Dislocation defect states in acoustic quadrupole topological insulators

Jiang Jing; Wang Xiao-Yun; Kong Peng; Zhao He-Ping; He Zhao-Jian; Deng Ke

doi:10.7498/aps.73.20240640

Abstract

Quadrupole topological insulator (QTI) is the first proposed higher-order topological phase of matter with quantized quadrupole moment but zero dipole moment. The QTI has expanded widely the traditional bulk-boundary correspondence, thereby the lower-dimensional topological boundary state can be observed. The recent interest has turned to the bulk-dislocation correspondence, which dominates the topological states localized to disclinations, and links the reciprocal-space topology of lattices with the appearance of dislocation states. Recently, many research groups have turned the studies of dislocation defects to classical wave systems. In these researches, the method of inducing dislocation defects is to remove a portion of the lattices of topological insulator and then rearrange the remaining lattices of the topological insulator. Through such a method, the micro structure of the lattices is changed, but it is difficult to realize in the actual operation. In this work, we study the dislocation defect states in acoustic QTIs. The acoustic QTI is designed by reversing the magnitude of the intracellular and extracellular coupling in the system, and the bulk energy bands and topological corner states are studied. Subsequently, by introducing partial trivial lattices into acoustic QTI structure, the dislocation bound states are generated in the corner formed by two different topological phases, which can be characterized by a 1/2 quantized fractional charge. The robustness of the topological dislocation states is verified by introducing the imperfection inside the system. Further, it is demonstrated that the dislocation positions can be designed at will. Without changing the microstructure of the lattice, we successfully modulate the line dislocation states and bulk dislocation states. The topological dislocation states studied in this work broaden the types of higher-order topological states in artificial structures, and provide new insights into the acoustic applications of higher-order topological insulators, such as sensing and high-performance energy harvesting.

Keywords:

Authors and contacts

1.
Department of Physics, Jishou University, Jishou 416000, China
2.
Hunan University of Finance and Economics, Changsha 410205, China
3.
School of Physics and electronic Science, Changsha University of Science and Technology, Changsha 410114, China

Corresponding author: Wang Xiao-Yun, wxyyun@163.com ; Kong Peng, kongpeng@jsu.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11964011).

