Quantum transport in a three-dimensional Weyl electron system

Quantum transport in a three-dimensional Weyl (massless Dirac) electron system with long-range Gaussian impurities is studied theoretically using a self-consistent Born approximation (SCBA). We find that the conductivity significantly changes its behavior at a certain critical disorder strength which separates the weak and strong disorder regimes. In the weak disorder regime, the SCBA conductivity mostly agrees with the Boltzmann conductivity, except for the Weyl point (the band touching point) at which the SCBA conductivity exhibits a sharp dip. In the strong disorder regime, the Boltzmann theory fails in all the energy regions and the conductivity becomes larger the disorder potential increases, contrary to the usual metallic behavior. At the Weyl point, the conductivity and the density of states are exponentially small in the weak disorder regime, and they abruptly rise at the critical disorder strength. The qualitative behavior near zero energy is well described by an approximate analytic solution of the SCBA equation. The theory applies to three-dimensional gapless band structures including Weyl semimetals.

Figure 1

Boltzmann conductivity [Eq. (12)] plotted as a function of the Fermi energy.

Figure 2

(Solid lines) Density of states at zero energy calculated by the SCBA. (Dotdashed lines) Approximate expression Eq. (40) with Eq. (38), where ${\varepsilon }_{\mathrm{c}}/{\varepsilon }_0$ is taken as $0.87$. The inset shows the density of states near $W_{\mathrm{c}}\approx 1.806$ on a smaller scale.

Figure 3

Density of states (a),(c) and the conductivity (b),(d) calculated by the SCBA, as a function of the Fermi energy. Panels (c) and (d) show the detailed plots near $\varepsilon =0$ of (a) and (b), respectively. The dot-dashed lines in panels (c) and (d) represent the approximate SCBA solution, Eqs. (40) and (46), respectively.

Figure 4

SCBA conductivity and the Boltzmann conductivity [Eq. (12)] at several $W$'s plotted for (a) wide and (b) narrow energy regions.

Figure 5

(a) SCBA conductivity (solid lines) and the Boltzmann conductivity (dashed lines) as a function of $1/W$, for some fixed Fermi energies. (b) A similar plot for small Fermi energies, $\varepsilon =0$ and $0.001{\varepsilon }_0$. Dot-dashed lines represent the approximate SCBA solution [Eqs. (48) and (50)] at $\varepsilon =0$.