# Implications of bond disorder in a S =1 kagome lattice

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### Crystal structure

The 100 K structure of [H2F]2[Ni3F6(Fpy)12][SbF6]2 was determined by single crystal X-ray diffraction (Table 1). The compound crystallizes in the trigonal space group R

$3¯$

. Evidence for the existence of H2F+ cations comes from the infrared spectrum (see Fig. S1) that shows a broad peak of medium intensity at 1730 cm−1. Aside from weak symmetric and asymmetric C-H stretches from Fpy above 3000 cm−1, no additional features were observed between 1800 and 3600 cm−1. The lack of O-H stretching and bending modes precludes H2O or H3O+ from the structure.

The Ni(II) ions occupy axially compressed octahedra comprised of two axial-F ions and four equatorial N atoms donated by Fpy ligands. The coordination sphere is characterized by three bond lengths: Ni-F = 2.000(1) Å, Ni-N = 2 × 2.101(1) and 2 × 2.145(1) Å. The neutral NiF2(Fpy)4 octahedra occupy the vertices of a 2D kagome lattice (Fig. 1a) with Ni···Ni distances of 7.18 Å. Adjacent kagome layers are well isolated due to intervening SbF6 anions. The central-F of the H2F+ has 3-fold symmetry but is positionally disordered over two sites F1C and F2C (labelling scheme in Fig. 1b) while H1C and H2C are 33% occupied over six positions. Two unique H···F distances of 0.95 and 1.40 Å occur whereas the shortest F···F distance is 2.352(1) Å. The latter distance is typical of poly-HF adducts which are shorter than F···O (≥2.60 Å)23,25 and within the sum of van der Waals radii for two fluorine atoms (2.94 Å).

Broadly described, the close proximity of F3 to H2F+ structurally resembles a H3F4 anion but with partial occupancies (Fig. 1b). The disordered nature of this species affords random H···F bonds and variable connectivity of NiF2(Fpy)4 octahedra throughout the 2D kagome network.

Existence of the fluoronium ions H2F+ and H3F2+ has been established by an X-ray study of separate crystalline phases of [H2F][Sb2F11] and [H3F2][Sb2F11]26. A terse structural description reported an F···F distance of 2.30(1) Å for the H3F2+ salt whereas H2F+ exhibited orientational disorder26. In either case, none of the protons were located from the X-ray diffraction study nor were any spectroscopic data reported. The presence of H2F+ in [H2F]2[Ni3F6(Fpy)12][SbF6]2 likely arises from the HSbF6 precursor27,28. Of note is that the occurrence of H2F+ in a room-temperature stable, solid-state environment is unprecedented.

The unique crystal structure of [H2F]2[Ni3F6(Fpy)12][SbF6]2 warranted detailed study by muon-spin relaxation (μ+SR), electron-spin resonance (ESR), pulsed-field magnetization, heat capacity, magnetic susceptibility and density-functional theory (DFT).

### Magnetic properties

In an applied field (H), the Hamiltonian in eq. 1 was used to model the single-ion properties of S = 1 Ni(II) in [H2F]2[Ni3F6(Fpy)12][SbF6]2,

$ℋ=DSˆz2+E(Sˆx2−Sˆy2)+μ0μBHTgSˆ,$

(1)

where g = diag (gxy, gxy, gz) is the g-tensor and Ŝ = (Ŝx, Ŝy, Ŝz) are the S = 1 spin operators29. A zero-field splitting (ZFS) lifts the degeneracy of the ms = 0 and ms = ±1 energy levels30, which is resolved into axial (D) and rhombic (E) components. In general, the local Ni(II) site symmetry dictates whether easy-plane (D > 0) or Ising-like (D < 0) anisotropy occurs. For the present compound, D and E were determined to be + 8.3 and 1.21 K, respectively. This D is smaller (and of opposite sign) than D = −13.6 K (Dstr = −12.3 pm) calculated using the methodology described in ref.31.

