  Two-dimensional magnetism and valleys have recently emerged as two
significant research areas, with intriguing properties and practical uses in
advanced information technology. Considering the importance of these two areas
and their couplings, controllable creations of both the magnetism and valley
polarization are highly sought after. Based on first-principles calculations,
we propose a new class of two-dimensional monolayers with a chemical formula of
MAZ$_3$, which is viewed as a 2H-MZ$_2$ trilayer passivated by the A-Z bilayer
on its one side. Taking VSiN$_3$ as an example, the MAZ$_3$ monolayers are
found to exhibit tunable magnetism and valleys. For the intrinsic VSiN$_3$
monolayer, it is a non-magnetic semiconductor, with multiple degenerate valleys
and trigonal warping near $K_\pm$ points in the band structure. Besides, the
bands have spin splittings owing to the spin-orbit coupling. Under a moderate
carrier doping, the monolayer becomes a Stoner ferromagnet, which enhances the
spin splittings of the valence band and generates valley splittings. Moreover,
the Berry curvature is valley contrasting, leading to distinct valley-spin
related anomalous Hall currents as the doping concentration increases. Our work
opens up new way to modulate the spin splittings and valley splittings via
electric means, and provides opportunities for exploring advanced spintronic
and valleytronic devices.
