Abstract
Geometric spin frustration in lowdimensional materials, such as the twodimensional kagome or triangular antiferromagnetic nets, can significantly enhance the change of the magnetic entropy and adiabatic temperature following a change in the applied magnetic field, that is, the magnetocaloric effect. In principle, an equivalent outcome should also be observable in certain highsymmetry zerodimensional, that is, molecular, structures with frustrated topologies. Here we report experimental realization of this in a heptametallic gadolinium molecule. Adiabatic demagnetization experiments reach ~200 mK, the first subKelvin cooling with any molecular nanomagnet, and reveal isentropes (the constant entropy paths followed in the temperaturefield plane) with a rich structure. The latter is shown to be a direct manifestation of the trigonal antiferromagnetic net structure, allowing study of frustrationenhanced magnetocaloric effects in a finite system.
Introduction
SubKelvin temperatures can be achieved via adiabatic demagnetization of paramagnetic salts^{1,2}. The underlying physics is the magnetocaloric effect (MCE) that can be evaluated by considering the adiabatic temperature change, which is when the system is driven on a constant entropy (S) curve (an isentrope):
where C is the heat capacity, T is the temperature and B is the applied magnetic field. For a paramagnet, the isentropes are straight lines in a T–B plane that run through the origin. Interacting spin systems can show a much richer response to magnetic fields and thus very different isentropes. Importantly, the cooling rate can massively outperform those of paramagnets in certain regions of the T–B plane^{3}. The simplest illustration is an antiferromagnetically coupled dimer of s=1/2 spin (Fig. 1) where extremes in the cooling rates (even changing sign) are found at the fieldinduced level crossing between singlet and triplet because the density of states (and hence the lowtemperature entropy) peaks at this field.
Such a crossing belongs to the broader class of quantum phase transitions where the groundstate characteristics of a system change (for example, nonmagnetic to magnetic, or from gapped to gapless) as a function of an external parameter such as magnetic field, pressure or doping^{4}. For MCE, the drastic changes in entropy across a fieldinduced quantum critical point can give very efficient lowtemperature magnetic cooling as recently demonstrated experimentally for a onedimensional (1D) antiferromagnetic (AF) s=1/2 chain^{3}. Geometric spin frustration can also give rise to regions of high density of states (and zerotemperature entropy), hence very high cooling rates should also be achievable, for example, when sweeping across the saturation field in such materials. The combination of these features in lowdimensional frustrated magnetic materials, for example, the famous 2D kagome or triangular AF lattices or the 1D sawtooth AF chain^{5,6,7,8,9}, makes them attractive targets for enhanced MCE and lowtemperature refrigeration. In fact, such effects should be also observable in certain 0D systems, that is, molecular clusters of spins in frustrated geometries^{10,11,12,13}. These are a subset of the broader class of molecules known as molecular nanomagnets.
The molecular cluster [Gd_{7}(OH)_{6}(thmeH_{2})_{5}(thmeH)(tpa)_{6}(MeCN)_{2}](NO_{3})_{2} (‘Gd_{7}’; H_{3}thme=tris(hydroxymethyl)ethane; Htpa=triphenylacetic acid) consists of a planar centred hexagon of weakly AFcoupled Gd(III) ions (Fig. 2; ref. 14), each of which has an electronic spin s=7/2. Hence, this topology is a finite ‘cutout’ of the 2D triangular AF lattice (Fig. 2). Here we model all the magnetic observables of Gd_{7}, including subKelvin susceptibility and heat capacity data. We then use this model to calculate the isentropes for Gd_{7}, revealing detailed structure in the T–B landscape due to the frustration. Finally, we follow these isentropes experimentally by direct measurement of the temperature in applied magnetic field cycles under quasiadiabatic conditions. The experimental data, reproduced by theoretical modelling, show the characteristics of frustrationenhanced MCE; moreover, we achieve cooling to ~200 mK—the first time subKelvin cooling has been achieved with a molecular nanomagnet.