### Muon-spin relaxation

The results of μ+SR measurements suggest that no significant change in static or dynamic magnetic properties occur between 0.019 to 10 K (see Fig. S2). This implies that the spin-exchange interactions between Ni(II) ions will be small which is coincident with the absence of long-range order. Thus, the magnetic behavior for T ≥ 0.019 K is largely determined by the single-ion properties of the Ni(II) ion.

### Electron-spin resonance

Fixed-frequency (ν) powder transmission ESR spectra measured at 435.2 GHz for 3.3 ≤ T ≤ 100 K (Fig. 2a) show a strong ESR transition (indicated by the derivative shape in differential intensity, dI/dH) at a resonance field of 6.3 T (labelled α). The weak T-dependence of the resonance field indicates the transition arises from Ni(II) single-ion spin excitations32. Upon cooling, two additional weak features emerge in the powder ESR spectra near 10 and 12 T (β and γ respectively). The intensity of each transition increases as T is reduced, indicating all three features originate from transitions from the ground state within the S = 1 multiplet32. Absence of spin-correlated behavior is in good agreement with the results of μ+SR. In ν-dependent measurements, the resonance field of the α transition is defined by its center where dI/dH = 0. The field position of the γ transition is indicated by the red arrows in Fig. 2b, while the β transition is relatively weak and absent from most spectra.

A representative experimental ESR spectrum at 3.3 K and 435.2 GHz is shown in Fig. 3a. The measured data is compared to a simulated spectrum (with D =  + 8.30 K, E = 1.21 K, gxy = 2.20 and gz = 2.17) as determined from the energy-level diagram for each Ni(II) site resulting from the Hamiltonian in eq. 1. The model reproduces the field positions and relative intensities of α, β, and γ transitions.

In this powder simulation the β and γ transitions are labeled as the ground state transitions with the field aligned parallel to the local x– and y-axis of the Ni(II) ions respectively (Fig. 3b). The α transition is attributed to an excitation with the field parallel to the local x-axis, but occurs between the two states of mx = −1 and mx =  + 1 with |Δmx| = 2 where mx is the spin projection onto its local x-axis. Typical ESR transitions for an applied field along x occur between states with |Δmx| = 1. However, when the applied field is comparable to the zero-field splitting D, a strong mixing between mx states is expected such that mx is no longer a good quantum number and the ESR selection rule (|Δmx| = 1) does not hold rigorously33.In this case the α transition is attributed to a half-field transition, exhibiting a large effective g factor (geff ≈ 4) that is almost twice that of a normal ESR transition (geff ≈ 2)32. Such a transition is typically observed in powder ESR spectra for pseudo-octahedral Ni(II) when the experimental frequency is comparable to D34.

If the ESR spectra are simulated instead with an easy-axis D = −8.30 K then the field of the α transition is reproduced; however, the β and γ transitions appear at much higher fields and the simulation cannot account for all transitions simultaneously. Therefore, the ESR results strongly favor the presence of an easy-plane anisotropy. The rhombic distortion term (E) is also essential to explain the ESR data since the separation between the β and γ transitions depend solely on E. When E = 0, β and γ morph into a single transition and the intensity of this transition would be similar to that of the α transition.

Comparing the observed peak positions of the frequency-dependent ESR transitions at 3.5 K to the expected peak positions from simulated ESR spectra (Fig. 3c), a good agreement was obtained when D = + 8.30 K, E = 1.21 K, gxy = 2.20 and gz = 2.17. These parameters are all well constrained by the data. The simulation also reveals that at high frequencies all half-field transitions are closely spaced in field and move slowly apart as frequency decreases. This may account for the observation that the transition labelled α becomes significantly broader at low frequency, exhibiting a complex fine structure for ν < 400 GHz (Fig. 2b).

