Abstract
The cyclisation of a short chain into a ring provides fascinating scenarios in terms of transforming a finite array of spins into a quasiinfinite structure. If frustration is present, theory predicts interesting quantum critical points, where the ground state and thus lowtemperature properties of a material change drastically upon even a small variation of appropriate external parameters. This can be visualised as achieving a very high and pointed summit where the way down has an infinity of possibilities, which by any parameter change will be rapidly chosen, in order to reach the final ground state. Here we report a mixed 3d/4f cyclic coordination cluster that turns out to be very near or even at such a quantum critical point. It has a ground state spin of S = 60, the largest ever observed for a molecule (120 times that of a single electron). [Fe_{10}Gd_{10}(Metea)_{10}(MeteaH)_{10}(NO_{3})_{10}]·20MeCN forms a nanotorus with alternating gadolinium and iron ions with a nearest neighbour Fe–Gd coupling and a frustrating nextnearest neighbour Fe–Fe coupling. Such a spin arrangement corresponds to a cyclic delta or sawtooth chain, which can exhibit unusual frustration effects. In the present case, the quantum critical point bears a ‘flatland’ of tens of thousands of energetically degenerate states between which transitions are possible at no energy costs with profound caloric consequences. Entropywise the energy flatland translates into the pointed summit overlooking the entropy landscape. Going downhill several target states can be reached depending on the applied physical procedure which offers new prospects for addressability.
Introduction
Coordination clusters (CCs) constructed from aggregations of paramagnetic metal ions which are cooperatively coupled may exhibit molecularbased slow relaxation of magnetisation leading to bistability, hysteresis and quantum tunnelling effects characteristic of socalled single molecule magnets.^{1} The intention is that such systems can be developed to provide information storage at a single molecule level giving a significantly enhanced capability for miniaturisation.^{2} One goal along this line is to create large ground state spins that are stabilised by an easyaxis anisotropy and whose magnetic multiplet is well separated from higherlying levels. However, nature usually does not favour 'giant spins' since electrons are fermions that prefer to pair into nonmagnetic singlets. Amongst other reasons this explains why highspin molecules remain notoriously rare. Mn_{12}acetate and Fe_{8}, both with S = 10 ground states, dominated early endeavours in the field of single molecule magnetism.^{3,4,5}
Setting aside anisotropy for a moment, a simpler goal might be to achieve the highest possible spin state residing on a single molecule. Such a giant spin goes way beyond what can be envisaged for a single ion – S = 7/2 is the highest spin attainable among the periodic table elements. Applications for giant molecular spins include providing contrast agents for magnetic resonance imaging applications through the influence of magnetic moments on relaxation processes, detectable using NMR, and molecules acting as local coolers by utilising the magnetocaloric effect (MCE) of a system, detectable using heat capacity measurements.
Finding routes to cooperatively couple open shell ions within a coordination cluster core can, indeed, lead to vastly higher spins than are imaginable for open shell single ions and a number of very high spin systems can be found in the literature^{6,7} including our mixed valent Mn^{II}/Mn^{III} Mn_{19} coordination cluster, which holds a record ground spin state of S = 83/2.^{8}
Recent research by us and others into mixed 3d/4f systems^{9,10,11} offers various means to increase both spin and total anisotropy. En route we synthesised the Gdcontaining isotropic member of a new series of cyclic CCs [Fe_{10}Ln_{10}(Metea)_{10}(MeteaH)_{10}(NO_{3})_{10}]·20MeCN, where Ln is Gd^{III} (1) or Y^{III} (2). As we will demonstrate below 1 turns out to be a fascinating spin system for two reasons. It possesses a ground state spin of S = 60, which is the largest spin observed for a single molecule to date. In addition, due to strong spin frustration the ground state is situated in very close proximity to a quantum phase transition, which strongly influences the physical properties of the molecule.
Results
Synthesis and structure
This cyclic coordination cluster system is synthesised from racemic N,Nbis(2hydroxyethyl)amino2propanol (MeteaH_{3}), Fe(NO_{3})_{3} and Gd(NO_{3})_{3}. The structures of some of these {Fe_{10}Ln_{10}} compounds have been published and described previously.^{12,13} Ten {FeGd(Metea)(MeteaH)(NO_{3})} units (Supplementary Figure S1) are linked by pairs of alkoxo bridges to form the 20membered elliptical nanotorus in 1 (Fig. 1a), in which the FeGd units are displaced alternately above and below the mean plane of the ring, describing a wavelike chain structure reminiscent of a Bohr closed standing wave (Fig. 1b). This nanoellipse has a major diameter of 28.4 Å and minor diameter of 26.3 Å, and is 12.7 Å thick, based on the van der Waals surfaces of the atoms (Fig. 1c). In other words, it represents a molecular realisation of a welldefined nanoparticle within the 1–3 nm size range.
