Quantum tunnelling of the magnetisation in single-molecule magnet isotopologue dimers

Low-temperature magnetic studies of two isotopologues dimers, with and without nuclear spins, reveal that, at very low temperatures, the nuclear spin facilitates the coupling to the phonon bath enhancing the direct relaxation process; observation reflected in the temperature and field dependence of the relaxation rates, whilst at higher temperatures the effect of the nuclear spins is less relevant.


Introduction
The highly anisotropic character of lanthanides and the strong effect of the ligands chelating the lanthanide ions resulted in the observation of mononuclear molecules exhibiting slow relaxation of the magnetisation, namely Single-Ion Molecule Magnets (SIMs), a subclass of Single-Molecule Magnets (SMMs). 1,2 The strong anisotropy and large energy barriers, along with quantum effects, have led to their proposed use in several technological applications ranging from data storage devices to quantum bits for quantum computers. 3 Depending on the desired application different characteristics are required. For example, large energy barriers to the reversal of the magnetisation (U eff ) and diminished Quantum Tunnelling of the Magnetisation (QTM) rates are necessary for data storage devices, whilst for the implementation of lanthanide-containing SMMs (Ln-SMMs) as qubits, an isolated electronic ground doublet state along with QTM provides access to the nuclear spins embedded in the lanthanide metal ion. 3a,4 On the data storage device side, a large separation between the ground state and the rst excited state would allow the molecule to preserve the stored information at high temperatures. In this regard, scientists have gained a deep insight into the general pre-requisites necessary for the design of molecules possessing large energy barriers; clearly exemplied by several molecules possessing extremely large U eff . 5 Unfortunately, despite the large U eff the magnetic properties of SMMs are oen hampered by the QTM, effect that allows the electronic spins to tunnel through the energy barrier following a non-thermally activated pathway. In turn, although large U eff can be obtained, in most cases the hysteresis loops of Ln-SMMs are practically closed at zero eld. 6 As consequence, out of the many Ln-SMM reported up today, molecules exhibiting large magnetic hysteresis remain scarce. 7 ‡ In spite the harmful effects for data storage device applications, QTM has been shown to play an important role in the successful implementation of SMMs in quantum information processing (QIP) schemes, where the nuclear spins embodied in the lanthanide are utilised as quantum registers. 4 In the nuclear spins scheme, the highly anisotropic character of the SMM isolates the ground doublet state, which is thenceforth coupled to the nuclear spins embedded in the lanthanide by the strong hyperne interaction; thus, the ground doublet state splits into (2I + 1) states, where I is the nuclear spin of the lanthanide. At some of these crossings QTM is active, consequently allowing the read-out and manipulation of the states of the qubit. 4 Remarkably, the multilevel character of the nuclear states contained in the lanthanides allows the operation of several states in a single unit. Systems possessing these characteristics are termed "qudits", where d represent the number of active states. 8 In both schemes, it is clear that QTM, as well as spin-lattice interactions, plays a major role on the magnetic behaviour of the molecular systems, 9 hence for the successful implementation of Ln-SMMs in any of these two applications, a deep understanding of relaxation effects is required. This has been evidenced by studies employing isotopically enriched lanthanide sources in mononuclear SMMs with moderate to high energy barriers, where nuclear spins are not entirely responsible for the observed fast tunnelling rates. 10,11 Herein, we study the effect of the nuclear spins on the dynamic properties of two isotopically enriched dysprosium dinuclear SMMs via AC magnetic susceptibility studies as well as single crystal m-SQUID data at sub-Kelvin temperatures. We nd the tunnelling probability to be equal for both isotopologue compounds; therefore, the effect of the nuclear spins is to span the avoided crossings over a larger eld range. Our results agree with recent reports of QTM studies of high energy barriers SMMs. 11 Nonetheless, although nuclear spins do not affect the QTM rate, we nd that these enhance the spin-phonon coupling, increasing the direct relaxation process in the SMMs.

