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Type-II Dirac points and edge transports in phononic crystal plates

Luo Quan-Bin Huang Xue-Qin Deng Wei-Yin Wu Ying Lu Jiu-Yang Liu Zheng-You

Type-II Dirac points and edge transports in phononic crystal plates

Luo Quan-Bin, Huang Xue-Qin, Deng Wei-Yin, Wu Ying, Lu Jiu-Yang, Liu Zheng-You
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  • The accidentally degenerate type-II Dirac points in sonic crystal has been realized recently. However, elastic phononic crystals with type-II Dirac points have not yet been explored. In this work, we design a two-dimensional phononic crystal plate in square lattice with type-II Dirac points for elastic waves. The type-II Dirac points, different from the type-I counterparts, have the tiled dispersions and thus the iso-frequency contours become crossed lines. By tuning structures to break the mirror symmetry, the degeneracies of the type-II Dirac points are lifted, leading to a band inversion. In order to have a further explanation, we also calculate the Berry curvatures of phononic crystals with opposite structure parameters, and it turns out that these two crystals hold opposite signs around the valley. The phononic crystal plates before and after the band inversion belong to different topological valley phases, whose direct consequence is that the topologically protected gapless interface states exist between two distinct topological phases. Topologically protected interface states are found by calculating the projected band structures of a supercell that contains two kinds of interfaces between two topological phases. Robustness of the interface transport is verified by comparing the transmission rate for perfect interface with that for defective interface. Moreover, owing to the special stress field distributions of the elastic plate waves, the boundaries of a single phononic crystal phase can similarly host the gapless boundary states, which is found by calculating the projected band structures of a supercell with a single phase, thus having two free boundaries on the edges. This paper extends the two-dimensional Dirac points and valley states in graphene-like systems to the type-II cases, and obtains in the same structure the gapless interface and boundary propagations. Owing to the simple design scheme of the structure, the phononic crystal plates can be fabricated and scaled to a small size. Our system provides a feasible way of constructing high-frequency elastic wave devices.
      Corresponding author: Wu Ying, phwuying@scut.edu.cn ; Lu Jiu-Yang, phjylu@scut.edu.cn
    [1]

    孙其诚, 何程, 卢明辉, 陈延峰 2017 物理学报 66 224203

    Sun Q C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 66 224203

    [2]

    Fleury R, Khanikaev A B, Alu A 2016 Nat. Commun. 7 11477

    [3]

    He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904

    [4]

    Deng W Y, Lu J Y, Li F, Huang X Q, Yan M, Ma J H, Liu Z Y 2019 Nat. Commun. 10 1769

    [5]

    Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368

    [6]

    Deng W Y, Huang X Q, Lu J Y, Li F, Ma J H, Chen S Q, Liu Z Y 2020 Sci. China-Phys. Mech. Astron. 63 287032

    [7]

    Zhang X J, Xiao M, Cheng Y, Lu M H, Christensen J 2018 Commun. Phys. 1 97

    [8]

    Huang X Q, Deng W Y, Li F, Lu J Y, Liu Z Y 2020 Phys. Rev. Lett. 124 206802

    [9]

    Wei Q, Zhang X W, Deng W Y, Lu J Y, Huang X Q, Yan M, Chen G, Liu, Z Y, Jia S T 2021 Nat. Mater. 20 812

    [10]

    Xue H R, Yang Y H, Gao F, Chong Y D, Zhang B L 2019 Nat. Mater. 18 108

    [11]

    Yang Y T, Lu J Y, Yan M, Huang X Q, Deng W Y, Liu Z Y 2021 Phys. Rev. Lett. 126 156801

    [12]

    Yang Z J, Gao F, Shi X H, Lin X, Gao Z, Chong Y D, Zhang B L 2015 Phys. Rev. Lett. 114 114301

    [13]

    Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016

    [14]

    Khanikaev A B, Fleury R, Mousavi S H, Alu A 2015 Nat. Commun. 6 8260

    [15]

    贾鼎, 葛勇, 袁寿其, 孙宏祥 2019 物理学报 68 224301

    Jia D, Ge Y, Yuan S Q, Sun H X 2019 Acta Phys. Sin. 68 224301

    [16]

    Ding Y J, Peng Y G, Zhu Y F, Fan X D, Yang J, Liang B, Zhu X F, Wan X G, Cheng J C 2019 Phys. Rev. Lett. 122 014302

    [17]

    He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124

    [18]

    Mousavi S H, Khanikaev A B, Wang Z 2015 Nat. Commun. 6 8682

    [19]