FullText HTML : translate this paragraph

References

[1]	Benalcazar W A, Bernevig B A, Hughes T L 2017 Phys. Rev. B 96 245115 Google Scholar
[2]	Benalcazar W A, Bernevig B A, Hughes T L 2017 Science 357 61 Google Scholar
[3]	Serra-Garcia M, Peri V, Susstrunk R, Bilal O R, Larsen T, Villanueva L G, Huber S D 2018 Nature 555 342 Google Scholar
[4]	Huang X Q, Lu J Y, Yan Z B, Yan M, Deng W Y, Chen G, Liu Z Y 2022 Sci. Bull. 67 488 Google Scholar
[5]	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
[6]	Biesenthal T, Maczewsky L J, Yang Z, Kremer M, Segev M, Szameit A, Heinrich M 2022 Science 376 1114 Google Scholar
[7]	Song Z D, Fang Z, Fang C 2017 Phys. Rev. Lett. 119 246402 Google Scholar
[8]	Xue H R, Yang Y H, Zhang B L 2022 Nat. Rev. Mater. 7 974 Google Scholar
[9]	Ezawa M 2018 Phys. Rev. Lett. 120 026801 Google Scholar
[10]	Xie B, Wang H X, Zhang X, Zhan P, Jiang J H, Lu M, Chen Y 2021 Nat. Rev. Phys. 3 520 Google Scholar
[11]	Pu Z H, He H L, Luo L C, Ma Q Y, Ye L P, Ke M Z, Liu Z Y 2023 Phys. Rev. Lett. 130 116103 Google Scholar
[12]	Ma Q Y, Pu Z H, Ye L P, Lu J Y, Huang X Q, Ke M Z, He H L, Deng W Y, Liu Z Y 2024 Phys. Rev. Lett. 132 066601 Google Scholar
[13]	Zhu W, Deng W Y, Liu Y, Lu J Y, Wang H X, Lin Z K, Huang X Q, Jiang J, Liu Z Y 2023 Rep. Prog. Phys. 86 106501 Google Scholar
[14]	Wu J E, Huang X Q, Lu J Y, Wu Y, Deng W Y, Li F, Liu Z Y 2020 Phys. Rev. B 102 104109 Google Scholar
[15]	胡军容, 孔鹏, 毕仁贵, 邓科, 赵鹤平 2022 71 054301 Google Scholar Hu J R, Kong P, Bi R G, Deng K, Zhao H P 2022 Acta Phys. Sin. 71 054301 Google Scholar
[16]	Mittal S, Orre V V, Zhu G, Gorlach M A, Poddubny A, Hafezi M 2019 Nat. Photonics 13 692 Google Scholar
[17]	Qi Y J, Qiu C Y, Xiao M, He H L, Ke M Z, Liu Z Y 2020 Phys. Rev. Lett. 124 206601 Google Scholar
[18]	Li C A, Wu S S 2020 Phys. Rev. B 101 195309 Google Scholar
[19]	Li C A, Fu B, Hu Z A, Li J, Shen S Q 2020 Phys. Rev. Lett. 125 166801 Google Scholar
[20]	Qi Y J, He H L, Xiao M 2022 Appl. Phys. Lett. 120 212202 Google Scholar
[21]	Ye L P, Qiu C Y, Xiao M, Li T Z, Du J, Ke M Z, Liu Z Y 2022 Nat. Commun. 13 508 Google Scholar
[22]	Yamada S S, Li T, Lin M, Peterson C W, Hughes T L, Bahl G 2022 Nat. Commun. 13 2035 Google Scholar
[23]	Ran Y, Zhang Y, Vishwanath A 2009 Nat. Phys. 5 298 Google Scholar
[24]	Grinberg I H, Lin M, Benalcazar W A, Hughes T L, Bahl G 2020 Phys. Rev. Appl. 14 064042 Google Scholar
[25]	Xue H R, Jia D, Ge Y, Guan Y J, Wang Q, Yuan S Q, Sun H X, Chong Y D, Zhang B L 2021 Phys. Rev. Lett. 127 214301 Google Scholar
[26]	Teo J C, Hughes T L 2017 Annu. Rev. Condens. Matter Phys. 8 211 Google Scholar
[27]	Peterson C W, Li T, Jiang W, Hughes T L, Bahl G 2021 Nature 589 376 Google Scholar
[28]	Liu Y, Leung S W, Li F F, Lin Z K, Tao X F, Poo Y, Jiang J H 2021 Nature 589 381 Google Scholar
[29]	Wei H, Tsui D, Paalanen M, Pruisken A 1988 Phys. Rev. Lett. 61 1294 Google Scholar
[30]	Roberts E, Behrends J, Béri B 2020 Phys. Rev. B 101 155133 Google Scholar
[31]	Liu F, Wakabayashi K 2017 Phys. Rev. Lett. 118 076803 Google Scholar
[32]	Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903 Google Scholar
[33]	Yang Y B, Li K, Duan L M, Xu Y 2021 Phys. Rev. B 103 085408 Google Scholar
[34]	Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401 Google Scholar
[35]	He L, Addison Z, Mele E J, Zhen B 2020 Nat. Commun. 11 3119 Google Scholar
[36]	Benalcazar W A, Li T, Hughes T L 2019 Phys. Rev. B 99 245151 Google Scholar
[37]	Weiner M, Ni X, Li M, Alù A, Khanikaev A B 2020 Sci. Adv. 6 eaay4166 Google Scholar
[38]	Wheeler W A, Wagner L K, Hughes T L 2019 Phys. Rev. B 100 245135 Google Scholar
[39]	Zhang X, Xie B Y, Wang H F, Xu X, Tian Y, Jiang J H, Lu M H, Chen Y F 2019 Nat. Commun. 10 5331 Google Scholar
[40]	Kang B, Shiozaki K, Cho G Y 2019 Phys. Rev. B 100 245134 Google Scholar
[41]	Ni X, Li M, Weiner M, Alù A, Khanikaev A B 2020 Nat. Commun. 11 2108 Google Scholar
[42]	Zheng S, Man X, Kong Z L, Lin Z K, Duan G, Chen N, Yu D, Jiang J H, Xia B 2022 Sci. Bull. 67 2069 Google Scholar
[43]	Kim H R, Hwang M S, Smirnova D, Jeong K Y, Kivshar Y, Park H G 2020 Nat. Commun. 11 5758 Google Scholar
[44]	Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F 2018 Phys. Rev. B 98 205147 Google Scholar
[45]	李荫铭, 孔鹏, 毕仁贵, 何兆剑, 邓科 2022 71 244302 Google Scholar Li Y M, Kong P, Bi R G, He Z J, Deng K 2022 Acta Phys. Sin. 71 244302 Google Scholar
[46]	Chen Y Y, Liu D Y, Wu Y, Yu P, Liu Y J 2023 Int. J. Mech. Sci. 239 107884 Google Scholar

