Received 00th January 20xx,Accepted 00th January 20xx
DOI: 10.1039/x0xx00000x
www.rsc.org/
Studies of Hysteresis and Quantum Tunnelling of the Magnetisation in Dysprosium(III) Single Molecule MagnetsFabrizio Ortu,a,† Daniel Reta,a,† You-Song Ding,b Conrad A. P. Goodwin,a Matthew P. Gregson,a Eric J. L. McInnes,a Richard E. P. Winpenny,a Yan-Zhen Zheng,b,* Stephen T. Liddle,a,* David P. Mills a,* and Nicholas F. Chilton a,*
We report magnetic hysteresis studies of three Dy(III) single-molecule magnets (SMMs). The three compounds are [Dy(tBuO)Cl(THF)5][BPh4] (1), [K(18-crown-6-ether)(THF)2][Dy(BIPM)2] (2, BIPM = C{PPh2NSiMe3}2), and [Dy(Cpttt)2][B(C6F5)4] (3), chosen as they have large energy barriers to magnetisation reversal of 665, 565, and 1223 cm-1, respectively. There are zero-field steps in the hysteresis loops of all three compounds, that remain in magnetically dilute samples and in samples that are isotopically enriched with 164Dy, which has no nuclear spin. These results demonstrate that neither dipolar fields nor nuclear hyperfine coupling are solely responsible for the quantum tunnelling of magnetisation at zero field. Analysing their vibrational modes, we find that the modes that most impact the first coordination sphere occur at the lowest energies for 1, at intermediate energies for 2 and at higher energies for 3, in correlation with the hysteresis coercive fields. Therefore, we suggest that the efficiency of quantum tunnelling of magnetisation is related to molecular flexibility.
Single-molecule magnets (SMMs) exhibit slow relaxation of their magnetisation and display magnetic memory effects at the molecular level. In principle this permits the use of individual molecules as bits in high-density data storage devices,1 however current generation SMMs require very low temperatures to retain their magnetic memory effect; typically, this is the liquid helium regime rather than that of liquid nitrogen which is cheap and plentiful. However, there
has been a recent step-change with the advent of Dy(III) bis-cyclopentadienyl metallocenium cations,2 which have led to SMMs with magnetic relaxation slow enough to observe hysteresis up to 80 K.3,4 For technological exploitation, the requirement is for a molecular memory that retains information on the timescale of years at such temperatures.
SMMs display slow magnetic relaxation because of an internal energy barrier to the inversion of their magnetic moment (Ueff), and increasing the size of this barrier is crucial for developing SMMs with higher operating temperatures. However, the magnetisation dynamics of monometallic lanthanide SMMs are multifaceted, and such compounds often display magnetic relaxation via pathways that circumvent the Ueff barrier.5–7 For example, an SMM with one of the largest Ueff
barriers8 does not show magnetic hysteresis at a temperature higher than the first SMM reported nearly a quarter of a century ago,9 and so maximising Ueff is clearly not the sole consideration for overcoming low operating temperatures.5
Excluding the recent Dy(III) metallocenium SMMs,2–4 the magnetic hysteresis loops of most high-barrier SMMs have a characteristic fingerprint, exhibiting a “waist-restricted” or “butterfly” shape (e.g. Fig. 2a). This highlights the key problem: there are significant magnetic memory effects (i.e. open hysteresis) at non-zero magnetic fields, but the hysteresis abruptly collapses at zero magnetic field. This efficient magnetic relaxation is often referred to as quantum tunnelling of the magnetisation (QTM), an effect that has been extensively studied in polynuclear transition metal SMMs,10–17
and also for some of the more recent lanthanide-based SMMs.18–21
b. Frontier Institute of Science and Technology, Xi’an Jiaotong University99 Yanxiang Road, Xi’an, Shaanxi 710054, ChinaEmail: [email protected]
† These authors contributed equally. Electronic Supplementary Information (ESI) available: experimental and computational procedures, magnetic data, vibrational data.See DOI: 10.1039/x0xx00000x
Figure 1. Molecular structures of complex ions in compounds 1 a), 2 b) and 3 c). Counter-ions omitted for clarity.
