  Intralayer deformation in van der Waals (vdW) heterostructures is generally
assumed to be negligible due to the weak nature of the interactions between the
layers, especially when the interfaces are found incoherent. In the present
work, graphene-phosphorene vdW-heterostructures are investigated with the
Density Functional Theory (DFT). The challenge of treating nearly
incommensurate (very large) supercell in DFT is bypassed by considering
different energetic quantities in the grand canonical ensemble, alternative to
the formation energy, in order to take into account the mismatch elastic
contribution of the different layers. In the investigated heterostructures, it
is found that phosphorene contracts by ~4% in the armchair direction when
compared to its free-standing form. This large contraction leads to important
changes in term of electronic properties, with the direct electronic optical
transition of phosphorene becoming indirect in specific vdW-heterostructures.
More generally, such a contraction indicates strong substrate effects in
supported or encapsulated phosphorene -neglected hitherto- and paves the way to
substrate-controlled stress- tronic in such 2D crystal. In addition, the
stability of these vdW-heterostructures are investigated as a function of the
rotation angle between the layers and as a function of the stacking
composition. The alignment of the specific crystalline directions of graphene
and phosphorene is found energetically favored. In parallel, several several
models based on DFT-estimated quantities are presented; they allow notably a
better understanding of the global mutual accommodation of 2D materials in
their corresponding interfaces, that is predicted to be non-negligible even in
the case of incommensurate interfaces.
