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We study the anomalous Hall effect in monolayers of transition-metaldichalcogenides 2H-$MX_2$ ($M$=Mo, W, and $X$=S, Se, Te) under a proximityeffect of ferromagnetic substrate. If a proximity-induced exchange field isintroduced, the spin-polarized energy bands in $K$ and $K'$ valleys are shiftedin the opposite directions, and it causes the Hall effect by breakingtime-reversal symmetry. The induced Hall effect is the most prominent in thevalence band which has a large intrinsic spin splitting. Moreover, we find thattilting the magnetization from the perpendicular direction gives rise to asensitive change in the Hall conductivity only in the electron side, and it isattributed to the mixing of the Berry phase by the in-plane field in the nearlydegenerate conduction bands.

Anomalous Hall effect in 2$H$-phase transition-metal dichalcogenide monolayers on ferromagnetic substrates