Tuning the spin texture of graphene with size-specific Cu n clusters: a first-principles study

The size dependent interaction of Cu n ( n = 1 − 5) clusters with pristine and defective (C-vacancy) graphene is studied by employing density functional theory. The computed binding energies are in the range of ∼ 0.5 eV for pristine graphene and ∼ 3.5 eV for defective graphene, indicating a much stronger interaction in the later system. The induced spin–orbit coupling interaction, due to the proximity of the Cu n cluster, is studied with non-collinear spin-polarized simulations. The clusters cause a spin splitting in the order of few meV. The resultant low energy bands spin textures are also computed, and a spin–valley coupling in the case of even atom clusters on pristine graphene is predicted, leading to the emergence of a spin lifetime anisotropy. For defective graphene, a complete out-of-plane spin texture and a large spin splitting of 40–100 meV is obtained for Cu n ( n = 1, 2, 3, 5) clusters due to local magnetic moment. On the other hand, for Cu 4 /defective graphene, having no net magnetic moment, the spin–valley coupling prevails close to the band edges.


Introduction
Ever since the discovery of graphene [1], a large family of 2D materials have been designed and studied, with each demonstrating unique properties for technological applications [2][3][4][5]. The 2D nature of these materials renders them highly sensitive to their environment, and allows to tune their properties through close proximity to other materials, e.g. to induce magnetic ordering [6,7], superconductivity [8][9][10], and topological states [11][12][13][14]. These effects could be combined and engineered for exploring new phenomena [15][16][17]. In particular, in the field of spintronics, graphene with its Dirac cone and long room temperature spin lifetime makes it a suitable candidate for passive spintronics. However, to realize innovative manipulation and generation of spin currents, its spin-orbit coupling (SOC) needs to be amplified by orders of magnitude. Several research efforts have been made to induce SOC by proximity, which include forming graphene-vdW heterostructures [18,19] and the adsorption of adatoms like hydrogen [20], fluorine [21] and heavy metals Os [13], Au [22,23] and In [14]. In all these cases, SOC induced band splitting is in the range of few meV. By their mere proximity the transition metals are capable of imprinting a unique spin texture in graphene electronic bands. So far not much effort has been made in studying the size dependent proximity effect of atomic clusters, presumably because the clusters may aggregate, and their size-selective fabrication and controlled deposition is challenging. Notably, in the work of Scheerder et al and Keijers et al, size specific adsorption of Au n clusters (n = 3 & 6) on graphene was experimentally demonstrated and the later studied the variation of the induced SOC strength for different cluster sizes [24,25]. Upon comparing with the spin-splitting obtained from density functional theory (DFT) simulations, they were able to identify the spin relaxation mechanism of the Au n clusters/graphene systems as being primarily the Elliot-Yafet (EY) like mechanism.
The additional degree of freedom, provided via the number of atoms in the clusters to tune the electronic properties of graphene, is studied in the present work. More specifically, the size-dependent interaction of Cu n clusters on pristine and defective (C-vacancy) graphene is investigated, using first-principles simulations based on DFT. Copper is traditionally employed as a substrate for the growth of graphene by chemical vapour deposition [26]. An earlier report suggested that graphene deposited on Cu(111) surface interact through weak van der Waals type bonding [27], which implies copper may induce proximity effects without drastically affecting graphene's properties. Graphene functionalized with copper adatoms was studied by Frank et al by employing ab initio and tight binding model calculations [28]. Their simulations predicted spin-orbit induced splitting in the order of a few meV, which is three orders of magnitude larger than the intrinsic SOC of pristine graphene of a few µeV [29].
The lowest energy ground state configuration for Cu n (n = 1-5) clusters adsorbed on pristine and defective graphene is first obtained. The induced properties can be altered significantly by the cluster orientation, therefore a site-specific search was made to find the geometrical configuration corresponding to the global minimum. Non-collinear band structure and spin texture calculations are performed to study the induced SOC effects. Apart from inducing spin splitting in the range of few meV, the adsorption of the clusters also strongly modifies the spin texture of the low energy bands of graphene. The breaking of the structural symmetry upon cluster adsorption induces Rashba and spin-valley coupling, resulting in a spin lifetime anisotropy between in-plane and out-of-plane polarized electrons. We show that the size of the adsorbed cluster imparts large variations in the spin texture of graphene monolayers, either with or without defects.

