Transient 3D heat flow analysis for integrated circuit devices using the transmission line matrix method on a quad tree mesh

Abstract

This paper presents a 3D transmission line matrix (TLM) implementation for the solution of transient heat flow in integrated semiconductor devices. The implementation uses a rectangular discontinuous mesh to allow for local mesh refinement. This approach is based on a quad tree meshing technique which can have a complex geometry using blocks of varying sizes. Each such block can have a maximum of two adjacent blocks on any vertical side and a maximum of four blocks on the top or bottom.The TLM implementation is based on a physical extraction of a resistance and capacitance network and then the creation of the appropriate TLM matrix. The formulation allows for temperature-dependent material parameters and a non-uniform time stepping.The simulator is first tested using a 2D example of a heat source in a rectangular region. Using this example the numerical error is determined and found to be less than 0.4%. Next, non-linearities are included, and a number of non-uniform time stepping algorithms are tested. Then, a 3D problem is also compared to an analytical solution and again the error is very small. Finally, an example of a full solution of heat flow in a realistic Si trench device is presented.