New Materials and Effects in Molecular Nanomagnets

Molecular magnets are a relatively new class of purely organic or metallo-organic materials, showing magnetism even without an external magnetic field. This interdisciplinary field between chemistry and physics has been gaining increased interest since the 1990s. While bulk molecular magnets are usually hard to build because of their molecular structures, low-dimensional molecular magnets are often easier to construct, down to dot-like (zero-dimensional) structures, which are investigated by different scanning probe technologies. On these scales, new effects such as superparamagnetic behavior or coherent switching during magnetization reversal can be recognized. Here, we give an overview of the recent advances in molecular nanomagnets, starting with single-molecule magnets (0D), typically based on Mn12, Fe8, or Mn4, going further to single-chain magnets (1D) and finally higher-dimensional molecular nanomagnets. This review does not aim to give a comprehensive overview of all research fields dealing with molecular nanomagnets, but instead aims at pointing out diverse possible materials and effects in order to stimulate new research in this broad field of nanomagnetism.


Introduction
When thinking about nanomagnets from a physicist's point of view, usually zerodimensional (0D) or one-dimensional (1D) magnets will come to mind. Such nanomagnets may be formed, e.g., from the ferromagnets iron, nickel, cobalt, or permalloy, or from the ferrimagnets of magnetite or nickel ferrite, to name just a few [1][2][3][4][5]. Ferromagnetic materials contain elementary magnets for which a parallel orientation is energetically favored, while ferrimagnets can be imagined as containing two antiparallely oriented ferromagnetic sublattices with different magnitudes of magnetization, in this way also resulting in a net magnetization. Diverse shapes can be thought of, from square or round nanodots [6,7] to magnetic nanowires, e.g., those used in the so-called Racetrack memory [8][9][10], to more complicated shapes, including 3D particles [11][12][13]. Combining different magnetic materials, e.g., a ferromagnetic and an antiferromagnetic one, can result in additional effects as a result of the surface interactions, such as the exchange bias [14][15][16][17].
Here, we give a brief overview of the principle of why even purely organic molecules can exhibit ferromagnetism, followed by sections reporting on recent advances in SMMs, SCMs, and other structures prepared from purely organic molecules or containing typical inorganic ions, such as Mn 12 and Fe 8 , concluding with possible applications, e.g., in data storage. The review is organized as follows: Starting with a brief overview of the magnetic In addition, a double exchange can occur in mixed valence systems (e.g., metal ions in different oxidation states, such as Mn III and Mn IV ), describing the fast hopping of unpaired electron between the ions, i.e., delocalization of the valences. This results in a ferromagnetic coupling, as the transferred spin must be parallel to the other Mn 4+ spins because of the Pauli principle [40,41].

Single-Molecule Magnets
Probably the most interesting applications of single-molecule magnets are for information storage and spintronic devices. While common magnetic nanoparticles reach the superparamagnetic limit for dimensions below 10-100 nm, single-molecule magnets with transition-metal ions can be used for data storage, while being much smaller [42]. For the first SMM reported, antiferromagnetic interactions between the spins of Mn IV and Mn III ions in the molecule [Mn12O12(CH3COO)16(H2O)4] were found to result in a large spin ground state with the spin quantum number S = 10, and in very long magnetization relaxation times, making such a bistable molecule highly interesting for data storage [43,44]. It should be mentioned that the coercive fields in this system, measured by hysteresis loops, depend not only on the temperature, but also on the field sweep rate [18]. Gatteschi et al. attributed the magnetic uniaxial anisotropy of this system to the magnetic anisotropy of the eight Mn +3 in the molecule, which show zero-field splitting due to the Jahn-Teller elongation of some bonds and the spin-orbit interactions [45]. In this system, hysteresis loops are also modified in comparison with "normal" loops, showing steps at some external magnetic fields due to the resonant tunneling between levels with identical energies in the two energetically favored orientations [46,47].
Another often investigated complex is Fe8 in the form of the cluster [Fe8O2(OH)12(tacn)6] 8+ , with (tacn) indicating a macrocyclic ligand. However, here, the relaxation is temperature-independent below a temperature of 0.36 K or 0.4 K, respectively, as opposed to Mn12, because tunneling is only possible between the lowest energy levels, i.e., the levels with magnetic quantum numbers ms = ±10 [48,49]. Similarly, [Mn4O3Cl(O2CCH3)3(dbm)3] with the mono-anion of dibenzoylmethane dbm − (usually In addition, a double exchange can occur in mixed valence systems (e.g., metal ions in different oxidation states, such as Mn III and Mn IV ), describing the fast hopping of unpaired electron between the ions, i.e., delocalization of the valences. This results in a ferromagnetic coupling, as the transferred spin must be parallel to the other Mn 4+ spins because of the Pauli principle [40,41].

