Emergent magnetic order and correlated disorder in formate metal-organic frameworks

Abstract

Magnetic materials with strong local interactions but lacking long-range order have long been a curiosity of physicists. Probing their magnetic interactions is crucial for understanding the unique properties they can exhibit. Metal-organic frameworks have recently gathered more attention as they can produce more exotic structures, allowing for controlled design of magnetic properties not found in conventional metal-oxide materials. Historically, magnetic diffuse scattering in such materials has been overlooked but has attracted greater attention recently, with advances in techniques. In this study, we investigate the magnetic structure of metal-organic formate frameworks, using heat capacity, magnetic susceptibility and neutron diffraction. In Tb(DCO₂)₃, we observe emergent magnetic order at temperatures below 1.2 K, consisting of two k-vectors. Ho(DCO₂)₃ shows diffuse scattering above 1.6 K, consistent with ferromagnetic chains packed in a frustrated antiferromagnetic triangular lattice, also observed in Tb(DCO₂)₃ above 1.2 K. The other lanthanides show no short- or long-range order down to 1.6 K. The results suggest an Ising-like one-dimensional magnetic order associated with frustration is responsible for the magnetocaloric properties, of some members in this family, improving at higher temperatures.

This article is part of the theme issue ‘Mineralomimesis: natural and synthetic frameworks in science and technology’.

1. Background

Magnetic materials have long been studied from a fundamental standpoint [1] and for their functional properties, including data storage and transfer [2], magnetic refrigeration [3] and as MRI contrast agents [2,4]. Ferromagnetism has been known about before modern science through the naturally occurring iron-containing mineral, magnetite [5], despite a lack of understanding of its underlying microscopic magnetic mechanisms. Strong magnetic interactions in oxide materials give rise to high ordering temperatures, but they prevent the appearance of more exotic states of matter. Such exotic states, such as spin liquids and other phases devoid of long-range order, are of great interest to the condensed matter community, for the unique phenomena they exhibit.

Metal-organic frameworks (MOFs) may be the ideal systems to search for such exotic states. MOFs have attracted much attention in recent years for the ability to exhibit a wide array of structures and properties, lending themselves to gas storage [6] where design and synthesis of highly porous frameworks is possible [7]. Recently however, magnetic studies of MOFs have emerged, highlighting the potential of magnetic MOFs in functional magnetic materials. MOFs have already been shown to exhibit a range of magnetic properties, with uses in chemical and magnetic sensors [8], magnetocalorics [9,10] and multiferroics that combine magnetism with ferroelectric order [11].

The ability for greater flexibility in the design of MOFs, allows for more specific topologies and magnetically isolated sheets and chains, opening up the means to low-dimensional and frustrated magnetism. In MOFs, the magnetic interactions are facilitated through the connecting ligand, or single anion where the strength and direction of these magnetic interactions depend on the precise nature of the linker. Magnetic interaction strength is typically inversely proportional to the size of the ligand and, therefore, in magnetic MOFs, linkers must be kept short to allow sufficient metal–metal magnetic interactions in the dimensions they are desired.

(a). Frustrated magnetism and magnetic systems

To fully understand how frustration can occur the models of magnetism must first be considered, and here the most prevalent models are shown. The simplest model for nearest-neighbour magnetic interactions is the Heisenberg model given by the Hamiltonian: $\hat{H}=-\sum _{(i,j)}{J}_{ij}{\mathit{S}}_{\mathit{i}}\cdot {\mathit{S}}_{\mathit{j}}$, where i and j are spin sites on a periodic lattice. S is a unit vector, representing the magnitude and direction of a magnetic moment, and J_ij is the strength of the interaction between neighbouring spins. In this simplest model, spins are considered to be three dimensions and can be oriented in any direction. Alternatively, spins can be fixed so they can only point up or down along a single axis, known as the Ising model [12], or in a easy plane—the XY model.

Below some temperature (T_c), the J exchange interactions are sufficient to overcome the thermal fluctuations in the material, and spins can align in a energetically favourable orientation. When the exchange integrals J_ij are negative, the spins align anti-parallel to nearest neighbours (antiferromagnetic; figure 1a); when exchange integrals are positive the spins are parallel (ferromagnetic). On a square lattice, with no next-nearest neighbours, either of these ground states can be satisfied.

