c). 90this system, in particular at the Γ point. Measurements presented in herewere taken at the Advanced Lightsource at the Lawrence Berkeley NationalLaboratory at the Merlin endstation. Data are all collected at 150 K, tomitigate the effects of charging of the insulating Sr2IrO4 sample. Photonenergy dependent measurements were done as the beamline specificationswould allow, in this case from 50 eV to 120 eV, enabling the observation ofalmost two full reciprocal lattice cells. The Fermi level was corrected by tak-ing measurements on electrically connected amorphous gold. For the innerpotential, a value of V0 = 11 eV was found by comparing the experimentaldata to the expected periodicity, in good agreement with the result in [152].To provide a guide for the presentation of the acquired photon energydependent data, we plot the band-structure of Sr2IrO4 in Fig. 4.2(e). Thespectrum displays the valence band states at Γ and X. The red lines indicatethe positions of cuts presented in other panels. We continue our overviewof kz-dependent effects by considering the constant energy cuts in Fig. 4.2.Figure 4.2(a) and (b) plot constant energy maps at Eb = 0.35, while (c)and (d) present maps at Eb = 0.55. Data for both σ- and pi-polarizationare shown ((a,c) and (b,d) respectively), with a clear difference in qualita-tive features between the two. Although it is challenging to find a clearperiodic structure, the modulated intensity changes, especially those thatrepeat along kz, are a clear sign of interlayer coupling. The intensity fluctu-ations arise due to the dipole matrix element (see Section 2.3.3), that variesas the composition of the initial state changes along the kz direction. Thephotoemission matrix element is however also dependent on the final state(Section 2.3.3), and there can be a global change to these effects that mayobfuscate the true periodic intensity fluctuations.We now consider the dispersion of the X-states. To this end we show con-stant energy maps at higher energy, E = −0.35 ((a) and (b) in Fig. 4.2. Anintensity pattern appears in these maps too, and furthermore, the perime-ter of these states moves inward and outward going between the Γ and Zpoints. As discussed in the previous section, this is an effect that can beexpected from the body centred crystal structure that Sr2IrO4 assumes. Itcauses a waving pattern between the different unit cells, which is discernible91in Fig. 4.2.A careful look at the state around Γ in Fig. 4.2(d) shows elliptical pock-ets that open and close with a periodicity twice that of the BZ. As discussedin Section 4.2.1, this is exactly the periodicity that a folded band in thebody centred tetragonal (BCT) structure would yield. Continuing our dis-cussion with the maps shown for σ-polarized light (Iσ), a careful inspectionof Fig. 4.2(a) and (c) yields that there is also a pattern of nodes, but itis shifted with respect to the pi-polarization (Ipi) maps. The closing of theelliptical pockets is highlighted more clearly in Fig. 4.2(f), where a seriesof momentum distribution curves (MDCs) is plotted, in which the merg-ing of two peaks can be clearly seen. The observation that this band isprobed differently by different light polarizations indicates a kz dependentsymmetry change of the bands (Iσ probes dxz, while Ipi probes dyz and dxy).Moreover, as the dispersion is different for each polarization, this may be in-dicative that this state actually encompasses multiple, closely spaced bands,with different symmetries, a statement that will be explored in more detailin the next section.4.2.3 Γ and X state dispersions.We now turn to discussing cuts in kz through the X and Γ points. Thesecuts allow better visualization of the amplitude of the oscillation in kz. Theresults are presented in Fig. 4.3, where (a) and (c) show cuts through theΓ point, while (b) and (d) show cuts through the X point, both shown forIpi ((a) and (b)) and Ipi ((c) and (d)) polarized light. Coloured markers areplotted at the peak maximum, extracted from fits these bands. We firstturn our attention to X, which shows no significant periodic dispersion.This is in line with predictions from the model presented in Section 4.2.1,that shows negligible dispersion is expected to occur at the X point in theBCT structure when pi-like hopping is dominant. Earlier photon energydependent work on Sr2IrO4 shows a small dispersion of this state [152].The amplitude of this oscillation is however small, and may be below ourdetectable limit. Meanwhile, (a) and (c) seem to show fluctuations that92Figure 4.3: Fits to kz dispersion along the Γ and X point. (a-d) Spec-tra along the Γ − Z (a,c) direction and the X (b,d) direction(indicated in Fig. 4.2 in panel (e)). Data are presented for pi(a,b) and σ polarization. Peak maxima extracted from fits areplotted in green and purple for pi and σ polarized light respec-tively. (e,f) EDC