Ab initio CASSCF-SO calculations
To develop a detailed picture of the electronic structure of 1 (I ¼ 5/2) and 2 (I ¼ 0) and to rationalise their magnetic properties, Complete Active Space Self-Consistent Field spin-orbit calculations of the CASSCF/SO-RASSI/SINGLE_ANISO [16][17][18][19] type were performed (see ESI † for details). Prediction of the electronic structure of the individual Dy(III) ions yields an isolated doublet ground state characterised by highly axial g tensors, i.e. g xx ¼ g yy z 0 and g zz z 20. The low-lying ligand eld states have the following order: m J ¼ AE15/2, AE13/2, AE11/2, AE9/2, with relative energies of 0, 188, 270, 310 K, respectively. The ensuing excited states are highly mixed and bunched over 380 to 730 K. Due to the site symmetry of the Dy(III) ions in 1 (I ¼ 5/2) and 2 (I ¼ 0) , the anisotropic magnetic axes are parallel (see Fig. 1a). In addition, the average values of the matrix elements of magnetic moment connecting the electronic states ( Fig. S4 †) show lower tunnelling rates between the ground doublet |AE15/2i state, while higher transition rates occur states at higher energy. In turn, the most probable thermally activated relaxation pathway would involve spin-phonon excitation to the rst, second and third excited doublets, followed by relaxation to the opposing ground state. The highly axial character of the ground state obtained by CASSCF agrees with the observed SMM behaviour for the non-isotopically enriched analogue and complexes here studied (vide infra). 12 Low-temperature m-SQUID studies CASSCF calculations predict 1 (I ¼ 5/2) and 2 (I ¼ 0) to be SMMs, with relaxation pathways active through the rst, second and third excited states. In order to understand the relaxation dynamics of the complexes, and to minimise the complexity of the possible relaxation pathways taking place in the SMMs, we rst investigate the nuclear spin effect on the magnetic properties of 1 (I ¼ 5/2) and 2 (I ¼ 0) via m-SQUID studies at very low temperatures, where thermally activated processes are expectedly less effective. m-SQUID measurements were performed on single crystals of 1 (I ¼ 5/2) and 2 (I ¼ 0) with the eld applied along the main anisotropic axis, employing the transverse method. 14 Hysteresis loops studies were performed at different sweep rates and temperatures ( Fig. 2 and S4 †). Well-resolved two-steps hysteresis loops were obtained for 1 (I ¼ 5/2) and 2 (I ¼ 0) with the width of the hysteresis loops increasing with decreasing temperatures and increasing sweep rates, conrming the SMM behaviour of the complexes.
The loops are very typical for two antiferromagnetically coupled Ising-like spins: around zero eld, the loops have a Sshape with two sharp tunnel steps at positive and negative elds. Above m 0 H Z ¼ AE0.3 T, the loops have a broad step, which is strongly eld-sweep-rate dependent and is a consequence of the direct relaxation process between the antiferromagnetic and ferromagnetic spin states. Additionally, the loops exhibit a small hysteresis at m 0 H Z ¼ 0, which comes from the fact that some of the molecules do not tunnel to the antiferromagnetic ground state but remain pinned to the ferromagnetic state. 15 Upon simple comparison of the hysteresis curves for 1 (I ¼ 5/2) and 2 (I ¼ 0) it can be observed that narrower loops are obtained for the nuclear spin bearing system, indicating the relaxation mechanism is more effective for this system. Note also that the loops for 1 (I ¼ 5/2) show a more temperature dependent behaviour than that of 2 (I ¼ 0) .
The mean exchange eld (H ex ) can be directly extracted from the inexion points in the hysteresis loops, leading to an effective exchange constant between the Ising spins of the Dy(III) ions: H ex ¼ J m J /g J m B where m J ¼ 15/2 and g J ¼ 4/3. The determined H ex (4.18 mK) is slightly larger than the one obtained from a purely point dipolar approximation: D dip zz ¼ 3.53 mK for a 163 Dy/ 163 Dy distance of 6.7964(4)Å and 164 Dy/ 164 Dy distance of 6.7971(4)Å, thus the interaction between the Dy(III) pairs is mainly of dipolar origin, with a small exchange contribution. Note that the shortest Dy/Dy distance is 9.9374(5)Å and 9.9306 (5)Å for 1 (I ¼ 5/2) and 2 (I ¼ 0) , respectively, therefore, intermolecular interactions are less relevant compared to intramolecular.
