Magnetic clusters as efficient EY-like spin-scattering centres in graphene

Abstract

The spin scattering induced by magnetic adsorbates on graphene was studied using a combination of transport measurements on a graphene field effect transistor decorated with atomically precise nickel clusters and first principles calculations. A comparative study before and after deposition of Ni₄ clusters unambiguously corroborated the contribution of the added scatterers. An investigation of the spin scattering parameters as a function of the applied voltage indicated a cluster-induced Elliot–Yafet like spin scattering mechanism. Density functional theory calculations were used in combination with a tight-binding model to quantify the strength of the spin–orbit coupling terms induced by the adsorbed clusters.

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1 Introduction

Controlled doping by impurities can provide semiconducting materials with the desired properties. In the case of two-dimensional materials, the adsorption of metal atoms and nanoparticles offers a similar platform to extend functionalities. For example, to transform graphene into a spin active material, a strong Rashba spin–orbit coupling can be induced by the proximity effect. Transition metal dichalcogenides (TMDs) and adatoms have the potential to enhance spin–orbit coupling (SOC),^1–3 thereby inducing spin anisotropy^4–7 and robust quantum spin Hall states, enabling gate manipulation of spin transport and the conversion of electrical currents into spin currents via the spin Hall effect (SHE) or vice versa (iSHE).^8–11 However, fundamental questions remain on the spin relaxation mechanisms in graphene. Both experimental and theoretical studies dealing with the influence of impurities and adatoms on spin scattering yield contradictory results on the dominant spin scattering mechanism.^12–16 In a recent proof-of-principle experiment, we investigated the spin transport parameters of graphene spin valves decorated with soft-landed, size-selected, few-atom Au_n clusters.¹⁷ A study as a function of cluster density allowed the derivation of cluster-size dependent spin and momentum scattering times.

Exploiting the material flexibility of cluster beam deposition, we redirect our attention to magnetic clusters to identify their impact on the spin transport of the hybrid structure. We focus on nickel clusters due to their relatively high intrinsic magnetic moments,¹⁸ which are predicted to not be completely quenched when adsorbed on perfect graphene.¹⁹ In order to select the most appropriate cluster size, the electronic structure of Ni_n (n = 2–5) clusters adsorbed on graphene was calculated. These calculations pinpoint Ni₄ as an interesting candidate to enhance the spin properties of the Ni_n/graphene system. Firstly, the Dirac cone is almost undistorted, preserving graphene's high carrier mobilities. Secondly, the Ni₄ cluster has a large doping efficiency, implying a strong charge transfer to graphene. Finally, non-collinear spin-polarized calculations of the Ni₄/graphene system indicate that the Ni₄ cluster gives rise to sizable spin splitting of the Dirac bands due to the induced spin–orbit coupling and exchange interaction.

To investigate the impact of the magnetic clusters on (spin) transport, a graphene spin valve is decorated with Ni₄ clusters at a density of 5.7 × 10¹² clusters per cm². Following the example by Zomer et al.,²⁰ we study the spin relaxation mechanism by characterizing the spin transport parameters at different values of applied backgate voltages before and after cluster deposition. As the spin scattering mechanism in pristine graphene (i.e. the virgin state) is debated in the literature,⁴ only a comparative study can unambiguously corroborate the contribution of the added scatterers. By performing Hanle spin precession measurements at different applied backgate voltages, the spin scattering time, τ_s, and the momentum scattering time, τ_p, are obtained. By comparing the ‘before’ and ‘after’ states of the device, the induced effects of the clusters are identified as an Elliot–Yafet (EY) like spin scattering mechanism.

