An analytical model for calculating transient temperatures of 3-D integrated system considering heat capacity variations

Abstract

Constant pressure heat capacity is an important property that affects thermal diffusivity when calculating the transient temperature of three-dimensional (3-D) integrated systems, but current analytical models usually do not consider the effect of heat capacity. Acknowledging the temperature-dependent nature of heat capacity for materials within the AlN-substrate-based 3-D integrated system, a novel anisotropic transient thermal analytical model that takes into account the heat capacity's temperature-dependent variation is proposed. Initially, the AlN substrate, integrated with a through ceramic via (TCV) array, is modeled as a homogenized structure utilizing the equivalent thermal conductivity (ETC) model. Subsequently, an anisotropic transient thermal analytical model based on Fourier series was established, and the heat capacity values of various materials in the analytical model were updated in real time. The analytical model was compared with the finite element method (FEM), showing a relative error of about 0.73 % in the transient peak temperature predictions, which indicates a high level of agreement with the FEM results. Furthermore, the time required to extract the transient peak temperature within a 10-s simulation using our analytical model was only 0.43 s, which is 885 times faster than the FEM. This demonstrates that the proposed model excels in computational precision and velocity, offering a reliable and efficient strategy for predicting the thermal management needs of 3-D integrated systems.