Solidification of ${}^4\mathrm{He}$ clusters adsorbed on graphene

We determined the ground state of ${}^4\mathrm{He}_N$ clusters adsorbed on one side of graphene for selected cluster sizes in the range from $N=20$ to $N=127$. For all investigated clusters variational and diffusion Monte Carlo simulations were performed at $T=0$ K, and in addition for a selected subset finite temperature path integral Monte Carlo. At $T=0$ K the liquid or solid character of each cluster was investigated by restricting the phase using corresponding importance sampling trial-wave functions. The ${}^4\mathrm{He}$-graphene interaction was modeled as a sum of individual ${}^4\mathrm{He}$-C interactions, where both isotropic and anisotropic models were tested; also the effect of the substrate-mediated McLachlan interaction was investigated. We have found homogeneous crystallization in models of anisotropic interactions, starting from clusters with $N=26$ atoms in simulations without the McLachlan interaction, and between $N=37$ and 61 when it is included. The atoms become increasingly delocalized as one moves from the center of the cluster to the perimeter, evidenced by the Lindemann parameter. On the other hand, in the case of the isotropic interaction model, a liquidlike structure is more favorable for all considered cluster sizes. We use a liquid-drop model to extrapolate the energy per particle to the $N\to \infty$ limit, and the results are compared with the values obtained in studies of bulk ${}^4\mathrm{He}$ on graphene. Low-temperature path integral Monte Carlo simulations are in agreement with ground-state results.

Figure 1

The figure shows the structure of lattice points for ${}^4\mathrm{He}_N$ clusters, which is energetically optimal, as determined by the VMC simulations.

Figure 2

Energy of ${}^4\mathrm{He}_N$ cluster as a function of the number of particles in the cluster for Aniso/0 model for liquidlike and solidlike trial-wave functions. The solidlike structure of the cluster starts to be preferred from $N=26$ atoms.

Figure 3

Chemical potential of ${}^4\mathrm{He}_N$ clusters as a function of the number of particles in the clusters. Local minima in $\mu$ for $N$ greater than 26 correspond to more stable solidlike configurations, which can be seen by inspecting the corresponding structures in Fig. 1.

Figure 4

Energy per particle as a function of $N^{-1/2}$. The top set of data are obtained for the Iso/ML model and the bottom one for the Iso/0 model. The fits to the liquid-drop model, shown by lines, are performed starting at $N=20$ atoms.

Figure 5

Lindemann parameter $\delta$ as a function of the distance from the center of the cluster $\rho$ is presented for clusters containing $N{}^4\mathrm{He}$ atoms (Aniso/0 model). For clusters with hexagonal shape the data points corresponding to atoms at the vertex sites are encircled with dash-doted lines, while those corresponding to the other edge atoms are encircled with solid lines.

Figure 6

Density distribution $d\left(\rho \right)$ of He with respect to the center of mass for the isotropic models. $N$ is the number of ${}^4\mathrm{He}$ atoms in the cluster.

Figure 7

Density distribution $d\left(\rho \right)$ of He with respect to the center of mass for the anisotropic models. $N$ is the number of ${}^4\mathrm{He}$ atoms in the cluster. The red dots on the $x$ axis mark the distances of the lattice sites from the center of mass of the cluster.

Figure 8

Density distribution $d\left(\rho \right)$ of $61{}^4\mathrm{He}$ atoms with respect to the center of mass for the Aniso/ML model for $N=61$. The full (red) and dashed (green) curves are the DMC and PIMC results, respectively. The dotted (blue) line shows the density of He atoms that participate in two-body or larger exchange loops.

Figure 9

Pair distribution function $P\left(\rho \right)$ for models with anisotropic interactions (left) and models with isotropic interactions (right). Clusters on the left have a solidlike structure and those on the right a liquidlike structure. $N$ is the number of atoms in the cluster, which become wider with increase of $N$.