Fundamental Properties of Phononic Crystal


Phononic crystals are periodic composite structures where the elastic and acoustic properties display a periodic variation in space. They are in general constituted by a periodic array of inclusions in a matrix. Due to their periodicity, they can exhibit absolute band gaps in which the propagation of elastic waves is prohibited in any direction of the space. More generally, tailoring their band structure allows the control and manipulation of elastic waves and pave the way to several functionalities ranging from sound isolation to filtering and signal processing, negative refraction and high resolution imaging, nanoscale thermal transport managing, quantum information processing. In this chapter, we discuss some basic properties of the phononic crystals, in particular the dependence of the band gaps with the nature of the constituent materials (solid or fluid), the contrast between the elastic properties of the constituents, the shape and the filling fraction of the inclusions, the crystal lattice. We discuss the existence of gaps resulting from Bragg interference or from local resonances. The localized modes associated with defects such as cavities and waveguides are presented and their functionalities in filtering and multiplexing applications are discussed.


Dispersion Curve Hollow Cylinder Finite Difference Time Domain Stop Band Filling Fraction 
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  1. 1.
    F. Bloch, Z. Phys. 52, 555 (1928)CrossRefMATHGoogle Scholar
  2. 2.
    For a review, see E.H. El Boudouti, B. Djafari Rouhani, A. Akjouj, L. Dobrzynski, Acoustic waves in solids and fluid layered materials. Surf. Sci. Rep. 64, 471 (2009)Google Scholar
  3. 3.
    S.M. Rytov, Sov. Phys. Acoust. 2, 6880 (1956)Google Scholar
  4. 4.
    M.M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141 (1993)CrossRefGoogle Scholar
  5. 5.
    M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022 (1993)CrossRefGoogle Scholar
  6. 6.
    M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites. Phys. Rev. B 49, 2313 (1994)CrossRefGoogle Scholar
  7. 7.
    M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure. J. Sound Vib. 158, 377 (1992)CrossRefGoogle Scholar
  8. 8.
    J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, M.S. Kushwaha, P. Halevi, Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy and some metallic systems. J. Phys.: Condens. Matter 7, 8759–8770 (1994)Google Scholar
  9. 9.
    For a review, see M. Sigalas, M.S. Kushwaha, E.N. Economou, M. Kafesaki, I.E. Psarobas, W. Steurer, Classical vibrational modes in phononic lattices: theory and experiment. Z. Kristallogr. 220, 765–809 (2005)Google Scholar
  10. 10.
    For a recent review, see Y. Pennec, J. Vasseur, B. Djafari Rouhani, L. Dobrzynski, P.A. Deymier, Two-dimensional phononic crystals: examples and applications. Surf. Sci. Rep. 65, 229 (2010)Google Scholar
  11. 11.
    E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987)CrossRefGoogle Scholar
  12. 12.
    J.D. Joannopoulos, R.D. Meade, J.N. Winn, Molding the Flow of Light (Princeton University Press, Princeton, 1995)MATHGoogle Scholar
  13. 13.
    I.E. Psarobas, A. Modinos, R. Sainidou, N. Stefanou, Acoustic properties of colloidal crystals. Phys. Rev. B 65, 064307 (2002)CrossRefGoogle Scholar
  14. 14.
    R. Sainidou, N. Stefanou, A. Modinos, Formation of absolute frequency gaps in three-dimensional solid phononic crystals. Phys. Rev. B 66, 212301 (2002)CrossRefGoogle Scholar
  15. 15.
    T. Still, W. Cheng, M. Retsch, R. Sainidou, J. Wang, U. Jonas, N. Stefanou, G. Fytas, Simultaneous occurrence of structure-directed and particle-resonance-induced phononic gaps in colloidal films. Phys. Rev. Lett. 100, 194301 (2008)CrossRefGoogle Scholar
  16. 16.
    C. Croënne, E.J.S. Lee, H. Hu, J.H. Page, Band gaps in phononic crystals: generation mechanisms and interaction effects. AIP Adv. 1, 041401 (2011)CrossRefGoogle Scholar
  17. 17.
    Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials. Science 289, 1734–1736 (2000)CrossRefGoogle Scholar
  18. 18.
    M. Torres, F.R. Montero de Espinosa, D. Garcia-Pablos, N. Garcia, Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects. Phys. Rev. Lett. 82, 3054 (1999)CrossRefGoogle Scholar
  19. 19.
    M. Kafesaki, M.M. Sigalas, N. Garcia, Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials. Phys. Rev. Lett. 85, 4044 (2000)CrossRefGoogle Scholar
  20. 20.
    A. Khelif, B. Djafari-Rouhani, J.O. Vasseur, P.A. Deymier, P. Lambin, L. Dobrzynski, Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal. Phys. Rev. B 65, 174308 (2002)CrossRefGoogle Scholar
  21. 21.
    A. Khelif, B. Djafari-Rouhani, J.O. Vasseur, P.A. Deymier, Transmission and dispersion relations of perfect and defect-contained waveguide structures in phononic band gap materials. Phys. Rev. B 68, 024302 (2003)CrossRefGoogle Scholar
  22. 22.
    A. Khelif, B. Djafari-Rouhani, V. Laude, M. Solal, Coupling characteristics of localized phonons in photonic crystal fibers. J. Appl. Phys. 94, 7944–7946 (2003)CrossRefGoogle Scholar
  23. 23.
    A. Khelif, A. Choujaa, B. Djafari-Rouhani, M. Wilm, S. Ballandras, V. Laude, Trapping and guiding of acoustic waves by defect modes in a full band-gap ultrasonic crystal. Phys. Rev. B 68, 214301 (2003)CrossRefGoogle Scholar
  24. 24.
    A. Khelif, A. Choujaa, S. Benchabane, B. Djafari-Rouhani, V. Laude, Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Appl. Phys. Lett. 84, 4400 (2004)CrossRefGoogle Scholar
  25. 25.
    S. Benchabane, A. Khelif, A. Choujaa, B. Djafari-Rouhani, V. Laude, Interaction of waveguide and localized modes in a phononic crystal. Europhys. Lett. 71, 570 (2005)CrossRefGoogle Scholar
  26. 26.
    Y. Pennec, B. Djafari Rouhani, J.O. Vasseur, H. Larabi, A. Khelif, A. Choujaa, S. Benchabane, V. Laude, Acoustic channel drop tunneling in a phononic crystal. Appl. Phys. Lett. 87, 261912 (2005)CrossRefGoogle Scholar
  27. 27.
    J.O. Vasseur, P.A. Deymier, G. Frantziskonis, G. Hong, B. Djafari Rouhani, L. Dobrzyński, Experimental evidence for the existence of absolute acoustic band gaps in two-dimensional periodic composite media. J. Phys.: Condens. Matter 10, 6051 (1998)Google Scholar
  28. 28.
    J.O. Vasseur, P.A. Deymier, B. Chenni, B. Djafari-Rouhani, L. Dobrzynski, D. Prevost, Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals. Phys. Rev. Lett. 86, 3012 (2001)CrossRefGoogle Scholar
  29. 29.
    F.R. Montero de Espinosa, E. Jimenez, M. Torres, Ultrasonic band gap in a periodic two-dimensional composite. Phys. Rev. Lett. 80, 1208 (1998)CrossRefGoogle Scholar
  30. 30.
    H. Larabi, Y. Pennec, B. Djafari-Rouhani, J.O. Vasseur, Multicoaxial cylindrical inclusions in locally resonant phononic crystals. Phys. Rev. E 75, 066601 (2007)CrossRefGoogle Scholar
  31. 31.
    E.N. Economou, M.M. Sigalas, Classical wave propagation in periodic structures: cermet versus network topology. Phys. Rev. B 48, 13434 (1993)CrossRefGoogle Scholar
  32. 32.
    M.M. Sigalas, E.N. Economou, Attenuation of multiple-scattered sound. Europhys. Lett. 36, 241 (1996)CrossRefGoogle Scholar
  33. 33.
    M.S. Kushwaha, Stop-bands for periodic metallic rods: sculptures that can filter the noise. Appl. Phys. Lett. 70, 3218 (1997)CrossRefGoogle Scholar
  34. 34.
