Spin-phonon coupling and magnetic relaxation in single-molecule magnets.

Electron-phonon coupling is important in many physical phenomena, e.g. photosynthesis, catalysis and quantum information processing, but its impacts are difficult to grasp on the microscopic level. One area attracting wide interest is that of single-molecule magnets, which is motivated by searching for the ultimate limit in the miniaturisation of binary data storage media. The utility of a molecule to store magnetic information is quantified by the timescale of its magnetic reversal processes, also known as magnetic relaxation, which is limited by spin-phonon coupling. Several recent accomplishments of synthetic organometallic chemistry have led to the observation of molecular magnetic memory effects at temperatures above that of liquid nitrogen. These discoveries have highlighted how far chemical design strategies for maximising magnetic anisotropy have come, but have also highlighted the need to characterise the complex interplay between phonons and molecular spin states. The crucial step is to make a link between magnetic relaxation and chemical motifs, and so be able to produce design criteria to extend molecular magnetic memory. The basic physics associated with spin-phonon coupling and magnetic relaxation was outlined in the early 20th century using perturbation theory, and has more recently been recast in the form of a general open quantum systems formalism and tackled with different levels of approximations. It is the purpose of this Tutorial Review to introduce the topics of phonons, molecular spin-phonon coupling, and magnetic relaxation, and to outline the relevant theories in connection with both the traditional perturbative texts and the more modern open quantum systems methods.

Accurate and Efficient Spin-Phonon Coupling and Spin Dynamics Calculations for Molecular Solids.

Molecular materials are poised to play a significant role in the development of future optoelectronic and quantum technologies. A crucial aspect of these areas is the role of spin-phonon coupling and how it facilitates energy transfer processes such as intersystem crossing, quantum decoherence, and magnetic relaxation. Thus, it is of significant interest to be able to accurately calculate the molecular spin-phonon coupling and spin dynamics in the condensed phase. Here, we demonstrate the maturity of ab initio methods for calculating spin-phonon coupling by performing a case study on a single-molecule magnet and showing quantitative agreement with the experiment, allowing us to explore the underlying origins of its spin dynamics. This feat is achieved by leveraging our recent developments in analytic spin-phonon coupling calculations in conjunction with a new method for including the infinite electrostatic potential in the calculations. Furthermore, we make the first ab initio determination of phonon lifetimes and line widths for a molecular magnet to prove that the commonplace Born-Markov assumption for the spin dynamics is valid, but such "exact" phonon line widths are not essential to obtain accurate magnetic relaxation rates. Calculations using this approach are facilitated by the open-source packages we have developed, enabling cost-effective and accurate spin-phonon coupling calculations on molecular solids.

Molecular magnetic hysteresis at 60 kelvin in dysprosocenium.

Lanthanides have been investigated extensively for potential applications in quantum information processing and high-density data storage at the molecular and atomic scale. Experimental achievements include reading and manipulating single nuclear spins, exploiting atomic clock transitions for robust qubits and, most recently, magnetic data storage in single atoms. Single-molecule magnets exhibit magnetic hysteresis of molecular origin-a magnetic memory effect and a prerequisite of data storage-and so far lanthanide examples have exhibited this phenomenon at the highest temperatures. However, in the nearly 25 years since the discovery of single-molecule magnets, hysteresis temperatures have increased from 4 kelvin to only about 14 kelvin using a consistent magnetic field sweep rate of about 20 oersted per second, although higher temperatures have been achieved by using very fast sweep rates (for example, 30 kelvin with 200 oersted per second). Here we report a hexa-tert-butyldysprosocenium complex-[Dy(Cpttt)2][B(C6F5)4], with Cpttt = {C5H2tBu3-1,2,4} and tBu = C(CH3)3-which exhibits magnetic hysteresis at temperatures of up to 60 kelvin at a sweep rate of 22 oersted per second. We observe a clear change in the relaxation dynamics at this temperature, which persists in magnetically diluted samples, suggesting that the origin of the hysteresis is the localized metal-ligand vibrational modes that are unique to dysprosocenium. Ab initio calculations of spin dynamics demonstrate that magnetic relaxation at high temperatures is due to local molecular vibrations. These results indicate that, with judicious molecular design, magnetic data storage in single molecules at temperatures above liquid nitrogen should be possible.

