An Analytical Model for Steady-State and Transient Temperature Fields in 3-D Integrated Circuits

Abstract

This paper proposes a noniterative analytical thermal model for 3-D integrated circuits (3-D-ICs) based on the solution of the governing energy equations using Green’s function, separation of variables, and superposition methods. The model is applicable to both transient and steady-state heat conduction problems. Any finite number of dies equally sized in plane can be taken into consideration in current analysis. In addition, the analytical solution presented here is very general, so that it takes into account the different interface boundary conditions between the dies. Importantly, the thermal properties (i.e., $k$ , $\rho $ , $c$ ) and thickness of different dies can be different, and the spatial and time-varying volumetric heat generation can be applied in each of the different dies. The results compared with the finite-element simulations show that the temperature fields predicted by the proposed model can offer an accurate and efficient solution to compute the temperature fields in the representative 3-D-ICs, with a maximum deviation between the two being around 2.0% in the transient heat conduction. In the steady-state case, the maximum deviation can reach even less than 1.0%.