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Single-molecule magnets (SMM) are a class of metalorganic compounds that show superparamagnetic behavior below a certain blocking temperature at the molecular scale. In this temperature range, SMMs exhibit magnetic hysteresis of purely molecular origin. Contrary to conventional bulk magnets and molecule-based magnets, collective long-range magnetic ordering of magnetic moments is not necessary.

Intramolecular coupling
The magnetic coupling between the spins of the metal ions is mediated by superexchange interactions and can be described by the following isotropic Heisenberg Hamiltonian:
 * $$\hat{\mathcal H}_{HB} = -\sum_{i<j} J_{i,j} \mathbf{S}_i \cdot \mathbf{S}_j,$$

where $$J_{i,j}$$ is the coupling constant between spin i (operator $$\mathbf{S}_i$$) and spin j (operator $$\mathbf{S}_j$$). For positive J the coupling is called ferromagnetic (parallel alignment of spins) and for negative J the coupling is called antiferromagnetic (antiparallel alignment of spins). The combination of these properties can lead to an energy barrier, so that at low temperatures the system can be trapped in one of the high-spin energy wells.
 * a high spin ground state,
 * a high zero-field-splitting (due to high magnetic anisotropy), and
 * negligible magnetic interaction between molecules.

Blocking temperature
Measurements take place at very low temperatures. The so-called blocking temperature is defined as the temperature below which the relaxation of the magnetisation becomes slow compared to the time scale of a particular investigation technique. A molecule magnetised at 2 K will keep 40% of its magnetisation after 2 months, and by lowering the temperature to 1.5 K this will take 40 years.

Future applications
As of 2008, there are many discovered types and potential uses.

Types
The archetype of single-molecule magnets is called "Mn12". It is a polymetallic manganese (Mn) complex having the formula [Mn12O12(OAc)16(H2O)4], where OAc stands for acetate. It has the remarkable property of showing an extremely slow relaxation of their magnetization below a blocking temperature. [Mn12O12(OAc)16(H2O)4]·4H2O·2AcOH, which is called "Mn12-acetate" is a common form of this used in research.

Single-molecule magnets are also based on iron clusters because they potentially have large spin states. In addition, the biomolecule ferritin is also considered a nanomagnet. In the cluster Fe8Br the cation Fe8 stands for [Fe8O2(OH)12(tacn)6]8+, with tacn representing 1,4,7-triazacyclononane.

The ferrous cube complex Fe4C40H52N4O12 (commonly called [Fe4(sae)4(MeOH)4]) was the first example of a single-molecule magnet involving an Fe(II) cluster, and the core of this complex is a slightly distorted cube with Fe and O atoms on alternating corners. Remarkably, this single-molecule magnet exhibits non-collinear magnetism, in which the atomic spin moments of the four Fe atoms point in opposite directions along two nearly perpendicular axes. Theoretical computations showed that approximately two magnetic electrons are localized on each Fe atom, with the other atoms being nearly nonmagnetic, and the spin–orbit-coupling potential energy surface has three local energy minima with a magnetic anisotropy barrier just below 3 meV.

History
Although the term "single-molecule magnet" was first employed in 1996, the first single-molecule magnet was reported in 1991. The complex Mn12O12(MeCO2)16(H2O)4 (Mn12Ac16), first described in 1980, exhibits slow relaxation of the magnetization at low temperatures. This manganese oxide compound features a central Mn(IV)4O4 cube surrounded by a ring of 8 Mn(III) units connected through bridging oxo ligands.

Importantly, Mn12Ac16 became the archetypal example of single-molecule magnets. Several other single-molecule magnets have later been made based on the polynuclear magnanese oxo clusters design. However, despite much studies, Mn12Ac16 has held the highest barrier to spin reversal (42 cm-1 or 61 K) and blocking temperature (~3 K) for over a decade since its discovery.

