From strong to weak interaction: reconciling SQUID and μSQUID-EPR data in anomalous Co(II) dimers

Abstract

Magnetic molecules have been proposed as scaffolds for novel quantum technologies, ranging from quantum sensing and quantum memory to multilevel quantum bits (qudits) and fault-tolerant quantum computation. Integration of magnetic molecules into cutting-edge applications hinges on a deep understanding and tunability of their spin states. To date, the strategic manipulation of the local environment of the ion and careful selection of the magnetic core have enabled the desired tunability and scalability of the spin states. For such goals, however, extracting the anisotropic parameters that dictate the characteristics of the Spin Hamiltonian is challenging, especially for molecules consisting of multiple magnetic cores. We address these challenges by studying two cobalt(II) dinuclear systems, complicated by inherent spin–orbit coupling. We explore the magnetic properties of these systems in two temperature regimes: (i) at sub-Kelvin temperatures employing single crystals at 30 mK using a unique μSQUID-EPR technique that examines the microwave absorption peaks in the magnetisation data and their variation with field angle and frequency; and (ii) in bulk employing convectional SQUID magnetometry above 2 K i.e., χ_MT(T) and M(H). Unexpectedly, sub-Kelvin temperature investigations reveal a negligible interaction, whereas the SQUID data reveal a much stronger interaction between the Co(II) ions. An understanding of these data is developed based on a strong coupling model and the coupling of two moieties with a spin-effective ground state.

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Introduction

Magnetic molecules (MMs) are attractive quantum objects, proposed as platforms for emerging quantum technologies.^1–4 Engineering the ligand field, magnetic cores, and incorporating nuclear spins of the magnetic ion can provide great control over the tunability and scalability of the spins. Employing such an approach, it has been possible to design MMs with the desired characteristics.^5,6 The remarkable control gained over these systems has opened the door to many new possibilities, such as high-density storage and quantum computing, among others.^1–4,7,8 On the data storage side, the significant control over the tunability has led to surpassing the liquid-nitrogen temperature limit,⁹ while it has also been possible to create dimeric units to operate as quantum gates^10–12 or for quantum error correction protocols,^13,14 the engineering of noise-resilient qubits,^15–17 and even the execution of a quantum algorithm.¹⁸ These advances make MMs excellent testbeds for more advanced quantum technology schemes.^2–4,7 For example, the multilevel character of these systems has been proposed as qudits,^3,19 which may outperform qubits in quantum information processing schemes. Moreover, they can be employed directly to perform quantum algorithms¹⁸ or quantum error correction protocols.^13,20,21

However, to successfully incorporate MMs into quantum technologies, a detailed understanding of the electronic characteristics of these systems is required. To this end, several techniques have proven resourceful in determining the physical parameters governing the quantum characteristics of the systems. Experimentally, above 2 K, the bulk magnetic parameters are commonly investigated through SQUID magnetometry,^22,23 Electron Paramagnetic Resonance (EPR) spectroscopy,^24–27 magnetic far infrared,^23,28 and torque magnetometry,^27,29 among others.³⁰ In a combinatorial approach, these techniques are often interconnected with wave function-based theoretical routes for the complete understanding of the spin system.^31–33 Furthermore, new theoretical approaches have shed light on the relaxation characteristics of these systems, hence providing new playgrounds to fine-tune their magnetic characteristics.^5,34 But even with these approaches, certain crucial parameters like those pertaining to the magnetic anisotropy and intra-molecular interaction proved difficult to obtain. Thus, moving forward, the development of improved techniques is critical. Recently, some of us have shown that by merging two highly sensitive techniques, such as μSQUID magnetometry and EPR spectroscopy, many of the highly elusive crystal field parameters for lanthanide-based MMs can be extracted.³⁵ These studies are performed at sub-Kelvin temperatures, allowing the precise determination of the spin Hamiltonian parameters of the ground states. Here, to deepen our understanding of MMs, and as a further exploitation of the sensitivity of the μSQUID-EPR technique, we investigate two cobalt Co(II)-complexes dimers in two regimes, that is, from room temperature to 2 K, through SQUID magnetometry and at 30 mK sub-Kelvin temperature, via μSQUID-EPR.

Co(II) in octahedral symmetry, is one of the most difficult-to-understand ions in the 3d transition metal row.^24,36–40 The magnetic anisotropy in Co(II) ions in octahedral symmetry and its inherent spin–orbit coupling, intrinsic to this ion in cubic symmetry, often precludes their spin Hamiltonian characterisation. Although the relaxation dynamics of these systems are far from ideal from a qubit/qudit perspective, a salient characteristic of the systems herein studied is that the magnetic hysteresis loops measured by μSQUID alone provide very little information about the magnetic characteristics of the systems. In contrast, the μSQUID-EPR technique allows direct exploration of the spin states by coupling microwaves to the crystal by a coplanar waveguide and in situ magnetisation measurement. The technique examines the microwave absorption peaks in the magnetisation data and their variation with field-angle and frequency. It enables the exploration of the field-angle and microwave-frequency (1–30 GHz) dependent absorption maps extracted from magnetic hysteresis loops at cryostat temperatures down to 30 mK. Simulation of the angular and frequency maps affords the entire spin Hamiltonian for both systems. A stark difference is found between the conventional SQUID magnetometry data and the μSQUID-EPR loops pertaining to the operative interaction in the dimers. Through detailed analysis, supported by theoretical calculations, we resolved the discrepancy by identifying the coexistence of two distinct components in the samples: one strongly coupled and the other weakly coupled. Such coexistence of multiple interactions has been observed often in multi-nuclear molecular systems,^41,42 however, using the angle and frequency dependent EPR study, here we are able to pinpoint the orientations and magnitude of interactions of the weakly interacting components, even though a strong interaction persists in the dimeric systems. As a result, we show how in a combinatorial approach, both magnetic and spectroscopic data allow the understanding of the magnetic characteristics of MMs.

