The classical and quantum dynamics of molecular spins on graphene

Controlling the dynamics of spins on surfaces is pivotal to the design of spintronic1 and quantum computing2 devices. Proposed schemes involve the interaction of spins with graphene to enable surface-state spintronics3,4, and electrical spin-manipulation4-11. However, the influence of the graphene environment on the spin systems has yet to be unraveled12. Here we explore the spin-graphene interaction by studying the classical and quantum dynamics of molecular magnets13 on graphene. While the static spin response remains unaltered, the quantum spin dynamics and associated selection rules are profoundly modulated. The couplings to graphene phonons, to other spins, and to Dirac fermions are quantified using a newly-developed model. Coupling to Dirac electrons introduces a dominant quantum-relaxation channel that, by driving the spins over Villain’s threshold, gives rise to fully-coherent, resonant spin tunneling. Our findings provide fundamental insight into the interaction between spins and graphene, establishing the basis for electrical spin-manipulation in graphene nanodevices.

Atomic force microscopy (AFM) confirms successful decoration of graphene by [Fe 4 (L) 2 (dpm) 6 ] (Fig.1b). Similar coverage was observed for single-and multi-layer graphene. Sufficiently low concentrations of [Fe 4 (L) 2 (dpm) 6 ] (C<10 −6 M) ensure decoration by individual molecules without appreciable aggregation, while a full layer is formed for C>10 −5 M (SI). Multiple techniques evidence that the deposited SMMs are structurally and electronically intact. In particular, the height distribution in AFM images peaks at 1.8±0.3 nm (Fig.1c), in close agreement with the size of a single [Fe 4 (L) 2 (dpm) 6 ] molecule. 20 In addition, Raman spectra ( Fig.1d) exhibit only a small up-shift of the graphene G-band by ~4 cm −1 after functionalization, indicating slight p-type doping 21 . This conclusion is supported by the small positive shift of the charge-neutrality-point observed in the resistance vs. gate voltage curves (Figs.1e and 1f), corresponding to a charge-transfer of only 0.08 e − per SMM. Furthermore, X-ray photoelectron spectra display the same characteristic Fe 2p peak positions of thick films of [Fe 4 (L) 2 (dpm) 6 ] (SI) and mass spectra evidence clear fingerprints of intact [Fe 4 (L) 2 (dpm) 6 ] at 1919, 1763 and 1736 amu ( Fig.1g and SI).
The static magnetic properties of the hybrids are indistinguishable from those of crystalline [Fe 4 (L) 2 (dpm) 6 ] (Fig.2). The temperature (T) dependence of the χ m T curves (where χ m is the static molar magnetic susceptibility, defined as the ratio between magnetization and the applied magnetic field, M/H) displays the same minimum at 100 K and a maximum around 6 K (Fig.2a, SI). The M vs H data also match within experimental error. The characteristic χ m T vs T curve can be fitted 20 by considering g=2.00±0.05 and two intramolecular exchange interactions between nearest-neighbour and next-nearest-neighbour Fe(III) ions (with exchange constants J NN =14.85±0.05 cm −1 and J NNN =0.09±0.05 cm −1 , respectively), in excellent agreement with previous investigations, confirming a total spin ground state S=5.
The low-temperature magnetic properties of SMMs are dominated by the presence of a large magnetic anisotropy 13 , which partially lifts the degeneracy of the ground multiplet and creates an energy barrier ΔE separating the |m s 〉 and |−m s 〉 states (Fig.3a). Two spinrelaxation mechanisms are thus possible: a classical overbarrier process, where multiple phonon absorptions allow overcoming ΔE, and a quantum tunneling (QT) mechanism, which takes place between degenerate levels at opposite sides of the barrier 13 . QT can be suppressed by applying an external magnetic field H that lifts the degeneracy of the |±m s 〉 levels. This allows examining the effect of graphene on both relaxation mechanisms separately.
The magnetization dynamics is probed by measuring the real (χ′) and imaginary (χ″) response to a small magnetic field oscillating at frequency ω. In a static field H=1 kOe (Fig.   3b), where only thermally-activated behaviour is present, isolated molecules on graphene show a faster magnetization dynamics than those in crystalline [Fe 4 (L) 2 (dpm) 6 ], for which Arrhenius analysis yields a barrier ΔE=9.7±0.7 cm −1 . In H=0 (Fig.3c), all |m s 〉 and |−m s 〉 sublevels fall back into resonance and QT is activated, thereby magnifying enormously the differences between the hybrids and crystals response. The T-dependence of the spin relaxation time, τ, (Fig.4a) shows that for crystalline [Fe 4 (L) 2 (dpm) 6 ] the thermallyactivated region persists down to 800 mK, below which temperature QT sets in. This contrasts strikingly with the spin relaxation in the hybrids, which is entirely dominated by QT, and the QT-rate is about one million times higher (τ −1 =10 5 Hz vs. 10 −1 Hz).
