A Transient Resistance/Capacitance Network-Based Model for Heat Spreading in Substrate Stacks Having Multiple Anisotropic Layers

Abstract

Heat spreading from local, time-dependent heat sources in electronic packages results in the propagation of temperature nonuniformities through the stack of material layers attached to the chip. Available models either predict the chip temperatures only in the steady-state or a single substrate for transient analysis, without the ability to predict the transient response in compound substrates having multiple anisotropic layers. We develop a transient resistance/capacitance network-based modeling approach capable of predicting the spatiotemporal temperature fields for this chip-on-stack geometry, accounting for in-plane heat spreading, through-plane heat conduction, and the effective convection resistance boundary conditions. The transient heat spreading resistance for an anisotropic substrate has been formulated by converting it to an effective isotropic substrate for which there is an available half-space solution. The transient heat spreading model for a step heat input to a single substrate is subsequently extended to any arbitrary transient heat input using a Fourier series and extending the half-space formulation from a step heat input to sinusoidal heat input. For obtaining the transient spreading resistance for heat inputs into multiple stacked substrates, a method has been outlined to convert the multisubstrate stack to a single substrate with effective isotropic properties by properly accounting for the in-plane and through-plane thermal conductivities, and the heat capacity. The estimates from the present model are validated with direct comparison to a finite volume numerical model for three-dimensional heat conduction. Case studies are presented to demonstrate the capabilities of the proposed network-based model and compare its estimates with the numerical conduction solution. In the presence of a step heat input, the results demonstrate that the model accurately captures the transient temperature rise across the multisubstrate stack comprising layers with different anisotropic properties. For a case where the rectangular stack is exposed to a sinusoidally varying heat input, the model is able to capture the general trends in the transient temperature fields in the plane where the heat source is applied to the multisubstrate stack. In summary, the developed resistance/capacitance network-based transient model offers a low-computational-cost method to predict the spatiotemporal temperature distribution over an arbitrary transient heat source interfacing with a multilayer stack of substrates.