Results

Structural characteristics of Co(II) dimers

Two Co(II) dimers were selected for the study. The first dimeric complex is a mixed valence tetranuclear cobalt complex with the formula [(μ_1,1-N₃){Co^II(L^r)(μ-O₂CR)Co^III(N₃)}₂]ClO₄·H₂O·1.5CH₃OH where H₂L^r = [(1,3-propanediyl)bis(iminomethylene)bis(6-methoxyphenol)], RCO₂H = 4-methyl-3-nitrobenzoic acid (Fig. 1a). Although the complex bears four cobalt ions, the non-magnetic nature of the two Co(III) centres renders a magnetic dimeric Co(II) system. From here on, we refer to this complex as 1Co₂ (synthetic details are given in section A.1 in the SI). Crystallographic studies show that 1Co₂ crystallises in the triclinic P1̄ space group, with two inversion-related molecules residing in the unit cell (Fig. S5). The Co(II)⋯N for both Co(II) ions in 1Co₂ are found to be 2.063(2) Å and 2.062(3) Å. Likewise, both Co(II) ions are bridged by an azide group, which can act as a spin carrier between the ions. The Co(II)⋯Co(II) distance is found to be 3.7862(5) Å Co(II)⋯Co(II), with a Co(II)⋯N⋯Co(II) bond angle of 133.2(1)°. Based on the large Co(II)⋯N⋯Co(II) bond angle, ferromagnetic interactions are deemed possible. In the packing, the shortest Co(II)⋯Co(II) distance is found to be 11.0439(6) Å; hence, intermolecular interactions are expected to be negligible.

Fig. 1 Crystal structure of Co(II) dimers: (a) Crystal structure for 1Co₂ and schematic diagram of Co ions and the intramolecular distances within the complex; (b) crystal structure for 2Co₂ and schematic representation of the two differently oriented molecules within the unit cell and the angle between the easy axes. The direction of the easy (green arrows) and hard (red arrows) axes is shown as determined from CASSCF calculations. The longer arrows indicate the dominant anisotropy axis in each case. Colour code: Co(II), blue; Co(III), violet; Cl, green; N, cyan; O, red; C, grey. Hydrogens were omitted for clarity.

The second system is a dinuclear system with the formula [Co^II(L)₂(Cl)₂]Cl₂ where L = N-(methoxy(pyrimidin-2-yl)(pyrimidine-2-carboxamido)methyl)(pyrimidine-2-carboxamide), here on termed as 2Co₂ (Fig. 1b) (details are given in section A.2 of the SI). 2Co₂ crystallises in a monoclinic unit cell with C2/c with four molecules residing in the unit cell, with two different orientations (Fig. S6). In contrast to 1Co₂, where a N₃⁻ ligand acts as a direct bridge between the Co(II) ions, in 2Co₂, the Co(II) ions are bridged by the L ligand, providing no direct contact between the metals. The Co(II)⋯Co(II) distance in this system is found to be 5.5632(7) Å, much longer than the one observed in 1Co₂. Detailed crystallographic parameters and further structural characteristics for both systems are provided in section B in the SI. For 2Co₂, we find that the shortest intermolecular Co(II)⋯Co(II) distance for the two differently oriented molecules is 9.7976(8) Å; thus, any operative intermolecular interaction should be larger than in 1Co₂.

Hamiltonian for Co(II) dimers

At a single ion level, the Co(II) ion is characterised by a ground state ⁴F separated by 10⁴ cm⁻¹ from the ⁴P first orbital excited state. The ⁴F splits into three states in a cubic crystal field, two of which are triplet states (⁴T_2g and ⁴T_1g), and a singlet state (⁴A_2g), with the ⁴T_2g lying lowest. Due to isomorphism, some mixing occurs between the ⁴P and the ⁴F(⁴T_1g). Moreover, strong first-order spin–orbit coupling between the 12-fold degenerate ground ⁴T_1g state (with S = 3/2 and L = 1) is the norm for Co(II) systems; hence, a Hamiltonian of the form of (1) is suited in such environments.in (1)λ is the Spin–Orbit Coupling (SOC), α is the orbital reduction factor described as α = Aκ, where κ represents the reduction of the orbital momentum by the delocalisation of the unpaired electrons. A represents the contribution of the upper ⁴T_1g(⁴P) into the ground ⁴T_1g(⁴F) state. Δ_ax and Δ_rh parameters account for the axial and rhombic distortions of the ideal O_h symmetry of Co(II), which split the ⁴T_1g(⁴P) ground manifold. These describe the separation between the singlet ⁴A₂ and doublet ⁴E states produced by splitting the ⁴T_1g(⁴P) state. A second-order SOC splits ⁴A₂ into two Kramers doublets, while the splitting of the ⁴E state renders four Kramers doublets. μ_B and g_e the Bohr magneton and the free electron g-value, respectively.

When considering a dimeric system, an additional parameter must be introduced into (1) to account for the interactions operating between the Co(II) ions, leading to:

where J_ex is an isotropic interaction acting upon the real spins.

