A Study of Topological Insulators in Three Dimensions by Gilad Amir Rosenberg B.Sc., Ben-Gurion University, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Physics) The University Of British Columbia (Vancouver) September 2012 c Gilad Amir Rosenberg, 2012 Abstract In this work we study four interesting effects in the field of topological insulators: the Witten effect, the Wormhole effect, topological Anderson insulators and the absence of bulk magnetic order in magnetically doped topological insulators. According to the Witten effect, a unit magnetic monopole placed inside a medium with q = 0 is predicted to bind a fractional electric charge. We con- 6 duct a first test of the Witten effect based on the recently established fact that the electromagnetic response of a topological insulator is given by the axion term q(e2=2ph)~B ~E, and that q = p for strong topological insulators. · We establish the Wormhole effect, in which a strong topological insulator, with an infinitely thin solenoid of magnetic half flux quantum carries protected gapless and spin filtered one-dimensional fermionic modes, which represent a distinct bulk manifestation of the topologically non-trivial insulator. We demonstrate that not only are strong topological insulators robust to disor- der but, remarkably, under certain conditions disorder can become fundamentally responsible for their existence. We show that disorder, when sufficiently strong, can transform an ordinary metal with strong spin-orbit coupling into a strong topo- logical ‘Anderson’ insulator, a new topological phase of matter in three dimensions. Finally, we lay out the hypothesis that a temperature window exists in which the surface of magnetically doped topological insulators is magnetically ordered but the bulk is not. We present a simple and intuitive argument why this is so, and a mean-field calculation for two simple tight binding topological insulator models which shows that indeed a sizeable regime such as described above could exist. This indicates a possible physical explanation for the results seen in recent experi- ments. ii Preface This thesis is based almost exclusively on extensive notes written by myself through- out my program. In addition, some short sections are based on notes written by my research supervisor Marcel Franz, and on publications authored by me, my super- visor and our collaborators Huaiming Guo (who was a postdoc at UBC at the time) and Gil Refael (Prof. at Caltech). Chapter 1 motivates the rest of the thesis by providing an overview of the re- search field of topological insulators. It is inspired and based on several review articles published on the subject. Chapter 2 provides an introduction to tight-binding Hamiltonians in second quantization and Chapter 3 presents the main model used in this work and its topo- logical classification. They are based on my notes, but some of the work is covered in basic Condensed Matter textbooks and papers. The model itself was my super- visor’s idea, and was based at first on symmetry arguments and a paper by Fradkin, Dagotto and Boyanovsky [1], but later we discovered it was used as a “regularized Bi2Se3 ” model in many papers around the same time or before. Section 2.7 is based on Ref. [2]. Chapter 4 is based on my notes while working on this project. The idea was my supervisor’s, I carried out the calculations and he wrote the paper which I then helped proof and edit, titled “Witten effect in a crystalline topological insulator”, published in Physical Review B, 82, 035105 (2010). Chapter 5 is based on my notes while working on this project. The idea was my supervisor’s, I carried out the calculations and he wrote the paper which I then helped proof and edit, titled “Wormhole effect in a strong topological insulator”, published in Physical Review B 82, 041104(R) (2010). Section 5.3 was done by iii my supervisor and is included for completeness. Chapter 6 is based on my notes while working on this project. We published a paper titled “Topological Anderson insulator in three dimensions” in Physical Review Letters 105, 216601 (2010). The sections in the paper on the Born ap- proximation and on the numerical simulation of the Witten effect in the quasi 1D system were based on my notes while working on the project, which are presented in this chapter. Section 6.3 follows Ref. [3] closely, filling in the gaps. Chapter 7 is based on my notes while working on this project. This project started with an idea of my supervisor’s of a possible explanation for some recent experiments. I carried out the calculations and wrote the paper (based on the notes), which was then edited and proofed by my supervisor: “Surface magnetic ordering in topological insulators with bulk magnetic dopants”, published in Physical Re- view B 85, 195119 (2012). Section 7.5 is based on notes written by my supervisor. The appendix is based on my notes. Sections A.1 and A.2 are known (but rarely explained in detail), Section A.3 is original, and Section A.4 follows [4] (Sections 5.3-5.4), filling in the gaps. iv Table of Contents Abstract . ii Preface . iii Table of Contents . v List of Tables . viii List of Figures . ix Glossary . xi Acknowledgments . xiii 1 Introduction to Topological Insulators . 1 1.1 Overview . 1 1.2 A brief history of topological insulators . 1 2 General Tight Binding Treatment . 13 2.1 Overview . 13 2.2 Real space representation . 13 2.3 Momentum space representation . 14 2.4 Diagonalizing the Hamiltonian . 16 2.5 Adding degrees of freedom . 17 2.6 Common terms . 17 2.7 Topological classification . 19 v 3 Model . 22 3.1 Overview . 22 3.2 Our cubic model . 22 3.3 Explicit calculation of q ...................... 36 4 The Witten Effect in a Topological Insulator . 39 4.1 Overview . 39 4.2 Axions . 40 4.3 Classical view of the Witten effect I . 41 4.4 Classical view of the Witten effect II . 45 4.5 Adding a single monopole . 46 4.6 Charge bound to a monopole . 48 4.7 Proposal for experimental realization . 52 4.8 Conclusions . 57 5 Wormhole Effect in a Strong Topological Insulator . 59 5.1 Overview . 59 5.2 Laughlin’s argument . 59 5.3 Analytical calculation . 61 5.4 Numerical study . 63 5.5 Experimental detection . 65 5.6 Conclusions . 66 6 Topological Anderson Insulators . 68 6.1 Overview . 68 6.2 Introduction . 68 6.3 TAI in 2D . 69 6.4 TAI in 3D . 74 6.4.1 General treatment . 74 6.4.2 Iterative solution . 79 6.4.3 Non iterative solution . 81 6.4.4 Analytical zero order Born approximation . 84 6.4.5 Witten effect in quasi 1D system . 87 vi 7 Topological Insulator with Magnetic Impurities . 92 7.1 Overview . 92 7.2 Experimental results . 92 7.3 Surface magnetic ordering in topological insulators . 93 7.3.1 Surface doping . 93 7.3.2 Bulk doping . 94 7.4 Determining parameters for Bi2Se3 . 95 7.5 2D surface mean field calculation . 100 7.6 3D bulk mean field calculation . 103 7.7 Adding the surfaces . 119 7.8 Perovskite lattice - bulk . 123 7.9 Perovskite lattice - z dependent case . 126 7.10 Conclusions . 130 8 Conclusions . 133 8.1 Summary . 133 8.2 Future work . 134 Bibliography . 136 A Appendix . 147 A.1 Derivatives in different sets of coordinates . 147 A.1.1 Complex plane . 147 A.1.2 Polar coordinates . 148 A.2 Exploiting rotational symmetry . 151 A.2.1 Spinless case . 151 A.2.2 Adding the spin . 157 A.3 Adding monopoles in practice: calculating the Peierls factors . 163 A.3.1 Adding a lattice of monopoles . 163 A.3.2 Adding a single monopole . 167 A.3.3 Adding a planar monopole . 170 A.4 Derivation of the self consistent Born approximation . 172 vii List of Tables Table 3.1 Time reversal invariant momenta and the parity eigenvalues for our model . 27 Table 7.1 Lattice constants for Bi2Se3 . 100 viii List of Figures Figure 1.1 Chiral edge state on the edge of a quantum Hall state . 3 Figure 1.2 Trivial and topological states . 4 Figure 1.3 Quantum Hall state vs. Quantum Spin Hall State . 5 Figure 1.4 Destructive interference leads to a suppression of backscattering 6 Figure 1.5 Energy spectrum of massless and massive Dirac fermions . 9 Figure 1.6 ARPES measurements for Bi2Se3 . 10 Figure 1.7 Local density of states: energy and momentum . 11 Figure 3.1 Phase diagram for our 3D topological insulator (TI) . 28 Figure 4.1 Charge density of a monopole embedded in a strong topologi- cal insulator (STI) . 49 Figure 4.2 Charge density of a monopole embedded in a STI vs. the radius 50 Figure 4.3 Log-log plot of the excess charge of a monopole embedded in a STI . 51 Figure 4.4 3D charge density plot for the layer just below the planer monopole for weak disorder . 54 Figure 4.5 2D charge density plot for the layer just below the planer monopole for weak disorder . 55 Figure 4.6 Excess charge for different values of disorder strength . 56 Figure 5.1 Laughlin’s argument for a TI . 61 Figure 5.2 Additional charge for a flux tube in a TI . 64 Figure 5.3 Additional charge for a flux tube in a TI . 65 ix Figure 6.1 Born approximation phase boundary lines with conductivity showing the 2D topological Anderson insulator (TAI) phase . 74 Figure 6.2 Born approximation phase boundary lines .