Some new trends in the design of single molecule magnets

Magnetization reversal barrier in clusters containing orbitally-degenerate 3d – ions: role of first-order single ion anisotropy

One of the first well-documented example of such SMM was the trigonal bipyramidal cyano-bridged cluster [Mn^III(CN)₆]₂[Mn^II(tmphen)₂]₃ (tmphen = 3,4,7,8–tetramethyl–1,10–phenanthroline) [23]. The molecular geometry of this Mn₅-cyanide cluster is shown in Fig. 3a. Five Mn ions form a trigonal bipyramid in which two Mn(III) ions (1 and 2) occupy the apical positions and possess the low-spin ground terms ³T_1g(t2g4) in a strong crystal field (CF) of the carbon octahedral surroundings (Fig. 3b), while the three Mn(II) ions (3–5) reside in the equatorial plane and have the high-spin ground terms 6A1g(t2g3eg2) in a weak CFs induced by the nitrogen octahedra (Fig. 3c). In addition to the spin value S = 1 the ground term ³T_1g(t2g4) of the Mn(III) ion carries first order orbital angular momentum. In fact the orbital triplet ³T_1g is in some sense equivalent to the atomic triply degenerate P state which can be associated with L = 1. One can state that the Mn₅-cyanide system drastically differs from the pure spin SMMs in which all constituent metal ions are characterized only by spin values. We will demonstrate that the presence of the unquenched orbital angular momentum can lead under some conditions to the appearance of rather strong uniaxial magnetic anisotropy which is responsible for the formation of the barrier for the reversal of magnetization [24], [25]. It is important to note that the presence of the unquenched orbital angular momentum in the ground term created by perfect octahedral ligand field does not carry magnetic anisotropy. Fortunately, the carbon surroundings of the Mn(III) ions are distorted along the trigonal Z axis (Fig. 3a). Instead of considering the real situation occurring in the Mn₅-cyanide cluster for which this trigonal distortion is rather weak, we will discuss here a hypothetical situation of rather strong trigonal distortion of the nearest surrounding of the Mn(III) ion. The analysis of such strong trigonal CF is quite instructive because it reveals with utmost clarity the role of orbital angular momentum in the formation of the barrier for magnetization reversal.

Fig. 3:

Molecular structure of the Mn₅-cyanide cluster in which only the manganese ions are indicated with their nearest ligand surroundings (a) and electronic configurations and ground terms of the Mn(III) (b) and Mn(II) (c) ions.

The trigonal CF acting on the Mn(III) ion (which is actually axial) can be described by the Hamiltonian:

where Δ is the axial CF parameter, L^Z is the Z-component of the orbital angular momentum operator and for the orbital triplet one should adopt L = 1. The trigonal field splits the ³T_1g state into the orbital singlet ³A_2g (orbital angular momentum projection M_L = 0) and the orbital doublet ³E_g(M_L = ±1) which is the ground state providing Δ < 0. We will focus on the case when the ground term is the trigonal orbital doublet ³E_g, which carries a residual orbital angular momentum. The splitting of the cubic ³T_1g – term (L = 1, S = 1, Fig. 4a) into the ground orbital doublet ³E_g(S = 1, M_L = ±1) and excited orbital singlet ³A_2g (S = 1, M_L = 0) is shown in Fig. 4b.

Fig. 4:

Splitting of the ground cubic ³T_1g – term (a) of the Mn(III) ion by the negative trigonal CF (b) and SOC (c) in the limit of strong trigonal CF.

In case of perfect octahedral surrounding of this ion the spin-orbit coupling (SOC) acts within the cubic ³T_1g(S = 1, L = 1) term and is described by the following Hamiltonian:

where λ is the SOC parameter that is negative for more than half-filled t_2g – shell which is just the case under consideration (Fig. 3b). The parameter κ is the orbital reduction factor (κ ≤ 1) accounting for the effects of covalence which decreases the electronic density on the metal center. The sign “minus” in the SOC Hamiltonian, eq. (2), and also in the orbital part, −μBκL^H, of the Zeeman interaction with the applied magnetic field H appears due to the fact, that the matrices of the operator L^ defined in the T41g(t2g4) basis differ in sign from the matrices of this operator defined in a pure atomic p – basis (the so-called T-P analogy) [38]. In the considered case of strong axial field when |Δ/λ| ≫ 1, it is reasonable to restrict the SOC by the truncated basis of the ground orbital doublet ³E_g(S = 1, M_L = ±1). In this case the matrix elements of the operators L^X and L^Y vanish within the orbital doublet ³E_g and so the SOC Hamiltonian acquires the following axial form:

