我們所認知的材料通常被分為導體或者是絕緣體，但有一種材料其本身是絕緣體但是其介面卻可以予許電流通過，此一材料即為拓樸絕緣體，近年來學者已經將拓樸絕緣體的概念引入週期排列結構的光子和聲子晶體，其能夠免疫缺陷、抑制後向散射的能力大大的提升人們對信號傳輸的能力。
本文利用雪花狀聲子晶體，來實現聲學量子能谷霍爾效應，首先建立聲子晶體的模型，並利用有限元素法計算其能帶結構，藉由改變散射柱的幾何去探討對狄拉克點頻率之影響，接下來進一步打破其空間對稱性，可以得到贋自旋方向不同的兩種拓樸不等價結構，接著利用石墨烯切割的兩種介面並以超晶胞法分析其邊體關係圖，可以在圖上找到相當乾淨的邊緣模態，並且可以觀察其能量集中的效果。最後再透過全波模擬驗證其效果，我們引入完美直線、無序、空腔和Z字形介面，透過不同模型去驗證受拓樸保護之邊緣模態免疫缺陷和抑制後向散射的能力。
本研究利用兩種不等價的介面，實現聲學能谷邊緣態的控制，利用其內部不受無序、缺陷等結構的影響，可以實現高集中、高穿透率的聲學傳播，有望可以設計分波器、濾波器等有潛力的聲學裝置。

The discoveries of topology insulators have extended to classical wave systems such as elastic waves and sound waves. In this thesis, we propose snowflake-like sonic crystals in order to realize the acoustic quantum valley Hall effect. The results show that snowflake-like sonic crystals with high symmetry will induce the Dirac cone at the K or K^' point. By breaking the spatial inversion symmetry, we can get two topologically distinct structures. Besides, we also demonstrate the vortex such as right-hand circularly polarized (RCP) or left-hand circularly polarized (LCP) exist in sonic crystals, which show a new degree of freedom in acoustic systems. Here we use two opposite pseudospins to create topology protected edge mode. Finally, in order to prove that edge modes can against defects and immune to backscattering, we deliberately introduce the perfect straight line, disorder, cavity, and z-shaped interfaces to investigate its influences by full wave simulation. Through two topologically distinct interfaces, we realize the control of acoustic valley edge states. By its intriguing characteristic, we can achieve high-concentration, high-transmission acoustic propagation, and it is expected to design potential acoustic devices such as signal splitters and filters.

[1] E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Physical review letters, vol. 58, no. 20, p. 2059, 1987.
[2] S. John, "Strong localization of photons in certain disordered dielectric superlattices," Physical review letters, vol. 58, no. 23, p. 2486, 1987.
[3] L. Brillouin, “Wave propagation in periodic structures,” 2nd ed. Dover, New York (1953)
[4] L. Cremer and H. Leilich, "Zur theorie der biegekettenleiter," On Theory of Flexural Periodic Systems), Arch. Elektr. Uebertrag, vol. 7, pp. 261-270, 1953.
[5] E. Yablonovitch, T. Gmitter, and K. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Physical review letters, vol. 67, no. 17, p. 2295, 1991.
[6] M. M. Sigalas and E. N. Economou, "Elastic and acoustic wave band structure," Journal of Sound Vibration, vol. 158, pp. 377-382, 1992.
[7] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, "Acoustic band structure of periodic elastic composites," Physical review letters, vol. 71, no. 13, p. 2022, 1993.
[8] M. S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, and B. Djafari-Rouhani, "Theory of acoustic band structure of periodic elastic composites," Physical Review B, vol. 49, no. 4, p. 2313, 1994.
[9] M. Kushwaha and P. Halevi, "Band‐gap engineering in periodic elastic composites," Applied Physics Letters, vol. 64, no. 9, pp. 1085-1087, 1994.
[10] M. S. Kushwaha, "Classical band structure of periodic elastic composites," International Journal of Modern Physics B, vol. 10, no. 09, pp. 977-1094, 1996.