Figure 1.

(a) Antiferromagnetically ordered square lattice, (b) a frustrated antiferromagnet on a triangular lattice and (c) antiferromagnetically coupled Heisenberg spins on a triangular lattice. (Online version in colour.)

Frustration can only occur if there are competing magnetic interactions, usually when some of the exchange paths are antiferromagnetic, if only nearest-neighbour interactions are considered. The classical example of these systems are Ising antiferromagnetic coupled spins arranged on a triangular lattice (figure 1b), first studied by Wannier in 1950 [13]. In this model system, it is energetically favourable for the spin to be anti-parallel, with respect to its nearest neighbour, but the crystallographic structure of the materials means there are two near neighbours competing for anti-parallel alignment.

In an Ising system, it is not possible to satisfy both simultaneously and so a frustrated ground state emerges with many degenerate energy levels. In a Heisenberg system, this frustration can be alleviated though other routes as shown in figure 1c. When frustration arises purely from the topology of the lattice, this is described as geometrically frustrated. While this is a convenient illustrative example, frustration is not limited to two dimensions, or even triangular lattices, for example in three dimensions this corresponds to a pyrochlore motif.

Frustration in materials can lead to exotic states including some of the most studied pyrochlore spin ice materials, Dy₂Ti₂O₇ and Ho₂Ti₂O₇, consisting of four ferromagnetically coupled lanthanide ions with strong Ising character [14]. Spin ices are named because of their analogy with water ice, with two spins facing the centre, and two facing outwards (figure 2a). This leads to some fascinating properties, such as the existence of magnetic monopoles quasi-particles [15], that cannot exist in isolation. Spin ice materials are an example of systems which can exhibit magnetically charge ordered states, when one of the spins is flipped. The garnet lattice consists of cations sitting on the vertices of corner sharing triangles, forming a highly frustrated three-dimensional system of interpenetrating rings. Frustration, in the benchmark magnetocaloric Gd₃Ga₅O₁₂ garnet (GGG), prevents long-range order forming, but shows spin-liquid behaviour between 140 mK and 5 K [16], where despite significant antiferromagnetic interactions the magnetic moments continue to fluctuate. Unlike a paramagnet however, the spins are correlated and do not form static order even at absolute zero.

Figure 2.

(a) A spin ice tetrahedra with two in, two out spin arrangement and (b) a kagome emergent charge ordered state. (Online version in colour.)

By magnetically isolating the layers in crystal lattices from one another, the magnetic order can be reduced to two dimensions, shown in figure 3. The highly studied two-dimensional kagome lattice can be described as a pattern of corner sharing triangles and the triangular lattice is a tessellation of edge sharing triangles with magnetic cations sitting on the vertices. The kagome lattice Dy₃Mg₂Sb₃O₁₄ has also shown evidence of emergent charge order (ECO) [17], which displays long-range magnetic order, on average, not present on the local scale due to frustration (figure 2b). This has been described as magnetic fragmentation i.e. the system can be thought of as in two states: a divergence free state (which gives rise to Bragg scattering) and a divergence full state (which is the origin to the diffuse scattering and the kagome and pyrochlore pinch points). The unequal in/out arrangement of spins on the triangle create magnetic dipoles, which can be considered analogous to electric charges. Like electric charges, the magnetic charges repel and attract opposites, and so are forced to arrange into the lowest possible energy state [17].

Figure 3.

Two two-dimensional geometrically frustrated crystal structures. (a) The triangular lattice-edge sharing triangles. (b) The kagome lattice-corner sharing triangles. (Online version in colour.)

For the two-dimensional case, the triangular lattice is the simplest case which can accommodate frustration, one such example is the yavapaiite mineral with the general structure AM(SO₄)₃. The alkali metal A cation layers isolate the layers of magnetic ions, so the ground state magnetic order is constrained to two dimensions. Magnetic order CsFe(SO₄)₂ and RbFe(SO₄)₂ relieves frustration through antiferromagnetically coupled spins rotated by 120° in the ab plane [18], which is only possible in this case because the spins are Heisenberg-like [19]. An Ising system, with a single easy axis, cannot relieve frustration through this method. Long-range order is suppressed due to competing interactions, but unlike a paramagnet the spins are strongly correlated with their immediate neighbouring spins [20].

