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Inspired by the observation of the extremely large anomalous Hall effect inthe absence of applied magnetic fields or uniform magnetization inCoNb$_3$S$_6$ [Nature Comm. 9, 3280 (2018); Phys. Rev. Research 2, 023051(2020)], we perform a first-principles study of this and related compounds ofthe $M$Nb$_3$S$_6$ type with different transition metal $M$ ions to determinetheir magnetic orders and the anomalous Hall conductivity (AHC). We find thatnon-coplanar antiferromagnetic ordering is favored relative to collinear orcoplanar order in the case of $M$=Co, Fe and Ni, while ferromagnetic orderingis favored in MnNb$_3$S$_6$ at low temperatures. The AHC in these materialswith non-coplanar spin ordering can reach about $e^2/h$ per crystalline layer,while being negligible for coplanar and collinear cases. We also find that theAHC depends sensitively on doping and reaches a maximum for intermediate valuesof the local spin exchange potential between 0.3 and 0.8 eV. Our AHC resultsare consistent with the reported Hall measurements in CoNb$_3$S$_6$ and suggesta possibility of similarly large anomalous Hall effects in related compounds.

First-principles study of magnetic states and the anomalous Hall conductivity of $M$Nb$_3$S$_6$ ($M$=Co, Fe, Mn, and Ni)