DOI: 10.1615/ICHMT.2004.IntThermSciSemin
ISBN Print: 978-9-61913-930-1
Boltzmann Transport Equation-based Thermal Modeling Approaches for 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 and nano scale 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- and nano-scales. This paper presents a review of the BTE -based solution methods widely employed in the literature. Particular attention is given to the problem of self-heating (hotspot) in sub-micron transistors. First, the solution approaches based on the gray formulation of the BTE are presented. In this class of solution methods, phonons are characterized by one single group velocity and relaxation time. Phonon dispersion is not accounted for in any detail. This is the most widely employed approach in the literature. 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. This includes a full phonon dispersion model developed recently by the authors. This full phonon dispersion model satisfies energy conservation, incorporates the different phonon modes, interactions between the different modes, and frequency dependence for both the phonon group velocity and relaxation times. Results which illustrate the differences between some of these models reveal the importance of developing models that incorporate substantial details of phonon physics.