Bonding of Gold Nanoclusters on Graphene with and without Point Defects

Hybrid nanostructures of size-selected nanoparticles (NPs) and 2D materials exhibit striking physical and chemical properties and are attractive for many technology applications. A major issue for the performance of these applications is device stability. In this work, we investigate the bonding of cuboctahedral, decahedral and icosahedral Au NPs comprising 561 atoms on graphene sheets via 103-atom scale ab initio spin-polarized calculations. Two distinct cases we considered: (i) the Au NPs sit with their (111) facets on graphene and (ii) the NPs are oriented with a vertex on graphene. In both cases, we compare the binding energies with and without a graphene vacancy under the NP. We find that in all cases, the presence of the graphene vacancy enhances the bonding of the NPs. Significantly, in the vertex-on-graphene case, the binding energy is considerably increased by several eVs and becomes similar to the (111) facet-on-graphene case. The strain in the NPs is found to be minimal and the displacement of the carbon atoms in the immediate neighborhood of the vacancy is on the 0.1 Å scale. The work suggests the creation of stable NP-graphene systems for a variety of electronic, chemical and photonic applications.


Introduction
The unique physical and chemical properties of 2D materials-graphene being the primary example-and of nanoparticles (NPs) can be further enhanced by combining the two in hybrid nanostructures. Indeed, NP-graphene composites show promise for electrochemical energy applications such as lithium batteries, supercapacitors and catalysts for various reactions including oxygen reduction, oxygen evolution and hydrogen evolution reactions [1] and a variety of bioapplications, including electronic, electrochemical and optical biosensors, bioimaging, photothermal therapies, drug delivery and tissue engineering [2]. Amongst NPs, Au is attractive for both bio [3,4] and chemical applications [5,6] due to its stability, biocompatibility and catalytic activity, and has been a well-established model material in nanocluster science for decades. During the past decade, several Au NP-graphene hybrid nanostructures have been developed and used in optoelectronic and sensing applications [7][8][9][10].
The triangular lattice of the (111) surface of the face-centered cubic (fcc) metals is a good match for the hexagonal honeycomb lattice of graphene, and the metals are accommodated on graphene with minimal strain depending on their lattice constant. An ab initio investigation reported weak bonding between graphene and Al, Ag, Cu, Au and Pt, with a binding energy of 0.03 eV per carbon atom for Au [11], and large equilibrium separations. Further experimental studies of Au-plated graphene, combining optical microscopy and electronic measurements, showed that the presence of Au on Van der Waals interactions were taken into account through the DFT-D3 method [24] with Becke-Jonson damping [25]. Under these settings, the lattice constant of the Au unit cell was found to be 4.101 Å and the a lattice constant of the graphene unit cell was found to be 2.468 Å. The choice of the correction method was made based on the results of Ref. [26] on the comparison of different van der Waals corrections to the bulk properties of graphite. The inclusion of van der Waals interactions is an important part of this work. They are known to play a role in the adsorption of small aromatic molecules on Au surfaces [27], the catalytic selectivity for the oxidation of organic molecules on metallic Au [28], the electronic and energetic properties of small Au NPs [29], and to be competing with covalent and ionic bonds for bonding on Au-S surfaces and Au NPs [30].
The imaging of the results was performed with the VESTA 3.4 visualization program (Tsukuba, Ibaraki, Japan) [31].
The dataset generated during the current study is available online at the Zenodo repository [32].

Nanoclusters with a Facet on Graphene
The (111) facets of the NPs were interfaced with graphene following the energetically favorable configuration of according to Ref. [12]: the atoms of these 21-atom triangular (111) facets are accommodated alternately on top of C atoms and in the middle of C hexagons. This particular configuration forces a lattice constant of 4.031 Å on the gold lattice-down from 4.101, Å for an initial strain of only 1.71% for a perfect pairing of the two surfaces ( Figure 2). Van der Waals interactions were taken into account through the DFT-D3 method [24] with Becke-Jonson damping [25]. Under these settings, the lattice constant of the Au unit cell was found to be 4.101 Å and the a lattice constant of the graphene unit cell was found to be 2.468 Å. The choice of the correction method was made based on the results of Ref. [26] on the comparison of different van der Waals corrections to the bulk properties of graphite. The inclusion of van der Waals interactions is an important part of this work. They are known to play a role in the adsorption of small aromatic molecules on Au surfaces [27], the catalytic selectivity for the oxidation of organic molecules on metallic Au [28], the electronic and energetic properties of small Au NPs [29], and to be competing with covalent and ionic bonds for bonding on Au-S surfaces and Au NPs [30].
