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

Path-integral Monte Carlo calculations have been performed to study the ${}^4\mathrm{He}$ adsorption on $\gamma$-graphyne, a planar network of benzene rings connected by acetylene bonds. Assuming the ${}^4\mathrm{He}$-substrate interaction described by a pairwise sum of empirical ${}^4\mathrm{He}$-carbon interatomic potentials, we find that unlike $\alpha$-graphyne, a single sheet of $\gamma$-graphyne is not permeable to ${}^4\mathrm{He}$ atoms in spite of its large surface area. One-dimensional density distributions computed as a function of the distance from the graphyne surface reveal a layer-by-layer growth of ${}^4\mathrm{He}$ atoms. A partially-filled ${}^4\mathrm{He}$ monolayer is found to exhibit different commensurate solid structures depending on the helium coverage; it shows a ${\mathrm{C}}_{2/3}$ commensurate structure at an areal density of 0.0491 $\AA {}^{-2}$, a ${\mathrm{C}}_{3/3}$ structure at 0.0736 $\AA {}^{-2}$, and a ${\mathrm{C}}_{4/3}$ structure at 0.0982 $\AA {}^{-2}$. While the promotion to the second layer starts beyond the ${\mathrm{C}}_{4/3}$ helium coverage, the first ${}^4\mathrm{He}$ layer is found to form an incommensurate triangular solid when compressed with the development of the second layer.

Figure 1

(a) A contour plot of the minimum ${}^4\mathrm{He}$-graphyne potential, $V_{min}(x,y)$, above each point $(x,y)$ on $\gamma$-graphyne and (b) the ${}^4\mathrm{He}$-graphyne potential as a function of the distance $z$ from the graphyne surface along different symmetry directions. The black dots in (a) represent the positions of carbon atoms on the $\gamma$-graphyne surface. In (b), the blue, the red, and the black line correspond to the threefold symmetry direction, the sixfold direction, and the direction of a saddle point [see white numbers and letter S in (a)], respectively. The length unit is Å and the potential energies are in units of Kelvin.

Figure 2

One-dimensional density of ${}^4\mathrm{He}$ atoms adsorbed on a single $\gamma$-graphyne sheet as a function of the distance $z$ (in Å) from the graphyne surface. Here $N$ represents the number of ${}^4\mathrm{He}$ adatoms per $3\times 2$ rectangular simulation cell with dimensions of $20.58\times 23.76{\AA }^2$, and the computations were done at a temperature of $0.5$ K.

Figure 3

Energy per ${}^4\mathrm{He}$ atom of the first ${}^4\mathrm{He}$ layer on $\gamma$-graphyne as a function of the helium coverage. The letters (a), (b), (c), and (d) correspond to the areal densities of 0.0491, 0.0736, 0.0859, and 0.0982 $\AA {}^{-2}$, respectively, for which two-dimensional ${}^4\mathrm{He}$ density plots are shown in Fig. 4. The energies were computed at a temperature of 0.5 K.

Figure 4

Two-dimensional density distributions of the first-layer ${}^4\mathrm{He}$ atoms adsorbed on a single $\gamma$-graphyne sheet at areal densities of (a) 0.0491, (b) 0.0736, (c) 0.0859, and (d) 0.0982 $\AA {}^{-2}$. The black dots represent the positions of the carbon atoms of graphyne. The length unit is Å and all contour plots are in the same color scale denoted by the color table in the upper right hand corner. The yellow dotted lines in (c) separate two different domains from each other. The PIMC calculations were done at a temperature of $0.5$ K.

Figure 5

The peak values of the static structure factor divided by the number of ${}^4\mathrm{He}$ atoms in the first helium layer on $\gamma$-graphyne, at three different commensurate helium coverages. Here $G_1$ represents the primitive vector of the reciprocal lattice of the corresponding commensurate solid. The solid (open) diamonds represent the PIMC data for the $3\times 2$ ($4\times 3$) rectangular simulation cell, and the statistical errors are smaller than the symbol sizes.