Observation of the influence of dipolar and spin frustration effects on the magnetocaloric properties of a trigonal prismatic {Gd7} molecular nanomagnet

Adiabatic demagnetisation experiments on a C3h symmetric molecular {Gd7} cluster reveal effects of intramolecular dipolar interactions and geometric spin frustration.


Introduction
A wide variety of applications have been envisioned exploiting the magnetic properties of mono-or poly-metallic lanthanide complexes. 1 The anisotropic lanthanides have been proposed as building blocks in quantum computers, 2 data storage 3 and spintronic devices. 4 Conversely, Gd(III) systems, where the lanthanide ion is isotropic and has a large spin (s ¼ 7/2), can have large magnetocaloric effects (MCE), 5 and hence could be good candidates for very low temperature magnetic cooling. 6 Almost all MCE studies of molecular clusters have been indirect, where magnetization or heat capacity data are analysed to derive magnetic entropy changes for a given change in applied magnetic eld. Some of us recently reported direct magnetocaloric experiments on a {Gd 7 } molecule where we achieved cooling to temperatures as low as $200 mK in adiabatic demagnetization experiments. 7 Moreover, we found that the temperature evolution of the sample on demagnetisation under quasi-adiabatic conditions (following paths of constant entropy or isentropes) was not linear as for a simple paramagnet, but showed specic regions of enhanced cooling, and even heating, on demagnetisation. That {Gd 7 } molecule has a centered-hexagonal array of metal ions. With antiferromagnetic exchange coupling, this triangle-based structure suffers geometric spin frustration. The latter is an important phenomenon in extended lattices, and is associated with having many degenerate ground state congurations, leading to exotic magnetic behaviours such as spin glasses and spin ice. 8 It can also lead to enhanced MCE because of the increased density of states at critical elds. 9 We showed that the observed isentrope structure in {Gd 7 } was a direct signature of spin frustration. 7 Here we report a new {Gd 7 } cage with a different geometrically frustrated structure and investigate the consequences on its MCE via adiabatic demagnetization experiments. Moreover, we observe the effects of internal dipolar interactions that are competitive with the exchange couplings, with further consequences for MCE applications.
1 crystallizes in the P 62c space group with the anionic {Gd 7 } cage lying on the 6 (S 3 ) axis, giving one sixth of the molecule in the asymmetric unit, and hence crystallographic C 3h symmetry. The metallic core consists of six Gd(III) ions forming a trigonal prism [Gd(2) and symmetry equivalents]. The prism encapsulates a central Gd(III) ion [Gd(1), Fig. 1]. Six CO 3 2À groups bridge the trigonal edges of the prism, also binding to the central Gd (3.211 binding in Harris notation); 11 these carbonates must arise from CO 2 xation. 12 There are three m 3 -OH, one bridging each edge between the triangular faces of the trigonal prism, also binding to the central Gd(III) ion. This highlights an alternative description of the cage as three {Gd 3 (m 3 -OH)} triangles sharing a single vertex [Gd(1), Fig. 1b]: the triangles are isosceles with Gd(1)/Gd(2,2 0 ) and Gd(2)/Gd(2 0 ) distances of 3.9548(1) and 3.9388(9)Å, respectively. Twelve pivalates complete the cage: two pivalates bridge each Gd(2)/Gd(2 0 ) with 2.11 and 2.21 coordination modes (the latter is disordered, lying on the mirror plane), and each Gd(2) vertex is capped by a 1.11 pivalate. Gd(1) is nine-coordinate with tricapped trigonal-prismatic geometry (continuous shape measure, CShM of 0.910, Table S1 †). 13 Gd(2) is eight-coordinate, but no clear unique description is favoured by CShM (Table S1 †). Six di-isopropyl ammonium cations provide charge-balance. Magnetic susceptibility (c M ) studies (applied magnetic eld, B 0 ¼ 0.1 T) were carried out on a polycrystalline sample of 1 in the temperature (T) range 2-300 K (Fig. 2a). At room-temperature the product c M T is 55.2 emu K mol À1 , consistent with seven Gd(III), 8 S 7/2 ions (c M T ¼ 55.1 emu K mol À1 for seven non-interacting s ¼ 7/2 with g ¼ 2.00). Upon cooling, c M T(T) remains constant down to ca. 10 K where it drops sharply, to 43.5 emu K mol À1 at 2 K. The magnetization (M) data for 1 at low temperature show rapid increases of M with applied magnetic eld, reaching a saturation value of 48.7 m B at 7 T and 2 K ( Fig. 2b and S1 †), consistent with complete polarization of the spin system (maximum possible M is 49.0 m B for seven s ¼ 7/2 with g ¼ 2.00).
