Adsorption of He isotopes on fluorographene and graphane: Fluid and superfluid phases from quantum Monte Carlo calculations

Monolayer films on graphite, remarkably diverse examples of two-dimensional matter, are now well understood in terms of semiempirical interactions. We explore the phase behavior of helium films on two variants of graphene: graphane (graphene coated with H, denoted GH) and graphene fluoride (GF). The behaviors predicted with quantum Monte Carlo differ qualitatively from those on graphite because of the different surface composition, symmetry, and spacing of the adsorption sites. On both substrates we find that the analog of the standard $\sqrt{3}\times \sqrt{3}$ $R$30${}^{\circ }$ commensurate state on graphite is unstable. Results include a superfluid ground state for ${}^4$He and a fluid ground state for ${}^3$He, neither of which has been found for the monolayer film on any substrate; these two-dimensional fluids are anisotropic because of the symmetry imposed by the honeycomb lattice of adsorption sites. In the case of ${}^4$He on GF the anisotropy is as large as if the superfluid were restricted to move in a multiconnected space, along the bonds of a honeycomb lattice. The superfluid transition temperature at the ground-state density of ${}^4$He on GF (GH) is of order 0.25 (1.1) K. At higher coverages both an incommensurate triangular solid and a commensurate state at filling factor 2/7 are found.

Figure 1

He on GF. (a) View of GF (GH) from above. Green circles represent F (H) atoms in the upper plane, dark gray circles represent half of the C atoms, and the other C atoms lie below the green circles. The other half of the F (H) atoms lie below the dark gray circles. The two blue circles represent two adsorption sites; one is on top of a C atom and the other has no atoms directly below. (b) Details of the geometry of the unit cell; $a_1$ and $a_2$ indicate the sides of the unit cell, $d$ is the C-C distance projected on the $x$-$y$ plane, and $b$ is the buckling displacement of the C atoms. Values for $a_1$, $a_2$, $d$, and $b$ have been obtained from a DFT computation (Ref. 12) and are reported in Appendix 7.

Figure 2

He on GF. Plot of the minimum value with respect to $z$ of $V_{\mathrm{ad}}\left(\vec{r}\right)$ in K as a function of $x$-$y$.

Figure 3

Energy barrier in GF, GH, and graphite along a line at an angle $\theta$ with the horizontal direction, following the height $z(x,y)$ giving the minimum of $V_{\mathrm{ad}}\left(\vec{r}\right)$. Plotted energy is relative to energy at the adsorption site.

Figure 4

(a) Energy of the first band along two directions of the first Brillouin zone for ${}^4$He (triangles) and ${}^3$He (circles) on GF. Data along $\Gamma$K beyond the Dirac point K give the results in the second Brillouin zone. The dashed lines are fits made with the tight-binding model on a honeycomb lattice with nearest-neighbor parameter ${\gamma }_1=3.63$ (5.695) for ${}^4$He (${}^3$He) and next-nearest-neighbor parameter ${\gamma }_2=-0.16$ ($-0.33$). Close to the K point the second band given by the tight-binding model is also plotted in order to show the presence of the the Dirac point in the spectrum. (b) Same as for (a) for ${}^4$He and ${}^3$He on GH. The fit parameters for the dashed lines are ${\gamma }_1=4.514$ (6.434) for ${}^4$He (${}^3$He) and ${\gamma }_2=-0.021$ ($-0.04$). More information can be found in Ref. 12.

Figure 5

$S\left(k\right)$ of ${}^4$He on GF in the $k_x$ or $(1,0)$ and $k_y$ or $(0,1)$ directions at coverage $x=1/6$ for indicated numbers of particles. Notice the logarithmic scale for $S\left(k\right)$. The dotted lines represent a guide to the eye.

Figure 6

(a) Energy per particle of ${}^4$He as function of the density $\rho$ on GF at $T=0$ K. (b) Same as for (a) for ${}^4$He on GH. In both cases, the used particle numbers ranged between $N=60$ and $N=120$. The dotted line represents a guide to the eye.