    王健, 吴世巧, 梅军 2017 物理学报 66 224301

    Wang J, Wu S Q, Mei J 2017 Acta Phys. Sin. 66 224301

    [20]

    Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901

    [21]

    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

    [22]

    Deng W Y, Huang X Q, Lu J Y, Peri V, Li F, Huber S D, Liu Z Y 2020 Nat. Commun. 11 3227

    [23]

    Peng Y G, Li Y, Shen Y X, Geng Z G, Zhu J, Qiu C W, Zhu X F 2019 Phys. Rev. Research. 1 033149

    [24]

    Fan H Y, Xia B Z, Tong L, Zheng S J, Yu D J 2019 Phys. Rev. Lett. 122 204301

    [25]

    Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401

    [26]

    Süsstrunk R, Huber S D 2015 Science 349 47

    [27]

    耿治国, 彭玉桂, 沈亚西, 赵德刚, 祝雪丰 2019 物理学报 68 227802

    Geng Z G, Peng Y G, Shen Y X, Zhao D G, Zhu X F 2019 Acta Phys. Sin. 68 227802

    [28]

    Wallace P R 1947 Phys. Rev. 71 622

    [29]

    Yan M Z, Huang H Q, Zhang K N, et al. 2017 Nat. Commun. 8 257

    [30]

    Sadeddine S, Enriquez H, Bendounan A, Das P K, Voborni I, Kara A, Mayne A J, Sirotti F, Dujardin G, Oughaddou H 2017 Sci. Rep. 7 44400

    [31]

    Chang T R, Xu S Y, Sanchez D S, et al. 2017 Phys. Rev. Lett. 119 026404

    [32]

    Das S, Amit, Sirohi A, Yadav L, Gayen S, Singh Y, Sheet G 2018 Phys. Rev. B 97 014523

    [33]

    Liang T, Gibson Q, Ali M N, Liu M H, Cava R J, Ong N P 2015 Nat. Mater. 14 280

    [34]

    Politano A, Chiarello G, Ghosh B, Sadhukhan K, Kuo C N, Lue C S, Pellegrini V, Agarwal A 2018 Phys. Rev. Lett. 121 086804

    [35]

    Wu X X, Li X, Zhang R Y, et al. 2020 Phys. Rev. Lett. 124 075501

    [36]

    Horio M, Matt C E, Kramer K, et al. 2018 Nat. Commun. 9 3252

    [37]

    Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Japan 74 1674

    [38]

    Ezawa M 2013 Phys. Rev. B 88 161406

    [39]

    Zhang F, MacDonald A H, Mele E J 2013 Proc. Natl. Acad. Sci. 110 10546

  • 图 1  具有第二类狄拉克点的声子晶体板 (a)正方晶格声子晶体板和单胞结构示意图; (b)声子晶体沿布里渊区高对称线的能带结构, 插图为第一布里渊区; (c)第二类狄拉克点附近的能带结构, 绿色平面为简并点所在的等频面

    Figure 1.  Phononic crystal plates with type-II Dirac point. (a) Schematics of the phononic crystal plates and the unit cell. (b) Dispersions of the phononic crystal plates along the high-symmetry lines. Inset: the first Brillouin zone. (c) Dispersions around the type-II Dirac point. Green plane shows the isofrequency plane at the frequency of the type-II Dirac point.

    图 2  镜面对称破缺以及拓扑相变 (a)镜面对称破缺声子晶体板($ \Delta h=2.5\;\;\mathrm{m}\mathrm{m} $)的能带结构, 插图为原胞示意图; (b) $ D $点本征频率随$ \Delta h $的变化关系, 插图给出了$ \Delta h=0 $前后上下能带对应的本征模态; (c), (d)分别为$ \Delta h=\pm 2.5\;\;\mathrm{m}\mathrm{m} $时第一条能带的贝里曲率分布

    Figure 2.  Breaking of mirror symmetry and topological phase transition: (a) Dispersions of the phononic crystal plate with $ \Delta h=2.5\;\mathrm{m}\mathrm{m} $, where the inset is the diagram of unit cell; (b) eigenfrequencies at $ D $ point versus $ \Delta h $, where the insets show eigenmodes before and after the band inversion; (c), (d) Berry curvature distributions of the first bands for phononic crystal plates with $ \Delta h=2.5\;\mathrm{m}\mathrm{m} $ and $ \Delta h=-2.5\;\mathrm{m}\mathrm{m} $, respectively.

