根據石墨烯在低能量的能帶性質，在經由電極閘調控費米能使得石墨烯形成pn接面後，載子通過接面使得其從類似電子(電洞)行為轉變成類似電洞(電子)行為，且由於電子與電洞的動量差一個負值，我們可以得出載子經過接面後會形成非典型的負折射路徑。在2015年時，有一篇關於量測石墨烯負折射實驗論文被發表在Nature Physics[1]，他們宣稱量測到負折射的訊號。但由於他們的結果不是很明顯，因此本論文一開始根據他們元件的參數與設計，再藉由數值分析的工具，Kwant、Gmsh 和FEniCS 來模擬這樣的設計在理想的情況下會得到什麼結果。此外，本論文更進一步根據他們的設計提出了一個改善結果的方法。最後，由於Veselago lens 的設計無法避免Klein tunneling的貢獻，本論文提出新的設計來去除Klein tunneling的貢獻，可做為量測石墨烯之電子負折射更加直接的設計。

Due to the gapless low-energy band structure, carriers in graphene transform from electron-like (hole-like) to hole-like (electron-like) after passing through the gate-induced pn junction. Because the momenta of these two kinds of carriers are opposite, the trajectory of carriers shows a negative refraction behavior after passing through the pn junction. In 2015, an experimental work reporting negative refraction in graphene was published in Nature Physics[1]; however, the signal is not clear in their result. This work begins with following their geometric design and parameters to check what results should be in the ideal conditions by numerical tools such as Kwant, Gmsh and FEniCS. A modified geometry showing much improved signals due to negative refraction is then investigated. The last part of this work is devoted to a new design for a direct observation of negative refraction without the contribution from the Klein tunneling, which actually dominates the signal in the previous design based on the Veselago lensing geometry.

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