With the knowledge of the low-lying magnetic properties of 1 (I ¼ 5/2) and 2 (I ¼ 0) , it is possible to understand the precise role of the absence/presence of the nuclear spins in both complexes. To begin with our analysis, we rstly focus on the low-temperature magnetic properties of the individual 2 (I ¼ 0) , as the lack of nuclear spins embodied in the 164 Dy(III) ions simplies the analysis. The single ion magnetic properties of the Dy(III) dimers are dominated by the spin-orbit coupling and the interaction with the surrounding ligands, leading to a separation of 188 K between the ground m J ¼ AE15/2 and the rst excited, m J ¼ AE13/ 2, multiplet (see CASSCF section). This allows us to describe the complex as two isolated Ising spins (s ¼ 1 2 ) coupled through an effective interaction J eff s 1z s 2z , where J eff is an effective coupling that can incorporate a small exchange contribution and s 1z,2z are the z-Pauli matrices. Thus, under the action of an external magnetic eld applied along the easy axis, the Hamiltonian is written as: (1) where g eff ¼ 20, and D is the effective tunnel splitting that arises from transverse interactions in the system. Fig. 3a shows the corresponding Zeeman diagram.
With this, we can start to understand the hysteresis loops of the 2 (I ¼ 0) complex ( Fig. 3a and 4). At H z ¼ À1 T (with O z chosen along the easy axis of the Dy(III) ions) the sample is polarised and all the spins are in the ground state |+15/2,+15/2i. As the magnetic eld is swept, the molecules remain in the ground state until the external eld compensates the bias eld, m 0 H r $ À35 mT, and the SMM makes a transition from the ferromagnetic to the antiferromagnetic order by quantum tunnelling. The effective coupling was xed to 4.18 mK, as described above. The height of the relaxations step (DM) is related to the tunnelling probability (p) through the relation: where M in is the initial magnetisation. The next transition happens at m 0 H z $ +35 mT where the molecules relax non- Fig. 2 Temperature dependence of the magnetisation of (a) 1 (I ¼ 5/2) and (b) 2 (I ¼ 0) at a field sweep rate of 0.070 T s À1 . The field was applied parallel to the easy axis of the magnetisation. Before each field sweep, a waiting time of more than 1000 s at AE1 T was used to thermally equilibrate the nuclear spin system with the thermal bath.
adiabatically from the state |+15/2,À15/2i to |À15/2,À15/2i, with the same probability, p. The above discussion is valid only for the idealised situation describing a system of isolated molecules. In a real crystal the molecules are coupled by weak dipolar (and sometimes exchange) interactions and collective effects, such as reshuffling of the internal elds, which have an important inuence on the relaxation process. 10f Therefore, in order to properly describe the dynamics of the ensemble of SMMs, a multi-body model should be employed. However, in a rst approximation, we can assume that the resonance elds of the molecules that tunnel follow a Gaussian distribution around the bias eld, m 0 H r , (DN $ exp(À(m 0 H z À m 0 H r )2/s 2 )), with the variance of this distribution depending linearly on the magnetisation of the sample: s(H) ¼ s 0 |M(H)| + s min . Using the above assumptions, we are able to t the magnetisation curves, employing a nonlinear least-square algorithm (green trace in Fig. 4), with the sole t parameter being the tunnelling probability, p, which for the sweeping rate of 2 mT s À1 is found to be p ¼ 0.74. The parameters s 0 and s off that describe the distribution of the resonance elds are chosen so that a simultaneous t of the magnetisation curves under different sweeping rates is obtained.
With a clear picture of the nuclear spin free system, now we are prepared to consider 1 (I ¼ 5/2) . The 163 Dy(III) isotope has a nuclear magnetic moment I ¼ 5/2 coupled to the electronic shell by the hyperne (A hyp Is) and quadrupolar interaction (P quad I z 2 ). Thus, the total Hamiltonian of the 1 (I ¼ 5/2) complex can be written as: with A hyp ¼ 107.1 mK and P quad ¼ 19.6 mK. The corresponding Zeeman diagram is shown in Fig. 3b. The analysis of the magnetisation curve of the 1 (I ¼ 5/2) complex is done in a similar fashion to the analysis of the 2 (I ¼ 0) complex, with two new assumptions related to the presence of the nuclear spin. We consider that the hyperne levels corresponding to the ground multiplet |+15/2,+15/2i are initially uniformly populated and the tunneling transitions are allowed only between the levels that conserve the nuclear spin, with a xed probability p. The resulting t is shown in Fig. 4 (orange trace), which yields the tunnelling probability, p ¼ 0.76, for the sweeping rate of 2 mT s À1 .
