傳統晶格結構如四方晶格、六角晶格，因其具有平移與旋轉對稱性(Translation and rotational symmetry)，可定義出布里淵區，當電子波函數經過布里淵區邊界時，會產生能量散射，使電子波函數無法與高階倒晶格點作用。而透過旋轉堆疊晶格後，可以破壞晶體對稱性，使其無法定義出布里淵區邊界，進而產生長距離作用 (Long range interaction)；由倒晶格基底彼此線性組合產生之倒晶格點互相作用，使低階倒晶格動量得以與高階倒晶格動量耦合，產生傳統晶格不具備的能帶結構。透過旋轉堆疊產生之結構可分為摩爾晶格(Moiré lattice)或準晶格(Quasilattice)。在晶體結構上，摩爾晶格對於旋轉錯位角(Twist angle)有嚴格要求，才可使其同時滿足平移及旋轉對稱性，使其得以合成摩爾布里淵區;而準晶格則不然，僅須滿足旋轉對稱性而不具備平移對稱性，因此其可以依據旋轉錯位的程度調製晶體的能帶結構。因此我們對旋轉錯位角如何影響能帶結構有著相當濃厚的興趣，在本文除了對摩爾晶格及準晶格的製備外，也嘗試建立模型對其能帶演化進行解釋。本論文為了簡化討論，我們將利用表面電漿子(Surface plasmons, SPs)其為表面傳遞的近場波之特性，進而將討論維度限制在二維平面，如此一來可同時簡化了製程的困難度以及模型建立的複雜度，因此在實驗上我們利用聚焦離子束蝕刻(Focused ion beam milling)技術在原子級平坦的磊晶銀(Epitaxial silver film)表面製備二維電漿子晶體作為我們探討摩爾晶格及準晶格的研究平台，並著重於旋轉錯位角上的調製，使我們同時探討摩爾晶格及準晶格的能帶演變。
文中，首先介紹關於晶格的基本定義，並區分摩爾晶格與準晶格之間的不同，以及其在光子學上的相關應用 (如:平帶(Flat band)、魔角雷射(Magic angle laser)以及能隙控制…等)。接著，介紹此平台主要傳播媒介:表面電漿子，其主要區分為表面電漿極化子(Surface plasmon polaritons, SPPs)以及局域性表面電漿子(Localized surface plasmons, LSPs)，緊接著我們將介紹週期排列的奈米結構，即電漿子晶體(Plasmonic crystals)，並針對六角晶格的色散曲線進行詳述，以利於我們會後續能處理更加複雜的摩爾晶格以及準晶格的能帶演變。在實驗上，透過角度分辨顯微鏡量測摩爾晶格及準晶格的角解析反射光譜，來觀測表面電漿極化子色散曲線並解釋能帶結構隨角度變化之關係，成功證明我們的理論建模與觀測結果相呼應，最後這種探討不僅限於表面電漿子領域，因此我們期待本文的晶格製備與分析建模方式，能提供給後者一個全新思路去分析複合晶體(如:超晶格、摩爾晶格以及準晶格)之能帶結構。
關鍵字: 表面電漿極化子、表面電漿子晶體、摩爾晶格、準晶格

Traditional lattice structures such as square and hexagonal lattices, characterized by their translational and rotational symmetry, define the boundaries of a Brillouin zone. When an electron wave function encounters these boundaries, energy scattering occurs, preventing interaction with higher-order reciprocal lattice points. However, by rotating and stacking these lattices, the crystal symmetry can be disrupted, eliminating the defined Brillouin zone boundaries and enabling long-range interactions. This interaction between reciprocal lattice points, generated through linear combinations of reciprocal lattice bases, allows for coupling between low and high-order reciprocal lattice momenta, resulting in band structures not seen in traditional lattices. The structures created by this rotation and stacking can be categorized into Moiré lattices or quasilattices. In crystallography, Moiré lattices require precise control over the twist angle to maintain both translational and rotational symmetry, enabling the formation of a Moiré Brillouin zone. Quasilattices, in contrast, only require rotational symmetry without translational symmetry, allowing the crystal’s band structure to be tuned according to the degree of rotational misalignment. This study focuses on how the twist angle affects the band structure. In addition to preparing Moiré and quasilattices, this thesis also attempts to model and explain their band evolution. For simplicity, the discussion is limited to two dimensions using surface plasmons (SPs), which are near-field waves propagating along surfaces. This approach simplifies both the manufacturing process and the complexity of model construction. Experimentally, we used focused ion beam milling to fabricate two-dimensional plasmonic crystals on atomically flat epitaxial silver films, serving as a platform to explore the band evolution in Moiré and quasilattices, with a focus on adjusting the twist angle. Initially, the thesis introduces the basic definitions of lattices and differentiates between Moiré lattices and quasilattices, as well as their applications in photonics, such as flat bands, magic angle lasers, and bandgap control. The primary propagation medium of this platform, surface plasmons, is categorized into surface plasmon polaritons (SPPs) and localized surface plasmons (LSPs). Subsequently, we discuss periodically arranged nanostructures, namely plasmonic crystals, detailing the dispersion curves of hexagonal lattices to facilitate handling the more complex band evolution of Moiré and quasilattices. Experimentally, angle-resolved microscopy was employed to measure the angle-resolved reflection spectra of Moiré and quasilattices, observing the dispersion curves of surface plasmon polaritons to elucidate the relationship between band structure and angular variation. Our theoretical modeling and experimental results correspond well, demonstrating the applicability of our approach not only in the realm of surface plasmons but also in offering new insights for analyzing the band structures of complex crystals such as superlattices, Moiré lattices, and quasilattices.
Keyword: Surface Plasmon Polariton, Surface Plasmon Polariton, Moiré Lattice, Quasicrystal

一、 摘要 I
二、 Abstract III
三、 目錄 V
四、 圖目錄 VII
一、 簡介 1
1.1研究動機 1
1.2晶格定義 5
1.2.1布拉菲晶格與倒晶格 5
1.2.2 摩爾晶格 7
1.3表面電漿子(Surface plasmons, SPs) 12
1.3.1表面電漿極化子(Surface plasmon polaritons, SPPs)11 12
1.3.2局域性表面電漿子(Localized Surface Plasmons, LSPs)11 15
1.4電漿子晶體(Plasmonic Crystals)與色散曲線 17
二、 儀器介紹與原理 21
2.1樣品製造 21
2.1.2熱蒸鍍機(Thermal evaporator) 21
2.1.2聚焦離子束蝕刻(Focused ion beam, FIB) 22
2.1.3原子層沉積(Atomic layer deposition, ALD) 23
2.2光學量測 24
2.2.1 傅立葉光學介紹 24
2.2.2角解析反射光譜(Angle-resolved reflectivity spectrum) 25
2.2.3 能量分布 26
三、 實驗流程與分析 27
3.1 樣品製備與光學量測 27
3.2 理論模擬 32
3.3 結果分析 38
四、 結論 43
五、 參考資料 44

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