Proximity-induced low-energy renormalization in hybrid semiconductor-superconductor Majorana structures

A minimal model for the hybrid superconductor-semiconductor nanowireMajorana platform is developed that fully captures the effects of the low-energy renormalization of the nanowire modes arising from the presence of the parent superconductor. In this model, the parent superconductor is an active component that participates explicitly in the low-energy physics, not just a passive partner that only provides proximity-induced Cooper pairs for the nanowire. This treatment on an equal footing of the superconductor and the semiconductor has become necessary in view of recent experiments, which do not allow a consistent interpretation based just on the bare semiconductor properties. The general theory involves the evaluation of the exact semiconductor Green's function that includes a dynamical self-energy correction arising from the tunnel-coupled superconductor. Using a tight-binding description, the nanowire Green's function is obtained in various relevant parameter regimes, with the parent superconductor being treated within the BCS-BdG prescription. General conditions for the emergence of topological superconductivity are worked out for single-band as well as multiband nanowires and detailed numerical results are given for both infinite and finite wire cases. The topological quantum phase diagrams are provided numerically and the Majorana bound states are obtained along with their oscillatory energy-splitting behaviors due to wave function overlap in finite wires. Renormalization effects are shown to be both qualitatively and quantitatively important in modifying the low-energy spectrum of the nanowire. The results of the theory are found to be in good qualitative agreement with Majorana nanowire experiments, leading to the conclusion that the proximity-induced low-energy renormalization of the nanowire modes by the parent superconductor is of fundamental importance in superconductor-semiconductor hybrid structures, except perhaps in the uninteresting limit of extremely weak superconductor-semiconductor tunnel coupling. Implications of the general theory for obtaining true zero-energy topological Majorana modes are pointed out.