SPECIAL ISSUE Sreekant V. J. Narumanchi Jayathi Y. Murthy Cristina H. Amon Boltzmann transport equation-based thermal modeling approaches for hotspots in microelectronics Received: 14 June 2004 / Accepted: 31 January 2005 / Published online: 25 November 2005 Ó Springer-Verlag 2005 Abstract Fourier diffusion has been found to be inade- quate for the prediction of heat conduction in modern microelectronics, where extreme miniaturization has led to feature sizes in the sub-micron range. Over the past decade, the phonon Boltzmann transport equation (BTE) in the relaxation time approximation has been employed to make thermal predictions in dielectrics and semiconductors at micro-scales and nano-scales. This paper presents a review of the BTE-based solution methods widely employed in the literature and recently developed by the authors. First, the solution approaches based on the gray formulation of the BTE are presented. The semi-gray approach, moments of the Boltzmann equation, the lattice Boltzmann approach, and the bal- listic-diffusive approximation are also discussed. Models which incorporate greater details of phonon dispersion are also presented. Hotspot self-heating in sub-micron SOI transistors and transient electrostatic discharge in NMOS transistors are also examined. Results, which illustrate the differences between some of these models reveal the importance of developing models that incor- porate substantial details of phonon physics. The impact of boundary conditions on thermal predictions is also investigated. Keywords Micro/nanoscale Sub-continuum thermal transport BTE Hotspot NMOS transistors SOI Nomenclature C Total volumetric heat capacity (J/m 3 K) C w Volumetric specific heat per unit frequency (Js/m 3 K) D(w) Phonon density of states (m 3 ) e total Total energy (J/m 3 ) f w Phonon distribution function  h Reduced Planck’s constant (= h/(2p), 1.054 · 10 34 Js) k B Boltzmann’s constant (1.38 · 10 23 J/K) K Thermal conductivity (W/m K) N LA , N TA Number of frequency bands in LA and TA branches N bands Total number of frequency bands (N LA + N TA + 1) N h , N / Number of h and / divisions in an oc- tant q vol Volumetric heat generation (W/m 3 ) ~ r Position vector (m) ^ s Unit direction vector t Time (s) T Temperature (K) v Phonon velocity (m/s) w Phonon frequency (rad/s) Greek symbols Dw Frequency width (rad/s) u Azimuthal angle c Band-averaged inverse relaxation time for interaction (s 1 ) s Relaxation time of a phonon (s) h Polar angle (degrees) Subscripts i ith frequency band ij Property specific to bands i and j L Lattice O Optical mode P Propagating mode S. V. J. Narumanchi National Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO, USA J. Y. Murthy School of Mechanical Engineering, Purdue University, 585 Purdue Mall, W. Lafayette, IN, USA C. H. Amon (&) Institute for Complex Engineered Systems and Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA E-mail: camon@cmu.edu Heat Mass Transfer (2006) 42: 478–491 DOI 10.1007/s00231-005-0645-6