Magnetic properties

Low-temperature 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 g-factor) 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 non-interacting 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 very-low-temperature 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, non-magnetic 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=aT3 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).

Figure 3: Magnetic properties of Gd 7 . (a) Magnetization (M) as a function of applied magnetic field (B) and temperature (T=2, 3, 4 K), and fits (solid lines) from spin Hamiltonian (2). (b) Molar magnetic susceptibility (χ), in the form of χT and χ (inset), as a function of temperature, measured in an applied field of 0.1 T, and fits (solid lines). (c) Molar heat capacity (C) of Gd 7 as a function of temperature at B=0 (black symbols) and 7 T (red), and for its diamagnetic analogue Y 7 (blue) in nil field giving the lattice (non-magnetic) contribution to C. Solid lines are the calculated magnetic contributions to C(B,T) from Hamiltonian (2). (d) Magnetic molar entropy, as obtained from C(T) data for B=0 (black) and 7 T (red). Solid lines are the calculated entropies from Hamiltonian (2). Arrows denote the magnetic entropy change, ΔS m (see Supplementary Fig. 1). The small deviations between theory and experiment seen in χ and C at very low temperatures indicate the onset of magnetic dipolar interactions. Full size image

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 87 requires exploiting group theoretical methods15,16 (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 weak-field susceptibility and zero-field heat capacity show slight deviations between calculated and experimental data. For instance, the calculated susceptibility reproduces the shallow two-peak structure, with the higher-temperature 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 mean-field 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 relations17: 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 quasi-adiabatic conditions, using the set-up 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 (non-adiabaticity) 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 sub-Kelvin cooling with such a species. Second, in contrast to the straight-line isentropes found for simple paramagnets, a rich structure is observed.

Figure 4: Experimental temperature evolution of Gd 7 on applied field changes. The different curves (which correspond to isentropes) are for different initial temperature and applied field conditions T 0 and B 0 , respectively; solid lines are guides to the eye. The magnetic entropy values are S/R=1.6, 2.9, 3.5, 4.4, 5.9 and 7.6, from bottom to top, respectively. Data are shown for the sub-Kelvin temperature regime (see Supplementary Fig. 4 for a wider temperature range). Full size image

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 point3. 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 size5,7, and also for very high-symmetry (cuboctahedral, icosidodecahedral) frustrated clusters10,11,12,13, that is, they arise as a function of finite-size 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 (87) 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 zero-temperature 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 low-lying 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 zero-temperature 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.