Research Papers

Comparison of Atomistic and Continuum Methods for Calculating Ballistic Phonon Transmission in Nanoscale Waveguides

[+] Author and Article Information
Drew A. Cheney

e-mail: dcheney@seas.upenn.edu

Jennifer R. Lukes

e-mail: jrlukes@seas.upenn.edu
Department of Mechanical Engineering and
Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 3, 2012; final manuscript received March 8, 2013; published online July 26, 2013. Assoc. Editor: Pamela M. Norris.

J. Heat Transfer 135(9), 091101 (Jul 26, 2013) (9 pages) Paper No: HT-12-1347; doi: 10.1115/1.4024355 History: Received July 03, 2012; Revised March 08, 2013

We compare two methods for the calculation of mode dependent ballistic phonon transmission in nanoscale waveguides. The first method is based on continuum acoustic waveguide theory and uses an eigenmode expansion to solve for phonon transmission coefficients. The second method uses lattice dynamics (LD)-computed mode shapes to excite guided phonon wavepackets in a nonequilibrium molecular dynamics (MD) simulation and calculates phonon transmission from the final distribution of system energy. The two methods are compared for the case of shear-horizontal (SH) phonons propagating in a planar waveguide with a T-stub irregularity, a geometry which has been proposed for the tuning of phonon transmission and nanostructure thermal conductance. Our comparison highlights advantages and disadvantages of the two methods and illustrates regimes when atomistic effects are prominent and continuum approaches are not appropriate.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Liu, W., and Asheghi, M., 2006, “Thermal Conductivity Measurements of Ultra-Thin Single Crystal Silicon Layers,” ASME J. Heat Transfer.128, pp. 75–83. [CrossRef]
Ju, Y. S., and Goodson, K. E., 1999, “Phonon Scattering in Silicon Films With Thickness of Order 100 nm,” Appl. Phys. Lett., 74, pp. 3005–3007. [CrossRef]
Li, D., Wu, Y., Kim, P., Li, S., Yang, P., and Majumdar, A., 2003, “Thermal Conductivity of Individual Silicon Nanowires,” Appl. Phys. Lett., 83, pp. 2934–2936. [CrossRef]
Chen, R., Hochbaum, A. I., Murphy, P., Moore, J., Yang, P., and Majumdar, M., 2008, “Thermal Conductance of Thin Silicon Nanowires,” Phys. Rev. Lett., 101, p. 105501. [CrossRef]
Hochbaum, A. I., Chen, R.Delgado, R. D., Liang, W., Garnett, E. C., Najarian, M., Majumdar, A., and Yang, P., 2008, “Enhanced Thermoelectric Performance of Rough Silicon Nanowires,” Nature, 451, pp. 163–167. [CrossRef]
Heron, J. S., Bera, C., Fournier, T., Mingo, N., and Bourgeois, O., 2010, “Blocking Phonons via Nanoscale Geometrical Design,” Phys. Rev. B, 82, p. 155458. [CrossRef]
Chang, C. W., Okawa, D., Garcia, H., Majumdar, A., and Zettl, A., 2007, “Nanotube Phonon Waveguide,” Phys. Rev. Lett., 99, p. 045901. [CrossRef]
Li, W. X., Chen, K. Q., Duan, W., Wu, J., and Gu, B. L., 2004, “Acoustic Phonon Mode Splitting Behavior of an Asymmetric Y-Branch Three Terminal Junction,” Applied Physics Letters85, pp. 822–824. [CrossRef]
Huang, W. Q., Chen, K. Q., Shuai, Z., Wang, L., Hu, W., and Zou, B. S., 2005, “Acoustic-Phonon Transmission and Thermal Conductance in a Double-Bend Quantum Waveguide,” J. Appl. Phys., 98, p. 093524. [CrossRef]
Tang, L. M., Wang, L. L., Chen, K. Q., Huang, W. Q., and Zou, B. S., 2006, “Coupling Effect on Phonon Thermal Transport in a Double-Stub Quantum Wire,” Appl. Phys. Lett., 88, p. 163505. [CrossRef]
Chen, K. Q., Li, W. X., Duan, W., Shuai, Z., and Gu, B. L., 2005, “Effect of Defects on the Thermal Conductivity in a Nanowire,” Phys. Rev. B, 72, p. 045422. [CrossRef]
Sun, Q., Yang, P., and Guo, H., 2002, “Four-Terminal Thermal Conductance of Mesoscopic Dielectric Systems,” Phys. Rev. Lett., 89, p. 175901. [CrossRef]
Huang, W. Q., Huang, B. Y., Yi, D. Q., Wang, M. P.Huang, G. F., and Wang, L. L., 2008, “Selective Transmission and Enhanced Thermal Conductance of Ballistic Phonon by Nanocavities Embedded in a Narrow Constriction,” J. Phys. D: Appl. Phys., 42, p. 015101. [CrossRef]
Huang, W. Q., Yi, D. Q., Huang, B. Y., Wang, M. P., Huang, G. F., and Wang, L. L., 2008, “Selective Transport of Ballistic Phonon Modes by an Acoustic Nanocavity in a Psi-Shaped Semiconductor Nanowire,” J. Appl. Phys., 104, p. 054309. [CrossRef]
Yang, P., Sun, Q., Guo, H., and Hu, B., 2007, “Thermal Transport in a Dielectric T-Shaped Quantum Wire,” Phys. Rev. B, 75, p. 235319. [CrossRef]
Liu, Z., Yu, X., and Chen, K., 2009, “Thermal Transport Associated With Ballistic Phonons in Asymmetric Quantum Structures,” Fron. Phys. China, 4, pp. 420–425. [CrossRef]
Roberts, N. A., and Walker, D. G., 2011, “Phonon Transport in Asymmetric Sawtooth Nanowires,” Proceedings of the ASME/JSME 2011 8th Thermal Engineering Joint Conference, Paper No. AJTEC2011-44.
Balandin, A., and Wang, K. L., 1998, “Significant Decrease of the Lattice Thermal Conductivity Due to Phonon Confinement in a Free-Standing Semiconductor Quantum Well,” Phys. Rev. B, 58, pp. 1544–1549. [CrossRef]