The operator L^XS^X+L^YS^Y mixes the ground orbital doublet ³E_g with the excited orbital singlet ³A_2g , but this mixing produces negligible effect if we deal with the limit of strong axial field when the ³E_g – ³A_2g – gap strongly exceeds the matrix elements of the SOC. The axial SOC splits the ³E_g-term into three equidistant non-Kramers doublets separated by the energy gaps κ| λ | as shown in Fig. 4c. This set of levels includes the ground non-Kramers doublet with M_L = ±1, M_S = μ1 and two non-Kramers doublets with M_L = ±1, M_S = 0 and M_L = ±1, M_S = ±1. A remarkable feature of the ground non-Kramers doublet of Mn(III) ion is that this state is magnetic in spite of the fact that the total angular momentum projection for this state is vanishing (M_L + M_S = 0). In fact, in the magnetic field parallel to C₃ axis the ground non-Kramers doublet is split into two Zeeman sublevels with the energies

On the contrary, this level is non-magnetic in a perpendicular magnetic field because the matrices of the operators S^X, S^Y, L^X and L^Y are vanishing within the ground Kramers doublet. This means that providing Δ < 0 the Mn(III) ion possesses uniaxial magnetic anisotropy with χ_|| > χ_⊥ which corresponds to the existence of the easy axis of magnetization. We will see that this negative single-ion magnetic anisotropy leads to the negative uniaxial magnetic anisotropy of the entire cluster and to the formation of the barrier for the reversal of magnetization.

It is seen from the structure of the Mn₅-cyanide cluster that the exchange interaction between Mn(III) and Mn(II) ions occurs through the cyanide bridge. The exchange coupling for each such Mn(III)–Mn(II) pair can be roughly described by the isotropic Heisenberg–Dirac–Van–Vleck (HDVV) Hamiltonian

which acts within the (L = 1, S = 1)_Mn(III)⊗(S = 5/2)_Mn(II) – manifold of the pair. Here we employ a simplifying assumption about isotropic HDVV form of the exchange Hamiltonian which was proposed by Lines in his consideration of the Co(II) clusters [39]. If the exchange coupling is weak as compared with the splitting caused by the SOC, the basis can be truncated by the manifold (M_L = ±1, M_S = 1)_Mn(III)⊗(S = 5/2)_Mn(II) involving only the ground non-Kramers doublet as illustrated by Fig. 5. It is seen that all matrix elements of the operators S^X[Mn(III)] and S^Y[Mn(III)] are vanishing within the ground doublet of the Mn(III) ion. Hence the initial isotropic exchange Hamiltonian of HDVV form, eq. (5), projected onto the ground manifold of the pair, is reduced to the following axially anisotropic effective Hamiltonian of the Ising form:

Fig. 5:

Energy scheme illustrating the effective exchange interaction acting within the ground (M_L = ±1, M_S = 1)_Mn(III)⊗ (S = 5/2)_Mn(II) – manifold of the Mn(III)–Mn(II) pair.

The effective exchange Hamiltonian for the entire Mn₅-cyanide cluster is given by:

Its eigenvalues are the following:

where M_S(1, 2) = M_S(1) + M_S(2) and M_S(3, 4, 5) = M_S(3) + M_S(4) + M_S(5).

Note that each eigenvalue can be characterized by the total angular momentum projection M_J of the cluster that is equal to M_J = M_S(1, 2) + M_L(1, 2) + M_S(3, 4, 5) = M_S(3, 4, 5) which follows from the definition M_L(1, 2) = M_L(1) + M_L(2) = −M_S(1) − M_S(2) = −M_S(1, 2). Alternatively, each level can be characterized by the microscopic magnetization μ¯Z that represents the expectation value of the operator μ^Z=μBge[S^Z(1, 2)+S^Z(3, 4, 5)]−μBκL^Z(1, 2). The microscopic magnetization is evidently equal to μ_B[(g_e + κ)M_S(1, 2) + g_eM_S(3, 4, 5)]. Figure 6 shows the energy pattern calculated with the aid of eq. (8) providing J < 0 (Fig. 6a) and J > 0 (Fig. 6b). The energies are shown as functions of the microscopic magnetization μ¯Z and the low-lying levels are also labelled by the projection M_J of the total angular momentum. The main features of both energy patterns in Fig. 6 is that in the low-lying groups of levels the energy is decreased with the increase of μ¯Z and so these groups of levels (shown by red in Fig. 6) form the barriers for the reversal of magnetization. Such energy patterns correspond to the existence of easy axis of magnetization for which χ_|| > χ_⊥ as in the case of pure spin SMMs described by spin Hamiltonian DSS^Z2 with D_S < 0. Note however, that the low-lying levels in Fig. 6 are equidistant and in this aspect the present energy patterns are drastically different from those described by the ZFS spin Hamiltonian DSS^Z2. In the last case the energy gaps between the levels spaced near the top of the barrier are smaller than those for levels situated near the bottom of the barrier. This difference arises from the fact that the ZFS describes the second-order anisotropy, while the magnetic anisotropy described by eq. (8) is the result of the unquenched orbital angular momentum. Being the first-order effect, the magnetic anisotropy associated with the unquenched orbital angular momentum is expected to be much stronger than the second-order anisotropy in spin clusters, and the corresponding barrier for magnetization reversal can be also much larger than the barrier in spin-clusters.

Fig. 6:

Energy patterns of Mn₅-cyanide cluster calculated with the aid of eq. (8) providing antiferromagnetic (a) and ferromagnetic (b) exchange coupling. The low-lying levels forming the barriers are shown in red.

