Determining the zero-field cooling/field cooling blocking temperature from AC susceptibility data for single-molecule magnets

Abstract

We present a general relationship between the magnetisation blocking temperature (T_B) measured using the zero-field cooling/field cooling technique (ZFC/FC) and the temperature-dependent spin relaxation time obtained from AC susceptibility and magnetisation decay measurements. The presented mathematical approach supplies ZFC/FC blocking temperatures at any heating rate (R_H), providing comparable values to those obtained experimentally, as demonstrated by testing 107 examples for reported single-molecule magnets (SMMs) where the ZFC/FC curve has been measured. This procedure is examined in further detail for a new single-molecule magnet, [Dy(OPAd₂Bz)₂(H₂O)₄Br]Br₂·4THF (1) (OPAd₂Bz: di(1-adamantyl)benzylphosphine oxide). For this compound, ZFC/FC measurements were made over a broad range of heating rates (0.01–5 K min⁻¹), which agreed with the general behaviour predicted from AC susceptibility data. We discuss how the demagnetisation mechanism determines the sensitivity of T_B with respect to the heating rate: T_B is mostly insensitive to R_H for Orbach relaxation, while there is a larger sensitivity for Raman-limited systems. Our conclusions provide a clear physical interpretation of ZFC/FC blocking temperatures, aiding in the proper contextualization of this figure of merit.

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Introduction

In the early 1990s, the discovery of single molecule magnets (SMMs) sparked a revolution in the field of molecular magnetism. These molecular transition metal coordination compounds exhibit a magnetic memory effect, which arises from blocking magnetisation via an anisotropic barrier (U_eff) for prolonged periods in the absence of an external magnetic field and below a critical temperature known as the blocking temperature (T_B).^1–5 SMMs have been extensively investigated due to their immense technological potential in molecular spintronics, ultra-high-density data storage, and quantum information technologies using spin qubits.^6–12

However, a significant challenge in the field remains, necessitating the achievement of a magnetic memory effect at practical/high temperatures while maintaining high thermal stability in the presence of air and humidity.^13,14 Mononuclear SMM complexes exhibit modulable magnetic anisotropy through chemical tuning of their coordination environment, representing the smallest nanomagnets which can be modulated by the rational selection of metal ions and ligands.^15–26 Notably, the Dy^III ion, with its unquenched orbital momentum and large magnetic anisotropy, has emerged as a leading candidate to revolutionize technology based on electron spin.^{15–20,27–29}

Experimental and computational studies have revealed that strong axial crystal fields, achieved through axial distribution of ligands, are crucial for Dy^III complexes to exhibit remarkable SMM behaviour.^{15–20,30–32} More recent approaches seek to attenuate vibrational displacements to hamper Raman relaxation.^33,34 For future systems, some theoretical works suggest exploring the surface deposition of SMMs and uncommon oxidation states.^35,36 In the quest to develop practical SMM applications, reaching a blocking temperature (T_B) above liquid nitrogen temperature (77 K) is crucial. There are only two molecules that have surpassed this crucial barrier; both of them are Dy compounds. Mononuclear Dy^III [Dy(C₅Me₅(Cp^iPr5))][B(C₆F₅)₄]³⁷ and mixed-valence Dy^IIIDy^II dinuclear [Dy₂I₃(Cp^iPr5)₂] metallocene³⁸ compounds have achieved landmark performances with U_eff = 2217 K and 2347 K, respectively, and T_B = 80 K in both cases. The uniaxial local symmetry stabilizes the largest m_J = ±15/2 ground state^39,40 in the former, and the collinearity of the local anisotropic axes and the strong 4f-radical coupling in the latter, results in a significant separation from the first and higher excited states.^38,41,42 However, compounds with such low coordination are unstable and the need for more stable systems with similar magnetic properties arises. Different high-order symmetry axes, such as those found in trigonal bipyramidal (D_3h),^43,44 square antiprismatic (D_4d),^45–49 sandwich,^37,50–54 and pentagonal bipyramidal (D_5h),^55–73 have been recommended to favour slower relaxation of magnetisation by reducing transverse anisotropy and suppressing quantum tunnelling of the magnetisation (QTM). The presence of a high-order symmetry axis also promotes the collinearity of anisotropic axes of the excited and ground states, leading to larger U_eff values.^74,75 Nonetheless, it is essential to engineer molecular vibrations to control the spin lifetime of SMM complexes,⁷⁶ as flexible lattices are responsible for fast relaxation. Addressing the requirements of high-temperature performance and thermal air and humidity stability, mononuclear Dy^III SMMs with D_5h geometry hold a fair position, displaying, in many cases, stability against these factors with U_eff and T_B values as high as 1162 K ⁶³ and 36 K,⁷⁰ respectively. Although some recent mononuclear Dy^III SMMs with D_5h geometry have been shown to be air-sensitive, the majority are stable.^60,61,73,77 Recently, the role of spin-vibrational coupling in designing high-performance pentagonal bipyramidal Dy^III SMMs was revealed using a combination of density functional theory (DFT) and complete active space self-consistent field (CASSCF) calculations.³³ There are about thirty examples of mononuclear Dy^III SMMs with D_5h geometry, but those employing axial bulky phosphine oxide type ligands (high electron density) and weak donor equatorial ones (e.g., water)^{56–59,62,64} are relatively scarce. These D_5h based SMMs represent a highly efficient approach for constructing new high-performance SMMs.