However, QTM should not occur for monometallic Dy(III) compounds, which are the most common high-barrier SMMs.2–
4,8,22–26 This is because the ground electronic state is, by design, a pure mJ = ±15/2 Kramers doublet. According to Kramers’ theorem for half-integer total angular momentum there can be no mixing between two doublet states in zero magnetic field (in this case the mJ = +15/2 and mJ = -15/2 states), thus preventing QTM. However, this idealised picture is clearly inconsistent with experimental data. One suggestion to explain the experimental observations has been that the presence of small transverse magnetic fields (dipolar or stray fields perpendicular to the main magnetic axis of the molecule) break the Kramers degeneracy in “zero” applied magnetic field, thus allowing QTM.27,28 Indeed, numerous experiments have shown that diluting Dy(III) SMMs in a diamagnetic matrix to reduce the dipolar fields can reduce zero-field QTM,25,29–32
however this does not completely prevent zero-field relaxation to give SMMs with significantly higher operating temperatures.
Another proposed source of QTM is hyperfine coupling of the electronic angular momentum J = 15/2 to the nuclear spin of the metal nucleus (161Dy and 163Dy have I = 5/2, ca. 44% total of natural abundance), which can also break the Kramers degeneracy. Indeed, experiments at mK temperatures have shown that QTM occurs at avoided crossings that arise from hyperfine coupling,18,20 and experiments with isotopically pure 161Dy (I = 5/2) vs. 164Dy (I = 0) have shown that the former has enhanced magnetic relaxation in the QTM regime.20,21,31,33–35 However, these experiments have also not been able to completely remove the zero-field QTM step, and, importantly, have thus far only been performed on SMMs with moderate Ueff values, for which thermally activated relaxation may be important even at low temperatures. Therefore, whether nuclear hyperfine coupling or transverse dipolar fields are the dominant causes of QTM in Dy(III) SMMs with very large Ueff
barriers has remained an open question.To directly probe the contribution of dipolar fields and
nuclear hyperfine coupling to zero-field QTM in high-performance SMMs, we selected three compounds with large Ueff barriers, viz. [Dy(tBuO)Cl(THF)5][BPh4] (1),26 [K(18-crown-6-ether)(THF)2][Dy(BIPM)2] (2, BIPM = C{PPh2NSiMe3}2),24 and [Dy(Cpttt)2][B(C6F5)4] (3, Cpttt = C5H2
tBu3-1,2,4),2 Fig. 1, with Ueff
barriers of 665, 565, and 1223 cm-1, respectively. We focus on magnetic hysteresis measurements as this is the crucial experiment that demonstrates the utility of a memory effect for an SMM. Magnetic relaxation over the Ueff barrier occurs via the Orbach mechanism and involves sequential direct single-phonon transitions between excited crystal field states;7,13 therefore, there must be phonon modes of the same energy as the difference between subsequent crystal field states to allow relaxation in this regime. Because the energies
of the first excited crystal field states in 1 – 3 (397, 168 and 485 cm-1, respectively2,24,26) are two orders of magnitude larger than kT at 2 K (ca. 1.4 cm-1), the Orbach mechanism should have no contribution to magnetic relaxation at this temperature. Furthermore, all three compounds show a clear step in their magnetisation hysteresis curves at zero magnetic field and 2 K (blue traces in Fig. 2) which directly indicates an efficient relaxation process with a strong field dependence. As there should only be a minor field dependence for the Raman and Orbach mechanisms,36,37 this clearly indicates a QTM regime at zero field and 2 K. However, the three compounds display markedly different QTM efficiencies at zero field, leading to coercive fields of ca. 0, 11 and 28 kOe, respectively, despite all having well-isolated mJ = ±15/2 ground states.
The magnetic sites in compounds 2 and 3 have rigorously collinear main magnetic axes (P-1 space groups), and those for compound 1 are at an angle of only 18° for nearest neighbours (based on the crystallographic O-Dy-Cl axes).26 Such alignment of the magnetic axes does not prevent transverse dipolar fields, and thus to examine the zero-field QTM, all three compounds were prepared with naturally abundant Dy at a ~5% dilution level in a matrix of the isostructural diamagnetic yttrium congener. While 5% dilution will not fully suppress the dipolar fields, we must strike a balance between dilution and signal-to-noise in or magnetometry experiments; 5% is a good compromise. Analysis of the Dy content gives 5.6%, 5.9% (see ESI) and 8%2 for 1-Dy@Y, 2-Dy@Y and 3-Dy@Y, respectively, which should result in average dipolar fields (stochastic point dipole calculations based on crystal structure30) of ca. 20(10), 9(4) and 17(6) Oe, respectively. Note that hysteresis data for dilute samples of 1 – 3Dy@Y have also been reported previously in the literature.2,24,26 These diluted samples (black traces in Figs. 2 and S1) show slightly slower zero-field relaxation compared to their concentrated samples (blue traces in Fig. 2). It is clear from these data that a significant zero-field step remains for compounds 1-Dy@Y and 2-Dy@Y, and therefore that transverse dipolar fields cannot be the sole cause of QTM.