traces corresponding to the data presented in(c) and (d) respectively, including the fits to the peaks.93Figure 4.4: EDCs for σ and pi polarizations at Γ (a) and X (b) athν = 60 eV (kz = 4.2 A˚−1).are significantly larger, with a structure that appears to resemble the sameperiodicity also found in the constant energy maps in Fig. 4.2. The fits showthat the peak positions change on the order of ∼ 100 meV. No kz dependentdata at Γ was presented in [152] (this work concludes that kz dispersion inthese layered materials is negligible). The models in Section 4.2.1 howeverindicate that such kz-dispersion would mostly arise at the Γ point, andnot the X point, which is consistent with the data presented here. Thedichotomy between the Γ and X point is highlighted in Fig. 4.3(e) and (f),which shows kz dependent energy distribution curve (EDC)s. It is clear thatwhereas the state at the X point does not disperse in kz significantly, thestate at Γ does.Interestingly, when comparing the two polarizations we find differentpeak maxima. This difference was alluded to in Section 4.2.2, and can beobserved on careful inspection of Fig. 4.3(a) and (c). The difference becomesmore evident when comparing the EDCs directly, as is done in Fig. 4.4. Thedata in Fig. 4.4 are taken at hν = 60 eV for both polarizations, at Γ (a) andX (b). While the peak positions of the EDCs at X line up well, those at Γare approximately 100 meV apart. The peak maxima occurring at differentenergies for the polarizations, implies that this peak is comprised of morethan one state with a small separation, with the individual states havingdifferent orbital symmetries. As the kz dispersion moves these states relative94to one another, different photon energies and polarizations give differentresults.The observation of multiple of states the Γ point is seemingly at oddswith a simple jeff = 1/2 model, which predicts a single state throughoutthe Brillouin zone. Moreover, these data imply that the inter-layer couplingis strong enough to observe kz-dispersion effects in photoemission, with anamplitude on the order of ∼ 100 meV. In particular when states are closetogether like the states observed at Γ, this can have quite profound effects onthese states (as for example demonstrated for Sr2Ru4 in Ref. 21). It turnsout that this has strong consequences for their spin-orbital entanglement aswill be discussed in the next section.4.3 Circularly polarized spin-ARPESWe have used circularly polarized spin-ARPES (CPSA) to directly probe theentangled spin-orbital nature of the bands. This technique has been formerlyused in [21, 23, 100, 101] and allows direct measurement of the various com-ponents of 〈L · S〉 by simultaneously selecting an orbital angular momentumusing circular light and a spin using a spin detector. A full overview of thetechnique is given in Section 2.4.2, where we discuss the various matrix ele-ments that give rise to the effect. In this section, we present data at variouspoints in the Brillouin zone to investigate the spin-orbital entangled natureof the ground state, and test the validity jeff = 1/2 scenario. Although datais presented all throughout the Brillouin zone, most attention is placed onresults at normal emission (Γ− Z, depending on kz), which are most easilyinterpreted, because the allowed final states are limited. The data presentedare intended to demonstrate by direct experiment the entangled nature ofthe bands, and will lead to the conclusion that Sr2IrO4 cannot be simplydescribed by an effective spin = 1/2 model.Measurements in Section 4.3 were performed at the VESPA endstation[94] at the Elettra Sincrotrone Trieste. The endstation is equipped witha Scienta DA30 electron spectrometer with electrostatic deflectors. Usingthese deflectors it is possible to measure a ∼ 30◦ solid angle without moving95Figure 4.5: kz dependent tight binding model of Sr2IrO4. In (a) thecolour encodes 〈L · S〉, while in (b) the colour encodes LzSz.The bands that are referred to in the text are labelled in (b).the sample. This way the date presented here was acquired without the needto rotate the sample. The spin was detected using a very low energy electrondiffraction (VLEED) type detector (explained in detail in Section 2.4.1).All data were collected at 150 K to mitigate the effects of charging of theinsulating Sr2IrO4. The Fermi energy of the spin-detectors was measuredusing evaporated gold films, and the experimental energy resolution wasmeasured to be 60 meV.4.3.1 Interpretation of CPSA results for Sr2IrO4To give a perspective to the resulting measurements, we briefly highlightthe results that are expected from an ab initio tight binding model thatincludes interlayer hopping. The model contains the full rotated I41/acdstructure of Sr2IrO4 for the Ir t2g orbitals. We use the Wannier Hamiltonianextracted from density functional theory (DFT) calculations as