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

doi:10.7498/aps.70.20210712

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.

Keywords:

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 and School of Physics and Technology, Wuhan University, Wuhan 430072, China

Corresponding author: Wu Ying, phwuying@scut.edu.cn ; Lu Jiu-Yang, phjylu@scut.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11890701), the Guangdong Basic and Applied Basic Research Foundation, China (Grant Nos. 2019B151502012, 2021B1515020086, 2021A1515010347), and the China Postdoctoral Science Foundation (Grant No. 2020M672615)

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References

[1]	孙其诚, 何程, 卢明辉, 陈延峰 2017 物理学报 66 224203 Google Scholar Sun Q C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 66 224203 Google Scholar
[2]	Fleury R, Khanikaev A B, Alu A 2016 Nat. Commun. 7 11477 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[7]	Zhang X J, Xiao M, Cheng Y, Lu M H, Christensen J 2018 Commun. Phys. 1 97 Google Scholar
[8]	Huang X Q, Deng W Y, Li F, Lu J Y, Liu Z Y 2020 Phys. Rev. Lett. 124 206802 Google Scholar
[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 Google Scholar
[10]	Xue H R, Yang Y H, Gao F, Chong Y D, Zhang B L 2019 Nat. Mater. 18 108 Google Scholar
[11]	Yang Y T, Lu J Y, Yan M, Huang X Q, Deng W Y, Liu Z Y 2021 Phys. Rev. Lett. 126 156801 Google Scholar
[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 Google Scholar
[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 Google Scholar
[14]	Khanikaev A B, Fleury R, Mousavi S H, Alu A 2015 Nat. Commun. 6 8260 Google Scholar
[15]	贾鼎, 葛勇, 袁寿其, 孙宏祥 2019 物理学报 68 224301 Google Scholar Jia D, Ge Y, Yuan S Q, Sun H X 2019 Acta Phys. Sin. 68 224301 Google Scholar
[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 Google Scholar
[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 Google Scholar
[18]	Mousavi S H, Khanikaev A B, Wang Z 2015 Nat. Commun. 6 8682 Google Scholar
[19]	王健, 吴世巧, 梅军 2017 物理学报 66 224301 Google Scholar Wang J, Wu S Q, Mei J 2017 Acta Phys. Sin. 66 224301 Google Scholar
[20]	Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[24]	Fan H Y, Xia B Z, Tong L, Zheng S J, Yu D J 2019 Phys. Rev. Lett. 122 204301 Google Scholar
[25]	Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401 Google Scholar
[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 Google Scholar Geng Z G, Peng Y G, Shen Y X, Zhao D G, Zhu X F 2019 Acta Phys. Sin. 68 227802 Google Scholar
[28]	Wallace P R 1947 Phys. Rev. 71 622 Google Scholar
[29]	Yan M Z, Huang H Q, Zhang K N, et al. 2017 Nat. Commun. 8 257 Google Scholar
[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 Google Scholar
[31]	Chang T R, Xu S Y, Sanchez D S, et al. 2017 Phys. Rev. Lett. 119 026404 Google Scholar
[32]	Das S, Amit, Sirohi A, Yadav L, Gayen S, Singh Y, Sheet G 2018 Phys. Rev. B 97 014523 Google Scholar
[33]	Liang T, Gibson Q, Ali M N, Liu M H, Cava R J, Ong N P 2015 Nat. Mater. 14 280 Google Scholar
[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 Google Scholar
[35]	Wu X X, Li X, Zhang R Y, et al. 2020 Phys. Rev. Lett. 124 075501 Google Scholar
[36]	Horio M, Matt C E, Kramer K, et al. 2018 Nat. Commun. 9 3252 Google Scholar
[37]	Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Japan 74 1674 Google Scholar
[38]	Ezawa M 2013 Phys. Rev. B 88 161406 Google Scholar
[39]	Zhang F, MacDonald A H, Mele E J 2013 Proc. Natl. Acad. Sci. 110 10546 Google Scholar

Cited By

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.

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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.

DownLoad: Full-Size Img PowerPoint

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.

DownLoad: Full-Size Img PowerPoint

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).

DownLoad: Full-Size Img PowerPoint

[1]	孙其诚, 何程, 卢明辉, 陈延峰 2017 物理学报 66 224203 Google Scholar Sun Q C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 66 224203 Google Scholar
[2]	Fleury R, Khanikaev A B, Alu A 2016 Nat. Commun. 7 11477 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[7]	Zhang X J, Xiao M, Cheng Y, Lu M H, Christensen J 2018 Commun. Phys. 1 97 Google Scholar
[8]	Huang X Q, Deng W Y, Li F, Lu J Y, Liu Z Y 2020 Phys. Rev. Lett. 124 206802 Google Scholar
[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 Google Scholar
[10]	Xue H R, Yang Y H, Gao F, Chong Y D, Zhang B L 2019 Nat. Mater. 18 108 Google Scholar
[11]	Yang Y T, Lu J Y, Yan M, Huang X Q, Deng W Y, Liu Z Y 2021 Phys. Rev. Lett. 126 156801 Google Scholar
[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 Google Scholar
[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 Google Scholar
[14]	Khanikaev A B, Fleury R, Mousavi S H, Alu A 2015 Nat. Commun. 6 8260 Google Scholar
[15]	贾鼎, 葛勇, 袁寿其, 孙宏祥 2019 物理学报 68 224301 Google Scholar Jia D, Ge Y, Yuan S Q, Sun H X 2019 Acta Phys. Sin. 68 224301 Google Scholar
[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 Google Scholar
[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 Google Scholar
[18]	Mousavi S H, Khanikaev A B, Wang Z 2015 Nat. Commun. 6 8682 Google Scholar
[19]	王健, 吴世巧, 梅军 2017 物理学报 66 224301 Google Scholar Wang J, Wu S Q, Mei J 2017 Acta Phys. Sin. 66 224301 Google Scholar
[20]	Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[24]	Fan H Y, Xia B Z, Tong L, Zheng S J, Yu D J 2019 Phys. Rev. Lett. 122 204301 Google Scholar
[25]	Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401 Google Scholar
[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 Google Scholar Geng Z G, Peng Y G, Shen Y X, Zhao D G, Zhu X F 2019 Acta Phys. Sin. 68 227802 Google Scholar
[28]	Wallace P R 1947 Phys. Rev. 71 622 Google Scholar
[29]	Yan M Z, Huang H Q, Zhang K N, et al. 2017 Nat. Commun. 8 257 Google Scholar
[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 Google Scholar
[31]	Chang T R, Xu S Y, Sanchez D S, et al. 2017 Phys. Rev. Lett. 119 026404 Google Scholar
[32]	Das S, Amit, Sirohi A, Yadav L, Gayen S, Singh Y, Sheet G 2018 Phys. Rev. B 97 014523 Google Scholar
[33]	Liang T, Gibson Q, Ali M N, Liu M H, Cava R J, Ong N P 2015 Nat. Mater. 14 280 Google Scholar
[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 Google Scholar
[35]	Wu X X, Li X, Zhang R Y, et al. 2020 Phys. Rev. Lett. 124 075501 Google Scholar
[36]	Horio M, Matt C E, Kramer K, et al. 2018 Nat. Commun. 9 3252 Google Scholar
[37]	Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Japan 74 1674 Google Scholar
[38]	Ezawa M 2013 Phys. Rev. B 88 161406 Google Scholar
[39]	Zhang F, MacDonald A H, Mele E J 2013 Proc. Natl. Acad. Sci. 110 10546 Google Scholar

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