Size effects on the mixed modes and defect modes for a nano-scale phononic crystal slab

doi:10.1007/s10483-023-2945-6

Abstract

Abstract: The size-dependent band structure of an Si phononic crystal (PnC) slab with an air hole is studied by utilizing the non-classic wave equations of the nonlocal strain gradient theory (NSGT). The three-dimensional (3D) non-classic wave equations for the anisotropic material are derived according to the differential form of the NSGT. Based on the the general form of partial differential equation modules in COMSOL, a method is proposed to solve the non-classic wave equations. The bands of the in-plane modes and mixed modes are identified. The in-plane size effect and thickness effect on the band structure of the PnC slab are compared. It is found that the thickness effect only acts on the mixed modes. The relative width of the band gap is widened by the thickness effect. The effects of the geometric parameters on the thickness effect of the mixed modes are further studied, and a defect is introduced to the PnC supercell to reveal the influence of the size effects with stiffness-softening and stiffness-hardening on the defect modes. This study paves the way for studying and designing PnC slabs at nano-scale.

Key words: band structure, phononic crystal (PnC), nonlocal strain gradient theory (NSGT), size effect

2010 MSC Number:

74J30
78M16

Jun JIN, Ningdong HU, Hongping HU. Size effects on the mixed modes and defect modes for a nano-scale phononic crystal slab. Applied Mathematics and Mechanics (English Edition), 2023, 44(1): 21-34.