    P. Lambin, A. Khelif, J.O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, Stopping of acoustic waves by sonic polymer-fluid composites. Phys. Rev. E 63, 066605 (2001)CrossRefGoogle Scholar
  35. 35.
    J.O. Vasseur, P.A. Deymier, A. Khelif, P. Lambin, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, N. Fettouhi, J. Zemmouri, Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: a theoretical and experimental study. Phys. Rev. E 65, 056608 (2002)CrossRefGoogle Scholar
  36. 36.
    Z. Liu, C.T. Chan, P. Sheng, A.L. Goertzen, J.H. Page, Elastic wave scattering by periodic structures of spherical objects: theory and experiment. Phys. Rev. B 62, 2446 (2000)CrossRefGoogle Scholar
  37. 37.
    J.H. Page, A.L. Goertzen, S. Yang, Z. Liu, C.T. Chan, P. Sheng, in Photonic Crystals and Light Localization in the 21st Century, ed. by C.M. Soukoulis (Kluwer, Dordrecht, 2001), p. 59CrossRefGoogle Scholar
  38. 38.
    M.S. Kushwaha, B. Djafari-Rouhani, Complete acoustic stop bands for cubic arrays of spherical liquid balloons. J. Appl. Phys. 80, 3191 (1996)CrossRefGoogle Scholar
  39. 39.
    R. Martinez-Sala, J. Sancho, J.V. Sanchez, V. Gomez, J. Llinares, F. Meseguer, Sound attenuation by sculpture. Nature 378, 241 (1995)CrossRefGoogle Scholar
  40. 40.
    D. Caballero, J. Sanchez-Dehesa, C. Rubio, R. Martinez-Sala, J.V. Sanchez-Perez, F. Meseguer, J. Llinares, Large two-dimensional sonic band gaps. Phys. Rev. E 60, 6316 (1999)CrossRefGoogle Scholar
  41. 41.
    Y. Pennec, B. Djafari-Rouhani, J. Vasseur, P.A. Deymier, A. Khelif, Tunable filtering and demultiplexing in phononic crystals with hollow cylinders. Phys. Rev. E 69, 046608 (2004)CrossRefGoogle Scholar
  42. 42.
    J.V. Sanchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-Dehesa, F. Meseguer, J. Llinares, F. Gálvez, Sound attenuation by a two-dimensional array of rigid cylinders. Phys. Rev. Lett. 80, 5325 (1998)CrossRefGoogle Scholar
  43. 43.
    Y. Pennec, B. Djafari-Rouhani, J.O. Vasseur, P.A. Deymier, A. Khelif, Transmission and dispersion modes in phononic crystals with hollow cylinders: application to waveguide structure. Phys. Stat. Sol. (c) 1(11), 2711–2715 (2004)CrossRefGoogle Scholar
  44. 44.
    A. Sato, W. Knoll, Y. Pennec, B. Djafari-Rouhani, G. Fytas, M. Steinhart, Anisotropic propagation and confinement of high frequency phonons in nanocomposites. J. Chem. Phys. 130, 111102 (2009)CrossRefGoogle Scholar
  45. 45.
    R. Lucklum, J. Li, Phononic crystals for liquid sensor applications. Meas. Sci. Technol. 20, 124014 (2009)CrossRefGoogle Scholar
  46. 46.
    M. Ke, M. Zubtsov, R. Lucklum, Sub-wavelength phononic crystal liquid sensor. J. Appl. Phys. 110, 026101 (2011)CrossRefGoogle Scholar
  47. 47.
    M.S. Kushwaha, B. Djafari-Rouhani, Giant sonic stop bands in two-dimensional periodic system of fluids. J. Appl. Phys. 84, 4677 (1998)CrossRefGoogle Scholar
  48. 48.
    M.S. Kushwaha, B. Djafari Rouhani, L. Dobrzynski, Sound isolation from cubic arrays of air bubbles in water. Phys. Lett. A 248, 252–256 (1998)CrossRefGoogle Scholar
  49. 49.
