First-principles multiscale modelling of charged adsorbates on doped graphene

Figure 1 shows the multiscale approach we developed to model the properties of adsorbates on doped graphene. The method consists of three key steps. First, we perform first-principles DFT calculations of the pristine graphene and the graphene with adsorbed species. From the former, we derive the pristine graphene lattice parameter a and the Fermi velocity v_F, which we use in the continuum and TB Hamiltonians of the second and third steps. The periodic supercell for the graphene with adsorbate must be sufficiently large to converge chemical interactions between the adsorbate and the substrate [29]. To sample different values of the carrier density and, thereby, chemical potential μ, we perform a series of calculations with different numbers of electrons in the supercell.

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Secondly, we parametrize a continuum screening model of the graphene with the adsorbed species. In this approach, the adsorbate is treated as a point impurity charge Z located above the graphene sheet, which is described by the Dirac Hamiltonian (i.e. without atomistic detail). To determine the charge transfer between the adsorbate and graphene, we compute the self-consistent (or screened) potential of the continuum model in the same supercell that was used for the DFT calculations and, for each value of μ, we fit the value of Z to achieve optimal agreement with the DFT self-consistent potential. Different continuum screening models may be used in this procedure, corresponding to different treatments of electron-electron interactions. In this work we employ non-linear Thomas–Fermi theory [25] (NLTF), but also include interband transitions through an effective dielectric function. The screened potential at a position $\mathbf{r}$ in the graphene plane resulting from an impurity at $\mathbf{R}=\left(0,0,{{z}_{\text{imp}}}\right)$ is given by

$$\begin{eqnarray}&&V_{\text{scr}}^{\text{cont}}\left(\mathbf{r};\mu \right)=Z{{v}_{{{z}_{\text{imp}}}}}\left(\mathbf{r}\right)+{\int}^{}{{\text{d}}^{2}}{{\mathbf{r}}^{\prime}}\left[n\left({{\mathbf{r}}^{\prime}}\right)-{{n}_{0}}\right]{{v}_{0}}\left(\mathbf{r}-{{\mathbf{r}}^{\prime}}\right),\end{eqnarray} \tag{ 1 }$$

where ${{n}_{0}}\left(\mu \right)=\mu |\mu |/\left(\pi v_{\text{F}}^{2}\right)$ and $n\left(\mathbf{r}\right)={{n}_{0}}\left(\mu -{{V}_{\text{scr}}}\left(\mathbf{r}\right)\right)$ denote the electron density of graphene without and with the adsorbate, respectively, and ${{v}_{z}}\left(\mathbf{r}\right)$ is the Coulomb interaction screened by graphene interband transitions, i.e. ${{v}_{z}}(q)=\epsilon _{\text{inter}}^{-1}(q)2\pi /q\times \exp \left(-qz\right)$ with $\epsilon _{\text{inter}}^{-1}(q)$ being the interband contribution to the graphene dielectric function in the random-phase approximation [30–32]. In the above equation, the first term captures the bare impurity potential and the second term is the induced potential arising from electronic screening. We denote this model, which reduces to the well-known random-phase approximation in the linear-response limit [30–32], as NLTF  +  inter. We further included additional screening due to graphene σ bands [33] and exchange-correlation effects [20], but found that they have a small effect and can be neglected.

Thirdly, we perform a TB calculation in a supercell that is sufficiently large to capture the decay of the screened impurity potential. We find that the supercells needed to converge the screened potential are several orders of magnitude larger than the supercells needed to converge chemical interactions. The screened potential calculated from the continuum model, with the value of Z that has been fitted to DFT, is used as an on-site term in the TB Hamiltonian. The TB calculation then yields the local density of states (LDOS), which can be compared to scanning tunneling experiments [10, 28].

In this section, we describe the application of the multiscale approach to isolated Ca adatoms on doped graphene.

2.2.1. Structure

2.2.2. First-principles calculations

2.2.3. Continuum screening calculations

2.2.4. Tight-binding calculations

The charge transfer Z is an important quantity for characterizing the interaction between an adsorbate and the substrate. Standard approaches for calculating Z partition the continuous electron density obtained from DFT into an adsorbate and a substrate contribution. For example, Chan et al [15] calculated the charge transfer for different adatoms on graphene, but found large discrepancies between two different methods of partitioning the electron density (0.18 e and 0.78 e for Ca adatoms). As shown in the inset of figure 2 and figure S2 in the supplementary material, we obtain an even broader range of values for Z, from 0.2 e to 1.4 e, depending on the particular charge partitioning method used. While each method provides a different value for Z, all of them find the charge transfer to be independent of the supercell charge $\Delta Q$, indicating that Ca states are neither filled nor depleted as the graphene doping level is varied.

