TECHNICAL PAPERS: Micro/Nanoscale Heat Transfer

Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation

[+] Author and Article Information
Ronggui Yang, Gang Chen, Marine Laroche

Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02139-4307

Yuan Taur

Department of Electrical & Computer Engineering, University of California, La Jolla, CA 92093-0407

J. Heat Transfer 127(3), 298-306 (Mar 24, 2005) (9 pages) doi:10.1115/1.1857941 History: Received October 30, 2003; Revised September 12, 2004; Online March 24, 2005
Copyright © 2005 by ASME
Your Session has timed out. Please sign back in to continue.


Taur, Y., Wann, Ch. H., and Frank, D. J., 1998, “25 nm CMOS Design Considerations,” Electron Devices Meeting, Tech. Digest., Intl., Dec. 6–9 pp. 789–792.
Tien, C. L., Majumdar, A., and Gerner, F. M., 1998, Microscale Energy Transport, Taylor & Francis, Washington, DC.
Lewis, E., 1984, Computational Methods of Neutron Transport, Wiley, New York.
Modest, M. F., 2003, Radiative Heat Transfer, 2nd ed., Academic Press, New York.
Tellier, C. R., and Tosser, A. J., 1982, Size Effects in Thin Films, Elsevier, Amsterdam.
Goodson,  K. E., and Ju,  Y. S., 1999, “Heat Conduction in Novel Electronic Films,” Annu. Rev. Mater. Sci., 29, pp. 261–293.
Chen,  G., 2000, “Phonon Heat Conduction in Low-Dimensional Structures,” Semicond. Semimetals, 71, pp. 203–259.
Chen,  G., 1996, “Nonlocal and Nonequilibrium Heat Conduction in the Vincinity of Nanoparticles,” ASME J. Heat Transfer, 118, pp. 539–545.
Sverdrup,  P. G., Ju,  Y. S., and Goodson,  K. E., 2001, “Sub-Continuum Simulations of Heat Conduction in Silicon-on-Insulator Transistors,” ASME J. Heat Transfer, 123, pp. 130–137.
Murthy,  J. Y., and Mathur,  S. R., 2003, “An Improved Computational Procedure for Sub-Micron Heat Conduction,” ASME J. Heat Transfer, 125, pp. 904–910.
Narumanchi,  S. V. J., Murthy,  J. Y., and Amon,  C. H., 2003, “Simulation of Unsteady Small Heat Source Effects in Sub-Micron Heat Conduction,” ASME J. Heat Transfer, 125, pp. 896–903.
Chen,  G., 2001, “Ballistic-Diffusive Heat Conduction Equations,” Phys. Rev. Lett., 86, pp. 2297–2300.
Chen,  G., 2002, “Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macroscales,” ASME J. Heat Transfer, 124, pp. 320–328.
Yang, R. G., and Chen, G., 2001, “Two-Dimensional Nanoscale Heat Conduction Using Ballistic-Diffusive Equations,” Proc. of Int. Mechanical Engineering Conference and Exhibition, New York, 369 , pp. 363–366.
Yang, R. G., Chen, G., and Taur, Y., 2002, “Ballistic-Diffusive Equations for Multidimensional Nanoscale Heat Conduction,” Proc. of International Heat Transfer Conference, Grenoble, France, Elsevier, Paris, Vol. 1, pp. 579–584.
Majumdar,  A., 1993, “Microscale Heat Conduction in Dielectric Thin Films,” ASME J. Heat Transfer, 115, pp. 7–16.
Chen,  G., 2003, “Diffusion-Transmission Condition for Transport at Interfaces and Boundaries,” Appl. Phys. Lett., 82, pp. 991–993.
Olfe,  D. B., 1967, “A Modification of the Differential Approximation for Radiative Transfer,” AIAA J., 5, pp. 638–643.
Modest,  M. F., 1989, “The Modified Differential Approximation for Radiative Transfer in General Three-Dimensional Media,” J. Thermophys. Heat Transfer, 3, pp. 283–288.
Ramankutty,  M., and Crosbie,  A., 1997, “Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures,” J. Quant. Spectrosc. Radiat. Transf., 57, pp. 107–140.
Sakami,  M., and Charette,  A., 2000, “Application of a Modified Discrete Ordinates Method to Two-Dimensional Enclosures od Irregular Geometery,” J. Quant. Spectrosc. Radiat. Transf., 64, pp. 275–298.
Pomraning, G. C., 1973, The Equation of Radiation Hydrodynamics, Pergamon, New York.
Chen,  G., 1997, “Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin-Film Structures,” ASME J. Heat Transfer, 119, pp. 220–229.
Chen,  G., 1998, “Thermal Conductivity and Ballistic Phonon Transport in Cross-Plane Direction of Superlattices,” Phys. Rev. B, 57, pp. 14958–14973.
Goodson,  K. E., 1996, “Thermal Conductivity in Nonhomgeneous CVD Diamond Layers in Electronic Microstructures,” ASME J. Heat Transfer, 118, pp. 279–336.
Joshi,  A. A., and Majumdar,  A., 1993, “Transient Ballistic and Diffusive Heat Transport in Thin Films,” J. Appl. Phys., 74, pp. 31–39.
Siegel, R., and Howell, J. R., 2001, Thermal Radiation Heat Transfer, 4th ed, Taylor & Francis, Washington DC.
Fiveland,  W. A., 1987, “Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically and Anisotropically Scattering Media,” ASME J. Heat Transfer, 109, pp. 809–812.
Truelove,  J. S., 1987, “Discrete-Ordinate Solutions of the Radiation Transport Equation,” ASME J. Heat Transfer, 109, pp. 1048–1051.
Raithby,  G. D., 1994, “Discussion of the Finite-Volume Method for Radiation, and Its Application Using 3D Unstructured Meshes,” Numer. Heat Transfer, Part B, 35, pp. 389–405.
Briggs,  L. L., Miller,  W. F., and Lewis,  E. E., 1975, “Ray-Effect Mitigation in Discrete Ordinate-Like Angular Finite Element Approximations in Neutron Transport,” Nucl. Sci. Eng., 57, pp. 205–217.
Chai,  J. C., Lee,  H. S., and Patankar,  S. V., 1993, “Ray Effect and False Scattering in the Discrete Ordinates Method,” Numer. Heat Transfer, Part B, 24, pp. 373–389.
Coelho,  P., 2002, “The Role of Ray Effect and False Scattering on the Accuracy of the Standard and Modified Discrete Ordinates Methods,” J. Quant. Spectrosc. Radiat. Transf., 73, pp. 231–238.
Murthy,  J., and Mathur,  S., 1998, “Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes,” J. Thermophys. Heat Transfer, 12, pp. 313–321.
Fiveland, W. A., 1991, “The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering,” Fundamentals of Radiation Heat Transfer, ASME, New York, HTD-Vol. 160, pp. 89–96.
Chai,  J. C., Patankar,  S. V., and Lee,  H. S., 1994, “Evaluation of Spatial Differencing Practices for the Discrete-Ordinates Method,” J. Thermophys. Heat Transfer, 8, pp. 140–144.


