C6v對稱之二維PT光子晶體中的量子自旋霍爾效應及其拓樸邊界態

Abstract

本研究利用C6v對稱之二維宇稱時間對稱性光子晶體(parity-time symmetric photonic crystals)來製造出拓樸邊界態(topological edge state)，藉由使在單位晶格(unit cell)內的介電質材料滿足宇稱時間對稱性不變量(PT invariant)的條件，來形成宇稱時間對稱性光子晶體，然而由頻帶結構(band structure)分析可以得知，當介電係數(electric permittivity)之虛部的參數小於臨界值時，此時為非破壞宇稱時間對稱性相(Unbroken PT phase)，具有全實數根的特徵值(eigenvalues)，而當電係數之虛部的參數大於臨界值時，此時則為破壞宇稱時間對稱性相(Broken PT phase)，將會有共軛虛數根的特徵值產生，頻帶將會形成平坦(flat band)的情況，而此時將可以由頻帶結構來得知獨特點(exceptional point)的位置。 當頻帶結構為非破壞宇稱時間對稱性相時，則可以藉由出現雙狄拉克錐(double Dirac cone)的參數，來尋找出合適的零解相(trivial phase)與能帶反轉(band inversion)後的拓樸相(topological phase)結構。然而藉由將不同拓樸性質的零解相與拓樸相結構合併，並且陣列成超晶格結構(supercell structure)，則可以在打破偽時間反演對稱性(pseudo time reversal symmetry)的情況下，保留真正的時間反演對稱性(time reversal symmetry)，進而實現量子自旋霍爾效應(quantum spin Hall effect)。由於不同拓樸性質的結合下，導致拓樸邊界態的形成，而此拓樸邊界態能夠在兩個不同拓樸性質的結構接合面上進行傳播，並且能夠在所設計具有鈍角與銳角的軌道上行進，同時也擁有抑制背向散射(backscattering)的能力。 關鍵字：宇稱時間對稱性、光子晶體、非破壞宇稱時間對稱性相、破壞宇稱時間對稱性相、量子自旋霍爾效應、拓樸邊界態

In this thesis, the two dimensional parity-time symmetric photonic crystal are used to construct topological edge state. Let dielectric materials of unit cell satisfy parity-time symmetric invariant to become parity time symmetric photonic crystals. However, from the analysis of band structure, when the imaginary part of electric permittivity is smaller than some critical value, it is a unbroken parity-time symmetric phase, and eigenvalues are all real. But, when the imaginary part of electric permittivity is greater than the critical value, it is a broken parity-time symmetric phase, and eigenvalues are imaginary conjugate pairs. The local frequency bands will merge into a flat band, and can learn about the position of exceptional points from band structure. When band structure is a parity-time symmetric phase, it can adjust structure parameter of double Dirac cone in the band structure to find out appreciative trivial phase and topological phase structure with band inversion. However, let the trivial phase and topological phase structure of the two different topological properties to be combined to form the supercell structure, it can keep the genuine time-reversal symmetry by breaking the pseudo time-reversal symmetry, and realize the quantum spin Hall effect. Because the lattice structure of two different topological properties are combined together, it can lead to form the topological edge state. The topological edge state can propagate in the interface of the two different structures, and it can march in the orbit with obtuse and acute angle. Meanwhile, it possess ability to suppress the backscattering. Keywords: parity-time symmetry, photonic crystals, Unbroken parity-time symmetric phase, Broken parity-time symmetric phase, quantum spin Hall effect, topological edge state