One-dimensional synthetic waterborne phononic crystals

Hu Chen-Yang; Liang Jia-Luo; Zheng Ri-Yi; Lu Jiu-Yang; Deng Wei-Yin; Huang Xue-Qin; Liu Zheng-You

doi:10.7498/aps.73.20240298

Abstract

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1. 引　言
2. 合成二维能谷拓扑声子晶体
3. 能谷陈数保证的拓扑界面态
4. 结　论

References

Cited By

Abstract

Underwater acoustics has wide applications in underwater communication, underwater positioning, underwater navigation, and so on. Inspired by the concept of topological physics, the study of topological states in waterborne phononic crystals provides a brand-new way for innovatively controlling underwater waves, which has both basic research value and important application prospects. In this work, we design a one-dimensional bilayer iron grid waterborne phononic crystal to realize a synthetic two-dimensional Dirac point by considering the relative lateral translation between the two layers as a synthetic dimension. Through changing the relative lateral translation, the double degenerate band opens a gap, which is characterized by the valley Chern number. As the band gap opens, closes and reopens, the bulk band undergoes a band inversion, that is, a topological phase transition from one valley topological phase to another. At the interface formed by two phononic crystals with distinct valley topological phases, the valley Chen number ensures the deterministic existence of the interface state. Experimental measurements are in good agreement with numerical simulations, both showing the bulk bands of waterborne phononic crystals at different valley topological phases and the interface state dispersion between them. The waterborne phononic crystal proposed in this work has a simple structure. With the help of the concept of synthetic dimension, it provides an effective way to study the topological properties of high-dimensional systems in low-dimensional real space systems, and gives new ideas for designing topological functional underwater acoustic devices. In addition, we can expand the real space system to two or even three dimensions, and introduce more synthetic dimensions to study the topological states and associated transport characteristics of higher-dimensional systems.

Keywords:

PACS:

43.40.+s	(Structural acoustics and vibration)
43.20.+g	(General linear acoustics)
43.35.+d	(Ultrasonics, quantum acoustics, and physical effects of sound)

Authors and contacts

1.
School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
2.
Key Laboratory of Artificial Micro- and Nanostructures of Ministry of Education, School of Physics and Technology, Wuhan University, Wuhan 430072, China

Corresponding author: Huang Xue-Qin, phxqhuang@scut.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12074128, 12222405) and the Basic and Applied Basic Research Foundation of Guangdong Province, China (Grant Nos. 2021B1515020086, 2022B1515020102).

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1. 引　言

受电子带隙材料的启发, 声子晶体的概念被提出^[1,2]. 声子晶体是质量密度和弹性参数在空间中周期性分布的人工结构材料, 具有超越天然材料的声学性质^[3–6]. 在电子体系中, 除了传统的易受杂质和无序散射的边缘态, 研究者们发现了一类新型的边缘态, 称为拓扑边缘态^[7–14]. 它由体能带的拓扑特性决定, 不受边缘处的杂质和无序影响, 更加易于传播. 然而, 在电子体系中, 由于体系相对比较复杂, 同时材料参数难以随意改变, 所以相关的物理现象常常难以被观测到. 对于具有周期性结构的声子晶体, 也存在着类似于电子体系的能带结构. 相对于电子体系, 声子晶体是一个宏观体系, 可操控性强, 更加易于实现和观测, 这就提供了一条全新的途径来研究新型的拓扑态. 声子晶体拓扑态的研究引起了广泛关注, 该研究领域正在蓬勃发展, 从起初只是伴随着电子体系的研究而发展, 到后来发现一些电子体系中尚未被发现的新拓扑现象^[15–18]. 值得指出的是, 在二维狄拉克点体系中, 通过打破对称性实现能谷拓扑绝缘体^[19,20]. 由不同能谷陈数的两体系构成的边界处, 存在着能谷陈数保证的界面态. 这种受拓扑保护的界面态在传播过程中不受缺陷和无序的影响, 展示了很多有趣的现象, 例如拓扑负折射、反常分流等^[21–24], 为声波的创新性调控提供了一种全新的手段. 声学拓扑绝缘体的研究主要集中在空气声和弹性波体系中, 关于水基声子晶体拓扑态的研究相对较少^[25–27]. 由于电磁波在水中传播衰减较快, 难以被用于水下通信及探测, 相对来说声波在水下衰减较小, 可以进行较长距离的传输. 利用声波在水下的特性, 可以进行水下测深, 甚至开展深海海底地形地貌的测量, 进行海洋探测、水下通讯和水下定位导航等工作. 因此, 对于水基声子晶体拓扑态的研究既有基础研究价值, 也有重要的应用前景.