2 DFT calculations of Ni_n/graphene

First-principles calculations were performed using the Vienna ab initio simulation package (VASP) with plane wave basis sets and projected-augmented wave (PAW) potentials.^21–23 First, the ground state geometrical configurations of the Ni_n clusters adsorbed on graphene were obtained by performing simulations with non-local vdW-DF2 exchange and correlation functionals.^24,25 The clusters were placed at different possible adsorption sites in the 5 × 5 graphene supercell with an interlayer spacing of 20 Å. This corresponds to a cluster density of 7.6 × 10¹³ cm⁻². The systems were relaxed until the forces between the atoms were less than 25 meV Å⁻¹ using a 5 × 5 × 1 k-point mesh with an energy cutoff of 520 eV. Furthermore, electronic structure calculations were performed on a denser mesh of 11 × 11 × 1 k-points. For Ni₃ and Ni₅ adsorbed on graphene, the Ni d-states are close to the Dirac point, resulting in a distortion of the Dirac cone in the spin-polarised band structure. As discussed below, the Ni₄ cluster preserves the Dirac cone and thus the high carrier mobility, while it has at the same time a large doping efficiency per cluster, implying a stronger charge transfer to graphene, with a high mobility and strong localized magnetic moment. Therefore, we direct our attention to Ni₄ clusters as a test bed to explore the physics at play when magnetic clusters are adsorbed on graphene.

The resultant total energies were compared to obtain the most stable configuration, which is shown in Fig. 1(a). The isolated Ni₄ cluster is tetrahedrally shaped and it prefers to adsorb with a single Ni atom on the bridge position (between two carbon atoms) with bond lengths of 2.119 Å and 2.299 Å. The Ni₄ adsorption energy is calculated to be −1.05 eV, indicating a stronger interaction than physisorption (a few hundred meV), but too weak for a real chemical bond. It can be seen from electronic structure calculations that Ni₄ donates electrons to graphene, resulting in a (Ni⁺C⁻) charge state type of attraction at the adsorption site. The adsorption of the Ni₄ cluster leads to the elongation of the C–C bond length by ∼0.01 Å. The adsorbed Ni₄ has a localised magnetic moment of 3.3μ_B, which is slightly reduced when compared to the magnetic moment of the isolated cluster (3.7μ_B).

Fig. 1 (a) The optimized geometrical structure obtained for Ni₄/graphene after relaxation. (b) Non-collinear band structure of the Ni₄/graphene system, where the red-colored dots represent the C-projected bands. The band structure is zoomed close to the Dirac point around the symmetry point K. The blue arrows mark the 4 Ni states, located close to the band extrema, which were used in the tight binding Hamiltonian.

The doping characteristics of the cluster are studied by performing open-shell electronic structure calculations. From the band structure of the Ni₄/graphene system, the Dirac cone is almost undistorted (see Fig. 1(b)) and the adsorption of the Ni₄ cluster preserves the high carrier mobility of graphene. The resultant band structure yields an n-type doping, as can be observed from the apparent shift of the Dirac cone below the Fermi level, which is consistent with the experimental results that are presented in section 3. From the shift of the Dirac point, one can compute the charge carrier doping density,²⁶ which corresponds to a doping efficiency of about 0.09 electrons per cluster. While the electron transfer was found to be independent of the supercell size, its absolute value does vary slightly with the size of the supercell (see the ESI† for details).

In the next step, we studied the induced spin-dependent effects of the Ni₄ cluster on the graphene layer. The induced spin splitting is estimated from the band splitting along the graphene band lines (i.e., only the C-projected band states are considered) around the Dirac point. The band gap amounts to 25 meV and the obtained values of the band line splitting vary along the M → K → Γ path with an average value of 48.5 meV for the valence band and 29.9 meV for the conduction band. These values vary slightly with the size of the supercell that is used in the computations, as discussed in the ESI.† The average value is computed by calculating the absolute energy difference between the spin-split C-projected band in the −0.6 eV to 0 eV energy range. The origin of the induced spin-splitting is discussed below.

