# Geometric frustration on the trillium lattice in a magnetic metal–organic
framework

Johnathan M. Bulled  Inorganic Chemistry Laboratory, University of Oxford,
South Parks Road, Oxford OX1 3QR, U.K.  Joseph A. M. Paddison  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, U.S.A.  Churchill College, University of Cambridge, Storey’s Way,
Cambridge CB3 0DS, U.K.  Andrew Wildes  Institut Laue-Langevin, BP156, 71
Avenue des Martyrs, 38000, Grenoble, France  Elsa Lhotel  Institut Néel, 25
Avenue des Martyrs, 38042 Grenoble, France  Simon J. Cassidy  Inorganic
Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR,
U.K.  Breogán Pato-Doldán  Department of Chemistry, University of Bergen, P.O.
Box 7803, N-5020 Bergen, Norway  L. Claudia Gómez-Aguirre  Department of
Fundamental Chemistry and CICA, Faculty of Sciences University of A Coruña,
15071 A Coruña, Spain  Paul J. Saines  School of Physical Sciences, University
of Kent, Canterbury CT2 7NH, U.K.  Andrew L. Goodwin  Inorganic Chemistry
Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR, U.K.

###### Abstract

In the dense metal-organic framework Na[Mn(HCOO)3], Mn2+ ions
($S=\frac{5}{2}$) occupy the nodes of a ‘trillium’ net. We show that the
system is strongly magnetically frustrated: the Néel transition is suppressed
well below the characteristic magnetic interaction strength; short-range
magnetic order persists far above the Néel temperature; and the magnetic
susceptibility exhibits a pseudo-plateau at $\frac{1}{3}$-saturation
magnetisation. A simple model of nearest-neighbour Heisenberg
antiferromagnetic and dipolar interactions accounts quantitatively for all
observations, including an unusual 2-$\mathbf{k}$ magnetic ground-state. We
show that the relative strength of dipolar interactions is crucial to
selecting this particular ground-state. Geometric frustration within the
classical spin liquid regime gives rise to a large magnetocaloric response at
low applied fields that is degraded in powder samples as a consequence of the
anisotropy of dipolar interactions.

Geometrically-frustrated magnets are of fundamental interest because their
macroscopic ground state degeneracies give rise to a number of exotic effects
Moessner_2006 ; Savary_2016 ; Han_2012 ; Starykh_2015 . The suppression of
magnetic ordering to temperatures far below the dominant magnetic interaction
energy scale means that small perturbations can have a profound effect on both
the magnetic ground state and its excitations Moessner_2006 . For obvious
reasons, a common motif in the interaction network of frustrated materials is
the presence of odd-membered rings—especially triangles. Yet despite the large
number of relevant topologies, much of the field has focused on the relatively
small set of structure types that are common amongst ceramic materials; _e.g._
, the pyrochlore, triangular, kagome, and face-centred cubic nets. The study
of frustration on less common lattices may therefore allow for the discovery
of novel magnetic phases and their corresponding physics Paddison_2015 ;
Gao_2007 ; Kikuchi_2015 .

The chemistry of metal–organic frameworks (MOFs) allows rational design of
network structures that can be difficult to realise in conventional ceramics
Zhou_2012 . MOFs are comprised of inorganic ‘nodes’ and organic ‘linkers’; by
choosing the connectivity and geometry of each component, the chemist has
remarkable control over the topology of the resulting network structure
Yaghi_2003 . Likewise, the particular combination of metal and ligand employed
then governs the relevant magnetic degrees of freedom, and the strength and
anisotropy of their interactions Saines_2018 ; Thoraninsdottir_2020 .

Figure 1:  (a) The trillium lattice, shown here as a $2\times 2\times 2$
supercell of the corresponding cubic unit cell Hopkinson_2005 . Each site has
six neighbours, arranged at the vertices of a three-bladed propellor (_cf_.
the $\beta$-Mn net OKeeffe_1977 ). (b) Representation of the crystal structure
of Na[Mn(HCOO)3]; Na, C, H and O atoms shown as green, black, white and red
spheres, respectively, and Mn coordination environments shown as filled
polyhedra ParedesGarcia_2009 . The Mn atoms decorate a trillium lattice.

