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Research Papers: Micro/Nanoscale Heat Transfer

Phonon Heat Conduction in Multidimensional Heterostructures: Predictions Using the Boltzmann Transport Equation

[+] Author and Article Information
Syed Ashraf Ali

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210

Sandip Mazumder

Fellow ASME
Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: mazumder.2@osu.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 6, 2015; final manuscript received April 24, 2015; published online June 2, 2015. Assoc. Editor: Laurent Pilon.

J. Heat Transfer 137(10), 102401 (Oct 01, 2015) (11 pages) Paper No: HT-15-1012; doi: 10.1115/1.4030565 History: Received January 06, 2015; Revised April 24, 2015; Online June 02, 2015

In this article, two models for phonon transmission across semiconductor interfaces are investigated and demonstrated in the context of large-scale spatially three-dimensional calculations of the phonon Boltzmann transport equation (BTE). These include two modified forms of the classical diffuse mismatch model (DMM): one, in which dispersion is accounted for and another, in which energy transfer between longitudinal acoustic (LA) and transverse acoustic (TA) phonons is disallowed. As opposed to the vast majority of the previous studies in which the interface is treated in isolation, and the thermal boundary conductance is calculated using closed-form analytical formulations, the present study also considers the interplay between the interface and intrinsic (volumetric) scattering of phonons. This is accomplished by incorporating the interface models into a parallel solver for the full seven-dimensional BTE for phonons. A verification study is conducted in which the thermal boundary resistance of a silicon/germanium interface is compared against the previously reported results of molecular dynamics (MD) calculations. The BTE solutions overpredicted the interfacial resistance, and the reasons for this discrepancy are discussed. It is found that due to the interplay between intrinsic and interface scattering, the interfacial thermal resistance across a Si(hot)/Ge(cold) bilayer is different from that of a Si(cold)/Ge(hot) bilayer. Finally, the phonon BTE is solved for a nanoscale three-dimensional heterostructure, comprised of multiple blocks of silicon and germanium, and the time evolution of the temperature distribution is predicted and compared against predictions using the Fourier law of heat conduction.

