Anisotropy of CoII transferred to the Cr7Co polymetallic cluster via strong exchange interactions

In the Cr7Co model-system the anisotropy of CoII is effectively transferred to the whole cluster through strong and anisotropic exchange interactions.


Introduction
Single-Molecule Magnets (SMMs) are the rst generation of molecular nanomagnets and have been considered as possible means to store information in single molecules since the early 1990s. 1,2 Indeed, the combination of a large magnetic moment and a strong easy-axis magnetic anisotropy creates in these systems a double-well potential with an energy barrier for the reorientation of the magnetization, resulting in magnetic bistability for the single molecule below the so-called "blocking temperature". Thus, increasing the temperature range where molecules display SMM behaviour, in order to build miniaturized memory units, is one of the main goals of the current research in molecular magnetism. Originally this was sought in exchange-coupled 3d-metal cage complexes, by exploiting highspin ions like Fe III , Mn II or Cr III to increase the total-spin of the molecular ground state. [3][4][5][6][7] However, the quenching of the orbital momentum and the resulting small single-ion anisotropy have strongly limited the maximum achievable barriers. Recently it has been shown that monometallic complexes based on 3d transition metal ions can exhibit appreciably enhanced magnetic anisotropy with certain axial coordination environments that minimize Jahn-Teller distortions, leading to an almost unquenched orbital momentum. 8 The second attempt has been to use the large anisotropy of rare-earth ions. Indeed, aer the discovery of SMM behaviour in mononuclear Tb III and Dy III complexes, 9,10 much focus has been on the design of highly-axial environments for single lanthanide ions, with the aim to reduce the efficiency of the tunnelling mechanism and to increase the blocking temperature of the molecule. [11][12][13][14][15] This has led to a remarkable recent report of a Dy III sandwich complex that retains magnetization to 60 K. 16 These advances with mononuclear lanthanide-based SMMs are important but distract from one crucial issue that has only been partially addressed, that is understanding magnetic exchange interactions in molecules containing highly anisotropic ions. This is important not only in dinuclear Ln III complexes, where the SMM behaviour results from the interplay between ligand elds and exchange interactions, 17,18 but also in more diverse areas ranging from quantum information to biology. For example, highly anisotropic Fe II centers are found in the active sites of proteins. 19 Thus, we have focused our attention towards polymetallic complexes containing Co II , as test systems to investigate anisotropic exchange interactions. Indeed, Co II in the proper coordination environment can be highly anisotropic thanks to its non-quenched orbital moment.
SMMs containing Co II ions have been largely investigated in the last years, thanks to their potential for the production of very large magnetic anisotropies and thus higher blocking temperatures. [20][21][22] However, the interpretation of magnetic data for polynuclear complexes containing Co II centres is oen difficult, due to their important orbital contribution. In particular, there is now a literature on Co II -based SMMs, but the precise nature of the magnetic exchange in these systems is still a crucial open question that need to be investigated. 20 There have been several papers studying homometallic complexes including Co II23,24 and lanthanides, 25 where exchange interactions have been investigated with different spectroscopic techniques, but studies where a highly anisotropic ion is included in a larger polymetallic complex remain rare. Anisotropic ions (such as Co II ) coupled with other magnetic ions are important building blocks to implement efficient and scalable quantum information schemes. 26 Moreover, this kind of compounds is expected to display rich physics associated with the unquenched orbital degrees of freedom. For instance, in antiferromagnetic highly anisotropic ring-like clusters the lowfrequency dynamics should be characterized by Néel Vector Tunneling. [27][28][29][30][31][32] Thus, it is important to understand in detail the anisotropic exchange interactions involving a Co II ion embedded in a large polymetallic cluster.