As mentioned previously, the Dy^III metallocene compounds [Dy(C₅Me₅(Cp^iPr5))][B(C₆F₅)₄] and [Dy₂I₃(Cp^iPr5)₂] show the highest blocking temperature reported to date. This value was measured as the maximum temperature at which magnetic hysteresis is observed (herein T_B–H). However, there are two other ways to quantify the blocking temperature: the temperature at which the magnetic relaxation time is equal to 100 s (T_B-100) and the maximum of the zero-field cooled (ZFC) magnetic susceptibility (T_B-ZFC/FC). For the dysprosium metallocene cation, these values are 67 and 52 K, respectively. This example indicates how different the blocking temperature can be depending on the measurement technique. Furthermore, key experimental parameters can also influence T_B significantly. In the case of magnetic hysteresis measurements, T_B–H varies with the field sweep rate, where faster sweep rate programs lead to higher blocking temperatures. Focusing on ZFC/FC experiments, T_B-ZFC/FC is sensitive to the heating rate, with faster heating rates leading to higher blocking temperatures. Unfortunately, the experimentally employed heating rate is often missing from reports in the literature, hindering a rigorous comparison between systems from different publications. Another issue is related to a misunderstanding of the definition of T_B-ZFC/FC, since some authors refer to this value as the temperature where the ZFC and FC curves diverge, which corresponds to the irreversibility temperature (T_irrev) and is higher than that of T_B-ZFC/FC.⁷⁸ Although T_B-100 usually does not depend strongly on the measurement conditions, the 100 s definition is arbitrary and it is not clear why the blocking temperature at this specific relaxation time is more informative of SMM behaviour than other thresholds. Blocking temperatures are convenient as “single-molecule magnet” performance metrics since they condense a complex magnetic relaxation dependence into a single figure. Moreover, the existence of a magnetic blocking temperature provides a clear definition of a “molecular magnet” that goes beyond the presence of slow relaxation of magnetisation. However, other demagnetisation parameters, such as magnetic coercivities or demagnetisation barriers (U_eff) are also relevant for the assessment of SMM performance.

In this paper, we propose a new approach for estimating T_B-ZFC/FC at any heating rate from the temperature-dependent spin relaxation time obtained from AC susceptibility and magnetisation decay measurements, to advance towards a more harmonized definition of T_B. In this way, T_B-ZFC/FC data from different studies can be better compared, as many examples of ZFC/FC experiments in the literature do not state the heating rate. Furthermore, the model allows for the estimation of T_B-ZFC/FC in cases where the ZFC/FC experiment was not done. Experimentally, performing the ZFC/FC experiment together with AC magnetometry measurements is not a practical problem, so the main use of our proposed method is not to avoid the ZFC/FC experiment but to provide a new tool to contextualize and estimate ZFC/FC blocking temperatures.

To validate this new model, we compared T_B-ZFC/FC values for 107 examples of SMMs from the literature for which ZFC/FC data were reported, including both lanthanide and transition metal systems. Furthermore, we conducted a detailed AC susceptibility and ZFC/FC study for a new Dy^III SMM with D_5h geometry. Concretely, we measured ZFC/FC curves for a wide range of heating rates, demonstrating that the presented model accurately predicted T_B-ZFC values and captured the heating rate dependence of the blocking temperature.

Results and discussion

Mathematical model for ZFC/FC blocking temperature determined from AC susceptibility

Our first goal is relating the blocking temperature measured by zero field cooling experiments with an expression in terms of the temperature dependent relaxation time. Experimentally, susceptibility is calculated as the ratio between the magnetic moment and the applied magnetic field. The isothermal (static) susceptibility is:where M_eq is the magnetic moment corresponding to the equilibrium population at a given temperature and magnetic field (B), g is the Landé factor, S is the spin and P_↑ − P_↓ is the population difference between spin up and down species. The expression for non-equilibrium susceptibility measured in the ZFC experiment is analogous to eqn (1) when the magnetic moment P_↑ − P_↓ term is out of equilibrium.

Combining the equilibrium and out of equilibrium expressions for magnetic susceptibility, χ_ZFC is defined as:

The numerator on the r.h.s. of eqn (2) is the difference between the equilibrium and out of equilibrium magnetic moments (ΔM).

The kinetic equation for a system after a perturbation is:

where the relaxation rate τ⁻¹ is temperature dependent. In the ZFC experiment, the sample is initially cooled in the absence of an external direct magnetic field. After reaching a cryogenic temperature, the magnetic moment is measured using a small magnetic field and the sample is heated at a rate of R_H (K s⁻¹):Hence, eqn (3) becomes:

The general solution of eqn (4) is:

where C₁ is the magnetisation at the beginning of the ZFC heating step; if all spins remain frozen when the magnetic field needed for the measurement of the heating curve is turned on, and the combination of eqn (2) and (5) yields the simple form:

In practice, the magnetic moment is not zero at the beginning of the heating step. This is the typical situation in ZFC/FC experiments and depends on the orientational distribution of magnetic moments, their alignment with respect to the external field and operational parameters, as the time needed to stabilize the initial temperature in each experiment. Since this parameter is sample and experiment dependent, we assume it is an effective constant, , with a value between 0 and 1. In this way, eqn (7) becomes:

where the term in parentheses is the fraction of relaxed magnetic moments (0 ≤ λ ≤ 1). Eqn (7) allows for the determination of the blocking temperature by the evaluation of the maximum of χ_ZFC for an arbitrary χ_T. Importantly, eqn (7) can be applied to different experimental conditions (such as the presence of magnetic fields) if these effects are present in the data that produced the τ⁻¹ function. Although all results from this paper can be obtained from eqn (7), we are also interested in an analytical expression for T_B. Thus, the maximum of χ_ZFC is expressed as:

For simplicity, we consider that the isothermal susceptibility reasonably follows the Curie law:

The assumption of the Curie law greatly simplifies the following equations but can be reversed if needed.

Hence, the blocking temperature is:

Eqn (12) can be expressed in a way that the l.h.s. coincides with the term in square brackets from eqn (7):

Thus, the blocking temperature can be estimated by knowing the heating rate, the temperature dependence of τ and . In summary, T_B can be obtained from either eqn (7) or (12), where the latter expression gives an explicit term for T_B by assuming the Curie law. We analysed how these two approaches performed for a selected group of experimental examples from the literature (see section 4 and Table S8 in the ESI‡) and concluded that both equations provided satisfactory results for T_B, especially for larger values of . In general, the Curie law assumption from eqn (13) gives slightly lower calculated T_B values than eqn (7), which employs the experimental dependence of χ. In conclusion, we recommend employing either of the equations with to obtain reliable estimations for T_B-ZFC/FC.

Synthesis and characterization of the Dy^III SMM with D_5h geometry

In view of previous considerations, we focused our efforts on the preparation of a mononuclear Dy^III coordination complex, [Dy(OPAd₂Bz)₂(H₂O)₄Br]Br₂·4THF (1) (OPAd₂Bz: di(1-adamantyl)benzylphosphine oxide), close to ideal pentagonal bipyramidal geometry, which fulfilled all the desired characteristics to obtain SMMs with high U_eff and T_B values, that is, a strong axial crystal field, humidity and air stability, and a rigid network to avoid rapid relaxation. A strong axial crystal field is created by the two bulky di(1-adamantyl)benzylphosphine oxide ligands located in the axial positions, which lead to a magnetic hysteresis that remains open up to 14 K, one of highest values for air/humidity stable SMMs synthesized to date. Four water molecules and one bromide anion are located in the equatorial positions; this is structurally different from previous compounds with axial phosphine oxide ligands, where five water molecules are located instead.^56–59,62 Furthermore, we provide detailed insights into the mechanism that governs the magnetic relaxation of complex 1 by using ab initio CASSCF based computational methods.

Complex 1 was prepared by the solvothermal reaction of the di(1-adamantyl)benzylphosphine oxide ligand with anhydrous DyBr₃ in anhydrous tetrahydrofuran and in a 2 : 1 molar ratio using a 23 mL Teflon-lined stainless steel container and keeping it at 100 °C for three days (see ESI‡ for further details). The resulting solution from the solvothermal reaction was allowed to evaporate at room temperature for several days, whereupon large colourless prismatic single-crystals of [Dy(OPAd₂Bz)₂(H₂O)₄Br]Br₂·4THF (1) were obtained, which were air-stable (Fig. 1).

Fig. 1 Perspective view of the molecular structure of [Dy(OPAd₂Bz)₂(H₂O)₄Br]Br₂·4THF (1). Code colours: dysprosium (cyan), oxygen (red), bromide (brown), phosphorus (orange), and carbon (grey). Hydrogen atoms are omitted for clarity. Blue dashed lines indicate hydrogen bond interactions.

The molecular structure of 1 consists of [Dy(OPAd₂Bz)₂(H₂O)₄Br]²⁺ cationic units, with are charge balanced by two free bromide anions, and four crystallization THF molecules (Figs. 1 and S3‡). The bromide anions and THF molecules interact with the cationic unit via hydrogen bonds. Within the cationic unit, the seven-coordinate Dy^III centre exhibits a pentagonal bipyramidal geometry (PBPY-7), very close to an ideal D_5h polyhedron, as supported by the continuous shape measurement analysis,⁷⁹ which provides an S(PBPY-7) value of 0.948 (where 0 corresponds to the ideal D_5h geometry) (Table S1, ESI‡). Two bulky OPAd₂Bz ligands occupy the axial positions, whereas four water molecules and one bromide ion are in the equatorial plane (Fig. 1). The axial Dy–O1 distances (2.210(3) Å) are shorter than the equatorial Dy–O2/O3 (average value of 2.365 Å) and Dy–Br1 (2.8860(6) Å) ones; this indicates that the cationic unit shows a compressed PBPY-7 geometry with an almost linear axial O1–Dy–O1 angle (176.73(16)°) and equatorial Br1–Dy–O2, O2–Dy–O3 and O3–Dy–O3 angles of 74.24(9)°, 71.88(13)° and 68.12(17)°, respectively, close to the ideal angle of 72° (Table S3, ESI‡). The coordinated bromide atom seems to generate certain steric repulsion with the water molecules close to it, which is reflected in a Br1–Dy–O2 angle greater than 72°. In turn, this brings about the closeness between these two water molecules and the other two, thus showing O2–Dy–O3 and O3–Dy–O3 angles less than 72°. The P–O1–Dy angle is also very close to linearity (173.67(19)°) and the angles between the equatorial and axial atoms are around 90° (Table S3, ESI‡). Specifically, the local symmetry of the DyO₆Br coordination sphere is C_2v, with the C₂ axis lying along the line connecting the coordinated bromide anion and the Dy^III ion (Fig. 1). Each coordinated water molecule interacts with one free bromide anion and one THF molecule via hydrogen bonds. The O2⋯Br2 and O3⋯Br2 donor–acceptor distances show respective values of 3.161(4) Å and 3.154(3) Å, whereas those for the O2⋯O4(THF) and O3⋯O5(THF) donor–acceptor distances are 2.802(6) Å and 2.774(19) Å, respectively. Moreover, the shortest Dy⋯Dy intermolecular distance for 1 is 12.1090(3) Å, which indicates that the [Dy(OPAd₂Bz)₂(H₂O)₄Br]²⁺ units are well separated in the structure. There are no π⋯π stacking interactions between the aromatic benzene rings of different units. The free bromide atoms establish van der Waals interactions with the benzene hydrogens inside the same unit (3.0386(5) Å), and with the hydrogens of the CH₂ (2.6475(5) Å) and adamantyl groups (2.7673(5) Å) of adjacent units (Fig. S3‡).

