The quantum nature of single-ion magnets, single-molecule magnets, and single-chain magnets has been manifested among other phenomena by magnetic hysteresis due to slow spin relaxation, competing with fast quantum tunneling at low temperatures. Slow spin relaxation, described by Arrhenius-type law with the effective barrier energies Ueff = 50 cm–1, was discovered 3 decades ago in paramagnetic Mn12-acetate complex of oxy-bridged mixed-valence manganese ions, below the blocking temperature TB = 3 K. In contrast to common magnetic materials, it is governed primarily by magnetic anisotropy, set by zero-splitting of spin states of a magnetic ion in a field of ligands, and spin-lattice coupling. The emerging studies on the border of coordination chemistry, physics of spin systems with reduced dimensionality, and nanotechnologies, were performed in search of routes for enhancement of Ueff and TB characteristics, in line with increase of operation temperature and quantum correlation time, mandatory for quantum applications. The best results with TB ∼ 80 K and Ueff ∼ 1261 cm–1, were obtained for DyIII single-ion magnet, so far. Numerous excellent research and review articles address particular activities behind this achievement. It follows, that present challenges are dictated by the rational development of novel, smart magnetic molecules, featured by butterfly cores, cyano-bridges, 2D metal-organic frameworks, and metal-free graphene nanoclusters, as well as stable free radicals, magnetized by spare electrons. These species are briefly considered here with respect to the unique experience of international collaborative activity, established by Prof. Juan Bartolomé.

Contents

1. Introduction 432 
2. Superparamagnetism in spin-nanoclusters 432 
2.1. Evolution of anisotropic magnetic state with field and temperature 432 
2.2. Activated magnetization reversal and quantum tunneling 432 
3. SMM complexes with metal center 435 
3.1. Transition-metal centers 435 
3.2. Rare-earth metal complexes 438 
3.3. Molecular spin-cluster arrangements for multifunctional quantum applications 440 
4. Spare electron magnetism of metal-free molecular prototypes 440 
4.1. Graphene with dangling bonds 440 
Conclusions 442 
References 442 
1. Introduction 432 
2. Superparamagnetism in spin-nanoclusters 432 
2.1. Evolution of anisotropic magnetic state with field and temperature 432 
2.2. Activated magnetization reversal and quantum tunneling 432 
3. SMM complexes with metal center 435 
3.1. Transition-metal centers 435 
3.2. Rare-earth metal complexes 438 
3.3. Molecular spin-cluster arrangements for multifunctional quantum applications 440 
4. Spare electron magnetism of metal-free molecular prototypes 440 
4.1. Graphene with dangling bonds 440 
Conclusions 442 
References 442 

Slow spin reversal, and eventually long-lived quantum-correlations, is a key parameter in quantum applications met by single-molecule magnets and their derivatives (see, the review articles, e.g., Refs. 1–3 and references therein). Naturally, its first-ever observation in a “guinea pig” complex of Mn12-acetate4,5 with the magnetization relaxation time of the order of two months at liquid helium temperatures sparked numerous studies of coordinate metal complexes, called since then a single-molecule magnet (SMM), if magnetic hysteresis manifested. Progress in this field is reflected in Fig. 1 with respect to milestones of the life-long activity of Juan Bartolomé and his team.1,6–40 Slow spin relaxation is, thus, a fingerprint of single-molecule magnetism. In fact, magnetization reversal of a spin cluster in a varying magnetic field occurs by common slow activated spin flipping in the energy-barrier potential, by energy exchange with the lattice13 (Fig. 2), and by rapid quantum tunneling through the barriers, thermally activated or via the ground state, unwanted in quantum applications. Their regulation and studies are summarized here for coordinate complexes of transition- and rare-earth metals. Break-through in their miniaturization and functionalization is illustrated by the works of the Zaragoza team with co-authors.13 Here, we consider two family members of molecular nanomagnets: (i) magnetized due to spin moments of transition (T) and rare-earth (RE) metal ions with unpaired electron shells or (ii) due to delocalized spins in stable organic free radicals and in graphene nanostructures.39 In fact, the longest quantum coherence time record (∼1 μs at ambient temperature and 68 μs at low temperature (qubit figure of merit QM = 3.400 compared to specified QM = 10.000) is reported for transition-metal 1CuD–1NiD SMM complex avoiding hydrogen in ligand and, therefore, its magnetic nuclei-spin contribution.41 

FIG. 1.

Raise of relaxation time τ = τ 0 exp ( U eff k B T ) control by Juan Bartolomés team in SMMs and beyond.1,6–40

FIG. 1.

Raise of relaxation time τ = τ 0 exp ( U eff k B T ) control by Juan Bartolomés team in SMMs and beyond.1,6–40

Close modal
FIG. 2.

(a) Schematic phase diagram of the Ising ferromagnet diluted by antiferromagnetic spin-spin couplings; TN(u) is the Nishimori line; F, P, and SG denote the ferromagnetic, paramagnetic phase and spin glass, respectively.48 Reprinted from O. S. Bakai, Fiz. Nizk. Temp. 49, 658 (2023) [Low Temp. Phys. 49, 601 (2023)]; (b) Magnetic phase diagram of archetypal Mn12-acetate. The solid dots show the critical magnetic field Bc at which the longitudinal magnetization is observed to vanish at each temperature. The open dots give the irreversibility field, below which spins do not attain equilibrium within the experimental time.55 Reprinted with permission from F. Luis, J. Campo, J. Gómez, G. J. McIntyre, J. Luzón, and D. Ruiz-Molina, Phys. Rev. Lett. 95, 227202 (2005).