Cited By

图 1 四极子拓扑绝缘体的紧束缚模型和对应的声学模型　(a)二维紧束缚模型结构示意图, 每个黄色的圆圈代表一个无自旋的电子轨道, 蓝色方框内表示一个原胞, 蓝线表示正耦合, 红线表示负耦合, 细线表示胞内耦合, 粗线表示胞间耦合; (b)声学四极子拓扑绝缘体的原胞结构, 黄色矩形柱为声学谐振腔, 代表无自旋的电子轨道, 两根耦合管的直接连接(红色标记)表示负耦合, 交叉连接(蓝色标记)表示正耦合; (c)声子晶体的能带结构(黑色曲线), 红色散点是根据紧束缚模型计算的, 可以看出, 由声子晶体与由紧束缚模型计算得到的P_z模平坦带吻合得很好; (d)由$ 4 \times 4 $个有限单元计算得到的声学四极子拓扑绝缘体的本征频谱, 在P_z模打开的带隙范围内观察到角态(红点)是四极子拓扑绝缘体为高阶拓扑绝缘体的特征之一

Figure 1. Tight-binding (TB) model and the corresponding acoustic model for QTI. (a) Schematic of the two-dimensional TB model. Each yellow circle represents a spinless electronic orbital, and the light shaded square sketches a four-site unit cell. The blue lines indicate the positive hoppings and red lines indicate negative hoppings. The thin lines represent intracellular hoppings and the thick lines represent the intercellular hoppings. (b) The unit cell structure of the corresponding acoustic model. The yellow rectangular pillars are the resonant cavities, representing spinless electronic orbitals. The straight link of two tubes (marked with red) indicates the negative coupling, and the crossing link (marked with blue) indicates the positive coupling. (c) The band structures for the acoustic crystal (black curves). The red scatter are calculated from TB model. It can be seen that the coupled bands from P_z modes in acoustic crystal agree well with these from TB model. (d) The band structures for a finite-size acoustic structure with $ 4 \times 4 $unit cells. The corner states (marked red) are observed within the gap of P_z bands, which is one of the characteristics of higher-order QTI.

DownLoad: Full-Size Img PowerPoint

图 2 线位错态　(a)调制线位错态的模型结构, 橙色矩形表示嵌入四极子拓扑绝缘体的平庸晶格, 黑色细线表示将原始的强耦合反转为弱耦合, 红色圆圈表示位错点; (b)线位错态的本征频谱, 可以观察到两个线位错态(实空间中的拓扑态)位于P_z模平坦带打开的带隙范围内(红点); (c), (d)本征频率分别为8298.2 Hz和8298.4 Hz的两个线位错态的本征场分布

Figure 2. Line-dislocation states. (a) The model structure to introduce the line-dislocation states. The orange rectangle represents the trival lattice, embedded into the QTI. Black thin lines indicate the weak couplings which are reversed from original strong couplings. The red circles are marked as the dislocation sites. (b) The band structures for the line-dislocation model. It can be observed that two line-dislocation states (topological states in real space) are located within the gap of P_z bands (marked as red). (c), (d) The eigen fields distribution for the two line-dislocation states with frequency of 8298.2 Hz and 8298.4 Hz, respectively.