The kagome lattice is well studied, for its high degree of frustration, its potential as a quantum spin-liquid (QSL) state [21] and the variety of minerals whose transition metal ions form such a lattice. QSLs form a novel class of matter where despite strong interactions between neighbouring spins, the system does not form any long-range order, due to (zero-point) quantum fluctuations [22]. The first experimentally observed QSL candidate was in the mineral herbertsmithite in 2006 [23]. Interest has re-emerged in QSLs since the discovery of Majorana fermions in the QSL α-RuCl₃ [24]—a type of particle that is its own anti-particle and does not obey the Dirac equations. First predicted in 1937 [25] but only recently experimentally observed, it has opened up new horizons in condensed matter physics. QSL candidates can also be found in the triangular lattice YbMgGaO₄, consisting of highly disordered correlated Ising spins [26].

(b). One-dimensional magnetism

When the exchange interactions between magnetic ions is restricted to one dimension the material behaves as a one-dimensional system. Such systems have attracted great interest for the potential to discover new phases of matter, attracting interest for quantum information transfer [27] and spin quasi-particles [28]. Long-range one-dimensional magnetic order cannot exist in an isolated system, with the exception of an Ising system exclusively at 0 K due to a spin gap [29]. For an isolated one-dimensional Ising ferromagnet, with a ground state of −nJ/2, where n (the number of magnetic ions in the chain) is very large, a defect in the chain has an energy cost of J/2. However, the entropy gain is equal to κ_blnn, thereby any defect introduced into the system by magnetic or temperature fluctuations [30], will induce a response throughout the chain.

Examples of low-dimensional magnetism do exist, but it is often difficult in mineral and oxide systems to properly isolate magnetic chains from interacting with each other. While this is far from an exhaustive list [31], some examples of this include the AFeX₃ family of compounds (where A = Rb or Cs and X = Cl or Br) consisting of stacked triangular lattices. These systems behave as Heisenberg quasi-one-dimensional spin chains, and in the case of CsFeCl₃ does not show any long-range three-dimensional order down to 0.8 K [32]. If chains are poorly isolated as is the case for RbFeCl₃, the material will transition to a three-dimensional ordered state [32].

In the extended perovskite ABX₃ family, other examples of quasi-one-dimensional magnetic compounds include CsNiF₃ and CsCuCl₃, with the latter being extensively studied. Both have their own associated fascinating ground state physics, but still consist of stronger Heisenberg ferromagnetic correlations in one dimension with weaker antiferromagnetic exchange between the ferromagnetic chains [33,34]. An interesting result in one-dimensional magnetism can be found in the compounds Sr₃CuIrO₆ and Sr₃CuPtO₆, which are one-dimensional Heisenberg ferromagnetic and antiferromagnetic, respectively. Mixing of the two materials results in a random ferro/antiferromagnetic paramagnetic state, described as a quantum spin chain paramagnet [35].

More fascinating examples of quasi-one-dimensional systems are the Ising one-dimensional systems. Ca₃Co₂O₆ has been extensively studied experimentally and theoretically as it undergoes a transition to one-dimensional magnetic state in zero field and contains unusual magnetic properties [36,37]. It has one-dimensional ferromagnetic intrachain correlations and weaker interchain antiferromagnetic interactions, and at low temperatures orders into longitudinal amplitude-modulated spin-density waves propagating along the c-axis. α-CoV₂O₆, a one-dimensional Ising ferromagnet [38], undergoes transitions into different phases under variable applied fields. Studies of this material also showed that it was an excellent magnetocaloric, within its operating temperature, with a steep magnetization curve as a result of overcoming the antiferromagnetic interchain correlations into a ferromagnetic state. It should be noted that these systems show quasi-one-dimensional Ising-like behaviour only because they are not truly isolated one-dimensional chains, and exhibit three-dimensional magnetic order.

For highly frustrated or low-dimensional materials, with only short-range order, the correlated disorder may materialize as structured diffuse scattering [20]. With modern advances in understanding of magnetic systems and new reverse Monte Carlo (RMC) techniques, it is now possible to probe the nature of this diffuse scattering from powder neutron diffraction experiments [39,40].

