Domain-wall skyrmion chain and domain-wall bimerons in chiral magnets

We construct domain-wall skyrmion chains and domain-wall bimerons in chiral magnets with an out-of-plane easy-axis anisotropy and without a Zeeman term coupling to a magnetic field. Domain-wall skyrmions are skyrmions trapped inside a domain wall, and they are present in the ferromagnetic (FM) phase of a chiral magnet with an out-of-plane easy-axis anisotropy. In this paper, we explore the stability of domain-wall skyrmions in the FM phase and in a chiral soliton lattice (CSL) or spiral phase, which is a periodic array of domain walls and antidomain walls arranged in an alternating manner. In the FM phase, the worldline of a domain-wall skyrmion is bent to form a cusp at the position of the skyrmion. We describe such a cusp using both an analytic method and numerical solutions, and find a good agreement between them for small Dzyaloshinskii-Moriya interactions. We show that the cusp grows toward the phase boundary with the CSL, and eventually diverges at the boundary. Second, if we put one skyrmion trapped inside a domain wall in a CSL, it decays into a pair of merons by a reconnection of the domain wall and its adjacent antidomain wall. Third, if we put skyrmions and antiskyrmions alternately in domain walls and antidomain walls, respectively, such a chain is stable.

Figure 1

Single domain-wall skyrmions using the moduli approximation. The top panels represent the profile of the magnetization vector given with the solutions (28) and (30), and the bottom panels their topological charge density. For those figures, we used $\kappa =1.0, {\mu }^2=3$, and $\vartheta =\alpha =\beta =0$. The name of the configurations, e.g., $\mathrm{DW}\times \mathrm{Kink}$, specifies the sign of $\mu$ and the sign in Eq. (28), where $\overline{\text{DW}}$ and $\overline{\text{Kink}}$ denote anti-DW and antikink, respectively.

Figure 2

The domain-wall skyrmion position measured from the domain wall itself as a function of $\left|\mu \right|/|\tilde{\kappa }|$.

Figure 3

Numerical solution of a domain-wall skyrmion, $Q=1$ skyrmion trapped in a domain wall, corresponding to the leftmost configuration, i.e., $\text{DW}\times \text{Kink}$ in Fig. 1. The left figure shows the topological charge density and the right represents the magnetization profile. We used for the simulation $\kappa =1.0, \mu =+\sqrt{3}$, and $\vartheta =0$.

Figure 4

Comparison of the numerical solution of the system (3) (orange) and analytic one in the effective theory (blue) for $\varphi$ and $X$. The parameters used are $\kappa =1.0, \mu =+\sqrt{3}$.

Figure 5

A bimeron in the CSL phase. We used the parameters $\kappa =1.1, \mu =1.0$, and $\vartheta =0.0$ for the numerical simulation. (a) shows the energy density, (b) the topological charge density, and (c) the magnetization vector with the same color code as Fig. 1. We can recognize two lumps with a symmetric form in (b). Since the total topological charge is unity, each lump should possess the charge of 1/2.

Figure 6

Quantities of the domain-wall skyrmion lattice solution with $\vartheta =0, \mu =1.0$, and $\kappa =1.1$. (a) shows the energy density, (b) topological charge density, and (c) the profile of the magnetization vector with the same color code as Fig. 1.We chose the plus sign in Eq. (45) and (47), and let $\alpha =\beta =0$.