不同的空间维度对体系拓扑态的研究起着重要的作用. 高空间维度提供更多的自由度, 有望产生更加丰富的效应. 然而, 空间维度的增加也为样品制备和实验测量带来了新的挑战. 合成维度是指在一个低维的实空间体系中, 把可调的额外参量等效成新的空间维度, 用于研究高维体系的特性, 这提供了一种在低维实空间中研究高维体系特性的有效途经. 借助合成维度的概念, 合成三维拓扑绝缘体^[28]、合成外尔半金属^[29]以及合成高阶拓扑绝缘体^[30]等拓扑态都相继在声学体系中被实现.

本文在水基背景中, 构造一维双层声子晶体, 把层间的相对平移量等效成一个新的空间维度, 发现该水基声子晶体中存在着狄拉克点色散. 通过连续改变相对平移量, 能隙发生打开到闭合到再次打开的现象, 从而实现了能带的翻转. 此时的能带翻转对应着不同能谷拓扑相的拓扑相变. 接着, 由不同能谷相的两个一维声子晶体构成边界, 在边界处发现能谷陈数保护的界面态. 数值仿真和实验测量结果非常一致, 都证实了界面态的存在.

2. 合成二维能谷拓扑声子晶体

水基声子晶体的结构示意图如所示, 由两个相同的一维铁栅B1和B2沿 $y$ 方向堆垛而成, 并被置于水中. 其中, 铁的质量密度 ${\rho }_{0}= 7800\;{\mathrm{k}}{\mathrm{g}}/{{\mathrm{m}}}^{3}$ , 压力波速度 ${c}_{{\mathrm{l}}}=6064\;{\mathrm{m}}/{\mathrm{s}}$ , 剪切波速度 ${c}_{{\mathrm{t}}}=3325\;{\mathrm{m}}/{\mathrm{s}}$ ; 水的质量密度 ${\rho }_{1}=1000\;{\mathrm{k}}{\mathrm{g}}/{{\mathrm{m}}}^{3}$ , 声速 ${c}_{2}=1490\;{\mathrm{ }}{\mathrm{m}}/{\mathrm{s}}$ . 右图展示了该结构的原胞, 周期 $a=3\;{\mathrm{m}}{\mathrm{m}}$ , 由上下两个正方形铁栅条组成, 两栅条之间的相对横向平移量为D, 栅条的边长 $L=2.4\;{\mathrm{m}}{\mathrm{m}}$ . 本文定义无量纲量S表示横向平移率, $S=D/a$ , S可以从0到1之间连续变化. $S=0$ 和1都表示两个一维结构沿y方向直接堆垛, 它们对应着完全相同的几何结构, 也就是说S是一个周期性变量, 周期是1. 这类似于本结构中沿x方向的布洛赫波矢 ${k}_{x}$ , 其可以从0到 $2{\mathrm{\pi }}/a$ 连续变化, 同时 ${k}_{x}=0$ 和 $2{\mathrm{\pi }}/a$ 都表示相同的本征态. 于是, 可以把S看成是一个合成维度, 它与 ${k}_{x}$ 有类似的作用. 由此, 我们就可以在这个一维体系中, 通过引入平移变量构造一个合成二维体系.

$图 1 一维水基声子晶体的结构示意图\r\nFig. 1. Schematic of the one-dimensional waterborne phononic crystal.$

图 1 一维水基声子晶体的结构示意图

Fig. 1. Schematic of the one-dimensional waterborne phononic crystal.

