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Angle-resolved photoemission and density functional theory studies of topological materials Zhu, Zhi-Huai 2013

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Angle-resolved photoemission anddensity functional theory studies oftopological materialsbyZhi-Huai ZhuB.Sc., Nanjing University, 2007M.Sc., The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)Septermber, 2013c? Zhi-Huai Zhu 2013AbstractTopological insulators (TIs), with a gapless surface state located in a largebulk band gap, define a new class of materials with strong application po-tential in quantum electronic devices. However, real TI materials have manycritical problems, such as bulk conductivity and surface instability, whichhinder us from utilizing their exotic topological states. Another fundamentalquestion in the TI field is what the realistic spin texture of the topologicalsurface states (TSSs) is; no conclusive answer has yet been reached, despiteextensive studies.We present two studies of doping the prototypical TI materials via in situpotassium deposition at the surface of Bi2Se3 and by adding magnetic impu-rities into the bulk Bi2Te3 during crystal growth. We show that potassiumdeposition can overcome the instability of the surface electronic properties.In addition to accurately setting the carrier concentration, new Rashba-likespin-polarized states are induced, with tunable, reversible, and highly stablespin splitting. Our density functional theory (DFT) calculations reveal thatthese Rashba states are derived from quantum well states associated with aK-induced 5 nm confinement potential. The Mn impurities in Bi2Te3 havea dramatic effect on tailoring the spin-orbit coupling of the system, mani-fested by decreasing the size of the bulk band gap even at low concentrations(2%-5%). This result suggests an efficient way to induce a quantum phasetransition from TIs to trivial insulators.We also explicitly unveil the TSS spin texture in TI materials. By acombination of polarization-dependent angle-resolved photoemission spec-troscopy (ARPES) and DFT slab calculations, we find that the surface statesare characterized by a layer-dependent entangled spin-orbital texture, whichbecomes apparent through quantum interference effects. We predict a wayiiAbstractto probe the intrinsic spin texture of TSS, and to continuously manipulatethe spin polarization of photoelectrons all the way from 0 to ?100% byan appropriate choice of photon energy and linear polarization. Our spin-resolved ARPES experiment confirms these predictions and establishes ageneric rule for the manipulation of photoelectron spin polarization. Thiswork paves a new pathway towards the long-term goal of utilizing TIs foropto-spintronics.iiiPrefaceThe work described in this thesis was conducted over the past four years dur-ing my Ph.D. program at the University of British Columbia (UBC). I didnot include all the work that I have accomplished, selecting only the topo-logical insulator related portions. A full list of my publications is provided inthe Appendix, including my other work contributed to group publications.All the work presented here has been or will be published with me as thefirst author. This is not to say that I produced this work alone; details ofmy contributions are below.Chapter 3 ? Ab initio tight-binding modelThis chapter is the supplementary work for Chapter 5. Under the directionof I. S. Elfimov and M. W. Haverkort, I developed the ab initio tight-binding(TB) model for Bi2Se3. M. W. Haverkort provided me with several Math-ematica packages that are particularly useful for figure plotting, which alsohelped me learn how to program in Mathematica. I wrote all the sourcecodes for the TB model calculations and produced all relevant data.Chapter 4, Section 4.1 ? Rashba spin-splitting control at thesurface of Bi2Se3This work has been published in Physical Review Letters 107, 186405 (2011)[1]. I performed the experiments by using the in-house angle-resolved pho-toemission spectroscopy (ARPES) system at UBC, a system I have playeda role in helping to maintain and characterize over the past five years. Allthe data presented in this chapter was physically taken by myself, with helpfrom other ARPES group members, in particular G. Levy and B. Ludbrook.ivPrefaceThe Bi2Se3 samples were grown by P. Syers, N. P. Butch and J. Paglione atthe University of Maryland, and A. Ubaldini at the University of Geneva.I also performed all the data analysis and wrote the manuscript. My su-pervisor, A. Damascelli, guided me regarding the direction of the work andedited the final version of the manuscript. A. Damascelli, I. S. Elfimov, andI contributed theoretical ideas to explain the ARPES data. I performed thedensity functional theory (DFT) calculations together with I. S. Elfimov.Chapter 4, Section 4.2 ? Tailoring spin-orbit coupling inMn-doped Bi2?xMnxTe3The work in this chapter has not been published yet. I performed theARPES experiments at UBC with assistance from Ivana Vobornik, a beam-line scientist at ELETTRA who also brought the samples to me for ARPESmeasurements. The samples were origionally grown by R. Cava?s groupat Princeton University. I performed all the data analysis and wrote themanuscript. A. Damascelli supervised the development of my scientific ideasfor this work.Chapter 5 ? Layer-by-layer entangled spin-orbital texture inBi2Se3This chapter has been published in Physical Review Letters 110, 216401(2013) [2]. I performed ARPES experiments with assistance from G. Levy.The single crystals were grown by P. Syers, N. P. Butch, J. Paglione andA. Ubaldini. I performed the theoretical calculations and data analysis,under the supervision of M. W. Haverkort and I. S. Elfimov. A. Damascellisupervised the overall project direction and planning. C. N. Veenstra helpedme in writing the code; he was also very helpful on editing my manuscript.I wrote the manuscript together with M. W. Haverkort and A. Damascelli.Chapter 6 ? Photoelectron spin polarization control in Bi2Se3This chapter has been submitted for publication. The spin-resolved ARPESdata was taken at Hiroshima Synchrotron Radiation Center by the group ofvPrefaceC.N. Veenstra, S. Zhdanovich, and I. I designed the experiments and led thedata acquisition during the beamtime. C. N. Veenstra was extremely helpfulfor discussions and planning. Together with S. Zhdanovich, he also preparedthe high-quality samples for this experiment. T. Okuda and K. Miyamoto,two beamline scientists, also constantly provided assistance with instrumentoperations and experiment planning. I performed the data analysis andwrote the manuscript together with A. Damascelli. A. Damascelli supervisedthe overall project direction and planning.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 SOC in systems with inversion symmetry . . . . . . . . . . . 21.2 SOC in systems without inversion symmetry . . . . . . . . . 41.3 Topological insulators . . . . . . . . . . . . . . . . . . . . . . 72 Angle-resolved photoemission spectroscopy . . . . . . . . . 112.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . 112.2 ARPES intensity calculation . . . . . . . . . . . . . . . . . . 132.2.1 General formulae . . . . . . . . . . . . . . . . . . . . 132.2.2 Initial states . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Final states and selection rules . . . . . . . . . . . . . 152.2.4 Measured spin polarization . . . . . . . . . . . . . . . 17viiTable of Contents3 Ab initio tight-binding model . . . . . . . . . . . . . . . . . . 193.1 Ab initio bulk tight-binding model . . . . . . . . . . . . . . . 213.2 Ab initio slab tight-binding model . . . . . . . . . . . . . . . 284 Impurities in 3D topological insulators . . . . . . . . . . . . 334.1 Rashba spin-splitting control at the surface of Bi2Se3 . . . . 334.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Experimental and calculation methods . . . . . . . . 364.1.3 In situ K deposition-induced Rashba-like states . . . 364.1.4 Quantum well states and conclusion . . . . . . . . . . 424.2 Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3 . . 454.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Material properties . . . . . . . . . . . . . . . . . . . 464.2.3 Doping-level-dependent ARPES spectra . . . . . . . . 474.2.4 Temperature effects on ARPES spectra . . . . . . . . 514.2.5 K deposition at the surface of Bi1.96Mn0.04Te3 . . . . 564.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 585 Layer-by-layer entangled spin-orbital texture in Bi2Se3 . . 605.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Polarization-dependent ARPES intensity pattern . . . . . . . 625.3 Layer-dependent entangled spin-orbital texture . . . . . . . . 685.4 TSS spin texture and photoelectron spin polarization . . . . 695.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.6 Supplemental material . . . . . . . . . . . . . . . . . . . . . . 735.6.1 Experimental and theoretical methods . . . . . . . . 735.6.2 ARPES intensity and interference effects . . . . . . . 745.6.3 Photoelectron spin polarization . . . . . . . . . . . . 765.6.4 TSS orbital character and spin polarization . . . . . . 795.6.5 Manipulation of ARPES spin texture . . . . . . . . . 805.6.6 Asymmetric ARPES from a simple TSS model . . . . 84viiiTable of Contents6 Photoelectron spin-polarization-control in Bi2Se3 . . . . . 906.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . 926.3 SARPES results and discussion . . . . . . . . . . . . . . . . 926.4 A two-layer model to describe interference effects in SARPES 1006.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108AppendixList of publications . . . . . . . . . . . . . . . . . . . . . . . . . 125ixList of Figures1.1 Effect of SOC on the band dispersion and Fermi surface ofSr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Rashba-type spin splitting bands of BiTeI . . . . . . . . . . . 61.3 Band inversion of bulk bands in Bi2Se3 . . . . . . . . . . . . . 81.4 Topological surface states and Rashba-split bands of Bi2Se3 . 92.1 Sketch of selection-rule-determined ARPES intensity maps . . 173.1 Procedural flow for constructing a tight-binding model . . . . 213.2 Comparison between the LMTO and WIEN2k band structures 233.3 Illustration of the partial content of the Bi2Se3 HAMR file . . 243.4 Isosurface of charge density from the Wannier orbitals pz . . 253.5 Wannier orbitals plotted in a crystal lattice . . . . . . . . . . 253.6 Comparison between the TB model and WIEN2k band struc-tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7 Crystal structure of Bi2Se3 . . . . . . . . . . . . . . . . . . . 293.8 Band inversion and topological surface states in Bi2Se3 . . . . 303.9 Real space distribution of quantum well states . . . . . . . . 314.1 Time evolution of the ARPES dispersions . . . . . . . . . . . 354.2 Evolution of the Bi2Se3 ???K? electronic dispersion with lowK deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Evolution of the Bi2Se3 ???K? electronic dispersion with heavyK deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Stable and reversible band dispersions . . . . . . . . . . . . . 404.5 Quantum well and surface slab states . . . . . . . . . . . . . . 42xList of Figures4.6 Energy splitting of quantum well states . . . . . . . . . . . . 444.7 ARPES dispersions measured at 6 K with freshly cleaved sam-ples of Bi2?xMnxTe3 . . . . . . . . . . . . . . . . . . . . . . . 474.8 Doping-level-dependent ARPES spectra of Bi2?xMnxTe3 . . . 494.9 Constant energy contours of Bi2?xMnxTe3 . . . . . . . . . . . 504.10 ARPES dispersions of Bi1.96Mn0.04Te3 measured below andabove Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.11 ARPES dispersions of Bi1.9Mn0.1Te3 measured below and aboveTc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.12 Doping-level-dependent MDC width of Bi2?xMnxTe3 . . . . . 554.13 Temperature-dependent MDC width of Bi2?xMnxTe3 . . . . 564.14 Evolution of ARPES dispersions of Bi1.96Mn0.04Te3 along ???K? as a function of K deposition time . . . . . . . . . . . . . . 574.15 ARPES constant energy contours of Bi1.96Mn0.04Te3 . . . . . 575.1 ARPES experimental geometry with linearly polarized light . 635.2 Linear polarization dependence of the measured ARPES in-tensity in momentum space . . . . . . . . . . . . . . . . . . . 645.3 Calculated ARPES polarization dependence . . . . . . . . . . 655.4 Layer-projected topological surface state . . . . . . . . . . . . 665.5 Layer-by-layer entangled spin-orbital texture . . . . . . . . . 675.6 Orbital-projected spin texture of TSS . . . . . . . . . . . . . 705.7 Calculated photoelectron spin polarization . . . . . . . . . . . 715.8 Energy dependence of the layer-integrated TSS orbital char-acters and spin polarization . . . . . . . . . . . . . . . . . . . 795.9 Calculated spin texture of photoelectrons . . . . . . . . . . . 815.10 Photon-energy dependence of the photoelectron spin polar-ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.11 Photon-energy dependence of the photoelectron spin polar-ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.12 Quantum interference effects on ARPES intensity patternfrom a simple model . . . . . . . . . . . . . . . . . . . . . . . 88xiList of Figures6.1 Spin orientation of the TSS in Bi2Se3 . . . . . . . . . . . . . . 936.2 Light-polarization-controlled photoelectron spin polarization . 946.3 Quantum interference effects on spin polarization of photo-electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.4 Photon-energy-dependent spin polarization of photoelectrons 98xiiList of Abbreviations2D two-dimensional3D three-dimensionalARPES angle-resolved photoemission spectroscopya.u. arbitrary unitsBZ Brillouin zoneCEC constant energy contourDC Dirac coneDFT density functional theoryDP Dirac pointEDC energy distribution curveESPRESSO Efficient SPin REsolved SpectroscopyFS Fermi surfaceFWHM full width at half maximumLMTO linear muffin-tin orbitalMDC momentum distribution curveNMTO order-N muffin-tin orbitalQL quintuple layerxiiiList of AbbreviationsQW quantum wellRB Rashba bandSARPES spin- and angle-resolved photoemission spectroscopySOC spin-orbit couplingSPECS SPECS Surface Nano Analysis GmbHSTM scanning tunneling microscopeTB tight-bindingTI topological insulatorTRS time-reversal symmetryTSS topological surface stateUBC University of British ColumbiaxivList of Symbols? the origin in momentum space of the bulk Brillouin zone.?? the origin in momentum space of the surface Brillouin zone.I photoemission intensity? strength of atomic spin-orbit couplingk electron momentumk|| electron momentum parallel to the sample surfacekx electron momentum projected along x?ky electron momentum projected along y?kz electron momentum projected along z?M matrix elementpx atomic px orbitalpy atomic py orbitalpz atomic pz orbitalPx spin polarization of photoelectrons along the x quantization axisPy spin polarization of photoelectrons along the y quantization axisPz spin polarization of photoelectrons along the z quantization axis?TSS wave function of topological surface statexvList of Symbols? Pauli spin matrices? index of the basis set of atomic orbitalsxviAcknowledgementsI would not have been able to accomplish the work presented in this thesiswithout the help of many people. First and foremost, I would like to thankmy advisor, Prof. Andrea Damascelli, for his mentorship and consistent sup-port. Andrea has provided me with strict scientific training that will guideme through all future work. He also gave me endless encouragement andprofessional advice for my graduate career. Working with Andrea enabledme to believe that I could always find the best resources to solve problems,either provided directly by him or through new opportunities that he createdfor me to work with other experts.I would like to give a special thank you to Dr. Ilya Elfimov, who taughtme how to do theoretical calculations. As a theoretical supervisor, Ilyahas consistently supported and actively contributed to all my work. I ap-preciated how Ilya always gave me priority to use clusters for super-largecalculations, and his tremendous effort in providing me with computationalresources. I would definitely like to continue collaborating with him in myfuture academic work.I also received great support from all my lab mates: Dr. Giorgio Levy,Christian Veenstra, Bart Ludbrook, Riccardo Comin, Sergey Zhdanovich,Ryan Wicks, Jonathan Rosen, and Michael Schneider. Thank you all. Iwould like to underscore my thanks to Giorgio again; when I first arrivedat UBC, he taught me the XAS, XPS, and ARPES techniques, and hasgiven me consistent support in my ARPES experiments. I also would like toespecially thank Christian, who helped me with professional English editingand also gave me great advice on programming. I was happy to conduct twosuccessful SARPES beamtime work with him, and his precise working stylewas very helpful.xviiAcknowledgementsDiscussion with Prof. George Sawatzky was always a valuable opportu-nity for me to learn new physics. The richness of his knowledge in physicsalways inspired new ideas for my research projects. It has been a wonderfulexperience to have George involved in most of my scientific discussions. Iwould really like to thank him for all his time and ideas.I have enjoyed working with Dr. Maurits Haverkort, and would like tothank him for leading me into the field of tight-binding related models.His expertise in theoretical modeling and insights into fundamental physicalproblems always greatly inspired me. I look forward to more collaborationswith him.Prof. Marcel Franz helped me with understanding the fundamentalphysics of topological insulators, the central topic of this thesis. I learned alot during our discussions. I would like to thank Marcel for his patience inanswering my questions, even when I brought up underdeveloped ideas.I would like to thank all my sample providers: Paul Syers, Dr. NicholasButch, Prof. Johnpierre Paglione at the University of Maryland, and AlbertUbaldini at the University of Geneva. I also would like to thank IvanaVobornik and Giancarlo Panaccione at ELETTRA for collaborating withme on the Mn-doped topological insulators project.I would like to thank Prof. Taichi Okuda for his warmest hospitality andall his help during my spin-resolved ARPES beamtime in Japan. I thankDr. K. Miyamoto and Siyuan Zhu for their help as well.Finally, I would like to thank my fiance?e, Vicky Li. She has been atremendous support throughout my entire Ph.D. career. She has enduredthe grueling aspects of my physics life and always gave me her supportwhenever things did not seem to be progressing as expected. She was alsothe first person to read all my drafts and helped me with the first round ofediting.xviiiChapter 1IntroductionIn atomic physics, electrons always move in an environment with an electricfield ?eE = ??V , due to the strong Coulomb potential of the atomic coreregions. Classically, the magnetic moment of electrons ?s does not coupleto the electric field. However, taking into account relativistic effects, theelectron sees in its rest frame a magnetic field, whose value is given by B =?v ?E/c (v/c)2, with v being the velocity of the electron and c the speedof light [3]. The interaction of the magnetic moment ?s with this magneticfield leads to a potential energy term, called spin-orbit coupling (SOC).A more rigorous, albeit less physically transparent, derivation of the SOCterm can be obtained from the Pauli equation by taking the nonrelativisticapproximation of the Dirac equation [4]. This approach gives rise to thePauli SOC term:HSOC = ?~4m20c2? ? p? (?V ), (1.1)where ~ is Planck?s constant, m0 is the mass of a free electron, c is thevelocity of light, p is the momentum operator, V is the Coulomb potentialof the atomic core, and ? = (?x, ?y, ?z) is the vector of Pauli spin matrices.It is clear that the strength of SOC depends on the potential gradient nearthe core of atoms, therefore heavier elements usually have stronger SOC.SOC makes the spin degree of freedom respond to its orbital environ-ment. In solids this yields such intriguing phenomena as a spin splitting ofelectron states in inversion-asymmetric systems even at zero magnetic field[5, 6], a Zeeman splitting that is significantly enhanced in magnitude overthat for free electrons, and the newly discovered topological insulator (TI)[7, 8]. This exotic physics has led SOC to become central to many inter-11.1. SOC in systems with inversion symmetryesting and technologically important phenomena, including ferromagnetism,spintronics, non-centrosymmetric superconductivity, and the quantum Halleffect.1.1 SOC in systems with inversion symmetryIn systems with space inversion and time-reversal symmetry (TRS), thespace inversion symmetry gives rise to the same energy state at two oppositemomenta for both spin up and spin down: E+/?(k) = E+/?(?k); but TRSflips the spin and results in a Kramers degeneracy: E+(k) = E?(?k). Theconsequence of the combined effect of inversion symmetry in space and timeis the spin degeneracy: E+(k) = E?(k). Although SOC will not inducespin-polarized states in systems with inversion symmetry, it can change theorbital and spin characteristics of the electronic states, which are generallydescribed by energy band structures. For example, in a tight-binding picturewithout spin, these electron states can be characterized by atomic orbitals,such as p-like states with orbital angular momentum l = 1. With SOC takeninto account, electronic states become mixed and will be characterized bythe total angular momentum j = 3/2 and j = 1/2. The energy splitting ofj = 3/2 and j = 1/2 states results in a gap equal to the strength of SOC andconsequently lifts the hybridization degeneracy existing at the band edges.It has been reported that the interplay between SOC and details of theband structure close to the Fermi energy is essential for understanding themicroscopic physics in transition-metal oxides, such as the mott-insulatorSr2IrO4 [9, 10], the p-wave superconductor Sr2RuO4 [11], and the paramag-netic Fermi liquid Sr2RhO4 [11, 12]. Here we take Sr2RuO4 as an example toshow the importance of the SOC effect in a system with inversion symmetry[13, 14], which is the work in which I have been involved and to which Ihave contributed, although details are not shown in the main body of thisthesis.1In Sr2RuO4, the calculated effective SOC is comparatively small (?eff ?90 meV at the ? point) with respect to the bandwidth (? 3 eV) of the Ru-t2g1Details of the experimental results and calculation methods can been found in Ref. [13].21.1. SOC in systems with inversion symmetryZ ? M X ?Binding energy (eV)-11(a)??? ??X(b) ???X-.5.501Figure 1.1: Effect of SOC on the band dispersion and Fermi surface ofSr2RuO4. (a) Band structure along the high-symmetry directions. (b) kz =0 Fermi surface, calculated both without (thin black) and with (thick, color-coded to show ?l ? s?) the inclusion of SOC. At the ? point, the latter givesrise to a ?eff ? 