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We have carried out a comprehensive study of the intrinsic anomalous Halleffect and spin Hall effect of several chiral antiferromagnetic compounds,Mn$_3X$ ($X$ = Ge, Sn, Ga, Ir, Rh and Pt) by $ab~initio$ band structure andBerry phase calculations. These studies reveal large and anisotropic values ofboth the intrinsic anomalous Hall effect and spin Hall effect. The Mn$_3X$materials exhibit a non-collinear antiferromagnetic order which, to avoidgeometrical frustration, forms planes of Mn moments that are arranged in aKagome-type lattice. With respect to these Kagome planes, we find that both theanomalous Hall conductivity (AHC) and the spin Hall conductivity (SHC) arequite anisotropic for any of these materials. Based on our calculations, wepropose how to maximize AHC and SHC for different materials. The bandstructures and corresponding electron filling, that we show are essential todetermine the AHC and SHC, are compared for these different compounds. We pointout that Mn$_3$Ga shows a large SHC of about 600 $(\hbar/e)(\Omega\cdotcm)^{-1}$. Our work provides insights into the realization of strong anomalousHall effects and spin Hall effects in chiral antiferromagetic materials.

Strong, anisotropic anomalous Hall effect and spin Hall effect in chiral antiferromagnetic compounds Mn$_3X$ ($X$ = Ge, Sn, Ga, Ir, Rh and Pt)