FIG. 2.

(a) Schematic phase diagram of the Ising ferromagnet diluted by antiferromagnetic spin-spin couplings; TN(u) is the Nishimori line; F, P, and SG denote the ferromagnetic, paramagnetic phase and spin glass, respectively.48 Reprinted from O. S. Bakai, Fiz. Nizk. Temp. 49, 658 (2023) [Low Temp. Phys. 49, 601 (2023)]; (b) Magnetic phase diagram of archetypal Mn12-acetate. The solid dots show the critical magnetic field Bc at which the longitudinal magnetization is observed to vanish at each temperature. The open dots give the irreversibility field, below which spins do not attain equilibrium within the experimental time.55 Reprinted with permission from F. Luis, J. Campo, J. Gómez, G. J. McIntyre, J. Luzón, and D. Ruiz-Molina, Phys. Rev. Lett. 95, 227202 (2005).

Close modal

Magnetically, SMMs represent the class of nanocluster superparamagnets with magnetic hysteresis, observed below the blocking temperature, similar to common magnetic domain walls, but different in nature. The comprehensive insight into relaxation properties of Co-based, nanoparticles, in part10–17 and references therein), helped to understand the processes in magnetic molecule nanoclusters. In fact, an arrangement of unpaired electrons on 5d-orbitals of cobalt ion enable both high-spin and low-spin states, in favor of high magnetic anisotropy, while half-integer spins rather support magnetic superexchange and fast relaxation due to quantum tunneling. The first example of slow magnetic relaxation in Co complexes was reported for a one-dimensional coordination polymer, designed with hepta-coordinate CoII nodes.42–44 

Long-known and studied molecular magnetism45–47 investigations have profited strongly today from progress in materials production, which allows to regulate magnetic properties by controlling the specified space dimensionality, anisotropy, and chemical composition. One of the most requested and studied features is a slow random spins relaxation, emerging after cooling in a bias magnetic field below the Curie and superparamagnet blocking temperatures.13 A similarity of the polyamorphism of orientationally disordered fullerite C60 and that, allegedly found in the disordered spin state,48,49 enables deeper insight into phase diagram and phase transitions in a magnetic cluster, described by Ising interaction model.50–52 It is argued that the cross-section point of the Nishimori line and Tc(u)53 is the multicritical point at which the paramagnetic, ferromagnetic, and random spin phases merge. In strongly anisotropic systems, like SMM nanoclusters, a competition of exchange interactions and critical fluctuation around transition temperature produce unusual regions in their HT diagrams54,55 and references within). It is difficult to expect anything like that in SCMs, though, because such transitions are forbidden in 1-dimension species, and they remain paramagnetic over the entire HT region. For an obvious comparison, the calculations-based and measured magnetic phase diagram is presented in Fig. 2 for the spin system with the quenched two-state disorder. The line TN(u) is the Nishimori line. At T < TN(u) ferromagnetism is absent and the paramagnetic state is transforming into a heterophase spin-glass state consisting of quenched orientationally disordered clusters. The intersection of the Nishimori curve TN(u) and Tc(0) determines the probabilities of 2 types of clusters formation. A lot of research efforts were directed distinguishing transitions into superparamagnetic and spin-glass states in uniaxial magnets by model approach50–53 and measuring HT diagrams and magnetic relaxation time across the transition to random spin state49 with references). A number of measuring techniques and approaches are in hand now Refs. 13, 56–60, and references therein). They allow, in part, to estimate the blocking temperature TB as one of the key parameters to characterize particles with uniaxial anisotropy, which is defined as the temperature at which the average time for a magnetic nanoparticle moment to escape from an energy well is equal to the characteristic measurement time for the system. Below this temperature, a particle will generally be thermally stable or “blocked” and above this temperature, it is classified as superparamagnetic (SPM) since thermal fluctuations are large enough to cause the time-averaged magnetization to be zero, when there is no external field. The transition from blocked to SPM behavior is usually taken to occur at a single critical energy barrier. This leads to the assumption of a single blocking temperature TB at which the transition instantneously occurs. However, given the stochastic nature of the thermally induced switching, the blocked/SPM transition may have a finite width. In highly anisotropic materials it is governed by the hierarchy of energy barrier heights for flipping spins, which are differently oriented with respect to field direction, in the regime of thermally activated motion.

The origin of magnetization reversal with switching off a magnetic field in the blocked state of a magnetic molecule nanocluster differs completely from that in any other type of magnet. It develops in a coordinate complex of magnetic ion with a specific configuration of a ground spin state S, which favors zero-field-splitting (ZFS) of the ms components due to lowering the ligand-field-induced symmetry. Magnetization reversal occurs in this case by spin flipping in a landscape of energy barriers between + ms and – ms components by fast quantum tunneling and slow, thermally activated relaxation. Rhombic parameter E mixes different ms and is, therefore, responsible for the quantum tunneling process, unwanted in quantum applications. The combined data set definitely establishes the transverse anisotropy terms responsible for the low-temperature quantum dynamics,61 which attracts attention to transverse magnetization component measurements. Appropriate techniques are described in Refs. 62–64.

