Stabilization and control of Majorana bound states with elongated skyrmions

We show that elongated magnetic skyrmions can host Majorana bound states in a proximity-coupled two-dimensional electron gas sandwiched between a chiral magnet and an $s$-wave superconductor. Our proposal requires stable skyrmions with unit topological charge, which can be realized in a wide range of multilayer magnets, and it allows quantum information transfer by using standard methods in spintronics via skyrmion motion. We also show how braiding operations can be realized in our proposal.

Figure 1

Spin density $\mathit{n}$. The arrows show the in-plane component and contours show the out-of-plane component.

Figure 2

Squared amplitude ${\left|\mathrm{\Psi }\right|}^2$, length in units of $a$ with $a=10$ nm for a skyrmion with a core radius of $R_c^y=50$ nm, $R_c^x=490$ nm, $\mathrm{\Delta }=0.25$ meV, $\tilde{B}=0.87,\tilde{J}=1$, and $\tilde{\mu }=0.2$. The white solid line is the border between topological and nontopological regions as determined by Eq. (3). The black solid lines indicate the boundaries of the ferromagnetic nanotrack.

Figure 3

(a) Squared amplitude ${\left|\mathrm{\Psi }\right|}^2$ in units of $1/a^2$ along the middle horizontal line for $R_c^x=490$ nm, using parameters given in the caption of Fig. 2. The large peaks correspond to the MBSs localized at the edges of the elliptic topological region. (b) The energy spectrum $E_n$ for $R_c^x=490$ nm. (c) Ground state $\left(E_0\right)$ and first excited energy levels $\left(E_1\right)$ as a function of the horizontal core radius $R_c^x$ in units of $a$. (d)–(f) Similar plots for ${\alpha }_R=2.5$ meV nm.

Figure 4

Hybridization energies ${\mathrm{\Gamma }}_{ij}$ of two MBSs hosted by two different skyrmions in the presence of Rashba SOC, as a function of the spatial separation between skyrmions along the $x$ axis, $d$ (skyrmion radius taken to be $R_c^x)$.

Figure 5

A double-braiding of two MBSs at the ends of two topological regions in a typical $p$-wave superconductor in the presence of a vortex, corresponding to two sequential elementary braiding operations given in Eq. (9). Note that the topological regions cross each other at intermediate times during the process.

Figure 6

Schematic depiction of the three Cooper pair boxes connected at a trijunction. Each Cooper pair box hosts a pair of MBSs and is connected to a bulk superconductor via split Josephson junctions. Coulomb coupling strength can be modulated by changing the applied magnetic flux ${\mathrm{\Phi }}_i$ at each junction.