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One-Dimensional Phononic Crystals

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Acoustic Metamaterials and Phononic Crystals

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 173))

Abstract

In this chapter, we discuss the vibrational properties of one-dimensional (1D) phononic crystals of both discrete and continuous media. These properties include the dispersion curves of infinite crystals as well as the confined modes and localized (surface, cavity) modes of finite and semi-infinite crystals. A general rule about the existence of localized surface modes in finite and semi-infinite superlattices with free surfaces is presented. We also present the calculations of reflection and transmission coefficients, particularly in view of selective filtering through localized modes. Most of the results presented in this chapter deal with waves propagating along the axis of the superlattice. However, in the last part of the chapter, we also discuss wave propagation out of the normal incidence and, more particularly, we demonstrate the possibility of omnidirectional transmission gap and selective filtering for any incidence angle. A comparison of the theoretical results with experimental data available in the literature is also presented and the reliability of the theoretical predictions is indicated.

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References

  1. M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett. 71, 2022 (1993)

    Article  CAS  Google Scholar 

  2. M. Sigalas, E.N. Economou, Solid State Commun. 86, 141 (1993)

    Article  CAS  Google Scholar 

  3. M.S. Kushwaha, Int. J. Mod. Phys. B 10, 977 (1996)

    Article  CAS  Google Scholar 

  4. L. Esaki, R. Tsu, IBM J. Res. Dev. 14, 61 (1970)

    Article  CAS  Google Scholar 

  5. L. Colvard, R. Merlin, M.V. Klein, A.C. Gossard, Phys. Rev. Lett. 45, 298 (1980)

    Article  CAS  Google Scholar 

  6. For a recent review, see: E.H.El Boudouti, B. Djafari-Rouhani, A. Akjouj and L. Dobrzynski, Surf. Sci. Rep. 64, 471 (2009)

    Google Scholar 

  7. J. Sapriel, B. Djafari-Rouhani, Surf. Sci. Rep. 10, 189 (1989)

    Article  CAS  Google Scholar 

  8. B. Jusserand, M. Cardona, in Light Scattering in Solids, ed. by M. Cardona, G. Güntherodt (Springer, Berlin, 1989), p. 49

    Chapter  Google Scholar 

  9. S. Tamura, Proceedings of the 6th International Conference on Phonon Scattering in Condensed Matter, Heidelberg, 1989 (World Scientific, Singapore, 1990), p.703

    Google Scholar 

  10. D.J. Lockwood, J.F. Young, Light Scattering in Semiconductor Structures and Superlattices (Plenum, New York, 1991)

    Google Scholar 

  11. S. Tamura, Phys. Rev. B 39, 1261 (1989)

    Article  Google Scholar 

  12. S. Mizuno, S. Tamura, Phys. Rev. B 45, 13423 (1992)

    Article  Google Scholar 

  13. M.J.S. Lowe, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 525 (1995)

    Article  Google Scholar 

  14. E.H. El Boudouti, B. Djafari-Rouhani, E.M. Khourdifi, L. Dobrzynski, Phys. Rev. B 48, 10987 (1993)

    Article  CAS  Google Scholar 

  15. E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, Phys. Rev. B 54, 14728 (1996)

    Article  Google Scholar 

  16. F. Garcia-Moliner, V.R. Velasco, Theory of Single and Multiple Interfaces. The Method of Surface Green Function Matching (World Scientific, Singapore, 1992)

    Google Scholar 

  17. B. Jusserand, D. Paquet, F. Mollot, F. Alexandre, G. LeRoux, Phys. Rev. B 35, 2808 (1987)

    Article  CAS  Google Scholar 

  18. D.J. Lockwood, R.L.S. Devine, A. Rodriguez, J. Mendialdua, B. Djafari-Rouhani, L. Dobrzynski, Phys. Rev. B 47, 13553 (1993)