Relaxation, on the other hand, is a rather slow process, described by the exponential law of Arrhenius type, independently of its nature, and can be determined at specified measuring time–temperature scales. It is most fascinating to observe the relaxation of spins themselves in a common macroscopic experiment, like magnetic susceptibility measurements. Description of spin relaxation stems from Néel expression for the average time of a particle to flip from one well to another, given by
τ N = τ 0 exp ( K V / k B T ) ,
(1)
where KV = Ueff is the height of the energy barrier, a product of the magnetic anisotropy energy density K and volume V, kB is the Boltzmann constant, T is the temperature and their product the thermal energy; and τ0 is the attempt period, characteristic for a material, typically taken as around 10−9–10–10 s. Obviously, the application of a magnetic field can change the height of the energy barrier. Experimentally, TB is defined from Eq. (1) by setting the time τN equal to the characteristic measuring time τm,
T B = K V / k B ln ( τ m / τ 0 ) .
(2)

In single-molecule magnets, the height of the energy barrier is determined by ZFS parameter D and spin number S as U eff = D S 2 for integer spin, and U eff = D ( S 2 1 / 4 ) for half-integer spin. Its value can be increased by choosing the molecule with appropriate bonds, their lengths and angles, which stabilize orbitals in the coordinate complex. For example, zero-splitting-parameter D was increased in such a way from 12 to –17 cm–1 for a four-coordinate complex Fe4+ and amidopirine ligand by reducing the N–Fe–N angle,65 correspondent to energy barrier of Ueff = 27 cm–1 for spin flipping. At the same time, the pioneering observations of magnetization reversal in spin chain magnet (SCM) (in Ref. 66 and references therein) suggest that in this case spin flipping follows the scenario, proposed by Glauber67 for a chain of Ising spins with a single type of exchange interactions. Easy manipulation of the interaction parameters in such a system renders an unambiguous advantage in their studies.

Experimentally, the correlation between the relaxation time τ and temperature T can be obtained from an Arrhenius plot ln τ versus 1/T and beyond.68–71 It has been probed by a broad variety of techniques at various τm scales, including quasi-static magnetic measurements, electron paramagnetic and nuclear resonances, neutron and Raman scattering, and complex magnetic susceptibility χAC. Interestingly, unusual magnetic relaxation in a single-molecule magnet with toroidal magnetic moments64 was observed, while on the time-scale of resonance measurements, the medium and slow processes are dominated by the direct Raman and in-direct Orbach relaxation pathways. Quantum tunneling in magnetism (QTM), otherwise, dictates the faster process. Among the applied techniques, the nanoparticles magnetization measurements, made in the field-cooled (FC) and zero-field-cooled (ZFC) regimes as a function of temperature, are used to characterize the particle size, size distribution and anisotropy energy. In order to elucidate the true origin of slow spin relaxation to equilibrium state change with field and temperature, magnetic phase diagrams are built using magnetization reversal measurements in varied direct magnetic field and temperature70,72,73 (Fig. 5) and compared with theoretical models.50,51 It should be noted here, that the collaborative research of Zaragoza and the Kharkov team revealed some tricky transitions in spin arrays at slow-relaxation (to be published). Quantum theory of one-dimensional spin systems, along with the quantum chemistry and first principle calculations, provide a consistent description of spin dynamics in a variety of molecule magnet clusters, which makes them easier to be tunable and even more attractive for quantum applications. The highest single ion magnetic anisotropy seems to be observed in molecules with central f metal from a Lanthanide group. However, lack of exchange in this case, determined by spin-orbit coupling, may result in higher relaxation times, unwelcome in quantum applications.

In this section, the background of slow relaxation in molecular nanoclusters of d and f metal single-molecule magnets (SMMs) and single-ion magnets (SIMs) is considered with respect to the magnetic properties of central ion within environment of bridging ligands. The latter play a decisive role, sometimes equally important to that of the magnetic metal ion(s).41,73 In SMM chemistry, bridging ligands that propagate strong ferromagnetic exchange interactions between the metal ions resulting in large spin ground states, which are well isolated from excited states, are preferable; however, antiferromagnetic coupling can also lead to SMM behavior. In SIM chemistry, ligands that create a strong axial crystal field are highly desirable for metal ions with oblate electron density, e.g., TbIII and DyIII, whereas equatorial crystal fields lead to SMM behavior in complexes based on metal ions with prolate electron density, e.g., ErIII.74 Three large classes of multifunctional molecular magnetic nanocluster materials are distinguished at present. The first class comprises the single-molecule magnets (SMMs) and the single-ion magnets (SIMs), the second the single-chain magnets (SCMs), and the third involves the multifunctional molecular magnetic materials. The focus of this section is on transition- and rare-earth metal SMMs and SIMs75,76 and references therein). This class of magnetic molecule nanoclusters contains a transition metal or rare-earth ion(s) as the source of their magnetic properties.