It was known in 2006 that the deliberate structural distortion of a Mn6 compound by the use of a bulky salicylaldoxime derivative switches the intra-triangular magnetic exchange from antiferromagnetic to ferromagnetic, resulting in an S = 12 ground state.

A record magnetization was reported in 2007 for [Mn(III) 6O2(sao)6(O2CPh)2(EtOH)4], with S = 12, D = −0.43 cm−1, and hence U = 62 cm−1, or 86 K at a blocking temperature of 4.5 K. This was accomplished by replacing acetate ligands (OAc) by the bulkier salicylaldoxime, thus distorting the manganese ligand sphere. It is prepared by mixing the perchlorate of manganese, the sodium salt of benzoic acid, a salicylaldoxime derivate and tetramethylammonium hydroxide in water and collecting the filtrate.

In 2003, it was reported that a single lanthanide ion complexed by phthalocyanine ligands, [TbPc2] − and [DyPc2]−, can exhibit single-molecule magnet behavior. The terbium complex exhibits a barrier as high as 230 cm−1, which arises from the ligand field splitting of the MJ states. Terbium complex exhibits open magnetic hysteresis at 1.7 K, indicative of magnetic blocking in a single lanthanide system.

In 2010, it was found that a mononuclear transition metal complex [Fe(tpaMes)]− exhibits slow magnetic relaxation behavior characteristic of single-molecule magnets, later leading to numerous studies of mononuclear transition metal single-molecule magnets.

In 2011, a N23− radical-bridged dinuclear dysprosium complex [K(18-crown-6)][{(N(SiMe3)2)2(THF)Dy}2(η2: η2-N2)] was reported to exhibit open magnetic hysteresis up to a record blocking temperature of 8.3 K. It was later superseded by a terbium analogue [K(18-crown-6)(THF)2][{(N(SiMe3)2)2(THF)Tb}2(η2: η2-N2)] which shows magnetic hysteresis at 14 K.

In 2013, it was shown that magnetic blocking can be achieved in a mononuclear transition metal complex [K(crypt-222)][Fe(C(SiMe3)3)2], where open waist-restricted hysteresis was observed up to 6.5 K.

In 2017, a mononuclear [Dy(Cpttt)2][B(C6F5)4] complex was reported to exhibit magnetic hysteresis up to 60 K, bringing the single-molecule magnet behavior close to the temperature of liquid nitrogen (78 K), where practical use of molecular magnets can be envisioned.

Detailed behavior
Molecular magnets exhibit an increasing product (magnetic susceptibility times temperature) with decreasing temperature and can be characterized by a shift both in position and intensity of the AC magnetic susceptibility.

Single-molecule magnets represent a molecular approach to nanomagnets (nanoscale magnetic particles). In addition, single-molecule magnets have provided physicists with useful test-beds for the study of quantum mechanics. Macroscopic quantum tunneling of the magnetization was first observed in Mn12O12, characterized by evenly spaced steps in the hysteresis curve. The periodic quenching of this tunneling rate in the compound Fe8 has been observed and explained with geometric phases.

Due to the typically large, bi-stable spin anisotropy, single-molecule magnets promise the realization of perhaps the smallest practical unit for magnetic memory, and thus are possible building blocks for a quantum computer. Consequently, many groups have devoted great efforts into synthesis of additional single-molecule magnets; however, the Mn12O12 complex and analogous complexes remain the canonical single-molecule magnet with a 50 cm−1 spin anisotropy.

The spin anisotropy manifests itself as an energy barrier that spins must overcome when they switch from parallel alignment to antiparallel alignment. This barrier (U) is defined as
 * $$U = S^2|D|,$$

where S is the dimensionless total spin state, and D the zero-field splitting parameter (in cm−1); D can be negative, but only its absolute value is considered in the equation. The barrier U is generally reported in cm−1 or in kelvins. The higher the barrier, the longer a material remains magnetized, and a high barrier is obtained when the molecule contains many unpaired electrons and when its zero-field splitting value is large. For example, in the "Mn(OAc)12" cluster the spin state is 10 (involving 20 unpaired electrons), and D = −0.5 cm−1, resulting in a barrier of 50 cm−1 (equivalent to 60 K).

