Abstract
The use of single molecule magnets in mainstream electronics requires their magnetic moment to be stable over long times. One can achieve such a goal by designing compounds with spinreversal barriers exceeding room temperature, namely with large uniaxial anisotropies. Such strategy, however, has been defeated by several recent experiments demonstrating underbarrier relaxation at high temperature, a behaviour today unexplained. Here we propose spin–phonon coupling to be responsible for such anomaly. With a combination of electronic structure theory and master equations we show that, in the presence of phonon dissipation, the relevant energy scale for the spin relaxation is given by the lowerlying phonon modes interacting with the local spins. These open a channel for spin reversal at energies lower than that set by the magnetic anisotropy, producing fast underbarrier spin relaxation. Our findings rationalize a significant body of experimental work and suggest a possible strategy for engineering room temperature single molecule magnets.
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Introduction
Single molecule magnets (SMMs) are molecules comprising a few magnetic ions, which show the properties of both bulk magnets and lowdimensional systems^{1}. For instance, SMMs can display magnetic hysteresis together with quantum tunnelling of the magnetization^{2}. The magnetism of SMMs can be characterized through relaxation experiments, where an ensemble of molecules is polarized along the direction of an external field and then the magnetization is monitored in time. At high temperature, the relaxation time, τ, exhibits an activated Arrheniuslike behaviour , with τ_{0} being the inverse attempt frequency, k_{B} the Boltzmann constant and U_{eff} the effective barrier for relaxation^{1}. Clearly, the use of SMMs in devices, such as nonvolatile memories or simply longliving quantum systems^{3}, requires the relaxation times to be long enough. This in turn means to design molecules with a large U_{eff}.
The spin relaxation effective barrier is related to the energies of the spin excited states. For instance, the Hamiltonian of a spin S system with uniaxial anisotropy and zerofield splitting, D, is , so that U_{eff}=DS^{2}. One has then two options for enhancing the relaxation barrier, either increasing the molecule total spin or designing compounds with very large magnetic anisotropy (large D)^{4}. The same argument, of course, applies to any magnetic material and in fact the design of small magnetic bits for data storage is accompanied by using hard magnetic materials. In SMMs, one can use lanthanide ions as magnetic centres and engineer the first coordination shell so to produce a crystal field maximizing the magnetic anisotropy^{5}. Such a strategy to enhance U_{eff} has produced some success^{6,7,8}, however, several discrepancies remain. While significant deviations from the Arrhenius behaviour at low temperature can be ascribed to tunnelling effects^{1,2}, there is now a consistent body of evidence showing that the hightemperature spin relaxation follows a thermally activated behaviour with an observed relaxation barrier significantly lower than that measured by, for instance, spectroscopic methods^{9,10,11}. Furthermore, such deviations are sometime more pronounced when D is large^{12}.
Overthebarrier activation is essentially a classical diffusive process that requires the system to absorb enough energy to be excited over a potential barrier. In SMMs, such energy is provided by the interaction of the spin with the environment, namely with the molecule vibrations. This is a spin–phonon relaxation mechanism, also known as Orbach relaxation^{13}. In a nutshell (see Fig. 1), a phonon with energy ħω=U_{0}, equal to the difference between the energy of the ground state, E_{0}, and that of the first excited state, E_{2}, is absorbed by the molecule. Such excited state can then relax back to the ground state or to a spinflipped one of energy E_{1}, which is typically quasidegenerate with the ground state. The phonon absorbed should have two requirements: to be resonant with the first available spin transition (with the energy barrier in Fig. 1), and to have nonvanishing spin–phonon coupling.