90 meV splitting.orbitals, which define the ?, ? and ? conduction bands. Nevertheless, itsinfluence always becomes important whenever bands would be degenerate inthe absence of SOC, either by symmetry or accidentally. This happens in thethree-dimensional Brillouin zone (BZ), as demonstrated in Fig. 1.1, where weshow a comparison of the ab initio tight-binding band structure and Fermisurface calculated with (color) and without (black) SOC included. In theabsence of SOC, by symmetry the dxz and dyz bands would be degeneratealong the entire kz momentum path from ? to Z. Additionally, there areaccidental degeneracies along the kz = 0 path from ? to X, where the bandsof dxz,yz and dxy character all cross each other. At all these locations SOCnaturally leads to a splitting and mixing of orbital character from all threebands.The predicted importance of SOC can be directly visualized via the ex-pectation value of l ? s, with l and s being the orbital and spin angularmomentum operators. A non-zero value of ?l ? s? indicates complex orbitaleigenstates that can be entangled with the spin, seen in Fig. 1.1. Using31.2. SOC in systems without inversion symmetrycircularly polarized light-combined spin- and angle-resolved photoemissionspectroscopy, we directly measured the value of the effective SOC to be130?30 meV. This was even larger than theoretically predicted and compa-rable to the energy splitting of the dxy and dxz,yz orbitals around the Fermisurface, resulting in a strongly momentum-dependent entanglement of spinand orbital character.The consequence of a strong spin-orbital entanglement may offer a reso-lution to conflicting experiments regarding the nature of the superconduct-ing pairing in Sr2RuO4, which has been an unsolved question for decades.A fundamental assumption of classifying superconductors as a realizationof singlet or triplet paired states is that one can write the wave function ofeach electron as a simple product of spatial and spin parts, which is not pos-sible in the case of a strong entanglement between spin and orbital. Thus,the classification of the Cooper pairs in terms of singlets or triplets fun-damentally breaks down, necessitating a description of the unconventionalsuperconducting state of Sr2RuO4 beyond these pure spin eigenstates.1.2 SOC in systems without inversion symmetryAs described in the previous section, spin degeneracy is a combined effectof inversion symmetry in space and time. However, if the system has noinversion centre, which is the usual case at the surface of solids due to ashape termination or in bulk systems with a zinc blende structure,2 SOCcan lift the spin degeneracy and lead to spin-split states.In semiconductors with a zinc blende structure, the asymmetry of theunderlying crystal structure (usually called bulk inversion asymmetry), to-gether with SOC, gives rise to a Dresselhaus term in the Hamiltonian of thesystem [5]:HD = ?(kx?x ? ky?y). (1.2)Note only the linear k term is kept here, but there is also a cubic term thatcan be important with a large k. Together with the kinetic energy term2For example, GaAs, InSb, and HgxCd1?xTe have a zinc blende structure.41.2. SOC in systems without inversion symmetryp2/2m, we obtain a pair of parabolic-like bands with an equal and oppositemomentum shift away from the high-symmetry point (such as at ? withk = 0), for example as shown in Fig. 1.2(d).In semiconductor heterostructures, although the system has an inversioncentre, the confinement potential near the surface region (usually calledstructure inversion asymmetry) breaks the inversion symmetry and can leadto a spin-split Rashba term [15]:HR = ?(kx?y ? ky?x), (1.3)with ? being the Rashba parameter proportional to the SOC strength ? andthe potential gradient along the z direction dV/dz (i.e., ? ? ?dV/dz). Theenergy dispersion of Rashba band is similar to that of the Dresselhaus state,making it a challenge to determine the absolute value of both contributionsin a single sample. However, there is a substantial difference in the spintexture in momentum space, which can be simply checked by calculatingthe expectation value of spin operators for both Hamiltonians (Eq. 1.2 andEq. 1.3). For Rashba spin splitting, the spin is perpendicularly locked withmomentum and forms two chiral spin textures that have an axial symme-try about the (001) axis along kz. On the other hand, Dresselhaus spinsplitting has a spin texture only with a reflection symmetry to the (100) or(010) axis. The symmetry difference between the spin orientations allowsan experimental way to distinguish them by measuring the electron?s spinprecession [16].Rashba spin splitting has been extensively studied due to the wide avail-ability of materials, such as the surface Shockley states in Au(111) [17], thetwo-dimensional electron gas state in noble-metal-based surface alloys [18],and heterostructures [19]. Since spin splitting would provide great oppor-tunities for practical spintronic applications, high-energy-scale Rashba spinsplitting is highly desirable for enhancing the coupling between electronspins and electricity relevant for spintronic functions. Efforts to achievegiant Rashba spin splitting (? 100 meV) have been conducted mainly intwo-dimensional (2D) systems [17, 18, 20]. Recent discoveries in three-51.2. SOC in systems without inversion symmetry2.01.51.00.50-0.50-0.1-0.2-0.3-0.4-0.5Binding energy (eV)(a) (b)(c) (d)0 0.2-0.2 L A Hk// (1/?)with/withoutSOCALHMK?kxkykzx yzBiTeIFigure 1.2: Rashba-type spin split bands of BiTeI. (a) Crystal structureof BiTeI. (b) Bulk Brillouin zone. (c) Band dispersion with Rashba-splitconduction band measured by ARPES. (d) Bulk band structure calculatedby DFT with (blue lines) and without (red dashed lines) SOC included.dimensional (3D) TIs [7, 8] have fueled further exploration of materials withstrong SOC. In Chapter 4, we will show that beside their exotic topologicalsurface states, TIs also provide a new platform to achieve large-energy-scaleRashba spin splitting. Another good example of a 3D material with gi-ant Rashba spin splitting is BiTeI [21?23], which we have also measured(Fig. 1.2). The huge spin splitting in BiTeI is derived from the large SOCof all the elements (Bi, Te, and I) and the material?s layered structure with61.3. Topological insulatorsits lack of natural termination [Fig. 1.2(a)], which leads to a polar surface.Our angle-resolved photoemission spectroscopy (ARPES) measured banddispersions show a pair of spin-splitting bands near the Fermi level andare supported by DFT calcualtions, as shown in Figs. 1.2(c) and (d). TheDFT calculation shows that only the inclusion of SOC will produce the spinsplitting state, indicating a SOC-driven Rashba-type spin splitting system.1.3 Topological insulatorsRecently, it has been discovered that band insulators with strong SOC canhave a conducting spin-polarized surface state located inside the large bulkband gap. These materials, characterized by subtle topological invariants ofband structure, rather than broken symmetries, define a new quantum phaseof matter called topological insulators (TIs) [7, 8, 24?28]. The topologicalinvariant is usually expressed as an integral involving the electron?s wavefunction in momentum space, and is a unique index characterizing the elec-tronic band structure of the material [26, 32]. Topological insulators havea non-trivial topological invariant that is different from the one of ordinaryinsulators (e.g. vacuum is a ordinary insulator). The discontinuity of theseinvariants in crossing the interface between topological and ordinary (e.g.vacuum) insulators demands an emergence of a new metalic state existingat the interface, in order to enable a transformation between the topologicaland ordinary insulating states. This exotic phase of matter, and associatedtopological surface state (TSS), have become a subject of intensive researchin the past five years, because they have been predicted to have strong ap-plication potential in quantum electronic devices [29, 30] and to give rise toplatforms for detecting new particles, such as majorana fermions [31].Generally, most insulators thus far known are ordinary insulators, suchas those classified as Mott insulators or band insulators. To find TIs inreal materials, we need to develop criteria for recognizing them from theirfundamental physical properties, such as bulk band structures. It is clearthat the existence of a bulk energy gap is a necessary condition to host agapless surface state. Another condition to be a TI is to have non-trival71.3. Topological insulators1.00.50-0.5-11.00.50-0.5-1Binding energy (eV)L Z F? L Z F?(a) (b)Figure 1.3: Band inversion of bulk bands in Bi2Se3. (a), (b) Bulk band struc-tures of Bi2Se3 without (a) and with (b) SOC included, colored accordingto the parity of the bands. Red represents a bonding state with ?+? parity,and blue represents an anti-bonding state with ??? parity. Insets are visualguides to the parities of the conduction and valence states at the ? point.A band inversion can been seen at the time-invariant ? point in (b) afterSOC was included in the calculations.Z2 invariants. Based on the Z2 invariant theory, we can identify systemswith inversion symmetry as TIs, from knowledge of the parity eigenvaluesof band states at time-reversal invariant points [32]. Therefore, the guidingprinciple to search for TI materials is that the material should have oppositeparity at valence and conduction bands, and a band inversion should occurwhen the strength of some parameters, such as SOC, is tuned.By using this method, we did analysis on the bulk band structure ofBi2Se3, which is the central material studied in this thesis. As shown inFig. 1.3, the bulk bands are colored according to their parity, here definedby bonding and anti-bonding states. Without SOC included, the paritiesof valence and conduction bands at ? point have opposite signs. However,their signs are inverted after the inclusion of SOC, indicating the occurrenceof band inversion. In the DFT slab model, a spin-polarized surface state alsoappears simultaneously after the band inversion is induced by SOC3, shown3A series of slab band structures with different SOC values can be seen in Section 3.2,Fig. 3.8, and spin polarization of the surface state is discussed in Chapter 5 and 6.81.3. Topological insulators1.61.20.80.40-0.4Binding energy (eV)1.61.20.80.40-0.4(a) (b)K M? K M?Figure 1.4: Topological surface states and Rashba-split bands of Bi2Se3. (a)Topological band structure with surface states shown in orange. (b) Coex-istence of TSS and Rashba-split bands, both shown in orange. The Rashba-split band appears only after a downward band bending was introduced byadding a potential profile near the surface region, e.g., V = ?0.3e?z/19.1 eV.in Fig. 1.4. By doing this analysis, we can deduce that Bi2Se3 is a strongtopological insulator material. In fact, Bi2X3 (X = Bi, Te) has become themost promising 3D TI, attracting a great deal of interest in the past fouryears [33?35].A fully spin-polarized topological surface state, an exotic quantum stateof matter, is determined only by the non-trivial topological invariant of thebulk system. This means that no specific surface potential environment isrequired to obtain the TSS, and also that the TSS is topologically protected.These features substantially contradict the Rashba spin splitting state asdescribed in the previous section. There, a confined potential is requirednear the surface region, and the size of Rashba spin splitting is proportionalto the potential gradient. We can demonstrate the distinction between theTSS and the Rashba state by tuning the surface potential in Bi2Se3. InFig. 1.4(a), when the atomic onsite potentials are the same, both in thebulk and at the surface, we obtain a TSS band standing alone inside ofthe bulk gap. In Fig. 1.4(b), when an additional potential profile is added91.3. Topological insulatorsnear the surface region, a pair of new bands with Rashba-like spin splittingappears, and the TSS band still remains. On the other hand, because ofthe robustness of the TSS, TIs have triggered extensive research activitiesand become one of the most interesting and active current topics in materialphysics.In this thesis, we provide a systematic study of the band structure andwave function of the realistic TSS in Bi2Se3 by using ARPES and DFT. Weaddress material issues regarding the surface instability and the impurityeffects in Chapter 4. Our light-polarization-dependent study on the ARPESintensity maps reveals the full complexity of the wave function of TSS inreal materials, which has a layer-by-layer entangled spin-orbital texture, asto be presented in Chapter 5. Based on the knowledge of the existence ofa layer-by-layer entangled spin-orbital texture, we theoretically predictedand experimentally demonstrated the manipulation of photoelectron spin-polarization for the TSS in Chapter 6.Although my Ph.D. research involved various SOC-related systems, in-cluding transition-metal oxides ruthenates and the kondo insulator SmB6,this thesis will be devoted only to presenting our work on topological mate-rials, so consistency can be maintained across the thesis.10Chapter 2Angle-resolvedphotoemission spectroscopy2.1 General principlesAngle-resolved photoemission spectroscopy (ARPES) is a momentum deter-mined, low-energy electron excitation measurement technique that has beenwidely used to investigate complex systems, yielding especially notable suc-cess in high-Tc superconductors [36] and topological insulators [7]. Duringan experiment, a low-energy beam of monochromatized radiation ? gener-ated by a gas-discharged lamp, a laser, or a synchrotron beamline ? is usedto illuminate the sample to emit electrons by the photoelectric effect. Theemitted electrons, named photoelectrons, travel to the sample surface in thesolid and eventually escape into a vacuum, as described by the one-step orthree-step model in Ref.[37]. In the vacuum, the emitted photoelectrons aredetected by an electron-energy analyzer, and both their kinetic energy andemission angles are measured. Thus, the momentum of a photoelectron pin the vacuum can be determined by p =?2mEkin together with its polar(?) and azimuthal (?) emission angles. In the solid, the total energy andmomentum conservation laws can be applied to the system of crystal andphotoelectron during the photoexcitation process. The fact that there is avery short electron escape length (? 5?10 A?) allows us to reasonably assumethat there are no interactions between the photoelectron and the remainingelectrons. When the photoelectron arrives at the surface, it has to consumea portion of the kinetic energy to overcome the perpendicular surface poten-tial before escaping into the vacuum. Thus, the perpendicular component of112.1. General principlesmomentum is not conserved, but the parallel component is still conserved:p|| = ~K|| =?2mEkin ? sin?, (2.1)where K is the momentum of the photoelectron inside the solid, with thesample surface assumed to be in the xy plane. By scanning the energyand momentum distribution of photoelectrons in real space, ARPES dataprovides a vivid picture to directly observe the Fermi contours and bandstructure in the solid. To associate the measured parameters with bandstructure, we would first write down the general expression of momentumin both angular space and k-space:K =1~?2mEkin(cos?sin?i?+ sin?sin?j? + cos?k?)= Kx + Ky + Kz= kx + ky + kz + G, (2.2)where the small k is the wave vector of a Bloch state in the Brillouinzone (BZ) and G is a reciprocal lattice vector. The polar angle ? and az-imuthal angle ? are the normal spherical coordinates. Therefore, the bandstructure can be obtained by plotting the intensity distribution of electronsin a binding energy and momentum plane. Considering that only the mo-mentum component parallel to the sample surface is conserved, a generalARPES measurement is not available to provide a 3D band structure, al-though some experimental methods have been developed to obtain 3D bandmapping [38?40]. We note that the relationship in Eq. 2.2 is correct only inthe plane of the sample surface.In ARPES one needs low-energy incoming photons to obtain high-energyand high-momentum resolution, which is easy to achieve by looking at themomentum differential ?K from Eq. 2.1:?K|| '?2mEkin/~2 ? cos? ???. (2.3)In Eq. 2.3, any contribution from a fine energy resolution ?Ekin is neglected.122.2. ARPES intensity calculationEq. 2.3 clearly shows that either using low photon energy (i.e. low Ekin) ordetecting electrons with a large emission angle (i.e. large ?) would improvethe momentum resolution, meaning a smaller ?K||. The lower photon en-ergy also brings another benefit: easier achievement of a higher energy res-olution. However, the drawback of working at low photon energies is theextreme surface sensitivity due to the associated short mean free path forthe unscattered photoelectrons (? 5?10 A? for 20?100 eV kinetic energy).The high surface sensitivity thus limits the application of ARPES for bulkproperties investigation and requires an atomically clear sample surface andan ultra-high vacuum condition (usually lower than 5?10?11 torr).The technical details of the in-house ARPES system at the Universityof British Columbia (UBC) can been found in our previous group members?Ph.D. theses [41, 42]. The establishment of a theoretical expression forthe ARPES spectra is described in Damascelli?s review papers [36, 37] andreferences therein. Here, we would rather focus on the calculation of ARPESintensity, which is used to produce all the results in Chapter 5.2.2 ARPES intensity calculation2.2.1 General formulaeThe photoemission intensity measured at a momentum k is proportionalto the transition probability for an optical excitation between the groundstate ?inital(k) and one of the possible final states ?final(k), which can bewritten using Fermi?s golden rule:I(k) =2pi~|??final|Hint|?initial?|2?(Efinal ? Einitial ? h?), (2.4)where Einital and Efinal are the initial- and final-state energies of the system,and h? is the photon energy used to excite photoelectrons. The interactionbetween electrons and photons is treated as a perturbation given by:Hint = ?e2mc(A ? p + p ?A) = ?emcA ? p, (2.5)132.2. ARPES intensity calculationwhere p is the electronic momentum operator and A is the electromagneticvector potential. Note that here we did not include the relativistic term inHint, which is ? ~e4m2c2 (?V ? s) ? A. In the dipole approximation and byusing the commutation relation ~p/m = ?i[r, H], we can write:I(k) ? |??final|? ? r|?initial?|2, (2.6)where ? is the unit vector along the polarization direction of the vectorpotential A. We call the term inside the square modulus the matrix element:M(k) = ??final(k)|? ? r|?initial(k)?. (2.7)Then we can rewrite the photoemission intensity as I(k) ? |M(k)|2.2.2.2 Initial statesHere we describe the wave function of the initial state ?initial based on atight-binding (TB) model. In this model, ?initial can be written as a linearcombination of the wave functions at each basis set.?initial =?i?i, (2.8)where i is the index of the basis set, which usually presents atomic orbitals,atom sites, and spin. Since the ARPES intensity is momentum k dependent,we separate the k part of the wave function from the spatial part in ?initialto explicitly show the momentum information:?initial =?iC?i (k)?i| ??+ C?i (k)?i| ??. (2.9)C?,?i is the k-dependent expansion coefficients obtained by diagonalizingthe TB Hamiltonian in k space; | ?? and | ?? are the eigenstates of thePauli matrix ?z, representing the spin basis; and ?i is the local spatial wave142.2. ARPES intensity calculationfunction of the atomic orbitals:? = Rn,l(r)Yl,m, (2.10)where l,m is the angular momentum, Rn,l(r) is the radial part of the Blochwave function, and Yl,m are the real spherical harmonics. Taking Bi2Se3 asan example, the wave function of the initial state near the Fermi level hasonly p orbital characters (4p for Se and 6p for Bi atoms), therefore l = 1and the spatial wave functions for the px, py, and pz orbitals are:?px =?12(Y1,?1 ? Y1,1)Rn,l(r);?py = i?12(Y1,?1 + Y1,1)Rn,l(r); (2.11)?pz = Y1,0Rn,l(r).2.2.3 Final states and selection rulesIn our model, the final states are treated as free electrons whose wave func-tion can be described by a plane wave ?final = eik?r, which can be expressedas a superposition of spherical waves:eik?r = 4pi??l?=0il?jl?(k ? r)m?=l??m?=?l?Y ?l?,m?(?k, ?k)Yl?,m?(?, ?), (2.12)where jl?(k ? r) is the spherical bessel function.Plugging Eq. 2.9, 2.10, and 2.12 into Eq. 5.3, we obtain:M(k) =?i(C?i (k)| ??+ C?i (k)| ??)Gil?,l(?k, ?k)Bin,l,l? , (2.13)where Gil?,l is the integral of the angular wave function and Bin,l is the integralof the radial wave funciton. Gil?,l contains the information of the light po-larization and photoemission channels, which determines the selection rules:152.2. ARPES intensity calculationGil?,l(?k, ?k) =? m?=l??m?=?l?Y ?l?,m?(?k, ?k)Yl?,m?(?, ?)? ? r?Yl,m(?, ?)d?, (2.14)We further expand the interaction term ? ? r? in Eq. 2.14 into spherical har-monics:? ? r? =?4pi3(zY1,0 +?x + iy?2Y1,1 +x + iy?2Y1,?1), (2.15)Based on Eq. 2.14 and 2.15 we derive the explicit expression of the an-gular integral:Gil?,l(?k, ?k) =?=1??=?1(x + iy?2??,?1 + z??,0 +?x + iy?2??,1)? Yl?,m+?(?k, ?k)C1,l,l??,m,m+?,(2.16)where ? is the delta function, and C1,l,l??,m,m+? is the Clebsch-Gordan co-efficient, an integral product of three spherical harmonics, as defined by?1, ?; l,m|l?,m + ??. Eq. 2.16 has a non-zero value only at l? = l ? 1, andthis is how we obtain the optical selection rule: l? = l ? 1.For the case of Bi2Se3, we have l = 1 because of the p orbitals and there-fore we will have two possible photoemission channels: p-to-s excitation forl? = 0 and p-to-d excitation for l? = 2. The probability of exiting photo-electrons from each channel depends on the magnitude of the radial integralBin,l,l? , which has the form:Bin,l,l? =?il?jl?(k ? r)Rn,l(r)r3dr. (2.17)For the surface state of Bi2Se3, we focus on electronic states near the ??point with ?k ? 4?, which corresponds to a nearly normal photoemission.The selection rule for p-to-s and p-to-d excitations basically has the samefunction in terms of the in-plane angle ?k for normal photoemission. There-162.2. ARPES intensity calculation?pzpxpx(a) (b) (c)(d) (e) (f)Figure 2.1: Sketch of selection-rule-determined ARPES intensity maps. (a)?(c) Topological Dirac states with pz orbital character (a), with in-planeorbitals tangential to the momentum (b), and with in-plane orbitals radialto the momentum (c). The incident light is polarized along the z directionfor (a), and along the x direction for (b) and (c). (d-f) Expected ARPESintensity maps according to the optical selection rule for cases shown in (a)to (c), respectively.fore, the selection rule for linearly polarized light can be simplified into thisdescription: the linearly polarized light only excites photoelectrons from or-bitals that orient along the same direction of the light polarization. Forexample: x/y/z-polarized light only excites photoelectrons from px/py/pzorbitals, respectively, as represented in Fig. 2.1.2.2.4 Measured spin polarizationSo far, we have shown general formulae used in our photoemission inten-sity calculations. In this subsection, we will focus on the spin-resolved172.2. ARPES intensity calculationphotoemission intensity and show how to calculate the spin polarizationof photoelectrons measured by spin- and angle-resolved photoemission spec-troscopy (SARPES). For simplicity, we define:Mi ? Gil?,l(?k, ?k)Bin,l,l , (2.18)with Mi representing the matrix element, which is constant for a certainlight polarization and experimental geometry when ?k ? 0. Therefore, wecan write the photoemission intensity as the sum of intensities from spin upand spin down channels:I(k) ? I?(k) + I?(k), (2.19)withI?(k) = |?iC?i (k)Mi|2I?