Disorder effects in topological insulator nanowires

Abstract

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 $N\ensuremath{\sim}{10}^{19}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ 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 $\mathrm{\ensuremath{\Gamma}}$ is smaller than the subband gap $\mathrm{\ensuremath{\Delta}}$. Here we calculate $\mathrm{\ensuremath{\Gamma}}$ for recently experimentally studied large-dielectric-constant ${({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{2}{\mathrm{Te}}_{3}$ 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 $\mathrm{\ensuremath{\Gamma}}$, leading to $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}3\mathrm{\ensuremath{\Delta}}$. 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 $N\ensuremath{\gtrsim}{10}^{18}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$.