In the presence of axial magnetic anisotropy D,77 the MS levels of a transition-metal complex with total spin S will split under zero magnetic field according to the Hamiltonian H ^ = D S ^ z 2. If the value of D is negative, the two ± MS levels of maximal projection along the z axis form a bi-stable ground state because they are degenerate. If we reverse the magnetic moment by converting −MS to + MS, this requires the traversal of a spin-inversion barrier. This barrier is U = S 2 | D | for integer S or U = ( S 2 1 / 4 ) | D | for non-integer S, and the system passes through the MS = 0 or the MS = 1/2 levels, respectively, at the height of the energy barrier. The existence of such a barrier can lead to the slow relaxation of the magnetic moment at very low temperatures upon removal of the external dc field. The presence of this barrier is often proven by the appearance of magnetic hysteresis of molecular origin as first observed for the iconic [Mn12O12(O2CMe)16(H2O)4] (Fig. 3) SMM,6,78,79 Clusters containing polynuclear molecules that exhibit such behavior have been named single-molecule magnets. The magnetic behavior of each of these clusters can be described as a giant anisotropic spin as a result of the exchange coupling between the spins of neighboring metal ions. Because of the magnetic bi-stability, these polynuclear molecules were proposed for use in magnetic memory devices since they can remain magnetized in one of the two spin states, thus giving rise to a “bit” of memory. The aim during the first decade of SMM research (1993–2003) was to prepare SMMs with memory effects at higher temperatures.75 Although synthetic inorganic chemists made many efforts to achieve this goal, the progress was little and the energy barriers that stabilize the magnetic bits against thermal fluctuations remained small. Another tremendously important consequence of the discovery of SMMs was the observation of quantum effects in mesoscopic magnets. At that time, physicists were looking for small magnetic particles, all identical to each other, to investigate if quantum effects could be observed in ensembles of such identical particles; however, the preparation of these collections proved difficult. Chemists solved the problem using a molecular approach to prepare identical cluster molecules in crystalline SMMs. A few years after the first characterization of SMM, scientists revealed that its crystals exhibit quantum tunneling of magnetization (QTM);16,17,80,81 this phenomenon is considered one of the milestones in the study of spin during the 20th century.75 Synthetic efforts were followed by advanced theoretical studies, and the latter provided strong evidence that the magnitude of D decreases as S increases; this implied that the construction of efficient SMMs with a large U cannot be achieved by only maximizing S and that control of D is equally important.78,79 With this in mind, let us consider pros and cons of transition metal ions in the core of spin nanocluster for quantum applications, exemplified by Mn and Co ions here. Transition metal ManganeseIII can in principle form complexes in three different spin states, depending on the ligands.82,83 One of the complexes that had been the subject of many investigations is the octahedral complex formed from the regular acetylacetonate. The electronic structure of Mn-(acac)III has been investigated in detail by a combination of photoemission spectroscopy, near-edge X-ray absorption fine structure (NEXAFS) spectroscopy, and theoretical calculations. The study could confirm that the electronic configuration can be described as t 2 g 3 e 1 g with S = 2 (S corresponds to the total spin angular momentum quantum number). Compounds in a diamagnetic low-spin state ( t 2 g 4 e 0 g with S = 0) are rare, but in spin–crossover compounds the paramagnetic intermediate-spin states ( t 2 g 4 e 0 g with S = 1) have been found to be accessible. Mn(mesacac)III displays the paramagnetism of the manganese ions over the whole measuring temperature range. 4.85 μB was an effective moment determined from a Curie–Weiss fit together with a negligible Curie–Weiss temperature. The reference value is 4.90 to 5.00 μB for high-spin Mn3+. The SMM, manganese-based complexes are happily represented by an archetypal mixed-valence cluster Mn12-acetate [Mn12O12(O2CMe)16(H2O)4] (Fig. 3). It is characterized by the highest anisotropy barrier (∼ 65 K) discovered in SMM so far; its core is composed of 4 Mn4+ ions (electron spin S = 3/2, green balls in Fig. 3), and 8 Mn3+ ions (S = 2, red balls in Fig. 3) in two inequivalent crystallographic sites. The intracluster superexchange interactions result in a total spin S = 10 for the cluster. Below T ∼ 3 K, the electron spins are effectively frozen along the anisotropy axis. Field dependence of magnetization and high-frequency electron paramagnetic resonance (HFEPR) data indicated that it has an S = 10 ground state. It arises due to antiferromagnetic interactions between the S = 3/2 spins of MnIV ions and the S = 2 spins of MnIII ions. An axial zero-field splitting was observed, which resulted in the splitting of the S = 10 state into 21 levels, with a spin projection quantum number ms, where – S ≤ Ms ≤ S. The energy of each level is read as E ( m s ) = M s 2 D, where the axial zero-field splitting parameter D = –0.7 K, and potential-energy barrier between the “spin-up” (Ms= –10) and “spin-down” (Ms= 10) orientations of the magnetic moment in an individual Mn12 molecule exists [Fig. 3(b)]. Consequently, an axial magnetic anisotropy is present and spin-flip proceeds with energy barrier, estimated as D ≈ –70 K. For a thermally activated process, the time for the reorientation of the magnetization depends exponentially on the height of the barrier. Then relaxation of magnetization at T = 2 K to 60% of magnetization of saturation takes about 2 months. It was shown to occur in an isolated molecule without implementation of long-range ordering observed in nano-sized magnetic domains of bulk magnets79 and references therein). Effect of ligand type in such complexes is illustrated by Fig. 4.

FIG. 3.

