Search for Majorana Fermions in Multiband Semiconducting Nanowires

We study multiband semiconducting nanowires proximity-coupled with an $s$-wave superconductor. We show that, when an odd number of subbands are occupied, the system realizes a nontrivial topological state supporting Majorana modes. We study the topological quantum phase transition in this system and calculate the phase diagram as a function of the chemical potential and magnetic field. Our key finding is that multiband occupancy not only lifts the stringent constraint of one-dimensionality but also allows one to have higher carrier density in the nanowire, and as such multisubband nanowires are better suited for observing the Majorana particle. We study the robustness of the topological phase by including the effects of the short- and long-range disorder. We show that there is an optimal regime in the phase diagram (“sweet spot”) where the topological state is to a large extent insensitive to the presence of disorder.

Figure 1

(a) Schematic plot of the quasi-1D nanowire proximity-coupled with an $s$-wave SC. The rectangular quantum well has the dimensions $L_z$, $L_y$, and $L_x$: $L_z\ll L_y\ll L_x$. The nanowire can be top gated to control chemical potential in it. The method for fabricating the proposed quantum well heterostructure based on InAs has been demonstrated; see, e.g., Refs. 12, 18.(b) Schematic plot of the lowest energy subbands due to the transverse confinement.

Figure 2

(a) Phase diagram for the two-band nanowire model as a function of the chemical potential $\tilde{\mu }$ and external magnetic field ${\tilde{V}}_x$. The width of the topological region is largest at the sweet spot: $\tilde{W}\approx 60$. Here the tilde denotes rescaled energy $\tilde{E}=E/m^{*}{\alpha }^2$ and ${\tilde{\Delta }}_{12}=4$. The light and dark regions correspond to topologically trivial and nontrivial phases. (b),(c) Quasiparticle excitation gap obtained by using Eq. (6) as a function of ${\tilde{V}}_x$ and $\tilde{\mu }$. The solid (red) and dashed (blue) lines correspond to ${\tilde{\Delta }}_{12}=4$ and ${\tilde{\Delta }}_{12}=1$, respectively. The closing of the gap for ${\tilde{\Delta }}_{12}=4$ (solid red line) is consistent with the phase diagram shown in (a). The quasiparticle excitation gap at the sweet spot strongly depends on the magnitude of ${\Delta }_{12}$. We assume here $m^{*}=0.04m_e$ with $m_e$ being electron mass and $\alpha =0.1\mathrm{eV}\AA$ yielding $m^{*}{\alpha }^2\approx 0.6\mathrm{K}$. We used realistic parameters $L_y=130\mathrm{nm}$ and ${\tilde{E}}_{\mathrm{sb}}=30$, ${\tilde{E}}_{\mathrm{bm}}=5$, and ${\tilde{\Delta }}_{11}={\tilde{\Delta }}_{22}=4$.

Figure 3

(a),(b) Energy spectrum ${\tilde{E}}_m$ for a finite-size nanowire obtained by numerical diagonalization of $H_{\mathrm{tot}}=H_{\mathrm{SM}}+H_{\mathrm{SC}}$ for $\tilde{\mu }=15$, ${\tilde{V}}_x=15$, and ${\tilde{V}}_x\approx 8$, respectively. Here $\tilde{E}=E/m^{*}{\alpha }^2$, and $m$ labels eigenvalues of $H_{\mathrm{tot}}$. Inset: Lowest-lying energy states. Majorana zero-energy modes are present in (a) and disappear in (b). (c),(d) Energy spectra for a given disorder realization with impurity potentials shown at the bottom of (c) and (d) corresponding to $\lambda =16\mathrm{nm}$ and $\lambda =260\mathrm{nm}$, respectively. The dark (light) colors denote positive (negative) $U_{0j}$. (c) The spectrum at the sweet spot for the short-range disorder shown at the bottom (${\tilde{U}}_0=100$). Inset: Lowest-lying eigenstates. The topological phase is robust against short-range disorder; i.e., the disorder affects extended states but leaves Majorana modes intact. (d) The spectrum at the sweet spot with long-range disorder for ${\tilde{U}}_0=100$ (squares) and ${\tilde{U}}_0=25$ (diamonds). The impurity potential $U_{\mathrm{imp}}$ is shown at the bottom of the panel. Inset: Lowest-lying eigenstates for ${\tilde{U}}_0=100$. The topological phase is stable as long as $U_0<W/2$. For ${\tilde{U}}_0=100$ additional Majorana modes are localized at the impurities, and the topological phase collapses by splitting into fragments of topological and nontopological regions; see the inset. Both types of disorder lead to the emergence of the additional subgap states localized at the ends. Here we used parameters specified in Fig. 2.