(c). Exotic magnetism in metal-organic frameworks

All examples discussed so far are dense oxides and minerals, but the greater flexibility in design of the size, shape and coordination of organic linkers in MOFs allow for more complex magnetic structure and the ability to tailor desired magnetic properties. Forming one- or two-dimensional magnetic systems in oxides is not trivial, exemplified by the scarce physical realizations of theoretical models in these exotic states of matter. MOFs are a good alternative because they can have longer distances between magnetic centres, and therefore can better isolate the magnetic sheets or chains from each other. However, these low-dimensional units can remain strongly coupled through oxygen atoms. It is possible to modify the ligand to achieve desired exchange interactions, making MOFs an ideal model systems for the study of new physics. Until recently, the field of magnetic MOFs have been largely overlooked and many studies remain limited to bulk magnetic property measurements. There are far fewer detailed studies of the microscopic magnetic phases in MOFs, which typically require neutron-based techniques to characterize in detail [41].

Examples of exotic magnetism in MOFs are present in the literature including chiral, two- and one-dimensional magnetic structures, in addition to long-range magnetically ordered systems [41]. Frustrated magnetism has been experimentally studied in magnetic MOFs in materials such as Co₃(OH)₂(sq)₂ · 3H₂O and M(tca)₂ (tca = tricyanomide) using neutron scattering, but these materials undergo phase transitions to three-dimensional long-range ordered structures at low temperature. Co₃(OH)₂(sq)₂ · 3H₂O has significant frustration within its CoO₆ ribbons, and unusually undergoes a phase transition upon dehydration [42]. In Mn(tca) strong frustration is observed and is relieved by the formation of an incommensurate three-dimensional structure [43].

Tb(DCO₂)₃ is an excellent example of a magnetic MOF with interesting properties at low temperatures and a physical realization of a stacked triangular lattice, forming chains of Tb ions (figure 4). The material perfectly combines frustration and one-dimensional magnetic order [44], making it the ideal model for the triangular lattice with equivalent J coupling between Tb ions, through the formate ligand. The MOF examples discussed so far all have three-dimensional ordered ground states but, for this Ising antiferromagnetic compound, the situation appears to be different. Above 1.6 K, it shows strong magnetic diffuse scattering as a result of short-range order emerging from ferromagnetically correlated chains, coupled through frustrated antiferromagnetic interactions on the triangular lattice. Below 1.6 K, it displays long-range Ising-like one-dimensional magnetic order along the c-axis with spins aligned in the chain direction, but shows no long-range order in the ab plane. The lack of long-range three-dimensional order appears to be suppressed by the frustrated interactions between chains leading to the emergence of a triangular Ising antiferromagnetic (TIA) state, with a large number of degenerate ground states [44]. The ferromagnetic correlations and antiferromagnetic frustration that persists, in the paramagnetic state, allow for high entropy changes in small applied magnetic fields contributing to a high efficiency as a magnetocaloric [9]. This dominant ferromagnetic intrachain coupling allow the moments to be more easily aligned with the applied field. The additional weaker frustrated antiferromagnetic interchain interactions lead to suppression of magnetic order, in the absence of a field, required for paramagnetic magnetocalorics. This study aims to expand on the studies of Tb(DCO₂)₃ thus far, exploring the magnetic structure between 0.28 and 1.95 K and physical property measurements to lower temperatures to establish how the TIA phase is modified. The other compounds in the Ln(DCO₂)₃ series are also probed by neutron diffraction for diffuse scattering, indicative of short-range magnetic correlations.

Figure 4.

Tb(HCO₂)₃ crystal structure. Tb³⁺ coordination environments are shown as purple polyhedra, and carbon, oxygen and hydrogen are shown in black, red and white, respectively. (Online version in colour.)

2. Experimental methods

Ln(HCO₂)₃ and Ln(DCO₂)₃ (where Ln = Ce, Pr, Nd, Tb-Er) samples were synthesized by slowly adding 2 g of Ln(NO₃)₃ · 6H₂O (99.9%, Sigma-Aldrich) to a solution of 4.75 ml of formic acid (97.5% Sigma-Aldrich), or d-formic acid (95.0% Sigma-Aldrich), respectively, with 0.25 ml of ethanol added. After several minutes of stirring, NO_x was released and the product precipitated out of solution. The products were collected by vacuum filtration, washed several times with ethanol and dried in a desiccator. It is necessary to deuterate neutron samples to minimize background noise caused by the incoherent scattering of hydrogen.