described inSection 3.A.3. Since we are interested in the kz dispersion, we keep out-of-plane hopping elements. We introduce distortions by rotating each atomfrom its local basis into the global basis by ±12 degrees, in a checkerboardpattern within the layers, using the method described in [148]. The structure96Figure 4.6: High binding energy CPSA (green) measurements at Γ(a) and X (b). The sum of the photoemission signals is plottedin grey. The features in (a) are labelled in the way they are bestrepresented by the bands in Fig. 4.5.of the rotations is defined over 2 layers, giving a unit-cell size of 4 atoms,as defined in [16]. At this point, we project out the eg orbitals to producethe final model. The calculated band structure is presented in Fig. 4.5along various points in the BZ, with the expectation value for 〈L · S〉 (a)and LzSz (b) for each state illustrated by the colour of the lines. A fewthings are important to note: firstly, the expectation value of 〈L · S〉 ishighly k-dependent, in the case of LzSz even changing sign for some of thebands. In particular, toward the X point the expectation of LzSz reducessignificantly, indicative of the itinerant nature of the Ir 5d orbitals, whichwas also pointed out in [153]. Secondly, the amplitude of the kz dispersionfor the bands labelled j1/2 and j3/2 is significant. Thirdly, the bands that aremost likely observed in ARPES at Γ are the bands closest to EF labelledj′3/2. At odds with our findings in Section 4.2, these bands do not havesignificant kz dispersion in this tight binding (TB) model.Turning now to the experimental results obtained using CPSA, in Fig. 4.6a dataset that was collected over a large binding energy range is presented.The CPSA is measured up to the oxygen states for the Γ (a) and X (b)97point. The data presented here help to identify features from the simula-tions, and serve as an introduction for later results. In both panels, theCPSA signal (see Section 2.4.2) is plotted in green, and the sum of the fourspin-polarization signals is plotted as the grey shaded area. The CPSA spec-trum at the X point is flat and featureless within the indicated errors (theintegrated signal can instead be observed to contain a significant amount ofstructure). This matches our initial expectation that spin-orbital entangle-ment is greatly reduced at X, and hopping terms dominate at that k-pointin the Hamiltonian. Meanwhile at the Γ point there is a clear CPSA signalwith a large number of peaks. Using the tight binding model presented inFig. 4.6 it is possible to interpret the origin of these peaks. At this pointit is good to reiterate the result from Section 2.4.2: at normal emission thesign of the CPSA is well defined, corresponding directly to LzSz. There-fore, any positive CPSA signal can be associated with a parallel spin-orbitalentanglement, and vice-versa. Recalling the analysis in Section 1.4, a purejeff = 1/2 state (dark blue in Fig. 4.5), should give rise to a positive peak,while pure jeff = 3/2 states (dark red in Fig. 4.5) should show up negative.Even though in Fig. 4.5 full entanglement is not always reached, in the fol-lowing we will refer to these states as j3/2-like and j1/2-like, as their CPSAsignal still produces the sign of the expectation value for a fully entangledstate.Comparing the CPSA spectrum in Fig. 4.6(a) with the spin-orbital en-tanglement in Fig. 4.5(b) at the Γ point, we match up the positive andnegative features in the CPSA with the blue and red bands respectively.Going from the high binding energy (large negative values) to lower bind-ing energies, we identify a negative peak around E = −2 eV, followed bya strong positive at E = −1 eV, belonging to the unfolded bands labelledj3/2 and j1/2 in Fig. 4.5) respectively. The sharpest feature in the CPSAspectrum closest to EF is observed to be positive. It seems most likely thatthis feature arises from folded band labelled j′3/2 in Fig. 4.5(b), however, thered (negative) and white (zero) bands would give rise to a negative signal in-stead of a positive, hence the sharp upturn of the signal around E = −0.5 eVseems unexplained by ab initio band structure. The positive signal around98E = −2.5 eV is also unexplained by the model presented in Fig. 4.5, butmay arise from hybridization of the Ir states with the O band. Since thisstate is at higher binding energy we will not emphasize investigating itsorigin. From these CPSA measurements it is evident that the spin-orbitalentanglement is strong, because a non-zero signal is measured all the wayup to energies of ∼ 3.5 eV. However, the variation of the CPSA signal atenergies close to the Fermi energy indicates that a description in terms ofa pseudo-spin 1/2 state may be challenged, as states vary drastically in anarrow region of energy.4.3.2 k dependent CPSAWe continue our discussion of the data with spectra collected at various(kx, ky) points