Within each FeGd unit, the ligand chelating the Gd^{III} ion retains the hydroxyl proton on its methyl substituted arm, whereas the ligand chelating the Fe^{III} ion does not, resulting in strong Hbonding between the two ligands (e.g., between O(3) and O(6) in Supplementary Figure S1). This forces the two ligands within such a unit to be of the same chirality, and in turn results in the ten ligands over one face of the ellipse to be of the R enantiomer, while those on the other side of the mean plane are S.^{12} This arrangement of the enantiomeric forms of the ligands means that the methyl groups can easily snuggle between the rest of the organic parts of the ligands to give a rather rigid ligand shell (Fig. 1c).
Magnetic properties
The χT product for 1 (Fig. 2a, χ = M/B) at 300 K is consistent with the value for 10+10 noninteracting Fe^{III} (s_{Fe} = 5/2) and Gd^{III} (s_{Gd} = 7/2) ions with g≈2.0. For 1 the monotonic increase of the susceptibility down to 3 K suggests the presence of ferromagnetic interactions between Fe^{III} and Gd^{III}. In addition, the maximum of the temperature dependence of the inphase susceptibility (χ'T) of 745 cm^{3}K/mol observed in 1 (Supplementary Figure S2) suggests a ground state with a large total spin S of at least 38, assuming g = 2. The field dependence of the magnetization (Fig. 2b) at low temperatures supports that the exchange interactions are ferromagnetic. However, the magnetization curve at fields lower than 20 kOe is offset compared with the Brillouin function calculated for a single spin S = 60 with g = 2.0. This is in line with the observation for the diamagnetic Y^{III} system 2 that there is a weak antiferromagnetic coupling between the Fe^{III} centres, which obviously only becomes significant at low temperatures and small fields, compare Supplementary Figure S3.
Theoretical calculations and interpretation
The structure of the metal skeleton together with the possible magnetic exchange interactions between the metal ions is depicted in Fig. 3. We denote the nearestneighbour exchange interactions between each Gd and its adjacent Fe centres as J_{1} and the nextnearestneighbour interactions between adjacent Fe ions as J_{2}. The nextnearestneighbour Gd–Gd interactions are approximated as zero, which appears reasonable for distant felements and a posteriori turns out to be compatible with all observables. In this configuration the iron atoms constitute the basal spins of the delta chain, compare Fig. 3, whereas the Gd atoms play the role of the apical spins. Depending on the sign and ratio of the two exchange interactions, the delta chain can exhibit several very unusual properties such as flat energy bands, extended magnetization plateaus, giant magnetization jumps as well as a quantum phase transition along with pronounced magnetocaloric effects—all driven by geometric frustration.^{14,15,16,17,18,19,20,21,22}
Before discussing what makes the present compound so special we set out to fit the magnetic observables with the following Hamiltonian containing two Heisenberg terms describing the interactions between the magnetic ions and a Zeeman term which models the interaction with the external magnetic field.
Although both Fe^{III} and Gd^{III} have halffilled electron shells they might nevertheless possess nonvanishing singleion anisotropy tensors depending on their coordination. In the present case these tensors circle around the ring structure and thus cancel to a large extent.^{23} This is experimentally supported by the fact that no hysteresis or ac signal has been found. Along this line, and in order to keep the numerically favourable SU(2) symmetry, we assume g_{Gd} = g_{Fe} = 2, which is a reasonable approximation both for Gd^{III} and Fe^{III}. But even when employing all symmetries, this large spin system with a Hilbert space dimension of 6.5 × 10^{16} cannot be treated with any existing exact method.^{24,25} The theoretical problem appears to be a challenge on its own. We thus explored four approximations to model the system, namely (1) HighTemperature Series Expansion (HTE), (2) Quantum MonteCarlo (QMC), (3) Classical MonteCarlo (CMC) and (4) the FiniteTemperature Lanczos Method (FTLM). None of these methods alone is capable of modelling the thermodynamic behaviour of all observables, but combined we can draw definitive conclusions.
