Phononic Band Gaps in One-dimensional Phononic Crystals with Nanoscale Periodic Corrugations at Interfaces. FDTD and PWM Simulations
Nowak Przemysław, Krawczyk Maciej
Surface Physics Division, Faculty of Physics, Adam Mickiewicz University
ul. Umultowska 85, 61-614 Poznań, Poland
e-mail: pnowak@atrom.au
Received:
Received: 31 December 2009; revised: 26 February 2010; accepted: 10 March 2010; published online: 29 April 2010
DOI: 10.12921/cmst.2010.16.01.85-95
OAI: oai:lib.psnc.pl:716
Abstract:
We present and apply two complementary calculation methods used in phononic crystal studies: the finite difference time domain (FDTD) method and the plain wave method (PWM). The FDTD technique allows to simulate the time dependence of a wave packet of vibrational modes propagating through a composite and to determine the transmission coefficient. The PWM method is used for the determination of the phononic dispersion relation in systems with discrete translational symmetry. We use both methods for investigating the effect of periodic interface perturbations on the spectrum of longitudinal vibrational modes in 1D phononic crystals composed of semiconducting materials. The material parameters in the composites under consideration are modulated in the nanoscale.
Key words:
finite difference time domain method, phononic crystals, plane wave method
References:
[1] E. Yablonovitch, Inhibited spontaneous emission in solidstate
physics and electronics. Phys. Rev. Lett. 58, 2059- 2062 (1987).
[2] E.N. Economou, M. Sigalas, Stop bands for elastic-waves in periodic composite-materials. J. Acoust. Soc. Am. 95, 1734-1740 (1994).
[3] M.S. Kushwaha, P. Halevi, Band-gap engineering in periodic elastic composites. Appl. Phys. Lett. 64, 1085-1087 (1994).
[4] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari- Rouhani, Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022-2025 (1993).
[5] M.S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites. Phys. Rev. B 49, 2313-2322 (1994).
[6] M. Sigalas, E.N. Economou, Elastic waves in plates with periodically placed inclusions. J. Appl. Phys. 75, 2845- 2850 (1994).
[7] M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure. J. Sound Vibr. 158, 377-382 (1992).
[8] M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141-143 (1993).
[9] R. Martinez-Sala, J. Sancho, J.V. Sanchez-Perez, J. Llinares, F. Meseguer, Sound attenuation by sculpture. Nature 378, 241 (1997).
[10] Y.Y. Chen, Z. Ye, Acoustic attenuation by two-dimensional arrays of rigid cylinders. Phys. Rev Lett. 87, 184301- 184305 (2001).
[11] Y.Y. Chen, Z. Ye, Theoretical analysis of acoustic stop bands in two-dimensional periodic scattering arrays. Phys. Rev. E 64, 36616-36622 (2001).
[12] D. Caballero, J. Sánchez-Dehesa, C. Rubio, R. Mártinez- Sala, J.V. Sánchez-Pérez, F. Meseguer, J. Llinares, Large two-dimensional sonic band gaps. Phys. Rev. E 60, R6316-
R6319 (1999).
[13] D. Caballero, J. Sánchez-Dehesa, R. Martínez-Sala, C. Rubio, J.V. Sánchez-Pérez, L. Sanchis, F. Meseguer, Suzuki phase in two-dimensional sonic crystals. Phys. Rev. B 64, 64303-64308 (2001).
[14] L. Sanchis, F. Cervera, J. Sánchez-Dehesa, J.V. Sánchez- Pérez, C. Rubio, R. Mártinez-Sala, Reflectance properties of two-dimensional sonic band-gaps crystals. J. Acoust. Soc. Am. 109, 2598-2605 (2001).
[15] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally Resonant Sonic Materials. Science 289, 1734-1736 (2000).
[16] 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, 4282-4285 (2001).
[17] F. Cervera, L. Sanchis, J.V. Sanchez-Perez, R. Martinez-Sala, C. Rubio, F. Meseguer, C. Lopez, D. Caballero, J. Sanchez-Dehesa, Refractive acoustic devices for airborne sound. Phys. Rev. Lett. 88, 23902-23906 (2002).
[18] L. Sanchis, A. Hakansson, F. Cervera and J. Sanchez- Dehesa, Acoustic interferometers based on two-dimensional arrays of rigid cylinders in air. Phys. Rev. B 67, 35422-1-35422-11 (2003).