To qualitatively study the strength of different SOC terms, we constructed a tight binding Hamiltonian model to reproduce the electronic structure of the Ni₄/graphene system close to the Dirac point. From the DFT band structure, four Ni states are located close to the band extrema. These states are indicated by blue arrows in Fig. 1(b). A simple model was constructed dealing only with the interaction between the Ni anchor atom (with four spin states) and the two graphene C atoms (with two spin states per site). The Hamiltonian is given by

Here, c⁺_iσ(c_iσ) and h⁺_Mσ(h_Mσ) correspond to the creation (annihilation) operator for an electron with spin σ at graphene lattice site i and cluster site M, respectively. ε₀ and t represent on-site and nearest-neighbor hopping energies between the carbon atoms. The onsite energy and local Zeeman coupling of the Ni atom are given by ε_Ni and J_Ni respectively. The adsorption of the Ni cluster modifies the onsite potential (ε_ads1) and induces a local magnetic interaction (J₁) at the closest (2.119 Å) adsorption site and its two nearest neighboring sites. J₂ corresponds to the Zeeman coupling at the carbon site 2.299 Å from the Ni₄ cluster. This C site is the third nearest neighbor of the closest adsorption site, but due to the bonding with the Ni atom, the on-site potential (ε_ads2) is slightly modified. The hopping from C to Ni (t_C–Ni) was set to 0.5 eV to fit the DFT band structure. The spin–orbit coupling Hamiltonian is derived based on ref. 27–29 as follows:

The first term represents spin-conserving next nearest neighbor hopping, with the intrinsic (λ_i) and valley-Zeeman (λ_VZ) spin–orbit terms, where and . The second term in eqn (2) represents the spin mixing nearest neighbor hopping, with λ_BR being the Bychkov–Rashba spin–orbit coupling term. The third term in eqn (2) corresponds to the spin mixing next nearest neighbor hopping, where λ^A/B_PIA is the strength of the pseudospin inversion asymmetry induced spin–orbit coupling, due to the sublattice symmetry breaking. In the above Hamiltonian, ξ_i is +1 (−1) for sublattice A (B), υ_mn is +1 (−1) for the clockwise (counterclockwise) hopping path to the next nearest neighbor site, and d_mn (D_mn) is the unit vector from site n to its nearest (next-nearest) site m. The single (double) bracket represents the sum of the first (second) nearest neighbor sites.

The tight binding model was fitted to the spin-split conduction and valence Dirac cones of the Ni₄/graphene system. The parameters were obtained by minimizing the energy difference in the DFT computed band lines and are provided in Table 1. The strength of the SOC is given by the magnitude of λ, and its sign corresponds to the direction of the spin splitting. The calculated electronic band structure, obtained from the tight binding model, is compared with the DFT simulations in Fig. 2. The tight binding model reproduces the DFT spin–orbit band structure close to the Dirac point. The tight-binding analysis suggests that the spin conserving valley-Zeeman-type SOC, represented by a large value of λ_VZ, dominates the SOC interaction in the Ni₄/graphene system.

Fig. 2 The band extrema of the Ni₄/graphene system obtained with the simple tight binding Hamiltonian (solid green line) in comparison with the DFT SO band structure (blue dots). The band extrema of pristine graphene computed using DFT (orange dots) and a simple tight binding Hamiltonian (solid grey line) are shown for reference purposes. Tight binding computations are done using a hopping parameter of −2.5 eV and a spin orbit coupling strength of 12 μeV.

Table 1 Tight binding parameters used to reproduce the DFT SO band structure of the Ni₄/graphene system. The two values for ε_Ni and J_Ni are due to the consideration of only two sites for the Ni anchor atom in the model

Orbital parameters (eV)
ε 0	t	ε ads1	ε ads2	ε Ni	J Ni	J 1	J 2
−2.16	−2.53	−5.94	−6.82	−2.79, −2.30	0.030, 0.023	−0.056	0.082

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Spin–orbit parameters (meV)
λ i	λ VZ	λ BR	λ ^APIA	λ ^BPIA
16.9	−59.8	4.22	1.60	−8.06

