我們探討了兩種可能在高溫下產生量子異常霍爾現象的方法，我們考慮的系統是蜂窩晶格。第一個方法是在外加磁場當中，我們會得到霍爾電壓與磁通量的方程式戴爾凡庭方程式(Diophantine equation), 我們還發現系統的霍爾電壓會受到自旋軌道耦合作用力的影響而改變。 第二個方法是考慮延展具有應變(strain)之晶格，系統會給出一個類磁場的效應，發現當應變的周期很大時，在K點會有蘭道能階的出現。在加入自旋軌道耦合作用力與因接觸磁性物質之近鄰效應所產生的磁性，系統可以在室溫產生量子異常霍爾現象。

In this thesis, we explore two methods that may generate Quantum Anomalous Hall effect in room temperature on a honey comb lattice.

In the first case, we consider the honeycomb
lattice in external magnetic fields. It is shown that the distribution of Chern numbers over bands in Hofstadter butterfly obeys the Diophantine
equation. Furthermore, we find that the distribution of Chern numbers can be altered by including the spin orbital interaction when its strength exceeds some threshold. It is argued that by appropriate using the spin orbit interaction, the Quantum Anomalous Hall in a subband of a honeycomb lattice in a magnetic field can be engineered.

In the second work, we propose to induce magnetization in the honeycomb lattice by placing the honeycomb
lattice in proximity to a ferromagnetic material. We find that under strain, pseudo magnetic fields generate flat bands similar to Landau levels in the honeycomb lattice. These flat bands possess non-vanishing Chern number and can exhibit Quantum Anomalous Hall effects in room temperatures.

front page (I)
Abstract in Chinese (IV)
Abstract in English (V)
list content (VI)
list figure (VIII)
list table (XII)

1 Introduction ........ 1
2 Berry phase and Hall conductance ..4
2.1 Berry phase . . .. . . . . . . ..4
2.1.1 The derivation of the Berry phase . . . 4
2.1.2 Berry curvature . . . . . . . . . . . . 7
2.2 Hall conductance . . . . . . . . . . . . 9
2.2.1 Quantization of Berry phase . . . . . . 9

3 Honeycomb structure in magnetic field ..... 11
3.1 Honeycomb lattice with Tight binding model in external magnetic field ...........11
3.2 Tight binding model and Kane-Mele interaction in magnetic field . ............ 19
3.3 Discussion . . . . . . . 25

4 Honeycomb structure with strained field .. . ..46
4.1 Honeycomb structure with strained field . . . . 46
4.1.1 Zero energy mode . . . . . . . . . . . . . 50
4.2 Honeycomb structure with strained field and Kane-Mele interaction ....... 55
4.3 Honeyconb structure with strained field and Kane-Mele interaction and Rashba interaction . . . . .. . 60
4.4 Discussion . . . . . . . . . . . . . . . 63

5 Conclusion .....80

Bibliography.... 81

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