Hall-Effect Sign Inversion in a Realizable 3D Metamaterial

Abstract

In 2009, Briane and Milton proved mathematically the existence of three-dimensional isotropic metamaterials with a classical Hall coefficient that is negative with respect to that of all of the metamaterial constituents. Here, we significantly simplify their blueprint towards an architecture composed of only a single-constituent material in vacuum or air, which can be seen as a special type of porosity. We show numerically that the sign of the Hall voltage is determined by a separation parameter between adjacent tori. This qualitative behavior is robust even for only a small number of metamaterial unit cells. The combination of simplification and robustness brings experimental verification of this striking sign inversion into reach. Furthermore, we provide a simple intuitive explanation of the underlying physical mechanism.

Alternate abstract:

Plain Language Summary

Intuition tells us that increasing the porosity of a bulk material should lead to effective material properties in between those of the bulk and air. Indeed, such intuition is sufficient for the majority of porous materials. Here, we discuss a striking exception: By constructing a porous metal or a doped semiconductor consisting of interlocking micron-scale tori, the effective Hall voltage can assume any value and can even reverse its sign with respect to that of the bulk material. This Hall voltage has experimental relevance because measuring the Hall voltage makes it possible to calculate magnetic fields.

An electrical conductor placed in a magnetic field will experience a potential difference across its surfaces; this difference is the so-called Hall voltage. Textbook wisdom states that the sign of the Hall voltage allows researchers to determine whether negatively charged electrons or positively charged holes dominate the electrical conduction in a material. We theoretically investigate Hall voltage in a porous metamaterial composed of interlocking tori with parameters similar to those of phosphorus-doped silicon. Each torus has a diameter of 20μm , and we investigate a variety of separations between adjacent tori. The sign inversion that we observe between the porous metamaterial and the bulk material from which it was constructed demonstrates that this wisdom must be adopted with caution. Our findings are applicable to even only a small sampling of the material’s unit cells.

Such three-dimensional porous materials or “metamaterials” like these have yet to be realized experimentally, but we describe a feasible route to making them using existing technologies like three-dimensional direct laser writing. We expect that our results will motivate future experimental verification of this theory despite the challenges of microfabrication.