Interplay between topology and dynamics in chiral magnetic skyrmions

It is shown that in a spin-valve consisting of Pt(5 nm)/Co(0.6 nm)/Cu (4 nm)/CoPt(4 nm) a perpendicular spin-polarized current, in the presence of the Dzyaloshinskii–Moriya interaction [1] and of a strong perpendicular anisotropy, induces the rotation of the spins from the hedgehog-like to the vortex-like texture in the topological droplet state and excites low-frequency topological modes. The topological character of these spin-wave modes results from the synchronized dynamics between the 360o rotation of the spin of the outer droplet domain from the hedgehog-like (Néel) to the vortex-like (Bloch) texture of the chiral magnetic skyrmion and the expansion/shrinking of the droplet core. A quantitative description of topological droplet modes is given according to an analytical model based on the linearization of the equation of motion including intrinsic positive Gilbert damping and negative damping related to the spin-transfer torque [2]. The analytical frequency of the topological droplet mode is expressed as the solution of a second-order algebraic equation written in terms of the magnetic parameters including the intrinsic Gilbert damping and the extrinsic damping related to the spin-polarized current. Micromagnetic simulations confirm the red-shift behaviour of the topological mode frequencies as a function of the current density predicted by the analytical model. The analysis of the symmetry properties of the linearized equations of motion demonstrates the non-reciprocal role of the spin polarized current on the topological mode dynamics. The interplay between topology and dynamics is discussed by introducing the notion of topological degeneracy according to which two topological droplet textures (hedgehog-like and vortex-like, respectively) having different ground-state energies are characterized by the same topological charge. This work was partially supported by MIUR-PRIN 2010-11 Project2010ECA8P3 "DyNanoMag". [1] T. Moriya, Phys. Rev. Lett. 4, 228 (1960). [2] R. Zivieri et al., “Topological skyrmion dynamics driven by spin-transfer torque” submitted