Research Papers: Conduction

Heat Conduction in Nanostructured Materials Predicted by Phonon Bulk Mean Free Path Distribution

[+] Author and Article Information
Giuseppe Romano

Department of Materials Science
and Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: romanog@mit.edu

Jeffrey C. Grossman

Department of Materials Science
and Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: jcg@mit.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 29, 2014; final manuscript received January 27, 2015; published online March 24, 2015. Assoc. Editor: Robert D. Tzou.

J. Heat Transfer 137(7), 071302 (Jul 01, 2015) (7 pages) Paper No: HT-14-1573; doi: 10.1115/1.4029775 History: Received August 29, 2014; Revised January 27, 2015; Online March 24, 2015

We develop a computational framework, based on the Boltzmann transport equation (BTE), with the ability to compute thermal transport in nanostructured materials of any geometry using, as the only input, the bulk cumulative thermal conductivity. The main advantage of our method is twofold. First, while the scattering times and dispersion curves are unknown for most materials, the phonon mean free path (MFP) distribution can be directly obtained by experiments. As a consequence, a wider range of materials can be simulated than with the frequency-dependent (FD) approach. Second, when the MFP distribution is available from theoretical models, our approach allows one to include easily the material dispersion in the calculations without discretizing the phonon frequencies for all polarizations thereby reducing considerably computational effort. Furthermore, after deriving the ballistic and diffusive limits of our model, we develop a multiscale method that couples phonon transport across different scales, enabling efficient simulations of materials with wide phonon MFP distributions length. After validating our model against the FD approach, we apply the method to porous silicon membranes and find good agreement with experiments on mesoscale pores. By enabling the investigation of thermal transport in unexplored nanostructured materials, our method has the potential to advance high-efficiency thermoelectric devices.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Majumdar, A., 2004, “Thermoelectricity in Semiconductor Nanostructures,” Science, 303(5659), pp. 777–778. [CrossRef] [PubMed]
Venkatasubramanian, R., Siivola, E., Colpitts, T., and O' Quinn, B., 2001, “Thin-Film Thermoelectric Devices With High Room-Temperature Figures of Merit,” Nature, 413(6856), pp. 597–602. [CrossRef] [PubMed]
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(7175), pp. 163–167. [CrossRef] [PubMed]
Song, D., and Chen, G., 2004, “Thermal Conductivity of Periodic Microporous Silicon Films,” Appl. Phys. Lett., 84(5), pp. 687–689. [CrossRef]
Yu, J.-K., Mitrovic, S., Tham, D., Varghese, J., and Heath, J. R., 2010, “Reduction of Thermal Conductivity in Phononic Nanomesh Structures,” Nat. Nanotechnol., 5(10), pp. 718–721. [CrossRef] [PubMed]
Tang, J., Wang, H.-T., Lee, D. H., Fardy, M., Huo, Z., Russell, T. P., and Yang, P., 2010, “Holey Silicon as an Efficient Thermoelectric Material,” Nano Lett., 10(10), pp. 4279–4283. [CrossRef] [PubMed]
Lee, J.-H., Galli, G. A., and Grossman, J. C., 2008, “Nano Si as an Efficient Thermoelectric Material,” Nano Lett., 8(11), pp. 3750–3754. [CrossRef] [PubMed]
Chen, G., 2005, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons, Oxford University Press, New York.
Casimir, H., 1938, “Note on the Conduction of Heat in Crystals,” Physica, 5(6), pp. 495–500. [CrossRef]
Majumdar, A., 1993, “Microscale Heat Conduction in Dielectric Thin Films,” ASME J. Heat Transfer, 115(1), pp. 7–16. [CrossRef]
Chen, G., 1998, “Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices,” Phys. Rev. B, 57(3), pp. 14958–14973. [CrossRef]
Cahill, D. G., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A., Maris, H. J., Merlin, R., and Phillpot, S. R., 2003, “Nanoscale Thermal Transport,” J. Appl. Phys., 93(2), pp. 793–818. [CrossRef]
Esfarjani, K., Chen, G., and Stokes, H. T., 2011, “Heat Transport in Silicon From First-Principles Calculations,” Phys. Rev. B, 84(8), p. 085204. [CrossRef]
Minnich, A. J., Chen, G., Mansoor, S., and Yilbas, B. S., 2011, “Quasiballistic Heat Transfer Studied Using the Frequency-Dependent Boltzmann Transport Equation,” Phys. Rev. B, 84(23), p. 235207. [CrossRef]
Hsieh, T.-Y., Lin, H., Hsieh, T.-J., and Huang, J.-C., 2012, “Thermal Conductivity Modeling of Periodic Silicon With Aligned Cylindrical Pores,” J. Appl. Phys., 111(12), p. 124329. [CrossRef]
Loy, J. M., Murthy, J. Y., and Singh, D., 2013, “A Fast Hybrid Fourier–Boltzmann Transport Equation Solver for Nongray Phonon Transport,” ASME J. Heat Transfer, 135(1), p. 011008. [CrossRef]
Minnich, A. J., Johnson, J., Schmidt, A., Esfarjani, K., Dresselhaus, M., Nelson, K. A., and Chen, G., 2011, “Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths,” Phys. Rev. Lett., 107(9), p. 095901. [CrossRef] [PubMed]
Minnich, A. J., 2012, “Determining Phonon Mean Free Paths From Observations of Quasiballistic Thermal Transport,” Phys. Rev. Lett., 109(20), p. 205901. [CrossRef] [PubMed]
Ziman, J. M., 2001, Electrons and Phonons: The Theory of Transport Phenomena in Solids, OUP, Oxford, UK.
Mingo, N., Stewart, D., Broido, D., Lindsay, L., and Li, W., 2014, “Ab Initio Thermal Transport,” Length-Scale Dependent Phonon Interactions, Springer, Berlin, pp. 137–173. [CrossRef]
Yang, F., and Dames, C., 2013, “Mean Free Path Spectra as a Tool to Understand Thermal Conductivity in Bulk and Nanostructures,” Phys. Rev. B, 87(3), p. 035437. [CrossRef]
Paraud, J. P. M., and Hadjiconstantinou, N. G., 2011, “Efficient Simulation of Multidimensional Phonon Transport Using Energy-Based Variance-Reduced Monte Carlo Formulations,” Phys. Rev. B, 84(20), p. 205331. [CrossRef]
Paraud, J. P. M., and Hadjiconstantinou, N. G., 2012, “An Alternative Approach to Efficient Simulation of Micro/Nanoscale Phonon Transport,” Appl. Phys. Lett., 101(15), p. 153114. [CrossRef]
Jeng, M.-S., Yang, R., Song, D., and Chen, G., 2008, “Modeling the Thermal Conductivity and Phonon Transport in Nanoparticle Composites Using Monte Carlo Simulation,” ASME J. Heat Transfer, 130(4), p. 042410. [CrossRef]
Chen, G., 2001, “Ballistic-Diffusive Heat-Conduction Equations,” Phys. Rev. Lett., 86(11), pp. 2297–2300. [CrossRef] [PubMed]
Romano, G., and Di Carlo, A., 2011, “Multiscale Electrothermal Modeling of Nanostructured Devices,” IEEE Trans. Nanotechnol., 10(6), pp. 1285–1292. [CrossRef]
Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G. L., Cococcioni, M., Dabo, I., Dal Corso, A., De Gironcoli1, S., Fabrisi, S., Fratesi, G., Gebauer, R., Gerstmann, U., Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N., Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C., Scandolo, S., Sclauzero, G., Seitsonen, A. P., Smogunov, A., Umari, P., and Wentzcovitch, R. M., 2009, “quantum espresso: A Modular and Open-Source Software Project for Quantum Simulations of Materials,” J. Phys.: Condens. Matter, 21(39), p. 395502. [CrossRef] [PubMed]
Slack, G. A., 1994, Handbook of Thermoelectrics, D. M. Rowe, ed., CRC Press, Boca Raton, FL, p. 407.
Hashin, Z., and Shtrikman, S., 1962, “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials,” J. Appl. Phys., 33(10), pp. 3125–3131. [CrossRef]
Esfarjani, K., Chen, G., and Stokes, H. T., 2011, “Heat Transport in Silicon From First-Principles Calculations,” Phys. Rev. B, 84(8), p. 085204. [CrossRef]
Ravichandran, N. K., and Minnich, A. J., 2014, “Coherent and Incoherent Thermal Transport in Nanomeshes,” Phys. Rev. B, 89, p. 205432. [CrossRef]
Davis, B. L., and Hussein, M. I., 2014, “Nanophononic Metamaterial: Thermal Conductivity Reduction by Local Resonance,” Phys. Rev. Lett., 112, p. 055505. [CrossRef] [PubMed]
Jain, A., Yu, Y.-J., and McGaughey, A. J., 2013, “Phonon Transport in Periodic Silicon Nanoporous Films With Feature Sizes Greater Than 100 nm,” Phys. Rev. B, 87(19), p. 195301. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Discretization of the solid angle in slices of ΔΩ. Because the system is two-dimensional, we consider only the upper hemisphere and then apply the symmetry. (b) The cumulative thermal conductivity of Si, the only input required by the MFP–BTE. The MFP range is discretized into a small number of MFPs, typically a few tenths, for which the MFP–BTE is solved. (c) An example of a triangular element of the simulation domain. The centroid is denoted by P, whereas the point at the middle of the side with normal n is denoted by B.