    Article  CAS  Google Scholar 

  19. A.S. Barker, J.L. Merz, A.C. Gossard, Phys. Rev. B 17, 3181 (1978)

    Article  CAS  Google Scholar 

  20. V. Narayanamurti, H.L. Stormer, M.A. Chin, A.C. Gossard, W. Wiegmann, Phys. Rev. Lett. 43, 2012 (1979)

    Article  CAS  Google Scholar 

  21. O. Koblinger, J. Mebert, E. Dittrich, S. Dottinger, W. Eisenmenger, P.V. Santos, L. Ley, Phys. Rev. B 35, 9372 (1987)

    Article  CAS  Google Scholar 

  22. D.J. Dielemen, A.F. Koenderink, M.G.A. van Veghel, A.F.M. Arts, H.W. de Wijn, Phys. Rev. B 64, 174304 (2001)

    Article  CAS  Google Scholar 

  23. S. Tamura, D.C. Hurley, J.P. Wolf, Phys. Rev. B 38, 1427 (1988)

    Article  Google Scholar 

  24. P.J. Shull, D.E. Chimenti, A. Safaeinili, J. Acoust. Soc. Am. 95, 99 (1994)

    Article  Google Scholar 

  25. A. Safaeinili, D.E. Chimenti, J. Acoust. Soc. Am. 98, 2336 (1995)

    Article  Google Scholar 

  26. H.T. Grahn, H.J. Maris, J. Tauc, B. Abeles, Phys. Rev. B 38, 6066 (1988)

    Article  CAS  Google Scholar 

  27. A. Yamamoto, T. Mishina, Y. Masumoto, M. Nakayama, Phys. Rev. Lett. 73, 740 (1994)

    Article  CAS  Google Scholar 

  28. A. Bartels, T. Dekorsy, H. Kurz, Appl. Phys. Lett. 72, 2844 (1998)

    Article  CAS  Google Scholar 

  29. A. Bartels, T. Dekorsy, H. Kurz, K. Köhler, Phys. Rev. Lett. 82, 1044 (1999)

    Article  CAS  Google Scholar 

  30. P. Sondhauss, J. Larsson, M. Harbst, G.A. Naylor, A. Plech, K. Scheidt, O. Synnergren, M. Wulff, J.S. Wark, Phys. Rev. Lett. 94, 125509 (2005)

    Article  CAS  Google Scholar 

  31. B. Djafari-Rouhani, L. Dobrzynski, O. Hardouin Duparc, R.E. Camley, A.A. Maradudin, Phys. Rev. B 28, 1711 (1983)

    Article  CAS  Google Scholar 

  32. T. Aono, S. Tamura, Phys. Rev. B 58, 4838 (1998)

    Article  CAS  Google Scholar 

  33. S. Mizuno, S. Tamura, Phys. Rev. B 53, 4549 (1996)

    Article  CAS  Google Scholar 

  34. M. Hammouchi, E.H. El Boudouti, A. Nougaoui, B. Djafari-Rouhani, M.L.H. Lahlaouti, A. Akjouj, L. Dobrzynski, Phys. Rev. B 59, 1999 (1999)

