Soft superconducting gap in semiconductor-based Majorana nanowires

We develop a theory for the proximity effect in superconductor–semiconductor–normal-metal tunneling structures, which have recently been extensively studied experimentally, leading to the observation of transport signatures consistent with the predicted zero-energy Majorana bound states. We show that our model for the semiconductor nanowire having multiple occupied subbands with different transmission probabilities through the barrier reproduces the observed “soft-gap” behavior associated with substantial subgap tunneling conductance. We study the manifestations of the soft-gap phenomenon both in the tunneling conductance and in local density of states measurements and discuss the correlations between these two quantities. We emphasize that the proximity effect associated with the hybridization between low-lying states in the multiband semiconductor and the normal-metal states in the lead is an intrinsic effect leading to the soft-gap problem. In addition to the intrinsic contribution, there may be extrinsic effects, such as, for example, interface disorder, exacerbating the soft-gap problem. Our work establishes the generic possibility of an ubiquitous presence of an intrinsic soft gap in the superconductor–semiconductor–normal-metal tunneling transport conductance induced by the inverse proximity effect of the normal metal.

Figure 1

Differential conductance versus voltage for different values of the barrier strength $Z$. The curves, illustrating the main result of the BTK theory, correspond to zero temperature and are normalized by multiplying $dI/dV$ with the normal-state resistance $R_N$ (see Ref. [43]). In the strong-barrier limit ($Z\to \infty$) $dI/dV$ is proportional to the density of states in the superconductor, but the two quantities become completely unrelated in the weak-barrier regime.

Figure 2

Contribution of a hybridized SM state with bare energy $E_0=0$ to the LDOS integrated over the semiconductor nanoribbon volume in a SC-SM-NM structure (lateral view shown in the inset). (a) SM-SC coupling (${\gamma }_{\mathrm{SC}}\ne 0$, ${\gamma }_{\mathrm{NM}}=0$). The sharp peaks correspond to Bogoliubov quasiparticles with energies given by the induced SC gap ${\Delta }_{\mathrm{ind}}\approx 0.5{\Delta }_0$. (b) SM-NM coupling (${\gamma }_{\mathrm{SC}}=0$, ${\gamma }_{\mathrm{NM}}\ne 0$). Each peak corresponds to a SM-NM hybrid state having most of its weight inside the NM and a small tail inside the SM ribbon. (c) Proximity-coupled NM-SM-SC structure (${\gamma }_{\mathrm{SC}}\ne 0$, ${\gamma }_{\mathrm{NM}}\ne 0$). The hybrid states with energies within the induced SC gap are responsible for the soft-gap phenomenon. The weight of the corresponding peaks depends on the SM-NM coupling strength ${\gamma }_{\mathrm{NM}}$. (d) Same as in panel (c) but for a system with large SC and NM components (with quasicontinuum spectra). The green circles are given by an effective theory obtained by integrating out the SC and NM degrees of freedom (see Sec. 3).

Figure 3

(a) Schematic representation of a SM Majorana wire proximity coupled to a NM and a SC (top view). A barrier potential $V_b\left(x\right)$ is applied in the region of the wire that is not covered by the NM or the SC. (b) Spectrum of the SM wire in the absence of coupling (${\gamma }_{\mathrm{SC}\left(\mathrm{NM}\right)}=0$). The chemical potential is placed near the bottom of the third band. (c) Local density of states integrated over the barrier region (i.e., around $x=1\mu m$) for a wire coupled to the NM (only). The smooth background is due to states from the low-energy bands that penetrate through the barrier and hybridize strongly with NM states. The peaks correspond to states from the top occupied band that are confined to the right side of the barrier and hybridize very weakly with the NM.

Figure 4

Density of states (inside the SM wire) at zero Zeeman field in the presence of both NM and SC couplings. (a) Total DOS inside the SM wire. (b) LDOS integrated over the region covered by the NM contact. (c) LDOS integrated over the SC region. (d) LDOS integrated over the barrier region. Note that the sharp peaks are due states that are confined to the SC region (and penetrate slightly inside the barrier region). Parameters: ${\gamma }_{\mathrm{SC}}=0.5$ meV, ${\gamma }_{\mathrm{NM}}=0.1$ meV, $\Gamma =0$, and $V_b^{\max }=5$ meV.

Figure 5

Differential conductance and LDOS (integrated over the barrier region—$DO{S}_V$) calculated using the double-chain model. The sharp features represent the conductance of a system with a single occupied band ($n_b=1$) at zero temperature, $T=0$ (filled black lines). Notice the quantization of the Majorana-induced zero-bias peak. At finite temperature ($T=65$ mK), the sharp features (including the Majorana peak) become suppressed (red line). No background that could be associated with the soft-gap phenomenon exists for $n_b=1$. $DO{S}_V$ (not shown) has a similar structure. When two SM bands are occupied ($n_b=2$), both $dI/dV$ (red line with yellow filling) and $DO{S}_V$ (blue line) show a zero-bias peak on top of a substantial smooth background. The Majorana peak is strongly suppressed at finite temperature (here $T=65$ mK), but the background is practically independent of $T$. In all these calculations the Zeeman field is $\Gamma =0.4$ meV.

Figure 6

(a) Magnetic field dependence of the LDOS integrated over the barrier region for a system described by Eqs. (8), (10), and (11) with ${\gamma }_{\mathrm{NM}}=0.1$ meV, ${\gamma }_{\mathrm{SC}}=0.5$ meV, and $L_S=1.1\mu m$. (b) Differential conductance for a double-chain wire with ${\Delta }_{\mathrm{ind}}=0.25$ meV, $L_S=1.1\mu m$, and $T=0.65$ mK. The curves (shifted for clarity) correspond to Zeeman fields from 0 (bottom) to $0.8$ meV (top). A Majorana peak emerges above a certain critical field and splits at high Zeeman fields. Note the close correspondence between the two types of calculations which involve systems described by different models and having different parameters.

Figure 7

Dependence of the LDOS integrated over the barrier region (left) and of $dI/dV$ (right) on the barrier height $V_b^{\max }$. Note that increasing the barrier height reduces the amplitude of the zero-bias peak as the Majorana bound state penetrates less inside the barrier region and decouples from the lead. In addition, varying the barrier potential changes the energy of the delocalized states that extend throughout the wire and, as a result, modifies the profile of the soft gap. The finite-energy sharp features present near the induced SC gap edge (${\Delta }_{\mathrm{ind}}\approx 0.25$ meV, left panel) are due to the states from the second highest occupied band that become confined inside the superconducting region. On the other hand, the smooth low-energy maximum corresponding to $V_b^{\max }=30$ meV is generated by a state from the second highest occupied band that crosses the chemical potential and is confined inside the normal segment of the wire. The interband spacings are 35 meV (left) and 8 meV (right) and the Zeeman field is $\Gamma =0.4$ meV.

Figure 8

Comparison between the differential conductance characterizing two hybrid systems with ${\Delta }_{\mathrm{ind}}=0.25$ meV, $L_S=2\mu m$, $\Gamma =0.4$ meV and different potential barriers (see inset). The two barriers have the same transparency for the Majorana band, but significantly different transparencies for the low-energy occupied band. This results in the two Majorana ZBPs having the same weight and the soft gap being strongly suppressed in the system with a narrow barrier.