声子晶体中的表面声波赝自旋模式和拓扑保护声传输

王一鹤; 张志旺; 程营; 刘晓峻

doi:10.7498/aps.68.20191363

摘要

声子晶体中声表面波的调控在表面波应用方面有重要意义, 拓扑声学理论为声子晶体表面波调控提供了新的思路. 本文通过在硬质基板上排布蜂窝状晶格的空气圆柱孔阵列实现了结构表面局域的声表面波传播, 并可在布里渊区K点上形成狄拉克锥. 基于能带折叠理论构造复合胞, 在复合胞布里渊区中心处实现了由二重简并偶极子态(p态)和四极子态(d态)组成的双狄拉克锥. 通过扩大或缩小复合胞内相邻单元的间距,可以打开双狄拉克锥, 将p态和d态分离, 形成完全带隙. 研究进一步发现, 带隙附近声压场中声能流沿顺时针或逆时针方向转动, 形成了表面声波的赝自旋态. 复合胞内单元间距的缩小到扩大可导致能带反转, 系统从平庸态转变为非平庸态, 并伴随着拓扑相的变化. 根据体态-边界态对应原则, 构造了受拓扑保护的表面声波波导, 实现了对声子晶体表面波的调控.

关键词:

Abstract

The manipulation of surface acoustic wave (SAW) in phononic crystal plays an important role in the applications of SAW. The introduction of topological acoustic theory has opened a new field for SAW in phononic crystals. Here we construct pseudospin modes of SAW and topological phase transition along the surface of phononic crystal. The local SAW propagation is realized by air cylindrical holes in honeycomb lattice arranged on rigid substrate, and the Dirac cone is formed at the K point of the first Brillouin zone. Furthermore, using the band-folding theory, double Dirac cones can be formed at the center Г_s point in the Brillouin zone of compound cell that contains six adjacent cylindrical air holes. The double Dirac cone can be broken to form two degenerated states and complete band gap by only shrinking or expanding the spacing of adjacent holes in the compound cell. It is found that the direction of energy is in a clockwise or counterclockwise direction, thus the pseudospin modes of SAW are constructed. The shrinkage-to-expansion of the compound cell leads to band inversion, and the system changes from trivial state to nontrivial state, accompanied by the phase transition. According to the bulk-boundary correspondence, the unidirectional acoustic edge states can be found at the interface between trivial system and nontrivial system. Then we can construct a topologically protected waveguide to realize the unidirectional transmission of surface waves without backscattering. This work provides a new possibility for manipulating the SAW propagating on the surface of phononic crystals and may be useful for making the acoustic functional devices based on SAW.

Keywords:

作者及机构信息

南京大学物理学院, 声学研究所, 南京　210093

通信作者: 程营, chengying@nju.edu.cn ; 刘晓峻, liuxiaojun@nju.edu.cn

基金项目: 国家重点研发计划(批准号: 2017YFA0303702)、国家自然科学基金(批准号: 119224071, 11834008, 11874215, 11674172, 11574148)和江苏省自然科学基金(批准号: BK20160018)资助的课题

Authors and contacts

Institute of Acoustics, School of Physics, Nanjing University, Nanjing 210093, China

Corresponding author: Cheng Ying, chengying@nju.edu.cn ; Liu Xiao-Jun, liuxiaojun@nju.edu.cn

Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0303702), the National Natural Science Foundation of China (Grant Nos. 119224071, 11834008, 11874215, 11674172, 11574148), and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20160018)

文章全文 : translate this paragraph

参考文献

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施引文献

图 1 (a)晶格在xy平面截面图及最小元胞(红色箭头)和复合胞(黄色箭头)的矢量表示; (b) 最小元胞的三维图; (c) 最小元胞第一布里渊区的能带图, 插图为晶格的第一布里渊区

Fig. 1. (a) Cross section of the phononic crystal lattice in xy-plane and the vector representation of the minimum cell (red arrows) and compound cell (yellow arrows); (b) three-dimensional view of the minimum cell; (c) band structure of the first Brillouin zone of the minimum cell. The inset shows the first Brillouin zone.