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3 Experimental results

3.1 Device characteristics

The device used as a starting point to fabricate the Ni₄/graphene hybrid is presented in Fig. 3. A more in-depth discussion of the sample fabrication process can be found in ref. 17. The graphene layer resides on a highly doped Si substrate terminated with a 300 nm SiO₂ top layer. The graphene is grown using platinum-based chemical vapor deposition (Pt-CVD) and subsequently transferred to the substrate.³⁰ The graphene's area of interest lies between the two magnetic contacts indicated by the width (W) and length (L) of the probed area, both of which are 10 μm. A backgate voltage V_g can be applied between the highly doped Si electrode and the graphene channel. To ensure a difference in magnetization reversal of the two Co electrodes at the center, the widths of these Co bars are set at 290 nm and 320 nm, respectively. The interface between these Co electrodes and graphene is made of 1 nm TiO_x to facilitate the spin injection and measurement. These left and right barriers behave as ohmic contacts in the applied voltage range and are roughly 425 Ω and 280 Ω, respectively. Conductivity measurements are performed by applying a current from #1 to #4 while measuring voltage over #2 and #3. Nonlocal measurements, i.e. the Hanle spin precession measurement, are carried out by applying a current from #2 to #1, while a nonlocal voltage is measured over #4 and #3. Both the conductivity and non-local transport measurements are performed at room temperature and a current of 10 μA is used. For such a low current, the effect of heating in these graphene devices can be ignored.³¹ During the Hanle spin precession measurements, after aligning the magnetization of the central Co electrodes, a magnetic field is applied perpendicular to the plane of the graphene. The measurements are done for field strengths ranging from −120 mT to +120 mT.

Fig. 3 (a) Scanning electron micrograph of the device. The scalebar represents 5 μm. The Co/TiO_x electrodes (false coloured in blue) are numbered #1–#6. The horizontal dark strip is the graphene flake (false coloured in green). (b) Sheet conductivity σ_S of graphene as a function of backgate voltage V_g of the virgin and Ni₄ decorated device.

Using a magnetron sputtering gas aggregation cluster source,³² a molecular beam of nickel clusters is formed. Ni₄⁺ clusters are selected with atomic precision using a quadrupole mass filter and finally soft-landed on graphene devices. With a retarding field, the average kinetic energy of the Ni₄⁺ clusters was determined to be 5.1 eV.³³ During deposition, the device was supplied a repelling voltage of +2.5 V, reducing the average landing energy of the clusters to 2.6 eV. Therefore, the cluster energy is tuned to the soft-landing regime (1 eV per atom (ref. 34)). The beam current I_B was recorded during the deposition, and after integration, the total cluster density was found to be n_c = 5.7 × 10¹² clusters per cm².

3.2 Impact of Ni₄ on charge transport properties

Fig. 3(b) presents the backgate dependent conductivity measurements of graphene in the virgin state and after Ni₄ cluster deposition. For the virgin state, these measurements indicate a charge neutrality point, V_CNP, of +0.5 V and an average mobility μ_av = (μ_e + μ_h)/2 of 2.7 × 10³ cm² V⁻¹ s⁻¹. The electron and hole mobilities (μ_e and μ_h, respectively) are extracted from linear least-squares fitting procedures with a 2 V interval around the steepest part of the conductivity curve σ_S(V_g).³⁵ The V_CNP becomes −18 V after cluster deposition. The shift of V_CNP corresponds to a doping efficiency of 0.15–0.25 electrons per Ni₄ cluster taking into account the 5.7 × 10¹² clusters per cm² deposited density. The hole mobility after cluster deposition decreased to 1.9 × 10³ cm² V⁻¹ s⁻¹, while the electron mobility became 2.3 × 10³ cm² V⁻¹ s⁻¹. The different influence of the Ni₄ clusters on the electron and hole mobilities, which is reflected in the enhanced asymmetry of the sheet conductivity curve after cluster deposition, can be attributed to the stronger Ni–graphene interaction in the hole Dirac cone than in the electron Dirac cone. This aligns with the calculated band structure (Fig. 1), where the energy difference of Ni states in the valence band with the Dirac point (∼0.2 eV) is slightly smaller than the corresponding energy difference of Ni states in the conduction band. Compared to the cases of Au₃ and Au₆,¹⁷ the results correspond to each other in a qualitative sense, where a decrease in mobility was seen as well following cluster deposition. However, quantitatively, the electron donation per cluster is larger than that of the Au₆ cluster by a factor of ∼3 and almost two orders of magnitude compared to the Au₃ cluster. Considering the Pauling electronegativity of Au and Ni (2.4 and 1.8), compared to 2.5 for C, Ni has a lower tendency to attract electrons and it therefore donates more electrons to graphene. Compared to the simulations, the experimentally obtained doping efficiency is twice as large. This discrepancy could be due to the fact that DFT simulations are performed on ideal free-standing graphene monolayers, i.e. neither defects nor substrate effects are taken into account. Next, the spintronic properties of the device are evaluated, both in the virgin state and after Ni₄ deposition.