下载: 全尺寸图片幻灯片

以 ${k}_{x}$ 和S为两周期性变量, 计算水基声子晶体的能带结构, 其三维能带结构如所示. 有趣的是, 当 ${k}_{x}={\mathrm{\pi }}/a$ 和 $S=0.5$ 时, 能带结构在 $f= 140\;{\mathrm{ }}{\mathrm{k}}{\mathrm{H}}{\mathrm{z}}$ 时存在一个二重简并点(黑色五角星). 为了清晰地展示该简并点附近的能带, , 分别展示了S不同时能带随 ${k}_{x}$ 的变化, 以及固定 ${k}_{x}={\mathrm{\pi }}/a$ , 能带随S的变化. 需要指出的是, 声波在背景水中传播模式的频率 $\omega$ 与波矢 ${k}$ 满足线性色散关系 $\omega ={c}_{2}{k}$ . 在二维情况下, 波矢 ${{k}}^{2}={k}_{x}^{2}+ {k}_{y}^{2}$ . 由于样品具有x方向的周期性, 用 ${k}_{x}$ 替代 ${k}$ , 可得 $\omega$ 与 ${k}_{x}$ 所满足的线性色散关系 $\omega ={c}_{2}{k}_{x}$ , 如中紫色实线(称作水线)所示. 如果 ${k}_{x}$ 在水线下方, 即 ${k}_{x} > \omega /{c}_{2}$ , 此时 ${k}_{y}$ 为纯虚数, 声波呈现在x方向传播, 而在y方向指数衰减的形式. 反之, 如果 ${k}_{x}$ 在水线上方, 则 ${k}_{y}$ 为实数, 声波可在水中自由传播. 因此, 水线是声波在水中传播模式与衰减模式之间的分界线.

$图 2 一维合成维度水基声子晶体的体能带　(a)三维能带结构, 黑色五角星表示狄拉克点; (b)不同的$ S $下能带结构随$ {k}_{x} $的变化, 蓝色和红色点表示$ S=0.3 $, 黑色点表示$ S=0.5 $, 紫色实线表示水线, A和B分别表示$ {k}_{x}={\mathrm{\pi }}/a $时上下两条能带的本征态; (c)态A, B关于$ S $的拓扑相图; (d), (e) $ S=0.3 $和$ S=0.7 $时的实验测量(彩色图)和数值仿真(黑色点)体能带\r\nFig. 2. Bulk band structure of the one-dimensional waterborne phononic crystal: (a) Three-dimensional band structure, black star represents the Dirac point; (b) bulk band structure as a function of $ {k}_{x} $ for different $ S $, blue and red dots indicate the case of $ S=0.3 $, while the black dots are for $ S=0.5 $, the purple solid line represents the water line, A and B denote the eigen states in the upper and lower bands at $ {k}_{x}={\mathrm{\pi }}/a $; (c) phase diagram of states A and B; (d), (e) measured and simulated bulk band structures for the cases of $ S=0.3 $ and $ S=0.7 $, respectively.$

图 2 一维合成维度水基声子晶体的体能带　(a)三维能带结构, 黑色五角星表示狄拉克点; (b)不同的

$S$

下能带结构随

${k}_{x}$

的变化, 蓝色和红色点表示

$S=0.3$

, 黑色点表示

$S=0.5$

, 紫色实线表示水线, A和B分别表示

${k}_{x}={\mathrm{\pi }}/a$

时上下两条能带的本征态; (c)态A, B关于

$S$

的拓扑相图; (d), (e)

$S=0.3$

和

$S=0.7$

时的实验测量(彩色图)和数值仿真(黑色点)体能带

Fig. 2. Bulk band structure of the one-dimensional waterborne phononic crystal: (a) Three-dimensional band structure, black star represents the Dirac point; (b) bulk band structure as a function of

${k}_{x}$

for different

$S$

, blue and red dots indicate the case of

$S=0.3$

, while the black dots are for

$S=0.5$

, the purple solid line represents the water line, A and B denote the eigen states in the upper and lower bands at

${k}_{x}={\mathrm{\pi }}/a$

; (c) phase diagram of states A and B; (d), (e) measured and simulated bulk band structures for the cases of

$S=0.3$

and

$S=0.7$

, respectively.