Figure 2. Normalized magnetization (M) vs. external magnetic field (H) hysteresis of 1 a), 2 b) and 3 c) at 2 K. Blue traces: naturally abundant Dy, undiluted; black traces: naturally abundant Dy, diluted ~5% in Y); red traces: ~96.6% 164Dy enriched, ~5% diluted in Y. For all data, except the blue traces for 1 and 2, sweep rates are 110(20) Oe s-1
for |Hext| > 20 kOe, 60(10) Oe s-1 for 10 kOe < |Hext| < 20 kOe, 38(8) Oe s-1 for 6 kOe < |Hext| < 10 kOe, and 20(4) Oe s-1 for |Hext| < 6 kOe. For the blue trace for 1 the data
Please do not adjust margins
Please do not adjust margins
Journal Name COMMUNICATION
To examine the contribution of the Dy nuclear spin to zero-field QTM, we prepared a third set of compounds with 96.8% isotopic purity 164Dy, again at a ~5% dilution level in the isostructural yttrium analogues; of the remaining 3.2% Dy isotopic distribution, only ca. 2.6% have I = 5/2 nuclear spin (see ESI). Note that hysteresis data for 1-164Dy@Y has been reported previously.26 Analysis of the Dy content gives 5.6%, 7.3% and 7.4% (see ESI) for 1-164Dy@Y, 2-164Dy@Y and 3-164Dy@Y, respectively. We confirm that these 164Dy compounds have been enriched with 164Dy by ICP-MS (see ESI) and that their relaxation dynamics are consistent with their naturally abundant parent compounds (Figs. S2 – S7). Comparison of the magnetic hysteresis traces for the paramagnetically dilute naturally abundant Dy samples (black traces in Fig. 2) with the paramagnetically dilute nuclear-spin-free 164Dy samples (red traces in Fig. 2) shows that there are only small changes to the zero-field step upon removal of the Dy nuclear spin (note that due to the statistical nature of preparing doped samples, the concentration of Dy is not identical between the Dy@Y and 164Dy@Y samples), with the most notable differences for compound 2-164Dy@Y. We observe a slight decrease in the magnetic relaxation rate for compounds 2-164Dy@Y and 3-164Dy@Y after the zero-field crossing at fields < 10 kOe for the 164Dy samples, which likely results from removal of avoided crossings due to hyperfine coupling.20
However, given the overall minor changes upon paramagnetic dilution and removal of the nuclear spin compared to the pure natural abundance compounds, together with the large differences in hysteresis between compounds 1 – 3, these data directly suggest that neither transverse dipolar fields nor hyperfine interactions are the dominant cause of QTM in these large-Ueff Dy(III) SMMs. [We note that in all experiments there are other ligand-based nuclei with non-zero nuclear spin (e.g. 1H) and therefore the super-hyperfine interaction may possibly contribute to QTM; but given the relatively small effect of the Dy nuclear spin and the “core-like” distribution of 4f electrons, we do not believe that more distant ligand-based nuclear spins will have a more
significant contribution to QTM.] This observation is confirmed by the recent study of McClain et al., showing that a series of dysprosocenium cations with varying substituents have markedly different zero-field QTM steps despite all possessing the same basic electronic structure.3
Experimental determination of the origin of the zero-field QTM for large-Ueff Dy(III) SMMs would require study of many more molecules under a wide range of conditions, however we can conclude from our results that the nature of the ligand environment encapsulating the Dy(III) ion is much more important than transverse dipolar fields or hyperfine coupling in determining zero-field QTM. Moving from compounds 1 – 3 the ligand environment becomes much more rigid, from seven monodentate ligands in 1, to two tridentate ligands in 2, to two η5-aromatic ligands in 3. Hence we suggest, empirically, that molecular flexibility is related to QTM efficiency. This is compatible with our recent finding of temperature dependence in the QTM for 1, which we postulated arises from the phonon collision rate of the lattice.26
Quantification of molecular rigidity is not simple. Here we have used density-functional theory (DFT, see ESI) to determine the normal modes of vibration, and produced histograms of the vibrational spectra (i.e. a pseudo vibrational density of states, pseudo-DOS) for the complex ions in 1 – 3 (Figs. S8 – S12). These do not show any significant differences in the low-energy region that would directly account for one molecule being more flexible than another. However, vibrational modes that involve movement in the first coordination sphere would seem strong candidates for contributing to relaxation via QTM. Therefore, we use the average displacement of the first coordination sphere atoms and of the Dy atom to determine which vibrational modes of each complex significantly perturb the magnetic ion, and thus likely have a bearing on QTM, and compare the energies of these modes for compounds 1 – 3.