Here we extend the concept of phononmediated spin relaxation to the more realistic case in which the phonons acquire a finite lifetime, namely to the situation where the spin couples to an anharmonic phonon bath. We will demonstrate that the finite linewidth of the phonons spectral distribution allows an Orbachtype spin relaxation mechanism even when the phonon is not resonant at the spin excitation energy (see Fig. 1). This has the effect of reducing the effective barrier for relaxation, consistently with the most recent experiments. Such mechanism is first illustrated with a model S=1 spin system, and then calculated with advanced electronic structure theory for a single crystal of [(tpa^{Ph})Fe]^{−} molecules (S=2), where the ligand H_{3}tpa^{Ph} corresponds to the tris((5phenyl1Hpyrrol2yl)methyl)amine. Our findings introduce new important design elements in the search for slowrelaxing SMMs, namely one has to maximize the magnetic anisotropy and at the same time engineer the molecule vibrations so to reduce the spin–phonon coupling or the phonon dissipation in the spectral range close to the molecule spin excitations.
Results
Spin–phonon dynamics theory
Let us start by writing down the equations of motion for the coupled spin–phonon system. The lowenergylying SMM spectrum is composed by a manifold of spin levels, whose degeneracy is removed by relativistic interactions. The total electronic Hamiltonian can then be written as , where is the nonrelativistic Born–Oppenheimer Hamiltonian and all the spin–orbit interactions have been included in . The ionic degrees of freedom are described by the normal modes of vibration through the Hamiltonian, , where is the phonon density operator and is the creation (annihilation) operator for a phonon of frequency ω_{α}. Molecular vibrations, either within a single molecule or through the relative motion of the molecules in a crystal, modulate the spin–orbit interaction. This results in a spin–phonon coupling Hamiltonian that in first approximation is linear in the ionic displacement, , and can be written as
The dynamics of the entire system (spin plus phonons) follow the time evolution of the total density operator, ,
where is the total Hamiltonian. Equation (2) can be simplified by assuming that the phonon dynamics is much faster than the spin one, as it is found for slowrelaxing SMMs. The Born–Markov approximation allows us to integrate out the phonons’ component of the density matrix and reduces the problem to a purely electronic one in the presence of a phonon bath. The dynamics of the spin degrees of freedom can then be studied through the firstorderreduced spin density operator, , as described by the diagonal elements of the Redfield equation^{14}
where , , a〉 is the eigenfunction of the spin system described by the Hamiltonian H_{0} with energy E_{a} and ω_{ab}=(E_{a}−E_{b})/ħ. Thus in equation (3) represents the kinetic rate of population transfer between the eigenstates a〉 and c〉.
The spectral representation of the Green’s function for the phonon bath, G(ω_{ij}, ω_{α}), is the central quantity of our discussion, since it contains the temperature dependence of the spin dynamics. A rigorous treatment of this quantity would require knowledge of the phonons dissipation process, namely of the anharmonic constants. This means solving the appropriate phonons equations of motion or approximating the relevant selfenergy if a perturbative approach is possible. Until now, only the zerothorder term of G has been introduced into the theory^{1,13}. This corresponds to the ‘undamped’ case, that is, to the harmonic approximation to bath dynamics. Here we make a step forward and we explicitly introduce phonons dissipation by modelling their spectral shape. Thus, G of each phonon assumes a Lorentzian shape with amplitude Δ_{α},
where is the mean phonon occupation number and β=1/k_{B}T. Note that the lifetime of the αth mode is τ_{α}=ħ/Δ_{α}.
The inset of Fig. 1c schematically illustrates the typical temperature dependence of Δ_{α}, which takes a small constant value for T<T* and then becomes linear above the threshold temperature, T* (refs 15, 16, 17). The spin dynamics then qualitatively depends on the value of U_{0} with respect to k_{B}T* (see Fig. 1c). For *, the phonons have a long lifetime (Δ_{α} is small) over the entire temperature range in which the spin dynamics is usually measured (k_{B}T<U_{0}). Therefore, at any relevant temperature, spin relaxation takes place through the standard Orbach’s mechanism for harmonic phonons, with τ following an Arrheniuslike activated temperature dependence and U_{eff}=U_{0}. This is a frequent situation found in lowanisotropy SMMs, where indeed the first excitation energy can be associated with the effective barrier for relaxation. In contrast, for U_{0}>k_{B}T*, the temperature dependence of Δ_{α} becomes important. At high temperature, there are significant deviations from a singleexponent Arrhenius behaviour, which can no longer describe relaxation across the entire temperature range. Importantly, if one insists in fitting the experimentally measured relaxation curve at high temperature with an Arrhenius plot, an effective barrier significantly lower than U_{0} will be estimated (see Fig. 1c, for *). This situation becomes relevant for SMMs with high anisotropy and already explains why one often measures .