The imaging of the results was performed with the VESTA 3.4 visualization program (Tsukuba, Ibaraki, Japan) [31].
The dataset generated during the current study is available online at the Zenodo repository [32].

Nanoclusters with a Facet on Graphene
The (111) facets of the NPs were interfaced with graphene following the energetically favorable configuration of according to Ref. [12]: the atoms of these 21-atom triangular (111) facets are accommodated alternately on top of C atoms and in the middle of C hexagons. This particular configuration forces a lattice constant of 4.031 Å on the gold lattice-down from 4.101, Å for an initial strain of only 1.71% for a perfect pairing of the two surfaces ( Figure 2). The most common and energetically favorable point defect of graphene is the single vacancy. We calculated the formation energy , for the neutral-charge vacancy according to: where , is the total energy of the 576-atom pristine graphene sheet and , is the total energy of the graphene sheet with the vacancy. We found a formation energy of 7.71 eV, in good agreement with previous studies [33]. The C atoms surrounding the vacancy showed the expected Jahn-Teller distortion, displayed in Figure 3a: two of the three atoms previously bonded to the missing C atom form a bond with a length of 2.002 Å, reduced from their initial distance of 2.468 Å in pristine graphene. The third atom moved further away. We found no out-of-plane displacements of the three3 C atoms. The two C atoms in the pair show magnetic moments of 0.06 , the third atom shows a magnetic moment of 0.50 . The charge is unevenly distributed between the three C atoms, with the two in the pair showing a charge of 2.40 and the third a charge of 2.58 .  The most common and energetically favorable point defect of graphene is the single vacancy. We calculated the formation energy E f ,vac for the neutral-charge vacancy according to: where E tot,sheet is the total energy of the 576-atom pristine graphene sheet and E tot,vac is the total energy of the graphene sheet with the vacancy. We found a formation energy of 7.71 eV, in good agreement with previous studies [33]. The C atoms surrounding the vacancy showed the expected Jahn-Teller distortion, displayed in Figure 3a: two of the three atoms previously bonded to the missing C atom form a bond with a length of 2.002 Å, reduced from their initial distance of 2.468 Å in pristine graphene. The third atom moved further away. We found no out-of-plane displacements of the three3 C atoms. The most common and energetically favorable point defect of graphene is the single vacancy. We calculated the formation energy , for the neutral-charge vacancy according to: where , is the total energy of the 576-atom pristine graphene sheet and , is the total energy of the graphene sheet with the vacancy. We found a formation energy of 7.71 eV, in good agreement with previous studies [33]. The C atoms surrounding the vacancy showed the expected Jahn-Teller distortion, displayed in Figure 3a: two of the three atoms previously bonded to the missing C atom form a bond with a length of 2.002 Å, reduced from their initial distance of 2.468 Å in pristine graphene. The third atom moved further away. We found no out-of-plane displacements of the three3 C atoms. The two C atoms in the pair show magnetic moments of 0.06 , the third atom shows a magnetic moment of 0.50 . The charge is unevenly distributed between the three C atoms, with the two in the pair showing a charge of 2.40 and the third a charge of 2.58 .  found that the lowest energy position is below the center of the (111) triangle, followed by the triangle edge, with an energy difference between the two of only 0.01 eV. The vacancy under the cluster's (111) facet vertex is 0.43 eV higher in energy. Based on these results, we proceeded with the relaxations of the different NP isomers with a vacancy positioned under the center of their (111) facet. The binding energies were calculated according to: , where , and are the total energies after the relaxations for the free NP, the graphene sheet (with or without the vacancy), and the NP-graphene combination, respectively. The results are shown in Table 1. The binding energy followed the icosahedral > decahedral > cuboctahedral order for both substrate configurations. We observed a general increase of the binding energy when the substrate vacancy is added on the scale of 0.17-0.24 eV for all the NPs.
