Studies of the Temperature Dependence of the Structure and Magnetism of a Hexagonal-Bipyramidal Dysprosium(III) Single-Molecule Magnet

Abstract

The hexagonal-bipyramidal lanthanide(III) complex [Dy(O^tBu)Cl(18-C-6)][BPh₄] (1; 18-C-6 = 1,4,7,10,13,16-hexaoxacyclooctadecane ether) displays an energy barrier for magnetization reversal (U_eff) of ca. 1000 K in a zero direct-current field. Temperature-dependent X-ray diffraction studies of 1 down to 30 K reveal bending of the Cl–Ln–O^tBu angle at low temperature. Using ab initio calculations, we show that significant bending of the O–Dy–Cl angle upon cooling from 273 to 100 K leads to a ca. 10% decrease in the energy of the excited electronic states. A thorough exploration of the temperature and field dependencies of the magnetic relaxation rate reveals that magnetic relaxation is dictated by five mechanisms in different regimes: Orbach, Raman-I, quantum tunnelling of magnetization, and Raman-II, in addition to the observation of a phonon bottleneck effect.

Short abstract

The temperature-dependent crystallography down to 30 K and field-dependent magnetic relaxation rate using both alternating- and direct-current magnetic measurements of a hexagonal-bipyramidal dysprosium(III) compound, [Dy(O^tBu)Cl(18-C-6)][BPh₄], were studied. Temperature-dependent bending of the Cl−Ln−O angle results in an approximate 10% decrease in the energy of the excited electronic states. Magnetic studies reveal that magnetic relaxation is dictated by five mechanisms in different regimes: Orbach, Raman-I, quantum tunnelling of magnetization, and Raman-II, in addition to the observation of a phonon bottleneck effect.

Introduction

Single-molecule magnets (SMMs) are touted for possible applications in high-density data storage¹ and in multifunctional magnetooptical materials,² but those can only occur if the blocking temperature (the temperature up to which magnetic memory is maintained in the absence of a magnetic field, T_B) can be brought closer to ambient conditions. One of the most important prerequisites for enhancing T_B is to raise the effective energy barrier for magnetization reversal (U_eff) of SMMs, which directly correlates to the magnetic anisotropy of the constituent metal ions.³ Hence, the dysprosium(III) ion with a ⁶H_15/2 ground-state term has attracted a great deal of interest for the construction of SMMs with high energy barriers. Recently, many Dy(III)-based SMMs with U_eff values of over 1000 K have been reported.⁴⁻¹⁵

Extending our recent work on equatorially supported axial SMMs,¹⁶⁻¹⁸ we had found that 1,4,7,10,13,16-hexaoxacyclooctadecane (18-C-6) ether can trap a Dy(III) ion in a very weak equatorial coordination environment and facilitate bonding to anionic axial donors.^16,19 However, the axial donors in that examples were a pair of chelating nitrates or borohydride, which do not confer strong magnetic anisotropy. To obtain stronger axiality, we decided to replace the chelating ligands with monodentate ligands. We had previously explored the use of tert-butoxide (O^tBu^–), or a combination of O^tBu^– and Cl^–, to construct nearly perfect pentagonal-bipyramidal Dy(III) SMMs, viz., [Dy(O^tBu)₂(py)₅][BPh₄] (py = pyridine)⁷ and [Dy(O^tBu)Cl(THF)₅][BPh₄] (THF = tetrahydrofuran),⁸ and so herein we employ O^tBu^– and Cl^– to form the hexagonal-bipyramidal complex [Dy(O^tBu)Cl(18-C-6)][BPh₄] (1). This complex is similar to other recent examples of Dy(III) SMMs with pseudo-C₆ symmetry, for example, [Dy(L^N6)(Ph₃SiO)₂][BPh₄] (L^N6 is a neutral Schiff base ligand formed from 2,6-diacetylpyridine and ethylenediamine)²⁰ and [Dy(L^E)(4-MeO-PhO)₂][BPh₄] [L^E is a Schiff base ligand derived from 2,6-pyridinedicarboxaldehyde and (1R,2R)-(+)-1,2-diphenylethylenediamine].¹⁸ Here, the neutral 18-C-6 ether ligand provides six O atoms that are weaker-field donors than the N atoms from the Schiff base ligands in the former two examples. We studied the temperature dependence of the crystal structure of 1 down to 30 K to show that there is significant bending of the O–Dy–Cl angle upon cooling from 273 to 100 K, which leads to an approximate 10% decrease in the energy of the excited electronic states according to ab initio calculations. We also studied the field and temperature dependencies of the magnetic relaxation rate of 1 to reveal the interplay of the Orbach, Raman-I, quantum tunnelling of magnetization (QTM), Raman-II, and phonon bottleneck mechanisms.