下载: 全尺寸图片幻灯片

展示了 $S=0.3$ (红色和蓝色点)和 $S= 0.5$ (黑色点)两种情况下的能带结构, 可见感兴趣的能带都位于水线之下, 即这些模式都局域在声子晶体的表面. $S=0.5$ 时, 体系在 ${k}_{x}={\mathrm{\pi }}/a$ 附近存在线性色散二重简并态; $S=0.3$ 时, 体系在 ${k}_{x}={\mathrm{\pi }}/a$ 附近打开带隙. 由此可见, S的变化伴随着带隙的闭合和打开. 为了进一步展示能带的演变过程, 固定 ${k}_{x}={\mathrm{\pi }}/a$ , 将 $S=0.3$ 时的下能态记作态B, 上能态记作态A, 展示了这两个本征态随着S的变化, 能带在 $S=0.5$ 附近同样呈现线性关系. 由能带在 ${k}_{x}$ 和S两个方向上都具有线性色散关系可判定上述的二重简并点是狄拉克点. 此外, 计算出 $S=0.3$ 和 $S=0.7$ 时, 上下两个本征态的声压场图. 从图中可以看到: $S=0.3$ 时处于上(下)能带和 $S=0.7$ 时处于下(上)能带的原胞本征场图是一致的, 均是态A (态B), 进一步佐证了体系在 $S=0.5$ 两侧经历了态A, 态B互换的能带翻转过程, 也就是发生拓扑相变. 这种一维合成维度水基声子晶体可以通过以下哈密顿量来定性描述:

$\begin{split}H\left(S,{k}_{x}\right)=\;&\{{v}_{{\mathrm{g}}}\left[1-{\mathrm{c}}{\mathrm{o}}{\mathrm{s}}\left({\mathrm{\pi }}+{k}_{x}a\right)\right]+\eta {\mathrm{cos}}\left({\mathrm{\pi }}S\right)\}{\sigma }_{x}\\ &+{v}_{{\mathrm{g}}}{\mathrm{s}}{\mathrm{i}}{\mathrm{n}}\left({\mathrm{\pi }}+{k}_{x}a\right){\sigma }_{y} , \end{split}$

其中, ${v}_{{\mathrm{g}}}$ 表示声波沿x方向的群速度, $\eta$ 表示不同模式之间的耦合, ${\sigma }_{x, y}$ 表示泡利矩阵. 体系的拓扑特性与该哈密顿量所描述的完全一致. 当 $S < 0.3$ 时, 上下能带中的能谷态A和态B的能谷陈数分别为 $\pm 1/2$ ; 当 $S < 0.7$ 时, 能带发生翻转, 对应能带的能谷陈数变成 $\mp 1/2$ ^[19].

为了实验验证该声子晶体的能带结构, 本文选取了平移率 $S=0.3$ 和 $S=0.7$ 的几何结构作为实验对象. 在前面的数值仿真中, 只需考虑纯二维体系就可以, 然而对于实际结构, 还需要考虑z方向的长度. 通过激光切割加工, 我们制作了两个不同平移率的样品. 样品沿z方向上的长度为20 cm, 远大于x方向的周期3 mm, 因此, 可以把该三维结构看成是一个有效的二维结构. 此外, 样品在x方向上包含25个周期. 实验时, 样品被不锈钢金属夹固定在水箱中心. 针式换能器(Precision Acoustics NH1000)作为激发源被放置在样品下表面左端的第1个凹槽处, 其连接在串联波形发生器(Keysight 33500B)上, 在垂直于结构表面发射高斯超声脉冲. 针式水听器(Precision Acoustics NH1000)作为探测器被安装在超声波三维测量系统(Precision Acoustics Ltd UMS 3)的电动台上进行近场扫描, 它连接在数字存储示波器(Keysight InfiniiVision DSOX4052A)上用于采集时域超声信号. 水听器在样品上表面以3 mm为步径沿x方向和以1 mm为步径沿z方向进行扫描, 收集二维近场超声信号. 接着对声场做二维傅里叶变换, 得到以 ${k}_{z}$ 为参量沿 ${k}_{x}$ 方向上的能带结构, 最后选取 ${k}_{z}=0$ 的能带结构. , 分别展示了实验测量的 $S=0.3$ 和 $S=0.7$ 的体能带结构, 其中, 彩色图为实验结果, 黑色点为数值仿真结果, 结果符合得很好. 两个样品中的能带结构都在137—142.5 kHz的频率范围于 ${k}_{x}={\mathrm{\pi }}/a$ 附近打开带隙, 这为接下来研究的界面态提供了可行性.