下载: 全尺寸图片幻灯片

图 2 (a) 复合胞的三维图; (b) 从Bz到Bz_s的折叠机制, 红色和黄色六边形区域代表了最小元胞和复合胞的第一布里渊区, 分别用Bz和Bz_{_s}表示; (c) 复合胞第一布里渊区的能带图

Fig. 2. (a) Three-dimensional view of the compound cell; (b) the folding mechanism from Bz to Bz_{_s}. Red and yellow hexagon region represents the first Brillouin region of minimum cells and compound cells, respectively represented by Bz and Bz_{_s}; (c) band structure of the compound cell.

下载: 全尺寸图片幻灯片

图 3 拓扑平庸　(a) R₁ = 0.32b和非平庸(c) R₁ = 0.345b复合胞的能带图; 插图给出了带隙频率下方能流顺时针流动时, 相应晶格在Г_s点附近的声压场分布. 拓扑平庸(b)和非平庸(d) 复合胞能带图中Г_s点两个双重简并态p态和d态的声压场分布图. 黑色箭头表示能流的运动方向. 能流顺时针转动, 对应向下赝自旋态, 用红色箭头表示; 能流逆时针转动, 对应向上赝自旋态, 用蓝色箭头表示

Fig. 3. Band structure of the compound cell for the case of (a) topologically trivial R₁ = 0.32b and (c) topologically nontrivial R₁ = 0.345b. The insets show the pressure field below the band gap around Г_s in corresponding lattice when the energy flow rotating clockwise. The pressure filed of the double degenerated state at Г_s point in the band structure of topologically (b) trivial and (d) nontrivial compound cell. The black arrows indicate the direction of energy flow. The energy flow rotating clockwise (anticlockwise) corresponds to the pseudospin-down state (pseudospin-up) represented by red (blue) arrow.

下载: 全尺寸图片幻灯片

图 4 (a) 条状超胞示意图(xy平面)和能带图; 条状超胞是由中间的10个拓扑非平庸复合胞和上下各5个拓扑平庸复合胞构成的三明治结构, 能带图中灰色区域为体态, 蓝线和红线表示边界态; (b) 图(a)中A点与B点的声场分布图. 中间的四张菱形彩色图分别为A点和B点的两个边界态. 黑色箭头表示表面声波能流的运动方向

Fig. 4. (a) Schematic diagram and band structure of the ribbon-shaped supercell (in xy-plane). The ribbon-shaped supercell is composed of 10 topologically nontrivial compound cells sandwiched by 5 topologically trivial compound cells on both sides. The gray areas in the band diagram represent the bulk modes and the blue and red lines indicate the edge modes; (b) pressure fields of points A and B in (a). The four diamond-shaped color graphs in the middle are the edge modes of point A and B, respectively. The black arrows indicate the direction of energy flow.

下载: 全尺寸图片幻灯片

图 5 频率为7630 Hz时　(a)拓扑平庸声子晶体(结构Ⅰ), (b) 受拓扑保护的直线表面波波导(结构Ⅱ), (c) 弯曲型拓扑保护表面波波导(结构Ⅲ)的声压绝对值分布; 红色菱形框内为三种结构在z = L处的xy平面上绝对值声压分布图; (d)结构Ⅰ(黑色虚线), 结构Ⅱ(红色实线)和结构Ⅲ(蓝色点划线)中的声波传输系数, 阴影部分表示复合胞超元胞带隙频率范围

Fig. 5. Absolute pressure field of (a) the topologically trivial phonon crystals (Structure Ⅰ), (b) linear type topologically protected surface wave waveguide (Structure Ⅱ), (c) bending type topologically protected surface wave waveguide (Structure Ⅲ) at f = 7630 Hz. The insets in red diamonds show the absolute pressure field of the three structures at z = L in xy-plane; (d) transmission coefficient of Structure Ⅰ (black dashed line), Structure Ⅱ (red solid line) and Structure Ⅲ(blue dotted line). The shaded areas represent the gap frequency range of the compound cell.

