在早期，關於量子霍爾效應的研究與討論主要是集中於電子氣體之傳導。而近幾年來研究發現光子晶體與聲子晶體這類的週期結構，藉由特殊方法可以在其界面處形成類似邊緣電流傳導現象，使得該方面之研究更成為最近學術界的熱門話題。
大多造成討論板上突出物之空間破壞方式，常用手段為改變突出物高度，而改變突出物高度即為改變重量在空間之分布，因此本文不僅討論改變柱高之手段亦討論了改變半徑之手段形成空間對稱性之破壞。本文以有限元素法求得二維週期突出物聲子晶體的能帶結構，証實了突出物高度對板波模態的影響，並且於某一高度該能帶結構可形成狄拉克錐。再來，利用圓柱幾何或材料方式造成空間反演對稱性之破壞，將其能帶結構中之狄拉克點破壞，形成一完全能隙。本文中另外進行了一般聲子晶體能帶結構之討論，探討各項幾何參數對於本文所提出模型之能帶結構以及藉由空間對稱性破壞所形成之完全能隙影響。並且演示該破壞空間對稱性之方法將引起能谷旋渦和拓樸相位反轉現象，產生拓樸相變的發生，進而將狄拉克點破壞形成兩拓樸不等價之能谷。該相變結果證實了藉由兩不等價之晶格所構成之界面存在著具拓樸保護之能谷邊緣模態。
在本文中討論了3種具拓樸保護之界面形式，並且透過穿透頻譜證明了該界面中之波傳不受路徑中之點缺陷以及尖銳角處的後向散射之影響。本文中在扶手椅界面中，我們成功在固體力學中證明類似於聲學中沒有無間隙的邊緣模態現象發生。利用各種不同界面之間的轉換和傳播特性，我們將有機會設計出高穿透且可忽略點缺陷和抑制的後向散射的聲學元件。

An elastic analog of quantum valley Hall topological insulator is introduced and analyzed. We investigate a phononic crystal with inversion symmetry breaking that consists of a honeycomb lattice of stubs deposited on the thin plate. It demonstrates that the topological properties can be adjusting by controlled the relative radius of adjacent stubs. By utilizing two kinds of distinct topological properties, we can obtain two types of domain walls. The results of numerical simulation show the robustness of topological-protected valley edge transports at the sharp corners and the point defect in the zigzag domain. Our works offer a prospect to design low-consumption acoustic components with topological edge state in the plate.

[1] Nobelprize.org（2016）。The Nobel Prize in Physics 2016- Scientific background: Topological phase transitions and topological phases of matter。民107年6月20日，取自：https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/adv
anced-physicsprize2016.pdf
[2] E. Yablonovitch,"Inhibited spontaneous emission in solid-state physics and electronics", Phys Rev Lett, Vol. 58, No. 20, pp. 2059-2062. (1987).
[3] S. John,"Strong localization of photons in certain disordered dielectric superlattices", Phys Rev Lett, Vol. 58, No. 23, pp. 2486-2489. (1987).
[4] E. Yablonovitch and T. J. Gmitter,"Photonic band structure: The face-centered-cubic case", Phys Rev Lett, Vol. 63, No. 18, pp. 1950-1953. (1989).
[5] L. Brillouin, “Wave propagation in periodic structures: electric filters and crystal lattices”, 2nd ed. Dover, New York (1953).
[6] M. Sigalas and E. Economou,"Elastic and acoustic wave band structure", Journal of sound and vibration, Vol. 158, No. 2, pp. 377-382. (1992).
[7] M. S. Kushwaha, P. Halevi, L. Dobrzynski and B. Djafari-Rouhani,"Acoustic band structure of periodic elastic composites", Phys Rev Lett, Vol. 71, No. 13, pp. 2022-2025. (1993).
[8] 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, No. 4, pp. 2313-2322.(1994).
[9] M. S. 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. 9, pp.977-1094.(1996).
[11] R. Martinezsala, J. Sancho, J. V. Sanchez, V. Gomez, J. Llinares and F. Meseguer,"SOUND-ATTENUATION BY SCULPTURE", Nature, Vol. 378, No. 6554, pp. 241-241. (1995).
[12] A. Bousfia, E. H. El Boudouti, B. Djafari-Rouhani, D. Bria, A. Nougaoui and V. R. Velasco,"Omnidirectional phononic reflection and selective transmission in one-dimensional acoustic layered structures", Surface Science, Vol. 482-485, No., pp. 1175-1180. (2001).