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References

Ju, Y. S., and Goodson, K. E., 1999, ‘‘Phonon Scattering in Silicon Films With Thickness of Order 100 nm,’’ Appl. Phys. Lett., 74(20), pp. 3005–3007. [CrossRef]
Tien, C. L., Majumdar, A., and Gerner, F. M., eds., 1998, Microscale Energy Transport, Taylor and Francis, Bristol, PA.
Zhang, Z., 2007, Nano/Microscale Heat Transfer, McGraw-Hill, New York.
Swartz, E. T., and Pohl, R. O., 1989, “Thermal Boundary Resistance,” Rev. Mod. Phys., 61(3), pp. 605–668. [CrossRef]
Zhao, H., and Freund, J. B., 2009, “Phonon Scattering at a Rough Interface Between Two fcc Lattices,” J. Appl. Phys., 105(1), p. 013515. [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]
Vincenti, W. G., and Kruger, C. H., 1977, Introduction to Physical Gas Dynamics, Kreiger Publishing, Malabar, FL.
Dames, C., and Chen, G., 2004, “Theoretical Phonon Thermal Conductivity of Si/Ge Superlattice Nanowires,” J. Appl. Phys., 95(2), pp. 682–693. [CrossRef]
Hopkins, P. E., 2009, “Multiple Phonon Processes Contributing to Inelastic Scattering During Thermal Boundary Conductance at Solid Interfaces,” J. Appl. Phys., 106(1), p. 013528. [CrossRef]
Hopkins, P. E, Duda, J. C., and Norris, P. M., 2011, “Anharmonic Phonon Interactions as Interfaces and Contributions to Thermal Boundary Conductance,” ASME J. Heat Transfer, 133(6), p. 062401. [CrossRef]
Duda, J. C., Norris, P. M., and Hopkins, P. E., 2011, “On the Linear Temperature Dependence of Phonon Thermal Boundary Conductance in the Classical Limit,” ASME J. Heat Transfer, 133(7), p. 074501. [CrossRef]
Stoner, R. J., and Maris, H. J., 1993, “Kapitza Conductance and Heat Flow Between Solids at Temperatures From 50 to 300 K,” Phys. Rev. B, 48(22), pp. 16373–16387. [CrossRef]
Lyeo, H.-K., and Cahill, D. G., 2006, “Thermal Conductance of Interfaces Between Highly Dissimilar Materials,” Phys. Rev. B, 73(14), p. 144301. [CrossRef]
Stevens, R. J., Zhigilei, L. V., and Norris, P. M., 2007, “Effects of Temperature and Disorder on Thermal Boundary Conductance at Solid–Solid Interfaces: Nonequilibrium Molecular Dynamics Simulations,” Int. J. Heat Mass Transfer, 50(19–20), pp. 3977–3989. [CrossRef]
Landry, E. S., and McGaughey, A. J. H., 2009, “Thermal Boundary Resistance Predictions From Molecular Dynamics Simulations and Theoretical Calculations,” Phys. Rev. B, 80(16), p. 165304. [CrossRef]
Duda, J. C., Beechem, T. E., Smoyer, J. L., Norris, P. M., and Hopkins, P. E., 2010, “Role of Dispersion on Phononic Thermal Boundary Conductance,” J. Appl. Phys., 108(7), p. 073515. [CrossRef]
Reddy, P., Castelino, K., and Majumdar, A., 2005, “Diffuse Mismatch Model of Thermal Boundary Conductance Using Exact Phonon Dispersion,” Appl. Phys. Lett., 87(21), p. 211908. [CrossRef]
Singh, D., Murthy, J. Y., and Fisher, T. S., 2011, “Effect of Phonon Dispersion on Thermal Conduction Across Si/Ge Interfaces,” ASME J. Heat Transfer, 133(12), p. 122401. [CrossRef]
Little, W. A., 1959, “The Transport of Heat between Dissimilar Solids at Low Temperatures,” Can. J. Phys., 37(3), pp. 334–349. [CrossRef]
Modest, M. F., 2013, Radiative Heat Transfer, 3rd ed., Academic, New York.
Prasher, R. S., and Phelan, P. E., 2001, “A Scattering-Mediated Acoustic Mismatch Model for the Prediction of Thermal Boundary Resistance,” ASME J. Heat Transfer, 123(1), pp. 105–112. [CrossRef]
Kazan, M., 2011, “Interpolation Between the Acoustic Mismatch Model and the Diffuse Mismatch Model for the Interface Thermal Conductance: Application to InN/GaN Superlattice,” ASME J. Heat Transfer, 133(11), p. 112401. [CrossRef]
Ni, C., and Murthy, J. Y., 2009, “Parallel Computation of the Phonon Boltzmann Transport Equation,” Numer. Heat Transfer, Part B, 55(6), pp. 435–456. [CrossRef]
Murthy, J. Y., Narumanchi, S. V. J., Pascual-Gutierrez, J. A., Wang, T., Ni, C., and Mathur, S. R., 2005, “Review of Multi-Scale Simulation in Sub-Micron Heat Transfer,” Int. J. Multiscale Comput. Eng., 3(1), pp. 5–32 [CrossRef].
Raithby, G. D., and Chui, E. H., 1990, “A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosures With Participating Media,” ASME J. Heat Transfer, 112(2), pp. 415–423. [CrossRef]
Chai, J. C., Lee, H. S., and Patankar, S. V., 1994, “Finite-Volume Method for Radiative Heat Transfer,” J. Thermophys. Heat Transfer, 8(3), pp. 419–425. [CrossRef]
Murthy, J. Y., and Mathur, S. R., 2002, “Computation of Sub-Micron Thermal Transport Using an Unstructured Finite-Volume Method,” ASME J. Heat Transfer, 124(6), pp. 1176–1181. [CrossRef]
Ali, S. A., Kollu, G., Mazumder, S., Sadayappan, P., and Mittal, A., 2014, “Large-Scale Parallel Computation of the Phonon Boltzmann Transport Equation,” Int. J. Therm. Sci., 86, pp. 341–351. [CrossRef]
Chen, G., 1997, “Size and Interface Effects on the Thermal Conductivity of Superlattices and Periodic Thin Film Structures,” ASME J. Heat Transfer, 119(2), pp. 220–229. [CrossRef]
Yang, R., and Chen, G., 2004, “Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposities,” Phys. Rev. B, 69(19), p. 195316. [CrossRef]
Majumdar, A., 1993, “Microscale Heat Transfer in Dielectric Thin Films,” ASME J. Heat Transfer, 115(1), pp. 7–16. [CrossRef]
Whitaker, S., 1983, Fundamental Principles of Heat Transfer, Krieger Publishing, Malabar, FL.
Mazumder, S., and Majumdar, A., 2001, “Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization,” ASME J. Heat Transfer, 123(4), pp. 749–759. [CrossRef]
Brockhouse, B. N., 1959, “Lattice Vibrations in Silicon and Germanium,” Phys. Rev. Lett., 2(6), pp. 256–258. [CrossRef]
Mittal, A., and Mazumder, S., 2010, “Monte Carlo Study of Phonon Heat Conduction in Silicon Thin Films Including Contributions of Optical Phonons,” ASME J. Heat Transfer, 132(5), p. 052402. [CrossRef]
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(4), pp. 373–389. [CrossRef]
Mittal, A., and Mazumder, S., 2011, “Generalized Ballistic-Diffusive Formulation and Hybrid SN-PN Solution of the Boltzmann Transport Equation for Phonons for Non-Equilibrium Heat Conduction,” ASME J. Heat Transfer, 133(9), p. 092402. [CrossRef]
Mittal, A., and Mazumder, S., 2011, “Hybrid Discrete Ordinates—Spherical Harmonics Solution to the Boltzmann Transport Equation for Phonons for Non-Equilibrium Heat Conduction,” J. Comput. Phys., 230(18), pp. 6977–7001. [CrossRef]
Saad, Y., 2003, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, PA. [CrossRef]
Srinivasan, S., and Miller, R. S., 2004, “Parallel Computation of the Boltzmann Transport Equation for Microscale Heat Transfer in Multilayered Thin Films,” Numer. Heat Transfer, Part B, 36, pp. 31–58. [CrossRef]
Lacroix, D., Joulain, K., and Lemonnier, D., 2005, “Monte Carlo Transient Phonon Transport in Silicon and Germanium at Nanoscale,” Phys. Rev. B, 72(6), p. 064305. [CrossRef]
Holland, M. G., 1963, “Analysis of Lattice Thermal Conductivity,” Phys. Rev., 132(6), pp. 2461–2471. [CrossRef]
Alan, J. McGaughey, private communication.
Sellan, D. P., Landry, E. S., Turney, J. E., McGaughey, A. J. H., and Amon, C. H., 2010, “Size Effects in Molecular Dynamics Thermal Conductivity Predictions,” Phys. Rev. B, 81(21), p. 214305. [CrossRef]

