Influence of substrate thickness on thermal impedance of microelectronic structures

The thermal impedance Zth(jω) has been calculated numerically, using the boundary element method, for a silicon substrate with a uniform heat source on top. The key feature is that the dynamic thermal behaviour is calculated directly in the frequency domain. The calculations were performed for a wide range of values for the thickness of the substrate. By representing the thermal impedance in a Nyquist plot (i.e. Im[Zth(jω)] vs. Re[Zth(jω)] with ω as parameter), mainly two circular arcs are observed. For the lower frequency arc, the impedance values as well as the frequency scale are found to be largely influenced by the substrate thickness. The arc corresponding to high frequencies on the other hand remains unchanged under thickness variations.Further analysis revealed an almost perfectly linear relationship between the thermal resistance Rth = Zth(jω = 0) and the substrate thickness, even when the heat source is not centred on the substrate. Both the slope and intersection value obtained from the curve fitting can be explained by a simple geometrical model including the fixed-angle heat spreading approximation, used since many years in the literature.