3.3 Spin transport characteristics of the virgin state

3.3.1 Hanle measurements and spin signal amplitude. Fig. 4(a) presents a selection of Hanle spin precession measurements performed at applied backgate voltages of V_g = −50 V, 0 V and 50 V. The background has been subtracted from the presented data (see the ESI of ref. 17 for details). It is clear that the amplitude of the nonlocal resistance (‘Amp’) is large at V_g = ±50 V, but low for the one performed around V_CNP. By extracting the amplitude for all Hanle curves at different applied backgate voltages by means of least squares fits (lines in panel (a)), the purple curve presented in panel (c) is obtained. It should be noted that the non-local resistance is lower at applied backgate voltages V_g close to V_CNP, which is around 0 V in the virgin state (see Fig. 3(b)). Combined with the linear V(I) observed for all contacts, the gate dependence of the nonlocal spin signal indicates spin injection and detection dominated by pin-holes in the barriers.³⁶ The background observed in the nonlocal measurements, similar to the one observed by Volmer and co-workers,³⁷ corroborates the pin-hole nature of the barriers. These pin-holes could inject and detect the spin in a non-uniform way.

Fig. 4 (a) Hanle spin precession measurements, after background subtraction, at backgate voltages V_g = −50 V, 0 V and 50 V in the virgin state. (b) Repeat of the measurement in panel (a) after Ni₄ deposition. (c) Amplitude of the nonlocal resistance at zero field, after background subtraction, as a function of backgate voltage. All data are corrected for back-gate creep (see the ESI of ref. 17 for details).

3.3.2 Spin transport parameters. The parameters extracted from the Hanle measurements are the conventional spin transport parameters, with the diffusion constant D and the spin lifetime τ_s. The graphene's V_g dependence on the fitted τ_s and D in the virgin state and after Ni₄ adsorption is presented in Fig. 5. Similar to the signal amplitude as a function of applied backgate voltage, the spin transport parameters show minima around V_g ≃ V_CNP. The relative (1 − σ) error on these parameters ranges from 10% to 30%, with the largest error around the charge neutrality point, where the S/N ratio is lower due to the diminished signal amplitude. These errors are estimated from χ² analysis. As the signal degrades even further after cluster deposition, the error on these fitting parameters around V_CNP becomes too large and therefore, the measurements are omitted in panels 5(b) and 5(d).