Powder neutron diffraction measurements were carried out on the high-resolution time-of-flight WISH diffractometer at the ISIS neutron source, Rutherford Appleton Laboratory [45]. Tb(DCO₂)₃ measurements were carried out between 0.28 K and 1.95 K, with the sample cooled using a³He Heliox sorption refrigerator. Ln(DCO₂)₃ (where Ln = Ce, Pr, Nd, Dy, Ho and Er) measurements were carried out between 1.6 K and 100 K, with the samples cooled using the standard Oxford Instruments WISH cryostat. Average structure neutron patterns were fitted in FULLPROF [46] and JANA2006 [47] using the Rietveld method [48]. The base temperature solution was determined using JANA2006 and the evolution of the magnetic moments with respect to temperature was fitted in FULLPROF. Diffuse neutron patterns were fitted with the reverse Monte Carlo program—SPINVERT [40], using a supercell of 52 × 54 × 55 ų. Heat capacity measurements were measured on a Quantum Design PPMS DynaCool with 14 T superconducting magnet and³He insert, between 400 mK and 14 K. The 400 mK–30 K magnetic measurements of the polycrystalline samples were performed using a Quantum Design MPMS SQUID magnetometer, with a³He insert in a 100 Oe DC magnetic field. For additional details, see the electronic supplementary material, appendix S1.

3. Results and discussion

(a). Magnetic properties of Tb(HCO₂)₃ framework

It has previously been reported that the Ln(HCO₂)₃ frameworks do not show any indication of long-range order down to 2 K, and follow Curie–Weiss behaviour, with Weiss temperatures of −0.6 K, −0.9 K, −6.1 K, −10.3 K and −16.0 K, for Ln = Gd - Er [9]. Since neutron diffraction measurements indicated the emergence of magnetic order in Tb(HCO₂)₃ at 1.6 K, we have probed the physical properties down to 0.4 K to identify the effect this order has on its physical properties. Zero-field cooled magnetic susceptibility data for Tb(HCO₂)₃ shows indications of magnetic transitions at ≈1.6 K and ≈0.5 K (figure 5). Heat capacity measurements shown in figure 5b and electronic supplementary material, figure S1 show a large peak in 1.68 K for zero applied field, this is suppressed in applied fields consistent with the onset of a magnetic transition. Despite the small change in susceptibility, associated with the TIA transition, the signal in heat capacity is much more significant. Both of these features are consistent with the emergence of the TIA, as a transition to a long-range one-dimensional order would result in a large entropy change, while the strong short-range antiferromagnetic coupling of the chains would prevent any great change in magnetic susceptibility. It is also possible that the signal at 1.6 K is an experimental artefact from the MPMS switching between high- and low-temperature modes. A clearer transition can be seen at ≈0.5 K indicating more complete, possibly long-range antiferromagnetic order [9,44]. A second large change in the heat capacity is observed below 600 mK, this may be the onset of long-range order but could also be the onset of a hyperfine Schottky anomaly.

Figure 5.

Magnetic measurements of Tb(HCO₂)₃. (a) Magnetic susceptibility in 100 Oe field below 30 K, close up below 1.6 K in insert and (b) C_mag/T in variable fields between 400 mK and 14 K. (Online version in colour.)

(b). Magneto-structural characterization of Ln(DCO₂)₃ frameworks

The interesting physical and magnetic measurements encouraged further investigation, so we have therefore performed neutron diffraction measurements in zero-field of the Ln(DCO₂)₃ frameworks. Rietveld fits to neutron diffraction data confirmed that all Ln(DCO₂)₃ samples used in this study were phase pure and did not exhibit significant exchange of D for H. They were all found to adopt the known rhombohedral R3m structure under ambient conditions (see electronic supplementary material, figures S2–S25) [9]. In this structure, the LnO₉ polyhedra are connected into face-sharing chains down the c-axis with these chains interconnected via the formate ligand into a triangular arrangement (figure 4). There are two different oxygen atoms in the structure, bonding to one and two Ln³⁺ cations, respectively, with the latter atom connecting the Ln³⁺ cations into chains. The triangle connections of the Ln³⁺ cations are not equilateral but instead feature two short bonds and one long bond, which rotate between adjacent layers along the c-axis.

