本篇論文利用數值模擬研究隨機位能對石墨烯帶的傳遞造成的影響。石墨烯帶的能帶結構有別於石墨烯，隨著石墨烯帶邊緣形狀及單位晶胞的原子數目造成不同的能帶結構，故我們首先介紹石墨烯帶的能帶結構及計算的緊束縛近似模型(tight-binding approximation model)。

再來我們將中間一段石墨烯帶加入隨機位能，兩端接上同寬度的石墨烯帶（長度無限），選取某一能量的電子從左邊入射，利用轉移矩陣方法(transfer matrix method)計算電子通過中間帶的穿透係數和反射係數。首先，我們檢查粒子流的守恆；然後研究穿透係數和入射電子能量、以及中間帶長度的關係。

最後，對於一個固定長度的中間帶，我們將穿透機率（|T^2 |）取隨機數的平均。這個平均值隨著中間帶的長度呈現指數衰減，衰減指數的倒數大約是電子局域化的長度。我們研究局域化的長度和入射電子能量、隨機位能的強度、以及石墨烯帶寬度和切割方向等變數的關係。

Using numerical simulation, we report the effects of an imposed random potential on the band structure of grapheme ribbon. The features of energy band structure of graphene ribbon depend on edge shape of the ribbon, zigzag or armchair. We calculate the energy band structure of a ribbon using tight-binding method.

We then study the transport properties of a ribbon with random potential. A sector of grapnene ribbon with random potential is connected by perfect ribbon from both sides. Using transfer matrix method, we study the transmission and reflection coefficients of an incident electron from the left-hand side. We first check the conservation of electron flux. Then the transmission is studied as function of energy of incident election, the width of the center sector, and the magnitude of the random potential.

For a ribbon of fixed width, we take the random average of the transmission probability（〈|T^2 | 〉）. From the dependence of the transition probability, we estimate the evanescence exponent. Reversion of this exponent gives an indication of the localization length. We finally report the localization length as functions of incident energy, magnitude of the random potential, and the width of ribbon for both zigzag and arm-chair graphene ribbons.

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