Topological Superconductivity Mediated by Skyrmionic Magnons

Topological superconductors are associated with the appearance of Majorana bound states, with promising applications in topologically protected quantum computing. In this Letter, we study a system where a skyrmion crystal is interfaced with a normal metal. Through interfacial exchange coupling, spin fluctuations in the skyrmion crystal mediate an effective electron-electron interaction in the normal metal. We study superconductivity within a weak-coupling approach and solve gap equations both close to the critical temperature and at zero temperature. Special features in the effective electron-electron interaction due to the noncolinearity of the magnetic ground state yield topological superconductivity at the interface.

Figure 1

(a) Plot of the electron energy ${\epsilon }_{\mathit{k}}$ showing the electron first Brillouin zone (EBZ) in black, the magnetic first Brillouin zone (MBZ) in orange, and the Fermi surface (FS) in white. Two choices of $\mu$ are shown, where the solid FS corresponds to $\mu /t=-5.9$ and the dashed FS corresponds to $\mu /t=-5.0$. (b) An illustration of the system under consideration. The itinerant electrons (blue arrows) in a normal metal (NM) are coupled to the spins (orange arrows) in a magnetic monolayer (MML). The MML is deposited on a heavy metal (HM) such that skyrmion crystal (SkX) ground states (GSs) are preferred. The (c) SkX1 and (d) SkX2 GSs are shown with periodic boundary conditions. Colors give the $z$ component of the spins, $m_{iz}$, and arrows show their in-plane component.

Figure 2

(a) Phase diagram and dimensionless coupling $\lambda$ in the superconducting state close to $T_c$. The vertical, black line shows the transition between SkX1 and SkX2 in the MML. The other black lines show the locations of the phase transitions with better resolution. In the green region, the unpolarized triplet gap ${\mathrm{\Delta }}_{\mathit{k}\uparrow \downarrow }^{E(s)}$ shows $p_x$-wave symmetry, while the other gaps are zero. In the orange and blue regions, the stated symmetry refers to the ${\mathrm{\Delta }}_{\mathit{k}\downarrow \downarrow }$ gap. The relevant gap symmetries are shown in (b). The parameters are $t/J=1000$, $\overline{J}/J=50$, $D/J=2.16$, $U/J=0.35$, and $S=1$.

Figure 3

Estimates of $T_c$ as a function of the easy-axis anisotropy $K$ in the MML for different chemical potentials $\mu$. We have set an upper cutoff of 10 K since larger $T_c$ means that $\lambda \ll 1$ is no longer valid. The parameters are $t/J=1000$, $D/J=2.16$, $U/J=0.35$, and $S=1$. In this figure only, $J=1\mathrm{meV}$ was set in order to find $T_c$ in kelvin.

Figure 4

Phase diagram at zero temperature showing $2\mathrm{\Delta }(0)/k_B{T}_c$ in color. Black regions show where two or more symmetries lead to convergence. Where applicable, the BCS result $2\mathrm{\Delta }(0)/k_B{T}_c=2\pi e^{-\gamma }$ is shown on the color bars. The parameters are $t/J=1000$, $D/J=2.16$, $U/J=0.35$, and $S=1$.