# Geometrically Constrained Skyrmions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Chiral Magnetization States in a Helimagnetic Rectangular Platelet

#### 2.2. Phase Diagram of the Magnetization States

#### 2.3. Geometrically Constrained Skyrmions

## 3. Discussion

## 4. Materials and Methods

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DMI | Dzyaloshinksii-Moryia interaction |

LLG | Landau-Lifshitz-Gilbert equation |

FEM | Finite Element Method |

BEM | Boundary Element Method |

GPU | Graphical Processing Unit |

3D | three-dimensional |

ChB | chiral bobber |

## Appendix A. Energy Minimization

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**Figure 1.**Non-trivial magnetization states forming in a rectangular FeGe platelet (310 $\mathrm{n}$$\mathrm{m}$ × 180 $\mathrm{n}$$\mathrm{m}$) of varying thickness (between 5 $\mathrm{n}$$\mathrm{m}$ and 60 $\mathrm{n}$$\mathrm{m}$) at different external field values. The color code describes the out-of-plane component ${m}_{z}$ of the normalized magnetization, and the isosurfaces indicate the regions where ${m}_{z}$ is equal to zero. Structures of this type appear at different film thicknesses as the external magnetic field is applied along the negative z direction and is varied between 0 $\mathrm{m}$$\mathrm{T}$ and 900 $\mathrm{m}$$\mathrm{T}$. The arrangement of these configurations in the image corresponds, roughly, to the order in which the preferential configurations appear upon increasing the field strength.

**Figure 2.**(

**a**) A single magnetic skyrmion in a 60 $\mathrm{n}$$\mathrm{m}$ thick platelet, stabilized by a 650 $\mathrm{m}$$\mathrm{T}$ field, can form as a metastable state. The skyrmion is located precisely at the center of the thin-film element. The cylindrical shape in the middle is the ${m}_{z}=0$ isosurface, which represents the core region of the skyrmion. (

**b**) The normalized magnetization components along a central cutline—oriented parallel to the y direction and shown as a grey line in panel (

**a**)—display the profile of the skyrmion. From the distance between two consecutive zero values of ${m}_{z}$, we obtain the skyrmion core diameter to be 30 $\mathrm{n}$$\mathrm{m}$. The Bloch character of the skyrmion is evidenced by the antisymmetric shape of the x (azimuthal) component and the vanishing y (radial) component of the magnetization.

**Figure 3.**(

**a**) Phase diagram displaying the lowest-energy magnetic configuration in the FeGe platelet as a function of the film thickness and the external field strength. At high fields and large film thickness, the quasi-saturated state is the ground state. By lowering the film thickness, the formation of skyrmion structures tends to become energetically favorable. (

**b**) Energy density as a function of the film thickness in the case of the skyrmion state (blue line) and the quasi-saturated state (red line) in an external field of 650 $\mathrm{m}$$\mathrm{T}$.

**Figure 4.**(

**a**) A skyrmion is formed at the base of the cylindrical pocket. At the inner cylinder surface of the cavities, the magnetization circulates on closed loops, thereby facilitating the formation of the skyrmion in the center. The semitransparent representation of the surfaces shows the formation of the skyrmion in both pockets, on the top and bottom surfaces. The magnetic structure is displayed by arrows on the sample surfaces. Some of the arrows have been removed in order to improve the visibility of the structure. (

**b**) View on the simulated skyrmion structure from inside the film. The skyrmion core connects the bases of the cylindrical pockets in the positive z direction, while the surrounding volume is magnetized in the negative z direction. The core of the skyrmion is delimited by a cylindrical isosurface ${m}_{z}=0$, shown here as a weak, transparent contrast in order to preserve the view on the central magnetic structure. Only a small subset of the calculated local magnetization vectors is displayed.

**Figure 5.**Geometrically constrained skyrmions in FeGe platelets. By introducing circular pockets at specific positions, skyrmions can be artificially stabilized at positions that they would otherwise not attain. Panels (

**a**–

**e**) show examples in which each pocket contains a skyrmion. The geometric control, however, is not unlimited. Attempts to pack skyrmions too closely or to place them too close to the sample boundary can fail. This is shown in panel (

**f**), where skyrmions are stabilized only in the three central pockets, while the two outermost pockets remain empty.

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Pathak, S.A.; Hertel, R.
Geometrically Constrained Skyrmions. *Magnetochemistry* **2021**, *7*, 26.
https://doi.org/10.3390/magnetochemistry7020026

**AMA Style**

Pathak SA, Hertel R.
Geometrically Constrained Skyrmions. *Magnetochemistry*. 2021; 7(2):26.
https://doi.org/10.3390/magnetochemistry7020026

**Chicago/Turabian Style**

Pathak, Swapneel Amit, and Riccardo Hertel.
2021. "Geometrically Constrained Skyrmions" *Magnetochemistry* 7, no. 2: 26.
https://doi.org/10.3390/magnetochemistry7020026