拓樸絕緣體由兩種拓樸不等價結構組成，可形成邊緣模態並將聲波侷限在拓樸介面上穩定傳輸，本文使用聲子晶體建構拓樸絕緣體，利用邊緣模態設計拓樸聲子晶體共振腔，使得聲波圍繞腔體傳播形成環形共振。
將具矩形散射柱的聲子晶體作正方晶格排列，使用有限元素軟體計算其能帶結構。改變散射柱旋轉角拉開雙狄拉克點得到拓樸不等價結構，以超晶胞法分析由兩拓樸不等價結構組成之介面，並於邊體關係圖中尋找能量集中於介面處之邊緣模態，接著透過全波模擬在結構中激發於邊緣模態頻率範圍之聲波，以直線與彎角的路徑驗證邊緣模態的魯棒性。
設計由拓樸不等價結構組成之拓樸聲子晶體共振腔，得到環形共振模態，計算品質因子與聲壓，討論不同旋轉角度之散射柱構成的拓樸聲子晶體腔對於共振頻率的影響，探討當拓樸聲子晶體腔含有缺陷時和在缺陷處擺放待測物是否影響共振頻率以及品質因子。與傳統缺陷共振腔比較，拓樸聲子晶體腔的品質因子是前者的1.5倍至3倍，集中聲壓是前者的1倍到1.5倍。最後在拓樸聲子晶體腔下方引入波導，並類比量子自旋霍爾效應，探討自旋波源之單向波傳行為以及透過波導耦合的共振腔之品質因子。

We propose a phononic crystal with rectangular scattering pillars arranged in a square lattice. The double Dirac cone is formed by modulating rectangular scattering pillars’ parameters. The rotation angle of the scattering pillars is changed to open the double Dirac cone and the topologically distinct structures are obtained. A supercell made by placing topologically distinct lattices adjacently. The bulk-edge correspondence is analyzed by the supercell method. The straight and bend paths are created to verify the robust edge mode. Topological phononic crystal resonant cavities are presented, which are composed of the topologically distinct structures. The quality factors and sound pressure of the topological cavities are analyzed. The influence of the topological cavity, which consist of scattering columns with different rotation angles, is investigated. The topological cavity with defects affects the resonance frequency and quality factor is discussed. The advantages of the topological cavity are the extremely high quality factors and the concentrated sound pressure larger than the defect cavity. A system of cavity-waveguide coupling is researched. The spin-locked unidirectional propagation caused by spin source are discussed finally.

[1] D. J. Thouless, F. D. M. Haldane and J. M. Kosterlitz, "Scientific background: Topological phase transitions and topological phases of matter" The Nobel Prize in Physics, 2016
[2] E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics." Physical Review Letters, Vol.58, pp.2059-2062 (1987)
[3] S. John, "Strong localization of photons in certain disordered dielectric superlattices." Physical Review Letters, Vol.58, pp.2486-2489 (1987)
[4] H. Zheng, S. Ravaine, "Bottom-up assembly and applications of photonic materials." Crystals, Vol.6, 54 (2016)
[5] L, Brillouin, Wave propagation in periodic structures, 2nd ed. Dover, New York (1953)
[6] L. Cremer, H. O. Leilich, “Zur theorie der Biegekettenleiter (On theory of flexural periodic systms),” Archiv der Elektrischen Ubertragung, Vol.7, pp.261-270 (1953)
[7] M. A. Heckl, "Investigations on the vibrations of grillages and other simple beam structures." The Journal of the Acoustical Society of America,Vol.36, pp.1335-1343 (1964)
[8] M. S. Kushwaha, P. Halevi, L. Dobrzynski and B. Djafari-Rouhani, "Acoustic band structure of periodic elastic composites." Physical Review Letters, Vol.71, pp2022-2025 (1993)
[9] M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski and B. Djafari-Rouhani, "Theory of acoustic band structure of periodic elastic composites." Physical Review B ,Vol.49, pp.2313-2322 (1994)
[10] M. S. Kushwaha, P. Halevi, "Band‐gap engineering in periodic elastic composites." Applied Physics Letters, Vol.64, pp.1085-1087 (1994)
[11] M. S. Kushwaha, "Classical band structure of periodic elastic composites." International Journal of Modern Physics B, Vol.10, pp.977-1094(1996)
[12] R. Martínez-Sala, J. Sancho, J. V. Sánchez, V. Gómez, J. Llinares and F. Meseguer, "Sound attenuation by sculpture." Nature, Vol.378, p.241 (1995)
[13] W. P. Yang, 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, 015011 (2007)
[14] A. Bousfia, E. H. El. Boudouti, B. Djafari-Rouhani, D. Bria, A. Nougaoui, V. R. Velasco "Omnidirectional phononic reflection and selective transmission in one-dimensional acoustic layered structures." Surface Science, Vol.482, pp.1175-1180 (2001)
[15] B. Manzanares-Martínez, J. Sánchez-Dehesa and A. Håkansson, "Experimental evidence of omnidirectional elastic bandgap in finite one-dimensional phononic systems." Applied Physics Letters, Vol.85, pp.154-156 (2004)