TrendMD

References

[1] MEHANEY, A. Phononic crystal as a neutron detector. Ultrasonics, 93, 37–42(2019)
[2] KHELIF, A., CHOUJAA, A., BENCHABANE, S., DJAFARI-ROUHANI, B., and LAUDE, V. Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Applied Physics Letters, 84, 4400–4402(2004)
[3] WU, T. T., WANG, W. S., SUN, J. H., HSU, J. C., and CHEN, Y. Y. Utilization of phononiccrystal reflective gratings in a layered surface acoustic wave device. Applied Physics Letters, 94, 101913(2009)
[4] LI, Z. N., WANG, Y. Z., and WANG, Y. S. Tunable three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial by initial stresses. Applied Mathematics and Mechanics (English Edition), 43(2), 167–184(2022) https://doi.org/10.1007/s10483-021-2808-9
[5] CIAMPA, F., MANKAR, A., and MARINI, A. Phononic crystal waveguide transducers for nonlinear elastic wave sensing. Scientific Reports, 7, 1–8(2017)
[6] HÅKANSSON, A., SÁCHEZ-DEHESA, J., and SANCHIS, L. Acoustic lens design by genetic algorithms. Physical Review B, 70, 214302(2004)
[7] YANG, S., PAGE, J. H., LIU, Z., COWAN, M. L., CHAN, C. T., and SHENG, P. Focusing of sound in a 3D phononic crystal. Physical Review Letters, 93, 024301(2004)
[8] SUKHOVICH, A., MERHEB, B., MURALIDHARAN, K., VASSEUR, J., PENNEC, Y., DEYMIER, P. A., and PAGE, J. Experimental and theoretical evidence for subwavelength imaging in phononic crystals. Physical Review Letters, 102, 154301(2009)
[9] ZHAO, C. Y., ZHENG, J. Y., SANG, T., WANG, L. C., YI, Q., and WANG, P. Computational analysis of phononic crystal vibration isolators via FEM coupled with the acoustic black hole effect to attenuate railway-induced vibration. Construction Building Materials, 283, 122802(2021)
[10] QIANG, C. X., HAO, Y. X., ZHANG, W., LI, J. Q., YANG, S. W., and CAO, Y. T. Bandgaps and vibration isolation of local resonance sandwich-like plate with simply supported overhanging beam. Applied Mathematics and Mechanics (English Edition), 42(11), 1555–1570(2021) https://doi.org/10.1007/s10483-021-2790-7
[11] LI, J., SHEN, C., HUANG, T. J., and CUMMER, S. A. Acoustic tweezer with complex boundaryfree trapping and transport channel controlled by shadow waveguides. Science Advances, 7, eabi5502(2021)
[12] JIANG, X., SHI, C., LI, Z., WANG, S., WANG, Y., YANG, S., LOUIE, S. G., and ZHANG, X. Direct observation of Klein tunneling in phononic crystals. Science, 370, 1447–1450(2020)
[13] ZHANG, X. and LIU, Z. Negative refraction of acoustic waves in two-dimensional phononic crystals. Applied Physics Letters, 85, 341–343(2004)
[14] HE, H., QIU, C., YE, L., CAI, X., FAN, X., KE, M., ZHANG, F., and LIU, Z. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. nature, 560, 61–64(2018)
[15] JIN, J., WANG, X., ZHAN, L., and HU, H. Strong quadratic acousto-optic coupling in 1D multilayer phoxonic crystal cavity. Nanotechnology Reviews, 10, 443–452(2021)
[16] JIN, J., JIANG, S., HU, H., ZHAN, L., WANG, X., and LAUDE, V. Acousto-optic cavity coupling in 2D phoxonic crystal with combined convex and concave holes. Journal of Applied Physics, 130, 123104(2021)
[17] MASRURA, H. M., KAREEKUNNAN, A., LIU, F., RAMARAJ, S. G., ELLROTT, G., HAMMAM, A. M., MURUGANATHAN, M., and MIZUTA, H. Design of graphene phononic crystals for heat phonon engineering. Micromachines, 11, 655(2020)
[18] CHAN, J., SAFAVI-NAEINI, A. H., HILL, J. T., MEENEHAN, S., and PAINTER, O. Optimized optomechanical crystal cavity with acoustic radiation shield. Applied Physics Letters, 101, 081115(2012)
[19] MACCABE, G. S., REN, H., LUO, J., COHEN, J. D., ZHOU, H., SIPAHIGIL, A., MIRHOSSEINI, M., and PAINTER, O. Nano-acoustic resonator with ultralong phonon lifetime. Science, 370, 840–843(2020)
[20] CHAFATINOS, D. L., KUZNETSOV, A. S., ANGUIANO, S., BRUCHHAUSEN, A. E., REYNOSO, A. A., BIERMANN, K., SANTOS, P. V., and FAINSTEIN, A. Polariton-driven phonon laser. Nature Communications, 11, 4552(2020)
[21] CUI, K. Y., HUANG, Z. L., WU, N., XU, Q. C., PAN, F., XIONG, J., FENG, X., LIU, F., ZHANG, W., and HUANG, Y. D. Phonon lasing in a hetero optomechanical crystal cavity. Photonics Research, 9, 937–943(2021)