3. 能谷陈数保证的拓扑界面态

接下来, 将研究由两个不同拓扑相的声子晶体构成的界面处的界面态. $S=0.3$ 和 $S=0.7$ 两个声子晶体下能带的能谷所对应的能谷陈数分别是–1/2 和1/2, 处于不同的能谷相. 它们之间的能谷陈数之差为1, 这保证由这两个声子晶体所构成的界面处一定存在着界面态. 利用有限元模拟软件计算界面态的本征声压场分布, 其中频率为142 kHz的本征场如所示. 此时, 声压场局域在界面处, 沿着远离界面的方向指数衰减. 为了与实验测量一致, 进行有限大小体系的数值模拟, 其中结构由两个分别为10个周期的 $S=0.3$ 和 $S=0.7$ 的声子晶体沿x方向拼接而成, z方向的长度为20 cm. 点源置于界面的凹槽处用于有效激励界面态, 首先记录三维声压场分布, 接着对声场做z方向的傅里叶变换, 从而得到以 ${k}_{z}$ 为参量的xy面内的二维声压场分布, 最后选取 ${k}_{z}=0$ 的声压场, 结果与二维数值仿真得到的本征声压场一致. 有限大小结构在频率为142 kHz的声压场分布如图3(c)所示, 这里只展示图3(b)中红色虚线区域内的声压场. 为了进行实验验证, 制作的样品如图3(a)右图所示, 水听器被放置在界面的凹槽处用于激励界面态. 网络分析仪连接水听器用于产生激励信号, 探测水听器连接在示波器上用于收集声信号. 与进行类似的数据处理, 可以得到频率为142 kHz的实验测量声压场, 如所示. 实验结果与数值计算结果吻合较好. 为了更好地展示界面态声压场的分布行为, 展示了固定 $y=0$ , 声压沿x方向上的分布情况, 可以清楚地发现声压场由界面处向两端呈指数衰减. 由此, 我们在一维合成水基声子晶体中实验证实界面态的存在.

$图 3 由两个不同能谷拓扑相水基声子晶体构成的界面处的界面态　(a)水下超声近场扫描实验平台示意图; (b)二维数值仿真得到的界面态本征声压场分布图; (c)三维有限大小结构数值仿真得到的界面态声压强分布图; (d)实验测量的界面态声压场分布图; (e)实验声压场在$ y=0 $沿着x方向的分布图. 图中声压场的频率均为142 kHz\r\nFig. 3. Interface states at the interface constructed by two waterborne phononic crystals with distinct valley topological phases: (a) Schematic diagram of the underwater ultrasonic near field scanning experimental platform; (b) simulated eigen pressure field distribution of the interface state in a pure two-dimensional structure; (c) simulated pressure field distribution of the interface state in a finite-size three-dimensional structure; (d) measured pressure field distribution of the interface state; (e) pressure field distribution along the x direction for $ y=0 $. All the frequencies of the pressure fields in this figure are 142 kHz.$

图 3 由两个不同能谷拓扑相水基声子晶体构成的界面处的界面态　(a)水下超声近场扫描实验平台示意图; (b)二维数值仿真得到的界面态本征声压场分布图; (c)三维有限大小结构数值仿真得到的界面态声压强分布图; (d)实验测量的界面态声压场分布图; (e)实验声压场在

$y=0$

沿着x方向的分布图. 图中声压场的频率均为142 kHz

Fig. 3. Interface states at the interface constructed by two waterborne phononic crystals with distinct valley topological phases: (a) Schematic diagram of the underwater ultrasonic near field scanning experimental platform; (b) simulated eigen pressure field distribution of the interface state in a pure two-dimensional structure; (c) simulated pressure field distribution of the interface state in a finite-size three-dimensional structure; (d) measured pressure field distribution of the interface state; (e) pressure field distribution along the x direction for

$y=0$

. All the frequencies of the pressure fields in this figure are 142 kHz.