In pristine graphene, all C atoms present the same charge with only minimal changes in the scale of 0.01 for the atoms in the proximity of the NP facet. No magnetism appears in any of the atoms of the model. In the defected graphene case, the magnetic moments of the three C atoms surrounding the vacancy do not change considerably. They are reduced by 0.01 for the lone C atom and by 0.03 for the C atoms in the pair. The charge distribution among these atoms is also greatly unaffected by the presence of the NP: the charge is only 0.01 lower for the pair and the same as before for the lone C atom.

Nanoclusters with a Vertex on Graphene
In the case where the cluster vertex points at the graphene substrate, we initially investigated the bonding site of the NPs with a vertex on pristine graphene and ran relaxations for two distinct positions: (i) the vertex of the NP sits on top of a C atom and (ii) the vertex of the NP lies in the center of the C-atom hexagon ( Figure 5). We found that the first case is energetically preferable by 0.33 eV, and proceeded with this particular positioning. The binding energies E binding were calculated according to: where E particle , E graphene and E graphene+particle are the total energies after the relaxations for the free NP, the graphene sheet (with or without the vacancy), and the NP-graphene combination, respectively. The results are shown in Table 1. The binding energy followed the icosahedral > decahedral > cuboctahedral order for both substrate configurations. We observed a general increase of the binding energy when the substrate vacancy is added on the scale of 0.17-0.24 eV for all the NPs. In pristine graphene, all C atoms present the same charge with only minimal changes in the scale of 0.01 e for the atoms in the proximity of the NP facet. No magnetism appears in any of the atoms of the model. In the defected graphene case, the magnetic moments of the three C atoms surrounding the vacancy do not change considerably. They are reduced by 0.01 µ B for the lone C atom and by 0.03 µ B for the C atoms in the pair. The charge distribution among these atoms is also greatly unaffected by the presence of the NP: the charge is only 0.01 e lower for the pair and the same as before for the lone C atom.

Nanoclusters with a Vertex on Graphene
In the case where the cluster vertex points at the graphene substrate, we initially investigated the bonding site of the NPs with a vertex on pristine graphene and ran relaxations for two distinct positions: (i) the vertex of the NP sits on top of a C atom and (ii) the vertex of the NP lies in the center of the C-atom hexagon ( Figure 5). We found that the first case is energetically preferable by 0.33 eV, and proceeded with this particular positioning. We explored two different symmetrical orientations for each NP isomer with a vertex on a vacancy, as shown in Figure 6. In the icosahedral and decahedral NPs, a pentagon is formed by the atoms of the atomic layer behind the vertex tip. In the orientation of Figure 6a the tip of this pentagon points towards the lone C3 atom; in the orientation of Figure 6b, it points away from it. For the cuboctahedral NP the orientations relate to the rectangle formed by the atoms of the second atomic layer of the cuboctahedral NP. The long edge of this rectangle is either parallel to the bond of atoms C1 and C2, orientation of Figure 6c, or anti-parallel to it, orientation of Figure 6d. The corresponding binding energy results after relaxation are shown in Table 2. For the decahedral and icosahedral NPs, the orientation of Figure 6b is slightly preferred energetically, possibly because three Au atoms (Au2, Au3, and Au6) sit directly on top of C atoms. However, the energy difference is minimal, only 0.001-0.002 meV. For the cuboctahedral NP, we observe an energy difference of 0.06 eV between the two orientations, with the orientation shown in Figure 6c being the We explored two different symmetrical orientations for each NP isomer with a vertex on a vacancy, as shown in Figure 6. In the icosahedral and decahedral NPs, a pentagon is formed by the atoms of the atomic layer behind the vertex tip. In the orientation of Figure 6a the tip of this pentagon points towards the lone C3 atom; in the orientation of Figure 6b, it points away from it. For the cuboctahedral NP the orientations relate to the rectangle formed by the atoms of the second atomic layer of the cuboctahedral NP. The long edge of this rectangle is either parallel to the bond of atoms C1 and C2, orientation of Figure 6c, or anti-parallel to it, orientation of Figure 6d. We explored two different symmetrical orientations for each NP isomer with a vertex on a vacancy, as shown in Figure 6. In the icosahedral and decahedral NPs, a pentagon is formed by the atoms of the atomic layer behind the vertex tip. In the