presented in Fig. 4.7, where CPSA traces are plotted fordifferent kx and ky values. A schematic representation of the BZ with thevarious measurement points indicated is plotted in Fig. 4.7(e). The datadisplay a reduction of the CPSA signal when moving away from normalemission, a property that is predicted by the TB simulation in Fig. 4.5.The yellow trace in Fig. 4.7(c,d) plots the CPSA signal measured in they-axis channel, which effectively measures the LzSy term, which is zero asexpected. Although CPSA has been successfully measured away from the Γpoint [90], it should be pointed out that effects reducing the CPSA signalaway from normal emission discussed in Section 2.4.2 may affect the signal.Nevertheless, such a drastic decrease of the CPSA signal throughout theBrillouin zone, highlights the change of the spin-orbital entanglement inreciprocal space.4.3.3 kz dependent CPSAWe now turn to a discussion of kz dependent CPSA, which has as an advan-tage that all measurements can be done at normal emission, and the CPSArelates directly to the spin-orbital entanglement. The results are presentedin Fig. 4.8, with an overview of the kz position of the points studied by spin-ARPES displayed in panel (a), plotted over the photon energy dependent99Figure 4.7: k-dependent CPSA measurements using hν = 64 eV. (a-d)CPSA measurements along kx, measured at the points indicatedin (e). The green curves represent the CPSA, with the greyshaded area the sum of the spin-signals. The yellow markers in(c) and (d) indicate the CPSA in the y channel. (f-h) CPSAmeasurements along the ky direction (locations also indicatedin (e)).100ARPES data previously presented in Fig. 4.2.Photon energies were chosen such that they span most of the BZ, or asthe beamline would allow. The data presented in Fig. 4.8(b) indicate theparallel signal (√I	↑ I⊕↓ ) in blue and the antiparallel signal (√I⊕↑ I	↓ ) in red.The CPSA signal can be found in panel (c) for the studied photon energies,with a grey line indicating the sum of the spin-polarization data. Whilethe absolute changes to the parallel and anti-parallel signal are small, theprogression of the CPSA signal is significant. The signal from the previouslydiscussed j3/2 (negative) and j1/2 (positive) bands is present in all curves.The part of the spectrum at E − EF & −0.5 eV that appears as a positivepeak in Fig. 4.6 at 64 eV can be seen to change sign as the photon energydecreases to 52 eV. The feature at E = −0.5 eV that causes this behaviouris precisely the feature that was observed to be dispersing in Section 4.2. Itshould be stressed that this is a significant result: the character of the spin-orbital entanglement changes from parallel to anti-parallel through changingkz, drastically changing the character of the states closest to EF .Sample ConsistencyTo show the consistency of the measurements in Fig. 4.8, we present inFig. 4.9 a collection of different samples measured at different times allshowing the same behaviour of both positive and negative 〈L · S〉. Thesedata provide convincing evidence that the results shown in this chapter areintrinsic sample properties, since the measurements can be repeated wellwithin the signal uncertainty.Considering the model presented in Fig. 4.5, such drastic dispersion andcomplete reversal of the spin-orbital entanglement is unexpected for thesestates: as such it seems that an ab initio derived model cannot explainthese effects. It is possible that a a multiplet of states should be consideredas in [154]. At this point it is also useful to point out that through theconstruction of the system in terms of jeff states, hybridization betweenthese spin-orbit states is possible through the original hopping elements, inboth the in- and out-of-plane channels. As the jeff orbitals are constructed101Figure 4.8: Photon energy dependent CPSA measurements. (a)Overview of of of the reciprocal space with green markers indi-cating the locations of the photon energy dependent measure-ments. The constant energy map is identical to the one in panel(c) of Fig. 4.2. (b) Plots of the parallel (blue) and anti-parallel(red) signal. (c) CPSA signal (green) plotted with the sum ofthe spin signals (grey).102Figure 4.9: CPSA repeatability. The two panels highlight the re-peatability on different samples and beamtimes for two photonenergies (64 eV in (a) and 51.7 eV in (b)). The different samplesare S1 and S2, with S2 measured on a second occasion as S2-2.from individual t2g states that overlap on neighbouring sites, hybridizationbetween the different jeff states persists. This hybridization may play a rolein the drastic changes that are observed upon changing kz. Regardless of theprecise origin of the change in sign of this state at −0.5 eV, the implicationseems clear: the small but significant kz dispersion in this system is strongenough to change the character of the highest electron