    图 3  声子晶体板的界面态传输 (a)在y方向上依次由ABA拼成的声子晶体板; (b) ABA结构的投影能带; (c)分别表示(b)中蓝色和红色五角星标记的位移本征场分布, 其中形变表示总位移, 彩色条表示z方向上的位移, 绿色虚线为边界所在位置; (d)含缺陷的BA界面态传输, 绿线为边界位置, 绿色五角星为位移沿z方向的偏振点源, 激发频率$ 22.3\;\mathrm{k}\mathrm{H}\mathrm{z} $; (e)蓝色点线和红色点线分别是无缺陷和存在缺陷时的两种边界态传输率; (f) AB和BA界面态的剪切应力分布

    Figure 3.  Interface state transports of phononic crystal plates. (a) Schematic of sandwich structure ABA successively consisting phononic crystal plates of phases A and B along the y direction. (b) Projected dispersions of the sandwich structure ABA. (c) Displacement field eigenmodes marked by the blue and red star in panel (b), where the deformation is the total displacement. The color bar is the displacement in z direction, and the green dotted line is the boundary position. (d) Interface state transports along the BA interface with defect (denoted by green line). Green star denotes the point source polarized along the z direction and operating at $ f=22.3\;\mathrm{k}\mathrm{H}\mathrm{z} $. (e) Transmissions for perfect and defective interfaces. (f) Shear stress distributions corresponding to AB and BA interface states.

    图 4  声子晶体板的边界态传输 (a)声子晶体B在自由边界下的投影能带, 插图为边界态位移本征场分布(仅存在于下边界); (b), (c)分别是无缺陷和存在缺陷时两种沿自由边界传播的边界态传输; (d)蓝色点线和红色点线分别是对应(b)和(c)情形的透射率

    Figure 4.  Boundary state transports of phononic crystal plates. (a) Projected dispersions of phononic crystal plates of phase B. Inset: the displacement field eigenmodes of the boundary state, locating at the bottom free boundary. (b), (c) Boundary state transports along the free boundaries without and with defect. (d) Transmissions for two distinct boundaries corresponding to (b) and (c).

  • [1]

    孙其诚, 何程, 卢明辉, 陈延峰 2017 物理学报 66 224203

    Sun Q C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 66 224203

    [2]

    Fleury R, Khanikaev A B, Alu A 2016 Nat. Commun. 7 11477

    [3]

    He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904

    [4]

    Deng W Y, Lu J Y, Li F, Huang X Q, Yan M, Ma J H, Liu Z Y 2019 Nat. Commun. 10 1769

    [5]

    Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368

    [6]

    Deng W Y, Huang X Q, Lu J Y, Li F, Ma J H, Chen S Q, Liu Z Y 2020 Sci. China-Phys. Mech. Astron. 63 287032

    [7]

    Zhang X J, Xiao M, Cheng Y, Lu M H, Christensen J 2018 Commun. Phys. 1 97

    [8]

    Huang X Q, Deng W Y, Li F, Lu J Y, Liu Z Y 2020 Phys. Rev. Lett. 124 206802

    [9]

    Wei Q, Zhang X W, Deng W Y, Lu J Y, Huang X Q, Yan M, Chen G, Liu, Z Y, Jia S T 2021 Nat. Mater. 20 812

    [10]

    Xue H R, Yang Y H, Gao F, Chong Y D, Zhang B L 2019 Nat. Mater. 18 108

    [11]

    Yang Y T, Lu J Y, Yan M, Huang X Q, Deng W Y, Liu Z Y 2021 Phys. Rev. Lett. 126 156801

    [12]

    Yang Z J, Gao F, Shi X H, Lin X, Gao Z, Chong Y D, Zhang B L 2015 Phys. Rev. Lett. 114 114301

    [13]

    Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016

    [14]

    Khanikaev A B, Fleury R, Mousavi S H, Alu A 2015 Nat. Commun. 6 8260

    [15]

    贾鼎, 葛勇, 袁寿其, 孙宏祥 2019 物理学报 68 224301

    Jia D, Ge Y, Yuan S Q, Sun H X 2019 Acta Phys. Sin. 68 224301

    [16]

    Ding Y J, Peng Y G, Zhu Y F, Fan X D, Yang J, Liang B, Zhu X F, Wan X G, Cheng J C 2019 Phys. Rev. Lett. 122 014302

    [17]

    He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124

    [18]

    Mousavi S H, Khanikaev A B, Wang Z 2015 Nat. Commun. 6 8682

    [19]

    王健, 吴世巧, 梅军 2017 物理学报 66 224301

    Wang J, Wu S Q, Mei J 2017 Acta Phys. Sin. 66 224301

    [20]

    Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901

    [21]

    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

    [22]

    Deng W Y, Huang X Q, Lu J Y, Peri V, Li F, Huber S D, Liu Z Y 2020 Nat. Commun. 11 3227

    [23]

    Peng Y G, Li Y, Shen Y X, Geng Z G, Zhu J, Qiu C W, Zhu X F 2019 Phys. Rev. Research. 1 033149

    [24]

    Fan H Y, Xia B Z, Tong L, Zheng S J, Yu D J 2019 Phys. Rev. Lett. 122 204301

    [25]

    Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401

    [26]

    Süsstrunk R, Huber S D 2015 Science 349 47

    [27]

    耿治国, 彭玉桂, 沈亚西, 赵德刚, 祝雪丰 2019 物理学报 68 227802

    Geng Z G, Peng Y G, Shen Y X, Zhao D G, Zhu X F 2019 Acta Phys. Sin. 68 227802

    [28]

    Wallace P R 1947 Phys. Rev. 71 622

    [29]

    Yan M Z, Huang H Q, Zhang K N, et al. 2017 Nat. Commun. 8 257

    [30]

    Sadeddine S, Enriquez H, Bendounan A, Das P K, Voborni I, Kara A, Mayne A J, Sirotti F, Dujardin G, Oughaddou H 2017 Sci. Rep. 7 44400

    [31]

    Chang T R, Xu S Y, Sanchez D S, et al. 2017 Phys. Rev. Lett. 119 026404

    [32]

    Das S, Amit, Sirohi A, Yadav L, Gayen S, Singh Y, Sheet G 2018 Phys. Rev. B 97 014523

    [33]

    Liang T, Gibson Q, Ali M N, Liu M H, Cava R J, Ong N P 2015 Nat. Mater. 14 280

    [34]

    Politano A, Chiarello G, Ghosh B, Sadhukhan K, Kuo C N, Lue C S, Pellegrini V, Agarwal A 2018 Phys. Rev. Lett. 121 086804

    [35]

    Wu X X, Li X, Zhang R Y, et al. 2020 Phys. Rev. Lett. 124 075501

    [36]

    Horio M, Matt C E, Kramer K, et al. 2018 Nat. Commun. 9 3252

    [37]

    Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Japan 74 1674

    [38]

    Ezawa M 2013 Phys. Rev. B 88 161406

    [39]

    Zhang F, MacDonald A H, Mele E J 2013 Proc. Natl. Acad. Sci. 110 10546

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  • Received Date:  14 April 2021
  • Accepted Date:  13 May 2021
  • Available Online:  07 June 2021
  • Published Online:  20 September 2021

Type-II Dirac points and edge transports in phononic crystal plates

    Corresponding author: Wu Ying, phwuying@scut.edu.cn
    Corresponding author: Lu Jiu-Yang, phjylu@scut.edu.cn
  • 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 and School of Physics and Technology, Wuhan University, Wuhan 430072, China

Abstract: The accidentally degenerate type-II Dirac points in sonic crystal has been realized recently. However, elastic phononic crystals with type-II Dirac points have not yet been explored. In this work, we design a two-dimensional phononic crystal plate in square lattice with type-II Dirac points for elastic waves. The type-II Dirac points, different from the type-I counterparts, have the tiled dispersions and thus the iso-frequency contours become crossed lines. By tuning structures to break the mirror symmetry, the degeneracies of the type-II Dirac points are lifted, leading to a band inversion. In order to have a further explanation, we also calculate the Berry curvatures of phononic crystals with opposite structure parameters, and it turns out that these two crystals hold opposite signs around the valley. The phononic crystal plates before and after the band inversion belong to different topological valley phases, whose direct consequence is that the topologically protected gapless interface states exist between two distinct topological phases. Topologically protected interface states are found by calculating the projected band structures of a supercell that contains two kinds of interfaces between two topological phases. Robustness of the interface transport is verified by comparing the transmission rate for perfect interface with that for defective interface. Moreover, owing to the special stress field distributions of the elastic plate waves, the boundaries of a single phononic crystal phase can similarly host the gapless boundary states, which is found by calculating the projected band structures of a supercell with a single phase, thus having two free boundaries on the edges. This paper extends the two-dimensional Dirac points and valley states in graphene-like systems to the type-II cases, and obtains in the same structure the gapless interface and boundary propagations. Owing to the simple design scheme of the structure, the phononic crystal plates can be fabricated and scaled to a small size. Our system provides a feasible way of constructing high-frequency elastic wave devices.

