Hg adatoms on graphene: A first-principles study

Figure 2 shows the adsorption energy of graphene-Hg system as a function of graphene-adatom distance, d, for the configuration 2 × 2 (θ = 1/8) when Hg is at the H site (as an example; B and T sites show equivalent behavior), obtained using each of the three used functionals (LDA, PBE and PBE+DFT-D3). Since adatoms on surfaces are subject to dispersion forces, commonly referred to as van der Waals interactions, which are not accounted for with PBE nor LDA, we included the DFT-D3 correction to the PBE functional. As it can be seen in the figure 2, the DFT-D3 correction results in a strengthening of the binding: lower energy minimum and corresponding distance d. LDA calculations also show a much stronger binding of the Hg atom to the graphene layer compared to PBE (and smaller equilibrium distance). This is however expected, since LDA does not provide an adequate description for weakly bound adatoms, and is known to overestimate binding energies. In this case, the LDA-overbinding coincidentally mimics the dispersion contribution to the binding that was missing from PBE [36].

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It is important to note that while the choice of functionals determines the energy balance of the system, it leaves the EFG generally unaffected. The EFG is a traceless symmetric rank 2 tensor so that the axis system can be defined in a way that the EFG tensor representation has only three non-vanishing diagonal components defined as $\vert V_{zz} \vert$ ≥ $\vert V_{yy} \vert$ ≥ $\vert V_{xx} \vert$. It is common to characterize the EFG tensor by its main component V_zz and the axial asymmetry parameter, η, defined as $\eta = \frac{V_{xx}-V_{yy}}{V_{zz}}$, taking into account that the three remaining degrees off freedom are characterized by the Euler angles that relate the principal axis system (where the EFG is diagonal) to the laboratory axis system. The EFG tensor is extremely sensitive to the charge distribution (and its symmetry, in particular) in the region of the nucleus, which does not appear to vary significantly when using the different functionals considered here. This is illustrated in figure 3, where the EFG parameters, V_zz and η, for Hg in the H site, are plotted as a function of the distance d: LDA and PBE calculations yield virtually identical EFG values at a given distance d( figure 3), whereas the binding energy at that distance can vary considerably (figure 2). For a distance $d = 3.48~\textrm{\AA}$, i.e. at the equilibrium position determined by the minimum of the energy with PBE+DFT-D3 functional, the EFG values at the Hg site are $V_{zz} = -3.0~{\textrm{V}}\ \textrm{\AA}^{-2}$ and η = 0 (the axial asymmetry parameter η is always zero for Hg at H site).

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In the following we focus on the results obtained using the PBE+DFT-D3 functional. To understand the stability of Hg adatoms on graphene, for different nominal concentrations, the adsorption energy was studied as a function of the distance d, for each one of the three high symmetry sites. Figure 4 shows the calculated adsorption energy as a function of d, for different θ, when Hg is at the H site (except for θ = 3/8, for which the Hg atoms are in mix of H and T sites). The adsorption energy is calculated as: $E_{{\textrm{ads}}} = E_{G+{\textrm{adatom}}}- (E_{G} + E_{{\textrm{adatom}}})$, where $E_{G+{\textrm{adatom}}}$ is the energy of the system graphene + adatom, and E_G and $E_{{\textrm{adatom}}}$ are the energy of graphene and the energy of the free adatom, respectively. The term $(E_{G} + E_{{\textrm{adatom}}})$ was calculated using a supercell containing both the graphene layer and the adatom separated by a distance of at least 7.5 Å ($1.5 \times c_\textrm{graphite}$) to minimize layer-adatom interaction, as it is possible to see from figures 2 and 4. Although with a qualitatively similar shape, the adsorption energy curves for the different Hg concentrations (θ) have different minima (i.e. different binding energies) at different distance values. For θ = 1/18 and θ = 1/32, however, the curves virtually coincide, indicating that a supercell 3 × 3 (θ = 1/18) is already large enough to adequately describe non-interfering, isolated Hg adatoms (i.e. further increasing the dilution would have a negligible effect).

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Figure 5 compiles the binding energy (defined as the absolute value of the minimum of the adsorption energy) and the corresponding equilibrium distance (d_eq ) obtained for Hg at H, T and B sites, for each of the θ values considered here. For each nominal concentration, both binding energy and corresponding distance depend on the position (H, T or B) occupied by the Hg adatom. The main conclusion to be extracted from these calculations is that the most stable configuration for Hg on graphene is as isolated adatoms (e.g. 188 meV for θ = 1/32) which is, in particular, more stable than the HTT monolayer (123 meV and θ = 3/8).

