Much to the confusion of many beginning physics students, electric current vectors are conventionally written as if they represented the flow of positive charge: The direction of the current is opposite to the direction in which electrons actually move. The convention has its origins in Benjamin Franklin’s one-fluid theory of electricity. Lacking evidence to the contrary, Franklin assumed that the phenomena he observed resulted from the movement of a positive “electric fire.” The theory was mostly serviceable: There’s often little to distinguish a positive charge moving in one direction from a negative charge moving in the other.
One way to tell the difference is via the Hall effect, the appearance of a transverse voltage when an electric current passes through a magnetic field. Charge carriers are deflected in the direction that corresponds to the cross product of the (conventionally written) current and the field. If the charge carriers are positive, they produce a voltage gradient in the same direction. If they’re negative, the gradient is in the opposite direction.
The Hall effect provides experimental evidence that currents in metals arise from the flow of negative charge. It also offers a way to distinguish between n-type semiconductors, whose charge carriers are also electrons, and p-type semiconductors, whose charge carriers are positively charged holes.
But the relationship between charge-carrier sign and Hall voltage is not always so simple, as Martin Wegener and his colleagues at the Karlsruhe Institute of Technology in Germany have now experimentally shown.1 Using an n-type semiconductor, the researchers crafted a microstructured metamaterial, shown in figure 1, that behaves like a p-type semiconductor—at least as far as the Hall effect is concerned.
Figure 1. Inspired by medieval armor. The metamaterial shown in this electron micrograph is a periodic array of linked hollow rings. Made of n-type zinc oxide, it exhibits the Hall signature of a p-type material. (Adapted from ref. 1.)
Figure 1. Inspired by medieval armor. The metamaterial shown in this electron micrograph is a periodic array of linked hollow rings. Made of n-type zinc oxide, it exhibits the Hall signature of a p-type material. (Adapted from ref. 1.)
Science mirrors art
The literature is full of examples of metamaterials with electromagnetic, acoustic, or mechanical properties that are qualitatively different from those of their constituents. For example, metamaterials can be designed to have both negative electric permittivity and negative magnetic permeability (or, more simply, a negative index of refraction), despite the fact that there are no such bulk materials in nature.
That tunability of electromagnetic properties can be exploited to create an invisibility cloak: a metamaterial shell that bends electromagnetic waves in a way that leaves no trace of the cloak itself or any objects it conceals. (See Physics Today, February 2007, page 19.) Unfortunately, the effect isn’t quite as striking in real life as it is in the fictional Star Trek or Harry Potter universes. Cloaking metamaterials are made up of tiny resonators that function only over a limited frequency range. To reveal an invisibility cloak, all you need to do is illuminate it with a different color light. Most other curious metamaterial behaviors also arise from internal resonances, so they’re also frequency dependent.
The Hall-effect inversion is different. Rather than relying on an oscillation of just the right frequency, it works with direct current. As pointed out by the University of Utah’s Graeme Milton, one of the mathematicians who predicted the effect, the fact that an n-type metamaterial can so effectively mimic a p-type base material “shows some limitations on what information we can gain about what’s really inside a three-dimensional body.”
Milton and his collaborator Marc Briane came up with the idea of Hall-effect inversion while working with Vincenzo Nesi on a different but related problem: the effective conductivity of a composite material under a static nonzero electric field but no magnetic field. They rigorously proved that it was possible for certain properties of the conductivity to change sign in a 3D composite but not in a 2D composite. The same turned out to be true for the Hall voltage.
Milton and Briane were exploring the electromagnetic properties of 2D arrays of interlocked rings resembling medieval chain-mail armor when they happened upon the website of a chain-mail artist named Dylon Whyte. Briane sought Whyte’s permission to reproduce one of his artistic images in a paper. In his reply, Whyte suggested a 3D interlocking ring structure for the mathematicians to consider. “That turned out to be precisely what we needed!” says Milton: He and Briane showed theoretically in 2009 that a version of Whyte’s structure, made entirely of n-type materials, had the Hall properties of a p-type semiconductor.2
Forging links
Briane and Milton’s metamaterial required three distinct n-type materials: one for the rings themselves, one to form bridges between linked rings, and one background material in which the whole structure was embedded. Creating such a metamaterial in the lab—on the micron scale and with the necessary precision of all the material interfaces—would have been unreasonably demanding.
But two years ago, while tinkering with numerical simulations of Hall-effect inversions, Wegener’s postdoc Muamer Kadic found that the structure could be simplified to require only one semiconducting material.3 (The unit cell of the simulated structure is shown on the cover of this issue.) That simplification brought fabrication within reach.
The fabrication effort, led by PhD student Christian Kern, began with 3D direct laser writing—a form of 3D printing—to make a scaffold structure out of a polymer material. The scaffold was then coated with a thin film of n-type zinc oxide, a wide-bandgap semiconductor. Removing the scaffold was unnecessary: Because it’s electrically insulating, it has no effect on the metamaterial’s electrical properties. The interlocked rings were essentially hollow tubes rather than solid tori—but according to simulations, that geometry doesn’t change the qualitative behavior.
Theory was borne out by experiment: The n-type ZnO metamaterial produced the Hall voltage of a p-type semiconductor. For a qualitative picture of what’s going on, consider the sketch in figure 2. With the current in the +x direction and the magnetic field in the +z direction, the Hall effect in an n-type material normally produces a positive potential on the +y side (toward the left in the figure) and a negative potential on the −y side. Indeed, that’s still the case locally for small regions of material, such as the vicinities of the small black arrows. However, because the currents and voltages are passed from ring to ring via the rings’ inner edges, the ring on the left picks up a negative potential and the ring on the right picks up a positive potential. Compounded across the 5-unit-cell width of the metamaterial, the effect gives a measurable inverted Hall voltage on the order of 50 µV from a current of 0.5 mA and a magnetic field of 0.83 T.
Figure 2. Why the Hall voltage flips. With the current I in the x direction and the magnetic field B in the z direction, as shown here, an n-type semiconductor produces a local Hall-voltage gradient in the +y direction. But because of the way the rings are linked, the ring on the +y side picks up a negative potential and the ring on the −y side picks up a positive potential. The simulated potential map on the right shows the same region in more detail. (Adapted from refs. 1 and 3.)
Figure 2. Why the Hall voltage flips. With the current I in the x direction and the magnetic field B in the z direction, as shown here, an n-type semiconductor produces a local Hall-voltage gradient in the +y direction. But because of the way the rings are linked, the ring on the +y side picks up a negative potential and the ring on the −y side picks up a positive potential. The simulated potential map on the right shows the same region in more detail. (Adapted from refs. 1 and 3.)
It’s not yet clear what applications, if any, the Hall-effect inversion might have. Just because the metamaterial behaves like a p-type semiconductor doesn’t mean it is one. “The charge carriers still carry negative charge,” says Kern, so the metamaterial can’t replace the p-type material in a semiconductor diode, for example. And even if it could, p-type semiconductors are readily available, so there’d be no obvious advantage to the replacement.
Still, Wegener and his group are pressing onward. They’re working on the technical challenges of interfacing their metamaterial with a silicon chip, to study the effect in more detail and possibly exploit it in a magnetic field sensor. And they’re looking into anisotropic metamaterials, which can produce a Hall-voltage gradient with a component parallel to the magnetic field—another effect not found in nature.