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

The hexagonal-bipyramidal lanthanide(III) complex [Dy(OtBu)Cl(18-C-6)][BPh4] (1; 18-C-6 = 1,4,7,10,13,16-hexaoxacyclooctadecane ether) displays an energy barrier for magnetization reversal (Ueff) 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-OtBu 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.

Ab Initio Prediction of High-Temperature Magnetic Relaxation Rates in Single-Molecule Magnets.

Organometallic molecules based on [Dy(CpR)2]+ cations (where CpR is a substituted cyclopentadienyl anion) have emerged as clear front-runners in the search for high-temperature single-molecule magnets. Within this family of structurally similar molecules, significant variations in their magnetic properties are seen, demonstrating the importance of understanding magneto-structural relationships to develop more efficient design strategies. Here we develop an ab initio spin dynamics methodology and show that it is capable of quantitative prediction of relative relaxation rates in the Orbach region. Applying it to all reported [Dy(CpR)2]+ cations allows us understand differences in their relaxation dynamics, highlighting that the main discriminant is the magnitude of the crystal field splitting, rather than differences in spin-vibrational coupling. We subsequently employ the method to predict relaxation rates for a series of hypothetical organometallic sandwich compounds, revealing an upper limit to the effective barrier to magnetic relaxation of around 2100-2200 K, which has been reached by existing compounds. Our conclusion is that further improvements to monometallic single-molecule magnets require moving vibrational modes off-resonance with electronic excitations.

Opening Magnetic Hysteresis by Axial Ferromagnetic Coupling: From Mono-Decker to Double-Decker Metallacrown.

Combining Ising-type magnetic anisotropy with collinear magnetic interactions in single-molecule magnets (SMMs) is a significant synthetic challenge. Herein we report a Dy[15-MCCu -5] (1-Dy) SMM, where a DyIII ion is held in a central pseudo-D5h pocket of a rigid and planar Cu5 metallacrown (MC). Linking two Dy[15-MCCu -5] units with a single hydroxide bridge yields the double-decker {Dy[15-MCCu -5]}2 (2-Dy) SMM where the anisotropy axes of the two DyIII ions are nearly collinear, resulting in magnetic relaxation times for 2-Dy that are approximately 200 000 times slower at 2 K than for 1-Dy in zero external field. Whereas 1-Dy and the YIII -diluted Dy@2-Y analogue do not show remanence in magnetic hysteresis experiments, the hysteresis data for 2-Dy remain open up to 6 K without a sudden drop at zero field. In conjunction with theoretical calculations, these results demonstrate that the axial ferromagnetic Dy-Dy coupling suppresses fast quantum tunneling of magnetization (QTM). The relaxation profiles of both complexes curiously exhibit three distinct exponential regimes, and hold the largest effective energy barriers for any reported d-f SMMs up to 625 cm-1 .

A Study of Magnetic Relaxation in Dysprosium(III) Single-Molecule Magnets.

Although the development of single-molecule magnets (SMMs) is rapid, there are only two families of high energy barrier (Ueff ) dysprosium(III) SMMs known so far: the cyclopentadienyl (Cp) family with a sandwich structure and the pentagonal-bipyramidal (PB) family with D5h symmetry. These high-barrier SMMs, which usually possess Ueff >500 cm-1 allow the separate study of the four magnetic relaxation paths, namely, direct, quantum tunnelling, Raman and Orbach processes, in detail. Whereas the first family is chemically more challenging to modify the Cp rings, it is shown herein that the latter family, with the common formulae [DyX1 X2 (Leq )5 ]+ , such as X1 /X2 =- OCMe3 , - OSiMe3 , - OPh, Cl- or Br- ; Leq =THF/pyridine/4-methylpyridine, can be readily fine-tuned with a range of axial and equatorial ligands by simple substitution reactions. This allows unambiguous confirmation that the Ueff mainly depends on the identity of X1 and X2 , rather than on Leq . More importantly, the fitted parameters are barrier dependent. If X1 is an O donor and X2 is a halide, 500

Bis-Monophospholyl Dysprosium Cation Showing Magnetic Hysteresis at 48 K.

