Skyrmion stacking in stray field-coupled ultrathin ferromagnetic multilayers

TL;DR

The paper develops a rigorous reduced-energy framework for stacks of skyrmions in stray-field‑coupled ultrathin ferromagnetic multilayers, deriving a two‑dimensional energy $E_N$ in the thin-layer, intralayer-exchange regime. Using a truncated Belavin–Polyakov skyrmion ansatz, it obtains a finite‑dimensional energy $F_N(\{\rho_n,\theta_n,\mathbf r_n\})$ and proves the existence of minimizers for fixed skyrmion centers. In the bilayer case without DMI, the authors fully characterize the global minimizers as two concentric Néel skyrmions with equal radii and anti‑parallel in‑plane components, with an explicit radius expressed via the Lambert $W$ function and zero center separation. The results are supported by micromagnetic simulations, revealing a strong, short-range attractive interaction that stabilizes bound skyrmion pairs and offering design insights for room‑temperature skyrmion devices.

Abstract

This paper explores the energy landscape of ferromagnetic multilayer heterostructures that feature magnetic skyrmions -- tiny whirls of spins with non-trivial topology -- in each magnetic layer. Such magnetic heterostructures have been recently pursued as possible hosts of room temperature stable magnetic skyrmions suitable for the next generation of low power information technologies and unconventional computing. The presence of stacked skyrmions in the adjacent layers gives rise to a strongly coupled nonlinear system, whereby the induced magnetic field plays a crucial stabilizing role. Starting with the micromagnetic modeling framework, we derive a general reduced energy functional for a fixed number of ultrathin ferromagnetic layers with perpendicular magnetocrystalline anisotropy. We next investigate this energy functional in the regime in which the energy is dominated by the intralayer exchange interaction and formally obtain a finite-dimensional description governed by the energy of a system of one skyrmion per layer as a function of the position, radius and the rotation angle of each of theses skyrmions. For the latter, we prove that energy minimizers exist for all fixed skyrmion locations. We then focus on the simplest case of stray field-coupled ferromagnetic bilayers and completely characterize the energy minimizers. We show that the global energy minimizers exist and consist of two stray field-stabilized Néel skyrmions with antiparallel in-plane magnetization components. We also calculate the energy of two skyrmions of equal radius as a function of their separation distance.

Figures (8)

• Figure 1: Schematics of the geometry of a multilayer system. The heterostructure consists of $N$ repeats of a sandwich of thickness $ad$ in the form of a layer of one non-magnetic material (NM a), followed by a layer of a ferromagnet (FM) of thickness $d$, followed by a layer of another non-magnetic material (NM b), from the bottom to the top.
• Figure 2: An example of a skyrmion configuration from \ref{['eq:prof']} with three distinct radii and centers in a ferromagnetic trilayer. The skyrmion in the bottom layer is of Néel type ($\theta_1 = 0$), the skyrmion in the top layer is of Bloch type ($\theta_3 = \pi/2$), and the skyrmion in the middle layer is of mixed type ($\theta_2 = \pi/4$).
• Figure 3: The plots of $F_{vv}(\alpha, 0)$ (a), $F_{ss}(\alpha, 0)$ (b), and $F_{vs}(\alpha, 0)$ (c).
• Figure 4: Plot of $F(\rho)$ when $\rho \in (0, \rho_0)$ with $\rho_0 = \bar{L}_0^{-1}$ for $\bar{\delta} = 0.25$.
• Figure 5: An illustration of the stray field interactions between the volume and surface charges in a bilayer for a skyrmion with anti-parallel in-plane magnetization components (i.e., with $m_1^\| = m_2^\|$ and $\mathbf{m}_1^\perp = -\mathbf{m}_2^\perp$): (a,c,e) clockwise rotation in the bottom layer; (b,d,f) anti-clockwise rotation in the bottom layer. The volume charge density $\uprho_\mathbf{m}^\mathrm{vol} = -\nabla_\perp \cdot \mathbf{m}^\perp$ in one layer is indicated in blue (negative) and red (positive), and its associated magnetic field lines are shown by the lines with arrows going from red to blue regions. Only the out-of-plane component $m_n^\|$ of the magnetization in the other layer that contributes to the volume-surface interaction is shown in (a-d), while the corresponding full magnetization profiles are shown in (e,f). In (a,c), the out-of-plane magnetic moments of one layer point agains the field lines from the other layer, resulting in a higher energy. In (b,d), the out-of-plane magnetic moments of one layer point along the field lines from the other layer, resulting in a lower energy. The total surface-surface interaction energies are identical in both cases, and the total volume-volume interaction energy is asymptotically zero.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof