Analytical Solution for Temperature Rise in Complex Multilayer Structures With Discrete Heat Sources

Temperature rise and thermal spreading resistance in multilayered structures are an important research topic in several branches of the thermal-fluid sciences, including thermal management of electronics and contact resistance. Previous work in developing analytical solutions for the temperature rise and thermal spreading resistance has been limited to relatively few layers and simple conditions at the interfaces. Recent development of multilayer epitaxial structures for high power electronics has led to the need for more general and flexible analytical solutions because numerical methods, such as the finite element method, are often computationally inefficient. This paper presents a closed-form, analytical solution for the temperature distribution in a rectangular structure with rectangular isoflux heat sources, any number of layers of arbitrary thermal conductivity, and perfect interfacial contact or finite interfacial conductance. Extensions are also presented for convective boundary conditions in the source and sink planes. The proposed analytical solution is demonstrated and validated to study gallium nitride (GaN)-based epitaxial structures in realistic device configurations. The capability to explore the parametric space in a computationally efficient manner provides the ability to understand the key dependencies of the temperature rise on the properties of the structure, such as the substrate thickness in GaN-on-diamond epitaxial structures. Our results show that reduction of the thickness of diamond substrates may actually increase the device temperature in a realistic device configuration due to the importance of thermal spreading within the first ~100 μm of the heat source.