Single-Molecule Magnets
Probably the most interesting applications of single-molecule magnets are for information storage and spintronic devices. While common magnetic nanoparticles reach the superparamagnetic limit for dimensions below 10-100 nm, single-molecule magnets with transition-metal ions can be used for data storage, while being much smaller [42]. For the first SMM reported, antiferromagnetic interactions between the spins of Mn IV and Mn III ions in the molecule [Mn 12 O 12 (CH 3 COO) 16 (H 2 O) 4 ] were found to result in a large spin ground state with the spin quantum number S = 10, and in very long magnetization relaxation times, making such a bistable molecule highly interesting for data storage [43,44]. It should be mentioned that the coercive fields in this system, measured by hysteresis loops, depend not only on the temperature, but also on the field sweep rate [18]. Gatteschi et al. attributed the magnetic uniaxial anisotropy of this system to the magnetic anisotropy of the eight Mn +3 in the molecule, which show zero-field splitting due to the Jahn-Teller elongation of some bonds and the spin-orbit interactions [45]. In this system, hysteresis loops are also modified in comparison with "normal" loops, showing steps at some external magnetic fields due to the resonant tunneling between levels with identical energies in the two energetically favored orientations [46,47].
Another often investigated complex is Fe 8 in the form of the cluster [Fe 8 O 2( OH) 12 (tacn) 6 ] 8+ , with (tacn) indicating a macrocyclic ligand. However, here, the relaxation is temperatureindependent below a temperature of 0.36 K or 0.4 K, respectively, as opposed to Mn 12 , because tunneling is only possible between the lowest energy levels, i.e., the levels with magnetic quantum numbers m s = ±10 [48,49]. Similarly, [Mn 4 O 3 Cl(O 2 CCH 3 ) 3 (dbm) 3 ] with the mono-anion of dibenzoylmethane dbm − (usually named Mn 4 ) with an S = 9/2 ground 4 of 21 state shows a temperature-independent relaxation below a temperature of 0.6 K for the same reason [50].
It should be mentioned that, as mentioned in Sections 2 and 3, the magnetic behavior of these clusters based on Mn 12 , Mn 4 , or Fe 8 is defined by the magnetic exchange, which is necessary for the high ground state spin. However, for lanthanide-based SMMs, magnetic exchange can even be disadvantageous and is not necessary to create an SMM. This should be taken into account for the lanthanide-based SMMs discussed next.
One disadvantage of the aforementioned single-molecule magnets is their very small blocking temperature, typically below the temperature of liquid helium. Ideally, magnetic properties should be reached above the temperature of liquid nitrogen, i.e., 77 K. This is one of the challenges diverse research groups have been working on during the last decades [51].
Recently, Li et al. reported on a relatively large blocking temperature of 9 K found in highly-axial lanthanide SMMs of the form [Dy(Cy 3 PO) 2 I 3 (CH 3 CN)]. The iodide ions have a large volume and low surface charge density, and serve as weak donors in this six-coordinate neutral molecule. The reduced ligand field strength results in a strongly axial crystal field in this SMM, leading to strong crystal-field splitting and high axial orientation [52].
A recipe for higher blocking temperatures was recently proposed by Chiesa 4 ] as a model complex to investigate the relaxation mechanisms that they found based on Orbach, Raman, and quantum tunneling of the magnetization processes, i.e., some of the special relaxation pathways mentioned in Section 2. Based on these results, they could predict these processes' temperature and field dependencies, showing that suppressing the Raman mechanism is important to increase the blocking temperature. In general, they concluded that a crystal with weakly-interacting rigid molecules was necessary to reach the requirements of the soft acoustic branches and high-energy optical modes [53], which was revealed from the phonon density of states (DOS) computed ab initio.
An experimental approach was chosen by Jin et al. They prepared two different disprosiacarboranes, [(C 2 B 9 H 11 ) 2 Dy(THF) 2 ][Na(THF) 5 ] (denoted as 1Dy) and [(THF) 3 (µ-H) 3 Li] 2 [{η 5 -C 6 H 4 (CH 2 ) 2 C 2 B 9 H 9 }Dy{η 2 :η 5 -C 6 H 4 (CH 2 ) 2 C 2 B 9 H 9 } 2 Li] (named 3Dy), finding opened hysteresis loops up to 6.8 K in the complex 3Dy ( Figure 2), i.e., a blocking temperature of 6.8 K and thus above the temperature of liquid helium [54]. named Mn4) with an S = 9/2 ground state shows a temperature-independent relaxation below a temperature of 0.6 K for the same reason [50]. It should be mentioned that, as mentioned in Sections 2 and 3, the magnetic behavior of these clusters based on Mn12, Mn4, or Fe8 is defined by the magnetic exchange, which is necessary for the high ground state spin. However, for lanthanide-based SMMs, magnetic exchange can even be disadvantageous and is not necessary to create an SMM. This should be taken into account for the lanthanide-based SMMs discussed next.
One disadvantage of the aforementioned single-molecule magnets is their very small blocking temperature, typically below the temperature of liquid helium. Ideally, magnetic properties should be reached above the temperature of liquid nitrogen, i.e., 77 K. This is one of the challenges diverse research groups have been working on during the last decades [51].
