Skyrmion dynamics in chiral ferromagnets under spin-transfer torque

We study the dynamics of skyrmions under spin-transfer torque in Dzyaloshinskii-Moriya materials with easy-axis anisotropy. In particular, we study the motion of a topological skyrmion with skyrmion number Q = 1 and a nontopological skyrmionium with Q = 0 using their linear momentum, virial relations, and numerical simulations. The nontopological Q = 0 skyrmionium is accelerated in the direction of the current flow and it either reaches a steady state with constant velocity, or it is elongated to infinity. The steady-state velocity is given by a balance between current and dissipation and has an upper limit. In contrast, the topological Q = 1 skyrmion converges to a steady state with constant velocity at an angle to the current flow. When the spin current stops the Q = 1 skyrmion is spontaneously pinned, whereas the Q = 0 skyrmionium continues propagation. Exact solutions for the propagating skyrmionium are identified as solutions of equations given numerically in a previous work. Further exact results for propagating skyrmions are given in the case of the pure exchange model. The traveling solutions provide arguments that a spin-polarized current will cause rigid motion of a skyrmion or a skyrmionium.