Results

Synthesis and Structures

The synthesis of 1 is similar to that described previously for [Dy(O^tBu)Cl(THF)₅][BPh₄];⁸ however, in this case, 18-C-6 has been added to the reaction so that the five equatorial THF molecules could be replaced by 18-C-6 [see the Supporting Information (SI) for details]. Complex 1 crystallizes in the orthorhombic Pbca space group and comprises the cationic complex [Dy(O^tBu)Cl(18-C-6)]⁺, one counterion [BPh₄]⁻, and a THF molecule in the asymmetric unit. The Dy(III) ion is coordinated by six O atoms from the 18-C-6 crown ether in an equatorial fashion, with one O^tBu^– anion and one Cl^– anion occupying the axial positions, affording a hexagonal-bipyramidal coordination geometry (Figure 1). The cationic complex [Dy(O^tBu)Cl(18-C-6)]⁺ is disordered over two positions with a 78:22 distribution at 273 K (we define the A component as the larger fraction at all temperatures); both disorder components are displaced by 0.27 Å in the Dy–18-C-6 equatorial plane and rotated 12.3° around the Cl–Dy–O axis. The [BPh₄]⁻ anion and THF molecules are also disordered over two positions with 55:45 and 61:39 distribution, respectively. The axial Dy–O bond length for the larger disorder component is 2.012(4) Å, which is the shortest Dy–O bond reported to date and significantly shorter than the equatorial Dy–O bonds (ca. 2.57 Å). The corresponding Dy–Cl distance is 2.608(5) Å and the Cl–Dy–O angle is 175.3(3)°, showing a near-linear alignment of charge on the axis. The equatorial O–Dy–O angles range from 58.97(17)° to 61.13(16)°, and the A–Dy–B (A is one of the axial donor atoms, and B is an O atom from 18-C-6) angles are ranging from 86.7(3)° to 97.9(3)°, and average to 90.0°, indicating a slightly waving equatorial plane. Thus, 1 has a coordination geometry approaching hexagonal-bipyramidal, hence making a nice structural comparison to the pentagonal-bipyramidal structure of [Dy(O^tBu)Cl(THF)₅][BPh₄].⁸ We have also prepared the isostructural yttrium complex [Y(O^tBu)Cl(18-C-6)][BPh₄] (see the SI for details) and subsequently prepared a 5% dilute Dy@[Y(O^tBu)Cl(18-C-6)][BPh₄] sample for magnetic studies.

Figure 1.

(a) Structure of [Dy(O^tBu)Cl(18-C-6)]⁺ in 1 (273 K). H atoms are removed for clarity. (b) Top and (c) side views of a coordination polyhedron. (d) Top and (c) side views of the disordered model for 1 (273 K). Color codes: Dy, dark green; O, red; Cl, light green; C, gray.

