Spiraling Fermi arcs in Weyl materials

In Weyl materials the valence and conduction electron bands touch at an even number of isolated points in the Brillouin zone. In the vicinity of these points the electron dispersion is linear and may be described by the massless Dirac equation. This results in nontrivial topology of the Berry connection curvature. One of its consequences is the existence of peculiar surface electron states whose Fermi surfaces form arcs connecting projections of the Weyl points onto the surface plane. Band bending near the boundary of the crystal also produces surface states. We show that in Weyl materials band bending near the crystal surface gives rise to a spiral structure of energy surfaces of arc states. The corresponding Fermi surface has the shape of a spiral that winds about the projection of the Weyl point onto the surface plane. The direction of the winding is determined by the helicity of the Weyl point and the sign of the band-bending potential. For close valleys the arc state morphology may be understood in terms of the avoided crossing of oppositely winding spirals.

Figure 1

Sketch of the energy dispersion of the bulk states (interior of the Dirac cones) and the surface arc states (inclined plane segment) in the absence of band bending.

Figure 2

Left: The spiral energy surface of bound states for band-bending potentials obtained from Eq. (7) with parameters $\gamma =4.5, z_0=0.01$. For clarity, only the part with $\theta \in (-\pi /2,\pi /10)$ is shown. The surface terminates at the Dirac cones of particlelike and holelike states of the continuum. Right: Parabolic energy surfaces of bound states for a conventional semiconductor.

Figure 3

Schematics of the band-bending potential $U\left(z\right)$ and Dirac cones close to the boundary.

Figure 4

Left: Fermi arc shape for the undoped Weyl semiconductor determined from Eq. (11). The spiral makes an infinite number of turns about the Weyl point. Right: Dashed lines—Fermi arc shapes for two nearby valleys of opposite helicity for a square well potential, Eq. (12). The spirals make a finite number of turns (equal to the number of the bound states) about the Weyl points (red dots). Solid lines—avoided crossing of the two spirals due to finite valley mixing.