Green's function formulation of conductivity of anisotropic two-dimensional materials containing metallic inclusions: Application to phosphorene

Abstract

A semi-analytical expression is presented for the effective thermal/electrical conductivity of two-dimensional anisotropic host solids, such as phosphorene, containing a metallic inclusion of arbitrary shape and size. The expression is derived by using a new, computationally efficient representation of the Green's function for the steady-state Laplace/Poisson equation. The representation is based upon a semi-discrete model of the solid, which yields an analytical formula for the Green's function in terms of its partial Fourier transform. The method is tested by comparing the results with the known exact results for effective conductivity in the limit of an isotropic host and low concentration of elliptic inclusions of different eccentricities and orientations and a square shaped inclusion. The calculated values are found to be in excellent agreement with the exact values. Numerical results are presented for (i) isotropic hosts each containing an elliptic inclusion at different inclinations, and (ii) phosphorene and two other hosts (one anisotropic, one isotropic), each containing a square shaped inclusion. The effective conductivity should be useful as a single parameter for design and characterization of two-dimensional composites.