Recently, Li et al. reported on a relatively large blocking temperature of 9 K found in highly-axial lanthanide SMMs of the form [Dy(Cy3PO)2I3(CH3CN)]. The iodide ions have a large volume and low surface charge density, and serve as weak donors in this sixcoordinate neutral molecule. The reduced ligand field strength results in a strongly axial crystal field in this SMM, leading to strong crystal-field splitting and high axial orientation [52].
A recipe for higher blocking temperatures was recently proposed by Chiesa et al. They used [Dy(C5H t 2Bu3-1,2,4)2][B(C6F5)4] as a model complex to investigate the relaxation mechanisms that they found based on Orbach, Raman, and quantum tunneling of the magnetization processes, i.e., some of the special relaxation pathways mentioned in Section 2. Based on these results, they could predict these processes' temperature and field dependencies, showing that suppressing the Raman mechanism is important to increase the blocking temperature. In general, they concluded that a crystal with weaklyinteracting rigid molecules was necessary to reach the requirements of the soft acoustic branches and high-energy optical modes [53], which was revealed from the phonon density of states (DOS) computed ab initio.
An  (Figure 2), i.e., a blocking temperature of 6.8 K and thus above the temperature of liquid helium [54]. Reprinted from [54], copyright 2020, with permission from Wiley.
Another approach was chosen by Spree et al. They investigated the fullerene-based SMMs DySc2N@C80 and Dy2ScN@C80, as well as DyLu2N@C80 and Dy2LuN@C80, and found blocking temperatures of 9.5 K (DyLu2N@C80) and 6.9 K (DySc2N@C80), combined with longer relaxation times, while the other complexes showed blocking temperatures of around 8 K [55]. Nie et al. found similar blocking temperatures of 9 K in the fullerene- SMM DyErScN@I h -C 80 [56]. A much higher blocking temperature of 24 K was observed by Velkos et al. in the azafullerene Tb 2 @C 79 N [57].
Much higher values were observed by Gould et al. who compared the divalent linear metallocenes Ln(Cp iPr5 ) 2 (with Ln = Tb, Dy) with the trivalent ones [Ln(Cp iPr5 ) 2 ][B(C 6 F 5 ) 4 ], and found longer relaxation times in Tb II than in TB III and longer relaxation times in Dy III than in Dy II , and a similar behavior for Tb II and Dy III , as could be expected from the identical number of unpaired electrons in the latter. Most importantly, in Tb(Cp iPr5 ) 2 they found a high blocking temperature of 52 K [59].
Other high blocking temperatures were found, e.g., in [(Cp iPr5 )Dy(Cp*)] + with Cp iPr5 = penta-iso-propylcyclopentadienyl and Cp* = pentamethylcyclopentadienyl, an SMM for which Guo  While pure metal SMMs have been investigated in detail [32,63], metal complexes such as the aforementioned ones are often highly interesting. In particular, 3d-4f metal complexes can add special properties to SMMs, such as a high-spin ground state and a large magnetic anisotropy [64]. The first SMM containing such a 3d-4f metal complex was [Cu II-LTb III (hfac) 2 ] 2 [65]; other 3d-4f complexes are, e.g., Mn 3 Ln or Fe 3 Ln with the lanthanides Gd or Dy [66]. More generally, typical paramagnetic 3d ions used in such complexes are transition metal ions such as Co II , Mn II , Mn III , Fe II , Fe III , Ni II , and Cu II , while 4f ions can be, e.g., lanthanide ions such as Dy III , Tb III , Gd III , or Er III [67]. Even diamagnetic 3d metal ions can sometimes be used to create SMMs [68].
Recently Wang et al. prepared 3d-4f clusters with Co III 2Dy III 4 and Co III 2Gd III 4, respectively, using chemical reactions and structural X-ray characterization. Their investigations revealed centrosymmetric hexanuclear metallic cores. Interestingly, they found ferromagnetic coupling for the Dy III ions and antiferromagnetic coupling for the Gd III ions, in spite of the Generally, many researchers have concentrated on the aforementioned 3d-4f metal complexes. Das et al. recently gave an overview of the structural variety of 3d-4f heterometallic polyoxometalates, including dimers, trimers, and tetramers, and the different polyanions, such as pure 3d-4f polyanions, 3d substituted anions with 4f linkers, 4f substituted anions with 3d linkers, and polyoxometalates with 3d and 4 d linkers. They concentrated not only on the possibilities of using such 3d-4f polyoxometalates as singlemolecule or single-ion magnets, but also for applications in catalysis [71].
Wang et al. prepared 3d-4f clusters with Co III 2 Dy III 4 and Co III 2 Gd III 4 , respectively, using chemical reactions and structural X-ray characterization. Their investigations revealed centrosymmetric hexanuclear metallic cores. Interestingly, they found ferromagnetic coupling for the Dy III ions and antiferromagnetic coupling for the Gd III ions, in spite of the othewise identical composition, with the lanthanide ions being coplanar, thus underlining the importance of the choice of the lanthanide [72]. On the other hand, Dy 4 was also reported in similar structures, working as a single-molecule magnet, but with different energy barriers and other physical properties, indicating the importance of the whole molecular structure, even in the case of structures that seemed very similar at first glance [73][74][75]. In these works, the structural single-crystal X-ray diffraction investigations were additionally supported by dc and ac magnetic susceptibility studies [73].
