Application of the Kirchhoff Transform to Thermal Spreading Problems With Convection Boundary Conditions

Thermal management and thermal analysis of microelectronic devices and packages are critical in ensuring the performance, reliability, and lifetime of today's electronic systems. When the thermal conductivity of a semiconductor or packaging material depends strongly on temperature, the use of a constant thermal conductivity value may significantly underestimate the temperature rise and thermal resistance. The Kirchhoff transform provides a convenient way of linearizing the heat conduction equation to use computationally efficient analytical solutions to calculate the device or package temperature. In the past, the application of the Kirchhoff transform has been restricted to temperature and heat flux boundary conditions in thermal spreading problems. In this paper, we developed an approximate solution for the application of the Kirchhoff transform to thermal spreading problems with convection in the sink plane and show the technique to be accurate to within 1% for relevant problems in device-level thermal analysis. The proposed technique is combined with a recently developed analytical solution for temperature rise in complex, multilayered structures in which a finite heat transfer coefficient in the sink plane needs to be considered. These analytical expressions and the Kirchhoff transform are valuable tools for accurately predicting the temperature in high-power, wide bandgap electronics, such as gallium nitride power amplifiers.