Experimentally determining the exchange parameters of quasi-two-dimensional Heisenberg magnets

Though long-range magnetic order cannot occur at temperatures T>0 in a perfect two-dimensional (2D) Heisenberg magnet, real quasi-2D materials will invariably possess nonzero inter-plane coupling J⊥ driving the system to order at elevated temperatures. This process can be studied using quantum Monte Carlo calculations. However, it is difficult to test the results of these calculations experimentally since for highly anisotropic materials in which the in-plane coupling is comparable with attainable magnetic fields J⊥ is necessarily very small and inaccessible directly. In addition, because of the large anisotropy, the Néel temperatures are low and difficult to determine from thermodynamic measurements. Here, we present an elegant method of assessing the calculations via two independent experimental probes: pulsed-field magnetization in fields of up to 85 T, and muon-spin rotation. We successfully demonstrate the application of this method for nine metal–organic Cu-based quasi-2D magnets with pyrazine (pyz) bridges. Our results suggest the superexchange efficiency of the [Cu(HF2)(pyz)2]X family of compounds (where X can be ClO4, BF4, PF6, SbF6 and AsF6) might be controlled by the tilting of the pyz molecule with respect to the 2D planes.


Introduction
Systems that can be described by the S = 1 2 two-dimensional (2D) square-lattice quantum Heisenberg antiferromagnet model [1]- [3] continue to attract considerable experimental [4,5] and theoretical [6]- [9] attention. Recent impetus has been added to this field by suggestions that antiferromagnetic fluctuations from S = 1 2 ions on a square lattice play a pivotal role in the mechanisms for superconductivity in the 'high T c ' cuprates [3], [10]- [14] and other correlated-electron systems [15]; moreover, 2D Heisenberg magnets have been suggested as possible test-beds for processes applicable to quantum computation [4,9].
Although long-range magnetic order cannot occur above T = 0 in a true 2D Heisenberg system [2,16], real materials that contain planes approximating to 2D Heisenberg systems [2,4,17,18] inevitably possess inter-plane coupling that can lead to a finite Neél temperature [4,17,19]. In this context, synthesis of coordination complexes containing ions such as Cu 2+ , neutral bridging ligands [4] and coordinating anion molecules [17,18] has proved fruitful in the production of a variety of 1D and 2D magnetic systems [17], [20]- [22]. The current paper describes high-field magnetization measurements on nine Cu-based (Cu 2+ , S = 1/2) quasi-2D Heisenberg magnets that employ pyrazine (pyz) as a neutral bridging ligand [17,18] 8 . The data show that the field (B)-dependent, low-temperature magnetization M(B) shows a characteristic sharp 'elbow' feature at the transition to saturation, with a concave curvature at lower B. Monte Carlo evaluations of a 2D Heisenberg square lattice with an additional inter-plane exchange coupling energy J ⊥ reproduce the data quantitatively; the degree of concavity depends on the effective dimensionality of the system, while the field at which the 'elbow' occurs is an accurate measure of the in-plane exchange energy J . Using these Table 1. The quasi-2D magnets studied in this work, along with their saturation fields B c and g-factors (here pyz is pyrazine, pyo is pyridine-N-oxide). Data for oriented single crystals are indicated by B (B parallel to 2D layers) and B ⊥ (B perpendicular to 2D layers); other data are for powders. In the latter cases, an average g was evaluated from single-crystal electron paramagnetic resonance (EPR) data using standard formulae [25]. The in-plane exchange energy is calculated using equation (2); typical uncertainties in the values of J resulting from uncertainties in g and B c are ±0.1 K. Neél temperatures T N were measured to±0.04KusingµSR [22],apartfromCu(pyz) 2 (ReO 4 ) 2 , where the transition wasobservedinheatcapacitydata(seefootnote8)(typicaluncertainty±0.1K). The anisotropy |J ⊥ /J | is calculated using equation (3). The magnetic properties of a number of other quasi-2D antiferromagnets based on copper-pyrazine coordination complexes can be found in [26].  2 forms linear copper-pyrazine chains, EPR measurements on this compound have shown that the dominant superexchange pathway is through the hydrogen bonds between the water molecules and fluorine ions, and hence almost perpendicular to the chains. The result is a quasi-2D antiferromagnet on a square lattice [23,24].
J values in conjunction with Neél temperatures deduced from muon-spin rotation (µSR), it is then possible to gain a good estimate of the exchange anisotropy |J ⊥ /J | for all of the magnets. Having established these findings using the whole range of compounds, we suggest that in magnets of the form [Cu(HF 2 )(pyz) 2 ]X, [18,19] where X is a non-coordinating counterion, J and J ⊥ may be influenced by the tilting of the pyz molecule with respect to the 2D planes.

