Implications of bond disorder in a S=1 kagome lattice

Strong hydrogen bonds such as F···H···F offer new strategies to fabricate molecular architectures exhibiting novel structures and properties. Along these lines and, to potentially realize hydrogen-bond mediated superexchange interactions in a frustrated material, we synthesized [H2F]2[Ni3F6(Fpy)12][SbF6]2 (Fpy = 3-fluoropyridine). It was found that positionally-disordered H2F+ ions link neutral NiF2(Fpy)4 moieties into a kagome lattice with perfect 3-fold rotational symmetry. Detailed magnetic investigations combined with density-functional theory (DFT) revealed weak antiferromagnetic interactions (J ~ 0.4 K) and a large positive-D of 8.3 K with ms = 0 lying below ms = ±1. The observed weak magnetic coupling is attributed to bond-disorder of the H2F+ ions which leads to disrupted Ni-F···H-F-H···F-Ni exchange pathways. Despite this result, we argue that networks such as this may be a way forward in designing tunable materials with varying degrees of frustration.


Muon-spin relaxation experiments.
In a muon-spin rotation (μ + SR) experiment, spin-polarised μ + ions are implanted in a sample such that the muons occupy interstitial positions in the crystal lattice. 1 The population of muons then decreases on the time-scale set by their 2.2 μs meanlifetime. By-products of the muon-decay are positrons, which are preferentially emitted parallel to the instantaneous direction of the μ + polarisation at the decay event. (S1) where α is an experimentally determined constant accounting for differences in the detectors. The asymmetry is proportional to the muon polarization along the beam direction.
Example asymmetry spectra for [H2F]2[Ni3F6(Fpy)12][SbF6]2, at several temperatures are shown in Fig. S1 (left). Little change is observed in the spectra between 0.019 and 10 K. In all cases, the signal shows a fast decay at early times and then relaxes roughly as an exponential function. An additional low-amplitude, low-frequency oscillation is super-imposed on the relaxing signal.
The fast relaxing component is likely due to paramagnetic muon sites and has been observed in many molecular magnets. 2 The slow exponential decay (with decay rate ~0.7 MHz) arises from the fluctuation of the electronic moments and is therefore sensitive to magnetism in the material. The weak oscillatory signal is commonly seen in fluorine compounds, indicative of Fμ + entangled states, 3 where the muon is strongly coupled to a spin-1/2 19 F nucleus and the oscillatory frequency ω/2π is related to the F-μ + separation.
The asymmetry can therefore be best fitted to a function with several relaxing components: where Afast and Aslow are the amplitudes of the fast and slow relaxing components with relaxation rates λfast and λslow, respectively; AFμ and λFμ are the amplitude and relaxation rate of the F-μ + signal and Dz(ω, t) is the F-μ + polarization function. 3 The amplitude Abg accounts for the non-relaxing contribution from the muons that stop within the sample holder and cryostat tail.
The extracted ω values and the slow relaxation rate λslow are plotted against temperature in Fig. S1(a) and (b). No significant features were observed throughout the measured temperature window, notably near 2.5 K where a peak was revealed by heat capacity measurements. The average value of ω/2π is 1.74 MHz, corresponding to a F-μ + separation of 1.09 Å within the unit cell. The lack of change in the relaxation rate λslow, which is proportional to the width of the local field distribution and the correlation time via λ  (B -B) 2 τ, 4 suggests that no significant change in static or dynamic magnetic properties is detected using muons in this material from 0.019 to 10 K. This suggests that in terms of the Hamiltonian involving interactions between Ni(II) ions will be small, which implies that for temperatures T  0.019 K, the magnetic behavior of the sample can be largely determined from the single-ion properties.

Modelling of bulk thermodynamic properties of powdered samples from statistical mechanics
Magnetization. For a given field strength, H, and orientation of the field with respect the hard-axis expressed in terms of the polar angles (θi; ϕj), the eigenvalues of eq. 1 (main text) are deduced by diagonalizing the Hamiltonian. Inserting these eigenvalues into a partition function at a fixed temperature, the magnetization at a particular field strength and orientation M(H, θi, ϕj), can be deduced (see e.g. Ref. 6).
The polar angles are incremented (in 20 evenly spaced steps of Δθ and Δϕ) such that, in total, 400 orientations of the field are considered. The average magnetization, 〈 ( )〉, at a particular applied field is approximated by: This was calculated up to μ0H = 10 T, at μ0ΔH = 0.1 T intervals. By using different temperatures in the partition function for this calculation, the temperature dependence of the magnetization was also be explored ( Fig. 5b, main text).