(i). Tb(DCO₂)₃.

Upon cooling below 1.6 K the neutron diffraction measurement performed on Tb(DCO₂)₃ show the appearance of extra reflections ascribable to the development of long-range magnetic ordering. This can be appreciated from the broad peak observed around 9 Å (figure 6b). The extra reflections can be indexed with a propagation vector k₁ = [0, 0, 1] belonging to the LD line (00γ) of the first Brillouin zone (BZ). The quantitative Rietveld refinement agrees with the P3m^′1 magnetic space group corresponding to the mLD2LE2 irreducible representation with order parameter (ξ₁, ξ₂) as previously reported [9].

Figure 6.

Rietveld fits to neutron diffraction patterns of Tb(DCO₂)₃ at variable temperatures from bank 2 and 9 of WISH along with the fitting statistics (a) 1.7 K, R_p = 6.45%, R_wp = 5.27% (b) 1.2 K, R_p = 6.51%, R_wp = 5.44% and (c) 0.28 K, R_p = 6.58%, R_wp = 5.41%. (Online version in colour.)

It is worth underlining that the observed propagation vector does not correspond to a special point of the BZ and the γ-value can, in principle, be incommensurate. The broad nature of the diffraction peaks does not allow us to determine with absolute certainty that the propagation vector is locked to the commensurate [0,0,1] value. Nevertheless, the comparison with similar systems in the literature and the good refinement of the diffraction data with the commensurate value led us to conclude that the real propagation vector is likely to be γ = 1, as previously reported [9]. The spin ordering can be described by two different global phases. One of these resembles a partially disordered antiferromagnet (PDA) in which each of the triangles has one chain ordered up, one down and the third remaining disordered (π/4). The other up–down–down model contains one TbO₉ chain ordered up, and two down with the latter having half the magnetic moment of the former (π/6) [9,49]. The two descriptions for the magnetic structure cannot be distinguished between, as they only vary in the phase (π/4 and π/6), which diffraction is not sensitive to. As reported previously, both of these can thought of as conventional crystallographic approximations of the TIA, which only has long-range one-dimensional order in the chains [44].

Patterns collected below 1.2 K revealed the presence of additional diffuse reflections at high d-space, the most significant Bragg-like peak can be seen at ≈ 10.5 Å. The broad nature of these features, comparable to those associated with the TIA phase and the retention of the diffuse suggest these are associated with a state that lacks conventional crystallographic order. We have, however, treated them with conventional crystallographic approaches, as in the absence of clearly established prediction of the scattering, we cannot completely exclude that the peak broadening is not a result of small magnetic domains. The peak width was fit with an anisotropic peak broadening model, and the broadening is an indication of a finite correlation length. When fit with a Warren-like peak shape function [50] the correlation length, considered to be associated with correlations in the ab plane, was found to be ≈ 52 Å.

We observed no reduction in the width of the magnetic peaks upon cooling below 1.6 K indicating the magnetic correlation length remains unchanged. These extra reflections can be indexed to a supercell violating the parent rhombohedral symmetry and doubled along the b-axis with a k₂-vector = [0, 0.5, 1] corresponding to the mF2 mode. These new reflections here have been interpreted as an additional component of the magnetic structure, but the possibility of magnetic phase separation in the material cannot be completely excluded. The lack of intensity on the satellites of the 00l reflections indicate that the moment remains along the c-axis only, in agreement with the Ising character of the Tb spins.

Using ISODISTORT [51], we have determined the commensurate structure can be described in the Pm^′ space group in which there are six independent Tb sites with moments dependent on the global phase, with the lattice parameters a = 3.96860(13) Å, b = 10.42077(15) Å, c = 18.0493(3) Å as shown in figure 7. Since the structure factors are insensitive to the choice of the global phase, it is not possible to distinguish between the two structures from the neutron diffraction alone.

Figure 7.

Two possible magnetic average structure solutions for Tb(DCO₂)₃ at 0.28 K. Unit cell shown as black box with magnetic moments in Pm^′ space group. The spin charges are shown as minus and plus signs. J₂ and J₃ are shown next to the corresponding interaction and J₁ not shown, goes into the plane. On the right of each figure is shown the projection of the structure in the ab plane and on the left of each figure the bc plane. Magnetic moments colour coded, but not to scale. (a) Global phase = π/4 and (b) global phase = π/6. (Online version in colour.)