DownLoad: Full-Size Img PowerPoint

图 3 体位错态　(a)调制体位错态的模型结构, 橙色矩形表示嵌入四极子拓扑绝缘体中的平庸晶格, 黑色实线表示弱耦合, 红色圆圈表示位错点; (b)调制体位错态的声子晶体拓扑绝缘体计算得到的本征频谱, 两个体位错态位于P_z模打开的带隙范围内(红点标记); (c), (d)频率分别为8145.4 Hz和8146.1 Hz的两个体位错态的本征场分布

Figure 3. The bulk-dislocation states. (a) The model structure to introduce the bulk-dislocation states. The orange rectangle represents the trival lattice, embedded into the QTI. Black thin lines indicate the weak couplings. The red circles are marked as the dislocation sites. (b) The band structures for the bulk-dislocation model. Two bulk-dislocation states are observed within the gap of P_z bands (marked as red). (c), (d) The eigen fields distribution for the two bulk-dislocation states with frequency of 8145.4 Hz and 8146.1 Hz, resepectively.

DownLoad: Full-Size Img PowerPoint

图 4 线位错态和体位错态的鲁棒性　(a)验证线位错态鲁棒性的声学模型, 黑色圆圈表示长和宽都为7.5 mm (原始腔的1.5倍)的更大谐振腔, 表明在声学四极子拓扑绝缘体内部引入了缺陷; (b)对应于(a)中模型的本征频谱, 插图展示了线位错态的本征场分布情况; (c)验证体位错态鲁棒性的声学模型, 在声学四极子拓扑绝缘体内引入长和宽都为7.5 mm (原始腔的1.5倍)的更大谐振腔(黑色圆圈标记); (d)对应于(c)的本征频谱, 插图为体位错态的本征场分布, 可以看出, 两种情况下的特征频率和本征场分布几乎保持不变

Figure 4. Robustness of line-dislocation and bulk-dislocation states. (a) The acoustic model to verify the robustness of line-dislocation states. The bold circle denotes a bigger resonant cavity with length of 7.5 mm (1.5 times of that of original cavity), indicating the introduction of imperfection inside acoustic QTI. (b) The corresponding the band structure for the model shown in (a), insets display the eigen fields for the dislocation states. (c) The acoustic model to verify the robustness of bulk-dislocation states. A bigger resonant cavity with length of 7.5 mm (1.5 times of that of original cavity) are introduced inside acoustic QTI (marked as bold circle). (d) The corresponding the band structure for the model shown in (c), insets display the eigen fields for the dislocation states. It can be seen that both the eigen frequency and eigen field keeps almost unchanged for the two cases.