### Pulsed-field magnetization

A small bump in the powder differential susceptibility (dM/dH) develops at ≈ 5 T as the sample is cooled below 2 K (Fig. 4a). The field at which this feature occurs is T independent. Given the easy-plane anisotropy of the Ni(II) ions, eq. 1 implies that for a field applied along the local z-axis (i.e., the hard-axis) a transition from an mz
= 0 to mz
= 1 ground state occurs at a critical field (Hc):

At T = 0, this level crossing induces a sharp step in the z-component of the magnetization (Mz
) resulting in a large peak in dMz
/dH. For T > 0, this step-like feature is thermally broadened due to occupation of energy levels above the ground state. If H is applied within the easy-plane the ground state does not cross any other spin-states (Fig. 3b) so Mx,y will resemble a Brillouin function, smoothly approaching saturation as the separation of energy levels becomes larger than kBT. Typical experimental powder M(H) curves are shown in Fig. S3 of the SI.

A simulation of the powdered differential susceptibility was performed by taking the D, E, gz
and gxy
values deduced from ESR, the eigenvalues of eq. 1 for various orientations of the field and deriving the magnetization from the resultant eigenvalues using statistical mechanics. A small bump in the simulated dM/dH (Fig. 4b) is reproduced at low Ts compared to the ZFS. This feature results from the level crossing for fields parallel to z and has small amplitude primarily due to the effects of powder averaging. Since the level crossing giving rise to the bump in dM/dH is only expected when D > 0, this behavior unambiguously points to easy-plane anisotropy in [H2F]2[Ni3F6(Fpy)12][SbF6]2.

### Heat capacity

The zero-field powder heat capacity Cmeas (Fig. 5a) displays a broad maximum at 2.5 K superimposed on a gradually rising background due to the lattice contribution, Clatt. To approximate Clatt, the data for T > 10 K were modelled (see SI) with one Debye and one Einstein phonon mode (solid red line). The spin entropy, Smag(T), was then calculated from the magnetic heat capacity, Cmag = Cmeas − Clatt, using:

$Smag(T)=∫0TCmag(T′)T′dT′,$

(3)

where Cmag = 0 at T = 0 is assumed. Smag approaches a high-T value of 8.3 ± 1.3 J/mol·K (Fig. 5a, inset), indicating the spin-entropy change associated with cooling through the broad maximum in Cmeas is consistent with Rln3. This is the full value expected for S = 1 moments cooled from a paramagnetic state to a single ground state, thus, the broad maximum is attributed to a Schottky anomaly arising from the easy-plane zero-field splitting of Ni(II) spin-states. The peak position in Cmag shifts to higher Ts and is reduced in height as the field is increased (Fig. 5b), which arises from the increased splitting of energy levels in an applied field.

The powder magnetic heat capacity was modeled with the eigenvalues of eq. 1 for various orientations of the field, using the D, E, gxy and gz values deduced from ESR. The simulated form of Cmag (Fig. 6c) reproduces several features of the measured data. A peak in the simulation moves to higher T as the field increases as a result of the aforementioned Zeeman splitting of energy levels. The simulated peak height is suppressed in fields up to ≈5 T and, thereafter, the change with increasing field is much smaller. This trend in peak height is very similar to that observed in the measured Cmag, while the field dependence of the T of the peak in Cmag is also well described by the calculation (see Fig. S4).

### Magnetic susceptibility

The powder molar susceptibility measured in a 0.1 T applied field, χmol (Fig. 6, points), increases upon cooling and begins to plateau below ≈2 K, as expected for independent Ni(II) ions with finite ZFS29. By considering fields along the i = x, y, and z directions in eq. 1, the susceptibility components, χi
, are determined to be

$χx=2μ0NAμB2gxy2D+E1−e−β(D+E)1+2e−βDcosh(βE),$

(4)

