Characteristics of Single-Molecule Magnets

Typically, single-molecule magnets (SMMs)^3-8 are clusters of exchange-coupled transition metal ions, with Mn12ac¹³ being the most prominent example.^1,3-8,14 They show slow relaxation of the magnetization due to magnetic bistability, meaning that they remain magnetized for a certain time after having switched off an external magnetic field.

Importantly, the characteristic magnetic properties are of purely molecular origin with negligible intermolecular interactions. The origin of magnetic bistability is the presence of an energy barrier for spin reversal which has to be overcome. For pure spin magnetism, i.e. for magnetic ions with completely quenched orbital angular momenta, this energy barrier is given by

∆𝐸 = |𝐷| ∙ 𝑆² (1)

for systems with integer electron spins S (non-Kramers systems) and by

∆𝐸 = |𝐷| ∙ (𝑆²−¹₄) (2)

for systems exhibiting half-integer spins S (Kramers systems).³ Thus, the energy barrier is determined by two factors: The spin S and the axial zero-field splitting (ZFS) parameter D. S is the ground state spin of the whole molecule and results from exchange coupling of the individual electron spins, usually via the bridging ligands (super-exchange). The strength of the exchange coupling is described by the coupling constant Jex which can be isotropic or anisotropic. In the isotropic case the Hamiltonian describing the interaction between two paramagnetic ions can be formulated as

ℋ_𝑒𝑥= −𝐽_𝑒𝑥𝑆̂₁∙ 𝑆̂₂ (3)

where Ŝ₁ and Ŝ₂ represent the spin operators for each of the two metal ions.³ According to equation (3), Jex is positive for ferromagnetic coupling and negative for anti-ferromagnetic coupling but several sign conventions can be found in literature, some of them also including a factor of 2.³⁹ Thus, care has to be taken when comparing data. In Mn12ac, ferrimagnetic

coupling between four Mn(IV) centers (S = 3/2) and eight Mn(III) centers (S = 2) results in a giant spin of S = (8 x 2) – (4 x 3/2) = 10 (Figure 1).¹ The sign and the magnitude of the exchange coupling not only depend on the metal centers themselves, but also on the nature of the bridging ligands and the relative orientation of the orbitals involved. Considering the extent of overlap of the spin-containing molecular orbitals based on the bridging geometry, the sign of the exchange coupling can be predicted by means of the so-called Goodenough-Kanamori rules.^40-42

The ZFS parameter D is a measure for the axial anisotropy of the system and describes the separation of the MS states within the spin ground state. MS is the magnetic spin quantum number for the coupled system and adopts values from –S to +S. In a completely isotropic system all MS states are degenerate, but axial distortion and second-order spin-orbit coupling lift this degeneracy resulting in an energy level structure which is commonly described by a double-well potential.^3-8 The value of D is expected to be high for systems with small energy gaps between the electronic ground term and admixing excited terms. The double-well potential for Mn12ac is schematically illustrated in Figure 2a. Here D is negative, meaning that the states with MS = ±S = ±10 are lowest in energy. They remain twofold degenerate, but are separated by the energy barrier E which was determined to 46 cm^-1.⁴³

In addition to axial field splitting, low-symmetry molecules exhibit rhombic zero-field splitting which is accounted for by the transverse ZFS parameter E. Rhombic distortion causes mixing of different MS states and in the case of non-Kramers systems all degeneracy can be lifted even in the absence of a magnetic field.

Figure 1: Spin structure in Mn12ac.¹ Four Mn(IV) centers (S = 3/2; shown in blue) couple ferrimagnetically to eight Mn(III) centers (S = 2; shown in red), resulting in an S = 10 ground state. Grey circles represent oxygen bridges.

Basic Concepts in Molecular Magnetism 5

Figure 2: Schematic illustration of the double-well potential for Mn12ac.⁶ a) In the absence of an external field the ±MS states are degenerate. b) If an external magnetic field is applied, the twofold degeneracy is lifted, resulting in an asymmetric shape.

The corresponding ZFS Hamiltonian is given by

ℋ_𝑍𝐹𝑆 = 𝐷(𝑆̂_𝑧²−1

3(𝑆(𝑆 + 1)) + 𝐸(𝑆̂_𝑥²− 𝑆̂_𝑦²) (4) where x, y and z label the three principal axes and are conventionally chosen such that 0  |E/D|  1/3.^3,44 The distortion described by the E term mixes only states which differ by

MS = ±2 (second-rank operators). For 3d systems with S  2 higher-rank terms are possible;

however, they are often neglected in order to avoid over-parametrization.

When an external magnetic field B is applied, the states will be further split by the Zeeman interaction and the double-well potential will become asymmetric (Figure 2b). The well corresponding to negative values for MS will be lowered in energy with respect to the other and will therefore be preferably populated: The molecule becomes magnetized and reaches its saturation magnetization at low temperatures and high fields when only the lowest-lying state is populated. The Zeeman splitting is described by the Hamiltonian

ℋ_{𝑍𝑒𝑒𝑚𝑎𝑛} = 𝜇_𝐵 ∑ 𝑔_𝑘,𝑞𝐵_𝑘𝑆̂_𝑞

𝑘,𝑞=𝑥,𝑦,𝑧 (5)

where µB denotes the Bohr magneton and g is the orientation-dependent Landé factor of the system.^3,38

When the external magnetic field is switched off, the system will relax, meaning that both wells will be populated equally again. This process requires overcoming the energy barrier by climbing up the ladder of MS states (multi-step Orbach relaxation).³ Thus, for high energy barriers and at sufficiently low temperatures, slow relaxation of magnetization will be observed. In Mn12ac magnetic relaxation can take up to several months.¹

In an ideal SMM, such a thermally activated multi-step Orbach relaxation would be the only pathway for magnetic relaxation and the temperature dependence of the relaxation time  could be solely described by an Arrhenius law

𝜏_{𝑂𝑟𝑏𝑎𝑐ℎ}= 𝜏₀∙ 𝑒^{−∆𝐸⁄𝑘𝐵𝑇} (6)

with the attempt time 0, the energy barrier E, the Boltzmann constant kB and the temperature T.^3,38 In real systems however, not only the Orbach process contributes to the magnetic relaxation, but also the Raman process, the direct process and quantum tunneling of magnetization.^3,38 These relaxation pathways give rise to effective energy barriers Ueff which are usually much smaller than the expected barrier E. A more detailed description will be given in section 2.1.3.

The quadratic dependence of the energy barrier on the cluster spin S in equations (1) and (2) motivated chemists to synthesize metal ion complexes with rather large ground state spins. For instance, new records were obtained in 2006 in a ferromagnetically coupled Mn19

aggregate exhibiting a spin of S = 83/2¹⁵ and recently in a Fe42 cluster with S = 45¹⁸. However, in contrast to expectations the energy barriers were very small. Meanwhile it is well-known that the cluster anisotropy constant D is not independent of the ground state spin S and that large spins tend to come along with low anisotropies, preventing high anisotropy barriers.^19,20 Attention therefore has turned to the design of metal ion complexes employing highly magnetically anisotropic metal centers.9,11,12,22-24,28,37 High anisotropy can be achieved by using metal centers with unquenched orbital angular momenta such as lanthanide(III) or octahedrally coordinated cobalt(II) ions (first-order spin-orbit interactions) or by ligand field design that leads to admixing of excited states with orbital angular momentum (second-order spin-orbit coupling).

Basic Concepts in Molecular Magnetism 7