Floquet second-order topological superconductor driven via ferromagnetic resonance

We consider a Floquet triple-layer setup composed of a two-dimensional electron gas with spin-orbit interactions, proximity coupled to an $s$-wave superconductor and to a ferromagnet driven at resonance. The ferromagnetic layer generates a time-oscillating Zeeman field which competes with the induced superconducting gap and leads to a topological phase transition. The resulting Floquet states support a second-order topological superconducting phase with a pair of localized zero-energy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet helical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an effective time-reversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases are stable against disorder as well as against parameter variations and are within experimental reach.

Figure 1

(a) Schematics of the Floquet triple-layer setup consisting of a 2DEG (blue layer) with SOI of strengths ${\alpha }_x$ and ${\alpha }_y$, proximity coupled to an $s$-wave SC (red layer) at the bottom and a FM at the top (green layer) which is resonantly driven by an external time-dependent magnetic field $\mathit{H}\left(t\right)$ to generate an oscillating magnetic field $\mathit{B}\left(t\right)=B_{\perp }cos\left(\omega t\right){\mathit{e}}_z+B_{\parallel }sin\left(\omega t\right){\mathit{u}}_{\parallel }$ in the 2DEG. (b) Band structure of the 2DEG in the isotropic regime ${\alpha }_x={\alpha }_y$. The chemical potential $\mu$ is fixed below the crossing point of the spin-split bands (indicated by the blue area). The driving frequency $\omega$ is tuned to achieve resonance at the smallest Fermi momentum (represented by the two green circles) between the two Floquet bands labeled by $\tau =\pm 1$.

Figure 2

(a) Energy spectrum $E\left(k_x\right)$ in the FHeTS phase with $t_{\mathrm{Z}}^{\perp }/{\mathrm{\Delta }}_{\mathrm{sc}}=3$ and ${\alpha }_x={\alpha }_y$. The helical edge modes are localized at the edges. (b) Phase diagram showing the gap to the first excited bulk state (in units of $E_{\mathrm{so}}$) as a function of the ratios ${\alpha }_y/{\alpha }_x$ and $t_{\mathrm{Z}}^{\perp }/{\mathrm{\Delta }}_{\mathrm{sc}}$. Blue lines indicate the phase boundaries. In the isotropic regime, ${\alpha }_x={\alpha }_y$, the phase transition occurs at the critical point ${\mathrm{\Delta }}_{\mathrm{sc}}=t_{\mathrm{Z}}^{\perp }$. When ${\alpha }_x\ne {\alpha }_y$, the critical point is transformed into a gapless Weyl phase, and the value $t_{\mathrm{Z}}^{\perp }/{\mathrm{\Delta }}_{\mathrm{sc}}$ required to reach the FHeTS increases up to the point of strong anisotropy beyond which the FHeTS cannot be reached. Remaining parameters in both simulations are $t_{\mathrm{Z}}^{\parallel }=0$, ${\mathrm{\Delta }}_{\mathrm{sc}}/E_{\mathrm{so}}=0.2$, $k_{\mathrm{so}}{L}_x=k_{\mathrm{so}}{L}_y=80$, and $\mu =-E_{\mathrm{so}}/2$.

Figure 3

(a), (b), (d) Probability density of the lowest-energy state in the FHOTS phase for $t_{\mathrm{Z}}^{\perp }/{\mathrm{\Delta }}_{\mathrm{sc}}=3$, $t_{\mathrm{Z}}^{\parallel }/{\mathrm{\Delta }}_{\mathrm{sc}}=2.5$, and ${\alpha }_x={\alpha }_y$. The inset shows the ten lowest eigenenergies. (a) For ${\mathit{u}}_{\parallel }=(1,1)/\sqrt{2}$, the in-plane Zeeman field $B_{\parallel }$ opens gaps in all edge modes such that zero-energy MCSs emerge at two opposite corners. (b) The vector ${\mathit{u}}_{\parallel }=(0,1)$ is parallel to the edges along which the system stays gapless. (c) Phase diagram showing the bulk gap (color coded in units of $E_{\mathrm{so}}$) as a function of the ratios $t_{\mathrm{Z}}^{\perp }/{\mathrm{\Delta }}_{\mathrm{sc}}$ and $t_{\mathrm{Z}}^{\parallel }/{\mathrm{\Delta }}_{\mathrm{sc}}$ with ${\mathit{u}}_{\parallel }=(1,1)/\sqrt{2}$. The critical point ${\mathrm{\Delta }}_{\mathrm{sc}}=t_{\mathrm{Z}}^{\perp }$ at $t_{\mathrm{Z}}^{\parallel }=0$ merges into a gapless Weyl phase at finite $t_{\mathrm{Z}}^{\parallel }$. As a result, higher values of $t_{\mathrm{Z}}^{\perp }/{\mathrm{\Delta }}_{\mathrm{sc}}$ are required to reach the topological phase. The various phase boundaries correspond to ${\alpha }_y/{\alpha }_x=0.8$ (red dotted), 0.9 (purple dashed), and 1.0 (blue solid). (d) The MCSs are stable against strong external perturbations and disorder: $\delta \omega ={\mathrm{\Delta }}_{\mathrm{Z}}=0.1E_{\mathrm{so}}$, $S_{\mu }=S_{\mathrm{Z}}^{\mathrm{stat}}=S_{\mathrm{Z}}^{\mathrm{dyn}}=0.5E_{\mathrm{so}}$. A single disorder realization is considered. The rest of the parameters are the same as in Fig. 2.

Figure 4

Setup of the FM layer to generate the desired oscillating magnetic field in the 2DEG. The 2DEG-SC heterostructure is placed in the vicinity of the FM. The system is subjected to an external magnetic field $\mathit{H}\left(t\right)={\mathit{H}}_0+\mathit{h}\left(t\right)$ (yellow lines) generating FM resonance (shown not to scale). This induces the precession of the FM magnetization $\mathit{M}\left(t\right)$ (black lines inside the FM) and demagnetizing field $\mathit{D}\left(t\right)$ (black lines outside of the FM). Close to the surface of the FM layer, the static components of $\mathit{D}\left(t\right)$ and $\mathit{H}\left(t\right)$ cancel out, so that the dynamics of the total field $\mathit{B}\left(t\right)=\mathit{H}\left(t\right)+\mathit{D}\left(t\right)$ in the 2DEG layer is close to a full ${360}^{\circ }$ rotation.