In tetranuclear 3d-4f systems containing DyFe(CN) 6 , TbFe(CN) 6 , HoFe(CN) 6 , DyCo(CN) 6 , TbCo(CN) 6 , or HoCo(CN) 6 , respectively, Wang et al. also found differences between the isomorphic compounds. Especially, the complex containing DyFe(CN) 6 showed a weak antiferromagnetic interaction between Dy and Fe, and no slow magnetic relaxation above 2 K, which would be typical for an SMM, while this effect was found in the complex containing DyCo(CN) 6 . Wang et al. attributed this finding to a larger intermolecular Dy-Dy distance for the second complex, based on the diamagnetic properties of the Co III ion, as opposed to the paramagnetic Fe III ion [76]. The authors concluded these results from magnetic susceptibility experiments, including real and imaginary parts, as a function of the temperature within the range of 2-20 K. Many other authors investigated these and other lanthanide-hexacyanidometallate SMMs [77][78][79][80].
Lun et al. also investigated Co III 2 combined with Dy, using structural characterization, in this case in the complex {[Co III 2 Dy 3 Na(CH 3 CH 2 COO) 6 (OH) 6 (NO 3 ) 4 (H 2 O) 2 ]·H 2 O} n . They found a trigonal bipyramidal geometry in the Co 2 Dy 3 core and a single-molecule magnetic behavior for the whole complex with an energy barrier of 60.3 K under a zero dc field [81].
Co II was investigated by Zhang et al. They prepared 3d-4f heterometallic complexes with Co II 2 -Ln III 2 units, applying Nd, Sm, Eu, Gd, Tb, Dy, and Er as the lanthanides. The structures of the complexes containing Er and all of the other lanthanides are depicted in Figure 4. Interestingly, they found, on the one hand, that the molecular structures of all complexes besides the one containing Er were identical, which was attributed to Er III being smaller than Dy III . On the other hand, a weak ferromagnetic interaction between the metal ions was found for Gd and Er, while the complex containing Dy again showed field-induced slow magnetic relaxation, typical for SMMs [82]. Appl Xi et al. also worked with Co II , in the form of Co II -Ln III with the lanthanides Y, Gd, Tb, Dy, and Ho, and found SMM behavior for the combination with Dy [83]. In these studies, the magnetic susceptibility method was complemented with fluorescence spectroscopy, revealing the characteristic energy levels of the Tb ions.
Again, another metal, Cr, was used by Yu et al. in different coordination polymers with the basic units of Gd4Cr4, Tb4Cr4, or Er4Cr4. They found a butterfly-like structure of the basic unit and each basic unit was connected with four Ln4Cr4 clusters in a 3D structure resembling a 1D honeycomb [84]. Apart from standard single-crystal X-ray diffractometry and magnetic susceptibility studies, they employed isothermal thermogravimetry/differential thermal analysis (TG/DTA) experiments to detect the magnetic entropy changes.
With Zn as the metal and Dy, Tb, or Er as the lanthanide, Fan et al. prepared heterometallic complexes with a windmill-like structure and a (Ln3Zn3) core. They found field-induced frequency-dependent signals, measured by field-induced frequencydependent signals, as a sign of SMM behavior for Dy and Tb, while the Er-based complex showed a slow magnetic relaxation [85].
Cu was combined with Dy, Tb, and Gd to prepare the tetranuclear Cu II 2Ln III 2 complexes. These complexes were isostructural and isomorphic, and showed ferromagnetic interactions between Cu II and Ln III in all cases, as well as SMM behavior, with Cu II Gd III exhibiting a magneto-caloric effect [86].
Quite a different approach was recently chosen by Darii et al., who combined Mn6 clusters with Dy III and a triazine-like ligand to create a large bean-shaped cluster [Mn26Dy6O16(OH)12(O2CCHMe2)42 with Mn II , Mn III , and Dy III ions, in this way enabling diverse magnetic interactions and leading to a high-nuclearity Mn-Dy cluster that still exhibits an SMM behavior [87].
Multi-decker systems of lanthanides with phthalocyanine molecules have been described by different groups. Gao et al. prepared sandwich-like complexes based on phthalocyanine molecules and a closed-macrocyclic Schiff base of the form [(Pc)2Ln3(L)(OAc)(OCH3)2] (with Ln 3+ = Dy 3+ or Er 3+ , H2Pc = phthalocyanine and H2L = closed-macrocyclic Schiff base molecules), and found a single-molecule magnetic behavior in the complex containing dysprosium [88]. In these sandwich structures, the strong coupling between the lanthanide ions resulted in an important role for the rare earth atom in the SMM properties [89], making these multi-decker structures highly interesting for diverse applications [90][91][92].
Going one step further, Patrascu et al. prepared coordination compounds with three different spin carriers (2p, in addition to 3d and 4f), resulting in three different exchange Hydrogen atoms are omitted for clarity. Reprinted from [82], copyright 2020, with permission from Elsevier.
