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標題: | 電磁等效材料之拓撲相變 Topological Phase Transitions in Electromagnetic Effective Media |
作者: | You-Zhong Yu 余佑中 |
指導教授: | 陳瑞琳 |
關鍵字: | 拓撲不變量,時間反轉對稱,等效材料, Topological Invariant,Time Reversal Symmetry,Effective Medium, |
出版年 : | 2017 |
學位: | 碩士 |
摘要: | 2004年隨著新興材料石墨烯的發現,Kane與Mele於隔年利用石墨烯的自旋軌道耦合設計二維拓撲材料促成拓撲絕緣體的誕生,不久後HgTe量子井的量子自旋霍爾效應於實驗室被觀察到使得實現拓撲表面態更為容易。2008年Haldane利用電子與光子晶體中的相似性,設計一個打破時間對稱性的拓撲光子晶體結構使表面波波傳只往單一方向傳播並被拓撲保護。
關於拓撲光子晶體,已知頻隙與旋性是形成拓撲邊緣態的關鍵。為了找到邊緣態,許多拓撲光學文獻發展出許多不同的幾何結構與不同對稱性的光子晶體,亦有文獻改變複雜的材料參數形成由等效材料組成的拓撲材料如:雙曲線對掌性超常材料、磁化電漿超常材料與規範場變換材料來設計達成表面態。 本文利用對稱的材料參數假設達成電磁場的去耦,透過此定義得到光的兩種模態。接著將此兩種模態的基底做線性變換改寫成旋性基底對應到的漢米爾頓用波向量與自旋1矩陣表示,得到具有兩種正負旋性的特徵波函數可計算幾何相位與貝瑞曲率,證實材料的拓撲不變量不為零。在體態的分析上對參考頻率做微擾其一階展開可求得近似的波數進而分析體態的頻帶,此外兩種電磁去耦模態對應的漢米爾頓其波數相反可分別計算出表面態。最後,將拓撲不變量非零的材料與拓撲不變量為零的材料相接,利用其切向電磁場在介面連續可以得到表面波的解,並以模擬驗證與理論相互吻合。 In 2004, with the discovery of graphene, Kane and Mele proposed a hexagonal model with the spin-orbit coupling of electron which led to the birth of topological insulators. Before long, HgTe quantum well with quantum spin Hall effect was validated in the experiment which confirmed the topological edge state. In 2008, Haldane designed the topological photonic crystal which break the time reversal symmetry by applying the similar concept between the electrons and photonic crystals. Haldane showed that the surface wave is unidirectional and protected by the topological invariant. With respect to the topological photonic crystals, frequency gaps and spin are the keys to the formation of the topological edge states. In order to find the edge state, many topological photonic research develop different geometries with symmetrical photonic crystals and resonators. There are also some papers that change the complex parameters of the effective material such as hyperbolic chiral metamaterial, magnetized plasma metamaterial and gauge field transformation medium to observe the edge state. In this paper, the symmetrical material parameters lead to the two decouple mode of the electromagnetic field. Then the linear transformation of the basis from the two modes is rewritten into the new wave function with the positive and negative spin which correspond to the Hamiltonian in term of the wave number and spin 1 matrix. Because the intrinsic spin properties of the light, we use the new basis to calculate the geometric phase and Berry curvature which confirmed that the topological invariants of the material are not zero. Next we do the expansion around the reference frequency to obtain the approximating wave number and then analyze the frequency band. The two electromagnetic decoupling modes correspond to two independent Hamiltonian in term of reverse wave number that can derive the surface states. Finally, if the material with non-zero topological invariant is connected with the material with zero topological invariant, the surface waves can be obtained by calculating the continuity of the tangential electromagnetic fields on the interface. And the simulation is consistent with the analytical method. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/76565 |
DOI: | 10.6342/NTU201701915 |
全文授權: | 未授權 |
顯示於系所單位: | 應用力學研究所 |
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