Generation of modulated magnetic structures based on cluster multipole expansion: Application to $\alpha$-Mn and $\mathrm{Co}{M}_3{\mathrm{S}}_6$

We present a systematic method to automatically generate symmetry-adapted magnetic structures for a given crystal structure and general propagation vector $\mathit{k}$ as an efficient approach of the analysis of complex modulated magnetic structures. The method is developed as an extension of the generation scheme based on the multipole expansion, which was demonstrated only for the propagation vector $\mathit{k}=\mathit{0}$ [M.-T. Suzuki et al., Phys. Rev. B 99, 174407 (2019)]. The symmetry-adapted magnetic structures characterized with an ordering vector $\mathit{k}$ are obtained by mapping the multipole magnetic alignments on a virtual cluster to the periodic crystal structure with the phase factor for the wave vector $\mathit{k}$. This method provides all magnetic bases compatible with irreducible representations under a $\mathit{k}$ group for a given crystal structure and wave vector $\mathit{k}$. Multiple-$\mathit{k}$ magnetic structures are derived from a superposition of single-$\mathit{k}$ magnetic bases related to the space group symmetry. We apply the scheme to deduce the magnetic structures of $\alpha$-Mn and $\mathrm{Co}{M}_3{\mathrm{S}}_6$ ($M=\mathrm{Nb}$, Ta), in which the large anomalous Hall effect has recently been observed in antiferromagnetic phases, and identify the magnetic structures inducing anomalous Hall effect without net magnetization. The physical phenomena originating from emergent multipoles in the ordered phases are also discussed based on the Landau theory.

Figure 1

(a) Ferromagnetic and (b) antiferromagnetic structures in the unit cell with $\mathit{k}=(\frac{1}{2},0,0)$ on a zigzag chain.

Figure 2

Virtual clusters in space group $P4bm$.

Figure 3

Transformation properties of atoms on $2b$ site in a crystallographic system with space group $P4bm$. The arrow represents primitive translation vector ${\mathit{R}}_{i_{\zeta }}^{\alpha }$ associated with each space group operator (see main text).

Figure 4

Outline of magnetic bases generation.

Figure 5

[(a), (c), and (e)] Single-$\mathit{k}$ and [(b), (d), and (f)] double-$\mathit{k}$ magnetic structures with $\mathit{k}=(\frac{1}{2},0,0)$ and $(0,\frac{1}{2},0)$ on the square lattice. The representation matrices for symmetry operations $\left\{E\right|\mathit{0}\}$, $\left\{C_{4z}\right|\mathit{0}\}$, and $\left\{C_{4z}\right|{\mathit{R}}_a\}$ with ${\mathit{R}}_a=(1,0,0)$ are also shown.

Figure 6

Triple-$\mathit{k}$ magnetic structures on the FCC lattice induced from magnetic structures with $\mathit{k}=(1,0,0)$ having (a) $A_{2g}^{-}$ and (b) $E_g^{-}$ representations.

Figure 7

(a) Crystal structure and (b) first Brillouin zone of $\alpha$-Mn. Mn-I, -II, -III, and -IV sites are separately shown in (c), (d), (e), and (f), respectively, where body-centered sites are not shown.

Figure 8

(a) Magnetic structure bases with $\mathit{k}=\mathit{0}$ on Mn-II sites in $\alpha$-Mn. (b) Comparison between magnetic basis having $A_2^{-}$-IRREP with $\mathit{k}=\mathit{0}$ and that with $\mathit{k}=(1,0,0)$. (c) Examples of magnetic structures without net magnetization compatible with finite AHE ${\sigma }_{xy}\ne 0$.

Figure 9

(a) Crystal structure of ${\mathrm{CoTa}}_3{\mathrm{S}}_6$ and (b) that on $ab$ plane with only Co atoms. (c) First Brillouin zone for ${\mathrm{CoTa}}_3{\mathrm{S}}_6$ and (d) that on $k_z=0$ plane, where several high-symmetry $\mathit{k}$ points are shown by dots.

Figure 10

Magnetic structure bases with single-$\mathit{k}(0,\frac{1}{2},0)$ in crystal structure of $\mathrm{Co}{M}_3{\mathrm{S}}_6$. The arrow and red (green) sphere represents magnetic moment in $ab$ plane and along $+z(-z)$ axis, respectively.

Figure 11

Triple-$\mathit{k}$ magnetic structures induced from single-$\mathit{k}$ magnetic bases with $B_1^{-}$ symmetry at $M$ points. (a) and (b) correspond to those from magnetic bases shown in Figs. 10 and 10, respectively.

Figure 12

Examples of linear combinations of triple-$\mathit{k}$ states shown in Fig. 11.