Identifying the vibrational modes that have the largest average displacements around the metal ion (≥ 0.02 Å, Figs. 3a and S13), we observe that these modes have the lowest energies for compound 1 and the highest energies for
Figure 2. Normalized magnetization (M) vs. external magnetic field (H) hysteresis of 1 a), 2 b) and 3 c) at 2 K. Blue traces: naturally abundant Dy, undiluted; black traces: naturally abundant Dy, diluted ~5% in Y); red traces: ~96.6% 164Dy enriched, ~5% diluted in Y. For all data, except the blue traces for 1 and 2, sweep rates are 110(20) Oe s-1
for |Hext| > 20 kOe, 60(10) Oe s-1 for 10 kOe < |Hext| < 20 kOe, 38(8) Oe s-1 for 6 kOe < |Hext| < 10 kOe, and 20(4) Oe s-1 for |Hext| < 6 kOe. For the blue trace for 1 the data
Figure 3. (a) Energies of vibrational modes as a function of average displacement of the Dy(III) and first coordination sphere atoms in 1 – 3. Modes with an average displacement of ≥ 0.02 Å are shown, full plot in Fig. S9. Inset shows the mean and standard deviation of the vibrational energies with average displacements ≥ 0.02 Å, as a function of the coercive field of the pure compounds. (b) Low-energy pseudo vibrational DOS only considering modes with average displacements ≥ 0.02 Å.
Please do not adjust margins
Please do not adjust margins
COMMUNICATION Journal Name
compound 3. Interestingly, there is a correlation between the average energies for these modes and the coercive field for compounds 1 – 3 (Fig. 3a, inset). Another way of analysing this data is to examine the pseudo-DOS of the selected modes (average displacements around Dy(III) ≥ 0.02 Å, Fig. 3b): this shows that there is a progression in the energies of modes that perturb the coordination sphere in the low energy region, in the sequence of 1 < 2 < 3; these results are consistent for different choices of bin size (Figs. S14 – S18). There are some outliers in this analysis (e.g. the mode at ~800 cm-1 for 1 or the mode at ~200 cm-1 for 3, Fig. 3a) but the general trend towards increased energies for compound 3 over 2, over 1, is evident.
In order to test the generality of this observation, we have performed the same analysis on three additional Dy-based SMMs with Ueff > 1000 K (Fig. S19).4,8,25 We observe a similar trend in the energy distribution of the selected vibrational modes (Figs. S20 and S21), where those with larger hysteresis tend to have larger pseudo-DOS at higher energies. Further, the average energy of these modes follows the same trend with coercive field, once the difference in magnetic field sweep rate is accounted for (Fig. S22).
The average energies of the modes with the largest average displacements are ca. 180, 550 and 890 cm-1 for 1 – 3 respectively. Our analysis here does not imply that QTM occurs via excitation or de-excitation of these vibrational modes; that is, there is no exchange of quanta on the scale of 100’s cm -1 to effect relaxation via QTM. Rather, considering each vibrational mode as a harmonic oscillator, the energy of the mode is related to the gradient of the energy with respect to displacement and thus related to the classical “stiffness” of the mode. Hence, these average energies imply that compound 3 is stiffer than compound 2, which in turn is stiffer than compound 1, with respect to the vibrational modes that affect the first coordination sphere.