(k) = |?iC?i (k)Mi|2. (2.20)The measured spin polarization vector (P ) for SARPES is defined byP = [Px, Py, Pz]:Px,y,z =I?x,y,z ? I?x,y,zI?x,y,z + I?x,y,z. (2.21)In Chapter 5, Section 5.6, we use the above formulae to further demon-strate quantum interference effects in ARPES, and show a manipulation ofphotoelectron spin polarization in SARPES, which is a substantial aspectof this thesis.18Chapter 3Ab initio tight-binding modelBesides using angle-resolved photoemission spectroscopy (ARPES) to con-duct my Ph.D. research projects, I also spent a substantial amount of time ondensity functional theory (DFT) calculations, the quantitative technique forcomputing ground-state properties of materials [43]. These two techniquesare a good combination, because ARPES is the most direct experimentalmethod to probe electronic structures, and DFT is the most accurate theo-retical method to calculate the electronic structures of complex systems.In this chapter, we describe the methodology behind the ab initio tight-binding (TB) model which was used for the work presented in Chapters 5and 6. The purpose of developing an ab initio TB approach is to allowus to quantitatively understand the experimental results using a minimalmodel to describe the electron wavefunction, i.e. in terms of the smallestpossible basis set. The key ingredients to construct this ab initio TB modelare the lattice onsite energies and the coupling strength between latticesites, which are obtained by performing a bulk band calculation by DFT.A minimal basis set, i.e. only involving p orbitals, is chosen during theextraction of those onsite and coupling parameters. The atomic spin-orbitcoupling (SOC) term is added as a free parameter in our TB model. TheSOC effect in the ab initio TB model is checked by comparing the results tothose from a DFT calculation inclusive of SOC. A detailed description forconstructing ab initio bulk and slab TB models is given in Sections 3.1 and3.2, respectively. Here, we start by discussing in details the advantages anddisadvantages of the ab initio TB model as compared to a standard DFTapproach.Today DFT has already become a standard method for ab initio calcula-tions in chemistry and solid state physics. DFT calculation gained popular-19Chapter 3. Ab initio tight-binding modelity over the past decades because it yields very accurate results for complexmaterials at low cost. Several packages are available to use for DFT calcu-lations in solid state physics, such as WIEN2k [44, 45], which I have usedmost of the time, VASP [46?49], LMTO [50], SIESTA [51], ABINIT [52],PWScf [53], and CRYSTAL [54]. However, there are some limitations tousing standard DFT packages. One well-known disadvantage is the lack ofproper treatment for many body interactions in strongly correlated systems.Although improved treatments of electron correlations have been developed,including the dynamical mean-field theory [55] and quantum Monte Carloapproaches [56], the application of these advanced methods is at present toocomputationally demanding to be generally applied to complex systems.Another limitation is the complexity of the method, which is unadjustableand involves many built-in parameters. Since our materials of interest hereare weakly correlated systems, the latter problem is more relevant to ourstudy. In this chapter, we will discuss and show how to overcome the lim-itations on computational size and unadjustable parameters by developingan ab initio TB model.This ab initio TB model becomes particularly useful in a way that al-lows us to study materials by using a minimal model but with ab initioaccuracy. Because the implementation of this method is highly determinedby the particular problem or application, at present no standard packagesare available. Nevertheless, we can follow some routines to construct theab initio TB model, as will be described in this chapter. The method ofthis model has several advantages compared to regular DFT calculations.First, the ab initio TB model can handle calculations of large-scale systems(? 200 atoms) without reaching computational limits. Second, and mostimportant, we can gain full access to the Hamiltonian of the system, andall the parameters are adjustable. Of course, there are also some drawbacksto this model. For example, it is not appropriate for studying problemswith low symmetries, such as electronic reconstructions induced by contin-uous transition of potential environments from the bulk to the surface. Itdoes, however, have great advantages for symmetry-related problems, suchas surface states in topological insulators (TIs). Since the surface and bulk203.1. Ab initio bulk tight-binding modelLMTO NMTO HAMR Bulk TB Slab TBWIEN2KReferenceadjustinputchoosebasisadjustinputHSOC etc.downfoldWIEN2KReferenceadjustinputFigure 3.1: Procedural flow for constructing a tight-binding (TB) model.Note that HAMR is the real space Hamiltonian containing the hoppingintegrals; HSOC is the local spin-orbit coupling Hamiltonian.potential environments are similar in TI materials, we can construct an abinitio TB model for a finite system, such as a slab model to mimic sur-face problems, based on the hopping parameters (representing the couplingstrength between lattice sites) obtained from a bulk DFT calculation. In thisway, we can easily break the computational limit encountered in standardDFT slab calculations. Here, I will take Bi2Se3 as an example to explicitlyshow this method step by step.3.1 Ab initio bulk tight-binding modelOur goal is to build a minimal model to describe the topological surfacestate (TSS) in Bi2Se3. Our approach should be generic and adaptable forother similar studies. We start with introducing how to construct an abinitio TB model for the bulk. In the next section, we will show how to builda slab model based on the bulk model without involving additional DFTcalculations.Now we give a short summary of the steps to construct an ab initio213.1. Ab initio bulk tight-binding modelTB Hamiltonian, as shown in Fig. 3.1. Step I, we perform a bulk DFTcalculation by using the linear muffin-tin orbital (LMTO) method [50]. Theinput parameters of the LMTO, such as muffin-tin radius and basis set, areadjusted according to the degree of agreement between the LMTO and theWIEN2k band structures. Note that we use the band structure calculatedby using the WIEN2k package as a reference. Step II, we use the optimizedinput file from the LMTO to perform the same calculation but employingthe order-N muffin-tin orbital (NMTO) method [57]. A minimal basis setis decided at this step, and the corresponding parameters are adjusted untilwe obtain good agreement between the band structures calculated with afull basis set and with a minimal basis set. Step III, we use the NMTObuilt-in function to downfold the bulk band structures into a minimal basisset formed by selected atomic orbitals. The downfold procedure gives usthe hopping integrals and stores them in the HAMR file. Step IV, we usethe hopping integrals in real space to construct a TB Hamiltonian in themomentum k-space for the bulk system. At this stage, we can add extra localHamiltonians, such as those coming from spin-orbit coupling (SOC), electricor magnetic fields, etc. Step V, based on the bulk TB Hamiltonian, we canconstruct a Hamiltonian for a freestanding TB slab system with sufficientthickness by a simple truncation. We obtain the energy eigenvalues andeigenstates of the system by using the matrix diagonalization as a functionof the wave vector k|| in the surface Brillouin zone (BZ). Since the basis setof the TB Hamiltonian is Wannier functions [58], the wave function of theelectronic states of the system is the linear combination of these Wannierfunctions. When the Wannier orbitals retain the same symmetry of theatomic orbitals, we can approximate the Wannier function by the atomicwave function.In Fig. 3.2, we show good agreement between the WIEN2k and optimizedLMTO band structures from the bulk Bi2Se3. In particular, the occupiedvalence bands are in excellent agreement. But there is a visible offset for theunoccupied conduction bands that is difficult to further improve in this caseand can contribute to the different potential approximations used in LMTO(which uses sphere potential) and WIEN2k (which uses full potential).223.1. Ab initio bulk tight-binding model-6-4-202Energy (eV)WIEN2kLMTOZ?LFL Z ? F(a)(b)Figure 3.2: Comparison between the linear muffin-tin orbital (LMTO) andWIEN2k band structures. (a) Brillouin zone of bulk Bi2Se3, with dashedlines connecting high symmetric momentum points. (b) Bulk band struc-tures calculated by using WIEN2k and optimized LMTO, respectively.Since the valence and conduction bands of Bi2Se3 mainly have p orbitalcharacters, we are able to downfold the band structures near the Fermi levelwith a minimal basis set {px, py, pz} from two Bi and three Se atoms, whichgives us a minimal TB model with 15 bands. The real space parametersfor the TB model contain the hopping integral ti,j,?,? ? between differentatom sites and connecting vectors Ri,j , as illustrated in Fig. 3.3. We usethe notations i, j to represent atom positions and ? to represent atomicorbitals. The Hamiltonian in k-space can be constructed by applying Fouriertransformation, written as:H0(k) =?i,j,?,? ?ti,j,?,? ?eik?Ri,ja?i,?aj,? ? . (3.1)The eigenstates of the Hamiltonian (Eq. 3.1) are a linear combination of233.1. Ab initio bulk tight-binding model  ======================================  Real Space hamiltonian (eV)  ======================================  INT=    1|Int.bet. Se1 0001 / Se1 0001                    ||Dist= 0.000000  Trans. vecs. :  0  0  0         ||Connecting vector:   0.00000  0.00000  0.00000  |                    py          pz          px      py                -4.51207     0.00000     0.00000pz                 0.00000    -4.29555     0.00000px                 0.00000     0.00000    -4.51207 ------------------------------------------------INT=    2|Int.bet. Bi  0004 / Se1 0001                    ||Dist= 1.272415  Trans. vecs. : -1  0  0         ||Connecting vector:  -1.00000  0.00000  0.78679  |                    py          pz          px      py                -0.42425     0.00000     0.00000pz                 0.00000     0.40199    -1.05214px                 0.00000    -1.09768     1.00482 ------------------------------------------------INT=    3|Int.bet. Bi  0004 / Se1 0001                    ||Dist= 1.272415  Trans. vecs. :  0 -1  0         ||Connecting vector:   0.50000 -0.86603  0.78679  |                    py          pz          px      py                 0.64755    -0.95062    -0.61881pz                -0.91118     0.40199     0.52607Figure 3.3: Illustration of the partial content of the Bi2Se3 HAMR file, whichis obtained by using a minimal basis set {px, py, pz}.243.1. Ab initio bulk tight-binding model90% 85% 70% 50%(a) (b) (c) (d)Figure 3.4: Isosurface of charge density from the Wannier orbitals pz, withfour isovalues (a)?(d): 0.9, 0.85, 0.7, and 0.5, respectively. Red and blueindicate the phase of the orbital.(a) px py pz(b) (c)Figure 3.5: (a)?(c) Wannier orbitals of px, py, and pz with an isovalue of90%, plotted in the crystal lattice of Bi2Se3.253.1. Ab initio bulk tight-binding modelWannier functions [58]:?(k) =?i,?Ci,? (k)wi,? . (3.2)One of the most important advantages of ab initio TB model is that itallows us to expand the Hamiltonian by adding other local terms, such asthe SOC term, without involving intensive computational resources. Thesimplest SOC expression is the atomic SOC, which is derived based on thesymmetry of atomic wave functions and is written as:HSOC = ?L ? S, (3.3)where ? is the SOC parameter, L is the orbital angular momentum, and Sis the spin angular momentum. Before we add this local term HSOC into theHamiltonian of Eq. 3.1, we need to check the spatial charge distribution ofthe Wannier orbitals and make sure that we can approximate them with apicture of the atomic orbitals. As shown in Fig. 3.4, the isosurface of chargedensity from pz, shown by a Wannier orbital, indeed remains in the atomicpz orbital shape, particularly when the isovalue that defines the percentageof charge density contained inside an isosurface is below 80%. PlottingWannier orbitals in the crystal lattice, as in Fig. 3.5, shows a quite extendedspatial charge distribution around each atom site, which is important forforming bonding and anti-bonding states in the Bi2Se3 system.To add the SOC effect into the bulk TB Hamiltonian, we have to doublethe basis set by considering spin up (?) and spin down (?) for each electron.Now the basis set at each atom site is:|px, ??, |py, ??, |pz, ??, |px, ??, |py, ??, |pz, ??. (3.4)By writing the SOC Hamiltonian (Eq. 3.3) in the basis set of atomic p or-263.1. Ab initio bulk tight-binding model-6-4-2024Energy (eV) without SOC with SOCL Z ? FWIEN2kTB model(a) (b)-6-4-2024WIEN2kTB modelL Z ? FFigure 3.6: Comparison between the TB model and WIEN2k band struc-tures. Bulk band structures of Bi2Se3 obtained from WIEN2k and from aTB model with (a) and without (b) spin-orbit coupling (SOC) included.bitals and real spins, we obtain:HSOC =?2???????????0 ?i 0 0 0 1i 0 0 0 0 ?i0 0 0 ?1 i 00 0 ?1 0 i 00 0 ?i ?i 0 01 i 0 0 0 0???????????(3.5)This is the SOC Hamiltonian we added as a local term into Eq. 3.1. TheSOC parameters for Bi and Se atoms are taken from Wittel?s spectral data:?Bi = 1.25 eV and ?Se = 0.22 eV [59]. In Fig. 3.6, we compare the bulk bandstructures calculated by WIEN2k and by the ab initio TB model with andwithout SOC included, respectively. In both cases, the overall agreementbetween the two methods is good, which means that the atomic SOC ap-proximation is sufficient to capture the realistic SOC effects in the Bi2Se3273.2. Ab initio slab tight-binding modelsystem. By combining Eq. 3.1 and Eq. 3.3, the TB Hamiltonian for the bulksystem becomes:Hbulk = H0 +?iHSOC. (3.6)3.2 Ab initio slab tight-binding modelNow that we have a TB Hamiltonian with SOC included for the bulk system,the slab TB Hamiltonian used to study surface states can be created byfollowing three steps: (1) we set kz in the bulk Hamiltonian to zero; (2)using the bulk unit cell as a unit block, which is one quintuple layer (QL)for Bi2Se3, as shown in Fig. 3.7, we repeatedly duplicate the unit block alongthe z direction and make a supercell with a finite thickness, such as 50 QLs;and (3) we simply truncate the supercell Hamiltonian by discarding thehopping integrals that are coming from any atom sites above(below) thetop(bottom) surface layer, i.e., outside of the supercell.The simple truncation of the effective TB model means that this ap-proach contains no surface-specific information, being based exclusively onthe bulk Wannier functions. The accuracy of this approach might be ques-tionable, due to the fact that TB parameters near the surface might beslightly different from those in the bulk because of the possible surface po-tential. Instead of applying the naive truncation, one can refine the proce-dure so as to incorporate the changes to the TB parameters near the surface.To do so, we can perform the bulk calculation and a thin slab calculationtogether. Upon aligning the on-site energies in the interior of this slab withthe bulk values, the changes to the TB parameters near the surface can beinferred. However, it has been found that the topological surface states areessentially the same with and without the surface potential corrections [60].On the other hand, the truncation method is very useful as an appraisal forillustrating the?topologically protected? surface states that arise as a man-ifestation of the bulk electronic structure [7]. An alternative strategy forcalculating the surface bands is to use Green?s function for the semi-infinitecrystal as a function of the atomic plane, which can be obtained via iterativemethods [61?63]. For simplicity, here we use a truncated-slab approach to283.2. Ab initio slab tight-binding model1st QL2nd QL3rd QLSe1BiSe2K?MZK?Mkykx Brillouin Zone? LFFigure 3.7: Left: Crystal structure of Bi2Se3 with a unit cell formed by fiveatomic layers, called one quintuple layer (QL). A structure of three QLs isshown here. Right: Bulk Brillouin zone (black) and surface Brillouin zone(red) in momentum space.293.2. Ab initio slab tight-binding modelK ? MEnergy (eV)Energy (eV)00.40.81.2-0.4-0.800.40.81.2-0.4-0.8(a) (b) (c)(d) (e) (f)?Bi= 0 eV?Se= 0 eV?Bi= 0.25 eV?Se= 0.04 eV?Bi= 0.38 eV?Se= 0.07 eV?Bi= 0.63 eV?Se= 0.11 eV?Bi= 0.89 eV?Se= 0.16 eV?Bi= 1.25 eV?Se= 0.22 eVK ? M K ? MK ? M K ? M K ? MFigure 3.8: Band inversion and topological surface states in Bi2Se3. (a)?(f) Band dispersions obtained from a 250-atomic-layers slab for Bi2Se3 withseveral SOC values. As the SOC strength increases, the gap between thebulk bands (grey) first closes [(a)?(c)] and then reopens [(d)?(f)], indicatinga band inversion induced by tuning the SOC. A gapless surface state (blue)appears simultaneously when the band inversion occurs.study the topological surface properties of topological insulators.Since the SOC value ? is a free parameter in the TB Hamiltonian, and inorder to exhibit how SOC induces topological surface states, we calculate theband dispersions as a function of SOC strength. In Fig. 3.8, we plot the banddispersions obtained with a 250-atomic-layer slab TB model. By varyingthe SOC value, we can see that the gap between the bulk bands first closeswith increasing SOC [Figs. 3.8(a)?(c)], then reopens once the SOC is largeenough [Figs. 3.8(d)?(f)]. As expected from the bulk electronic structure oftopological insulators, a band inversion occurs after the gap reopens fromzero. Meanwhile, a gapless surface state simultaneously appears when theband inversion occurs. The topological surface state becomes more evident303.2. Ab initio slab tight-binding model50 150100 200 50 150100 20050 150100 200Atomic layer index i Atomic layer index iLayer weight (a.u.)Layer weight (a.u.)QW1 QW2QW3 QW450 150100 200Figure 3.9: Real space distribution of quantum well (QW) states obtainedfrom a 250-atomic-layer slab model. These four QW states shown here arefrom the first four conduction bands.as the band inversion becomes stronger by further increasing the SOC values.The mechanism of the SOC-driven band inversion can be understood byanalyzing the energy levels at ? point in the bulk band structure of Bi2Se3[33, 64]: SOC results in new eigenstates expressed by the total angularmomentum and induces energy repulsion between these energy states. Thelowest-energy conduction band is pushed down to a lower energy level withincreasing SOC values, while the highest-energy valence band is pushed upto a higher energy level. Consequently, in a small SOC range, these twobands get closer and eventually touch at the ? point with increasing SOC.If we keep increasing the value of SOC, these two bands will cross each otherand reopen a gap because of the strong hybridization between them. Afterthe bulk gap reopens, the characteristic of the bands sitting right above and313.2. Ab initio slab tight-binding modelbelow the Fermi level, here referring to parity as described by Z2 invarianttheory [32], are inverted as compared to the zero SOC system. Now thesystem has also been translated from a trivial insulator to a topologicalinsulator after the band inversion is induced by a large enough SOC.Quantum well (QW) states also always exist for the slab model of Bi2Se3,as shown in Fig. 3.9. These states are characterized by the bulk electronicstructure, but they have spatial distribution profiles consistent with theconfined potential of a finite slab model. As will be discussed in Chapter 4,these QW states can evolve into Rashba-like states with spin splitting whenan additional potential is added into the surface region.32Chapter 4Impurities in 3D topologicalinsulators4.1 Rashba spin-splitting control at the surfaceof Bi2Se3The electronic structure of Bi2Se3 is studied by angle-resolved photoemissionand density functional theory. We show that the instability of the surfaceelectronic properties, observed even in ultrahigh-vacuum conditions, can beovercome via in situ potassium deposition. In addition to accurately settingthe carrier concentration, new Rashba-like spin-polarized states are induced,with a tunable, reversible, and highly stable spin splitting. Ab initio slab cal-culations reveal that these Rashba states are derived from 5-quintuple-layerquantum-well states. While the K-induced potential gradient enhances thespin splitting, this may be present on pristine surfaces due to the symmetrybreaking of the vacuum-solid interface.4.1.1 IntroductionTopological insulators, with a gapless topological surface state (TSS) locatedin a large bulk bandgap, define a new quantum phase of matter [7, 26, 28, 65].Their uniqueness, and their strong application potential in quantum elec-tronic devices, stem from the TSS combination of spin polarization andprotection from backscattering [66, 67]. Bi2Se3 is a three dimensional topo-logical insulator, as theoretically proposed [33] and experimentally verifiedby angle-resolved photoemission spectroscopy (ARPES) and other surfacesensitive techniques [34, 35, 68]. Unfortunately, despite great effort in con-334.1. Rashba spin-splitting control at the surface of Bi2Se3trolling the Se stoichiometry and with it the bulk carrier concentration [69],unintentional and uncontrolled doping seems to lead to a bulk conductivitythat masks the surface electronic properties [70].ARPES studies also have shown that cleaved sample surfaces and sub-surfaces become progressively more electron doped over time ? even in ultra-high vacuum conditions ? by either gas adsorption, or formation/migrationof defects and vacancies [71, 72]. Lastly, the TSS might become inacces-sible and/or be completely deformed through a hybridization with trivialstates induced by gas molecule adsorption when exposed to air, hinderingmost attempts of material processing and characterization, as well as devicefabrication.Developing new approaches to stabilize and control the surface of thesesystems is arguably the most critical step towards the exploitation of theirtopological properties. Some success has been obtained in inducing electronand hole surface doping by a combination of in situ processing, such asmaterial evaporation and radiation exposure [73, 74]. The same TSS has alsobeen fabricated on nanoribbons, which have large surface-to-volume ratio[75]. From a different perspective, carefully doped topological insulators canprovide a platform to study the interplay between TSS and bulk electrondynamics, which has important implications for TSS control and exploringtopological superconductivity [76].In this section, we present a systematic ARPES study of the evolutionof the surface electronic structure of Bi2Se3 as a function of time and in situpotassium evaporation. The deposition of submonolayers of potassium al-lows us to stabilize the otherwise continually evolving surface carrier concen-tration. It also leads to a more uniform surface electronic structure, in whichwell-defined Rashba-like states emerge from the continuum of parabolic-likestates that characterizes the as-cleaved, disordered surfaces. This approachprovides a precise handle on the surface doping, and also allows tuningthe spin splitting of the Rashba-like states. Our density functional theory(DFT) slab calculations reveal that the new spin-split states originate fromthe bulk-like quantum-well (QW) states of a 5-quintuple-layer (5QL) slab,as a consequence of the K-enhanced inversion symmetry breaking already344.1. Rashba spin-splitting control at the surface of Bi2Se3present for the pristine surface of Bi2Se3.Time after cleave (hr)?EDP  (meV)0 353025201510550100(a)MinMaxk|| (1/?) k|| (1/?)Binding energy (eV)DPDP0.00.10.20.30.40.50.20.0-0.2 0.20.0-0.2(b)(c)0Cleaved at6 K300 K6 K cleave: 3 hrs 6 K cleave: 34 hrsFigure 4.1: (a),(b) Time evolution of the ARPES dispersion of Bi2Se3 at5?10?11 torr and T = 6 K: (a) 3 hours after cleaving; (b) 34 hours aftercleaving. (c) Exponential fit of the shift of the Dirac point (DP) bindingenergy position versus time for 6 and 300 K cleaves (both measured at 6 K);the fit result, ?EDP ? e?t/? , gives a mean lifetime with ? = 23 hours and? = 11 hours for the 6 and 300 K cleaves, respectively.354.1. Rashba spin-splitting control at the surface of Bi2Se34.1.2 Experimental and calculation methodsARPES measurements were performed at UBC with 21.2 eV linearly polar-ized photons on an ARPES spectrometer equipped with a SPECS Phoibos150 hemispherical analyzer and UVS300 monochromatized gas dischargelamp. Energy and angular resolution were set to 10 meV and ?0.1?. Bi2Se3single crystals, grown from the melt (with carrier density n'1.24?1019 cm?3[69]) and by floating zone, were aligned by Laue diffraction then cleaved andmeasured at pressures better than 5?10?11 torr and 6 K, unless otherwisespecified. No difference was observed for samples grown with different meth-ods. Potassium was evaporated at the sample temperature of 6 K, with a6.2 A evaporation current for 30 second intervals [77, 78]. DFT calcula-tions were performed using the linearized augmented-plane-wave method inthe WIEN2k package [44], with structural parameters from Ref. [33]. Weconsidered stoichiometric slabs terminated by a Se layer on both sides, rep-resenting natural cleavage planes within this material. Spin-orbit coupling(SOC) is included as a second variational step using scalar-relativistic eigen-functions as a basis [44]; exchange and correlation effects are treated withinthe generalized gradient approximation [79].4.1.3 In situ K deposition-induced Rashba-like statesThe time evolution of the as-cleaved Bi2Se3 surface is shown in Fig.4.1.As typically observed by ARPES, and contrary to what is predicted byDFT for fully stoichiometric Bi2Se3 (Fig. 4.6), even immediately after a 6 Kcleave the Fermi level is not in the bulk gap; instead it crosses both TSS andparabolic continuum of bulk-like states. The pronounced time dependenceof the data is exemplified by the variation of the Dirac point (DP) bindingenergy (?EDP ), which increases from ? 