Anisotropic charge and heat conduction through arrays of parallel elliptic cylinders in a continuous medium

Abstract

Arrays of circular pores in silicon can exhibit a phononic bandgap when the lattice constant is smaller than the phonon scattering length, and so have become of interest for use as thermoelectric materials, due to the large reduction in thermal conductivity that this bandgap can cause. The reduction in electrical conductivity is expected to be less, because the lattice constant of these arrays is engineered to be much larger than the electron scattering length. As a result, electron transport through the effective medium is well described by the diffusion equation, and the Seebeck coefficient is expected to increase. In this paper, we develop an expression for the purely diffusive thermal (or electrical) conductivity of a composite comprised of square or hexagonal arrays of parallel circular or elliptic cylinders of one material in a continuum of a second material. The transport parallel to the cylinders is straightforward, so we consider the transport in the two principal directions normal to the cylinders, using a self-consistent local field calculation based on the point dipole approximation. There are two limiting cases: large negative contrast (e.g., pores in a conductor) and large positive contrast (conducting pillars in air). In the large negative contrast case, the transport is only slightly affected parallel to the major axis of the elliptic cylinders but can be significantly affected parallel to the minor axis, even in the limit of zero volume fraction of pores. The positive contrast case is just the opposite: the transport is only slightly affected parallel to the minor axis of the pillars but can be significantly affected parallel to the major axis, even in the limit of zero volume fraction of pillars. The analytical results are compared to extensive FEA calculations obtained using Comsol™ and the agreement is generally very good, provided the cylinders are sufficiently small compared to the lattice constant.