[11] M. Plihal, A. Shambrook, A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Optics communications, vol. 80, no. 3-4, pp. 199-204, 1991.
[12] M. Plihal and A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Physical Review B, vol. 44, no. 16, p. 8565, 1991.
[13] X. Wang, X.-G. Zhang, Q. Yu, and B. Harmon, "Multiple-scattering theory for electromagnetic waves," Physical Review B, vol. 47, no. 8, p. 4161, 1993.
[14] P. Bell, J. Pendry, L. M. Moreno, and A. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Computer physics communications, vol. 85, no. 2, pp. 306-322, 1995.
[15] A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method. Artech house, 2005.
[16] A. Bousfia, E. El Boudouti, B. Djafari-Rouhani, D. Bria, A. Nougaoui, and V. Velasco, "Omnidirectional phononic reflection and selective transmission in one-dimensional acoustic layered structures," Surface science, vol. 482, pp. 1175-1180, 2001.
[17] B. Manzanares-Martínez, J. Sánchez-Dehesa, A. Håkansson, F. Cervera, and F. Ramos-Mendieta, "Experimental evidence of omnidirectional elastic bandgap in finite one-dimensional phononic systems," Applied physics letters, vol. 85, no. 1, pp. 154-156, 2004.
[18] M. Ruzzene and A. Baz, "Control of wave propagation in periodic composite rods using shape memory inserts," Journal of Vibration and Acoustics, vol. 122, no. 2, pp. 151-159, 2000.
[19] T. Chen, M. Ruzzene, and A. Baz, "Control of wave propagation in composite rods using shape memory inserts: theory and experiments," Journal of Vibration and control, vol. 6, no. 7, pp. 1065-1081, 2000.
[20] J.-Y. Yeh, "Control analysis of the tunable phononic crystal with electrorheological material," Physica B: Condensed Matter, vol. 400, no. 1-2, pp. 137-144, 2007.
[21] W.-P. Yang and L.-W. Chen, "The tunable acoustic band gaps of two-dimensional phononic crystals with a dielectric elastomer cylindrical actuator," Smart materials and structures, vol. 17, no. 1, p. 015011, 2007.
[22] R. Martínez-Sala, J. Sancho, J. V. Sánchez, V. Gómez, J. Llinares, and F. Meseguer, "Sound attenuation by sculpture," nature, vol. 378, no. 6554, p. 241, 1995.
[23] E. L. Thomas, T. Gorishnyy, and M. Maldovan, "Phononics: Colloidal crystals go hypersonic," Nature materials, vol. 5, no. 10, p. 773, 2006.
[24] J. Wen, G. Wang, D. Yu, H. Zhao, and Y. Liu, "Theoretical and experimental investigation of flexural wave propagation in straight beams with periodic structures: Application to a vibration isolation structure," Journal of Applied Physics, vol. 97, no. 11, p. 114907, 2005.
[25] Z.-G. Huang, "Silicon-based filters, resonators and acoustic channels with phononic crystal structures," Journal of Physics D: Applied Physics, vol. 44, no. 24, p. 245406, 2011.
[26] F. Wu, Z. Hou, Z. Liu, and Y. Liu, "Point defect states in two-dimensional phononic crystals," Physics Letters A, vol. 292, no. 3, pp. 198-202, 2001.
[27] T. Miyashita, "Experimentally study of a sharp bending wave-guide constructed in a sonic-crystal slab of an array of short aluminum rods in air," in IEEE Ultrasonics Symposium, 2004, 2004, vol. 2: IEEE, pp. 946-949.
[28] X. Zhang, Z. Liu, Y. Liu, and F. Wu, "Defect states in 2D acoustic band-gap materials with bend-shaped linear defects," Solid state communications, vol. 130, no. 1-2, pp. 67-71, 2004.
[29] H.-C. Chan and G.-Y. Guo, "Tuning topological phase transitions in hexagonal photonic lattices made of triangular rods," Physical Review B, vol. 97, no. 4, p. 045422, 2018.
[30] X. Wu et al., "Direct observation of valley-polarized topological edge states in designer surface plasmon crystals," Nature communications, vol. 8, no. 1, p. 1304, 2017.