Specic heat (C) measurements under different applied magnetic elds were also carried out for 1 (Fig. 2c). Above ca. 5 K, C is dominated by lattice phonon modes of the crystal, which can be described by the Debye model (dotted line) and simplify to C/T 3 ¼ 0.05 J kg À1 K À1 at the lowest temperatures. The strong eld dependence of C at low temperature, and the ability to fully magnetise the system, suggests 1 as a good candidate for a cryogenic magnetic refrigerant. Therefore, we have evaluated the entropy (Fig. 2d) and MCE ( Fig. 2e and f) of 1 indirectly from C(B 0 ,T) and M(B 0 ,T), using known procedures. 5c We obtain a maximum magnetic entropy change of ÀDS m ¼ 30.8 J kg À1 K À1 at 2.1 K for a eld change DB 0 ¼ (7-0) T. Hence, under these experimental conditions, we are accessing a large proportion (83%) of the full magnetic entropy content, viz. S m ¼ nR ln(2s + 1) ¼ 36.9 J kg À1 K À1 (n ¼ 7, s ¼ 7/2, R is the gas constant). The latter is reached in zero eld for T above ca. 3 K (Fig. 2d), while nil magnetic entropy (i.e. magnetic saturation) is reached at 7 T below ca. 2 K. We can access this high fraction of the magnetic entropy because of the small antiferromagnetic exchange between Gd(III) ions, coupled with spin frustration, giving a high density of low-lying states.
The magnetic data have been modelled using a Heisenberg spin Hamiltonian (1) consistent with the C 3h symmetry (Fig. 2a,  insert): s i denote the individual spin operators (i ¼ 1 is the central Gd) andŜ z denotes the z-component of the total spin operator. Since the Heisenberg Hamiltonian possesses several symmetries, a complete matrix diagonalization of the system is possible despite the huge 8 7 matrix dimension. 14 J 1 is the interaction between the central Gd ion (S 1 ) and all others; J 2 is the interaction on the edges between triangular faces of the trigonal prism; J 3 is the interaction on the edges of the triangular faces. The best agreement between experimental data and calculations for c M T(T) and M(B 0 ) is achieved with J 1 ¼ J 2 ¼ À0.02 cm À1 and J 3 ¼ 0 cm À1 , assuming g ¼ 2.00 ( Fig. 2a and b). Note that these parameters, with J 3 ¼ 0, would support the description of 1 as three triangles sharing a vertex as depicted in Fig. 1(a and  b). This topology results in geometric spin frustration because not all the antiferromagnetic interactions can be satised, 15 and results in very high degeneracies of states. For example, for these parameters in this Heisenberg model, the three lowest energy eigenstates possess total spins of S ¼ 13/2, 11/2 and 15/ 2, each with their 2S + 1 degeneracy, and lie within 0.005 cm À1 of each other. However, using this model to calculate the zero-eld specic heat as a function of temperature gives a peak in C(T) at ca. T ¼ 0.1 K which is not observed experimentally (Fig. 2c). This suggests that our model, at least below 2 K, is inadequate. The isotropic exchange interactions found using eqn (1) are very weak, in fact signicantly smaller than in other Gd(III) cages we have studied (e.g. J z À0.06 cm À1 in our planar {Gd 7 } that also contains m 3 -OH bridges). 7 Hence we calculated the intramolecular dipolar interactions to test the validity of neglecting them. In fact for the shortest Gd/Gd distances (on the edges between triangular faces of the trigonal prism) the dipolar interaction tensor has elements that range in magnitude from 0.014 to 0.033 cm À1 , and are thus of the same order as the isotropic J values found using (1). Hence they must be included in our model. However, quantum mechanical calculations with anisotropic Hamiltonians for spin systems of this size need millions of CPU hours on supercomputers, 16 hence we have performed simulations with a ctitious cluster having the same magnetic skeleton as 1, but replacing the s ¼ 7/2 spins with spins s ¼ 3/2.
Such simulations ( Fig. S2 and S3, ESI †) show that the dipolar interactions inuence the magnetic properties only mildly for T > 2 K, i.e. they do not signicantly inuence our modelling of the experimental c M T(T) and M(B 0 ) curves, and hence we would obtain the same best values for the isotropic J as above. However, inclusion of dipolar interactions has drastic effects on the calculated magnetic properties below 1 K. This is because of the degeneracy breaking of the otherwise degenerate low-lying states. Specic heat is particularly sensitive to rearrangement of low-lying levels and, given that we have C(T) data down to 0.3 K, this provides a sensitive test.
Full matrix diagonalization of the anisotropic spin Hamiltonian (i.e. including the dipolar interactions) is not possible for the full spin system of 1. Therefore, in order to mimic the effect of the dipolar eld on the magnetocalorics of 1 for the full spin system, we have introduced a ctitious effective internal eld B eff that, by means of its Zeeman splitting, smears out the otherwise sharply peaked density of states. From calculation of C(T) we nd that B eff ¼ 0.5 T moves the heat capacity at zero applied eld (B 0 ¼ 0) to give good agreement with experiment (Fig. 2c); for an external eld of B 0 ¼ 1 T, we nd B eff ¼ 0.2 T gives good results. For the larger applied elds such a correction is not necessary.