To explain these observations we subdivide the spin anisotropy into different contributions, with each term having the symmetry of a corresponding tesseral harmonic. This approach produces an intuitive but rigorous picture of the spin behaviour via the Hamiltonian: (1) where the first term describes the Zeeman interaction between the molecular spin S and H (μ B is the Bohr magneton, g is the Landé factor) and the second models the spin anisotropy via the spin operators that correspond to different harmonics, each weighted by a coefficient . These operators, called Stevens operator-equivalents 13,22 contain the j th power of the spin ladder operators, and , and the (i-j) th power of (see SI for explicit expressions and a pictorial representation of the associated harmonics). With equation (1) we can model the effect of any external electric field, e.g. the electron density of an underlying substrate, by introducing in the sum a term of corresponding symmetry 12,22 and discarding all terms not belonging to the point group of the spin environment. Stevens operatorequivalents have an effect on both classical and quantum relaxation channels: terms with j=0 contain powers of only, and contribute to ΔE but do not mix the |m s 〉 eigenstates; terms with j≠0, on the contrary, mix states differing in m s by a multiple of j, opening the QT relaxation channel. , the magnetization curves of both hybrids and crystals (Fig.2b) are perfectly reproduced. Moreover, the barrier value extracted from the dynamics provide , again leading to an excellent overall agreement for the leading anisotropy term. Axial distortion of the SMMs on graphene can thus be excluded.
We now discuss how the graphene environment alters the spin-relaxation (Fig.4b). Dipolar and hyperfine interactions introduce a local dynamic magnetic field distribution that alters the QT rate by broadening the QT resonances 13,25 (Fig.3a). The widths of these distributions, as determined 25,26 from dilution and nuclear resonance experiments in crystals, are ~200 and 1 Oe, respectively. In the hybrids, intermolecular dipolar interactions are strongly reduced due to the sizeable separation between the molecules and magnetic shielding by graphene, which is predicted to be extremely efficient 27 . On the contrary, hyperfine interactions arise from the nuclei inside the molecules themselves, and are unaffected by the substrate. Overall, this imposes an intrinsic upper limit of ~10 Hz to the QT-rate of isolated [Fe 4 (L) 2 (dpm) 6 ] (Fig.4a). 13,25,26 Any additional acceleration of QT must arise from interaction with the graphene layer.
To evaluate the effect of graphene vibrations, we extend the established theoretical framework 23,24 to two-dimensional phonon baths (SI). In principle, the effect of phonons on each molecular group is to induce a time-dependent modification (or "modulation") of any term of the magnetic energy. However, it can be shown 13 that, at low temperatures, the modulation of the isotropic exchange does not induce transitions between the |m s 〉 sublevels of the lowest exchange multiplet. Therefore, as we are interested in temperatures much lower than the Debye temperature, the only significant contribution will arise from the modulation of the anisotropy energy induced by long-wavelength acoustic phonons. The transition probabilities can be obtained 23,24 by adding to equation (1) the spin-phonon perturbation term , where ω, q and λ are the phonon frequency, wavevector and polarization, ħ the Planck constant, m c the unit cell mass, V q,λ the spin-phonon interaction potential, C q,λ the phonon creation operator, with the sum running over the Brillouin zone. For intact molecules, similar local distortions are expected in both crystals and hybrids and, provided that is essentially unaltered, as in the present case, V q,λ will also be similar in both materials. On the contrary, the different phonon densities of three-and two-dimensional baths will strongly alter the relaxation dynamics. For the hybrids, the relevant phonon bath can be described by the long-wavelength spectrum of the acoustic branches of graphene, which has an analytical expression 21 . Under these assumptions we obtain (see SI) the following relation: (2) where c and h subscripts denote crystals and hybrids, R is a parameter proportional to the experimentally-observable phonon-induced relaxation rate, ρ the mass density, C the sound velocity and k B is Boltzmann constant. This model provides excellent agreement with the data at H≠0, where the relaxation is expected to be dominated by spin-phonon interactions (Fig.3b), simultaneously accounting for the behaviour in both crystals and hybrids. Notably, no fitting parameters are necessary, and full, quantitative agreement is attained by using the same set of coefficients and introducing the known speed of sound of graphene 21 , C h =2·10 4 m/s, for the hybrids. It is noteworthy that the behaviour of crystalline [Fe 4 (L) 2 (dpm) 6 ] is restored in hybrids with large SMM aggregates, for which the vibrational bath more closely resembles the three dimensional case (SI). These observations underscore the relevance of graphene vibrational modes for classical spin relaxation.
