Zero-Field Quantum Tunnelling of the Magnetisation in a Series of High Energy-Barrier Dysprosium(III) Single-Molecule Magnets Fabrizio Ortu,1,§ Daniel Reta,1,§ You-Song Ding,2 Conrad A. P. Goodwin,1 Matthew P. Gregson,1 Eric J. L. McInnes,1 Richard E. P. Winpenny,1 Yan-Zhen Zheng,2,* Stephen T. Liddle,1,* David P. Mills1,* and Nicholas F. Chilton1,* 1School of Chemistry, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K. 2Frontier Institute of Science and Technology, and State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, 99 Yanxiang Road ， Xi’an, Shaanxi 710054, China. *For correspondence: [email protected]; [email protected]; [email protected]; [email protected] §These authors contributed equally.

Abstract

Energy barriers to magnetisation reversal (Ueff) in single-molecule magnets (SMMs) ttt have vastly increased recently, but only for the dysprosocenium SMM [Dy(Cp )2] ttt t [B(C6F5)4] (Cp = C5H2 Bu3-1,2,4) has this translated into a considerable increase in magnetic hysteresis temperatures. The lack of concomitant increases in hysteresis temperatures with Ueff values is due to efficient magnetic relaxation at zero-field, referred to as quantum tunnelling of the magnetisation (QTM); however, the exact nature of this phenomenon is unknown. Recent hypotheses suggest that both transverse dipolar magnetic fields and hyperfine coupling play a significant role in t this process for Dy(III) SMMs. Here, by studying the compounds [Dy( BuO)Cl(THF)5]

