To Close or Not to Close: The Fate of the Superconducting Gap Across the Topological Quantum Phase Transition in Majorana-Carrying Semiconductor Nanowires

We investigate theoretically the low-energy physics of semiconductor Majorana wires in the vicinity of a magnetic field-driven topological quantum phase transition (TQPT). The local density of states at the end of the wire, which is directly related to the differential conductance in the limit of point-contact tunneling, is calculated numerically. We find that the dependence of the end-of-wire local density of states on the magnetic field is nonuniversal and that the signatures associated with the closing of the superconducting gap at the Majorana TQPT are essentially invisible within a significant range of experimentally relevant parameters. Our results provide a possible explanation for the recent observation of the apparent nonclosure of the gap at the Majorana TQPT in semiconductor nanowires.

Figure 1

Top: Total DOS as a function of $B$ for $\mu =\Delta /2$. The closing of the bulk gap is clearly visible at the TQPT, $B=B_c\approx 0.2\mathrm{T}$. Middle: LDOS at the end of the wire as a function of $B$. The strong features associated with $\Delta \sim 250\mu \mathrm{eV}$ are only weakly dependent on $B$. For $B>B_c$, a peak associated with the Majorana bound state is present at zero energy. The LDOS shows no visible signature of the bulk gap closing at the TQPT. Bottom: LDOS at the middle of the wire. Note the closure of the gap at the TQPT and the absence of the zero-energy Majorana peak. Traces for selected values of $B$ are provided in the Supplemental Material [21].

Figure 2

Top: Spectrum of an infinite nonsuperconducting wire. The lowest three bands (six subbands) cross the chemical potential at Fermi wave vectors $k_F^{n\pm }>0.1{\mathrm{nm}}^{-1}$, while the top occupied band has $k_F^{4\pm }\ll 0.1{\mathrm{nm}}^{-1}$. The fifth band is unoccupied. Bottom: BdG spectrum for three different values of $B$. The induced SC pair potential is $\Delta =0.25\mathrm{meV}$. The three lower-energy bands depend weakly on $B$ and have nearly overlapping contributions characterized by six minima at energies $\approx 250\mu \mathrm{eV}$. By contrast, the top occupied band is strongly $B$ dependent (three distinct curves at $k_x<0.02{\mathrm{nm}}^{-1}$) and is characterized by a gap that closes at the critical field $B_c\approx 0.2\mathrm{T}$. Inset: Relevant portion of the phase diagram showing the topological SC phase [light gray (yellow)] and the topologically trivial phase (white). Figure 1 corresponds to a cut with $\mu =\Delta /2$ (path I), Fig. 4 is for $\mu =2\Delta$ (path III), while Fig. 5 corresponds to $B(\mu )=B_c(\mu )/2$ (path II).

Figure 3

The lowest-energy BdG states associated with the Majorana band near the TQPT for different values of $\mu$. The dark gray (red) lines correspond to $B=0.9B_c$, while the light gray (yellow) curves are for $B=1.1B_c$. The insets show the BdG spectra for an infinite wire near $k_x=0$. In the topological SC phase ($B>B_c$), the lowest-energy state is the MF state. In the nontopological SC phase ($B<B_c$), there is a crossover from the region with $\mu >{\mu }_c\approx \Delta$ characterized by localized lowest-energy states (top panel) to the region with $\mu <{\mu }_c\approx \Delta$, where the states have mostly extended character (middle and bottom panels). The corresponding BdG spectra change from a structure with two minima ($\mu >{\mu }_c$) to a single minimum at $k_x=0$ ($\mu <{\mu }_c$).

Figure 4

Dependence of the LDOS at the end of the wire on $B$ for $\mu =2\Delta$. The gap-edge feature stemming from the localized end state clearly reveals the gap closing at the TQPT. For $B>B_c$ the finite-energy states are delocalized and, consequently, there is no feature showing the reopening of the gap. Traces for selected values of $B$ are provided in the Supplemental Material [21]

Figure 5

Total DOS (top) and LDOS at the end of the wire (bottom) as functions of $\mu$. The dependence on the chemical potential is along path II (see inset of Fig. 2). The gap is smaller than the pair potential $\Delta =250\mu \mathrm{eV}$, as shown by the DOS (top panel). In the LDOS, a feature associated with the gap edge becomes visible only in the localized region, $\mu >{\mu }_c\approx 250\mu \mathrm{eV}$ (blue dots in the inset of Fig. 2).