In this work we exploit the octagonal heterometallic ring [NH 2 Me 2 ][Cr 7 CoF 8 (O 2 CC t Bu) 16 ] (hereaer Cr 7 Co ring) as a model system to understand how the insertion of an anisotropic Co ion strongly coupled to the Cr ones affects the anisotropy of the whole molecule. Cr-Cr exchange interactions and the Cr zero-eld splittings have been already determined in the isostructural Cr 8 and Cr 7 M (M ¼ Ni, Mn, Zn) compounds. [32][33][34][35] Thus, the combination of Electron Paramagnetic Resonance (EPR) and Inelastic Neutron Scattering (INS) techniques with ab initio and spin Hamiltonian calculations, 36 allows us to investigate in detail the anisotropy of the Co II ion embedded in the antiferromagnetic ring and the Cr-Co anisotropic exchange interactions. By combining EPR measurement on the isotructural Ga 7 Co compound with ab initio calculations, we demonstrate that the Co II ion in this environment is highly anisotropic and we are able to determine the spectroscopic splitting tensor g Co . We have then performed INS measurements on both powder samples of Cr 7 Co at different temperatures and on single-crystal samples with different applied magnetic elds. Neutron scattering intensities on single-crystal samples have also been measured as a function of energy transfer and neutron momentum transfer (Q x , Q y , Q z ) with the 4-dimensional INS (4D-INS) technique. 32,37 Guided by ab initio calculations based on the density-functional theory + many-body approach (DFT + MB), 36 and thanks to the thorough set of INS measurements described above, we determine the Cr-Co isotropic coupling J and anisotropic exchange tensor D. In agreement with ab initio calculations, we nd that Cr 7 Co is characterised by a strong and highly anisotropic Cr-Co exchange interaction, which effectively transmits the anisotropy of Co II to the whole molecule. As discussed in the last part of the paper, these results suggest a promising way to reach the strong coupling regime between photons and individual molecules, 38 a crucial step for building a scalable molecule-based quantum information architecture. The structure of Cr 7 Co is an octagon of metals bridged on each M/M edge by one uoride and two pivalate ligands ( Fig. 1). Therefore each metal is six-coordinate, with a F 2 O 4 coordination sphere. At the centre of the ring is a dimethylammonium cation where the protonated N atom is involved in hydrogen-bonding to bridging uorides. Using this cation and solvents the Cr 7 Co ring crystallises in a tetragonal crystal system with a four-fold rotation axis passing through the N-atom of the cation. This four-fold axis leads to the Co II ion being disordered about the eight metal sites of the ring, and to disorder of the cation.

Spin Hamiltonian model
The Cr 7 Co ring contains seven isotropic Cr III ions, behaving as pure spin s i ¼ 3/2, and one anisotropic Co II ion, that can be described at low temperatures as an effective spin s Co ¼ 1/2, with highly anisotropic g tensor and exchange interactions 40 Fig. 1 Laboratory/unit cell reference frame for the Cr 7 Co ring (black) and local reference frames of the anisotropic exchange tensor D local (light blue) for the s Co -s 7 interaction and of the spectroscopic splitting tensor g local (red), assuming the Co ion on site 8.
(see the following Section). The magnetic properties of Cr 7 Co can therefore be described by the following spin Hamiltonian (with the Co ion on site 8): The rst two terms in eqn (1) describe the dominant isotropic exchange interactions, while the third term corresponds to the axial single-ion zero-eld-splitting term acting on Cr ions (with the z-axis perpendicular to the plane of the ring). The fourth and h terms take into account the anisotropic exchange interaction of the Co II ion with the two neighbouring Cr ions (labelled as s 1 and s 7 respectively), which is described by the traceless diagonal tensor D local , referred to its local principal axis on each Cr-Co bond, with diagonal elements D yy , D zz and D xx ¼ À(D yy + D zz ). The last terms in eqn (1) describe the Zeeman interaction with an applied magnetic eld, where the spectroscopic splitting tensor is assumed isotropic for Cr III (g Cr ¼ 1.98) and anisotropic for the Co II ion (g local ). The local principal axis for both D local and g local tensors have been determined by the DFT + MB approach described below and are depicted in Fig. 1. The matrices R 8,1À7 and R 0 8 in eqn (1) transform the D local and g local tensors from their local reference frames to the laboratory/ unit cell reference system (see Fig. 1). At last, the delocalization of the Co II ion along the ring in the crystal (equal probability on each site) is taken into account in modeling the experimental results. Cr-Cr exchange interactions and Cr zero-eld splittings have already been determined in the isostructural Cr 8 and Cr 7 M (M ¼ Ni, Mn, Zn) compounds, yielding J Cr-Cr ¼ 16.9 K and d Cr ¼ À0.44 K. [33][34][35] Hence, here we can focus on Co II and Cr-Co interactions to extract J Cr-Co , D local and g local by combining EPR and INS data with the spin Hamiltonian and ab initio calculations.