By contrast, the different phonon bath cannot explain the striking enhancement of QT at H=0 (Fig.4a), which requires introducing the coupling to graphene electrons. The latter is expressed, within Stevens' formalism 13,22 , by including the perturbation integral 〈m s (ψ M )| V I |ψ G 〉, where molecular and graphene electronic states (ψ M and ψ G , respectively) interact via the potential V I , altering the 〈m s | spin state. The wavefunction ψ G has threefold symmetry, and the corresponding point-group 22 permits non-vanishing integral terms only for coefficients with j a multiple of 3, independently of the form of V I . In the perturbation treatment of the spin-graphene interaction, all such terms have an identical effect on the level mixing and the QT rate of the lowest |m s 〉 levels, differing only in the perturbation order. To address QT, we merge all terms into an effective overall graphene effect represented by the leading-order term , which directly admixes only the m s =±3 states (direct mixing of lower levels would require lower symmetry terms, such as ). By introducing a term into the simulations, while maintaining all other parameters, both the ac-susceptibility peak (Fig.3c) and the Tdependence of τ (Fig.4a) are excellently reproduced. A pictorial representation of the spin anisotropy in the crystals and hybrids (Fig.4c) highlights the significant weight of the additional term and the signature of the threefold rotational symmetry of graphene on the spin anisotropy. The explicit operator form of the microscopic interaction potential V I is debated. One possibility is the Heisenberg form , where , , are the spin operators of the Dirac-fermion, nevertheless the nature of the underlying scattering processes remains debated. For [Fe 4 (L) 2 (dpm) 6 ], a direct quantitative comparison among the different proposals is intricate, because of the cumbersome ab initio evaluation of the amplitude cross-sections. Most importantly, spin-flip processes as efficient as observed for the hybrids are only predicted when scattering due to the presence of two K points is included in the theory 28 , and the values here obtained can be used to validate the theoretical models proposed for spin-Dirac fermion interactions 29 .
Strongly-perturbative QT of spins, predicted by Villain decades ago 13 , strictly requires pronounced electronic perturbations combined with minimal dipolar interactions. These conditions are hard to fulfil in crystals, such that the quantum spin dynamics of Villain's regime has so far eluded observation. Graphene hybrids offer the ideal conditions by simultaneously suppressing dipolar interactions and electronically enhancing the transverse anisotropy terms. For suppressed dipolar fields, weak perturbations (as in crystals) lead to a non-monotonic behaviour in the τ vs T plot, where standard tunnelling is followed by a slow descent into Villain's QT regime (SI). In contrast, the perturbation term introduced by graphene is large enough to directly drive the spins into Villain's regime (Fig.4d), leading to a monotonous temperature dependence of τ. The associated threshold yields a lower bound of 0.3 μeV for the interaction between SMM spins and the Dirac electrons, lower than the estimated spin-orbit coupling in graphene.
To conclude, our observations set the theoretical and experimental basis for the exploration of the magnetic effects introduced by localized spins in graphene. The clear roles of the vibrational and electronic interactions, here unravelled, can now be exploited in the design of spin-nanomechanical and nanospintronic devices. The strong graphene-induced terms allows envisaging coherent electric spin manipulation, via electrically-tuned spin mixing, allowing molecular spin control in ways hitherto reserved to other qubits 30 . The observed interaction already introduces a spin-level mixing comparable to those used for qubit manipulations 30 , and the use of molecular magnets with even higher spin-orbit coupling (e.g. rare-earths) could enhance such effects even further. Similar effects, with the same symmetry, can also be envisaged for other systems, e.g. SMMs on reconstructed Au surfaces. 19 Villain's tunneling regime, being by definition coherent, opens also novel perspectives for quantum operations. Eventually, the observation of substrate effects on the quantum dynamics constitutes an important methodological tool, which can be applied to study other spin-substrate interactions, providing unique details on symmetry-breaking quantum effects.

Methods
[Fe 4 (L) 2 (dpm) 6 ] was synthesized as previously reported 20 . Graphene was mechanically exfoliated from highly-oriented pyrolytic graphite onto Si/SiO 2 substrates. Surface hybrids were obtained by immersion in a solution of [Fe 4 (L) 2 (dpm) 6 ] in 1,2dichloroethane:dichloromethane (9/1, v/v) at room temperature, followed by washing with isopropanol. Bulk samples of the hybrids were prepared by first sonicating 8.4 mg of graphite powder in 20 ml of the solvent mixture for 48 hrs, followed by adding 11.5 mg of [Fe 4 (L) 2 (dpm) 6 ] in 20 ml of the same mixture, followed by stirring for 45 min and final filtering through PTFE membranes (0.2 μm pore size). Magnetic measurements were corrected for the diamagnetism of the sample and sample holder, as independently determined. A home-built micro-SQUID susceptometer inside a dilution refrigerator was used for zero-field measurements, with extremely small ac field amplitudes (50 mOe) and the data corrected by the bare graphene signal. AFM images were acquired with a Bruker Icon system in tapping mode, using Olympus AC200TS cantilevers. MALDI-TOF spectra were recorded using a laser pulsed at ~4 Hz, while the XPS spectra were obtained with a monochromatized Al-Kα source (1486.6 eV), see SI for details. Electronic devices were fabricated using a PMMA resist, e-beam lithography and thermally evaporated Au/Ti.

Supplementary Material
Refer to Web version on PubMed Central for supplementary material.    6 ], as obtained from equation (1). Phonons determine transition probabilities (green arrows) to thermally overcome the barrier. Quantum tunneling happens between admixed levels (red arrows), while hyperfine and dipolar interactions create energy distributions (orange). Interaction with Dirac electrons increases the tunnel splittings below |3〉 and |−3〉 (red areas, exaggerated for clarity). b) Temperature and frequency dependence of the imaginary component of the dynamic susceptibility (χ″) in a magnetic field H=1 kOe for