Universal analytical solution to the temperature rise of substrate with on-chip hotspot

While gallium nitride (GaN) is widely used in microwave devices, the die-level heat flux increases dramatically from hundreds of to thousands of watts per square centimeter. Nowadays, integrated with high thermal conductivity substrate is commonly accepted to enhance heat spreading effect. Based on three-dimensional steady-state conduction, universal analytical solution of substrate temperature rise caused by hotspot is obtained and the exact solution is calculated through software Matlab. Also, we compare the calculation results using finite element simulation, semi-infinite solution, simplified solution and exact solution. Finally, a series of simplified fifth-order polynomial equations are established to estimate such temperature rise and both the influence of substrate thickness and hotspot size on such temperature rise are studied. As predicted, the results computed by the exact solution agree well with the fitting formula and numerical solution. It turns out that an accurate, convenient closed-form equation can be provided to evaluate the substrate heat spreading effect.