Results
Magnetic properties
Lowtemperature magnetic data of Gd_{7} are summarized in Fig. 3. The magnetization (M) saturates to the maximum possible 49/2 gμ_{B} (where g is the electronic gfactor) per molecule at 2 K, showing that the full magnetic entropy is accessible (Fig. 3a). The χT product, where χ is the molar magnetic susceptibility, has the value calculated for noninteracting Gd(III) ions at room temperature (56.2 e.m.u. K mol^{−1}) and decreases only slowly on cooling down to ~50 K before decreasing rapidly on further cooling (Fig. 3b), denoting a dominant AF interaction. That Gd_{7} has a richer physics than a simple paramagnet is manifested in the verylowtemperature susceptibility, which goes through two shallow maxima, at 1–2 K and at 0.2–0.3 K (Fig. 3b, inset). Above 4 K, the molar heat capacity (C) in zero applied field is dominated by lattice phonon modes of the crystal, that is, nonmagnetic contributions (Fig. 3c). This is confirmed from C(T) data on the isostructural and diamagnetic yttrium analogue [Y_{7}(OH)_{6}(thmeH_{2})_{5}(thmeH)(tpa)_{6}(MeCN)_{2}](NO_{3})_{2} (‘Y_{7}’), which overlay those of Gd_{7} at higher temperatures. The phonon heat capacity can be described by the Debye model, which simplifies to a C/R=aT^{3} dependence (R is the gas constant), where a=1.35 × 10^{−2} K^{−3} for Gd_{7} and Y_{7}, at the lowest temperatures. The magnetic contribution to the C(T) data for Gd_{7} consists of a broad hump that shifts to higher temperature on increasing the applied magnetic field (Fig. 3c).
Magnetic modelling
We have modelled all these magnetic data assuming the simple Heisenberg spin Hamiltonian:
where J_{1} is the exchange interactions between nearest neighbours on the hexagon (spins 1–6), and J_{2} is the interactions between each of these spins and the central Gd (spin 7). The huge matrix dimension of 8^{7} requires exploiting group theoretical methods^{15,16} (and the approximate C_{6} molecular symmetry) for full matrix diagonalization. We find J_{1}=−0.090(5) K, and J_{2}=−0.080(5) K with g=2.02 reproduces all the experimental magnetic observables (Fig. 3). Only at the very lowest temperatures, the weakfield susceptibility and zerofield heat capacity show slight deviations between calculated and experimental data. For instance, the calculated susceptibility reproduces the shallow twopeak structure, with the highertemperature feature agreeing well but the lower temperature one calculated to be at ~0.05 K rather than the experimental 0.2–0.3 K. Most likely, these discrepancies are due to weak magnetic dipolar interactions, which are not incorporated in the theoretical model. Dipolar interactions modify the structure of energy levels and can determine (on the meanfield level) an internal field; both become relevant in proximity of absolute zero and zero applied field.
Experimental evaluation of the MCE
The MCE can be evaluated indirectly for a given applied field change from the experimental C(B,T) (for example, Fig. 3d) and M(B,T) data via Maxwell’s relations^{17}: values for Gd_{7} derived from these two observables are in very good agreement (Supplementary Fig. 1). Here we have also performed direct experimental measurements of the MCE for continuous field variations, that is, the temperature evolution via magnetization–demagnetization cycles that we perform under controlled quasiadiabatic conditions, using the setup and protocols described in Supplementary Note 1 and ref. 18. Supplementary Fig. 2 displays a representative full magnetic field cycle, and Supplementary Fig. 3 a representative demagnetization process from an initial temperature T_{0}=0.50 K and field B_{0}=2 T. We show both the raw temperature data and those for an ideal adiabatic process, that is, corrected for unavoidable thermal losses (nonadiabaticity) that have been evaluated independently (see Supplementary Note 1). By this method, we experimentally follow isentropes in the T–B plane for different B_{0} and T_{0} (up to 3 T and 3 K, respectively; Fig. 4; Supplementary Fig. 4). The general trend is a decrease in T as B is decreased, as expected. There are two important results from these adiabatic demagnetization experiments. First, we achieve temperatures as low as ~200 mK. Despite many indirect MCE studies on molecular nanomagnets, this is the first direct experimental demonstration of subKelvin cooling with such a species. Second, in contrast to the straightline isentropes found for simple paramagnets, a rich structure is observed.
On demagnetization from B_{0}=3 T, a minimum (at 2.2 T) is found in the isentropes, that is, the sample cools rapidly (large positive slope) then heats (negative slope), strongly reminiscent of the behaviour observed recently for a 1D AF chain at a quantum critical point^{3}. On decreasing the field further, the T(B) curves go through a second minimum (at ~0.7 T). As far as we are aware, such multiple peak behaviour has not been observed previously. However, secondary minima have been predicted theoretically for ideal frustrated 2D lattices as a function of decreasing size^{5,7}, and also for very highsymmetry (cuboctahedral, icosidodecahedral) frustrated clusters^{10,11,12,13}, that is, they arise as a function of finitesize effects.
