Topologically protected edge states of elastic waves in phononic crystal plates

Zheng Zhou-Fu; Yin Jian-Fei; Wen Ji-Hong; Yu Dian-Long

doi:10.7498/aps.69.20200542

Abstract

The topologically protected edge states of elastic waves in phononic crystal plates have the outstanding characteristics in wave manipulation such as the strong suppression of back-scattering and defect immunity, which can be used for controlling vibration and noise, detecting the structural damage, conducting the material nondestructive test and other engineering practices, and therefore have received much attention. But for plate structures, the propagation of elastic waves is complicated due to the coexistence and coupling of different types of wave modes, resulting in a challenge in designing topologically protected states. In this paper, a simple phononic crystal plate with triangular holes is designed for elastic wave manipulation based on topologically protected edge states. The band structure characteristics of the unit cell are studied by varying the rotation angle θ of the triangular holes around their geometric centers from the initial positions. It is found that the band structure of the initial unit cell with rotation angle θ = 0° has two pairs of degenerate modes. At $ \theta = \pm 33^\circ $, a double Dirac cone appears at the center Γ point of the Brillouin zone without requiring the lattices to fold, and a band inversion occurs on both sides of $ \pm 33^\circ $ which can be characterized as a topological phase transition. The elastic band gap and two kinds of pseudospin states with clockwise or counterclockwise circulating mechanical energy flux patterns in the band structure are found by calculating the projected band structures of a supercell which is composed of phononic crystals with different topological phases. Based on this finding, different constructions of phononic waveguide are used for implementing the numerical analysis to demonstrate the back-scattering immunity of the edge states when disorder, tortuosity and cavity are introduced into the waveguide. Unidirectional robust propagation and multichannel waveguide switch due to the pseudospin-dependent one-way edge modes are also validated with numerical models. The phononic crystal plate presented in this paper provides a simple realizable method of designing the topologically protected elastic edge states.

Keywords:

Authors and contacts

Laboratory of Science and Technology on Integrated Logistics Support, College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China

Corresponding author: Yin Jian-Fei, nmhsyjf@nudt.edu.cn ; Wen Ji-Hong, wenjihong@vip.sina.com ;

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References

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Cited By

图 1 声子晶体板结构、元胞及其布里渊区示意图　(a) 具有三角形穿孔的声子晶体板结构; (b) 声子晶体板的元胞; (c) 晶格的第一布里渊区和不可约布里渊区(红色区域)

Figure 1. Schematic diagram of phononic crystal plate structure, its unit cell and corresponding Brillouin zone: (a) The phononic crystal plate structure with triangular through-holes; (b) the unit cell of the phononic crystal plate; (c) the first Brillouin zone and irreducible Brillouin zone (red region) of the lattice.

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图 2 三角形穿孔声子晶体板在不同旋转角度下的能带结构与本征态z方向位移场分布　(a) $\theta =0^\circ $, ${p_ \pm }$模位于${d_ \pm }$模下方, 左侧插图为${P_z}=0$点所对应的水平剪切模振型, 右侧插图为${p_ \pm }$和${d_ \pm }$模态的位移场分布和振型, 能带结构中用不同颜色表示${P_z}$极化指标; (b) $\theta =33^\circ $, 偶然简并形成双狄拉克锥, 右侧插图为${p_ \pm }$和${d_ \pm }$模态的位移场分布; (c) $\theta =60^\circ $, ${p_ \pm }$模位于${d_ \pm }$模上方

Figure 2. Band structure and displacement field distribution (DFD) in z-direction at eigenstates of the phononic crystal plate with triangular holes with different rotation angle: (a) $\theta =0^\circ $, ${p_ \pm }$ modes are below ${d_ \pm }$ modes. The left DFD demonstrates mode shape of shear horizontal mode corresponding to the ${P_z}=0$ points, while the right group of DFDs illustrate the mode shapes of ${p_ \pm }$ and ${d_ \pm }$ modes. The color of the points on the dispersion curves corresponds to the ${P_z}$ polarization index; (b) $\theta =33^\circ $, a double Dirac cone is formed, and the right DFDs show eigenstates distributions of ${p_ \pm }$ and ${d_ \pm }$ modes; (c) $\theta =60^\circ $, ${p_ \pm }$ modes are above ${d_ \pm }$ modes.

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图 3 晶格参数变化对布里渊区中心Γ点的偶极模态和四极模态能带特征频率的影响　(a) 三角形穿孔旋转角度θ的影响; (b) 三角形穿孔边长l的影响

Figure 3. Effect of lattice parameters on eigenfrequencies of the dipole modes and quadrupole modes at the center Γ point of Brillouin zone: (a) The effect of the rotation angle θ of the triangular holes; (b) the effect of the side length l of the triangular holes.

