Evidence for an impurity band in the anomalous low carrier density superconductor Pb$_{1-x}$Tl$_x$Te

The narrow-band-gap semiconductor PbTe superconducts with a Tc an order of magnitude greater than comparable low-density metals, but only when it is doped with the specific element thallium. Here we present a comprehensive study of the evolution of the Fermi surface of hole doped PbTe as derived from Shubnikov de Haas quantum oscillations, combined with measurements of the evolution of the electrical transport and specific heat. We compare cases for dopants that cause superconductivity (thallium) and those that don’t (sodium), and identify the presence of a Tl-impurity band as the key feature associated with the presence of superconductivity. These results give the clearest description of the fermiology of PbTe to date, relevant for understanding both to its superconducting ground state and its high thermoelectric figure of merit.