1.Key Laboratory of Microgravity, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China 2.University of Chinese Academy of Sciences, Beijing 100049, China 3.Wuhan Second Ship Design and Research Institute, Hubei 430064, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11602269, 11972034, 11802213), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB22040301), and the Research Program of Beijing, China (Grant Nos. Z161100002616034, Z171100000817010)
Received Date:23 September 2019
Accepted Date:27 October 2019
Available Online:01 January 2020
Published Online:20 January 2020
Abstract:The discovery of quantum Hall effect and quantum spin Hall effect has set off a new research upsurge in condensed matter physics. As is analogous to electronic systems, many novel optical and acoustic control devices have been designed by using the defects- immune and backscatter suppression of topological edges in photonic crystals and phononic crystals, which greatly enriches the current physical world and arouses more research enthusiasm. With the study of acoustic topological structure, it has been found that the realization of good reconfigurability, good compatibility against manufacturing defects, and compact acoustic topological insulators may become a promising development direction. This imposes higher requirements on the topological band gap width of the current acoustic topological structure. At the same time, the restriction on the using of the same primitive unit cells in previous researches does not reveal the implementation of aperiodic double Dirac cone topological insulators. Here in this work we present a tunable, two-dimensional broadband composite honeycomb lattice structure for airborne sound. Firstly, We construct a hexagonal structure and then take a circle with a radius of r1 in the center. Then the circle is anisotropically scaled with the scaling factor s, which means that the x direction of the circle is expanded by $\sqrt s $ times, and the y direction is reduced by $1/\sqrt s $ times to form an ellipse. Then, we perform a translation and rotation transformation on the ellipse, and finally construct a “triangular-like” petal pattern at each vertex of the hexagon. Secondly, we place a circle with a radius of r2 in the center to achieve the unit cell of the phononic crystal. This cell has two variables. One is the rotation angle θ of the petal pattern around its centroid, and the other is the scaling factor s. We find that there is a quadruple degenerate state at Γ with s = 1.2 and θ = ±33°. On both sides of ±33°, changing θ will induce an inverted band and a topological phase transition. At the same time, the relative band gap of the structure increases gradually. When θ is 0° and 60°, the structures are two topologically distinct broadband phononic crystals with relative band widths of 0.39 and 0.33, respectively. Calculated by the finite element software Comsol, the edge states existing in the band gap are found, and the backscattering immunity characteristics of the topological edges to defects such as right angle, Z-angle, disorder, and cavity are confirmed. For the first time we construct a aperiodic double Dirac cone acoustic topological insulators with different values of s and change their defect immunity. The research system is rich in function, and its relative bandwidth can even exceed 0.5 for a certain s value, which significantly exceeds the bandwidth of the known structure, and lays a good foundation for miniaturized acoustic wave devices taking full advantage of acoustic topological edges. Meanwhile, the realization of aperiodic topological insulators shows that the system can be used more flexibly for acoustic structure design. Keywords:topological phase transition/ broadband structure/ aperiodic double Dirac cone topological insulator
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2.1.体系的设计
CHL的元胞如图1(a)所示, 其最简布里渊区如图1(b)中的绿色区域Γ-M-K所示. 晶格常数a = 43 mm, 晶格基矢a1 = ai, ${{{a}}_2} = - 1/2 ai + \sqrt 3 /2 aj$. 本文首先构建一个六角形基本结构, 然后在晶格中心取一半径r1 = 6 mm的圆, 再将此圆进行s = 1.2的缩放, 即形成一个椭圆结构. 然后将椭圆绕着自己的中心旋转30°使得椭圆的水平轴指向六角形的一个角点b. 再将此旋转后的椭圆沿着顶点b与六角形中心o的连线移动, 移动距离为ob线段长度的0.8倍. 由于移动的距离没有达到ob的长度, 因此椭圆的中心并没有与b点重合. 再将椭圆绕着b点旋转120°, 240°, 将三个相交的椭圆组合成一个整体, 形成一个“类三角”的花瓣图形. 然后将“花瓣”绕着o点旋转到六边形的六个顶点. 然后在o点重新放置一个半径r2 = 10 mm的圆, 即可得到本体系的晶格元胞. 此元胞“花瓣”的顶点朝向均与晶格的高对称方向重合, 结构整体具有C6v对称性. 元胞中的“花瓣”形与位于中心的圆柱形结构使用的是硬质散射体, 其与空气的阻抗失配很大, 因此在使用有限元软件Comsol进行模拟仿真的时候可以忽略结构中剪切波的影响, 研究结构中的纵波传播特性. 图 1 (a)正六边形表示晶格的元胞, 其中a1, a2是晶格基矢. 在六边形顶点的蓝色“花瓣”形结构与位于中心的圆形结构表示位于空气中的硬质散射体; (b)晶格的最简布里渊区Γ-M-K; (c)晶格结构的示意图 Figure1. (a) The hexagon represents the cell of the lattice, where a1, a2 is the lattice basis vector, the blue “petal” shape at the apex of the hexagon and the circular structure at the center represent the hard scatterers surrounded by air; (b) the irreducible Brillouin zone Γ-M-K; (c) schematic diagram of crystal structure.
22.2.能带反转 -->
2.2.能带反转
根据量子系统中的规则, 对于具有C6v对称性的晶格结构, 在布里渊区的中心Γ点的本征态有2个二维不可约表示: ${E_1}$和${E_2}$. 二重简并的偶极子态, 对应于${E_1}$不可约表示, 具有奇宇称, 简称为p态, 如图2(a)中上面两个插图所示. 二重简并的四极子态, 对应于${E_2}$不可约表示, 具有偶宇称, 简称为d态, 如图2(a)中下面两个插图所示. 对于本文的声子晶体, 通过将每个“花瓣”绕着自己的中心旋转一定的角度θ, 发现在θ为±33°时Γ处的第2—5能带发生简并, 形成一个四重简并态, 即此时p态与d态形成简并. 改变θ发现p态与d态在特定角度会发生反转, 如图2(c)所示. 体系的p态与d态只是在θ为±33°时偶然地发生简并. 其他角度时p态与d态分开, 但是p态与d态仍然保持为双重简并的状态. 在0—33°(–33°—0)的范围内d态所对应的频率低(高)于p态所对应的频率. 在33°—60° (–60°—–33°)的范围内d态所对应的频率高(低)于p态所对应的频率. 也就是说体系在±33°两侧, 经历了p, d态互换的能带反转过程. 这种能带反转现象意味着拓扑相变的发生. 图 2 CHL不同转角时的频带图与其拓扑相变 (a) θ = 0°, 其中下面两幅插图表示d态的声压场分布, 上面两幅插图表示p态的声压场分布; (b) θ = 60°, 插图表示p, d态的声压场分布; (c)结构的拓扑相图, 表示随着转角变化, 频带发生反转; (d)不同转角下带隙的相对带宽 Figure2. The band structures with different θ and its topological phase transition of the CHL: (a) θ = 0°, in which the lower two illustrations show the sound pressure field distribution of the d state, and the upper two illustrations show the sound pressure field distribution of the p state; (b) θ = 60°, in which the upper and lower illustrations show the sound pressure field distribution of the d and p states, respectively; (c) the topological phase diagram of the structure, indicating that the frequency band is reversed as the rotation angle changes; (d) the relative bandwidth of the band gap at different θ.
选择θ = 60°的5层平庸型声子晶体与θ = 0°的5层非平庸型声子晶体沿着ky方向拼接起来, 将其构成的边界称为Ⅰ型边界, 并计算了该结构沿kx的投影带结构, 如图3(a)所示; 选择θ = 60°的10层平庸型声子晶体与θ = 0°的10层非平庸型声子晶体沿着kx方向拼接起来, 将其构成的边界称为Ⅱ型边界, 如图3(b)所示. 从两个不同方向的投影能带图中都发现了存在于体带隙范围内的边界态, 而且从边界处的能流分布中可以发现其存在“自旋与动量锁定”的特性. 从边界态中可以看出: 1)由于边界上C6v对称性被破坏, 导致边界态并没有完全占据体态的带隙范围, Ⅰ型边界中6278—6448 Hz范围依然存在禁带, Ⅱ型边界中6181—6612 Hz是禁带; 2)图中青色箭头大小和方向代表声波能流的大小和方向(黑色箭头是能流方向的示意图), 可以看出对于Ⅰ型边界在kx为–0.04π/a的位置低频点处的能流是逆时针向右传播的, 高频点处的能流是顺时针向左传播的. 而其能流的方向与0.04π/a处的能流的方向刚好相反. 对于Ⅱ型边界, 在ky为$ - 0.04{\text{π}}/\left( {a\sqrt 3 } \right)$的位置低频点处的能流是顺时针向左传播, 高频点处的能流是逆时针向右传播, 与$0.04{\text{π}}/\left( {a\sqrt 3 } \right)$处的能流的方向刚好相反. 所以, 本研究利用不同转角的CHL构造了一种类似于电子系统中QSHE的螺旋边界态. 而且, 由于此两种声子晶体具有较大的相对宽带, 所以使用较少的本体系即可观察到受到拓扑保护的边界态. 图 3 (a) Ⅰ型边界沿kx方向的投影带结构, 图中的灰色区域表示体态, 红色点线表示边界态, 两侧插图表示θ = 60°的5层平庸型声子晶体与θ = 0°的5层非平庸型声子晶体沿着ky方向拼接起来, 构成的超胞及a1, a2, b1, b2点的声压分布; (b) Ⅱ型边界沿ky方向的投影带结构, 两侧插图表示θ = 60°的10层平庸型声子晶体与θ = 0°的10层非平庸型声子晶体沿着kx方向拼接起来, 构成的超胞及c1, c2, d1, d2点的声压分布, 插图中的黑色弧形箭头表示边界处的能流方向 Figure3. (a) The projection band structure of the type I edge along the kx direction. The gray area in the figure represents the bulk state, and the red dotted line represents the edge state. The illustrations on both sides