Thermal impedance of multi-finger microelectronic structures: exact analytical model

Abstract

An exact analytical expression for the complex thermal impedance Z of multi-finger microelectronic components is presented in this paper. The integral transform technique has been used to obtain this expression and solve the three dimensional heat conduction equation directly in the frequency domain. Calculations were first performed for a single-finger on a single-layer structure in order to compare the results with those available in the literature and hence validate the solution. Generally, the comparison shows good agreement between our results and those given in most publications. When the structures are composed of several layers, the thermal impedance changes with the thermal conductivities and the thicknesses of the different layers. It is also affected by the thermal contact resistance between the layers. Some results illustrate the influence of these parameters. The case of a multi-finger component is then treated and the influence of distances between fingers is investigated. For all cases, the Nyquist diagram (i.e. Im(Z) versus Re(Z) for different pulsation values ω) is plotted. Mainly two zones are observed: one for the high frequencies and the other for the lower ones. The substrate dimensions are found to largely influence the scale of the low frequency zone whereas the distance between the fingers influences the higher one. Finally, the solution is applied to a multi-finger device in contact with a heat sink.