Fig. 5 Spin transport parameters of the graphene spin valve. (a) Spin lifetime τ_s and (c) diffusion constant D as a function of V_g for the device in the virgin state. (b) Spin lifetime τ_s and (d) diffusion constant D as a function of V_g for the device after Ni₄ deposition. All data shown are averaged for the gate up and down measurements after V_CNP creep correction.¹⁷

Since the Elliot–Yafet (EY) and D'yakonov Perel (DP) mechanisms have very different dependencies on the Fermi level E_F and the momentum scattering lifetime τ_p, studying the spin transport parameters at different V_g values provides information about the scattering mechanism at play. As introduced by Zomer et al.,²⁰ the spin scattering rate can be written as Matthiessen's rule of summation of the scattering rates caused by EY and DP simultaneously: τ_s⁻¹ = τ_s,EY⁻¹ + τ_s,DP⁻¹, assuming that individual scattering probabilities are independent of each other. Inserting the formulas for EY and DP, with Δ_EY being the spin–orbit coupling strength of the EY interaction³⁸ and Δ_DP being the spin–orbit coupling strength of the DP interaction,³⁹

into Matthiessen's rule gives:

Fig. 6 presents the left hand side term, , as a function of E_F²τ_p². τ_s is directly obtained from the Hanle fit, E_F is measured from V_CNP and the applied gate voltage, and τ_p is calculated from the Hanle fit-parameter D using the relation D = v_F²τ_p/2, where v_F ≈ 10⁶ m s⁻¹. Using this presentation of the data, eqn (5) implies that a linear fit estimates the slope as 4Δ_DP²/ħ² and the offset as Δ_EY². It should be noted that not all data extracted from the Hanle fit are presented here. Although the S/N ratio close to the V_CNP is still reasonable for the characterization of the virgin state, the signal decreased too much after deposition in order to obtain reliable fitting parameters. Therefore, for consistency, the data close to the V_CNP that could not be measured in the decorated state also was left out of the fit and figure in the virgin state.

Fig. 6 Hanle fit parameters (τ_s and τ_p = 2D/v_F²) before and after Ni₄ cluster deposition presented according to eqn (5) as the blue and red circles for the virgin and Ni₄ decorated device, respectively. The offset and slope of the linear fits plotted are related to the Δ_EY and Δ_DP coupling strengths.

The linear fit performed on the virgin state measurement provides an EY SOC Δ_EY = 0.97 ± 0.17 meV and a DP SOC Δ_DP = 0.053 ± 0.002 meV. It can be seen from these values that both scattering mechanisms have a non-zero contribution to the total spin scattering. Compared to the exfoliated graphene samples measured in Zomer et al.,²⁰ where the EY coupling strength lies in a range of 1–2 meV and DP in a range of 0.05–0.1 meV, the experimental values obtained here are of similar magnitude. Graphene devices on a SiO₂ substrate also showed similar coupling strengths.¹⁵ Using the EY and DP scattering strengths measured from the linear fit on the virgin state data (Fig. 6), the spin scattering rates of the two mechanisms can be compared directly. For example, at 50 V from the charge neutrality point (E_F = 0.22 eV), the scattering rate of DP is 1.5 ns⁻¹, while that of the EY mechanism is 0.35 ns⁻¹. While for the largest applied backgate voltages, V_g = ±75 V, corresponding to |E_F| ≈ 0.27 eV, the DP induced scattering rate is almost an order of magnitude higher than the EY induced scattering rate. This difference in the spin scattering rate can be calculated from eqn (3) and (4) using the SOC obtained from the fit and a typical τ_p of 50 fs. Only when |E_F| ≲ 0.13 eV (V_g − V_CNP ≲ 18 V), the EY induced scattering rate is larger than the DP induced scattering rate. However, in contrast to the Fermi level regimes away from the Dirac point, where the measured τ_s (Fig. 5(a and b)) corresponds to the sum of both scattering mechanisms, this sum fails to capture the behaviour around the Dirac point. As the EY spin scattering rate is proportional to E_F⁻², the scattering rate tends to infinity as E_F → 0. Thermal broadening and charge puddles are not taken into account in this formula, which could explain the discrepancy with the experimental data.