S=1 embedded in a stochastic anharmonic phonons bath
A simple S=1 model will help us in discussing the microscopic mechanism behind spin relaxation in the presence of inelastic phonons. Since calculating Δ_{α} would require an outofreach molecular dynamics simulations (from several ps to ns) for a crystal containing hundreds of atoms, we have taken here a stochastic approach. According to Kubo’s model^{18}, the phonons lineshape corresponds to the amplitude of the Gaussian probability distribution of the αmode’s energy fluctuations. In the canonical ensemble (NVT), this quantity is evaluated as
with . Such temperature dependence of is in qualitative agreement with both the experimental and theoretical homogeneous linewidth temperature behaviour^{15,17,19}. This choice of Δ corresponds to the case where the phonons rapidly become over damped as T increases and it can be considered as opposite to the harmonic limit.
Consider now the threelevel system of Fig. 1b, where the states 0〉, 1〉 and 2〉 are, respectively, the ground state, its spinreversed state and the excited state. The degeneracy between 0〉 and 1〉 is assumed to be slightly lifted for instance by a magnetic field. Assume also that there is only one phonon of frequency ω that can couple to the spin transition i→j with strength V_{ij}. Such phonon is then coupled anharmonically to a phonon’s bath so to acquire a finite linewidth Δ. Note that this model applies to the S=1 case and also to any situation with a lowenergylying almost degenerate doublet separated by an excited state. If one solves equation (3) for the threelevel system in the harmonic limit (Δ→0) (see Supplementary Note 2), the relaxation time will be , that is, relaxation will proceed through an Arrhenius behaviour with U_{eff}=U_{0} and a preexponential factor inversely proportional to the spin–phonon coupling coefficient V. Importantly, in this limit one requires the phonon to be resonant at the excitation energy U_{0}. Note that the same situation holds true for small U_{0} and , that is, when T is sufficiently low that Δ is practically temperature independent (T<T*).
In contrast, for an anharmonic phonon with linewidth described by equation (5), the relaxation time for ħω>k_{B}T is calculated as in equation (38) of Supplementary Note 2:
As expected, there is no relaxation, τ→∞, if the spin–phonon coupling vanishes, since no other mechanism is considered here (for example, quantum tunnelling). A much more surprising fact is that, for V_{02}≠0, the relaxation time follows an activated temperature dependence, but the effective barrier is uniquely determined by the phonon frequency. In particular, when the phonon is resonant with the spin level, we find , meaning that relaxation follows an Arrhenius behaviour with the effective barrier being half of the excitation energy U_{eff}=U_{0}/2. Note that the harmonic picture cannot be recovered from equation (6), due to the assumptions made in its derivation (see Supplementary Note 2). However, in this case of anharmonic phonons, and in stark contrast with the harmonic case often discussed in literature^{13,20,21}, the resonance between the phonon energy and the splitting of the spin levels is not a necessary condition for the relaxation, which can take place even off resonance with the more complex temperature dependence described by equation (6).
