Critical Transport in Weakly Disordered Semiconductors and Semimetals

Motivated by Weyl semimetals and weakly doped semiconductors, we study transport in a weakly disordered semiconductor with a power-law quasiparticle dispersion ${\xi }_{\mathbf{k}}\propto k^{\alpha }$. We show, that in $2\alpha$ dimensions short-correlated disorder experiences logarithmic renormalization from all energies in the band. We study the case of a general dimension $d$ using a renormalization group, controlled by an $ϵ=2\alpha -d$ expansion. Above the critical dimensions, conduction exhibits a localization-delocalization phase transition or a sharp crossover (depending on the symmetries of the Hamiltonian) as a function of disorder strength. We utilize this analysis to compute the low-temperature conductivity in Weyl semimetals and weakly doped semiconductors near and below the critical disorder point.

Figure 1

Critical behavior of disorder in materials with a power-low quasiparticle dispersion ${\xi }_{\mathbf{k}}\propto k^{\alpha }$ in $d$ dimensions. Above the $\alpha =d/2$ line the effects of disorder grow at low energies (the strong-disorder regime). Materials below the line exhibit a critical point between the weak-disorder and strong-disorder regimes.

Figure 2

Conductivity of a Weyl semimetal at small finite doping $\mu$ as a function of the disorder strength and temperature. The dashed parts of the $\sigma (\gamma ,T)$ curves correspond to the strong-disorder regime and may be affected by the interference effects at large length scales not studied here [9].

Figure 3

Diagrams for the renormalization of the disorder strength.