我們研究了先前提出的在雙層石墨烯內的量子線能谷濾波器 [Phys Rev B 88, 125422 (2013)]，在線性區域 (small source drain bias) 時的電子傳輸。其電子的穿透率是藉由迭代格林函數法在緊束縛模型下計算出的。能谷濾波器的濾波效應是藉由研究在兩個並接的濾波器元件內電子的穿透率來說明的。對於每個濾波器，我們考慮兩種結構：結構一和結構二濾波器，這兩種結構的濾波器是用不同的能帶結構將電子侷限在量子線內。在現在的實驗技術下，結構二的濾波器比結構一更可能實際做出。每一個濾波器是由一個橫向施加的電場操控的，我們研究了將兩個有加橫向電場的濾波器並置後，兩個濾波器的電場同向或反向的這兩種結構。我們發現當濾波器內部的能谷極化程度夠大時，兩個並置的濾波器在兩者電場平行下的穿透率，和在兩者電場反向下的穿透率有很大的對比。在穿透率解釋上，我們運用了一個兩個能帶的模型來輔助我們解釋數值的結果。這個研究定量地預測了實驗上可觀測到的穿透率結果，並且對於這些結果提供了物理解釋。

We investigate the electron transport in the linear regime in previously proposed quantum wire valley filters [Phys Rev B $\bf{88}$, 125422 (2013)] in bilayer graphene. The electron transmission rates are calculated with a tight-binding model by recursive Green's function method. The filtering effects are illustrated by the study of electron transmission through two back-to-back filters. For each filter, two kinds of structure - called type-I and type-II devices are considered which use different band alignments to provide electron confinement in the quantum wire, with the type-II device being more easily accessible by the present experimental gating technology. Each filter is controlled by a transverse electric field, and we study the two configurations where the two filters are transversely biased by parallel and antiparallel fields, respectively. A sharp contrast of transmission rates is found between parallel and anti-parallel configurations when each filter can generate a sizable valley polarization. The numerical results are explained with an auxiliary two band model. This work gives quantitative predictions to the electron transport in the filters together with some physical understandings about the numerical results.

1 Introduction 4
2 Methodology 9
2.1 The envisioned device structure . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Recursive Green's function . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Numerical results 29
3.1 Type-I transmission rates . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Type-II transmission rates . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Interlter spacing eect on the transmission rates . . . . . . . . . . . . . 45
4 Conclusion 49
A Derivation of current operator 51
B Two band model: explanation of transmission rates 53

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