Abstract
We theoretically consider the substrate-induced Majorana localization length renormalization in nanowires in contact with a bulk superconductor in the strong tunnel-coupled regime, showing explicitly that this renormalization depends strongly on the transverse size of the one-dimensional nanowires. For metallic (e.g. Fe on Pb) or semiconducting (e.g. InSb on Nb) nanowires, the renormalization effect is found to be very strong and weak, respectively, because the transverse confinement size in the two situations happens to be 0.5 nm (metallic nanowire) and 20 nm (semiconducting nanowire). Thus, the Majorana localization length could be very short (long) for metallic (semiconducting) nanowires even for the same values of all other parameters (except for the transverse wire size). We also show that any tunneling conductance measurements in such nanowires, carried out at temperatures and/or energy resolutions comparable to the induced superconducting energy gap, cannot distinguish between the existence of the Majorana modes or ordinary subgap fermionic states since both produce very similar broad and weak peaks in the subgap tunneling conductance independent of the localization length involved. Only low temperature (and high resolution) tunneling measurements manifesting sharp zero bias peaks can be considered to be signatures of Majorana modes in topological nanowires.
Highlights
Majorana fermions (MF), which were proposed theoretically 80 years ago as real solutions of the Dirac equation in the context of understanding neutrinos[1], have recently found their incarnations in solid state systems[2,3,4,5] as zero-energy localized excitations in topological superconductors (TS)
The question arises whether the MF localization length formula is modified from the simple coherence length formula, or equivalently, whether the Fermi velocity and/or the appropriate nanowire superconducting gap are renormalized by the substrate. This issue was discussed by Sau et al.[27] and Stanescu et al.[28] some years ago in the context of 2D sandwich structures involving semiconductor/superconductor and topologicalinsulator/superconductor heterostructures, and very recently by Peng et al.[29] in the context of 1D ferromagnetic nanowire on superconductor hybrid structures used in the recent Princeton STM experiment21. (The actual system theoretically considered by Peng et al.[29] is a helical magnetic adatom chain, not a ferromagnetic chain, on a superconducting substrate.) In the first part of the current work, we theoretically study the MF localization question in depth for 1D topological nanowire hybrid systems, discussing the substrate-induced MF renormalization for both semiconductor and ferromagnetic nanowires on an equal footing, comparing and contrasting the two situations
Our results for the temperature dependence of MF induced zero-bias conductance peaks (ZBCP) in the ferromagnetic nanowires agree completely with the results presented in Ref.[31], but what is new in our current work is showing that non-MF subgap states may lead to tunneling conductance features which are indistinguishable from the corresponding MF features in high temperature experiments
Summary
Majorana fermions (MF), which were proposed theoretically 80 years ago as real solutions of the Dirac equation in the context of understanding neutrinos[1], have recently found their incarnations in solid state systems[2,3,4,5] as zero-energy localized excitations in topological superconductors (TS). The actual MF localization length question is of great importance to the experimental observations in[21] since the estimated TS gap in the Fe adatom chains on Pb substrates studied therein is very small (∼ 0.1 meV) leading to a rather long coherence length (or equivalently MF localization length) of > 100 nm (assuming no substrateinduced renormalization) which would be much larger than the typical length of the adatom chains (5 − 50 nm) used in Ref.[21] In such a situation, the TS system is not in the exponentially protected regime at all, and the two end MFs should hybridize strongly leading to ordinary uninteresting fermionic states at high energies. Since the state of the arts low-temperature STM experiments are routinely carried out at ∼ 100 mK or below[42] (a temperature regime accessible since the early 1990s43), we urge future STM low temperature experiments (< 300 mK) in Fe chains on superconducting Pb substrates to settle the important question of the existence or not of MFs in this system
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