Two-dimensional surface charge transport in topological insulators

Abstract

We construct a theory of charge transport by the surface states of topological insulators in three dimensions. The focus is on the experimentally relevant case when the Fermi energy ${\ensuremath{\epsilon}}_{F}$ and transport scattering time $\ensuremath{\tau}$ satisfy ${\ensuremath{\epsilon}}_{F}\ensuremath{\tau}/\ensuremath{\hbar}⪢1$ but ${\ensuremath{\epsilon}}_{F}$ lies below the bottom of the conduction band. Our theory is based on the spin density matrix and takes the quantum Liouville equation as its starting point. The scattering term is determined to linear order in the impurity density ${n}_{i}$ and explicitly accounts for the absence of backscattering while screening is included in the random-phase approximation. The main contribution to the conductivity is $\ensuremath{\propto}{n}_{i}^{\ensuremath{-}1}$ and has different carrier density dependencies for different forms of scattering while an additional contribution is independent of ${n}_{i}$. The dominant scattering angles can be inferred by studying the ratio of the transport time to the Bloch lifetime as a function of the Wigner-Seitz radius ${r}_{s}$. The current generates a spin polarization that could provide a smoking-gun signature of surface state transport. We also discuss the effect on the surface states of adding metallic contacts.