Supplement Number 1

See the supplementary materials for the phonon dispersion of the pristine SMoSiN$_{2}$ monolayer; the calculated electronic structure of the pristine CrCl$_{3}$ monolayer; the electronic band structures of the pristine SMoSiN$_{2}$ monolayer; the magnified electronic structure of the valence band and conduction band of the pristine SMoSiN$_{2}$ monolayer at the $K/K^{\prime}$ valley, and its Berry curvature distribution at the valley; the electronic band structure of the bottom-stacked SMoSiN$_{2}$/CrCl$_{3}$ vdW heterostructure; the magnified electronic structure of the valence band and conduction band at the valley for the top-stacked and bottom-stacked SMoSiN$_{2}$/CrCl$_{3}$ vdW heterostructure; the work function, difference of electron density, and band alignment of the top-stacked and bottom-stacked vdW heterostructures; the Berry curvature of the vdW heterostructure; electron belonging to the Mo atom of the top-stacking with respect to the in-plane sliding along $\vec{a}$ direction; the valley splittings of $\Delta_{V}$ and $\Delta_{C}$ of the top-stacking with respect to the in-plane sliding; for the top-stacking, the fitted curve of magnetic moments of the Mo atoms and the charge transfer between the two sublayers varied with respect to the vertical strain engineering; the electron belonging to the S atom varied with respect to the strain engineering.