凝態物理中量子霍爾效應(Quantum Hall effect)被Klitzing於1980年所發現，於1988年F.D.M.Haldane提出了拓樸相變與拓樸絕緣體的模型，2005年C.L.Kane與E.J.Mele提出量子自旋霍爾效應(Quantum Spin Hall effect)，以Haldane Model為架構，利用自旋軌道耦合(SL coupling)，打開Nontrivial能隙，不須外加磁場與也不須破壞時間反演對稱性，2007年Xiao、Yao等人提出量子谷霍爾效應，於石墨烯系統透過破壞晶格反轉對稱性打開Nontrivial能隙，於K與K’谷發現谷自旋具有內在磁矩且具有谷自旋相依的單向傳輸。
本論文將光子晶體與量子自旋霍爾效應結合作為二維TM介電質實驗主架構，主要目的是找出金屬的表面電漿拓樸邊緣態。採用六角形晶格二維結構會產生簡併的Dirac point特性，透過改變六角形晶格內空氣三角柱的間距產生不同強度的自旋軌道耦合並產生光子偽自旋，空氣三角柱從收縮到擴張的過程中發生能帶反轉與拓樸相變。在矽中挖空氣三角孔洞是TM模態下電場在空氣介質下共振能量較強。將不同拓樸不變量的結構組合形成邊界，能帶結構中能隙裡有兩條曲線為拓樸邊緣態。在邊界上以不同偽自旋激發可觀察到電磁波被束縛於邊界並單向傳輸，也因體-邊緣對應拓樸邊緣態不太被缺陷影響。將材質由介電質換成金屬後，由於金屬在量子自旋霍爾效應二維TM結構下無法產生完全打開的能隙，在三維空間之中將六角形晶格與量子谷霍爾效應結合，在谷自由度之下以電磁場對偶性組建偽自旋態，從六角形晶格中劃分出片狀金屬薄膜與網狀金屬薄膜，因六角形晶格C_6v對稱性會在第一布里淵區的K、K’點形成Dirac point。片狀與網狀金屬薄膜堆疊形成金屬互補超穎表面，片狀與網狀金屬薄膜的交互作用使K、K’點打開了能隙並組建了偽自旋。將未翻轉與翻轉後的金屬互補超穎表面組合形成邊界，在邊界以偽自旋激發後可觀察電磁波被束縛於邊界並單向傳輸。

The Quantum Hall effect in condensed matter physics was discovered by Klitzing in 1980. In 1988, F.D.M. Haldane proposed a model for topological phase transitions and topological insulators. In 2005, C.L. Kane and E.J. Mele introduced the Quantum Spin Hall effect, based on the Haldane model. Spin-orbit coupling was utilized to open a nontrivial bandgap, without the need for an external magnetic field or the breaking of time-reversal symmetry. In 2007, Xiao and Yao et al. proposed the Quantum Valley Hall effect, which is observed in graphene systems by breaking the lattice inversion symmetry and opening a nontrivial bandgap. In the K and K' valleys, valley spins possess an intrinsic magnetic moment and support valley-spin-dependent unidirectional transport.
The main objective of this thesis is to explore the surface plasmon topological edge states in metal structures. In a two-dimensional metal structure, only transverse magnetic (TM) mode supports surface plasmon modes. Therefore, it is preferable to first study topological insulators in two-dimensional TM structures using dielectric materials. The first part of this thesis studies the photonic topological insulator using photonic crystals with the Quantum Spin Hall effect as the main framework for two-dimensional TM dielectrics. A hexagonal lattice with six triangular air hole placed identical to triangular lattice structure generates the double degenerated Dirac point in the  point of the Brillion zone. This can be realized by band folding between the hexagonal and rhombic Brillion zone. By further altering the spacing between the air triangular holes within the hexagonal lattice, different strengths of spin-orbit coupling are induced, resulting in the band gap opening and the generation of Pseudospin for TM modes. During the transition of contraction to expansion of the spacing between air triangular holes, band inversion and topological phase transitions occur. Combining structures with different topological invariants forms a boundary which exhibit a special dispersion relation within the bandgap. There are two curves representing the topological edge states in the bandgap. Exciting different Pseudospins at the boundary allows for the observation of electromagnetic waves being confined to the edge with unidirectional propagation.
When the material is changed from dielectric to metal, a complete bandgap cannot be achieved in the Quantum Spin Hall effect for a two-dimensional TM structure. Therefore, a three-dimensional metallic structure is a better choice to achieve plasmonic topological insulator. Hexagonal lattice with Quantum Valley Hall effect are constructed using EM duality to form Pseudospin states with the valley degree of freedom. EM duality is to utilized patch and complementary aperture structure where both have the same dispersion relation but different TE/TM characteristics respectively. Due to the C_6v symmetry of the hexagonal lattice, Dirac points are formed at the K and K' points in the first Brillouin zone. Stacking the patch and aperture creates a metal complementary metasurface. The interaction between the patch and aperture opens a bandgap at the K and K' points and forms Pseudospin states. By combining the dual-layered metal complementary metasurface and its flipped structure side by side, a boundary is formed. Exciting with pseudo-spin at the boundary allows for the observation of electromagnetic waves being confined to the edge with unidirectional propagation. The extinction ratio for unidirectional propagation is estimated to be a factor of 25. Such edges states are tested by introducing different type of defects and found to be vulnerable by spin-flip scattering.

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