石墨烯中時變位能的量子傳輸特性

Abstract

電子在石墨烯(Graphene)材料上擁有獨特的介觀傳輸性質，其色散關係(energy dispersion relation)在接近費米能量時呈線性關係，因此電子在這個能量區域的性質如同無質量的狄拉克 費米子(massless Dirac fermions)。由相對論的量子理論，無質量的粒子，具有Klein 穿隧效應 (Klein tunneling)，即粒子無法被一時間穩定的位能障 (time-independent potential barrier)所束 縛。 本論文分析在石墨烯上電子向時變位能(time-dependent)區垂直入射的傳輸行為，並透過緊束縛 模型(tight-binding model)使得本結果涵蓋非線性色散區域。在低能量區域本結果獲得與前人使 用有效哈密頓(effective Hamitonian)的預測一致：Klein 穿隧效應在時變位能作用下時仍然成 立，時變的位能亦無法束縛粒子；另外，分析不同位能寬度的傳輸行為，我們發現了一奇特現 象─中央帶再聚集(central band refocusing)效應，即在某些特定的時變位能寬度時，電子只能 從中央帶穿隧。在高能量區域我們觀察到不同能谷(inter-valley)之間的散射：電導驟降(dip structure)、旁帶(sideband)特性以及非典型Fabry-Perot 干涉特性。 我們也探討時變位能對具有能隙的石墨烯(gapped graphene)傳輸的影響，發現同能谷 (intra-valley)散射亦能產生電導驟降以及非典型Fabry-Perot 干涉特性。

Electron in graphene has a unique mesoscopic transport property due to its linear dispersion relation when the Fermi energy falls within the low energy regime. The electron behaves as a massless Dirac fermion. Most well-known characteristic of a massless Dirac fermion is the Klein-tunneling, where the particle cannot be blocked or trapped by static barriers.

This thesis focuses on the transport property, in general, and the Klein-tunneling characteristics, in particular, for a graphene acted upon by a time-modulated potential. For the clarity of the physics involved, our consideration is limited to the case of normal incidence. We use tight-binding model for the description of the graphene so that our results cover nonlinear dispersion regime for the electrons. In the low energy regime, we reproduce the Klein-tunneling results in a time-modulating potential case, which has been predicted by Tahir et.al. recently. In addition, we find an exotic central band refocusing phenomenon, where the transmission will be dominated by the central band (the elastic channel) at specific values of the length L of the time-modulated region. This L- periodic phenomenon is explained by a peculiar interference condition that is made possible by the linear energy dispersion relation and the chirality of the particle. Furthermore, we find dip structures in the total transmission in both the high-energy region, and in the low-energy regime of a gapped graphene. These dip structures signify the breakdown of the Klein-tunneling, and is shown to result from coherent hopping to or from the band edge via the emitting or absorbing of photons provided by the time-modulated potential. The band edge has a singular effective density of states as long as the transverse momentum is conserved. Finally, by staying on the dip structures, the total transmission is found to exhibit another L-periodic phenomenon which we can identify as a non-typical Fabry-Perot resonance.