(a) Schematic structure of the mixed valence Mn12 core [MnIV4MnIII8(O)12]16 (in [Mn12O12(O2CCH3)16(H2O)4]⋅4H2O⋅2CH3CO2H, Complex 1), showing the relative positions of spin down MnIII+ (red balls), spin up MnIV+ (green balls), and μ3–O2– — oxy-bridges (blue balls). The four central MnIV atoms are weakly ferromagnetically coupled, and the remaining MnIII⋅⋅⋅MnIV and MnIII⋅⋅⋅MnIII exchange interactions are all antiferromagnetic, with the former much stronger than the latter. As a result, the stronger MnIII⋅⋅⋅MnIV interactions overcome the weaker MnIII⋅⋅⋅MnIII ones within each triangular MnIII2MnIV subunit of the core, aligning the spins of the outer MnIII atoms all parallel, and thus antiparallel to the central MnIV atoms; this gives an S = 16 − 6 = 10 ground state. (b) Energy level scheme for the ground S = 10 multiplet. Three routes are possible between the wells, either the thermal activation path above the barrier, a thermal activated tunneling via an excited level or a quantum tunneling by the ground state.4,13 E. Bartolomé, A. Arauzo, J. Luzón, J. Bartolomé, and F. Bartolomé, Handbook Magn. Mater. 26, 1 (2017) with permission of Elsevier Books.

FIG. 3.

(a) Schematic structure of the mixed valence Mn12 core [MnIV4MnIII8(O)12]16 (in [Mn12O12(O2CCH3)16(H2O)4]⋅4H2O⋅2CH3CO2H, Complex 1), showing the relative positions of spin down MnIII+ (red balls), spin up MnIV+ (green balls), and μ3–O2– — oxy-bridges (blue balls). The four central MnIV atoms are weakly ferromagnetically coupled, and the remaining MnIII⋅⋅⋅MnIV and MnIII⋅⋅⋅MnIII exchange interactions are all antiferromagnetic, with the former much stronger than the latter. As a result, the stronger MnIII⋅⋅⋅MnIV interactions overcome the weaker MnIII⋅⋅⋅MnIII ones within each triangular MnIII2MnIV subunit of the core, aligning the spins of the outer MnIII atoms all parallel, and thus antiparallel to the central MnIV atoms; this gives an S = 16 − 6 = 10 ground state. (b) Energy level scheme for the ground S = 10 multiplet. Three routes are possible between the wells, either the thermal activation path above the barrier, a thermal activated tunneling via an excited level or a quantum tunneling by the ground state.4,13 E. Bartolomé, A. Arauzo, J. Luzón, J. Bartolomé, and F. Bartolomé, Handbook Magn. Mater. 26, 1 (2017) with permission of Elsevier Books.

Close modal
FIG. 4.

Semi-logarithmic plot of the zero-field relaxation time in Mn12-acetate (●) and Mn12 2-Cl benzoate (◯). The straight lines are Arrhenius fits.6 Reprinted from Marco Evangelisti and Juan Bartolomé, J. Magn. Magn. Mater. 221, 99 (2000) with permission of Elsevier.

FIG. 4.

Semi-logarithmic plot of the zero-field relaxation time in Mn12-acetate (●) and Mn12 2-Cl benzoate (◯). The straight lines are Arrhenius fits.6 Reprinted from Marco Evangelisti and Juan Bartolomé, J. Magn. Magn. Mater. 221, 99 (2000) with permission of Elsevier.

Close modal

Turning back to fingerprints of SMM, we should remember, that molecules are classified as single-molecule magnet when magnetic hysteresis and slow spin relaxation in varying magnetic field are manifested. It is due to splitting of bi-stable spin ground state in a field of ligands, which set symmetry and propagate magnetic ion coupling. Orbital splitting of Jahn–Teller Mn3+ ions by crystal field of octahedral symmetry, produced by oxygen ligand environment, similar to that in a magnetic molecule Mn12-ac, with Mn+ centers that are coordinated in octahedral geometry oxygen atoms of μ-oxo and μ-carboxylate bridge ligands, was earlier observed in its crystalline counterpart LaMnO3, actively studied and exploited for several decades, by our collaborative team, in part, because of colossal negative magneto-resistance (CMR) effect found in its mixed-valence derivatives (Fig. 5). It provides a practical platform for tailoring spin-orbit and spin-phonon interactions, responsible for the large characteristic attempt times, τ0 in Eq. (1), as well as negative magnetic anisotropy, in the counterpart SMM. As such, slow-relaxation of spin-glass type, observed in a cation-deficient single crystal of LaMnO3+δ, is featured by close resemblance with that in SMM at temperatures below TB. It is interesting to note also, that a ferromagnetic ordering high-temperature state of LaMnO3 is surprisingly transformed into a paramagnetic one with cooling below 720 °C84 due to the Jahn–Teller orbital ordering, followed by a succession of ordering transitions at further cooling, resulting in antiferromagnetic ordering at temperatures below ∼100 K, and most likely, inhomogeneous (bi-stable) ground state.85 These remarks stem from a comparative study of acetate ligand substituted by benzoate one6 and isolated Mn12-ac complexes,86,87 which confirmed, that SMM, magnetic reversal, is not affected much, resuming to be their intrinsic property Fig. 4. Such robustness of central Mn12 complexes is good news for their functionalization, e.g., by a trendy route towards the liquid crystal phase.

