A theory on skyrmion size and profile

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Skyrmions, topological objects originally used to describe resonance states of baryons [1], were observed in magnetic systems that involve Dzyaloshinskii- Moriya interaction (DMI). Magnetic skyrmions are believed to be poten- tial information carriers in future high density data storage and informa- tion processing devices [2, 3]. Although much knowledge about magnetic skyrmions has been accumulated after intensive studies including skyrmion generation, the dependence of skyrmion size (R) on material parameters such as exchange energy, magnetic anisotropy energy, and DMI strength is still poorly understood at a quantitative, or even qualitative level. Here we show that the skyrmion profiles agree well with Walker-like 360° domain wall formula [4, 5]. By minimizing the energy, we obtain the analytical expres- sions of the skyrmion size R and wall width w as functions of exchange constant A, DMI coefficient D, anisotropy constant K, and external magnetic field B. These results agree perfectly with micromagnetic simulations and are consistent with experiments. We consider a 2D film with Heisenberg exchange interaction (of exchange constant A), interfacial Dzyaloshinskii- Moriya interaction (DMI) (of DMI coefficient D), perpendicular easy-axis anisotropy (of anisotropy constant K), and a perpendicular magnetic field (of field strength B). We first numerically verify that an isolated Neel skyrmion is rotational symmetric and the magnetization profile along any diameter can be well fitted by a Walker-like 360° domain wall profile with two charac- teristic lengths, the skyrmion size and the skyrmion wall width, as shown in Fig. 1. The skyrmion profile can be described by $\Theta (r)$ and $\Phi (\varphi)$, where $\Theta $ and $\Phi $ are polar and azimuthal angles of magnetization m, and $r, \varphi $ are radial and angle coordinates in space. The total energy E is a functional of $\Theta (r)$ and $\Phi (\varphi)$. Then by substituting the 360° domain wall profile into the energy functional, we obtain an expression of total energy E as a function of R and $w$. The equilibrium R and $w$ can be obtained by minimizing the total energy, and the obtained results almost perfectly agree with the numerical simula- tion results as shown in Fig. 2. The exchange and DMI energies come from the spatial magnetization variation rate. For a skyrmion, the magnetization variation rates in the radial and tangent directions scale respectively as 1/w and 1/R. The exchange energy is then proportional to skyrmion wall area of $Rw$ multiplying the square of the magnetization variation rates $1/ \mathrm {R}^{2}+1/ \mathrm {w}^{2}$. Near the skyrmion wall region, the magnetization variation rate along the tangent direction is perpendicular to the magnetization and does not contribute to the DMI energy. The DMI energy is then proportional to wall area $Rw$ multiplying the magnetization variation rate along radial direction $(1/w$). The anisotropy energy is mainly from the skyrmion wall area. The Zeeman energy of the skyrmion comes from the inner domain proportional to its area of $\pi (R- cw ) ^{2}$, where $c$ is a coefficient depending on the magnetiza- tion profile, and from the wall area proportional to its area of $Rw$. Thus, by considering the physical meaning of each energy term as well as reasonable mathematical approximations, we obtain a simple analytical formula for the total energy in terms of R and w. The approximate formula also gives correct qualitative parameter dependence and good quantitative agreement with the simulations. The results are not limited to interfacial DMI. For bulk DMI, the only difference is Bloch-type skyrmions are preferred The radial profile as well the skyrmion size does not change.