A two-dimensional disordered magnetic metamaterial

TL;DR

This paper investigates how frequency disorder affects energy transport in a 2D array of split-ring resonators, revealing saturation and sub-diffusive behaviors depending on disorder type and strength.

Contribution

It extends the 1D random dimer model to 2D magnetic metamaterials, analyzing disorder effects on eigenstates and energy transport.

Findings

Uncorrelated disorder causes MSD saturation at all disorder levels.

Correlated tetramer disorder exhibits a disorder window with sub-diffusive MSD behavior.

Outside the disorder window, MSD saturates similarly to uncorrelated disorder.

Abstract

We study the effect of a resonant frequency disorder on the eigenstates and the transport of magnetic energy in a two-dimensional (square) array of split-ring resonators (SRRs). In the absence of disorder, we find the dispersion relation of magneto-inductive waves and the mean square displacement (MSD) in closed form, showing that at long times the MSD is ballistic. When disorder is present, we consider two types: the usual Anderson distribution (uncorrelated monomers) and $2 \times 2$ units assigned at random to lattice sites (correlated tetramers). This is a direct extension to two dimensions of the one-dimensional random dimer model (RDM). For the uncorrelated case, we see saturation of the MSD for all disorder widths, while for the correlated case we find a disorder window, inside which the MSD does not saturate at long times, with an asymptotic sub-diffusive behavior $MSD\sim t^{0.26}$. Outside this disorder window, the MSD shows the same kind of saturation as in the monomer case. We conjecture that the sub-diffusive behavior is a remanent of a weak resonant transmission of a 2D plane wave across a tetramer unit.