DownLoad: Full-Size Img PowerPoint

map

[1]	Benalcazar W A, Bernevig B A, Hughes T L 2017 Phys. Rev. B 96 245115 Google Scholar
[2]	Benalcazar W A, Bernevig B A, Hughes T L 2017 Science 357 61 Google Scholar
[3]	Serra-Garcia M, Peri V, Susstrunk R, Bilal O R, Larsen T, Villanueva L G, Huber S D 2018 Nature 555 342 Google Scholar
[4]	Huang X Q, Lu J Y, Yan Z B, Yan M, Deng W Y, Chen G, Liu Z Y 2022 Sci. Bull. 67 488 Google Scholar
[5]	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
[6]	Biesenthal T, Maczewsky L J, Yang Z, Kremer M, Segev M, Szameit A, Heinrich M 2022 Science 376 1114 Google Scholar
[7]	Song Z D, Fang Z, Fang C 2017 Phys. Rev. Lett. 119 246402 Google Scholar
[8]	Xue H R, Yang Y H, Zhang B L 2022 Nat. Rev. Mater. 7 974 Google Scholar
[9]	Ezawa M 2018 Phys. Rev. Lett. 120 026801 Google Scholar
[10]	Xie B, Wang H X, Zhang X, Zhan P, Jiang J H, Lu M, Chen Y 2021 Nat. Rev. Phys. 3 520 Google Scholar
[11]	Pu Z H, He H L, Luo L C, Ma Q Y, Ye L P, Ke M Z, Liu Z Y 2023 Phys. Rev. Lett. 130 116103 Google Scholar
[12]	Ma Q Y, Pu Z H, Ye L P, Lu J Y, Huang X Q, Ke M Z, He H L, Deng W Y, Liu Z Y 2024 Phys. Rev. Lett. 132 066601 Google Scholar
[13]	Zhu W, Deng W Y, Liu Y, Lu J Y, Wang H X, Lin Z K, Huang X Q, Jiang J, Liu Z Y 2023 Rep. Prog. Phys. 86 106501 Google Scholar
[14]	Wu J E, Huang X Q, Lu J Y, Wu Y, Deng W Y, Li F, Liu Z Y 2020 Phys. Rev. B 102 104109 Google Scholar
[15]	胡军容, 孔鹏, 毕仁贵, 邓科, 赵鹤平 2022 71 054301 Google Scholar Hu J R, Kong P, Bi R G, Deng K, Zhao H P 2022 Acta Phys. Sin. 71 054301 Google Scholar
[16]	Mittal S, Orre V V, Zhu G, Gorlach M A, Poddubny A, Hafezi M 2019 Nat. Photonics 13 692 Google Scholar
[17]	Qi Y J, Qiu C Y, Xiao M, He H L, Ke M Z, Liu Z Y 2020 Phys. Rev. Lett. 124 206601 Google Scholar
[18]	Li C A, Wu S S 2020 Phys. Rev. B 101 195309 Google Scholar
[19]	Li C A, Fu B, Hu Z A, Li J, Shen S Q 2020 Phys. Rev. Lett. 125 166801 Google Scholar
[20]	Qi Y J, He H L, Xiao M 2022 Appl. Phys. Lett. 120 212202 Google Scholar
[21]	Ye L P, Qiu C Y, Xiao M, Li T Z, Du J, Ke M Z, Liu Z Y 2022 Nat. Commun. 13 508 Google Scholar
[22]	Yamada S S, Li T, Lin M, Peterson C W, Hughes T L, Bahl G 2022 Nat. Commun. 13 2035 Google Scholar