    • 声子晶体由声介质材料在空间中周期性排列构成, 可以调控在其中传播的声波和弹性波, 形成类似电子在天然晶体中的能带结构[1]. 声子晶体的宏观特性使其几何结构和单元间的耦合强度均可精确调控, 从而可以人工设计丰富多样的声子晶体能带[2-8]. 声子晶体借助这一优势, 结合凝聚态物理中的拓扑能带理论, 产生了一系列重要的发现, 实现了多种声学拓扑态.

      拓扑声子晶体根据其体能带带隙的存在与否可以分为声学拓扑绝缘体和声学拓扑半金属[7,8]; 而根据其边缘态的维度和自身维度的比较又可分为一阶拓扑态和高阶拓扑态[9-11]. 经典的声学(一阶)拓扑绝缘体有多种实现机制, 包括: 利用声学环流打破时间反演对称性的声学陈数拓扑绝缘体[12-16], 利用能带反转机制实现的声学量子自旋拓扑绝缘体[17-19], 利用空间镜面或反演对称性破缺实现的声学能谷拓扑绝缘体[20,21], 以及通过引入赝自旋-轨道耦合的声学自旋陈数拓扑绝缘体[5,22]等. 不同于高阶声学拓扑态中的角态和棱态[23-25], 上述的声学绝缘体的显著特征是, 体能带具有非平庸的拓扑特性, 从而可以诱导出沿着边缘传播的手征性或螺旋性边缘态[21,26,27], 具有良好的声学传输特性. 声学拓扑态的实现, 不仅证明声子晶体作为理想平台可以用来展示并研究新奇的拓扑物理现象, 同时拓展了传统的声学理论, 为设计基于拓扑声物理的声学器件奠定了基础.

      狄拉克点最早在二维石墨烯中受到关注[28-30]: 石墨烯中的电子输运在其费米面附近呈现锥状的线性色散关系, 其低能方程符合洛伦兹不变性下的无质量狄拉克方程, 因此得名. 当狄拉克锥的倾斜程度足够大时, 费米面的几何形状将会由点状变成交叉的线状, 被称为(二维)第二类狄拉克锥. 而此前的狄拉克锥则被称为第一类狄拉克锥. 具有第二类狄拉克锥的半金属具有高导电率[31,32]、各向异性的磁导率[33,34]等特点. 在声学体系中, 能带具有第一类狄拉克锥的声子晶体已得到广泛研究, 最近第二类狄拉克锥也在空气介质的声子晶体中实现[35]. 相较于空气声子晶体, 弹性波声子晶体具有较强的抗干扰能力和较低的传播损耗等优势, 但弹性波声子晶体的设计却因为难以类比到紧束缚模型而更加困难.

      基于上述背景, 本文将研究弹性声子晶体中的第二类狄拉克锥及相关的弹性波传输现象. 采用一种结构简单的二维声子晶体板, 板上传播的弹性波受正方晶格排布的矩形凸台阵列调制, 可以实现存在于第一布里渊区边界上的第二类狄拉克锥色散. 受镜面对称保护, 在微调矩形凸台高度时, 第二类狄拉克点不会被破坏打开带隙, 而是沿布里渊区边界微小移动[35,36]. 调节引入的结构参数$ \Delta h $, 声子晶体板会因为镜面对称破缺而在第二类狄拉克点能带简并处打开带隙; 进一步利用符号相反的$ \Delta h $, 可以实现声子晶体的能带反转, 构建不同的谷相. 将两个不同谷相的声子晶体拼接, 可以在其界面上实现拓扑界面态. 更进一步, 基于镜面对称性构建了弹性波在单一绝缘相声子晶体自由边界上的拓扑传输.