[31] J. Lu et al., "Observation of topological valley transport of sound in sonic crystals," Nature Physics, vol. 13, no. 4, p. 369, 2017.
[32] Z. Zhang, Y. Tian, Y. Cheng, Q. Wei, X. Liu, and J. Christensen, "Topological acoustic delay line," Physical Review Applied, vol. 9, no. 3, p. 034032, 2018.
[33] Y. Yang, Z. Yang, and B. Zhang, "Acoustic valley edge states in a graphene-like resonator system," Journal of Applied Physics, vol. 123, no. 9, p. 091713, 2018.
[34] C. Brendel, V. Peano, O. Painter, and F. Marquardt, "Snowflake phononic topological insulator at the nanoscale," Physical Review B, vol. 97, no. 2, p. 020102, 2018.
[35] Y. Li and J. Mei, "Double Dirac cones in two-dimensional dielectric photonic crystals," Optics express, vol. 23, no. 9, pp. 12089-12099, 2015.
[36] H. Dai, B. Xia, and D. Yu, "Dirac cones in two-dimensional acoustic metamaterials," Journal of Applied Physics, vol. 122, no. 6, p. 065103, 2017.
[37] H. Dai, M. Qian, J. Jiao, B. Xia, and D. Yu, "Subwavelength acoustic topological edge states realized by zone folding and the role of boundaries selection," Journal of Applied Physics, vol. 124, no. 17, p. 175107, 2018.
[38] S. H. Mousavi, A. B. Khanikaev, and Z. Wang, "Topologically protected elastic waves in phononic metamaterials," Nature communications, vol. 6, p. 8682, 2015.
[39] S.-Y. Yu et al., "A monolithic topologically protected phononic circuit," arXiv preprint arXiv:1707.04901, 2017.
[40] J. W. Strutt and B. Rayleigh, The theory of sound. Dover, 1945.
[41] D. Torrent, D. Mayou, and J. Sánchez-Dehesa, "Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates," Physical Review B, vol. 87, no. 11, p. 115143, 2013.
[42] R. K. Pal and M. Ruzzene, "Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect," New Journal of Physics, vol. 19, no. 2, p. 025001, 2017.
[43] T. Kariyado and Y. Hatsugai, "Manipulation of dirac cones in mechanical graphene," Scientific reports, vol. 5, p. 18107, 2015.
[44] Y.-T. Wang and S. Zhang, "Elastic spin-Hall effect in mechanical graphene," New Journal of Physics, vol. 18, no. 11, p. 113014, 2016.
[45] 維基百科編者. (2019-05-08UTC15:28:50+00:00 （UTC）). 莫比烏斯帶.
[46] The 2016 Nobel Prize in Physics-Press Release.” from
http://www.nobelprize.org/nobel_prizes/physics/laureates/ 016/press.html.
[47] 維基百科編者. (2016-10-15UTC16:38:17+00:00 （UTC）). 拓撲絕緣體.
[48] 知乎（2014-06-07）。什麼是分數量子霍爾效應？-每日頭條。民107年6月20日，取自：https://kknews.cc/zh-mo/science/yn6r2k.html
[49] Y. Hatsugai, "Chern number and edge states in the integer quantum Hall effect," Physical review letters, vol. 71, no. 22, p. 3697, 1993.
[50] T. W. Liu and F. Semperlotti, “Tunable Acoustic Valley–Hall Edge States in Reconfigurable Phononic Elastic Waveguides”, Physical Review Applied, Vol. 9, No.1, p. 014001. (2018)
[51] J.-J Chen, S.-Y. Huo, Z.-G. Geng, H.-B. Huang and X.-F. Zhu, “Topological valley transport of plate-mode waves in a homogenous thin plate with periodic stubbed surface”, AIP Advances, Vol. 7, No.11, p. 115215. (2017)
[52] J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, "Dirac cones in two-dimensional artificial crystals for classical waves," Physical Review B, vol. 89, no. 13, p. 134302, 2014.