Even-odd effect and Majorana states in full-shell nanowires

Full-shell nanowires (semiconducting nanowires fully coated with a superconducting shell) have been recently presented as an alternative means to create Majorana zero modes. In contrast to partially coated nanowires, it has been argued that full-shell nanowires do not require high magnetic fields and low densities to reach a putative topological regime. Here we present a theoretical study of these devices taking into account all the basic ingredients, including a charge distribution spread across the section of the nanowire, required to qualitatively explain recent experimental results (Vaitiekėnas et al., arXiv:1809.05513). We derive a criterion, dependent on the even-odd occupation of the radial subbands with zero angular momentum, for the appearance of Majorana zero modes. In the absence of angular subband mixing, these give rise to strong zero-bias anomalies in tunneling transport in roughly half of the system's parameter space under an odd number of flux quanta. Due to their coexistence with gapless subbands, the zero modes do not enjoy generic topological protection. Depending on the details of subband mixing in realistic devices, they can develop a topological minigap, acquire a finite lifetime, or even be destroyed.

Figure 1

(a) Experimental results for differential tunneling conductance $dI/dV$ vs bias voltage $V$ and magnetic flux $\mathrm{\Phi }$ (or equivalently magnetic field $B$) into a solid-core full-shell superconductor-semiconductor nanowire in the destructive Little-Parks regime, taken from Ref. [46]. Zero-bias anomalies, revealing the presence of zero modes in the LDOS, are observed in the first Little-Parks lobes with $n=\pm 1$ fluxoid around the shell. (b),(c) Numerical simulation of the local density of states (LDOS) (in arbitrary units) in semi-infinite solid-core full-shell nanowires, both in the destructive (b) and weak/moderate (c) Little-Parks regime, showing similar phenomenology. Zero modes are absent around integer flux in simpler hollow-core full-shell nanowires (d),(e). Simulation parameters: $\mathrm{\Delta }=0.2$ meV, $\alpha =20$ meV nm; (b),(c) ${\lambda }_N=38$ nm, ${\lambda }_S=35$ nm, $R=80$ nm, and $R^2/\left(d\xi \right)=1.72$ (b) and 4.35 (c); (d),(e) ${\lambda }_N={\lambda }_S=61$ nm, $R=43$, $d\approx 0$, and $R/\xi =0.47$ (d) and 0.67 (e).

Figure 2

(a)–(c) LDOS at the end of a semi-infinite solid-core, full-shell nanowire as a function of energy $eV$ and flux $\mathrm{\Phi }$ in the first Little-Parks lobe (destructive regime). The Fermi wavelength of the superconducting shell ${\lambda }_S$ is fixed and the semiconducting core's ${\lambda }_N$ increases from (a) to (c). Log-scale line cuts of the LDOS at $\mathrm{\Phi }={\mathrm{\Phi }}_0$ are shown on the right, resolved by sectors of different angular momentum $m_j$. The line cuts show that black regions in the density plots have a small nonzero LDOS background coming from gapless $m_j\ne 0$ subbands, not visible with the chosen color scaling (as also happens in the experiment of Ref. [46]). Note the appearance of a quasibound zero mode throughout the lobe in (b) and (c) that was split in (a). The contributions to the LDOS from sectors with different $m_j$ show that the zero mode corresponds to the $m_j=0$ sector (in red). It arises as one $m_j=0$ subband undergoes an inversion at $k_z=0$, as shown in the Nambu band structure in panels (d)–(f), thus becoming topologically nontrivial when considered on its own. The emergence of the $m_j=0$ Majorana zero mode correlates with an odd occupancy of the $m_j=0$ radial subbands in the normal phase, panels (g)–(i). Parameters are as in Fig. 1 except for $R=100$ nm and ${\lambda }_S=24$ nm.

Figure 3

(a),(b) Phase diagram of a $\mathrm{\Phi }={\mathrm{\Phi }}_0$ solid-core, full-shell nanowire of radius $R$, vs $R/{\lambda }_{N,S}$, where ${\lambda }_{N,S}$ are Fermi wavelengths in the semiconductor core and superconductor shell, respectively. Panel (a) focuses on low densities while (b) shows a wider range. The thick orange lines in (a) mark the boundaries of regions with a topologically nontrivial $m_j=0$ subband with Majoranas. These are computed using exact tight-binding simulations with finite $\mathrm{\Delta }$. Blue regions in (a),(b) correspond to an odd occupancy of the $m_j=0$ normal-phase radial subbands, computed using wave matching at the core-shell boundary for $\mathrm{\Delta }=\alpha =0$. Black and gray lines correspond to two analytical approximations for the even-odd boundaries; see Eqs. (B10) and (B11). (c)–(f) LDOS as a function of energy $eV$ and subband-mixing strength $\eta$, starting at different points in the phase diagram, colored squares in (a). Note that the color scale is zoomed in around zero conductance with respect to Figs. 1 and 2 to resolve the small background LDOS. (d) Line cuts of (c) that emphasize the opening and subsequent band inversion of a minigap induced by mixing, which in our model eventually results in the destruction of the Majorana state. Parameters: (a) $R=100$ nm, $d=100$ nm, $\mathrm{\Delta }=0.2$ meV, $\alpha =10$ meV nm (orange curve); (b) $R=65\mathrm{nm}$, $d=28$ nm.

Figure 4

The LDOS at the end of a semi-infinite full-shell nanowire at $\mathrm{\Phi }={\mathrm{\Phi }}_0$, truncated to the $m_j=0,\pm 1$ subspace, as a function of the coupling $\eta$ between $m_j$ bands. The coupling is restricted to within 10 nm of the end of the nanowire, where the termination of the superconducting shell exposes the semiconductor to external perturbations. The $m_j=0$ Majorana state is coupled by the local $\eta$ to the gapless $m_j\ne 0$ bands in the nanowire bulk, which leads to broadening and decay in the limit of a semi-infinite nanowire.