By employing eqn (2) it is possible to account for the interactions between the Co(II) ions up to room temperature; however, below 50 K, solely the two lowest doublets are populated (⁴E). Thus, the zero-field splitting (ZFS) phenomenological method is suited, where S = 3/2.^{24,36–40,43,44} This allows us to simplify the Hamiltonian by writing it in terms of spin without the requirement of orbital angular momentum . For distorted octahedral symmetry of the Co(II) sites, this approach is appropriate. The Hamiltonian is hence:

where D_i, E_i are respectively axial and equatorial anisotropy constants are reduced from the D tensor diagonalised in (x_i, y_i, z_i) frame, i.e., D_i = 3D_zi/2, E_i = (D_xi − D_yi)/2. B^T is the magnetic field vector (μ₀H_x, μ₀H_y, μ₀H_z)^T, while g_i are the g-tensors. D is defined based on the largest anisotropy constant along the z-axis and the smallest along the x-axis, i.e. |D_z| > |D_y| > |D_x|, and 0 ≤ E/D < 1/3.

To deal with such a two-body Hamiltonian, generally, a frame can be chosen that simplifies/diagonalises the interaction tensor J_ex, while two sets of Euler rotations transform into individual ionic frames (Co_i = 1, 2). Assuming that the g and D tensors are simultaneously diagonalised in the transformed frame for each ion {(x₁, y, z), (x₂, y₂, z₂)}, the problem involves several independent parameters. Notice that eqn (2) also contains more parameters than (3) when considering single crystal data, and Euler angles between the spins’ anisotropy-frames; thus, for both cases, several parameters should be obtained via first principles calculations based on crystallographic data as initial guesses.

Sub-Kelvin investigations – μSQUID and μSQUID-EPR

Temperature and sweep-dependent M(H) curves with the field applied along the easy axes were measured employing a μSQUID array and the transverse method⁴⁵ (Fig. S9). While the M(H) curves at 30 mK temperature for 1Co₂ are completely closed (Fig. S9b), the loops for 2Co₂ show mild opening only at high sweeping rates (Fig. S9d). The behaviour indicates fast relaxation in both samples; hence, neither satisfies the conventional definition of single-molecule magnets, that is, neither system possesses a large barrier and/or diminished quantum tunneling rates that would yield slow relaxation of the magnetisation. The small hysteresis observed in 2Co₂ might arise from phonon bottleneck effects.⁴⁶ Furthermore, any feature in the M(H) curves, which may indicate intramolecular coupling between Co(II) ions, is smeared out above 100 mK temperature (Fig. S9a and c). In conclusion, solely based on the μSQUID results, no further information can be extracted for these systems.

To gain a deeper insight into the physical parameters of 1Co₂ and 2Co₂, we utilise the μSQUID-EPR technique, i.e. we collect the M(H) data using μSQUID while applying constant frequency (ν) microwave pulses using a co-planar waveguide fabricated on the μSQUID chip (upper inset in Fig. 2a). The resonant microwave absorption produces several sharp peaks, directly observed in the M(H) loops, at the experimental ‘resfields’. The latter is defined as the field magnitudes at which the energy difference between two specified spin states matches the energy of the applied microwave E(ν). At a fixed ν, the direction of the field (θ) within the SQUID plane is varied with respect to the crystallographic frame to yield the angular maps of the peaks (ΔM(H,θ)); while sweeping over ν at a fixed angle gives the frequency maps (ΔM(H,ν)) of the peaks.

Fig. 2 μSQUID-EPR studies: M(H) curves measured by μSQUID in the presence of microwave excitation with a few fixed frequencies for 1Co₂ (a) and 2Co₂ (b). The M(H) loops without microwave excitation are also shown in the same panels for comparison. (a) The absorption peaks in 1Co₂ are barely visible, but peak subtraction from the baseline curve, renders transitions sufficiently larger than the measurement noise (lower inset). The upper inset shows a μSQUID-EPR chip with a single crystal of a molecular magnet and a coplanar waveguide. (b) For 2Co₂ the absorption peaks are large, hence, directly visible in the M(H) loops. The μSQUID-EPR studies were conducted with a power (p), width (w) and delay [p ∼ −6 dBm, w = 10 μs, d = 300 μs] and different frequencies ranging between 1–30 GHz.

Fig. 2 shows the M(H) curves in 1Co₂ and 2Co₂, obtained at 30 mK bath temperature, for a microwave excitation with frequencies ranging between 1–30 GHz. The observed frequency dependence of the peak positions confirms resonant transitions between the spin states of the molecule. For 1Co₂, these absorption peaks are small and barely visible in the M(H,ν) curves (Fig. 2a). Nevertheless, upon baseline subtraction, the peaks appear sufficiently larger than the measurement noise (Fig. 2a lower inset). For 2Co₂, the peaks are rather large and visible directly in the M(H,ν) curves shown for the positive fields in Fig. 2b. The inset of Fig. 2b indicates that the observed peaks can be explained by considering resonant transitions between certain states of the Zeeman diagram, simulated for an appropriate angle of the applied field and specific parameters for the given molecule (vide infra).