The same offresonance mechanism described above applies to any other spin transition. For instance, see Supplementary equation (29) and Supplementary equation (30) of Supplementary Note 2. If V_{02}=0, relaxation can still take place by a phononinduced offresonant transition between 0〉 and 1〉. This gives a relaxation time
which again depends on the phonon frequency alone. Note, however, that such ‘direct’ relaxation mechanism, at variance with the Orbach’s one, is usually observed at very low temperature where the condition T>T* is hardly achievable and a closer agreement with the harmonic scenario is therefore expected. Clearly, if both V_{01} and V_{02} do not vanish, there will be competition between the different relaxation channels and in general τ will take a more complex form. Yet, our analysis clearly demonstrates that in addition to the anisotropy barrier, also the phonon frequency and the phonon lifetime become a relevant energy scale of the problem. In some cases, it is the dominant one. Our mechanism goes beyond previous attempts at explaining the subbarrier relaxation measured in some SMMs^{22,23,24}, where a phenomenologically finite linewidth was added to the spin excited state energy levels. Such early approaches, in fact, although able to explain U_{eff}<U_{0}, fail in relating the spin relaxation process to a physically sound dissipation mechanism. Furthermore, large barrier reductions require massive linewidths, in stark contrast to those measured spectroscopically. Indeed, a typical experimental electron paramagnetic resonance linewidth for SMMs is of the order of 1 cm^{−1} or less^{25}.
It should be stressed that the effects due to the phonon linewidth broadening induced by the phonon–phonon interactions are effective regardless the nature of the density of states of the lattice vibrations. Indeed, even a single phonon interacting with the spin is sufficient to induce a reduction of U_{eff}. However, the broadening of the phonons’ spectral shape also enables more phonons, close in energy, to be operative at the same time, making the vibrational density of states another important figure of merit for the interpretation of the spin dynamics.
Ab initio spin dynamics
A full quantitative analysis of our mechanism now requires the calculation of the phonon spectrum and the spin–phonon coupling for all possible modes and all possible spin transitions. We then abandon simple models for an ab initio description, and investigate the magnetization dynamics of the SMM [(tpa^{Ph})Fe]^{−} (ref. 25). This SMM exhibits a S=2 ground state with large uniaxial anisotropy (see Fig. 2a) and shows slow relaxation rates in an external applied magnetic field^{25}. This specific SMM has been chosen due to its well experimentally characterized structural and magnetic properties. The relaxation features showed by [(tpa^{Ph})Fe]^{−} are rather general and, given the large magnetic anisotropy (D=−27.5 cm^{−1} (ref. 25)), they readily translate to most appealing single ion SMMs, comprising either a transition metal or a lanthanide ion. Moreover, this SMM has one of the most compact unit cell in the SMMs family, which makes it easier the implementation of our demanding computational framework. The [(tpa^{Ph})Fe]^{−} crystal has symmetry and the unit cell contains two SMMs and two Na^{+} cations, coordinated by three dimethoxyethanes. Since [(tpa^{Ph})Fe]^{−} possesses an orbital nondegenerate ground state, we can separate the electronic and pure spin degrees of freedom, and describe it with an effective spin Hamiltonian, , where is a cartesian component of the spin operator^{26}. We simulate the [(tpa^{Ph})Fe]^{−} spectrum with the multideterminant wavefunction active space SCF scheme (CASSCF)^{27,28}. The 2S+1=5 lowestlying S=2 CASSCF energy roots have been used to fit the spin Hamiltonian , as provided by the Orca package (see ‘Methods’ section, and Supplementary Note 4 for a discussion on the validity of the spin Hamiltonian approach). The resulting energy ladder is reported in Fig. 2b, where a 1500 Oe external magnetic field has been introduced through a Zeeman term (g_{e}β is the gyromagnetic ratio). The external field is here introduced merely to be consistent with experimental conditions, optimized to reduce tunnelling relaxation, particularly efficient in zero field. All the matrix elements derivatives that appear in equation (3) can be written in terms of (see Supplementary Note 3) as
where S_{n}〉 is an eigenstate of . The righthand side expression in equation (8) makes it possible to implement spin–phonon coupling calculations through numerical differentiation of the spin Hamiltonian coefficients D_{ij}, which in turn are calculated by CASSCF method.
