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基于声子晶体拓扑特性构造的弹性波拓扑态在波调控方面具有背散射抑制和路径缺陷免疫等优异特性, 受到广泛关注. 本文设计了一种声子晶体板结构, 通过在初始元胞中引入具有一定旋转角度的三角形穿孔实现对称性破缺, 从而构造四重简并态. 与现有利用能带“区域折叠”进行构造的方法相比, 该方法简化了声子晶体的元胞构型. 元胞的主要变量为三角形穿孔围绕其中心旋转角度
$\theta $ , 研究发现, 旋转角度$\theta =0^\circ $ 时, 元胞能带结构存在两个二重简并态, 调整旋转角度到$ \pm 33^\circ $ 时, 布里渊区中心Γ点处出现四重简并态, 并发现旋转角度越过$ \pm 33^\circ $ 时均会发生能带反转, 这表明调整晶体结构参数$\theta $ 使得体系经历拓扑相变. 利用具有不同拓扑相的声子晶体组成超元胞, 并通过计算其投影能带, 发现能带结构中存在弹性波带隙以及不同赝自旋方向的两种边界态. 在此基础上, 构造多种不同类型的弹性声子晶体板, 验证了拓扑边界态对弹性波传播的强背散射抑制、缺陷免疫单向传播和多波导通道开关特性. 本文中所设计的弹性声子晶体板具有结构简单、特性易调的特点, 为利用拓扑态实现弹性波调控提供了一个可行方案.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:
- topological phase transition /
- elastic wave-guide /
- elastic topological edge states /
- phononic crystals
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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)
Fig. 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.
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Fig. 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.
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
Fig. 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.
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Fig. 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.
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Fig. 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.
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
Fig. 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
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
Fig. 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.
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
Fig. 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 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
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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
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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
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[28] Souslov A, van Zuiden B C, Bartolo D, Vitelli V 2017 Nat. Phys. 13 1091
Google Scholar
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Google Scholar
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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
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Google Scholar
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Google Scholar
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Google Scholar
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