Xi et al. also worked with Co II , in the form of Co II -Ln III with the lanthanides Y, Gd, Tb, Dy, and Ho, and found SMM behavior for the combination with Dy [83]. In these studies, the magnetic susceptibility method was complemented with fluorescence spectroscopy, revealing the characteristic energy levels of the Tb ions.
Again, another metal, Cr, was used by Yu et al. in different coordination polymers with the basic units of Gd 4 Cr 4 , Tb 4 Cr 4 , or Er 4 Cr 4 . They found a butterfly-like structure of the basic unit and each basic unit was connected with four Ln 4 Cr 4 clusters in a 3D structure resembling a 1D honeycomb [84]. Apart from standard single-crystal X-ray diffractometry and magnetic susceptibility studies, they employed isothermal thermogravimetry/differential thermal analysis (TG/DTA) experiments to detect the magnetic entropy changes.
With Zn as the metal and Dy, Tb, or Er as the lanthanide, Fan et al. prepared heterometallic complexes with a windmill-like structure and a (Ln 3 Zn 3 ) core. They found fieldinduced frequency-dependent signals, measured by field-induced frequency-dependent signals, as a sign of SMM behavior for Dy and Tb, while the Er-based complex showed a slow magnetic relaxation [85].
Cu was combined with Dy, Tb, and Gd to prepare the tetranuclear Cu II 2 Ln III 2 complexes. These complexes were isostructural and isomorphic, and showed ferromagnetic interactions between Cu II and Ln III in all cases, as well as SMM behavior, with Cu II Gd III exhibiting a magneto-caloric effect [86].
Quite a different approach was recently chosen by Darii 42 with Mn II , Mn III , and Dy III ions, in this way enabling diverse magnetic interactions and leading to a high-nuclearity Mn-Dy cluster that still exhibits an SMM behavior [87].
Multi-decker systems of lanthanides with phthalocyanine molecules have been described by different groups. Gao et al. prepared sandwich-like complexes based on phthalocyanine molecules and a closed-macrocyclic Schiff base of the form [(Pc) 2 Ln 3 (L)(OAc)(OCH 3 ) 2 ] (with Ln 3+ = Dy 3+ or Er 3+ , H 2 Pc = phthalocyanine and H 2 L = closed-macrocyclic Schiff base molecules), and found a single-molecule magnetic behavior in the complex containing dysprosium [88]. In these sandwich structures, the strong coupling between the lanthanide ions resulted in an important role for the rare earth atom in the SMM properties [89], making these multi-decker structures highly interesting for diverse applications [90][91][92].
Going one step further, Patrascu et al. prepared coordination compounds with three different spin carriers (2p, in addition to 3d and 4f), resulting in three different exchange interactions between each pair of spin carriers. By adding a 2p radical (Rad) to gain a three-dimensional Co II Dy III Rad derivative, they reached slow relaxation below 30 K and attributed the improved magnetic properties mainly to the Co II -Rad interaction [93], while Zn II Dy III Rad showed a significantly lower energy barrier and the linear Co II Dy III Rad compound showed no SMM behavior at all [94]. A broad overview of the possible metal centers for single-molecule and single-ion magnets, from 3d and 5d to 4f and 5f, is given in a recent review by Feng and Tong [95].
Lunghi et al. performed theoretical investigations that examined the under-barrier relaxation at high temperatures, which prohibited the use of SMMs at room temperature [96]. While anharmonic phonons were found to be responsible for the previously mentioned effect, spin-phonon coupling led to spin relaxation in the prototypical mononuclear SMM [(tpa Ph )Fe] − [97,98]. Ab initio calculations of single-molecule magnets have been performed by Vonci et al. [99] and Gransbury et al. [100].
For spin-based devices based on SMMs, controlling the spin relaxation is of the utmost importance. Sorensen et al. reported on the effect of a non-linear to pseudo-linear change in the crystal field symmetry in a dysprosium complex, leaving the residual chemistry unaltered, and found a strong reduction in the tunnel splitting at very low temperatures in the milliKelvin range [101]. Switching the magnetic anisotropy reversibly was enabled in lanthanide complexes as a function of external magnetic field and temperature [102,103].
Redox-active tetrathiafulvalene (TTF)-based ligands, on the other hand, allowed for designing coordination lanthanide complexes in different oxidation states, resulting in different magnetic properties for such SMMs [104]. An overview of such TTF-based ligands is given in [105].

