球形复合柱表面波声子晶体的带隙特性仿真

谭自豪; 孙小伟; 宋婷; 温晓东; 刘禧萱; 刘子江

doi:10.7498/aps.70.20210165

摘要

设计了一种由镍球与环氧树脂垫层组成的复合柱沉积在铌酸锂基体上构成的表面波声子晶体结构, 采用有限元法计算了其能带结构和位移矢量场. 结果表明: 与具有相同晶格常数的倒圆锥形表面波声子晶体结构相比, 研究结构可以在更低的频率范围打开更宽的声表面波完全带隙, 且随着复合柱半径增大, 镍球体与压电基体的硬边界之间形成限制腔模, 相邻高阶带隙间存在能量的耦合以及振动模式的继承; 此外, 温度场的引入可以实现带隙的主动调控, 带隙频率范围随着温度升高向低频移动; 通过增加复合柱体的层数, 多振子结构与行波发生多极共振耦合, 可在高阶能带间打开完全带隙. 本文的研究结果为微米级表面波声子晶体结构在100 MHz以下频率范围的带隙特性优化提供了理论支持.

关键词:

Abstract

In the study of acoustic characteristics of micro-scale surface phononic crystal, the band gap characteristics below 100 MHz need to be further optimized. In this work, a piezoelectric surface phononic crystal with a composite column composed of nickel balls and epoxy backing is proposed. The finite element method is used to calculate the band gap characteristics and displacement vector field of the model. The influence of column radius on the band structure is studied, and meanwhile, the effect of the multi-layer composite column structure on the band gap is discussed via increasing the number of elements in the composite column, while the reason for the opening of the high-order band gap is analyzed in detail by combining the vibration mode. Furthermore, the temperature adjustability of the band gap is further studied. The results show that the spherical composite column deposition structure can open a wider complete band gap of surface acoustic wave in a lower frequency range than the existing inverse conical surface phononic crystal structure with the same lattice constant (Hsu J C, Lin F S 2018 Jpn. J. Appl. Phys. 57 07LB01). The restricted cavity mode is easily formed between the hard boundaries with the increase of column radius, which provides a possible way for low-order vibration modes to open high-order band gaps. There exist mode inheritance and energy coupling between adjacent modes, which leads the band gap to flatten and anti-flatten. Moreover, the real-time adjustment of band gap frequency by external temperature field can be realized via introducing the temperature-sensitive material epoxy resin into the structure. The band gap frequency range can be effectively reduced by increasing the number of composite cylinder layers, while the multi-vibrator structure can generate multipole resonance coupling with traveling wave and finally open a complete band gap between high-order frequency bands. This work provides a theoretical reference for analyzing the low-frequency band gap mechanism of micron-scale surface phononic crystal.

Keywords:

作者及机构信息

1.
兰州交通大学数理学院, 兰州　730070

2.
兰州城市学院物理系, 兰州　730070

通信作者: 孙小伟, sunxw_lzjtu@yeah.net

基金项目: 国家自然科学基金(批准号: 51562021)、甘肃省重点人才项目(批准号: 2020RCXM100)、甘肃省自然科学基金重点项目(批准号: 20JR5RA427, 20JR5RA211)、甘肃省高等学校创新基金项目(批准号: 2020A-039)和兰州市人才创新创业项目(批准号: 2020-RC-18)资助的课题

Authors and contacts

1.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China

2.
Department of Physics, Lanzhou City University, Lanzhou 730070, China

Corresponding author: Sun Xiao-Wei, sunxw_lzjtu@yeah.net

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51562021), the Key Talent Foundation of Gansu Province, China (Grant No. 2020RCXM100), the Key Program of the Natural Science Foundation of Gansu Province, China (Grant Nos. 20JR5RA427, 20JR5RA211), the Innovation Fund Project of Colleges and Universities in Gansu Province, China (Grant No. 2020A-039), and the Talent Innovation and Entrepreneurship Project of Lanzhou, China (Grant No. 2020-RC-18)

