The vortices of two-dimensional chiral $p$-wave superconductors are predicted to exhibit some exotic behaviors; one of their curious features is the existence of two types of vortices (each vortex has vorticity either parallel or antiparallel to the Cooper pairs chirality) and the robustness of the antiparallel vortices against nonmagnetic Born-like impurities. In this work, we study the impurity effect on the vortex of the chiral $p$-wave superconductors through the quasiclassical Greens function formalism. We take account of impurities via the self-consistent $t$-matrix approximation so that we can deal with strong as well as Born-like (i.e., weak) scatterers. We found that the spectrum is heavily broadened when the phase shift $delta_0$ of each impurity exceeds a critical value $delta_{text{c}}$ above which the impurity band emerges at the Fermi level. We also found a quantitative difference in the impurity effects on the two types of vortex for $delta_0<delta_{text{c}}$. Part of the numerical results for $delta_0<delta_{text{c}}$ can be understood by a variant of the analytical theory of Kramer and Pesch for bound states localized within vortex cores.
Download