    2.   声子晶体板中的第二类狄拉克锥
    • 图1(a)所示, 本文在一个金属薄板上构建矩形凸台阵列, 薄板厚度$ t=1\;\mathrm{m}\mathrm{m} $. 矩形凸台以正方晶格分布, 晶格常数$ a=10.0\;\mathrm{m}\mathrm{m} $, 凸台的高度、长度和宽度分别为$ {h}_{0}=4.0\;\mathrm{m}\mathrm{m} $, $ b=8.0\;\mathrm{m}\mathrm{m} $$c= $$ 6.0\;\mathrm{m}\mathrm{m}$. 薄板和凸台均为具有以下参数的钢材料: 密度$ \rho =8.0\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $, 杨氏模量$ E=193.0\;\mathrm{G}\mathrm{P}\mathrm{a} $, 泊松比$ \nu =0.26 $. 本文的模拟计算由数值仿真软件COMSOL完成. 在计算声子晶体板的单体色散时, 使用固体结构模块, 在原胞边界上设置Floquet-Bloch周期性边界条件, 得到沿着布里渊区高对称线$\varGamma \text{-} Y \text{-} M \text{-} X \text{-} \varGamma$的能带结构(见图1(b)), 可以看到, 在第一布里渊区边界上($ YM $之间, $ {k}_{y}=\mathrm{\pi }/a $)存在第一和第二条能带的简并点(点D), 频率$ f=25.0\;\mathrm{k}\mathrm{H}\mathrm{z} $. D点附近的两条能带线性交叉, 且沿$ {k}_{x} $方向上两条能带斜率符号一致(都大于零), 该简并点因此为第二类狄拉克点. 为进一步展示第二类狄拉克点的能带结构, 在图1(c)中展示了$ D $点附近能带结构随${k}_{x} \text{-} {k}_{y}$的变化, 其中蓝色和红色曲面分别是第一和第二条能带, 它们形成两个严重倾斜的圆锥. 绿色平面标志简并点所在的频率, 该等频面恰好截取第一和第二条能带, 截线在$ D $点附近呈线性交叉(${k}_{x} \text{-} {k}_{y}$面上的蓝色和红色曲线). 简并点所在的等频面为线性交叉而非一个孤立的点, 正是第二类狄拉克点相对于第一类狄拉克点的不同之处. 由于时间反演对称性的存在, 第一布里渊区中和$ D $点相对的时间反演对称位置$ D' $点上还存在另外一个第二类狄拉克点. 破坏空间对称性, 将同时打开$ D $$ D' $点处的第二类狄拉克点简并, 得到不同的弹性波能谷拓扑相.

      Figure 1.  Phononic crystal plates with type-II Dirac point. (a) Schematics of the phononic crystal plates and the unit cell. (b) Dispersions of the phononic crystal plates along the high-symmetry lines. Inset: the first Brillouin zone. (c) Dispersions around the type-II Dirac point. Green plane shows the isofrequency plane at the frequency of the type-II Dirac point.

    3.   镜面对称破缺导致的能带反转
    • 第一布里渊区边界上的第二类狄拉克点由于受$ y $方向上的镜面对称性保护, 在仅改变矩形凸台的高度$ {h}_{0} $时, 其只会沿边界微小移动. 为了打开第二类狄拉克点处的能带简并, 需要破缺体系在$ y $方向上的镜面对称性. 这里选择一种简单的方案, 把矩形凸台沿$ y $方向分为宽度相等的两部分, 对这两部分的凸台分别增加和减去高度$ \Delta h $. 于是, $ \Delta h\ne 0 $对应于镜面对称性破缺的情形, 第二类狄拉克点简并被打开. 图2(a)展示了$ \Delta h=2.5\;\mathrm{m}\mathrm{m} $时声子晶体板的能带结构, 其在$ YM $上存在从$ 21.5\;\mathrm{k}\mathrm{H}\mathrm{z} $$ 22.2\;\mathrm{k}\mathrm{H}\mathrm{z} $的频率带隙. 对参数$ \Delta h $进行扫描, 可以观察$ D $点的本征频率和本征模态随$ \Delta h $的变化, 从而分析系统的拓扑相变过程. 结果如图2(b)所示, 在$ \Delta h < 0 $$ \Delta h=0 $再到$ \Delta h > 0 $的过程中, $ D $点经历了从带隙打开到闭合到再打开的变化; 从$ \Delta h=0 $前后能带本征场图的变化可以看到, 上下能带的振动模态带隙闭合前后发生了反转.

      Figure 2.  Breaking of mirror symmetry and topological phase transition: (a) Dispersions of the phononic crystal plate with $ \Delta h=2.5\;\mathrm{m}\mathrm{m} $, where the inset is the diagram of unit cell; (b) eigenfrequencies at $ D $ point versus $ \Delta h $, where the insets show eigenmodes before and after the band inversion; (c), (d) Berry curvature distributions of the first bands for phononic crystal plates with $ \Delta h=2.5\;\mathrm{m}\mathrm{m} $ and $ \Delta h=-2.5\;\mathrm{m}\mathrm{m} $, respectively.

