Stable $2R$ van der Waals heterostructures of NbS${}_2$ and $M$Se${}_2$ for $M$=Mo and W

In this letter, we investigate the stable and commensurate van der Waals heterostructures of metallic and semiconducting $1H$ transition-metal dichalcogenides, NbS${}_2$ and MoSe${}_2$ (WSe${}_2$), which possess almost the same lattice constant of the pristine honeycomb structure. In the most stable structure, the metallic and semiconducting layers are stacked in a similar manner to $3R$ stacking but the period is a pair of a metallic layer and a semiconducting layer. The heterostructure aligns the spin-polarization in each valley among all layers and induces spin-selective charge transfer between the metallic and semiconducting layers. Especially in hetero-trilayers, the electronic spin is conserved due to mirror symmetry along the out-of-plane axis in contrast to the $3R$ stacking structure. A drastic enhancement of spin Hall effect is numerically shown as an example of electronic spin transport phenomena in the hetero-trilayers.