    Article  CAS  Google Scholar 

  35. H.J. Trodahl, P.V. Santos, G.V.M. Williams, A. Bittar, Phys. Rev. B 40, R8577 (1989)

    Article  Google Scholar 

  36. W. Chen, Y. Lu, H.J. Maris, G. Xiao, Phys. Rev. B 50, 14506 (1994)

    Article  CAS  Google Scholar 

  37. B. Perrin, B. Bonello, J.C. Jeannet, E. Romatet, Physica B 219–220, 681 (1996)

    Article  Google Scholar 

  38. B. Bonello, B. Perrin, E. Romatet, J.C. Jeannet, Ultrasonics 35, 223 (1997)

    Article  CAS  Google Scholar 

  39. N-W Pu, J. Bokor, Phys. Rev. Lett. 91, 076101 (2003)

    Article  CAS  Google Scholar 

  40. N-W Pu, Phys. Rev. B 72, 115428 (2005)

    Article  CAS  Google Scholar 

  41. E.M. Khourdifi, B. Djafari-Rouhani, Surf. Sci. 211/212, 361 (1989)

    Article  Google Scholar 

  42. D. Bria, E.H. El Boudouti, A. Nougaoui, B. Djafari-Rouhani, V.R. Velasco, Phys. Rev. B 60, 2505 (1999)

    Article  CAS  Google Scholar 

  43. E.M. Khourdifi, B. Djafari-Rouhani, J. Phys. Condens. Matter 1, 7543 (1989)

    Article  Google Scholar 

  44. D. Bria, E.H. El Boudouti, A. Nougaoui, B. Djafari-Rouhani, V.R. Velasco, Phys. Rev. B 61, 15858 (2000)

    Article  CAS  Google Scholar 

  45. K.-Q. Chen, X.-H. Wang, B.-Y. Gu, Phys. Rev. B 61, 12075 (2000)

    Article  CAS  Google Scholar 

  46. S. Mizuno, Phys. Rev. B 65, 193302 (2002)

    Article  CAS  Google Scholar 

  47. S. Tamura, H. Watanabe, T. Kawasaki, Phys. Rev. B 72, 165306 (2005)

    Article  CAS  Google Scholar 

  48. G.P. Schwartz, G.J. Gualtieri, W.A. Sunder, Appl. Phys. Lett. 58, 971 (1991)

    Article  Google Scholar 

  49. M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, V. Thierry-Mieg, Phys. Rev. Lett. 89, 227402 (2002)

    Article  CAS  Google Scholar 

  50. P. Lacharmoise, A. Fainstein, B. Jusserand, V. Thierry-Mieg, Appl. Phys. Lett. 84, 3274 (2004)

    Article  CAS  Google Scholar 

  51. N.D. Lanzillotti Kimura, A. Fainstein, B. Jusserand, Phys. Rev. B 71, 041305(R) (2005)

    Article  CAS  Google Scholar 

  52. P.X. Zhang, D.J. Lockwood, J.M. Baribeau, Can. J. Phys. 70, 843 (1992)

    Article  CAS  Google Scholar 

  53. P.X. Zhang, D.J. Lockwood, H.J. Labbe, J.M. Baribeau, Phys. Rev. B 46, 9881 (1992)

    Article  Google Scholar 

  54. M. Trigo, A. Fainstein, B. Jusserand, V. Thierry-Mieg, Phys. Rev. B 66, 125311 (2002)

    Article  CAS  Google Scholar 

  55. A. Bousfia, E.H. El Boudouti, B. Djafari-Rouhani, D. Bria, A. Nougaoui, V.R. Velasco, Surf. Sci. 482–485, 1175 (2001)

    Article  Google Scholar 

  56. D. Bria, B. Djafari-Rouhani, A. Bousfia, E.H. El Boudouti, A. Nougaoui, Europhys. Lett. 55, 841 (2001)

    Article  CAS  Google Scholar 

  57. D. Bria, B. Djafari-Rouhani, Phys. Rev. E 66, 056609 (2002)

    Article  CAS  Google Scholar 

  58. B. Manzanares-Martinez, J. Sanchez-Dehesa, A. Hakansson, F. Cervera, F. Ramos-Mendieta, Appl. Phys. Lett. 85, 154 (2004)

    Article  CAS  Google Scholar 

  59. G. Wang, D. Yu, J. Wen, Y. Liu, X. Wen, Phys. Lett. A 327, 512 (2004)

    Article  CAS  Google Scholar 

  60. L.C. Parsons, G.T. Andrews, Appl. Phys. Lett. 95, 241909 (2009)

    Article  CAS  Google Scholar 

  61. G.N. Aliev, B. Goller, D. Kovalev, P.A. Snow, Appl. Phys. Lett. 96, 124101 (2010)

    Article  CAS  Google Scholar 

  62. N. Gomopoulos, D. Maschke, C.Y. Koh, E.L. Thomas, W. Tremel, H.J. Butt, G. Fytas, Nano Lett. 10, 980 (2010)

    Article  CAS  Google Scholar 

  63. P.M. Walker, J.S. Sharp, A.V. Akimov, A.J. Kent, Appl. Phys. Lett. 97, 073106 (2010)

    Article  CAS  Google Scholar 

  64. T. Berstermann, C. Brggemann, M. Bombeck, A.V. Akimov, D.R. Yakovlev, C. Kruse, D. Hommel, M. Bayer, Phys. Rev. B 81, 085316 (2010)