[23]	Ran Y, Zhang Y, Vishwanath A 2009 Nat. Phys. 5 298 Google Scholar
[24]	Grinberg I H, Lin M, Benalcazar W A, Hughes T L, Bahl G 2020 Phys. Rev. Appl. 14 064042 Google Scholar
[25]	Xue H R, Jia D, Ge Y, Guan Y J, Wang Q, Yuan S Q, Sun H X, Chong Y D, Zhang B L 2021 Phys. Rev. Lett. 127 214301 Google Scholar
[26]	Teo J C, Hughes T L 2017 Annu. Rev. Condens. Matter Phys. 8 211 Google Scholar
[27]	Peterson C W, Li T, Jiang W, Hughes T L, Bahl G 2021 Nature 589 376 Google Scholar
[28]	Liu Y, Leung S W, Li F F, Lin Z K, Tao X F, Poo Y, Jiang J H 2021 Nature 589 381 Google Scholar
[29]	Wei H, Tsui D, Paalanen M, Pruisken A 1988 Phys. Rev. Lett. 61 1294 Google Scholar
[30]	Roberts E, Behrends J, Béri B 2020 Phys. Rev. B 101 155133 Google Scholar
[31]	Liu F, Wakabayashi K 2017 Phys. Rev. Lett. 118 076803 Google Scholar
[32]	Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903 Google Scholar
[33]	Yang Y B, Li K, Duan L M, Xu Y 2021 Phys. Rev. B 103 085408 Google Scholar
[34]	Wang H X, Lin Z K, Jiang B, Guo G Y, Jiang J H 2020 Phys. Rev. Lett. 125 146401 Google Scholar
[35]	He L, Addison Z, Mele E J, Zhen B 2020 Nat. Commun. 11 3119 Google Scholar
[36]	Benalcazar W A, Li T, Hughes T L 2019 Phys. Rev. B 99 245151 Google Scholar
[37]	Weiner M, Ni X, Li M, Alù A, Khanikaev A B 2020 Sci. Adv. 6 eaay4166 Google Scholar
[38]	Wheeler W A, Wagner L K, Hughes T L 2019 Phys. Rev. B 100 245135 Google Scholar
[39]	Zhang X, Xie B Y, Wang H F, Xu X, Tian Y, Jiang J H, Lu M H, Chen Y F 2019 Nat. Commun. 10 5331 Google Scholar
[40]	Kang B, Shiozaki K, Cho G Y 2019 Phys. Rev. B 100 245134 Google Scholar
[41]	Ni X, Li M, Weiner M, Alù A, Khanikaev A B 2020 Nat. Commun. 11 2108 Google Scholar
[42]	Zheng S, Man X, Kong Z L, Lin Z K, Duan G, Chen N, Yu D, Jiang J H, Xia B 2022 Sci. Bull. 67 2069 Google Scholar
[43]	Kim H R, Hwang M S, Smirnova D, Jeong K Y, Kivshar Y, Park H G 2020 Nat. Commun. 11 5758 Google Scholar
[44]	Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F 2018 Phys. Rev. B 98 205147 Google Scholar
[45]	李荫铭, 孔鹏, 毕仁贵, 何兆剑, 邓科 2022 71 244302 Google Scholar Li Y M, Kong P, Bi R G, He Z J, Deng K 2022 Acta Phys. Sin. 71 244302 Google Scholar
[46]	Chen Y Y, Liu D Y, Wu Y, Yu P, Liu Y J 2023 Int. J. Mech. Sci. 239 107884 Google Scholar