Experimental details
The quasi-2D magnets studied in this work are listed in table 1. The samples are produced in single or polycrystalline form via aqueous chemical reaction between the appropriate CuX 2 salts and stoichiometric amounts of the ligands; further details are given in [18,27] and footnote8,wherestructuraldataderivedfromx-raycrystallographyarealsofound.Forsome compounds, it was possible to grow crystals large enough to permit measurements with a single orientated sample (table 1). In other cases, the materials were either polycrystalline or the crystals too small for accurate orientation; therefore, samples composed of many randomly orientated microcrystals, effectively powders were used. In addition to the characterization describedin [18](seefootnote8)sampleg-factorsweremeasured [28]usingstandardEPR techniques [25,29].
The pulsed-field magnetization experiments used a 1.5 mm bore, 1.5 mm long, 1500-turn compensated-coil susceptometer, constructed from 50 gauge high-purity copper wire [30,31]. When a sample is within the coil, the signal is V ∝ (dM/dt), where t is the time. Numerical integration is used to evaluate M [30]. The sample is mounted within a 1.3 mm diameter ampoule that can be moved in and out of the coil [30]. Accurate values of M are obtained by subtracting empty coil data from that measured under identical conditions with the sample present.
Fields were provided by the 65 T short-pulse and 100 T multi-shot magnets at NHMFL Los Alamos [32] and a 60 T short-pulse magnet at Oxford. The susceptometer was placed within a 3 He cryostat providing T s down to 0.4 K. B was measured by integrating the voltage induced in a ten-turn coil calibrated by observing the de Haas-van Alphen oscillations of the belly orbits of the copper coils of the susceptometer [31].
In cases where sufficient quantities of materials were available, Neél temperatures T N were measured using the zero-field µSR technique described in [22] (see also [19,27]). Muons are a useful probe of magnetic order in anisotropic spin systems, where more conventional measurement techniques often encounter several difficulties. These difficulties stem from the build-up of spin correlations at temperatures above T N , which reduce the amount of entropy available to be ejected upon magnetic ordering (and hence reduce the response of the specific heat at T N ), and the quantum renormalization of the magnetic moment due to spin fluctuations (which hampers susceptibility and neutron measurements) [22]. In general, µSR measurements are unaffected by these issues and have been shown to be sensitive to magnetic order in several anisotropic molecular magnets [19,22,27]. Thus, µSR is vital for accurately determining the transition temperatures of the quasi-2D systems described here.

Experimental results
Typical M(B) data are shown in figure 1; all compounds studied (table 1) behaved in a very similar fashion. At higher T , M(B) is convex, showing a gradual approach to saturation at high B. However, as T → 0, the M(B) data become concave at lower B, with a sharp, 'elbow'-like transition to a constant saturation magnetization M sat at higher B; no further changes in M occur to fields of 85 T with the current materials. We label the field at which the 'elbow' occurs B c . As shown in figure 1(b), B c depends on the crystal's orientation in the field. However, in such cases, the M data become identical to within experimental accuracy when plotted as M/M sat versus g B, where g is the g-factor appropriate for that direction of B (figure 1(c) and table 1). This suggests that the g-factor anisotropy is responsible for the observed angle dependence of B c .

