Shape of a skyrmion

Abstract

We propose a method of determining the shape of a two-dimensional magnetic skyrmion, which can be parameterized as the position dependence of the orientation of the local magnetic moment, by using the expansion in terms of the eigenfunctions of the Schrödinger equation of a harmonic oscillator. A variational calculation is done, up to the next-to-next-to-leading order. This result is verified by a lattice simulation based on Landau-Lifshitz-Gilbert equation. Our method is also applied to the dissipative matrix in the Thiele equation as well as two interacting skyrmions in a bilayer system. and θ NNLO (ρ) can be expressed in terms of these constant numbers and J, B and D. The detailed calculation is given in the Appendix A. It is found that a ≈ 0.2021 + B J (−0.4364 B J + 0.1449 D J 2 ) B J 2 + 0.01388 B J D J 2 − 0.1148 D J 4 , b ≈ −0.09900 + B J (0.2026 B J − 0.06728 D J 2 ) B J 2 + 0.01388 B J D J 2 − 0.1148 D J 4 , ω NNLO ≈ B D 0.9594 B J 2 − 0.02628 B J D J 2 − 0.09704 D J 4 B J 2 + 0.04050 B J D J 2 − 0.1237 D J 4 2 . (16) B J 0.582005 D J 2 − 1.75302 B J B J 2 + 0.013875 B J D J 2 − 0.114832 D J 4 .

The relation between the radii and the densities of magnetic skyrmions

“…In principle, θ(r) can be expanded using any Hilbert space. Since the wave-function of the ground state of the harmonic oscillator and the numerical solution of the Euler-Lagrange equation of a skyrmion are close in shape [17], we use the Hilbert space of harmonic oscillator to expand θ(r). We do not require q¢ = r 0 ( ) as in [17] because q¢ ¹ r 0 ( ) is allowed by the Euler-Lagrange equation, therefore the eigen-functions of odd energy levels are also included.…”

“…Since the wave-function of the ground state of the harmonic oscillator and the numerical solution of the Euler-Lagrange equation of a skyrmion are close in shape [17], we use the Hilbert space of harmonic oscillator to expand θ(r). We do not require q¢ = r 0 ( ) as in [17] because q¢ ¹ r 0 ( ) is allowed by the Euler-Lagrange equation, therefore the eigen-functions of odd energy levels are also included. To impose the boundary conditions θ(0) = π and θ(R) = 0, θ(r) to the next-to-next-to leading order can be written as where f n are eigen-functions of harmonic oscillator, ω and c are parameters to be determined.…”

“…Assuming the skyrmions are distributed homogenously, then N ≈ A/πR 2 , and p » s N A. It has been found that, by using the leading order ansatz θ LO , to minimize F yields ω = w 0 b 2 /d 2 where w 0 = 0.768548 is a constant [17]. By using equation (6), the average radius r s can be written as…”

“…A prerequisite for the use of skyrmions in devices is the knowledge of the relationship between the size of a skyrmion and parameters such as exchange strength, DMI strength and the strength of external magnetic field. Such a relationship can be investigated by solving the Euler-Lagrange equation of a skyrmion, for example numerically [16] or by using an ansatz [9], or by using the harmonic oscillation expansion [17], or by an asymptotic matching [18]. It has been noticed that the radius of a skyrmion in the skyrmion phase is much smaller than that of an isolated skyrmion [17].…”

“…Such a relationship can be investigated by solving the Euler-Lagrange equation of a skyrmion, for example numerically [16] or by using an ansatz [9], or by using the harmonic oscillation expansion [17], or by an asymptotic matching [18]. It has been noticed that the radius of a skyrmion in the skyrmion phase is much smaller than that of an isolated skyrmion [17]. Both the radii of an isolated skyrmion and the skyrmions in the skyrmion lattice were studied quantitatively in [19].…”