The resonant microwave absorption peaks can be directly mapped into angular and frequency maps by either varying the applied field direction or the microwave frequency, as shown in Fig. 3. Fig. 3a and b show the angle-dependent map of the resonant peaks (ΔM(H,θ)) at ν = 8 GHz, for 1Co₂. Two intersecting ellipses with nearly the same amplitudes of the long and the short axes are noticeable, providing information about the underlying spin Hamiltonian parameters characterising the system. Fig. 3c shows a frequency-dependent map of the resonant peaks ΔM(H,ν) exhibiting linear behaviour. This is obtained for the field applied along the magnetic X-axis as indicated by the white dashed line in Fig. 3a. Furthermore, the angle-dependent map highlights the anisotropy of the system, revealing immediate signatures of two differently oriented spins in the molecule with a tilting angle between the anisotropy axes. Moreover, in the frequency-dependent map, a non-zero intercept of the linear absorption lines to the X-axis and any distortion from the linear behaviour can usually be associated with intra-molecular interactions. In contrast, the maps (Fig. 3c) show no clear evidence of distortion or large intercept, indicating nearly no interaction in 1Co₂.

Fig. 3 Angular and frequency μSQUID-EPR for 1Co₂: EPR absorption maps and fits. The angle-dependent absorption map, i.e., M–M_base(H_x,y) for fixed frequency ν = 8 GHz is shown in panel (a). The corresponding fitted map is shown in panel (b) with the yellow dots as the maxima found by the peak-finder from the experimental maps. At the primary stage, the interaction is assumed to be J = 0; hence, the transitions (2,3) and (1,3) superpose with (0,1) and (0,2), respectively. The frequency-dependent absorption map, i.e. M–M_base(ν, H_x) and its simulation, for a fixed direction of the applied field, are shown in (c) and (d), respectively. The dashed line in (a) shows the direction of the applied field corresponding to (c), while the dashed lines in (c) show the frequency of the applied microwave corresponding to (a); (e) shows the Zeeman diagram for a 0° angle of the applied field as demonstrated by the white dashed line in (a); (f) Zeeman diagram with a −30° angle as shown by the red dashed line in (b). The resfields for ν = 8 GHz excitation (vertical bars) are indicated in each Zeeman diagram.

The angular and frequency-dependent maps for 2Co₂ are shown in Fig. 4a–c. In contrast to 1Co₂, two ellipses with one almost exterior to the other are observed. Fig. 4d and e show the frequency-dependent maps of the resonant peaks (ΔM(H,ν)) for 2Co₂, obtained at two different angles as indicated in Fig. 4c. Despite the apparent differences with 1Co₂, the angle-dependent map in 2Co₂ also indicates two magnetic cores in the system with their anisotropy axis tilted with respect to each other. The lack of significant variation in the shape of absorption lines in Fig. 4a–c and no obvious distortion from linear behaviour in the frequency maps in Fig. 4d and e suggests that the interactions are likewise negligible.

Fig. 4 Angular and frequency μSQUID-EPR for 2Co₂: EPR absorption maps for 2Co₂ with its fittings: The angle-dependent absorption maps, i.e., M–M_base(H_x,y) for fixed frequency ν = 20 GHz, 10 GHz and 15 GHz are shown in (a), (b) and (c) respectively. The corresponding fitted lines (grey and green) are also shown with the maxima (yellow dots) found by the peak finder from the experimental maps. The frequency-dependent absorption map, i.e. M–M_base(ν), H_x with its simulated curve a for a fixed direction of applied field θ = 0° and 150° are shown in (d) and (e) respectively. The red and orange dashed lines in (c) indicate the direction of the applied field corresponding to (d) and (e). Panels (i), (ii) show the Zeeman diagrams, simulated for these two angles, indicating resfields for ν = 15 GHz excitation (vertical bars) to clarify the observed angular dependence in (c).

High temperature regime – SQUID magnetometry

DC magnetic susceptibility data (2–300 K) were collected on restrained powdered samples of 1Co₂ and 2Co₂ using a SQUID magnetometer, applying a magnetic field of 1 kOe. M(H) were also carried out in the temperature and field range of 2–5 K and 0 to 5 T, respectively. The room temperature χ_MT value for complex 1Co₂ was found to be 5.16 cm³ mol K⁻¹ (Fig. S7), in good agreement with the expected value for two uncoupled Co(II) ions characterised by an S = 3/2 spin (i.e., 5.13 cm³ K mol⁻¹ for two Co(II) with g = 2.59). Upon cooling, the χ_MT value increases slowly as the temperature is decreased down to ∼18 K, where it more rapidly drops down to 2.9 cm³ mol K⁻¹ at 2 K. The upturn signals operative ferromagnetic interactions, while the downturn could be ascribed to the depopulation of excited levels. For 2Co₂, we find that the room temperature χ_MT value is also close to the expected for two uncoupled Co(II) ions characterised by an S = 3/2 spin with a value of 6.1 cm³ K mol⁻¹ (cf., 6.3 cm³ K mol⁻¹ for two Co(II) with g = 2.59) (Fig. S8). Upon cooling, the χ_MT product decreases gradually, reaching a value of 2.3 cm³ K mol⁻¹ at 2 K. The downturn in 2Co₂ can be a consequence of the depopulation of anisotropic levels and/or antiferromagnetic interactions. The relatively similar χ_MT value at 2 K signals a similar low-temperature energy manifold.

Magnetisation studies likewise reveal no saturation of the ground state at 2 K up to 5 T for 1Co₂ and 2Co₂, with the highest M(H) value at the lowest temperature and highest field reaching 4.1 and 3.1μ_B, respectively.