3.4 Spin transport characteristics of the Ni₄ cluster decorated state

3.4.1 Hanle measurements and spin signal amplitude. After performing the analysis on the virgin state of the graphene spin valve, the spin transport parameters of graphene with a deposited density of 5.7 × 10¹² Ni₄ clusters per cm² are examined. As discussed in section 3.2, the charge neutrality point shifts from +0.5 V to −18 V after Ni₄ cluster deposition (see Fig. 3(b)). Fig. 4(b) presents a few Hanle curves measured at different backgate voltages (−50 V, 0 V and 50 V). The amplitudes of the Hanle curves after deposition are smaller than those of the virgin sample. Moreover, since the V_CNP shifted to lower voltages after cluster deposition, the data measured at constant applied backgate voltages no longer have equal amounts of charge carriers due to the clusters acting as dopants. As such, the Hanle curves obtained at −50 V and 50 V are no longer identical. Considering Fig. 4(c), the V_g corresponding to minimum spin signal amplitude has shifted towards the new charge neutrality point.

3.4.2 Spin transport parameters. Fig. 6 presents the same analysis as was done for the virgin state of the device, but now for the decorated state shown as the red dots. Again, eqn (5) is used to extract the EY and DP SOC. The best fit parameters of the linear fit result in Δ_EY = 1.68 ± 0.09 meV and Δ_DP = 0.055 ± 0.001 meV. Comparing these coupling strengths with those in the virgin state, it is clear that the EY SOC increased significantly, while the DP SOC remains constant within the fit uncertainty. In summary, the backgate dependence of the spin transport parameters revealed a DP dominant spin relaxation mechanism in the virgin state away from the Dirac point. After the Ni₄ cluster deposition, an increase in the EY SOC coupling strength is found, while the DP SOC remained constant. This result indicates that the Ni₄ clusters introduced spin scattering via the EY mechanism. Next, the induced SOC by the Ni₄ clusters Δ_EY,Ni4 is further examined.

3.5 Ni₄ cluster-induced SOC in graphene

A comparison of the experimentally obtained SOC strengths with those found in the DFT calculations in section 2 requires isolating the Ni₄ contribution from the total coupling strength. We estimate the value for Δ_EY,Ni4, representing the SOC strength associated with the Ni₄ clusters at certain backgate voltages. Two highly doped regimes are chosen: high p-doping and high n-doping, evaluated at V_g − V_CNP = −50 V and V_g − V_CNP = +50 V. In these high doping regimes, the spin lifetime could be extracted from the Hanle spin precession experiments with low uncertainty. The carrier densities in these regimes are n = 3.59 × 10¹² holes per cm² and n = 3.59 × 10¹² electrons per cm², respectively. In order to obtain Δ_EY,Ni4, we consider eqn (6), representing the terms contributing to the total spin scattering rate:

The first term on the right hand side represents the spin scattering contribution due to the virgin scatterers in the EY mechanism. The second term expresses the Ni₄ cluster EY scattering contribution. The last term represents the DP scattering contribution. The required data extracted from the measurements are collected in Table 2. Finally, inserting these quantities and their errors in the equations, the Ni₄ cluster induced SOC EY coupling strength is Δ_EY,Ni4 = 4.0 ± 0.5 meV evaluated at V_g − V_CNP = −50 V and Δ_EY,Ni4 = 4.3 ± 0.5 meV at V_g − V_CNP = +50 V (or −/+0.21 eV from the Dirac point, respectively).

Table 2 Overview of the spin and electronic transport properties, evaluated at the charge densities n = 3.59 × 10¹² holes per cm² and n = 3.59 × 10¹² electrons per cm². The momentum scattering time τ_p is found via the relation with v_F ≈ 10⁶ m s⁻¹

n c	Virgin state	Ni4 cluster decorated state
V CNP	0.5 V	−18 V
n	3.59 × 10^12 holes per cm^2
E F	−0.221 eV
V g	−49.5 V	−68 V
σ S(n)	1.38 mS	1.13 mS
τ p(σS,n)	53 ± 1 fs	43 ± 1 fs
τ s	560 ± 60 ps	330 ± 70 ps
n	3.59 × 10^12 electrons per cm^2
E F	0.221 eV
V g	50.5 V	32 V
σ S(n)	1.51 mS	1.30 mS
τ p(σS,n)	58 ± 1 fs	50 ± 1 fs
τ s	500 ± 50 ps	360 ± 60 ps

