Skyrmions and anomalous Hall effect in a Dzyaloshinskii-Moriya spiral magnet

Monte Carlo simulation study of a classical spin model with Dzyalosinskii-Moriya interaction and the spin anisotropy under the magnetic field is presented. We found a rich phase diagram containing the multiple spin spiral (or Skyrme crystal) phases of square, rectangular, and hexagonal symmetries in addition to the spiral spin state. The Skyrme crystal states are stabilized by a spin anisotropy or a magnetic field. The Hall conductivity ${\sigma }_{xy}$ is calculated within the $sd$ model for each of the phases. Applying a magnetic field induces nonzero uniform chirality and the anomalous Hall conductivity simultaneously. The field dependence of ${\sigma }_{xy}$ is shown to be a sensitive probe of the underlying magnetic structure. Relevance of the present results to several recent experiments on MnSi is discussed.

Figure 1

Low temperature $\left(T=0.01\right)$ phase diagram of the spin model in Eq. (2) (2D version) with $K=\sqrt{6}$, $A_1=0.5$, and $\mathbf{H}=H\widehat{z}$. Phase boundaries are drawn on the basis of MC simulations at a large number of $\left(A_2,H\right)$ locations. Spin configurations are abbreviated as SS (spiral spin), and SC (spin crystal). Full spin polarization (SP) results at high field. The Skyrme crystal phases are further classified as ${\text{SC}}_1$ (unequal Bragg intensities), ${\text{SC}}_2$ (equal Bragg intensities), and ${\text{SC}}_h$ (hexagonal Bragg spots). The corresponding Bragg patterns are schematically shown. The Bragg peak at $\mathbf{k}=0$ emerges due to the field-induced uniform magnetization. On the vertical axis $H$ is divided by an arbitrary energy scale $1.5A_1+A_2$ for improved clarity of the plot.

Figure 2

(Color online) A plot of the spin configuration projected on the $xy$ plane $\left(S_i^x,S_i^y\right)$ in the three spin crystal ground states: (a) ${\text{SC}}_1$ at $\left(A_2,H\right)=\left(2.0,0.0\right)$, (b) ${\text{SC}}_2$ at $\left(A_2,H\right)=\left(3.0,0.0\right)$, and (c) ${\text{SC}}_h$ at $\left(A_2,H\right)=\left(0.0,2.0\right)$. At the bottom left of each figure are the plots of the Bragg intensity ${\left|⟨{\mathbf{S}}_{\mathbf{k}}⟩\right|}^2$ showing two $\left({\text{SC}}_1,{\text{SC}}_2\right)$ and three $\left({\text{SC}}_h\right)$ sets of modulation vectors. Shown at the bottom right are the plots of the local chirality ${\chi }_{\mathbf{r}}$. Bright (dark) regions correspond to Skyrmions (anti-Skyrmions).

Figure 3

(Color online) Upper panels: plot of uniform magnetization $⟨M⟩$ (blue circle) and uniform chirality $⟨\chi ⟩$ (red triangle) with varying field strength $H$, $\mathbf{H}=H\widehat{z}$, for (a) SS at $\left(A_1,A_2\right)=\left(0.5,0.0\right)$, (c) ${\text{SC}}_1$ at $\left(A_1,A_2\right)=\left(0.5,2.0\right)$, and (e) ${\text{SC}}_2$ at $\left(A_1,A_2\right)=\left(0.5,3.0\right)$, at three temperatures $T=0.1$ (top six figures), 0.5 (middle), and 1.1 (bottom). Lower panels: $⟨{\sigma }_{xy}⟩$ averaged over 100 MC-generated spin configurations are shown in (b), (d), and (f) at the corresponding temperatures and field strengths. Thermal fluctuations are indicated as error bars. Thermal-averaged $⟨{\sigma }_{xx}⟩$ are shown in gray for comparison.

Figure 4

(Color online) (a) Plot of the energies of spiral spins and Skyrme crystal spins in 2D for $J=1$, $K=\sqrt{6}$, $A_1=0.5$, and varying $A_2$. (b) Plot of the energies of spiral spins and Skyrme crystal spins in 3D for $J=1$, $K=3$, $A_1=0.5$, and varying $A_2$. In both instances the Skyrme crystal spin configuration becomes the lowest in energy for $A_2$ exceeding a certain critical value.