Duplication, Collapse, and Escape of Magnetic Skyrmions Revealed Using a Systematic Saddle Point Search Method

Various transitions that a magnetic Skyrmion can undergo are found in calculations using a method for climbing up the energy surface and converging onto first order saddle points. In addition to collapse and escape through a boundary, the method identifies a transition where the Skyrmion divides and forms two Skyrmions. The activation energy for this duplication process can be similar to that of collapse and escape. A tilting of the external magnetic field for a certain time interval is found to induce the duplication process in a dynamical simulation. Such a process could turn out to be an important avenue for the creation of Skyrmions in future magnetic devices.

Figure 1

An illustration of a method for climbing up the energy surface from an initial state minimum to a first order saddle point. The system contains only a single spin (see [23]) and because only the orientation of the magnetic vector can change, the energy surface can be mapped onto a sphere. The local energy minima (blue) are separated by an energy ridge, where points of low energy correspond to first order saddle points (black $\times$). The red curves illustrate saddle point search paths starting from various points. In this illustration, the system is made to follow the gradient of the energy until the lowest eigenvalue of the Hessian becomes negative (at the white dashed line). Beyond that point, the system is displaced along the effective force given by Eq. (1). This brings the system to a first order saddle point on the energy surface.

Figure 2

(Top, left) Minimum energy configuration of the Skyrmion. (Top, right) Minimum energy paths for the three types of transitions found: duplication, collapse, and escape. The reaction coordinate is the scaled total displacement along the path. The energy is given in units of the exchange coupling constant $J$. (Middle row) Saddle point configurations found for escape through a boundary (left), radial collapse (middle), and duplication (right). (Bottom) The translational, breathing, and elliptical eigenmodes of the Skyrmion in the minimum energy configuration, each leading to the saddle point displayed directly above.

Figure 3

Snapshots from a dynamical simulation where the duplication of a Skyrmion is induced by applying a magnetic pulse over 200 ps, giving a total field that is tilted with respect to the normal of the plane (see [23] for parameter values). The labels on top of the frames give the time and magnetic field of the pulse (which adds to a constant field in the direction of normal to the plane). The pulse is applied at time $t=0$ and lasts until $t=200\mathrm{ps}$. At the end of the pulse, an elongated defect is formed that later splits up (at $t=209\mathrm{ps}$) to form two Skyrmions.