In situ temperature-dependent single-crystal X-ray diffraction (XRD) was used to monitor the structural changes in 1 upon cooling (Tables S1–S3). Upon cooling, the disorder fraction changes to 59:41 at 200 K, followed by a return to 75:25 at 30 K (Figure 2a). There is a slight lengthening of the axial Dy–O and Dy–Cl bonds upon cooling to 30 K, by around 0.03 and 0.08 Å, respectively, with a simultaneous shortening of the equatorial Dy–O bonds by 0.02 Å in the same temperature range. Most importantly, the axial Cl–Dy–O angle changes significantly for both disorder components (Figures 2b,c); the angles of both components decrease by 3 or 5° from 273 to 200 K, and upon further cooling to 30 K, the angle for B component increases to 175.1(6)°, while the angle for A component continues to decrease, reaching 165.9(2)°. Unsurprisingly, the equivalent isotropic displacement parameters of both disorder components of the cation decrease with the temperature. By continuous shape measures (CSM), the local geometry of the Dy(III) ion in complex 1 was calculated for all temperatures and components (Table S5).²¹ The smallest values obtained (red column in Table S5) are always for hexagonal-bipyramidal geometry, with the CSM increasing upon cooling due to bending of the O–Dy–Cl angle.

Figure 2.

(a) Disorder fraction of the A component as a function of the temperature and (b) O–Dy–Cl angle in 1 as a function of the temperature for each disorder fraction (A and B). Lines are a guide to the eye. (c) Temperature-dependent structures of [Dy(O^tBu)Cl(18-C-6)]⁺ (A component) upon cooling (left to right) with thermal ellipsoids drawn at the 70% probability level and equatorial atoms corresponding to 18-C-6 represented as sticks for clarity. Color codes: Dy, light green; O, red; C, gray; Cl, dark green. H atoms are omitted for clarity.

Electronic Structure Calculations

The electronic structure of 1 was determined by complete-active-space self-consistent-field spin–orbit (CASSCF-SO) calculations with OpenMolcas, version 19.11,²² using the X-ray structures of both disorder components at each temperature. The low-lying states for 1 across the different temperatures are quite consistent and are composed of >99% m_J = ±¹⁵/₂ for the ground state, > 98% m_J = ±¹³/₂ for the first excited state, and >95% m_J = ±¹¹/₂ for the second excited state, while the third excited state is much more mixed (Tables S6–S16). In most cases, thermal evolution of the molecular structures has a negligible influence on the energy-level scheme and magnetic anisotropy.²³ However, for complex 1, the change to the molecular structure is significant, and there is a clear change of the excited energy levels above 100 K (Figure S1). For example, the gaps to the first, second, and third excited states in the largest disorder components are 472, 809, and 996 cm^–1 at 273 K, which decrease to 417, 721, and 894 cm^–1 at 100 K. Below 100 K, there is only a small change in the structure (Figure 2); thus, the electronic energies are relatively unaffected from 80 to 30 K.

Magnetic Measurements

Variable-temperature direct-current (dc) magnetic susceptibility data for 1 were collected under a 0.1 T field in the temperature range of 2–300 K (Figure S2). The value of χT at 300 K is 14.13 emu K mol^–1, in good agreement with the expected value of 14.17 emu K mol^–1 for a free Dy(III) ion (S = ⁵/₂, L = 5, g = ⁴/₃). Upon cooling, χT is weakly temperature dependent above 5 K, at which point it shows a sharp decrease to 8.96 emu K mol^–1 at 2 K (Figure S2, inset), which may be due to slow magnetic relaxation, intermolecular antiferromagnetic interactions, and/or Zeeman depopulation. The field dependence of the magnetization for 1 was measured at several temperatures between 2 and 125 K up to 7 T, reaching a maximum value of 5.32 μ_B at 2 K (Figure S2). There is a slight sigmoidal shape to the magnetization data at the lowest temperatures at fields of <1 T, suggesting some magnetic hysteresis at low temperature. Indeed, we find that 1 shows butterfly-like hysteresis loops that are open between 2 and 4 K (Figure 3); the large loss of magnetization at zero field is indicative of efficient QTM at low temperatures.^24,25

Figure 3.