indicate that the five-layer trivial phononic crystal with θ = 60° and the five-layer nontrivial phononic crystal with θ = 0° are spliced along the ky direction to form the supercell and the sound pressure distribution at a1, a2, b1, b2; (b) the projection band structure of the type II edge along the ky direction. The illustrations on both sides indicate that a 10-layer trivial phononic crystal with θ = 60° and a 10-layer nontrivial phononic crystal with θ = 0° are spliced together in the kx direction to form the supercell and the sound pressure distribution at points c1, c2, d1, d2. The black curved arrow in the illustration indicates the energy flow direction at the edge.
22.5.拓扑边界的免疫缺陷特性 -->
2.5.拓扑边界的免疫缺陷特性
拓扑边界的一个重要特点就是其对缺陷具有免疫性. 拓扑边界对直角、“Z”形角与存在于边界的缺失、乱序等具有免疫性, 使得受到拓扑保护的边界态能绕过这些缺陷几乎没有反射地进行传播. 图4(a)模拟的是使用幅值为1、f = 6900 Hz的平面波从左侧入射到一个具有直角与Z形角的边界上的情况, 从声压分布图中发现声波能沿着θ = 60°与θ = 0°的声子晶体构成的边界顺利地传播过去. 图4(b)接着在图4(a)的传播路径上引入乱序与缺失的缺陷, 发现声波也可以绕过这这些缺陷继续向下传播. 图4(c)是对比实验, 其表示的是声波入射到单独由θ = 60°的声子晶体构成的与图4(a)、图4(b)相同大小的结构上时, 声压的分布情况. 图4(d)是图4(a)和图4(c)结构的声强透射谱. 声强的探测位置选在边界的出口附近, 如图4(b)中黄色区域内黑线(output)所示, 具体位置为灰色区域下0.081a, 宽度为1.5a. 从图4(d)可以看出: 对于图4(c)结构, 其声强透射率在体态与带隙范围相差比较大, 特别是在其带隙内声能几乎不能透过去; 在图4(a)和图4(b)中, 声波依然能够沿着边界传播到出口; 不同频率的透射声波能量略有不同. 由于图4(a)和图4(b)中的缺陷不同, 导致两者的透射率存在差异. 本文所设计的边界态是具有相对带宽33.4%的宽带结构, 所以边界态十分稳定, 能够免疫文中的那些缺陷. 这种宽带结构无论是在声波隔离还是在声波操控方面都具有很大的优势, 为实现声波的灵活控制打下了良好的基础. 图 4 幅值为1, f = 6900 Hz的平面波从左侧入时结构的声压分布及透射谱 (a)由θ = 60°与θ = 0°的声子晶体的拼接结构组成的混合声子晶体, 两种声子晶体的相接触的边界称为拓扑边界, 左侧的青色箭头表示平面波入射, 从图中声压分布可以看出: 声波能够绕过直角与Z形角沿着边界进行传播; (b)在图(a)的基础上继续引入乱序与缺失的缺陷, 声波依然能够绕过这些缺陷传播; (c)由θ = 60°声子晶体单独构成的结构, 声波不能传播; (d)图(a)—图(c)结构的声强透射谱 Figure4. The sound pressure distribution and transmission spectrum of the structure when a plane wave is incident from the left side with amplitude 1 Pa and f = 6900 Hz: (a) A mixed phononic crystal composed of a spliced structure of phononic crystals with θ = 60° and θ = 0°, and the edge between the two phononic crystals is called the topological edge. The cyan arrow on the left side indicates the plane wave incidence. It can be seen from the sound pressure distribution that the sound wave can propagate around the right angle and the Z-angle along the edge; (b) introduce disorder and cavity on the basis of Fig. (a), sound waves can still propagate around these defects; (c) a structure consisted of phononic crystals with θ = 60° alone where sound waves cannot propagate; (d) sound intensity transmission spectra of the structures of Fig. (a)—Fig. (c).