removal state fromaligned parallel to anti-parallel. This makes a description in terms of apseudo-spin 1/2 model impossible, and signals that even a low energy modelshould consider excitations of the jeff = 3/2 states too.4.3.4 Slab Simulation of Sr2IrO4In order to better understand the results from our CPSA measurements, weperform matrix element simulations using the chinook package [90].To capture the effects arising from the kz dependence, including possiblephotoelectron interference, we construct a slab model, in a similar way asdescribed in Section 4.3.1. We use the same base Hamiltonian, but for thesimulations in the following section, we keep the full 5-orbital 5d manifold103for all the iridium atoms. We proceed to construct a slab along the c−axisof the crystal, consisting of 6 unit cells (12 atomic layers), resulting in abasis size of 12× 2× 10 = 240 orbitals.In order to simulate the gap, we have added an antiferromagnetic order-ing through an Sz term, using the Hamiltonian:H =∑i,ν,σdmiSzi,ν . (4.2)Here, i and ν are the atom and orbital index, Szi,ν the spin, mi the mag-netization, and d a parameter (related to the Coulomb interaction) thatwe set to d = 0.2 eV for an appropriate gap size. We set mi according tom1 = m2 = −m3 = −m4 = 1, where atom 1 and atom 3 are in the samelayer, and atom 2 and atom 4 are in the same layer, forming an antiferro-magnet, with a checkerboard pattern.The band structure for this system is presented in Fig. 4.10(b). Althoughall bands are flat along the Γ − Z axis, the effective kz dispersion for thisslab model is captured by the series of bands at different binding energiesformed by the inter-layer coupling. In the limit of an infinite sized crystalalong the c-axis, the original dispersion would be recovered.We proceed to calculate a simulated spectrum for various polarizations.As an example, the spectrum obtained using σ-polarization is shown inFig. 4.10(a), which is in reasonable agreement with the experimental datapresented in Fig. 4.2(e). Clearly visible are the hole like pockets centred atX, and the broad band around E = −1 eV.What is not captured well in this simulation is the intensity of the foldedband at Γ, around E = −0.5 eV, which appears more intense in experiments.Considering the discussion around unfolding in Section 2.2, for such a bandto appear with high intensity in an experiment, there needs to be a signifi-cant potential to break the symmetry, in this model arising from rotationsand antiferromagnetism. It is possible that the antiferromagnetism as it istreated in these simulations does not capture the full extent of symmetrybreaking potential.We continue to calculate the CPSA signal, by simulating Iσ, and taking104Figure 4.10: Slab model for Sr2IrO4 including AFM order. (a) Cal-culated ARPES spectrum using σ-polarization from the slabmodel band structure plotted in (b). (c) Calculated CPSAspectrum at Γ. The negative and positive peaks arise from thej3/2 and j1/2 like bands. (d) The CPSA spectrum calculatedfor the as a function of the out-of-plane momentum kz.105the geometric mean as:√I⊕,↓I	,↑ −√I	,↓I⊕,↑√I⊕,↓I	,↑ +√I	,↓I⊕,↑ + , (4.3)with  a small constant to avoid zero-division, that we set as 0.01·max (Iσ,),chosen to give the best agreement with experiments. The resulting CPSAcurve at the Γ point is shown in Fig. 4.10(c), which can be see to match someof the aspects of the data presented in Fig. 4.8: the negative and positivesignals belonging to the jeff = 3/2 and jeff = 1/2 states. A small peak closeto EF is also visible, a combination of the downturn arising from the foldedjeff = 3/2 band, with a positive signal from the jeff = 1/2 above the Fermienergy, which is not seen in experiment. The image presented in Fig. 4.10(d)explores the kz dependence of the CPSA signal for this slab model. While achange in kz is clearly visible, Fig. 4.10 fails to reproduce the dramatic signchange seen in experiment.In the rest of this section, we will investigate various properties that caninfluence the CPSA signal at Γ as a function of photon energy (and therebykz), to get in idea of the physical mechanism that could cause the switchingof signal observed in Section 4.3.Strength of Spin-Orbit CouplingThe first parameter we consider is spin-orbit coupling (SOC). AlthoughSOC coupling is an atomic property with a nominal value (λIr ∼ 0.4 eV[20]), actual observations seem to differ [44, 106, 148], which is possiblyexplained by the presence of the Coulomb interaction [132, 133], magnifyingthe Hamiltonian terms associated with SOC. We present simulations forλ ∈ {0.35, 0.45, 0.65} eV in Fig. 4.11(a-c). Similar to the results presentedin Fig. 4.10, the unfolded jeff states are visible most clearly, with a smallsignal coming from the folded band close to Eb. This state is expected togive a negative (red) CPSA signal, but in panels b and c, the intensity fromthe unfolded jeff = 1/2 state overpowers this small signal. This negative(red) signal can be seen in panel a, where the spin-orbit coupling is smaller,106Figure 4.11: CPSA simulations for different values of λSOC , each asa function of kz, as simulated for the antiferromagnetic slabmodel. The values for λSOC are indicated in each panel in thetop right corner.which results in the jeff = 1/2 being further away from the folded jeff = 3/2band. These simulations demonstrate that a reasonable change of spin-orbitcoupling is unable to reproduce a switch in sign as we find in our data.The Mean Free PathIn order to investigate the possibility of photoelectron interference causinga rapid change in photoemission intensity dependent on kz as seen for topo-logical insulators in Ref. 96, we calculate the CPSA signal for various sizesof the mean free path. By modulating the intensity coming from layers us-ing an exponential decay as is implemented in chinook we can investigatewhether a particular value for the mean free path could explain the observedresults. The results are presented in Fig. 4.12, where kz dependent CPSAcurves are plotted for various values of the mean free path. An interestingeffect appears in these simulations: the amplitude of the kz dispersion isdependent on the value of the mean free path in this slab model: when thevalue is so low that only a single layer is probed (e.g. 1 A˚ in (a)), the lack ofprobed coupling between layers, results in only a flat band being observed.The larger the mean free path, the more layers are probed and the strongerthe apparent kz dispersion (b,c). It is clear that aside from the observed107Figure 4.12: CPSA simulations for different values of the photoelec-tron mean free path. CPSA signal plotted as a function ofkz is presented for 1.0 (a), 4.8 (b) and 20.0 A˚(c). (d) Thedependence at Γ as a function of mean free path.variation of the kz dispersion, no big changes to the sign and magnitude ofthe CPSA signal occur, so we conclude that the mean free path can be ruledout as a source of observations presented in Fig. Magnitude of the kz hopping terms.Finally, we have modified the out-of-plane hopping elements to see if thatcould yield the desired result. To test that hypothesis, we rescaled all out-of-plane hopping elements by a fixed factor and simulated the CPSA signalas a function of kz. The results are presented in Fig. 4.13, where the effectis clearly visible: the dispersion of the states, seen as a shift of the peaks, isclearly enhanced at higher factors. While no changes in sign are observed foran enhancement of 2, for an enhancement of 4, the sign of the CPSA signal108Figure 4.13: CPSA simulations with enlarged kz hopping elements.The hopping elements are multiplied by a factor 1 (a), 2 (b)or 4 (c).changes at kz =pic . While this has so far been the only parameter that canbe seen to change the sign as a function of kz, there are two issues with this:an enhancement of 4 seems too large to be physically justifiable, as it wouldput the out-of-plane hopping at ∼ 0.1 eV similar to the in-plane hopping(∼ 0.3 eV). Furthermore, although a change in sign is observed, the peakitself does not change direction. It seems that this increased kz hopping isin fact moving the jeff = 1/2 state closer, thereby adding a background tothe state closest to EF .4.4 ConclusionThe results presented both for photon energy dependent ARPES and CPSAmeasurements have a clear consequence. Firstly, from these data it is ev-ident that Sr2IrO4 is certainly not two-dimensional, as interlayer couplingis strong enough to cause significant kz dispersion likely caused by the ex-tended character of the Ir 5d orbitals. Secondly, while spin-orbit coupling inSr2IrO4 is undoubtedly very strong, the suggestion that a pseudo-spin 1/2model is able to describe the ground state for this system is challenged by thereversal of spin-orbital entanglement that we observe. The photon energydependent results suggest that there are multiple states in close proximityaround the Γ-point, that move with respect to each other, dependent on kz.109The CPSA data confirm this result, and adds that the spin-orbital entangle-ment changes from parallel to anti-parallel. To appropriately describe theoccupied states of Sr2IrO4, it is therefore necessary to take into account thefull, multi-orbital Mott physics as the bands are not separated well enoughto fully project out the jeff = 3/2 states.110Chapter 5ConclusionThis thesis has covered various topics relating to Sr2IrO4 and spin-orbitcoupling. While most efforts were focussed specifically on Sr2IrO4, theseconclusions have further reaching effects. We readdress here the main ques-tions of this thesis: firstly, the role of spin-orbit coupling in a correlatedrelativistic insulator such as Sr2IrO4 and secondly, whether Sr2IrO4 can bedescribed by a pseudo-spin 1/2 model.5.1 Spin-orbit controlled metal