Figures

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Fig. 1

Schematic representation of a diffuse interface

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Fig. 2

The dispersion relationships for the silicon/germanium system: only acoustic polarization branches are shown

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Fig. 3

Spectral transmittance of acoustic phonons from silicon to germanium predicted using two different variations of the diffuse mismatch model: (a) DDMM and (b) IPT model. The predicted transmittance by both models is independent of temperature.

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Fig. 4

Unstructured stencil showing geometric connectivity and a discrete direction (line of sight) of propagation of phonons

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Fig. 5

Schematic representation of a face straddled by two cells with two different materials, the intensities on the two sides of the interface, and their integral representations

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Fig. 6

Schematic representation of the geometry and boundary conditions used for the verification study

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Fig. 7

Steady state temperature distributions predicted using solution of the BTE with two different interface models and two different configurations: (a) Si(hot)/Ge(cold) and (b) Ge(hot)/Si(cold)

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Fig. 8

Computed MFPs for acoustic phonons in silicon and germanium at two different temperatures: (a) 300 K and (b) 500 K. The computations were performed using the dispersion relationships shown in Fig. 2, and the scattering time-scale expressions given by Eq. (19).

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Fig. 9

Schematic representation (top and side views) of the geometry and boundary conditions used for the three-dimensional heterostructure. The light gray areas represent a silicon substrate. The dark gray and dotted regions represent germanium blocks embedded within the silicon substrate.

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Fig. 10

Predicted temporal evolution of temperature (K) in the three-dimensional heterostructure: (a) BTE solution and (b) Fourier law solution. The top views are at the midplane (z = 0.25 μm), and the size views are at the central plane (y = 5 μm).

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