orientation of Figure 6a the tip of this pentagon points towards the lone C3 atom; in the orientation of Figure 6b, it points away from it. For the cuboctahedral NP the orientations relate to the rectangle formed by the atoms of the second atomic layer of the cuboctahedral NP. The long edge of this rectangle is either parallel to the bond of atoms C1 and C2, orientation of Figure 6c, or anti-parallel to it, orientation of Figure 6d. The corresponding binding energy results after relaxation are shown in Table 2. For the decahedral and icosahedral NPs, the orientation of Figure 6b is slightly preferred energetically, possibly because three Au atoms (Au2, Au3, and Au6) sit directly on top of C atoms. However, the energy difference is minimal, only 0.001-0.002 meV. For the cuboctahedral NP, we observe an energy difference of 0.06 eV between the two orientations, with the orientation shown in Figure 6c being the The corresponding binding energy results after relaxation are shown in Table 2. For the decahedral and icosahedral NPs, the orientation of Figure 6b is slightly preferred energetically, possibly because three Au atoms (Au2, Au3, and Au6) sit directly on top of C atoms. However, the energy difference is minimal, only 0.001-0.002 meV. For the cuboctahedral NP, we observe an energy difference of 0.06 eV between the two orientations, with the orientation shown in Figure 6c being the lower energy one. The binding energies of the NPs with a vertex on pristine graphene and on a graphene vacancy are compared in Table 2. For all three structural motifs, the presence of the vacancy strengthens the bonding of the NPs on graphene by several eVs. Moreover, although few Au atoms interact with the C atoms-mainly the 1 atom at the NP tip and, secondly, the 4/5 atoms of the layer after it, the binding energies are comparable to the (111) facet-on-graphene cases. The addition of the NP at the vacancy site was found to remove the Jahn-Teller distortion in the C atoms surrounding the vacancy. An example of a vertex-on-graphene case is shown in Figure 3b. The three C atoms are pushed away from the cluster tip and the distances between them are on average 2.70 Å. The charge is equally distributed among the three C atoms and the magnetic moments disappear. In general, the in-plane displacement of the C atoms in the neighborhood of the vacancy is on the 0.1 Å scale. In the vertex-on-graphene case, the C atoms around the vacancy are displaced out of plane by 0.1-0.3 Å. The NPs easily absorb the strain induced on the Au surface in the facet-on-graphene case and, except for the first and second layers, are almost unaffected. This minimal strain of the NPs is in agreement with previous studies [16]. In the facet-on-graphene case, the equilibrium distance between the NPs and the graphene sheet is 3.30 Å, the same value previously reported in the literature [11]. In the vertex on graphene vacancy case, the corresponding distance is 1.60 Å, close to the value calculated in the literature for atomic adsorption of Au on a single graphene vacancy [34].
The charge density difference ∆ρ, shown in Figure 7 for a decahedral NP on a vacancy, was calculated according to: lower energy one. The binding energies of the NPs with a vertex on pristine graphene and on a graphene vacancy are compared in Table 2. For all three structural motifs, the presence of the vacancy strengthens the bonding of the NPs on graphene by several eVs. Moreover, although few Au atoms interact with the C atoms-mainly the 1 atom at the NP tip and, secondly, the 4/5 atoms of the layer after it, the binding energies are comparable to the (111) facet-on-graphene cases. The addition of the NP at the vacancy site was found to remove the Jahn-Teller distortion in the C atoms surrounding the vacancy. An example of a vertex-on-graphene case is shown in Figure 3b. The three C atoms are pushed away from the cluster tip and the distances between them are on average 2.70 Å. The charge is equally distributed among the three C atoms and the magnetic moments disappear. In general, the in-plane displacement of the C atoms in the neighborhood of the vacancy is on the 0.1 Å scale. In the vertex-on-graphene case, the C atoms around the vacancy are displaced out of plane by 0.1-0.3 Å. The NPs easily absorb the strain induced on the Au surface in the facet-ongraphene case and, except for the first and second layers, are almost unaffected. This minimal strain of the NPs is in agreement with previous studies [16]. In the facet-on-graphene case, the equilibrium distance between the NPs and the graphene sheet is 3.30 Å, the same value previously reported in the literature [11]. In the vertex on graphene vacancy case, the corresponding distance is 1.60 Å, close to the value calculated in the literature for atomic adsorption of Au on a single graphene vacancy [34].