$χy=2μ0NAμB2gxy2D−E1−e−β(D+E)1+2e−βDcosh(βE)$

(5)

and

$χz=2μ0NAμB2gz2Ee−βDsinh(βE)1+2e−βDcosh(βE),$

(6)

where β = 1/kBT. The measured data were fitted over the full T range to the powder-average model

$χmol=(1−ρ)(χx+χy+χz3)+ρχpara+χ0,$

(7)

where a paramagnetic impurity term χpara accounts for a slight increase in χmol on cooling below 2 K and a temperature independent paramagnetic term, χ0, models the linear response of χmol(T)T at high Ts (Fig. 6c). Assigning a g-factor of 2.2 to a S = 1 impurity phase and approximating gxy
gz = g, the fitted parameters are: g = 2.14(1), D = +8.05(1) K, E = 1.73(3) K, ρ = 0.42(1)% and χ0 = 3.2(6) × 10−9 m3mol−1. The model (Fig. 6a and b, line) accounts for the measured data over the full T-range measured; the concentration of paramagnetic impurity is small, the g-factor agrees with the powder average ESR value to within 5% while χ0 is comparable to that observed in similar octahedral Ni(II) complexes35 and accurately models the linear behavior of χmol(T)T at high Ts (Fig. 6c).

Additional susceptibility studies using a 0.01 T dc magnetic field yielded no additional features, while ac susceptibility measurements below 10 K revealed no out-of-phase component down to 2 K for the ac frequencies in the range 15 ≤ f ≤ 1500 Hz and in applied dc fields of 0 and 0.1 T (see Fig. S5 in the SI). These measurements confirm both the weak nature of the Ni(II) spin exchange interactions and that the dominant anisotropy in this temperature range is easy-plane in nature: an easy-axis anisotropy would give rise to a slow-relaxation of the magnetization.

### Exchange interactions and DFT

The unexpectedly weak nature of the Ni-Ni spin-exchange interactions were investigated by considering two alternative coupling scenarios based on Ni(II) ions arranged on a triangular array as realized in the rhombohedral unit cell of [H2F]2[Ni3F6(Fpy)12][SbF6]2. Accurate energies36 for these candidate structures and their magnetic configurations were calculated using density-functional theory (DFT) as implemented in the CASTEP package37 with PBE pseudo-potentials38. Further details of the DFT calculations are described in the SI (see also Figs S6–S9).

We first consider an idealized configuration whereby three equivalent Ni ions interact (we call this the ‘3-link’ model; Fig. 7a). The model, which is not intended to be realistic, does not involve any structural disorder, but provides a useful comparison. It comprises triangles where all three Ni(II) ions are connected equivalently through an intact H3F4 moiety. The central F-ion of the moiety in each triangle is situated close to the plane formed by the triangle, and all exchange pathways have the same exchange constant J. Since all triangles are equivalent, these tile to give the usual kagome lattice. Including spin degrees-of-freedom we find that the ground state for this configuration is FM.

The second model is intended to simulate the disorder revealed by the X-ray results. In this model, two Ni ions in a triangle strongly interact owing to superexchange via a connecting H2F3 moiety with the other Ni ion only weakly interacting (we refer to this as a ‘2-link’ geometry; Fig. 7b). This forms a triangle with one bond expected to provide a relatively strong interaction (with exchange constant J||) and two that are weaker in comparison (with exchange constant J|). Since the simulation of true disorder is difficult in DFT, we opt for a compromise approach in which calculations were performed on different ordered configurations of these triangles, since such triangles would likely be the building blocks of the disordered structure. Several relative orientations of two of these triangles, along with different relative positions of the connecting moiety, have been examined in order to find the most energetically stable. We find that the lowest energy configuration is formed from the triangles shown in Fig. 7b, where the central F-ion is displaced out of the plane in (i) and remains roughly in-plane in (ii). When tiled these give the configuration shown in Fig. 7c. We note that the next-lowest energy structural configuration for this material was found to occur 1220 K higher in energy than the lowest energy configuration described above. Examination of the energies of different spin states for this configuration led us to predict an antiferromagnetic (AFM) (or possibly ferrimagnetic) structure, part of which is shown in Fig. 7c. This comprises alternating stripes of up and down spins which form parallel to the direction in which nearest-neighbor Ni(II) ions are weakly connected, with spins that are strongly connected always AFM aligned. Note that since we use collinear (Ising) spins in these calculations, this magnetic structure is likely to approximate the true ground state, but its stability relative to the ferromagnetic (FM) state (which lies ≈ 6.3 K higher in energy) is sufficient to indicate a degree of antiferromagnetism in the 2-link structure.

