Influence of the surface dipole layer and Pauli repulsion on band energies and doping in graphene adsorbed on metal surfaces

The synthesis of single-layer graphene sheets on metal surfaces can be carried out routinely nowadays. The energetic alignment of the graphene band structure, including the position of the Dirac point relative to the Fermi level of the metal, and subsequently, the doping level of the graphene sheet, depends crucially on the graphene-metal distance and the specific metal considered. These dependencies are studied with density-functional theory considering as typical metal surfaces Au(111), Ni(111), and Au/Ni(111). In the latter case, a single layer of gold is intercalated between the Ni(111) surface and the graphene sheet. We show that the energetic positions of eigenstates of helium adsorbed on a Au(111) surface exhibit a behavior with the adsorption distance qualitatively comparable to that of bands of physisorbed graphene. In both cases, the distance dependence of the energy of adsorbate bands can be explained by the effect of the surface dipole layer on the adsorbate bands and by electrostatic interactions caused by small charge rearrangements due to Pauli repulsion between metal surface and graphene. These charge rearrangements are neither caused by a charge transfer nor by chemical interactions due to conventional orbital interaction but have the effect to reduce the overlap of the surface charge density of the metal with the charge density of the adsorbate. The latter effect is known as pillow effect from molecules adsorbed on metal surfaces. Charge transfer between graphene and the metal substrate does occur but has an opposite effect to the surface dipole layer and Pauli repulsion, i.e., reduces the effect of the latter. For very large adsorption distances, this charge transfer vanishes in such a way that the Dirac point of graphene aligns with the metal Fermi energy. It is shown that the amount and character of graphene doping can be controlled by tuning the graphene-metal distance. For a proper description of the involved electrostatic potentials, a finite-slab correction had to be applied to them in order to take into account the finite size of the metal slabs used to model the substrate.

Figure 1

Band structures of a helium atom adsorbed on a (3 $\times$ 3) Au(111) surface at an adsorption distances of 2.0 (left) and 6.0 $\AA$ (right). Contribution of helium states to bands is indicated by red spheres. The radius of the spheres indicates the magnitude of helium contributions to the respective bands.

Figure 2

(Top) Shifts $\Delta {\epsilon }_{1s}\left(d\right)$ of the energy of the helium $1s$ band for adsorption distances $d(\text{He}-\text{Surf})$ between 2.0 and 6.0 $\AA$ with respect to the helium $1s$-band energy at a reference distance of 6.0 $\AA$. The helium atom is adsorbed over a top site of a (3 $\times$ 3) unit cell of a gold (blue) or nickel (green) (111) surface. (Middle) $\Delta {\epsilon }_{1s}\left(d\right)$ together with the two contributions that constitute this shift, the contribution $\Delta {\tilde{\epsilon }}^{\text{S}}\left(d\right)$ of the surface potential and the contribution $\Delta {\tilde{\epsilon }}^{\text{CDD}}\left(d\right)$ of the charge density difference (CDD). The sum of both contributions is denoted $\Delta {\tilde{\epsilon }}_{1s}\left(d\right)$. The inlay shows a top view on the (3 $\times$ 3) Au(111) surface. Gold atoms are colored lime (topmost layer), cyan (second layer), and orange (third layer). Adsorption on a top site refers to adsorption over a lime colored atom, while adsorption on a fcc-hollow site corresponds to adsorption over an orange colored atom. (Bottom) According shifts $\Delta {\tilde{\epsilon }}^{\text{S}}\left(d\right)$, $\Delta {\tilde{\epsilon }}^{\text{CDD}}\left(d\right)$, $\Delta {\tilde{\epsilon }}_{1s}\left(d\right)$, and $\Delta {\epsilon }_{1s}\left(d\right)$ for a helium atom adsorbed on a (3 $\times$ 3) surface cell of Ni(111).

Figure 3

Electrostatic potential of Au(111) (blue) and Ni(111) (green) slabs, respectively. For a given value of the $z$ coordinate perpendicular to the surface, the potentials are averaged over the corresponding $xy$ planes parallel to the surface. The topmost layer of the slab is located at $z=0$. The inset shows the two potentials in the region $2.0\le z\le 6.0\AA$, relevant for adsorption. The vacuum level for both cases is denoted ${\epsilon }^{\text{vac}}$.

Figure 4

(Top) Charge density difference ${\varrho }_d^{\text{CDD}}$ between the combined systems He/Au(111) (left) and He/Ni(111) (right) and the sum of its constituents calculated individually, at two adsorption distances $d=2.0\AA$ (orange) and $d_{\text{ref}}=6.0\AA$ (cyan). The helium atom is adsorbed above a top site in a (3 $\times$ 3) metal (111) unit cell. Atom positions are indicated by gray, orange, and cyan circles. For $d=2.0$ $\AA$, electronic charge is accumulated slightly above the surface and depleted around the adsorbate. (Bottom) Electrostatic potentials $v_d^{\text{CDD}}$ that correspond to the charge density differences ${\varrho }_d^{\text{CDD}}$ at the two considered adsorption distances. Both ${\varrho }_d^{\text{CDD}}$ as well as $v_d^{\text{CDD}}$ are averaged over the corresponding $xy$ planes parallel to the surface for a given value of the $z$ coordinate perpendicular to the surface. At $z=0$, the topmost layer of the slab is located.