[13] 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).
[14] M. S. Kushwaha,"Stop-bands for periodic metallic rods: Sculptures that can filter the noise", Applied Physics Letters, Vol. 70, No. 24, pp. 3218-3220. (1997).
[15] Z. Liu, C. T. Chan and P. Sheng,"Three-component elastic wave band-gap material", Physical Review B, Vol. 65, No. 16, p. 165116. (2002).
[16] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. Chan and P. Sheng,"Locally resonant sonic materials", Science, Vol. 289, No. 5485, pp. 1734-1736. (2000).
[17] 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, Vol. 42, No. 4, pp. 477-482. (2005).
[18] 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).
[19] 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. (2008).
[20] 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).
[21] M. Ruzzene and A. M. Baz, in SPIE's 7th Annual International Symposium on Smart Structures and Materials. (International Society for Optics and Photonics, (2000)), pp. 389-407.
[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, No. 7, pp. 1065-1081. (2000).
[23] K. Bertoldi and M. C. Boyce,"Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures", Physical Review B, Vol. 77, No. 5, (2008).
[24] E. L. Thomas, T. Gorishnyy and M. Maldovan,"Phononics: Colloidal crystals go hypersonic", Nat. Mater., Vol. 5, No. 10, pp. 773-774. (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, No. 11, p. 114907. (2005).
[26] M. Sigalas,"Elastic wave band gaps and defect states in two-dimensional composites", The Journal of the Acoustical Society of America, Vol. 101, No. 3, 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, No. 3, pp. 198-202. (2001).
[28] T. Miyashita, in Ultrasonics Symposium, 2004 IEEE. (IEEE, (2004)), vol. 2, pp. 946-949.
[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, No. 1-2, pp. 67-71. (2004).
[30] H. Lamb, “On waves in an elastic plate”, Proceedings of the Royal Society of London. Series A, Vol. 93, No.648, pp. 114-128. (1917)
[31] Viktrov, “Rayleigh and Lamb waves: physical theory and applications”, Plenum Press, New York (1967)
[32] Achenbach, “Wave propagation in elastic solids”, 1st ed. Elsevier, New York (2012)
[33] 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)
[34] C. Hsu and T. T. Wu, “Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates”, Physical review B, Vol. 74, No.14, p. 144303. (2006)
[35] F. L Hsiao, A. Khelif, H. Moubchir, A. Choujaa, C.C. Chen and V. Laude, “Waveguiding inside the complete band gap of a phononic crystal slab”, Physical Review E, Vol. 76, No.5, p. 056601. (2007)
[36] J. Vasseur, P. A. Deymier, B. Chenni, B. Djafari-Rouhani, L. Dobrzynski and D. Prevost, “Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals”, Physical Review Letters, Vol. 86, No.14, p. 3012-3015. (2001)
[37] Y. Pennec, B. Djafari-Rouhani, H. Larabi, J. O. Vasseur,and A. C. Hladky-Hennion, “Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate”,Physical Review B,Vol.78,104105.(2008).
[38] Bonello, B., C. Charles and F. Ganot, “Lamb waves in plates covered by a two-dimensional phononic film”, Applied physics letters, Vol. 90, No.2, p. 021909. (2007)
[39] Y. Li, T. Chen, X. Wang, Y. Xi,and Q. Liang, “Enlargement of locally resonant sonic band gap by using composite plate-type acoustic metamaterial”, Physics Letters A,Vol. 379, No. 5, pp. 412-416. (2015).
[40] T. T. Wu, Z. G. Huang, T. C. Tsai,and T. C. Wu,“Evidence of completeband gap and resonances in a plate with periodic stubbed surface”, Applied Physics Letters,Vol. 93, No. 11, 111902. (2008).
[41] Yu, T. Chen,and X. Wang,“Band gaps in the low-frequency range based on the two-dimensional phononic crystal plates composed of rubber matrix with periodic steel stubs”,Physica B, Vol. 416,pp. 12-16. (2013).
[42] P. Jiang, “Low-frequency band gap and defect state characteristics in a multi-stub phononic crystal plate with slit structure”,Journal of Applied Physics, Vol. 121, No. 1, 015106. (2017).