Discussion
The most recent HTE code of sixth order for mixed spin systems^{26,27,28} yields exchange interactions J_{1} = 1.0 K between Fe and Gd ions and J_{2} = −0.65 K between adjacent Fe ions. If one considers the exchange between adjacent Gd ions, it turns out to be virtually zero. Dipolar interactions do not play a role for temperatures T ≥ 2.0 K.^{29} The resulting fit to the susceptibility is depicted by a black solid curve in Fig. 2a. The HTE diverges for smaller temperatures since the power series in 1/T terminates at some power, here six. QMC calculations,^{30,31} on the other hand, can deliver thermodynamic functions of huge spin systems as long as these systems are not frustrated. If a system is geometrically frustrated, as in the present case, QMC still works for high enough temperatures. Therefore, we compared our QMC results with those from HTE as well as with the data. As can be seen in Fig. 2a, the blue curve for QMC exactly matches the HTE curve above about 15 K. For lower T the QMC results do not converge, and are thus not shown. Also CMC calculations should yield good results since the spin quantum numbers of 5/2 for Fe^{III} and 7/2 for Gd^{III} are large and thus a classical approximation is appropriate. The red curve in Fig. 2a demonstrates that this expectation is indeed met. In addition to this good approximation at high temperatures, the maximum as well as the lowtemperature susceptibility are also nicely reproduced with the same exchange constants. At the low temperatures at which the magnetization was measured, hightemperature methods are not applicable and CMC fails since classical spins have a length of √(s(s + 1)), which is not compatible with the saturation magnetization. QMC, although limited due to frustration, can be applied for highenough magnetic fields since in the presence of Zeeman splitting only a few lowlying levels are accessible. The blue curves in Fig. 2b show that the parameterization in terms of J_{1} and J_{2} indeed goes through the data points for 2 and 4 K, and for 4 K even to smaller fields since the higher temperature improves convergence.
The opposite is true in terms of a modelling with the FiniteTemperature Lanczos Method (FTLM).^{32,33} The method provides very good approximations for thermodynamic functions up to Hilbert space dimensions of 10^{10}. So far, the largest system treated using this approach was a Gd^{III}_{12} cluster.^{34} However, for 1 the Hilbert space is much larger and thus we only included subspaces with total magnetic quantum number M > 45. This works well at low enough temperatures and large enough fields, since the large ferromagnetic ground state with S = 60 and the Boltzmann factor both favour an improvement in the approximation. The result is depicted by the red curves in Fig. 2b. The horizontal dotted line marks the lowest magnetization down to which the calculation is approximately still valid. FTLM does not work for χT vs T, since one would need to consider all Msubspaces in this case.
The final result of our endeavour is that in 1 the exchange interactions are J_{1} = 1.00 K between neighbouring Fe and Gd ions and J_{2} = −0.65 K between adjacent Fe ions with an estimated uncertainty of ±0.02 K. This immediately results in a ground state with the maximal possible total spin of S = 60, as shown in the scheme of lowlying levels (Supplementary Figure S4). The reason for the large ground state spin is that the net interaction is ferromagnetic. This might not seem immediately obvious, but can be understood in terms of the quantum phase transition we now describe. For a ferromagnetic J_{1} > 0 and J_{2} = 0 it is obvious that all spins must be aligned to the maximum possible total spin (Fig. 4). However, with a competing J_{2} < 0 that is increasing in magnitude the ground state will change if the ratio α = J_{2}/J_{1} assumes the critical value α_{c} = s_{Gd}/(2s_{Fe}) = 0.7.^{35,36,37}
Since J_{2}/J_{1} = 0.65 in 1, we are just on the ferromagnetic side of the quantum critical point (QCP). Owing to the special deltachain structure of 1, at such a QCP the ground state becomes massively degenerate,^{17,35,36,37,38,39} of the order of 10^{4}, see also Supplementary Figure S5. The ratio α = 0.65 for 1 is smaller than the critical value of α_{ c } = 0.7, but very close. Near the critical point the former ground state manifold now represents a huge number of very lowlying excited states, thereby establishing an extra lowenergy scale. As a result, one may find additional ultralow temperature features in the specific heat (Supplementary Figure S6). In addition, one can observe that the specific heat is massively increased at lowtemperature; the same holds for the magnetocaloric effect.^{35,36,37} Thus, the molecule, although not directly at the quantum critical point, acts as an observer of the quantum critical phase at elevated temperature (black arrow in Fig. 4).
Figure 5 shows the experimental specific heat per Fe–Gd pair, normalised to the gas constant (R = 8.314 J/molK) for various applied magnetic fields, which consists of the magnetic as well as the lattice contribution. The magnetic contribution is unusually large even at 20 K, and the Schottky anomaly is progressively shifted toward higher temperatures by the application of an external magnetic field (compare also Supplementary Figures S6, S7 and additional explanations in the Supplementary Information). As the quantum calculations for the similar but smaller model system Fe_{6}Gd_{6}, which in contrast to Fe_{10}Gd_{10} can be fully treated by FTLM, demonstrate (Fig. 5 and Supplementary Figure S7, solid curves), the magnetic contribution to the heat capacity indeed assumes very large values, even at elevated temperatures, resulting from the unusually high density of lowlying energy levels. Although the theoretical model system Fe_{6}Gd_{6} is smaller, it mimics the physical behaviour of 1 extremely well. This also holds for the specific heat at low temperatures. The essentially constant behaviour for B = 0, 0.5, and 1 T is a fingerprint of the additional lowenergy scale (Supplementary Figure S5), and is reproduced for these as well as for larger magnetic fields, where the specific heat drops. We attribute the small discrepancies between model calculations and experimental data to finite size effects as well as to dipolar interactions, which are not incorporated in the model.
