Thermal Analysis and Optimization of Substrates With Directionally Enhanced Conductivities

Abstract

An approximate analytical solution for the thermal resistance of the axisymmetric chip-on-substrate problem is presented for a substrate with a direction dependent (orthotropic) thermal conductivity. The substrate may be convectively cooled on either, or both, of its planar surfaces. The solution reveals substrate geometries with low maximum substrate temperatures. These optimal substrate sizes are mapped for Biot numbers typical of microelectronic applications. The effects of varying the radial and axial substrate conductivities are investigated. In general, radial conductivity enhancement is beneficial for bottom-side and both-side convective cooling of thin substrates, and for top-side cooling of all substrates. For thin substrates, radial conductivity enhancement provides comparable thermal performance to an equivalent isotropic conductivity enhancement. For electronic packaging applications thin substrates are desirable and radial conductivity enhancement is more beneficial than axial conductivity enhancement.