The trillium lattice is an obvious target for applying MOF chemistry to
frustrated magnetism: it is a chiral network of corner-sharing triangles that
is intrinsically predisposed to geometric frustration [Fig. 1(a)]
Hopkinson_2005 . Remarkably few magnets are known to adopt this lattice, but
those that do—mostly intermetallics—exhibit a wide range of interesting
physics. In MnSi, for example, magnetic skyrmions arise from competition
between ferromagnetism and the Dzyalloshinskii-Moriya interaction
Muhlbauer_2009 ; Hopkinson_2007 ; EuPtSi and EuPtGe are strongly-correlated
spin liquids Franco_2017 ; and CeIrSi is an Ising trillium spin-ice candidate
Redpath_2010 ; Kneidinger_2019 . Theory tells us that even the simple nearest-
neighbour Heisenberg antiferromagnet (nnHAF) on this lattice should support a
classical spin-liquid (CSL) phase over a wide temperature range Hopkinson_2005
, with 120∘ helical order (“Y” phase; $\mathbf{k}_{\rm
ord}=[\frac{1}{3},0,0]$) in the frustrated magnetic ground state Isakov_2008 .
To best of our knowledge, there has not been any experimental realisation of
this model system.

It was in this context that we developed an interest in sodium manganese(II)
formate, Na[Mn(HCOO)3]: a dense MOF in which magnetic Mn2+ ($S=\frac{5}{2}$)
ions decorate a trillium lattice [Fig. 1(b)] ParedesGarcia_2009 . The system
crystallises in the chiral cubic space group $P2_{1}3$ and has a nearest
neighbour Mn–Mn distance of $\sim$5.6 Å. Formate ions connect neighbouring
Mn2+ cations to give the Mn–O–C–O–Mn pathway that supports magnetic
superexchange Inoue_1970 ; Scatena_2020 ; Wang_2004 ; Walker_2017 . Bulk
thermodynamic measurements have shown the system to exhibit Curie-Weiss
behaviour—thus no sign of magnetic ordering—for $T>2$ K, with an
antiferromagnetic coupling strength $J\sim 1$ K Aston_2017 .

In this Letter, we show that Na[Mn(HCOO)3] exhibits three hallmarks of
geometric frustration: (i) the suppression of its Néel temperature $T_{\rm N}$
far below the characteristic energy scale of its magnetic interactions; (ii)
the strongly-correlated disorder indicative of a CSL phase; and (iii) a
magnetisation pseudo-plateau indicative of complex field-dependent behaviour.
The magnetic ground-state is different to that expected for the trillium nnHAF
model, but can be rationalised by accounting for the weaker but ever-present
dipolar interactions (‘nnHAF+D’). We show that the resulting nnHAF+D model:
(i) accounts quantitatively for the magnetic susceptibility above $T_{\rm N}$
using an interaction strength $J$ that is consistent with $T_{\rm N}$; (ii)
correctly predicts both the short-range magnetic correlations of the CSL phase
and the magnetic ground-state, as observed in neutron-scattering measurements;
and (iii) can be used to rationalise the existence and temperature-dependence
of the measured magnetisation pseudo-plateau. A characteristic magnetocaloric
effect associated with this pseudo-plateau emerges.

We begin by reporting the low-field magnetic behaviour of Na[Mn(HCOO)3],
expanding on the earlier measurements of Ref. 25. We prepared and
characterised a sample ($\sim 0.4$ g) of Na[Mn(HCOO)3] as described in si .
The low-field magnetic susceptibility, measured for a 10 mg fraction of this
sample, is shown as its inverse in Fig. 2(a). For $T\gtrsim 2$ K, the magnetic
behaviour is Curie-Weiss-like and antiferromagnetic; the corresponding fit
over the range $2.0<T<4.5$ K gives $\Theta_{\rm CW}=-2.3(3)$ K, which is
consistent with the value obtained in Aston_2017 . A Néel transition—seen here
for the first time for this system—is observed at $T_{\rm N}=0.223(18)$ K. The
corresponding frustration parameter $f=|\Theta_{\rm CW}|/T_{\rm N}\sim 10$
indicates strong frustration Savary_2016 .