Method
DFT simulations, as implemented in the plane wave code Vienna ab initio simulation package (VASP), were used to compute the ground state atomic configurations and electronic structures of the graphene/Cu n (n = 1-5) systems [30][31][32][33]. The Cu n clusters were adsorbed on a 5 × 5 graphene supercell. The z-axis (out-of-plane) of the supercell was set to 20 Å to avoid interlayer interactions with the neighbouring cells. Different adsorption sites were considered for the adsorption of the atomic clusters and the systems were relaxed until the forces between the atoms were less than 25 meV Å −1 . The spin-polarised DFT calculations were performed using a 11 × 11 × 1 k-point mesh to sample the Brillouin zone [34]. The energy cut-off was set to 520 eV and the width of the Gaussian smearing was set to 0.05 eV. The nonlocal vdW-DF2 exchange-and correlation functionals of Lee et al [35][36][37][38] were employed in these simulations.
The binding energy of the clusters adsorbed on pristine/defective graphene was estimated using the following relation: where E g is the total energy of the pristine/defective graphene without Cu cluster. E Cun is the total energy of cluster with n atoms. E g + Cun is the total energy of the cluster adsorbed pristine/defective graphene system. The spin texture plots were computed using a 21 × 21 Γ-centered mesh and the spin expectation values were post-processed using VASPKIT and pyprocar [39,40].

Structural optimization
The most stable geometries of the isolated copper clusters (with lowest total energies) were first obtained, and the results are in good agreement with the work of Calaminici et al for neutral Cu n clusters [41]. Next, the Cu n clusters were relaxed on graphene in different initial configurations to find the most stable geometries. The obtained geometries are shown in figure 1(f), and the results are summarized in table 1. These results were obtained for single cluster adsorbed graphene supercell, which would be equivalent to a cluster number density of 7.58 × 10 13 cm −2 . The clusters are adsorbed with a modest binding energy of −0.3 to −0.8 eV at about 2.2-2.6 Å above the pristine graphene plane, except Cu 5 which is located at a distance of 3.5 Å from the graphene. The adsorption of the cluster induces a slight distortion of the graphene plane, i.e. the bond lengths between the C atoms on which the cluster absorbs and their nearest neighbours is elongated by about 0.01-0.015 Å. The distortion of graphene indicates that the cluster adsorption could  induce charge transfer and SOC interactions, which will be discussed below. Isolated odd-numbered clusters possess a magnetic moment of ∼1 µ B due to the presence of an unpaired s state. Upon adsorption, the magnetic moment is slightly reduced, due to the interaction of the Cu n cluster with the graphene plane. For Cu 3 /graphene, the cluster moment is reduced to 0.79 µ B , indicating charge transfer interaction with graphene, and yielding a comparatively larger doping efficiency.
The computed band structures of the Cu n /graphene systems along the high symmetry points of the Brillouin zone are shown in figures 1(a)-(e). In general, one can observe a hybridization energy gap in the band structure around the Dirac cone (along M-K-Γ) due to the overlapping copper and carbon orbitals. Especially the Cu d-states (red coloured dots) hybridize with the graphene valence band. In the case of Cu 1 /graphene, the flat band observed at the Fermi level corresponds to a mid-gap impurity-like state. A similar band structure was obtained for a Cu adatom on graphene in the work of Frank et al [28]. An odd-even oscillation is manifested in the charge transfer process as well. The doping density (n s ) was estimated from the shift of the Dirac cone from the Fermi level (∆E F ), where v F is the Fermi velocity. For Cu 3 /graphene, n-type doping is predicted with a doping efficiency of 0.145 e per cluster, and for a Cu 5 cluster on graphene, it results in p-type doping with an efficiency of 0.022 h per cluster. In the case of even-numbered clusters, no doping is observed, but the cluster-induced structural asymmetry between the graphene sublattices results in the opening of an energy gap at the Dirac point, which is 21.9 meV and 26.9 meV for Cu 2 /graphene and Cu 4 /graphene respectively. These band gaps originate from the breaking of symmetry since the clusters are adsorbed in a top configuration on one of the two sublattices, leading to a symmetric separation of the Dirac cones [20]. The observed oscillation in graphene's doping is also related to the pairing of Cu 4 s electrons. In the case of odd-numbered clusters, the unpaired 4 s states (cyan coloured dots) are close to the Fermi level, resulting in the redistribution of charge, i.e. the 4 s states are involved in the charge transfer. For the even numbered clusters, the paired 4 s states are well below the Fermi level, and predominantly hybridize with the graphene valence band states.