    V. Leroy, A. Bretagne, M. Fink, H. Willaime, P. Tabeling, A. Tourin, Design and characterization of bubble phononic crystals. Appl. Phys. Lett. 95, 171904 (2009)CrossRefGoogle Scholar
  50. 50.
    T. Still, M. Oudich, G.K. Auernhammer, D. Vlassopoulos, B. Djafari-Rhouani, G. Fytas, P. Sheng, Soft silicone rubber in phononic structures: correct elastic moduli. Phys. Rev. B 88, 094102 (2013)CrossRefGoogle Scholar
  51. 51.
    C. Goffaux, J.P. Vigeron, Theoretical study of a tunable phononic band gap system. Phys. Rev. B 64, 075118 (2001)CrossRefGoogle Scholar
  52. 52.
    M. Torres, F.R. Montero de Espinosa, J.L. Aragon, Ultrasonic wedges for elastic wave bending and splitting without requiring a full band gap. Phys. Rev. Lett. 86, 4282 (2001)CrossRefGoogle Scholar
  53. 53.
    J.O. Vasseur, P.A. Deymier, M. Beaugeois, Y. Pennec, B. Djafari-Rouhani, D. Prevost, Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide. Z. Kristallogr. 220, 824 (2005)CrossRefGoogle Scholar
  54. 54.
    B. Djafari Rouhani, L. Dobrzynski, O. Hardouin Duparc, R.E. Camley, A.A. Maradudin, Sagittal elastic waves in infinite and semi-infinite superlattices. Phys. Rev. B 28, 1711 (1983)CrossRefGoogle Scholar
  55. 55.
    B. Djafari Rouhani, A.A. Maradudin, R.F. Wallis, Rayleigh waves on a superlattice stratified normal to the surface. Phys. Rev. B 29, 6454 (1984)CrossRefGoogle Scholar
  56. 56.
    Y. Tanaka, S. Tamura, Surface acoustic waves in two-dimensional periodic elastic structures. Phys. Rev. B 58, 7958 (1998) and Acoustic stop bands of surface and bulk modes in two-dimensional phononic lattices consisting of aluminum and a polymer. Phys. Rev. B 60, 13294 (1999)Google Scholar
  57. 57.
    T.-T. Wu, Z.-G. Huang, S. Lin, Surface and bulk acoustic waves in two-dimensional phononic crystals consisting of materials with general anisotropy. Phys. Rev. B 69, 094301 (2004)CrossRefGoogle Scholar
  58. 58.
    T.-T. Wu, L.-C. Wu, Z.-G. Huang, Frequency band-gap measurement of two-dimensional air/silicon phononic crystals using layered slanted finger interdigital transducers. J. Appl. Phys. 97(094916) (2005)Google Scholar
  59. 59.
    S. Benchabane, A. Khelif, J.-Y. Rauch, L. Robert, V. Laude, Evidence for complete surface wave band gap in a piezoelectric phononic crystal. Phys. Rev. E 73, 065601 (2006)CrossRefGoogle Scholar
  60. 60.
    B. Bonello, C. Charles, F. Ganot, Velocity of a SAW propagating in a 2D phononic crystal. Ultrasonics 44, 1259–1263 (2006)CrossRefGoogle Scholar
  61. 61.
    S. Benchabane, O. Gaiffe, R. Salut, G. Ulliac, Y. Achaoui, V. Laude, Observation of surface-guided waves in holey hypersonic phononic crystal. Appl. Phys. Lett. 98, 171908 (2011)CrossRefGoogle Scholar
  62. 62.
    V. Laude, M. Wilm, S. Benchabane, A. Khelif, Full band gap for surface acoustic waves in a piezoelectric phononic crystal. Phys. Rev. E 71, 036607 (2005)CrossRefGoogle Scholar
  63. 63.
    B. Manzanares-Martinez, F. Ramos-Mendieta, Surface elastic waves in solid composites of two-dimensional periodicity. Phys. Rev. B 68, 134303 (2003)CrossRefGoogle Scholar
  64. 64.
    R. Sainidou, B. Djafari Rouhani, J.O. Vasseur, Surface acoustic waves in finite slabs of three-dimensional phononic crystals. Phys. Rev. B 77, 094304 (2007)CrossRefGoogle Scholar
  65. 65.