The analysis of the magnetic behavior of the Mn₅-cyanide cluster performed in [24] shows that the real situation in this system is far from the above described idealized picture corresponding to the strong trigonal CF limit. Indeed the best fit value |Δ| = 251 cm⁻¹ of the negative trigonal CF parameter only slightly exceeds the found value κ|λ| ≈ 144 cm⁻¹ of the splitting caused by the SOC. Nevertheless, even under these much less favorable conditions the low-lying energy levels calculated for this cluster with the best-fit parameters Δ = −251 cm⁻¹ and J = −3.8 cm⁻¹ were shown to form the magnetization reversal barrier [24]. This means that the conclusion about the appearance of the barrier in the limit of strong negative trigonal CF proves to be also valid for the arbitrary negative value of Δ that is compatible with the observed SMM behavior of the Mn₅-cyanide cluster.

Magnetization reversal barrier in cyano-bridged 3d-4d-3d single molecule magnets

Recently new [Mn(LN5Me)(H2O)]2[Mo(CN)7]⋅6H2O cyanide compound exhibiting distinct SMM behavior has been reported [36]. This Mn₂Mo-cyanide compound is based on the central pentagonal bipyramidal [Mo^III(CN)₇]⁴⁻ heptacyanometalate of D_5h– symmetry coupled with two terminal Mn(II) ions via superexchange (Fig. 7a). The effective barrier found for this compound, U_eff ≈ 40.5 cm⁻¹, is the record for cyanide-bridged SMMs, and the relaxation time at 1.8 K has been found to be τ ≈ 2.73×10⁶ s ≈ 1 month.

Fig. 7:

Molecular structure of the Mn₂Mo – cyanide cluster (a) and electronic configuration and ground term of the Mo(III) ion in [Mo^III(CN)₇]⁴⁻ heptacyanometalate (b).

First, we will focus on the analysis of strong single-ion anisotropy of the Mo(III) ion and demonstrate that this anisotropy can result in the appearance of the magnetization reversal barrier. In the CF of D_5h symmetry the one-electron 4d-level is split into the ground orbital doublet (d_ZX, d_ZY) [40], excited doublet (d_XY, d_X²−Y²) and the upper singlet (d_Z²). The gap between the two low lying doublets is found to be around 30000 cm⁻¹. Since the CF induced by the carbon atoms is rather strong, the three 4d-electrons of the Mo ion occupy the lowest orbitals d_ZX and d_ZY thus giving rise to the ground low-spin orbital doublet E″21, which means that along with spin S = 1/2 the angular momentum projection M_L = ±1 can be attributed to this term (Fig. 7b).

The SOC acting within this state has an axial form −κλL^ZS^Z(the sign “minus” appears as a result of T-P analogy). This interaction splits the M_L = ±1, S = 1/2 level into two Kramers doublets as shown in Fig. 8a. Since we deal with the more than a half-filled electronic shell (three electrons per two orbitals) the SOC parameter for the ground orbital doublet is negative, and hence the ground Kramers doublet possesses M_L = ±1, M_S = 1/2, while for the excited doublet M_L = ±1, M_S = 1/2. The SOC gap κ| λ | ranges from 600 to 1000 cm⁻¹ [41] which is significantly smaller than the CF gap between the ground and excited CF terms. For this reason the above described strong axial CF limit is quite well justified for this complex.

Fig. 8:

Energy scheme illustrating the SOC splitting of the ground E″21- term of the Mo(III) ion (a), and effective exchange interaction acting within the ground (M_L = ±1, M_S = 1/2)_Mo(III)⊗ (S = 5/2)_Mn(II) – manifold of the Mo(III)–Mn(II) pair (b).

Let us consider now the exchange-coupled Mo(III)–Mn(II) pair. As in the case of Mn₅-cyanide cluster we will assume that the initial exchange Hamiltonian is of isotropic HDVV form. Although this is not fully true in the present case due to orbitally-dependent exchange (see Section ”Remarks concerning the role of orbitally-dependent exchange”), we will retain here this simplifying assumption because our aim is just to illustrate the role of the orbital contributions in the formation of the magnetization reversal barrier rather than to provide exact quantitative description of the magnetic anisotropy in such systems. Projecting the HDVV exchange Hamiltonian onto the truncated ground (M_L = ±1, M_S = 1/2)_Mo(III)⊗(S = 5/2)_Mn(II) – manifold of the Mo–Mn pair (Fig. 8b) we obtain axially anisotropic Ising Hamiltonian of the same form as the Hamiltonian, eq. (6), found for the Mn(III)–Mn(II) – pair. Then the effective exchange Hamiltonian for the entire Mn₂Mo-cyanide cluster is the following:

where S^Z(Mn2) is the Z-component of the spin operator for the pair of the terminal Mn(II)-ions. This gives the following set of eigenvalues:

where M_S(Mo) = ±1/2 and M_S(Mn₂) takes on the values 0, ±1, ±2, ±3 and ±5. Each level in eq. (10) can be characterized by the value of the total angular momentum projection