Finally, we would like to comment on the magnetocalorics. Figure 6 displays the isothermal entropy change, one figure of merit among magnetocaloric properties,^{40,41} calculated for the model system Fe_{6}Gd_{6}. As one can see, the entropy changes for ΔB = 7 T are very similar for the actual α = 0.65 of 1 (red) and the critical value of α_{ c } = 0.7 (dashed red). The very steep rise at the lowest temperatures signals that this compound should be a powerful cooler at liquid helium temperatures^{40,41,42} with impressive MCE figures for a single molecule unit [ΔS = S(7 T)S(0 T) > 20 R at T≈3 K]. This behaviour is compared in Fig. 6 with the respective ones of analogous systems without the nextnearest interaction J_{2}. The case with only a ferromagnetic J_{1} is shown by the black curve, whereas the case with only an antiferromagnetic J_{1} is shown as the blue curve. Whereas the latter antiferromagnetic ring would be obviously useless as a refrigerant, the ferromagnetic ring has its optimal working conditions at much higher temperatures compared to our frustrated delta chain cluster.
In summary, the combined use of several theoretical methods confirms the record ground state spin of S = 60 for 1. Furthermore, our calculations explain its near quantum critical behaviour, which is experimentally evident in the massively enhanced specific heat. Cyclisation has thus yielded a very unusual quantum spin system with large future potential.^{13,43} How we can direct the molecule to move across both sides of the quantum critical point, is a future challenge. Possibilities include: using chemical means, through gating,^{44} and by applying pressure. Being able to move such a system so that it lies near the quantum critical point, either to the left or right of it, as well as directly on its sharp peak offers a chance to move beyond bistable systems (binary logic) as typically envisaged in our field and thence 'towards new directions in terms of control and tunability for molecular spintronics' as suggested by Roch et al. ^{44}
Methods
Magnetic measurements were obtained using a Quantum Design SQUID magnetometer MPMSXL in the temperature range 1.8–300 K. Measurements were performed on polycrystalline samples constrained in eicosane. Magnetisation isotherms were collected at 2, 3, 5 K between 0 and 7 T. Alternating curent (ac) susceptibility measurements were performed with an oscillating field of 3 Oe and ac frequencies ranging from 1 to 1500 Hz. The data were corrected for the diamagnetic contribution of sample holder and substance. Specific heat data have been obtained with a commercial PPMS ^{3}He system from quantum design. Heat capacity was measured on pressed pellets of microcrystals of approx. weight of 1–2 mg by using the twotau relaxation method. Theoretical calculations haven been performed with selfwritten HTE, complete diagonalization as well as FTLM program codes. QMC calculations have been done with the ALPS package using the directed loop stochastic series expansion program 'dirloop_sse'. All data are available on request from the authors.
Data availability
Supplementary information accompanies the paper on the npj Quantum Materials website (https://doi.org/10.1038/s4153501800827). Crystallographic details are given in the Supplementary Information, or in CCDC 1557819 available from https://www.ccdc.cam.ac.uk/structures/. All other data are available on request from the authors.
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Acknowledgements
We acknowledge the Synchrotron Light Source ANKA for provision of instruments at the SCD beamline. J.S. thanks for computing time at the Leibniz Rechenzentrum in Garching. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) funded transregional collaborative research centre SFB/TRR 88 '3MET'. J.S. thanks the DFG for funding (SCHN 615/201, SCHN 615/231). J.R. as well thanks the DFG for funding (RI 615/212). We acknowledge support for the Article Processing Charge by the Deutsche Forschungsgemeinschaft and the Open Access Publication Fund of Bielefeld University.
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A.K.P. formulated the project. A.B. performed the synthesis. G.B. and C.E.A. measured the dataset for the crystal structure, and C.E.A. solved and refined the structure. Y.L. performed the magnetic susceptibility measurements and N.M. performed the initial evaluation of the chain model. M.A. measured the lowtemperature highfield specific heat. J.S. and J.R. carried out the theoretical calculations. J.S. and A.K.P. wrote the paper and all authors contributed to discussions and the final manuscript.
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Baniodeh, A., Magnani, N., Lan, Y. et al. High spin cycles: topping the spin record for a single molecule verging on quantum criticality. npj Quant Mater 3, 10 (2018). https://doi.org/10.1038/s4153501800827
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DOI: https://doi.org/10.1038/s4153501800827
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