Figure 7

Upper panel: Local density $\rho (x,y)$ (in units of Å${}^{-2}$) as function of $x$-$y$ for ${}^4$He at equilibrium density on GF. Lower panel: Local density $\rho (x=0,y)$ (in units of Å${}^{-2}$) in the unit cell (with side $a=4.486$ Å in the GF case and $a=4.347$ Å in the GH case) along the $y$ direction for ${}^4$He at equilibrium density on GF and GH; note the logarithmic scale used for $\rho (x=0,y)$. Error bars are below the symbol size; lines are guides to the eye.

Figure 8

(a) Static structure factor, $S(k_x,k_y)$, at equilibrium density on the $k_x$-$k_y$ plane for $N=96$ atoms of ${}^4$He on GF. (b) The same for ${}^4$He on GH at equilibrium density. $k_x$ and $k_y$ axis are expressed in Å${}^{-1}$. These static structure factors are characteristic of a modulated liquid; sharp peaks represent the first two stars of Bragg peaks due to the substrate, while the broad peaks indicate anisotropic short-range order.

Figure 9

Off-diagonal density matrix ${\rho }_1\left(r\right)$ on GF and GH at ${\rho }_{1/6}$ and at ${\rho }_{\mathrm{eq}}$.

Figure 10

The superfluid fraction ${\rho }_s/\rho$ at ${\rho }_{\mathrm{eq}}$ as function of temperature for a system of $N=20$ atoms of ${}^4$He on GH and a system of $N=26$ atoms of ${}^4$He on GF. The transition temperature can be roughly estimated in the range 1.0–1.2 K for ${}^4$He GH and 0.2–0.3 K for ${}^4$He GF. In the inset the superfluid fraction can be read as the long-$\tau$ limit of Eq. (2).

Figure 11

Local density (in Å${}^{-2}$ units) on the $x$-$y$ plane of the $2/7$ phase of ${}^4$He on GF (a) and on GH (b) compared with the geometry of the substrate. Small red balls are centered on the position of fluorine atoms and the small green ones on the carbon atoms. Thin white lines enclose the unit cell of the commensurate 2/7 phase.

Figure 12

Static structure factor on the $k_x$-$k_y$ plane of the $2/7$ phase of $N=112$ atoms of ${}^4$He on GF (a) and on GH (b). $k_x$ and $k_y$ axes are expressed in Å${}^{-1}$. Red arrows point to the peaks corresponding to the density modulation imposed by the adsorption potential.

Figure 13

(a) Energy per particle of ${}^4$He as function of the density $\rho$ on GF at $T=0$ K. (b) Same as for (a) for ${}^4$He on GH. The filled diamonds mark the energy of the 2/7 phase on both substrates; the stars represent the energies of incommensurate phases before completion. The dashed line represents a guide to the eye.

Figure 14

Local density along the $z$ direction of ${}^4$He on GF (with $N=111$) and ${}^4$He on GH (with $N=79$) at a density beyond the promotion density. The occupations of the first and the second layers are clearly visible as two peaks. The area under the peaks represents the number of ${}^4$He atoms in the corresponding layer.

Figure 15

Static structure factor on the $k_x$-$k_y$ plane of the incommensurate solid phase of $N=86$ atoms of ${}^4$He on GF at density $\rho =0.123$ Å${}^{-2}$. $k_x$ and $k_y$ axis are expressed in Å${}^{-1}$. Red arrows point to the peaks corresponding to the density modulation imposed by the adsorption potential.

Figure 16

Static structure factor on the $k_x$-$k_y$ plane of the incommensurate solid phase of $N=66$ atoms of ${}^4$He on GH at density $\rho =0.102$ Å${}^{-2}$. $k_x$ and $k_y$ axes are expressed in Å${}^{-1}$. Red arrows point to the peaks corresponding to the density modulation imposed by the adsorption potential.

Figure 17

(a) Ground-state energy as a function of density of mass 3 bosons on GF (circles), fermionic ${}^3$He GF obtained via the fermionic correlations method (triangles), and fermionic ${}^3$He GF obtained by approximating the Bose-Fermi gap with the kinetic energy of the free fermion gas (diamonds). (b) Same as for (a) for ${}^3$He on GH. In both cases, the system under study was composed of $N=18$ atoms of ${}^3$He.