Grahic Jump Location
Fig. 2

(a) Phonon dispersion along the 001 direction computed by first-principles calculations, (b) the MFPs for different phonon polarizations, (c) the cumulative thermal conductivity at room temperature, and (d) comparison between the FD–BTE and the MFP–BTE method for a unit cell of size L = 10 nm. After a few iterations, the two methods lead to the same thermal conductivity.

Grahic Jump Location
Fig. 3

(a) The phonon suppression function S(Λ), normalized to 1. For very small MFPs, S˜ reaches its maximum values and becomes flat, meaning that we are reaching the diffusive regime. For very large MFPs, the suppression function goes into the ballistic regime, namely, S ≈ 1/Λ. In the inset, the magnitude of the thermal flux is shown. (b) The derivative of the suppression function |S'(Λ)| superimposed on the cumulative thermal conductivity of Si. The maximum of |S'(Λ)| is obtained for a MFP that equals the pore–pore distance.

Grahic Jump Location
Fig. 4

(a) Effective temperature map for the L = 100 nm case. The normalized heat flux is ensured by applying a difference of temperature to the unit cell. (b) Normalized magnitude of thermal flux. Most of the heat travels in the space between pores. (c) Cumulative for different sizes of the unit cell L, ranging from the nanoscale to the macroscale. (d) PTC versus the unit cell size. For macroscales, the effective thermal conductivity reaches the diffusive regime.

Grahic Jump Location
Fig. 5

(a) The cumulative thermal conductivity for a mesoscale porous-Si membrane [4] and Si nanomesh [5]. Phonon classical size effects strongly depend on the limiting dimension that is the smallest between the pore size and the sample thickness. (b) The magnitude of the thermal flux for the periodic structure. Phonons travel between pores along the direction of the gradient of the temperature, due to phonon-pore scattering. (c) A cut of magnitude of the thermal flux map. Most of the heat is concentrated toward the middle of the sample due to the diffuse scattering of phonons with the top and bottom surfaces.



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