Correlated gyrotropic motion of skyrmion clusters in ultrathin ferromagnetic nanodisks

We report the correlated gyrotropic motion of two-dimensional skyrmion clusters in ferromagnetic nanodisks. Based on the massless Thiele equation, we calculate the gyrotropic eigenmodes for the multi-skyrmion system. Micromagnetic simulations are performed to verify theoretical predictions with a good agreement. By applying an in-plane magnetic field pulse on four-skyrmion system, four gyrotropic modes have been excited. One of them has all skyrmions moving in-phase; two modes have the face-to-face skyrmions pair moving out-of-phase; the rest one has a pair of face-to-face skyrmions moving in-phase with side skyrmions rotating out-of-phase. Meanwhile, we identify a reversal of phase and rotation direction originating from the complex interaction between skyrmions for five-skyrmion system. By applying an out-of-plane magnetic field pulse, we observe the correlated breathing modes for four-skyrmion system with the face-to-face skyrmions breathing in-phase or out-of-phase, and all skyrmions breathing consistently in-phase. In particular, for the five-skyrmion system, a hybridization mode has been clarified with side skyrmions breathing and central skyrmion rotation. Our results provide useful insights for controlling the dynamic behavior of multiple skyrmions that are indispensable in the future skyrmion-based spintronic devices.