By comparing the energies of the magnetic configurations we may calculate the Heisenberg exchange constants for each model. Using the simplified Hamiltonian,

$ℋ$

= Σi,i+âJi,i+âSi
· Si,i+â, where i denotes a site and â the hopping vector to a nearest-neighbor, we find that the 3-link geometry has a FM exchange constant J = −8 K along every nearest-neighbor exchange pathway, while the 2-link geometry has AFM exchange J|| = 0.4 K along the strongly interacting bonds and J| = 0.2 K along the weak ones. Note that the precise values of the exchange constants are subject to a systematic uncertainty of order ~0.1 K owing to the limit of what present exchange functionals used in DFT calculations can reliably calculate in this case, so the precise values for the 2-link geometry should be treated with caution. Despite this, it is clear that the value of the 3-link exchange constant is significantly larger than is observed experimentally, while the 2-link exchange constants are much smaller and thus consistent with the measured data.

Although we believe the system to possess a XY-like spin anisotropy, the Ising-like magnetic model considered here allows us to evaluate the magnetic exchange energy scales at work in the system. Our approach involves determining the energy differences between ordered spin configurations that differ by a number of reversed spins. The value of the magnetic exchange derived therefore assumes that nearest neighbor spins obey Si
·Si,i+â =  ± 1. If this is not the case, then the error in this assumption is absorbed into the value of J that is derived. Indeed, the single-ion parameters extracted from the ESR and magnetometry data indicate that, because of a relatively strong easy-plane anisotropy and a staggering of the octahedra between adjacent nickel ions, the spins will not be collinear at low temperatures. In any case, we believe the model provides a valuable insight into the relative energy scales that determine the magnetism of the compound.

Spin density plots (see Fig. S10) suggest an explanation as to why J is much larger than J|| and J|. In the FM spin configuration of the 3-link structure, we find delocalized spin on the central F ion that links the Ni(II) ions, suggesting that overlap between spins on different Ni(II) ions stabilizes this state. However, for an AFM configuration based on this model, any delocalization on the F-ion is negligible, suggesting that much less overlap between spins is taking place. The energy cost of flipping Ni-spins is then large leading to the large J-value. For the 2-link geometry we discover that there is some spin delocalization along the strong bond in both the lowest-energy AFM and the higher-lying FM configurations. This implies that the energy required to flip a spin is less than in the 3-link case. In the case of the weak bonds, there is very little delocalization between spins for either configuration, and so spins on neighboring sites have little influence over each other. In that case, very little energy is required to reverse the orientation of a spin.

In summary, the DFT calculations indicate that the preferred structure of this material is such that one exchange pathway in each spin-triangle is structurally distinct from the other two and that the resulting spin-density gives rise to much reduced exchange energies between spins, compared to the case in which all pathways are identical. This result then explains the experimental observation that the effective exchange J in this material must be significantly lower than those measured in related Ni(II) coordination compounds. We attribute the reduced J in [H2F]2[Ni3F6(Fpy)12][SbF6]2 to the distribution of disordered exchange paths, as determined from our 100 K X-ray measurements, though in all probability, persisting to even lower temperatures. The low value of the exchange constant explains why no long-range AFM order is observed in the muon-spin relaxation experiments