Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction

We study the modification of micromagnetic configurations in nanostructures, due to the Dzyaloshinskii-Moriya interaction (DMI), that appear at the interface of an ultrathin film. We show that this interaction leads to micromagnetic boundary conditions that bend the magnetization at the edges. We explore several cases of ultrathin film nanostructures that allow analytical calculations (one-dimensional systems, domain walls, cycloids, and skyrmions), compare with fully numerical calculations, and show that a good physical understanding of this type of micromagnetics can be reached. We particularly focus on skyrmions confined in circular nanodots and show that edges allow for the isolation of single skyrmions for a large range of the DMI parameter.

Figure 1

Magnetization rotation at the edges of an ultrathin film with interface DMI. (a) Magnetization profile in a stripe infinite in the $\widehat{y}$ direction and with a 100-nm width in the $\widehat{x}$ direction, with initial magnetization along the $\widehat{z}$ axis and for $D=3$ mJ/m${}^2$ ($\xi =10.67$ nm). (b) Variation of $m_x$ at the structure boundary versus $\Delta /\xi$. The calculation has been stopped at $D=D_c=3.6$ mJ/m${}^2$ as beyond this value, cycloids start to develop in the sample and $C$ in Eq. (5) is not zero. The continuous line is the solution (numerical integration) of Eq. (4) for different strengths of the DMI. In (a) and (b), symbols correspond to numerical calculations: for the open symbols, the local dipolar coupling approximation is used whereas, for the full symbols, the full dipolar energy is included. Note that in (a), both results are hardly distinguishable.

Figure 2

(a) Variation of the cycloid period $L$ as a function of the anisotropy-free period $L_0$ [result of Eqs. (13) and (14)]. (b) Shape of the cycloid (perpendicular magnetization component $m_z$) in the presence of anisotropy (the dashed line is the reference cycloid with no anisotropy) for (b1) $D/D_c=2.5$ ($D=9$ mJ/m${}^2$, $L_0=22.34$ nm) and (b2) $D/D_c=1.1$ ($D=4$ mJ/m${}^2$, $L_0=50.26$ nm).

Figure 3

Results of numerical integration of Eq. (16) for a 100-nm-diameter nanodot with $D=4.5$ mJ/m${}^2$ ($D/D_c=1.2$). The open symbols are the results of full numerical calculations [with local dipolar energy approximation (Ref. 28)], given for comparison. In (b), the variation of $\theta$ shows the chirality imposed by DMI in the micromagnetic configuration. For this set of parameters, three solutions are found: quasiuniform (black), skyrmion (red), and $2\pi$ (green) and $3\pi$ (blue) rotation states.

Figure 4

(a) Variation of the skyrmion profile for different values of $D$ (from 2 to 7 mJ/m${}^2$), for a $R=50$ nm nanodot. (b), (c) Variation of the skyrmion profile versus the dot radius $R$ for $D=3$ and 4.5 mJ/m${}^2$ ($D/D_c=0.83$ and 1.25), respectively. In (b), the skyrmion radius is independent of the dot radius, except for very small radius which compresses the skyrmion. In (c), the skyrmion radius is limited by the dot radius and thus increases with $R$. Note that for all these profiles, although $\theta \left(x\right)$ is not represented, it corresponds to a monotonic increasing function as in Fig 3b, thus to chiral solutions. (d) Variation of the skyrmion core radius $R_s$ versus $D$ for different dot radius. The radius is defined at the $m_z=0$ line. The line is the solution for an infinite thin film and the dashed line is the approximate solution described in the text [Eq. (18)].

Figure 5

Variation of the energy of the different states versus $D$, in a 100-nm-diameter dot. Note that each line does not cover the full explored $D$ range, as we only plot the solution where a (meta)stable solution has been found.