Magnetic-field instability of Majorana modes in multiband semiconductor wires

We investigate the occurrence of Majorana modes in semiconductor quantum wires in close proximity with a superconductor and when both Rashba interaction and magnetic field are present. We consider long, but finite, multiband wires (namely, planar wires with dimensions $L_x\gg L_y$). Our results demonstrate that interband mixing coming from the Rashba spin-orbit term hybridizes Majorana pairs originating from different transverse modes while simultaneously closing the effective gap. Consequently, multiple Majorana modes do not coexist in general. On the contrary, Majorana physics is robust provided that only one single transverse mode contributes to a Majorana pair. Finally, we analyze the robustness of Majorana physics with respect to magnetic orbital effects.

Figure 1

Sketch of a finite nanowire of dimensions $(L_x,L_y)$ in close proximity with an s-wave superconductor and in presence of a tilted magnetic field.

Figure 2

Results neglecting Rashba mixing, ${\alpha }_y=0$ in Eq. (4), and without magnetic orbital effects, for field orientation along $x$. (a) Spectrum of eigenvalues as a function of Zeeman energy ${\Delta }_B\equiv g{\mu }_{B}B/2$. Only the twelve eigenvalues closer to zero are displayed; higher or lower eigenvalues than the ones displayed correspond to bulk excitations and are not shown. Parameters: $L_y=150\mathrm{nm}$, $L_x=3\mu \mathrm{m}$, $\Delta =0.225\mathrm{meV}$, $\mu =0$, ${\alpha }_x=0.045\mathrm{eV}\mathrm{nm}$. (b) Probability density for the Majorana-like edge modes corresponding to the second and first transverse modes for a Zeeman energy of $2\mathrm{meV}$ in panel (a).

Figure 3

Same as Fig. 2 including Rashba mixing. The same parameters have been used, except for ${\alpha }_y=0.6{\alpha }_x$ in panel (a) and ${\alpha }_y={\alpha }_x$ in panel (b). The densities of panel (c) are for the three lowest modes when ${\Delta }_B=4\mathrm{meV}$.

Figure 4

(a) Energy of the lowest eigenvalue as a function of the field tilting angle (circles) and of the gap with the next state (squares). We have used a Zeeman energy of 0.45 meV. (b) Same as Fig. 2 for a field tilting angle of $1^{\circ }$ out of the plane. Same parameters as Fig. 3b.