Symmetry-changing commensurate-incommensurate solid transition in the ${}^4\mathrm{He}$ monolayer on 6,6,12-graphyne

Path-integral Monte Carlo calculations have been carried out to investigate physical properties of a ${}^4\mathrm{He}$ monolayer adsorbed on a single 6,6,12-graphyne sheet, which is one of the graphyne families possessing a rectangular symmetry. To characterize elusive quantum phases of an adsorbed ${}^4\mathrm{He}$ monolayer on 6,6,12-graphyne, we model the ${}^4\mathrm{He}$-graphyne interaction by the pairwise sum of empirical ${}^4\mathrm{He}\text{−}\mathrm{C}$ interatomic potentials. At partially filled ${}^4\mathrm{He}$ coverages, we identify three commensurate solids of the $C_{3/4},C_{4/4}$, and $C_{6/4}$ structures from the two-dimensional density distribution. These solids show the rectangular symmetry inherited from the symmetry of 6,6,12-graphyne, which were confirmed with the analysis of their static structure factors. At high helium coverages near its completion, the ${}^4\mathrm{He}$ monolayer is predicted to exhibit a transition from a rectangular commensurate structure to a triangular incommensurate structure, after going through inhomogeneous structures mixed with domains of triangular and rectangular orderings. This symmetry-changing transition has not been observed in ${}^4\mathrm{He}$ monolayers adsorbed on other carbon substrates.

Figure 1

Structures of (a) $\alpha$-graphyne, (b) $\gamma$-graphyne, and (c) 6,6,12-graphyne. The blue parallelograms represent the primitive cells of the respective graphyne structures.

Figure 2

One-dimensional density distributions for ${}^4\mathrm{He}$ atoms adsorbed on a single 6,6,12-graphyne sheet as a function of the vertical distance $z$ from the graphyne surface. Here $N$ represents the number of ${}^4\mathrm{He}$ atoms per $4\times 3$ rectangular simulation cell with the size of $27.44\times 28.38{\AA }^2$.

Figure 3

Two-dimensional density distributions for the first-layer ${}^4\mathrm{He}$ atoms adsorbed on the 6,6,12-graphyne surface at areal densities of (a) 0.0462, (b) 0.0616, (c) 0.0925, and (d) $0.113{\AA }^{-2}$, which correspond to the number of the first-layer ${}^4\mathrm{He}$ atoms $N=36$, 48, 72, and 88, respectively, per $4\times 3$ simulation cell. The black dots represent the locations of the carbon atoms in 6,6,12-graphyne. The computations were done at a temperature of 0.5 K and the length unit is Å. The white arrows in (a)–(c) represent a set of primitive lattice vectors for each solid while the white dotted lines in (d) separate two domains of triangular and rectangular orderings. All contour plots are in the same color scale denoted by the color bar on the right-hand side.

Figure 4

PIMC energy per ${}^4\mathrm{He}$ atom for the first ${}^4\mathrm{He}$ layer on the 6,6,12-graphyne surface as a function of its areal density. The vertical dotted lines correspond to the areal densities for the $C_{3/4},C_{4/4},C_{6/4}$ phases and the incommensurate (IC) structure whose density distributions are presented in Fig. 3. The computations were performed at $T=0.5$ K and statistical errors are smaller than the symbol sizes.

Figure 5

Static structure factors divided by the number of ${}^4\mathrm{He}$ atoms as a function of the magnitude of the wave vector $k$ for the first ${}^4\mathrm{He}$ layer on 6,6,12-graphyne. The red, blue, and green symbols represent the PIMC results for the $C_{3/4},C_{4/4}$, and $C_{6/4}$ commensurate densities, respectively. The indices in parentheses represent the coordinates of the wave vector in terms of the reciprocal primitive vectors of the rectangular graphyne lattice. The computations were done at a temperature of 0.5 K with the $4\times 3$ rectangular simulation cell. The statistical errors are smaller than the symbol sizes.

Figure 6

Schematic phase diagram of the first ${}^4\mathrm{He}$ layer adsorbed on the 6,6,12-graphyne surface as a function of an areal density.