Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds ${\mathrm{Mn}}_{3}X$ ($X=\mathrm{Ge}$, Sn, Ga, Ir, Rh, and Pt)

We have carried out a comprehensive study of the intrinsic anomalous Hall effect and spin Hall effect of several chiral antiferromagnetic compounds ${\mathrm{Mn}}_{3}X$ ($X$ = Ge, Sn, Ga, Ir, Rh and Pt) by ab initio band structure and Berry phase calculations. These studies reveal large and anisotropic values of both the intrinsic anomalous Hall effect and spin Hall effect. The ${\mathrm{Mn}}_{3}X$ materials exhibit a noncollinear antiferromagnetic order which, to avoid geometrical frustration, forms planes of Mn moments that are arranged in a Kagome-type lattice. With respect to these Kagome planes, we find that both the anomalous Hall conductivity (AHC) and the spin Hall conductivity (SHC) are quite anisotropic for any of these materials. Based on our calculations, we propose how to maximize AHC and SHC for different materials. The band structures and corresponding electron filling, that we show are essential to determine the AHC and SHC, are compared for these different compounds. We point out that ${\mathrm{Mn}}_3\mathrm{Ga}$ shows a large SHC of about 600 $(\hslash /e){\left(\mathrm{\Omega }\text{cm}\right)}^{-1}$. Our work provides insights into the realization of strong anomalous Hall effects and spin Hall effects in chiral antiferromagnetic materials.

Figure 1

Schematic illustrations of (a) the anomalous Hall effect and (b) the spin Hall effect from the viewpoint of spin-dependent Mott scattering. (c) The anomalous Hall effect in collinear FM, collinear AFM, chiral FM, and chiral AFM systems. ${\rho }_{xy},M,{\lambda }_{\mathrm{SOC}}$, and $S_i(S_j\times S_k)$ represent the anomalous Hall resistivity, magnetization, strength of SOC, and the scalar spin chirality, respectively.

Figure 2

Crystal lattice and magnetic structure for ${\mathrm{Mn}}_{3}X$ ($X$ = Ga, Ge, Sn, Rh, Ir, and Pt). (a) ${\mathrm{Mn}}_{3}X$ ($X$ = Ga, Ge, and Sn) display a hexagon lattice with Mn forming a Kagome sublattice stacking along the $c$ axis. The gray plane indicates the $M_y$ mirror plane of the symmetry operation $\left\{M_y\right|\tau =c/2\}$. The crystallographic $a$ and $c$ axes align with the $x$ and $z$ directions, respectively, with the $b$ axis lying inside the $xy$ plane. (b) Top view of the Mn Kagome-type lattice showing triangular and hexagonal arrangements of the Mn moments. Arrows represent the Mn magnetic moments, presenting a noncollinear AFM configuration. The mirror plane position is indicated by a black line. (c) The crystal structure of ${\mathrm{Mn}}_{3}X$ ($X$ = Rh, Ir, and Pt) has an FCC lattice. Three mirror planes are shown in gray. Here a mirror reflection combined with a time-reversal symmetry operation preserves the magnetic lattice. (d) The Mn sublattice also forms a Kagome-type configuration in ${\mathrm{Mn}}_{3}X$ ($X$ = Rh, Ir, and Pt), thereby forming a noncollinear AFM phase, but which is slightly different from the ${\mathrm{Mn}}_3\mathrm{Ge}$ family. The projections of three mirror planes are indicated by black lines. To match the hexagonal lattice conveniently, the Kagome plane is set as the $xy$ plane and the plane normal as the $z$ axis. Here $x$ is along the crystallographic $\left[1\overline{1}0\right]$, $y$ along $\left[11\overline{2}\right]$, and $z$ along [111].

Figure 3

Electronic band structure for (a) ${\mathrm{Mn}}_3\mathrm{Ga}$, (d) ${\mathrm{Mn}}_3\mathrm{Ge}$, and (g) ${\mathrm{Mn}}_3\mathrm{Sn}$. Energy dependent AHC ${\sigma }_{zx}$ for (b) ${\mathrm{Mn}}_3\mathrm{Ga}$, (e) ${\mathrm{Mn}}_3\mathrm{Ge}$, and (h) ${\mathrm{Mn}}_3\mathrm{Sn}$. Energy dependent SHC tensor elements of ${\sigma }_{xy}^z$ and ${\sigma }_{yx}^z$ for (c) ${\mathrm{Mn}}_3\mathrm{Ga}$, (f) ${\mathrm{Mn}}_3\mathrm{Ge}$, and (i) ${\mathrm{Mn}}_3\mathrm{Sn}$. ${\mathrm{Mn}}_3\mathrm{Ga}$ would have the same number of valence electrons as does ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$ if the Fermi level is shifted up to the blue dashed line in (a)–(c).