As a result, we observe that the magnitude of the tunnelling probability (p) for both compounds does not change (the small difference may originate in the difference in size and shape of the sample). At very low temperatures (T < 0.3 K), the nuclear spins have the sole role of broadening the relaxation steps. At higher temperatures, the hysteresis loops of the 1 (I ¼ 5/2) complex show a stronger temperature dependence than the 2 (I ¼ 0) ones (Fig. 2). This suggests that the spin lattice relaxation processes [20][21][22] are greatly enhanced by the presence of the nuclear spin in the 1 (I ¼ 5/2) compound (vide infra).

High-temperature static and dynamic magnetic studies
In order to get further insight into the role played by the nuclear spins in the relaxation process of the two isotopologues, we turn to direct current (DC) and alternating current (AC) susceptibility measurements. Static magnetic measurements were carried out employing restrained polycrystalline samples of 1 (I ¼ 5/2) and 2 (I ¼ 0) under an applied eld of 1000 Oe, while the reported AC measurements are performed on polycrystalline samples and under an oscillating eld of 3.5 Oe. The room temperature c M T for the complexes shows similar values, i.e. 28.6 and 28.4 cm 3 mol À1 K for 1 (I ¼ 5/2) and 2 (I ¼ 0) , respectively. The values bode well with the expected ones for two isolated Dy(III), i.e. 28.3 cm 3 K mol À1 for two Dy(III) with J ¼ 15/2 and g J ¼ 4/3 (see Fig. S6 †). Upon cooling, the c M (T) prole for both complexes stays practically constant down to ca. 80 K when it starts decreasing. Below 5 K c M (T) rapidly drops to a minimum value of 17.1 cm 3 K mol À1 for 1 (I ¼ 5/2) and 15.4 cm 3 K mol À1 for 2 (I ¼ 0) , indicative of depopulation of crystal eld levels and antiferromagnetic Fig. 3 First field derivative for a field sweep from À1 T to +1 T of the data in Fig. 2 for (a) 2 (I ¼ 0) and (b) 1 (I ¼ 5/2) . Bottom panels in (a) and (b) are the simulated Zeeman diagram with the field parallel to the easy axes, employing (1) (for 2 (I ¼ 0) ) and (2) (for 1 (I ¼ 5/2) ) and parameters described in the text. Fig. 4 Fits of the magnetisation curves of 1 (I ¼ 5/2) (orange trace) employing eqn (2) and 2 (I ¼ 0) (green trace) employing eqn (1). The tunnelling probabilities were found to be p ¼ 0.76 and 0.74, for 1 (I ¼ 5/2) and 2 (I ¼ 0) , respectively.
interactions. As can be observed in Fig. S5, † employing the ab initio results and the lines model, we are able to reproduce very well the c M T(T) (see ESI † for details).