300 to 400 meV over 34 hours at5?10?11 torr and 6 K [Fig. 4.1(c)]. An exponential fit of ?EDP versus timeindicates that the lifetime value is 23 hours, e.g., EDP ' 433 meV wouldbe reached 46 hours after cleaving. At variance with the time dependenceof the TSS, the bottom of the parabolic continuum shifts down by only30 meV in 34 hours, which provides evidence against the pure surface nature364.1. Rashba spin-splitting control at the surface of Bi2Se3Binding energy (eV)?K/4Momentum k|| (1/?)Binding energy (eV)0.60.40.20.00 min 0.5 min 1.0 min 1.5 min-0.1 0.10 -0.1 0.10 -0.1 0.10 -0.1 0.10(a1) (a2) (a3) (a4)0.6 00.3 0.6 00.3 0.6 00.3 0.6 00.3(b1) (b2) (b3) (b4)K/4MinMaxFigure 4.2: Evolution of the Bi2Se3 ???K? electronic dispersion with low Kdeposition, upon subsequent 0.5 min K-evaporation steps: (a1)?(a4) ARPESimage plots, (b1)?(b4) corresponding energy distribution curves (EDCs).The sample was kept at 5?10?11 torr and 6 K.of the continuum. One should note that the pristine position of DP dependsalso on the cleave temperature: on a sample cleaved at 300 K we found a70 meV deeper starting position for the DP, although the saturation valueis approximately the same as that of the 6 K cleave [Fig. 4.1(c)].In our ARPES study, the surface time evolution resulted only in thedeepening of Dirac cone (DC) and bulk continuum, as a consequence of thesample gaining electrons. Other effects, such as the reported appearance of a2-dimensional electron gas, were not observed [80]. More substantial changesare induced by the in situ evaporation of potassium on the cleaved surfaces,374.1. Rashba spin-splitting control at the surface of Bi2Se3Binding energy (eV)?K/4Momentum k|| (1/?)Binding energy (eV)0.60.40.20.0K/42.0 min 2.5 min 3.0 min 4.0 minMinMax-0.1 0.10-0.1 0.10 -0.1 0.10 -0.1 0.10(a5) (a6) (a7) (a8)0.6 00.3 0.6 00.3 0.6 00.3 0.6 00.3(b5) (b6) (b7)  (b8)Figure 4.3: Evolution of the Bi2Se3 ???K? electronic dispersion with heavy Kdeposition, upon subsequent 0.5 min K-evaporation steps: (a5)?(a8) ARPESimage plots, (b5)?(b8) corresponding energy distribution curves (EDCs).The sample was kept at 5?10?11 torr and 6 K.also performed at 6 K to guarantee the highest stability. As a function ofK-deposition time, three stages can be identified: Stage I ? for moderate Kdeposition [up to 1 minute, Figs. 4.2(ab1)?(ab3)], the DP moves to higherbinding energy by electron doping and a sharper parabolic state appearsat the edge of the bulk continuum, reminiscent of the proposed 2DEG [80].Stage II: for intermediate K deposition [from 1 to 3 minutes, Figs. 4.3(ab4)?(ab7)], the electron doping further increases and two pairs of sharp parabolicstates appear, with an equal and opposite momentum-shift away from the?? point, as in a Rashba type [15] splitting [these states are labelled RB1384.1. Rashba spin-splitting control at the surface of Bi2Se3and RB2 in Fig. 4.4(a)]. First RB1 develops from the newly formed sharpparabolic state identified in stage I, followed by RB2 which develops closerto the Fermi energy. Interestingly, the appearance of the sharp RB1 andRB2 features is accompanied by a suppression of the bulk-like continuum.This emergence of a coherent quasiparticle dispersion from a continuum ofincoherent spectral weight indicates that the evaporation of potassium leadsto a progressively more uniform surface and subsurface structure. Stage III:for heavy K deposition [beyond 3 minutes, Fig. 4.3(ab8)], the bottom of RB1and RB2 as well as EDP are not changing, indicating that the sample cannotbe doped any further. The only noticeable effect is a small decrease of spinsplitting for RB1 (by 0.015 A??1) and conversely an increase for RB2 (by0.01 A??1), perhaps stemming from a change in hybridization between thetwo Rashba pairs. As a last remark, during the entire K-deposition processthe band velocity of the TSS close to the DP is 3.2 ? 0.3 eVA?, consistentwith previous reports [81].Before analyzing quantitatively the evolution of the various states uponK deposition, we address the question of the stability of this new surfaceversus time and temperature cycling. In Figs. 4.4(a)?(c) we compare theARPES data from a 3 minutes K-evaporated surface, as measured right af-ter deposition and 30 hours later (during which the sample was kept at 6 K).Other than a smaller than 10 meV shift of the bottom of RB1 [Fig. 4.4(c)],all spectral features including the TSS have remained exactly the same overthe 30 hours interval. This is a remarkable stability, especially when com-pared to the 365 meV shift induced by the initial K deposition [Fig. 4.4(e)],and to the more than 100 meV shift observed versus time without any activesurface processing [Fig. 4.1(c)]. This approach might provide a new path toovercome the general instability and self-doping problem of the surface ofBi2Se3, which represents one of the major shortcomings towards the fabri-cation of topological devices. Temperature effects were studied by slowlywarming up the sample, in which case K atoms diffuse and eventually leavethe surface, reverting the material back to an earlier stage with lower Kcoverage. Indeed, as one can see by comparing Fig. 4.4(d) to Fig. 4.2(a4), asample initially K evaporated for 4 minutes at 6 K, and then measured at394.1. Rashba spin-splitting control at the surface of Bi2Se3T= 220 K      0.60.40.20.00.80.60.40.20.00.8T= 6 K, fresh K k|| (1/?)DCRB1RB2DPT= 6 K, 30 hrs laterk|| (1/?)-0.2 0.0 0.2 -0.2 0.0 0.2-0.2 0.0 0.2k|| (1/?)Binding energy (eV)RB?E (eV)n2D (?1013 cm-2)?kF (?10-2 ?-1)DCRB1K deposition time (min)0 1 2 3 4Fresh K30 hrs later(a) (b)(c) (d)(e)(f)(g)Binding energy (eV)Min Maxouterinner0.00.10.20.30.424681001.50.51.00.0Figure 4.4: Stable and reversible band dispersions. ARPES ??? K? banddispersion from Bi2Se3 taken (a) immediately after a 3 min K evaporation,and (b) 30 hours later (the sample was kept at 5?10?11 torr and 6 K thewhole time). As also emphasized by the comparison of the corresponding?? point EDCs in (c), the evaporated surface is highly stable. (d) Banddispersion measured at 220 K after a slow 36 hours warming up on a sampleinitially K evaporated for 4 min at 6 K; the comparison with the data inFig. 4.3(a8) reveals the suppression of the K-induced carrier doping. (e-g) Evolution vs. K-evaporation time of: (e) binding energy variation forDP (?EDP ) and bottom of RB1 (?ERB1), as defined in (a); (f) sheetcarrier density for DC (nDC2D ) and RB1 (nRB12D ); (g) variation of the DCFermi wavevector (?kDCF ) and of the Rashba band splitting at EF (?kRB1F ).Empty symbols in (e)?(g) are for T = 6 K and filled ones for T = 220 K.404.1. Rashba spin-splitting control at the surface of Bi2Se3220 K after a gradual 36 hours warming up, exhibits ARPES features simi-lar to those obtained directly after a 1.5 minutes K deposition at 6 K. Thisimplies that K deposition on Bi2Se3 is also reversible, making it possible tofine tune surface doping, position of the DP, and Rashba spin splitting.We summarize in Figs. 4.4(e)?(g) the K-evaporation evolution of variousparameters characterizing the ??? K? dispersion of DC and Rashba states(empty symbols identify 6 K data, and the filled ones 220 K data). As evi-dent in Fig. 4.4(e) from the variation of EDP and bottom of RB1, the highestpossible doping level is achieved ?3 minutes into the K deposition, corre-sponding to ?EDP ' 365 meV and ?ERB1 ' 150 meV (note that RB2 isnot plotted due to its later appearance and fewer data points; after ?3minutes K deposition ?ERB2'65 meV ). The K-induced change in surfaceelectron density for the various states can be estimated from the relationn2D = AFS/ABZ AUC between the area of Fermi surface, Brillouin zone,and unit cell, without accounting for spin degeneracy given that all relevantstates are spin split. Because at these electron fillings all FS?s are hexagonal,this reduces to n2D = k2F /2?3pi2, where kF is the Fermi wavevector alongthe ???K? direction of the BZ (as in Fig. 4.2 and Fig. 4.3).After 3 minutes K evaporation the total sheet carrier density is ntot2D '3.64? 1013 cm?2 (0.162 electron/BZ), corresponding to the sum of the con-tributions from DC, and inner-and-outer RB1 and RB2 (1.43, 1.60, and0.61 ? 1013 cm?2 respectively). This value is to be compared to ntot2D '3.87 ? 1012 cm?2 before K deposition (0.017 electron/BZ), which howeveronly accounts for the DC, given the impossibility of estimating the contri-bution from the parabolic continuum.As a last point, from the ?kF data presented in Fig. 4.4(g), and thedispersion of spin-split Rashba bands:E?(k?) = E?? +~2k2?2m?? ?Rk?, (4.1)we can estimate the Rashba parameter ?R = ~2?k?? /2m? for RB1 and RB2.The latter, which depends both on the value of spin-orbit coupling (SOC)and the gradient of the potential ?V/?z [82], reflects the size of the spin414.1. Rashba spin-splitting control at the surface of Bi2Se31.00.5-0.50K ? M K ? M K ? M K ? M2QL 4QL 6QL 8QLBinding enrgy (eV)(a) (b) (c) (d)Figure 4.5: (a)?(d) Quantum well (QW) and surface slab states calculatedwith various thicknesses: 2QL, 4QL, 6QL, and 8QL, respectively. The QWnumber (shown in red) plus one surface state (shown in bule) equals to theQL number of the slab thickness.splitting in momentum space and is here controlled directly by the amount ofK deposited on the as-cleaved surfaces. The largest RB1 splitting is observedafter 2.5-3 minutes K evaporation and is anisotropic: ?kF '0.066A??1 along??-K?, and 0.080A??1 along ??-M? . The spin splitting of RB2 increases slowlyduring the whole K evaporation process and has an isotropic ?kF '0.02A??1for 3 minutes K evaporation. Fitting the Rashba-like band dispersions along??-K? to Eq. 4.1, for RB1 (RB2) we obtain E?? = 295?10 meV (172 meV),m? = 0.28?0.02me (0.19me), and ?R = 0.79?0.03 eVA? (0.35 eVA?). Thevalue of ?R for RB1 is more than twice the Rashba splitting of the Au(111)surface state (?R ' 0.33 eVA?), and also larger than the one of the Bi(111)surface state (?R'0.56 eVA?) [18].4.1.4 Quantum well states and conclusionDFT calculations for bulk Bi2Se3, as well as slabs with varying number ofQL?s (Fig. 4.5 and Fig. 4.6), provide a detailed explanation for our obser-vations and some interesting insights. Each QL consists of 2 Bi and 3 Selayers alternating along the c axis, with one Se layer in the middle of theQL and the other two on either side. This forms a non-polar structure witha natural cleavage plane between two adjacent Se layers belonging to dif-424.1. Rashba spin-splitting control at the surface of Bi2Se3ferent QL?s. As shown in Fig. 4.6(b) for the particular case of a 5QL slabin addition to the TSS-DC there are 4 QW states, for a total of 5 statesmatching the number of QL?s. As evidenced by the comparison with thefully kz?projected bulk results in Fig. 4.6(a), where the TSS is missing dueto the absence of the surface, the slab QW states exhibit the same charac-ter and energy as the Bi-Se conduction band. However, they are discretein nature due to quantum confinement, and span a narrower energy rangethan the corresponding bulk bandwidth WB = 520 meV. The effective slabbandwidth WQL, defined as the energy difference between top and bottomQW states, is asymptotically approaching the bulk WB value [Fig. 4.6(c)];for a proper correspondence with the bulk electronic structure a rather largenumber of QL?s is needed (i.e., more than 10 QL). Interestingly, the splittingbetween the DP and the different QW states is extremely sensitive to thenumber of QL?s. For 5QL we obtain 346 meV QW1-DP and 126 meV QW2-QW1 splittings [Fig. 4.6(c)], which closely match the 3 minutes K-depositionvalues 380?50 meV and 123?6 meV for RB1-DP and RB2-RB1, respectively[as defined from the EDC?s at the ?? point in Fig. 4.4(a)].This analysis leads to several important conclusions: (i) The RB1 andRB2 states that emerge from the parabolic continuum are of the same Biand Se character as those obtained from bulk Bi2Se3 calculations in thesame energy range. However, because of the observed lack of kz dispersion[80] and the almost perfect match comparing the energy of RB1-RB2 withQW1-QW2 from 5QL slab calculations [as seen in Fig. 4.6(c)], these statesshould be more appropriately thought of as the quantum-confined analogof those bulk states associated with a band-bending over a 5QL subsurfaceregion (47.7 A?). While this subsurface region is disordered on the as-cleavedsurfaces (either in its depth and/or carrier concentration), which causesa continuum of states, the disorder is suppressed upon K evaporation asevidenced by the appearance of the well defined RB1 and RB2 features. (ii)Potassium, in addition to doping carriers, also induces a change in ?V/?z,which in turn provides a very direct control knob on both band-bendingdepth and spin splitting of the Rashba states. (iii) In light of the extent ofthe subsurface band-bending region, these quantum-confined states should434.1. Rashba spin-splitting control at the surface of Bi2Se3(b)-0.20Energy (eV)(a)0.80.40.6?E (eV)Exp: 3 minK depositionWB = 520 meVQW2QW1DCQW3QW45QL slabK/4 M/4?QW2-QW1QW3-QW1QW4-QW1QW5-QW1QW6-QW1QW7-QW10.10.30.40.50(c)Number of QL653 94 7 8WQLBulkK/4 M/4?0.2Figure 4.6: Energy splitting of quantum well (QW) states. DFT results for(a) kz-projected bulk and (b) 5QL slab of Bi2Se3 (EF is at energy 0). (c)Energy difference of the QW slab states with respect to QW1, calculated forvarious thicknesses. The QW number increases with QL number, defining abandwidth WQL that asymptotically approaches be fully kz-projected bulkWB = 520 meV. The 3 minutes K-deposition RB2-RB1 splitting of 123?6 meV is accurately reproduced by the 5QL results.affect more than just surface sensitive experiments. For instance, Rashbaspin-split states might have to be accounted for in the interpretation oftransport data even from pristine surfaces, although with a much smallersplitting induced solely by the symmetry breaking vacuum-solid interface.444.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe34.2 Tailoring spin-orbit coupling in Mn-dopedBi2?xMnxTe34.2.1 IntroductionSince topological surface states are chiral and spin-polarized, they shouldbe immune to localization as long as the disorder potential does not vio-late time reversal symmetry (TRS). Breaking the TRS of the topologicalsurface state (TSS) can lead to a gap opening at the Dirac point. Moreimportantly, breaking the TRS is key to realizing novel physics phenomenasuch as the axion electrodynamics and the spin-galvanic effect [30]. In thetwo-dimensional topological insulator (TI) HgTe, it has been shown thata magnetic field can break the TRS and open a gap in the surface state[83]. In three-dimensional TIs, an efficient way to break TRS is to develop along-range magnetic order by doping the system with magnetic impurities.In the previous section, we demonstrated that one way to study impurityeffects is through in situ surface doping via adsorption of impurities onpristine TI materials. Extensive studies have been conducted along this lineby evaporating magnetic impurities on the surface of topological insulators[84?87]. However, none of these studies showed any indication of an openinggap at the Dirac point, which we would expect from a broken TRS inducedby magnetic impurities. Success has instead been achieved using anotherexperimental method, which is to add magnetic impurities into the sampleduring crystal growth, i.e., bulk doping. It has been reported that magneticdopants break the TRS and open a gap at the Dirac point in Bi2Se3 [35].Besides their relevance for controlling the TRS in the system, a dilutionof the system?s SOC by impurities with small SOC can give rise to newquantum phases that could be essentially important for applications of thesystem, such as tuning the system through a quantum phase transition fromthe topological to the non-topological phase [88, 89].In this section, we focus on the doping evolution of the electronic statesof Bi2?xMnxTe3 (with x = 0, 0.04, and 0.1, respectively). The motivation ofthe study is to examine how the ferromagnetic phase affects the topological454.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3surface states, such as opening a gap at the Dirac point by breaking theTRS. Unfortunately, the Dirac point of Bi2Te3 is buried inside the valenceband [33], resulting in an invisible Dirac point. This characteristic of thematerial makes it impossible to study the broken TRS by examining thepresence of a gap in Bi2Te3. Nevertheless, we can study other effects causedby the ferromagnetic ordering and the almost zero SOC of Mn impurities.We will show that the Mn impurities can decrease the bulk band gap bydiluting the effective SOC of the system. Furthermore, we show that thetemperature effect on the width of momentum distribution curves (MDCs)indicates the possibility of ferromagnetic domain formation, albeit with ashort range at temperatures below 12 K.4.2.2 Material propertiesThe Mn-doped Bi2Te3 samples used in our study were grown by R.J. Cava?sgroup at Princeton University. The Mn concentration was analytically de-termined by doing elemental analysis, and samples with x = 0, 0.04, and0.1 were used for our ARPES study. The temperature-dependent magneticsusceptibilities measured in an applied field of 1 kOe showed a ferromagnetictransition for x ? 0.04 at T ? 12 K. The ordered ferromagnetic moment canreach a magnitude of ? 4?B per mol-Mn, with the c axis as an easy axisof magnetization. STM topographic images of the in situ cleaved surfaceof Bi1.91Mn0.09Te3 did not show Mn clusters and indicated that the systemas a true dilute ferromagnetic semiconductor [90]. X-ray absorption spectra(XAS) on the Mn L2,3 edge suggested a 2+ valence state for the Mn impu-rities, in contrast to the 3+ of Bi atoms in Bi2Te3 [91]. This means that Mnimpurities will lead to holes in the crystal and make the system hole-doped.The local density of states measured by STM showed a 150 meV upwardshift in chemical potential that is consistent with the p-type character of theMn dopants [90].464.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe30 0.2 0.4-0.2-0.4 0 0.2 0.4-0.2-0.4 0 0.2 0.4-0.2-0.400.20.4Binding energy (eV)k// (1/?)(a) (b) (c)x=0 x=0.04 x=0.1Figure 4.7: (a)?(c) ARPES dispersions measured at 6 K with freshly cleavedsamples of Bi2?xMnxTe3: x = 0 (a), 0.04 (b), and 0.1 (c).4.2.3 Doping-level-dependent ARPES spectraWe performed ARPES measurements on Bi2?xMnxTe3 crystals with x = 0,0.04, and 0.1, respectively. The samples were cleaved at low temperature,T = 6 K, with a pressure lower than 5? 10?11 torr. The orientations of thesamples were checked by Laue diffraction peaks and were aligned in the ?to K direction parallel to the entrance slit of the electronic analyzer.In Fig. 4.7, we show ARPES dispersions measured with in situ freshlycleaved samples. The Fermi levels of all three samples (indicated by whitedashed lines) cross the valence band below the Dirac point. This p-typecharacter of Bi2Te3 samples is in contrast to the n-type Bi2Se3 samples,which always have a Fermi level crossing the bulk conduction bands, asshown in Fig. 4.1. However, the as-cleaved surface of both systems, Bi2Te3and Bi2Se3, has the same instabilities from becoming progressively electron-doped, as seen from the dynamics of the electronic states for Bi2Te3 inRef.[72] and for Bi2Se3 in Fig. 4.1. Because of the surface instability, it isdifficult to identify the Mn impurities as hole dopants simply by comparingthe Fermi level positions.After about 18 hours of sample cleavage and with the assistance of asmall amount of potassium deposition on the sample surface, we were able474.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3to lower the topological surface state to cross the Fermi level, as shown inFig. 4.8. The surface state remained in both the pure and the Mn-dopedBi2Te3. No noticeable changes were observed by comparing the ARPESspectra Figs. 4.8 (a) to (c) and (d) to (f). First-principles model calculationssuggested that the dispersion of the surface state should depend considerablyon the Mn magnetization. With magnetization of Mn either along the c-axisor in the ab-plane, the dispersion will become more linear, as shown by DFTcalculations in Ref. [92]. We attempted to fit the ARPES spectra by usingthe surface band dispersion, taking into account the hexagonal warping effectas proposed by Fu [93]. The fitting results may suggest that the warpingterm was indeed suppressed by increasing the Mn concentrations, meaninga better linear dispersion at a higher Mn doping level (results are not shownhere). However, the fitting results are extremely sensitive to the fittingprocedures, such as the choice of fitting energy window and the estimationof the Dirac point position. All these parameters can bring large uncertaintyto the fitting results and thus lead to an inconclusive outcome.By comparing Figs. 4.8 (a) to (c) or (d) to (f), we learned that a visiblechange in the electronic dispersion induced by Mn impurities is the dopinglevel dependence of the bulk energy gap size. Based on the EDC profile at ??point, we define the bulk energy gap size by measuring the energy distancebetween the Dirac point4 and the bottom of the bulk conduction band. Forpure Bi2Te3, we obtained a gap of ? = 300?5 meV. For Mn-doped samples,? = 260?10 meV for x = 0.04 and ? = 200?20 meV for x = 0.1. Thedecrease in the bulk gap in Mn-doped Bi2Te3 can be also seen through theconstant energy contours (CECs). In Fig. 4.9, we plot CECs cutting throughfour different energy levels with reference to the Dirac point. It is clear thatup to 0.25 eV above the Dirac point, the bulk conduction band still did notappear in the CECs for pure samples; however, it appeared in the CECs ofthe Mn-doped samples as a nondispersive feature with high-intensity aroundthe ? point.In TIs, the bulk energy gap size is directly related to the strength of the4The Dirac point position is estimated by assuming that the surface bands are linearlydispersed near the maxima valence band and will cross each other at the Dirac point.484.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe30 0.1-0.1 0 0.1-0.1 0 0.1-0.1Binding energy (eV)0 0.1-0.1 0 0.1-0.1 0 0.1-0.100.20.400.20.4Binding energy (eV)k// (1/?)(a) (b) (c)x=0 x=0.1x=0.04(d) (e) (f)x=0 x=0.1x=0.040.3 eV0.26 eV0.2 eVKK ? KK ? KK ?MM ? MM ? MM ?Figure 4.8: (a)?