Stacking effects on the electronic and optical properties of bilayer transition metal dichalcogenides MoS 2 , MoSe 2 , WS 2 , and WSe 2

Employing the random phase approximation we investigate the binding energy and Van der Waals (vdW) interlayer spacing between the two layers of bilayer transition metal dichalcogenides ${\mathrm{MoS}}_{2}$, ${\mathrm{MoSe}}_{2}$, ${\mathrm{WS}}_{2}$, and ${\mathrm{WSe}}_{2}$ for five different stacking patterns, and examine the stacking-induced modifications on the electronic and optical/excitonic properties within the GW approximation with a priori inclusion of spin-orbit coupling and by solving the two-particle Bethe-Salpeter equation. Our results show that for all cases, the most stable stacking order is the high symmetry $A{A}^{\ensuremath{'}}$ type, distinctive of the bulklike $2H$ symmetry, followed by the $AB$ stacking fault, typical of the $3R$ polytypism, which is by only 5 meV/formula unit less stable. The conduction band minimum is always located in the midpoint between K and $\ensuremath{\Gamma}$, regardless of the stacking and chemical composition. All $M{X}_{2}$ undergo an direct-to-indirect optical gap transition going from the monolayer to the bilayer regime. The stacking and the characteristic vdW interlayer distance mainly influence the valence band splitting at K and its relative energy with respect to $\ensuremath{\Gamma}$, as well as, the electron-hole binding energy and the values of the optical excitations.