All the spin–phonon models employed to date to study the dynamics of SMMs collapse the phonon spectrum in a single mode (the Debye model) so to obtain an handy relation for the relaxation time^{1}, and only recently the validity of this approximation has been discussed^{29}. Here the Debye assumption is no longer required as all spin–phonon coupling coefficients are calculated, meaning that all the potential spin relaxation channels are taken into account. The only limitation in our case is practical and it is dictated by the size of the system that we can simulate. We consider a periodic crystal and calculate only the unit cell gammapoint normal modes. These consist of all short wavelength intramolecular modes and the limited number of intermolecular ones compatible with having two molecules per unit cell.
Next, we solve the master equation, equation (3), in time for different temperatures, finding an exponential decay of the magnetization at any T. The relaxation time is then extracted by fitting the time evolution of the magnetization component parallel to the external magnetic field with the function M_{z}(t)=(M_{z}(0)−M_{z}(∞))e^{−t/τ}+M_{z}(∞). The results of our calculations are reported as black dots in Fig. 3, where we present the logarithm of τ as a function of the inverse temperature 1/T. Clearly, there is no single exponential dependence of τ over the entire temperature range, indicating that nonresonant spin–phonon relaxation channels contribute to the dynamics, as discussed for the simple S=1 case. In particular, we notice that ln(τ) is approximately linear with 1/T for T<5 K (1/T>0.2) and then displays a strong deviation. The linear slope extracted for T<5 K gives us U_{eff}=19.7 cm^{−1}, which is in excellent agreement with the experimental value, U_{eff}=26 cm^{−1}, measured in a similar temperature range^{25}. Notably, such value is much smaller than our calculated excitation energy of 94 cm^{−1} (corresponding to the first excited twofold degenerate state with , see Fig. 2b), demonstrating that phononinduced underbarrier relaxation can drastically reduce the spin lifetime of a SMM.
To isolate the dominant relaxation mechanism at different temperatures, we perform the same exercise done for the simple S=1 model, namely we solve the master equation first by retaining only the transitions between the almost degenerate ground state doublet (blue curve in Fig. 3), and then by neglecting these and considering only the remaining ones. For T<5 K nonresonant intradoublet transitions dominate completely the dynamics and provide the largest contribution to τ. An Arrhenius fit of the blue curve essentially returns the same value, U_{eff}=19.7 cm^{−1}, obtained when fitting the complete one for T<5 K. Although several phonon modes may contribute to τ, we observe that U_{eff}=19.7 cm^{−1} is remarkably close to half of the energy of the lowest phonon mode (∼36 cm^{−1}). This makes us concluding that the lowT relaxation of [(tpa^{Ph})Fe]^{−} is dominated by direct doublet transitions via the lowest phonon mode.
In contrast, when only phononactivated transitions through the excited states are considered, we obtain the red curve of Fig. 2, which follows closely the complete curve in the intermediate range 5 K<T<10 K and can be fitted with an Arrhenius plot and U_{eff}=55.7 cm^{−1}. Thus, in this intermediate T range, the relaxation is driven by nonresonant phononactivated processes. These contribute to the barrier with ħω, again suggesting that it is the ħω=36 cm^{−1} mode to dominate the dynamics. This normal mode of vibration is complex in nature being a delocalized representation of a large number of unit cell degrees of freedom. Finally, when the temperature is further enhanced (this is a regime never investigated experimentally since the τs are too short) several modes participates to the dynamics and a single Arrhenius fit is no longer possible. A numerical demonstration of such multimode relaxation process is provided in the inset of Fig. 3 where we show calculations including all the transitions and an increasingly large number of phonon modes. Notably the relaxation times get smaller as more modes are available, meaning that more activated relaxation channels become possible.