[43] J. V. Jos, “40 years of Berezinskii-Kosterlitz-Thouless theory”, 1st ed. World Scientific, Singapore (2013)
[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, No.7065, pp. 201-204. (2005)
[45] X. Yin, Z. Ye, J. Rho, Y. Wang and X. Zhang, “Photonic spin Hall effect at metasurfaces”, Science, Vol. 339, No.6126, pp. 1405-1407. (2013)
[46] S. Raghu and F. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals”, Physical Review A, Vol. 78, No.3, p. 033834. (2008)
[47] P. Wang, L. Lu and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves”, Physical review letters, Vol. 115, No.10, p. 104302. (2015)
[48] R. K. Pal, M. Schaeffer and M. Ruzzene, “Helical edge states and topological phase transitions in phononic systems using bi-layered lattices”, Journal of Applied Physics, Vol. 119, No.8, p. 084305. (2016)
[49] Y. Liu, Y. Xu, S. C. Zhang and W. Duan, “Model for topological phononics and phonon diode”, Physical Review B, Vol. 96, No.6, p. 064106. (2017)
[50] 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, No.4, pp. 369-374. (2017)
[51] J. Lu, C. Qiu, M. Ke and Z. Liu, “Valley vortex states in sonic crystals”, Physical review letters, Vol. 116, No.9, p. 093901. (2016)
[52] 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, No.9, p. 094106. (2017)
[53] 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)
[54] X. Ni, M. A. Gorlach, A. Alu and A. B. Khanikaev, “Topological edge states in acoustic Kagome lattices”, New Journal of Physics, Vol. 19, No.5, p. 055002. (2017)
[55] J.-J. Chen, B. Bonello and Z.-L. Hou, “Plate-mode waves in phononic crystal thin slabs: Mode conversion”, Physical Review E, Vol. 78, No.3, p. 036609. (2008)
[56] S. H. Mousavi, A. B. Khanikaev and Z. Wang, “Topologically protected elastic waves in phononic metamaterials”, Nature communications, Vol. 6, p. 8682. (2015)
[57] S.Y. Huo, J. J. Chen, H. B. Huang and G. L. Huang, “Simultaneous multi-band valley-protected topological edge states of shear vertical wave in two-dimensional phononic crystals with veins”, Scientific reports, Vol. 7, No.1, p. 10335. (2017)
[58] S.Y. Huo, J. J. Chen, H. B. Huang, “Topologically protected edge states for out-of-plane and in-plane bulk elastic waves”, Journal of Physics: Condensed Matter, Vol. 30, No.14, p. 145403. (2018)
[59] 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)
[60] S.-Y. Yu, C. He, Z. Wang, F.-K. Liu, X.-C. Sun, Z. Li, M.-H. Lu, X.-P. Liu and Y.-F. Chen, “A Monolithic Topologically Protected Phononic Circuit”, arXiv preprint arXiv:1707.04901, pp. 1-17 (2017)
[61] 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)
[62] 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)
[63] 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)
[64] R. Skomski, P. A. Dowben, M. S. Driver and J. A. Kelber, “Sublattice-induced symmetry breaking and band-gap formation in graphene”, Materials Horizons, Vol. 1, No.6, pp. 563-571. (2014)
[65] T.-T. Wu and Z.-G. Huang, “Level repulsions of bulk acoustic waves in composite materials”, Physical Review B, Vol. 70, No.21, p. 214304. (2004)
[66] B. Morvan, A. C. Hladky-Hennion, D. Leduc and J. L. Izbicki, “Ultrasonic guided waves on a periodical grating: Coupled modes in the first Brillouin zone”, Journal of applied physics, Vol. 101, No.11, p. 114906. (2007)
[67] P. Lima, A. Fazzio and A. J. da Silva, “Interfaces between buckling phases in silicene: Ab initio density functional theory calculations”, Physical Review B, Vol. 88, No.23, p. 235413. (2013)
[68] J. Mei, Y. Wu, C. T. Chan and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals”, Physical Review B, Vol. 86, No.3, p. 035141. (2012)
[69] Nobelprize.org（2016）。The Nobel Prize in Physics 2016- Press Release。民107年6月20日，取自：https://www.nobelprize.org/nobel_prizes/physics/laureates
/2016/press.html
[70] H. Zheng and S. Ravaine, “Bottom-up assembly and applications of photonic materials”, Crystals, Vol. 6, No.5, pp. 54-78. (2016)
[71] 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