A simple and approximate analytical model for the estimation of the thermal resistances in 3D stacks of integrated circuits

The purpose of this paper is to describe a simple analytical model, useful for system architects, which describes the thermal resistances between a hot spot and the cold heat sink in steady state conditions with constant material properties, while taking into account the impact of the lateral spreading. The total thermal resistance Rth_total is modeled as dependent on Rth_1D, the thermal resistance in the Z direction, perpendicular to the layers, and Rth_3D, the effects of the lateral spreading of the heat (the X and Y directions) from the hot spot. In a 1^st step, the different layers are homogenized to obtain an equivalent anisotropic thermal conductivity per layer. It must be noticed that the presence of copper TSV increases the thermal conductivities in the Z direction but decreases this conductivity in XY directions, due to the very negative effect of the SiO2 layer surrounding the Cu. These conductivities are used in a 2^nd step to compute Rth_1D, combining the individual thermal resistance of each layer. The 3^rd step estimates Rth_3D by a simple analytical model. The few adjustable coefficients are fitted on several thousand FEM simulations.