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在本文中, 构建了一种易于实现的二维声子晶体: 截面为正方形的铁柱以三角晶格形式排列在水中. 研究发现, 在此声子晶体的布里渊区中心点有半狄拉克点出现: 其带结构沿Y方向是线性的, 但沿着X方向却是二次型的. 若散射体绕中心轴旋转角度 = 45, 则半狄拉克点的二次型带结构则会转至Y方向, 与X相互垂直. 接着, 本文采用k p 微扰法系统研究了在不同旋转角 值下, 简并点附近的带结构特点, 并在此基础上分析了半狄拉克点的出现原因. 在半狄拉克点附近, 以布洛赫简并态为基矢, 文中构造了一个有效哈密顿量, 根据它能准确计算贝利相位, 并发现其值为零. 此外, 通过有限元仿真, 还研究了在半狄拉克点频率附近声波沿着不同方向穿过该声子晶体的透射现象. 本文可以为经典体系中半狄拉克点色散关系的起源、有关传播性质的研究以及其在声子晶体的应用提供理论参考.A two-dimensional phononic crystal (PC) composed of a triangular array of square iron cylinders embedded in water is designed, in which the accidental degeneracy of the Bloch eigenstates is utilized to realize a semi-Dirac point at the Brillouin zone center. In the vicinity of the semi-Dirac point, the dispersion relation is linear along the Y direction but quadratic along the X direction. Rotating the iron cylinders around their axis by 45 and slightly tuning the side length of the cylinders, a new semi-Dirac point can be realized at the Brillouin zone center, where the dispersion relation is quadratic along the Y direction but linear along the X direction. To gain a deeper understanding of the semi-Dirac point, a k p perturbation method is used to investigate this peculiar dispersion relation and study how the semi-Dirac point is formed. The linear slopes of dispersion relations along any direction around the semi-Dirac point can be accurately predicted by the perturbation method, and the results agree very well with the rigorous band structure calculations. Furthermore, the mode-coupling integration between the degenerate Bloch eigenstates is zero in one direction but non-zero in the perpendicular direction, and this is the ultimate reason for the forming of a semi-Dirac point. With the help of the perturbation method, an effective Hamiltonian can be constructed around the semi-Dirac point, so that the Berry phase can be calculated, which is found to be zero. Actually, the different values of Berry phase indicate an important distinction between the semi-Dirac points and Dirac points. In addition, the acoustic wave transmission through the corresponding PC structure has been studied, and a switch-like behavior of the transmittance is observed along different directions. Along some particular direction, there exist deaf bands around the semi-Dirac point, and these bands cannot be excited by the externally incident plane waves due to the mismatch in mode symmetry. But the situation is different along the other direction, where the bands are active ones and therefore can be excited by the incident plane waves. Actually, such properties of the bands can be easily changed as long as the iron cylinders are rotated around their axis. The work described in this paper is helpful to the understanding of semi-Dirac point in phononic crystals and suggests possible applications in diverse fields.
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Keywords:
- semi-Dirac points /
- phononic crystals /
- effective Hamiltonian /
- Berry phase
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[1] Castro N A H, Guinea F, PeresN M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[2] Rowlands D A, Zhang Y Z 2014 Chin. Phys. B 23 37101
[3] Jung J, Raoux A, Qiao Z H, Mac-Donald A H 2014 Phys. Rev. B 89 205414