To conclude, whilst magnetic dilution and 164Dy enrichment influence zero-field QTM at 2 K, their effect is much smaller than the differences in QTM between the molecules, which all possess the same, well-isolated, electronic ground state. Based on these results, we hypothesise that the efficacy of zero-field QTM, and thus the size of the coercive field and magnetic hysteresis loops, is directly related to the rigidity of the ligand environment of Dy(III) SMMs. Therefore, we propose that QTM could be reduced by employing more rigid ligands with constrained metal-ligand vibrational modes, and that isotopic enrichment need not be the main focus for the development of high-temperature SMMs. To fully develop this new molecular design criterion, theoretical approaches for the ab initio determination of magnetic relaxation via spin-phonon coupling need to be extended,2,38–40 as there is currently no microscopic theory for how molecular vibrations facilitate QTM. Because the two states of the ground mJ = ±15/2 Kramers doublet of Dy(III) SMMs cannot be mixed by spin-phonon coupling in first order under the crystal field approximation, vibrationally-driven QTM must involve higher-order spin-phonon coupling, multi-phonon processes or a breakdown of the crystal field approximation; ab initio spin-dynamics models have not yet been able to treat these
effects.2,38–40 With a theory of vibrationally-driven QTM and computational tools to explore it, more rigorous guidelines can be developed for the control of the pertinent vibrational features by chemical means, to drive the burgeoning generation of high-temperature SMMs.
We thank the EPSRC (grant EP/P002560/1 for F.O. and D.R., and Doctoral Prize Fellowship for C.A.P.G.), Ramsay Memorial Trust (fellowship to N.F.C.), NSFC (grant 21620102002), The University of Manchester, and the EPSRC National EPR Facility for generously supporting this work.
Conflicts of interestThere are no conflicts to declare.
Notes and references1 R. Sessoli, Nature, 2017, 548, 400–401.2 C. A. P. Goodwin, F. Ortu, D. Reta, N. F. Chilton and D. P. Mills,
Nature, 2017, 548, 439–442.3 K. R. McClain, C. A. Gould, K. Chakarawet, S. Teat, T. J. Groshens,
J. R. Long and B. G. Harvey, Chem. Sci., 2018, 9, 8492–8503.4 F.-S. Guo, B. M. Day, Y.-C. Chen, M.-L. Tong, A. Mansikkamäki
and R. A. Layfield, Science, 2018, 362, 1400–1403.5 K. S. Pedersen, J. Dreiser, H. Weihe, R. Sibille, H. V. Johannesen,
M. A. Sørensen, B. E. Nielsen, M. Sigrist, H. Mutka, S. Rols, J. Bendix and S. Piligkos, Inorg. Chem., 2015, 54, 7600–7606.
6 S. T. Liddle and J. van Slageren, Chem. Soc. Rev., 2015, 44, 6655–6669.
7 L. Escalera-Moreno, J. J. Baldoví, A. Gaita-Ariño and E. Coronado, Chem. Sci., 2018, 9, 3265–3275.
8 Y.-S. Ding, N. F. Chilton, R. E. P. Winpenny and Y.-Z. Zheng, Angew Chem Int Ed, 2016, 55, 16071–16074.
9 R. Sessoli, D. Gatteschi, A. Caneschi and M. A. Novak, Nature, 1993, 365, 141–143.
10 J. R. Friedman, M. P. Sarachik, J. Tejada and R. Ziolo, Phys. Rev. Lett., 1996, 76, 3830–3833.
11 D. Gatteschi and R. Sessoli, Angew. Chem. Int. Ed., 2003, 42, 268–297.
12 M. Soler, W. Wernsdorfer, K. Folting, M. Pink and G. Christou, J. Am. Chem. Soc., 2004, 126, 2156–2165.
13 D. Gatteschi, R. Sessoli and J. Villain, Molecular Nanomagnets, Oxford University Press, 2006.
14 C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli and D. Gatteschi, Phys. Rev. Lett., 1997, 78, 4645–4648.
15 M. Mannini, F. Pineider, C. Danieli, F. Totti, L. Sorace, Ph. Sainctavit, M.-A. Arrio, E. Otero, L. Joly, J. C. Cezar, A. Cornia and R. Sessoli, Nature, 2010, 468, 417–421.
