張之威博士等人發表了「超材料的光學墨比爾斯對稱」 (Phys. Rev. Lett. 105, 235501 (2010))，其中提到將兩個隙環共振器(SRR) 合併在一起，導入一電磁波，其磁場會出現對稱態 (磁場方向相同) 與反對稱態 (磁場方向相反) 兩種態，比較兩者的能量大小，定義出正與負的交互作用，使得三個隙環共振器合併時，會有不同本徵頻率。此文將探討以緊密結合理論 (Tight Binding) 計算二維晶格排列的振子頻譜。在此我們探討了二維四角晶格、三角晶格、蜂巢狀結構晶格。我們發現，在考慮四方晶格及蜂巢狀結構在只有最近鄰交互作用下，隨著最近鄰的交互作用正負號的改變，其相同本徵態所對應的頻帶會互相對調。在蜂巢狀晶格中，考慮最近鄰有交互作用外，另外選定只有半數次近鄰有正的交互作用，則頻譜結構出現禁絕帶。在上述情形下，我們可以在兩種具有相反偶合的材料介面上產生邊緣態現象。

Dr. Chih-Wei Chang et al. published “Optical Mobius Symmetry in Metamaterials” at PRL (Phys. Rev. Lett. 105, 235501 (2010)). In the report putting two SRRs together and incident a light will have symmetry mode (same direction magnetic field) and anti-symmetry mode (opposite direction magnetic field). Comparing with each other’s energy will derive the positive and negative interaction. It will lead that when putting three SRRs together, the eigenvalue will different between positive or negative interaction. Therefore, this article is going to study 2D crystal resonators, which using tight-binding method to calculate the frequency spectrum (frequency and K space) to observe the property of band structure. In this article, we measured the 2D tetragonal lattice, triangular lattice and honeycomb lattice. We measure that when the interactions, the tetragonal and honeycomb lattice’s nearest-neighbor’s, is changing sign. The band with same eigenstate will switch each other. There will be completed forbidden gap when honeycomb lattice with positive interaction at A-site, and positive or negative interaction between nearest-neighbor resonators. Following the above case, we can create an edge state along the interface of two materials with opposite nearest-neighbor coupling.

論文審定書 i
誌謝 ii
中文摘要 iii
Abstract iv
第一章 緒論 1
1-1 SRR的正負交互作用 1
1-2 Tight Binding 2
1-3 Lagrange equation 3
第二章 人造物質中的莫比爾斯對稱 4
第三章 振子模型與Tight Binding Method 7
3-1 四方晶格最近鄰有交互作用 7
3-2 四方晶格一對角線次近鄰有交互作用 8
3-3 四方晶格所有次近鄰有交互作用 9
3-4 三角晶格最近鄰有交互作用 10
3-5 三角晶格次近鄰有交互作用 12
3-6 蜂巢結構最近鄰有交互作用 13
3-7 蜂巢結構A-site次近鄰有交互作用 14
3-8 蜂巢結構所有次近鄰有交互作用 15
第四章 結果與討論 17
4-1 四方晶格結果與分析 17
4-2 三角晶格結果與分析 26
4-3 蜂巢狀晶格結果與分析 29
第五章 結論 36
參考文獻 37

[1] Chi-Wei Chang, Ming Liu, Sunghyun Nam, Shuang Zhang, Yongmin Liu, Guy Bartal, Xiang Zhang, “Opticcal Mobius Symmetry in Metamaterials”, Phys. Rev. Lett. 105, 235501 (2010)
[2] Kittel, “Introduction to Solid State Physics 8th Edition”, Wiley, 2004, New York
[3] A. B. Movchan, S. Guenneau, “Split-ring resonators and localized modes”, Phys. Rev. B 70, 125116 (2004)
[4] Ruey-Lin Chern, “Magnetic and Surface Plasmon Resonances for Periodic Lattices of Plasmonic Split-ring Resonators”, Phys. Rev. B 78, 085116 (2008)
[5] Ruey-Lin Chern, “Large Magnetic Resonance Band Gaps for Split Ring Structures with High Internal Fractions”, OSA 16, 20186 (2008)
[6] John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, Robert D. Meade, “Photonic Crystals - Molding the Flow of Lights 2Ed”, Princeton Press 2008, New Jersey
[7] Shou-cheng Zhang, “Viewpoint: Topological states of quantum matter”, Physics 1, 6 (2008)