Multiterminal conductance at the surface of a Weyl semimetal

Weyl semimetals are a new paradigmatic topological phase of matter featuring a gapless spectrum. One of its most distinctive features is the presence of Fermi arc surface states. Here, we report on atomistic simulations of the dc conductance and quantum Hall response of a minimal Weyl semimetal. By using scattering theory we show that a quantized Hall conductance with a nonvanishing longitudinal conductance emerges associated to the Fermi arc surface states with a remarkable robustness to high concentrations of defects in the system. Additionally, we predict that a slab of a Weyl semimetal with broken time-reversal symmetry bears persistent currents fully determined by the system size and the lattice parameters.

Figure 1

(a) Bulk dispersion of the semimetal in the topological phase with only two Weyl nodes with parameters $m=1.0$, and $t_x=t_y=t_z=1/2$. The inset shows a cell of the real space lattice with nearest neighbor hoppings ${\gamma }_x$, ${\gamma }_y$, ${\gamma }_z$. (b) First Brillouin zone of the Weyl semimetal showing the high symmetry points where the dispersion in (a) was calculated. We show also the Weyl nodes of opposite chirality and the planes where the Chern numbers were obtained.

Figure 2

Energy dispersions for the model discussed in the text. (a) and (b) correspond to a system with 100 lattice sites in the $z$ direction (and infinite in $x$ and $y$). (a) shows the dispersion along the $k_y=0$ line in the 2D Brillouin zone, while (b) shows it along the $k_x=\pi /a$ line. (c) and (d) show the energy dispersion for a Weyl semimetal wire with a cross section of $30\times 30$ lattice sites [with the same parameters as in (a) and (b)]. (c) corresponds to a wire with the cross section in the $x\text{−}y$ plane (an $x\text{−}z$ wire has identical dispersion) while (d) corresponds to a wire with the cross section in the $y\text{−}z$ plane. The insets show the geometry of the wire in each case.

Figure 3

Transport response at the surface of the $yz$ face of a rectangular cuboid. Six electrical contacts in an H configuration access the $yz$ plane as shown in (g). (a) Two-terminal conductance $G$, (b) longitudinal conductance ${\sigma }_L$, and (c) Hall conductance ${\sigma }_H$, are plotted against the Fermi energy $E_f$ set on the leads. Panels (d), (e), and (f) show the same magnitudes as their counterparts on the left but averaged over 15 independent realizations of disorder as a function of the concentration of defects (vacancies) and for $E_F=0.001$. The bars depict the fluctuations around the average values. The setups are represented in the schemes in (g); note that the contacts are two dimensional.

Figure 4

In this figure we explore how $G$ (a), ${\sigma }_L$ (b), and ${\sigma }_H$ (c), determined from probes on the $yz$ face as in the previous figure, change as the dimension of the Weyl semimetal along $x$, $N$, is varied. Here, we set $E_F=0.001$.

Figure 5

Transport response at the surface of the $xy$ face of a rectangular cuboid. Six electrical contacts in an H configuration access the $xy$ plane. (a) Longitudinal conductance ${\sigma }_L$ is plotted against the Fermi energy $E_F$ set on the leads for two different orientations of the cuboid geometry, thus measuring either ${\sigma }_{yy}$ or ${\sigma }_{xx}$ as shown in panel (b).

Figure 6

Scheme showing the current density flow in a finite sample of Weyl semimetal made from $10\times 10\times 10$ lattice sites, at $E_F=0$. The arrows point in the direction of the current flow, and their size and color intensity are proportional to the magnitude of the current density.

Figure 7

(a) Maximum surface current for a cube of $N\times N\times N$ lattice sites as a function of $m$, the time reversal breaking parameter. (b) Maximum surface current as a function of the cube lateral dimension $N$ at fixed $m=1.0$. (c) Values of the Chern number $\nu$ for the planes intersecting $k_x=0$ and $k_x=\pi /a$ in the Brillouin zone, as a function of $m$.