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Phononic Crystals pp 23–50Cite as

Fundamental Properties of Phononic Crystal

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Abstract

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.

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References

  1. F. Bloch, Z. Phys. 52, 555 (1928)

    Article  MATH  Google Scholar 

  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. S.M. Rytov, Sov. Phys. Acoust. 2, 6880 (1956)

    Google Scholar 

  4. M.M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141 (1993)

    Article  Google Scholar 

  5. M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022 (1993)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  7. M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure. J. Sound Vib. 158, 377 (1992)

    Article  Google Scholar 

  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. 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. 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. E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987)

    Article  Google Scholar 

  12. J.D. Joannopoulos, R.D. Meade, J.N. Winn, Molding the Flow of Light (Princeton University Press, Princeton, 1995)

    MATH  Google Scholar 

  13. I.E. Psarobas, A. Modinos, R. Sainidou, N. Stefanou, Acoustic properties of colloidal crystals. Phys. Rev. B 65, 064307 (2002)

    Article  Google Scholar 

  14. R. Sainidou, N. Stefanou, A. Modinos, Formation of absolute frequency gaps in three-dimensional solid phononic crystals. Phys. Rev. B 66, 212301 (2002)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  31. E.N. Economou, M.M. Sigalas, Classical wave propagation in periodic structures: cermet versus network topology. Phys. Rev. B 48, 13434 (1993)

    Article  Google Scholar 

  32. M.M. Sigalas, E.N. Economou, Attenuation of multiple-scattered sound. Europhys. Lett. 36, 241 (1996)

    Article  Google Scholar 

  33. M.S. Kushwaha, Stop-bands for periodic metallic rods: sculptures that can filter the noise. Appl. Phys. Lett. 70, 3218 (1997)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 59

    Chapter  Google Scholar 

  38. M.S. Kushwaha, B. Djafari-Rouhani, Complete acoustic stop bands for cubic arrays of spherical liquid balloons. J. Appl. Phys. 80, 3191 (1996)

    Article  Google Scholar 

  39. R. Martinez-Sala, J. Sancho, J.V. Sanchez, V. Gomez, J. Llinares, F. Meseguer, Sound attenuation by sculpture. Nature 378, 241 (1995)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  45. R. Lucklum, J. Li, Phononic crystals for liquid sensor applications. Meas. Sci. Technol. 20, 124014 (2009)

    Article  Google Scholar 

  46. M. Ke, M. Zubtsov, R. Lucklum, Sub-wavelength phononic crystal liquid sensor. J. Appl. Phys. 110, 026101 (2011)

    Article  Google Scholar 

  47. M.S. Kushwaha, B. Djafari-Rouhani, Giant sonic stop bands in two-dimensional periodic system of fluids. J. Appl. Phys. 84, 4677 (1998)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  51. C. Goffaux, J.P. Vigeron, Theoretical study of a tunable phononic band gap system. Phys. Rev. B 64, 075118 (2001)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  60. B. Bonello, C. Charles, F. Ganot, Velocity of a SAW propagating in a 2D phononic crystal. Ultrasonics 44, 1259–1263 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  63. B. Manzanares-Martinez, F. Ramos-Mendieta, Surface elastic waves in solid composites of two-dimensional periodicity. Phys. Rev. B 68, 134303 (2003)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  73. Z. Hou, F. Wu, Y. Liu, Phononic crystals containing piezoelectric material. Solid State Commun. 130(11), 745–749 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  77. K. Bertoldi, M.C. Boyce, Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures. Phys. Rev. B 77, 052105 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  80. M. Maldovan, E.L. Thomas, Simultaneous localization of photons and phonons in two-dimensional periodic structures. Appl. Phys. Lett. 88, 251907 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  83. M. Eichenfield, J. Chan, R.M. Camacho, K.J. Vahala, O. Painter, Optomechanical crystals. Nature 462, 78 (2009)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  88. K. Imamura, S. Tamura, Negative refraction of phonons and acoustic lensing effect of a crystalline slab. Phys. Rev. B 70, 174308 (2004)

    Article  Google Scholar 

  89. X. Zhang, Z. Liu, Negative refraction of acoustic waves in two-dimensional phononic crystals. Appl. Phys. Lett. 85, 341 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

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  1. Institut d’Electronique, de Microélectronique et de Nanotechnologie, Université de Lille 1, IMEN Bat P5, Villeneuve d’Ascq, 59655, France

    Yan Pennec & Bahram Djafari-Rouhani

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  1. Yan Pennec
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  1. Institut FEMTO-ST, Centre National de la Recherche Scientifique, Besançon Cedex, France

    Abdelkrim Khelif

  2. School of Electrical Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA

    Ali Adibi

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Pennec, Y., Djafari-Rouhani, B. (2016). Fundamental Properties of Phononic Crystal. In: Khelif, A., Adibi, A. (eds) Phononic Crystals. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9393-8_2

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