三维集成电路稳态解和瞬态解的建模与热分析

Modeling and Thermal Analysis of Steady and Transient Solutions for 3-Dimensional Integrated Circuits

针对三维集成电路(3D IC)热效应问题,提供了三维集成电路热模型的稳态解析解和瞬态解析解,这些解适用于N层(N≥2)模型;热源可以非均匀分布,瞬态时热源的大小可随时间变化。求解的过程使用了分离变量法、格林函数法和本征函数正交性。通过瞬态解可以获得任意时刻模型的温度场,稳态解是与时间无关的函数,通过它可以直接计算出稳态热传导的温度场。使用所获得的解析解在MATLAB中计算得到的温度场和COMSOL仿真温度场进行比较,结果表明,稳态解计算3层和5层3D IC模型得到的结果和COMSOL仿真结果最大误差在1%左右;使用所获得的瞬态解计算3层和5层3D IC模型得到的结果和COMSOL仿真结果最大误差不超过3%。

Aiming at the thermal effect of 3-dimensional integrated circuits(3D IC),steady and transient analytic solutions for the 3D IC model which is N-tier(N≥2)are proposed;heat sources can be nonuniform and their values can vary with time in transient case.Solutions are solved by separation of variables,Green's function and orthogonality of eigenfunctions.The temperature field of the model at any time can derive from the transient solution;steady temperature field can be solved directly by the steady solution which is a time-independent function.Temperature fields,obtained by analytical solutions with MATLAB,are compared with temperature fields simulated by COMSOL.The results show that the maximum error between the results calculated through the steady solution for 3-tier and 5-tier model and the results simulated by COMSOL is about 1%;the maximum error between the results calculated through the transient solution for 3-tier and 5-tier model and the results of COMSOL is less than 3%.