SadhuJ., and SinhaS., 2011, “Room-Temperature Phonon Boundary Scattering Below the Casimir Limit,” Phys. Rev. B, 84(11), p. 115450. [CrossRef]
Tamura, H., and Ando, T., 1991, “Conductance Fluctuations in Quantum Wires,” Phys. Rev. B, 44, pp. 1792–1800. [CrossRef]
Landauer, R., 1957, “Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction,” IBM J. Res. Dev., 1, pp. 223–231. [CrossRef]
Rego, L. G. C., and Kirczenow, G., 1998, “Quantized Thermal Conductance of Dielectric Quantum Wires,” Phys. Rev. Lett., 81, pp. 232–235. [CrossRef]
Galán, J. M., and Abascal, R., 2003, “Elastodynamic Guided Wave Scattering in Infinite Plates,” Int. J. Numer. Methods Eng., 58, pp. 1091–1118. [CrossRef]
Cho, Y., 2000, “Estimation of Ultrasonic Guided Wave Mode Conversion in a Plate With Thickness Variation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 47, pp. 591–603. [CrossRef]
Koshiba, M., Hasegawa, K., and Suzuki, M., 1987, “Finite-Element Solution of Horizontally Polarized Shear Wave Scattering in an Elastic Plate,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 34, pp. 461–466. [CrossRef]
Al-Nassar, Y. N., Datta, S. K., and Shah, A. H., 1991, “Scattering of Lamb Waves by a Normal Rectangular Strip Weldment,” Ultrasonics, 29, pp. 125–132. [CrossRef]
Song, W. J., Rose, J. L., Galán, J. M., and Abascal, R., 2005, “Ultrasonic Guided Wave Scattering in a Plate Overlap,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 52, pp. 892–903. [CrossRef]
Schelling, P. K., Phillpot, S. R., and Keblinski, P., 2002, “Phonon Wave-Packet Dynamics at Semiconductor Interfaces by Molecular-Dynamics Simulation,” Appl. Phys. Lett., 80, pp. 2484–2486. [CrossRef]
Becker, B., Schelling, P. K., and Phillpot, S. R., 2006, “Interfacial Phonon Scattering in Semiconductor Nanowires by Molecular-Dynamics Simulation,” J. Appl. Phys., 99, p. 123715. [CrossRef]
Kondo, N., Yamamoto, T., and Watanabe, K., 2006, “Phonon Wavepacket Scattering Dynamics in Defective Carbon Nanotubes,” Jpn. J. Appl. Phys., 45, pp. L963–L965. [CrossRef]
Zuckerman, N., and Lukes, J. R., 2008, “Acoustic Phonon Scattering From Particles Embedded in an Anisotropic Medium: A Molecular Dynamics Study,” Phys. Rev. B, 77, p. 094302. [CrossRef]
Tian, Z. T., White, B. E., and Sun, Y., 2010, “Phonon Wave-Packet Interference and Phonon Tunneling Based Energy Transport Across Nanostructured Thin Films,” Appl. Phys. Lett., 96, p. 263113. [CrossRef]
Zhao, H., and Freund, J. B., 2005, “Lattice-Dynamical Calculation of Phonon Scattering at Ideal Si–Ge Interfaces,” J. Appl. Phys., 97, p. 024903. [CrossRef]
Young, D. A., and Maris, H. J., 1989, “Lattice-Dynamical Calculation of the Kapitza Resistance Between FCC Lattices,” Phys. Rev. B, 40, pp. 3685–3693. [CrossRef]
Landry, E. S., 2009, “Thermal Transport by Phonons Across Semiconductor Interfaces, Thin Films, and Superlattices,” Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA.
Wang, J. S., Wang, J., and Lü, J. T., 2008, “Quantum Thermal Transport in Nanostructures,” Eur. Phys. J. B, 62, pp. 381–404. [CrossRef]
Wang, J., and Wang, J. S., 2006, “Mode-Dependent Energy Transmission Across Nanotube Junctions Calculated With a Lattice Dynamics Approach,” Phys. Rev. B, 74, p. 054303. [CrossRef]
Zhang, W., Mingo, N., and Fisher, T., 2007, “Simulation of Phonon Transport Across a Non-Polar Nanowire Junction Using an Atomistic Green's Function Method,” Phys. Rev. B, 76, p. 195429. [CrossRef]
Huang, Z., Fisher, T. S., and Murthy, J. Y., 2010, “Simulation of Thermal Conductance Across Dimensionally Mismatched Graphene Interfaces,” J. Appl. Phys., 108, p. 114310. [CrossRef]
Huang, Z., Fisher, T. S., and Murthy, J. Y., 2010, “Simulation of Phonon Transmission Through Graphene and Graphene Nanoribbons With a Green's Function Method,” J. Appl. Phys., 108, p. 094319. [CrossRef]
Zhang, W., Fisher, T. S., and Mingo, N., 2007, “Simulation of Interfacial Phonon Transport in Si–Ge Heterostructures Using an Atomistic Green's Function Method,” ASME J. Heat Transfer, 129, pp. 483–491. [CrossRef]
Li, W. X., Chen, K. Q., Duan, W., Wu, J., and Gu, B. L., 2004, “Acoustic Phonon Transport Through a T-Shaped Quantum Waveguide,” J. Phys.: Condens. Matter, 16, p. 5049. [CrossRef]
Yang, P., Sun, Q., Guo, H., and Hu, B., 2007, “Thermal Transport in a Dielectric T-Shaped Quantum Wire,” Phys. Rev. B, 75, p. 235319. [CrossRef]
Solie, L. P., and Auld, B. A., 1973, “Elastic Waves in Free Anisotropic Plates,” J. Acoust. Soc. Am., 54, pp. 50–65. [CrossRef]
Mingo, N., 2003, “Calculation of Si Nanowire Thermal Conductivity Using Complete Phonon Dispersion Relations,” Phys. Rev. B, 68, p. 113308. [CrossRef]
Dove, M. T., 1993, Introduction to Lattice Dynamics, Cambridge University Press, Cambridge, UK.
Stillinger, F. H., and Weber, T. A., 1985, “Computer Simulation of Local Order in Condensed Phases of Silicon,” Phys. Rev. B, 31, pp. 5262–5271. [CrossRef]
Allen, M. P., and Tildesley, D. J., 1987, Computer Simulation of Liquids, Clarendon Press, Oxford, UK.


