Calculation and analysis of thermal impedance of microelectronic structures from analytical models

This article deals with thermal impedances of microelectronic components that are useful in Simulation Programs with Integrated Circuit Emphasis (SPICE). In devices like heterojunction bipolar transistors, the active regions thicknesses are often much smaller than the substrates thicknesses. The devices can then be thermally assimilated to heat densities located on top of solid media. In addition to that, when the other dimensions of the heat sources are also much smaller than the substrates dimensions, it is reasonable to consider that the substrate is semi-infinite. First, the expression of the thermal impedance Z of a circular shape heat source centered on top of a half space is presented. For this purpose, the integral transform technique has been used to solve the tri-dimensional heat conduction equation in the frequency domain. The original expression is explicit, exact and allows obtaining results very quickly. After that, the case of a circular heat source on top of a cylinder is treated. A complete analysis of the substrate dimensions influence on the thermal impedance is done. It is based on the impedance decomposition into the one-dimensional impedance and the spreading impedance. By comparing these impedances with that obtained for the heat source on top of the semi-infinite medium, the threshold pulsation at which the thermal impedance of the finite medium differs from the thermal impedance of the half space is extracted. Moreover the geometrical criteria resulting in an error of less than 2% between the spreading impedance of the finite medium and the semi-infinite one are extracted. When these criteria are observed the impedance can be calculated using two perfectly known impedances: the spreading impedance of the semi-infinite medium and the one-dimensional impedance. The results are plotted on the Nyquist diagram, providing a compact representation. Finally the assumption of a circular shape heat source to approximate the thermal impedance of a square shape heat source is validated by evaluating the associated error. The calculation times have been compared to confirm the interest of using this hypothesis.

► The paper provides an exact analytical solution for the thermal impedance in frequential regime.
► The proposed solution is simple to use and allows obtaining accurate results under a short computational time.
► This tool can be easily used by engineers to design electronic components and their supports.