16 W. Wernsdorfer and R. Sessoli, Science, 1999, 284, 133–135.17 M. Moragues-Cánovas, é. Rivière, L. Ricard, C. Paulsen, W.
Wernsdorfer, G. Rajaraman, E. K. Brechin and T. Mallah, Adv. Mater., 2004, 16, 1101–1105.
18 N. Ishikawa, M. Sugita and W. Wernsdorfer, Angew. Chem. Int. Ed., 2005, 44, 2931–2935.
19 N. Ishikawa, M. Sugita and W. Wernsdorfer, J. Am. Chem. Soc., 2005, 127, 3650–3651.
20 E. Moreno-Pineda, M. Damjanović, O. Fuhr, W. Wernsdorfer and M. Ruben, Angew. Chem. Int. Ed., 2017, 56, 9915–9919.
21 E. Moreno-Pineda, G. Taran, W. Wernsdorfer and M. Ruben, Chem. Sci., 2019, 10.1039.C9SC01062A.
22 Y.-C. Chen, J.-L. Liu, L. Ungur, J. Liu, Q.-W. Li, L.-F. Wang, Z.-P. Ni, L. F. Chibotaru, X.-M. Chen and M.-L. Tong, J. Am. Chem. Soc., 2016, 138, 2829–2837.
23 S. K. Gupta, T. Rajeshkumar, G. Rajaraman and R. Murugavel, Chem Sci, 2016, 7, 5181–5191.
24 M. Gregson, N. F. Chilton, A.-M. Ariciu, F. Tuna, I. F. Crowe, W. Lewis, A. J. Blake, D. Collison, E. J. L. McInnes, R. E. P. Winpenny and S. T. Liddle, Chem Sci, 2016, 7, 155–165.
25 J. Liu, Y.-C. Chen, J.-L. Liu, V. Vieru, L. Ungur, J.-H. Jia, L. F. Chibotaru, Y. Lan, W. Wernsdorfer, S. Gao, X.-M. Chen and M.-L. Tong, J. Am. Chem. Soc., 2016, 138, 5441–5450.
26 Y.-S. Ding, K.-X. Yu, D. Reta, F. Ortu, R. E. P. Winpenny, Y.-Z. Zheng and N. F. Chilton, Nat. Commun., 2018, 9, 3134.
27 D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B, 1997, 56, 11102–11118.
28 M. N. Leuenberger and D. Loss, Phys. Rev. B, 2000, 61, 1286–1302.
29 S.-D. Jiang, B.-W. Wang, G. Su, Z.-M. Wang and S. Gao, Angew. Chem. Int. Ed., 2010, 49, 7448–7451.
30 N. F. Chilton, S. K. Langley, B. Moubaraki, A. Soncini, S. R. Batten and K. S. Murray, Chem. Sci., 2013, 4, 1719–1730.
31 F. Pointillart, K. Bernot, S. Golhen, B. Le Guennic, T. Guizouarn, L. Ouahab and O. Cador, Angew. Chem. Int. Ed., 2015, 54, 1504–1507.
32 T. Pugh, N. F. Chilton and R. A. Layfield, Angew Chem Int Ed, 2016, 55, 11082–11085.
33 Y. Kishi, F. Pointillart, B. Lefeuvre, F. Riobé, B. L. Guennic, S. Golhen, O. Cador, O. Maury, H. Fujiwara and L. Ouahab, Chem. Commun., 2017, 53, 3575–3578.
34 G. Huang, X. Yi, J. Jung, O. Guillou, O. Cador, F. Pointillart, B. L. Guennic and K. Bernot, Eur. J. Inorg. Chem., 2018, 2018, 326–332.
35 J. F. Gonzalez, F. Pointillart and O. Cador, Inorg. Chem. Front., 2019, 6, 1081–1086.
36 A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, 1970.
37 L. T. A. Ho and L. F. Chibotaru, Phys. Rev. B, 2018, 97, 024427.38 L. Escalera-Moreno, N. Suaud, A. Gaita-Ariño and E. Coronado, J.
Phys. Chem. Lett., 2017, 8, 1695–1700.39 A. Lunghi, F. Totti, R. Sessoli and S. Sanvito, Nat. Commun.,
2017, 8, 14620.40 A. Lunghi, F. Totti, S. Sanvito and R. Sessoli, Chem. Sci., 2017, 8,