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4. 结　论

本文利用双层结构之间的层间相对平移构成一个合成维度, 在一维水基声子晶体中实现了合成二维狄拉克点. 通过相对平移的变化, 使得狄拉克点打开带隙形成两个不同能谷拓扑相. 实验证实由不同能谷拓扑相构成的界面处存在界面态. 本文所提出的合成维度方法可以推广到二维或三维体系中, 并引入更多的合成维度, 用于研究更高维度体系的拓扑物理及其相关的输运现象, 为探索高维拓扑态提供了一种有效的途经. 此外, 本文设计的水基声子晶体, 有望为拓扑功能性水声器件的设计提供新思路.

References

[1]	Sigalas M M, Economou E N 1992 J. Sound Vib. 158 377 Google Scholar
[2]	Kushwaha M S, Halevi P, Dobrzynski L, Djafari-Rouhani B 1993 Phys. Rev. Lett. 71 2022 Google Scholar
[3]	Yang S, Page J H, Liu Z, Cowan M L, Chan C T, Sheng P 2004 Phys. Rev. Lett. 93 024301 Google Scholar
[4]	Zhang X, Liu Z Y 2004 Appl. Phys. Lett. 85 341 Google Scholar
[5]	Zhang S, Xia C G, Fang N 2011 Phys. Rev. Lett. 106 024301 Google Scholar
[6]	Mei J, Ma G C, Yang M Y, Yang Z Y, Wen W J, Sheng P 2012 Nat. Commun. 3 756 Google Scholar
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期刊类型引用(1)

梁谦，曹江涛，郝会乾. 弹性支撑条件下声子晶体角梁结构振动特性分析研究. 河北省科学院学报. 2024(05): 45-49 .

百度学术

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图 1 一维水基声子晶体的结构示意图

Figure 1. Schematic of the one-dimensional waterborne phononic crystal.

DownLoad: Full-Size Img PowerPoint

一维合成维度水基声子晶体的体能带　(a)三维能带结构, 黑色五角星表示狄拉克点; (b)不同的 $S$ 下能带结构随 ${k}_{x}$ 的变化, 蓝色和红色点表示 $S=0.3$ , 黑色点表示 $S=0.5$ , 紫色实线表示水线, A和B分别表示 ${k}_{x}={\mathrm{\pi }}/a$ 时上下两条能带的本征态; (c)态A, B关于 $S$ 的拓扑相图; (d), (e) $S=0.3$ 和 $S=0.7$ 时的实验测量(彩色图)和数值仿真(黑色点)体能带

Bulk band structure of the one-dimensional waterborne phononic crystal: (a) Three-dimensional band structure, black star represents the Dirac point; (b) bulk band structure as a function of ${k}_{x}$ for different $S$ , blue and red dots indicate the case of $S=0.3$ , while the black dots are for $S=0.5$ , the purple solid line represents the water line, A and B denote the eigen states in the upper and lower bands at ${k}_{x}={\mathrm{\pi }}/a$ ; (c) phase diagram of states A and B; (d), (e) measured and simulated bulk band structures for the cases of $S=0.3$ and $S=0.7$ , respectively.

DownLoad: Full-Size Img PowerPoint

由两个不同能谷拓扑相水基声子晶体构成的界面处的界面态　(a)水下超声近场扫描实验平台示意图; (b)二维数值仿真得到的界面态本征声压场分布图; (c)三维有限大小结构数值仿真得到的界面态声压强分布图; (d)实验测量的界面态声压场分布图; (e)实验声压场在 $y=0$ 沿着x方向的分布图. 图中声压场的频率均为142 kHz

Interface states at the interface constructed by two waterborne phononic crystals with distinct valley topological phases: (a) Schematic diagram of the underwater ultrasonic near field scanning experimental platform; (b) simulated eigen pressure field distribution of the interface state in a pure two-dimensional structure; (c) simulated pressure field distribution of the interface state in a finite-size three-dimensional structure; (d) measured pressure field distribution of the interface state; (e) pressure field distribution along the x direction for $y=0$ . All the frequencies of the pressure fields in this figure are 142 kHz.

