Two-dimensional surface charge transport in topological insulators

TL;DR

This paper develops a quantum theory for charge transport on the surface of three-dimensional topological insulators, analyzing impurity effects, scattering mechanisms, and how these influence conductivity and relaxation times.

Contribution

It provides a comprehensive quantum Liouville equation-based model for surface state conductivity, including impurity scattering and polarization effects, with detailed analysis of different scattering contributions.

Findings

Conductivity is inversely proportional to impurity density.

Different scattering mechanisms lead to distinct carrier density dependencies.

The ratio of relaxation times reveals dominant scattering angles.

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 electron doping is such that the Fermi energy $\epsilon_F$ and transport scattering time $\tau$ satisfy $\epsilon_F \tau/\hbar \gg 1$, but sufficiently low that $\epsilon_F$ 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 accurately to linear order in the impurity density. We consider scattering by charged impurities and short-range scatterers such as surface roughness. We calculate also the polarization function in topological insulators, emphasizing the differences from graphene. We find that the main contribution to the conductivity is $\propto n_i^{-1}$, where $n_i$ is the impurity density, and will have different carrier density dependencies for different forms of scattering. Two different contributions to this conductivity are traced to the scalar and spin-dependent terms in the Hamiltonian and their relative weight depends on the doping density. Our results contain all contributions to the conductivity to orders zero and one in the impurity density. We discuss also a way to determine the dominant scattering angles by studying the ratio of the transport relaxation time to the Bloch lifetime as a function of the Wigner-Seitz radius $r_s$. We also discuss the effect on the surface states of adding metallic contacts.