Comparison with calculated results
We have calculated theoretical isentropes from the entropy function S(T,B) based on the parameters from spin Hamiltonian (2) (see Fig. 5c). We have done this for the experimental entropies that belong to the isentropes shown in Fig. 4 to allow a direct comparison, and for a lower entropy to emphasize the shape of the isentropes. The agreement with the experimental curves is remarkable, showing the double minimum in T(B) and consequent multiple cooling regimes. The agreement becomes poorer for the lowest temperatures and small fields because the aforementioned dipolar interactions become relevant. The latter, which are not included in our model, ultimately limit the base temperature reached by adiabatic demagnetization. Analysing the Zeeman diagram is difficult because of the massive (8^{7}) number of levels; in Fig. 5a, we plot the excitation energies (E*=E_{i}−E_{0}, where E_{i} and E_{0} are the energies of the ith and ground Zeeman states, respectively, at that field) to make the changes in density of states in certain field ranges more visible. The zerotemperature saturation field is ~2.9 T (that is, above which the ground state is singly degenerate and the magnetic entropy is nil; Fig. 5b). Below this saturation field, there is a high degeneracy of lowlying states (high entropy), hence rapid magnetic cooling is observed on demagnetizing towards 2.5 T (positive slope isentrope; Fig. 5c). Between about 2.2 and 1.4 T, the density of states is much lower (Fig. 5a), giving a plateau in the zerotemperature magnetization curve (Fig. 5b), hence demagnetizing into this region decreases the entropy and leads to heating (negative slope isentrope; Fig. 5c). Below 1.4 T, the density of states increases again, and we are back in a region of cooling.
Discussion
Several frustrated antiferromagnets, including 2D kagome and triangular lattices and certain 0D polytopes, have been predicted to show plateaus in their zerotemperature magnetization curves together with regions of lower densities of states^{5,7,10,11,12,13}. The uneven distributions of intervals between groundstate level crossings is a clear signature of frustration^{13}, and is the reason for the peaks observed in the isentrope distribution. This frustration arises because J_{1}≈J_{2}, and test calculations show that the isentrope peaks are quickly destroyed by smaller values of J_{2}/J_{1} (hence, weakening the frustration; Supplementary Fig. 5).
Insight into the microscopic origin of the zeroKelvin magnetization plateau in Gd_{7} is gained from evaluating the groundstate nearestneighbour spin–spin correlation functions as a function of the applied field (Fig. 6), evaluated by numerical differentiation of the groundstate energy with respect to J_{1} (S_{12}) or J_{2} (S_{17}). Calculation from the groundstate eigenfunctions is prohibitive given the enormous Hilbert space. The S_{12} function, that is, for neighbouring spins on the hexagon, grows from a fully antiparallel alignment (maximum negative S_{12}) at B=0 to a saturated parallel alignment (maximum positive S_{12}) at B=1.4 T. The S_{17} function, that is, for a spin on the hexagon correlated with the central spin, starts at a small negative value and becomes more negative with increasing B, reaching a fully antiparallel alignment at B=1.4 T. S_{17} is then constant until B=2.2 T after which it increases, reaching full parallel alignment at B=2.9 T (and saturation of the magnetization at 49/2 gμ_{B} per molecule). Hence, the magnetization plateau between 1.4 and 2.2 T corresponds to a region of stability for the spin configuration with all the spins on the hexagon fully aligned parallel with each other but fully antiparallel to the central spin, consistent with the calculated plateau magnetization of 35/2 gμ_{B} per molecule (Fig. 5b). In fact, the Gd_{7} structural motif is one of the smallest fragments of the triangular AF net that would be predicted to show such effects. For example, the smallest possible frustrated fragment—an equilateral triangle—has no such ‘metastable’ intermediate spin configuration, hence no magnetization plateau and a much simpler isentrope structure (Supplementary Fig. 6).
Many molecular nanomagnets have now been proposed for lowtemperature magnetic refrigeration (see, for example, refs 17, 19, 20, 21, 22, 23, 24, 25, 26), even in principle to the singlemolecule level^{27}, due to the high magnetic degeneracies that can be built in by appropriate choice of metal ion and a favourable exchange coupling scheme. Almost all these studies have relied on indirect MCE measurements from magnetization or heat capacity data, which are analysed to predict some maximum magnetic entropy change for a maximum field change (typically 0–5 T on a conventional SQUID magnetometer) and certain initial temperature. Such indirect analyses can give impressive headline figures but ignore the details of the exchange coupling (other than, for example, being ‘weak’, hence giving large quasidegeneracies in zero field). Hence, they are blind to the structure and true beauty of the isentropes that are a function of the exchange couplings. Here we have revealed the richness of the isentropes in Gd_{7} via direct MCE studies, including the first experimental achievement of subKelvin cooling with a molecular nanomagnet, with experimental and theoretical results in excellent agreement. Our results show that it is possible to design the cooling power of molecular materials by choosing an appropriate topology of magnetic couplings between the interacting spins, hence exploiting the great control of the latter given by molecular coordination chemistry.