[1]	Han Dong-Hai, Zhang Guang-Jun, Zhao Jing-Bo, Yao Hong. Low-frequency bandgaps and sound isolation characteristics of a novel Helmholtz-type phononic crystal. Acta Physica Sinica, 2022, 71(11): 114301. doi: 10.7498/aps.71.20211932
[2]	Gao Hui-Fen, Zhou Xiao-Fang, Huang Xue-Qin. Zak phase induced interface states in two-dimensional phononic crystals. Acta Physica Sinica, 2022, 71(4): 044301. doi: 10.7498/aps.71.20211642
[3]	Li Yin-Ming, Kong Peng, Bi Ren-Gui, He Zhao-Jian, Deng Ke. Valley topological states in double-surface periodic elastic phonon crystal plates. Acta Physica Sinica, 2022, 71(24): 244302. doi: 10.7498/aps.71.20221292
[4]	. Zak phase induces interface states in two-dimensional phononic crystals. Acta Physica Sinica, 2021, (): . doi: 10.7498/aps.70.20211642
[5]	Zheng Zhou-Fu, Yin Jian-Fei, Wen Ji-Hong, Yu Dian-Long. Topologically protected edge states of elastic waves in phononic crystal plates. Acta Physica Sinica, 2020, 69(15): 156201. doi: 10.7498/aps.69.20200542
[6]	Geng Zhi-Guo, Peng Yu-Gui, Shen Ya-Xi, Zhao De-Gang, Zhu Xue-Feng. Topological acoustic transports in chiral sonic crystals. Acta Physica Sinica, 2019, 68(22): 227802. doi: 10.7498/aps.68.20191007
[7]	Jia Ding, Ge Yong, Yuan Shou-Qi, Sun Hong-Xiang. Dual-band acoustic topological insulator based on honeycomb lattice sonic crystal. Acta Physica Sinica, 2019, 68(22): 224301. doi: 10.7498/aps.68.20190951
[8]	Sun Wei-Bin, Wang Ting, Sun Xiao-Wei, Kang Tai-Feng, Tan Zi-Hao, Liu Zi-Jiang. Defect states and vibration energy recovery of novel two-dimensional piezoelectric phononic crystal plate. Acta Physica Sinica, 2019, 68(23): 234206. doi: 10.7498/aps.68.20190260
[9]	Wang Jian, Wu Shi-Qiao, Mei Jun. Topological phase transitions caused by a simple rotational operation in two-dimensional acoustic crystals. Acta Physica Sinica, 2017, 66(22): 224301. doi: 10.7498/aps.66.224301
[10]	Chen Ze-Guo, Wu Ying. Multiple topological phases in phononic crystals. Acta Physica Sinica, 2017, 66(22): 227804. doi: 10.7498/aps.66.227804
[11]	Cao Hui-Xian, Mei Jun. Semi-Dirac points in two-dimensional phononic crystals. Acta Physica Sinica, 2015, 64(19): 194301. doi: 10.7498/aps.64.194301
[12]	Hou Li-Na, Hou Zhi-Lin, Fu Xiu-Jun. Defect state of the locally resonant phononic crystal. Acta Physica Sinica, 2014, 63(3): 034305. doi: 10.7498/aps.63.034305
[13]	Gao Guo-Qin, Ma Shou-Lin, Jin Feng, Kim Tong-Beum, Lu Tian-Jian. Acoustic band gaps in finite-sized two-dimensional solid/fluid phononic crystals. Acta Physica Sinica, 2010, 59(1): 393-400. doi: 10.7498/aps.59.393
[14]	Chen Sheng-Bing, Han Xiao-Yun, Yu Dian-Long, Wen Ji-Hong. Influences of different types of piezoelectric shunting circuits on band gaps of phononic beam. Acta Physica Sinica, 2010, 59(1): 387-392. doi: 10.7498/aps.59.387
[15]	Li Xiao-Chun, Gao Jun-Li, Liu Shao-E, Zhou Ke-Chao, Huang Bo-Yun. Disorder effect on the focus image of phononic crystal panel with negative refraction. Acta Physica Sinica, 2010, 59(1): 376-380. doi: 10.7498/aps.59.376
[16]	Wang Wen-Gang, Liu Zheng-You, Zhao De-Gang, Ke Man-Zhu. Resonant tunneling of acoustic waves in 1D phononic crystal. Acta Physica Sinica, 2006, 55(9): 4744-4747. doi: 10.7498/aps.55.4744
[17]	Cai Li, Han Xiao-Yun. Study of the band-structure and the uncoupled modes in two-dimensional phononic crystals with the multiple-scattering theory. Acta Physica Sinica, 2006, 55(11): 5866-5871. doi: 10.7498/aps.55.5866
[18]	Zhao Fang, Yuan Li-Bo. Defect states of homogeneity dislocation structures in two-dimensional phononic crystal. Acta Physica Sinica, 2006, 55(2): 517-520. doi: 10.7498/aps.55.517
[19]	Li Xiao-Chun, Yi Xiu-Ying, Xiao Qing-Wu, Liang Hong-Yu. Defect states in three-component phononic crystal. Acta Physica Sinica, 2006, 55(5): 2300-2305. doi: 10.7498/aps.55.2300
[20]	Zhao Fang, Yuan Li-Bo. Characteristics of the band structure in two-dimensional phononic crystals with complex lattices. Acta Physica Sinica, 2005, 54(10): 4511-4516. doi: 10.7498/aps.54.4511

Catalog

Vol. 73, Issue 15 - 2024-08-05

Metrics

Abstract views: 1358
PDF Downloads: 59
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板