insulatortransition in Sr2IrO4With the suggestion of Sr2IrO4 as a relativistic Mott insulator [25], spin-orbitcoupling was tentatively added to the canonical phase diagram describingcorrelated insulators [6]. However, while the insulating state in Sr2IrO4 hasbeen believed to be stabilized by spin-orbit coupling, direct evidence has sofar been lacking. In Chapter 3, a transition into a metallic state inducedby a dilution of spin-orbit coupling was presented, establishing SOC as afundamental parameter in the field of multi-orbital correlated insulators.Driving the metal-insulator transition in this material by SOC is not onlythe first demonstration of such a SOC controlled metal-insulator transition(MIT), but moreover provides direct evidence for the essential role of SOCin stabilizing the insulating state in Sr2IrO4.111In particular, we have substituted Ir in Sr2IrO4 with Ru and Rh tomake well-defined changes to the effective value of spin-orbit coupling whichdrives the metal-insulator transition. Ru and Rh, having significantly lowerspin-orbit coupling than Ir, both drive a MIT but at surprisingly differ-ent critical concentrations. It is found that the dilution of spin-orbit cou-pling, controlled by the impurity potential associated with the two differentspecies, is what explains the dichotomy in critical concentrations. For Rh,no such impurity potential is present and spin-orbit coupling is reduced ef-fectively. For Ru however, a potential of ∼ 0.25 eV hinders such dilution ofSOC and the rate of the transition is significantly retarded. ARPES mea-surements, interpreted using matrix element analysis confirm the predictedresults: spin-orbit coupling is effectively reduced when Rh is substituted,but not for Ru.A corollary to our results is the method with which spin-orbit couplingwas observed. The entanglement of t2g states by spin-orbit coupling mixesorbitals of different symmetries, causing a well-defined intensity changethrough the photoemission dipole matrix element. This method is notunique to Sr2IrO4 and may find applications in other t2g systems wherespin-orbit coupling is of importance.Furthermore, our observation of impurity potential controlled SOC mix-ing is well explained by a simple two-site tight binding model without anyspecific considerations of electronic structure. The generic result is that theimpurity potential controls hybridization between the sites, and the degreeof hybridization controls the amount of spin-orbit coupling dilution. Sincethese results are obtained and explained on such a minimal model, with nospecifically tuned crystal parameters, we expect these results to be validgenerally for spin-orbitally coupled multi-species systems in which orbitalshybridize. These results are particularly helpful in elucidating pursuits tomodify spin-orbit coupling by valence ion substitution. This strategy hasbeen successful in GaAs using Bi substitution [135] for example. It may alsoclarify interpretation of attempts to enhance spin-orbit coupling in graphenethrough adatom deposition [123, 124, 141–143]. Although theory predictslarge gap appearing at the Dirac cone of graphene when heavy elements112are deposited [123, 124], experimentally such a gap is not observed [142].It is possible that an impurity potential, or low hopping integrals preventhybridization with the graphene lattice, and thereby preclude substantialenhancement of spin-orbit coupling. Furthermore it could help to enlightenthe physics of attempts to drive a transition to a trivial state in topologicalinsulators [136–140]. The results presented here about spin-orbit couplingset clear constraints on how to to think about these experiments, and makeexplicit suggestions on how to continue them. It not only shows how spin-orbit coupling can be successfully diluted, it also explains directly how tomake use of the impurity potential and hybridization to fine-tune the ob-tained results.5.2 Spin- and kz-resolved photoemission onSr2IrO4Since the discovery of the cuprate superconductors [89, 150], an intense effortin the field has been made to observe superconductivity in other transitionmetal oxides. Sr2IrO4 appears to be a prime candidate: not only is the struc-ture identical to that of La2CuO4, it hosts an unexpected (Mott) insulatingphase, with an antiferromagnetic ordering. Moreover, it was suggested thatthe ground state in Sr2IrO4 is pseudo-spin 1/2, because strong spin-orbitcoupling could entangle the t2g states into a filled jeff = 3/2 and half-filledjeff = 1/2 state. The combination of these properties (two-dimensional,square transition-metal oxide lattice, half filled pseudo-spin 1/2 Mott insu-lator) are the quintessential ingredients attributed to high-temperature su-perconductivity. Yet, although some promising observations have been made[44–47], no signs of bulk-superconducting behaviour have been detected. InChapter 4 the