Topological s-Wave Pairing Superconductivity with Spatial Inhomogeneity: Midgap-State Appearance and Robustness of Superconductivity

We study the quasiparticle spectrum of 2D topological s-wave superconductors in the presence of spatial inhomogeneity. Solving the real-space Bogoliubov-de Gennes equations, we examine excitations within a superconducting gap amplitude, i.e., the appearance of midgap states. The model of spatial inhomogeneity is to add potential to a uniform system. Two types of setting, i.e., line-type (a chain of impurities) and point-type (a single impurity) potentials are examined. The line-type setting shows the link of midgap states with gapless surface modes in topological superfluid. The point-type one shows that quasiparticles with a midgap energy are much easily excited by an impurity, increasing a Zeeman magnetic field with a topological number unchanged. Thus, we obtain insights into the robustness of the 2D topological superconductors against nonmagnetic impurities. Moreover, we derive an effective theory applicable to high magnetic fields. The effective gap function is a mixture of chiral p-and s-wave characters. The former is predominant when the Zeeman magnetic field increases. Therefore, we claim that the chiral p-wave character of the effective gap function creates midgap states.