Figure 2:  (a) Temperature dependence of the inverse magnetic susceptibility
of Na[Mn(HCOO)3]. Data shown as black solid circles; Curie-Weiss fit as a
black line; and computed values for the nnHAF+D model as a red line. The inset
highlights the region near $T_{\rm N}$. (b) The magnetic diffuse neutron
scattering functions measured at 20 and 1.5 K; open and filled circles denote
data collected using neutron wavelengths of 3.1565 and 4.873 Å, respectively.
The 20 K values have been shifted vertically by one unit, and the Mn2+
magnetic form factor is shown as a dashed line. (c) Magnetic ordering in the
nnHAF+D model from MC simulations. The degree of red/blue colouring is taken
from the strength of magnetic scattering associated with $\mathbf{q}_{\rm
ord}=\langle\frac{1}{2},0,0\rangle$ and $\langle\frac{1}{3},0,0\rangle$,
respectively. Experimental $T_{\rm N}$ and $D/J$ values are shown as vertical
and horizontal lines. (d) The experimental magnetic neutron scattering
function at 100 mK (black markers) and that calculated directly from nnHAF+D
simulations at $D/J=0.119$ (red line). Blue and red tick-marks denote the
allowed magnetic reflection positions for the Y and 2-$\mathbf{k}$ phases,
respectively. (e,f) Representations of the (e) Y and (f) 2-$\mathbf{k}$
magnetic structures.

Evidence for strong magnetic correlations within the CSL regime comes from the
presence of structured magnetic diffuse scattering in polarised neutron
scattering measurements. We used the D7 instrument at the ILL Stewart_2009 ;
dois , operating in XYZ polarisation mode, to separate the magnetic scattering
for a powder sample of Na[Mn(HCOO)3] from the nuclear and spin-incoherent
components. Our measurements at 20 K ($T\gg J$) and 1.5 K ($T\sim J$) are
shown in Fig. 2(b), where we have placed the scattering intensities on an
absolute scale by Rietveld refinement of the nuclear scattering. At 20 K, the
magnetic scattering function $S_{\rm mag}(Q)$ is featureless and follows
closely the Mn2+ magnetic form factor Brown_2004 . By contrast, within the CSL
regime at 1.5 K we observe clear oscillations in $S_{\rm mag}(Q)$
characteristic of short-range magnetic order.

The magnetic scattering at 100 mK ($T<T_{\rm N}\ll J$) reflects long-range
magnetic order, but the magnetic Bragg diffraction pattern cannot be indexed
in terms of the $\mathbf{k}_{\rm ord}=[\frac{1}{3},0,0]$ ordering vector
expected from theory [Fig. 2(d,e)] Hopkinson_2005 ; Isakov_2008 . Hence the
physics governing selection of this magnetic ground-state must be more complex
than accounted for by the simple nnHAF model of Refs. 13; 19.

An obvious omission of this model is consideration of the always-present
magnetic dipolar interactions; note that no significant single-ion anisotropy
is expected for the high-spin Mn2+ ($d^{5}$) configuration in an octahedral
crystal field. Using the crystallographic Mn$\ldots$Mn separation and taking
$S=\frac{5}{2}$ for Mn2+, the relevant dipolar energy scale in Na[Mn(HCOO)3]
is $D=0.118$ K. Since $D\ll J$, one might ordinarily have expected the nnHAF
ground-state to be robust to this additional interaction. Yet, in the absence
of the corresponding theory (to the best of our knowledge), it fell to us to
test this assertion by introducing dipolar interactions into the nnHAF model.

