Screening, Friedel oscillations and low-temperature conductivity in topological insulator thin films

In thin topological insulator films, the top and bottom surfaces are coupled by tunneling, which restores backscattering and strongly affects screening. We calculate the dielectric function in the random phase approximation obtaining a closed-form result. Unlike independent TI surfaces, the dielectric function of thin films exhibits a valley as a function of wavenumber $q$ and tunneling, as well as a cusp at $q=2k_F$, with $k_F$ the Fermi wave vector. As a result of the cusp, Friedel oscillations decay with distance $r$ as $\sin(2k_Fr)/(2k_Fr)^2$. We determine the longitudinal conductivity $\sigma$ in the first Born approximation at low temperatures where screened impurities provide the dominant scattering mechanism. At high electron densities $n_e$, $\sigma \propto n_e$, while at low densities $\sigma \propto n_e^{3/2}$.