Single-molecule magnets (SMMs) have potential applications in high-density data storage, but magnetic relaxation times at elevated temperatures must be increased to make them practically useful. Bis-cyclopentadienyl lanthanide sandwich complexes have emerged as the leading candidates for SMMs that show magnetic memory at liquid nitrogen temperatures, but the relaxation mechanisms mediated by aromatic C5 rings have not been fully established. Here we synthesize a bis-monophospholyl dysprosium SMM [Dy(Dtp)2][Al{OC(CF3)3}4] (1, Dtp = {P(CtBuCMe)2}) by the treatment of in-situ-prepared "[Dy(Dtp)2(C3H5)]" with [HNEt3][Al{OC(CF3)3}4]. SQUID magnetometry reveals that 1 has an effective barrier to magnetization reversal of 1760 K (1223 cm-1) and magnetic hysteresis up to 48 K. Ab initio calculation of the spin dynamics reveals that transitions out of the ground state are slower in 1 than in the first reported dysprosocenium SMM, [Dy(Cpttt)2][B(C6F5)4] (Cpttt = C5H2tBu3-1,2,4); however, relaxation is faster in 1 overall due to the compression of electronic energies and to vibrational modes being brought on-resonance by the chemical and structural changes introduced by the bis-Dtp framework. With the preparation and analysis of 1, we are thus able to further refine our understanding of relaxation processes operating in bis-C5/C4P sandwich lanthanide SMMs, which is the necessary first step toward rationally achieving higher magnetic blocking temperatures in these systems in the future.

Light Lanthanide Metallocenium Cations Exhibiting Weak Equatorial Anion Interactions.

As the dysprosocenium complex [Dy(Cpttt )2 ][B(C6 F5 )4 ] (Cpttt =C5 H2 tBu3 -1,2,4, 1-Dy) exhibits magnetic hysteresis at 60 K, similar lanthanide (Ln) complexes have been targeted to provide insights into this remarkable property. We recently reported homologous [Ln(Cpttt )2 ][B(C6 F5 )4 ] (1-Ln) for all the heavier Ln from Gd-Lu; herein, we extend this motif to the early Ln. We find, for the largest LnIII cations, that contact ion pairs [Ln(Cpttt )2 {(C6 F5 -κ1 -F)B(C6 F5 )3 }] (1-Ln; La-Nd) are isolated from reactions of parent [Ln(Cpttt )2 (Cl)] (2-Ln) with [H(SiEt3 )2 ][B(C6 F5 )4 ], where the anion binds weakly to the equatorial sites of [Ln(Cpttt )2 ]+ through a single fluorine atom in the solid state. For smaller SmIII , [Sm(Cpttt )2 ][B(C6 F5 )4 ] (1-Sm) is isolated, which like heavier 1-Ln does not exhibit equatorial anion interactions, but the EuIII analogue 1-Eu could not be synthesised due to the facile reduction of EuIII precursors to EuII products. Thus with the exception of Eu and radioactive Pm this work constitutes a structurally similar family of Ln metallocenium complexes, over 50 years after the [M(Cp)2 ]+ series was isolated for the 3d metals.

Uncertainty estimates for magnetic relaxation times and magnetic relaxation parameters.

