Catching relativistic electrons

Low-energy electrons have been found to mimic relativistic high-energy particles in cadmium arsenide. This defines the first stable '3D Dirac semimetal', which holds promise for fundamental-physics exploration and practical applications.

In classical Newtonian mechanics, an object's energy varies as the square of its velocity or momentum (Fig. 1a) — a rule that car drivers should treat with respect. Photons, neutrinos and other light, fast-moving particles are governed instead by Einstein's theory of relativity: their energy scales linearly with their momentum, with fixed velocity equal to the slope of the increase. Such relativistic high-energy particles hold the key to fundamental understanding of our Universe. But where do electrons — which determine the more practical properties of the materials immediately around us — fit into this picture? Electrons move very fast, but their motion is not primarily relativistic in conventional solids. However, in a paper published in Physical Review Letters, Borisenko et al.¹ report the discovery of relativistic motion of low-energy electrons in cadmium arsenide (Cd₃As₂). Taken together with similar findings described in three independent papers, by Neupane et al.², Liu et al.³ and Jeon et al.⁴, this result paves the way for future relativistic electronics.

Figure 1: Energy–momentum spectra of electrons.

a, Classical objects and electrons exhibit a parabolic relationship between their energy (E) and momentum (p). b, Two-dimensional (2D) Dirac fermions, such as electrons in graphene, have valence (blue) and conduction (purple) energy bands with a linear energy–momentum relationship. These touch at a point called the Dirac point in the 3D parameter space formed by E, p_x and p_y. Shown here is a 2D slice of the 3D space. c, A 3D slice of the 4D (E, p_x, p_y, p_z) energy–momentum relationship of 3D Dirac fermions such as those discovered^1,2,3,4 in Cd₃As₂, with two Dirac points along a special high-symmetry axis (p_z).

The realization that low-energy electrons can mimic high-energy relativistic particles occurred a decade ago with the isolation of two-dimensional (2D) carbon in the form of graphene⁵. This material has dual significance for the exploration of fundamental physics and for revolutionary applications; it has prompted more than 100,000 publications, some 7,000 patent applications and a 2010 Nobel prize. Electrons in graphene are described as massless Dirac fermions because they have half-integer spin, which makes them fermions, and their linear energy–momentum relationship obeys Dirac's famous wave equation, which first united quantum mechanics and special relativity almost a century ago. Graphene is also a semimetal, meaning that its Fermi energy (the dividing line between filled and empty electronic states) sits ideally at its 'Dirac point' — where its valence and conduction energy bands meet (Fig. 1b) — and may be easily tuned using an applied voltage. The resultant charge carriers may be either electrons or holes (the absence of electrons) and have high mobility: a measure of inverse electrical resistivity per carrier, which increases with carrier velocity but decreases with carrier scattering.

Graphene's moderately high carrier velocity of about 10⁵ metres per second, combined with the reduced intrinsic scattering possibilities caused by the small carrier density inherent to a Dirac semimetal, can give a mobility up to 140 times that of silicon — the material of choice for most electronic applications. Therefore, graphene offers promise for making novel, high-efficiency electronic devices. However, graphene is challenging to fabricate and manipulate in large sheets, and its mobility is extremely susceptible to scattering from environmental defects because graphene is all surface.

A second kind of 2D Dirac semimetal arises from another relativistic effect of electrons called spin–orbit coupling — the interaction between an electron's spin and the induced magnetic field from the electron's orbital motion. Spin–orbit coupling is generally small for materials that are made up of light atoms such as carbon, but for materials containing heavy atoms such as bismuth and cadmium, the interaction can be significant; for example, it can invert the valence and conduction bands in the bulk of an insulator. This inversion can lead to surface Dirac fermions that are topologically protected — surface carriers that are robust against some local disorder and have their spin locked to their momentum (that is, the carrier's momentum determines its spin).

These 'topological insulators'^6,7 provoked tremendous excitement in recent years about possible applications such as low-energy-consumption spintronic devices, which manipulate the spin rather than the charge of electrons, for high-performance computing. But despite their name, existing topological insulators have excess conducting bulk electrons, which overwhelm the surface Dirac fermions and foil their use.