Note that the geometries of the isolated Cu n clusters are also similar to the Au n clusters, reported in our previous work [23], since both elements share a similar electronic configuration. However, compared to the Au n /graphene systems, where the Au n clusters are weakly chemisorbed (−0.6 to −1.7 eV), the lower binding energies between Cu n and graphene rather suggest a physisorption process. The physisorption of the clusters results in less distortion of the graphene, with preserved high electron and hole mobilities. Nevertheless, the physiosorbed clusters can induce SOC in graphene, due to the proximity effect and inversion symmetry breaking, which will be discussed in the next section.

SOC and spin texture
The induced SOC effect due to symmetry reduction and proximity effects were studied by performing non-collinear spin-polarized calculations on cluster adsorbed graphene systems. The induced SOC lifts the spin degeneracy, resulting in a band gap opening and spin-splitting of the conduction and valence bands. The obtained results are summarized in table 2 and the energy band structures are provided in the Supplementary Information (figure S2). The observed band gap openings are in the range of ∼15-20 meV (except for Cu 5 ∼ 0.1 meV). The reduced spin-splitting for Cu 5 /graphene could be related to the longer cluster-graphene distance of 3.53 Å. In order to confirm this, we made a single point calculation for a reduced cluster-graphene distance of 2.3 Å (this configuration is 2.38 eV higher in energy than the lowest energy configuration). This resulted in larger spin-splitting of ∼6 meV. Therefore, the spin splitting depends on the cluster size, distance, and adsorption site.
The adsorption of the cluster breaks the inversion symmetry of graphene along the z-axis, leading to a Rashba SOC and momentum-dependent spin splitting [18,19]. Apart from the Rashba-effect, the broken inversion symmetry also leads to a difference in strength of the pseudospin inversion asymmetry (PIA) SOC for graphene's A and B sublattices. PIA-SOC is manifested by the momentum modulated spin splitting of the bands, i.e. the k-linear spin splitting of bands [19,42]. When the sublattice symmetry is broken, a different intrinsic SOC arises for the two sublattices, which couples spin and valley, resulting in valley-Zeeman SOC. In contrast to the Rashba-induced splitting, the valley-Zeeman effect leads to the out-of-plane spin polarization, independent of the momentum direction, with opposite signs at the K and K ′ valleys, to preserve the time reversal symmetry. The interplay of the above-mentioned SOC terms can be observed in the Cu n /graphene systems and their strengths can be qualitatively highlighted by the spin texture.
In the case of odd-atom clusters with a local magnetic moment, the magnetic anisotropic energy (MAE) of these system can be determined by computing the total energy difference between the out-of-plane (z-axis) and in-plane (x-axis) spin direction (E(001) − E(100)). The obtained energies are in the range of 27.5 µeV, −1.52 µeV & −116.5 µeV, respectively, for Cu 1 , Cu 3 and Cu 5 adsorbed on graphene. The easy axis orients along the x-axis for Cu 1 and along the z-axis for Cu 3 & Cu 5 clusters. Though MAE ranges within a few µeV, the spins are coupled locally to the magnetic moment of the cluster (∼1 µ B ), leading to the Zeeman-type band splitting, with each subsequent band polarised with spins in opposite directions. Therefore, we observe a momentum independent spin texture (see supplementary information figure S3). This does not imply the absence of Rashba SOC, which still should be present due to the breaking of the inversion symmetry, but the overall spin polarization vector is dominated by the Zeeman SOC.