[16] M. S. Kushwaha, "Stop-bands for periodic metallic rods: Sculptures that can filter the noise." Applied Physics Letters, Vol.70, pp.3218-3220 (1997)
[17] Z. Liu, C. T. Chan and P. Sheng, "Three-component elastic wave band-gap material." Physical Review B, Vol.65, 165116 (2002)
[18] Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, P. Sheng, "Locally resonant sonic materials." Science, Vol.289, pp.1734-1736 (2000)
[19] X. Zhang, Z. Liu and Y. Liu, "The optimum elastic wave band gaps in three dimensional phononic crystals with local resonance." The European Physical Journal B-Condensed Matter and Complex Systems, Vol.42, pp.477-482 (2004)
[20] J. Y. Yeh, "Control analysis of the tunable phononic crystal with electrorheological material." Physica B: Condensed Matter, Vol.400, pp.137-144 (2007)
[21] M. Ruzzene, A. Baz, "Control of wave propagation in periodic composite rods using shape memory inserts." Journal of Vibration and Acoustics, Vol.122, pp.151-159 (2000)
[22] 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, pp.1065-1081 (2000)
[23] K. Bertoldi, M. C. Boyce, "Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures." Physical Review B, Vol.77, 052105 (2008)
[24] E. L. Thomas, T. Gorishnyy and M. Maldovan, "Colloidal crystals go hypersonic." Nature Materials, Vol.5, pp773-774 (2006)
[25] 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, 114907 (2005)
[26] M. M. Sigalas, "Elastic wave band gaps and defect states in two-dimensional composites." The Journal of the Acoustical Society of America, Vol.101, pp.1256-1261 (1997)
[27] F. Wu, Z. Hou, Z. Liu and Y. Liu, "Point defect states in two-dimensional phononic crystals." Physics Letters A, Vol.292, pp.198-202 (2001)
[28] 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." IEEE Ultrasonics Symposium, Vol.2, pp.946-949 (2004)
[29] 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, pp.67-71 (2004)
[30] X. Li, Z. Liu, "Coupling of cavity modes and guiding modes in two-dimensional phononic crystals." Solid State Communications, Vol.133, pp.397-402 (2005)
[31] X. Li, Z. Liu, "Bending and branching of acoustic waves in two-dimensional phononic crystals with linear defects." Physics Letters A, Vol.338, pp.413-419 (2005)
[32] L. Y. Wu, L. W. Chen and C. M. Liu, "Acoustic pressure in cavity of variously sized two-dimensional sonic crystals with various filling fractions." Physics Letters A, Vol. 373, pp.1189-1195 (2009)
[33] L. Y. Wu, L. W. Chen and C. M. Liu, "Experimental investigation of the acoustic pressure in cavity of a two-dimensional sonic crystal." Physica B: Condensed Matter, Vol.404, pp.1766-1770 (2009)
[34] L. Y. Wu, L. W. Chen and C. M. Liu, "Acoustic energy harvesting using resonant cavity of a sonic crystal." Applied Physics Letters, Vol.95, 013506 (2009)
[35] L. Y. Wu, L. W. Chen, "Wave propagation in a 2D sonic crystal with a Helmholtz resonant defect." Journal of Physics D: Applied Physics, Vol.43, 055401 (2010)
[36] J. K. Asbóth, L. Oroszlány and A. Pályi, "A short course on topological insulators." Lecture Notes in Physics, Vol.919 (2016)
[37] M. Z. Hasan, C. L. Kane, "Colloquium: topological insulators." Reviews of Modern Physics, Vol.82, pp.3045-3067 (2010)
[38] J. Jiao, T. Chen, H. Dai and D. Yu, "Observation of topological valley transport of elastic waves in bilayer phononic crystal slabs." Physics Letters A, Vol.383, 125988 (2019)
[39] B. Z. Xia, T. T. Liu, G. L. Huang, H. Q. Dai, J. R. Jiao, X. G. Zang, D. J. Yu, S. J. Zheng and J. Liu, "Topological phononic insulator with robust pseudospin-dependent transport." Physical Review B, Vol.96, 094106 (2017)
[40] Y. Yang, Z. H. Hang, "Topological whispering gallery modes in two-dimensional photonic crystal cavities." Optics Express, Vol.26, pp.21235-21241 (2018)
[41] Y. Gong, S. Wong, A. J. Bennett, D. L. Huffaker and S. S. Oh, "Topological insulator laser using valley-Hall photonic crystals." ACS Photonics, Vol.7, pp.2089-2097 (2020)
[42] Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li, Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, Y. Chong and Q. J. Wang, "Electrically pumped topological laser with valley edge modes." Nature, Vol.578, pp.246-250 (2020)