The enhanced MCE we observe in certain regions of the T–B plane for Gd_{7} also confirms longstanding predictions about unusually large cooling rates in frustrated spin 0D polytopes as well as lowdimensional extended spin lattices^{5,6,7,8,9,10,11,12,13}. Indeed, the Gd_{7} molecule is a cutout of the triangular AF lattice, with imposed geometric spin frustration giving exact or near degeneracies at certain applied magnetic fields, and serves as a finitesize realization of these predictions. Such finite systems are useful in their own right, as demonstrated here, but also enable exact numerical analysis, hence giving insight into the behaviour of infinitely extended systems. If bigger molecular fragments of the triangular AF net could be prepared (such molecules are known for some dblock ions, see refs 28, 29), this would allow fascinating insight into the transition from discrete to bulk behaviour in frustrated systems.
Methods
Materials
[Gd_{7}(OH)_{6}(thmeH_{2})_{5}(thmeH)(tpa)_{6}(MeCN)_{2}](NO_{3})_{2} (‘Gd_{7}’) was prepared as reported previously^{14}. Its diamagnetic and isostructural analogue [Y_{7}(OH)_{6}(thmeH_{2})_{5}(thmeH)(tpa)_{6}(MeCN)_{2}](NO_{3})_{2} (‘Y_{7}’) was prepared by an identical method but with substitution of the appropriate metal precursor. Solvothermal reaction of Y(NO_{3})_{3}·6H_{2}O (0.085 g, 0.22 mmol) with H_{3}thme (0.11 mmol), Htpa (0.11 mmol) and NEt_{3} (0.165 mmol) in MeCN (8 ml) at 100 °C for 12 h, followed by slow cooling (0.05 °C min^{−1}) to room temperature, gave colourless crystals of the product in ~40% yield. The formulation is confirmed by elemental analysis, powder Xray diffraction (Supplementary Fig. 7) and a singlecrystal unit cell determination, which show that Y_{7} is isostructural with Gd_{7}. Elemental analysis (%) for Y_{7}C_{154}H_{164}N_{4}O_{42} (found:calculated): C 53.26:54.96; H 4.45:4.91; N 1.74:1.66.
Measurements
Magnetization measurements down to 2 K and heat capacity measurements using the relaxation method down to 0.3 K were carried out on powdered crystalline samples by means of commercial setups for the 0–9 T magnetic field range. Susceptibility measurements were extended down to 0.1 K with a homemade susceptometer, installed in a ^{3}He^{4}He dilution refrigerator. Direct MCE measurements were performed on a pressed pellet sample mounted on a sapphire plate attached to a Cernox resistance thermometer, attached by wires to a controlled thermal bath. Each MCE measurement started with the sample at zero applied magnetic field and at temperature T_{0}, and comprised: (a) gradual application of a magnetic field, up to a maximum B_{0}; (b) relaxation until the sample reached the thermal equilibrium with the bath; (c) gradual demagnetization down to B=0; and (d) relaxation at zero field until the sample reached thermal equilibrium at T_{0}. During the whole procedure, the temperature T and applied magnetic field B were recorded continuously. See Supplementary Note 1 for full details.
Additional information
How to cite this article: Sharples, J. W. et al. Quantum signatures of a molecular nanomagnet in direct magnetocaloric measurements. Nat. Commun. 5:5321 doi: 10.1038/ncomms6321 (2014).
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Acknowledgements
We thank the EPSRC (UK) for funding. This work was supported by the Deutsche Forschungsgemeinschaft through Research Unit 945 and grant INST 215/3631, and by the Spanish MINECO through grant MAT201238318C0301 and MAT201344063R.
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J.W.S. made and characterized the materials, under the supervision of D.C. and E.J.L.M. E.P. and M.E. designed and performed the quasiadiabatic magnetocaloric and magnetic experiments. J.S. modelled the magnetic data. E.J.L.M., J.S. and M.E. wrote the manuscript with further contributions from all authors.
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Supplementary Figures 17, Supplementary Note 1 and Supplementary References (PDF 929 kb)
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Sharples, J., Collison, D., McInnes, E. et al. Quantum signatures of a molecular nanomagnet in direct magnetocaloric measurements. Nat Commun 5, 5321 (2014). https://doi.org/10.1038/ncomms6321
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