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研究生：	康文豪 Kang, Wun-Hao
論文名稱：	一維石墨烯超晶格與鄰近化石墨烯中的橫向磁聚焦 Transverse Magnetic Focusing in Spin-Orbit-Coupled Graphene and One-Dimensional Graphene Superlattices
指導教授：	劉明豪 Liu, Ming-Hao
學位類別：	博士 Doctor
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2026
畢業學年度：	114
語文別：	英文
論文頁數：	154
中文關鍵詞：	橫向磁聚焦、鄰近化石墨烯、石墨烯超晶格、作用量
外文關鍵詞：	Transverse magnetic focusing, Proximitized graphene, Graphene superlattices, Classical action
相關次數：	點閱：11 下載：0
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橫向磁聚焦測量（TMF）利用垂直於平面的磁場收集注入的載子，已被廣泛用於探測材料的能量色散和研究邊界散射。此外，石墨烯因其良好的遷移率，是一種極具潛力的二維電子裝置材料。為了拓展其應用，人們開始研究功能化石墨烯，包括自旋軌道耦合石墨烯和石墨烯超晶格。本論文回顧並深入探討了我們已發表論文中關於TMF量子模擬的研究，重點關注於近鄰化石墨烯的自旋分裂訊號和Rashba角，以及一維石墨烯超晶格中狄拉克點的複製。此外，在最後一章中，我們闡述如何利用最小作用量原理和劇變理論來描述TMF振盪。

Transverse magnetic focusing measurement (TMF), which collects injected carriers under an out-of-plane magnetic field, has been widely used to probe the energy dispersion of materials and study scattering on boundaries. Additionally, graphene is a promising two-dimensional material for electronic devices due to its high mobility. To broaden its application, functional graphene has been intensively investigated recently, including spin-orbit-coupled graphene and graphene superlattices. In this thesis, we review and thoroughly discuss the quantum simulation of TMF in proximitized graphene, focusing on the spin-splitting signal and Rashba angle, as well as one-dimensional graphene superlattices that demonstrate the cloning of zero modes, all of which were published. Additionally, we illustrate the method for describing TMF oscillations using the principle of least action and catastrophe theory in the last chapter.

摘要 i
Abstract ii
誌謝iii
Acknowledgements iv
Table of Contents vi
List of Tables ix
List of Figures x
Nomenclature xii
Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Graphene 2
1.3 Spin-orbit coupling in graphene-based materials 4
1.3.1 Adatom method 6
1.3.2 Proximity effect 7
1.4 Superlattices 7
1.5 Ballistic and equilibrium transport 8
1.6 Transverse magnetic focusing 8
Chapter 2 Methodology 11
2.1 The tool for the quantum transport simulation - Kwant 11
2.1.1 Two-terminal transmission 14
2.1.2 Probability density and spin density 15
2.1.3 Probability current density 16
2.2 Four-probe resistance 17
2.2.1 Landauer-Büttiker formalism 17
2.2.2 Probability current density and probability density 18
2.3 Peierls substitution 21
2.3.1 Commonly used gauges 21
2.3.2 Multi-terminal technique 23
2.4 Periodic boundary condition 26
2.5 Semiclassical method in Newtonian Mechanics 28
2.6 Electrostatic Simulation 30
Chapter 3 Spin-Orbit Coupling in Proximitized Graphene 33
3.1 Spin-orbit coupling 33
3.2 The tight-binding model 34
3.3 The effective Hamiltonian 37
3.4 The energy dispersion 39
3.5 Carrier density 40
Chapter 4 Spin Splitting Signal in Proximitized Graphene 45
4.1 Experimental results 45
4.2 Spin splitting signal 47
4.3 Peak analysis 48
4.4 Probability current density 54
4.5 The discrepancy from the result of Subnikov-de Haas oscillations 59
Chapter 5 Emergence of Radial-Component in Spin-Orbit Coupled Graphene 60
5.1 The symmetry response method 61
5.2 Rotating in-plane magnetic field method 66
Chapter 6 One-Dimensional Graphene Superlattices 68
6.1 The one-dimensionally periodic electric potential 68
6.2 The mini-band structures 70
6.3 The transverse magnetic focusing 73
6.4 The probability current densities 76
Chapter 7 Semiclassical Method in Hamiltonian Mechanics 79
7.1 The action from the Hamilton-Jacobi equation 79
7.2 Brief introduction to the Catastrophe theory 82
7.3 The first split peak 84
7.4 The second split peaks and other peaks 87
7.5 The peaks on the transmission map and the interface to the quantized transmission regime 91
Chapter 8 Concluding Remark 93
8.1 Summary of the thesis 93
8.2 Future work 95
References 97
Appendix A Derivations 104
A.1 Proximitized graphene 104
A.1.1 Relativistic correction 104
A.1.2 Effective Hamiltonian - low energy expansion at K/K' point 110
A.1.3 Radial Rashba SOC 115
A.1.4 Energy dispersion from effective Hamiltonian 117
A.2 One-dimensional graphene superlattices 118
A.2.1 Continuum model - Plane wave expansion method 118
A.2.2 The energies of cloned main Dirac cones 120
A.3 Semiclassical method in Hamiltonian mechanics 121
A.3.1 The W(y) in the generating function 121
A.3.2 The position from the generating function 122
A.4 Others 124
A.4.1 Probability current density of Kwant 124
A.4.2 Gauge transform 125
A.4.3 Difference of the magnetic field of the split first peak 126
A.4.4 Strength independent non-split second peak 127
Appendix B More Numerical Results 128
B.1 Armchair-oriented proximitized graphene with various combinations of spinorbit coupling 128
B.2 Probability current density and probability density for four-probe resistance 134
Appendix C Codes 138
C.1 Build the sample in Kwant 138
                                    

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