Hall-effect metamaterials and “anti-Hall bars”

In his letter in the July 2017 issue of Physics Today (page 13 ), Ramesh Mani points to the connection between part of one unit cell of our three-dimensional chainmail-like Hall-effect metamaterial ¹ (see Physics Today, February 2017, page 21 ) and his earlier work on planar “anti-Hall bars.” ² We were not aware of his work and thank Mani for pointing it out to us. However, the conclusions he derives in his comment are misleading.

He argues that the change in Hall-voltage sign “should be attributed to a change in effective geometry rather than to a change in sign of the Hall coefficient.” That viewpoint completely ignores the idea of metamaterials and composites, as described by homogenization theory. ^{3 , 4} Indeed, as emphasized by Mani, the Hall coefficient of the host material does not change when one introduces voids into it. However, the geometry or structure inside the metamaterial unit cell determines the effective Hall coefficient of the metamaterial crystal.

What does the metamaterial community generally mean by effective material parameters? Suppose, in the sense of a black box, an experimentalist cannot look into the unit cell of an artificial crystal but can perform experiments on the crystal. He or she may change the strength and direction of the applied static magnetic field, the amplitude and direction of the injected electrical current, the pickup of the Hall voltage, and the size of the sample, measured by the number of unit cells in any one direction.

For our 3D metamaterial, the experimentalist would conclude that all observations are perfectly consistent with a sign reversal of the Hall coefficient—that is, the effective Hall coefficient—with respect to that of the bulk host material. In sharp contrast, that statement is not true for a single planar anti-Hall bar.

Wiring up many individual Hall elements into a 3D, electrically isotropic metamaterial crystal has been the main aim of our work. ¹ It is demanding: In the resulting 3D chainmail-like geometry, ⁵ which has been inspired by the work of Marc Briane and Graeme Milton, ³ all connections are made from the same semiconductor material as the Hall elements. Thus local Hall voltages appear wherever a current is flowing, and, strictly speaking, one cannot even make a distinction between the connections and the local Hall elements inside the metamaterial.

Our work should not be misunderstood to mean that we simply somehow wanted to change the sign of the Hall voltage. That would be trivial indeed: One could simply interchange the two wires picking up the Hall voltage.