(c) Doping-level-dependent ARPES spectra ofBi2?xMnxTe3, measured along ?? ? K? with x = 0, 0.04, and 0.1, re-spectively. (d)?(f) Similar data but measured along ?? ? M? . The bulkenergy gap size of Mn-doped Bi2Te3 is labeled in (a)?(c) and has a value0.3 eV (x= 0), 0.26 eV (x= 0.04), and 0.2 eV (x= 0.1), respectively.SOC of the systems. As shown in Fig. 3.8 in Section 3.2, band inversion leadsfirst to closing the gap and then to reopening it as the SOC of the systemincreases. We quantitatively tracked the gap size as a function of the SOCstrength from our ab initio tight-binding model described in Chapter 3, andobtained a relationship between the bulk energy gap size ? and the strengthof the SOC ? after the surface state appears, written as:? = 0.33?? 0.18 (Unit: eV). (4.2)494.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe300.2-0.200.2-0.200.2-0.2Eb=0.25 eV 0.2 eV 0.1 eV 0 eV (DP)(a) (b) (c) (d)(e) (f) (g) (h)(i) (j) (k) (l)0.20-0.2 0.20-0.2 0.20-0.2 0.20-0.2ky (1/?)k x (1/?)Figure 4.9: (a)?(c) Constant energy contours of Bi2?xMnxTe3 with x= 0(a)?(d), x= 0.04 (e)?(h), and x= 0.1 (i)?(l). Here the zero binding energyis set at the Dirac point (DP) for convenience.Note that this relationship is obtained from the Bi2Se3 model. Bi2Te3 differsfrom Bi2Se3 mainly due to the higher SOC in Te atoms (0.49 eV) comparedto Se atoms (0.22 eV). However, Bi atoms have much larger SOC at 1.25 eV,so the total SOC of the system will mainly depend on the Bi atoms [59].Therefore, it is still useful to compare the experimental data to this lin-ear relationship with a ratio of 0.33 between between the gap size and thestrength of the SOC. Taking the sample with x= 0.1 as an example, we canapproximate the SOC of Mn to be zero, and thus a 5% Mn substitution of Bicould result in a 62.5 meV reduction in SOC. Naively, the bulk energy gapsize would decrease only by 21 meV if we made the estimate based on Eq. 4.2.However, we obtained a 100 meV reduction from ARPES data. Therefore,504.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3the disparity between theoretical prediction and experimental observationindicates that although the Mn concentration is low (only 2?5%), the ef-fective content of the impurity can be at least four times higher, which isactually similar to what has been reported for Bi-induced large relativisticcorrection in GaAs1?xBix semiconductors [94]. More importantly, when theSOC of the system is diluted, the impurities can turn the topological systeminto a trivial insulator even at low concentrations.4.2.4 Temperature effects on ARPES spectraMagnetic susceptibility measurements showed a development of ferromag-netism in Mn-doped Bi2Te3 at temperatures below 12 K [90]. Therefore,Mn-doped Bi2Te3 is expected to have broken TRS at T ?12 K. The bro-ken TRS may affect the topological protection and could potentially opena gap at the Dirac point of the topological surface band. Extensive studieshave been carried out to look for an open gap in topological surface bandsusing magnetic impurities through either bulk magnetic doping in crystalsand thin films [35, 95, 96] or in situ surface doping adsorption of impuritieson pristine TI materials [84?86]. Most of these efforts focused on Bi2Se3crystals, because there is a well-defined Dirac point in the surface bands.However, for Bi2Te3 the low-energy region of the surface band is buriedinside the bulk valence bands, making it difficult to identify a clear Diracpoint. Nevertheless, Mn impurities will always introduce disorder into thesystem, which could enhance the scattering rate of the TSS. How magneticimpurities affect the scattering channels for the TSS has so far only beenstudied through the adsorption of various magnetic impurities on the surfaceof topological insulators [85]. For bulk magnetic impurities-doped systems,this question has not been studied yet. Here we study the scattering rateof the TSS of Mn-doped Bi2Te3 by looking at the width change in themomentum distribution curve (MDC) above and below the ferromagnetictransition temperature Tc = 12 K.As we discussed in Section 4.2.3, the as-cleaved sample has a Dirac sur-face state above the Fermi level. The instability of the sample surface514.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe30 0.2-0.200.20.4Binding energy (eV)0 0.2-0.200.20.4Binding energy (eV)k// (1/?)5 K21 K5 K21 Kk//= 000.20.4Intensity (a.u.)5 K21 KEb= 0.02 eV0.20-0.2Intensity (a.u.)5 K21 KEb= 0.35 eVIntensity (a.u.)Eb (eV)k// (1/?)k// (1/?)0.20-0.2(a)(b)(c)(d)(e)Figure 4.10: (a), (b) ARPES dispersions of Bi1.96Mn0.04Te3, measured below(5 K) and above (21 K) Tc. The ferromagnetic transition temperature Tc is12 K [90]. (c) EDCs at the ?? point, measured at 5 K and 21 K. (d), (e)MDCs above and below the Dirac point, measured at 5 K and 21 K.can progressively electron dope the system either by absorbing residual gasmolecules or by diffusing Te vacancies. The uncontrolled electron dopingmoves the Fermi level towards the bulk conduction band, and allows us toaccess the TSS in ARPES after keeping the cleaved sample in an ultra-highvacuum for a certain period without degradation. Fig. 4.10 and Fig. 4.11show the ARPES spectra obtained 38 hours later after the sample cleavage524.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe30 0.1-0.100.20.4Binding energy (eV)0 0.1-0.100.20.4Binding energy (eV)k// (1/?)5 K20 K5 K21 Kk//= 000.20.4Intensity (a.u.)Eb= 0.07 eV0.10-0.1Intensity (a.u.)Eb= 0.25 eVIntensity (a.u.)Eb (eV)k// (1/?)k// (1/?)0.10-0.1(a)(b)(c)(d)(e)Figure 4.11: (a), (b) ARPES dispersions of Bi1.9Mn0.1Te3, measured belowand above Tc. The ferromagnetic transition temperature Tc is 12 K [90]. (c)EDCs at the ? point, measured at 5 K and 21 K. (d), (e) MDCs above andbelow the Dirac point, measured at 5 K and 21 K.of Bi2?xMnxTe3 with x= 0.04 and 0.1, respectively. The Fermi level hasbeen shifted upwards by about 200 meV, and the linearly dispersed TSS canbe clearly observed (comparing Figs. 4.10 and 4.11 to Fig. 4.7). To exam-ine the response of electronic structures to the ferromagnetic transition, wecompare the ARPES spectra in terms of dispersion maps, EDC and MDCprofiles that were measured at T = 5 K (below Tc) and T = 21 K (above534.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3Tc). The comparisons indicated subtle changes at temperatures below andabove Tc, in particular for the MDC profiles. To quantitatively identify thechanges, we will show the temperature dependence of the MDC width byfitting the MDCs with a Voigt profile, which is a convolution of Lorentz andGaussian profiles.Generally, there are three scenarios that we would expect for the tem-perature dependence of MDC width: case I, if the scattering is mainly dueto the electron-phonon interaction, the MDC width should become narrowerwith lower temperatures; case II, since the Mn-doped Bi2Te3 has a ferro-magnetic ordering formed below Tc, if this ferromagnetic ordering is of longrange, the scattering between electrons and disordered impurities could besuppressed and the MDC width would also become smaller at temperaturesbelow Tc; and case III, if the ferromagnetic ordering is a local ordering withshort range, then perturbation of impurities to the system could becomestronger due to the development of disordered ferromagnetic domains, sim-ilar to the local antiferromagnetic ordering induced by Mn impurities inSr3Ru2O7 [97]. In this case, the disorder of local domains would enhancethe scattering rate and result in a wider MDC width. In addition, if the localmagnetic field broke the TRS, the ferromagnetic transition would open newscattering channels by destroying the chiral spin texture of the TSS; thenwe would expect broader MDCs at T ? Tc, and the scattering rate couldmonotonically increase with increasing Mn concentration.Firstly, we look at the doping level dependence of MDC widths for freshlycleaved samples. As shown in Fig. 4.12. the MDC width of the bulk valenceband is plotted as a function of binding energy for x= 0, 0.04, and 0.1. Wesee that the MDC width becomes larger as the Mn doping level increases,indicating a more disordered system for a higher Mn concentration. Byfitting the dispersions near the Fermi level that were shown in Figs. 4.10(a),(b) and Figs. 4.11 (a), (b), we obtained the MDC width of the TSS for2% and 5% Mn-doped Bi2Te3, respectively. The fitting results from theright-hand-side branch of the TSS are plotted in Fig. 4.13. For x= 0.04, theMDC width at T = 5 K is clearly larger than at 22 K and 32 K. However,if we compare the MDC width at two temperatures (22 K and 32 K) that544.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe30.250.200.150.10FWHM (1/?)0.30.20.10Binding energy (eV)x = 0x = 0.04x = 0.1Figure 4.12: Doping-level-dependent MDC width of Bi2?xMnxTe3, mea-sured at 6 K. The MDC width was obtained by fitting the data shown inFig. 4.7.are both higher than Tc, the difference is rather small. These results werereplicable and were independent of the temperature history. A temperaturecycle was performed by cooling down and then warming up the crystals; asshown in Fig. 4.13(a), we obtained almost overlapping data points for thesame temperatures (5 K or 22 K) but measured during different temperaturecycles.The increase of MDC width at temperatures below Tc can be attributedto either the development of local ferromagnetic ordering or the broken TRS.As described earlier, if the enhancement of scattering rate at temperaturesbelow Tc was predominantly due to the broken TRS, we would expect toobserve a monotonic increase in MDC width with increasing Mn concentra-tion at T ? Tc. However, the difference in MDC width for x= 0.1 above554.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe30.160.140.120.100.080.06FWHM (1/?)806040200Energy (meV) 5 K 22 K 5 K 22 K 32 K0.180.160.140.120.100.0812080400Energy (meV)20 K 5 K x=0.04 x=0.1(a) (b)Figure 4.13: (a, b) Temperature-dependent MDC width of Bi2?xMnxTe3with x= 0.04 (a) and 0.1 (b). The width of the MDC is obtained by fittingthe data shown in Fig. 4.10 and Fig. 4.11. For the x= 0.04 sample, a tem-perature cycle measurement was performed, shown by two data sets in thesame color.and below Tc [Fig. 4.13(b)] is rather small compared to that for x= 0.04[Fig. 4.13(a)], meaning that there is a weakly broken TRS effect on the scat-tering rate; on the other hand, if the enhanced scattering rate is mainlydue to the onset of local ferromagnetic ordering, then the smaller changein MDC width at higher Mn doping levels can be understood by consid-ering that with increasing Mn concentration, the system will start to formlong-range ferromagnetic ordering, and therefore the effect of disordered do-mains with short-range ordering will be suppressed. In conclusion, (i) thesubtle temperature- and doping-level-dependence of MDC width might sug-gest that the ferromagnetic phase induced by Mn impurities enhanced thescattering of the TSS by forming disordered domains and (ii) the evidencefor broken TRS is not clear in Mn-doped Bi2Te3.4.2.5 K deposition at the surface of Bi1.96Mn0.04Te3In Section 4.1, we described work involving in situ K deposition on Bi2Se3.Here we use the same experimental method but apply it to 2% Mn-doped564.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3-1.0-0.8-0.6-0.4-0.20.0eV-0.4 -0.2 0.0 0.2 0.4k, 1/?-1.0-0.8-0.6-0.4-0.20.0eV-0.4 -0.2 0.0 0.2 0.4k, 1/?-1.0-0.8-0.6-0.4-0.20.0eV-0.4 -0.2 0.0 0.2 0.4k, 1/?-1.0-0.8-0.6-0.4-0.20.0eV-0.4 -0.2 0.0 0.2 0.4k, 1/?0 0.2-0.2Binding energy (eV)0.60.40.00 0.2-0.2 0 0.2-0.2 0 0.2-0.2Momentum k|| (1/?)(a) (b) (c) (d)1.5min 3.5min 5min 7minFigure 4.14: (a)?(d) Evolution of ARPES dispersions of Bi1.96Mn0.04Te3along ???K? as a function of K deposition time. Potassium was deposited onthe sample at 21 K.0.20.10.0-0.1-0.2ky-0.2 -0.1 0.0 0.1 0.2kx0.20.10.0-0.1-0.2ky-0.2 -0.1 0.0 0.1 0.2kx0.20.10.0-0.1-0.2ky-0.2 -0.1 0.0 0.1 0.2kx0.20.10.0-0.1-0.2ky-0.2 -0.1 0.0 0.1 0.2kx0.20.10.0-0.1-0.2ky-0.2 -0.1 0.0 0.1 0.2kx0.20.10.0-0.1-0.2ky-0.2 -0.1 0.0 0.1 0.2kx0 0.2-0.2 0 0.2-0.2 0 0.2-0.200.2-0.200.2-0.2ky (1/?)k x (1/?)k x (1/?)ky (1/?) ky (1/?)(a) (b) (c)(d) (e) (f)0.4 eV 0.3 eV0 eV (DP)0.1 eV0.2 eV-0.1 eVFigure 4.15: (a)?(f) ARPES constant energy contours of Bi1.96Mn0.04Te3after seven minutes of K deposition. The cutting energies refer to the Diracpoint.Bi2Te3. The purpose of this section is to demonstrate that potassium depo-sition on Bi2Te3 can also induce Rashba-like QW states with spin splitting.Fig. 4.14 shows the ARPES dispersions measured after varying the K de-574.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3position time. The modification of the surface electronic structure is similarto what we have discussed in Section 4.1.3. Quantizations of both the bulkconduction band and the valence band states are observable after 1.5 min-utes of K deposition. The valence band evolves into QW states, presenting aset of bands with an M shape. The bulk conduction band lacked dispersionand now becomes quantized by forming pairs of parabolic bands with spinsplitting, which are QW states with Rashba-like spin splitting.Previous DFT calculations and a photon-energy-dependent ARPES studyof Bi2Te3 [98] have indicated that the bulk states showed triangular CECsbecause of the crystal lattice symmetry, and the surface states gave hexago-nal CECs due to the warping effect [93]. In Fig. 4.15, we show the evolutionof CECs as they move away from the Dirac point and reveal the binding-energy-dependent bulk and surface character of the bands. Near the Diracpoint, the surface states are degenerate with the bulk valence band statesand therefore we observe triangular CECs. The surface states appear aloneat 0.1 eV above the DP, with hexagonal CECs. The CECs at high energy(i.e., 0.4 eV above DP) again show the coexistence of surface states (outsidecontour) and bulk featured states (inner contour), which are spin-splittingQW states here. The last point to emphasize is that since the Rashba-likespin-split states are from QW states with bulk features, the CECs of thesestates have a triangular star shape rather than a hexagonal or circular shape,although their band dispersion looks like a two-dimensional (2D) electrongas with parabolic dispersions, as shown in Fig. 4.14(d) and Fig. 4.15(a).4.2.6 ConclusionIn this section, we used ARPES to study the electronic structures of Mn-doped Bi2Te3 with different doping levels, as well as below and above theferromagnetic transition temperature Tc = 12 K. A notable effect of Mn im-purities is the dilution of the system?s SOC, evidenced by the decrease inthe bulk energy gap size with increasing Mn concentrations. Our analysisshowed that the effective doping level can be four times higher than thephysical Mn concentrations in the crystals. This suggested that a quantum584.2. Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3phase transition from non-trivial topological to trivial topological insulatorscan be induced by impurities even at low concentrations (? 5%); similarphenomena have also been observed in (Bi1?xInx)2Se3 [99]. A further the-oretical investigation about impurity-induced topological phase transitionis required and would have a fundamental impact on topologically tunablephysics and devices. The wider MDC width at lower temperatures may sug-gest an onset of local ferromagnetic ordering at T ? 12 K. The evidence ofbroken TRS due to ferromagnetic ordering is not conclusive from this study,and could be further investigated by performing fine temperature-dependentARPES and SARPES.59Chapter 5Layer-by-layer entangledspin-orbital texture in Bi2Se3With their spin-helical metallic surface state, topological insulators definea new class of materials with strong application potential in spintronics.Technological exploitation depends on the degree of spin polarization ofthe topological surface state (TSS), assumed to be 100% in phenomenolog-ical models. Yet in Bi2Se3, an archetypical topological insulator material,spin- and angle-resolved photoemission spectroscopy (SARPES) detects aspin polarization ranging from 20 to 85%, a striking discrepancy which un-dermines the applicability of real topological insulators. Here we show ?studying Bi2Se3 by polarization-dependent ARPES and density-functionaltheory slab calculations ? that the TSS Dirac fermions are characterized bya layer-dependent entangled spin-orbital texture, which becomes apparentthrough quantum interference effects. This explicitly solves the puzzle of theTSS spin polarization in SARPES, and suggests how 100% spin polariza-tion of photoelectrons and photocurrents can be achieved and manipulatedin topological-insulator-based devices by using linearly polarized light.5.1 IntroductionTopological insulators (TIs) define a new state of matter in which strongspin-orbit coupling (SOC) leads to the emergence of a metallic topologicalsurface state (TSS) formed by spin-nondegenerate Dirac fermions [7, 24?28]. To capture the physics of TIs, a spin-momentum locking with 100%spin polarization is usually assumed for the TSS in time-reversal invariantmodels [26?28]. A fully spin-polarized TSS has tantalizing properties, such605.1. Introductionas robust edge modes and exotic quasiparticle excitations, making TIs goodcandidates for spintronic applications [30]; and also a playground for engi-neering Majorana fermions, and exploiting their non-Abelian statistics intopological quantum computation [31]. The successful realization of topo-logical insulating behavior in quantum wells [66, 83] and crystalline mate-rials [33?35] brings us closer to the practical implementation of theoreticalconcepts built upon novel topological properties.To this end, the fundamental question is that of the effective spin-polarization of the TSS Dirac fermions in real materials and devices, i.e., howrealistic is the hypothesis of a 100% spin polarization. The large discrepancyin the degree of TSS spin polarization determined for Bi2Se3 by SARPES? ranging from 20 to 85% [73, 88, 100?102] ? highlights the complexity ofreal TIs. First principle density-functional theory (DFT) calculations in-dicate that the TSS spin polarization in members of the Bi2X3 materialfamily (X = Se, Te) can be substantially reduced from 100% [103, 104]. Inaddition, based on general symmetry arguments, it was shown that the spinpolarization direction of photoelectrons in SARPES can be very differentfrom that of the TSS wave function [105]. However, the role played by theintrinsic properties of the TSS wave function in defining the highest spinpolarization that could be achieved, for instance in d.c. and photoinducedelectrical currents, has remained elusive.We report here that the TSS many-layer-deep extension into the mate-rial?s bulk ? in concert with strong SOC ? gives rise to a layer-dependent,entangled spin-orbital texture of the Dirac fermions at the surface of Bi2Se3.This enables the precise control of both in- and out-of-plane spin polariza-tion of the photocurrent in SARPES ? all the way from 0 to ?100% ? byvarying energy, polarization, and angle of incidence of the incoming photons.More generally, the layer-by-layer variation in spin-orbital entanglement isof fundamental importance to spintronic studies and applications, whoseoutcome will depend critically on how one couples to the TSS. A remark-able consequence, which we will specifically exploit in this study, is thatone can gain exquisite sensitivity to the internal structure of the TSS wavefunction, ?TSS, via quantum interference effects using ARPES. In partic-615.2. Polarization-dependent ARPES intensity patternular, the spin-orbital texture is captured directly in the linear-polarizationdependence of the ARPES intensity maps in momentum space, and can befully resolved with the aid of ab initio DFT slab-calculations.5.2 Polarization-dependent ARPES intensitypatternLet us start our discussion from the Bi2Se3 ARPES results in Fig. 5.1,measured with ? and pi linearly-polarized 21.2 eV photons. Based on ourexperimental geometry [Fig. 5.1(a)] and the photoemission selection rules,?-polarized light probes mainly the in-plane px and py orbitals, whereaspi-polarized light a combination of both in-plane and out-of plane (pz) or-bitals: the observed 80% reduction in overall intensity by switching from pi-to ?-polarization indicates that the TSS has a dominant pz orbital charac-ter. As for the evolution of the ARPES intensity around the Dirac cone,in ?-polarization [Figs. 5.2(a)?(c)] we observe a twofold pattern at both 0.1and 0.2 eV above the Dirac point (DP), consistent with a previous report[106], although somewhat asymmetric with respect to the ky = 0 plane [seein particular Fig. 5.2(c)]; this suggests a tangential alignment of the in-planepx,y orbitals with respect to the Dirac constant-energy contours. Converselyin pi-polarization [Figs. 5.2(d)?(f)] we observe a strongly asymmetric pat-tern at 0.1 eV above the DP, which evolves into a triangular pattern whilestill retaining some asymmetry at 0.2 eV; this is in stark contrast with theuniform distribution of intensity along the Dirac contour expected for thedominant out-of-plane pz orbitals. Finally, at ?0.1 eV below the DP, a tri-angular pattern is observed for both polarizations [see insets of Figs. 5.2(a)and (d)].The asymmetry in ARPES intensity between ?k? is particularly evi-dent in pi-polarization at 0.1 eV in Fig. 5.2(d) and in the band dispersionof Fig. 5.1(b). This finding, which might seem in conflict with the time-reversal invariance of the TSS, provides fundamental clues on the structureof ?TSS. Time-reversal invariance requires the state at +k with (pseudo)625.2. Polarization-dependent ARPES intensity patternky (?-1)Energy (eV)00.20.40 0.1-0.1(a) (b)?xzy ?h??? 45o?e-MKhighlowsampleFigure 5.1: ARPES experimental geometry with linearly polarized light.(a) Schematics of the experimental geometry, with pi (horizontal) and ?(vertical) linear polarizations, and horizontal photoelectron emission plane.(b) ARPES dispersion measured along K?????K? with pi polarization; thezero of energy has been set at the Dirac point (DP) for convenience.spin up to be degenerate with the state at ?k with (pseudo) spin down, i.e.,to have the same real-orbital occupation numbers. This so-called Kramersdegeneracy, together with the ARPES selection rules for linearly polarizedlight, forbids intensity patterns which are different at ?k. We emphasizehere that this restriction can be rephrased in terms of purely in-plane mo-mentum coordinates, i.e., ?k?, only for a perfect 2-dimensional TSS with adelta-function-like density, for which kz plays no role. Thus the observationof an imbalance in ARPES intensity at ?k?, together with the establishedtime-reversal invariance of TIs, necessarily implies that ?TSS must have afinite extent ? albeit not a dispersion [80] ? along the third dimension. Whiledetails will become clear when discussing our DFT results in Fig. 5.4 andFig. 5.5, we anticipate that this ? together with the strong SOC ? leadsto a complex layer-dependent spin-orbital entanglement in Bi2Se3, whichbecomes apparent in ARPES through photoelectron interference effects.By performing ARPES intensity calculations [107]5 for TSS and bulk5Details of the ARPES calculations can be found in Section 5.6.2.635.2. Polarization-dependent ARPES intensity patternIntensityk x (?-1)ky (?-1)0 0.1-0.10 0.1-0.100.1-0.1?0.2 eV?0.2 eV?0.1 eV-0.1 eV?0.1 eV-0.1 eV0 ? 2?? (rad.)1.00.5000.1-0.11.00.50(b) (c)(d) (e)k x (?-1)(f)0.1 eV0.2 eV?Intensityky (?-1)0.1 eV0.2 eV?(a)Figure 5.2: Linear polarization dependence of the measured ARPES inten-sity in momentum space. (a), (b) Constant energy ARPES maps from above(0.1 and 0.2 eV) and below (-0.1 eV, inset) the DP, measured with ? polar-ization; (d), (e) same for pi polarization. (c), (f) Normalized variation of the?