Density functionals combined with van der Waals corrections for graphene adsorbed on layered materials

Standard density functionals like Perdew-Burke-Ernzerhof (PBE) or Strongly Constrained and Appropriately Normed (SCAN) need a correction to account for long-range van der Waals (vdW) interaction. The damped Zaremba-Kohn model (dZK) [J. Tao, H. Tang, A. Patra, P. Bhattarai, and J. P. Perdew, Phys. Rev. B 97, 165403 (2018)] starts from a formula for the vdW interaction of a distant atom with a solid surface, both with known dielectric properties, damps this formula at short range, and then treats an adsorbed molecule or atomic layer as a collection of renormalized atoms. We extend this model to graphene adsorbed on semiconducting layered materials [bulk graphite and hBN, and multilayer transition metal dichalcogenides (TMD)] by including the ${C}_{4}$ asymptotic term and multiple electrostatic image effects due to the two surfaces of the substrate slabs in the vdW calculations. The resulting SCAN-vdW-dZK and PBE-vdW-dZK give approximately the same results for the systems considered here, in agreement with available reference values. The predicted binding energies are roughly 25% lower than those from SCAN+rVV10 (revised Vydrov and Van Voorhis 2010), and $\ensuremath{\sim}15%$ lower than those from PBE+rVV10. Since SCAN+rVV10 usually overbinds, the predicted binding energies by SCAN-vdW-dZK and PBE-vdW-dZK are expected to be closer to the true values. The predicted equilibrium binding distances from SCAN-vdW-dZK and PBE-vdW-dZK are slightly larger ($\ensuremath{\sim}0.1\phantom{\rule{0.16em}{0ex}}\AA{}$) than those from SCAN+rVV10, and close to those from SCAN. The binding energy depends upon the number of substrate layers more strongly in vdW-dZK than in rVV10. The ${C}_{4}$-term contributions can be 40% of the total vdW interactions, and the ${C}_{5}$ term contributes about 10%. The effects of images and the back surfaces of slabs can contribute about 4--10%. The vdW interaction energy power laws from the vdW-dZK model for graphene adsorbed on multilayer $\mathrm{Mo}{\mathrm{S}}_{2}$ show slowly varying decay to the pairwise exponent \ensuremath{-}4 with increasing separation $D$, very similar to those obtained from the random-phase approximation and renormalization group approaches by Ambrosetti et al. Both the PBE-vdW-dZK and SCAN-vdW-dZK give a greater increase in interlayer binding energy when the TMD substrate changes from monolayer to four-layer for the graphene/TMD adsorption systems. This is consistent with the relevant electron-energy-loss spectroscopy experimental results showing increased dielectric response from the substrate with increasing substrate layer number.