下载: 全尺寸图片幻灯片

map

[1]	Ricca R L, Berger M A 1996 Phys. Today 49 28 Google Scholar
[2]	Vonklitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[3]	Thouless D J, Kohmoto M, Nightingale M P, Dennijs M 1982 Phys. Rev. Lett. 49 405 Google Scholar
[4]	Laughlin R B 1983 Phys. Rev. Lett. 50 1395 Google Scholar
[5]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[6]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757 Google Scholar
[7]	Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R J, Hasan M Z 2008 Nature 452 970 Google Scholar
[8]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[9]	Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61 Google Scholar
[10]	Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057 Google Scholar
[11]	Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904 Google Scholar
[12]	Raghu S, Haldane F D M 2008 Phys. Rev. A 78 033834 Google Scholar
[13]	Wang Z, Chong Y, Joannopoulos J D, Soljačić M 2009 Nature 461 772 Google Scholar
[14]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905 Google Scholar
[15]	Fang Y T, He H Q, Hu J X, Chen L K, Wen Z 2015 Phys. Rev. A 91 033827 Google Scholar
[16]	Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901 Google Scholar
[17]	Gao F, Xue H R, Yang Z J, Lai K F, Yu Y, Lin X, Chong Y D, Shvets G, Zhang B L 2018 Nat. Phys. 14 140 Google Scholar
[18]	Fleury R, Sounas D L, Sieck C F, Haberman M R, Alu A 2014 Science 343 516 Google Scholar
[19]	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
[20]	Khanikaev A B, Fleury R, Mousavi S H, Alu A 2015 Nat. Commun. 6 8260 Google Scholar
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[24]	Wei Q, Tian Y, Zuo S Y, Cheng Y, Liu X J 2017 Phy. Rev. B 95 094305 Google Scholar
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[27]	王健, 吴世巧, 梅军 2017 66 224301 Google Scholar Wang J, Wu S Q, Mei J 2017 Acta Phys. Sin. 66 224301 Google Scholar
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[32]	Dai H Q, Qian M Y, Jiao J R, Xia B Z, Yu D J 2018 J. Appl. Phys. 124 175107 Google Scholar
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[34]	Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901 Google Scholar
[35]	Xia J P, Jia D, Sun H X, Yuan S Q, Ge Y, Si Q R, Liu X J 2018 Adv. Mater. 30 1805002 Google Scholar
[36]	Zhang Z W, Cheng Y, Liu X J, Christensen J 2019 Phy. Rev. B 99 224104 Google Scholar
[37]	Zhang Z W, Lopez M R, Cheng Y, Liu X J, Christensen J 2019 Phys. Rev. Lett. 122 195501 Google Scholar
[38]	Zhang Z W, Cheng Y, Liu X J 2018 Sci. Rep. 8 16784 Google Scholar
[39]	Zhang Z W, Tian Y, Cheng Y, Wei Q, Liu X J, Christensen J 2018 Phys. Rev. Appl. 9 034032 Google Scholar
[40]	Jia D, Sun H X, Xia J P, Yuan S Q, Liu X J, Zhang C 2018 New J. Phys. 20 093027 Google Scholar
[41]	Gao Z, Yang Z J, Gao F, Xue H R, Yang Y H, Dong J W, Zhang B L 2017 Phys. Rev. B 96 201402 Google Scholar
[42]	吴世巧 2018 博士学位论文 (广州: 华南理工大学) Wu S Q 2018 Ph. D. Dissertation(Guangzhou: South China University of Technology) (in Chinese)
[43]	Yang Y H, Gao Z, Xue H R, Zhang L, He M J, Yang Z J, Singh R, Chong Y D, Zhang B L, Chen H S 2019 Nature 565 622 Google Scholar
[44]	Ge H, Ni X, Tian Y, Gupta S K, Lu M H, Lin X, Huang W D, Chan C T, Chen Y F 2018 Phys. Rev. Appl. 10 014017 Google Scholar
[45]	He H L, Qiu C Y, Ye L P, Cai X X, Fan X Y, Ke M Z, Zhang F, Liu Z Y 2018 Nature 560 61 Google Scholar
[46]	Yang Y H, Sun H X, Xia J P, Xue H R, Gao Z, Ge Y, Jia D, Yuan S Q, Chong Y D, Zhang B L 2019 Nat. Phys. 15 645 Google Scholar
[47]	Torrent D, Sánchez-Dehesa J 2012 Phys. Rev. Lett. 108 174301 Google Scholar
[48]	Ma C R, Gao S X, Cheng Y, Liu X J 2019 Appl. Phys. Lett. 115 053501 Google Scholar
[49]	孙晓晨 2017 博士学位论文 (南京: 南京大学) Sun X C 2017 Ph. D. Dissertation (Nanjing: Nanjing University) (in Chinese)

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声子晶体中的表面声波赝自旋模式和拓扑保护声传输