    A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, V. Laude, Complete band gaps in two-dimensional phononic crystal slabs. Phys. Rev. E 74, 046610 (2006)CrossRefGoogle Scholar
  66. 66.
    J.O. Vasseur, P. Deymier, B. Djafari Rouhani, Y. Pennec, A.C. Hladky-Hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates. Phys. Rev. B 77, 085415 (2008)CrossRefGoogle Scholar
  67. 67.
    Y. Pennec, B. Djafari Rouhani, H. Larabi, J. Vasseur, A.C. Hladky-Hennion, Low frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate. Phys. Rev. B 78, 104105 (2008)CrossRefGoogle Scholar
  68. 68.
    T.T. Wu, Z.G. Huang, T.-C. Tsai, T.C. Wu, Evidence of complete band gap and resonances in a plate with periodic stubbed surface. Appl. Phys. Lett. 93, 111902 (2008)CrossRefGoogle Scholar
  69. 69.
    T.C. Wu, T.T. Wu, J.C. Hsu, Waveguiding and frequency selection of Lamb waves in a plate with a periodic stubbed surface. Phys. Rev. B 79, 104306 (2009)CrossRefGoogle Scholar
  70. 70.
    Y. Pennec, B. Djafari Rouhani, H. Larabi, A. Akjouj, J.N. Gillet, J.O. Vasseur, G. Thabet, Phonon transport and waveguiding in a phononic crystal made up of cylindrical dots on a this homogeneous plate. Phys. Rev. B 80, 144302 (2009)CrossRefGoogle Scholar
  71. 71.
    S. Mohammadi, A.A. Eftekhar, W.D. Hunt, A. Adibi, High-Q micromechanical resonators in a two-dimensional phononic crystal slab. Appl. Phys. Lett. 94, 051906 (2009)CrossRefGoogle Scholar
  72. 72.
    C.M. Reinke, M.F. Su, R.H. Olsson, I. El-Kady, Realization of optimal bandgaps in solid–solid, solid–air, and hybrid solid–air–solid phononic crystal slabs. Appl. Phys. Lett. 98, 061912 (2011)CrossRefGoogle Scholar
  73. 73.
    Z. Hou, F. Wu, Y. Liu, Phononic crystals containing piezoelectric material. Solid State Commun. 130(11), 745–749 (2004)CrossRefGoogle Scholar
  74. 74.
    X.-Y. Zou, Q. Chen, B. Liang, J.-C. Cheng, Control of the elastic wave bandgaps in two-dimensional piezoelectric periodic structures. Smart Mater. Struct. 17, 015008 (2008)CrossRefGoogle Scholar
  75. 75.
    Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, W.-H. Huang, Elastic wave band gaps in magnetoelectrostatic phononic crystals. Wave Motion 46(47) (2009)Google Scholar
  76. 76.
    J.-F. Robillard, O. Bou Matar, J.O. Vasseur, P.A. Deymier, M. Stippinger, A.C. Hladky-Hennion, Y. Pennec, B. Djafari-Rouhani, Tunable magnetoelastic phononic crystals. Appl. Phys. Lett. 95, 124104 (2009)CrossRefGoogle Scholar
  77. 77.
    K. Bertoldi, M.C. Boyce, Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures. Phys. Rev. B 77, 052105 (2008)CrossRefGoogle Scholar
  78. 78.
    Z.-G. Huang, T.-T. Wu, Temperature effect on the bandgaps of surface and bulk acoustic waves in two-dimensional phononic crystals. IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 52, 365 (2005)CrossRefGoogle Scholar
  79. 79.
    A. Sato, Y. Pennec, N. Shingne, T. Thurn-Albrecht, W. Knoll, M. Steinhart, B. Djafari-Rouhani, G. Fytas, Tuning and switching the hypersonic phononic properties of elastic impedance contrast nanocomposites. ACS Nano 4, 3471 (2010)CrossRefGoogle Scholar
  80. 80.