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Above, we considered the relative stability of the Hg adatoms in various high-symmetry configurations. In this section we examine the energy barriers governing the migration of Hg adatoms in the dilute regime (i.e. the migration paths between H, B and T sites). In a practical scenario, adatoms are randomly deposited on a graphene surface, evolving from dilute to more concentrated configurations. Therefore, these migration barriers can also be regarded as governing the agglomeration of dilute Hg adatoms into other configurations (e.g. HTT configuration). Figure 6 shows how the adsorption energy varies along straight H$\longleftrightarrow$T and H$\longleftrightarrow$B paths (with d optimized at each intermediate position). From these calculations, we derive H$\longleftrightarrow$T and H$\longleftrightarrow$B migration barriers of ∼8.0 and ∼5.6 meV, respectively, i.e. the overall migration barrier for isolated Hg adatoms on graphene is therefore ∼5.6 meV. These can be considered low migration barriers, when compared to other metal adatoms such as Au (7 meV), Ag (10 meV), Cr (22 meV), Pd (80 meV) and Al (166 meV) [28, 37]. From these migration barriers, one can estimate a characteristic temperature above which the adatoms become mobile (i.e. start to diffuse and eventually agglomerate). Within an Arrhenius model, the rate Λ of thermally activated jumps between H sites is given by $\Lambda = \nu_{0}e^{-\frac{E_{a}}{k_{B}T}}$, where ν₀ is the attempt frequency (which we take as $\nu_{0} = 10^{12}~\textrm{s}^{-1}$, of the order of the lattice vibrations), and k_B and T the Boltzmann constant and the temperature. The average period between jumps is therefore given by Λ⁻¹, from which we can estimate the threshold temperatures for surface diffusion of Hg adatoms, for a given length and time scales. For example, the temperature corresponding to one jump between adjacent H sites is given by $T = {-\frac{E_{a}}{k_{B}\textrm{ln}(t\nu_{0})}}$, where t is the time interval in question. For 1 min (t = 60 s) and E_a  = 5.6 meV (corresponding to H$\longrightarrow$B$\longrightarrow$H), we obtain T = 2.2 K. In other words, isolated (dilute) Hg adatoms on an ideal (defect-free) graphene surface can only be observed if the deposition and subsequent characterization are performed at very low temperature, of the order of few K. Increasing temperature will activate surface diffusion, and Hg adatoms may become trapped in defective regions (e.g. graphene edges, Stone Wales defects, vacancies, grain boundaries), form clusters, etc [28, 38, 39].

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Figure 7 shows the band structure and density of states (DOS) for graphene and for graphene+Hg for the cases of the HTT configuration and for θ = 1/8 and θ = 1/32. The system remains gapless even when we consider SO-coupling in the calculations. Aside from the emergence of Hg-related bands, the main observation is the shift of the Fermi level with respect to the Dirac point of graphene, which depends on Hg concentration (summarized in figure 8). In the more dilute regime (θ = 1/32 and θ = 1/18), there is a small shift of the Fermi level to below the Dirac point, indicating a slight acceptor behavior of the isolated Hg adatoms. For an adatom concentration of θ = 1/18, this shift becomes even more pronounced. However, when the concentration further increases to θ = 3/8, in the HTT configuration, the effect reverses, with a shift of the Fermi level to above the Dirac point, i.e. suggesting a donor behavior. This reversal can be understood as due to the change from isolated adatoms to the formation of a Hg monolayer. This duality of regimes is clearly shown in figure 10, where it is possible to see that the charge densities are qualitatively the same within the same Hg coordination regime (adatom or monolayer), regardless of the Hg concentration. The observed donor behavior of the Hg monolayer is consistent with the simple analytical model proposed in reference [40] for the heavy elements Ag, Au, and Pt, characterizing the graphene-metal interface solely in terms of the work function: the work function of Hg (∼4.5 eV [41]) is below the ∼5.4 eV threshold where n-type behaviour is observed [40]. The formation of the Hg monolayer also renders the entire system (graphene plus Hg monolayer) metallic: the Hg monolayer introduces energy bands that cross the Fermi level (i.e. the Hg monolayer is metallic), which in addition is shifted upwards in energy, i.e. to a region of higher DOS in the Dirac cone of graphene (i.e. the graphene layer itself becomes more metallic).