Monte Carlo simulations of magnetization data
The magnetic properties of the materials in table 1 are well described by S = 1 2 Cu 2+ spins on a square lattice. The layers are arranged in a tetragonal structure [18]; coupling between the layers depends on the packing of the molecules between. To model data such as those in figure 1, we assume that the interaction between the spins is purely Heisenberg-like [1], resulting in the Hamiltonian where J (J ⊥ ) is the strength of the intra-(inter-) planar coupling and h = gµ B B is the Zeeman energy provided by the (uniform) magnetic induction B. The first (second) summation refers to summing over all nearest-neighbors parallel (perpendicular) to the 2D x y-plane. The dependence of M as a function of B is studied for a range of values of J ⊥ /J (from J ⊥ = 0 (completely decoupled layers) to J ⊥ /J = 1) using large-scale numerical simulations. Note that the orientation of B only affects h through the anisotropy of the g-factor, in agreement with the discussion of figure 1(c) above. The stochastic series expansion (SSE) method [33]- [35] is a finite-T quantum Monte Carlo (QMC) technique based on importance sampling of the diagonal matrix elements of the density matrix e −βH , where the inverse T is represented by β, and H is given by equation (1). Using the 'operator-loop' cluster update [34], the autocorrelation time for system sizes considered here (up to ≈3 × 10 4 spins) is at most a few Monte Carlo sweeps even at the critical T [37] for the onset of magnetic order. Estimates of ground state observables are obtained by using sufficiently large values of β. We have further found that the statistics of the data obtained can be significantly improved by the use of a tempering scheme [37]- [39]. We use parallel tempering [38,39], where simulations are run simultaneously on a parallel computer, using a fixed value of J ⊥ and different, but closely spaced, values of h = gµ B B over the entire range of fields up to saturation. Along with the usual Monte Carlo updates, we attempt to swap the values of fields for SSE configurations (processes) with adjacent values of h at regular intervals (typically after every Monte Carlo step, each time attempting several hundred swaps) according to a scheme that maintains detailed balance in the space of the parallel simulations. This has favorable effects on the simulation dynamics, and reduces the overall statistical errors (at the cost of introducing correlations between the errors, of minor significance here). Implementation of tempering schemes in the context of the SSE method is discussed in [40].  numerical curves, indicating a high degree of anisotropy. We shall give further justification for this assertion below. From equation (1) it is seen that gµ B B c = 4J + 2J ⊥ , and so for such highly anisotropic magnets the model predicts the ratio g B c /|J | to take values in the range from 5.95 T K −1 (J ⊥ = 0) to 6.10 T K −1 (J ⊥ /J = 1 16 ). Since the experimental uncertainties involved in the location of B c are ∼1-2%, and the errors in g are ∼1%, no significant loss in accuracy occurs if we employ the mean value,