Meanwhile, new ideas were brewing, suggesting that 3D Dirac semimetallic states could exist in the bulk of a solid material. It was known that such states could occur under finely tuned conditions, such as the exact concentration of bismuth at which spin–orbit coupling becomes strong enough to invert the bulk energy bands in antimony–bismuth alloys⁸ (Sb_1−xBi_x). But more recent theoretical work predicted the robust occurrence of such states in pure materials that have certain crystalline symmetries: first, unstable BiO₂ (ref. 9), then air-sensitive Na₃Bi (ref. 10) and, finally, the stable compound Cd₃As₂ (ref. 11). Furthermore, when time-reversal or spatial-inversion symmetries are broken — for example, by application of a magnetic field or pressure — each Dirac point can split into two copies at which the electrons become Weyl fermions¹², which have opposite chirality (spin orientation with respect to their direction of motion). These Weyl fermions could enable robust spintronics in three dimensions.

Cd₃As₂ has been known for more than 50 years¹³ for its extraordinary carrier mobility, which is larger than that of suspended graphene and among the highest of any bulk semiconductor. Thanks to the recent studies by Borisenko et al.¹, Neupane et al.² and Liu et al.³ — who all conducted experiments on Cd₃As₂ using a technique called angle-resolved photoemission spectroscopy (ARPES) — we now understand that the high mobility arises from high-velocity 3D Dirac semimetallic states.

During ARPES experiments, monochromatic light is incident on a sample and electrons can absorb a photon and escape from the material. To unveil the full 3D energy–momentum relationship of electrons within Cd₃As₂, a challenging but crucial step was to precisely measure the energy and momentum of emitted electrons while tuning the photon energy through a wide range. The data^1,2,3clearly show a linear energy–momentum relationship, with two Dirac points along a crystal axis of four-fold rotational symmetry (Fig. 1c). This result proves that electrons in this material are 3D massless Dirac fermions as predicted¹¹. Measurement of the energy–momentum slope gives electron velocity as high as about 10⁶ m s⁻¹ (ref. 2), but with a tenfold discrepancy between the three studies^1,2,3, which could be due to differences in sample quality or the angle of the exposed surface. Liu et al. additionally demonstrated that the carrier concentration in Cd₃As₂ could be finely tuned by 'doping' the surface of the material with potassium atoms³, making it a flexible platform for future studies.

Most recently, Jeon et al.⁴ used a scanning tunnelling microscope to confirm Cd₃As₂ as a 3D Dirac semimetal down to atomic length scales, and to visualize how dopant atoms scatter carriers primarily in the valence band, preserving the mobility of carriers in the high-velocity conduction band. Furthermore, Jeon and colleagues applied a magnetic field, which is not possible in an ARPES experiment. Although the field breaks the time-reversal symmetry that would be necessary to split the Dirac fermions into the more exotic chiral Weyl fermions, its orientation in this experiment also breaks the four-fold rotational symmetry of the crystal that was necessary to realize the Dirac fermions in the first place. This means that the first glimpse of Weyl fermions will need to wait for a follow-up experiment in which the magnetic field has a different orientation.

The work on Cd₃As₂ (refs 1,2,3,4), together with the lower-mobility Na₃Bi reported earlier this year¹⁴, confirms the existence of motion of Dirac fermions inside 3D materials. Despite its exciting new physics, the application potential of Cd₃As₂ is limited by its small band-inversion energy — the relativistic nature is not robust at room temperature⁴. Furthermore, Cd₃As₂ is not exactly something you want in your drinking water. Nevertheless, given the new understanding that robust Dirac fermions can arise in solids from general crystalline symmetries and strong spin–orbit coupling, there are probably numerous 3D Dirac semimetals yet to be discovered⁹. Immediate research priorities include magnetic-field and pressure control to isolate chiral Weyl fermions in existing materials, realization of these materials as thin films to access a phenomenon known as the quantum spin Hall effect to visualize the spatial flow of surface Dirac fermions¹⁵, and computational modelling to predict new materials and heterostructures with larger band-inversion energies¹⁶. Then exotic applications, such as a 'chiral battery' or a 'quantum amplifier' of magnetic field, may be on the horizon¹⁷.