Magnetization hysteresis data of polycrystalline 1 at temperatures in the range of 2 ≤ T (K) ≤ 6 and fields of −1 ≤ B (T) ≤ 1 at an average sweep rate of 9.5 Oe s^–1.

Variable-frequency alternating-current (ac) susceptibility measurements for 1 show significant frequency dependence up to 80 K in both in-phase (χ′) and out-of-phase (χ″) signals (Figure 4), and fitting these data with a generalized Debye model²⁶ allows us to extract the characteristic relaxation time τ and the distribution parameter α and, hence, determine the estimated standard deviations (ESDs) for the relaxation times (Table S17).²⁷ We note that, although we see two clear disorder components in the XRD data, we do not observe two distinct relaxation times for these species, and thus they must be similar enough so that we cannot resolve them with ac susceptometry;²⁸ hence, the use of the generalized Debye model, which includes a distribution parameter, is the most appropriate route. To access slower relaxation times, we employed dc magnetization decay techniques; however, like for [Dy(O^tBu)Cl(THF)₅][BPh₄],⁸ relaxation is too fast at zero field, and thus we performed field-dependent decay experiments at 2, 3, and 4 K (Tables S30–S32 and Figures S17–S19). These data can be modeled with a stretched exponential to obtain τ, where the β parameter can be interpreted as a distribution and thus used to obtain ESDs.²⁹ We have subsequently performed field-dependent ac experiments to probe the field dependence at higher temperatures, fitting these data in the same way as that for zero field (Figures S3–S12 and Tables S18–S27). It is important to recognize here that these measurements are all below 80 K, and thus the proportion of the two disorder components (ca. 75:25) and the structure of the disorder components are more-or-less constant in this regime (Figure 2).

Figure 4.

Cole–Cole plot of the in-phase (χ′) and out-of-phase (χ″) ac susceptibility signals under zero dc field at temperatures from 9.8 K (blue) to 80.0 K (orange) for 1. Low-temperature data are shown more clearly in the lower panel. Solid lines are best fits.

We note here that all magnetic experiments are conducted using polycrystalline samples; while this has no impact on the relaxation rates extracted from the ac measurements in zero dc field, dc decay and in-field ac relaxation measurements are, in principle, anisotropic. However, in this case, because the sample has such strong easy-axis magnetic anisotropy, the results are strongly biased by contributions from the easy-axis direction. This is because the magnetization of an isolated Ising-like Kramers doublet (an appropriate approximation for 1 under the conditions of our measurements) is proportional to M ∝ g_z² cos(θ)², which peaks when θ = 0 corresponds to the magnetic field along the easy axis.

Discussion

The temperature dependence of the zero-field relaxation rates reveals that magnetic relaxation is dominated by a power-law-like process between 10 and 60 K, most likely arising from a two-phonon first-order Raman mechanism (Raman-I, Figure 5).³⁰ Above 60 K, the relaxation rate increases sharply with temperature, suggestive of an exponential temperature-dependent regime due to an Orbach mechanism, while below 10 K, the relaxation rates are nearly independent of temperature, suggesting a QTM mechanism with a rate of ca. τ_QTM^–1 ≈ 10 s^–1. These data were modeled with eq 1 using the CC-FIT2 code,²⁷

