Effective medium theory for Van-der-Waals heterostructures

Abstract

We derive the electromagnetic medium equivalent to a collection of all-dielectric nano-particles (enjoying high refractive indices) distributed locally non-periodically, precisely the medium is periodic with a unit cell composed of a cluster of multiple nano-particles, in a smooth domain Ω. Such distributions are used to model well-known structures in material sciences as the Van-der-Waals heterostructures. Since the nano-particles are all-dielectric, then the permittivity remains unchanged with respect to the background while the permeability is altered by this effective medium. This permeability is given in terms of three parameters. The first one is the polarization tensors of the used nano-particles. The second one is the averaged Magnetization matrix $\left|{\Omega }_0{|}^{-1}{\int }_{{\Omega }_0}\underset{x}{\nabla }{\int }_{{\Omega }_0}\underset{y}{\nabla }{\Phi }_0\right(x,y)\cdot I_{3}dydx$, where ${\Phi }_0(x,y):=\frac{1}{4\pi |x-y|}$, $I_3$ is the identity matrix and ${\Omega }_0$ is the unit cell. The third one is $\nabla \nabla {\Phi }_0(z_i,z_j)$, where $z_i$'s are locations of the local nano-particles distributed in the unit cell. This last tensor models the local strong interaction of the nano-particles. To our best knowledge, such tensors are new in both the mathematical and engineering oriented literature. This equivalent medium describes, in particular, the effective medium of 2 dimensional type Van-der-Waals heterostructures. These structures are 3 dimensional which are built as superposition of identical (2D)-sheets each supporting locally non-periodic distributions of nano-particles. An explicit form of this effective medium is provided for the particular case of honeycomb heterostructures.

At the mathematical analysis level, we propose a new approach to derive the effective medium when the subwavelength nano-particles are distributed non-periodically. The first step consists in deriving the point-interaction approximation, also called the Foldy-Lax approximation. The scattered field is given as a superposition of dipoles (or poles for other models) multiplied by the elements of a vector which is itself solution of an algebraic system. This step is done regardless of the way how the particles are distributed. As a second step, which is the new and critical step, we rewrite this algebraic system according to the way how these nano-particles are locally distributed. The new algebraic system will then fix the related continuous Lippmann-Schwinger system which, in its turn, indicates naturally the equivalent medium.