[BPh4] (1), [K(18-crown-6-ether)(THF)2][Dy(BIPM)2] (2, BIPM = C{PPh2NSiMe3}2), ttt and [Dy(Cp )2][B(C6F5)4] (3), we show conclusively that neither of these processes are the main contributor to zero-field QTM for Dy(III) SMMs, and suggest that its origin instead owes to molecular flexibility. By analysing the vibrational modes of the three molecules, we show that the modes that most impact the magnetic ion occur at the lowest energies for 1, at intermediate energies for 2 and at higher energies for 3, in correlation with their ability to retain magnetisation. Therefore, we conclude that SMM performance could be improved by employing more rigid ligands with higher- energy metal-ligand vibrational modes. Introduction Single-molecule magnets (SMMs) are molecules that show slow relaxation of their magnetisation and thus can exhibit magnetic memory effects at the molecular level. This in principle permits the possibility of using individual molecules as bits in high- density data storage devices1, 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. This has remained the key roadblock, frustrating technological viability and exploitation. Therefore, one of the most important aims in this area is to raise the temperature at which the memory effect persists. 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 was postulated to be 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 2 that circumvent the Ueff barrier. For example, the current record holder for the largest 3 Ueff barrier does not show magnetic hysteresis at a temperature higher than the first 4 SMM reported nearly a quarter of a century ago , and so maximising Ueff is clearly not the sole consideration for overcoming low operating temperatures. Excluding the recent dysprosocenium SMM that shows magnetic hysteresis up to 60 K (ref. 5), the magnetic hysteresis loops of most high-barrier SMMs have a characteristic fingerprint, exhibiting a “waist-restricted” or “butterfly” shape, exemplified in Figure 2a. This directly 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 manganese-based SMMs6–10, and also for some of the more recent lanthanide-based SMMs11–13. However, QTM should not occur for monometallic Dy(III) compounds, which are the most common high-barrier SMMs3,5,14–17. 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 the mJ = +15/2 and mJ = -15/2 states (which is required for efficient relaxation) in zero magnetic field. 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 QTM18,19. Indeed, numerous experiments have shown that diluting Dy(III) SMMs in a diamagnetic matrix to reduce the dipolar fields can reduce zero-field QTM17,20–23, however, this approach has not completely prevented efficient zero-field relaxation to yield SMMs with significantly higher operating temperatures. Another proposed source of QTM is hyperfine coupling of the electronic angular momentum J = 15/2 to the non-zero nuclear spin of the metal nucleus ( 161Dy and 163Dy have I = 5/2, comprising approximately 44% of the naturally abundant Dy isotopes), which can also break the Kramers degeneracy. Indeed, experiments at mK temperatures have shown that QTM occurs at avoided crossings that arise from hyperfine coupling12,13, 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 regime24,25. 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. Herein we have synthesised dilute paramagnetic samples of three Dy(III) SMMs with 164 large Ueff barriers, employing natural abundance Dy and enriched Dy, and compared their magnetic hysteresis profiles to the undiluted natural abundance Dy analogues in order to directly probe the contribution of dipolar fields and nuclear hyperfine coupling to the zero-field QTM step at low temperatures. We focus on magnetic hysteresis as this is the crucial experiment that demonstrates the utility of a memory effect for an SMM. We find only a small effect on the zero-field step in the hysteresis loop upon both paramagnetic dilution and isotopic enrichment with nuclear-spin-free 164Dy, and conclude 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 magnetic relaxation. Results In order to address the question of the origin of QTM in high-performance SMMs, we t 26 selected the compounds [Dy( BuO)Cl(THF)5][BPh4] (1) , [K(18-crown-6-ether)(THF)2] 16 ttt ttt [Dy(BIPM)2] (2, BIPM = C{PPh2NSiMe3}2) , and [Dy(Cp )2][B(C6F5)4] (3, Cp = t 5 -1 C5H2 Bu3-1,2,4) , Figure 1, with Ueff barriers of 665, 565, and 1223 cm , respectively. Magnetic relaxation via the Orbach mechanism involves sequential direct single- phonon transitions between excited crystal field states27, and therefore there must be phonon modes of the same energy as the difference between subsequent crystal field states. Because the energies of the first excited crystal field states in the three compounds (397, 168 and 485 cm-1, respectively5,16,26) are at least 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 Figure 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 mechanisms28,29, 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. Figure 1. Molecular structures of complex ions in compounds 1 (a), 2 (b) and 3 (c). Counter-ions omitted for clarity. In order to examine the contribution of transverse dipolar fields to 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. The diluted samples (black traces in Figure 2) show slightly slower zero-field relaxation compared to their concentrated samples (blue traces in Figure 2). It is clear from these data that a significant zero-field step remains for compounds 1 and 2, and therefore that transverse dipolar fields cannot be the sole cause of QTM. This is a significant outcome towards the applicability of SMMs in high-density data storage, where the molecules would have to be tightly packed to realise high-density storage devices. To examine the contribution of the Dy nuclear spin to zero-field QTM, we prepared a third set of compounds with 96.80% isotopic purity 164Dy, again at a ~5% dilution level in the isostructural yttrium analogues. We confirm that these 164Dy compounds 164 have been enriched with Dy by ICP-MS (see Methods) and that their Ueff barriers are consistent with their naturally abundant parent compounds (Figs. S1 – S3). Comparison of the magnetic hysteresis traces for the paramagnetically dilute naturally abundant Dy samples (black traces in Figure 2) with the paramagnetically dilute nuclear-spin-free 164Dy samples (red traces in Figure 2) shows that there are only small changes to the zero-field step upon removal of the Dy nuclear spin, with the most notable differences for compound 2. We observe a slight decrease in the magnetic relaxation rate for compounds 2 and 3 after the zero-field crossing at fields < 10 kOe for the 164Dy samples, which likely results from removal of avoided crossings due to hyperfine coupling13. 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 large-Ueff Dy(III) SMMs. However, 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. However, given the relatively small effect of the Dy nuclear spin, we do not believe that more distant ligand-based nuclear spins will have a more significant contribution to QTM, especially given the well- localised distribution of 4f electrons.

Figure 2. Normalised magnetisation (M) vs. external magnetic field (H) hysteresis of 1 (a), 2 (b) and 3 (c). 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 trace for 2, sweep rates are 110(20) Oe s-1 -1 -1 for |Hext| > 20 kOe, 60(10) Oe s for 10 kOe < |Hext| < 20 kOe, 38(8) Oe s for 6 kOe -1 < |Hext| < 10 kOe, and 20(4) Oe s for |Hext| < 6 kOe. For the blue trace for 2 the data is taken from ref. 16, having a sweep rate of 35 Oe s-1.