M J = M S ( Mo ) + M L ( Mo ) + M S ( Mn 2 ) = M S ( Mn 2 ) − M S ( Mo )

because M_L(Mo) = ‒ 2M_S(Mo) for the ground Kramers doublet of the Mo(III) ion. Alternatively, each level can be characterized by the microscopic magnetization μ¯Z for which we find

μ ¯ Z = μ B [ g e M S ( Mo ) − κ M L ( Mo ) + g e M S ( Mn 2 ) ] = 2 μ B [ 2 M S ( Mo ) + M S ( Mn 2 ) ]

providing g_e = 2 and κ = 1. The equidistant low lying levels form the barrier for magnetization reversal whose height is determined by the magnitude of the exchange parameter (levels shown in red in Fig. 9).

Fig. 9:

Energy patterns of Mn₂Mo-cyanide cluster calculated with the aid of eq. (10) providing antiferromagnetic exchange coupling. The low-lying levels forming the barrier are shown in red.

The two examples so far considered clearly show that strong negative single-ion magnetic anisotropy associated with the first-order orbital angular momentum can give rise to the appearance of considerable magnetization reversal barrier composed of equidistant energy levels. The magnitude of this barrier is maximal in the strong axial CF limit in which case it is determined by the strength and the sign of the exchange coupling.

Remarks concerning the role of orbitally-dependent exchange

In the above considered examples we have assumed that the exchange interaction is of the isotropic HDVV form. In general, the HDVV Hamiltonian is applicable when the ground levels of the coupled ions are orbitally non-degenerate, for instance, in the case of high-spin d⁵ ions in octahedral CF (6A1g(t2g3eg2) ‒ term) or, half-filled subshell (4A2g(t2g3)). Alternatively, the HDVV Hamiltonian is valid when the low-symmetry CF removes the degeneracy and in this way stabilizes orbital singlet while the remaining components of the initial orbitally degenerate level are high enough. When the orbitally degenerate terms of the constituent ions are involved, the HDVV Hamiltonian, is, in general, inapplicable and a more general, so-called orbitally dependent effective Hamiltonian should be employed (see detailed discussion in Ref. [42]). Keeping this in mind, we note here that some cases (indicated in Ref. [42]) may prove the exception. In particular, the HDVV Hamiltonian can serve as a good approximation when the kinetic exchange interaction arises mainly from electron transfer within orbitally-nondegenerate sub-shells of the orbitally-degenerate metal ions. This approximation also works well when the number of equal hopping parameters contributing to the kinetic exchange proves to be equal to the total orbital multiplicity of the system [42]. In such cases we arrive at the Ising-type effective exchange Hamiltonian by projecting the isotropic HDVV exchange Hamiltonian onto the ground manifold of the pair involving the ground non-Kramers (case of the Mn₅-cyanide cluster) or Kramers (case of the Mn₂Mo-cyanide cluster) doublet of the orbitally-degenerate ion.

Turning back to the case of the Mn₂Mo-cyanide cluster, we can see that for strictly linear Mo–CN–Mn bridging group we have two equal transfer integrals of π-type t(d_XZ, d_XZ) = t(d_YZ, d_YZ) so that this number of transfer parameters is just equal to the total orbital multiplicity of the system (Fig. 10). As to the σ-transfer it is seen from Fig. 10 that it involves only the orbitally nondegenerate sub-shell of the Mo(III) ion and so this transfer also results in the HDVV – type contribution. Therefore, for the system with linear Mo–CN–Mn bridging group the kinetic exchange Hamiltonian is given by:

Fig. 10:

Kinetic mechanism of Mo(III)–CN–Mn(II) superexchange showing the origin of the HDVV – type exchange coupling in the case of linear geometry.

As was mentioned such HDVV-type Hamiltonian is reduced to the effective Hamiltonian of the Ising form, eq. (9), that is responsible for the formation of the magnetization reversal barrier as shown in Fig. 9. In Ref. [43] such Ising-type effective Hamiltonian was derived for the Mo–Mn pair directed along the C₅ axis of the Mo^III(CN)₇ by considering the kinetic exchange constrained within the ground Kramers-doublet space. In a more general sense, the procedure includes the derivation of the Hamiltonian acting in the full space with subsequent projecting of this Hamiltonian onto the ground manifold as described here. It is clear that both these approaches are equivalent and lead to the Ising-type effective Hamiltonian for the linear system.