Discussion

For a comprehensive understanding of the experimental sub-Kelvin μSQUID-EPR data, it is necessary to visualise the spin Hamiltonian (eqn (3)) in terms of the Zeeman diagram and its variation with the direction of the applied external magnetic field for a hypothetical case (Fig. 5a and b). Fig. 5c and d shows the angular variation of the simulated ‘resfields’, for three specific transitions. For this, a hypothetical Co(II) dimer was chosen, each ion characterised by an S = 3/2 state and a large D_i and E_i values (with a D_i/E_i = 1/4) and isotropic g-values. The simulations were carried out employing Easyspin.⁴⁷ For simplicity, at first, we assume no interaction between the ions while simulating Fig. 5. The model, however, considers the possibility of an intra-tilt between the anisotropy axis frames of Co(II) ions. The Euler angles (Fig. 5c and d) for each ion were chosen such that the largest anisotropy axis (in this case, the hard axis) between the two makes a 60° angle, while another 20° rotation in the perpendicular plane (in this case, the easy plane) follows. Due to the large D_i and E_i values, considering only the four lowest states in the ground multiplet is sufficient to describe the system at sub-Kelvin temperatures (vide supra). These four states (zoomed region in Fig. 5b) can be rationalised as originating from two states in the ground doublet for a single S = 3/2 system. The Zeeman diagram for a fixed direction of field contains only a small representation of the spin Hamiltonian, as it varies drastically with the angle of the applied field. For instance, the magnitude of resfield for a 15 GHz quantum of excitation between the ground state to the 1^st excited state is shown by the red arrow in Fig. 5b for a specific direction of field. Varying the direction of the field (θ, φ) with respect to the molecular frame for this excitation yields the red surface in panels Fig. 5c and d. Similarly, the angular variation of the resfields for 15 GHz between the ground state and 2^nd and 3^rd excited states (blue and green arrows in Fig. 5b, respectively) is displayed by the blue and greenish surface in Fig. 5c and d. The direction of the field in Fig. 5b is indicated by the arrows in Fig. 5c.

Fig. 5 Angular variation of the resonance field for Co(II) dimer: (a) Zeeman diagram for a hypothetical spin dimer with an angle between their anisotropy axis, modelled using S_1,2 = 3/2 with D_1,2 = +100 cm⁻¹, E_1,2 = +25 cm⁻¹, Euler angles [0 −60 10], [0 0 −10] and g = 2. For simplicity the case of zero interaction (J = 0) is shown here while the cases for non-zero J with different orders of magnitude are analysed in the SI. At sub-Kelvin temperatures, solely the ground multiplet is expectedly to be populated. (b) Zoomed region of the ground multiplet and the resonance fields for ν = 15 GHz. The red arrow is the resfield for the excitation to the first excited state, while the blue and green arrows are the excitation to the second and third excited states. (c and d) Angular dependence of the resfield for ν = 15 GHz and comprising excitations to the first, second and third excited states for S_1,2 = 3/2 dimer shown using two projections. The non-colinear arrangements between the spin results in the tilted ellipsoid shapes in the resfields.

Due to non-colinear arrangements in the hypothetical Co(II) dimer, the angle between the hard axes of the ions results in two tilted ellipsoids in the 3D angular plot of the resonance fields. Note that the largest values of resfields indicate the hard axis, and the smallest values are obtained along the easy axis. Fig. 5c shows the anisotropy in such molecular systems consisting of two transition metals with S = 3/2. It is worth noting that the magnitude of the resfield ellipsoids heavily depends on the g values, while the ellipticity is dictated by E/D ratio. Furthermore, we chose both D and E to be positive, which for small E would have resulted in prolate ellipsoids—indicating a hard axis and easy plane. If both D and E were negative, a small E would instead lead to oblate ellipsoids, signifying a unique easy axis and hard plane. However, when E approaches D/3, an easy, hard, and intermediate anisotropy axis becomes prominent. For E ∼ D/3, a negative D, E and a positive D, E scenario do not differ drastically as the easy and the hard axis closely compete to be the dominant anisotropy axis (see Fig. S10). These different categories are encountered in this study. Experimentally, the angle between two participating spins can be visualised in Fig. 5c and d as an angle between the two ellipsoids. Interestingly, the intersecting ellipsoid resfield surfaces may also appear as outer and inner ellipses depending on the plane of 2D projection, see Fig. 5d.

Based on this hypothetical model and the sub-Kelvin μSQUID-EPR experimental data obtained for 1Co₂ and 2Co₂, it can be inferred that the magnetic interactions within the dimeric units are minimal or effectively negligible. These findings are in stark contrast to the high-temperature SQUID magnetometry data, where clearly large ferromagnetic and antiferromagnetic interactions are visible. To quantify the interaction observed in high-temperature SQUID data, we employ Hamiltonians (2) and/or (3), leading to a strong ferromagnetic interaction of J_ave ∼ 4 cm⁻¹ and ∼ −6 cm⁻¹ for 1Co₂ and 2Co₂, respectively (see SI section C.1 for details).

