在1980年Von Klitzing發現量子霍爾效應(Quantum Hall effect)邊緣的電子會隨著邊緣傳遞。在1988由英國物理學家Duncan Haldane提出的理論模型來描述拓樸絕緣體中的電子結構。2005 年，Kane與 Mele 進一步推廣 Haldane 的模型，並考慮被 Haldane 忽略的電子自旋發現了被稱為量子自旋霍爾效應（Quantum Spin Hall Effect，QSHE），引入了自旋軌道交互作用並成功打開了非平凡(nontrivial)的能隙且無需外加磁場來破壞時間反演對稱。在二維拓樸絕緣體在材料中內部對電流是絕緣的，但在材料邊緣卻會出現沿著特定方向環繞的電流或自旋流，即順時針或逆時針轉，而且這種電流或自旋流不受障礙物、缺陷的單向傳播的狀態稱為螺旋邊緣態。
2007 年， Xiao、Yao 等人提出量子谷霍爾效應，於蜂窩狀石墨烯晶格透過破壞晶格反轉對稱性打開 Nontrivial 能隙，由於 K 與 K’谷分裂後發現自旋谷具有內在磁矩且會分別在相鄰K和K’相互自旋且方向相反的單向傳輸現象。2016 G. Shvet 提出光子晶體谷霍爾效應， 2019 Sievenpiper 提出TE/TM 雙層互補結構的電磁波谷霍爾效應。
本篇研究將將蜂窩狀光子晶體與量子谷霍爾效應來作為拓樸絕緣體，將蜂窩形的晶格分為Patch 與Aperture PEC金屬薄膜互補的超穎表面光子晶體結構，且在倒空間的第一布里淵區的K、K’點形成Dirac point打開能隙且各自仿建偽自旋。將翻轉與為翻轉的互補的蜂窩形光子晶體結構於邊界交界處激發，來觀察電磁波單一方向傳遞與結合共振腔不受障礙 物、缺陷、轉彎的影響，最後因為PEC金屬互補結構可以沿著邊緣態前進且轉彎特性，結合環形共振腔特性並探討分析。

In 1980, Von Klitzing discovered that electrons at the edge of the quantum Hall effect (QHE) would travel along the edge. In 2005, Kane and Mele discovered the Quantum Spin Hall Effect (QSHE) involving electron spins. They introduced spin-orbit interaction and successfully opened a non-trivial energy gap without needing an external magnetic field to break time-reversal symmetry. This paper uses the Finite-Difference Time-Domain (FDTD) method to simulate the three-dimensional topological insulator of PEC (Perfect Electric Conductor) metal materials. A honeycomb lattice structure is employed to separate the Patch metal film and Aperture metal film, which are complementary structures. Within these films, the respective Ez and Hz modes are Px and Py complementary, with the upper and lower energy bands also being complementary and possessing opposite modal characteristics. Due to the interaction between the Patch and the Aperture metal films, the Dirac point at the K point is opened. Pseudospin Up/Down excitations are used to observe the modal characteristics. The Pseudospin Up excitation of the Upper Band and Lower Band exhibits counterclockwise (CCW) handedness, while the Pseudospin Down excitation exhibits clockwise (CW) handedness for both bands. Similarly, the spin properties of the Pseudospin Up excitation for both Upper and Lower Bands are CW, and the Pseudospin Down excitation spin properties are CCW. By combining flipped and non-flipped complementary metal structures, and exciting the junction with Pseudospin Up/Down, a distinct edge state dispersion curve was observed within the energy gap: Pseudospin Up towards the positive direction (+kx) and Pseudospin Down towards the negative direction (-kx). The existence of edge states was confirmed using perfect matching layers in the three-dimensional x, y, and z directions, showing that they remain bound to the boundary regardless of the turning direction. However, if the boundary symmetry is disrupted, the intensity of the energy field significantly diminishes.
Finally, the turning characteristics were combined with the resonant cavity structure to observe the characteristics of the resonant cavity. Different ratios and gaps were compared, revealing that various structures exhibit different edge states and resonant cavity characteristics.

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