In more general situations the HDVV model proves to be inapplicable and the kinetic exchange becomes essentially orbitally dependent. Actually, this means that the coupling between ions cannot be expressed in terms of spin variables only. The exchange interaction between orbitally degenerate ions includes three different kinds of contributions: spin–spin term (resembling the HDVV Hamiltonian), orbital–orbital coupling involving matrices acting in orbital spaces and spin–orbital terms including products of orbital matrices of one of the ion and the spin operators of other ion (see detailed discussion in Ref. [42]). It is appropriate to emphasize that it is not only a formal mathematical complication of the theory of the magnetic exchange. The most significant observation is that the orbitally-dependent parts of the exchange interaction give rise to a strong magnetic anisotropy which is an inherent property of the systems comprising orbitally degenerate ions. In this sense, it is worth to mention again that the HDVV Hamiltonian is fully isotropic. The most general form of the orbitally-dependent exchange Hamiltonian for the A-B pair of metal ions, in which the ion A possesses unquenched orbital angular momentum while another ion B has only spin, is the following [42]:

where R^1orb(A) and R^2orb(A) are the orbital operators (combination of orbital matrices) for the site A whose explicit forms depend on the overall symmetry of the pair (determining the allowed transfer pathways), the electronic configurations of the orbitally-degenerate ions and the symmetry of the local CFs acting on these ions. Upon projecting this Hamiltonian onto the ground manifold we obtain the effective Hamiltonian whose most general form is the following:

Providing linear geometry of the A-bridge-B group we have only two different parameters J_ZZ ≡ J_||, J_XX = J_YY = J_⊥ and so the effective Hamiltonian is axially symmetric:

This form of the Hamiltonian directly follows from the axial point symmetry of the pair, meanwhile the relative values of two exchange parameters depend on the nature of interacting ions (their electronic configurations and terms). Thus, we have seen that for linear Mo–Mn pair J_⊥ = 0 and so the system exhibits Ising-type anisotropy.

In the case of a bent geometry of the A-bridge-B group the Hamiltonian is, in general, triaxial and has the form of eq. (13). This situation is apparently unfavorable for the creation of magnetization reversal barrier (especially if the difference between J_XX and J_YY parameters is large) because it promotes a fast quantum tunneling of magnetization effectively decreasing the barrier. Fortunately, in some important special cases the actual symmetry of the kinetic exchange Hamiltonian proves to be higher than the point symmetry of the A-bridge-B group. This occurs, for example, in the case of Mn₂Mo-cyanide cluster which is shown in Fig. 7a. For this system, in spite of bent geometry of the bridging Mo(III)–CN–Mn(II) – group, the effective Hamiltonian was found to be of axial form, eq. (14), with |J_|||≫|J_⊥| [37] (close to the Ising limit). In this case M_J remains good quantum number and the low-lying levels form the barrier which however has irregular structure unlike the barrier shown in Fig. 9.

Less favorable situation when the bent geometry leads to the triaxial anisotropy is exemplified by the Os(III)–CN–Mn(II) superexchange in trimeric cluster (NEt₄)[Mn₂(5-Brsalen)₂(MeOH)₂Os(CN)₆] (Fig. 11a) that was shown to exhibit SMM behavior with U_eff ≈ 13 cm⁻¹ [32]. For hypothetical linear system shown in Fig. 11b both F^1orb(A) and F^2orb(A) operators are proportional to L^Z2(A)(eq. (16) in Ref. [42]) provided that only the π-transfer contributes to the kinetic exchange. The SOC splits the low-spin 2T2g(t2g5)-term of the Os(III) ion in a perfect octahedral CF (Fig. 11c) into the Kramers doublets with J = 1/2 (ground doublet) and J = 3/2 (excited doubet) as shown in Fig. 11d, where J is the quantum number of the total angular momentum of the Os ion. Projecting the terms ~L^Z2(Os) and

L ^ Z 2 ( Os ) S ^ ( Os ) S ^ B ( Mn ) on the ground (J = 1/2)_Os(III)⊗(S = 2)_Mn(III) – manifold we arrive at the Ising-type effective Hamiltonian (see eq. (52) in Ref. [42]). Unfortunately, in reality the cluster exhibits bent geometry (Fig. 11a) and such bending opens additional transfer pathways giving rise to the triaxial effective exchange Hamiltonian [34]. This is the reason why in spite of stronger 5d-3d superexchange, as compared with the 4d-3d one, the Mn(III)₂Os(III)-cyanide cluster exhibits lower magnetization reversal barrier as compared with the Mn(II)₂Mo(III)-cyanide cluster.

Fig. 11:

Molecular structure of the Mn(III)₂Os(III)-cyanide cluster (a), hypothetical linear Mn(III)₂Os(III) – trimer (b), electronic configuration and ground term of the Os(III) ion in strong octahedral ligand field of carbon atoms (c) and spin-orbital splitting (d).

One should also mention that in some cases, in spite of Ising-type effective exchange interaction, the magnetization reversal barrier does not appear. Thus, the trigonal bipyramidal cyanide-bridged Ni^II₃Os^III₂ cluster with linear bridging groups Os(III)–CN–Ni(II) does not behave as SMM due to the fact that anisotropy axes associated with different Os(III)–Ni(II) pairs, exhibiting Ising exchange, are non-collinear and hence the overall anisotropy of the cluster is not of the Ising type [44].