    Article  CAS  Google Scholar 

  65. I.E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, V. Laude, Phys. Rev. B 82, 174303 (2010)

    Article  CAS  Google Scholar 

  66. A.J. Kent, R.N. Kini, N.M. Stanton, M. Henini, B.A. Glavin, V.A. Kochelap, T.L. Linnik, Phys. Rev. Lett. 96, 215504 (2006)

    Article  CAS  Google Scholar 

  67. A. Bousfia, Ph.D. Thesis, University Mohamed I, Oujda, Morocco (2004)

    Google Scholar 

  68. H. Kato, Phys. Rev. B 59, 11136 (1999)

    Article  CAS  Google Scholar 

  69. S. Mizuno, Phys. Rev. B 63, 035301 (2000)

    Article  CAS  Google Scholar 

  70. D. Zhao, W. Wang, Z. Liu, J. Shi, W. Wen, Physica B 390, 159 (2007)

    Article  CAS  Google Scholar 

  71. L.M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1981)

    Google Scholar 

  72. S.M. Rytov, Phys. Acoust. 2, 68 (1956)

    Google Scholar 

  73. M. Schöenberg, Wave Motion 6, 303 (1984)

    Article  Google Scholar 

  74. M. Rousseau, J. Acoust. Soc. Am. 86, 2369 (1989)

    Article  Google Scholar 

  75. M.A. Biot, J. Acoust. Soc. Am. 28, 168 (1962)

    Article  Google Scholar 

  76. B. Gurevich, J. Acoust. Soc. Am. 106, 57 (1999)

    Article  Google Scholar 

  77. B. Gurevich, Goephysics 67, 264 (2002)

    Google Scholar 

  78. R. Ciz, E.H. Saenger, B. Gurevich, J. Acoust. Soc. Am. 120, 642 (2006)

    Article  CAS  Google Scholar 

  79. T.J. Plona, K.W. Winkler, M. Schoenberg, J. Acoust. Soc. Am. 81, 1227 (1987)

    Article  CAS  Google Scholar 

  80. C. Gazanhes, J. Sageloli, Acustica 81, 221 (1995)

    Google Scholar 

  81. R. James, S.M. Woodley, C.M. Dyer, F. Humphrey, J. Acoust. Soc. Am. 97, 2041 (1995)

    Article  Google Scholar 

  82. M. Shen, W. Cao, Appl. Phys. Lett. 75, 3713 (1999)

    Article  CAS  Google Scholar 

  83. H. Sanchis-Alepuz, Y.A. Kosevich, J. Sánchez-Dehesa, Phys. Rev. Lett. 98, 134301 (2007)

    Article  CAS  Google Scholar 

  84. G. Monsivais, R. Rodrguez-Ramos, R. Esquivel-Sirvent, L. Fernández-Alvarez, Phys. Rev. B 68, 174109 (2003)

    Article  CAS  Google Scholar 

  85. R. Rodriguez-Ramos, G. Monsivais, J.A. Otero, H. Calás, V. Guerra, C. Stern, J. Appl. Phys. 96, 1178 (2004)

    Article  CAS  Google Scholar 

  86. E.E. Mendez, F. Agullo-Rueda, J.M. Hong, Phys. Rev. Lett. 60, 2426 (1988)

    Article  CAS  Google Scholar 

  87. L. Dobrzynski, Surf. Sci. Rep. 6, 119 (1986)

    Article  Google Scholar 

  88. L. Dobrzynski, Surf. Sci. Rep. 11, 139 (1990)

    Article  Google Scholar 

  89. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996)