We investigate both the dynamic temperature dependence of the susceptibility under a constant frequency, c(T;n), and the frequency dependence under a xed temperature, c(n;T). The c(T;n) characteristics reveal that both compounds exhibit an SMM behaviour. That is, a maximum around 18 K in the out of phase component of the c(T;n) is observed for both SMMs at the highest frequency available of 1512 Hz and it shis to lower temperatures as the frequency is decreased (see Fig. S7 †). Noticeable differences between the two isotopologues are better seen in the frequency dependence of the susceptibility, thus we will rst focus on these measurements. Fig. 5a and b show the out of phase component of c(n;T) under a zero DC applied magnetic eld for 1 (I ¼ 5/2) and 2 (I ¼ 0) , respectively. For 1 (I ¼ 5/2) , at the lowest temperature of 2 K, the maximum is centred around 7 Hz, and stays practically constant until reaching 5 K. Above 5 K the maximum in c(n;T) is clearly temperature dependent, shiing swily up to 18 K. In contrast, for the 2 (I ¼ 0) analogue, at the lowest temperature of 2 K, the maximum lies below our minimum working frequency of 0.1 Hz, while for temperatures between 4 K and 18 K the relaxation shows a strong temperature dependence. In order to compare the characteristic relaxation times of the two compounds at different temperatures we successfully t the susceptibility measurements using the generalised Debye model: c(n) ¼ c S + (c T À c S )/(1 + (2ipn) 1Àa ), where c T and c S are the isothermal and adiabatic susceptibilities, respectively, s is the relaxation time, and a indicates the distribution of relaxation times. The obtained temperature dependence of the relaxation times (s(1/T)) is shown in Fig. 5c with the parameter a taking values between 0.02 < a < 0.37 for 1 (I ¼ 5/2) , and 0.02 < a < 0.24 for 2 (I ¼ 0) . The wide distribution of a and its decrease with temperature indicates the presence of multiple relaxation channels that affects the relaxation time (more so for 1 (I ¼ 5/2) than for the 2 (I ¼ 0) complex). The big difference between the relaxation time of 1 (I ¼ 5/2) and 2 (I ¼ 0) at low temperatures (T < 5 K) can be understood qualitatively by considering the effect of the nuclear spin on the processes that dominate the relaxation of the molecular spins in this temperature range.
First, for a polycrystalline sample, the presence of nuclear spins increases the fraction of molecules that can relax through quantum tunnelling. That is, the relaxation of 2 (I ¼ 0) through QTM takes place only when the bias local eld satises the resonance condition (H z z H r ), while for 1 (I ¼ 5/2) the hyperne splitting leads to level anticrossings that are spread in the region of AE75 mT (Fig. 5b) and thus a larger fraction of molecules is found at resonance at any given time. Second, the hyperne interaction in 1 (I ¼ 5/2) results in broader electronic levels and thus in a stronger coupling between the molecular spins and the vibrational acoustic modes (in a rst approximation, the lifetime of an energy level is related to its width by the Heisenberg uncertainty principle, s $ ħ/DE). In the intermediate temperature range (2 K < T < 5 K), this leads to an increase in the rate of single phonon processes (direct relaxation) that dominates the spin-lattice relaxation dynamics. The stronger spin-phonon coupling for 1 (I ¼ 5/2) as compared to 2 (I ¼ Fig. 5 Experimental frequency dependent magnetic susceptibility data at zero applied DC (H DC ) field and varied temperatures (c M 00 (n)) for (a) 1 (I ¼ 5/2) and (b) 2 (I ¼ 0) . Panel (c) and (d) shows the Arrhenius analysis for the s(T) data for 1 (I ¼ 5/2) (orange) and 2 (I ¼ 0) (green), with the results obtained from fitting the c M 00 (n) to a single Debye process at (c) H DC ¼ 0 and (d) H DC ¼ +30 mT. Field dependent study of the relaxation times (s(H)) for 1 (I ¼ 5/2) (orange) and 2 (I ¼ 0) (green) at (a) 5 K and at (b) 14 K. s were obtained after fitting the c M 00 (n) to a single Debye process. The s(H) data comprised field ranging between À30 mT to +500 mT and was collected with an oscillating field of 3.5 Oe. 0) is also seen in the temperature dependence of the hysteresis loops obtained with the m-SQUID technique (Fig. 3).