Magnetization reversal barrier for two-coordinate Fe(I)-complex: role of low coordination number

Recently the remarkable two-coordinate Fe(I)-complex [Fe(C(SiMe₃)₃)₂]⁻ (Fig. 12a) have been reported in Ref. [58]. This system exhibits slow magnetic relaxation below 29 K in the absence of the direct current (dc) field, a large effective spin-reversal barrier U_eff ≈ 226 cm⁻¹ and magnetic blocking below 4.5 K. Based on the ab initio calculations [58] the three main factors responsible for the pronounced SMM behavior of this complex have been mentioned: the axial CF splitting imposed by the low coordination number, the SOC of the ground CF multiplet, and the advantage provided by the properties of the basis states with respect to the time reversal (Kramers theorem). The last hinders quantum tunneling of magnetization that becomes possible only if the hyperfine coupling with the nuclear spin of iron ion is included.

Fig. 12:

Molecular structure of the complex [Fe(C(SiMe₃)₃)₂]⁻ (a), electronic configuration and the ground ⁴E-term of this complex (b), and spin-orbital splitting of the ⁴E-term (c).

The axial CF leads to the orbital scheme depicted in Fig. 12b [58]. The distribution of seven electrons over these orbitals gives rise to the ground orbital doublet ⁴E (3d⁷) whose components possess the same symmetry properties as the dX2−Y2 and d_XY orbitals. Since the linear combinations of these orbitals φ3d±2=(dX2−Y2±idXY)/2 are characterized by the angular momentum projection m_l = ±2 the term ⁴E can be regarded as the state with M_L = ±2. This term undergoes a further splitting caused by the SOC which is described by the axial operator acting within the ⁴E-term:

where λ = −ζ/(2S) ≈ −120 cm⁻¹. The SOC splits the ⁴E-term into four equidistant Kramers doublets characterized by the absolute value of the projection M_J of the total angular momentum as shown in Fig. 12c. The energy plotted as function of the expectation value of the operator μ^Z=μB(κL^Z+geS^Z) with κ = 1 and g_e = 2 is shown in Fig. 13. It is seen that the three lowest doublets can be regarded as the magnetization reversal barrier.

Fig. 13:

Energy pattern of the complex [Fe(C(SiMe₃)₃)₂]⁻ composed of eigenvalues of the Hamiltonian, eq. (15), calculated with κ = 1. The low-lying levels forming the barrier are shown in red.

However the experimentally observed value for U_eff proves to be smaller than the height of the barrier in Fig. 13 and it is quite close to the energy gap 2| λ | between the ground doublet with |M_J| = 7/2 and the first excited doublet with |M_J| = 5/2. This is probably due to the phonon assisted quantum tunneling between the components M_J = −5/2 and M_J = 5/2 of the first excited doublet as schematically shown in Fig. 13. Anyway the found value of U_eff is the largest one observed so far for the SMMs based on 3d – ions, thus showing that low coordination number plays a constructive role in the design of SIMs because it increases the axial magnetic anisotropy giving rise to the high magnetization reversal barrier.

Single-ion magnets containing 4f-ions with unquenched orbital angular momenta

In spite of the large number of nd-type SIMs reported to date, the SIMs and SMMs based on complexes of 4f-ions with unquenched orbital angular momenta seem to be more promising for practical applications. There are two reasons for that. First, 4f-complexes can exhibit (under appropriate symmetry conditions) much stronger magnetic anisotropy [53] than nd-complexes. The second reason relates to the much weaker spin-phonon interactions in 4f-complexes because 4f – shell is shielded and 4f-orbitals are less extended. As a consequence, 4f-complexes can show much slower spin-phonon relaxation as compared with the nd-complexes.

The SIM behavior of lanthanide complexes was first discovered [45], [46] for two phthalocyanine double-decker complexes [Pc₂Ln]⁻ TBA⁺ (Ln = Tb or Dy; TBA⁺ = tetrabutylammonium cation) (Fig. 14a). They were found to exhibit characteristic temperature and frequency dependences of the alternating current (ac) magnetic susceptibility, which manifestations are indicative of SMMs. These compounds were the first lanthanide metal complexes functioning as SMMs. Moreover, they were the first mononuclear metal complexes showing SMM properties. One of the most remarkable features of these mononuclear lanthanide SMMs is that they show slow relaxation of magnetization in temperature ranges that are significantly higher than those for the discovered transition-metal SMMs. Thus, the [Pc₂Tb] and [Pc₂Dy] complexes exhibit out-of-phase ac susceptibility χ″M peaks at 40 and 10 K with a 10³ Hz ac field, respectively [45], [46]. Later on the Ho complex with the same structure was also shown to exhibit SMM behavior. Such characteristic features of SMMs as hysteresis and resonant quantum tunnelling of magnetization were found in the Tb, Dy and Ho complexes [47], [48].

Fig. 14:

(a) Molecular structure of the anion of bis(phthalocyaninato)-lanthanide, and (b) Ln(III) ion surrounded by eight nitrogen atoms showing approximate D_4d symmetry of the complex at skew angle φ = 45°.

