Lifshitz transition in dirty doped topological insulator with nematic superconductivity

We study the effects of the Lifshitz transition from the closed to open Fermi surface in dirty topological insulators with nematic superconductivity near the critical temperature. We solve linearized Gor'kov equations and find that the nematic superconductor with an open Fermi surface has a lower critical temperature and is more susceptible to disorder than the superconductor with the closed Fermi surface. We propose that correspondence between the critical temperature and stability against the disorder is a general feature of the superconductivity. We investigate the effects of the Lifshitz transition on the competition between superconducting phases in a topological insulator. The open Fermi surface is beneficial for the nematic order parameter ${\mathrm{\ensuremath{\Delta}}}₄$ in competition with orbital-triplet ${\mathrm{\ensuremath{\Delta}}}₂$ and disfavors nematic state over the $s$-wave order parameter. We study Meissner currents in both clean and dirty limits. We found that transition from closed to open Fermi surface increases anisotropy of Meissner currents. Finite disorder suppresses superconducting density stronger than critical temperature. We compare our results with existing experimental data.