Majorana fermions in superconducting nanowires without spin-orbit coupling

We study nanowires with proximity-induced $s$-wave superconducting pairing in an external magnetic field that rotates along the wire. Such a system is equivalent to nanowires with Rashba-type spin-orbit coupling, with strength proportional to the derivative of the field angle. For realistic parameters, we demonstrate that a set of permanent magnets can bring a nearby nanowire into the topologically nontrivial phase with localized Majorana modes at its ends. This occurs even for a magnetic field configuration with nodes along the wire and alternating sign of the effective Rashba coupling.

Figure 1

Sketch of a quantum wire subjected to a spatially varying magnetic field due to a set of magnetic gates deposited on the substrate.

Figure 2

Sketch of two realistic systems having a series of permanent magnets with either alternating (left) or parallel (right) magnetization directions. The top panels show the field lines, and it is clearly seen that the field changes direction a number of times along the wire. The bottom panels show the amplitude of the magnetic field $B/B_0$ in red (light gray), as well as the effective induced spin-orbit coupling ${\alpha }_{\mathrm{eff}}$ in blue (dark gray) (calculated with effective mass $m=0.014m_e$). The overall scale for the magnetic field $B_0$ is set by the magnetization of the magnets. The magnets have widths 600 nm, heights 330 nm, and the gap between them is 200 nm. For the left configuration the distance to the wire is 100 nm, and 50 nm for the parallel configuration.

Figure 3

Phase diagram for the superconducting nanowire subject to the magnetic field in the two configurations shown in Fig. 2, calculated for a 5-$\mu$m-long wire with $m=0.014m_e$ and $\Delta =0.3$ meV. The topologically nontrivial phase occurs when the determinant of the reflection changes from 1 to $-1$, which is shown in the phase diagram in (a) for changing magnetic field $B$ and chemical potential $\mu$ and also in the lower panels along the dashed (red) line $\mu =0.3\Delta$. The lower panels show the lowest positive eigenvalue $E_0$, which goes to zero after the topological phase transition, as well as the next positive eigenvalue $E_1$. After the transition the difference between the two is equal to the gap of the topologically nontrivial state. Both configurations in Fig. 2 are seen to exhibit a phase transition. In (b) and (c) the lines in red (light gray) show the value of the determinant of the reflection matrix.

Figure 4

The localized Majorana modes and their spin texture. The top panel shows the total weight of the end Majorana modes, ${\gamma }_a$ and ${\gamma }_b$, while the lower panel depicts the local spin direction for the configuration of alternating magnetizations (Fig. 2 left). The length of the arrows in the inset corresponds to the amplitude of the wave function. Notice that the $x$ axis of the inset is rescaled.