[22] MERCADÉ, L., PELKA, K., BURGWAL, R., XUEREB, A., MARTÍNEZ, A., and VERHAGEN, E. Floquet phonon lasing in multimode optomechanical systems. Physical Review Letters, 127, 073601(2021)
[23] SAFAVI-NAEINI, A. H., ALEGRE, T. M., CHAN, J., EICHENFIELD, M., WINGER, M., LIN, Q., HILL, J. T., CHANG, D. E., and PAINTER, O. Electromagnetically induced transparency and slow light with optomechanics. nature, 472, 69–73(2011)
[24] EICHENFIELD, M., CAMACHO, R., CHAN, J., VAHALA, K. J., and PAINTER, O. A picogramand nanometre-scale photonic-crystal optomechanical cavity. nature, 459, 550–555(2009)
[25] EICHENFIELD, M., CHAN, J., CAMACHO, R. M., VAHALA, K. J., and PAINTER, O. Optomechanical crystals. nature, 462, 78–82(2009)
[26] ZHENG, C. Y., ZHANG, G. Y., and MI, C. W. On the strength of nanoporous materials with the account of surface effects. International Journal of Engineering Science, 160, 103451(2021)
[27] ZHAO, Z. N. and GUO, J. H. Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals. Applied Mathematics and Mechanics (English Edition), 42(5), 625–640(2021) https://doi.org/10.1007/s10483-021-2721-5
[28] ERINGEN, A. C. Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science, 10, 425–435(1972)
[29] MINDLIN, R. D. and ESHEL, N. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures, 4, 109–124(1968)
[30] GURTIN, M. E. and MURDOCH, A. I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics Analysis, 57, 291–323(1975)
[31] YANG, F., CHONG, A., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, 2731–2743(2002)
[32] CHEN, A. L. and WANG, Y. S. Size-effect on band structures of nano-scale phononic crystals. Physica E, 44, 317–321(2011)
[33] CHEN, A. L., YAN, D. J., WANG, Y. S., and ZHANG, C. Z. Anti-plane transverse waves propagation in nano-scale periodic layered piezoelectric structures. Ultrasonics, 65, 154–164(2016)
[34] YAN, D. J., CHEN, A. L., WANG, Y. S., and ZHANG, C. Size-effect on the band structures of the transverse elastic wave propagating in nano-scale periodic laminates. International Journal of Mechanical Sciences, 180, 105669(2020)
[35] ZHENG, H., ZHANG, C. Z., WANG, Y. S., SLADEK, J., and SLADEK, V. Band structure computation of in-plane elastic waves in 2D phononic crystals by a meshfree local RBF collocation method. Engineering Analysis with Boundary Elements, 66, 77–90(2016)
[36] ZHENG, H., ZHANG, C. Z., WANG, Y. S., SLADEK, J., and SLADEK, V. A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals. Journal of Computational Physics, 305, 997–1014(2016)
[37] ZHENG, H., ZHANG, C. Z., WANG, Y. S., CHEN, W., SLADEK, J., and SLADEK, V. A local RBF collocation method for band structure computations of 2D solid/fluid and fluid/solid phononic crystals. International Journal for Numerical Methods in Engineering, 110, 467–500(2017)
[38] ZHENG, H., ZHANG, C. Z., and YANG, Z. S. A local radial basis function collocation method for band structure computation of 3D phononic crystals. Applied Mathematical Modelling, 77, 1954–1964(2020)
[39] ZHENG, H., ZHOU, C. B., YAN, D. J., WANG, Y. S., and ZHANG, C. Z. A meshless collocation method for band structure simulation of nano-scale phononic crystals based on nonlocal elasticity theory. Journal of Computational Physics, 408, 109268(2020)
[40] QIAN, D. H., WU, J. H., and HE, F. Y. Electro-mechanical coupling band gaps of a piezoelectric phononic crystal Timoshenko nanobeam with surface effects. Ultrasonics, 109, 106225(2021)
[41] LIU, W., CHEN, J. W., LIU, Y. Q., and SU, X. Y. Effect of interface/surface stress on the elastic wave band structure of two-dimensional phononic crystals. Physics Letters A, 376, 605–609(2012)
[42] ZHEN, N., WANG, Y. S., and ZHANG, C. Bandgap calculation of in-plane waves in nano-scale phononic crystals taking account of surface/interface effects. Physica E: Low-dimensional Systems and Nanostructures, 54, 125–132(2013)