Grahic Jump Location
Schematic drawing of device geometry simulated in this paper: (a) confined surface heating at y=0,T1, and T0 represents emitted temperature in case I and equivalent equilibrium temperature in case II; and (b) a nanoscale heat source embedded in the substrate, which is similar to the heat generation and transport in the MOSFET device
Grahic Jump Location
Numerical solution scheme of the ballistic-diffusive equations
Grahic Jump Location
Local coordinate used in phonon Boltzmann transport simulation
Grahic Jump Location
Directions of phonon transport in two-dimensional planes as given by different combinations of the directional cosines and corresponding differencing schemes used for the BTE solver
Grahic Jump Location
Comparison of the transient temperature and heat flux in y direction at the centerline of the geometry for Kn=10 based on emitted temperature condition: (a) temperature and (b) heat flux qy*
Grahic Jump Location
(a) Comparison of steady-state temperature distribution at the centerline using the Fourier theory, the Boltzmann equation, and the ballistic-diffusive equations for different Knudsen numbers. (b) Comparison of the heat flux qy* at the centerline for Kn=0.1 at t*=100.
Grahic Jump Location
Comparison of transient temperature and heat flux distribution at the centerline using the Fourier theory, the Boltzmann equation, and the ballistic-diffusive equations based on thermalized temperature boundary conditions: (a) and (b) heat flux qy*, and (c) temperature
Grahic Jump Location
The ballistic and diffusive component contributions to the total temperature and heat flux at the centerline: (a) temperature and (b) heat flux qy*
Grahic Jump Location
Comparison of two-dimensional temperature rise distribution after the device is operated for 10 ps: (a) the Boltzmann equation, (b) the Ballistic-diffusive equations, and (c) the Fourier law
Grahic Jump Location
(a) Comparison of the heat flux obtained by the Boltzmann equation, the ballistic-diffusive equations, and the Fourier law at the centerline. (b) Comparison of the peak temperature rise inside the device obtained by the Boltzmann equation, the Ballistic-diffusive equations and the Fourier law.



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