拓撲晶格絕緣體薄膜的k・p模型

Abstract

在二十一世紀初，實驗上發現了石墨烯這樣的材料，而在能帶理論中他有著能帶差距在動量空間原點為零的特性，也因此，在這種非金屬材料下電子可以導電。而在最近幾年，有一個新型材料，我們稱為拓撲絕緣體。一開始此種材料是從量子自旋霍爾效應發現的，也就是在二維拓撲絕緣體中，電子可以在其邊緣運行。而當我們拓展到三維材料上，就只能在表面導電，我們將這些可導電且在表面的態稱為表面態。拓撲絕緣體的零能帶差是由時間反演對稱所保護的，而在最近發現一種由晶格對稱保護的拓撲絕緣體，我們稱之為拓撲晶格絕緣體。在實驗上，拓撲絕緣體的材料已被發現是 Pb 1-x Sn x Te 這種材料，在此我們將單純探討 x = 1，即SnTe 此種拓墣絕緣體。

本篇論文將會呈現出擬合 SnTe此種拓撲絕緣體參數的過程，諸如費米速度，特別注意的是，我們探討的是 (001) 的晶體方向。再來我們會用數學模型來近似表面混成項對應於厚度的關係，以便於調整薄膜厚度 ; 同時我們會把不同基底下的態做連結，接著我們會完整推導出如何從塊材模型到薄膜模型，以及指出在厚度較大時，薄膜模型跟塊材模型的能帶關係，以及為什麼不能省略掉非導帶和價帶的原因。最後，當我們得到以上的資訊後，我們將可利用 k・p模型建構出一個雙層薄膜模型。

In the 21th century, scientists found a material called graphene, whose dispersion gap is zero at k = 0. Therefore, this non-metal material can conduct. In recent years, there is a revolutionary material called Topological Insulator. At first, this kind of material was found from quantum spin hall effect. i.e., it is found in 2D topological insulator and electrons can move in the edge. As we expand this concept to 3D, the electrons can only move in the surface, and we call it surface states. The zero gap points of topological insulator are protected by time-reversal symmetry. Recently, there is another material whose zero gap points are not protected by time-reversal symmetry but crystal symmetry. And we call this topological crystalline insulator (TCI). In experiment, TCIs are observed in 3D material Pb 1-x Sn x Te. Here, we will simply discuss the SnTe.

This thesis will present the fitting process for parameters of TCIs SnTe such as Fermi velocity, in the (001) direction. Later, we will use Yukawa potential to approximate the Hybridization terms versus the number of layers, and so we can adjust the number of layers. Also, we will connect the relation between different bases. Then we will derive the reduced process from bulk to thin film model. Also, we will point out the dispersion relation between bulk and thin film, and why we can not ignore non-conduction and non-valence band. Finally, when we information above , we can use k・p model to construct a bi-layer thin film model.