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 magnetic relaxation. Moving from compounds 1 – 3, we observe that the ligand environment becomes chemically much more rigid, from seven monodentate ligands in 1, to two tridentate ligands in 2, to two pentadentate aromatic ligands in 3. However, quantification of the ‘rigidity’ of a molecule is not a simple task, nor is it known how molecular flexibility permits QTM. Therefore, in a first step to determine the influence of molecular flexibility on QTM, we attempted to find if there are more low-energy vibrational modes (i.e. a more flexible molecule) that significantly perturb the magnetic ion for 1 than for 2, and if both have more than for compound 3. Therefore, we calculated the normal modes of vibration with density functional theory (DFT, see Supplementary Information) and examined how these modes influence the Dy(III) ion as a function of energy. Histograms of the vibrational spectra (i.e. a pseudo vibrational density of states (DOS)) for the complex ions in 1 – 3 (Figs. S4 – S8) do not show any significant differences in the low-energy region that would directly account for one molecule being more flexible than another; if anything, the low-energy pseudo-DOS is larger for 2 than for 1 or 3, which are similar. However, not all vibrational modes will significantly contribute to QTM and there is no distinction between modes in the pseudo-DOS. While some theoretical approaches for the ab initio determination of magnetic relaxation via spin-phonon coupling have been explored recently5,30-32, there is currently no microscopic theory for how molecular vibrations facilitate QTM. Importantly, the two states of the ground mJ = ±15/2 Kramers doublet of Dy(III) SMMs cannot be directly mixed by spin-phonon coupling in first order under the crystal field approximation, and therefore vibrationally-driven QTM must involve higher-order spin-phonon coupling, multiphonon processes or a breakdown of the crystal field approximation; preliminary ab initio spin-dynamics models cannot yet treat these effects5,30-32. However, vibrational modes that involve movement in the first coordination sphere and of the dysprosium atom must contribute in some way to relaxation via QTM. Therefore, in the absence of a model for vibrational-QTM and ab initio simulations thereof, we use the average displacement of the first coordination sphere atoms to determine which vibrational modes of each complex significantly perturb the magnetic ion, and thus have a bearing on QTM, and compare the energies of these modes for compounds 1 – 3.

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. Identifying the vibrational modes that have the largest average displacements around the metal ion (Fig. 3a), we observe that these modes have the lowest energies for compound 1 and the highest energies for 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, only taking into account these modes with average displacements ≥ 0.02 Å (Fig. 3b). This analysis shows that there is a clear 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. S10 – S14). Discussion Here we have explored the contribution of dipolar fields and hyperfine coupling to the zero-field QTM in three large-Ueff SMMs at low temperature, through magnetic dilution and 164Dy-enrichment, respectively. While there is an influence of these perturbations on the QTM step, their effect is much smaller than the differences between the molecules, which all possess the same 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. Analysis of the vibrational modes for 1 – 3 finds that such modes are at consistently lower energies for the most flexible molecule 1 with exclusively monodentate ligands, and at higher energies for molecule 2 with two tridentate ligands. Compound 3 is a very unusual case without conventional coordination bonds, and vibrational modes that distort the first coordination sphere are at very high energies due to the necessary deformation of the aromatic ring. We propose that QTM should be able to be reduced by employing more rigid ligands with constrained metal-ligand vibrational modes, and that isotopic enrichment does not need to be the main focus for the development of high-temperature SMMs. To fully develop this new molecular design criterion, a complete theory of vibrationally- driven QTM from first-principles is required, exposing how specific vibrational features are able to couple the Kramers ground states in some SMMs, but not in others. Only then can more rigorous guidelines be developed for the control of the pertinent vibrational features by chemical means, to drive the burgeoning generation of high-temperature SMMs.

Acknowledgements We thank the EPSRC (grant EP/P002560/1 and Doctoral Prize Fellowship for C.A.P.G.), Ramsay Memorial Trust (fellowship to NFC), NSFC (grant 21620102002), The University of Manchester, and the EPSRC National EPR Facility for generously supporting this work.

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