Disorder effects in topological insulator nanowires

Three-dimensional topological insulator (TI) nanowires with quantized surface subband spectra are studied as a main component of Majorana bound states (MBS) devices. However, such wires are known to have large concentration $N \sim 10^{19}$cm$^{-3}$ of Coulomb impurities. It is believed that a MBS device can function only if the amplitude of long-range fluctuations of the random Coulomb potential $\mathrm{\Gamma }$ is smaller than the subband gap $\mathrm{\Delta }$. Here we calculate $\mathrm{\Gamma }$ for recently experimentally studied large-dielectric-constant (Bi${}_{1-x}$Sb${}_x$)${}_2$Te${}_3$ wires in a small-dielectric-constant environment (no superconductor). We show that provided by such a dielectric-constant contrast, the confinement of electric field of impurities within the wire allows more distant impurities to contribute into $\mathrm{\Gamma }$, leading to $\Gamma \sim 3\Delta$. We also calculate a TI wire resistance as a function of the Fermi level and carrier concentration due to scattering on Coulomb and neutral impurities, and do not find observable discrete subband-spectrum related oscillations at $N\gtrsim {10}^{18}$ cm${}^{-3}$.