The use of alternating current (AC) magnetometry to measure magnetic relaxation times is one of the most fundamental measurements for characterising single-molecule magnets (SMMs). These measurements, performed as a function of frequency, temperature and magnetic field, give vital information on the underlying magnetic relaxation process(es) occurring in the material. The magnetic relaxation times are usually fitted to model functions derived from spin-phonon coupling theories that allow characterisation of the mechanisms of magnetic relaxation. The parameters of these relaxation models are then often compared between different molecules in order to find trends with molecular structure that may guide the field to the next breakthrough. However, such meta-analyses of the model parameters are doomed to over-interpretation unless uncertainties in the model parameters can be quantified. Here we determine a method for obtaining uncertainty estimates in magnetic relaxation times from AC experiments, and provide a program called CC-FIT2 for fitting experimental AC data as well as the resulting relaxation times, to obtain relaxation parameters with accurate uncertainties. Applying our approach to three archetypal families of high-performance dysprosium(iii) SMMs shows that accounting for uncertainties has a significant impact on the uncertainties of relaxation parameters, and that larger uncertainties appear to correlate with crystallographic disorder in the compounds studied. We suggest that this type of analysis should become routine in the community.

Post-B3LYP Functionals Do Not Improve the Description of Magnetic Coupling in Cu(II) Dinuclear Complexes.

The accuracy of post-B3LYP functionals is analyzed using an open-shell database of Cu(II) dinuclear complexes with well-defined experimental values of the magnetic coupling constants. This database provides a sound open-shell training set to be used to improve the fitting schemes in defining new functionals or when reparametrizing the existing ones. For a large set of representative hybrid exchange-correlation functionals, it is shown that the overall description of moderate-to-strong antiferromagnetic interactions is significantly more accurate than the description of ferromagnetic or weakly antiferromagnetic interactions. In the case of global hybrids, the most reliable ones have 25-40% Fock exchange with SOGGA and PBE0 being the most reliable and M06 the exception. For range-corrected hybrids, the long-range corrected CAM-B3LYP and ωB97XD provide acceptable results, and M11 is comparable but more erratic. It is concluded that the reliability of the calculated values is system- and range-dependent, and this fact introduces a serious warning on the blind use of a single functional to predict magnetic coupling constants. Hence, to extract acceptable magnetostructural correlations, a "standardization" of the method to be used is advised to choose the optimal functional.

Synthesis and Electronic Structures of Heavy Lanthanide Metallocenium Cations.

The origin of 60 K magnetic hysteresis in the dysprosocenium complex [Dy(Cpttt)2][B(C6F5)4] (Cpttt = C5H2tBu3-1,2,4, 1-Dy) remains mysterious, thus we envisaged that analysis of a series of [Ln(Cpttt)2]+ (Ln = lanthanide) cations could shed light on these properties. Herein we report the synthesis and physical characterization of a family of isolated [Ln(Cpttt)2]+ cations (1-Ln; Ln = Gd, Ho, Er, Tm, Yb, Lu), synthesized by halide abstraction of [Ln(Cpttt)2(Cl)] (2-Ln; Ln = Gd, Ho, Er, Tm, Yb, Lu). Complexes within the two families 1-Ln and 2-Ln are isostructural and display pseudo-linear and pseudo-trigonal crystal fields, respectively. This results in archetypal electronic structures, determined with CASSCF-SO calculations and confirmed with SQUID magnetometry and EPR spectroscopy, showing easy-axis or easy-plane magnetic anisotropy depending on the choice of Ln ion. Study of their magnetic relaxation dynamics reveals that 1-Ho also exhibits an anomalously low Raman exponent similar to 1-Dy, both being distinct from the larger and more regular Raman exponents for 2-Dy, 2-Er, and 2-Yb. This suggests that low Raman exponents arise from the unique spin-phonon coupling of isolated [Ln(Cpttt)2]+ cations. Crucially, this highlights a direct connection between ligand coordination modes and spin-phonon coupling, and therefore we propose that the exclusive presence of multihapto ligands in 1-Dy is the origin of its remarkable magnetic properties. Controlling the spin-phonon coupling through ligand design thus appears vital for realizing the next generation of high-temperature single-molecule magnets.

Calix[n]arene-based polyradicals: enhancing ferromagnetism by avoiding edge effects.