[19] A. Khelif, P.A. Deymier, B. Djafari-Rouhani, J.O. Vasseur, L. Dobrzynski, Two-dimensional phononic crystal with tunable narrow pass band: application to a waveguide with selective frequency. J. Appl. Phys. 94, 1308-1311 (2003).
[20] B. Liang, B. Yuan, J.-C. Cheng, Acoustic diode: rectification of acoustic energy flux in one-dimensional systems. Phys. Rev. Lett. 103, 104301-104305 (2009).
[21] D.O. Cahill, W.K. Ford, K.E. Goodson, Nanoscale thermal transport. J. Appl. Phys. 93, 793 (2003).
[22] S. Murad, I.K. Puri, Thermal transport through superlattice solid-solid interfaces. Appl. Phys. Lett. 95, 051907 (2009).
[23] S.T. Huxtable, A.R. Abramson, C.-L. Tien, A. Majumdar, C. Labounty, X. Fan, G. Zeng, J.E. Bowers, A. Shakouri, Thermal conductivity of Si/SiGe and SiGe/SiGe superlattices. Appl. Phys. Lett. 80, 1737-1739 (2002).
[24] J.-Y. Duquesene, Thermal conductivity of semiconductor superlattices: Experimental study of interface scattering. Phys. Rev. B 79, 153304 (2009).
[25] G. Chen, Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices. Phys. Rev. B 57, 14958-14973 (1998).
[26] M. Kafesaki, M.M. Sigalas, N. Garcia, The finite difference time domain method for the study of two-dimensional acoustic and elastic gap materials. Kluwer Academic Publishers, Dordrecht 2001.
[27] M. Qiu, S. He, FDTD algorithm for computing the offplane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions. Phys. Lett. A 278,
348-354 (2001).
[28] A. Taflove, Computational Electrodynamics; The Finite-Difference Time-Domain Method. Artech House, London 1995.
[29] D. Garcia-Pablos, M. Sigalas, F.R. Montero de Espinosa, M. Torres, M. Kafesaki, N. García, Theory and experiments on elastic band gaps. Phys. Rev. Lett. 84, 4349-4352
(2000).
[30] S. Mizuno, S. Tamura, Theory of acoustic-phonon transmission in finite-size superlattice systems. Phys. Rev. B 45, 734-741 (1992).
[31] J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, P.A. Deymier, Acoustic band gaps in fibre composite materials of boron-nitride structure. J. Phys.: Condens. Matter 9, 7327-
7333 (1997).
[32] 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 composite and some metallic systems. J. Phys.: Condens. Matter 6, 87599-8770 (1994).
[33] GNU Scientific Library (GSL): http://www.gnu.org/software/gsl/.
We present and apply two complementary calculation methods used in phononic crystal studies: the finite difference time domain (FDTD) method and the plain wave method (PWM). The FDTD technique allows to simulate the time dependence of a wave packet of vibrational modes propagating through a composite and to determine the transmission coefficient. The PWM method is used for the determination of the phononic dispersion relation in systems with discrete translational symmetry. We use both methods for investigating the effect of periodic interface perturbations on the spectrum of longitudinal vibrational modes in 1D phononic crystals composed of semiconducting materials. The material parameters in the composites under consideration are modulated in the nanoscale.
Key words:
finite difference time domain method, phononic crystals, plane wave method
References:
[1] E. Yablonovitch, Inhibited spontaneous emission in solidstate
physics and electronics. Phys. Rev. Lett. 58, 2059- 2062 (1987).
[2] E.N. Economou, M. Sigalas, Stop bands for elastic-waves in periodic composite-materials. J. Acoust. Soc. Am. 95, 1734-1740 (1994).
[3] M.S. Kushwaha, P. Halevi, Band-gap engineering in periodic elastic composites. Appl. Phys. Lett. 64, 1085-1087 (1994).
[4] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari- Rouhani, Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022-2025 (1993).
[5] M.S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites. Phys. Rev. B 49, 2313-2322 (1994).
[6] M. Sigalas, E.N. Economou, Elastic waves in plates with periodically placed inclusions. J. Appl. Phys. 75, 2845- 2850 (1994).
[7] M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure. J. Sound Vibr. 158, 377-382 (1992).
[8] M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141-143 (1993).
[9] R. Martinez-Sala, J. Sancho, J.V. Sanchez-Perez, J. Llinares, F. Meseguer, Sound attenuation by sculpture. Nature 378, 241 (1997).
[10] Y.Y. Chen, Z. Ye, Acoustic attenuation by two-dimensional arrays of rigid cylinders. Phys. Rev Lett. 87, 184301- 184305 (2001).