Cobalt-based structures were specially addressed in the studies of nanoparticle dynamics,11 CMR and SMM species due to a very promising anisotropy due to its distinguished high-spin ground state and, favorable for ZFS induced negative magnetic anisotropy (Fig. 6). Their ample study promoted a purposeful approach to the creation of coordinate Co complexes of SMM with the obtained record values Ueff = 450 cm–1. We should emphasize an important effort11 on the study of Co nanoparticles when capped by different metals. It was shown that the relaxation dynamics were sensitive to the different spin-orbit coupling constants of the 4d and 5d metals covering the nanoparticles.

FIG. 5.

Crystal field splitting of the Mn ion d levels and electronic occupation in an octahedral oxygen environment. The degeneracy of the eg and t2g levels is lifted by an in-plane contraction and out-of-plane elongation in the bulk oxygen octahedron [(a) CMR manganite], negative axial ZFS in the magnetic molecule [83 (b), reprinted from Sayani Majumdar and Sebastiaan Dijken, J. Phys. D: Appl. Phys. 47, 034010 (2013)] (b) prototypical Mn12-ac, for a ground spin state S = 10, transversal Hamiltonian is zero. The splitting of the ground doublet MS = ±10 by Zeeman interactions is highlighted in the figure [83(c)]. Reproduced from A. Zabala-Lekuona, José Manuel Seco, and E. Colacio, Coord. Chem. Rev. 441, 213984 (2021) with permission of Elsevier. Inset: Structure of Mn12-acetate viewed along the c axis. Large solid spheres represent Mn3+ ions (outer ring) and Mn4+ ions (inner core). All Mn atoms have a distorted octahedral coordination geometry.86 

FIG. 5.

Crystal field splitting of the Mn ion d levels and electronic occupation in an octahedral oxygen environment. The degeneracy of the eg and t2g levels is lifted by an in-plane contraction and out-of-plane elongation in the bulk oxygen octahedron [(a) CMR manganite], negative axial ZFS in the magnetic molecule [83 (b), reprinted from Sayani Majumdar and Sebastiaan Dijken, J. Phys. D: Appl. Phys. 47, 034010 (2013)] (b) prototypical Mn12-ac, for a ground spin state S = 10, transversal Hamiltonian is zero. The splitting of the ground doublet MS = ±10 by Zeeman interactions is highlighted in the figure [83(c)]. Reproduced from A. Zabala-Lekuona, José Manuel Seco, and E. Colacio, Coord. Chem. Rev. 441, 213984 (2021) with permission of Elsevier. Inset: Structure of Mn12-acetate viewed along the c axis. Large solid spheres represent Mn3+ ions (outer ring) and Mn4+ ions (inner core). All Mn atoms have a distorted octahedral coordination geometry.86 

Close modal
FIG. 6.

(a), (b) Schematic crystal field splitting and occupations of the 3d electrons in tetrahedral and octahedral coordination environments in anion Сomplex 1 of magnetic molecule. (c) Cobalt is shown in blue, oxygen in red, sulfur in yellow, nitrogen in violet, and carbon in grey. Hydrogen atoms have been omitted for clarity. (d) Molecular orbital diagram showing the calculated d orbital splitting for Сomplex 1. Horizontal lines depict orbital energies while arrows pointing up or down stand for single electron spins.88 Reprinted from Yvonne Rechkemmer, Frauke Breitgoff, Margarethe Meeret et al., Nat. Commun. 7, 10467 (2016).

FIG. 6.

(a), (b) Schematic crystal field splitting and occupations of the 3d electrons in tetrahedral and octahedral coordination environments in anion Сomplex 1 of magnetic molecule. (c) Cobalt is shown in blue, oxygen in red, sulfur in yellow, nitrogen in violet, and carbon in grey. Hydrogen atoms have been omitted for clarity. (d) Molecular orbital diagram showing the calculated d orbital splitting for Сomplex 1. Horizontal lines depict orbital energies while arrows pointing up or down stand for single electron spins.88 Reprinted from Yvonne Rechkemmer, Frauke Breitgoff, Margarethe Meeret et al., Nat. Commun. 7, 10467 (2016).

Close modal

The seminal must-read review13 on magnetic relaxation of lanthanide-based molecular magnets gives a comprehensive base for understanding and controlling the relaxation processes in molecule-spin nanoclusters. Unpaired electrons in the deeper, inner f orbitals of rare-earth metals are better shielded from interaction with those from the other shells and, therefore, sustain pronounced magnetic anisotropy and spin-orbit coupling. Consequently, lanthanides and actinides look like excellent candidates for expected long quantum correlation times to incorporate in quantum devices. In fact, best results with TB ∼ 80 K and Ueff = 1261 cm–1 were achieved on chloralilate (Cl2An2−) bridged DyIII,89 with slow magnetic relaxation, induced by weak magnetic field (Fig. 7), so far. However, the approach based on increasing the negative magnetic anisotropy, as well as the number of spins, is not so straightforward in general, because high anisotropy may and does facilitate growing contribution from quantum tunneling and decreasing TB. For this reason, many promising manufacturing efforts led to the disappointing result, which have demonstrated the crucial importance of preliminary computational analysis, with input-measured parameters. At this end, the Zaragoza team has performed a targeted complex investigation of rare-earths (Tb, Nd, Dy, …) coordinate complexes for quantum applications (plot in Table I).

