Theory of the Hall effect in three-dimensional metamaterials

TL;DR

This paper develops a homogenization theory for 3D metamaterials to accurately predict their effective electrical and Hall properties, including novel architectures with sign reversal of the Hall coefficient and magnetic material integration.

Contribution

It introduces a comprehensive homogenization framework for 3D metamaterials, visualizes critical structural features, and proposes new designs with unique Hall effect behaviors.

Findings

Consistent with previous numerical and experimental results.

Identifies critical structural regions affecting effective parameters.

Proposes new metamaterial architectures with Hall coefficient sign reversal.

Abstract

We apply homogenization theory to calculate the effective electric conductivity and Hall coefficient tensor of passive three-dimensionally periodic metamaterials subject to a weak external static homogeneous magnetic field. We not only allow for variations of the conductivity and the Hall coefficient of the constituent material(s) within the metamaterial unit cells, but also for spatial variations of the magnetic permeability. We present four results. First, our findings are consistent with previous numerical calculations for finite-size structures as well as with recent experiments. This provides a sound theoretical justification for describing such metamaterials in terms of effective material parameters. Second, we visualize the cofactor fields appearing in the homogenization integrals. Thereby, we identify those parts of the metamaterial structures which are critical for the observed effective metamaterial parameters, providing a unified view onto various previously introduced single-constituent/multiple-constituent and isotropic/anisotropic architectures, respectively. Third, we suggest a novel three-dimensional non-magnetic metamaterial architecture exhibiting a sign reversal of the effective isotropic Hall coefficient. It is conceptually distinct from the original chainmail-like geometry, for which the sign reversal is based on interlinked rings. Fourth, we discuss two examples for metamaterial architectures comprising magnetic materials: Yet another possibility to reverse the sign of the isotropic Hall coefficient and an approach to conceptually break previous bounds for the effective mobility.