In this spirit, we carried out a series of classical Monte Carlo (MC)
simulations related to those of Ref. 19 but with MC energies now calculated
using the nnHAF+D expression

$E_{\rm MC}=J\sum_{\langle
i>j\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+D\sum_{i>j}\frac{\mathbf{S}_{i}\cdot\mathbf{S}_{j}+3(\mathbf{S}_{i}\cdot\hat{\mathbf{r}}_{ij})(\mathbf{S}_{j}\cdot\hat{\mathbf{r}}_{ij})}{(r_{ij}/r_{1})^{3}}.$
(1)

Here, $\mathbf{S}_{i}$ represents the vector spin orientation at site $i$,
$\hat{\mathbf{r}}_{ij}$ is the unit-vector pointing from spin $i$ to spin $j$,
and $r_{1}$ is the nearest-neighbour separation. The first sum is taken over
distinct nearest-neighbour pairs $i,j$. We used Ewald summation as implemented
in Ref. 5 to treat the long-range dipolar interactions. Here, as elsewhere, we
subsume the spin magnitude within the constants $J$ and $D$. Our MC
configurations respresented $6\times 6\times 6$ supercells of the $P2_{1}3$
unit-cell shown in Fig. 1(a); full details of these simulations and our
methods for extracting physical observables are given in si .

The type of magnetic order driven by Eq. (1) at low temperatures turns out to
be surprisingly sensitive to the value of $D/J$ [Fig. 2(c)]. For $D\gtrsim
0.08J$, the magnetic ground-state switches from the helical $\mathbf{k}_{\rm
ord}=[\frac{1}{3},0,0]$ order of the nnHAF model to a 2-$\mathbf{k}$ state
with $\mathbf{k}_{\rm ord}\in\langle\frac{1}{2},0,0\rangle$ [Fig. 2(f)]. This
sensitivity is evident also from mean–field calculations (see si ). The
magnetic Bragg diffraction pattern of this new 2-$\mathbf{k}$ ground-state
shows a remarkable similarity to our 100 mK measurement for Na[Mn(HCOO)3]
[Fig. 2(d)], which suggests that the nnHAF+D model may indeed capture the key
physics at play in this magnetic MOF.

In order to test the effectiveness of the model against all our experimental
measurements, we required an accurate estimate of the Heisenberg exchange
strength $J$. We found that for $D\ll J$, the behaviour of Eq. (1) at $T\geq
J/2$ was essentially indistinguishable to the nnHAF model ($D=0$; see si ).
Consequently, we estimated $J$ by determining the $T\geq J/2$ susceptibility
for the $D=0$ case, and then carrying out a least-squares fit of $J$ to best
match experiment (see si ). In this way we obtain $J=0.96(2)$ K, which is
consistent with the estimate given in Ref. 25. Moreover, the resulting
relative energy scale $D/J=0.119$ locates Na[Mn(HCOO)3] within the regime for
which 2-$\mathbf{k}$ order is expected [Fig. 2(c)].

The results of our nnHAF+D MC simulations carried out using this combination
of $J$ and $D$ are shown as the red solid lines in Fig. 2(a) and (b).
Considering first the magnetic susceptibility data, our simulations capture at
once the Curie-Weiss behaviour for $T\gtrsim 2$ K, the departure from Curie-
Weiss behaviour for $T\lesssim 2$ K, and the Néel transition itself. We find
$T_{\rm N}=0.27(1)$ K, in good agreement with experiment. Likewise for our
magnetic neutron scattering data: the $S_{\rm mag}(Q)$ functions calculated
directly from MC configurations generated at the corresponding temperatures
Blech_1964  follow experiment closely (see si  for the corresponding spin
correlation functions). And, of course, the MC ground-state is a
2-$\mathbf{k}$ structure for which $S_{\rm mag}(Q)$ is consistent with the 100
mK neutron magnetic diffraction pattern [Fig. 2(d)].

So the nnHAF+D Hamiltonian provides a good representation of the spin physics
at play in Na[Mn(HCOO)3]. Perhaps the largest discrepancy between experiment
and calculation is the small difference in $T_{\rm N}$ [see inset to Fig.
2(a)]. This discrepancy may reflect the relevance of yet weaker interactions,
such as single-ion anisotropy (usually safely ignored for high-spin Mn2+)
and/or antisymmetric exchange interactions, as allowed by the chiral crystal
structure. We have not explored either case here further. We do note, however,
that the dipolar interactions of our nnHAF+D model induce an Ising-like spin
anisotropy consistent with the crystallographic point symmetry of the Mn site
(_cf_. Paddison_2015 ; see si ). Consequently, an easy-plane single-ion term
in an extended Hamiltonian may compete with the dipolar interactions, in turn
reducing $T_{\rm N}$.