FIG. 7.

(a) Splitting of f orbitals of Tm3+ in Oh and D3 configurations and their occupations in the ground state. (b) Rare-earth complexes for slow magnetic relaxation [89(b)]. Reprinted form R. Ishikawa, S. Michiwaki, T. Noda, K. Katoh, M. Yamashita, and S. Kawata, Magnetochemistry 5, 30 (2019).

FIG. 7.

(a) Splitting of f orbitals of Tm3+ in Oh and D3 configurations and their occupations in the ground state. (b) Rare-earth complexes for slow magnetic relaxation [89(b)]. Reprinted form R. Ishikawa, S. Michiwaki, T. Noda, K. Katoh, M. Yamashita, and S. Kawata, Magnetochemistry 5, 30 (2019).

Close modal
TABLE I.

Illustrious milestones.

 
 

Each cluster containing polynuclear molecules that exhibit magnetic hysteresis and named single-molecule magnets can be described as a giant anisotropic spin as a result of the exchange coupling between the spins of neighboring metal ions. Because of the magnetic bi-stability, these polynuclear molecules were proposed for use in magnetic memory devices since they can remain magnetized in one of the two spin states, thus giving rise to a “bit” of memory. In multinuclear SMMs, the magnetic core of each complex comprises multiple ions with unpaired electrons, which are coupled to each other through intramolecular exchange interactions. This coupling leads to a large magnetic moment for each molecule and separates the energy spectrum into respective spin multiplets. For a traditional polynuclear SMM system, the exchange interaction is sufficient to isolate the ground state spin multiplet from higher lying multiplets. The ground state of multiplets is then modeled as a state with a perfectly fixed magnetic moment vector. The organic ligands that surround the magnetic core serve to isolate each molecule from surrounding neighbors. Thus, intermolecular exchange interactions are rather weak and there is no long range order observed, which is proved by the absence of lambda peaks in heat capacity measurements, but, however, could be successfully employed.66 Progress in understanding the relaxation properties of SMM, SIM, and SCM, in line with permanently upgraded quantum technologies, provides opportunities for their effective implementation in quantum devices. For that, they should be stabilized in favorable conditions for application configuration. The milestones on the route from the discovery to the application of magnetic molecule nanoclusters are illustrated in Table I with respect to the Zaragoza team activities, and publications in this Special Issue. Sufficient ingredients of success include the size, surface, stability, operating temperatures, correlation times, and, for sure, prognostic optimization at the stage of development. As such SCM and SIM look to the winners, though SMMs do not give up. Smartly assembled in either open or closed networks, liquid crystals, all these entities become eligible for future application. Achievements in this field are enormous and immediately generalized not only in excellent review articles but in regular research papers as well. Therefore, the main practical configurations presented in Table I are simply a guide for the eye, because of unprecedentedly rapid progress in their perfection.90 

So, the functionalization of magnetic molecules requires considerable attention to their non-metallic part in the form of ligands configuration and composition, as propagators of magnetic interactions and mediators of spin-lattice coupling. It is carefully examined whether the magnetic molecules are located, either at the selected surface,87 or within metallofulleren,91 polyoxometalate complexes,92 and SCM93–95 metal-organic frameworks (MOFs), even driving them into the liquid crystalline phase.96 Moreover, their bridging properties with respect to spin-lattice coupling, energy barriers distribution, and relaxation time74,89,93 and other excellent works), have been investigated.

At the same time, the metal-free molecule spins raise increasing attention in quantum applications. Their magnetism stems from unpaired electron spins on the molecular orbitals. Zeeman Hamiltonian of the interaction of electrons with the external magnetic field should then be complemented by the orbital and Van Vleck contributions.96 The largest ever observed spin coherence time T2 ∼ 0.2Ms at 170 K for a molecular system was thus registered97 in a metal-free configuration of nitrogen, trapped within a carbon C60 cage in a liquid CS2. It obeys Arrhenius temperature dependence, indicating the spin relaxation is driven primarily by an Orbach process. Similarly, pure diamond,98 especially with the nitrogen-vacancy (NV) color centers,99 and Si100 spark big expectations for quantum memory units. In fact, there are several entities of multifunctional non-metallic magnetic molecules, selected for quantum applications, which are sometimes borrowed from biotechnologies. The organic stable free radical DPPH is among these selections101 Special Issue). Valence, unpaired-electron population may be enhanced by electron doping, with respect to Pauling’s concept of dangling bonds,102 under electron irradiation. It provided a base for our collaborative activity,103,104 aimed at nanotubes production in layered transition-metal dichalcogenide under electron irradiation, similarly to the caged structures in graphene. The developed theoretical approach was successfully applied to the description of the evolution of electron states near the Fermi energy level with the introduction of vacancy complexes, zigzag boundaries, and interstitial atoms105–111 (Fig. 8). The Jacobi-matrix technique, used in this work allows to avoid translational symmetry consideration and is therefore particularly appropriate for crystals with defects, which as was shown in Refs. 112 and 113, may facilitate magnetism.

FIG. 8.

