An exact recursion relation solution for the steady-state surface temperature of a general multilayer structure

A recursion relation technique has been used in the past to determine the surface potential from the multilayer electrical Laplace equation. This has provided for a vastly simplified evaluation of the electrical spreading resistance and four-probe resistance. The isomorphism of the multilayer Laplace equation and the multilayer steady-state heat flow equation suggests the possibility of developing a recursion relation applicable to the multilayer thermal problem. This recursive technique is developed and is shown to provide the surface temperature of the multilayer steady-state heat flow equation. For the three-layer case, the thermal recursion relation readily yields the surface results,which are identical with those presented by Kokkas and the TXYZ thermal code. This recursive technique can be used with any number of layers while incurring only a small increase in computation time for each added layer. For the case of complete, uniform top surface coverage by a heat source, the technique gives rise to the generalized one-dimensional thermal resistance result. An example of the use of the new recursive method is provided by the preliminary calculations of the surface temperature of a buried oxide (SOI, SIMOX) structure containing several thicknesses of the surface silicon layers. This new technique should prove useful in the investigation and understanding of the steady-state thermal response of modern multilayer microelectronic structures