Electrical Manipulation of Majorana Fermions in an Interdigitated Superconductor-Ferromagnet Device

We show that a topological phase supporting Majorana fermions can form in a two-dimensional electron gas (2DEG) adjacent to an interdigitated superconductor-ferromagnet structure. An advantage of this setup is that the 2DEG can induce the required Zeeman splitting and superconductivity from a single interface, allowing one to utilize a wide class of 2DEGs including the surface states of bulk InAs. We demonstrate that the interdigitated device supports a robust topological phase when the finger spacing $\lambda$ is smaller than half of the Fermi wavelength ${\lambda }_F$. In this regime, the electrons effectively see a “smeared” Zeeman splitting and pairing field despite the interdigitation. The topological phase survives even in the opposite limit $\lambda >{\lambda }_F/2$, although with a reduced bulk gap. We describe how to electrically generate a vortex in this setup to trap a Majorana mode, and predict an anomalous Fraunhofer pattern that provides a sharp signature of chiral Majorana edge states.

Figure 1

Schematic of the proposed interdigitated superconductor-ferromagnet 2DEG architecture. The device supports a robust topological phase when the finger spacing $\lambda$ is smaller than half of the Fermi wavelength (i.e., $\lambda <{\lambda }_F/2$). A topological phase can also appear in the regime $\lambda >{\lambda }_F/2$, although generally with a suppressed gap. Table  provides values of ${\lambda }_F$ for select 2DEGs when the chemical potential is set to $\mu =0$.

Figure 2

Bulk quasiparticle spectrum versus $k_x$ in a uniform structure (dashed curve) and interdigitated device with ${\lambda }_F/\lambda =2.6$ (solid curve). In both cases we use parameters $k_y=0$, $\mu =0$, ${\overline{V}}_z=1.5\overline{\Delta }$, and $m{\alpha }^2=3\overline{\Delta }$. The arrows indicate the pairing gap at the Fermi momentum $k_F\equiv \frac{2\pi }{{\lambda }_F}\approx 2m\alpha /{\hslash }^2$ and the degeneracy gap opened at $k_x=\frac{\pi }{\lambda }$ due to the interdigitation. Note that the excitation spectra for the uniform and interdigitated systems differ appreciably only at rather higher energies here.

Figure 3

Quasiparticle spectrum in various regimes for an interdigitated device with periodic boundary conditions along $y$ but open boundary conditions along $x$. In all parts we take $\mu =0$ and $m{\alpha }^2=3\overline{\Delta }$, while the finger spacing and Zeeman energy vary as (a) ${\lambda }_F/\lambda =2.5$, ${\overline{V}}_z=0.5\overline{\Delta }$, (b) ${\lambda }_F/\lambda =2.5$, ${\overline{V}}_z=2\overline{\Delta }$, and (c) ${\lambda }_F/\lambda =1.5$, ${\overline{V}}_z=2\overline{\Delta }$. A trivial state appears in (a) while the larger Zeeman field in (b) drives a topological phase supporting chiral Majorana edge states within the bulk gap. Interestingly, the topological phase and associated Majorana edge states survive even in (c) despite the relatively small ratio of ${\lambda }_F/\lambda$.

Figure 4

Phase diagrams for $\mu =0$ and spin-orbit energies (a) $m{\alpha }^2=3.2\overline{\Delta }$ and (b) $m{\alpha }^2=1.3\overline{\Delta }$. The horizontal axis represents the ratio of the Fermi wavelength ${\lambda }_F$ to the finger spacing $\lambda$, while the vertical axis is the Zeeman energy normalized by pairing strength. The shading indicates the bulk gap $\delta$ normalized by $\overline{\Delta }$. Red dashed lines denote the boundary between topological and trivial phases.

Figure 5

(a) Scheme to electrically stabilize a vortex binding a Majorana zero mode. Here the singular phase winding is induced by current flowing from contact 1 to contact 2, rather than from a magnetic field. In (b) we illustrate the probability density extracted from the near-zero-energy mode generated by a current-induced vortex at the center of the device (parameters are $\mu =0$, $m{\alpha }^2=1.3\overline{\Delta }$, ${\overline{V}}_z=2\overline{\Delta }$, and $\lambda ={\lambda }_F/4$). The large central peak corresponds to the Majorana bound to the vortex, which hybridizes weakly with the outer Majorana running along the perimeter.

Figure 6

(a) Long Josephson junction formed by adjacent topological superconductors. A magnetic field $\overrightarrow{B}$ orients perpendicular to the plane and uniformly penetrates through the junction. (b) The solid black curve represents the magnitude of the tunneling current arising from coupled Majorana zero modes at the edge. This contribution exhibits zeros at even multiples of the flux quantum in sharp contrast to the Fraunhofer pattern exhibited by ordinary $s$-wave superconductor junctions (red dashed curve). The dotted blue curve represents the anomalous Fraunhofer pattern that would arise in an experiment due to the Majorana-mediated component and a parallel conventional current contribution.