Characterization of the Thermal Behaviour in ICs

A complete four-dimensional solution T(x, y, z, t) to the classical dynamic heat-flow equation, for the case where the power sources lie on the silicon surface, is presented. The solution is found by: (i) applying multiple finite Fourier transforms with respect to x and y; (ii) Laplace-transforming with respect to time; (iii) solving the resulting differential equations for the z variable and determining the Fourier components; (iv) performing the inverse Laplace transforms for all the Fourier terms and summing them. The solution has been used to simulate the thermal behaviour of several different IC power configurations.