Ab initio DFT + MB results
Ab initio calculations based on the DFT + MB approach described in ref. 36 have been used as a guide for tting the spin Hamiltonian parameters. In contrast with conventional DFT approaches, relying on approximations of the exchangecorrelation functional (the most popular being LDA/GGA or their extensions, such as the hybrid B3LYP), here strong electron-electron correlations are explicitly included in the description of the low energy 3d electrons. This is achieved by constructing a system specic generalized Hubbard model (see details in the ESI †), explicitely including a Coulomb tensor, as well as inter-site electron hopping, crystal eld splittings and spin orbit effects. The parameters of this model are deduced ab initio, by means of DFT based calculations either in the localdensity (one-electron terms) or in the constrained-localdensity approximations (Coulomb tensor). In a second step, the spin Hamiltonian is obtained by a canonical transformation of the Hubbard model, without any a priori assumption on its form.
This method ensures an accurate description of strongly correlated systems (such as MNMs), as shown also by a recent study of a family of Cr 7 M compounds isostructural to Cr 7 Co. 35 Co II (3d 7 electronic conguration, S ¼ 3/2) is embedded in a distorted octahedral cage of ligands (O and F). In perfect octahedral symmetry, spin-orbit coupling would split the twelve-fold degenerate (l ¼ 1, S ¼ 3/2) ground multiplet into a doublet, a quartet and a sextet. 40 Here the distorted octahedral environment leads to a sequence of Kramers doublets, the lowest two being separated by about 180 K according to ab initio calculations (see details in the ESI †). Thus, at low temperatures we can restrict to the lowest-energy doublet and describe it as an effective spin 1/2. In this subspace, we obtain the spin Hamiltonian in eqn (1), with the principal axes of the D local and g local tensors sketched in Fig. 1. Our rst-principles calculations predict an isotropic Cr-Co exchange constant of J Cr-Co ¼ 21 K and anisotropic exchange with D zz ¼ À11.2 K and D yy ¼ 10.4 K leading to (in K): These results show two directions where the Cr-Co anisotropic exchange is strong but opposite in sign and a third direction where it is almost one order of magnitude weaker. Finally, we obtain g local ¼ (5.4, 3.3, 2.9). These parameters have been used as a guide for the nal determination of the spin Hamiltonian based on EPR and INS data. DFT + MB calculations also indicate the presence of a signicant Dzialoshinski-Moriya (D-M) interaction. However, here this interaction acts only at the second-order through the S-mixing 41 and its effects are much weaker than those of anisotropic exchange. Thus, in the following we only consider the isotropic and anisotropic parts of the exchange interaction to keep the model with the minimum number of parameters.

EPR and INS spectroscopy
EPR measurements have been performed on powder samples of the isotructural compound Ga 7 Co (see the ESI †). Since Ga III is diamagnetic, these measurements have allowed us to selectively target the Co II ion embedded in the same environment as in the Cr 7 Co ring. In Fig. S2 of the ESI † we report Q-and W-band EPR spectra, whose main features can be reproduced by assuming a very anisotropic g local for Co II , with principal values (6.8, 2.9, 2.7). The results obtained with the DFT + MB method are in good agreement with the EPR ndings and are used to determine the principal axis of the spectroscopic splitting tensor for the Co II ion, reported in Fig. 1, yielding g x 00 Co ¼ 6:8, g y 00 Co ¼ 2:9 and g z 00 Co ¼ 2:7. These orientations are conrmed by the good agreement with single-crystal INS measurements in applied elds (see below). As a rst step to investigate the Cr 7 Co ring, we have surveyed the full energy spectrum of a non-deuterated powder sample on the time-of-ight spectrometer IN5 at the Institute Laue Langevin in Grenoble. 42 Measurements have been performed at three different temperatures, T ¼ 1.5, 6, 15 K, with two different incident neutrons energies E i , 1.9 and 8 meV (see Fig. S3 in the ESI †). The high-resolution/low-energy-transfer spectra show two cold transitions around 0.5 meV, whereas higher-energy data show magnetic excitations up to 5 meV, with three cold peaks at 1.8, 3.7 and 4.8 meV. The same measurements at T ¼ 1.5 K have also been performed on a single-crystal sample of Cr 7 Co, where we have exploited the position-sensitive detectors of IN5 to implement the four-dimensional inelastic neutron scattering (4D-INS) technique. 