Heat Capacity.
In analogous manner to calculation of the magnetization above, the heat capacity at a particular temperature and field orientation, Cmag(T, θi, ϕj) was determined from the eigenvalues of eq. 1. 6 The direction of the applied field was incremented and Cmag recalculated. By considering 400 different field orientations with respect to the hard axis, the average 〈 mag ( )〉 at a given field and temperature is estimated from: This calculation was performed for different fixed magnitudes of the applied field in the range 0 ≤ μ0H ≤ 9 T, and the result is compared to the measured data in Fig. 6 (main text).
Using the D, E, gz and gxy parameters determined from ESR, the simulated heat capacity exhibits a broad maximum whose field-dependence is in good agreement with that observed in [H2F]2[Ni3F6(Fpy)12][SbF6]2 (Fig. S2).      4 Density functional theory 4

.1 Structural and Magnetic Models
To clarify the structure of the material and investigate the magnetic interactions between spins on different Ni ions, we consider six models for the structure of the compound Several possible orientations of the SbF 6 clusters are possible given the crystallographic data, but we have chosen a single orientation (as shown for a sample model in Fig. S6) since we are interested in the pattern of bonds between Ni ions. We will discuss the models in terms of the two triangles of Ni ions that can be seen in Figure S6, closest and furthest from the origin.
The first two models we consider have Ni ions in each triangles equivalently connected by a H 3 F − 4 moiety, leading to equal superexchange interactions between all spins which we label J. (We call these '3-link' models below.) As shown in Fig. S7, the difference between the two models is whether the central F ion is displaced from the plane of the triangle towards the center of the cell (called 'out-of plane' hereafter) or not (called 'in-plane'). For these 3-link models, we examine both ferromagnetic (FM) and ferrimagnetic (i.e. one spin opposite to the other two in the basic triangles) states. We expect the ferrimagnetic configurations to lead to overall antiferromagnetic (AFM) magnetic structures when the whole material is considered, so label these AFM below. The labels (A) to (C) in Fig. S7(a) show equivalent Ni ions; we can line up the ions in an in-plane triangle with their identically labeled vertices with out-of-plane triangles (and vice-versa) in order to construct a kagome lattice.
In the next four models, in-plane and out-of-plane configurations have two Ni ions linked by a H 2 F − 3 moiety and another not. We refer to these models as the '2-link' models. Here we have two exchange couplings, J || between strongly linked spin sites and J | between weakly linked sites. The in-plane and out-of-plane triangle and symmetric and asymmetric arrangements of bonds give the four possible combinations shown in Fig. S8, where J || couplings are labeled with double lines and the J | couplings with single lines. These models have the three spin states shown in Fig. S9. As before, the labels from (A) to (C) in Fig. S9 indicate how a kagome lattice is constructed from the triangles.

Density Functional Theory Calculation
Results for different magnetic configurations in each model were obtained using density functional theory (DFT) calculations performed on rhombohedral unit cells using CASTEP [1] version 16.1 with accurate [2] on-the-fly generated ultrasoft PBE [3] pseudopotentials using the experimentally determined crystal structure and lattice parameters. An energy convergence tolerance of 1 × 10 −10 eV, a plane wave basis cutoff of 396 eV and a Grid Scaling (which fixes the standard grid size relative to the diameter of the cutoff sphere) of 1.75 were used. It was found that this converged the energy difference between the asymmetric, out-of-plane 2-link model's AFMs and AFMw magnetic configurations to within 0.01 meV. Energy calculations for the asymmetric, out-of-plane 2-link AFMs, AFMw and FM cases were repeated in order to check for consistency. The energy differences between the 3-link models are given in Table 1(a): the FM state of the in-plane structure is found to be the most stable. Table 1(b) gives the relative separation between the lowest lying energy states of each 2-link structure; the AFMs state of the asymmetric, out-of-plane structure is the lowest-lying state.

3-link asymmetric model
We begin with the Hamiltonian of the Ising-type system: The eigenvalues of the spin operatorŜ in this case are −1, 0 and 1. From these we can calculate the energies of the AFM and FM states: This is the value for the Ising exchange constant; in order to obtain the exchange constants for the Heisenberg model that are quoted in the main text we follow [6] and divide the Ising value by the number of magnetic sites in the cell, which is 3.

2-link asymmetric model
Starting with the Hamiltonian of the asymmetric Ising system we find the following values of the energy per unit cell for each state of interest: From these values we obtain the following simultaneous equations: This system is overdetermined, but as Eq. (11) is a linear combination of Eq. (9) and Eq. (10) it is possible to solve exactly to obtain the following for the exchange constants: Again, these are the values for the Ising exchange constants and so to obtain the Heisenberg values we follow [6] and divide by the number of magnetic sites in the cell, which is 3.

Spin Density Plots from Calculations
Fig. S10 shows plots of the spin density across the linking moieties of the Ni triangles for the in-plane 3-link and asymmetric, out-of-plane 2-link models. For the asymmetric, out-of-plane 2-link model we see for that both the AFMs and FM configurations [ Fig. S10(a) and (b) respectively], the presence of the H ions has led to spin delocalisation from the F ions neighboring the Ni ions to the central F ion. In the AFMs case, we see two separate isosurfaces of opposite spin, and in the FM case a single isosurface of the majority spin. The F neighboring the unlinked Ni ion has a larger spin density. For the in-plane 3-link model, we see no spin density at the central F ion in the AFM case [ Fig. S10(c)], which implies that any spin delocalisation there is negligible. In the FM case [ Fig. S10(d)], we see a small, triple-lobed isosurface indicating that spin has been transferred from all three Ni-neighboring F ions to the central ion via the H ion links. Note also that the corresponding lobes are smaller for the in-plane 3-link configurations than the asymmetric, out-of-plane 2-link configurations. This is consistent with the unbroken network of superexchange that exists in the in-plane 2-link model: spin may delocalise between all Ni ions without interruption, whereas it only delocalises along links between members of the same trimer in the asymmetric, out-of-plane 2-link case.