Fast transient thermal simulation of packages using alternating-direction-implicit method

Abstract

Effective thermal management is one of the most important issues for system-level integration technique. In contrast to traditional 2-D integrated system, complicated and multiscale components of the 2.5-D system significantly increase the modeling complexity. What is more, the super-linear relationship between temperature and power consumption further complicates thermal simulation. Among existing numerical methods for the transient thermal simulation, the finite difference method (FDM) and finite element method (FEM) are the most popular ones. Nevertheless, when simulating thermal problems with a large number of unknowns arising from system-level integration, both of the two conventional methods become time-consuming. In this work, an efficient transient thermal simulation of 2.5-D integrated system is carried out with the equivalent thermal model and the alternating-direction-implicit (ADI) method. The equivalent thermal conductivities of TSV interposer and bump layers are extracted properly. With the ADI technique, the heat conduction equations in the matrix form are derived at three sub-time steps. The resulting tri-diagonal equations can be solved efficiently with linear computational complexity and memory requirement. The power consumption, which includes the temperature-insensitive dynamic power and temperature-dependent leakage power, is taken into account in the modeling as well. In thermal simulation, the temperature rise enhances the leakage power, which in turn increases the surrounding temperature. Thus, the thermal profile updates temperature-dependent leakage power in an iterative way until convergence. The validity and high-efficiency of the proposed method are demonstrated by the numerical results.