Grahic Jump Location
Fig. 2

Description of supercell. In this figure Ncell = 6. Blue atoms at the upper and lower boundaries are held fixed.

Grahic Jump Location
Fig. 4

Illustration of wavepacket width for kx = 0.250 (1/σ). Wider wavepackets better approximate phonons of single wavenumber.

Grahic Jump Location
Fig. 5

Dispersion relations of four different atomistic waveguide leads simulated: (a) 2 cells wide, (b) 4 cells wide, (c) 6 cells wide, and (d) 8 cells wide. Only the highlighted branches (SH1-heavy solid line, SH2-heavy dashed line) are investigated in this study.

Grahic Jump Location
Fig. 6

Transmission dependence on wave packet width for 2 cell waveguide. Upper set of data points correspond to SH1 at kx = 0.159 (1/σ). Lower set of data points correspond to SH1 at kx = 0.111 (1/σ). Marker shapes correspond to different domain sizes: triangles (Lwaveguide = 259.2 σ), crosses (Lwaveguide = 518.4 σ), circles (Lwaveguide = 2073.8 σ), squares (Lwaveguide = 2592.2 σ).

Grahic Jump Location
Fig. 1

T-stub-waveguide geometry

Grahic Jump Location
Fig. 3

Molecular dynamics computational geometry

Grahic Jump Location
Fig. 7

SH1 transmission dependence on wave packet width for 6 cell waveguide. Results are shown across a range of wavenumber and compared with the continuum result obtained from technique presented in Sec. 2.

Grahic Jump Location
Fig. 8

Snap-shots of z-displacement field for different values of wavenumber for incident SH1 mode. Extrema in transmission correspond to formation of standing waves in T-stub region.

Grahic Jump Location
Fig. 9

Transmission of SH1 mode for different sized waveguide leads. For all cases, hII* = 2.0 and d* = 1.0. Agreement with continuum is best for the thicker waveguide leads (6 cell and 8 cell) and at lower frequencies.

Grahic Jump Location
Fig. 10

Transmission of SH2 mode for different sized waveguide leads



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In