[43] K. V. Klitzing, G. Dorda and M. Pepper, "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance." Physical Review Letters, Vol.45, pp.494-497 (1980)
[44] Y. Zhang, Y. W. Tan, H. L. Stormer and P. Kim "Experimental observation of the quantum Hall effect and Berry's phase in graphene." Nature, Vol.438, pp.201-204 (2005)
[45] R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman and A. Alù, "Sound isolation and giant linear nonreciprocity in a compact acoustic circulator." Science, Vol.343, pp.516-519 (2014)
[46] X. Ni, C. He, X. C. Sun, X. P. Liu, M. H. Lu, L. Feng and Y. F. Chen, "Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow." New Journal of Physics, Vol.17, 053016 (2015)
[47] L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner and W. T. M. Irvine, "Topological mechanics of gyroscopic metamaterials." Proceedings of the National Academy of Sciences, Vol.112, pp.14495-14500 (2015)
[48] P. Wang, L. Lu and K. Bertoldi, "Topological phononic crystals with one-way elastic edge waves." Physical Review Letters, Vol.115, 104302 (2015)
[49] C. L. Kane, E. J. Mele, "Quantum spin Hall effect in graphene." Physical Review Letters, Vol.95, 226801 (2005)
[50] J. E. Hirsch, "Spin Hall effect." Physical Review Letters, Vol.83, pp1834-1837 (1999)
[51] S. Murakami, N. Nagaosa and S. C. Zhang, "Dissipationless quantum spin current at room temperature." Science, Vol301, pp.1348-1351 (2003)
[52] B. A. Bernevig, S. C. Zhang, "Quantum spin Hall effect." Physical Review Letters, Vol.96, 106802 (2006)
[53] B. A. Bernevig, T. L. Hughes and S. C. Zhang, "Quantum spin Hall effect and topological phase transition in HgTe quantum wells." Science, Vol.314, pp.1757-1761 (2006)
[54] W. Feng, C. C. Liu, G. B. Liu, J. J. Zhou and Y. Yao, "First-principles investigations on the berry phase effect in spin–orbit coupling materials." Computational Materials Science, Vol.112, pp.428-447 (2016)
[55] O. A. Pankratov, S. V. Pakhomov and B. A. Volkov, "Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe." Solid State Communications, Vol.61, pp93-96 (1987)
[56] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi and S. C. Zhang, "Quantum spin Hall insulator state in HgTe quantum wells." Science, Vol.318, pp.766-770 (2007)
[57] Y. Li, Y. Wu and J. Mei, "Double Dirac cones in phononic crystals." Applied Physics Letters, Vol.105, 014107 (2014)
[58] C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu and Y. F. Chen, "Acoustic topological insulator and robust one-way sound transport." Nature Physics, Vol.12, pp.1124-1129 (2016)
[59] L. H. Wu, X. Hu, "Scheme for achieving a topological photonic crystal by using dielectric material." Physical Review Letters, Vol.114, 223901 (2015)
[60] Y. Chen, F. Meng and X. Huang, "Creating acoustic topological insulators through topology optimization." Mechanical Systems and Signal Processing, Vol.146, 107054 (2021)
[61] J. Lu, C. Qiu, L. Ye, X. Fan, M. Ke, F. Zhang and Z. Liu, "Observation of topological valley transport of sound in sonic crystals." Nature Physics, Vol.13, pp.369-374 (2017)
[62] J. Lu, C. Qiu, M. Ke and Z. Liu, "Valley vortex states in sonic crystals." Physical Review Letters, Vol116, 093901 (2016)
[63] Y. Yang, Z. Yang and B. Zhang, "Acoustic valley edge states in a graphene-like resonator system." Journal of Applied Physics, Vol.123, 091713 (2018)
[64] X. Ni, M. A. Gorlach, A. Alu and A. B. Khanikaev, "Topological edge states in acoustic Kagome lattices." New Journal of Physics, Vol.19, 055002 (2017)
[65] 張泰榕，拓樸材料與拓樸能帶理論-台灣物理學會-物理雙月刊，2017
[66] Z. G. Chen, R. Y. Chen, R. D. Zhong, J. Schneeloch, C. Zhang, Y. Huang, F. Qu, R. Yu, Q. Li, G. D. Gu and N. L. Wang, "Spectroscopic evidence for bulk-band inversion and three-dimensional massive Dirac fermions in ZrTe5." Proceedings of the National Academy of Sciences, Vol.114, pp.816-821 (2017)
[67] N. Nagaosa, "A new state of quantum matter." Science, Vol.318, pp.758-759 (2007)
[68] C. L. Kane, "Topological band theory and the ℤ2 invariant." Contemporary Concepts of Condensed Matter Science, Vol.6, pp.3-34 (2013)