In the case of even-atom clusters (with no net magnetic moment), due to the coupling of spin and valley, we expect a scenario which is presented in figure 2(a), where the effect of the Rashba and Zeeman contributions on the band structure are schematically drawn. In Rashba (Zeeman) SOC the spin-split bands are polarised oppositely along the in-plane (out-of-plane) direction and their Fermi surface splits into two concentric circle. In the case of combined Rashba and Zeeman SOC, both in-plane and out-of-plane spin components are present, and their resulting spin texture depends on the competition of both terms. For Cu 2 and Cu 4 adsorbed graphene, we expect this particular scenario of Rashba + Zeeman SOC. Figure 2(b) shows the spin-splitted band structure of Cu 4 /graphene, where the induced SOC spin splits the band (projected along the S x component) which shows the presence of both Rashba and Zeeman SOC. In the event of dominant Zeeman SOC, the two concentric circles would be non-equidistantly spaced, as shown by the fermi surface of Rashba + Zeeman SOC (which is observed for Cu 3 on defective graphene, discussed in In the case of the Cu 2 cluster-graphene system, the in-plane components of the spin present a chiral spin texture, with a finite out-of-plane S z component. The in-plane spin components follow the typical Rashba spin texture, due to the breaking of graphene's inversion symmetry. On the other hand, the out-of-plane spin component is driven by the presence of valley-Zeeman and, which arises due to the breaking of sublattice symmetry, and is characterised by the valley dependent polarization of the S z component. The out-of-plane component of the spin on a constant energy surface is shown in the top and bottom inset of figure 3(a) along the K and K ′ valley, respectively. The observed 'hedgehog-type' spin texture [43] could be used for potential spintronic applications involving the valley degree of freedom [44,45].
Interestingly, the in-plane spin component in the Cu 4 /graphene system follows a similar chiral texture as in Cu 2 /graphene, due to the prevalence of inversion symmetry breaking in both systems. However, the Cu 4 /graphene spin texture displays only a very small out-of-plane spin component S z as evidenced in figure 3(b) and indicated by the (light blue) colourmap near the K and K ′ valley. This suggests the dominance of the in-plane spin components due to Rashba SOC term over the out-of-plane SOC terms. Consequently, the size of the cluster plays a significant role in favouring certain component of the spins. In comparison with our earlier work on Au n /graphene systems [23], the induced splitting in Cu n /graphene is smaller but chiral spin texture was not observed for Au 4 /graphene (since the in plane distortion in Au 4 /graphene is twice as large as in Cu 4 /graphene). This provides the opportunity to engineer Cu n cluster size dependent spin polarizers, as discussed below.

Spin lifetime anisotropy
In the literature, the two majorly discussed spin relaxation mechanisms for graphene are the EY and the Dyakonov-Perel (DP) mechanisms [46][47][48]. In the EY mechanism, each momentum scattering carries a finite probability for spin flip, whereas in the DP mechanism, the electron spin undergoes Larmor's precession in the presence of an SO effective magnetic field [49,50]. Both mechanisms have different dependences on the momentum lifetime, and their dominance depends on the energy range and carrier density. Assuming that the DP mechanism plays an important role near the Dirac cone, similar to the case of SOC-induced proximity effects between graphene and topological insulators [19], one can extract information about the spin-lifetime anisotropy from the spin texture.