We have previously shown that we can observe the effects of spin frustration in the shape of isentropes (paths of constant entropy) in temperature-eld plots measured by direct MCE experiments. 7 Hence, we have performed such measurements for 1 (see ESI † and ref. 6b and 7 for the experimental procedure) and also used these to test the inuence of dipolar effects on the MCE response. An example of the temperature evolution of 1 on demagnetization under controlled quasi-adiabatic conditions is in Fig. S4 †: in this example, demagnetizing from B 0 ¼ 1.5 T results in cooling of the sample from an initial temperature T 0 ¼ 0.6 K to a nal temperature T ad ¼ 0.13 K, which is signicantly lower than the temperatures achieved in our earlier experiments with the planar {Gd 7 } cage. 7 Fig. 3a shows data for different B 0 and T 0 , in the form of isentropes (see Fig. S5 † for further data). Unlike a simple paramagnet, where the isentropes would be linear, we nd minima in the T-B plots at ca. 0.3 and 1.0 T (the latter is shallow but real). These minima mean there are regions of enhanced and retarded cooling (even warming) on demagnetization. These minima are due to the frustration, and they are most pronounced at the lowest temperatures where quantum effects are dominant, smearing out on warming as many states become Boltzmann populated and the system behaves more classically. Insight into their origin comes from a Zeeman diagram and zero-Kelvin magnetization plot calculated from the Heisenberg Hamiltonian (1) and the isotropic exchange values above (Fig. S6 †). Above 1.1 T, there is a unique singly degenerate ground state (nil entropy), i.e. M is saturated at 49 m B . Below this eld there is a high degeneracy of states (high entropy) and the sample cools on demagnetising into this regime. Between ca. 0.6 and 0.3 T there is a low (single) degeneracy of states and M(B) plateaus at 35 m B ; hence on demagnetising into this region the material passes from high to low entropy which would lead to heating. Below 0.3 T a high degeneracy of states is again met, as the system undergoes a series of level crossings to lower M, and we are back in a cooling regime. The 35 m B plateau in M(B) is consistent with a meta-stable spin conguration where all the outer Gd(III) spins are fully aligned with each other but fully opposed to the central Gd(III) spin (an S ¼ 35/2 state).
Calculated isentropes based on this Heisenberg model are in good qualitative agreement with the experimental curves (Fig. 3), correctly nding two anomalies in the T-B curves, but: (i) the theoretical isentropes give the anomalies at ca. B 0 ¼ 0.1 T and 0.9 T, i.e. there is a signicant eld shi particularly for the lower eld feature, and (ii) the features, particularly at 0.9 T, are much more pronounced than in the experimental data. Both points can be explained by the neglect of the dipolar interactions in the calculations. In terms of the shi in the lower eld minimum, the effect of the dipolar eld will be greatest for the lowest applied magnetic elds (and lowest temperatures). This is also apparent from our modelling of the experimental C(T) data (see above). We also observed such a discrepancy for the centred-hexagonal {Gd 7 } system. 7 Ultimately, any source of magnetic ordering, including dipolar elds, can shi such features in eld. This also limits the lowest attainable temperature upon demagnetization.
The broadening of the isentrope minimum at 0.9 T implies a less regular spin structure than arises from the Heisenberg model, and this cannot be modelled by an additional static eld to mimic the dipolar eld. Hence, in order to examine this effect of the dipolar interactions we have calculated the T-B isentropes for the ctitious cluster of s ¼ 3/2 spins, enabling exact calculation (see above), with and without dipolar interactions (Fig. S3, ESI †). Inclusion of the dipolar interaction leads to a pronounced smearing of the minimum. Although these results cannot be compared quantitatively to 1 (the different spin leads to different numbers and densities of states, hence to T-B minima at different elds), the shapes of the curve are remarkably similar to the experimental data for 1.

Conclusions
Summarizing, we have found a second example where signatures of geometric spin frustration have been observed in adiabatic demagnetization experiments of a molecular nanomagnet. They give rise to minima in temperature-applied magnetic eld curves, in turn giving regions of enhanced cooling at critical applied magnetic elds. In contrast to the previous example, 7 the isotropic exchange interactions are comparable to the intra-molecular dipolar interactions and the latter cannot be neglected. Their effect is to smear out the frustration signatures and to dampen the enhanced cooling rates. Moreover, although in the present investigation a temperature T ad lower than 100 mK could be achieved (Fig. 3), in general dipolar interactions limit the temperatures that can be reached in such experiments. These results highlight the importance of the relative magnitudes of the interaction parameters in molecular clusters in terms of their use for MCE. The antiferromagnetic interactions are necessary for spin frustration which gives rise to large entropy changes, but if the interactions are too large then the full magnetic entropy will not be available on (de)magnetization. If, on the other hand, the interactions are too weak, then the MCE will be limited by dipolar interactions. Most studies of the MCE in molecular systems have simply relied on indirect determination of MCE parameters, and are blind to these effects.