    M. Maldovan, E.L. Thomas, Simultaneous localization of photons and phonons in two-dimensional periodic structures. Appl. Phys. Lett. 88, 251907 (2006)CrossRefGoogle Scholar
  81. 81.
    S. Mohammadi, A.A. Eftekhar, A. Khelif, A. Adibi, Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs. Opt. Express 18, 9164 (2010)CrossRefGoogle Scholar
  82. 82.
    Y. Pennec, B. Djafari-Rouhani, E.H. El Boudouti, C. Li, Y. El Hassouani, J.O. Vasseur, N. Papanikolaou, S. Benchabane, V. Laude, A. Martinez, Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs. Opt. Express 18, 14301 (2010)CrossRefGoogle Scholar
  83. 83.
    M. Eichenfield, J. Chan, R.M. Camacho, K.J. Vahala, O. Painter, Optomechanical crystals. Nature 462, 78 (2009)CrossRefGoogle Scholar
  84. 84.
    Y. Pennec, B. Djafari-Rouhani, C. Li, J.M. Escalante, A. Martinez, S. Benchabane, V. Laude, N. Papanikolaou, Band gaps and cavity modes in dual phononic and photonic strip waveguides. AIP Adv. 1, 041901 (2011)CrossRefGoogle Scholar
  85. 85.
    S. Amoudache, Y. Pennec, B. Djafari Rouhani, A. Khater, R. Lucklum and R. Tigrine, Simultaneous sensing of light and sound velocities of fluids in a two-dimensional phoXonic crystal with defects, J. Appl. Phys. 115, 134503 (2014)Google Scholar
  86. 86.
    R. Lucklum, Y. Pennec, A. Kraych, M. Zubstov, B. Djafari Rouhani, in Phoxonic Crystal Sensor, SPIE Photonics Europe, Photonic Crystal Materials and Devices X, Brussels, Belgium, April 16–19, 2012, Proc. SPIE-Int. Soc. Opt. Eng., 8425 (2012) 84250N-1-8, ISBN 978-0-8194-9117-6. doi: 10.1117/12.922553
  87. 87.
    S. Yang, J.H. Page, Z. Liu, M.L. Cowan, C.T. Chan, P. Sheng, Focusing of sound in a 3D phononic crystal. Phys. Rev. Lett. 93, 024301 (2004)CrossRefGoogle Scholar
  88. 88.
    K. Imamura, S. Tamura, Negative refraction of phonons and acoustic lensing effect of a crystalline slab. Phys. Rev. B 70, 174308 (2004)CrossRefGoogle Scholar
  89. 89.
    X. Zhang, Z. Liu, Negative refraction of acoustic waves in two-dimensional phononic crystals. Appl. Phys. Lett. 85, 341 (2004)CrossRefGoogle Scholar
  90. 90.
    A. Sukhovich, L. Jing, J.H. Page, Negative refraction and focusing of ultrasound in two-dimensional phononic crystals. Phys. Rev. B 77, 014301 (2008)CrossRefGoogle Scholar
  91. 91.
    A. Sukhovich, B. Merheb, K. Muralidharan, J.O. Vasseur, Y. Pennec, P.A. Deymier, J.H. Page, Experimental and theoretical evidence for subwavelength imaging in phononic crystals. Phys. Rev. Lett. 102, 154301 (2009)CrossRefGoogle Scholar
  92. 92.
    J. Bucay, E. Roussel, J.O. Vasseur, P.A. Deymier, A.-C. Hladky-Hennion, Y. Pennec, K. Muralidharan, B. Djafari-Rouhani, B. Dubus, Positive, negative, zero refraction, and beam splitting in a solid/air phononic crystal: theoretical and experimental study. Phys. Rev. B 79(214305) (2009)Google Scholar
  93. 93.
    P.E. Hopkins, C.M. Reinke, M.F. Su, R.H. Olsson III, E.A. Shaner, Z.C. Leseman, J.R. Serrano, L.M. Phinney, I. El-Kady, Reduction in the thermal conductivity of single crystalline silicon by phononic crystal patterning. Nano Lett. 11, 107 (2011)CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut d’Electronique, de Microélectronique et de Nanotechnologie, Université de Lille 1Villeneuve d’AscqFrance

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