As distinguished from nd-ions, in lanthanides the SOC acts as the leading interaction, stabilizing the ground terms ^2S+1L_J(4fⁿ) of the free 4f-ions, e.g. ⁷F₆(4f⁸) for Tb(III) and ⁶H_15/2(4f⁹) for Dy(III). These terms undergo further splitting by the CF giving rise to the energy patterns of the Stark sublevels. Although the CF in lanthanides is much weaker than in nd-complexes, this interaction is of primary importance because it can lead to the appearance of considerable magnetic anisotropy responsible for the formation of the magnetization reversal barriers. The nearest coordination environment of 4f ion in the [Pc₂Tb] and [Pc₂Dy] complexes is represented by eight nitrogen atoms forming two parallel plains rotated with respect to each other by the angle φ (Fig. 14b). In the cases of [Pc₂Tb] and [Pc₂Dy] complexes the skew angle φ is close to 45° and hence, in the octacoordinated lanthanides LnN₈ the local symmetry can be approximately considered as D_4d.

In the case of D_4d symmetry the effective CF Hamiltonian acting within the ground ^2S+1L_J(4fⁿ) term is written in the operator equivalent form as follows:

where Akq〈rk〉 are the ligand field parameters, O^kq are the irreducible tensor operators (Stevens operators [69]) defined with respect to the quantization Z axis (C₄ axis of the complex), finally α, β and γ are the Stevens coefficients [70] given in Table 1 for three lanthanide ions. The operators O^k0 have the following forms:

The analysis of the absorption and fluorescence spectra of the lanthanide ions represents the most direct way to detect the Stark structure and thus to find the set of the CF parameters Akq〈rk〉. In the present case, however, neither fluorescence nor absorption spectra associated with lanthanide centers are obtainable because of the low-lying energy levels of Pc quenching the lanthanide fluorescence, and the extremely intense Pc absorption bands concealing the lanthanide bands. For this reason it was proposed in Ref. [45] to search the sets of the CF parameters, which simultaneously reproduce the temperature dependence of the dc magnetic susceptibility for a powder sample and the paramagnetic shifts of ¹H NMR spectra. The patterns of the Stark sublevels calculated in this way for [Pc₂Tb] and [Pc₂Dy] complexes are shown in Fig. 15 in which the energy is plotted as function of the microscopic magnetization μ¯Z=μBgJMJ, where g_J = 3/2 and 4/3 for Tb(III) and Dy(III) ions, respectively. Note that M_J is a good quantum number for the system of D_4d –symmetry and so each level (except the upper singlet level for Tb complex) represents a doublet |±M_J〉.

Fig. 15:

Patterns of the Stark sublevels for [Pc₂Tb]⁻ TBA⁺ (a) and [Pc₂Dy]⁻ TBA⁺ (b) calculated with the sets of parameters A20〈r2〉, A40〈r4〉 and A60〈r6〉 found in Ref. [45]. The levels forming the barriers are shown in red.

It is seen from Fig. 15a that the lowest doublet of [Pc₂Tb] possesses the largest |M_J| value (M_J = ±6) and the energy gap between this doublet and the first excited doublet with M_J = ±5 exceeds 400 cm⁻¹. This situation is similar to that occurring in SMMs based on the transition metal spin clusters, but the magnetization reversal barrier separating sublevels M_J = +6 and M_J = −6 proves to be much higher as compared to the barriers in spin clusters, U_eff ≈ 230 cm⁻¹ for the [Pc₂Tb] complex. The Stark sublevels for the Dy^III ion are distributed more evenly, so the SIM behavior of [Pc₂Dy] complex is less pronounced than that for [Pc₂Tb]. Indeed, it follows from Arrhenius analysis that U_eff ≈ 230 cm⁻¹ for the [Pc₂Tb] complex, while for [Pc₂Dy] complex much smaller barrier of U_eff ≈ 28 cm⁻¹ was found. In addition to the complexes of Tb and Dy, the [Pc₂Ho] complex was also shown to exhibit SIM properties.

Later on it was demonstrated that the concept of SIMs can be extended to other families of mononuclear 4f – complexes. Thus, the polyoxometalate complexes encapsulating some of 4f – ions were shown to exhibit SIM behavior for coordination sites close to the antiprismatic D_4d symmetry [49], [50]. Here we will mention one such family, namely [Ln(W₅O₁₈)₂]⁹⁻ (Ln^III = Tb, Dy, Ho, and Er) whose structure is shown in Fig. 16a. Among these complexes only Er shows slow relaxation of magnetization above 2 K, while the Tb, Dy and Ho complexes do not exhibit distinct SIM behavior at T>2 K. Thus the pattern of the Stark sublevels calculated with the parameters A20〈r2〉=−36.8 cm−1, A40〈r4〉=−89 cm−1, and A60〈r6〉=−5.2 cm−1 found by fitting the dc magnetic properties of the [TbW₁₀O₃₆]⁹⁻ complex [50] shows no magnetization reversal barrier (Fig. 16b) as distinguished from the energy levels of the [Pc₂Tb] complex which were shown to form the barrier (Fig. 15a). On the other hand, in contrast to the system [ErW₁₀O₃₆]⁹⁻, the [Pc₂Er] complex does not exhibit SIM properties.

