Solitonlike magnetization textures in noncollinear antiferromagnets

We show that proper control of magnetization textures can be achieved in noncollinear antiferromagnets. This opens the versatile toolbox of domain-wall manipulation in the context of a different family of materials. In this way, we show that noncollinear antiferromagnets are a good prospect for applications in the context of antiferromagnetic spintronics. As in many noncollinear antiferromagnets, the order parameter field takes values in SO(3). By performing a gradient expansion in the energy functional we derive an effective theory that accounts for the physics of the magnetization of long-wavelength excitations. We apply our formalism to static and dynamic textures such as domain walls and localized oscillations, and identify topologically protected textures that are spatially localized. Our results are applicable to the exchange-bias materials ${\text{Mn}}_{3}X$, with $X=\text{Ir},\text{Rh},\text{Pt}$.

Figure 1

(a) Different configurations of a given triangle achieved through different rotations around the out-of-plane axes. An arbitrary configuration is encoded by a smooth distribution of such rotations. (b) Kagome lattice in the (111) plane in ${\text{Mn}}_3\text{Ir}$ where Mn atoms are at each corner of a basis triangle. The basis vectors ${\mathbf{n}}_1=(0,1,0), {\mathbf{n}}_2=(\sqrt{3}/2,-1/2,0)$, and ${\mathbf{n}}_3=(-\sqrt{3}/2,-1/2,0)$ defined at every point in the lattice are shown. The vectors ${\mathbf{e}}_i$ point towards the nearest neighbors of each site. These vectors are used in the gradient expansion in the continuum approximation and are defined as ${\mathbf{e}}_1=(cos\pi /3,sin\pi /3,0), {\mathbf{e}}_2=(cos\pi /3,-sin\pi /3,0)$, and ${\mathbf{e}}_3=(-1,0,0)$.

Figure 2

(a) Homogenous state with all the spins pointing toward the center of the triangles. This state is degenerated with the state with all the spins pointing away from the center of each triangle. (b) Dispersion relations for the spin-wave spectrum of the homogeneous state. Solid lines correspond to the case with anisotropy and describe two branches one being a flat band with zero group velocity and another with a Klein-Gordon-like dispersion. Dashed line represents the dispersion relations of the isotropic case.

Figure 3

(a) Typical shape of a domain wall in kagome lattice. (b) Numerical fit of the soliton solution, black dots are the result of numerical simulation, red line corresponds to fitted solution $\varphi =2{tan}^{-1}[exp(x/W)]$. (c) Width dependence on anisotropy. Black dots are numerical results while red dashed line is the fitted curve which has a slope equal to 1/2.

Figure 4

(a) Contraction of the width $W$ of a moving DW as function of speed $v$. The simulations were performed setting the easy axis anisotropy by $K=0.025J$, and the hard axis anisotropy by $K_z=0.1J$. The results of the Landau-Lifshitz equation shows perfect agreement with the Lorentz contraction factor $\sqrt{1-{(v/c)}^2}$, full line, that can be inferred from the sine-Gordon equation. (b) Time evolution of the orientation $\varphi$ in the case of a breather state with frequency $\omega =0.25$. The results correspond to the solution of the Landau-Lifshitz equation with the same parameters as in Fig. 4. The color code is the same as the one used for the DW. (c) Snapshot at time (1) and (2) showing the orientation of the local moments in the texture.

Figure 5

(a) Cartoon of a lump texture, given by the parametrization $\mathcal{R}(\eta ,\widehat{z}){\mathbf{n}}_{\mathbf{r}}$, where $\eta \left(r\right)=2arctan[exp(r-R)/W]$. This is an example of a trivial two-dimensional texture. While this solution is stable, it has no topological protection at all so it can be continuously deformed to the homogeneous state. (b) Cartoon of a class 1 disgyration. This solution is topologically protected because of the nontrivial homotopy ${\pi }_1\left[\text{SO}\left(3\right)\right]$, then it is not possible to reach the homogeneous state adiabatically. As this state is not localized, its energy grows with the size of the system.