To resolve this contradiction, we explore the hypothetical dimer model in both the regimes of large and small interactions. Introducing a small interaction between the ions results in the opening of anti-level crossings or mixing of states in the Zeeman diagram, mostly near zero field with small distortions of these resfield surfaces. If the interaction is significantly smaller than D, E and a half integer spin is concerned, the exact role of interaction can be clearly distinguished from the effect of transverse parameters E on the frequency scans, as shown in the simulations in Fig. S11 with and without a small interaction. This simulation also highlights the mentioned small distortion of the resfield surfaces. On the other hand, when interaction is comparable to the magnitude of D, E, it is often experimentally impossible to distinguish it from the role of E. Nevertheless, we extend the hypothetical model to simulate the case of large interactions and investigate the Zeeman diagram with possible EPR transitions in such a case. When an isotropic large interaction of the order ∼±5 cm⁻¹ (comparable with DC SQUID results) is introduced, the Zeeman diagram has been found to drastically change, giving an effective coupled spin-like scenario with no EPR transitions below <50 GHz microwave frequency (see Fig. S12).

To address this conundrum and to choose a reliable route of analysis, we delve into the structural and anisotropic parameters obtained via CASSCF/SO-RASSI/SINGLE_ANISO calculations.^31,48 Crystallographically, 1Co₂ possesses one molecular orientation in the unit cell (two molecules related by inversion symmetry), while 2Co₂ involves two molecular orientations at a large angle (Fig. S5, S6 and S13). The case of 1Co₂ turned out to be quite convoluted. As there is no angle between two neighbouring molecules’ spins, the two observed angles cannot be explained by intermolecular orientations. We also noticed that the μSQUID-EPR peaks were very weak compared to 2Co₂. This points to the possibilities involving impurity/defect molecules in the lattice. For 1Co₂, four Co sites with two non-magnetic Co(III) at the periphery and two Co(II) near the centre of the molecule are observed. In this scenario, with two neighbouring Co(II) ions, strongly interacting, our hypothetical model with large coupling (Fig. S12) would render no EPR transition (below < 50 GHz) as well. However, considering the charge balance intact within the molecule, the most plausible impurity/defect scenario is a swapping of ions between a Co(II) and a Co(III) sites. A model involving Co(II) (S = 3/2) only at 2^nd and 4^th site (and parallelly Co(II) only at 1^st and 3^rd site, see Fig. S13 and schematic in Fig. 1a) seemed most convincing to explain the μSQUID-EPR peaks (Fig. 3), especially after noticing that the CASSCF simulations for such a model yields angles that very well match with that obtained from the μSQUID-EPR angular maps on 1Co₂ (vide infra). This would mean that a small fraction of molecules exhibits a very weakly interacting S = 3/2 dimer (at 2^nd and 4^th site, or 1^st and 3^rd site), accounting for the μSQUID-EPR maps. While the majority of 1Co₂ molecules (strong interaction) with Co(II) at 2^nd and 3^rd site account for the observed in DC SQUID data.

The impurity in 1Co₂ was modelled employing eqn (3) by considering the Co(II) to be located at 2^nd and 4^th sites, with the input of CASSCF at these locations. However, as the ligands at these two sites are not identical, the CASSCF yields very different anisotropic parameters between these two sites. We obtain D₁ = 33.03 cm⁻¹, E₁ = 5.51 cm⁻¹ for ‘site 2’ and D₂ = −71.04 cm⁻¹, E₂ = −21.42 cm⁻¹ for ‘site 4’. The Euler angles for the largest anisotropy axis (for 2^nd and 4^th sites), as obtained by CASSCF simulations (Fig. S13 and Table S8) were Co¹ ≈ [−41 49 60]° and Co² ≈ [70 42 131]°. Using fixed D_1,2, E_1,2 and these angles as initial values, we finally fit the μSQUID-EPR maps as shown in Fig. 3b and d. The fitting yields g₁ = [2.54, 2.63, 2.40], g₂ = [3.43, 2.27, 2.24] (with Δg_max = 0.07); Euler angles (with respect to the lab frame H_x,y,z) for Co¹ = [−41.6 49.4 59.5]° and the Euler angles for Co² = [64.0 41.8 130.7]° (with Δθ_max = 1°). The excellent matching with the CASSCF obtained angles clearly indicates that the measurement plane was very closely related to the ‘a–c’ crystallographic plane (see Fig. S13: crystallographic plane projected on the μSQUID plane). The case for Co(II) at 1^st and 3^rd sites was also analysed (Fig. S15) and found to complete the observed features in the angular maps in Fig. 3. In addition, a small intermetallic interaction (as expected assuming only dipolar contribution between next nearest sites) is estimated as the dominant contributor to the broadening of the peaks (see section G.2 in SI). The estimated interaction between the Co(II) placed at 2^nd and 4^th sites: J ∼ 0.003 cm⁻¹ was of the same order of magnitude compared to purely dipolar interaction ∼0.007 cm⁻¹, considering the distance and tilt between 2^nd and 4^th sites (Table S9).⁴⁹ The two ellipses visible in Fig. 3a and b have nearly similar major and minor axes, indicating that both the hard axes of the two Co(II) ions in 1Co₂ lie symmetrically close to the measurement plane, a situation comparable to the line cuts of the surfaces shown in Fig. 5c. On the other hand, the frequency map in Fig. 3c and d displays a linear behaviour indicative of the Zeeman splitting, with a major contribution from (0,1); however, the resfields for the transition (0,2) also lie very close at this angle (grey and green lines in Fig. 3d). Some additional features, which were very weakly visible in the frequency maps in Fig. 3c, could be explained as well, as the case for Co(II) at 1^st and 3^rd sites was also included (see Fig. S15). The weak EPR signal in 1Co₂ leads to a large margin of error in this analysis; however, it points to the impurities, which are only a small fraction of the molecules. X-band EPR and μSQUID-EPR intensity analysis (phenomenological) both hint at ∼1.8% impurity present in 1Co₂, supporting our model (see sections I, J in SI for details). Although studies of the impurity-free compound would be ideal, the fact that the Co(III) sites are formed in situ during the reaction conditions precludes the synthesis of such systems.

