Boltzmann transport equation-based thermal modeling approaches for hotspots in microelectronics

Abstract

Fourier diffusion has been found to be inadequate 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 ballistic-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 incorporate substantial details of phonon physics. The impact of boundary conditions on thermal predictions is also investigated.