[43] ZHEN, N., WANG, Y. S., and ZHANG, C. Surface/interface effect on band structures of nanosized phononic crystals. Mechanics Research Communications, 46, 81–89(2012)
[44] LIU, W., LIU, Y. Q., SU, X. Y., and LI, Z. Finite element analysis of the interface/surface effect on the elastic wave band structure of two-dimensional nanosized phononic crystals. International Journal of Applied Mechanics, 6, 1450005(2014)
[45] ZHANG, S. Z., HU, Q. Q., and ZHAO, W. J. Surface effect on band structure of magneto-elastic phononic crystal nanoplates subject to magnetic and stress loadings. Applied Mathematics and Mechanics (English Edition), 43(2), 203–218(2022) https://doi.org/10.1007/s10483-022-2806-7
[46] ZHANG, G., GAO, X. L., and DING, S. Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mechanica Sinica, 229, 4199–4214(2018)
[47] ZHANG, G. and GAO, X. L. Elastic wave propagation in 3-D periodic composites: band gaps incorporating microstructure effects. Composite Structures, 204, 920–932(2018)
[48] JIN, J., HU, N., and HU, H. Investigation of size effect on band structure of 2D nano-scale phononic crystal based on nonlocal strain gradient theory. International Journal of Mechanical Sciences, 219, 107100(2022)
[49] ASPELMEYER, M., KIPPENBERG, T. J., and MARQUARDT, F. Cavity optomechanics. Reviews of Modern Physics, 86, 1391–1452(2014)
[50] CHAN, J., ALEGRE, T., SAFAVI-NAEINI, A. H., HILL, J. T., KRAUSE, A., GRÖBLACHER, S., ASPELMEYER, M., and PAINTER, O. Laser cooling of a nanomechanical oscillator into its quantum ground state. nature, 478, 89–92(2011)
[51] SAFAVI-NAEINI, A. H., ALEGRE, T., CHAN, J., EICHENFIELD, M., WINGER, M., LIN, Q., HILL, J. T., CHANG, D. E., and PAINTER, O. Electromagnetically induced transparency and slow light with optomechanics. nature, 472, 69–73(2011)
[52] ERINGEN, A. C. Theory of nonlocal electromagnetic elastic solids. Journal of Mathematical Physics, 14, 733–740(1973)
[53] BARRETTA, R., CANADIJA, M., LUCIANO, R., and DE SCIARRA, F. M. Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams. International Journal of Engineering Science, 126, 53–67(2018)
[54] ROMANO, G. and BARRETTA, R. Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Composites Part B: Engineering, 114, 184–188(2017)
[55] LIM, C. W., ZHANG, G., and REDDY, J. N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313(2015)
[56] EL-JALLAL, S., OUDICH, M., PENNEC, Y., DJAFARI-ROUHANI, B., LAUDE, V., BEUGNOT, J. C., MARTINEZ, A., ESCALANTE, J. M., and MAKHOUTE, A. Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities. Physical Review B, 88, 205410(2013)
[57] JIANG, S., HU, H. P., and LAUDE, V. Ultra-wide band gap in two-dimensional phononic crystal with combined convex and concave holes. Physica Status Solidi-Rapid Research Letters, 12, 1700317(2018)
[58] SHAAT, M. and ABDELKEFI, A. New insights on the applicability of Eringen’s nonlocal theory. International Journal of Mechanical Sciences, 121, 67–75(2017)
[59] ESEN, I. Response of a micro-capillary system exposed to a moving mass in magnetic field using nonlocal strain gradient theory. International Journal of Mechanical Sciences, 188, 105937(2020)
[60] DJAFARI-ROUHANI, B., EL-JALLAL, S., OUDICH, M., and PENNEC, Y. Optomechanic interactions in phoxonic cavities. AIP Advances, 4, 124602(2014)
[61] EL-JALLAL, S., OUDICH, M., PENNEC, Y., DJAFARI-ROUHANI, B., MAKHOUTE, A., ROLLAND, Q., DUPONT, S., and GAZALET, J. Optomechanical interactions in two-dimensional Si and GaAs phoxonic cavities. Journal of Physics: Condensed Matter, 26, 015005(2014)
[62] ROLLAND, Q., OUDICH, M., EL-JALLAL, S., DUPONT, S., PENNEC, Y., GAZALET, J., KASTELIK, J. C., LEVEQUE, G., and DJAFARI-ROUHANI, B. Acousto-optic couplings in two-dimensional phoxonic crystal cavities. Applied Physics Letters, 101, 061109(2012)
[63] TANG, H. S., LI, L., HU, Y. J., MENG, W. S., and DUAN, K. Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Structures, 137, 377–391(2019)
[64] LU, L., GUO, X., and ZHAO, J. Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12–24(2017)