Meron configurations in easy-plane chiral magnets

We demonstrate the existence and study in detail the features of chiral bimerons which are static solutions in an easy-plane magnet with the Dzyaloshinskii-Moriya interaction. These are skyrmionic textures with an integer topological charge, and they present essential analogies to the meron configurations introduced in the context of quark confinement in the O(3) nonlinear $\sigma$ model. We employ a Möbius transformation to show that, for weak chirality, bimeron configurations approach Belavin-Polyakov solutions characterized by tightly bound vortex and antivortex parts of the same size. Stronger chirality induces a larger difference in the vortex and antivortex sizes and also a detachment of merons, suggesting the possibility for a topological phase transition. Exploiting the fact that bimerons of opposite topological charges may exist in the same material, we demonstrate numerically a mechanism to generate meron pairs.

Figure 1

Ground states and phase transitions for an easy-plane chiral ferromagnet vs model parameter $\lambda$. We have the polarized state for $\lambda <{\lambda }_{\text{NF}}$, a nonflat spiral for ${\lambda }_{\text{NF}}<\lambda <{\lambda }_{\text{F}}$, and a flat spiral for $\lambda >{\lambda }_{\text{F}}$.

Figure 2

Static bimeron solution of the model given in Eq. (2). Vectors show the projection of the magnetization $(m_1,m_2)$ on the plane. Contour plots for $m_3$ are colored, where red indicates $m_3>0$ and blue indicates $m_3<0$. The center of the bimeron, defined to be at the point where $m_1=-1$, is at the origin. The skyrmion number is $Q=1$. (a) A bimeron for parameter value $\lambda =0.4$. The centers of the vortex and antivortex, defined to be at the point where $m_3=\pm 1$, are located on the $y$ axis, at $y=1.66$ and $-1.01$, respectively, in units of ${\ell }_{\mathrm{w}}$. (b) A bimeron for $\lambda =0.32$. The vortex center is at $y=0.71$ and the antivortex at $y=-0.56$.

Figure 3

Distances of the vortex (upper, blue line) and the antivortex (lower, red line) centers from the bimeron center as a function of the parameter $\lambda$. Points present numerical results, connected by solid lines. For small $\lambda$, the vortex and the antivortex are progressively placed more symmetrically on either side of the bimeron center. For $\lambda \ge 0.5$, at the point of transition to the nonflat spiral, the vortex is expected to get completely detached.

Figure 4

The energy of the bimeron as a function of the parameter $\lambda$. Results from numerical simulations are given by circles. The dotted line shows the asymptotic result as obtained analytically in Eq. (10) for the axially symmetric skyrmions, and it is shown for comparison.

Figure 5

(a) Vector and contour plots for the Möbius-transformed configuration given in Eq. (13) for the bimeron solutions of Fig. 2 for (a) $\lambda =0.4$, and (b) $\lambda =0.32$. The transformation has produced a skyrmion. Blue and red indicate $m_3>0$ and $m_3<0$, respectively. The antivortex has been mapped to the center of the skyrmion and the vortex to its periphery. The contours in the red region and in the central part of the blue region are approximately circles.

Figure 6

(a) The dotted line shows the path for $v=0$ on the $z$ plane for $\lambda =0.40$ (left) and $\lambda =0.32$ (right). (b) The relative scaling $\gamma$ [see Eq. (14)] is calculated along the $u$ axis (for $v=0$) for $\lambda =0.40$ and $0.32$. The value of $\gamma$ obtained at $u=0$ indicates the antivortex radius $R_2=\gamma R<R$, and the value at $\left|u\right|\to \infty$ indicates the vortex radius $R_1=R/\gamma <R$. They are both found to be smaller than the radius $R$ given in Eq. (9) for the exchange model.

Figure 7

Contour plots for the topological density $q$, defined in Eq. (6), of the bimeron solutions shown in Fig. 2 for (a) $\lambda =0.4$ and (b) $\lambda =0.32$. The same number of contours is plotted in both figures.

Figure 8

A bimeron solution of the model given in Eq. (2) for $\lambda =0.4$, where the vortex points down and the antivortex points up. Contour plots for $m_3$ are colored, where red indicates $m_3>0$ and blue indicates $m_3<0$. The center of the bimeron, defined to be at the point where $m_1=-1$, has been placed at the origin. The skyrmion number is $Q=1$.

Figure 9

Spin torque is applied on an almost polarized state. The following snapshots are shown. (a) A vortex-antivortex is generated, after the application of spin torque with $\beta =-8$ in a circular region with radius 2 for 20 time units. Following this stage, the polarization is reversed by setting $\beta =8$, and we show snapshots at (b) $t=20.6$, where a second vortex-antivortex pair with polarity down and a third one with polarity up are created, (c) $t=22.0$, where a bimeron has been formed, and (d) $t=28.0$, where only a single bimeron has remained in the system.