Magnetic measurements

The DC magnetic properties of 1 were studied over the 2–300 K temperature range under an applied magnetic field of 0.5 T and magnetisation was studied over the field range of 0–7 T at temperatures between 2 and 7 K, see Fig. S4.‡ It is worth noting that the obtained curves clearly show the typical features of magnetisation blocking in an efficient mononuclear Dy^III-SMM (sharp decrease of χ_MT at low temperature, sinusoidal behaviour of magnetisation at low field, divergence between FC and ZFC magnetic susceptibilities at low temperature, and magnetic hysteresis; see below and the magnetic studies section in the ESI‡).

The ZFC/FC magnetic susceptibilities were collected at two magnetic fields and over a wide range of heating rates to evaluate T_B and compare it with the proposed model (see Fig. 2). For the small field, 50 Oe, at very low heating rates, the ZFC/FC curves separate slightly at very low temperatures (Fig. S5‡). When increasing the heating rate, the ZFC and FC curves differ more and the temperatures at which the curves diverge rise to larger values. The maximum of the ZFC curve signalling T_B goes from 2.5 K at 0.01 K min⁻¹ to 5 K at 5 K min⁻¹ (Table S4‡). When increasing the field to 500 Oe, the blocking temperatures rise significantly; this is probably due to the suppression of relaxation through QTM. At very low heating rates (0.01 K min⁻¹), the blocking temperature is around 5 K and reaches 8 K at 2 K min⁻¹. Faster heating rates did not provide values for the blocking temperature since the ZFC curve had no clear maximum (see Figs. 2 and S6‡).

Fig. 2 Temperature dependence of χ_M under ZFC conditions at different heating rates and with an applied field of 50 Oe (above) and 500 Oe (below). FC data are omitted for a better appreciation of the ZFC curves and their dependence on the heating rate. ZFC/FC curves for all data are available as ESI (Fig. S5 and S6‡).

Alternating current (AC) magnetic susceptibility measurements were performed to study the slow relaxation of magnetisation. To obtain the maximum number of relaxation times and be able to study a larger temperature range, two pieces of equipment were employed to study the 1–10 000 Hz frequency range: SQUID MPMS XL and PPMS-9 instruments (see details in the ESI‡). At zero external DC field, the in-phase and out-of-phase components of the AC susceptibility show frequency-dependent peaks (Figs. S7 and S8‡) with well-defined maxima in the vs. T plot over the 20–40 K range for higher frequencies (Fig. S8‡), indicating a high magnetisation reversal barrier. The vs. frequency plot displays temperature-dependent maxima over the 19–27 K range (Fig. S9‡). The relaxation times were extracted from fitting of the frequency dependence of at different temperatures using the generalized Debye model. The extracted relaxation times (τ) are collected in Table S5.‡

Magnetisation decay experiments were performed to evaluate relaxation times at lower temperatures. Exponential decay of the magnetisation was clearly observed until 8 K. The obtained data were fitted using a stretched exponential function (Fig. S11‡); this is commonly employed to obtain relaxation times from magnetisation decay measurements.^80,81 The obtained τ values are collected in Table S6‡ and represented in Fig. 3 as the ln(τ⁻¹) vs. temperature plot together with the obtained values from the AC susceptibility measurements. At very low temperature, there is a constant region indicative of quantum tunnelling relaxation. The onset of the Raman regime is discernible from magnetisation decay data, the trend in which matches with the lowest points measured by AC susceptibility. The transition between Raman and Orbach demagnetisation is clearly visible at around 18 K. As the three magnetic relaxation mechanisms are identified in the τ⁻¹(T) curve, the following equation is employed:

τ⁻¹ = τ_QTM⁻¹ + CTⁿ + τ₀⁻¹e^U_eff/kT (14)

where the effective demagnetisation barrier was adjusted to a value of 427.7 K (297.3 cm⁻¹), with a preexponential factor (τ₀) of 4.66 × 10⁻¹¹ s. Raman and tunnelling were fitted to C = 2.64 × 10⁻⁷ K⁻ⁿ s⁻¹, n = 5.28 and τ_QTM = 142.7 s, respectively.