Discussion
Finally, let us now critically discuss the results just obtained. In general, we expect the physics shown here for [(tpa^{Ph})Fe]^{−} to be quite common in highanisotropy SMMs. The most recurrent experimental observation is that of a singleexponent Arrheniuslike relaxation, with an activation barrier significantly smaller than that expected from the molecule excitation spectrum. Here we have clearly shown that identifying U_{eff} with U_{0} is no longer justified, since U_{eff} is also determined by the specific molecule phonon structure and by the spin–phonons coupling. In fact, we have demonstrated that offresonance phonon modes always contribute to lower U_{eff} with respect to U_{0}, as observed experimentally.
Intriguingly, in literature there are also reports of SMMs displaying different relaxation regimes at different temperatures. For instance, the lowT relaxation time of Dy^{3+} is defined by a doubleexponent Arrhenius curve with the two fitted effective barriers (334 and 94 K) being lower than the first accessible spin exited state (430 K)^{30}. This behaviour suggests the existence of two different spin relaxation mechanisms which, in light of our discussion, could be attributed to the effect of different normal modes. In fact, we believe that the most common situation is the one presented here, namely the one in which the relaxation time can be associated to different barriers at different temperatures. In experiments, the temperature range is usually narrow, so that a singleexponent Arrhenius behaviour is often observed. Notably underbarrier relaxation has been recently explained on the basis of a secondorder Raman effect^{9,31,32}, but here we clearly show that one does not necessarily need to claim secondorder effects to reconciliate theory with experiments.
Besides the temperaturedependent component, the magnitude of τ is also defined by a preexponential coefficient (see equation (6)), τ_{0}, which in turn depends on the matrix elements of the spin–phonon coupling Hamiltonian and the actual phonon frequencies. This complex dependence over the system details makes it difficult to predict the relaxation time on the sole basis of the molecule excitation spectrum and indeed calls for a more complete treatment, going also beyond the Debye model^{29,33}. Nevertheless, a few general designing rules for increasing U_{eff} may be provided by our analysis. For instance, the direct relaxation between the two quasidegenerate ground states can be slowed down by either increasing the smallest phonon frequency and by reducing the spin–phonon coupling coefficients. The first strategy translates in designing more structurally rigid SMMs, while the second one is readily accomplished with the usual quantum tunnelling reduction methods such as by employing highly axial symmetry Kramer ions.
At high temperature, the goal is instead that of quenching the Orbach’s mechanism. Although complicated to exploit, a possible strategy may be that of modulating the spin excited state energy to obtain a nonresonant condition with the phonon spectrum. In this context, interesting solutions may be offered by 3D and on surface 2D crystal engineering techniques^{34,35}. However, the selective engineering of the spin–phonon coupling coefficients could be the most effective way to slow down the Orbach’s relaxation rate and more detailed studies will be needed. Our finding are in line with the recent research trend that see the engineering of SMM/phonon spectra as a major challenge for the design of the nextgeneration magnetic molecular materials^{9,32,36,37,38}. Here we have shown that the vibrational features of SMMs are the central figures of merit in the design of new systems. Indeed, frequency magnitude and lifetime of phonons, along with the spin–phonon coupling coefficients, are all quantities that directly enter in the spin equation of motions and strongly affect the relaxation timescales.
In conclusion, we have here recast the basic concepts of spin–phonon relaxation in a fully quantum mechanical formalism. This eliminates previously adopted approximations aiming at simplifying the problem and provides a general theoretical framework amenable to ab initio methods. Our approach requires the calculation of the fullphonon spectrum, the spin Hamiltonian and the spin–phonon coupling, all quantities that enter a master equation for the spin dynamics. We have shown that the phonons dissipation, treated here at a stochastic level, enables offresonance spin relaxation, whose magnitude and temperature dependence are determined by the electronic and vibrational details of the specific SMM. A fully quantitative analysis for [(tpa^{Ph})Fe]^{−} demonstrates that the relaxation time has a complex temperature behaviour, where different relaxation mechanisms dominate over different energy ranges. Our work suggests several designing strategies for reducing the spin relaxation and thus for engineering longliving SMMs. Furthermore, we remark that spin–phonon interaction is common to any magnetic material and the theory outlined here does not apply just to SMMs. Moreover, as other classes of magnets show a long magnetization lifetime, it is possible to speculate that they could also be affected by similar spinbath dynamical features as those reported here.