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图 4 (a) 由5个TTCs与5个TNCs组成的超元胞; (b) 该超元胞的能带结构图, 其中红色和蓝色点代表边界态, A和B点为波矢${k_x} = \pm 0.2{\text{π}}/a$时对应的边界态, 灰色圆圈表示水平剪切模; (c) 图(b)中A和B点对应的z方向位移场分布, 放大图显示了边界态处的机械能量通量方向

Figure 4. (a) Supercell composed of 5 TTCs and 5 TNCs; (b) the band structure of the supercell in (a), red and blue dots represent the edge states, A and B dots represent the pseudospin states in ${k_x} = \pm 0.2{\text{π}}/a$, and the gray circles represent the shear horizontal modes; (c) the DFDs in z-directiona corresponding to points A and B in (b), the enlarged figure shows the mechanical energy flux direction.

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图 5 由TTCs和TNCs构成的不同通道的声子晶体板　(a) 黑色虚线表示$\theta =25^\circ $的TTCs与$\theta =0^\circ $的TTCs构成的边界, 蓝色虚线表示$\theta =0^\circ $的TTCs与$\theta =60^\circ $的TNCs构成的边界, 并设置A, B, C三个激励点(红色点处), 下图为z方向振动激励下的位移场分布; (b) 蓝色虚线表示$\theta =0^\circ $的TTCs与$\theta =60^\circ $的TNCs构成的“Z”字形边界

Figure 5. Phononic crystal plate composed of TTCs and TNCs with different waveguide channels. (a) The black dashed line represents the edge formed by $\theta =25^\circ $ TTCs and $\theta =0^\circ $ TTCs, and the blue dashed line represents the edge formed by $\theta =0^\circ $ TTCs and $\theta =60^\circ $ TNCs. Three excitation points A, B and C are set at red points, and the DFDs under the vibration excitation in z-direction are shown in below. (b) The blue dashed line represents the zigzag edge formed by $\theta =0^\circ $ TTCs and $\theta =60^\circ $ TNCs.

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图 6 由TTCs和TNCs构成的存在缺陷的声子晶体板　(a) “Z”字形通道中存在三角形穿孔缺失(红色点为激励位置), 下图为该声子晶体板在z方向振动激励下的位移场分布情况; (b) “Z”字形通道中存在乱序缺陷

Figure 6. Defective phononic crystal plate composed of TTCs and TNCs: (a) The zigzag channel with missing triangular holes (the red point is the excitation position), and the DFDs under the z-direction vibration excitation are shown in below; (b) the zigzag channel with disordered triangular holes

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图 7 利用多点激励策略实现弹性波的单向传播(其对应的位移场分布情况清晰地说明了基于拓扑保护边界态下弹性波单向传播效果)　(a)在多点激励下产生赝自旋+模态; (b) 在多点激励下产生赝自旋–模态

Figure 7. One-way propagation of elastic wave is realized by using multi-point excitation strategy, and the corresponding DFDs clearly show the one-way propagation phenomenon of elastic wave based on the topological protected edge states: (a) The pseudospin + state is generated by the strategy; (b) the pseudospin － state is generated by the strategy.

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图 8 基于拓扑保护边界态的多通道波导开关　(a) 声子晶体板上蓝色虚线为不同类型晶体构成的四个波导通道, 并在板左侧设置蓝色激励点, 其中蓝色或红色圆形箭头表示弹性波从激励点出发沿该通道传播时的赝自旋方向; (b) 激励点为左侧蓝点时的位移场分布, 放大图显示了边界态处的机械能量通量方向; (c) 在晶体板上侧设置红色激励点; (d) 激励点为上侧红点时的位移场分布

Figure 8. Multichannel waveguide switch based on topologically protected edge states: (a) The phononic crystal plate with an excitation point (blue point) on the left side and four waveguide channels (blue dashed lines) which formed by different types of crystals, in which the blue or red circular arrow indicates the pseudospin direction of elastic wave propagating along the channel from the excitation point; (b) the DFDs when the excitation point is the left blue point and the enlarged figure shows the mechanical energy flux direction; (c) the phononic crystal plate with an excitation point (red point) on the upper side; (d) the DFDs when the excitation point is the upper red point.