Single-Chain Magnets
Single-chain magnets (SCMs) have been investigated since the early 2000s [108][109][110][111][112]. Similar to SMMs, they show a magnetic hysteresis loop and slow relaxation of the magnetization at low temperatures [113]. It should be mentioned, however, that the slow relaxation dynamics in SCMs are different from the quantum tunneling found in SMMs; here, there are spins in the chain flip, a process that may propagate and even result in collective reversal of short chains [113]. In this regard, SCMs are much closer to purely inorganic magnetic nanowires or nanostrips, as mentioned in the introduction. A systematic overview of the possible strategies to prepare such SCMs was given in 2010 by Sun et al. (Figure 5) [22]. While the ferromagnetic spin arrangement is not often found because of the complex dipole-dipole interactions in the chain, it can still be found in different spin structures, ferromagnetically coupling anisotropic spin carriers. The ferrimagnetic spin state is often more stable, and the first reported SCM also showed ferrimagnetic coupling. Spin-canted chains show a weak ferromagnetic order, and SCMs based on this principle are not often observed [22]. In this simple approach, interchain interactions, chirality, photo-switchable states or other highly interesting possibilities, have not yet been taken into account [114].  Nowadays, research on SCMs is, on the one hand, still focused on fully understanding the theoretical basics, and, on the other hand, practical, application-related interests such as high blocking temperatures and very slow relaxation are also being investigated. Nowadays, research on SCMs is, on the one hand, still focused on fully understanding the theoretical basics, and, on the other hand, practical, application-related interests such as high blocking temperatures and very slow relaxation are also being investigated.
Two magnetic phase transitions were found by Nadeem et al. in a Ni 4 O 4 -cubanebased network of 1D linear chains of Ni II ions, crosslinked via Ni 4 O 4 cubanes. While magnetic ordering occurred below 23.9 K, a structural phase transition was recognized at 2.8 K through the use of susceptibility measurements [115].
Yang et al. prepared azido-bridged homospin coordination polymers based on Fe II and Co II , more specifically [Fe 2 (Bzp) 2 (N 3 ) 4 ] n and [Co 4 (Bzp) 4 (N 3 ) 8 ·(MeOH) 2 ] n , respectively, with bzp = 2-benzoylpyridine, forming neutral chains with azido-bridges. They found ferromagnetic intrachain interactions in both SCMs and broad hysteresis loops [117]. Cyanobridges were used to prepare a [Mo(CN) 7 ] 4− based SCM in which the magnetic anisotropy stemmed from the anisotropic magnetic exchange of the cyanide ligands of the Mo III units and the Mn II spins, resulting in relatively high blocking temperature and large coercive fields of up to 15 kOe [118]. Jiang et al. also used cyanide bridges in an Fe 2 Co 4,2-ribbon single-chain magnet [119]. Several other studies on cyanide-bridged SCMs have been performed by Ohkoshi and Sieklucka with their groups [120][121][122], as well as by other research groups [123][124][125].
Other lanthanides may also offer interesting SCM properties. Yang et al. prepared [LnCu(hfac) 5 NIT-Ph-p-OCH 2 trz·0.5C 6 H 14 ] n with Ln = Er, Ho, Yb, resulting in ladder-like chain structures, as depicted in Figure 6 for the case of Er. Here, inserting Ho into the complex resulted in single-chain magnet behavior, as indicated by the frequency-dependent out-of-phase AC magnetic susceptibility [126].
In the so-called "butterfly molecule" [Fe3Y(µ3-O)2(CCl3COO)8(H2O)(THF)3], the Fe 3+ ions form an Fe3 cluster with a strong intracluster exchange and a S = 5/2 total spin without long-range magnetic ordering down to 20 mK. At low temperatures, a quantum tunneling process was revealed, as is usual in SMMs, and this single-chain magnet was described as combining single-molecule magnetic anisotropy with a spin-spin correlation along the chains [127].
In the so-called "butterfly molecule" [Fe 3 Y(µ3-O) 2 (CCl 3 COO) 8 (H 2 O)(THF) 3 ], the Fe 3+ ions form an Fe 3 cluster with a strong intracluster exchange and a S = 5/2 total spin without long-range magnetic ordering down to 20 mK. At low temperatures, a quantum tunneling process was revealed, as is usual in SMMs, and this single-chain magnet was described as combining single-molecule magnetic anisotropy with a spin-spin correlation along the chains [127].
Another shape of the chain was found by Houard et al. in nitronyl-nitroxide organic radical chains. In [Tb(hfac) 3 NIT-O-Hexyl] n latter structures, with hfac − = hexafluotoacetylacetonate, chiral curled chains were formed, building supramolecular tubes with a diameter of 4.5 nm and large coercive fields, as visible in Figure 7 [129].
ions form an Fe3 cluster with a strong intracluster exchange and a S = 5/2 total spin without long-range magnetic ordering down to 20 mK. At low temperatures, a quantum tunneling process was revealed, as is usual in SMMs, and this single-chain magnet was described as combining single-molecule magnetic anisotropy with a spin-spin correlation along the chains [127].
Besides the aforementioned examples, many other variations of SCMs can be found in the recent literature, based on diverse physical principles, combining a broad variety of materials and shapes. Here, as well as in SMMs, many more highly interesting new findings can be expected in the next years.

Theoretical Aspects and Methods
Besides the aforementioned theoretical aspects in the evaluation of new materials and understanding of the measured phenomena, here, a brief overview is given on the recent theoretical aspects and methods used to model molecular magnets.
