Stacking effects on the electronic and optical properties of bilayer transition metal dichalcogenidesMoS2,MoSe2,WS2, andWSe2

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.