1

where U_eff is the effective energy barrier and 10^A = τ₀^–1 is the attempt frequency of the Orbach process, 10^R = C, n are the Raman-I parameters, and 10^Q₁ = τ_QTM^–1 is the QTM rate. The result of the fit yields the parameters A = 10(3) log₁₀ [s^–1], U_eff = 1000(500) K, R = −2(1) log₁₀ [s^–1 K^–n], n = 2.8(6), and Q₁ = 1.1(5) log₁₀ [s^–1]. The very large uncertainties associated with the Orbach parameters are due to the limited data in this regime and inclusion of the ESDs in the fitting process, which makes the assumption of a random-error model that is not strictly accurate;²⁷ similarly large uncertainties have been observed previously in a similar context.³¹ Nonetheless, CASSCF-SO calculations using the temperature-dependent geometries from XRD show that the second excited Kramers doublets have transverse g values that become significant (∼0.2) and a main magnetic axis that starts to deviate significantly (∼10°) from that of the ground state (Tables S6–S16), suggesting that it is through this state that relaxation occurs, which would indicate a U_eff value of ca. 720 cm^–1 or 1040 K (at around 80 K, where we experimentally observe the Orbach regime), which is consistent with our experimental data. Alternatively, we could fix U_eff from CASSCF-SO calculations as 1040 K and fit the other parameters to give very similar results, albeit with smaller uncertainties: A = 9.8(1) log₁₀ [s^–1], R = −1.8(9) log₁₀ [s^–1 K^–n], n = 2.8(5), and Q₁ = 1.1(5) log₁₀ [s^–1]. Indeed, instead of the power-law form of the two-phonon Raman-I process, there have been recent theoretical works invoking optical phonons that lead to exponential forms.³²⁻³⁴ We have attempted a fit where the power-law Raman-I term is replaced by the expression 4ℏωΓ|a|²e^{–ℏω/k_BT}/(ℏ⁴ω⁴δ² + 4ℏ²ω²Γ²), where ℏω is the energy of the Raman-I active mode, Γ is the phonon line width, a is the effective spin–phonon coupling strength, and δ is the intra-Kramers doublet splitting.³² We fix Γ = 10 cm^–1 because we have found this order-of-magnitude to be correct for phonons involved in magnetic relaxation³⁵ and δ = 0.01 cm^–1 because this must be ≪1 cm^–1 (in any case, the rates are not as sensitive to these two parameters) and fit the zero-field ac data set to obtain a reasonable fit to the experimental data (Figure S13) with parameters A = 8(1) log₁₀ [s^–1], U_eff = 800(200) K, Q₁ = 1.4(3) log₁₀ [s^–1], ℏω = 90(30) cm^–1, and a = 2000(1000) cm^–1. While the fit is passable, the effective spin–phonon coupling strength is orders of magnitude too strong: these parameters are usually on the order of 1 cm^–1,^36,37 and hence we do not think this model is the most correct interpretation of the data in this case.

Figure 5.

Temperature dependence of the zero-field relaxation rate of 1 (black) from the ac susceptibility data, fitted (red) to a model containing Orbach (orange), Raman-I (green), and QTM (blue) contributions. The purple data points are taken from the minimum relaxation rate at each temperature from the dc decay data, and the line is a best fit of the form τ^–1 = 10^R′T^n′. The pink data point is the minimum relaxation rate at 2 K from the dc decay data on 5% Dy@[Y(O^tBu)Cl(18-C-6)][BPh₄]. Error bars are ESDs derived from the distribution of relaxation times in the generalized Debye model for the ac data²⁷ or from the stretched exponential model for the dc data.²⁹

Plotting the field dependence of the relaxation rate extracted from the dc decay measurements at 2–4 K shows profiles similar to those observed for [Dy(O^tBu)Cl(THF)₅][BPh₄];⁸ the relaxation rate is initially slowed down by the application of a field to a minimum of around 0.1 T, followed by an increase in the relaxation rates at higher fields (Figures 6 and S22 and S23). The decrease in the relaxation rates at low field is due to suppression of the QTM mechanism, which dominates at zero field and below 10 K,^8,38 while the increase at higher fields could be due to the onset of either a direct single-phonon relaxation mechanism⁸ or a second-order Raman mechanism (Raman-II),³⁰ both of which have the same power-law form of the field dependence for experiments on powder samples (albeit with different expected exponents).³⁹

Figure 6.

