Tunneling conductance for Majorana fermions in spin-orbit coupled semiconductor-superconductor heterostructures using superconducting leads

It has been recently pointed out that the use of a superconducting (SC) lead instead of a normal metal lead can suppress the thermal broadening effects in tunneling conductance from Majorana fermions, helping reveal the quantized conductance of $2{e}²/h$. In this paper we discuss the specific case of tunneling conductance with SC leads of spin-orbit coupled semiconductor-superconductor (SM-SC) heterostructures in the presence of a Zeeman field, a system which has been extensively studied both theoretically and experimentally using a metallic lead. We examine the $dI/dV$ spectra using a SC lead for different sets of physical parameters including temperature, tunneling strength, wire length, magnetic field, and induced SC pairing potential in the SM nanowire. We conclude that in a finite wire the Majorana splitting energy $\mathrm{\ensuremath{\Delta}}E$, which has nontrivial dependence on these physical parameters, remains responsible for the $dI/dV$ peak broadening, even when the temperature broadening is suppressed by the SC gap in the lead. In a finite wire the signatures of Majorana fermions with a SC lead are oscillations of quasi-Majorana peaks about bias $V=\ifmmode\pm\else\textpm\fi{}{\mathrm{\ensuremath{\Delta}}}_{\text{lead}}$, in contrast to the case of metallic leads where such oscillations are about zero bias. Our results will be useful for analysis of future experiments on SM-SC heterostructures using SC leads.