Graphene nanoribbon with effects: Local electron densities of states evolution with a distance from zigzag boundary (3 upper panels) and vacancy (2 bottom panels), calculated by procedure, described in Refs. 104–109.

FIG. 8.

Graphene nanoribbon with effects: Local electron densities of states evolution with a distance from zigzag boundary (3 upper panels) and vacancy (2 bottom panels), calculated by procedure, described in Refs. 104–109.

Close modal

Since Nobel-prize-winning observations of graphene, it was shown, that it has many potentially useful properties, but is usually not magnetic when pristine. However, theoretical predictions suggest that the edges of graphene sheets should become magnetic when they have a zigzag arrangement of carbon atoms. Observing this effect has been challenging because of the difficulties of detecting the predicted minute magnetic signal and because it is hard to fabricate defect-free edges that have the required shape. In Refs. 39 and 114 a method was reported for making nanometre-wide graphene ribbons in solution, and thereby for producing nanoribbons with well-defined zigzag edges “decorated” with organic radical molecules that bear electron spins desired as a quantum property of electrons that is associated with magnetism. The authors’ results provide solid evidence of magnetism at graphene edges and show that edge spins have potentially useful quantum dynamics. As a single-layer network of carbon atoms, graphene has outstanding electrical and mechanical properties. Graphene ribbons with nanometre-scale widths (nanoribbons) should exhibit half-metallicity and quantum confinement. Magnetic edges in graphene nanoribbons have been studied extensively from a theoretical standpoint because their coherent manipulation would be a milestone for spintronic and quantum computing devices. However, experimental investigations have been hampered because nanoribbon edges cannot be produced with atomic precision and the graphene terminations that have been proposed are chemically unstable. In Refs. 39 and 114, both of these problems are addressed, by using molecular graphene nanoribbons functionalized with stable spin-bearing radical groups. They observe the predicted delocalized magnetic edge states and test theoretical models of the spin dynamics and spin-environment interactions. Comparison with a non-graphitized reference material enables us to clearly identify the characteristic behavior of the radical-functionalized graphene nanoribbons. They also quantify the parameters of spin-orbit coupling, define the interaction patterns, and determine the spin decoherence channels. Even without any optimization, the spin coherence time is in the range of microseconds at room temperature, and allows for quantum inversion operations between edge and radical spins. Our approach provides a way of testing the theory of magnetism in graphene nanoribbons experimentally. The observed coherence times open up encouraging prospects for the use of magnetic nanoribbons in quantum spintronic devices.

It is clearly understood now that the central problem posed by a quantum molecular spins application is an increase in their reversal times in a cycled magnetic field at moderate temperatures. It is obviously achieved by suppression of fast, quantum tunneling processes, on one hand, and enhancement of slow, thermally activated, relaxation, originated from the zero-field splitting of ground state, on the other. As such, the emergent negative magnetic anisotropies, responsible for barrier energy, attempt frequencies, and temperature range of spin relaxation, are regulated by the choice of molecule structure with respect to central magnetic ion, bonds composition, length and angles. Despite unconventional progress in barrier energy and blocking temperature enhancement from 50 cm–1 and 3 K, up to, respectively, 1261 cm–1 and 80 K in the frame of this approach, it appeared, that an increase of magnetic anisotropy does not decide the problem on its own, because growth of fast component, and moreover, decrease of blocking temperature may occur at the same time. Deep insight into the intra-molecular interactions, supported by diverse experimental and quantum calculation techniques, is therefore required. To this end, we significantly addressed the findings of the Zaragoza team, because of their unique experience, ultra-low temperature techniques, software, and advanced synthesis facilities, gained during decades by the investigations led-by Juan Bartolomé of magnetic nanoparticles. It was targeted, in part, at modulating their relaxation times by composition, surface anisotropy, and treatment. They were among the first researchers of slow dynamics in single-molecule magnets. All that allowed them to build an utmost effective plot for search of optimal slow-relaxation scenario. It is accomplished by means of the complex relaxation-time measurements at different measuring-time scales on a rational selection of, primarily, metal-organic coordinated complexes. It enabled the distinct separation of multiple contributions, such as spin-lattice coupling, ordering processes, and hierarchy of energy barriers. The outcome provides the optimum ratio of high- and low-spin states, molecules composition, and space configurations for longevity of quantum correlated states, in MOFs, in particular, with emerging directions for their further improvement. Thus, a critical assessment of recent developments, with a surface dominance in mind, gives a complementary approach for tuning the molecular-spin behaviors, by means of (i) purposeful incorporation of defects, with electron irradiation or capping, which is exemplified by graphene with vacancies and zigzag boundaries here; (ii) modulation of inter-molecule interaction for further slowing down the spin relaxation processes by a spin-glass scenario. Many aspects of advanced magnetic-molecule nanoclusters require further insight. We cannot pass by the enormous potentialities of J. Bartolomé team to make further breakthrough in the field. Hopefully, the unique experience of ILTPE team, e.g., in investigation of low-dimensional spin systems and quantum bites (qubits) will be in demand.

The ILTPE side greatly appreciates the unprecedented effort of Fernando Bartolomé and the authors of this Special Issue to share their experience and beautiful results despite the risks of collaboration with a country at war. Greatest and sincere gratitude to Juan Bartolomé for his great support and inspiration.

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Supplementary Material