Field dependence of the relaxation rate for 1 at 2 K from dc decay experiments. Black points are experimental data fitted (red) to the sum of the QTM (green), Raman-I (purple), and Raman-II (blue) processes. Error bars are ESDs derived from the stretched exponential model for the dc data.²⁹

For [Dy(O^tBu)Cl(THF)₅][BPh₄], we found that the temperature dependence of the minimum relaxation rate from the field-dependent dc data extrapolates to match very closely the power-law Raman-I region in the zero-field ac data.⁸ Performing the same analysis here, we observe that the minimum relaxation rates at 2, 3, and 4 K indeed show a power-law temperature dependence (Figure 5, purple data); however, the characteristic parameters are rather different from those found from the zero-field ac data for the Raman-I process [R′ = −5(1) log₁₀ [s^–1 K^–n] and n′ = 4(2) compared to R = −2(1) log₁₀ [s^–1 K^–n] and n = 2.8(6)]. The observed minimum relaxation rates at 2–4 K are ca. 100 times slower than those expected from the Raman-I process observed in the ac data (Figure 5), and because the overall relaxation rate is the sum of all relaxation processes, this cannot be due to the addition of another relaxation mechanism (which could only increase the rates). Excluding a sudden structural change below 10 K (which we cannot probe with our XRD apparatus), the other possibility is that these data are reporting a phonon bottleneck,⁴⁰ where the lattice phonon bath is “heated” by spin relaxation such that the measured relaxation rates are no longer the true spin–lattice rates but rather limited by extrinsically slower lattice-cryostat rates.⁴¹⁻⁴³ This situation can arise when the crystallites are too large or the concentration of the paramagnetic ions is too high, such that the mean free path of the phonons involved in the relaxation is much shorter than the crystallite dimensions (that is, the phonons continually scatter from paramagnetic sites and the lattice cannot equilibrate with the cryostat).⁴¹⁻⁴³ This situation results in relaxation rates that have a power-law temperature dependence that is slower than the true underlying spin–lattice relaxation rate.^44,45 To confirm that this occurs due to a phonon bottleneck regime, we performed magnetization decay and in-field ac measurements on a dilute 5% Dy@[Y(O^tBu)Cl(18-C-6)][BPh₄] sample. This experiment increases the distance between paramagnetic ions and is traditionally used to reduce internal dipolar fields and, hence, slow down relaxation via QTM; however; in this case, if we are in the phonon bottleneck regime, we would expect the relaxation rates to increase upon dilution. This is exactly what we see in both the ac and dc experiments (Figures S25 and S26), and thus we confirm that the slower-than-expected Raman-I-like relaxation below 10 K in nonzero field in 1 arises from a phonon bottleneck. We note, however, that the dilute sample is still itself in the phonon bottleneck regime (Figures 5 and S26), although to a lesser extent than that for 1.

The field-dependent data from dc decay measurements can be fitted with a model accounting for the three regimes, eq 2,

2

where 10^Q₁ is the zero-field QTM rate, Q₂ and p express the field dependence of the QTM term, X = 10^R′T^n′ is the field-independent phonon bottleneck rate for that temperature, and G and q are the Raman-II/direct parameters. By fixing Q₁ = 1.1(5) log₁₀ [s^–1] from the ac data, we find the best-fit values for Q₂, p, X, G, and q for the 2–4 K data in Table 1; note that Q₂, p, X, G, and q are temperature-dependent quantities. The temperature dependence of X is consistent with the phonon bottleneck regime parametrization as R′ = −6(2) log₁₀ [s^–1 K^–n] and n′ = 6(5) (Figure S24); note that these numbers differ from those above because the experimental rates in this region are actually best modeled as the sum given in eq 2 rather than by a 10^R′T^n′ term alone. The QTM regime should be roughly quadratic in the field (i.e., p ≈ 2) based on simple models,⁴⁶ but we have previously found that this is not always true and that it is also temperature-dependent;⁸ for [Dy(O^tBu)Cl(THF)₅][BPh₄], the field exponent was ca. 4, and in this case, we find they are around 3. We also find that the Q₂ parameter is correspondingly smaller in this case, around 7–8 log₁₀ [T^–p], compared to ca. 11.5 log₁₀ [T^–p] for [Dy(O^tBu)Cl(THF)₅][BPh₄].⁸ Similar to [Dy(O^tBu)Cl(THF)₅][BPh₄], we find the field dependence of the relaxation rate above 0.2 T to be τ^–1 ∝ B², which is in better agreement with a Raman-II process³⁰ than with a direct process.³⁹

