Analytical Green’s function of the multidimensional frequency-dependent phonon Boltzmann equation

Thermal phonon transport at length scales comparable to mean free paths is governed by the Boltzmann equation, which is challenging to solve due to its high dimensionality. Here, we present an analytical Green's function for the frequency-dependent, multidimensional Boltzmann equation under the relaxation-time approximation. The new analytical solution is valid from diffusive to ballistic transport regimes and rigorously includes frequency dependence of phonon properties. We demonstrate that our result enables simple closed-form solutions for a number of multidimensional problems for which the only prior solution methods have been computationally expensive numerical simulations.

[1]  Phase-controlled, heterodyne laser-induced transient grating measurements of thermal transport properties in opaque material , 2011, 1109.6685.

[2]  Hafner,et al.  Ab initio molecular dynamics for open-shell transition metals. , 1993, Physical review. B, Condensed matter.

[3]  Hafner,et al.  Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium. , 1994, Physical review. B, Condensed matter.

[4]  G. Kresse,et al.  Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set , 1996 .

[5]  A constitutive equation for nano-to-macro-scale heat conduction based on the Boltzmann transport equation , 2011 .

[6]  Yuan Taur,et al.  Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation , 2005 .

[7]  E. Pop.  Energy dissipation and transport in nanoscale devices , 2010, 1003.4058.

[8]  Wu Li,et al.  ShengBTE: A solver of the Boltzmann transport equation for phonons , 2014, Comput. Phys. Commun..

[9]  Gang Chen,et al.  Heat transport in silicon from first-principles calculations , 2011, 1107.5288.

[10]  Mahan,et al.  Nonlocal theory of thermal conductivity. , 1988, Physical review. B, Condensed matter.

[11]  Natalio Mingo,et al.  Thermal conductivity of diamond nanowires from first principles , 2012 .

[12]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[13]  R. Englman.  The Transport Theory of Temperature Waves in Insulators , 1958 .

[14]  Austin J. Minnich,et al.  Transport regimes in quasiballistic heat conduction , 2014 .

[15]  Nicolas Hadjiconstantinou,et al.  An alternative approach to efficient simulation of micro/nanoscale phonon transport , 2012, 1206.3316.

[16]  R. Peierls,et al.  Zur kinetischen Theorie der Wärmeleitung in Kristallen , 1929 .

[17]  Kenneth M. Case,et al.  Elementary solutions of the transport equation and their applications , 1960 .

[18]  Laurent Chaput,et al.  Direct solution to the linearized phonon Boltzmann equation. , 2013, Physical review letters.

[19]  Natalio Mingo,et al.  Lattice thermal conductivity of silicon from empirical interatomic potentials , 2005 .

[20]  A. Majumdar.  Microscale Heat Conduction in Dielectric Thin Films , 1993 .

[21]  Bekir Sami Yilbas,et al.  Quasiballistic heat transfer studied using the frequency-dependent Boltzmann transport equation , 2011 .

[22]  M. Kanatzidis,et al.  High-performance bulk thermoelectrics with all-scale hierarchical architectures , 2012, Nature.

[23]  Andrew G. Glen,et al.  APPL , 2001 .

[24]  Gang Chen,et al.  Applied Physics Reviews Nanoscale Thermal Transport. Ii. 2003–2012 , 2022 .

[25]  Maik Moeller,et al.  Introduction to Electrodynamics , 2017 .

[26]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[27]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[28]  Arun Majumdar,et al.  Transient ballistic and diffusive phonon heat transport in thin films , 1993 .

[29]  Cristina H. Amon,et al.  Simulation of Unsteady Small Heat Source Effects in Sub-Micron Heat Conduction , 2001 .

[30]  P. Zweifel,et al.  An exact solution of equations of radiative transfer for Local Thermodynamic Equilibrium in the non-gray case. Picket fence approximation☆ , 1966 .

[31]  Richard W Siegel,et al.  A new class of doped nanobulk high-figure-of-merit thermoelectrics by scalable bottom-up assembly. , 2012, Nature materials.

[32]  Gang Chen,et al.  Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticles , 1996 .

[33]  Nicolas Hadjiconstantinou,et al.  Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations , 2011, 1109.3910.

[34]  D. Cahill.  Thermal conductivity measurement from 30 to 750 K: the 3ω method , 1990 .

[35]  M. Williams.  The Energy-Dependent Milne Problem with a Simple Scattering Kernel , 1964 .

[36]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[37]  Qing Hao,et al.  Frequency-dependent Monte Carlo simulations of phonon transport in two-dimensional porous silicon with aligned pores , 2009 .

[38]  G Chen,et al.  Ballistic-diffusive heat-conduction equations. , 2001, Physical review letters.

[39]  Kresse,et al.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. , 1996, Physical review. B, Condensed matter.