The charge density difference ∆ , shown in Figure 7 for a decahedral NP on a vacancy, was calculated according to: Where , and , are the electron charge densities obtained from static runs of the relaxed models of the NP on graphene, the graphene sheet and the free NP, respectively. We observed a small charge transfer to the two C atoms bonded in the Jahn-Teller distortion of 0.14 and from the third lone C atom of 0.04 . There is also a charge transfer to the Au atom at the tip of the NP of 0.23-0.24 . Visible in Figure 7a, is the uniform charge distribution between the three C atoms interacting with the NP tip. The charge density difference in Figure 7b illustrates the striking fact that only one Au atom essentially participates in the bond between the NP and the graphene sheet. This also explains the lack of any strain or deformations on the whole NP in the vertex-on-graphene cases.  Where ρ cluster+graphene , ρ cluster and ρ graphene , are the electron charge densities obtained from static runs of the relaxed models of the NP on graphene, the graphene sheet and the free NP, respectively. We observed a small charge transfer to the two C atoms bonded in the Jahn-Teller distortion of 0.14 e and from the third lone C atom of 0.04 e. There is also a charge transfer to the Au atom at the tip of the NP of 0.23-0.24 e. Visible in Figure 7a, is the uniform charge distribution between the three C atoms interacting with the NP tip. The charge density difference in Figure 7b illustrates the striking fact that only one Au atom essentially participates in the bond between the NP and the graphene sheet. This also explains the lack of any strain or deformations on the whole NP in the vertex-on-graphene cases.
The electronic properties of graphene are affected by the presence of the NPs. In Figure 8, the partial density of states (DOS) of graphene sheets sampled at the Γ-point with and without Au NPs is shown. The greater effect is observed in the facet-down case. In Figure 8a a comparison between pristine graphene and pristine graphene interfaced with the (111) facet of an icosahedral Au NP is shown. All peaks are "shifted" towards higher energies by~0.2 eV when the NP is added. When an icosahedral NP is placed with a vertex on a graphene vacancy, the levels before and after the peak of the Fermi level are again "shifted" towards higher energies by~0.1 eV (Figure 8b). The electronic properties of graphene are affected by the presence of the NPs. In Figure 8, the partial density of states (DOS) of graphene sheets sampled at the Γ-point with and without Au NPs is shown. The greater effect is observed in the facet-down case. In Figure 8a a comparison between pristine graphene and pristine graphene interfaced with the (111) facet of an icosahedral Au NP is shown. All peaks are "shifted" towards higher energies by ~0.2 eV when the NP is added. When an icosahedral NP is placed with a vertex on a graphene vacancy, the levels before and after the peak of the Fermi level are again "shifted" towards higher energies by ~0.1 eV (Figure 8b). Finally, the energetic ordering of the NPs is unchanged by the interaction with the graphene sheet. The stability of the bare NPs follows the icosahedral > decahedral > cuboctahedral trend reported previously in previous theoretical studies of regular clusters in the literature [35], with an energy difference of 2.97 eV to the decahedral and 6.83 eV to the cuboctahedral NP from the icosahedral isomer, respectively. These energy differences become 3.68 eV and 4.25 eV, respectively, when the NPs are accommodated with their (111) facet on graphene (3.51 eV and 4.49 eV with a C vacancy under the NP facet); and to 3.26 eV and 7.14 eV when the NPs are pinned with their vertices in a graphene vacancy. We note that the experimental measurements of the isomers' relative stability differ from the theory predictions for regular clusters [36].