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For an intuitive way to understand eqn (6) and how the obtained Δ_EY,Ni4 relates to the Δ_EY, one has to consider the complete (spin) scattering picture of our device. When it scatters due to a scattering centre already present in the virgin sample there is about a 1 in 52 000 chance that the spin will scatter as well, since . When there is a momentum scattering event with the Ni₄ cluster, this probability is much higher: about 1 in 3000 as , with the coupling strength taken in the hole regime. In this picture, the Δ_EY provided by the linear regression in Fig. 6 can be regarded as the ‘average’ scattering probability behaving like the EY mechanism, weighted over the clusters and the other momentum scatterers. As seen in Table 2, the momentum lifetime decreases from τ_p,0 = 53 fs to τ_p = 43 fs after Ni₄ deposition in the hole dominated regime and from τ_p,0 = 58 fs to τ_p = 50 fs in the electron dominated regime. The relative contribution of the cluster momentum scatterers to the total momentum scattering equals τ_p,Ni4⁻¹/τ_p⁻¹ = (τ_p⁻¹ − τ_p,₀⁻¹)/τ_p⁻¹ = 19% in the hole doped regime and 14% in the electron doped regime. This indicates that about 1 in 6 momentum scattering events are caused by the Ni₄ cluster. Therefore, the measured increase from Δ_EY = 0.97 ± 0.17 meV to Δ_EY = 1.68 ± 0.09 meV can be attributed to a Ni₄ EY coupling strength of Δ_EY,Ni4 = 4.0 ± 0.5 meV and Δ_EY,Ni4 = 4.3 ± 0.5 meV for the hole and electron charge carrier regimes, respectively.

The spin–orbit coupling strengths obtained from the tight binding model for the spin-mixing terms λ_BR and λ^A/B_PIA (which act like a second-order Rashba SOC) are in close agreement with the experimental values obtained for Δ_EY,Ni4. It should be noted that the presence of magnetic impurities can lead to an efficient spin relaxation mechanism due to resonant spin scattering induced by impurities, as shown by Kochan et al.⁴⁰ The presence of a larger spin–orbit coupling and exchange interaction induced by the adsorbed Ni₄ cluster could thus turn the clusters into local spin scatterers, which enhances the spin flip probability for each scattering event. Consequently, Ni₄ clusters act as charge and magnetic dopants in graphene, having a strong impact on its spin transport properties while possibly maintaining graphene's high charge carrier mobility due to the weak distortion of the Dirac cone induced by the adsorbed clusters.

4 Conclusions

We performed an in-depth analysis of Ni₄ clusters on graphene using both DFT calculations and spin transport experiments. Before and after deposition analysis reveals an EY-like spin scattering mechanism induced by the Ni₄ clusters with an EY SOC strength of Δ_EY,Ni4 = 4.0 ± 0.5 meV. The overall SOC can only be ‘engineered’ up to a maximum of 4.0 ± 0.5 meV due to the nature of the EY mechanism by decorating graphene with Ni₄ clusters. When the cluster density is increased, the contribution of the cluster to the total momentum scattering moves closer towards 100%. The experimental values obtained for Δ_EY,Au3 and Δ_EY,Au6 were 5.7 meV and 3.9 meV.¹⁷ These values are comparable to the one obtained for Δ_EY,Ni4 in the current work and as such, the magnetic character of nickel cannot be pinpointed as a much stronger source of spin scattering. In order to increase the SOC, different sizes or elements need to be examined and the experimental protocol presented in this work is ideally suited for performing such studies.

Data availability

The data that support the findings of this study are available from corresponding author JVdV upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the Research Foundation-Flanders (FWO, Belgium), grant no. G0D5619N.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4nr01478b

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