Fig. 3 Logarithmic plot of τ⁻¹ vs. temperature for 1 (in the inset, τ⁻¹ vs. temperature, as in eqn (14)). The values at lower temperatures correspond to those obtained from magnetisation decay and the values at higher temperatures are the ones derived from fitting of the AC susceptibility data to a generalized Debye function. The blue line corresponds to fitting of the data with eqn (14).

For completeness, the field dependent magnetisation measurements at different temperatures were acquired (Fig. 4), with a sweep rate of 20 mT s⁻¹. Compound 1 shows a clear magnetic hysteresis, which remains open up to 14 K; this is one of highest values for air/humidity stable SMMs synthesized to date. The butterfly shape of the hysteresis loop arises from a faster relaxation around zero field and a slower relaxation at intermediate fields. This compound retains a large magnetisation that falls only when H < 20 mT, which can be attributed to unsuppressed quantum tunnelling of the magnetisation due to symmetry deviation and hyperfine and dipole interactions. When the temperature increases, the hysteresis loop narrows as the relaxation speeds up and results in smaller coercive fields and remanent magnetisation. The hysteresis loop shows a coercive field of 1 T and a remanent magnetisation of 2μ_B at 3 K at a sweep rate of 20 mT s⁻¹,

Fig. 4 Magnetic hysteresis measurements/hysteresis plot for 1 at a sweep rate of 20 mT s⁻¹.

Ab initio calculations

Ab initio calculations based on the experimental X-ray structural data were performed to provide insights into the mechanism that governed the magnetic relaxation of complex 1. In particular, multiconfigurational CASSCF calculations implemented in the ORCA 5.0.3 program package^82–84 and the CASSCF/RASSI-SO/SINGLE_ANISO approach using OpenMOLCAS^85–87 were carried out. Both programs agree in the general description of the double well potential associated with the ground ⁶H_15/2 multiplet. To avoid redundancy in the discussion, we present the ORCA results in the manuscript while OpenMOLCAS data are presented in the ESI‡ for comparison.

The eight computed Kramers’ doublets (KDs) for 1, corresponding to the ⁶H_15/2 ground state of the Dy^III ion, span an energy range of about 674 cm⁻¹ (Table S7‡). The computed temperature dependence of χ_MT reproduces the experimental temperature dependence of χ_MT rather well (Fig. S4‡). The ground KD (KD1) is a pure m_J = |±15/2〉 state that is highly anisotropic (g_zz = 19.86) with negligible transverse components (g_xx ∼ g_yy < 1 × 10⁻³), thus establishing a strong magnetic anisotropic axis. These g-values suggest strongly suppressed QTM within the ground KD (Fig. 5), which is consistent with the relatively large experimental value for τ_QTM (142.7 s). The anisotropic g_zz axis is almost collinear with the pseudo-C₅ axis lying along the axial O–Dy–O bonds (the deviation between the g_zz axis and O–Dy–O direction is 1.5°, see Fig. S12‡ and Fig. 5 bottom). This strong uniaxial magnetic anisotropy is consistent with the weak ligand field from aqua and bromine ligands in the equatorial plane and the strong donor ligands in the axial positions.

Fig. 5 Top: ab initio magnetisation blocking barrier for 1, where the Kramers’ doublets (KDs) are represented as dark blue bars and tunnelling relaxation times (τ_QTM) between the connecting pairs are indicated as orange lines and the values are represented on the log₁₀ scale. Bottom: molecular structure of 1 with the calculated orientation of the main magnetic axis of the ground Kramers’ doublet (KD1) (black line). Colour code: Dy (green), O (red), P (orange), Br (brown) and C (grey).

The first excited state (KD2) lies 275.2 cm⁻¹ above the ground state. This KD2 is also axial in nature, with g_zz = 16.92, g_xx = 0.13, and g_yy = 0.27. In this case, the g_z tensor passes through the O–Dy–O direction again and presents a deviation of 6.4° with respect to the g_zz anisotropic axis of the ground state. The transverse components of KD2 can be large enough to promote magnetic relaxation via the first excited state, giving a calculated magnetisation barrier, U_cal, of 275.2 cm⁻¹ (396 K), which is close to the experimental energy barrier, U_eff, of 427.7 K. The next excited state (KD3) is close in energy (336.6 cm⁻¹, 484 K) and shows large transverse components of g (g_xx = 0.42 and g_yy = 1.63). Tunnelling relaxation times were calculated according to an ab initio model based on the spin–dipolar interaction.⁸⁸ For the ground state (KD1), the calculated tunnelling time is 4.6 × 10⁻² s while for the first excited state (KD2) it is much faster (9.1 × 10⁻⁷ s), which is consistent with the large transverse g-tensors obtained for the latter, confirming that magnetic relaxation for 1 occurs via the first excited state (KD2).