32 Thanks to this technique it is possible to measure the INS cross-section not only as a function of energy, but also as a function of the three component of the momentum transfer vector Q (see also the ESI †). The information accessible with this technique enables us to characterize the eigenstates of the system involved in the detected excitations. 43 INS single-crystal spectra as a function of the energy-transfer, integrated over the full Q-space are reported in Fig. 2 and show the same magnetic transitions measured on the powder sample. In this conguration we have been able to obtain one crucial additional information on the Cr 7 Co energy spectrum, by detecting a low-energy transition below 0.1 meV (see the shoulder in Fig. 2). In order to check its magnetic origin, we have extracted the dependence of the INS intensity of this excitation on the two horizontal wavevector components Q x À Q y , integrated over the full experimental Q z range (À0.2 to 0.2 A). The so-obtained experimental intensity map ( Fig. 3-a) shows clear Q-dependent modulations whose pattern of maxima and minima clearly identify a magnetic transition. The intensity maps have been extracted also for the 0.5 meV peak (Fig. 3-c) and for the 1.8 meV excitation (Fig. S4 of the ESI †). These maps show the same pattern of intensity modulations in the explored Q range and all the maxima in each map have the same intensity, demonstrating the same occupation probability for Co II of all the ring sites. Indeed, the differences in the intensity pattern of each excitation are averaged-out by the delocalization of Co II along the ring (intensity maps calculated with the Co ion sitting on one single site are reported in Fig. S5 of the ESI †).
INS experiments on IN5 provide information on the lowlying energy levels of Cr 7 Co up to 6 meV and measurements in temperature allow us to distinguish between transitions involving or not the ground state. However, they are not sufficient to unambiguously identify the Cr-Co exchange parameters. Indeed, IN5 data can be reproduced with a high-anisotropy model (in agreement with DFT + MB results), but also with a weak-anisotropy one. In both cases the antiferromagnetic exchange interactions lead to a S ¼ 1 ground multiplet. In the high-anisotropy case both the 0.1 meV and 0.5 meV INS peaks in Fig. 2 are due to intramultiplet transitions within the ground S ¼ 1 manifold and therefore a strong and rhombic anisotropic Cr-Co exchange is responsible for the large 0.5 meV splitting of the ground multiplet. In this case the cold transition at 1.8 meV is the lowest-energy intermultiplet transition, yielding a large J Cr-Co value, of the same order as J Cr-Cr . Conversely, if we assume  that the 0.5 meV peak is the lowest intermultiplet transition, we have to consider a weak Cr-Co anisotropic exchange leading to a small splitting of the S ¼ 1 ground state multiplet of only 0.1 meV. In order to solve this puzzle, we have performed a targeted INS experiment in applied magnetic eld. We have measured a single crystal sample of Cr 7 Co on the LET 44 spectrometer at the ISIS facility, Rutherford Appleton Laboratory (Didcot, UK), at T ¼ 1.8 K and with magnetic elds B ¼ 0 T, 2.5 T, 5 T and 7 T (experimental details are given in the ESI †). The behaviour of the INS excitations as a function of the applied magnetic eld enables us to distinguish the two situations and to evaluate the anisotropy of the molecule. Fig. 4-a shows the high-resolution spectra collected on the LET spectrometer with an incident neutron wavelength E i ¼ 1.5 meV and with different applied magnetic elds, while the higher energy-transfer spectra with E i ¼ 3 meV are reported in Fig. 4-b. The cold transitions at 0.5 meV and 1.8 meV are clearly visible in the zero-eld spectra. A decisive information about the anisotropy of the Cr 7 Co has been provided by the magnetic-eld dependence of the position of the lowest-energy cold peak, detected at 0.1 meV at zero-eld (see Fig. 2). This transition is not visible in the B ¼ 0 T and B ¼ 2.5 T LET spectra, as it is underneath the elastic peak. Nevertheless, the peak moves to higher energies by increasing the magnetic eld and becomes clearly visible at 0.3 meV with B ¼ 7 T (see Fig. 4-a). This lowest-energy INS excitation corresponds to a transition within the S ¼ 1 ground multiplet in both the highanisotropy and weak-anisotropy model. The small observed energy shi (from 0.1 meV to only 0.3 meV) with the application of a strong a magnetic eld of 7 T can be explained only by the high-anisotropy model. Indeed, a much larger eld-induced shi is expected in the weak-anisotropy one, almost twice as large as the one detected by INS measurements (energy levels as a function of the applied magnetic eld calculated with the weak-anisotropy model are reported in Fig. S7 of the ESI †).
The parameters of the spin Hamiltonian in eqn (1) have been determined by tting all the INS data reported in Fig. 2 and 4, which are very well-reproduced with J Cr-Co ¼ 19 AE 2 K and a strong and rhombic Cr-Co anisotropic exchange term with D zz ¼ À10 AE 1 K and D yy ¼ 13.0 AE 1 K leading to (in K) As in the DFT + MB results, there are two directions where the Cr-Co anisotropic exchange is strong but opposite in sign and a third one where it is almost one order of magnitude weaker. The spin Hamiltonian results also yield a strong Cr-Co isotropic exchange interaction and are therefore in good agreement with DFT + MB predictions. It is worth to note that this model also reproduces the IN5 intensity maps in Fig. 3 and S4 of the ESI. † 32,37,43 The calculated low-lying energy levels and their magnetic eld dependence are reported in Fig. 5.

Discussion
The analysis of the EPR and INS experimental data supported by our DFT + MB calculations have allowed us to characterize the anisotropy of the Cr 7 Co ring, which has proven to be an ideal test system to study anisotropic exchange interactions. Our results demonstrate that the insertion of an anisotropic Co ion strongly coupled to the Cr ones determines the anisotropy of the whole molecule. Indeed, the strong and rhombic Cr-Co anisotropic exchange leads to a splitting of the S ¼ 1 ground multiplet D ¼ 0.5 meV, which is ve times larger than the splitting of the S ¼ 1 ground manifold of the parent compound Cr 7 Mn. 33 Since Mn II is essentially isotropic in Cr 7 Mn, the comparison between these two experimental splittings is practically a direct comparison between the effects of Cr-Co anisotropic exchange and of the seven Cr single-ion anisotropies. The ZFS due to the Cr ions is very similar in both Cr 7 Mn and Cr 7 Co and leads to a very small splitting of the S ¼ 1 ground state. Indeed, if we neglect the Cr-Co anisotropic exchange, we obtain a splitting of about 0.1 meV in Cr 7 Co, practically identical to Cr 7 Mn. Thus, by inserting only one anisotropic ion we obtain a ve times larger splitting of the ground state, conrming that the experimentally observed splitting is mostly due to the anisotropic exchange between Co II and the neighbouring Cr ions. The anisotropy of Cr 7 Co is also responsible for a strong S-mixing 41 in the eigenstates of the molecule and therefore the total-spin S is not a really good quantum number for the full spin Hamiltonian in eqn (1). For instance, the strong Cr-Co anisotropic exchange leads to a mixing between the S ¼ 1 ground multiplet and excited multiplets of about 10% (mainly with the lowest S ¼ 2 manifold). This strong S-mixing signicantly affects the physics of Cr 7 Co leading, for instance, to oscillations of the total spin 45,46 for specic eld values. Large anticrossings associated with these oscillations are induced by a magnetic eld applied perpendicular to the plane of the ring (see Fig. S8 of the ESI †). In addition, S-mixing can play an important role in the dynamics of the Néel vector. 28 It is worth noting that the effective anisotropic exchange interaction in eqn (1) originates from the combined effect of the Co II ion zero-eld splitting and of a real Cr-Co anisotropic exchange. In fact, our DFT + MB results show that the g Co tensor, which reects the single-ion anisotropy of the Co II , and the D local tensor, accounting for the effective Cr-Co anisotropic exchange, have different local principal axis (see Fig. 1). In order to better understand the effects of the Cr-Co anisotropic exchange on the magnetic behaviour of the molecule at low temperatures, we have applied second-order perturbation theory to project the Cr-Co anisotropic exchange interactions onto an effective S ¼ 1 multiplet. 41 With this procedure we have determined the principal axis of the effective anisotropy tensor acting on the S ¼ 1. The easy-axis of the effective anisotropy lies within the plane of the ring and points in the radial direction from the centre of the ring towards the Co ion, whereas one of the two hard directions is perpendicular to the plane of the ring.