Given the sensitivity of the nnHAF model to perturbations in the zero-field
limit, we might expect Na[Mn(HCOO)3] to be especially sensitive to an applied
magnetic field. To explore this point we carried out a series of high-field
magnetisation measurements at finely-spaced temperatures within the ordered
and CSL regimes ($200<T<350$ mK) using applied fields $0<\mu_{0}H<1.5$ T. Our
results are summarised in Fig. 3(a), where we plot the susceptibility
$\chi_{\rm m}={\rm d}(M/M_{\rm sat})/{\rm d}\mu_{0}H$ as a function of
$\mu_{0}H$ and $T$. The most obvious feature we observe is a valley in
$\chi_{\rm m}$ centred on $\mu_{0}H\sim 0.6$ T that corresponds to a pseudo-
plateau in the field-dependence of the magnetisation at $M/M_{\rm
sat}=\frac{1}{3}$ [see inset to Fig. 3(a)]. This feature is strongest at low
temperatures, but persists with varying prominence throughout the temperature
range of our measurements. We proceed to show that the origin of this feature
is the suppression of magnetic fluctuations within a new ordered phase that is
stabilised at finite field.

Figure 3:  (a) Experimental field and temperature dependence of the magnetic
susceptibility of Na[Mn(HCOO)3]. The inset shows the field-dependent
magnetisation at 100 mK; experimental data in black and low-field linear fit
in red. Note the pseudo-plateau at $M/M_{\rm sat}\simeq\frac{1}{3}$. This
feature corresponds to the horizontal blue region at $\sim$0.6 T in the main
plot. (b) Magnetic susceptibility from MC simulations, with phase boundaries
shown as solid lines. (c,d) The corresponding field-induced change in magnetic
entropy from (c) experiment and (d) MC simulations, which has a local maximum
within the UUD phase. Within the regime $T_{\rm N}<T<T^{\ast}$, it is possible
to cycle between disordered and ordered states under the application and
removal of a low magnetic field. Hence the cycling regime illustrated by white
circles in panel (d) offers a magnetocaloric cooling strategy.

To help understand this field-dependant behaviour, we explored the effect of
introducing a Zeeman term into the original nnHAF model:

$E=J\sum_{\langle
i>j\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\mu_{0}\mathbf{H}\cdot\sum_{i}\mathbf{S}_{i}.$
(2)

Here we are intentionally neglecting dipolar interactions for two key reasons:
(i) the corresponding energy scale in this system is an order of magnitude
smaller than that of Heisenberg exchange, and (ii) our powder measurements
represent an average over crystallite orientations, the effect of which is
computationally difficult to take into account when including anisotropic
interactions. We will discuss in due course the most important consequences of
this approximation. Nevertheless we carried out a new set of MC simulations
driven by the modified Hamiltonian (2), focusing on the same range of
$\mu_{0}H$ and $T$ values explored experimentally.

The simulation results are shown in Fig. 3(b) and reflect both similarities
and differences to experiment. Perhaps most importantly, the susceptibility
minimum near $M/M_{\rm sat}\sim\frac{1}{3}$ appears strongly for temperatures
below 290 mK, which indicates that a magnetisation pseudo-plateau is intrinsic
to the Hamiltonian (2). In the simulations, this susceptibility ‘valley’ is
flanked by two sharp maxima that demarcate a new phase region centred at
finite field. This phase is stable for temperatures beyond $T_{\rm N}$ to a
critical value $T^{\ast}\sim 290$ mK. The susceptibility maxima marking the
phase boundary are not resolved in our experimental data, but may have been
washed out as a result of orientational averaging. The zero-field nnHAF
Hamiltonian (2) is isotropic, which means the system’s response is invariant
to the direction of applied field and hence there is no need to take into
account crystallite orientation; the presence of anisotropic dipolar
interactions in practice introduces an orientational dependence over which our
experimental measurements are integrated.

