Fig. 1: The “3 + 1” types of MX2 bilayers (M = Mo, W; X = S, Se): classification, interlayer-interaction-induced band edge change, and (super)cells.
a, b Three intrinsic types, (i) to (ii) to (iii), and one derived type-(iv) from types (i) and (ii) with tuning in real and/or energy spaces: type-(i), homobilayers with larger interlayer hopping integral (|t|) and zero energy difference (EDiff = 0) between layers, which prefer indirect gaps; type-(ii), heterobilayers with the same X atoms between layers which have larger |t| while EDiff ≠ 0; type-(iii), heterobilayers with different X atoms with supercells (see text), which have smaller |t| and EDiff ≠ 0, and prefer direct gaps; type-(iv) is the derived type from types (i) and (ii) by decreasing |t| and/or increasing EDiff between layers (see text). c Sketches for the interlayer-interaction-induced band edge change from monolayer to bilayer [ΔE in (b)] using VB@Γ (ΔEΓ) as the example for homobilayers (i) and heterobilayers (ii). d, e Supercells used for type-(iii) heterobilayers and for type-(iv) homo(hetero)-bilayers with 38.2° twist angle between layers as an example. In (d, e) supercells of the two layers are shown with a relative rotation, while in (a)(iii) and (a)(iv) the supercells are aligned and the atomic structure between layers is rotated relatively.