[4] Wang X X, Bian G, Wang P, Chiang T C 2015 Phys. Rev. B 91 125103
[5] Zhang Y P, Yin Y H, Lu H H, Zhang H Y 2014 Chin. Phys. B 23 027202
[6] Wen J, Guo H, Yan C H, Wang Z Y, Chang K, Deng P, Zhang T, Zhang Z D, Ji S H, Wang L L, He K, Ma X C, Chen X, Xue Q K 2014 Chin. Phys. Lett. 31 116802
[7] Li W F, Guo M, Zhang G, Zhang Y W 2014 Phys. Rev. B 89 205402
[8] Lin S Y, Chen M, Yang X B, Zhao Y J, Wu S C, Felser C, Ya B H 2015 Phys. Rev. B 91 094107
[9] Zhang D, Lin L Z, Zhu J J 2014 Chin. Phys. Lett. 31 028102
[10] Torrent A, Dehesa J S 2012 Phys. Rev. Lett. 108 174301
[11] Zhang X D, Liu Z Y 2008 Phys. Rev. Lett. 101 264303
[12] Lu J Y, Qiu C Y, Xu S J, Ye Y T, Ke M Z, Liu Z Y 2014 Phys. Rev. B 89 134302
[13] Chen Z G, Ni X, Wu Y, He C, Sun X C, Zheng L Y, Lu M H, Chen Y F 2014 Sci. Rep. 4 4613
[14] Sun L, Gao J, Yang X D 2013 Opt. Express 2121542
[15] Huang X Q, Lai Y, Hang Z H, Zheng H H, Chen C T 2011 Nature Materials 10 1038
[16] Sepkhanov R A, Bazaliy Y B, Beenakker C W J 2007 Phys. Rev. A 75 063813
[17] Mei J, Wu Y, Chan C T, Zhang Z Q 2012 Phys. Rev. B 86 035141
[18] Wu Y 2014 Opt. Express 22 001906
[19] Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 12066
[20] Wang X, Jiang H T, Yan C, Deng F S, Sun Y, Li Y H, Shi Y L, Chen H 2014 Europhys. Lett. 108 14002
[21] Deng F S, Sun Y, Wang X, Xue R, Li Y, Jiang H T, Shi Y L, Chang K, Chen H 2014 Opt. Express 22 23605
[22] Cao H X, Mei J 2014 Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition Montreal, Quebec, Canada November 14-20, 2014, 37422
[23] Xiao X B, Yang S Y A, Liu Z F, Li H L, Zhou G H 2015 Sci. Rep. 5 7898
[24] Yang D Z, Si M S, Zhang G P, Xue D X 2014 Europhys. Lett. 107 20003
[25] Zhou X F, Dong X, Oganov A R, Zhu Q, Tian Y J, Wang H T 2014 Phys. Rev. Lett. 112 085502
[26] Wang Q, Shen R, Sheng L, Wang B G, Xing D Y 2014 Phys. Rev. A 89 022121
[27] Feng Y, Wang Z J, Chen C Y, Shi Y G, Xie Z J, Yi H M, Liang A J, He S L, He J F, Peng Y Y, Liu X, Liu Y, Zhao L, Liu G D, Dong X L, Zhang J, Chen C T, Xu Z Y, Dai X, Fang Z, Zhou X J 2014 Sci. Rep. 4 5385
[28] Ortix C, Yang L P, Brink J V D 2012 Phys. Rev. B 86 081405
[29] Banerjee S, Singh R R P, Pardo V, Pickett W E 2009 Phys. Rev. Lett. 103 016402
[30] Zhai F, Wang J 2014 Appl. Phys. Lett. 116 063704
[31] Zhai F, Mu P Y, Chang K 2011 Phys. Rev. B 83 195402
[32] Cheng C, Wu F G, Zhang X, Yao Y W 2014 Acta Phys. Sin. 63 024301(in Chinese) [程聪, 吴福根, 张欣, 姚源卫 2014 63 024301]
[33] Hou L N, Hou Z L, Fu X J 2014 Acta Phys. Sin. 63 034305(in Chinese) [侯丽娜, 侯志林, 傅秀军 2014 63 034305]
[34] Zhang X J, Wu Y 2014 Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition Montreal, Quebec, Canada November 14-20, 2014, 37421
[35] Li Y, Wu Y, Chen C, Mei J 2013 Opt. Express 21 7699
[36] Berry M V 1984 Proc. R. Soc. A 392 45
[37] Kafesaki M, Economou E N 1999 Phys. Rev. B 60 11993
[38] Dresselhaus M S, Dresselhaus G, Jorio A 2008 Group Theory: Application to the Physics of Condensed Matter(Berlin Herdelberg: Springer-Verlag) pp209-235
[39] Wu Y, Li J, Zhang Z Q, Chan C T 2006 Phys. Rev. B 74 085111
[40] Sakurai J J 1994 Modern Quantum Mechanics (Boston: Addsion-Wesley, Reading, MA) pp465-480
[41] Sakoda K 2005 Optical Properties of Photonic crystals (Second Edition) (Berlin Herdelberg: Springer-Verlag) pp94-95
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