Commensurate-incommensurate solid transition in the $^{4}$He monolayer on a single $\gamma $-graphyne sheet

We have performed path-integral Monte Carlo calculations to study $^{4}$He adsorption on $\gamma $-graphyne. Assuming the $^{4}$He-substrate interaction described by a pairwise sum of empirical helium-carbon interatomic potentials, we find that unlike $\alpha $-graphyne [1], a single sheet of $\gamma $-graphyne is not permeable to $^{4}$He atoms despite its large surface area. One-dimensional density distribution shows layer-by-layer growth of $^{4}$He on $\gamma $-graphyne. Partially-filled $^{4}$He monolayers are found to exhibit different commensurate structures depending on the helium coverage; it shows a C$_{2/2}$ commensurate structure at the areal density of 0.0491 {\AA}$^{-2}$, a C$_{3/2}$ structure at 0.0736 {\AA}$^{-2}$, and a C$_{4/2}$ structure at 0.0982 {\AA}$^{-2}$. After going through various domain structures, the $^{4}$He monolayer is completed at the areal density of 0.115 {\AA}$^{-2}$ where $^{4}$He adatoms form an incommensurate triangular solid. Possible superfluid response of the $^{4}$He monolayer on $\gamma $-graphyne is now under investigation. \\[4pt] [1] Y. Kwon, H. Shin, and H. Lee, Phys. Rev. B \textbf{88}, 201403(R) (2013).