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Effect of disorder on Majorana localization in topological superconductors: a quasiclassical approach

Two dimensional topological superconductors (TS) host chiral Majorana modes (MMs) localized at the boundaries. In this work, within quasiclassical approximation we study the effect of disorder on the localization length of MMs in two dimensional spin-orbit (SO) coupled superconductors. We find nonmonotonic behavior of the Majorana localization length as a function of disorder strength. At weak disorder, the Majorana localization length decreases with an increasing disorder strength. Decreasing the disorder scattering time below a critical value $τ_c$, the Majorana localization length starts to increase. The critical scattering time depends on the relative magnitudes of the two ingredients behind TS: SO coupling and exchange field. For dominating SO coupling, $τ_c$ is small and vice versa for the dominating exchange field.