Enhanced Zero-Bias Majorana Peak in the Differential Tunneling Conductance of Disordered Multisubband Quantum-Wire/Superconductor Junctions

A recent experiment Mourik et al. [Science 336, 1003 (2012)] on InSb quantum wires provides possible evidence for the realization of a topological superconducting phase and the formation of Majorana bound states. Motivated by this experiment, we consider the signature of Majorana bound states in the differential tunneling conductance of multisubband wires. We show that the weight of the Majorana-induced zero-bias peak is strongly enhanced by mixing of subbands, when disorder is added to the end of the quantum wire. We also consider how the topological phase transition is reflected in the gap structure of the current-voltage characteristic.

Figure 1

(a) Setup of multisubband quantum nanowire (NW) with gate-induced tunnel barrier (G) and proximity coupled $s$-wave superconductor (S). As in Ref. 1 we consider the conductance between the normal lead (N) and the superconductor. Subband mixing is induced through disorder in the short segment of length $L$ between the tunnel barrier and superconductor. (b) Normal-state dispersion in the absence of disorder for four subbands with $B=1\mathrm{meV}$ and $m{\alpha }^2/2=50\mu \mathrm{eV}$.

Figure 2

(a) Zero-bias conductance peak at zero temperature in a quantum wire with $B=0.5\mathrm{meV}$ and $N=4$ transverse subbands, one of which is in the topological phase with barrier transmission $T_4=0.01$. The three nontopological subbands have transmissions $20T_4$, $10T_4$, and $4T_4$. The red curves show the conductance for four different disorder configurations with $l=10L$. The black dashed line shows the peak shape for the clean wire. (b) Same as in (a), but for a temperature $T=60\mathrm{mK}$, larger than the zero-temperature peak width.

Figure 3

Probability distribution of the zero-bias peak width $\Gamma$ in the presence of disorder in the nanowire segment $-L<x<0$ between the superconducting part and the barrier for a multichannel wire with the same choice of parameters as in Fig. 2. With increasing disorder, $\Gamma$ increases on average (red and green curve) due to subband mixing. For $L\gg l$ Anderson localization reduces the overall transparency of the junction, causing $\Gamma$ to decrease again in the case of very strong disorder (blue curve).

Figure 4

Ensemble average of the contribution ${\Gamma }_1$ from disorder-induced subband mixing to the width $\Gamma$ of the zero-bias peak as a function of disorder strength in the segment $-L<x<0$. The peak width is normalized by the normal state conductance $G_B\equiv (2e^2/h)N{T}_B$ to focus on the effects of subband mixing and to eliminate changes in the overall transparency by Anderson localization. Inset: Contribution ${\Gamma }_2$ to the peak width from lateral spin-orbit coupling for a rectangular barrier (red crosses) and a Gaussian barrier (blue dashed line). In both figures the parameters of the barrier potential have been chosen such that only $T_1$ differs appreciably from zero.

Figure 5

(a) Differential conductance versus bias voltage in a clean multichannel nanowire for increasing $B$ from 0 to 0.5 meV (750 mT in InSb) in steps of 0.02 meV with the realistic parameters [1] ${\alpha }_y=0$, $T=60\mathrm{mK}$ ($k_{B}T=5\mathrm{meV}$), $L=10\mathrm{nm}$, and $T_N=0.01$. The $B>0$ traces are offset vertically for clarity. The formation of a Majorana fermion is reflected in the emergence of a zero-bias peak. The corresponding closing of the topological gap is hardly discernible due to the low transparency of the topological channel. For $B=0$ there are coherence peaks at the proximity induced gap $\Delta =0.25\mathrm{meV}$. For larger Zeeman fields the bulk gap of the lower channels is decreased consistently with expectations. (b) Same as (a) but with weak disorder in the region $-L<x<0$ adjacent to the barrier. All traces are calculated for the same disorder configuration with a scattering length $l=10L$. The zero-bias peak and the signature of the topological gap closing are considerably enhanced.