Discussion
We investigated the energetics and the electronic, magnetic, and structural effects of graphene vacancies on Au NP-graphene hybrid nanostructures in this work. To do so we used large-scale models consisting of more than 10 3 atoms in ab initio calculations that took into account spinpolarisation and van der Waals interactions. Our results showed that defects enhance the bonding of Au561 clusters with three atomic structures on graphene, particularly when the clusters sit with a vertex on the vacancy. The resulting binding energies in the last case were similar to the facet-ongraphene case, despite the involvement of only 1 Au atom in the bond with graphene, compared with 21 in the parallel-facet case. All of the resulting nanostructures showed a small strain, a symmetrical displacement of the C atoms in the immediate neighborhood of the vacancy, a uniform distribution of the charge between these C atoms, and a minimal strain on the NPs. The increased binding energy and the competition between facet-down and vertex-down are expected to be even more pronounced in clusters smaller than 561 atoms, since fewer atoms belong to the (111) facets. In contrast, the Finally, the energetic ordering of the NPs is unchanged by the interaction with the graphene sheet. The stability of the bare NPs follows the icosahedral > decahedral > cuboctahedral trend reported previously in previous theoretical studies of regular clusters in the literature [35], with an energy difference of 2.97 eV to the decahedral and 6.83 eV to the cuboctahedral NP from the icosahedral isomer, respectively. These energy differences become 3.68 eV and 4.25 eV, respectively, when the NPs are accommodated with their (111) facet on graphene (3.51 eV and 4.49 eV with a C vacancy under the NP facet); and to 3.26 eV and 7.14 eV when the NPs are pinned with their vertices in a graphene vacancy. We note that the experimental measurements of the isomers' relative stability differ from the theory predictions for regular clusters [36].

Discussion
We investigated the energetics and the electronic, magnetic, and structural effects of graphene vacancies on Au NP-graphene hybrid nanostructures in this work. To do so we used large-scale models consisting of more than 10 3 atoms in ab initio calculations that took into account spin-polarisation and van der Waals interactions. Our results showed that defects enhance the bonding of Au 561 clusters with three atomic structures on graphene, particularly when the clusters sit with a vertex on the vacancy. The resulting binding energies in the last case were similar to the facet-on-graphene case, despite the involvement of only 1 Au atom in the bond with graphene, compared with 21 in the parallel-facet case. All of the resulting nanostructures showed a small strain, a symmetrical displacement of the C atoms in the immediate neighborhood of the vacancy, a uniform distribution of the charge between these C atoms, and a minimal strain on the NPs. The increased binding energy and the competition between facet-down and vertex-down are expected to be even more pronounced in clusters smaller than 561 atoms, since fewer atoms belong to the (111) facets. In contrast, the number of atoms in the vertex is independent of the NP size. We generally expect the vertex-on-graphene binding energies to remain constant with the size of the NPs, and the facet-on-graphene binding energies to increase with the size of the NPs.
We believe that a closer look into the electronic properties of these nanostructures would be of great interest for future studies and critical for their application in electronic devices. The methodology of this paper can also be applied for the examination of the properties of another novel hybrid nanostructure of NPs and graphene: graphene-covered gold NPs [37,38]. Such a study should consider the structural changes that might appear in both graphene and the nanoparticles [39]. Experimental information, such as high-resolution electron microscopy images, would be of the utmost importance for replicating the realized nanoparticles in such a theoretical study.
Funding: This work was financially supported by the Engineering and Physical Sciences Research Council through the fellowship EP/K006061/2 and by the European Union's Horizon 2020 program through the CritCat project under Grant Agreement No. 686053. TP received financial support from the European Union's Horizon 2020 program and the Welsh Government through the Marie Skłodowska-Curie Actions Sêr Cymru II COFUND fellowship No. 663830-SU165. We acknowledge the support of the Supercomputing Wales project, which is part-funded by the European Regional Development Fund (ERDF) via Welsh Government.