Interlayer correlation between two ${}^4\mathrm{He}$ monolayers adsorbed on both sides of $\alpha$-graphyne

Path-integral Monte Carlo calculations have been performed to study the ${}^4\mathrm{He}$ adsorption on both sides of a single $\alpha$-graphyne sheet. For investigation of the interlayer correlation between the upper and the lower monolayers of ${}^4\mathrm{He}$ adatoms, the ${}^4\mathrm{He}$-substrate interaction is described by the sum of the ${}^4\mathrm{He}\text{−}\mathrm{C}$ interatomic pair potentials for which we use both Lennard-Jones and Yukawa-6 anisotropic potentials. When the lower ${}^4\mathrm{He}$ layer is a ${\mathrm{C}}_{4/3}$ commensurate solid, the upper-layer ${}^4\mathrm{He}$ atoms are found to form a kagome lattice structure at a Mott-insulating density of $0.0706{\AA }^{-2}$ and a commensurate solid at an areal density of $0.0941{\AA }^{-2}$ for both substrate potentials. The correlation between upper- and lower-layer pseudospins, which were introduced in Kwon et al. [Phys. Rev. B 88, 201403(R) (2013)] for two degenerate configurations of three ${}^4\mathrm{He}$ atoms in a hexagonal cell, depends on the substrate potential used; with the substrate potential based on the anisotropic Yukawa-6 pair potentials, the Ising pseudospins of both ${}^4\mathrm{He}$ layers are found to be antiparallel to each other whereas the parallel and antiparallel pseudospin alignments between the two ${}^4\mathrm{He}$ layers are nearly degenerate with the Lennard-Jones potentials. This is attributed to the difference in the interlayer distance, which is $\sim 4\AA$ with the Yukawa-6 substrate potential but as large as $\sim 4.8\AA$ with the Lennard-Jones potential.

Figure 1

One-dimensional density distributions of ${}^4\mathrm{He}$ atoms adsorbed on both sides of $\alpha$-graphyne as a function of the vertical coordinate $z$ perpendicular to the graphyne surface, which were computed with the LJ substrate potential. Here $N_{\mathrm{up}}$ and $N_{\mathrm{dn}}$ represent the number of ${}^4\mathrm{He}$ atoms per $3\times 2$ simulation cell in the upper and the lower ${}^4\mathrm{He}$ layers, respectively. An additional $12 {}^4\mathrm{He}$ atoms per simulation cell are involved to form the zeroth layer around $z=0$ where one ${}^4\mathrm{He}$ atom is embedded at every hexagon center (see Ref. [8]).

Figure 2

Contour plots of two-dimensional density distributions of ${}^4\mathrm{He}$ atoms adsorbed on the upper side of $\alpha$-graphyne for upper-layer areal densities of (a) $0.0706{\AA }^{-2}$ and (b) $0.0941{\AA }^{-2}$ (red: high; blue: low). The black dots correspond to the carbon atoms, and the white stars represent the peak positions of the lower-layer ${}^4\mathrm{He}$ density distribution, which form a ${\mathrm{C}}_{4/3}$ commensurate solid. The computations were performed at $T=0.5\mathrm{K}$ with the LJ substrate potential, and the length unit is in angstroms.

Figure 3

Contour plot of two-dimensional density distribution of the upper ${}^4\mathrm{He}$ layer at the areal density of $0.0687{\AA }^{-2}$, which corresponds to one less ${}^4\mathrm{He}$ atoms per our $3\times 2$ rectangular simulation cell than the Mott-insulating density (red: high; blue: low). The black dots correspond to the carbon atoms, and the white stars represent the density peaks of the lower-layer ${\mathrm{C}}_{4/3}$ commensurate solid. The computations were performed at $T=0.5\mathrm{K}$ with the LJ substrate potential, and the length unit is in angstroms.

Figure 4

Static structure factors divided by the number of ${}^4\mathrm{He}$ atoms as a function of wave vector $k$ (a) for the upper helium layer on a single $\alpha$-graphyne sheet whose lower side is coated with a ${\mathrm{C}}_{4/3}$ commensurate ${}^4\mathrm{He}$ solid and (b) for the first ${}^4\mathrm{He}$ layer on bilayer $\alpha$-graphyne without the lower ${}^4\mathrm{He}$ layer. Both ${}^4\mathrm{He}$ layers in (a) and (b) are at the Mott-insulating density of $0.0706{\AA }^{-2}$. The configuration of ${}^4\mathrm{He}$ atoms in (a) is the same as in Fig. 2 whereas the helium configuration in (b) is identical to the one for Fig. 2 of Ref. [8]. The black and the red symbols in (a) represent the data obtained from the PIMC calculations for the $3\times 2$ simulation cell at $T=0.5$ and 0.25 K, respectively, whereas the blue ones in (a) show the PIMC data for the $4\times 3$ simulation cell at $T=0.5\mathrm{K}$. The statistical errors are smaller than the symbol sizes.

Figure 5

One-dimensional density distributions of ${}^4\mathrm{He}$ atoms adsorbed on both sides of $\alpha$-graphyne as a function of the vertical coordinate $z$ perpendicular to the graphyne surface, which were computed with the Yukawa-6 substrate potential. Here $N_{\mathrm{up}}$ and $N_{\mathrm{dn}}$ represent the number of ${}^4\mathrm{He}$ atoms per $3\times 2$ simulation cell in the upper and the lower ${}^4\mathrm{He}$ layers, respectively.

Figure 6

(a) Contour plots of two-dimensional density distributions of ${}^4\mathrm{He}$ atoms adsorbed on the upper side of $\alpha$-graphyne for upper-layer areal densities of (a) $0.0706{\AA }^{-2}$ and (b) $0.0941{\AA }^{-2}$ (red: high; blue: low). The black dots correspond to the carbon atoms, and the white stars represent the peak positions of the ${}^4\mathrm{He}$ density distribution of the lower layer whose density is $0.0941{\AA }^{-2}$ for both cases. The computations were performed at $T=0.5\mathrm{K}$ with the Yukawa-6 substrate potential, and the length unit is in angstroms.

Figure 7

Contour plot of two-dimensional density distribution of the upper ${}^4\mathrm{He}$ layer at the areal density of $0.0687{\AA }^{-2}$, which corresponds to one less ${}^4\mathrm{He}$ atoms per our $3\times 2$ rectangular simulation cell than the Mott-insulating density (red: high; blue: low). The black dots correspond to the carbon atoms, and the white stars represent the density peaks of the lower-layer ${\mathrm{C}}_{4/3}$ commensurate solid. The computations were performed at $T=0.5\mathrm{K}$ with the Yukawa-6 substrate potential, and the length unit is in angstroms.