In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii–Moriya interaction (DMI). Our analysis includes both a stationary $\mathrm{\Gamma }$-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional (3D) model with the 2D thin-film limit.

在本文中，我们研究了在大块 Dzyaloshinskii-Moriya 相互作用 (DMI) 存在下微磁能泛函的薄膜极限。我们的分析包括一个固定的$\mathrm{\Gamma }$-微磁能的收敛结果，以及相关Landau-Lifshitz-Gilbert方程的渐近行为的识别。特别是，我们证明，在极限模型中，DMI 项的一部分表现得像磁矩在薄膜法线上的投影，这有助于增加由静磁自能引起的形状各向异性。最后，我们讨论了一种用于逼近时间相关情况的收敛有限元方法，并使用它对原始三维 (3D) 模型与二维薄膜极限进行数值比较。