    Google Scholar 

  90. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953)

    Google Scholar 

  91. S.-Y. Ren, Electronic States in Crystals of Finite Size: Quatum Confinement of Bloch Waves (Springer, New York, 2006). Chap.4

    Book  Google Scholar 

  92. S.-Y. Ren, Phys. Rev. B 64, 035322 (2001)

    Article  CAS  Google Scholar 

  93. S.Y. Ren, Ann. Phys. (N.Y.) 301, 22 (2002)

    Article  CAS  Google Scholar 

  94. S.-Y. Ren, Y.-C. Chang, Phys. Rev. B 75, 212301 (2007)

    Article  CAS  Google Scholar 

  95. Y. El Hassouani, E.H. El Boudouti, B. Djafari-Rouhani, H. Aynaou, Phys. Rev. B 78, 174306 (2008)

    Article  CAS  Google Scholar 

  96. E.H. El Boudouti, Y. El Hassouani, B. Djafari-Rouhani, H. Aynaou, Phys. Rev. E 76, 026607 (2007)

    Article  CAS  Google Scholar 

  97. A.C. Hladky-Hennion, G. Allan, J. Appl. Phys. 98, 054909 (2005)

    Article  CAS  Google Scholar 

  98. B. Djafari-Rouhani, E.H. El Boudouti, A. Akjouj, L. Dobrzynski, J.O. Vasseur, A. Mir, N. Fettouhi, J. Zemmouri, Vacuum 63, 177 (2001)

    Article  CAS  Google Scholar 

  99. Y. El Hassouani, H. Aynaou, E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, V.R. Velasco, Phys. Rev. B 74, 035314 (2006)

    Article  CAS  Google Scholar 

  100. M.L.H. Lahlaouti, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, M. Hammouchi, E.H. El Boudouti, A. Nougaoui, B. Kharbouch, Phys. Rev. B 63, 035312 (2001)

    Article  CAS  Google Scholar 

  101. K.R. Subbaswamy, A.A. Maradudin, Phys. Rev. B 18, 4181 (1978)

    Article  CAS  Google Scholar 

  102. B. Djafari-Rouhani, E.M. Khourdifi, in Light Scattering in Semiconductor Structures and Superlattices (Ref. [11]) p 139.

    Google Scholar 

  103. J. He, B. Djafari-Rouhani, J. Sapriel, Phys. Rev. B 37, 4086 (1988)

    Article  Google Scholar 

  104. V. Lemos, O. Pilla, M. Montagna, C.F. de Souza, Superlattices Microstruct. 17, 51 (1995)

    Article  CAS  Google Scholar 

  105. J. Christensen, A.I. Fernandez-Dominguez, F. de Leon-Perez, L. Martin-Moreno, F.J. Garcia-Vidal, Nat. Phys. 3, 851 (2007)

    Article  CAS  Google Scholar 

  106. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988)

    Google Scholar 

  107. T.E. Hartman, J. Appl. Phys. 33, 3427 (1962)

    Article  Google Scholar 

  108. C. Spielmann, R. Szipöcs, A. Stingl, F. Krausz, Phys. Rev. Lett. 73, 2308 (1994)

    Article  CAS  Google Scholar 

  109. P. Pereyra, Phys. Rev. Lett. 84, 1772 (2000)

    Article  CAS  Google Scholar 

  110. S. Esposito, Phys. Rev. E 64, 026609 (2001)

    Article  CAS  Google Scholar 

  111. G. Nimtz, A.A. Stahlhofen, Ann. Phys. (Berlin) 17, 374 (2008)

    Article  CAS  Google Scholar 

  112. J.P. Dowling, Science 282, 1841 (1998)

    Article  CAS  Google Scholar 

  113. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, E.L. Thomas, Science 282, 1679 (1998)