The effect of the f-orbital splitting on the observed magnetic anisotropy can be investigated in further detail by considering the individual contributions of each f_±n orbital block to the demagnetisation barrier, as shown previously by some of us.⁸⁹ In the case of 1, the high axiality of the coordination environment dictates orbital splitting where the most destabilized orbital corresponds to the f₀ (f_z³) orbital, lying along the z-axis with an energy of 854 cm⁻¹ (violet level in Fig. 6, top). Interestingly, the most stable orbital block is not the one lying on the xy plane (f_{y(3x²−y²)} and f_{x(x²−3y²)} orbitals, collectively named f_±3) but the next block, the functions of which have a first-order dependency with respect to z (f_xyz and f_z(x²−y²) orbitals, collectively named f_±2) (red levels in Fig. 6, top). Fig. 6 (bottom) compares the CASSCF energies of the m_J sublevels of the ground ⁶H_15/2 multiplet with the ones derived from the orbital energies weighted by their contributions to each sublevel in a ligand field stabilization energy (LFSE) approach.⁸⁹ The agreement between both data sets is satisfactory as both present an isolated ground doublet, separated by ca. 300 cm⁻¹ from a dense pack of seven doublets spanning around 300–400 cm⁻¹. The pattern of the double well resembles the letter “M”, like the pattern derived from CASSCF (Fig. 5, top). Considering that magnetic relaxation probably proceeds via the first excited doublet, the contribution of each orbital block to anisotropy can be estimated from the LFSE energy difference between the m_j = ±13/2 and m_j = ±15/2 states, which is 2/3*(E_f±1 − E_f±2). In this case, the average energy of the f_±1 and f_±2 orbital blocks is 9.5 cm⁻¹ and 523.7 cm⁻¹, resulting in a gap of 342.8 cm⁻¹.

Fig. 6 Top: f-orbital energy splitting obtained from AILFT calculations; blue, red, orange, and violet levels correspond to f_±3, f_±2, f_±1 and f₀ orbitals, respectively. One orbital for each block is depicted next to the corresponding level, where atoms not belonging to the immediate coordination environment of the Dy^III ion are omitted for clarity. Bottom: LFSE and CASSCF energy levels (in cm⁻¹) for 1 are represented in blue and black, respectively.

Model assessment

The accuracy of the presented model can be evaluated for 1 since the temperature dependence of the relaxation time successfully fits to the combination of tunnelling, Raman and Orbach contributions. Employing eqn (7), the ZFC curve can be simulated, and the maximum is obtained numerically. The effect of different demagnetisation mechanisms can be analysed by modifying the τ(T) curve to include diverse combinations of Orbach, tunnel, and Raman mechanisms. Our analysis starts with the ZFC/FC determined using a field of 500 Oe. At this field, the tunnelling mechanism should be mostly suppressed so only the Orbach and Raman mechanisms are considered in the integration of τ⁻¹(T). It is important to stress that the heating step in ZFC/FC measurements is done with an external magnetic field, which can affect demagnetisation parameters, especially tunnelling. Thus, care must be taken to account for this effect. Fig. 7 presents the simulated ZFC curves for all the experimentally determined heating rates (0.01–5 K min⁻¹). Pleasingly, the positions of the ZFC maxima are similar to the temperature range of the experiment (vide supra). In the same way as the analysis of the experimental ZFC/FC curves, data associated with the slower heating rates show ZFC curves where T_B-ZFC/FC is clearly discernible, while the faster heating rates show a less clear maximum.

Fig. 7 Simulated ZFC/FC curve for the sum of the Orbach and Raman mechanisms, using the demagnetisation parameters fitted for 1. is assumed to be one and the adiabatic susceptibility is represented in black with a fixed value of χ_T = 14.17 cm³ K mol⁻¹.

The former simulations are now repeated by incorporating tunnelling relaxation into the integration of τ⁻¹(T). The reference experimental data are now in the ZFC/FC curve measured at 50 Oe, which should present a smaller quenching of tunnelling than the 500 Oe results. Fig. 8 shows the simulated curves under these conditions. The comparison between Fig. 7 and 8 clearly indicates that tunnelling is efficient at lowering T_B-ZFC/FC for 1, in the same way that ZFC/FC data at 500 Oe show higher blocking temperatures than the 50 Oe results. This highlights an important and probably overlooked experimental parameter that affects the blocking temperatures measured by ZFC/FC experiments: the external field necessary to record the magnetic moment along the temperature program, which adds to the importance of reporting the heating rate for these experiments.

Fig. 8 Simulated ZFC/FC curve for the sum of tunnelling, Orbach and Raman mechanisms, using the demagnetisation parameters fitted for z. is assumed to be one and the adiabatic susceptibility is represented in black with a fixed value of χ_T = 14.17 cm³ K mol⁻¹.

A more quantitative comparison between the experimental and calculated blocking temperatures is presented in Fig. 9. As mentioned earlier, the model can capture the effect of tunnelling in 1, which diminishes the blocking temperature by around 5 K. The agreement is quantitative at low heating rates but departs for faster heating rates (see the red dashed line in Fig. 9). Small modifications to the tunnelling value reveal a high sensitivity of the blocking temperature, where correcting the tunnelling time by a factor of 0.25 provides a simulated curve that agrees with the 50 Oe experiment over the complete range of heating rates (dotted–dashed red line in Fig. 9). This factor might seem a drastic correction, but it is significantly lower than the error bars of the magnetisation decay measurements calculated from the β parameter, according to Chilton and coworkers.⁹⁰ In this case, the range of relaxation times is even larger since the magnetisation decay and AC susceptibility are combined. The 500 Oe data also agree with simulations, where the low heating rate results agree perfectly and depart by a couple of K at faster rates. Overall, the performance of the model is satisfactory since it allows us to predict T_B-ZFC/FC accurately and capture the heating rate dependency.

