Theory of magnetic field stabilized compact skyrmions in thin-film ferromagnets

We present a micromagnetic theory of compact magnetic skyrmions under applied magnetic field that accounts for the full dipolar energy and the interfacial Dzyaloshinskii-Moryia interaction (DMI) in the thin-film regime. Asymptotic analysis is used to derive analytical formulas for the parametric dependence of the skyrmion size and rotation angle, as well as the energy barriers for collapse and bursting, two processes that lead to a finite skyrmion lifetime. We demonstrate the existence of a regime at low DMI, in which the skyrmion is stabilized by a combination of nonlocal dipolar interaction and a magnetic field applied parallel to its core, and discuss the conditions for an experimental realization of such field-stabilized skyrmions.

Figure 1

Leading order skyrmion energy $E_{\rho ,\theta ,L}$ vs radius $\rho$ after minimization over $\theta$ and $L$ for $\kappa =0.35$, $\delta =0.7$, $h=0.008$, and $Q=1.591$ [see (2.27) in the Supplemental Material 32].

Figure 2

Scaled skyrmion radius ${\overline{\rho }}_{\mathrm{sk}}$ for the intermediate thickness regime with $\overline{\delta }=0.7$ (a) and ultrathin regime with $\overline{\delta }=0$ (b) as a function of $\overline{\kappa }$ and $\overline{h}$. The convention for the field direction is shown schematically in the insets, where in the skyrmion image, the red and blue colors represent, respectively, down and up magnetizations and white corresponds to in-plane magnetization. For $\overline{h}<0$, the field is applied antiparallel to the skyrmion core (red arrow in the inset), while for $\overline{h}>0$ the field is parallel (blue arrow in the inset).

Figure 3

Skyrmion collapse $\mathrm{\Delta }{E}_{\mathrm{coll}}$ (a) and bursting $\mathrm{\Delta }{E}_{\mathrm{burst}}$ (b) energy barriers (in units of $Ad$) for the intermediate thickness regime with $\overline{\delta }=0.7$. Also shown is the effective barrier $\mathrm{\Delta }E=min(\mathrm{\Delta }{E}_{\mathrm{coll}},\mathrm{\Delta }{E}_{\mathrm{burst}})$, which illustrates the optimal stability region against collapse and bursting for $\overline{\delta }=0.7$ (c) and for the ultrathin film with $\overline{\delta }=0$ (d).

Figure 4

The skyrmion characteristics in the low DMI regime for $d=5$ nm, $A=20$ pJ/m, $M_{\mathrm{s}}={10}^5$ A/m, and $K_{\mathrm{u}}=6346$ J/${\mathrm{m}}^3$ corresponding to $Q=1.01$: (a) skyrmion radius $r_{\mathrm{sky}}$ and (b) skyrmion rotation angle $|{\theta }_0|$ from (6) and (5); (c) skyrmion radius and (d) skyrmion rotation angle from mumax3 simulations on a $2048\times 2048{\mathrm{nm}}^2$ square box with mesh size $2\times 2\times 5$ nm. The dashed line is the line of zero bursting barrier defined in (10).

Figure 5

The magnetization profiles along the $x$ axis with the skyrmion at the origin obtained from the mumax3 simulations for the parameters indicated by the respective dots in Fig. 4. The simulation datapoints are shown with the solid black dots, while the Belavin-Polyakov profiles obtained by fitting their radius and angle to the simulation data are shown with the solid blue lines.