For 2Co₂, however, a different scenario is found: a large μSQUID-EPR signal (and large X-band EPR intensity compared to 1Co₂, see section J in SI) in this sample rules out the possibilities that involve modelling with an impurity molecule. We further notice that for 2Co₂ CASSCF calculations yield parallel anisotropy axes between the two Co(II) within a molecule, but two directions within the packing (schematic in Fig. 1b and Fig. S13), hence confirming that the visible two orientations in the μSQUID-EPR angular maps (Fig. 4) are due to the inter-molecular (packing) angle. Due to the parallel anisotropy axis and large interaction within a molecule as predicted by SQUID measurements, each molecule could be represented by an effective spin. The non-isotropic nature and the magnitude of the interaction (not infinitely large, but comparable to D, E) prohibit an exact total spin representation. We also found that such effective spin (one per molecule) model with appropriate parameters can mimic the low-lying states of the more general model, i.e. a 4-spin Hamiltonian (strong intra and weak inter molecular coupling) considering two molecules at an angle. Hence, in the upcoming discussion, we model 2Co₂ as an effective spin S = 3/2 dimer exhibiting intermolecular angles and small intermolecular interactions (also see section G.2 in SI).

For the 2Co₂, the ΔM(H,θ) maps (Fig. 4a–c) show two ellipses with different ellipticities, one appearing almost like a circle. Based on the angular and frequency maps, the intermolecular interactions can be ignored for the primary stage of analysis. Furthermore, the two ellipses with different ellipticities shown in Fig. 4a–d indicate that the measurement plane is not symmetric concerning the two molecular orientations. This situation can be compared to the line cuts of the surfaces in the theoretical model in Fig. 5d. Fitting the angular maps and the frequency maps leads to the final set of parameters for this system as: D_1,2 = −68.64 cm⁻¹, E_1,2 = −21.52 cm⁻¹, g₁ = [3.75, 3.01, 2.35], g₂ = [3.73, 3.14, 2.50], (with Δg_max = 0.08); Euler angles (with respect to the lab frame H_x,y,z) for the two orientations of molecules = [48, 53, −25], [91, −58, 9] (with Δθ_max = 1°) leading to an angle between the two easy axis ∼100°. The obtained value is close to that calculated from CASSCF for the molecules residing in the unit cell (∼88°). In this case, it is not possible to unambiguously determine the measurement plane of the crystal. Similarly to 1Co₂, a weak intramolecular Co–Co interaction is necessary to describe the weakly visible lines in the absorption maps (Fig. S16e and f). The estimated intermolecular interaction J ∼ 0.02 cm⁻¹ (anisotropic J = [0.020, −0.015, −0.005]) was larger than the purely dipolar interaction with the order of magnitude ∼0.004 cm⁻¹ considering the intermolecular distances and tilt (Table S9).⁴⁹ This possibly indicates that the orientation of pyrazine part in the ligands allows a mild π–π intermolecular exchange interaction (see Fig. S6).

Since both systems are eventually modelled with two weakly interacting spin 3/2, the possible EPR transitions can be commonly marked by comparing with the hypothetical case in Fig. 5(b) associated with eqn (3). In this Zeeman diagram, six transitions are expected for a constant frequency of excitation: one occurring between the zero and 3^th state (0,3) (green arrow in Fig. 5b), a second transition between the zero and 2^nd state (0,2) (blue arrow in Fig. 5b), a third transition between the zero and 1^st (0,1) (red arrow in Fig. 5b). Due to the symmetry of the Zeeman diagram and the absence of intermolecular interaction, two other transitions occur at the same field, i.e., a transition between the 1^st and 3^rd state (1,3) and a transition between the 2^nd and 3^rd state (2,3), with the same resonance field as transitions (0,3) and (0,2), respectively. A 6^th transition would also be expected between the 1^st and 2^nd states (1,2); however, it would occur in a much larger field than our experimentally accessible conditions. At our working temperature, it is safe to conclude that the strongest peaks would be associated only with (0,1) and (0,2), corresponding to transitions from the ground state.

Once the fitting parameters from the angular and frequency maps for both complexes are obtained, the exact Zeeman diagrams for two specific angles could finally be simulated. Fig. 3e, f and 4c (panels) show the Zeeman diagram for two appropriate angles with respect to the crystal, which represents the two directions in the measurement plane (0° and −30° with respect to the magnet H_x axis) indicated by white and red dashed lines (Fig. 3) and red and orange dashed lines (Fig. 4). In these diagrams, a vertical bar is fitted to the gap between any two states in resonance with the microwave energy (shown only for the negative fields in Fig. 4c). The results show that the experimental peaks are exactly at the resfields for transitions (0,1) and (0,2) and explain the angular evolution of the observed peak positions.

It is worth noting that despite the qualitative similarities in the experimental results and theoretical modelling between 1Co₂ and 2Co₂, the origin of the two weakly interacting spins (oriented at an angle) in these two systems is very different. The results in 1Co₂ highlight that a strongly coupled system, which is usually EPR silent, can exhibit some weak resonant absorptions in the presence of impurities involving magnetic ions at larger distances. It also signifies that weakly coupled systems are interesting for their multiple states addressable within the usual microwave frequency range.⁵⁰ On the other hand, the results in 2Co₂ showcase the role of intermolecular tilt angles and interactions on the addressable quantum states in single-crystalline molecular systems, and the capability of the angle and frequency resolved μSQUID-EPR technique to identify these details.