[11] Y.Y. Chen, Z. Ye, Theoretical analysis of acoustic stop bands in two-dimensional periodic scattering arrays. Phys. Rev. E 64, 36616-36622 (2001).
[12] D. Caballero, J. Sánchez-Dehesa, C. Rubio, R. Mártinez- Sala, J.V. Sánchez-Pérez, F. Meseguer, J. Llinares, Large two-dimensional sonic band gaps. Phys. Rev. E 60, R6316-
R6319 (1999).
[13] D. Caballero, J. Sánchez-Dehesa, R. Martínez-Sala, C. Rubio, J.V. Sánchez-Pérez, L. Sanchis, F. Meseguer, Suzuki phase in two-dimensional sonic crystals. Phys. Rev. B 64, 64303-64308 (2001).
[14] L. Sanchis, F. Cervera, J. Sánchez-Dehesa, J.V. Sánchez- Pérez, C. Rubio, R. Mártinez-Sala, Reflectance properties of two-dimensional sonic band-gaps crystals. J. Acoust. Soc. Am. 109, 2598-2605 (2001).
[15] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally Resonant Sonic Materials. Science 289, 1734-1736 (2000).
[16] 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, 4282-4285 (2001).
[17] F. Cervera, L. Sanchis, J.V. Sanchez-Perez, R. Martinez-Sala, C. Rubio, F. Meseguer, C. Lopez, D. Caballero, J. Sanchez-Dehesa, Refractive acoustic devices for airborne sound. Phys. Rev. Lett. 88, 23902-23906 (2002).
[18] L. Sanchis, A. Hakansson, F. Cervera and J. Sanchez- Dehesa, Acoustic interferometers based on two-dimensional arrays of rigid cylinders in air. Phys. Rev. B 67, 35422-1-35422-11 (2003).
[19] A. Khelif, P.A. Deymier, B. Djafari-Rouhani, J.O. Vasseur, L. Dobrzynski, Two-dimensional phononic crystal with tunable narrow pass band: application to a waveguide with selective frequency. J. Appl. Phys. 94, 1308-1311 (2003).
[20] B. Liang, B. Yuan, J.-C. Cheng, Acoustic diode: rectification of acoustic energy flux in one-dimensional systems. Phys. Rev. Lett. 103, 104301-104305 (2009).
[21] D.O. Cahill, W.K. Ford, K.E. Goodson, Nanoscale thermal transport. J. Appl. Phys. 93, 793 (2003).
[22] S. Murad, I.K. Puri, Thermal transport through superlattice solid-solid interfaces. Appl. Phys. Lett. 95, 051907 (2009).
[23] S.T. Huxtable, A.R. Abramson, C.-L. Tien, A. Majumdar, C. Labounty, X. Fan, G. Zeng, J.E. Bowers, A. Shakouri, Thermal conductivity of Si/SiGe and SiGe/SiGe superlattices. Appl. Phys. Lett. 80, 1737-1739 (2002).
[24] J.-Y. Duquesene, Thermal conductivity of semiconductor superlattices: Experimental study of interface scattering. Phys. Rev. B 79, 153304 (2009).
[25] G. Chen, Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices. Phys. Rev. B 57, 14958-14973 (1998).
[26] M. Kafesaki, M.M. Sigalas, N. Garcia, The finite difference time domain method for the study of two-dimensional acoustic and elastic gap materials. Kluwer Academic Publishers, Dordrecht 2001.
[27] M. Qiu, S. He, FDTD algorithm for computing the offplane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions. Phys. Lett. A 278,
348-354 (2001).
[28] A. Taflove, Computational Electrodynamics; The Finite-Difference Time-Domain Method. Artech House, London 1995.
[29] D. Garcia-Pablos, M. Sigalas, F.R. Montero de Espinosa, M. Torres, M. Kafesaki, N. García, Theory and experiments on elastic band gaps. Phys. Rev. Lett. 84, 4349-4352
(2000).
[30] S. Mizuno, S. Tamura, Theory of acoustic-phonon transmission in finite-size superlattice systems. Phys. Rev. B 45, 734-741 (1992).
[31] J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, P.A. Deymier, Acoustic band gaps in fibre composite materials of boron-nitride structure. J. Phys.: Condens. Matter 9, 7327-
7333 (1997).
[32] 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 composite and some metallic systems. J. Phys.: Condens. Matter 6, 87599-8770 (1994).
[33] GNU Scientific Library (GSL): http://www.gnu.org/software/gsl/.
