目前在我們的實驗室中，已可以將光子晶體三維空間中的14種布拉非晶格，透過 Yee's scheme 的離散方法離散馬克斯威爾方程，再分析其廣義特徵值問題，透過零空間免除法的技巧和快速傅立葉轉換的方式，來求得此特徵值問題較小的特徵值，進而繪製出各種晶體的能帶結構圖。然而卻只能計算單一晶格單一材料，無法計算兩種或多種材料，經過周期性排列堆疊而成的超晶格。因此此篇論文透過擴增其晶格生長方向向量、材料的移動判別法等，讓我們能夠建構並計算兩種以上材料堆疊而成的超晶格結構，將單晶格與超晶格做對照，能帶結構圖中有相同的能帶間隙和增根結果，輔合我們的預期和其固定的物理性質。透過此超晶格計算建構方法，我們能夠模擬不同材料、不同介電系數堆疊而成的超晶格，而我們主要的目標，是能建構出一種內部為絕緣體，材料交界表面卻可導電的特殊材料，並計算模擬出其的表面態情形。最後因晶格的堆疊而導致矩陣計算量以幾何倍數增加，因此我們也透過GPU的轉換使用，來提升其高效能計算速率。

In our laboratory, we can already use the Yee’s scheme to discretize the Maxwell’s equation of the 14 Bravais lattices in the 3 dimensional photonic crystal, and then by the method of null space free as well as fast Fourier transformation, we can analyze its generalized eigenvalue problem to derive the smaller eigenvalue of this eigenvalue problem, and hence, plot the band-structure of the crystals. However, by doing so, we can only compute the crystals with unit-cells and cannot compute the super-cells containing two or more materials that is stacked periodically. Therefore, the main purpose of this thesis is to construct and compute the super-cell crystals stacked by two or more materials by scaling the lattice vector of the crystal and using the moving determinant to compare the band-gap and the eigenvalues from the band-structure of the unit-cell and the super-cell to observe whether it corresponds with the physical properties of what we expected. By using the super-cells computation method, we can modal super-cells with different material and dielectric coefficient. Our main goal is to construct a material where it is non-conductive in the inner part and conductive in the interface of the surface, and also modal its surface condition. Last, since the matrix computation increases exponentially with the number of the stacking, we enhance the efficiency by the GPU.

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