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参考文献

[1]	Yu N F, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333 Google Scholar
[2]	Ni X J, Emani N K, Kildishev A V, Boltasseva A, Shalaev V M 2012 Science 335 427 Google Scholar
[3]	Grady N K, Heyes J E, Chowdhury D R, Zeng Y, Reiten M T, Azad A K, Taylor A J, Dalvit D A R, Chen H T 2013 Science 340 1304 Google Scholar
[4]	Cummer S A, Christensen J, Alu A 2016 Nat. Rev. Mater. 1 16001 Google Scholar
[5]	Xie Y B, Wang W Q, Chen H Y, Konneker A, Popa B-I, Cummer S A 2014 Nat. Commun. 5 5553 Google Scholar
[6]	Xia J P, Zhang X T, Sun H X, Yuan S Q, Qian J, Ge Y 2018 Phys. Rev. Appl. 10 014016 Google Scholar
[7]	丁昌林, 董仪宝, 赵晓鹏 2018 物理学报 67 194301 Google Scholar Ding C L, Dong Y B, Zhao X P 2018 Acta Phys. Sin. 67 194301 Google Scholar
[8]	Li Y, Assouar B M 2016 Appl. Phys. Lett. 108 063502 Google Scholar
[9]	Chen C R, Du Z B, Hu G K, Yang J 2017 Appl. Phys. Lett. 110 221903 Google Scholar
[10]	Wang X L, Luo X D, Zhao H, Huang Z Y 2018 Appl. Phys. Lett. 112 021901 Google Scholar
[11]	Wu X X, Au-Yeung K Y, Li X, Roberts R C, Tian J X, Hu C D, Huang Y Z, Wang S X, Yang Z Y, Wen W J 2018 Appl. Phys. Lett. 112 103505 Google Scholar
[12]	Lu Y J, Ge Y, Yuan S Q, Sun H X, Liu X J 2020 J. Phys. D: Appl. Phys. 53 015301 Google Scholar
[13]	Pennec Y, Djafari-Rouhani B, Larabi H, Vasseur J O, Hladky-Hennion A C 2008 Phys. Rev. B 78 104105 Google Scholar
[14]	Wu T T, Huang Z G, Tsai T C, Wu T C 2008 Appl. Phys. Lett. 93 111902 Google Scholar
[15]	Wu T C, Wu T T, Hsu J C 2009 Phys. Rev. B 79 104306 Google Scholar
[16]	Jin Y, Fernez N, Pennec Y, Bonello B, Moiseyenko R P, Hémon S, Pan Y D, Djafari-Rouhani B 2016 Phys. Rev. B 93 054109 Google Scholar
[17]	Jin Y, EI Boudouti E H, Pennec Y, Bonello B, Moiseyenko R P, Hémon S, Pan Y D, Djafari-Rouhani B 2017 J. Phys. D 50 425304 Google Scholar
[18]	Jin Y, Pennec Y, Pan Y D, Djafari-Rouhani B 2016 Crystals 6 64 Google Scholar
[19]	曾伟, 王海涛, 田贵云, 胡国星, 汪文 2015 物理学报 64 134302 Google Scholar Zeng W, Wang H T, Tian G Y, Hu G X, Wang W 2015 Acta Phys. Sin. 64 134302 Google Scholar
[20]	Oudich M, Li Y 2017 J. Phys. D 50 315104 Google Scholar
[21]	Oudich M, Djafari-Rouhani B, Bonello B, Pennec Y, Hemaidia S, Sarry F, Beyssen D 2018 Phys. Rev. Appl. 9 034013 Google Scholar
[22]	周振凯, 韦利明, 丰杰 2013 物理学报 62 104601 Google Scholar Zhou Z K, Wei L M, Feng J 2013 Acta Phys. Sin. 62 104601 Google Scholar
[23]	钱莉荣, 杨保和 2013 物理学报 62 117701 Google Scholar Qian L R, Yang B H 2013 Acta Phys. Sin. 62 117701 Google Scholar
[24]	Benchabane S, Khelif A, Rauch J Y, Robert L, Laude V 2006 Phys. Rev. E 73 065601 Google Scholar
[25]	Yudistira D, Boes A, Graczykowski B, Alzina F, Yeo L Y, Sotomayor Torres C M, Mitchell A 2016 Phys. Rev. B 94 094304 Google Scholar
[26]	Ash B J, Worsfold S R, Vukusic P, Nash G R 2017 Nat. Commun. 8 174 Google Scholar
[27]	Hsu J C, Lin F S 2018 Jpn. J. Appl. Phys. 57 07LB01 Google Scholar
[28]	Coffy E, Euphrasie S, Addouche M, Vairac P, Khelif A 2017 Ultrasonics 78 51 Google Scholar
[29]	Cheng Y, Liu X J, Wu D J 2011 J. Acoust. Soc. Am. 129 1157 Google Scholar
[30]	Liu H, Huo S Y, Feng L Y, Huang H B, Chen J J 2019 Ultrasonics 94 227 Google Scholar
[31]	Li Z, Li Y M, Kumar S, Lee H P 2019 J. Appl. Phys. 126 155102 Google Scholar
[32]	郝娟, 周广刚, 马跃, 黄文奇, 张鹏, 卢贵武 2016 物理学报 65 113101 Google Scholar Hao J, Zhou G G, Ma Y, Huang W Q, Zhang P, Lu G W 2016 Acta Phys. Sin. 65 113101 Google Scholar
[33]	Yudistira D, Pennec Y, Rouhani B D, Dupont S, Laude V 2012 Appl. Phys. Lett. 100 061912 Google Scholar
[34]	Zhang D B, Zhao J F, Bonello B, Li L B, Wei J X, Pan Y D, Zhong Z 2016 AIP Adv. 6 085021 Google Scholar
[35]	Graczykowski B, Alzina F, Gomis-Bresco J, Sotomayor Torres C M 2016 J. Appl. Phys. 119 025308 Google Scholar
[36]	Bian Z G, Zhang S, Zhou X L 2019 Mech. Adv. Mater. Struct. 28 1663321 Google Scholar