use of photon energy dependent ARPES has demonstratedthat the kz dispersion in this compound is significant (∼ 100 meV) sincein this energy window multiple states are observed, having distinct orbitalsymmetry for horizontally and vertically polarized light.We furthermore study the spin-orbital entanglement directly by perform-ing CPSA measurements. The results show that the spin-orbital entangle-113ment is strong at energies throughout the Ir d manifold at normal emission.However, results also show that the spin-orbital entanglement reduces sig-nificantly away from normal emission toward the X-points, a feature thatis explained by considering the increased itinerancy for those states. Atnormal emission, the state closest to EF shows a complete reversal of thespin-orbital entanglement when changing kz. This result agrees with the ob-servations from photon energy dependent ARPES, and implies that Sr2IrO4cannot simply be described by a jeff = 1/2 model. Not only does the spin-orbital entanglement change significantly with (kx,ky), but the completereversal of spin-orbital entanglement can only be explained if the jeff = 3/2are explicitly taken into account. These observations challenge the validityof the treatment of Sr2IrO4 as a pseudo-spin 1/2 model. The results pre-sented here instead require that models constructed for Sr2IrO4, take intoaccount all three t2g orbitals, and suggest that Sr2IrO4 should be considereda “relativistic multi-orbital Mott insulator”, rather than a “jeff = 1/2 Mottinsulator”.Our results may furthermore shed light on the necessary ingredients forsuperconductivity in transition metal oxides: the studies suggesting super-conductivity in Sr2IrO4 all take as a starting point either a pseudo-spin1/2 model or strong (λ > t) spin-orbit coupling [35, 40, 41] limited to twodimensions. Our finding that Sr2IrO4 is not in fact spin 1/2, nor fully two-dimensional suggests that those two properties are key pieces in the puzzleof high-temperature superconductivity.5.3 ConclusionThis thesis has focussed on variations of a single compound with an ex-tensive amount of work spent towards understanding the ground state ofSr2IrO4 and has therefore made significant strides in the understanding ofrelativistic correlated insulators. It has been shown that the ground stateof Sr2IrO4 is strongly influenced by spin-orbit coupling, and that spin-orbitcoupling should be considered as a fundamental parameter of multi-orbitalMott physics. However, it has also been clearly shown that SOC is not114strong enough to describe Sr2IrO4 as a pseudo-spin 1/2 model, and it isimportant to consider the full manifold of t2g orbitals. 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Commun. 8(1), 686, 2017. → page101133Appendix AOverview of expectationvalues spin-orbit entangledstatesHere, tables are provided for frequently used states, including their repre-sentations in other relevant bases, as well as expectation values that areoften referenced and used.134state Y m` Lz Sz Jz LxSx LzSz L · S|5/2, 5/2〉 |2, ↑〉 2 12 52 0 1 1j = 5/2 |5/2, 3/2〉√45 |1, ↑〉+√15 |2, ↓〉 1210 310 32 410 210 1|5/2, 1/2〉√610 |0, ↑〉+√410 |1, ↓〉 410 110 12 610 − 210 1j = 3/2 |3/2, 3/2〉√15 |1, ↑〉 −√45 |2, ↓〉 1810 − 310 32 − 410 − 710 −32|3/2, 1/2〉√410 |0, ↑〉 −√610 |1, ↓〉 610 − 110 12 − 610 − 310 −32Table A.1: Table of expectation values for the spin-orbit coupled ` = 2 states.135state Y m` Lz Sz Jz LxSx LzSz L · Sj = 3/2 |3/2, 3/2〉 |1, ↑〉 1 12 32 0 36 12|3/2, 1/2〉√23 |0, ↑〉+√13 |1, ↓〉 26 16 12 26 −16 12j = 1/2 |1/2, 1/2〉√13 |0, ↑〉 −√23 |1, ↓〉 46 −16 12 −26 −26 -1Table A.2: Table of expectation values for the spin-orbit coupled ` = 1 states.136state Y m` Lz Sz Jz LxSx LzSz L · Sjeff = 3/2 |3/2, 3/2〉 i |2,−1, ↑〉 -1 12 −12 0 −12 −12|3/2, 1/2〉√23 |0eff , ↑〉+ i√13 |2,−1, ↓〉 −26 16 −16 −26 16 −12|3/2,−1/2〉√23 |0eff , ↓〉 − i√13 |2, 1, ↓〉 26 −16 16 −26 16 −12|3/2,−3/2〉 −i |2, 1, ↓〉 1 −12 12 0 −12 −12jeff = 1/2 |1/2, 1/2〉√13 |0eff , ↑〉 − i√23 |2,−1, ↓〉 −46 −16 −56 26 26 1|1/2,−1/2〉√13 |0eff , ↓〉+ i√23 |2, 1, ↑〉 46 16 56 26 26 1Table A.3: Table of expectation values for the spin-orbit coupled t2g (`eff = 1) states.137t2g constructionjeff = 3/2 1√2(|dyz, ↑〉+ i |dxz, ↑〉)1√6(|dyz, ↓〉+ i |dxz, ↓〉 − 2 |dxy, ↑〉)1√6(− |dyz, ↑〉+ i |dxz, ↑〉 − 2 |dxy, ↓〉)1√2(− |dyz, ↓〉+ i |dxz, ↓〉)jeff = 1/2 1√3(|dyz, ↓〉+ i |dxz, ↓〉+ |dxy, ↑〉)1√3(− |dyz, ↑〉+ i |dxz, ↑〉+ |dxy, ↓〉)Table A.4: Construction of the jeff states in terms of the t2g orbitals.138state Y m` Lz Sz Jz LxSx LzSz L · S t2gj+ |j+, ↑〉 |2, 1, ↑〉 1 12 32 0 12 12 1√2 (|dxz, ↑〉+ i |dyz, ↑〉)|j+, ↓〉 |2,−1, ↓〉 -1 -12 −32 0 12 12 1√2 (|dxz, ↓〉 − i |dyz, ↓〉)j− |j−, ↑〉 |2, 1, ↓〉 1 -12 12 0 -12 -12 1√2 (|dxz, ↓〉+ i |dyz, ↓〉)|j−, ↓〉 |2,−1, ↑〉 -1 12 −12 0 -12 -12 1√2 (|dxz, ↑〉 − i |dyz, ↑〉)Table A.5: Table of expectation values for the spin-orbit coupled dxz and dyz states.139


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