- (c) and pi-polarization (f) ARPES intensity, along the Dirac contours,plotted as a function of the in-plane angle ?.wave functions from our DFT slab-calculations, we accurately reproducethe data6. As shown in Figs. 5.3(a)?(f), we obtain very different intensitiesat ?k? in excellent agreement with the results for both ? and pi polariza-tions. Specifically, we reproduce the quasi-twofold pattern in ? polariza-tion, stemming from the spatial configuration of px,y orbitals [Figs. 5.3(a)and (b)]; the quasi-threefold pattern away from the DP [Fig. 5.3(e)], whichoriginates from the hybridization between TSS and bulk states [108]7; andalso the triangular patterns at -0.1 eV [insets of Figs. 5.3(a) and (d)]. Note6Note that the TSS calculations have actually been performed for 0.12 and 0.23 eV,in order to account for the DP?conduction-band gap renormalization observed betweenexperiment (?260 meV) and DFT (300 meV).7Note that the threefold pattern is associated with the kz dispersion of valence- andconduction-band bulk states.645.2. Polarization-dependent ARPES intensity patternIntensityk x (?-1)00.1-0.1ky (?-1)0 0.1-0.1 0 0.1-0.1? (rad.)0 2?+1-100.1 eV 0.2 eV 0.3 eV0 0.1-0.1ky (?-1)k x (?-1)RCP-LCP RCP-LCP RCP-LCP1.00.5000.1-0.11.00.500 0.1-0.1 0 0.1-0.100.1-0.1(g) (h) (i)k x (?-1)ky (?-1)0.1 eV 0.2 eV? ?BVB0.1 eV0.2 eV?Intensityky (?-1) ky (?-1)0.1 eV 0.2 eV? ?BVB0.1 eV0.2 eV?(a) (b) (c)(d) (e) (f)?Figure 5.3: Calculated ARPES polarization dependence. (a), (b) Calculatedconstant-energy ?-polarization ARPES maps for TSS (0.1 and 0.2 eV) andbulk valence band (BVB, -0.1 eV in the inset); (c) corresponding variationof the ARPES intensity versus the in-plane angle ?. (d)?(f) Same data asin (a)?(c), but now for pi-polarization. (g)?(i) Calculated constant-energycircular dichroism ARPES patterns at 0.1, 0.2, and 0.3 eV above the DP;insets: patterns obtained by rotating the sample by 90 ? about the normal.655.2. Polarization-dependent ARPES intensity pattern0.51.01.5-0.50Energy (eV)(a)K M?010201005pzpx, yBiSe1 Se2Atomic layer index i  1 5 10 15Projected TSS (%)(b)Figure 5.4: Layer-projected topological surface state. (a) Electronic disper-sion from our 250-atomic-layer-slab DFT model, with TSS in orange andbulk states in green (?TSS is composed of those states that exceed a 20%projection onto the top five atomic layers in real space). (b) Percentage con-tribution of pz and px,y orbitals to ?TSS at 0.15 eV above the DP, resolvedlayer-by-layer, for the top 15 atomic layers of the slab (Se1 is the naturalcleavage plane).that the ARPES intensity visible at the ?? point in Fig. 5.2(b) and (e), butnot reproduced by our calculations, originates from the scattering-inducedbroadening of the bulk conduction band8. As a final test of the robustnessof our DFT analysis of ?TSS, we have calculated constant-energy circulardichroism ARPES patterns, which are also in excellent agreement with pre-vious studies [111, 112].8In the experiment, the bulk conduction band is significantly broadened by disorder-induced scattering, which leads to intensity leaking down to lower binding energies insidethe gap, an effect not accounted for in our calculations. We also note that while thissame disorder affects to some degree the linewidth and electronic filling of the TSS, itsmain features ? such as dispersion and spin-orbital texture close to the Dirac point ? areretained, as a manifestation of topological protection [109, 110].665.2. Polarization-dependent ARPES intensity pattern?Sy?px, y(a)(b)(c)p totalSe1BiSe2BiSe1i=1i=2i=3i=4i=5(d)+1-10pz0.15 eVkxkyFigure 5.5: Layer-by-layer entangled spin-orbital texture. (b)?(d)Layer- and orbital-projected charge density along the 0.15 eV k-spacecontour indicated in (a): their surfaces are defined by r(?, ?) =??,? ? Z? (?, ?)Z? ?(?, ?)?a?i,?,kai,? ?,k?, where i and ? are the layer and localorbital basis indexes, and Z the cubic harmonics; their surface color rep-resents the expectation value of the Sy operator. While the pz channel islayer independent (b), a strong layer-dependent spin-orbital entanglement isobserved for px,y (c). The total layer-resolved TSS texture (d) is obtained byadding all p orbital contributions according to their relative, layer-dependentweight from panel [Fig. 5.4(b)].675.3. Layer-dependent entangled spin-orbital texture5.3 Layer-dependent entangled spin-orbitaltextureTo gain a microscopic understanding of the properties of ?TSS we presentour DFT results for a 250-atomic-layer slab of Bi2Se39 in Fig. 5.4(a), withbulk states in green and TSS in orange. The in- and out-of-plane p orbitalprojections in Fig. 5.4(b) confirm that ?TSS indeed has a large pz (70%)character ? although px,y (30%) is also significant ? and most importantlythat ?TSS extends deep into the solid. Even though the orbital weight decaysexponentially with the distance from the surface, as expected for a surfacebound-state, ?TSS extends approximately 2 quintuple layers (QL) belowthe surface (? 2 nm), with ?75% contribution from the 1st QL and ?25%from the 2nd QL. Note also the interesting layer dependence of the orbitalcharacter: while for most layers the main component is the out-of-plane pz,for the 5th for example the in-plane px,y is actually dominant.As a consequence of the relativistic SOC, which directly connects orbitalto spin flips via the l?s? terms of the spin-orbit operator l ? s = lzsz +(l+s?+ l?s+)/2, the strongly layer-dependent orbital occupation becomesentangled with the spin polarization of ?TSS. To visualize this entanglement,in Figs.5.5(b)?(d) we present the layer- and orbital-projected charge densityalong the 0.15 eV Dirac contour indicated in Fig.5.5(a), colored according tothe expectation value of the Sy operator. The pz-projected charge density,being associated with a single orbital, cannot be entangled and has thelayer-independent spin helicity shown in Fig. 5.5(b)10. In contrast, a stronglayer-dependent spin-orbital entanglement is observed for px,y because theeigenstates can be a linear combination of px,?, py,?, and similar states,resulting in a complex set of charge-density isosurfaces. These surfaces showtwo overall spatial configurations having opposite spin helicity, which areoriented tangentially and radially with respect to the Dirac cone contour11,9See methods in Section 5.6.1.10Note that in the ky direction the charge density is white because a state with spin inthe x direction is written as a linear combination of states with equal amount of spin upand spin down along the y direction.11Interestingly the in-plane spin-orbital entanglement is reversed for states below the685.4. TSS spin texture and photoelectron spin polarizationas seen in Fig. 5.5(c). In Fig. 5.5(d) we show the total layer-dependent chargedensity obtained by adding in- and out-of-plane contributions according totheir relative weights in Fig. 5.4(b); from this it is clear that while the pzorbitals dominate, the in-plane px,y orbitals lead to a substantial spin-orbitalentanglement of the combined ?TSS.We will now highlight the interplay between photoelectron interferenceand the measured ARPES intensity. While a complete derivation ? inclu-sive of selection rules, Bi and Se cross sections, as well as photoelectronescape depth ? is given in Section 5.6.1 and 5.6.2, for the purpose of thisdiscussion we approximate the ARPES intensity as I?|?eik?r|A?p|?TSS?|2,expressed in terms of plane-wave photoelectron final states for simplicity.By writing ?TSS as linear combination of the layer-dependent eigenstates,?TSS =?i,? ?i??i,k?with i and ? being layer and spin indexes, the ARPESintensity then becomes I ??? |?i e?ikzzi?eik??r? |A ? p|?i??i,k??|2. Here thee?ikzzi phase term accounts for the photoelectron optical path differencestemming from the TSS finite extent into the bulk. Because both e?ikzziand ??i,k? vary from layer to layer [the latter via the relative orbital contentas shown in Fig. 5.7(b)], the photoemission intensity is dominated by inter-ference between the ??i,k? eigenstates, and can in fact be regarded as theFourier transform of the layer-dependent ?TSS. We also note that, becausethe phase of photoelectrons is defined by additive kz and k? contributions,reversing the sign of either kz or k? will change the ARPES intensity, i.e.,I(kz) 6= I(?kz) and especially I(k?) 6= I(?k?) as observed experimentally.5.4 TSS spin texture and photoelectron spinpolarizationThe spin-orbital entanglement also leads to complex in- and out-of-planespin-texture, as shown in Figs. 5.6(a)?(d) where the layer-integrated spinpatterns of individual and total p orbitals are presented. While for pz wefind the in-plane helical spin texture expected for the TSS this is not the caseDP: while the spin helicity remains the same as above the DP, the orbital texture switchesbetween tangential and radial configurations.695.4. TSS spin texture and photoelectron spin polarization0.7-0.7?K K?MM?K K?Sz??n?S    ?/npx?px ?S    ?/npy?py?K K ?K K?S    ?/npz?pz ?S         ?/ntotal?total00.2-0.2?Sz??n0?MM(a) (b)(c) (d)Figure 5.6: Orbital-projected spin texture of TSS. (a)?(d) Spin texture ofthe Dirac cone upper branch in Bi2Se3 obtained from the expectation valueof the layer-integrated, orbital-projected spin operators, normalized to theorbital occupation [shown in Fig. 5.8(a)]; in-plane and out-of-plane spin com-ponents are represented by arrows and colors, respectively. Note that (a),(b) and (c), (d) have different color scales but that the arrow scaling remainsthe same, with the largest arrows representing full polarization; also, movingaway from ?? corresponds to moving along the Dirac dispersion away fromthe DP (?0.4 eV at the map edge).705.4. TSS spin texture and photoelectron spin polarization= ?30o20o10o0o40o50o60o70o80o90oh? (eV)Photoelectron Py20 40 60 80 10001.00.5-0.50-1.0?-polarizatione KMkykxPy=1Px=Pz=0? independenth? (eV)Photoelectron P 20 40 60 80 10001.00.5-0.50-1.0?-polarizatione KMkxky(a)(b)Figure 5.7: Calculated photoelectron spin polarization. (a), (b) Predictionfor the photoelectron spin polarization (P ) measured in SARPES as a func-tion of photon energy and incidence angle; two experimental geometries areexamined in pi polarization for the same k point located at 0.15 eV along???M?, as indicated in the sketches [in (a) only the P component along y isshown].for the px and py orbitals, which exhibit patterns opposite to one another.Combining all contributions [?~Stotal?/ntotal in Fig. 5.6(d)], the TSS out-of-plane spin texture vanishes in the vicinity of the DP; most important, thein-plane spin polarization is reduced from 100% to 75% at the DP, and to715.5. Conclusion60% at 0.4 eV above the DP. Note that this is also critically dependent onthe relative px,y orbital content of ?TSS, which increases from 25% to 45%over the same energy range12.Photoelectron interference also severely affects the photoelectron spinpolarization Px,y,z = (I?x,y,z?I?x,y,z)/(I?x,y,z+I?x,y,z) measured in SARPES.This exhibits a strong dependence on photon energy, polarization, and an-gle of incidence, which in general prevents the straightforward experimentaldetermination of the intrinsic spin-texture of Bi2Se3. While comprehensiveresults are presented in Section 5.6.5, in Figs. 5.7(a) and (b) we show as anexample the same k point along ???M? measured in two different geometries,probing selectively py,z (a) and px,z (b) orbitals. In Fig. 5.7(a), because?~Spy? and ?~Spz? (the spin polarization of the py and pz orbitals) are an-tiparallel at this specific k point, Py varies between ?100% upon changing?, and oscillates wildly as a function of photon energy (with the exceptionof 0 ? and 90 ?, which probe py and pz separately). However, if the sampleis rotated by 90 ? as in Fig. 5.7(b) ?~Spx? becomes parallel to ?~Spz? and themeasured Px,y,z are all independent of photon energy and incidence angle,allowing the detection of the intrinsic spin polarization. We note that thisbehavior is consistent with reported SARPES results [102]: for the situationof Fig. 5.7(b), Py > 80% was obtained, close to our 100% expectation; along???K?, Py was observed to vary from 25% at h? = 36 eV to ?50% at 70 eV,while we obtain +20?10% and ?40?15%, respectively.5.5 ConclusionWe have shown that the spatial extent of the TSS into the solid and itsstrong layer-dependent spin-orbital entanglement are responsible ? via pho-toelectron interference ? for the apparent time-reversal symmetry breakingin ARPES and the large discrepancy in the estimated TSS spin-polarizationfrom SARPES. This is of critical importance for many applications andfundamental studies of TIs. For instance we note that the photoelectroninterference responsible for I(k?) 6= I(?k?) in ARPES also provides an12See Fig. 5.8 in Section 5.6.4.725.6. Supplemental materialexplanation for the so-far puzzling observation of spin-polarized electricalcurrents photoinduced by linearly-polarized light [113], which also is associ-ated with an imbalance in the number of photoelectrons removed at ?k?. Inaddition, exploiting photoelectron interference in SARPES provides a waynot only to probe the intrinsic spin texture of TIs, but also ? and mostimportantly ? to continuously manipulate the spin polarization of photo-electrons and photocurrents all the way from 0 to ?100% by an appropriatechoice of photon energy, polarization, and angle of incidence.5.6 Supplemental material5.6.1 Experimental and theoretical methodsAngle-resolved photoemission spectroscopy (ARPES) was performed at UBCusing a SPECS Phoibos 150 hemispherical analyzer and a monochromatizedand linearly polarized UVS300 gas-discharge lamp. Energy and angular reso-lutions were set to 10 meV and ?0.1?. We used 21.2 eV photons, whose close-to-100% linear polarization can be rotated to any angle without changingsample orientation; in this study we focused on experiments for horizontal(pi) and vertical (?) polarizations. Bi2Se3 single crystals were grown from themelt at the University of Maryland [with carrier density n'1.24?1019 cm?3[69]], and by floating zone at the University of Geneva. The samples wereprealigned ex situ by conventional Laue diffraction, and cleaved and mea-sured at pressures better than 5?10?11 torr and at a constant temperatureof 6 K [1].The bulk electronic structure of Bi2Se3 was calculated using the tight-binding order-N muffin-tin orbital (TB-NMTO) [50, 57] and full-potentialWIEN2k [44] density functional theory codes; we find excellent agreementbetween the two methods. The TB-NMTO approach is used to down-foldthe ab initio Hamiltonian to a 15-band model involving only the p orbitalsof Bi and Se. This allows us to extract on-site energies and hopping param-eters which are used to construct a 250-atomic-layer thick slab TB model(i.e., 50 quintuple layers), with atomic spin-orbit coupling (SOC) included735.6. Supplemental materialas a local term for Bi and Se orbitals [1.25 eV and 0.22 eV, respectively [59]].To understand the microscopic origin of the ARPES intensity patterns inFig. 5.1, we have performed photoemission intensity calculations for bothlinearly and circularly polarized light. Following an established approach[107], photoelectron final states are treated as spin-degenerate plane waves;however, to account for ARPES matrix elements, these plane waves havebeen expanded in spherical harmonics and Bessel functions around eachatom. Since the initial states have mainly p orbital character, conservationof angular momentum only allows excitations into s and d-like free-electronstates. Under these selection rules, photoelectrons from px, py, and pz or-bitals can be excited by x, y, and z polarized light respectively [37]; all otherexcitations are forbidden. Finally, the Bi and Se atomic cross sections [e.g.,2.7 for Bi 6p and 8.0 for Se 4p at photon energy 21.2 eV [114]] also havebeen taken into account in calculating photoemission intensities, as well asthe finite escape depth of the photoelectrons. Also note that throughout thechapter, the coordinate system is consistent with the crystal structure: thekx axis is along ???M?, the ky axis is along ???K?, and the z axis is along thesample normal [001].5.6.2 ARPES intensity and interference effectsIn the following sections we will show how we calculated the data presentedin Sections 5.2 and 5.4. We will begin, in this section, by calculating the ex-plicit ARPES intensity based on our ab initio TB model13. In the followingSection 5.6.3, we will calculate the spin-polarization of photoelectrons, andhow that relates to the TSS ground-state spin-polarization; importantly wewill note how they can differ due to interference terms, and also depend onthe relative orbital occupations. In Section 5.6.4 we will present these rel-ative orbital occupations, while in Section 5.6.5 we resolve the interferenceterms and explicitly present the spin texture patters of photoelectrons forsome example experimental configurations. In Section 5.6.6, we will demon-strate how certain aspects of these results, namely the asymmetry in in-13An introduction about the ab initio TB model is given in Chapter 3.745.6. Supplemental materialtensity between k? and ?k?, can be understood through a simple modelsystem.Based on Fermi?s golden rule, the photoemission intensity can be writtenas [37, 107]:I ? |??f |A ? p|?i?|2, (5.1)where p is the electron momentum operator, A the electromagnetic vectorpotential, and ?i and ?f the initial- and final-state wave functions. We usethe dipole approximation in the calculations such that A ?p is approximatedby r. Here we focus on the photoemission of the topological surface states(TSS). Therefore, ?i = ?TSS which is the wave function of the TSS andcan be written as a linear combination of atomic wave function in our abinitio TB model:?TSS =?i,?,?C?i,??i,? . (5.2)Here i is the atomic layer index along the z axis of the slab with the surfacelayer at i = 1, the orbital basis is given by ? ? {px, py, pz}, ? is the spinindex which is ? or ?, and ?i,? are the atomic wave functions of orbital? centered around the atomic layer i. The photoelectron final states aretreated as free-electron-like, whose wave function can be described by aplane wave ?f = eik?r. We can therefore define the matrix element termas:Mi,? ? ?eik?r|A ? p|?i,? ?. (5.3)As discussed in section 5.3, the TSS is not a perfect two-dimensionalstate with a delta-function-like density in the z direction. Instead, it ex-tends more than 2 nm deep into the bulk along the z direction. We takeinto account the spatial extent of the wave function along z by assigningan atomic-layer-dependent phase to photoelectrons: eikzzi , with zi being theposition of the atomic layer i along z and kz =?2me~2 (h? ? EB)? k2x ? k2yis the momentum of photoelectrons along z, which depends on photon en-ergy h? and the initial-state binding energy EB. Note that the phase ofthe photoelectrons is determined by their kinetic energy inside the materialrather than in the vacuum; for this reason we do not consider the work755.6. Supplemental materialfunction here. The finite photoelectron escape depth is also considered inour calculation, by including an exponential attenuation factor dependenton the mean free path (?) of the photoexcited electrons; we used ? = 7 A?,although no substantial change in our results was observed in the 5?10 A?range. In order to show the effects of photon energy and escape depth, weredefine Eq. 5.3 as:Mi,? ? e?ikzzie?zi/(2?)?eik??r? |A ? p|?i,? ?. (5.4)with k = {kx, ky, kz} and k? = {kx, ky}. Finally, Eq. 5.1 can be written asthe sum of the intensity from up and down spin channels:I ???=?,?|?i,?C?i,?Mi,? |2. (5.5)The latter we can expanded to obtain the explicit form of the ARPES in-tensity:I =?i,?(C?i,??C?i,? + C?i,??C?i,? )|Mi,? |2+?i 6=i?,? 6=? ?(C?i,??C?i?,? ? + C?i,??C?i?,? ?)M?i,?Mi?,? ? .(5.6)Here?i,? C?i,??C?i,? + C?i,??C?i,? = 1 for the normalized TSS wave functionand the sum?i 6=i?,? 6=? ? represents the interference between different termsin the basis set ? i.e., orbitals in the same or different atomic layers.5.6.3 Photoelectron spin polarizationThere is a clear analytical relationship between the photoelectron spin po-larization measured by SARPES and the TSS ground-state spin polarizationobtained from the expectation value of spin operators applied on the TSSwave function. In a simple system with a single orbital and a single atomiclayer, the photoelectron spin polarization is given by the TSS ground-statespin polarization. For a system with multiple orbitals and atomic layers,765.6. Supplemental materialthe interference terms become important and lead to a deviation from thesingle-orbital and single-atomic-layer system. In the following part of thissection, we will derive the relationship in the multi-orbital and -atomic layersystem with the interference term included.SARPES measures the spin polarization along different quantizationaxes, which here are the x , y and z directions as defined in Fig. 5.1(a),Section 5.2. The photoelectron spin polarization vector (P ) is defined asP = [Px, Py, Pz] where:Px,y,z =I?x,y,z ? I?x,y,zI?x,y,z + I?x,y,z. (5.7)Hereafter, we define ? (?) ??z (?z) and use the usual spin relations:| ?? =1?2(| ?x?+ | ?x?) =1?2(| ?y?+ | ?y?),| ?? =1?2(| ?x? ? | ?x?) =1?2(?i| ?y?+ i| ?y?). (5.8)By using Eq. 5.5, 5.7, and 5.8, we can calculate Px,y,z:Px =?i,?(C?i,??C?i,? + C?i,??C?i,? )|Mi,? |2I+?i 6=i?,? 6=? ?(C?i,??C?i?,? ? + C?i,??C?i?,? ?)M?i,?Mi?,? ?I,Py =?i,?i(?C?i,??C?i,? + C?i,??C?i,? )|Mi,? |2I+?i 6=i?,? 6=? ?i(?C?i,??C?i?,? ? + C?i,??C?i?,? ?)M?i,?Mi?,? ?I,(5.9)775.6. Supplemental materialPz =?i,?(C?i,??C?i,? ? C?i,??C?i,? )|Mi,? |2I+?i 6=i?,? 6=? ?(C?i,??C?i?,? ? ? C?i,??C?i?,? ?)M?i,?Mi?,? ?I.In order to clarify the relationship between the photoelectron spin po-larization (Eq. 5.10) and the TSS ground-state spin polarization, we canexpress the photoelectron spin polarization in terms of expectation value ofspin operators, with the spin operator defined as:Si,? ; i?,? ?? = |?i,? ???i?,? ? |??, (5.10)where ? ? {x, y, z} and ?x,y,z are the Pauli spin matrices. Using Eq. 5.2and Eq. 5.10, one can write down the expression for the layer- and orbital-projected expectation value of spin operators:?Si,? ; i?,? ?x ? = C?i,??C?i?,? ? + C?i,??C?i?,? ? ,?Si,? ; i?,? ?y ? = i(?C?i,??C?i?,? ? + C?i,??C?i?,? ?), (5.11)?Si,? ; i?,? ?z ? = C?i,??C?i?,? ? ? C?i,??C?i?,? ? .The spin-polarization vector of the TSS ground state, defined as ?STSS? =[?Sx?, ?Sy?, ?Sz?], is the sum of the expectation value of spin operators shownin Eq. 5.10 with i = i? and ? = ? ?: ?STSS? ? =?i,? ?Si,?? ?. When i 6= i? and? 6= ? ?, the spin operator in Eq. 5.10 represents the interference effect in thephotoelectron spin polarization.Plugging Eq. 5.11 into Eq. 5.10, we can now rewrite the photoelectronspin polarization in terms of the expectation values of spin operators asdefined in Eq. 5.10:P? =?i,? ?Si,?? ?|Mi,? |2 +?i 6=i?,? 6=? ??Si,? ; i?,? ?? ?M?i,?Mi?,? ?I. (5.12)785.6. Supplemental materialEnergy (eV)P00.20.41.00.60.8(a)Orbital occupationEnergy (eV)0.1 0.3-0.1(b)? K? K-0.1 0.2 -0.1 0.2DPPin-plane? M? MPout-of-plane0-1.01.0-0.50.5-0.1 0.2 -0.1 0.2Energy (eV)(c)Pin-planePout-of-planenpzpx, y n?S    ?/npx?px?S    ?/npy?py?S    ?/npz?pz?S        ?/ntotal?totalFigure 5.8: Energy dependence of the layer-integrated TSS orbital charactersand spin polarization. (a) Energy dependence of the px,y and pz relativecontributions to the TSS wave function. (b)?(c) cuts through Fig. 5.6 in theSection 5.4 showing the relative out-of-plane (left panel) and in-plane (rightpanel) spin polarization (P ) of individual p orbitals as a function of energyalong ??? M? (b) and ??? K? (c) [obtained from the expectation value oflayer-integrated, orbital-projected spin operators, normalized to the orbitaloccupation shown in (a)].This shows the relationship between the photoelectron and the TSS ground-state spin polarization. We can see that the matrix element Mi,? and theinterference term?i 6=i?,? 