One of the most interesting problems in molecular magnets is decoherence. Takahashi et al. investigated environmental decoherence and showed that the theory for insulating electronic spin systems could predict environmental decoherence in molecular quantum magnets well, based on phonons, nuclear spins, and intermolecular dipolar interactions [134]. The theory of environmental decoherence was experimentally first verified in the Fe 8 complex by spin-echo measurements and was compared with calculations from the Hamiltonian model used to predict the field splitting [134]. Decoherence is highly important as it modifies the equations describing the time-dependent evolution of a magnetic moment in a varying magnetic field, i.e., the Landau-Zener expression, which allows for calculating the spin reversal probability in a constantly swept magnetic field [135,136]. The equation is limited to time scales much smaller than the dephasing time, as pointed out for a single-molecule spin transistor containing the single-ion magnetic molecule TbPc 2 (with Pc = phthalocyanine), by comparing the experimental results with calculations based on a Hamiltonion description of the interplay between the time-dependent magnetic field and a constant tunneling term [137]. Mirzoyan and Hadt used density functional theory (DFT) and time-dependent DFT to calculate decoherence due to spin-phonon coupling in spin 1 2 transition molecular qubits based on Cu(II) and V(IV), using the ORCA program to generalize the ligand field theory (LFT) and to enable a comparison of the experimental results with those of LFT, DFT, and time-dependent DFT [138]. A broad overview of such DFT calculations used for SMMs was given by Postnikov et al., who reported on recent interesting systems based on Ni 4 , Co 4 , Fe 4 , Mn 10 clusters, and V 15 , and discussed the anisotropies in different single-molecule magnets [139]. Taran et al. recently showed that a phenomenological Lindblad operator was suitable to model spin-flip probability for small probing currents for different temperatures, cooling times, and sweeping rates in a diluted crystal of TbPc 2 lanthanide single ion molecular magnets [140]. Spin dephasing was studied by propagation of unitary quantum spin dynamics in a reduced Hilbert space to model spin dephasing in vanadyl-based molecular qubits VO(acac) 2 and VO(dmit) 2 (with acac = acetylacetonate and dmit = 1,3-dithiole-2-thione-4,5-dithiolate), applying ORCA to compute some not experimentally available parameters [141]. Hu et al. used the Kubo-Anderson model to connect hyperfine coupling, calculated using DFT, to the coherence time for different vanadyl complexes [142]. It should be mentioned that besides environmental decoherence, there is also an intrinsic decoherence, which is actually a breakdown of quantum mechanics and is much harder to understand [143].
Besides the aforementioned problem of decoherence or coherence time, several more specific questions have risen with respect to molecular magnets and were solved by different theoretical approaches. Liu et al., e.g., used an analytical theory to correlate the tunnel splitting of a single molecular spin with the height of the quantum steps found in the hysteresis loops of the molecular magnets Fe 8 , Mn 12 , and Mn 4 [144]. Bhandary et al. combined DFT using the VASP code with second order perturbation theory to calculate a strain-induced spin change in iron porphyrin embedded in a graphene lattice [145]. Baker 6 ] complexes, with teaH 3 = triethanolamine and R = meta-CN, para-CN, meta-CH 3 , para-NO 2 , and para-CH 3 [148]. Besides the aforementioned modules, post-Hartree-Fock ab-initio calculations are nowadays often performed with the modules CASSCF (complete active space self-consistent field), RASSI, and SINGLE-ANISO [99,[149][150][151][152].
Finally, it should be mentioned that several open-source software solutions are available to calculate the magnetic properties of SMMs and SCMs. Mannini et al. [153], e.g., report on using NWChem, an open-source solution for large-scale molecular simulations that is well suited for magnetic systems [154]. Prsa et al. [155] used Monte Carlo simulations with the ALPS framework [156] to investigate lanthanide-based SMMs. The same software was used by Baniodeh et al. [157]. Another one is CP2K [158], a software used to calculate molecular dynamics, which was used, e.g., by Burgess et al. [159]. Tandon [162].
As this short overview shows, most problems are theoretically investigated by DFT calculations, while there are several additional theoretical approaches used in combination with DFT or solely to understand the special experimental findings.

Potential Challenges in the Application of Molecular Nanomagnets
The application of molecular magnets means bringing them into an environment with which they can interact via their ligands. In case of using molecular nanomagnets, e.g., as magnetic labels for biosensing or as hyperthermia agents, self-assembling of clusters or chains cannot be excluded under the application of an external magnetic field, which would change magnetic states and magnetization dynamics compared with single molecular nanomagnets. Aggregates may be formed by magnetizable nanoparticles depending on their biological environment, mimicking the surrounding natural tissue properties. Such sometimes desired, sometimes unrequested processes are known from magnetic gels [163][164][165] and have to be taken into account during the development of modern compact medical devices based on molecular nanomagnets [166][167][168].
As Domingo et al. pointed out in their review [166], SMMs forming self-assembled monolayers on surfaces show different magnetic properties, particularly different anisotropies and anisotropy axes, compared with the single SMMs that are often theoretically investigated. They describe different studies aimed at orienting the easy magnetization axis of SMM clusters on surfaces, either by self-organization or by partial functionalization through ligand substitution with binding surface groups. This means, on the other hand, that the ligand shell defines the orientation of SMMs on surfaces or during self-assembly in biomedical applications [169], so that this parameter has to be taken into account to reach the desired magnetic properties and to avoid possible interference effects between neighboring SMMs in such a cluster, as compared with single SMMs.