施引文献

图 1 球形复合柱表面波声子晶体单胞结构及第一布里渊区示意图

Fig. 1. Schematic diagram of the unit cell and the first Brillouin zone of the surface phononic crystal with spherical composite column.

下载: 全尺寸图片幻灯片

图 2 用于计算所设计表面波声子晶体结构插入损失的半无限结构

Fig. 2. The semi-finite structure for calculating the insertion loss of designed surface phononic crystal.

下载: 全尺寸图片幻灯片

图 3 球形复合柱半径不同取值时所设计模型的能带结构以及对比模型的能带结构　(a)左图为柱体半径r = 0.3a时的能带结构图, 右图为对应的插入损失谱; (b)参考文献[26]中对比模型半径r = 0.32a时的能带结构图; (c)柱体半径r = 0.4a时的能带结构; (d)柱体半径r = 0.5a时的能带结构

Fig. 3. Band structures of the designed model with different radius of spherical composite column and the comparison model: (a) Designed model with cylinder radius r = 0.3a in the left and its insertion loss spectrum in the right; (b) comparison model with cylinder radius r = 0.32a from Ref. [26]; (c) designed model with cylinder radius r = 0.4a and (d) r = 0.5a, respectively.

下载: 全尺寸图片幻灯片

图 4 (a)—(j)分别为图3(a)中A — F点附近的振动模态. 红色代表振动部分, 振动位移的大小如图例所示

Fig. 4. (a)–(j) are the vibration modes at the marked points A–F in Fig. 3, respectively. The vibration part corresponds to the red and the magnitude of vibration displacement is shown in the legend.

下载: 全尺寸图片幻灯片

图 5 当r = 0.5a时, 所设计表面波声子晶体的第7—10条能带的结构图

Fig. 5. The band structure of the seventh to tenth bands of designed surface phononic crystal when r = 0.5a.

下载: 全尺寸图片幻灯片

图 6 (a)—(r)分别为图5中A—R标记点处的振动模态.

Fig. 6. (a)–(r) are the vibration modes at the marked points A–R in Fig. 5, respectively.

下载: 全尺寸图片幻灯片

图 7 (a)参考文献[36]中随温度的变化对环氧树脂弹性模量的影响; (b) r = 0.4a时, 前6条能带随温度的变化

Fig. 7. (a) Effect of temperature change on elastic modulus of epoxy resin in the Ref. [36]; (b) change of the first six bands with temperature when r = 0.4a.