Inspection of our MC simulations shows the new field-stabilised phase to be an
‘up–up–down’ (UUD) state in which the moments of two thirds of the Mn2+ ions
align parallel to the applied field, and that of the other third opposes the
field. Complete UUD order gives $M/M_{\rm sat}=\frac{1}{3}$, which is why the
susceptibility drops around this value. We note that the field-dependent
behaviour of Heisenberg antiferromagnets on other tripartite lattices (_e.g._
triangular Kawamura_1985 ; Seabra_2011  and Shastry-Sutherland (SSL)
Moliner_2009 ) are also known to involve a transition from their zero-field
ground-state to an ‘UUD’ phase, such as we see here Kawamura_1985 ;
Seabra_2011 ; Moliner_2009 . The similarity to SSL physics turns out to be a
geometric feature of the trillium net; a discussion of the mapping between the
two systems is given in si .

There is a special significance of the proximity of (disordered) CSL and
(ordered) UUD phases for $T_{\rm N}<T<T^{\ast}$. Within this regime, an
applied field drives a disorder/order transition, resulting in a sudden loss
of entropy to be expelled as heat— _i.e._ a magnetocaloric effect
Kitanovski_2015 . Moreover, since the UUD state has $M/M_{\rm
sat}\sim\frac{1}{3}$, the field required to generate this entropy change is
small by comparison to that required to induce transitions in conventional
antiferromagnets. As such we expect an attractive low-field magnetocaloric
entropy change in Na[Mn(HCOO)3].

The magnitude of this magnetocaloric response was calculated from both
experiment and simulation according to the relation Kitanovski_2015

$\Delta S_{\rm mag}(H,T)=\mu_{0}\int_{0}^{H}\frac{\partial
M_{\parallel}(H^{\prime},T)}{\partial T}{\rm d}H^{\prime}.$ (3)

Here $M_{\parallel}$ is the magnetisation parallel to the direction of applied
field. Our results are shown in Fig. 3(c,d). For temperatures $T>T^{\ast}$,
the magnetic entropy change behaves essentially as one expects for a
paramagnet: field increasingly reduces the magnetic degrees of freedom, such
that entropy reduces smoothly and monotonically. For $T<T^{\ast}$, however,
there is a local maximum in magnetic entropy change that coincides with UUD
order. In our simulations, the most negative value of $\Delta S_{\rm mag}$ we
obtain (_viz._ $-12$ J kg-1 K-1) is actually competitive with that of systems
used commercially for magnetocaloric cooling Numazawa_2003 ; Mukherjee_2017 .
The experimental value is less extreme by a factor of about four because
orientational averaging over crystallite orientations makes the
CSL$\rightarrow$UUD transition smooth. One expects a sharper transition in
single-crystal measurements that no longer integrate over crystallite
orientation, which in turn would amplify the magnetocaloric entropy change
attainable in this system. One way or the other, the basic principle would be
to cycle between CSL and UUD states by applying and removing field, dumping
the heat generated as entropy during the CSL$\rightarrow$UUD transition, then
forcing the system to cool its environment on recovery of the higher-entropy
CSL phase [Fig. 3(d)].

From a materials design perspective, avenues for optimising trillium
magnetocalorics are straightforwardly identified—after all, the phase
behaviour we identify here depends only on the magnitudes of $S$ and $J$, and
the trillium edge length, which in turn influences $D$. Because a large $D/J$
washes out the magnetocaloric effect in powder samples, optimising the
isotropic coupling $J$ is likely better than optimising $S$. One possibility
for doing so is to use an alternative bridging ion (_e.g._ azide) with
stronger superexchange.

We conclude by highlighting the importance of geometric frustration within the
CSL phase for generating a strong magnetocaloric effect at relatively low
applied fields. A number of the most important magnetocaloric materials, such
as gadolinium gallium garnet, are also strongly frustrated Saines_2015 ;
Schiffer_1994 ; Zhitomirsky_2003 ; Schmidt_2007 , and we see our analysis here
as providing a robust link between frustration and this useful physical
property. A corollary is that non-magnetic frustration in pseudospin systems
may favour strong barocaloric responses at low pressures, offering a new
design strategy for efficient solid-state cooling devices Lloveras_2019 ;
Aprea_2020 ; Simonov_2020 .