In the DP mechanism, the precession frequency is given by ω = ∆/h, where ∆ is the spin splitting of the bands, and the direction of the effective field B (k) = ωS k depends on the momentum direction (due to Rashba SOC). The scattering of electrons influences the spin precession and leads to spin relaxation at a rate given by τ −1 x , which randomizes the x-component of the effective magnetic field. In general, it is assumed that τ * z = τ * y = τ * x = τ p , with τ p the momentum relaxation time. The spin lifetime anisotropy is defined as the ratio of out-of-plane to in-plane spin relaxation time, which is given by [19] where S 2 x = ⟨S x ⟩ 2 is the squared average of the x component of the spin (similarly for the z components). The sum is considered over all four bands near the Fermi surface. Due to the spin-valley coupling in our particular case, there is an additional intervalley scattering time τ iv to the in-plane rate, Therefore, one gets a spin life-time anisotropy ζ in units of τ iv τp . The computed spin lifetime anisotropy for the conduction band of Cu 2 /graphene is shown in figure 4(a) (in the valence band, the flat band due to the contribution of the Cu cluster states comes into play). The anisotropy term becomes larger as one moves closer to the Dirac point, in the range of 10 ( τ iv τp ). Away from the Dirac point, the anisotropy is much reduced, but typically remains larger than 1 till the energy range of Table 3. The binding energy, bond length, magnetic moment, and spin splitting of Cun/defective graphene (n = 1-5). Only the spin splitting obtained at the Dirac point is provided. CB and VB represent conduction band and valence band, respectively. 0.2 eV from the Fermi level, indicating longer lifetime for spins polarised in the out-of-plane direction. At higher energy the spin texture is dictated mainly by the Rashba term. Therefore, the average spin lifetime anisotropy collapses below 1. As the anisotropy depends on the ratio of τ iv τp , and assuming τ iv = (5 − 10) τ p [19], this leads to a substantial difference in spin lifetime for the out-of-plane polarised spins.
Opposed to Cu 2 /graphene, the computed anisotropy of Cu 4 /graphene is smaller than 1 ( figure 4(b)), with the out-of-plane spin relaxation being about three times as fast as the in-plane component. This could be a result of the stronger valley-Zeeman term in Cu 2 /graphene with sizable S z component (colormap in figure 3) in comparison to Cu 4 /graphene. Spin lifetime anisotropies have been experimentally observed for graphene/MoSe 2 and graphene/WS 2 heterostructures due to the spin proximity effects [51][52][53]. These results again highlight that the size of the adsorbed cluster is an important 'parameter' that can tune the direction of the spin polarised current in graphene.

Geometry
The presence of defects can influence the adsorption of a metal cluster and its induced SOC interaction. We performed simulations on graphene with a single carbon vacancy. Upon introducing a single carbon vacancy in graphene, the Cu n clusters prefer to adsorb on the defect site, with a higher binding energy in the range of 3 eV (see table 3), indicating chemisorption. The enhanced interaction between the cluster and graphene leads to the deformation of the carbon bonds near the adsorption site, as shown in figures 6(c) and (d). The absence of a single carbon atom in the graphene lattice induces a local magnetic moment (<2 µ B ). As pointed out in the work of Chen et al, the presence of a single vacancy releases four electrons to occupy four localized states, due to a Jahn-Teller distortion [54]. Notably, no net magnetic moment is predicted for the Cu 4 cluster adsorbed on defective graphene. The magnetic moment of the system after the adsorption of the Cu n cluster is given in supplemental information table 3.

SOC and spin texture
The computed spin polarised electronic band structures of the Cu n /defective graphene systems are summarized in figure 5. In the case of a single C-vacancy in graphene, the Fermi level lies within the valence band (figure S6), but upon the adsorption of the cluster, the Fermi level shifts closer to the Dirac point (figure 5), indicating charge transfer between the cluster and defective graphene. The MAE of clusters with a local magnetic moment were obtained using the same procedure as mentioned in section 3.1.2 for pristine graphene. In the case of defective graphene, the magnetic moment prefers to orient along the z-axis, with a MAE of −26.2 µeV, −413.6 µeV, −1.17 µeV & −18.5 µeV for Cu n (n = 1,2,3&5) adsorbed graphene, respectively. The strength of the computed MAE tends to increase with the magnetic moment, which in turn depends on the charge transfer interaction between the cluster and graphene.
The magnitude of the spin-splitting of the conduction and valence bands, as obtained by the non-collinear spin calculations, are provided in table 3. The spin splitting values in defective graphene are much larger than in pristine graphene (table 2) due to the significantly enhanced graphene-cluster interaction. The adsorption of a Cu n cluster on a C-vacancy also breaks the inversion symmetry along the in-plane direction. This allows for the presence of an out-of-plane component in the spin polarization vectors. The computed band structures and the corresponding spin textures are shown in figure 6 for Cu 3 and Cu 4 clusters adsorbed on defective graphene.
