^{1}, S. P. Kang

^{1}, Y. Z. Wu

^{2}and C. Won

^{1}

### Abstract

We demonstrate that a magnetic vortex generated by the Dzyaloshinskii–Moriya interaction is stable state in a spatially confined system. The properties of the locally confined magnetic vortex structure are investigated by performing Monte-Carlo simulations and theoretical calculations. The results reveal the relationship between the confinement size and the magnetic vortex size. We obtain the structural size of the most stable magnetic vortex as well as the critical size above which the magnetic vortex becomes unstable. The field required to flip the vortex core was estimated theoretically and compared with simulation results. The thermal stability of the magnetic vortex is also discussed.

This research was supported by a Grant from the National Research Foundation of Korea funded by the Korean Government (2012R1A1A2007524).

I. INTRODUCTION

II. RESULTS AND DISCUSSION

III. CONCLUSION

### Key Topics

- Rotating flows
- 41.0
- External field
- 13.0
- Magnetic anisotropy
- 11.0
- Coercive force
- 6.0
- Ground states
- 6.0

## Figures

(a) Magnetic LH structure obtained from the computational simulations under the Hamiltonian in Eq. (2) without an external field (*h* = 0). (b) Magnetic skyrmion lattice structure obtained at . (c) Magnetic vortex structure in the circular boundary without an external field. The length scale corresponds to approximately 12 nm in Fe_{0.5}Co_{0.5}Si. Arrows indicate the in-plane magnetization direction and brightness scale corresponds to out-of-plane magnetization.

(a) Magnetic LH structure obtained from the computational simulations under the Hamiltonian in Eq. (2) without an external field (*h* = 0). (b) Magnetic skyrmion lattice structure obtained at . (c) Magnetic vortex structure in the circular boundary without an external field. The length scale corresponds to approximately 12 nm in Fe_{0.5}Co_{0.5}Si. Arrows indicate the in-plane magnetization direction and brightness scale corresponds to out-of-plane magnetization.

Simulated magnetic vortex in a system with a radius of 1.2 . (a) The magnetism as a function of *ρ* in a circular structure of radius *r*. The magnetism is perpendicular at the core and rotates around the boundary. The in-plane component of the magnetism increases linearly with distance from the center. We define *R* as the position where the in-plane magnetism extrapolates to 1. The solid lines from Eq. (3) are used for the extrapolation. For*r* = 1.2 , *R* is obtained at 1.5 . (b) The graph and a grayscale plot (inset figure) of the energy density distribution of the magnetic vortex. The energy density from the simulation (solid dots) agrees well with the results from Eq. (4) (solid line), and it is lower than the (dashed line).

Simulated magnetic vortex in a system with a radius of 1.2 . (a) The magnetism as a function of *ρ* in a circular structure of radius *r*. The magnetism is perpendicular at the core and rotates around the boundary. The in-plane component of the magnetism increases linearly with distance from the center. We define *R* as the position where the in-plane magnetism extrapolates to 1. The solid lines from Eq. (3) are used for the extrapolation. For*r* = 1.2 , *R* is obtained at 1.5 . (b) The graph and a grayscale plot (inset figure) of the energy density distribution of the magnetic vortex. The energy density from the simulation (solid dots) agrees well with the results from Eq. (4) (solid line), and it is lower than the (dashed line).

The size of the magnetic structure *R* and averaged energy density as a function of the structure size *r*. The computational simulation data is indicated by the solid dots, and the results from Eqs. (6) and (7) are shown as the solid line. Dashed line in the *R-r* graph means the asymptote of Eq. (6) . The vortex structure breaks at . The inset figures show the magnetism configuration at , and .

The size of the magnetic structure *R* and averaged energy density as a function of the structure size *r*. The computational simulation data is indicated by the solid dots, and the results from Eqs. (6) and (7) are shown as the solid line. Dashed line in the *R-r* graph means the asymptote of Eq. (6) . The vortex structure breaks at . The inset figures show the magnetism configuration at , and .

(a) The coercivity field along the normal direction as a function of *r* at *T* = 0, 0.01, 0.1, and 0.2T_{C}. The dashed line is the maximum coercivity at a fixed temperature, which approaches as the temperature increases. (b) The hysteresis loops at for *T* = 0, 0.01, 0.1, 0.2, and 0.3T_{C}. The coercivity field exists up to *T* = 0.3T_{C}.

(a) The coercivity field along the normal direction as a function of *r* at *T* = 0, 0.01, 0.1, and 0.2T_{C}. The dashed line is the maximum coercivity at a fixed temperature, which approaches as the temperature increases. (b) The hysteresis loops at for *T* = 0, 0.01, 0.1, 0.2, and 0.3T_{C}. The coercivity field exists up to *T* = 0.3T_{C}.

The phase map for the energy ground state after the introduction of uniaxial anisotropy along *z* direction. The dotted circle indicates that the ground state has zero total in-plane magnetization ( ) and non-zero out-of-plane magnetization ( ). The solid circle indicates that the ground state has non-zero in-plane magnetization. The half-filled circle indicates that the ground state is a mixed state. The inset figures show the details of magnetization configurations. The sizes of structure in the inset figures are set to be same, though their physical sizes are different.

The phase map for the energy ground state after the introduction of uniaxial anisotropy along *z* direction. The dotted circle indicates that the ground state has zero total in-plane magnetization ( ) and non-zero out-of-plane magnetization ( ). The solid circle indicates that the ground state has non-zero in-plane magnetization. The half-filled circle indicates that the ground state is a mixed state. The inset figures show the details of magnetization configurations. The sizes of structure in the inset figures are set to be same, though their physical sizes are different.

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