Fig. 9 Comparison of experimental (solid lines) and simulated (dashed or dotted–dashed lines) blocking temperatures. Data corresponding to 50 Oe and 500 Oe measurements are depicted in red and blue, respectively.

The demagnetisation mechanism limiting the blocking temperature can be estimated by repeating the same simulations and including each demagnetisation mechanism separately. The tunnelling mechanism limits T_B to 2–3 K for slow heating rates while it keeps most of the magnetic moment frozen at faster temperature programs, without reaching a maximum over the simulated temperature range (2–20 K) (see Fig. S14,‡ left). The Raman-only simulation is similar to the one for the Orbach + Raman results from Fig. 8, indicating that Raman relaxation is responsible for the simulated blocking between 4 and 12 K. The Orbach-only plot shows blocking temperatures over a higher temperature range (14–17 K). Hence, it does not determine the value of T_B in this case (see Fig. S14,‡ right).

To assess the broader accuracy of the presented approach, a literature search for experimental examples of T_B-ZFC/FC was performed. Such values can also be obtained for any single-molecule magnet system by introducing the spin relaxation parameters at the web page https://tbsim.ee.ub.edu. By reconstructing the τ(T) curve using fitting parameters, we can calculate T_B using eqn (7). Unfortunately, the heating rate is not always reported, so we have assumed lower and higher limit values for this parameter when R_H information is missing (0.2 K min⁻¹ and 5 K min⁻¹ to account for slow and fast sweeping rates).

Table S9‡ compares experimental and predicted values for ZFC/FC blocking temperatures for 97 lanthanide SMMs^{13,14,37,38,41,50,51,60,61,63,65,91–136} and ten transition metal complexes^137–146 reported in the literature. The lanthanide benchmark set contains mononuclear and polynuclear complexes based on Dy^III, Tb^III, Er^III and Ho^III ions, heteronuclear complexes containing Dy^III or Tb^III ions and transition metals, including diluted systems. The ten transition compounds are Mn₁₂ complexes, a trinuclear Mn₂Mo complex, and tetranuclear and mononuclear iron and cobalt complexes. The fitting parameters for AC susceptibility data range from systems involving combinations of Orbach, Raman, and tunnelling, Orbach and Raman, Orbach and tunnelling, and Raman and tunnelling to Orbach-only terms. Besides, the list includes complexes exhibiting two or three relaxation processes, where one of them (the main process) is considered to calculate T_B. As observed in this list, some authors report T_irrev (indicated with an asterisk) instead T_B-ZFC/FC, and others report both values. As expected, T_irrev is always higher than the T_B-ZFC/FC value. In some cases, the difference between T_irrev and T_B-ZFC/FC values exceeds 10 K (e.g. 78 K and ∼55 K, respectively, for complex 3 in Table S9‡). Another issue to highlight is the lack of information about the heating rate (R_H). From the 107 complexes in Table S9,‡ R_H values were only included in the original papers for 37 systems, and only for 3 transition metal complexes (see Fig. 10). Reported R_H values range from 0.189 to 5 K min⁻¹, demonstrating that there is no consensus on this important parameter that defines the T_B value. In most cases, the reported T_B-ZFC/FC value lies in the range of T_B obtained with R_H values of 0.2 K min⁻¹ and 5 K min⁻¹, or close to these values (see Fig. 11). Similarly, when R_H is reported, the obtained T_B value at this heating rate is close to the reported T_B-ZFC/FC value (see Fig. 10), demonstrating the accuracy and broad applicability of the presented approach.

Fig. 10 Comparison of the experimental zero-field cooling/field cooling blocking temperatures and the ones calculated using eqn (12). Only 38 cases (35 lanthanide and 3 transition metal compounds depicted in red and green, respectively) with a reported heating rate are considered (see Fig. 11 for the other systems). The black line represents perfect agreement between both values.

Fig. 11 Comparison of the experimental T_B-ZFC/FC blocking (black dots) or T_irrev irreversible (blue dots, values in Table S9‡ indicated with an asterisk) temperatures for the 62 lanthanide systems (for clarity, the 7 transition metal systems are in the inset) without a reported heating rate. The x-axis indicates the complex number in Table S9.‡ The green bar corresponds to the limit values calculated using eqn (12) using with R_H = 0.2 K min⁻¹ and 5 K min⁻¹.

The large amount of data collected in Table S9‡ allow us to investigate if a temperature at a fixed relaxation time like T_B-100 can be useful as a descriptor for ZFC/FC blocking temperatures. Fig. S15‡ presents the relaxation time at the experimental T_B-ZFC/FC value for the data presented in Table S9‡ as a function of the heating rate. Clusters of data are observed around the most common R_H values (0.2–0.4 K min⁻¹, 2 K min⁻¹ and 5 K min⁻¹). In all cases, a high dispersion of relaxation times at the experimental T_B-ZFC/FC value is observed, especially for slow heating rates. In the case of the cluster located at 0.2–0.4 K min⁻¹, the maximum and minimum relaxation times are 282 s and 1 s. Thus, the experimental T_B-ZFC/FC value cannot be related with a unique treshold value of the relaxation time, even at a fixed heating rate.