Figure 5

Band structures of graphene adsorbed on Ni(111) in a top-fcc adsorption geometry for nickel graphene distances of $d_{\text{0}}=2.12$ $\AA$ (left) and $d_{\text{ref}}=9.0$ $\AA$ (right). The contribution of carbon orbitals to bands are colored red ($2s$), orange ($2p_x,2p_y$), and yellow ($2p_z$). The points in the band structures whose energy shifts are analyzed in detail are marked by blue squares in the panel to the right.

Figure 6

(Top) Energy shifts $\Delta {\epsilon }_{\sigma }^{\Gamma }\left(d\right)$, $\Delta {\epsilon }_{\sigma }^{\text{K}}\left(d\right)$, $\Delta {\epsilon }_{\pi }^{\Gamma }\left(d\right)$, and $\Delta {\epsilon }_{\pi }^{\text{K}}\left(d\right)$ of the graphene $\sigma$ and $\pi$ band at $\mathbf{K}$ and $\mathit{\Gamma }$ as observed in the calculated band structure for a graphene ML on Ni(111) in a top-fcc adsorption geometry for adsorption distances from $d=2.0$ $\AA$ to $d=6.0$ $\AA$. In addition corresponding modeled shifts $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma }\left(d\right)$ and $\Delta {\tilde{\epsilon }}_{\pi }^{\Gamma }\left(d\right)$ at $\mathit{\Gamma }$ are displayed that originate from the nickel surface potential and from charge redistribution upon adsorption, see text for details. (Bottom) Energy shift $\Delta {\epsilon }_{\sigma }^{\Gamma }\left(d\right)$ of the graphene $\sigma$ band at $\mathit{\Gamma }$ together with the contributions $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma ,\text{S}}\left(d\right)$ and $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma ,\text{CDD}}\left(d\right)$ that constitute this shift and that are caused by the surface potential $v_d^{\text{S}}$ and the potential $v_d^{\text{CDD}}$ of the charge density difference due to adsorption, respectively. The sum of both contributions is denoted $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma }\left(d\right)$. The optimized equilibrium adsorption distance is indicated by a dashed gray line.

Figure 7

(Top) Charge density difference ${\varrho }_d^{\text{CDD}}$ between the combined system graphene/Ni(111) and the sum of its constituents Ni(111) and graphene calculated individually, at two adsorption distances $d_{\text{0}}=2.12\AA$ (orange) and $d_{\text{ref}}=9.0\AA$ (cyan). The graphene sheet is adsorbed above a Ni(111) (1 $\times$ 1) unit cell in a top-fcc adsorption geometry. Atom positions are indicated by gray, orange, and cyan circles. For the equilibrium adsorption distance $d_{\text{0}}$, electronic charge is accumulated at the surface and depleted next to the adsorbate in the region towards the surface. (Bottom) Electrostatic potentials $v_d^{\text{CDD}}$ that correspond to the charge density differences ${\varrho }_d^{\text{CDD}}$ at the two considered adsorption distances. In addition, charge transfer potentials $v_d^{\text{CT}}$, obtained by calculating a simple plate capacitor model based on the Bader charges of graphene $\Delta q^{\text{G}}$ and the metal $\Delta q^{\text{M}}=-\Delta q^{\text{G}}$ are depicted as red and blue dashed line, respectively (see Sec. 4 for details). The planes of the model plate capacitor are indicated by the dark gray and red and the light gray and blue dotted lines for the two considered adsorption distances, respectively. Both ${\varrho }_d^{\text{CDD}}$ as well as $v_d^{\text{CDD}}$ are averaged over the corresponding $xy$ planes parallel to the surface for a given value of the $z$ coordinate perpendicular to the surface. At $z=0$, the topmost layer of the slab is located.

Figure 8

(Top) Energy shifts $\Delta {\epsilon }_{\sigma }^{\Gamma }\left(d\right)$, $\Delta {\epsilon }_{\sigma }^{\text{K}}\left(d\right)$, $\Delta {\epsilon }_{\pi }^{\Gamma }\left(d\right)$, and $\Delta {\epsilon }_{\pi }^{\text{K}}\left(d\right)$ of the graphene $\sigma$ and $\pi$ bands at $\mathbf{K}$ and $\mathit{\Gamma }$ as observed in the calculated band structure for a graphene ML adsorbed on a Au(111) surface for adsorption distances from $d=2.25\AA$ to $d=6.0\AA$. In addition, corresponding modeled shifts $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma }\left(d\right)$ and $\Delta {\tilde{\epsilon }}_{\pi }^{\Gamma }\left(d\right)$ at $\mathit{\Gamma }$ are displayed that originate from the gold surface potential and from charge redistribution upon adsorption, see text for details. (Bottom) Energy shift $\Delta {\epsilon }_{\sigma }^{\Gamma }\left(d\right)$ of the graphene $\sigma$ band at $\mathit{\Gamma }$ together with the contributions $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma ,\text{S}}\left(d\right)$ and $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma ,\text{CDD}}\left(d\right)$ that constitute this shift and that are caused by the surface potential $v_d^{\text{S}}$ and the potential $v_d^{\text{CDD}}$ of the charge density difference due to adsorption, respectively. The sum of both contributions is denoted $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma }\left(d\right)$. The optimized equilibrium adsorption distance is indicated by a dashed gray line.

