We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii–Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate the existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree $$\pm \,1$$ ± 1 harmonic maps from the two-dimensional sphere to itself.

中文翻译：

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超薄铁磁薄膜中 Skyrmions 的定量描述和度数 $$\pm \,1$$ 谐波映射从 $${\mathbb {R}}^2$$ 到 $${\mathbb {S}}^2 $$

我们将超薄铁磁薄膜中的斯格明子表征为适用于具有垂直磁各向异性和界面 Dzyaloshinskii-Moriya 相互作用的准二维材料的降低微磁能的局部极小值。最小化是在合适的二维磁化配置类别中进行的，可防止能量达到负无穷大，同时不对配置的空间尺度施加任何限制。当能量由交换能量支配时，我们首先证明了模型参数的明确范围内最小化器的存在。然后我们研究共形极限，其中只有交换能量存在，并将斯格明子的渐近轮廓识别为从平面到球体的 1 次谐波映射，连同它们的半径，角度和能量。我们分析的一个副产品是从二维球体到自身的度数 $$\pm \,1$$ ± 1 谐波映射的定量刚性结果。