###### Acknowledgements.

The authors gratefully acknowledge financial support from the E.R.C. (Grant
788144), E.P.S.R.C. (Grant EP/T027886/1) and the Leverhulme Trust (Grant
RPG-2018-268). JAMP’s work at Cambridge was supported by Churchill College,
Cambridge. JAMP’s work at ORNL was supported by the Laboratory Directed
Research and Development Program of Oak Ridge National Laboratory, managed by
UT–Battelle, LLC for the U.S. Department of Energy. JMB and ALG gratefully
acknowledge assistance from Philip Welch (Oxford). Low temperature
magnetization experiments were funded by the European Union’s Horizon 2020
Research and Innovation Programme, under Grant Agreement no 824109.

## References

  * (1) R. Moessner and A. P. Ramirez, Phys. Today 59, 24 (2006).
  * (2) L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502 (2016).
  * (3) T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm and Y. S. Lee, Nature 492, 406 (2012).
  * (4) O. A. Starykh, Rep. Prog. Phys. 78 052502 (2015).
  * (5) J. A. M. Paddison, H. Jacobsen, O. A. Petrenko, M. T. Fernández-Díaz, P. P. Deen and A. L. Goodwin, Science 350, 179 (2015).
  * (6) E.-Q. Gao, N. Liu, A.-L. Cheng and S. Gao, Chem. Commun. 24, 2470 (2007).
  * (7) H. Kikuchi, K. Kunieda, T. Asano, Y. Fujii, Y. Inagaki, A. Matsuo and K. Kindo, Phys. Proc. 75, 653 (2015).
  * (8) H.-C. Zhou, J. R. Long, and O. M. Yaghi, Chem. Rev. 112, 2, 673 (2012).
  * (9) O. M. Yaghi, M. O’Keeffe, N. W. Ockwig, H. K. Chae, M. Eddaoudi and J. Kim, Nature 423, 705 (2003).
  * (10) P. J. Saines and N. C. Bristowe, Dalton Trans. 47, 13257 (2019).
  * (11) A. E. Thoraninsdottir and T. D. Harris, Chem. Rev. 120, 8716, (2020).
  * (12) M. O’Keeffe and S. Andersson, Acta Cryst. A33, 914 (1977).
  * (13) J. M. Hopkinson and H.-Y. Kee, Phys. Rev. B 74, 224441 (2006).
  * (14) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii and P. Böni, Science 323, 915 (2009).
  * (15) J. M. Hopkinson and H.-Y. Kee, Phys. Rev. B 75, 064430 (2007).
  * (16) D. G. Franco, Y. Prots, C. Geibel and S. Seiro, Phys. Rev. B 96, 014401 (2017).
  * (17) T. E. Redpath and J. M. Hopkinson, Phys. Rev. B 82, 014410 (2010).
  * (18) F. Kneidinger, I. Zeiringer, A. Siderenko, E. Bauer, H. Michor, P. Rogl and J. G. Sereni, Phys. Rev. B 100, 134442 (2019).
  * (19) S. V. Isakov, J. M. Hopkinson and H.-Y. Kee, Phys. Rev. B 78, 014404 (2008).
  * (20) V. Paredes-García, A. Vega, M. A. Novak, M. G. F. Vaz, D. A. Souza, D. Venegas-Yazigi and E. Spodine, Inorg. Chem. 48, 4737 (2009).
  * (21) M. Inoue and M. Kubo, Inorg. Chem. 9, 2310 (1970).
  * (22) R. Scatena, R. D. Johnson, P. Manuel and P. Macchi, J. Mater. Chem. C 8, 12840 (2020).
  * (23) X.-Y. Wang, L. Gan, S.-W. Zhang and S. Gao, Inorg. Chem. 43, 4615 (2004).
  * (24) H. C. Walker, H. D. Duncan, M. D. Le, D. A. Keen, D. J. Voneshen and A. E. Phillips, Phys. Rev. B 96, 094423 (2017).