Comparisons of model and data
in what follows. To check the model prediction, a fit of the T -dependent low-field susceptibility following the method of [41] 9 was used to determine J independently for a selection of compounds ( figure 3(a)); the values obtained are compared with g B c in figure 3(b) 10 . As can be seen, the points lie close to the line g B c /|J | = 6.2 ± 0.2 T K −1 , in good agreement with the predicted value (equation (2)). As noted above, it is possible to determine the value of 8 B c to a very good accuracy (±1-2%); once the dimensionality of the magnet in question is seen to fall within the limits J ⊥ /J ≈ 0 − 1 16 using comparisons such as those in figure 2(b), equation (2) almost certainly presents the most accurate method for evaluating J . Intralayer exchange energies J derived in this way from measured values of g and B c are in table 1.
Thus, we have developed a method for determining both J and T N in anisotropic magnetic systems. In addition, an estimate of the anisotropy |J ⊥ /J | can be made using the results of quantum Monte Carlo simulations of quasi-2D Heisenberg antiferromagnets [42]. This study found where both |J | and T N are measured in K; the resulting values are given in table 1. Note that equation (3) figure 2) that suggested 0 |J ⊥ /J | 1 16 for all of the compounds.
We now discuss whether the non-coordinating X counterions can play a direct role as exchange pathways between the Cu 2+ ions. First of all, the molecules ClO 4 and BF 4 have radically different electronic orbitals, and yet the in-plane exchange energies for the magnets containing these counterions are very similar (see the first three rows of table 1). Moreover, as mentioned above, the X counterions are not within the 2D Cu 2+ planes, but at body-center positions within the 'cubes' (figure 4). The Cu-X separation is therefore roughly the same for the in-plane and inter-plane directions; if the X counterions played a direct role as exchange pathways, one might expect that the [Cu(HF 2 )(pyz) 2 ]X compounds would have |J ⊥ /J | values that were somewhat larger (i.e. more isotropic) than the values ∼10 −3 -10 −2 that are actually observed (see table 1). Therefore, instead of playing a direct role in the exchange, it is more likely that it is an counterion size effect on the crystal structure that is affecting the exchange pathways. The size of the counterion in the direction perpendicular to the planes differs by a factorofapproximately2atroomtemperatureforthetetrahedralandoctahedralcounterions (seefootnote8andfigure4).ThisleadstoamodestchangeintherespectiveinterlayerCu-Cu distances, d ⊥ .Forexample,thereisa∼5%differencebetweentheX=BF 6 (d ⊥ ≈6.62Å)and the X = SbF 6 (d ⊥ ≈6.95Å)compounds(seefootnote8).Thistrendininter-planedistanceis opposite to that might be expected given that J ⊥ appears to be enhanced in the compounds with octahedral counterions (see table 1), and it certainly cannot by itself account for the large change in the in-plane exchange energy.
The structural difference that may be well responsible for the factor of two change in J is the configuration of the pyz molecules within the Cu-(pyz)-Cu linkages. Figure 4 (upper) shows [Cu(HF 2 )(pyz) 2 ]SbF 6 , one of the systems with octahedral counterions; the structures of X = AsF 6 , PF 6 are very similar. At room temperature, geometry of the octahedral counterions allowsthepyzligandstostandupvirtuallyperpendiculartotheabplanes(seefootnote8 and figure 4). By contrast, the planes of the pyz ligands in the compounds with the smaller tetrahedral counterions, [Cu(HF) 2 (pyz) 2 ]BF 4 (figure 4 (lowest)) and [Cu(HF) 2 (pyz) 2 ]ClO 4 , are not perpendicular to the ab-planes, but are tilted away by 31.6 • (X = BF 4 ) or 25.8 • (X = ClO 4 ) inapatternthatpreservesthefour-foldsymmetryoftheCu 2+ sites(seefootnote8).This suggests that it may be the orientation dependence of the pyz ligand that produces the factor of two differences in J , with the more perpendicularly disposed pyzs (figure 4) presenting a more efficient exchange pathway within the layers.
Although we cannot rule out the possibility of a structural distortion occurring in these compounds as the temperature is reduced, EPR measurements show no evidence for changes in the local symmetry of the magnetic Cu 2+ ion on cooling to T N . Therefore, at present it seems unlikely that any such structural reorientation could lead to the observed disparity between the compounds with counterions of different symmetries.

Summary
Magnetization experiments have been carried out on nine metal-organic Cu-based 2D Heisenberg magnets. These systems exhibit a low-T magnetization that is concave as a function of field, with a sharp 'elbow' transition to a constant saturation value at a critical field B c . Monte Carlo simulations including interlayer exchange quantitatively reproduce the data; the concavity indicates the effective dimensionality and B c is an accurate measure of the in-plane exchange energy J . Taken in conjunction with Neél temperatures derived from µSR, the values of J may be used to obtain quantitative estimates of the exchange anisotropy, |J ⊥ /J |.
We suggest that in metal-organic magnets of the form [Cu(HF 2 )(pyz) 2 ]X, where X is a non-coordinating counterion molecule, the sizes of J and J ⊥ may be controlled by the tilting of the pyz molecule with respect to the 2D planes. Thus, it may be possible to use molecular architecture to design magnets with very specific values of J , tailored to a particular desired property.
Another way of looking at these structures is that they are pseudo-perovskite AB L 3 , where the A is the counterion (BF 4 , ClO 4 , etc), Cu is the B cation and L are the coordinated bridges, pyz and HF 2 . In light of this, one possible application would be a metal-organic magnet designed to simulate the antiferromagnetic interactions germane to cuprate superconductivity [3], [10]- [14], but with exchange energies small enough to permit manipulation of the magnetic groundstate using standard laboratory fields.