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In contrast to these approaches, no partitioning of the charge density is required in our multiscale method. Instead, Z is obtained by imposing that the screened impurity potential of the continuum screening model reproduces the self-consistent DFT potential. In the fitting procedure, we enforce a μ-independent value of Z by performing a global fit to all doping levels at once with equal weighting. This is justified by the charge stability of the impurity discussed previously. It should be noted that performing individual fits for each doping level indeed results in a fairly constant Z for all charge states except $\Delta Q=2$ e, in which case the value is enhanced by up to  ∼50%; we believe this to be because the graphene system is very close to being undoped and, hence, is the least well-described by our models as the screening is dominated in this case by the interband transitions [40].

Table 1 gives the value of Z obtained from the new multiscale approach using different approximations for the description of electron-electron interactions. The NLTF  +  inter method, which offers the most accurate description of electron-electron interactions, yields Z  =  1.6 e. The same value is obtained from the NLTF method indicating that inclusion of interband transition does not change the value of Z. In contrast, the linear-response LTF and LTF  +  inter approaches yield significantly smaller values for Z.

Table 1. Charge transfer Z obtained from the new multiscale approach, i.e. by fitting the screened impurity potential of a continuum screening model to the self-consistent DFT potential, and the corresponding value of the fitness metric $\mathcal{F}$ of the model averaged over all DFT doping levels. We also include the value of the adatom charge estimated from the DFT DOS (see figure 2).

Model	Z (e)	$\mathcal{F}$ (eV)
LTF	0.3	$8.9\times {{10}^{-3}}$
LTF  +  inter	1.3	$8.4\times {{10}^{-3}}$
NLTF	1.6	$9.3\times {{10}^{-3}}$
NLTF  +  inter	1.6	$7.4\times {{10}^{-3}}$
Linear-bands fit to DFT (figure 2)	$1.6\pm 0.3$	—

Table 1 also gives the value of the fitness function $\mathcal{F}$ averaged over all DFT doping levels. It is interesting to note that there is almost no difference in this value between the different approaches¹; all of them reproduce the DFT screened potential with high accuracy (<10 meV), but with significantly different Z values. This observation was explained by Katsnelson [25] who pointed out that the screened potential of a charged impurity in doped graphene has the same functional form in linear and non-linear Thomas–Fermi theory, but with a different prefactor corresponding to different effective values of Z.

An alternative method for calculating Z which also does not require a partitioning of the charge density is shown in the main panel of figure 2. For each value of the supercell charge $\Delta Q$, we determine μ from the spectrum of Kohn–Sham energies (open squares) and then fit a linear-bands model of doped graphene of the form $\mu \left(\Delta Q\right)=-\text{sgn}\left[\Delta Q-Z\right]{{v}_{\text{F}}}\sqrt{| \Delta Q-Z|\pi /A}$ (dashed line), where $\Delta Q-Z$ is the total free carrier charge and A is the area of the supercell. This method yields $Z=1.6\pm 0.3$ e, significantly higher than previously suggested values [15], but in excellent agreement with our multiscale method.

Figure 3 shows the LDOS obtained from the large-scale TB simulation with the screened potential from the NLTF  +  inter model. The main panel shows a representative example for the case of n-doped graphene (at a chemical potential $\mu =0.12$ eV).

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As observed in previous studies employing an unscreened impurity potential [10, 19], the LDOS near the adatom exhibits a broken electron-hole symmetry. This behavior can be understood within a simple TF-based picture, in which the LDOS in the presence of the adsorbate is given by that of unperturbed graphene, but shifted in energy by the value of the screened impurity potential (shown by the dotted line in figure 3). While the shifted LDOS agrees well with the full result for energies more than $\pm \sim$0.5 eV from the Dirac point, there are significant deviations within this range. In particular, the position of the Dirac point remains pinned at its unperturbed value, which is a consequence of the linear dispersion of the bands from graphene's chiral Dirac fermions.

The inset of figure 3 shows the decay of the LDOS at a fixed energy towards its unperturbed value far from the adatom (corresponding to experimental $\text{d}I/\text{d}V$ linescans [10]). The decay length decreases with increasing $|\mu |$. This is in agreement with very recent experimental measurements [28]. The chemical potential only enters the simulation through its effect on the screened potential; this demonstrates the impact of screening on measured properties of the system and highlights the importance of an accurate description of electron-electron interactions.