6=? ? can make the photoelectron spin polarizationdeviate from the TSS ground-state spin polarization.5.6.4 TSS orbital character and spin polarizationAs shown in Section 5.4, Fig. 5.6, px and py orbitals have almost oppositein-plane spin textures. Therefore, the in-plane spin polarization from thepx,y channel is almost zero over a large energy window, which results in aless than 100% TSS in-plane spin polarization. This orbital-dependent spintexture makes the relative occupation of px,y and pz critical to determinethe value of the TSS in-plane spin polarization. Upon moving away fromthe Dirac point (DP), the px,y occupation increases from 25% to 45% andthe pz occupation correspondingly decreases, as shown in Fig. 5.8(a). Thein-plane and out-of-plane spin polarization of individual orbitals is shown inFig. 5.8(b) [along ??? M?) and Fig. 5.8(c) (??? K?]. We can see that the TSS795.6. Supplemental materialspin polarization for the total of p orbitals can never reach 100%; instead itdecreases from 75 to ? 60% while energy is increasing from 0 to 0.4 eV.5.6.5 Manipulation of ARPES spin textureIn Section 5.4, Fig. 5.7, we show that the photoelectron spin polarizationcan strongly depend on photon energy and experimental geometry. Here,in Fig. 5.9, we explicitly present the spin texture patterns of photoelectronsas measured by SARPES using four different light polarizations and fivephoton energies. The strong deviations between photoelectron and TSSground-state spin textures can be seen by comparing Figs. 5.9(b)?(d) toFigs. 5.6(a)?(d) in Section 5.4. One can also observe a remarkably strong andnontrivial photon energy dependence for the experimentally determined spintexture of photoelectrons. The only exception is the result obtained withpi-polarization at a 90? incidence angle [Fig. 5.9(a)]: in this case one onlyprobes initial states with pz orbital character, whose spin texture is nearlylayer-independent.In Fig. 5.10, we focus on the photoelectron spin polarization vector attwo k points under two experimental geometries, as a function of photonenergy and incidence angle of pi-polarized light. The photoelectrons excitedunder different experimental geometries are composed of electrons with dif-ferent spin orientations, depending on their orbital source. Even at thesame k point, the photoelectron spin polarization will change if we changethe experimental geometry, as it can be seen by comparing Fig. 5.10(a) toFig. 5.11(a) or Fig. 5.10(b) to Fig. 5.11(b). Moreover, the photoelectron spinpolarization can be non-zero along directions which are expected to be zerobased on the spin polarization of the TSS ground state. For instance, Py andPz in Fig. 5.10(a), Px and Pz in Fig. 5.10(b), Py and Pz in Fig. 5.11(a) areexpected to be zero from the spin polarization of the TSS ground state; how-ever, the photoelectron spin polarization of these components is not zero andhas a very strong photon energy dependence as a result of interference effectsbetween photoelectrons from different orbitals. Only the photoelectron spinpolarization shown in Fig. 5.11(b) directly presents the TSS ground-state805.6. Supplemental materialKMkxKMkxKMkx10 eV17 eV21 eV30 eV50 eV1-1Pz0h? ?K K ?K K ?K K ?K Ke- (px, z)45oe- (py)e- (pz) e- (p total)45o?MM?MM?MM?MM?-polarized ?-polarized ?-polarized RC-polarized(a) (b) (c) (d)KMkxFigure 5.9: Calculated spin textures of photoelectrons excited from the Diraccone upper branch in Bi2Se3, for different photon energies and polarizations.Arrows and colors are used to describe the in-plane and out-of-plane photo-electron spin polarization, respectively; note that moving away from the ??point in these maps correspond to moving along the Dirac dispersion awayfrom the DP.815.6. Supplemental materialTe KMkykxab(a) Px Py PzpzpypzpyPhotoelectron P1.00.5-0.50-1.0ba(b)Photoelectron P1.00.5-0.50-1.020 40 60 800 20 40 60 800 20 40 60 800h? (eV)?30o20o10o0o40o50o60o70o80o90oFigure 5.10: Photon-energy dependence of the photoelectron spin polariza-tion at two k points. The two measured k points are labeled by red dotsin the schematics of experimental geometry shown on the top panels. Themiddle panels with green arrows and blue/red/white squares are used toschematically present the in-plane (green arrows) and out-of-plane (filledcolor with red: ?; white: 0; blue: +) spin polarization of the only twoatomic orbitals which emit photoelectrons based on selection rules. Thebottom three panels show the three components of the photoelectron spinpolarization as a function of photon energy, with different incidence angles(?) of the pi-polarized light shown with color.825.6. Supplemental materialTe KMkxkyP y =1P z =0P x=0bapzpxpzpxba? independent ? independent ? independent?30o20o10o0o40o50o60o70o80o90oPxPyPz(a)Photoelectron P1.00.5-0.50-1.0(b)Photoelectron P1.00.5-0.50-1.020 40 60 800 20 40 60 800 20 40 60 800h? (eV)Figure 5.11: Photon-energy dependence of the photoelectron spin polariza-tion at two k points. The two measured k points are labeled by red dotsin the schematics of experimental geometry shown on the top panels. Themiddle panels with green arrows and blue/red/white squares are used toschematically present the in-plane (green arrows) and out-of-plane (filledcolor with red: ?; white: 0; blue: +) spin polarization of the only twoatomic orbitals which emit photoelectrons based on selection rules. Thebottom three panels show the three components of the photoelectron spinpolarization as a function of photon energy, with different incidence angles(?) of the pi-polarized light shown with color.835.6. Supplemental materialspin polarization owing to the fact that ? thanks to the choice of geometryand light polarization ? all photoelectrons from different layers and selectedorbitals have the ground states with the same expectation value of spin op-erators; this eliminates possible deviations induced by matrix element andinterference effects.Note that our calculated spin polarization of photoelectrons is in quan-titative agreement with reported SARPES results [102]. For the situationshown in Figs. 5.11(a) and (b), along ???M?, Py ? 80% was reported, closeto 100% obtained from the calculation; along ???K?, Py was reported to be25% at h? = 36 eV and ?50% at h?=70 eV at ky = ?0.11A??1, while ourcalculation gives 20%?10% and ?40%?15%, respectively. The uncertaintyof our calculated results is estimated based on the uncertain ratio betweenp? to? s and p? to? d excitations.5.6.6 Asymmetric ARPES from a simple TSS modelWhile ab initio density functional theory (DFT) calculations ? as presentedabove ? are required in order to describe the complex layer-dependent wavefunction of the TSS in realistic TI materials, and especially to quantitativelyreproduce the experimental data, it is illuminating to explore certain funda-mental aspects with as simple a model as possible. Here we will develop amost basic time-reversal-symmetric model to qualitatively capture interfer-ence effects on the photoemission intensity patterns, based on the solution ofthe effective TSS Hamiltonian [33, 64]. First we will express the wave func-tion of the TSS in a basis set of px,y,z orbitals to account for the entangledspin-orbital texture. Then we will generalize it to a model wave function fortwo atomic layers, to allow for a layer-dependent spin-orbital texture. Fi-nally we use it to calculate the asymmetric ARPES using pi-polarized lightincident in the yz plane, as one of the examples treated in the main contentof this Chapter.The model Hamiltonian for the three-dimensional topological insulatorsbelonging to the Bi2Se3 family of materials has been fully derived by Zhanget al. [33, 64]. In the basis set [?+ 12,?? 12] formed by the total angular845.6. Supplemental materialmomentum with Jz = ?12 , the effective Hamiltonian for the TSS of Bi2Se3near the ?? point can be written as [33, 64, 115]:HTSS = ?(?xky ? ?ykx), (5.13)where ? is the constant coefficient containing the strength of spin-orbit cou-pling (SOC), ?x,y are the spin Pauli matrices, and kx,y represent the electronmomentum. The eigenstates of this model Hamiltonian are [33, 64, 115]:?? =1?2[?ie?i?|? 12?+ |?? 12?], (5.14)with ? defined by k?ei? = kx + iky. The exact k?-dependent form of ?? 12is determined by the material details. As verified based on the DFT cal-culations for Bi2Se3, near the ?? point (i.e. away from the bottom of theconduction band) the k? dependence of ?? 12is very weak, and is dominatedby the zeroth-order term in k?. Therefore, here we express ?? 12in terms ofpx,y,z orbitals by only retaining the zeroth-order term, and the basis becomes[115]:?? 12= ?|pz, ? (?)? ???2[|px, ? (?)? ? i|py, ? (?)?], (5.15)where ? and ? are again material-dependent. Finally, the eigenstates of themodel Hamiltonian (Eq. 5.13) in the [px, py, pz] basis, with spin up (?) andspin down (?), become [33, 64, 115]:(??????)=??2(?ie?i?1)|pz? ??2(?1?ie?i?)|px?+?2(?i?e?i?)|py?. (5.16)The orbital-dependent spin texture for this model wave function ? whichqualitatively reproduces the behavior obtained through a complete DFTslab calculations [2] ? can be obtained by calculating the expectation value855.6. Supplemental materialof spin operators for individual p orbitals [33, 64, 115]:?Spx? ? = ??22[sin?, cos?, 0],?Spy? ? = ??22[sin?, cos?, 0], (5.17)?Spz? ? = ??2[sin?,? cos?, 0],where? refers to the upper- and lower-branch of the Dirac cone, respectively.Here we see that the px, py and pz orbitals are associated with different spintextures, which are also similar to the results of the full DFT calculations,as shown in Fig. 5.6 [2].As for the interpretation of experimental ARPES data in this Chapter, aswell as the SARPES data presented in the following Chapter 6, we emphasizethat the wave function presented in Eq. 5.16 alone is not sufficient to giverise to interference-induced asymmetric intensity pattern, which has beenobserved in ARPES experiments. Our ab initio DFT calculations indicatethat, in order to describe the unusual ARPES data, we need to account forthe interference between photoelectrons with different optical path lengths,i.e. a model with at least two atomic layers is required. We construct thisby generalizing the wave function of the single-layer system (as described byEq. 5.16) to a two-layer system. To ensure that the two-layer model has anentangled spin-orbital texture similar to the one obtained by DFT [2] andalso by the effective TSS model (Eq. 5.17), the spin-related phase informa-tion of each of the individual orbitals are assumed to be layer-independent;however, we note that the details of the px,y,z orbital superposition in theTSS wave function can be layer-dependent. This way, the wave functionof the two-layer model will have a layer-dependent spin-orbital texture, asreported by DFT calculations in realistic materials [2]. Following this strat-egy, we rewrite the J = 1/2 basis states by introducing a layer-dependentorbital character through the coefficients ?i and ?i:?? 12=2?i=1?i|pz, ? (?)? ??i?2[|px, ? (?)? ? i|py, ? (?)?]. (5.18)865.6. Supplemental materialWe note once again that ?i and ?i are material-determined coefficients, andtheir value can be estimated with the aid of ab initio DFT calculations.Combining Eq. 5.14 and Eq. 5.18, the two-layer model wave function with alayer-dependent spin-orbital texture can be obtained:(??model??model)=2?i=1?i?2(?ie?i?1)|pz? ??i2(?1?ie?i?)|px?+?i2(?i?e?i?)|py?.(5.19)As an example, we study the ARPES intensity based on the experimentalgeometry using pi-polarization, as shown in Section 5.2, Fig. 5.1(a), for theupper Dirac cone states. Considering the selection rules for pi-polarization,the photoelectrons are excited only from pz and py orbitals. Following thestrategy in Section 5.6.2, but using Eq. 6.3 as the initial state, we can writethe photoemission intensity as:I ??????2?i=1ie?i?e?ikzzi?i?2M?pz ? ie?ikzzi ?i2M?py?????2+?????2?i=1e?ikzzi?i?2M?pz + e?i?e?ikzzi?i2M?py?????2.(5.20)Here, M?pz,y are matrix elements defined by M?pz,y ? ?eik??r? |A ? p|pz,y?,which are real numbers here, and the two sums on the left and right are theintensity from the up and down spin channels, respectively. The imbalancein ARPES intensity at ?k? is equivalent to the intensity difference between? and ?+ pi, which can be calculated from Eq. 5.20:Ik? ? I?k? ? (?1?2 ? ?2?1) sin(kz(z1 ? z2)) sin(?)M?pzM?py . (5.21)From this expression for the asymmetry of ARPES intensity between ?k?in pi-polarization along emission plane yz, we find that there are three keyrequirements to get Ik? 6= I?k? :1. an atomic-layer-dependent orbital character, as shown by (?1?2??2?1);2. an out-of-plane phase difference between different atomic layers, from875.6. Supplemental material(a) (b) (c)Figure 5.12: Quantum interference effects on ARPES intensity pattern froma simple model described in Eq. 5.21. (a) A monolayer system with multipleorbitals at each atom site. (b) A multiple-layer system with a single typeof atom orbital. (c) A multiple-layer system with layer-dependent orbitalcharacters.sin(kz(z1?z2)); and3. an in-plane phase difference ? between the pz and py orbitals.It is by combining these basic aspects with the detailed layer-by-layer de-scription of the spin-orbital texture in Bi2Se3 that we can accurately evaluatethe photoemission intensity and spin polarization from this real material.Based on Eq. 5.21, we can simply predict the ARPES intensity patternfor systems with different structures, as shown in Fig. 5.12. For a monolayersystem with a single lattice structure, both ?2 and ?2 in Eq. 5.21 are zero.Therefore, we will not expect to observe an asymmetric intensity patterninduced by the quantum interference effect, even if there is one or multi-ple orbitals at each atom site [Fig. 5.12(a)]. For a system with a minimaltwo-layered structure, but with only one type of orbital character, the ?in Eq. 5.21 is zero. Again, we will not expect to see any asymmetric in-tensity pattern that is merely induced by the quantum interference effect[Fig. 5.12(b)]. However, if the system has more than one sub-lattice and885.6. Supplemental materialalso has multiple orbital characters at each atom site, such as the Bi2Se3system we discuss here, all three requirements listed for Eq. 5.21 are satisfied,and we expect an asymmetric ARPES intensity pattern due to the quan-tum interference effect [Fig. 5.12(c)]. A similar argument can be extendedto understand the ARPES intensity patterns of other systems, such as thesurface state of Cu (111) [116] and graphene [117], in which the asymmetricintensity patterns have also been experimentally observed. In particular,the asymmetric ARPES intensity in graphene has attracted a great dealof interest, and the quantum interference effects have also been mentioned,with consideration of graphene?s two sub-lattice structures [117?121].89Chapter 6Photoelectronspin-polarization-control inBi2Se3We study the manipulation of the photoelectron spin-polarization in Bi2Se3by spin- and angle-resolved photoemission spectroscopy. General rules areestablished that enable controlling the spin-polarization of photoemittedelectrons via light polarization, sample orientation, and photon energy. Wedemonstrate the ?100% reversal of a single component of the measuredspin-polarization vector upon the rotation of light polarization, as well asits full three-dimensional manipulation by varying the experimental configu-ration and photon energy. While a material-specific DFT analysis is neededfor a quantitative description, a minimal phenomenological two-layer modelqualitatively accounts for the SARPES response based on the interplay ofoptical selection rules, photoelectron interference, and the complex internalstructure of the topological surface state. It follows that photoelectron spin-polarization control is generic for systems with a layer-dependent, entangledspin-orbital texture.6.1 IntroductionThe central goal in the field of spintronics is to realize highly spin-polarizedelectron currents and to be able to actively manipulate their spin polar-ization direction. Topological insulators (TIs), as a new quantum phase ofmatter with a spin-polarized topologically-protected surface state [7, 24?28],906.1. Introductionhold great promise for the development of a controllable ?spin generator? forquantum spintronic applications [122]. A possible avenue is via the spin Halleffect and the spin currents that appear at the boundaries of TI systems, andthe electric-field-induced magnetization switching that can be achieved atthe interface between a TI and a ferromagnet [30]. In addition, it has beenexperimentally demonstrated that a spin-polarized photocurrent can be gen-erated from the topological surface state (TSS) of Bi2Se3 by using polarizedlight, suggesting the possibility of exploiting TIs as a material-platform fornovel optospintronic devices [113, 123, 124].All these exciting developments fundamentally rely on the spin proper-ties of the TSS, which have been extensively studied by density functionaltheory (DFT) [33, 103, 104] and spin- and angle-resolved photoemissionspectroscopy (SARPES) [73, 88, 100?102, 125?127]. On the theoretical side,we have shown that the TSS in real materials is not a simple two-dimensionalstate. Rather, it has a layer-dependent spin-orbital entangled structure ?extending over 10 atomic layers (? 2 nm) ? challenging the hypothesis of100% spin-polarization for the TSS Dirac fermions [2]. Our DFT work alsosuggested a new pathway to control the spin polarization of photoelectronsvia photon energy and linear polarization [2]; although this is consistentwith some experimental observations by SARPES [125?127], no conclusiveunderstanding of the phenomenon and its governing principles has yet beenachieved. This is of critical importance for future applications, and will re-quire a full examination of the photoelectron spin-polarization response inspecifically designed SARPES experiments.In this chapter ? guided by a detailed DFT analysis of the TSS entangledspin-orbital texture ? we present a systematic SARPES study to elucidatethe dependence of the photoelectron spin on light polarization, experimen-tal geometry, and photon energy. We demonstrate a reversal of the spinpolarization from ?100% to +100% upon switching from pi to ? polarizedlight. By changing the sample geometry and tuning photon energy we canmanipulate the photoelectron spin polarization in three dimensions. Whilea material-specific DFT analysis is needed for the complete quantitative de-scription, here we introduce a minimal and fully-general phenomenological916.2. Experimental methodstwo-layer model that qualitatively captures the unusual SARPES response,based on the combined effect of TSS spin-orbital texture, optical selectionrules, and photoelectron interference. This paves the way to generatingfully controllable spin-polarized photocurrents in TI-based optospintronicdevices.6.2 Experimental methodsSARPES experiments were performed at the Hiroshima Synchrotron Radia-tion Center (HSRC) on the Efficient SPin REsolved Spectroscopy (ESPRESSO)endstation [128, 129], with 50 meV and ? 0.04 A??1 energy and momen-tum resolution, respectively. This spectrometer can resolve both in-plane(Px,y) and out-of-plane (Pz) photoelectron spin-polarization components.These are obtained from the relative difference between the number ofspin-up and spin-down photoelectrons, according to the relation Px,y,z =(I?x,y,z?I?x,y,z)/(I?x,y,z +I?x,y,z). Samples were oriented by Laue diffractionand cleaved in-situ at ? 7 ? 10?11 torr; all measurements were performedat 30 K once the surface evolution had mostly stabilized [1], using 21 eVphotons unless otherwise specified.6.3 SARPES results and discussionIn Bi2Se3, the TSS wavefunction is composed of both in-plane (px,y) andout-of-plane (pz) orbitals. As a consequence of spin-orbit coupling, the spintexture associated with each orbital is remarkably different, and has beenreferred to as entangled spin-orbital texture [2, 115]. In Fig. 6.1, we sketchthe orbital-dependent in-plane spin polarization of the upper-branch Diracfermions (with the out-of-plane spin component neglected). We see that thewell-known TSS chiral spin texture is actually derived only from the out-of-plane pz orbitals [Fig. 6.1(c)]; instead, the individual px and py spin config-urations are not chiral14, and are also opposite to one another [Figs. 6.1(a)14The total in-plane orbital states do instead possess a chiral spin texture, with helicitydependent on the details of the linear combination of px and py orbitals [2].926.3. SARPES results and discussionkykx(a) (b) (c)pxpypz?x?y?zKMFigure 6.1: Spin orientation of the TSS in Bi2Se3. (a)?(c) In-plane spintexture as obtained separately for the px (a), py (b), and pz (c) orbital con-tributions to the topological surface state (TSS). Red/blue arrows indicatethe light electric field (pi/? polarization) that must be used to excite pho-toelectrons from each of the individual orbitals, according to the electricdipole selection rules.and 6.1(b)]. By comparing the spin orientation of in-plane and out-of-planeorbitals, we learn that they can be parallel, anti-parallel, or even perpendic-ular to each other, in dependence of the specific momentum-space location.For example, px and pz spin polarizations are parallel along the ???M? di-rection (i.e., the kx axis), but antiparallel along ???K? (i.e., the ky axis). Asfor probing these different orbital-dependent configurations, we note that ?based on the ARPES optical selection rules ? photoelectrons are emittedfrom a given px,y,z orbital if the photon electric field has a non-zero com-ponent ?x,y,z along the corresponding direction [130]. Thus, using linearlypolarized photons with electric field parallel to the kx/ky/kz directions, wecan probe the px/py/pz spin textures individually in SARPES (Fig. 6.1).Fig. 6.2 demonstrates the ?100% manipulation of photoelectron spin-polarization upon switching the light polarization from pi to ? in SARPES.When we measure the energy distribution curve (EDC) at kx=0.07 A??1 withpi polarization [photon electric field in the xz plane, as in Figs. 6.2(a,b)], weobserve a peak only in the spin-down y-channel at the TSS upper-branchbinding energy at ?0.1 eV [green curve in the top panel of Fig. 6.2(c)]. Thus936.3. SARPES results and discussion00.20.40.6 00.20.40.60-0.51-10.5Intensity (a.u.)PyEnergy (eV) Energy (eV)yzx?hv??MK?e-Energy (eV)00.20.40.60 0.1-0.1kx (1/?)(a)(b)(c) (d)????kx = 0.07 ?-1kx = 0.07 ?-1I?I?I?I?Figure 6.2: Light-polarization-controlled photoelectron spin polarization.(a) Schematics of the experimental geometry, with pi (horizontal) and ?(vertical) linear polarization also indicated. (b) ARPES dispersion of TSSDirac fermions measured along the M?????M? direction with pi polarization.(c) The top panel shows SARPES EDCs, with spin quantization axis alongthe y direction, measured with pi polarization along the gray-bar highlightedin (b); the corresponding spin polarization curve Py is shown in the lowerpanel (the TSS is located at 0.1 eV in this data taken at kx = 0.07 A??1). (d)SARPES data analogous to those in (c), now measured with ? polarization.946.3. SARPES results and discussionwe obtain Px,z ' 0 and remarkably Py ' ?100% for the spin-polarizationvector components, as highlighted in the bottom panel of Fig. 6.2(c) by thegreen arrow at 0.1 eV (note that the positive Py value at ?0.4 eV originatesfrom the TSS bottom branch and its reversed spin helicity [2, 115]). Mostimportantly, when light polarization is switched from pi to ?, while Px,zremain zero Py suddenly becomes +100% at 0.1 eV, as shown in Fig. 6.2(d).We note that a spin polarization as high as ?100% is rarely reported inprevious SARPES studies of Bi2Se3 [73, 88, 100?102, 125]; this is achievedin this study owing to the high efficiency of the spin polarimeter and theperfect alignment within the photoelectron emission plane of both the lightpolarization and sample ???M? direction, which eliminates the interference-induced deviations to be discussed below. In Fig. 6.2 we outline how theexperimental configuration and the entangled spin-orbital texture (Fig. 6.1)lead to spin-polarization switching: pi polarization excites photoelectronsfrom px and pz orbitals only, both of which are ?100% spin polarized alongy for all positive ?