The effect is also observed by hysteresis experienced when magnetization is measured in a magnetic field sweep: on lowering the magnetic field again after reaching the maximal magnetization the magnetization remains at high levels, and it requires a reversed field to bring magnetization back to zero.

Recently, it has been reported that the energy barrier U is slightly dependent on Mn12 crystal size/morphology, as well as the magnetization relaxation times, which varies as function of particle size and size distributions.

Multinuclear Transition Metal Single-Molecule Magnets
Most multinuclear transition metal single-molecule magnets are transition metal clusters where several metal centers are engaged in magnetic exchange interactions within the cluster. The majority of transition metal clusters consist of metal centers coupled together through diamagnetic bridging ligands via superexchange interaction. The first single-molecule magnet Mn12Ac16 operates by this principle, where four Mn(IV) centers in the Mn4O4 cubane core are coupled antiferromagnetically to eight peripheral Mn(III) centers, resulting in an S = 10 ground state. Single-molecule magnet behavior arises from the presence of zero field splitting in the ground state of the cluster, which splits the 2S+1 magnetic states (MS) according to the zero field splitting Hamiltonian

$$\hat{\mathcal H} = D(S_z^2 - \frac{1}{3}S(S+1))+E(S_x^2-S_y^2) $$

Where S is the total spin quantum number, and Sx/y/z the projection of total spin onto x/y/z axis, respectively. The axial and transverse zero field splitting parameters D and E, respectively, describe the magnetic anisotropy of the molecular magnets.

In the simple case when the system is axially symmetric, the transverse anisotropy E is zero, and the energy of the spin wavefunctions can be written as

$$ E(M_S) = D M_S^2 $$

When D < 0, this equation gives rise to a quantized, concave down parabola energy landscape, which is commonly referred to as the double-well potential. The thermal barrier to spin reversal, U, is defined as the energy separation between the ground MS and the highest MS states, of which the spin has to overcome in order to revert its orientation. In non-Kramers ions (system with integer S), $$U=|D|S^2$$, and in Kramers ions (system with half-integer S), $$U=|D|(S^2-1/4)$$.

Despite the quadratic relationship of U on S, it was shown that D is intrinsically proportional to S−2, thus this equation implies that U is virtually constant despite how large S is. Indeed, the advancement in the area of multinuclear transition metal single-molecule magnets are rather slow, where it took more than a decade to increase the initial thermal barrier in Mn12Ac16 of 42 cm−1 to 60 cm−1 as was realized in [Mn6O2(sao)6(O2CPh)2(EtOH)4]. And the record set in 2007 has been standing until now.

In addition to superexchange-mediated magnetic coupling in most transition metal clusters, other exchange mechanisms have been sought in order to improve the relatively weak superexchange coupling in polynuclear transition metal single-molecule magnets. Radical-bridged ligands were found to exhibit much stronger coupling interactions mediated by direct exchange mechanism. An azophenine radical-bridged diiron complex [(TPyA)2Fe2(L)]+ exhibits strong antiferromagnetic interaction between the radical ligand and the two iron centers, with magnetic coupling constant estimated more than 900 cm−1, the highest reported to date. The resulting dinuclear complex exhibits an effective barrier to spin reversal of 50 cm−1. Direct exchange through direct orbital overlap has also been shown to give rise to very strong magnetic coupling. The hexanuclear iron complex [NBu4][Fe6(L)2(py)2] where six iron centers form a single valence orbital manifold exhibits an S = 19/2 ground state that persists to 300 K. The cluster shows a spin reversal barrier of 42.5 cm−1 and magnetic blocking at 2.9 K.