Thermal stability of metastable magnetic skyrmions: Entropic narrowing and significance of internal eigenmodes

We compute annihilation rates of metastable magnetic skyrmions using a form of Langer's theory in the intermediate-to-high damping (IHD) regime. For a Néel skyrmion, a Bloch skyrmion, and an antiskyrmion, we look at two possible paths to annihilation: collapse and escape through a boundary. We also study the effects of a curved versus a flat boundary, a second skyrmion, and a nonmagnetic defect. We find that the skyrmion's internal modes play a dominant role in the thermally activated transitions compared to the spin-wave excitations and that the relative contribution of internal modes depends on the nature of the transition process. Our calculations for a small skyrmion stabilized at zero field show that collapse on a defect is the most probable path. In the absence of a defect, the annihilation is largely dominated by escape mechanisms, even though in this case the activation energy is higher than that of collapse processes. Escape through a flat boundary is found more probable than through a curved boundary. The potential source of stability of metastable skyrmions is therefore found not to lie in high activation energies, nor in the dynamics at the transition state, but comes from entropic narrowing in the saddle point region which leads to lowered attempt frequencies. This narrowing effect is found to be primarily associated with the skyrmion's internal modes.

Figure 1

Typical energy surface of a system possessing a metastable local minimum $A$ and a stable global minimum $M$ separated by a saddle point $S$. The reaction coordinate is represented by a yellow line and corresponds to the path of minimun energy connecting $A$, $S$, and $M$.

Figure 2

Spin maps of the collapse mechanisms. The index in the top left-hand corner corresponds to the image index of the GNEB method. The saddle point corresponds to the state preceding the flipping of the core spin. (a) Néel skyrmion. (b) Antiskyrmion. (c) Bloch skyrmion. (d) Néel skyrmion in the presence of a nonmagnetic defect. (e) Néel skyrmion in the presence of a second skyrmion.

Figure 3

Spin maps of the escape mechanisms. The conventions are the same as in Fig. 2. On all the subfigures, the saddle point is im. 11, which is the state where the skyrmion sits tangent to the boundary. (a) Néel skyrmion. (b) Antiskyrmion. (c) Bloch skyrmion. (d) Néel skyrmion escaping through a curved boundary.

Figure 4

Interpolated energy profiles along the normalized reaction coordinate. Each dot corresponds to an image of the system on the energy surface. The insets show a closeup of the spin configuration at the saddle point. (a) Collapse of a single skyrmion. The energy increases slowly as the skyrmion shrinks (ims. 1–8). Past the barrier top (im. 8), it annihilates by breaking the radial symmetry which is accompanied by a brutal decrease in the energy. (b) Escape through a flat boundary. The energy rises as the skyrmion gets closer to the edge (ims. 9–11). The top of the barrier is the state tangent to the boundary (im. 11). Past that point, the skyrmion disappears through the edge. This is accompanied by a rapid drop in the energy, with a notable slowdown halfway through the process as half the skyrmion has disappeared (im. 13). (c) Collapse in the presence of a single non-magnetic defect. The skyrmion shrinks in size as the core moves towards the defect. This costs very little energy (ims. 1–5). Past the saddle point, the skyrmion collapses on the defect. (d) Collapse in the presence of another skyrmion. The skyrmions get closer to each other, which at first costs little energy (ims. 1–3) until a critical distance is reached where the collapse of the upper skyrmion is initiated. The saddle point is the same as that of the first mechanism for the upper skyrmion, while the other remains stable (im. 4).

Figure 5

All energy curvatures at $A$ (in blue) and $S$ (in red) ordered by increasing amplitudes for the case of the collapse of a single skyrmion. The other mechanisms exhibit the same profiles which we interpreted as the dispersion of spin-wave excitations, with the exception of the first few eigenvalues. These are found below the main curve and are shown on Fig. 6.

Figure 6

For the first few eigenvalues of each annihilation mechanism, we show the following: (a) eigenvalues at $A$ and $S$, (b) ratio of individual eigenvalues in semilogarithmic scale, and (c) ${\mathrm{\Omega }}_{0,i}$ in semilogarithmic scale. The inset figure shows all eigenvalues. The red dotted line marks the separation between localized and collective eigenmodes. The $x$ axis is the same for all subfigures.

Figure 7

Eigenmodes associated to the $\theta$ variable for a single skyrmion in a simulated system of $50\times 50$ spins. The blue and green color scheme is associated with metastable states and the blue and red one with saddle points. Negative amplitudes are plotted in blue and positive ones in green/red. The range of the color map is adjusted on each plot so that zero-amplitude fluctuations coincide with white. Modes are designated via the $i$-index of the corresponding ordered eigenfrequencies of Fig. 6. (a) Metastable single-skyrmion state with localized modes $i=1,\cdots ,7$ and collective modes $i>7$. (b) Saddle point of the collapse with localized modes $i=1,\cdots ,5$ and collective modes $i>5$. (c) Saddle point of the escape with localized modes $i=1,\cdots ,6$ and collective modes $i>6$.

Figure 8

Eigenmodes associated to the $\theta$ variable. The color code is the same as in Fig. 7. Additional mode profiles can be found in Ref. [21] for each of the following mechanisms. [(a) and (b)] Internal and collective modes of two coupled skyrmions on a $50\times 50$ lattice at $A$ and $S$. [(c) and ( d)] Modified internal modes for the collapse on a defect at $A$ and $S$. (e) Modified internal modes for the escape through a curved boundary at $S$.

Figure 9

Calculated change in configurational entropy induced when the system goes to the saddle point $\mathrm{\Delta }S/k_B={\displaystyle \frac{S_S-S_A}{k_B}}$ as defined in Eq. (10) over a broad range of temperatures for all mechanisms. The highest entropic barrier corresponds to the most negative $\mathrm{\Delta }S$, i.e., the collapse involving a second skyrmion.