Effects of large induced superconducting gap on semiconductor Majorana nanowires

With the recent achievement of extremely high-quality epitaxial interfaces between InAs nanowires and superconducting Al shells with strong superconductor-semiconductor tunnel coupling, a new regime of proximity-induced superconductivity in semiconductors can be explored where the induced gap may be similar in value to the bulk Al gap (large gap) with negligible subgap conductance (hard gap). We propose several experimentally relevant consequences of this large-gap strong-coupling regime for tunneling experiments, and we comment on the prospects of this regime for topological superconductivity. In particular, we show that the advantages of having a strong spin-orbit coupling and a large spin $g$ factor in the semiconductor nanowire may both be compromised in this strongly coupled limit, and somewhat weaker interface tunneling may be necessary for achieving optimal proximity superconductivity in the semiconductor nanowire. We derive a minimal, generic theory for the strong-coupling hard-gap regime obtaining good qualitative agreement with the experiment and pointing out future directions for further progress toward Majorana nanowires in hybrid semiconductor-superconductor structures.

Figure 1

Schematic illustration of the differences between hard and soft gaps in the context of proximity superconductivity. The soft gap tunnel conductance has several features common across several experiments: a clear but broad maximum (at $\pm {\mathrm{\Delta }}_1$) associated with the coherence peak, and a V-shaped subgap conductance with a minimum at zero energy where the conductance is still a substantial fraction of the normal state conductance. For comparison, we show a characteristic hard gap tunnel conductance, similar to that seen in Ref. [26]. Here there are very sharp coherence peaks at the induced gap scale $\pm {\mathrm{\Delta }}_2$, with negligible subgap conductance. The measured zero-field zero-bias conductance scales in the hard gap [26] and the soft gap [15] systems differ by an order of magnitude (being 0.03 and 0.3 roughly in the units of $e^2/h$ for Refs. [26] and [15], respectively).

Figure 2

(a) Expected tunneling densities of states in the Maki formalism at values $\alpha$ corresponding to the values $B/B_c$ in Fig. 4 of Ref. [26]. Inset is a schematic drawing of the tunneling experiment geometry from Ref. [26]. An InAs core (green) is symmetrically coated with Al (gray); part of the coating is etched away and the wire is contacted by a normal metal (yellow). An additional superconducting lead contacts the shell. The magnetic field is aligned along the wire axis. (b) Pair potential $\mathrm{\Delta }$ and spectral gap $E_g$ as a function of the pair-breaking parameter $\alpha$ in Maki's theory. As $\alpha$ goes to ${\mathrm{\Delta }}_{00}/2$, superconductivity is ultimately destroyed (indicated by the vanishing pair potential), but prior to this the gap in the density of states vanishes. The symbols are estimated from tunneling conductance data reported in Refs. [42] (triangles) and [26] (squares). Below (b) is a plot of the same quantities for the purely Zeeman-driven transition, relevant to topological superconductivity.

Figure 3

Local density of states integrated over a small region near the end of a SM wire–SC structure. The weight function decays exponentially on a scale ${\delta }_{\mathrm{SC}}=8a$ and ${\delta }_{\mathrm{SM}}=10a$, with $a$ the lattice spacing. The low-energy states are confined within the proximity-coupled structure by a Gaussian potential barrier of height $V_b$. The SM-SC coupling is controlled by the hopping parameter $t_{\gamma }$. The values of the parameters are in units of ${\mathrm{\Delta }}_0$. Red (dark gray) and yellow (light gray) correspond to contributions to the integrated LDOS from wire sites and SC sites, respectively. From top to bottom the panels correspond to weak, intermediate, and strong coupling regimes. Increasing the potential barrier height reduces the relative weight of the SM wire contributions, including the induced gap feature.

Figure 4

Poles of the single-particle Green's function on the quantum dot, corresponding to spin-split Bogoliubov quasiparticle and quasihole states. For weak coupling to the superconductor, $\gamma /{\mathrm{\Delta }}_0\ll 1$, these appear near the energies associated with the regular spin-split states of $H_0$ at $\pm b$. As the coupling is made stronger, superconductivity dominates and the effect of the original spin splitting is weakened.

Figure 5

Effective Hamiltonian parameters with increasing QD-SC coupling. (a) The effective Zeeman energy $b_{\mathrm{eff}}$ and (b) the induced pair potential ${\mathrm{\Delta }}_{\mathrm{ind}}$. In both cases the solid red line is obtained from numerical solution of the BdG equation, while the dashed black line corresponds to the static approximation $\omega \to 0$.