Figure 4

(a),(b) Berry curvature projected onto the $k_3\text{−}{k}_1$ plane for ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively. The Berry curvature in ${\mathrm{Mn}}_3\mathrm{Ge}$ is dominated by the four areas that are highlighted in yellow in (a). (c) Energy dispersion of ${\mathrm{Mn}}_3\mathrm{Ge}$ along $k_2$ with ($k_3,k_1$) fixed at the Berry curvature dominated point (0.127, 0.428), identified as the black dashed circle marked in (a). The band gaps are very small near $k_2=0.5$ and 1, which are noted by the small black dashed circles. (d) Band structure of ${\mathrm{Mn}}_3\mathrm{Sn}$ for the same reciprocal space cut as in (c). The band gaps are much larger in ${\mathrm{Mn}}_3\mathrm{Sn}$, as denoted by the larger black dashed circles. (e) The evolution of the Berry curvature ${\mathrm{\Omega }}^y$ of ${\mathrm{Mn}}_3\mathrm{Ge}$ corresponding to the band structure given in (c). The small band gaps around $k_2=0.5$ and 1 make larger contributions to the Berry curvature. (f) The magnitude of Berry curvature along the same path in ${\mathrm{Mn}}_3\mathrm{Sn}$ is negligibly small compared to that in ${\mathrm{Mn}}_3\mathrm{Ge}$.

Figure 5

(a) Angle-dependent spin current $Js$ arising from a charge current $J$ along the $x$ axis (the crystallographic $a$ lattice vector) in ${\mathrm{Mn}}_3\mathrm{Ge}$. $Js$ rotates inside the $yz$ plane. The largest spin Hall conductivity is when $Js$ is along the $y$ axis ($\theta =0^{\circ }$). The blue arrows represent the spin polarization directions of $Js$. (b) Schematic of $Js$ and $J$ with respect to the lattice orientation.

Figure 6

(a),(b) Spin Berry curvature projected onto the $k_3\text{−}{k}_1$ plane for ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively. The two compounds have similar distributions of the projected spin Berry curvature. (c),(d) Band structures of ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively, along $k_2$. The coordinates of $k_3$ and $k_1$ are fixed at (0.31, 0.15) and (0.29, 0.24) for ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively, as noted by the black dashed circles in (a) and (b). Small band gaps exist around the $k_2=1$ point for both ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, as marked by the black dashed circles. (e),(f) Corresponding spin Berry curvature evolutions along $k_2$ for ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively. The large spin Berry curvature mainly originates from the small band gaps in the band structures.

Figure 7

Electronic band structure for (a) ${\mathrm{Mn}}_3\mathrm{Rh}$, (d) ${\mathrm{Mn}}_3\mathrm{Ir}$, and (g) ${\mathrm{Mn}}_3\mathrm{Pt}$. Energy dependent AHC element of ${\sigma }^z$ for (b) ${\mathrm{Mn}}_3\mathrm{Rh}$, (e) ${\mathrm{Mn}}_3\mathrm{Ir}$, and (h) ${\mathrm{Mn}}_3\mathrm{Pt}$. Energy dependent SHC tensor elements ${\sigma }_{xx}^x,{\sigma }_{yz}^x$, and ${\sigma }_{xy}^z$ for (c) ${\mathrm{Mn}}_3\mathrm{Rh}$, (f) ${\mathrm{Mn}}_3\mathrm{Ir}$, and (i) ${\mathrm{Mn}}_3\mathrm{Pt}$. ${\mathrm{Mn}}_3\mathrm{Pt}$ would have the same number of valence electrons as ${\mathrm{Mn}}_3\mathrm{Rh}$ and ${\mathrm{Mn}}_3\mathrm{Ir}$ if the Fermi level shifts down to the blue dashed line in (g)–(i).

Figure 8

(a) Angle-dependent spin Hall conductivity for ${\mathrm{Mn}}_3\mathrm{Ir}$. The charge current $J$ flows along $x$ (i.e., $\left[1\overline{1}0\right]$) and the resulting spin current $Js$ is shown in the $yz$ plane. The SHC is largest when $Js\left|\right|\left[001\right]$ and smallest when $Js\left|\right|\left[111\right]$. (b) Schematic of $J$ and $Js$ within the crystal structure.