An alternative approach to describe the system is by an effective spin dimer (S_eff: 1/2, 1/2) with anisotropic g values compensating the effects of D, E anisotropies to the ground multiplet. This approach leads to a nearly similar quality of fitting; however, it is a simplified view of these systems and does not provide complete information about the spin Hamiltonian (see SI, Fig. S17). Due to the limited accessibility to exact D, E values with our present μSQUID-EPR scheme, the use of the CASSCF results drastically enhances the fitting reliability and decreases the fitting times. Despite these limitations, finally we show that our models are reliable at least within the scope of low frequency EPR by measuring conventional X-band EPR spectrum on the powder sample of each system (Fig. S20) and simulating these with the same parameters used on corresponding μSQUID-EPR maps. Note that the access to these parameters, with fine details such as intermetallic and intermolecular angles, and pinpointing the type of crystal impurity discussed in this study, is commonly difficult, and the number of parameters often precludes the correct and comprehensive understanding of multi-spin systems.

Conclusion

In this study, two Co(II)-based dimeric spin systems were thoroughly characterised by reconciling conventional magnetic measurements down to T = 2 K with frequency- and angle-resolved μSQUID-EPR spectroscopy conducted at 30 mK. Initially, a stark contrast was found between conventional SQUID data exhibiting strong interactions in the dimer and μSQUID-EPR maps suggesting weak interactions. The contradiction was resolved by investigating the μSQUID-EPR maps with inputs from CASSCF calculations, to find that the weak interaction was associated with spin components at large distances with a tilt between their anisotropy axis, coexisting with the strongly coupled Co(II) dimers. To resolve the puzzle of contrasting results, the Zeeman diagrams for hypothetical spin dimers, encompassing different regimes of interaction, were investigated. This offered a broad theoretical perspective on multi-spin systems and emphasised how weak coupling yields multiple spin states addressable with conventional microwave frequency ranges (see Fig. S18). This behaviour differs in strongly coupled systems, where the large energy separation between molecular states limits accessibility. As a result, when dipolar interactions are weak, new states become available for selective manipulation.⁵⁰

A critical takeaway from this study is that when intermetallic (or intermolecular) coupling is weak and the anisotropy axes are tilted relative to one another, simultaneous diagonalisation of the spin Hamiltonian or the use of effective total spin approaches becomes infeasible. Hence, while analysing EPR transitions on such a single molecular crystal, it is important to consider inter-molecular tilt angles among the participating spins, as well as possible impurities, as these can even dominate the observed peaks.⁵⁰ Notably, our μSQUID-EPR maps demonstrate the capability to resolve inter-molecular tilt angles and interactions as subtle as 0.01 cm⁻¹, as shown in 2Co₂ It is worth noticing that the SQUID and μSQUID-EPR techniques appear not as conflicting but rather as complementary, i.e., during the fitting of SQUID data, the weak interactions can be assumed negligible in comparison to the strong interactions; conversely, strong interactions exceed the measurement limits of μSQUID-EPR techniques. Hence our approach exemplifies how the integration of magnetisation and spectroscopic measurement techniques can transcend the limitations of individual methods, prevent misinterpretation and provide a comprehensive view of complex molecular spin systems.

Author contributions

The idea was conceived by S. P., S. C., M. D., and W. W. The project was supervised by W. W. R. S. S., S. C., J. N. G. and M. D. synthesised the samples. G. N. and E. M.-P. carried out the DC SQUID data collection, analysis and processing. S. P., A. S. and W. W. carried out the μSQUID and μSQUID-EPR data collection. B. B., F. E. and M. G. B. D. carried out crystallographic studies, data reduction and refinement. E. M.-P. carried out the CASSCF calculations and interpretation. The data was analysed and interpreted by S. P. and E. M.-P. with input from all co-authors. The manuscript was written by S. P. and E. M.-P. with input from all co-authors.

Conflicts of interest

The authors declare no conflict of interest.

Data availability

All the magnetic data was processed employing Origin Pro 2023. The μSQUID-EPR data was all processed using an in-house built software based on Python. The simulation of the μSQUID-EPR was carried out employing an in-house code which works in conjunction with Easyspin implemented in MATLAB.⁴⁷

The data supporting this article has been included as part of the SI: cif files, further synthetic details, structural and magnetic plots. See DOI: https://doi.org/10.1039/d5qi01387a.

CCDC 2387470 and 2388164 contain the supplementary crystallographic data for this paper.^51a,b

Acknowledgements

We acknowledge the DFG-CCR 1573 “4f for future” (project B4) and the Karlsruhe Nano Micro Facility (KNMF, https://www.kit.edu/knmf) for the provision of access to instruments at their laboratories. E. M.-P. thanks the Alexander von Humboldt Fellowship for experienced researchers for support. W. W. thanks the German Research Foundation (DFG) for the Gottfried Wilhelm Leibniz-Award, ZVN-2020_WE 4458-5. The authors are grateful to Dr. Alexey Alfonsov, IFW Dresden, and Prof. Rüdiger Klingeler, Heidelberg University, for X-band EPR measurements and discussion.

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