Table 1. Best-Fit Parameters for the Field Dependence of Relaxation in 1 Measured by dc Decay between 2 and 4 K.

temperature (K)	G(log10[s–1 T–q])	q	Q2(log10[T–p])	p	X (s–1)
2	–2(1)	2(2)	8(2)	3(1)	–4(2)
3	–2(1)	2(4)	7(5)	3(3)	–3(1)
4	–2(1)	1(5)	7(2)	2(1)	–2.3(6)

We can also use field-dependent ac measurements to examine the magnetic relaxation mechanisms; these data show a decrease in the relaxation rate with the applied field, much like the 2–4 K dc decay data, but there is only a very small increase in the rate by 0.5 T, which appears to become more significant at lower temperatures (Figures S17 and S18). The data measured above 10 K are not yet in the QTM-dominated region (Figure 5), and so the decrease in the rate is not due to the suppression of QTM but rather to an inherent field dependence of the Raman-I process.⁴⁷ The data in this regime can be fit using the Brons–van Vleck model, eq 3,^{47,48,44,49−51}

3

where 10^Y = 10^RTⁿ is the zero-field Raman-I relaxation rate and e and f represent the competition between the internal and external fields,^30,49 with the parameters given in Table 2 for the data above 10 K. We find that the values of 10^Y are in excellent agreement with the zero-field Raman-I relaxation rates and that there is no change in the e or f terms as a function of the temperature, giving us confidence that the data in this regime indicate a true Raman-I mechanism. In SI units, the values of 10^e and 10^f are ca. 1000 T^–2, which are significantly larger than those found for a series of vanadyl compounds, which are on the order of ca. 10–100 T^–2,^44,49,51 owing to the significantly larger ground-state magnetic moment of a Dy(III) SMM than that for S = ¹/₂ vanadyl compounds. While the field-dependent ac data recorded at and below 10 K can be fit using the Brons–van Vleck model, the agreement becomes progressively worse as the temperature is lowered due to onset of the QTM mechanism (Figure S18); it is not possible to obtain better agreement with the data using eq 2 because of the competition of numerous field- and temperature-dependent processes in addition to the onset of the phonon bottleneck effect. We also note that the Brons–van Vleck model is not able to model the field-dependent relaxation rate data from dc decay experiments between 2 and 4 K.

Table 2. Best-Fit Parameters for the Field Dependence of Relaxation in 1 Measured by ac Susceptibility between 12 and 20 K.

temperature (K)	Y(log10 [s–1])	e(log10 [Oe–2])	f(log10 [Oe–2])
20	1.9(3)	–5(1)	–5(1)
18	1.8(3)	–5(1)	–5(1)
16	1.7(4)	–5(1)	–5(1)
14	1.7(4)	–5(1)	–5(1)
12	1.6(4)	–5(1)	–5(1)

