Dynamics and energetics of emergent magnetic monopoles in chiral magnets

The formation and destruction of topologically quantized magnetic whirls, i.e., the so-called skyrmions, in chiral magnets is driven by the creation and motion of singular hedgehog defects. These Bloch points can be identified with emergent magnetic monopoles and antimonopoles. We investigate how the energetics of and forces between monopoles and antimonopoles influence their creation rate and dynamics. We study a single skyrmion line defect in the helical phase using both micromagnetic simulations and a Ginzburg-Landau analysis. Monopole-antimonopole pairs are created in a thermally activated process, largely controlled by the (core) energy of the monopole. The force between monopoles and antimonopoles is linear in distance and described by a string tension. The sign and size of the string tension determines the stability of the phases and the velocity of the monopoles.

Figure 1

(a) Initial magnetic configuration: a skyrmion oriented along a $(-1,-1,2)$ direction embedded into a helical phase. (b)–(d) The skyrmion unwinds by the creation of a monopole-antimonopole pair which move apart [cf. Fig. 2].

Figure 2

Examples of trajectories of MPs (black) and AMPs (red) after a quench to $B=0$ ($T=0.7$) obtained from sLLG simulations (vertical axis: coordinate parallel to the skyrmion orientation; numbers: winding numbers of the magnetic texture). (a) A MP-AMP pair is created in the middle of the sample and moves to the edge (cf. Fig. 1). (b) An AMP is created at the surface and moves to the bottom. (c), (d) Events with both pair creation and pair annihilation.

Figure 3

(a) Average MP velocity $v$ (right legend) and average bulk creation rate $\Gamma$ (left legend), as a function of the temperature $T$ for $B=0$ (error bars: standard deviation of the mean). (b) $v$ and $\Gamma$ as a function of the applied magnetic field $B$ for $T=0.7$ and $T=0.8$.

Figure 4

Energy of a MP-AMP pair as a function of their distance obtained from minimization of the GL free energy ($t_0=-1.6$) for $B=0.14$ to $0.22$ (bottom to top). For large distances (here we use units where the pitch of the helical phase is $2\pi$), the force is linear and therefore described by a string tension (see Fig. 5).

Figure 5

Energy $E_0$ of a MP-AMP pair, sum of their core energies $2E_c$ (both left axis), and string tension $T_S$ (right axis) as a function of the magnetic field ${\mathbf{B}}_z$ ($t_0=-1.6$). At the dashed line, the string tension changes sign.