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[1]	Fang X, Wen J, Bonello B, Yin J, Yu D 2017 Nat. Commun. 8 1288 Google Scholar
[2]	Xiao Y, Wen J, Wen X 2012 J. Phys. D: Appl. Phys. 45 195401 Google Scholar
[3]	Ma G, Sheng P 2016 Sci. Adv. 2 e1501595 Google Scholar
[4]	Fang X, Wen J, Benisty H, Yu D 2020 Phys. Rev. B 101 104304 Google Scholar
[5]	陆智淼, 蔡力, 温激鸿, 温熙森 2016 物理学报 65 174301 Google Scholar Lu Z M, Cai L, Wen J H, Wen X S 2016 Acta Phys. Sin. 65 174301 Google Scholar
[6]	陈毅, 刘晓宁, 向平, 胡更开 2016 力学进展 46 382 Google Scholar Chen Y, Liu X N, Xiang P, Hu G K 2016 Adv. Mech. 46 382 Google Scholar
[7]	Guenneau S, Movchan A, Pétursson G, Ramakrishna S A 2007 New J. Phys. 9 399 Google Scholar
[8]	Zhu J, Christensen J, Jung J, Martin-Moreno L, Yin X, Fok L, Zhang X, Garcia-Vidal F J 2011 Nat. Phys. 7 52 Google Scholar
[9]	Zhang Z, Tian Y, Cheng Y, Wei Q, Liu X, Christensen J 2018 Phys. Rev. Appl. 9 34032 Google Scholar
[10]	Zhang Z, Tian Y, Wang Y, Gao S, Cheng Y, Liu X, Christensen J 2018 Adv. Mater. 30 1803229 Google Scholar
[11]	Lu J, Qiu C, Ye L, Fan X, Ke M, Zhang F, Liu Z 2017 Nat. Phys. 13 369 Google Scholar
[12]	Wen X, Qiu C, Lu J, He H, Ke M, Liu Z 2018 J. Appl. Phys. 123 91703 Google Scholar
[13]	Mei J, Chen Z, Wu Y 2016 Nat. Phys. 6 32752 Google Scholar
[14]	Jia D, Sun H, Xia J, Yuan S, Liu X, Zhang C 2018 New J. Phys. 20 93027 Google Scholar
[15]	He C, Ni X, Ge H, Sun X, Chen Y, Lu M, Liu X, Chen Y 2016 Nat. Phys. 12 1124 Google Scholar
[16]	裴东亮, 杨洮, 陈猛, 刘宇, 徐文帅, 张满弓, 姜恒, 王育人 2020 物理学报 69 024302 Google Scholar Pei D, Yang T, Chen M, Liu Y, Xu W, Zhang M, Jiang H, Wang Y 2020 Acta Phys. Sin. 69 024302 Google Scholar
[17]	Mousavi S H, Khanikaev A B, Wang Z 2015 Nat. Commun. 6 8682 Google Scholar
[18]	Li J, Wang J, Wu S, Mei J 2017 AIP Adv. 7 125030 Google Scholar
[19]	Huo S, Chen J, Huang H 2018 J. Phys.: Condens. Matter 30 145403 Google Scholar
[20]	Zhang Q, Chen Y, Zhang K, Hu G 2020 Phys. Rev. B 101 14101 Google Scholar
[21]	Yu S, He C, Wang Z, Liu F, Sun X, Li Z, Lu H, Lu M, Liu X, Chen Y 2018 Nat. Commun. 9 3072 Google Scholar
[22]	Yan M, Lu J Y, Li F, Deng W Y, Huang X Q 2018 Nat. Mater. 17 993 Google Scholar
[23]	Huo S, Chen J, Feng L, Huang H 2019 J. Acoust. Soc. Am. 146 729 Google Scholar
[24]	Wang J, Mei J 2018 Appl. Phys. Express 11 57302 Google Scholar
[25]	Yin J, Ruzzene M, Wen J, Yu D, Yue L 2018 Sci. Rep. 8 6806 Google Scholar
[26]	Wang P, Lu L, Bertoldi K 2015 Phys. Rev. Lett. 115 104302 Google Scholar
[27]	Khanikaev A B, Fleury R, Mousavi S H, Alù A 2015 Nat. Commun. 6 8260 Google Scholar
[28]	Souslov A, van Zuiden B C, Bartolo D, Vitelli V 2017 Nat. Phys. 13 1091 Google Scholar
[29]	Zhang Z, Tian Y, Cheng Y, Liu X, Christensen J 2017 Phys. Rev. B 96 241306 Google Scholar
[30]	Miniaci M, Gliozzi A S, Morvan B, Krushynska A, Pugno N M 2017 Phys. Rev. Lett. 118 214301 Google Scholar
[31]	Graff K F 1991 Wave Motion in Elastic Solids (New York: Dover publications) pp431−463
[32]	Ganti S S, Liu T, Semperlotti F 2020 J. Sound Vib. 466 115060 Google Scholar
[33]	Ma G, Xiao M, Chan C T 2019 Nat. Rev. Phys. 1 281 Google Scholar
[34]	何程, 卢明辉, 陈延峰 2017 物理 46 12 Google Scholar He C, Lu M H, Chen Y F 2017 Physics 46 12 Google Scholar
[35]	Chaunsali R, Chen C, Yang J 2018 Phys. Rev. B 97 54307 Google Scholar
[36]	Nanthakumar S S, Zhuang X, Park H S, Nguyen C, Chen Y, Rabczuk T 2019 J. Mech. Phys. Solids 125 550 Google Scholar
[37]	Zhang Z, Wei Q, Cheng Y, Zhang T, Wu D, Liu X 2017 Phys. Rev. Lett. 118 84303 Google Scholar
[38]	Deng Y, Ge H, Tian Y, Lu M, Jing Y 2017 Phys. Rev. B 96 184305 Google Scholar
[39]	Xia B, Liu T, Huang G, Dai H, Jiao J, Zang X, Yu D, Zheng S, Liu J 2017 Phys. Rev. B 96 94106 Google Scholar
[40]	Vila J, Pal R K, Ruzzene M 2017 Phys. Rev. B 96 134307 Google Scholar
[41]	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 Google Scholar
[42]	Li Y, Wu Y, Mei J 2014 Appl. Phys. Lett. 105 14107 Google Scholar

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