The solution with global phase π/4 results in a modification of the PDA structure in which the k₂ propagation modulates the moment in the chains resulting in couples of large moment Tb chains ( ± 7.68(3) μ_B and ±6.19(3) μ_B) antiferromagnetically coupled separated by chain with a small moment ( ± 0.74(3) μ_B). For the high temperature magnetic structure, this phase choice gives a PDA solution closest to the TIA model proposed [44], and therefore is our preferred model. The π/6 solution is a slight modification of the up (1/2)down (1/2)down structure on a triangular lattice in which the moments are modulated along the b-axis of the parent structure [49]. This phase choice gives the most symmetric solution with the least spread of the Tb moment size with refined lengths of 4.75(3), 7.27(3), 8.75(4) and 3.26(3) μ_B.

Both solutions find the moments on each triangle sum to be ±0.75 μ_B, which arrange into an emergent antiferromagnetic-like state, absent from nearest neighbour interactions alone. This is analogous to the ECO observed in the kagome ECO states [17,52], which has two observed experimental signatures; non-zero entropy and the presence of magnetic diffuse and Bragg-like peaks in the neutron diffraction measurements [53]. Such exotic magnetic states are elusive and have so far only been reported in oxide materials. Based on observing similar features, this may be the first example of such a state present in a MOF. The emergence of antiferromagnetic order, associated with k₂, is consistent with the observation of antiferromagnetic-like transitions in the physical property data. The emergent order present suggests that the one-dimensional order of the TIA phase is retained in this lower temperature phase.

The evolution of the magnetic moments, with respect to temperature, associated with the two k-vectors is shown in figure 8. While the high temperature transition is consistent with the physical property measurements we attribute the low-temperature transition discrepancy to temperature equilibration issues, in particular that the physical property measurements were measured on cooling and neutron diffraction patterns on warming.

Figure 8.

Evolution of the ordered magnetic moments associated with k₁ and k₂, in Tb(DCO₂)₃, with respect to temperature. The left- and right-hand y-axes indicate the magnetic moment associated with k₁ and k₂, respectively. (Online version in colour.)

(ii). Ho(DCO₂)₃.

In all the samples studied, excluding Tb(DCO₂)₃, the only material which shows evidence of structured magnetic scattering is Ho(DCO₂)₃. Ho(DCO₂)₃ begins to exhibit diffuse scattering below 10 K, while remaining paramagnetic to 1.6 K, see the electronic supplementary material, figures S2–S25 for neutron diffraction patterns. The other samples did not show any magnetic diffuse scattering above 1.5 K. The diffuse in Ho(DCO₂)₃ has been fit well with SPINVERT, and the stereographic projections of the refined spin orientations indicate that the spins are preferentially aligned along the c-axis (figure 9a), consistent with large spin–orbit coupling and the Ising-like magnetic anisotropy, seen in Tb(DCO₂)₃ [9]. Attempts were therefore made to fit the diffuse scattering data with Ising spins constrained to point along the c-axis. The Ising refinements also yielded a reasonable fit (figure 9a), though of somewhat lower quality than the Heisenberg-like refinement (χ² = 2.53 and 8.29), expected as the Ising refinement is more highly constrained compared to Heisenberg fits. This result suggests that the spins have a strong, although probably not purely, Ising character. These results are consistent with observed Ho³⁺ anisotropy seen in the spin-ice materials Ho₂Ti₂O₇ and Ho₂Sn₂O₇ [54]. We would therefore expect to see a gap in the inelastic neutron spectra of Ho(DCO₂)₃ and Tb(DCO₂)₃, between the ground and first excited state.

Figure 9.

(a) RMC fit to magnetic diffuse with an Ising model, data points in black, fit in red and the difference in blue, and stereographic projections in insert. Stereographic projections of the spin orientations averaged over 100 RMC Heisenberg-like fits to diffuse neutron scattering data from Ho(DCO₂)₃ at 1.6 K. The relative spin density, ρ(θ, ϕ), is defined as ρ(θ, ϕ) = ln[(ρ(θ, ϕ)/Nd(cosθ) dϕ)] and (b) correlation lengths at various temperature with spin correlations and fit of Ho(DCO₂)₃ at 1.6 K in insert with chain correlations highlighted in black. (Online version in colour.)