This study has allowed us to understand the nature of the Cr-Co anisotropic exchange interactions in the Cr 7 Co ring and their effects on the magnetic behaviour of the molecule. Moreover, it also represents an important starting point for the design of new systems where strong exchange interactions transmit the large anisotropy of Co II ions to the whole molecule. As a rst attempt, more than one Co ion can be embedded in the antiferromagnetic ring to further increase the anisotropy of the cluster. 47 Having fully characterized the Cr-Co anisotropic exchange, we are able to design the molecule in order to maximize the anisotropy-induced splitting of the ground states (which will be an S ¼ 2 with two Co II ions within the ring). For instance, Co ions on opposite sites (e.g. sites 1 and 5 in Fig. 1) will produce a splitting about 40% larger than that obtained when the two Co ions have one Cr III in between (e.g. sites 6 and 8 in Fig. 1). § The inclusion of anisotropic ions strongly coupled to highspin ones (like Mn II , Fe III or Cr III ) in polymetallic clusters is important also in view of exploiting MNMs for quantum information processing (QIP). In particular, a crucial milestone would be to reach the so-called strong coupling between a single magnetic molecule and the quantized magnetic eld (photons) of coplanar superconducting resonators. 38 Indeed, this would allow the local control and read out of the molecule magnetic state 38,52 and to implement QIP schemes similar to those used for superconducting qubits. 53,54 The large spinphoton coupling needed to reach the strong coupling regime could be achieved with a molecule characterized by a large total spin and by a strong easy-plane anisotropy. 38 These are oen conicting requirements, but the present results demonstrate that the inclusion of one or more anisotropic 3d ions in a high-spin polymetallic complex is a promising way of achieving this very important goal. At last, anisotropic ions coupled with molecular qubits can be used as a switch of the effective qubit-qubit interaction in the implementation of quantum gates. 26

Conclusions
We have studied the Cr 7 Co ring as a model system to understand how the insertion of an anisotropic Co ion strongly coupled to the Cr ones affects the anisotropy of the whole molecule. Since the ring is isostructural with the other previously studied Cr 7 M AF rings, we have been able to focus on Co II single-ion anisotropy and on Cr-Co anisotropic exchange interactions. This characterization has been possible thanks to a combined use of EPR and INS techniques with spin Hamiltonian and DFT + MB calculations. EPR measurements on Ga 7 Co, where Ga III is diamagnetic, have allowed us to determine the strong single-ion anisotropy of Co II embedded in the ring. Then, the Cr 7 Co energy spectrum has been surveyed with a broad set of INS experiments. In-eld INS measurements have been crucial to determine the Cr-Co anisotropic exchange parameters and therefore the anisotropy of the molecule. DFT + MB results have been used as a guide for tting the spin Hamiltonian parameters and their agreement with experiments is good, especially considering the presence of an highly-anisotropic ion like Co and the many anisotropic terms in the spin Hamiltonian.
We have found strong anisotropic exchange interactions between Co II and the neighbouring Cr ions. Thus, our results demonstrate that the anisotropy of Co II is efficiently transmitted to the anisotropy of the whole Cr 7 Co polymetallic cluster through strong effective anisotropic exchange interactions.
This study is also a starting point for the design of new systems, where strong exchange interactions transmit the large anisotropy of Co II ions to the whole molecule. On the one hand, a rich physics is expected in these systems, due to unquenched orbital degrees of freedom (e.g. Néel Vector Tunneling in the low-frequency dynamics). On the other hand, the combination of high-spin ions strongly coupled to a few very-anisotropic ions like Co II represents a promising route for building scalable quantum information architectures.

Conflicts of interest
There are no conicts to declare.