下载: 全尺寸图片幻灯片

图 8 当r = 0.4a, h = 0.2a时, 多层球形复合柱表面波声子晶体的模型结构示意图及其能带结构　(a)两层球形复合柱模型结构示意图; (b)两层球形复合柱模型的能带结构图; (c)球形复合柱层数分别为三层、四层和五层时的能带结构及其模型示意图

Fig. 8. Model structures and band structures of the surface phononic crystals with multi-layer spherical composite columns when r = 0.4a and h = 0.2a: (a) Schematic of the two-layer spherical composite column model structure; (b) band structures of the designed model with two-layer spherical composite column; (c) band structures of the designed model with three, four and five layers, respectively.

下载: 全尺寸图片幻灯片

图 9 两层球形复合柱模型的能带结构中k = X点处的振动模态

Fig. 9. Vibration modes at the points k = X of the band structure of the designed model with two-layer.

下载: 全尺寸图片幻灯片

表 1 所设计模型的弹性垫层材料参数

Table 1. Material parameters of elastic cushion of the designed model.

材料	密度 ρ/(kg·m^–3)	杨氏模量 E/(10¹⁰ Pa)	剪切模量 μ/(10¹⁰ Pa)
环氧树脂	1180	0.435	0.159

下载: 导出CSV

表 2 所设计模型的基体与散射体材料参数

Table 2. Material parameters of matrix and scatterer of the designed model.

材料	密度 ρ/(kg·m^–3)	弹性常数 c_ij/(10¹⁰ N·m²)						压电常数 e/(C·m^–2)				介电常数 ε/(10^–11 F·m^–1)
材料	密度 ρ/(kg·m^–3)	c₁₁	c₁₂	c₁₃	c₁₄	c₃₃	c₄₄	e₁₅	e₂₂	e₃₁	e₃₃	ε₁₁	ε₃₃
铌酸锂	4700	20.3	5.3	7.5	0.9	24.5	6.0	3.7	2.5	0.2	1.3	39.0	25.7
镍	8905	29.8	13.4				8.2

