石墨烯除了電荷、自旋的自由度之外，還有與生俱來的額外自由度－能谷，因此在石墨烯上發展能谷量子元件是可行的。我們在2016年研究中[1]雙層石墨烯單一介面的橫向量子結構，發現能隙或電位能的不連續對能谷有篩選性。在此研究，我們更進一步討論能谷電子在經過多個介面的傳輸行為，尤其是經過量子能障的結構。在反轉對稱性被破壞的Bernal-stacking雙層石墨烯中，數值模擬顯示在單一量子能障的系統，能谷的極化(篩選性)會和能障的高度、寬度正相關；而在雙重量子能障的系統，能夠使用共振傳輸的能谷相較於一般穿透傳輸的能谷更具有優勢。另外，在低溫時能谷電流的極化率，可以達到百分之十左右。總而言之，斜向入射的電子在經過單一量子能障或是雙重量子能障都能夠提供有效的能谷電流源。

Valley-based quantum devices are feasible in graphene due to the existence of an inherent carrier degree of freedom - valley pseudospin. As shown previously, a valley contrast emerges when obliquely incident electrons are transmitted through graphene-based lateral quantum structures with a single interface where discontinuity of bandgaps or potentials occurs[1]. In this work, we extend the previous study to graphene structures with multiple interfaces, especially to those consisting of quantum barriers, and investigate the intriguing behavior of valley polarization therein. Here, we focus on the Bernal-stacking bilayer graphene with broken inversion symmetry. Our results demonstrate that in the case of single barrier structures the valley polarization is positively correlated to the barrier width and the barrier height. In the case of double barrier structures, the carrier transport through resonant states can lead to sizable valley polarization, which can reach around ten percent or more at low temperatures. Overall, the transmission of obliquely incident electrons through single or double quantum barriers serves as an effective way to generate valley-polarized electron sources.

Chapter 1 Motivation and Introduction --- P.1
Chapter 2 Theoretical method and model --- P.3
Chapter 3 Results and discussions --- P.9
Chapter 4 Conclusion - P.21

[1] F.-W. Chen, M.-Y. Chou, Y.-R. Chen, and Y.-S. Wu, Physical Review B 94, 075407 (2016).
[2] M.-K. Lee, N.-Y. Lue, C.-K. Wen, and G. Wu, Physical Review B 86, 165411 (2012).
[3] G. Wu and N.-Y. Lue, Physical Review B 86, 045456 (2012).
[4] A. E. e. Galashev and O. R. Rakhmanova, Physics-Uspekhi 57, 970 (2014).
[5] E. V. Castro, K. Novoselov, S. Morozov, N. Peres, J. L. Dos Santos, J. Nilsson, F. Guinea, A. Geim, and A. C. Neto, Physical review letters 99, 216802 (2007).
[6] J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, Nature communications 6 (2015).
[7] S. Y. Zhou, G.-H. Gweon, A. Fedorov, P. First, W. De Heer, D.-H. Lee, F. Guinea, A. Neto, and A. Lanzara, arXiv preprint arXiv:0709.1706 (2007).
[8] E. McCann and V. I. Fal’ko, Physical Review Letters 96, 086805 (2006).
[9] A. Voskoboynikov, S. S. Liu, and C. Lee, Physical Review B 59, 12514 (1999).