在反轉對稱性被破壞的石墨烯中，若外加一個側向電場，帶有K跟K'能谷的電子會各自往垂直於電場的反方向運動，造成一個淨能谷電流，此現象稱為「能谷霍爾效應」。在過去T.Ando等人曾計算，當費米能級位於能隙裡面時，能谷霍爾電導率為$e^{2}/h$。此篇工作運用緊束縛模型，探討在扶手式石磨烯奈米緞帶的準一維結構中是否仍會有此效應。我們發現，奈米緞帶的邊界會摧毀能谷電流，但當我們在通道跟邊界間引入一個有限的能障部分分離電子跟邊界時，電流就會回復。對此計算我們也提出一個簡單的物理模型，用邊界散射的物理機制來解釋我們的數值結果。

The intrinsic valley Hall conductivity is quantized
into $e^2/2h$ and has opposite sign between the two valleys within the gap in the ideal case of gapped monolayer graphene. Thus, there will be a valley current in the direction transverse to an applied in-plane electric field in bulk graphene. In this work, we present a numerical calculation of valley current in armchair graphene nanoribbons based on the tight-binding model and offer a physical picture to explain the influence of edges on the valley current. This method may be generalized to other nanoribbons
with different edges.

Abstract 1
1 Introduction 5
1.1 Electronic Structure of Monolayer Graphene . . . . . . . . . . . . . . . 6
1.2 Dirac Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Electronic Structure of Graphene Nanoribbons . . . . . . . . . . . . . . 11
1.4 Valley Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Methods 16
2.1 Tight Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Standing Wave Basis . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Valley Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Results and Discussions 29
3.1 Two-State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Effect of Reversing kx . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Effect of Reversing the In-plane Electric Field . . . . . . . . . . . . . . 40
4 Conclusion 42
Appendices 43
Appendix A Standing Wave Basis 43
A.1 Orthonormality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Appendix B Time Reversal Symmetry 45
B.1 Band Structure and Group velocities . . . . . . . . . . . . . . . . . . . 46
B.2 Valley Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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