DownLoad: Full-Size Img PowerPoint

[1]	Sigalas M M, Economou E N 1992 J. Sound Vib. 158 377 Google Scholar
[2]	Kushwaha M S, Halevi P, Dobrzynski L, Djafari-Rouhani B 1993 Phys. Rev. Lett. 71 2022 Google Scholar
[3]	Yang S, Page J H, Liu Z, Cowan M L, Chan C T, Sheng P 2004 Phys. Rev. Lett. 93 024301 Google Scholar
[4]	Zhang X, Liu Z Y 2004 Appl. Phys. Lett. 85 341 Google Scholar
[5]	Zhang S, Xia C G, Fang N 2011 Phys. Rev. Lett. 106 024301 Google Scholar
[6]	Mei J, Ma G C, Yang M Y, Yang Z Y, Wen W J, Sheng P 2012 Nat. Commun. 3 756 Google Scholar
[7]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[8]	Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057 Google Scholar
[9]	Wieder B J, Bradlyn B, Cano J, Wang Z, Vergniory M G, Elcoro L, Soluyanov A A, Felser C, Neupert T, Regnault N, Bernevig B A 2021 Nat. Rev. Mater. 7 196 Google Scholar
[10]	Lodge M S, Yang S A, Mukherjee S, Weber B 2021 Adv. Mater. 33 2008029 Google Scholar
[11]	Gilbert M J 2021 Commun. Phys. 4 70 Google Scholar
[12]	Bernevig B A, Felser C, Beidenkopf H 2022 Nature 603 41 Google Scholar
[13]	Chang C Z, Liu C X, MacDonald A H 2023 Rev. Mod. Phys. 95 011002 Google Scholar
[14]	He Q L, Hughes T L, Armitage N P, Tokura Y, Wang K L 2022 Nat. Mater. 21 15 Google Scholar
[15]	Zhang X, Xiao M, Cheng Y, Lu M H, Christensen J 2018 Commun. Phys. 1 97 Google Scholar
[16]	Ma G C, Xiao M, Chan C T 2019 Nat. Rev. Phys. 1 281 Google Scholar
[17]	Xue H R, Yang Y, Zhang B L 2022 Nat. Rev. Mater. 7 974 Google Scholar
[18]	Yves S, Ni X, Alu A 2022 Ann. New York Acad. Sci. 1517 63 Google Scholar
[19]	Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901 Google Scholar
[20]	Lu J Y, Qiu C Y, Ye L P, Fan X Y, Ke M Z, Zhang F, Liu Z Y 2017 Nat. Phys. 13 369 Google Scholar
[21]	Gao F, Xue H R, Yang Z Y, Lai K F, Yu Y, Lin X, Chong Y D, Shvets G, Zhang B L 2018 Nat. Phys. 14 140 Google Scholar
[22]	Zhang Z W, Tian Y, Wang Y H, Gao S X, Cheng Y, Liu X J, Christensen J 2018 Adv. Mater. 30 1803229 Google Scholar
[23]	Yan M, Lu J Y, Li F, Deng W Y, Huang X Q, Ma J H, Liu Z Y 2018 Nat. Mater. 17 993 Google Scholar
[24]	Zhu Z, Huang X Q, Lu J Y, Yan M, Li F, Deng W Y, Liu Z Y 2019 Phys. Rev. Appl. 12 024007 Google Scholar
[25]	Wu S, Wu Y, Mei J 2018 New J. Phys. 20 023051 Google Scholar
[26]	Wu X X, Fan H Y, Liu T, Gu Z M, Zhang R Y, Zhu J, Zhang X 2022 Nat. Commun. 13 6120 Google Scholar
[27]	Shen Y, Qiu C, Cai X, Ye L, Lu J Y, Ke M, Liu Z Y 2019 Appl. Phys. Lett. 114 023501 Google Scholar
[28]	Wang W, Chen Z G, Ma G C 2021 Phys. Rev. Lett. 127 214302 Google Scholar
[29]	Liu J J, Li Z W, Chen Z G, Tang W Y, Chen A, Liang B, Ma G C, Cheng J C 2022 Phys. Rev. Lett. 129 084301 Google Scholar
[30]	Chen H, Zhang H K, Wu Q, Huang Y, Nguyen H, Prodan E, Zhou X M, Huang G L 2021 Nat. Commun. 12 5028 Google Scholar

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1. 梁谦，曹江涛，郝会乾. 弹性支撑条件下声子晶体角梁结构振动特性分析研究. 河北省科学院学报. 2024(05): 45-49 . 百度学术
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