Methods
Density functional theory calculations
All the structural optimization and Hessian calculations have been performed with the CP2K software^{39} at the level of density functional theory with the PerdewBurkeErnzerhof (PBE) functional including Grimme’s D3 van der Waals corrections^{40,41}. Other dispersion forces correction schemes have been tested and compared with each other in describing the unit cell structure. Among the available dfD, dfD2, rvv10 and D3 methods, Grimme’s one has been found to be the most accurate^{42}. A doublezeta polarized MOLOPT basis set and a 600 Ry of planewave cutoff have been used for all the atomic species. The same setup has been used to evaluate the Γpoint phonons of the entire periodic unit cell. A finite differences algorithm for the evaluation of the Hessian matrix with a 0.001 a.u. differentiation step has been used.
Ab initio magnetic properties calculations
The ORCA software^{43} has been employed for computing of all the magnetic properties. Calculations of the D anisotropy tensor have been performed at the CASSCF level with a def2TZVP basis set for Fe and N, def2SVP for C and H and a def2TZVP/C auxiliary basis set for all the elements. This choice has been carefully tested and it correctly reproduces both spin–orbit corrected spectra and the D derivatives calculated by employing the def2TZVP basis set for all the atomic species. A (6,5) active space have been chosen as recommended in literature and the spin–orbit contribution has been included through quasidegenerate perturbation theory. Although CASSCF calculations could be done only on isolated molecules, we take into account electrostatic effects by embedding the single [(tpa^{Ph})Fe]^{−} molecule in a 21 × 21 × 21 lattice of RESP point charges^{44} mimicking the periodicity of the crystal. Although the crystal is made of charged molecules, its imposes a quenching of the electrostatic fields and indeed the inclusion of the point charges does not produce any sizable effect.
Ab initio mapping of the spin Hamiltonian
Due to the presence of lowlying excited electronic states, the spin Hamiltonian formalism is here only partially correct and it could not exactly fit the entire lowerlying spectrum. Other parameterizations have been tested, including for instance the fourthorder operators in the fit, but, although the agreement with the CASSCF results improve (RSS passes from 689 to 6.0 × 10^{−9}), it does not change the results that we have discussed. To evaluate all the coefficients in equation (3), we numerically derived the D anisotropy tensor by modulating the equilibrium molecular structure of one [(tpa^{Ph})Fe]^{−} molecule along all normal modes by ±0.2 units.
Data availability
All the relevant data discussed in the present paper are available from the authors on request.
Additional information
How to cite this article: Lunghi, A. et al. The role of anharmonic phonons in underbarrier spin relaxation of single molecule magnets. Nat. Commun. 8, 14620 doi: 10.1038/ncomms14620 (2017).
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Acknowledgements
This work was supported by the European Research Council Quest and Advanced Grant MolNanoMaS (no. 267746) projects. We also acknowledge the MOLSPIN COST action CA15128. Computational resources were provided by the Trinity Centre for High Performance Computing (TCHPC) and the Irish Centre for HighEnd Computing (ICHEC). A special acknowledgement is due to Prof Roberto Righini from the University of Florence and LENS for the many useful discussions.
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All the authors conceived the project, analysed/discussed the results and wrote the manuscript. A.L. performed all the calculations and developed the theory jointly with S.S.
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Lunghi, A., Totti, F., Sessoli, R. et al. The role of anharmonic phonons in underbarrier spin relaxation of single molecule magnets. Nat Commun 8, 14620 (2017). https://doi.org/10.1038/ncomms14620
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DOI: https://doi.org/10.1038/ncomms14620
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