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In 2015, physicists discovered a new topological state of matter known as a Weyl semimetal. Electronic excitations in such a material do not behave as nearly free electrons, like in most metals. Instead, they can effectively be described as massless relativistic particles with properties similar to photons. Beside their fundamental interest, the exotic behavior of Weyl fermion materials motivates the pursuit of electronic or optical devices with novel functionalities. Romilly Hills at Loughborough University, UK, and collaborators have now theoretically investigated a range of devices that could be made using such materials. In particular, they propose that a structure made of multiple layers of Weyl semimetals would behave as a medium whose effective refractive index is negative, which could focus a diverging electron beam onto an extremely small region. Such a property could lead to a “superlens” that boosts the spatial resolution of scanning tunneling microscopes (STMs).

In 984, the Persian scientist Ibn Sahl showed that the bending of light as it passes from one medium to another is governed by the two media’s indices of refraction—a relation known more familiarly today as Snell’s law. Typical materials, like water or air, have a positive index, meaning that the angle of transmission, as measured between the normal and the interface between the two media, is always positive (see Fig. 1, left). In this case, light refraction creates familiar optical effects like the bending of a straw in a glass of water. In 1967, however, the physicist Viktor Veselago showed that a negative index of refraction would lead to unconventional optical properties: the transmission angle would become negative (see Fig. 1, right) such that diverging light rays would be focused. In other words, a slab of negative-index material acts as a lens. In 2000, John Pendry suggested that such a Veselago lens, or “hyperlens,” could bypass the limits posed by diffraction, allowing optical microscopes to probe subwavelength objects .

Realizing a medium with negative refraction is challenging. For photons, this behavior has been achieved using artificial metamaterials that work at microwave and visible wavelengths. The main proposal of Hills et al. is that Weyl semimetals could be used to realize a Veselago lens for electrons. The result builds on earlier work on graphene, the one-atom-thick carbon layer whose electrons also behave as massless relativistic particles. In 2007, Cheianov et al. predicted that electrons propagating through a potential barrier in graphene behave as light waves that have been transmitted into a medium with a negative refractive index. This occurs because the group velocity of charged massless relativistic particles can be parallel or antiparallel to their momentum, depending on their charge. Hence, when an electron wave is transmitted through an np junction, that is, from an n-type (electron dominated) region into a p-type (hole dominated) region, the group velocity switches sign. At the same time, the momentum tangential to the interface must be preserved. Together, these two conditions imply that the ratio of the incident to transmitted momenta is negative, resulting in an effective refraction index that’s negative. Cheianov et al.’s proposal was experimentally realized in 2015, providing a two-dimensional electronic realization of the Veselago lens.

Hills and collaborators now show that the mechanism described by Cheianov et al. also works in a three-dimensional Weyl semimetal. The team first analyzes a heterostructure made of two Weyl semimetals, such as NbAs and NbP, calculating the transmission probability of an electron current impinging on the interface at a certain angle (see Fig. 2). The resulting transmission curve is similar to that seen in graphene, indicating a negative refractive index behavior. This result confirms a previous theoretical study on layered Weyl semimetal structures, which, however, was primarily focused on how an applied magnetic field affects electron transmission.

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