For systems without a reported heating rate, the limiting values of the blocking temperature were calculated considering R_H values of 0.2 K min⁻¹ and 5 K min⁻¹, as shown in Fig. 11. In practically all cases, the T_B-ZFC/FC value falls within the calculated range. Furthermore, the T_irrev values tend to appear in the highest part or above the bars associated with the limiting R_H values, in agreement with the fact that the irreversibility temperature (T_irrev) must be higher than T_B-ZFC/FC.⁷⁸

Conclusions

This work discusses the various metrics for quantifying blocking temperatures and the challenges associated with each method. The newly proposed approach for estimating T_B-ZFC/FC from AC susceptibility parameters offers a practical way to compare the performance of SMMs in the literature and estimate T_B-ZFC/FC without the need for additional measurements. Thus, the T_B-ZFC/FC blocking temperature for a given heating rate can be estimated from the relaxation time data, which are the most common magnetic characterization data available for SMMs. In this context, the synthesis and characterization of the Dy^III SMM with D_5h geometry, [Dy(OPAd₂Bz)₂(H₂O)₄Br]Br₂·4THF (1), enabled the predictions of the blocking temperature to be tested over a broad range of heating rates and two different static magnetic fields. Furthermore, complex 1 is an interesting example of an air- and humidity-stable SMM with high U_eff and T_B values, adding to the restricted list of examples of SMMs with D_5h geometry.

Magnetic measurements confirm the SMM behaviour of complex 1, with a ZFC/FC blocking temperature in the range of 2.5–8 K, depending on the heating rate and the magnitude of the probe magnetic field. The presence of the magnetic hysteresis at zero-field with a large coercive field and remanent magnetisation values further supports the SMM properties of the complex. Additionally, AC magnetic susceptibility measurements reveal a high magnetisation reversal barrier, indicating slow relaxation of magnetisation. Ab initio calculations provide insights into the electronic structure and relaxation mechanism of complex 1. The calculations confirm the strong uniaxial magnetic anisotropy and reveal that magnetic relaxation occurs primarily via the first excited state (KD2).

In summary, the study of complex 1 exemplifies the potential of stable SMMs with D_5h geometry and axial ligands for achieving high-performance SMMs. The proposed approach for estimating T_B-ZFC/FC provided accurate values of the ZFC/FC blocking temperature of 1, properly captured the heating rate and probe field dependence of T_B-ZFC/FC and highlighted their importance. The study also identified the Raman and tunnelling mechanisms as the ones determining the blocking temperature. The possibility of relating the spin relaxation mechanisms directly to the blocking temperature opens the way for a better understanding of T_B tuning, which is a fundamental parameter of SMMs. Furthermore, the proposal successfully predicted blocking temperatures for many examples of SMMs in the literature, offering a valuable tool for characterising and comparing SMMs in future research.

Author contributions

Y. G. and M. A. P.: investigation, data analysis; D. A.: software; M. M. Q. M., E. C., S. G. C., E. R. and D. A.: conceptualisation, investigation, data analysis, validation, supervision, resources, funding acquisition; all authors contributed to the writing and editing of the article.

Data availability

The data supporting this article have been included as part of the ESI.‡ Estimation of the blocking temperature for any single-molecule magnet system can be obtained by introducing the spin relaxation parameters at the web page https://tbsim.ee.ub.edu.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

D. A. thanks FONDECYT Regular 1210325 for financial support. Powered@NLHPC: this research was partially supported by the supercomputing infrastructure of the NLHPC (ECM-02). Financial support from the Ministerio de Ciencia e Innovación (projects PID2022-138090NB-C21, PID2021-122464NB-I00, TED2021-129593B-I00, CNS2023-144561 and Maria de Maeztu CEX2021-001202-M), the Junta de Andalucía (FQM-195 and FQM-337), FEDER/Junta de Andalucía (projects I + D + i P20_00692, C-EXP-140-UGR23 and M.1.B.B TA_000722, Programas Operativos FEDER 2014-2020 y 2021-2027, Consejería de Economía, Conocimiento, Empresas y Universidad), and the University of Granada (project I + D + i PPJIA2020.10) is greatly appreciated. The authors also acknowledge the Centro de Servicios de Informática y Redes de Comunicaciones (CSIRC) for computational time and facilities. M.M.Q.M. thanks the Ministerio de Ciencia e Innovación for a Ramón y Cajal contract (the publication is part of the project PID2022-138090NB-C21 and grant RYC2021-034288-I funded by MCIN/AEI/10.13039/501100011033 and by the European Union “NextGenerationEU”/PRTR”). E. R. also acknowledges the Generalitat de Catalunya for ICREA Academia and 2021-SGR-00286 grants, and for computational resources at CSUC.

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Footnotes

† We want to dedicate this paper to Prof. Miquel Julve from the Universitat de Valencia, who recently passed away.

‡ Electronic supplementary information (ESI) is available. CCDC 2372498. For ESI and crystallographic data in CIF or other electronic formats, see DOI: https://doi.org/10.1039/d4qi03259d.

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