Recent Trends in Molecular Nanomagnets
Besides the aforementioned topics, which are often related to basic research, some recent trends should be mentioned that are currently in the focus of research in this area, often aiming at multifunctional molecular magnets.
One of these topics is related to luminescent SMMs based on lanthanides for possible application in quantum computing, bio-labeling, or LEDs. Jia et al. recently reviewed the design strategies for such luminescent lanthanide-based SMMs, pointing out the necessary combination of a strongly UV absorbing ligand environment that efficiently populates excited states in the metal ion, with a filled-up coordination sphere of the Ln(III) ions to reduce radiation-less deactivation processes [170]. Yi [172]. Wang et al. suggested using a near-infrared emissive SMM as a highly sensitive luminescent thermometer [173]. A review of luminescent Schiff-base lanthanide SMMs can be found in [174].
Other authors have concentrated on the electric properties of SMMs. Long et al. investigated the magneto-electric coupling in a paramagnetic ferroelectric lanthanide complex and found a strong interaction between both electric and magnetic properties, making this material possibly useful for data storage or spintronics applications [175]. The electro-conductive properties of a Dy(III) double-decker SMM were investigated by Sato et al. and Katoh et al. who found conductivity at temperatures below the spin blocking temperature and magneto-resistance in the form of a magnetic hysteresis curve, possibly enabling the utilization of such systems as spin valves [176].
Even ferroelectricity was found in SMMs [177,178] and SCMs [179,180]. As this short overview shows, multifunctional molecular nanomagnets enable combining magnetic with different optical or electrical properties, making them highly useful for diverse recent applications.

Reviews on Special Sub-Topics
While this review gives an overview of the most recent results in the research on molecular nanomagnets, including SCMs and SMMs, diverse reviews on sub-topics can be found in the literature. For a deeper understanding and further reading, here, we give an overview of some recently published reviews.
Slota and Bogani evaluated how SMMs can be used in molecular spintronics and provided a possibility to combine quantum transport with paramagnetic spectroscopy [181]. Spin dynamics in SMMs were reviewed by Aravena and Ruiz, focusing on theoretical calculations of the spin relaxation [182].
Zhu et al. concentrated on lanthanide ions in SMMs and the effects of different external stimuli, leading to structural transformations and thus magnetic relaxation [183]. Similarly, Kalita et al. reviewed SMMs based on lanthanide complexes, concentrating on two-coordinate and pentagonal bipyramidal Ln(III) complexes [184]. Tian and Zheng reviewed molecular wheels containing lanthanides [185]. Chorazy et al., on the other hand, reviewed octacyanidometallates, not only in terms of their applications as molecular magnets, but also in terms of further functionalities, making them suitable as magnetic coolers, photomagnets, ionic conductors, etc. [186]. Guan et al. reviewed SMMs based on metallo-fullerenes [187]. Cyanido-bridged coordination polymers were the focus of a recent review of Reczynski et al., in particular dealing with the possibilities of tuning the optical properties of the corresponding SMMs [188]. Coronado concentrated on metal-organic frameworks and 2D materials in molecular magnetism [189]. The special field of molecular transistors in which SMMs can be used was reviewed by Hao et al. [190].
Shao and Wang gave a relatively broad overview of high-performance SMMs, from a basic understanding of the principles until recent research from a chemical perspective [51]. Similarly, Perlepe et al. concentrated on the chemistry of different ligands, but without offering prior basic knowledge [191].

Conclusions
A review of the most recent developments in single-molecule magnets and singlechain magnets reveals the broad range of the different materials, binding mechanisms, molecular or chain structures, etc., of these classes of molecular nanomagnets, suggesting new findings in the basic research and understanding of the principles of magnetism in these systems. While common inorganic magnets are still better understood than molecular magnets, the latter propose a large field of possible applications that may be enabled by their strongly varying physical properties.
Future directions of research should, on the one hand, aim at investigating more molecular systems in theory and experiments, to develop new SMMs and SCMs with high anisotropy barriers, blocking magnetization reversal, and hysteretic behavior at temperatures higher than room temperature, so as to enable practical utilization in data storage and other applications. On the other hand, the aforementioned multifunctional molecular magnets belong to recent trends that offer even more applications, some of which may not have been taken into account yet.
In general, taking into account the studies published over the last few decades, there is no simpler way to reach these goals than by carefully investigating the parameters of the well-known and newly developed SMMs and SCMs, so as to iteratively improve the desired properties.
We hope that this overview of some of the typical, as well as a few highly special, properties of SMMs and SCMs will inspire more researchers to investigate this fascinating class of materials.
Author Contributions: Conceptualization, T.B. and A.E.; writing-original draft preparation, A.E. and T.B.; writing-review and editing, both authors; visualization, both authors. Both authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.