22.6.非周期双狄拉克锥型拓扑绝缘体 -->
2.6.非周期双狄拉克锥型拓扑绝缘体
保持“花瓣”形整体结构不变, 改变r1 = 6 mm圆的缩放参数s, 并使用不同s的结构构造非周期双狄拉克锥型拓扑绝缘体. 首先研究在保持角度θ = 0°不变, s从0.7增加到1.7的情况下, 元胞布里渊区中心点Γ处d, p态的频率变化, 结果如图5(a)所示. 随着s的增加d, p态之间的带隙宽度逐渐增加, 并且没有相变发生. 此时, 计算可知, s为1.5, 1.6时, 相对带宽达到了0.52, 0.56, 超过了0.5. 然后保持s不变, 改变θ, 发现这些结构在θ为±33°两侧发生相变, 与s = 1.2时的拓扑变化相似. s = 0.8与s = 1.0的相图如图5(b)和图5(c)所示. 图 5 (a) Γ处d, p态对应的频率值随s的变化情况; (b) s = 0.8时结构的拓扑相图; (c) s = 1.2时结构的拓扑相图 Figure5. (a) The frequency corresponding to the d and p states with the changes of the parameter s at Γ; (b) the topological phase diagram of the structure at s = 0.8; (c) the topological phase diagram of the structure at s = 1.2.
使用s为0.8, 1.0, 1.2三种结构构造一个由“45 × 5”个混合声子晶体构成的非周期声拓扑绝缘体. 其结构如图6(a)所示, 右下角的插图表示组成, 其中A, B, C分别表示s为0.8, 1.0, 1.2三种不同的声子晶体, 数字1, 2分别代表θ为0°, 60°两种不同的转角. 研究中使用f = 6900 Hz的平面波从右侧入射, 图6(a)表示声压幅值的分布, 从图中可以看出声压主要分布在不同s参数声子晶体的水平拼接位置附近, 离开此位置声压迅速衰减, 这点从图6(b)中也能得到. 由此得出此处构造的非周期性结构能够限制特定频率的声波沿着边界传播; 图6(b)表示x = 20a处沿着y方向的声压幅值分布, 其值经过最大值归一化; 图6(c)是在图6(a)的基础上进一步引入缺失与乱序的缺陷时结构声压幅值的分布, 从中可以看出声波能够绕过缺陷继续向前传播. 在先前的工作中, 由于使用相同原始单位单元的限制, 并未实现非周期双狄拉克锥型拓扑绝缘体的构建. 本节使用不同s参数这种灵活简便、易于设计的方式实现了非周期的声拓扑绝缘体的设计, 为声拓扑绝缘体的构建提供了新颖、多变的“原材料”. 图 6 非周期拓扑绝缘体结构的声压分布 (a)为由s为0.8, 1.0, 1.2三种声子晶体构造的非周期声拓扑绝缘体组成及其在右侧f = 6900 Hz声波入射下的声压幅值分布. 中间的横虚线表示水平拼接位置, 竖虚线表示竖直拼接位置, 第一、二幅插图代表θ为0°, s为0.8, 1.0时, 晶格的元胞. 三种声子晶体的具体位置如第三幅插图所示, 其中A, B, C分别表示s为0.8, 1.0, 1.2三种结构, 数字1, 2分别代表θ为0°, 60°两种不同的转角, 右侧的青色箭头代表平面波入射; (b)表示x = 20a处沿着y方向的声压幅值分布, 其值经过最大值归一化; (c)在图(a)的基础上进一步引入乱序与缺失的缺陷时结构声压幅值的分布 Figure6. Sound pressure distribution of aperiodic topological insulator structure: (a) A periodic acoustic topological insulator composed of three phononic crystal structures with s = 0.8, 1.0, and 1.2 and its sound pressure amplitude distribution when sound wave with f = 6900 Hz is incident from the right side. The horizontal dashed line in the middle indicates the horizontal stitching position, and the vertical dashed line indicates the vertical stitching position. The first and second insets represent the lattice cells with θ = 0° and s of 0.8 and 1.0, respectively. The specific positions of the three phononic crystals are shown in the third illustration, where A, B, and C respectively represent s = 0.8, 1.0, and 1.2, and the numbers 1 and 2 represent θ = 0° and 60°, respectively. The cyan arrow on the right indicates the incident plane wave; (b) the sound pressure amplitude distribution along the y direction at x = 20a, and its value is normalized by the maximum value; (c) the distribution of the sound pressure amplitude when introducing disorder and cavity on the basis of Fig. (a).