Through-bond interacting organic polyradicals, rendered by customizable capacities of the state-of-the-art synthetic routes, are ideal systems to investigate spin topologies. Relying on Rajca and co-workers' synthetic efforts, hereby we investigate the role of borders in the stability of the high-spin ground state in a series of realistic linear and ring-like arylmethyl polyradical derivatives. We show that, compared to their linear counterpart, the absence of borders in a ring-like arrangement of arylmethyl radicals imposes a larger number of spin-alternation rule violations, which strongly stabilizes the high-spin ground state. In addition, the structural flexibility of the investigated compounds translates into the existence of various structural energy minima for which the ferromagnetic ground state is always maintained. In view of the present results we propose these rings as possible candidates for the development of enhanced high spin single molecule toroics.

Modulated spin dynamics of [Co2] coordination helicates via differential strand composition.

Coordination supramolecular chemistry provides a versatile entry into materials with functionalities of technological relevance at the nanoscale. Here, we describe how two different bis-pyrazolylpyridine ligands (L1 and L2) assemble with Co(II) ions into dinuclear triple-stranded helicates, in turn, encapsulating different anionic guests. These constructs are described as (Cl@[Co2(L1)3])3+, (SiF6@[Co2(L1)(L2)3])2+ and (ClO4@[Co2(L2)3])3+, as established by single-crystal X-ray diffraction. Extensive magnetic and calorimetric measurements, numerical treatments and theoretical calculations reveal that the individual Co(II) centers of these supramolecular entities exhibit field-induced slow relaxation of magnetization, dominated by direct and Raman mechanisms. While the small variations in the spin dynamics are not easily correlated with the evident structural differences among the three species, the specific heat measurements suggest two vibronic pathways of magnetic relaxation: one that would be associated with the host lattice and another linked with the guest.

Vibronic effects on the quantum tunnelling of magnetisation in Kramers single-molecule magnets.

Single-molecule magnets are among the most promising platforms for achieving molecular-scale data storage and processing. Their magnetisation dynamics are determined by the interplay between electronic and vibrational degrees of freedom, which can couple coherently, leading to complex vibronic dynamics. Building on an ab initio description of the electronic and vibrational Hamiltonians, we formulate a non-perturbative vibronic model of the low-energy magnetic degrees of freedom in monometallic single-molecule magnets. Describing their low-temperature magnetism in terms of magnetic polarons, we are able to quantify the vibronic contribution to the quantum tunnelling of the magnetisation, a process that is commonly assumed to be independent of spin-phonon coupling. We find that the formation of magnetic polarons lowers the tunnelling probability in both amorphous and crystalline systems by stabilising the low-lying spin states. This work, thus, shows that spin-phonon coupling subtly influences magnetic relaxation in single-molecule magnets even at extremely low temperatures where no vibrational excitations are present.

Unparalleled selectivity and electronic structure of heterometallic [LnLn'Ln] molecules as 3-qubit quantum gates.

Heterometallic lanthanide [LnLn'] coordination complexes that are accessible thermodynamically are very scarce because the metals of this series have very similar chemical behaviour. Trinuclear systems of this category have not been reported. A coordination chemistry scaffold has been shown to produce molecules of type [LnLn'Ln] of high purity, i.e. exhibiting high metal distribution ability, based on their differences in ionic radius. Through a detailed analysis of density functional theory (DFT) based calculations, we discern the energy contributions that lead to the unparalleled chemical selectivity of this molecular system. Some of the previously reported examples are compared here with the newly prepared member of this exotic list, [Er2Pr(LA)2(LB)2(py)(H2O)2](NO3) (1) (H2LA and H2LB are two ß-diketone ligands). A magnetic analysis extracted from magnetization and calorimetry determinations identifies the necessary attributes for it to act as an addressable, conditional multiqubit spin-based quantum gate. Complementary ab initio calculations confirm the feasibility of these complexes as composite quantum gates, since they present well-isolated ground states with highly anisotropic and distinct g-tensors. The electronic structure of 1 has also been analyzed by EPR. Pulsed experiments have allowed the establishment of the quantum coherence of the transitions within the relevant spin states, as well as the feasibility of a coherent control of these states via nutation experiments.

