# Mechanism of Skyrmion Attraction in Chiral Magnets near the Ordering Temperatures

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## Abstract

**:**

## 1. Introduction

## 2. Phenomenological Theory

## 3. Solutions for High-Temperature Isolated Skyrmions

#### 3.1. Equations

#### 3.2. Asymptotic Behaviour of Skyrmion Solutions

#### 3.3. The Internal Structure of Confined Isolated Skyrmions

## 4. The Properties of Confined Isolated Skyrmions

#### 4.1. Collapse of Skyrmions at High Fields

#### 4.2. Inter-Skyrmion Attraction

#### 4.3. Phenomenon of Skyrmion Confinement

## 5. Three-Dimensional Attracting Skyrmions within the Conical Phase of Cubic Helimagnets

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(color online) (

**a**) The diagram on plane $(a,h)$ showing the regions with different types of skyrmion–skyrmion interaction according to the analysis of IS asymptotic behaviour: I—repulsive interaction between isolated skyrmions; II—attractive inter-skyrmion interaction; III—the region of skyrmion confinement. Solid line is defined by Equation (20): the turning points have the following coordinates—p ($-0.75,\sqrt{2}/4$), q ($0.06,0.032\sqrt{5}$), and u (−0.5, 0). Above the dotted line ${h}_{0}$ no isolated skyrmions can exist, since they collapse. (

**b**) Dependences of angular $\theta $ and longitudinal m order parameters on polar coordinate $\rho $ for isolated skyrmion in region I ($a=-1,\phantom{\rule{0.166667em}{0ex}}h=0.4$). (

**c**) $\theta \left(\rho \right)$ and $m\left(\rho \right)$ for isolated skyrmion in region II ($a=0.21,\phantom{\rule{0.166667em}{0ex}}h=0.05$). Shaded parts of oscillating profiles emphasize “wrong” and “right” rotational senses according to DMI. The oscillations are damped and end at the level of the homogeneous state.

**Figure 2.**(color online) Increasing magnetic field applied to isolated skyrmion ($a=-0.5$) localizes profiles $\theta \left(\rho \right)$ (

**a**) and leads to the disappearance of isolated skyrmions by squeezing out modulus ${m}_{1}$ in the center (

**b**). (

**c**) Snapshots showing skyrmion transformation into the homogeneous state. The modulus ${m}_{1}$ in the skyrmion center gradually decreases, passes through zero and appears on the other side being aligned along the field. Although one can obtain solutions for the whole process, skyrmions collapse once the modulus approaches zero value.

**Figure 3.**(color online) (

**a**) The skyrmion–skyrmion interaction energy ${\epsilon}_{\mathrm{int}}$ plotted in dependence on the distance L between the centers of two isolated skyrmions. The energy ${\epsilon}_{\mathrm{int}}$ exhibits a number of local minima which imply attracting skyrmion interaction; (

**b**) Dependencies of the modulus m (red line) and ${m}_{z}$-component of the magnetization (black line) in the cross-section of two interacting isolated skyrmions for $a=0.21,\phantom{\rule{0.166667em}{0ex}}h=0.048$ corresponding to the first (deepest) minimum of the interaction energy ${\epsilon}_{\mathrm{int}}$; (

**c**) color plots of ${m}_{z}$-component of the magnetization showing skyrmion pair configurations corresponding to the local minima of ${\epsilon}_{\mathrm{int}}$.

**Figure 4.**(color online) Magnetic structure of nonaxisymmetric skyrmions within the conical phase of bulk cubic helimagnets near the ordering temperature. Color plots of the out-of-plane magnetic moment, ${m}_{z}(x,y)$ (

**a**), and the modulus, $m(x,y)$ (

**b**), plotted for one cross-section with the fixed coordinate z. In (

**c**), an alternative way of representing the internal structure of a nonaxisymmetric skyrmion is used. After the magnetization components corresponding to the conical phase have been extracted, the 3D-model of the modulus m represents a cylinder-like core centered around the magnetization opposite to the field and exhibiting some reduced value and two coils with the magnetization along the field and enhanced modulus value. (

**d**) two-dimensional cross-cuts across the skyrmion center exhibiting distributions ${m}_{z}\left(y\right)$ and $m\left(y\right)$. After averaging over all energy profiles within one conical period, the energy distribution (

**e**) exhibits the following composite parts: the core with the positive energy density, the ring with the negative energy density, and a number of additional rings with alternating energy density.

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**MDPI and ACS Style**

Leonov, A.O.; Rößler, U.K.
Mechanism of Skyrmion Attraction in Chiral Magnets near the Ordering Temperatures. *Nanomaterials* **2023**, *13*, 891.
https://doi.org/10.3390/nano13050891

**AMA Style**

Leonov AO, Rößler UK.
Mechanism of Skyrmion Attraction in Chiral Magnets near the Ordering Temperatures. *Nanomaterials*. 2023; 13(5):891.
https://doi.org/10.3390/nano13050891

**Chicago/Turabian Style**

Leonov, Andrey O., and Ulrich K. Rößler.
2023. "Mechanism of Skyrmion Attraction in Chiral Magnets near the Ordering Temperatures" *Nanomaterials* 13, no. 5: 891.
https://doi.org/10.3390/nano13050891