本論文提出一系列的研究針對二維拓撲絕緣體與三維拓撲晶態絕緣體之物理性質與潛在的應用。具備時間返演不變性的二維拓撲絕緣體，為已知的量子自旋霍爾絕緣體，潛藏著拓撲保護的螺旋邊緣態，帶有不可消滅且(幾乎)量化的自旋霍爾響應。尤其是被預測為二維拓撲絕緣體的四族元素化合物，我們研究其對應的電子性質並發現數個顯卓的特徵，如塊材能譜中出現場致自旋極化的非簡併狄拉克錐。基於此族材料的等效最緊束縛模型，透過數值計算的發現，提出兩個可行的自旋電子學元件：高效能的自旋篩選器和由不均勻電場產生自旋電流的三進位制邏輯元件。另外，對於鏡像對稱保護的拓撲晶態絕緣體，如我們所關注的材料—碲(硒)化鉛錫，經第一原理計算，在(001)表面上拓撲保護的表面態，已經預測了一些值得注意的特性。為了證實並彰顯這些特徵，我們首先理論上計算表面態間受雜質散射造成的準粒子干涉圖。我們的結果顯示似狄拉克錐的能譜和不尋常的軌道結構，並且都符合掃描穿隧能譜實驗的觀測結果。其次，我們運用區塊重整化群的分析，並預估當逼近表面能譜中的凡霍夫奇點附近時，受電子-電子交互作用驅使的P 波超導序會脫穎而出。最後，我們探討了表面態上破壞鏡像對稱的應力所產生的效應。如下的發現拓展了拓撲晶態絕緣體在未來電子工程與自旋電子學之應用的可能性: 1)不需破壞時間返演對稱性而出現如刺蝟狀的自旋結構，2)季曼場下可經應力調變的陳數，與3)受應力與電子關聯性的相互影響而產生的競爭序。

This dissertation brings together a series of studies on physical properties and potential applications in both two-dimensional (2D) topological insulators (TIs) and three-dimensional (3D) topological crystalline insulators (TCIs). Time-reversal invariant 2D TIs, also known as quantum spin Hall (QSH) insulators, can harbor topology-protected helical edge states with nonvanishing (almost) quantized spin Hall response. Particularly in a predicted family of 2D TIs synthesized by group IVA elements, we study its corresponding electronic properties and find a number of salient features such as the presence of field-induced spin-polarized, non-degenerate Dirac cones in the bulk spectrum. Based on our findings, we propose two possible spintronic applications: A high-efficiency spin filter and a ternary logic device via an inhomogeneous electric field generated spin current, which are both demonstrated by our numerical calculations in an effective tight-binding model appropriate for such family of materials. In addition, the mirror-symmetry protected TCIs, such as our focused materials, Pb_(1-x)Sn_x(Te,Se), have been predicted to have several remarkable features for the topology-protected surface states on (001) surface via first-principles calculations. To verify and sharpen these features, we first theoretically compute the quasiparticle interference patterns due to impurity scattering among the surface states. Our results reveal their Dirac-like spectra and nontrivial orbital texture, which are consistent with scanning tunneling spectroscopy seen in the experiment. Second, we perform patch renormalization group analysis and predict the emergence of electron-electron interaction-driven p-wave superconducting order around van Hove singularities appearing in the surface spectrum. Finally, we consider the effects of broken-mirror-symmetry strains on the surface states. The discoveries of hedgehog-like spin texture without breaking time-reversal symmetry, a strain-tunable Chern number with a given Zeeman field, and resultant competing orders due to the interplay between strain and electron correlations extend the possibilities of future electronic and spintronic applications based on TCIs.

Acknowledgments ii
摘要 iii
Abstract iv
LIST OF FIGURES viii
LIST OF TABLES xi
1 Introduction to topological materials 1
2 Basic electronic properties of 2D topological insulators via IVA group elements 6
3 Potential applications: Silicene as an example 20
3.1 High efficiency spin filter 21
3.1.1 Geometry of device 21
3.1.2 Numerical results of spin-polarized of transport 21
3.1.3 Transport properties 24
3.1.4 Discussion and future perspectives 25
3.2 Field-tunable local transport channels 28
3.2.1 Generation of spin currents via domain wall states 28
3.2.2 Discussion on ternary logic device 30
4 Basic electronic properties of 3D topological crystalline insulators 33
4.1 Quasi-particle interference patterns 39
4.2 Interaction-driven p-wave superconductivity 45
4.2.1 Low-energy effective model at saddle points 46
4.2.2 Patch renormalization group analysis 47
4.2.3 RG equations 49
4.2.4 Competing orders and phase diagram 52
4.2.5 Discussion 56
4.3 Strain effects: Gap opening and hedgehog spin textures 59
5 Potential applications of TCIs: Pb_(1-x)Sn_x(Te,Se) as an example 64
5.1 Tuning Chern number via the interplay between strain and a Zeeman field 64
5.2 Strain-tunable broken-symmetry orders via electron correlations 67
5.2.1 System without strain 70
5.2.2 System under moderate strain 72
5.2.3 System with large strain 73
5.3 Discussion 74
6 Conclusion 75
References 77
APPENDICES 95
A Spin Chern number in silicene 95
B Derivation of energy dispersion of domain wall states 97
C Iterative Green’s function method for electric transportation 103
D Investigation of the efficiency of spin filter in various phases 107
D.1 Case 1: λ_v = 0.7t, h = 0.1t 107
D.2 Case 2: λ_v = 0, h = 0.52t 108
D.3 Case 3: λ_v = 0.6t, h = 0.1t 108
E Investigation of effects of defects in spin filter 110
F Simulation of QPI patterns using the T-matrix approach 114
F.1 T-matrix formulation 116
F.2 Simulation results 117
G Derivation of RG flow equations 122
G.1 Brief introduction of Shankar’s RG process 122
G.2 The explicit derivations for RG flow equations 126
G.2.1 RG flow equations from 1/8 ∫〈g^2_1〉 127
G.2.2 RG flow equations from 1/2 ∫〈g^2_3〉 131
G.2.3 RG flow equations from 1/2 ∫〈g^2_5〉 134
G.2.4 RG flow equations from 1/2 ∫〈g_1g_3〉 140
G.2.5 RG flow equations from 1/2 ∫〈g_1g_5〉 143
G.2.6 RG flow equations from ∫〈g_3g_5〉 146
G.2.7 Summary of RG flow equations 150
H Derivation of susceptibilities in particle-particle and particle-hole channels 151
H.1 Particle-hole susceptibility χ^(ph)_0 with transfer momentum q = 0 152
H.2 Particle-hole susceptibility χ^(ph)_(Q͂_1) with transfer momentum Q͂_1 153
H.3 Particle-hole susceptibility χ^(ph)_(Q͂_2) with transfer momentum Q͂_2 153
H.4 Particle-particle susceptibility χ^(pp)_0 with total momentum q = 0 156
H.5 Particle-particle susceptibility χ^(pp)_(Q͂_1) with total momentum Q͂_1 157
H.6 Particle-particle susceptibility χ^(pp)_(Q͂_(13)) with total momentum Q͂_(13) 157
I Derivation of out-of-plane spin component with xz-mirror symmetry breaking 160
J Derivation of mean-field theory 164
Autobiography 170

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