下载: 导出CSV

[1]	Yu N F, Genevet P, Kats M A, Aieta F, Tetienne J P, Capasso F, Gaburro Z 2011 Science 334 333 Google Scholar
[2]	Ni X J, Emani N K, Kildishev A V, Boltasseva A, Shalaev V M 2012 Science 335 427 Google Scholar
[3]	Grady N K, Heyes J E, Chowdhury D R, Zeng Y, Reiten M T, Azad A K, Taylor A J, Dalvit D A R, Chen H T 2013 Science 340 1304 Google Scholar
[4]	Cummer S A, Christensen J, Alu A 2016 Nat. Rev. Mater. 1 16001 Google Scholar
[5]	Xie Y B, Wang W Q, Chen H Y, Konneker A, Popa B-I, Cummer S A 2014 Nat. Commun. 5 5553 Google Scholar
[6]	Xia J P, Zhang X T, Sun H X, Yuan S Q, Qian J, Ge Y 2018 Phys. Rev. Appl. 10 014016 Google Scholar
[7]	丁昌林, 董仪宝, 赵晓鹏 2018 物理学报 67 194301 Google Scholar Ding C L, Dong Y B, Zhao X P 2018 Acta Phys. Sin. 67 194301 Google Scholar
[8]	Li Y, Assouar B M 2016 Appl. Phys. Lett. 108 063502 Google Scholar
[9]	Chen C R, Du Z B, Hu G K, Yang J 2017 Appl. Phys. Lett. 110 221903 Google Scholar
[10]	Wang X L, Luo X D, Zhao H, Huang Z Y 2018 Appl. Phys. Lett. 112 021901 Google Scholar
[11]	Wu X X, Au-Yeung K Y, Li X, Roberts R C, Tian J X, Hu C D, Huang Y Z, Wang S X, Yang Z Y, Wen W J 2018 Appl. Phys. Lett. 112 103505 Google Scholar
[12]	Lu Y J, Ge Y, Yuan S Q, Sun H X, Liu X J 2020 J. Phys. D: Appl. Phys. 53 015301 Google Scholar
[13]	Pennec Y, Djafari-Rouhani B, Larabi H, Vasseur J O, Hladky-Hennion A C 2008 Phys. Rev. B 78 104105 Google Scholar
[14]	Wu T T, Huang Z G, Tsai T C, Wu T C 2008 Appl. Phys. Lett. 93 111902 Google Scholar
[15]	Wu T C, Wu T T, Hsu J C 2009 Phys. Rev. B 79 104306 Google Scholar
[16]	Jin Y, Fernez N, Pennec Y, Bonello B, Moiseyenko R P, Hémon S, Pan Y D, Djafari-Rouhani B 2016 Phys. Rev. B 93 054109 Google Scholar
[17]	Jin Y, EI Boudouti E H, Pennec Y, Bonello B, Moiseyenko R P, Hémon S, Pan Y D, Djafari-Rouhani B 2017 J. Phys. D 50 425304 Google Scholar
[18]	Jin Y, Pennec Y, Pan Y D, Djafari-Rouhani B 2016 Crystals 6 64 Google Scholar
[19]	曾伟, 王海涛, 田贵云, 胡国星, 汪文 2015 物理学报 64 134302 Google Scholar Zeng W, Wang H T, Tian G Y, Hu G X, Wang W 2015 Acta Phys. Sin. 64 134302 Google Scholar
[20]	Oudich M, Li Y 2017 J. Phys. D 50 315104 Google Scholar
[21]	Oudich M, Djafari-Rouhani B, Bonello B, Pennec Y, Hemaidia S, Sarry F, Beyssen D 2018 Phys. Rev. Appl. 9 034013 Google Scholar
[22]	周振凯, 韦利明, 丰杰 2013 物理学报 62 104601 Google Scholar Zhou Z K, Wei L M, Feng J 2013 Acta Phys. Sin. 62 104601 Google Scholar
[23]	钱莉荣, 杨保和 2013 物理学报 62 117701 Google Scholar Qian L R, Yang B H 2013 Acta Phys. Sin. 62 117701 Google Scholar
[24]	Benchabane S, Khelif A, Rauch J Y, Robert L, Laude V 2006 Phys. Rev. E 73 065601 Google Scholar
[25]	Yudistira D, Boes A, Graczykowski B, Alzina F, Yeo L Y, Sotomayor Torres C M, Mitchell A 2016 Phys. Rev. B 94 094304 Google Scholar
[26]	Ash B J, Worsfold S R, Vukusic P, Nash G R 2017 Nat. Commun. 8 174 Google Scholar
[27]	Hsu J C, Lin F S 2018 Jpn. J. Appl. Phys. 57 07LB01 Google Scholar
[28]	Coffy E, Euphrasie S, Addouche M, Vairac P, Khelif A 2017 Ultrasonics 78 51 Google Scholar
[29]	Cheng Y, Liu X J, Wu D J 2011 J. Acoust. Soc. Am. 129 1157 Google Scholar
[30]	Liu H, Huo S Y, Feng L Y, Huang H B, Chen J J 2019 Ultrasonics 94 227 Google Scholar
[31]	Li Z, Li Y M, Kumar S, Lee H P 2019 J. Appl. Phys. 126 155102 Google Scholar
[32]	郝娟, 周广刚, 马跃, 黄文奇, 张鹏, 卢贵武 2016 物理学报 65 113101 Google Scholar Hao J, Zhou G G, Ma Y, Huang W Q, Zhang P, Lu G W 2016 Acta Phys. Sin. 65 113101 Google Scholar
[33]	Yudistira D, Pennec Y, Rouhani B D, Dupont S, Laude V 2012 Appl. Phys. Lett. 100 061912 Google Scholar
[34]	Zhang D B, Zhao J F, Bonello B, Li L B, Wei J X, Pan Y D, Zhong Z 2016 AIP Adv. 6 085021 Google Scholar
[35]	Graczykowski B, Alzina F, Gomis-Bresco J, Sotomayor Torres C M 2016 J. Appl. Phys. 119 025308 Google Scholar
[36]	Bian Z G, Zhang S, Zhou X L 2019 Mech. Adv. Mater. Struct. 28 1663321 Google Scholar

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[20]	齐共金, 杨盛良, 白书欣, 赵恂. 基于平面波算法的二维声子晶体带结构的研究. 物理学报, 2003, 52(3): 668-671. doi: 10.7498/aps.52.668

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