At zero field, there is a crossover from the Raman-I regime to the QTM regime around 10 K, and we find that a phonon bottleneck effect is revealed below 4 K when the QTM process is quenched (Figures 5 and 6). However, all we can say from these data is that the phonon bottleneck process becomes important at some point between 4 and 20 K. Examination of the temperature dependence of the ac data recorded in a 0.2 T field (roughly corresponding to the minimum rate in the range 6–20 K) allows us to pin down the crossover point between the Raman-I and bottleneck regimes to ca. 12 K (Figure 7); this crossover point does not seem to be affected by dilution because even the 5% Dy@[Y(O^tBu)Cl(18-C-6)][BPh₄] dilute sample remains in a phonon bottleneck regime under these conditions. The power-law fits of these data, and of those at other fields, are in excellent agreement with the Raman-I regime from the zero-field ac data and the phonon bottleneck regime from the dc data (Figures S14–S16 and Tables S27 and S28). At the largest field investigated (0.5 T), we find that the temperature dependence of the phonon bottleneck regime becomes similar to that of the Raman-I mechanism (Figure S16d), owing to the onset of the Raman-II mechanism. We note that model parameters arising from fits to these data are subject to large errors, which is due to the assumption that the ESDs account for a randomly distributed variable, which in this case is not true: these ESDs arise from a distribution of relaxation rates that likely arises from a distribution of local structures.²⁷ Indeed, we observe that the whole distribution of rates undergoes the same change in slope in this regime, indicating a real change, which is belied by the fits of the data and which assumes a random-error model.

Figure 7.

Temperature dependence of the relaxation rate of 1 (black) from the ac susceptibility data in a 0.2 T field, fitted to two power-law processes for the Raman-I (τ^–1 = 10^RTⁿ, green) and phonon bottleneck (τ^–1 = 10^R′T^n′, blue) regimes. Error bars are ESDs derived from the distribution of relaxation times in the generalized Debye model for the ac data.²⁷

Conclusion

Herein we have reported a new hexagonal-bipyramidal Dy(III) SMM with a large energy barrier to magnetic relaxation. Temperature-dependent XRD experiments reveal a significant bending of the axial Cl–Ln–OBu^t angle and changes in the disorder fractions between 298 and 80 K; ab initio calculations suggest that this bending should lead to an approximate 10% reduction in the U_eff barrier in the Orbach region; however, this effect is not observed in the limited temperature range that we observe here. In contrast, relatively small changes are observed between 80 and 30 K. A thorough temperature- and field-dependent study of the relaxation dynamics reveals the interplay between the Orbach, Raman-I, QTM, and Raman-II relaxation mechanisms, in addition to the observation of a phonon bottleneck effect.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.inorgchem.1c02779.

• Synthesis, crystallography data (Tables S1–S4), CSM calculations (Table S5), ab initio calculation data (Tables S6–S16 and Figure S1), static magnetic data (Figure S2), dynamic magnetic data (Tables S17–S27 and Figures S3–S12) from the ac susceptibility data, temperature dependence of the relaxation rate from the ac susceptibility data (Tables S28 and S29 and Figures S13–S16), field dependence of the relaxation rate from the ac susceptibility data (Figures S17 and S18), magnetization decay fitting parameters (Tables S30–S32 and Figures S19–S21), field dependence of the relaxation rate from dc decay experiments (Figures S22 and S23), temperature dependence of the X parameter from eq 2 to extract phonon bottleneck parameters (Figure S24), and magnetic relaxation rate for a diluted sample (Figures S25 and S26) (PDF)

Author Present Address

^# Kimika Fakultatea, Euskal Herriko Unibertsitatea & Dostia International Physics Center (DIPC), Donostia, Euskadi, IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

Author Contributions

Y.-S.D. synthesized the complexes and performed magnetic measurements. I.V.-Y. conducted low-temperature crystallography. W.J.A.B., D.R., and N.F.C. modeled the magnetic data. Y.-Q.Z. and M.J.G. performed CASSCF-SO calculations. Y.-Z.Z., N.F.C., and R.E.P.W. jointly directed the study. The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

The authors declare no competing financial interest.

Notes

Raw research data can be found here: 10.48420/16573880 (magnetism); 10.48420/16573874 (CASSCF-SO).