Spin correlations 〈S₀ · S_r〉, averaged over 100 RMC refinements show that dominant spin correlations in this material are ferromagnetic along the chain direction, with weaker antiferromagnetic interchain interactions. We have extracted correlation lengths with the function 〈S₀ · S_r〉 = A_exp( − r/ϵ), where A is the Ising-like anisotropy, r is the correlation distance and ϵ is the correlation length. As the temperature was raised and the correlations became significantly weaker, the best fit to data was found with un-physical A values, for this reason A was fixed to the value −0.72—determined for the lowest temperature fit. For correlation fits, see electronic supplementary material, figure S26.

At 1.6 K, the ferromagnetic chain correlation length is 7.3(6) Å with weaker antiferromagnetic inter-chain correlations of −0.018(2) and −0.034(2) observed for Ho³⁺ cations separated by 6.10 and 6.57 Å, found in triangles, respectively. This indicates that the correlations in Ho(DCO₂)₃ are weaker than Tb(DCO₂)₃ for a given temperature [44]. In the Ln(DCO₂)₃ series Tb(DCO₂)₃ and Ho(HCO₂)₃ were both noted to have their magnetocaloric properties optimized for use above 4 K, these are also the only two compounds to show structured diffuse scattering above 1.6 K. Ho(HCO₂)₃ may be a worse magnetocaloric than Tb(HCO₂)₃ due to its weaker ferromagnetic correlations, needing greater changes in magnetic fields to align spins with the field. The decrease in correlation lengths in Ho(DCO₂)₃ follows the same exponential law seen in the Tb(DCO₂)₃ framework and the relative strength of the ferromagnetic intrachain and anitferromagnetic interchain correlations in the two compounds is the same. We therefore suggest at sufficiently low temperatures, it is likely that the Ho(DCO₂)₃ framework will feature similar exotic magnetic states to that of Tb(DCO₂)₃.

4. Conclusion

This work reports the physical properties and lower temperature magnetic structure of Tb(DCO₂)₃, and the magnetic correlations of Ho(DCO₂)₃ in the paramagnetic state. The magnetic phase observed in Tb(DCO₂)₃ below 1.2 K appears to display emergent order, when interpreted via conventional crystallographic approaches, in the long-range structure. This result is robust to the two possible magnetic structures which vary only in the phase of their scattering and as such it may be the first example of such a state in an MOF. The broadness of the Bragg-like peaks and retention of the magnetic diffuse scattering is consistent with the retention of one-dimensional order in the TIA phase of Tb(DCO₂)₃ in this lower temperature phase. Ho(DCO₂)₃ shows significant magnetic diffuse scattering at 1.6 K, indicating the inherent magnetic frustration in this system. Fits to this suggest that it has similar magnetic correlations to that seen in the paramagnetic phase of Tb(DCO₂)₃ which suggests it may exhibit similar exotic phases at lower temperatures. The stronger one-dimensional ferromagnetic correlations in Tb(DCO₂)₃ likely explain why this is the most efficient magnetocaloric material of the two compounds. This highlights the great potential for exotic magnetic states to emerge in MOFs, including new emergent phases.

Data accessibility

Electronic supplementary material is available online and neutron data can be downloaded from doi:10.5286/ISIS.E.RB1710081.

Author's contributions

R.J.C.D. carried out data analysis, and measurements, and drafted the manuscript. F.O. and P.M. provided assistance with Rietveld refinements and the neutron diffraction measurements. P.M. and S.E.D. collected physical property data, and assisted with analysis. P.J.S. conceived and designed the study and drafted the manuscript. All authors read and approved the manuscript.

Competing interests

We declare we have no competing interests.

Funding

The authors thank the Science and Technologies Facilities Council for access to the ISIS facility at Harwell. R.J.C.D. would like to acknowledge the University of Kent for financial support through the provision of a Vice-Chancellors scholarship. P.J.S. would like to thank the Leverhulme Trust for funding via RPG-2018-268. P.M. and S.E.D. acknowledge funding from the Winton Programme for the Physics of Sustainability. Low temperature MPMS measurements were carried out on the EPSRC Advanced Materials Characterization Suite (EP1M0052411).