      为了进一步表征弹性声子晶体能带的拓扑特性, 分别计算了$ \Delta h=\pm 2.5\;\mathrm{m}\mathrm{m} $两种情况下第一条能带的贝里曲率在布里渊区中的分布, 如图2(c)图2(d)所示. 贝里曲率定义为$\varOmega \left({\boldsymbol{k}}\right)=\nabla \times $$ \left\langle {{u}_{{\boldsymbol{k}}}|\mathrm{i}{\nabla }_{{\boldsymbol{k}}}|{u}_{{\boldsymbol{k}}}} \right\rangle$, 其中$ \left\langle {{u}_{{\boldsymbol{k}}}|\mathrm{i}{\nabla }_{{\boldsymbol{k}}}|{u}_{{\boldsymbol{k}}}} \right\rangle $被称为贝里联络, 表示为动量空间中的梯度算符在布洛赫态下的期望. 计算时将布里渊区离散化为许多单元, 从而把贝里曲率表示为贝里联络沿单元边界的求和再除以单元面积[37]. 贝里曲率在$ D $点和$ D' $点分别出现大小相等符号相反的峰值, 把这两个点及其附近的区域称为$ D $$ D' $能谷. 在低能近似下, 对贝里曲率在$ D $能谷所在半个布里渊区内积分(除以$ 2\mathrm{\pi } $), 得到对应于两种情形下$ D $能谷的谷陈数${C}_{D}= \dfrac{1}{2\mathrm{\pi } } \displaystyle\int \varOmega \left({\boldsymbol{k}}\right){\mathrm{d}}^{2}{\boldsymbol{k}}=\mp 1/2$. 时间反演对称性的存在使第一条能带$ D $$ D' $能谷的总谷陈数为零.

    4.   声子晶体板的弹性波传输
    • 具有相反符号$ \Delta h $的弹性声子晶体属于不同的能谷相. 将$ \Delta h=-2.5\;\mathrm{m}\mathrm{m} $的声子晶体(记为A)和$ \Delta h=2.5\;\mathrm{m}\mathrm{m} $的声子晶体(记为B)沿$ y $方向以ABA的方式拼接, 排布示意图如图3(a)所示, 实际计算时$ y $方向共有46个周期单元, 两端设为连续边界条件. 如此组成的ABA结构存在AB和BA两种不同界面, 计算得到的投影能带如图3(b)所示, 其中灰色部分为体能带在界面方向上的投影, 蓝线和红线分别对应AB界面和BA界面上的界面态色散. 两条界面态均具有无带隙的特征, 即色散贯穿整个体能带的带隙频率范围. 图3(c)分别给出了投影能带中蓝、红五角星标记的界面态位移本征场及其在界面上的局部放大图. AB和BA界面上的界面态具有明显不同的场分布特征, AB界面态在界面处$ z $方向上的位移为零, 而BA界面态在界面处$ z $方向上的位移极大, 这样的场分布是由于两个界面都具有镜面对称性, 且AB和BA界面态分别具有奇宇称和偶宇称的镜面对称性.

      Figure 3.  Interface state transports of phononic crystal plates. (a) Schematic of sandwich structure ABA successively consisting phononic crystal plates of phases A and B along the y direction. (b) Projected dispersions of the sandwich structure ABA. (c) Displacement field eigenmodes marked by the blue and red star in panel (b), where the deformation is the total displacement. The color bar is the displacement in z direction, and the green dotted line is the boundary position. (d) Interface state transports along the BA interface with defect (denoted by green line). Green star denotes the point source polarized along the z direction and operating at $ f=22.3\;\mathrm{k}\mathrm{H}\mathrm{z} $. (e) Transmissions for perfect and defective interfaces. (f) Shear stress distributions corresponding to AB and BA interface states.