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In this section, we discuss how the calculated EFG parameters, which strongly depends on the local charge distribution, can be used in an experimental setting to identify the spatial configurations of Hg adatoms on graphene. This involves comparing the EFG values calculated for multiple possible configurations to those measured experimentally using hyperfine techniques. Figure 9 shows the EFG parameters, V_zz and η, calculated at the Hg nucleus, for the different nominal concentrations and different high-symmetry sites, at the equilibrium distance, d_eq , between Hg and graphene. Table 1 compiles all the relevant parameters obtained in these calculations: binding energy and the corresponding distance d_eq , and the EFG parameters, V_zz and η. The calculated EFG values show some variation when comparing different high-symmetry sites, and are in particular strongly dependent on concentration θ, i.e. the V_zz strongly increases in the monolayer-type configurations (θ = 3/8 and θ = 1/2) due to the Hg coordination, making EFG a clear indicator of the concentration and position of Hg on graphene. We evaluated the role of the different orbitals in the obtained EFG values. Significant contributions are observed for p-p, d-d and s-d orbitals. Although total V_zz changes significantly from the highest nominal concentration (θ = 1/2) to the dilute regime (θ = 1/32), contributions from the p-p orbitals always dominate (73% and 87% for the two extremes, respectively). Figure 10 shows the charge distribution and the charge isolines when Hg adatoms are at equilibrium distance d_eq for the metallic configuration HTT and isolated adatoms with θ = 1/18. There are noticeable differences between the charge distribution of these two regimes (isolated versus monolayer) which help understand the drastically different V_zz . On one hand, the small V_zz for θ = 1/18 agrees with the nearly perfectly circular isolines of the charge distribution representing the outer electronic shells of the Hg adatoms. This means that for diluted Hg coverages (here θ = 1/8, θ = 1/18 and θ = 1/32) the V_zz will have low values due to the mostly spherical charge distribution of the Hg closed 6s² shell.

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On the other hand, the high V_zz for the HTT monolayer is a consequence of the non-spherical charge distribution around Hg, mostly due to the interaction with the Hg nearest neighbors.

PAC spectroscopy is a particularly suited technique to measure the EFG parameters which can be compared to the calculated values presented here. For PAC experiments on Hg, the $^{199~\textrm{m}}$Hg/¹⁹⁹Hg isomeric decay, with an intermediate 5/2⁻ state with 2.45 ns half-life, is well established [42, 43]. PAC is a time differential statistical measurement where the observable is the decay histogram of two consecutive γ-rays in coincidence. Taking into account the short half-life of the probing state $^{199{\textrm{m}}}$Hg of 2.45 ns, no more than 20 ns (approximately 8 half-lives) can be resolved with enough statistics. Therefore, we estimate a limit such that half a period of the characteristic PAC perturbation function can still be interpreted, i.e. 20/2 ns, leading to a minimum value of V_zz to be experimentally determined of $100~\textrm{V}\ \textrm{\AA}^{-2}$. The experimental time resolution is below 0.8 ns being the precision of the V_zz mainly dependent of the error of the quadrupole moment of about 9% [42]. This implies that based on PAC measurements of V_zz , it would be possible to distinguish the two regimes of Hg adatom concentration discussed above, i.e. between isolated Hg adatoms and the onset of aggregation into a metallic Hg monolayer. Such experiments can be performed, for example, using the ASPIC setup at the ISOLDE facility at CERN [29], where adatoms can be deposited at low temperature, down to liquid He, and PAC measurements can be performed in-situ as a function of temperature. Based on the calculations presented here, one can expect that, upon deposition at low temperature, one would first observe the low V_zz associated with isolated Hg adatoms on H sites. Increasing temperature would eventually activate diffusion and consequently part of the Hg will also be in segregated states such as the θ = 1/2 and θ = 3/8 cases. This different populations of Hg arrangements would result in the observation of both the low and large V_zz values associated with the isolated and metallic Hg layers, respectively. This type of experiments has been previously performed at ASPIC to investigate structural and magnetic properties of metal surfaces (e.g. Ni and Pd) [29]. The calculations presented here show the potential of hyperfine techniques in the context of adatoms on two-dimensional materials. One of the advantages of this approach is that it allows to locally probe the adatom without affecting it, i.e. avoiding the influence of external probes, such as the tip of a scanning probe microscope or the intense electron beam of a transmission electron microscope. Additionally, a PAC spectrum can be measured with as little as 10¹⁰ probe atoms, corresponding to a coverage of 0.01% for a typical 5 × 5 mm² sample, i.e. the approach can be applied down to extreme levels of adatom dilution. Moreover, measurements are compatible with applied electric or magnetic fields [29].