  * (25) J. C. Aston and P. J. Saines, Z. Anorg. Allg. Chem. 643, 287 (2017).
  * (26) See supporting information, which includes Refs. Paulsen_2001 ; RodriguezCarvajal_1993 ; Metropolis_1953 ; Newman_1999 ; Paddison_2012 ; Paddison_2013 ; Paddison_2016
  * (27) J. R. Stewart, P. P. Deen, K. H. Andersen, H. Schober, J.-F. Barthelemy, J. M. Hillier, A. P. Murani, T. Hayes and B. Lindenau, J. Appl. Cryst. 42, 69 (2009).
  * (28) https://doi.ill.fr/10.5291/ILL-DATA.5-32-862 and https://doi.ill.fr/10.5291/ILL-DATA.DIR-235
  * (29) P. J. Brown, in International Tables for Crystallography (Kluwer, Dordrecht, 2004), Vol. C, pp. 454–460.
  * (30) I. A. Blech and B. L. Averbach. Physics 1, 31 (1964).
  * (31) H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn. 54, 4530 (1985).
  * (32) L. Seabra, T. Momoi, P. Sindzingre and N. Shannon, Phys. Rev. B 84, 214418 (2011).
  * (33) M. Moliner, D. C. Cabra, A. Honecker, P. Pujol and F. Stauffer, Phys. Rev. B 79, 144401 (2009).
  * (34) A. Kitanovski, J. Tušek, U. Tomc, U. Plaznik, M. Ožbolt and A. Poredoš in Magnetocaloric Energy Conversion: From Theory to Applications (Springer International Publishing, Cam, 2015), pp. 23–37.
  * (35) T. Numazawa, K. Kamiya, T. Okano and K. Matsumoto, Phys. B 329-333, 1656 (2003).
  * (36) P. Mukherjee, A. C. Sackville Hamilton, H. F. J. Glass and S. E. Dutton, J. Phys.: Cond. Matt. 29, 405808 (2017).
  * (37) M. E. Zhitomirsky, Phys. Rev. B 67, 104421 (2003).
  * (38) B. Schmidt, P. Thalmeier and N. Shannon, Phys. Rev. B 76, 125113 (2007).
  * (39) P. Schiffer, A. P. Ramirez, D. A. Huse, and A. J. Valentino, Phys. Rev. Lett. 73, 2500 (1994).
  * (40) P. J. Saines, J. A. M. Paddison, P. M. M. Thygesen and M. G. Tucker, Mater. Horiz. 2, 528 (2015).
  * (41) P. Lloveras, A. Aznar, M. Barrio, Ph. Negrier, C. Popescu, A. Planes, L. Mañosa, E. Stern-Taulats, A. Avramenko, N. D. Mathur, X. Moya and J.-Ll. Tamarit, Nat. Commun. 10, 1803 (2019).
  * (42) C. Aprea, A. Greco, A. Maiorino and C. Masselli, Energy 190, 116040 (2020).
  * (43) A. Simonov and A. L. Goodwin, Nat. Rev. Chem. 4, 657 (2020).
  * (44) C. Paulsen, in Introduction to Physical Techniques in Molecular Magnetism: Structural and Macroscopic Techniques–Yesa 1999, edited by F. Palacio, E. Ressouche and J. Schweizer (Servicio de Publicaciones de la Universidad de Zaragoza, Zaragoza, 2001).
  * (45) J. Rodriguez-Carvajal, Physica B 192, 55 (1993).
  * (46) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. Phys. 21, 1087 (1953).
  * (47) M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999).
  * (48) J. A. M. Paddison and A. L. Goodwin, Phys. Rev. Lett. 108, 017204 (2012).
  * (49) J. A. M. Paddison, J. R. Stewart and A. L. Goodwin, J. Phys.: Cond. Matt. 25, 454220 (2013).
  * (50) J. A. M. Paddison, H. S. Ong, J. O. Hamp, P. Mukherjee, X. Bai, M. G. Tucker, N. P. Butch, C. Castelnovo, M. Mourigal and S. E. Dutton, Nat. Commun. 7, 13842 (2016).