    Article  CAS  Google Scholar 

  114. D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Appl. Phys. A Mater. Sci. Process. A 68, 25 (1999)

    Article  CAS  Google Scholar 

  115. D. Bria, B. Djafari-Rouhani, E.H. El Boudouti, A. Mir, A. Akjouj, A. Nougaoui, J. Appl. Phys. 91, 2569 (2002)

    Article  CAS  Google Scholar 

  116. X.-H. Wang, K.-Q. Chen, G. Ben-Yuan, J. Appl. Phys. 92, 5113 (2002)

    Article  CAS  Google Scholar 

  117. S. Mizuno, Phys. Rev. B 68, 193305 (2003)

    Article  CAS  Google Scholar 

  118. H. Kato, H.J. Maris, S.I. Tamura, Phys. Rev. B 53, 7884 (1996)

    Article  CAS  Google Scholar 

  119. B. Manzanares-Martínez, F. Ramos-Mendieta, Phys. Rev. B 76, 134303 (2007)

    Article  CAS  Google Scholar 

  120. Y. Tanaka, M. Narita, S.-I. Tamura, J. Phys. Condens. Matter 10, 8787 (1998)

    Article  CAS  Google Scholar 

  121. K. Imamura, Y. Tanaka, S. Tamura, Phys. Rev. B 65, 174301 (2007)

    Article  CAS  Google Scholar 

  122. A.L. Shuvalov, A.S. Gorkunova, J. Sound Vib. 243, 679 (2001)

    Article  Google Scholar 

  123. U. Fano, Phys. Rev. 124, 1866 (1961)

    Article  CAS  Google Scholar 

  124. Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Science 289, 1734 (2000)

    Article  CAS  Google Scholar 

  125. S. Yang, J.H. Page, Z. Liu, M.L. Cowan, C. Chan, P. Sheng, Phys. Rev. Lett. 88, 104301 (2002)

    Article  CAS  Google Scholar 

  126. C. Goffaux, J. Sánchez-Dehesa, A.L. Yeyati, P. Lambin, A. Khelif, J.O. Vasseur, B. Djafari-Rouhani, Phys. Rev. Lett. 88, 225502 (2002)

    Article  CAS  Google Scholar 

  127. Z. Liu, C.T. Chan, P. Sheng, Phys. Rev. B 71, 014103 (2005)

    Article  CAS  Google Scholar 

  128. Y.A. Kosevich, C. Goffaux, J. Sánchez-Dehesa, Phys. Rev. B 74, 012301 (2006)

    Article  CAS  Google Scholar 

  129. M.L. Ladrón de Guevara, F. Claro, P.A. Orellana, Phys. Rev. B 67, 195335 (2003)

    Article  CAS  Google Scholar 

  130. E.H. El Boudouti, T. Mrabti, H. Al-Wahsh, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Phys. Condens. Matter 20, 255212 (2008)

    Article  CAS  Google Scholar 

  131. H. Al-Wahsh, E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, T. Mrabti, L. Dobrzynski, Phys. Rev. B 78, 075401 (2008)

    Article  CAS  Google Scholar 

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Acknowledgments

This work was realized within the framework of a scientific convention between the universities of Lille 1 and Oujda.

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Authors and Affiliations

  1. LDOM, Département de Physique, Université Mohamed I, 60000, Oujda, Morocco

    EI Houssaine EI Boudouti

  2. IEMN, UMR-CNRS 8520, UFR de Physique, Université de Lille 1, 59655, Villeneuve d’Ascq, France

    Bahram Djafari-Rouhani

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  1. EI Houssaine EI Boudouti
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Correspondence to EI Houssaine EI Boudouti .

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  1. , Dept. of Materials Science and Engineeri, University of Arizona, East James E. Rogers Way 1235, Tucson, 85721-0012, Arizona, USA

    Pierre A. Deymier

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Boudouti, E.H.E., Djafari-Rouhani, B. (2013). One-Dimensional Phononic Crystals. In: Deymier, P. (eds) Acoustic Metamaterials and Phononic Crystals. Springer Series in Solid-State Sciences, vol 173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31232-8_3

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