Figure 9

(Top) Charge density difference ${\varrho }_d^{\text{CDD}}$ between the charge density of the combined system graphene/Au(111) and the sum of the charge densities of its constituents Au(111) and graphene calculated individually, at two adsorption distances $d_{\text{0}}=3.64\AA$ (orange) and $d_{\text{ref}}=9.0\AA$ (cyan). The graphene sheet is adsorbed above a Au(111) (2 $\times$ 2) unit cell. Atom positions are indicated by gray, orange, and cyan circles. Electronic charge is accumulated at and slightly above the surface and depleted next to the adsorbate in the region towards the surface. (Bottom) Electrostatic potentials $v_d^{\text{CDD}}$ that correspond to the charge density differences ${\varrho }_d^{\text{CDD}}$ at the two considered adsorption distances. In addition, charge transfer potentials $v_d^{\text{CT}}$, obtained by calculating a simple plate capacitor model based on the Bader charges of graphene $\Delta q^{\text{G}}$ and the metal $\Delta q^{\text{M}}=-\Delta q^{\text{G}}$ are depicted as red and blue dashed line, respectively (see Sec. 4 for details). The planes of the model plate capacitor are indicated by the gray and red and the gray and blue dotted lines for the two considered adsorption distances, respectively. Both ${\varrho }_d^{\text{CDD}}$ as well as $v_d^{\text{CDD}}$ are averaged over the corresponding $xy$ planes parallel to the surface for a given value of the $z$ coordinate perpendicular to the surface. At $z=0$, the topmost layer of the slab is located.

Figure 10

(Top) Energy shifts $\Delta {\epsilon }_{\sigma }^{\Gamma }\left(d\right)$, $\Delta {\epsilon }_{\sigma }^{\text{K}}\left(d\right)$, $\Delta {\epsilon }_{\pi }^{\Gamma }\left(d\right)$, and $\Delta {\epsilon }_{\pi }^{\text{K}}\left(d\right)$ of the graphene $\sigma$ and $\pi$ bands at $\mathbf{K}$ and $\mathit{\Gamma }$ as observed in the calculated band structure for a graphene ML adsorbed on a Au/Ni(111) surface for adsorption distances from $d=2.25\AA$ to $d=6.0\AA$. In addition, corresponding modeled shifts $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma }\left(d\right)$ and $\Delta {\tilde{\epsilon }}_{\pi }^{\Gamma }\left(d\right)$ at $\mathit{\Gamma }$ are displayed that originate from the Au/Ni(111) surface potential and from charge redistribution upon adsorption, see text for details. (Bottom) Energy shift $\Delta {\epsilon }_{\sigma }^{\Gamma }\left(d\right)$ of the graphene $\sigma$ band at $\mathit{\Gamma }$ together with the contributions $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma ,\text{S}}\left(d\right)$ and $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma ,\text{CDD}}\left(d\right)$ that constitute this shift and that are caused by the surface potential $v_d^{\text{S}}$ and the potential $v_d^{\text{CDD}}$ of the charge density difference due to adsorption, respectively. The sum of both contributions is denoted $\Delta {\tilde{\epsilon }}_{\sigma }^{\Gamma }\left(d\right)$. The optimized equilibrium adsorption distance is indicated by a dashed gray line.

Figure 11

The potential $v_{d,n}^{\text{CDD}}$ along the $z$ direction, the direction perpendicular to the surface, for graphene adsorbed on Ni(111) at adsorption distances of $d=2.0\AA$ (solid lines) and $d_{\text{ref}}=9.0\AA$ (dashed lines), for slabs of $n=$ 6, 9, and 12 layers of nickel, respectively. The potentials $v_{d,n}^{\text{CDD}}$ were averaged in the $xy$ plane. (Top) Situation before applying the finite-slab correction. $\Delta v_n^{\text{fs}}$ and $\Delta v_n^{\text{G}}$ with $n=$ 6, 9, 12 denote the differences of the potentials $v_{d,n}^{\text{CDD}}$ for adsorption distances $d=2.0\AA$ and $d=d_{\text{ref}}$ in the region $z\le 0$ and in the region of the graphene sheet and further outside, respectively. Both differences $\Delta v_n^{\text{fs}}$ and $\Delta v_n^{\text{G}}$ depend on the slab size $n$. (Bottom) Potentials $v_{d,n}^{\text{CDD}}$ after applying the finite-slab correction, i.e., after shifting the potentials for $d=2.0\AA$ by $\Delta v_n^{\text{fs}}$. After applying the finite-slab correction, the differences $\Delta {\tilde{v}}_n^{\text{G}}$ are independent of the slab size and so is the contribution $\Delta {\tilde{\epsilon }}_d^{\text{CDD}}$ to the energetic shifts of the graphene bands.