Ultrahard magnetism from mixed-valence dilanthanide complexes with metal-metal bonding.

Magnetic effects of lanthanide bonding Lanthanide coordination compounds have attracted attention for their persistent magnetic properties near liquid nitrogen temperature, well above alternative molecular magnets. Gould et al. report that introducing metal-metal bonding can enhance coercivity. Reduction of iodide-bridged terbium or dysprosium dimers resulted in a single electron bond between the metals, which enforced alignment of the other valence electrons. The resultant coercive fields exceeded 14 tesla below 50 and 60 kelvin for the terbium and dysprosium compounds, respectively. ­JSY

Slow magnetic relaxation in Fe(II) m-terphenyl complexes.

Two-coordinate transition metal complexes are exciting candidates for single-molecule magnets (SMMs) because their highly axial coordination environments lead to sizeable magnetic anisotropy. We report a series of five structurally related two-coordinate Fe(II) m-terphenyl complexes (4-R-2,6-Xyl2C6H2)2Fe [R = tBu (1), SiMe3 (2), H (3), Cl (4), CF3 (5)] where, by changing the functionalisation of the para-substituent (R), we alter their magnetic function. All five complexes are field-induced single-molecule magnets, with relaxation rates that are well-described by a combination of direct and Raman mechanisms. By using more electron donating R groups we were able to slow the rate of magnetic relaxation. Our ab initio calculations predict a large crystal field splitting (>850 cm-1) and sizeable zero-field splitting parameters (ca. -60 cm-1, |E| < 0.2 cm-1) for 1-5. These favourable magnetic properties suggest that m-terphenyl ligands have untapped potential as chemically versatile ligands able to impose highly axial crystal fields.

Eight coordinated mononuclear dysprosium complexes of heptadentate aminophenol ligands: the influence of the phenol substituents and the ancillary donors on the magnetic relaxation.

The mononuclear complexes [Dy(3Br,5Cl-H3L1,1,4)(D)]·solvate (D = H2O, solvate = 0.25MeOH, 1W·0.25MeOH; D = Py without solvate, 1Py), and [Dy(3NO2,5Br-H3L1,1,4)(H2O)] (2W) were isolated. The crystal structures of 1W·0.25MeOH, 1Py and 2W·2CH3C6H5 show that the DyIII ion is octacoordinated, in N4O4 or N5O3 environments, with distorted geometries, between square antiprism, biaugmented trigonal prism and triangular dodecahedral. A similar environment for the metal ion is shown in the chiral crystals of the diamagnetic yttrium analogue [Y(3Br,5Cl-H3L1,1,4)(MeOH)] (3M), which were spontaneously resolved. Magnetic analyses of the three dysprosium complexes, and their diluted analogous 1W@Y, 1Py@Y and 2W@Y, reveal that none of them seem to relax through an Orbach mechanism at Hdc = 0. However, the three complexes show Orbach relaxation under Hdc = 1000 Oe, and 1Py is the in-field SIM with the highest energy barrier among these complexes, with a Ueff value of 358 K. Analysis of ac magnetic data shows that the electron-withdrawing substituents on the phenol rings of the aminophenol ligands, as well as the auxiliary oxygen donors from water ligands, reduce the energy barriers of the complexes, which is attributed to a charge reduction in the coordinating atoms of the aminophenol donor. Ab initio calculations support the experimental results.

A Cost-Effective Semi-Ab Initio Approach to Model Relaxation in Rare-Earth Single-Molecule Magnets.

We discuss a cost-effective approach to understand magnetic relaxation in the new generation of rare-earth single-molecule magnets. It combines ab initio calculations of the crystal field parameters, of the magneto-elastic coupling with local modes, and of the phonon density of states with fitting of only three microscopic parameters. Although much less demanding than a fully ab initio approach, the method gives important physical insights into the origin of the observed relaxation. By applying it to high-anisotropy compounds with very different relaxation, we demonstrate the power of the approach and pinpoint ingredients for improving the performance of single-molecule magnets.