+kx? locations [Figs. 6.1(a,c)]; this gives Py ? ?100% inSARPES, consistent with the experiment in Fig. 6.2(c). On the contrary,in ? polarization photoelectrons originate only from the py orbitals, whichat +kx locations are +100% spin-polarized along the y direction, i.e. Py?+100% as detected in Fig. 6.2(d).By rotating light polarization between ? and pi, we would observe a con-tinuous change of Py between ?100%, as experimentally verified by Jozwiaket al. [125]. However, here we argue that the manipulation of the photo-electron spin polarization by light stems from the TSS orbital-dependentspin texture combined with optical selection rules, rather than being dueto the relativistic photon-electron interaction [105, 125] which is generallya weak perturbation compared to the non-relativistic electric-dipole term.The SARPES response is indeed most unusual for configurations differentfrom the one in Fig. 6.2(a) ? which is unique in that electrons photoemittedby either pi or ? light all have the same spin polarization even if originatingfrom multiple orbitals. This is shown in Figs. 6.3(c?e) where we examinethe photoelectron spin polarization at the same +kx point15, for the two15In this experiment, performed along the ???M? direction at |kx|= 0.04 A??1, the TSS956.3. SARPES results and discussion(c) (d) (e)(f) (g) (h)0-0.51-10.50-0.51-10.500.20.40.6Energy (eV)00.20.40.6Energy (eV)Spin polarization00.20.40.6Energy (eV)PxPzPyPxPzPyKMKMKMSpin polarization(a)pxpzpypzInterferencexzyzInterference(b)Figure 6.3: Quantum interference effects on spin polarization of photoelec-trons. (a),(b) Schematics of photoelectron interference effects for two exper-imental configurations: (a) pi-polarization incident in the xz plane probespx and pz orbitals with the same spin state (see also Fig. 6.1); (b) when inci-dent in the yz plane, pi-polarization probes py and pz orbitals with oppositespin states (Fig. 6.1). (c)?(e) Spin polarization curves at the +kx pointas sketched in (e), measured for (a)/(b) configurations (red/blue curves).(f)?(h) Spin polarization curves at ?kx as sketched in (h) for the (b) con-figuration. Note that the TSS is located at 0.25 eV in this data measured at|kx| = 0.04 A??1.966.3. SARPES results and discussionconfigurations of Figs. 6.3(a,b). Case I ? ??xz : photoelectrons are emit-ted from px,z orbitals in the same spin state [Fig. 6.3(a)], and as before weobserve a close to ?100% Py16 and zero Px,z [red symbols in Figs. 6.3(c?e)]. Case II ? ??yz : photoelectrons are emitted from py,z orbitals withmixed spin states [Fig. 6.3(b)], and are no longer fully polarized along Py.Instead Py decreases and an unexpected Px ' 74% appears [blue symbolsin Figs. 6.3(c-e) and sketch in 6.3(e)]. Another interesting aspect is thatwhile both Py and Pz17 switch sign at opposite momenta ?kx, as expectedfrom time-reversal symmetry [Figs. 6.3(g,h)], the Px retains the same value[Figs. 6.3(f) and sketch in 6.3(h)].To understand the unexpected results of Fig. 6.3 ? seemingly inconsistentwith the TSS time-reversal invariance ? we need to consider photoelectron-interference effects specific for SARPES. To this end, we express the mea-sured spin polarization vector ~P in terms of the expectation value of gener-alized spin operators18:P? =?i,? ?Si,? ; i,?? ?|Mi,? |2Itotal+?i 6=i?, ? 6=? ??Si,? ; i?,? ?? ? eikz(zi?zi? )M?i,?Mi?,? ?Itotal,(6.1)where ??{x, y, z}, ? ?{px, py, pz}, i is the atomic-layer index (the TSS layer-dependent structure is a key factor here [2]); Mi,? ? ?eik??r? |A ?p|?i,? ? is thematrix element of the optical transition between an atomic wavefunction oforbital ? centered around the atomic layer i and a free-electron final state;the kz part of the latter has been factorized in the phase term eikz(zi?zi? ),which accounts for the optical path difference for photoelectrons from dif-ferent layers; and Itotal is the sum of intensity from spin-up and spin-downchannels. The generalized spin operator in the expectation value ?Si,? ; i?,? ?? ?is defined as:Si,? ; i?,? ?? = |?i,? ???i?,? ? |??, (6.2)is located at 0.25 eV below EF (or, equivalently, at 0.1 eV above the Dirac point).16The slight reduction from Py =?100% measured initially is due to the sample surfacedegradation during the three continuous days of experiments after cleaving.17Although Pz =0 for the TSS upper branch, the sign change of Pz can be observed forthe TSS lower branch.18See Section 5.6.3 for the derivation of Eq. 6.1.976.3. SARPES results and discussion(a) (b)0-0.51-10.56040200 80hv (eV)Spin polarization 6040200 80PxPyhv (eV)(c)6040200 80Pzhv (eV)45oKMkxky(d)kz0-0.51-10.5Spin polarization Figure 6.4: Photon-energy-dependent spin polarization of photoelectrons.(a)?(c) Solid blue lines: calculated photon-energy-dependence of the photo-electron spin-polarization-vector components, as obtained at the ?ky pointfor pi-polarized light incident in the xz plane as shown in the sketch in (d).The solid red triangular symbols are SARPES data from this work; the openred triangles are from Ref.[102, 125].where ?x,y,z are the Pauli spin matrices. The crucial point is that in Eq. 6.1the i 6= i?, ? 6= ? ? off-diagonal terms account for the interference effects. Ifthe initial states ?i,? and ?i?,? ? being probed all have the same spin expec-tation value, then ?Si,? ; i,?? ?= ?Si,? ; i?,? ?? ? and P? = 100% for the ? componentcorresponding to the spin quantization axis, as in Case I of Fig. 6.3(a). How-ever, when the initial states being simultaneously probed have different spinstates, as in Case II of Fig. 6.3(b), non-trivial effects should be expectedfor the measured spin polarization due to the contribution of the Si,? ; i?,? ??interference term.To qualitatively demonstrate that Eq. 6.1 describes the SARPES results986.3. SARPES results and discussionin Fig. 6.3, in Section 6.4 we generalize the effective single-layer TSS wavefunction derived by Zhang et al. [33, 64, 115] into a two-layer one for px,y,zorbitals, as the minimal model-wave function needed to capture interferenceeffects19. For the upper branch of the Dirac-cone this becomes:?=2?i=1?i?2(ie?i?1)|pz???i2(?1ie?i?)|px?+?i2(?ie?i?)|py?, (6.3)where ?i and ?i are layer-dependent coefficients, and the in-plane phase? (defined as the angle between k and the +kx direction) reproduces theorbital-dependent spin texture shown in Fig. 6.1. To further simplify theproblem we assume ? without loss of generality ? that ?1 =?2 =0, ?2 =?3/2,and ?1 = 1/?2; this choice matches the 1:3 overall in-plane/out-of-planeorbital weight ratio calculated by DFT for Bi3Se2 [2]. Then, for ??yz (CaseII ), the initial-state components being probed reduce to:?pz =?32(ie?i?1)and ?py =?24(?ie?i?). (6.4)At ?kx (?=0 and pi, respectively), the intrinsic spin polarization is ?100%(?100%) along the ky direction for the pz (py) orbital, as in Fig. 6.1. Bymeans of Eq. 6.1, we can now calculate the photoelectron spin-polarizationvector ~P as seen at ?kx in SARPES, obtaining:~P (?kx) ? (sin ?kz ,?0.6,? cos ?kz), (6.5)where ?kz = kz(z1?z2). We see that, although the spin polarization of eachindividual initial state is purely along y, the photoelectron spin polarizationcan have non-zero components along x and/or z, if z1?z2 6=0. This highlightsthe need for a minimal two-atomic-layer model. Also note that the explicitpresence of kz leads to photon-energy-dependence (more below), and allPx,y,z components oscillate sinusoidally with different phases, upon varyingkz ; this is responsible for the maximal Px and minimal Pz in Figs. 6.3(f)?(h).19Details are shown in Section 6.4.996.4. A two-layer model to describe interference effects in SARPESFinally, Eq. 6.5 confirms the fact that only Py and Pz components reversetheir signs, while Px retains the same value at ?kx, again as observed in ourSARPES data in Figs. 6.3(f)?(h).While our two-layer model reproduces the SARPES results qualitatively,we stress that the quantitative description must be based on the complete?10-layer TSS wave function obtained for Bi2Se3 by DFT [2]. To this end, inFig. 6.4 we present the photon-energy-dependence of the photoelectron spinpolarization at ?ky, for ??xz. We find good agreement between our DFT-based results and SARPES data from this and published work [102, 125],conclusively demonstrating that the photon-energy-controlled photoelectronspin polarization stems from interference effects acting in concert with theTSS layer-dependent spin-orbital texture.6.4 A two-layer model to describe interferenceeffects in SARPESIn Section 5.6.6, we constructed a two-layer model based on the effectiveHamiltonian of the TSS to demonstrate the asymmetric ARPES intensity.Here we use the same two-layer-model wave function (Eq. 6.3) to qualita-tively show that quantum interference effects can allow a manipulation ofphotoelectron spin polarization in SARPES experiments. We take the ex-periment in Figs. 6.3(f)?(h) as an example, where the spin polarization wasmeasured at both ?kx positions with pi-polarized light incident in the yzplane. Since pi polarization incident in the yz plane would only excite pho-toelectrons from the py and pz orbitals, the initial state wave function of theTSS upper-branch can be reduced to the py and pz terms in Eq. 6.3, andcan thus be rewritten as:(??initial??initial)=2?i=1?i?2(ie?i?1)|pz?+?i2(?ie?i?)|py?, (6.6)where ?i and ?i represent the layer-dependent orbital characters. At the +kxpoint, ? = 0, we calculate the measured spin polarization of photoelectrons1006.4. A two-layer model to describe interference effects in SARPESby using Eq. 6.1 and Eq. 6.6:Px =2(?2?1 ? ?1?2) sin(kzz1 ? kzz2)MpyMpzItotal,Py =(?21 + ?22)M2py ? (?21 + ?22)M2pzItotal+2(?1?2 ? ?1?2) cos(kzz1 ? kzz2)MpyMpzItotal, (6.7)Pz = ?2(?1?1 + ?2?2) + 2(?2?1 ? ?1?2) cos(kzz1 ? kzz2)ItotalMpyMpz .We can see that although the initial state is fully spin polarized along they direction, with a value of 100% for the py orbital and ?100% for the pzorbital (Eq. 5.17), the photoelectron spin polarization measured by SARPEScan have non-zero components along the x and/or z direction, as long as?1 6=?2, ?1 6=?2, and z1 6=z2. Also, all the components of the photoelectronspin polarization can be controlled by tuning the photon energy (h?) becausewe have kz =?2me~2 (h? ? EB)? k2x ? k2y, with EB being the binding energy.Similarly, for the ?kx point, ? = pi, we obtain the photoelectron spinpolarization:Px =2(?2?1 ? ?1?2) sin(kzz1 ? kzz2)MpyMpzItotal,Py = ?(?21 + ?22)M2py ? (?21 + ?22)M2pzItotal?2(?1?2 ? ?1?2) cos(kzz1 ? kzz2)MpyMpzItotal, (6.8)Pz =2(?1?1 + ?2?2) + 2(?2?1 ? ?1?2) cos(kzz1 ? kzz2)ItotalMpyMpz .Comparing Eq. 6.7 to Eq. 6.8, we find that both Py and Pz change theirsigns when moving from +kx to ?kx; on the contrary, Px maintains thesame value, consistent with our SARPES data shown in Figs. 6.3(f)?(h).These model results demonstrate that a layer-dependent entangled spin-orbital texture is key to observe and manipulate the photoelectron spinpolarization in SARPES experiments (as shown in Fig. 6.3 and Fig. 6.4).1016.5. ConclusionAt variance with the observed behavior ? for single atomic-layer model ?all the results would be photon-energy independent because the kz term inEq. 6.7 and Eq. 6.8 vanishes when z1?z2 = 0; in addition ? for systems withlayer-independent orbital character ? also the value of the (?2?1 ? ?1?2)term becomes zero leading to a constant photoelectron spin polarization inEq. 6.7 and Eq. 6.8.In Section 6.3, we chose a simpler situation to illustrate the effects ofinterference on the photoelectron spin polarization. This simplified situationis obtained by using these parameters: ?1 = 0, ?1 = 1/?2, ?2 =?3/2, and?2 = 0, and also by assuming Mpy = Mpz .6.5 ConclusionIn this chapter, we have explained the underlying mechanism of the manipu-lation of photoelectron spin polarization in TIs, as a consequence of the TSSentangled spin-orbital texture, optical selection rules, and quantum interfer-ence. This is responsible also for the significantly different ARPES intensi-ties observed at ?kx in Fig. 6.2(b), which implies that a net spin-polarizedcurrent can be photoinduced by linearly polarized light [113]. Thus, ourSARPES study demonstrates how to generate a spin-polarized photocurrentin Bi2Se3 and manipulate its absolute spin polarization by linearly polarizedlight, a key step in TI-based optospintronics. We argue that all these phe-nomena could be valid in other spin-orbit coupled systems, as long as theinitial states are characterized by a layer-dependent entangled spin-orbitaltexture.102Chapter 7ConclusionThis thesis presented systematic work on Bi2Se3 and Bi2Te3, materials thathave been referred to as the hydrogen atom of three-dimensional topo-logical insulators (TIs). Using angle-resolved photoemission spectroscopy(ARPES), spin-resolved ARPES (SARPES), and density functional the-ory (DFT) calculations, various topics have been studied, ranging from TImaterial problems and engineering surface electronic structures, to the fun-damental physics of realistic surface states and their possible applicationsin spintronics. In regards to the technical aspect, this thesis presented anexample of a tight combination of experimental and theoretical work. Thepower of this combination allowed us to gain significant insights into thefundamental principles of the systems we were studying. Here we will reca-pitulate the main conclusions and their context.Rashba spin-splitting control at the surface of Bi2Se3The material problem is an obstacle to further advance topological-insulator-based devices. Bi2Se3, known as a prototypical TI candidate, does nothave a truly insulated bulk ? due to the unintentional and uncontrolleddoping that leads to a Fermi level crossing the bulk conduction band ratherthan staying inside the bulk gap. Another critical material problem is theunstable surface of the crystal, even in ultra-high vacuums. One approach toreset the chemical potential of materials is via doping, which is also relatedto another important question about surface or bulk impurity effects onthe surface electronic structure. Because of the polarized spin texture ofthe topological surface state (TSS), no back-scattering is allowed, and thesurface state is expected to be immune to localized weak disorders, as long103Chapter 7. Conclusionas the time-reversal symmetry (TRS) of the system is preserved.Our first experiment was to investigate the effect of surface impurities onTSS by in situ potassium deposition on the surface of Bi2Se3. We have sys-tematically studied the ARPES evolution of the surface electronic structureas a function of potassium evaporation. The deposition of submonolayers ofpotassium allows us to stabilize the otherwise continuously evolving surfacecarrier concentration. Thus, our approach provides a technique to overcomethe problem of general instability and self-doping at the surface of TIs.On the other hand, the potassium deposition leads to a more uniformsurface electronic structure, in which well-defined Rashba-like states emergefrom the continuum of parabolic-like states that characterizes the as-cleaved,disordered surfaces. The spin splitting of Rashba-like states can be con-trolled by the amount of potassium deposition, and the entire process isreversible by tuning the temperatures. These results suggest that TIs canbe a platform to host and manipulate the coexistence of non-trivial spinsplitting states (TSS) and conventional spin splitting states (Rashba type),both of which are substantially important for spintronic applications. OurDFT slab calculations reveal that the new spin-split states originate fromthe bulk-like quantum well states of a five-quintuple-layer slab, as a conse-quence of the K-enhanced inversion symmetry breaking already present forthe pristine surface.This work is one of the earliest studies on engineering the surface elec-tronic structures of TIs. The demonstration of generating and preciselycontrolling a large energy-scale spin splitting shows a promising avenue forthe future development of TI-based spintronic devices.Tailoring spin-orbit coupling in Mn-doped Bi2?xMnxTe3The beauty of TIs is in its spin-polarized TSS and its topological protection.The latter is valid under the condition that the system has TRS. BreakingTRS therefore becomes a way to switch on and off the topological protec-tions, and can be achieved by applying an external magnetic field or by themagnetic moment of impurities. Doping TIs with magnetic impurities has104Chapter 7. Conclusionbeen viewed as an efficient way to break the TRS by forming long-rangemagnetic orders. ARPES can be one of the most direct methods to monitorthe TRS breaking effects, such as by looking for an opening gap at the Diracpoint of the TSS spectra.We investigated magnetically doped Bi2?xMnxTe3 to observe a responsein the TSS spectra to the development of ferromagnetic order at low temper-atures. Instead of focusing on the gap opening at the Dirac point, we raisedanother fundamental question about how impurities affect the spin-orbitcoupling (SOC) of the systems, a critical parameter to sustain the appear-ance of TSS. The size of the bulk gap shows a strong dependence of Mndoping levels, even below 5%. Our ab initio tight-binding (TB) model calcu-lations show that the bulk energy gap size is directly linked to the strength ofthe SOC in the system. Therefore, our observation indicates that althoughthe Mn concentration is low (only 2-5%), the effective content of impuritiescan be ?4 times higher. More importantly, through diluting the SOC of thesystem, impurities can turn a topological insulator into a trivial insulatoreven at a low concentration.A temperature-dependent study crossing the ferromagnetic transitionTc = 12 K suggests a surprising result that the ferromagnetic order mightbe of a short range. This raises a question regarding TRS behaviour in aTI system where the magnetic order is of short range. Further theoreticalinvestigations are required to address this question.Layer-by-layer entangled spin-orbital texture in Bi2Se3In this work, we address one of the most fundamental outstanding questionsin the field of TIs: What is the degree of spin polarization of Dirac fermionsin real TI materials? In phenomenological models, a 100% spin polarizationis assumed for TSS. DFT calculations indicate that the TSS spin polariza-tion of the Bi2X3 material family (X = Se, Te) can be substantially reducedfrom 100%. Extensive SARPES studies also have attempted to answer thisquestion. However, the large discrepancy in the degree of spin polarizationdetermined from the TSS by SARPES - ranging from 20 to 85% - challenges105Chapter 7. Conclusionall the theoretical predictions. Thus, this question has become a puzzle andundermines the applicability of real TIs in spintronic devices.By a combination of polarization-dependent ARPES and DFT slab cal-culations, we are the first to explicitly solve this puzzle. We have foundthat the surface state Dirac fermions in Bi2Se3 are characterized by a layer-dependent entangled spin-orbital texture, which becomes apparent throughquantum interference effects. Quantum interference between photoelectronsnot only affects the measured ARPES intensity but also, and more impor-tantly, causes disagreement between the measured spin polarization and theintrinsic spin texture of TSS. We have theoretically demonstrated how thelight polarization, photon energy, and experimental setup can affect the spinpolarization of photoelectrons. We also proposed how to probe the intrinsicspin texture of TSS and suggested a way to continuously manipulate the spinpolarization of photoelectrons all the way from 0 to ?100% by an appro-priate choice of photon energy, linear polarization, and angle of incidence.These discoveries are key to the understanding and exploitation of TIs.To accomplish this work, we developed an ab initio slab TB model andimplemented spin-resolved photoemission intensity calculations. These cal-culations involve very technical details and are not typically performed inthe ARPES community; however, we think the generic value of the methodcan benefit future ARPES studies in other spin-orbit coupled systems. Forthis reason, we devote Chapter 2, Chapter 3, and Section 5.6 in Chapter 5 toall the details about how to construct ab initio TB models and use them tocalculate ARPES and SARPES data.Photoelectron spin-polarization-control in Bi2Se3The goal of this SARPES work is to experimentally unveil the layer-by-layer entangled spin-orbital texture of the TSS in Bi2Se3. As predictedby our theoretical calculations, the entangled spin-orbital texture resultsin a strong photoelectron interference effect, which deviates the measuredspin polarization of photoelectrons from the intrinsic spin polarization of theTSS, and therefore provides us with a unique opportunity to manipulate the106Chapter 7. Conclusionspin polarization of photoelectrons. The manipulation of photoelectron spinpolarization challenges the basic assumption of spin conservation commonlyused in SARPES, but at the same time opens a new possibility to advanceTI applications in opto-spintronics.Our SARPES data has successfully demonstrated a generic rule for ma-nipulating photoelectron spin polarization. We have shown that the spin po-larization of photoelectrons can be flipped between ?100% and also rotatedin three dimensions with an appropriate choice of linear light polarizationand photon energy. The underlying principle of manipulating spin polariza-tion is to selectively excite photoelectrons from orbitals that have differentspin orientations, which can be provided by an initial state with an entan-gled spin-orbital texture. Quantum interference between photoelectrons is akey mechanism to achieve three-dimensional manipulation and also generatespin-polarized photocurrents or even pure spin currents. 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Lett. 109,266406 (2012).5. Z.-H. Zhu, G. Levy, B. Ludbrook, C.N. Veenstra, J.A. Rosen, R.Comin, D.Wong, P. Dosanjh, A. Ubaldini, P. Syers, N.P. Butch, J.125Appendix . List of publicationsPaglione, I.S. Elfimov and A. Damascelli, ?Rashba spin-splitting con-trol at the surface of the topological insulator Bi2Se3?, Phys. Rev.Lett. 107, 186405 (2011).6. G. Panaccione, U. Manju, F. Offi, E. Annese, I. Vobornik, P. Torelli,Z.H. Zhu, M.A. Hossain, L. Simonelli, A. Fondacaro, P. Lacovig,A. Guarino, Y. Yoshida, G. A. Sawatzky, and A. Damascelli, ?Depthdependence of itinerant character in Mn-substituted Sr3Ru2O7?, NewJ. Phys. 13, 053059 (2011).Papers in preprint7. Z.-H. Zhu, G. Levy, A. Nicolaou, X.-H. Zhang, N.P. Butch, P. Syers,J. Paglione, I.S. Elfimov and A. Damascelli,?Polarity-driven surfacemetalicity in SmB6?, submitted (2013).8. Z.-H. Zhu, C.N. Veenstra, S. Zhdanovich, M. P. Schneider, T. Okuda,K. Miyamoto, S.-Y. Zhu, H. Namatame, M. Taniguchi, M.W. Harverkort,I.S. Elfimov and A. Damascelli,?Photoelectron spin-polarization-controlin the topological insulator Bi2Se3?, submitted (2013).9. C.N. Veenstra, Z.-H. Zhu, B. Ludbrook, A. Nicolaou, M. Raichle,B. Slomksi, G. Landolt, S. Kittaka, Y. Maeno, J.H. Dil, I.S. Elfi-mov, M.W. Harverkort and A. Damascelli,?Observation of strong spin-charge entanglement in Sr2RuO4?, submitted, arXiv e-prints, 1303.5444(2013).126

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