      无带隙界面态的存在源自能谷态的体-边对应关系[38,39], 色散位于$ {k}_{x}=\mathrm{\pi }/a $左侧和右侧的界面态分别由$ D $能谷和$ {D}' $能谷诱导. 对于$ D $能谷, AB界面左侧(沿$ x $正方向看)的声子晶体A的谷陈数$ {C}_{D}^{\mathrm{A}}=-1/2 $, 右侧B的谷陈数$ {C}_{D}^{\mathrm{B}}=+1/2 $, 于是$ \Delta {C}_{D}^{\mathrm{A}\mathrm{B}}={C}_{D}^{\mathrm{A}}-{C}_{D}^{\mathrm{B}}=-1 $, 即在AB界面上存在$ D $能谷投影点附近群速度为负的界面态; 同理, 对于BA界面, $ \Delta {C}_{D}^{\mathrm{B}\mathrm{A}}={C}_{D}^{\mathrm{B}}-{C}_{D}^{\mathrm{A}}=1 $, 该界面上可以存在由$ D $能谷诱导的群速度为正的界面态. $ {D}' $能谷的情形可由$ D $能谷时间反演直接得到. 受能谷拓扑保护, 界面态对于界面上的弯折和缺陷具有一定的抗反射特性. 我们在BA界面上引入了一个小的弯折, 破坏了$ x $方向上的晶格平移对称性. 图3(d)展示了在外界激励下的弹性波传输情况, 其中的计算区域由$ 30\times 30 $个周期单元拼接而成, 四周均设置为低反射边界条件, 绿线表示BA界面所在位置. 在BA界面左端(五角星处)放置$ z $方向偏振的弹性波源, 可以激发群速度向右的由$ D $能谷诱导的弹性波界面态($ {D}' $能谷诱导的界面态群速度向左, 无法与左端的激励源耦合), 该界面态很好地跨过缺陷区域在界面上向右传播. 同时由于体带隙的存在, $ z $向偏振点源激发出界面态被很好地局域在边界处. 图3(e)给出了不同频率激发下无缺陷和存在缺陷的两种不同界面的透射率, 可以看到, 在体带隙频率范围内两种边界具有很高的透射率, 两条曲线几乎重合表明了该界面态具有一定的抗缺陷反射能力.

    • 和界面态的位移场分布类似, 界面上的镜面对称性对弹性应力分布同样有约束. 对于薄板中的弹性波, BA界面态在界面处底板上沿$ z $方向的剪切应力分布为零, 即$ {\sigma }_{yz}=0 $(图3(f)左下), 而AB界面态在界面处底板上的剪切应力则具有非零分布(图3(f)右上). BA界面态在界面处独特的应力分布启发我们是否可以只利用声子晶体B构建沿自由边界传播的表面态. 因此考察了具有自由边界的单一能谷相声子晶体. 声子晶体B组成的超胞($ y $方向上15个周期单元, 两端为自由边界)的投影能带图如图4(a)所示, 其中插图为五角星标记点的本征场. 可以看到, 边界态仅局域在下边界传播, 并且单一能谷相声子晶体的边界态能带和图3(b)中的BA界面态能带十分相似, 也具有无带隙的特征. 这表明声子晶体B的下边界, 由于提供了BA界面态所需的零剪切应力条件, 从而可以很好地支持边界态的传输; 另一方面, 由于AB界面态的剪切应力分布与自由边界完全不同, 声子晶体B的上边界在不施加外应力的自由边界下无法支持边界态的存在.

      Figure 4.  Boundary state transports of phononic crystal plates. (a) Projected dispersions of phononic crystal plates of phase B. Inset: the displacement field eigenmodes of the boundary state, locating at the bottom free boundary. (b), (c) Boundary state transports along the free boundaries without and with defect. (d) Transmissions for two distinct boundaries corresponding to (b) and (c).

      这里的边界态可以在外界激励下沿着声子晶体的自由边界传播, 并且同样具有一定的抗反射能力. 构建的声子晶体B是由20 × 15个单胞组成, 如图4(b)所示, 左右边界为低反射边界, 其余均为自由边界, 在左下角$ z $向偏振的声源激发下, 弹性波局域在下底边, 沿自由边界向右传输. 为表征边界态传输的稳定性, 在自由边界上去除两个单胞用以构建缺陷, 如图4(c)所示. 可以发现, 同一频率下的弹性波沿自由边界绕过该缺陷近乎无反射的向前传播. 在图4(d)中进一步给出了不同频率下, 两种不同边界的弹性波透射. 可以看到, 弹性波的透射率均接近于1且两条曲线高度重合, 这表明边界态也具有较好的抗缺陷散射能力. 利用边界态传输, 使得我们可以脱离界面的限制, 只用单一声子晶体相实现稳定的传输效果, 为声子晶体弹性波传输迈向实际应用提供了新的方案.

    5.   结 论
    • 本文在正方晶格声子晶体板中实现了弹性波的第二类狄拉克锥色散, 通过调节几何参数破坏体系的狄拉克点简并, 实现了由第二类狄拉克点诱导的声学能谷态, 并构建了不同能谷态之间的受拓扑保护的无带隙界面态. 不仅如此, 利用弹性界面态的特殊应力分布, 在单一能谷相声子晶体板的自由边界上同样构建了无带隙边界态. 通过数值模拟验证了这两种不同机制产生的边缘传输都具有稳定的抗反射传输特性. 我们的研究丰富了弹性波的调控方式, 为新型声学拓扑器件的设计提供了参考.

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