For temperatures larger than 5 K, the relaxation times of the two isotopologues are very close to each other and are well tted by the Arrhenius law: s ¼ s 0 exp(ÀU eff /k B T). The ts shown in Fig. 5c lead to similar effective energy barriers: U eff ¼ 81.7(1) K for 1 (I ¼ 5/2) and U eff ¼ 81.0(1) K for 2 (I ¼ 0) . As observed, the experimental U eff is approximately half the separation between the ground state and rst excited state obtained via CASSCF calculations, highlighting the importance of anharmonic phonons in the relaxation of complexes here studied. 22 To investigate further the differences between the dynamic magnetic properties of the two isotopologues, s was examined in detail by eld dependent studies, i.e. s(H) at a xed temperature of 5 K with elds ranging from À30 mT to +500 mT (Fig. 5e). First, it should be noticed that the difference in the magnitude of the relaxation times of the two compounds is preserved for elds with an amplitude smaller than +25 mT. Also, for 2 (I ¼ 0) , a modulation of s(H) with a local maximum at zero and a minimum at around +30 mT is observed because when applying a small external eld, the fraction of molecules that are found at resonance and can relax through QTM is increased. The polycrystalline nature of the sample is responsible for shiing the minimum to a smaller eld value (m 0 H min z 30 mT) as compared to the resonance eld of about +35 mT observed for a monocrystal. At the same time, no such modulation is seen for 1 (I ¼ 5/2) because of the multiple hyperne crossings and stronger spin-lattice coupling results in practically uniform relaxation rates. As we increase the eld past +25 mT, a signicant decrease in the relaxation rate is observed as the molecules are gradually shied out of resonance and already at +30 mT the relaxation of the two compounds becomes very similar (Fig. 5d). At higher elds, m 0 H z > 200 mT, the relaxation is again enhanced due to the direct relaxation process (see also the m-SQUID measurements in Fig. 3).
Interestingly, for elds larger than 100 mT the relaxation of 2 (I ¼ 0) is faster than that of 1 (I ¼ 5/2) . This is unexpected and explaining it will require further investigations. However, in order to conrm that this is due to the presence/absence of nuclear spins we measured s(H) at 14 K (Fig. 5f), where nuclear spin effects are expected to be less important and indeed the difference in s(H) characteristics of the two isotopologues is greatly reduced, thus nuclear spin effects are of less relevance at higher temperatures.

Conclusions
Two dinuclear 163/164 Dy isotopologues have been synthesised and structurally and magnetically characterised. Both complexes are SMMs, however, they show marked differences in the dynamic properties as revealed by AC and m-SQUID studies at sub-Kelvin temperatures. m-SQUID loops reveal an interaction between the two Dy(III) ions, leading to S-shaped hysteresis loops characteristic of antiferromagnetically coupled Ising spins. A closer inspection of the temperature dependent hysteresis loops shows that the relaxation is slower in 2 (I ¼ 0) than in 1 (I ¼ 5/2) . This can be mainly ascribed to the absence of nuclear spins in 2 (I ¼ 0) . Fitting the hysteresis loops reveals that the tunnelling rate in both complexes is equal, therefore, tunnelling does not solely play a role in the relaxation dynamic of both complexes. Instead, the larger spectrum of hyperne states in 1 (I ¼ 5/2) allows a better coupling to acoustic phonons, consequently enhancing the direct relaxation process at sub-Kelvin temperatures compared to 2 (I ¼ 0) , possessing no hyper-ne states. Note that phonons modulate the electric eld of the magnetic ions, therefore inducing direct relaxation process. 20,23 Now, if we consider 2 (I ¼ 0) , the absence of nuclear spins states leads to a ground doublet state with no hyperne-split levels. In contrast, in 1 (I ¼ 5/2) the hyperne level splits the electronic state in (2I + 1) 2 states, thus a total of 36 states (for I ¼ 5/2) comprise the ground doublet. At very low temperatures, where direct relaxation is important, the number of available phonon modes with energy corresponding to the difference between the spins state is therefore larger for 1 (I ¼ 5/2) than for 2 (I ¼ 0) .
The difference in the relaxation rate of the two compounds, at low temperatures and small magnetic elds, is also clearly evidenced in AC measurements. Note that, the nuclear spin effects are more important at lower temperatures than at higher as revealed by the s(H) at 5 and 14 K. In both Dy 2 isotopologues here studied, we show that the tunnelling probabilities are not affected by the nuclear spins and play a minor role in well-performing SMMs, where the operating temperatures are rather large, in agreement with recent studies. 11 Finally, we argue that although we and others nd that tunnelling is not affected by the nuclear spin presence/absence, the hyperne level broadening still plays an important role for SMMs with moderate energy barriers (more importantly at very low temperatures), since it facilitates the spinphonon coupling, thus enhancing the direct relaxation process. These nding must be contemplated for Ln-SMMs proposed for very low temperature applications, such as quantum bits, where the utilisation of the nuclear states embedded in the lanthanide ions can be used as quantum bits. For example, the indirect coupling of the nuclear states, via the electronic states, would increase the number of nuclear states available for the realisation of complex quantum algorithms. 24

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