Large Contribution of Fermi Arcs to the Conductivity of Topological Metals

Surface-state contributions to the dc conductivity of most homogeneous metals exposed to uniform electric fields are usually as small as the system size is large compared to the lattice constant. In this Letter, we show that surface states of topological metals can contribute with the same order of magnitude as the bulk, even in large systems. This effect is intimately related to the intrinsic anomalous Hall effect, in which an applied voltage induces chiral surface-state currents proportional to the system size. Unlike the anomalous Hall effect, the large contribution of surface states to the dc conductivity is also present in time-reversal invariant Weyl semimetals, where the surface states come in counterpropagating time-reversed pairs. While the Hall voltage vanishes in the presence of time-reversal symmetry, the twinned chiral surface currents develop similarly as in the time-reversal-broken case. For this effect to occur, the relaxation length associated with scattering between time-reversed partner states needs to be larger than the separation of contributing surfaces, which results in a characteristic size dependence of the resistivity and a highly inhomogeneous current-density profile across the sample.

Figure 1

(a) Weyl semimetal with two Weyl cones (green) and straight Fermi-arc surface states (blue) illustrated in mixed momentum-real space. (b) A Weyl semimetal in slab geometry (finite in the $y$ direction) with an imposed potential gradient in the $x$ direction. The enlarged view illustrates the chemical-potential gradients of surface and bulk states and the resulting current flow in the case of strong surface-bulk scattering $l_{sb}\ll W$ (left) and weak surface-bulk scattering $l_{sb}\gg W$ (right).

Figure 2

Resistivity as a function of the slab width for different values of ${\sigma }_0^{xy}/{\sigma }_0^{xx}$. With decreasing $W/l_{sb}$ the resistivity goes to zero since the current is increasingly conducted via surface states, which dissipation decreases.

Figure 3

(a) TR-symmetric Weyl semimetal built out of two TR copies of the previous model [cf., Fig. 1] separated by $\mathrm{\Delta }k$ and coupled by scattering (quantified by the relaxation length $l_{s\overline{s}}$ and $l_{b\overline{b}}$). (b) Potential gradients and current flow of the two subsystems for $l_{sb}\ll W\ll \overline{l}=\min (l_{s\overline{s}},\sqrt{2D{l}_{b\overline{b}}/v})$. Both subsystems exhibit the anomalous Hall effect of a TR-broken Weyl semimetal [cf., Fig. 1]. The Hall voltages cancel each other but the surface-state currents add up.

Figure 4

Resistivity $\rho$ versus slab width $W$ for different ratios $l^{*}/l_{sb}$ at ${\sigma }_0^{xy}/{\sigma }_0^{xx}=1$. The other parameters were chosen as $l_{s\overline{s}}=l_{b\overline{b}}$ and $2D/v=l_{sb}$ so that $\overline{l}=\min (l^{*},l_{s\overline{s}})=l^{*}$. If $l_{\mathrm{sb}}<\overline{l}$, the anomalous contribution of surface states leads to a drop from $\rho /{\rho }_0=1$ for $\overline{l}\ll W$ to a plateau at $\rho /{\rho }_0=1/[1+({\sigma }_0^{xy}/{\sigma }_0^{xx}{)}^2]$ (green dotted line) for $l_{sb}\ll W\ll \overline{l}$.