Cluster multipole theory for anomalous Hall effect in antiferromagnets

We introduce a cluster extension of multipole moments to discuss the anomalous Hall effect (AHE) in both ferromagnetic (FM) and antiferromagnetic (AFM) states in a unified framework. We first derive general symmetry requirements for the AHE in the presence or absence of the spin-orbit coupling by considering the symmetry of the Berry curvature in $\mathit{k}$ space. The cluster multipole (CMP) moments are then defined to quantify the macroscopic magnetization in noncollinear AFM states as a natural generalization of the magnetization in FM states. We identify the macroscopic CMP order which induces the AHE. The theoretical framework is applied to the noncollinear AFM states of ${\mathrm{Mn}}_3\mathrm{Ir}$, for which an AHE was predicted in a first-principles calculation, and ${\mathrm{Mn}}_{3}Z\left(Z=\mathrm{Sn},\mathrm{Ge}\right)$, for which a large AHE was recently discovered experimentally. We further compare the AHE in ${\mathrm{Mn}}_{3}Z$ and bcc Fe in terms of the CMP. We show that the AHE in ${\mathrm{Mn}}_{3}Z$ is characterized by the magnetization of a cluster octupole moment in the same manner as that in bcc Fe characterized by the magnetization of the dipole moment.

Figure 1

Transformation of the Berry curvature under the symmetry operators. (a) Operators which reverse the sign of the Berry curvature and (b) the threefold rotation along the $z$ axis.

Figure 2

Time-reversal operation and global spin rotation for (a) a coplanar spin configuration and (b) a noncoplanar spin configuration in the triangular system.

Figure 3

AFM structures characterized by the cluster octupole moments of the $D_{6h}$ IREPs in Table 4.

Figure 4

(a) Crystal structure of ${\mathrm{Mn}}_3\mathrm{Ir}$. (b) The AFM structures characterized by the cluster octupole moments of $T_{1g}$ CMP with the relation $T_x^{\alpha }=T_y^{\alpha }=T_z^{\alpha }$, which has been recognized experimentally, and of $T_{2g}$ CMP with $T_x^{\beta }=T_y^{\beta }=T_z^{\beta }$ in the Mn cluster of ${\mathrm{Mn}}_3\mathrm{Ir}$.

Figure 5

(a) Crystal structure of ${\mathrm{Mn}}_{3}Z(Z=\mathrm{Sn},\mathrm{Ge})$ with the Mn clusters defined by the space group (see text). (b) Spin configuration of the Mn atoms in ${\mathrm{Mn}}_3\mathrm{Sn}$. The AFM1 and AFM2 spin structures are experimentally realized depending on the direction of magnetic fields along the $x$ and $y$ directions, respectively [45]. The lowest-rank CMP moments characterizing the spin configurations are also shown for each Mn cluster.

Figure 6

Local magnetic moment dependence of the cluster octupole moments for the Mn clusters in ${\mathrm{Mn}}_3\mathrm{Sn}$ and ${\mathrm{Mn}}_3\mathrm{Ge}$. The parameters for the crystal structures are described in Sec. 4a. $T_x^{\gamma }$ and $T_{xyz}$ ($T_y^{\gamma }$ and $T_z^{\beta }$) are for AFM1 (AFM2) in Fig. 5.

Figure 7

BZ and the energy bands for (a) the FM state of bcc Fe and (b) the AFM1 state of ${\mathrm{Mn}}_3\mathrm{Sn}$. The red and green lines are the energy bands obtained from the first-principles calculations and from the Wannier interpolation, respectively.

Figure 8

Local magnetic moment (cluster dipole moment) dependence of the AHC in the FM states of bcc Fe. The solid line indicates the local magnetic moment obtained by the first-principles calculation.

Figure 9

Fe $d$ spin-projected DOS for the FM states of bcc Fe with the magnetic moments (a) $2.22{\mu }_B$ and (b) $2.86{\mu }_B$ corresponding to $\lambda =1.0$ and 2.0, respectively. (c) and (d) ${\mathrm{\Omega }}_{\mathrm{sum}}\left(\mathit{k}\right)$ (see text) on the $k_y=0$ plane corresponding to the states in (a) and (b), respectively. ${\mathrm{\Omega }}_{\mathrm{sum}}$ vectors are colored by the weight of ${\mathrm{\Omega }}_{\mathrm{sum}}^z$, which contributes to the AHC.

Figure 10

Mn cluster $T_x^{\gamma }$ octupole moment dependence of the AHC in AFM1 states of ${\mathrm{Mn}}_3\mathrm{Sn}$ and ${\mathrm{Mn}}_3\mathrm{Ge}$. The solid and dashed lines indicate the local magnetic moments obtained with the first-principles calculations for ${\mathrm{Mn}}_3\mathrm{Sn}$ and ${\mathrm{Mn}}_3\mathrm{Ge}$, respectively.

Figure 11

Projected DOS for the AFM states of ${\mathrm{Mn}}_3\mathrm{Sn}$ with the Mn $T_x^{\gamma }$ CMP (local magnetic moment): (a) $11.8{\AA }^2{\mu }_B\left(1.28{\mu }_B\right)$ and (b) $31.2{\AA }^2{\mu }_B\left(3.39{\mu }_B\right)$, corresponding to $\lambda =0.20$ and 1.0, respectively. (c) and (d) ${\mathrm{\Omega }}_{\mathrm{sum}}\left(\mathit{k}\right)$ (see text) on the $k_z=0$ plane corresponding to the states in (a) and (b), respectively. ${\mathrm{\Omega }}_{\mathrm{sum}}$ vectors are colored by the weight of ${\mathrm{\Omega }}_{\mathrm{sum}}^x$, which contributes to the AHC.