Fig. 3: DFT results for the interlayer interaction between a hydrogen-passivated silicon layer and (supported) graphene.

a Plane-average charge density differences upon the formations of tip and monolayer graphene complex (Si-mono) or tip and monolayer graphene on Cu substrate complex (Si-mono@Cu), Δρ (Δρ is defined as the charge density of the coupled system subtracts those of the isolated components). The vertical positions of the atomic layers in a are marked by the dashed lines. b Surface charge density difference (0.00015 e/A³). Charge accumulation and depletion regions are labeled by yellow and green isosurfaces, respectively. c Binding energy densities as a function of interlayer distance d, defined as the vertical separation between the top graphene and the bottom hydrogen layer of the passivated silicon layer shown in b. The short and long dashed lines denote the pairwise and PBE energies, respectively. d Adhesive pressures as a function of d. e Ratios of equilibrium binding energies (left) and critical adhesive pressures (right) of the tip-mono@Cu interfaces relative to those of the tip-bulk interface, calculated by the pairwise and MBD theories. The error bar represents the standard deviation of the experiment.