In the case of Cu 3 , we observe a complete out-of-plane polarization, with negligible in-plane spin component. The observed spin texture indicates a dominant Zeeman SOC term, originating from the lack of in-plane inversion symmetry and the presence of a local magnetic moment [55]. A small Rashba SOC is also present, due to the symmetry reduction in the out-of-plane direction. From the Fermi contour in figures 6 (a) and (d), an uneven shift of the spin-split bands can be observed (along the k y direction), confirming the manifestation of a combination of Rashba and Zeeman SOC ( figure 2(a)). Similar spin textures were observed for Cu n (n = 1,2,5) adsorbed systems, i.e. the polarisation of the split bands is in opposite direction, and they align precisely in the out-of-plane direction. These results are provided in the supplementary information (see figure S5).
On the other hand, the net magnetic moment of the Cu 4 /defective-graphene system is zero. In this case, the Rashba SOC term becomes dominant (like in the Cu 4 /pristine graphene system) and allows for the presence of the in-plane component of the spin polarization vectors. The cumulative effect of in-plane and out-of-plane inversion symmetry breaking also enhances the band splitting, as compared to the Cu 4 /pristine graphene system (2 meV). In addition, since the in-plane spin components are momentum dependent, and the out-of-plane components are coupled with the valley, the spin life-time anisotropy ζ can be computed, as mentioned in section 3.1.3. Note that the results are presented here for the valence band, due to the prevalence of Cu states at the conduction band edge. The obtained results are plotted in figure 7. The  anisotropy is largest closer to the band edge and decreases at higher energies, reaching a value which is higher than 1, thereby favouring out-of-plane spin polarisation. It could be seen that at energies between −0.17 eV to −0.25 eV the anisotropy drops to ∼0.8 ( τ iv τp ). The underlying reason for these minima is unclear. Note, however, that intervalley scattering is caused by structural defects like vacancies, therefore it could be in the same range as the momentum scattering, leading to an isotropic spin relaxation.

Conclusions
First-principles calculations, based on DFT, were performed to study the interaction of few atom Cu n clusters with pristine and defective (C-vacancy) graphene. The simulations indicate that Cu n clusters are physiosorbed on pristine graphene with binding energies in the range of ∼0.5 eV. The presence of a C-vacancy on the graphene supercell leads to a much stronger interaction with the Cu clusters, as well as the deformation of the graphene monolayer near the vacancy site. Non-collinear calculations were performed to study the induced SOC by the proximity of the Cu n clusters. The induced SOC strength shows a variation with the size of the adsorbed cluster, with different spin splitting values for the low energy bands. The computed spin texture depends on the local magnetic moment induced by the cluster. For clusters on pristine graphene without net magnetic moment, like Cu 2 and Cu 4 , a Rashba-type chiral spin texture is observed with a significant spin-valley coupling. Similarly, in the case of a Cu 4 cluster adsorbed on a single vacancy graphene site (with net zero magnetic moment), the spin-valley coupling is also predicted and is found to arise from the interplay between the Rashba and in-plane Zeeman terms. For other clusters adsorbed on defective graphene, a complete out-of-plane spin texture is predicted, due to breaking of the in-plane inversion symmetry.
The breaking of the sub-lattice symmetry and z/−z inversion symmetry could induce a spin lifetime anisotropy, within the Dyakonov-Perel regime. From our simulations, we expect a larger spin lifetime for the out-of-plane spin component for the Cu 2 /graphene system. In the case of Cu 4 /graphene, the scenario is inversed with the out-of-plane spin component being relaxed twice as fast as the in-plane component. The difference in behaviour is due to different dominating SOC terms in each system, which in turn depends on the size and orientation of the adsorbed cluster. This offers a possibility to design spin polarizers by precisely depositing size-selected clusters on graphene. The presence of defects and disorder can significantly impact the spin texture and spin lifetime, e.g. a Cu 4 cluster on defective graphene showed a different trend in comparison to the pristine graphene counterpart. These findings highlight that the size of the deposited Cu clusters can have a significant impact on graphene's electronic and spin properties. Therefore, proper engineering along with ab initio calculations could guide towards maximizing the use of atomic clusters as a tool to functionalize graphene-based devices.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.