- 學位論文
二維方形量子線網絡的電子穿透問題
Electronic Transmission in Two Dimensional Square Quantum Wire Networks
摘要
近十年來，奈米材料已應用在計算機科學中，其著重於開發奈米材料元件以取代傳統電子元件。例如藉由電子在奈米材料系統的傳輸來傳遞量子資訊。我們亦可藉由改變奈米材料的結構以改變此系統的導電率，使奈米材料作為濾波器。由 Landauer-Büttiker formula 可知，系統的導電率正比於電子的透射率。因此計算出系統的電子透射率即可知其導電率。一般而言是利用轉換矩陣 (transfer matrix) 的方法求解透射率，然而轉換矩陣的方法在一些情況下有其不便之處。 在本文，我們考慮電子在一維量子線及二維方形量子線網絡的穿透問題。此量子線系統包括三種邊界條件─固定、自由及週期性邊界條件。我們利用 Green 函數的方法從量子線系統的本徵值及本徵函數解出電子的透射率，並依各種邊界條件舉實例說明及分析「透射率對電子波數 (wave vector) 和連結鏈長度的乘積」的分佈圖形。 研究指出 Green 函數的方法在網絡具週期性結構及連結於少量輸入、輸出導線之情況下將優於轉換矩陣的方法。從透射率圖形的分析，我們發現高透射率典型上較易發生於方形量子線網絡的輸入及輸出端位在對角的接點時。我們也發現當輸入、輸出端分別位在網絡的對邊上時，網絡的平均透射率的圖形呈現量子化現象。另外，我們發現當量子線網絡的規模愈大，其透射率的圖形振盪愈趨劇烈。此因電子波在網絡中的干涉路徑愈趨複雜所致。
並列摘要
In the last decade, the applications of nanostructure are evident in computer technology. It focuses on the development of nanostructure to replace the traditional electronic devices. For example, we can transfer a quantum bit by the transport of electrons through a nanostructure system. We also can change the conductance of the system by controlling the structure so that the nanostructure can function as a filter. By the Landauer-Büttiker formula, the conductance is proportional to the transmittance of electron. Hence the conductance can be obtained by calculating the transmittance. Usually, the transmittance is solved by the transfer matrix method. However the method has a disadvantage in several situations. In the thesis, we consider the electronic transmission in the one-dimensional quantum wire lattices and the two-dimensional square quantum wire networks with three kinds of boundary conditions such as fixed boundary condition, free boundary condition and periodic boundary condition. We further employ the Green’s function method to solve the electronic transmittance with the help of the eigenvalues and the eigenfunctions of the quantum wire system. An example is given in each case for illustration purpose. We analyze the pattern of the transmittance versus the multiple of the wave vector of electron and the length of linking bond. We show that the Green’s function method has an advantage over the transfer matrix method for the regular network with less input/output leads. From the analysis of the transmittance pattern, high-transmittance is typically more likely to happen in a square network with the input/output leads connected at the opposite corners. We also find that the pattern of the average transmittance exhibits a quantized phenomenon when multiple inputs and outputs are in the opposite sides of the lattice. Besides, the transmittance oscillates more frequently as the network size goes larger and larger because the interference of waves becomes more severe.