Fig. 16:

Polyhedral view of [TbW₁₀O₃₆]⁹⁻ polyoxoanion (a), and patterns of the Stark sublevels calculated for this complex with the set of parameters A20〈r2〉=−36.8 cm−1, A40〈r4〉=−89 cm−1, and A60〈r6〉=−5.2 cm−1 found in Ref. [50].

The difference between the magnetic properties of these two classes of complexes belonging to the same symmetry lies in the fact that the nearest ligand surrounding of the lanthanide encapsulated in polyoxomatalate represents axially compressed square antiprism, meanwhile in phthalocyaninato complexes such antiprism is axially elongated. One can thus conclude that the axially elongated sites promote the SIM behavior in cases of Tb and Dy complexes, as exemplified by the double-decker bis-(phthalocyaninato) complexes, while axially compressed sites in the [Ln(W₅O₁₈)₂]⁹⁻ complexes are favorable to obtain Er(III)-based SIM. This can be realized by considering the distribution of the point charges around the 4f – ion [71]. In the framework of the point charge model one can use the following expressions for the CF parameters:

where R_i, θ_i and φ_i are the polar coordinates of the i^th ligand of the nearest surrounding of the lanthanide ion, −Z_ie is the charge of the i^th ligand, Y_p0(θ_i, φ_i) are the spherical harmonics, σ_p are the shielding parameters, finally c_p0(p = 2, 4, 6) are the following numerical factors:

Substituting the explicit expressions for spherical harmonics and taking into account that in present case θ₁ = θ₂ = θ₃ = θ₄ ≡ θ, θ₅ = θ₆ = θ₇ = θ₈ ≡ π ‒ θ and R₁ = R₂ = R₃ = R₄ = R₅ = R₆ = R₇ = R₈≡R one can obtain the following final expressions for the CF parameters:

These formulas show that the energies of the D_4d complex in the framework of the point charge CF are fully determined by the two structural factors, namely, by the distance R between the Ln(III) ion and the atoms of nearest ligand surrounding and on the polar angle θ. Alternatively, one can characterize the geometry of the complex by other two values, namely, by in-plane distance d_in, and the interplane distance d_pp (see Fig. 17). Simple geometrical consideration gives the following dependence between these two sets of values:

Fig. 17:

Two sets of parameters determining the geometrical structure of the D_4d complexes. Only one ligand plane is shown.

It is seen from eqs. (20) and (21) that providing undistorted square in antiprismatic geometry (din=dpp, cos(θ)=1/3) the parameterA20〈r2〉=0. For axially elongated sites (din < dpp, cos(θ)>1/3) we find that A20〈r2〉>0 and, hence, the term A20〈r2〉αO^20 tends to stabilize the doublet with large |M_J| value for lanthanides with negative Stevens coefficient α (i.e. for Tb and Dy, see Table 1) and the doublet with small |M_J| in case of ions with positive α (i.e. for Er ion). As the rule, the A20〈r2〉αO^20 contribution dominates except for the case of d_in = d_pp when A20〈r2〉 is small and the energy pattern is mainly determined by the contribution A40〈r4〉βO^40. Thus, we arrive at the conclusion that axial elongation is favourable for SIM behaviour of Tb and Dy complexes in agreement with what is observed for double-decker bis-(phthalocyaninato) complexes. In contrast, for axially compressed sites din>dpp, cos(θ) < 1/3, A20〈r2〉 < 0 and hence the SIM behaviour is expectable for ions with positive Stevens coefficient α (i.e. for the Er ion, Table 1). This takes place, for example, in the [Er(W₅O₁₈)₂]⁹⁻ complex. Note that the parameters A20〈r2〉, A40〈r4〉 and A60〈r6〉 do not depend on the skew angle φ and so the above arguments are also valid for the case of C₄ symmetry when φ≠45° (and φ≠0, 90°) and also for the case of cubic O_h symmetry when φ = 0 or 90°. Note that in the latter case A20〈r2〉=0 and so the cubic geometry (O_h symmetry) is less suitable for obtaining SIMs than the geometry of antiprism (D_4d symmetry). It is also notable that at φ≠45° the CF Hamiltonian includes along with the terms A20〈r2〉αO^20, A40〈r4〉βO^40 and A60〈r6〉βO^60 also two additional off-diagonal terms A44〈r4〉βO^44 and A64〈r6〉γO^64 which mix the M_J states with |ΔM_J  = 4 promoting thus quantum tunnelling of magnetization. The analysis based on the point charge model shows that the parameter A44〈r4〉 is vanishing only providing φ = 45° and reaches the maximal value for cubic geometry (φ = 0 or 90°), while the parameter A64〈r6〉=0 for D_4d and O_h symmetries.

The application of the point charge model for similar analysis of other typical environments, such as triangular dodecahedron and trigonal prism, has allowed to establish the following simple rules which are advantageous for the rational design of SIMs based on 4f-complexes [71]:

1. As a general rule, to form an energy barrier leading to slow spin relaxation the pattern of the Stark sublevels should exhibit ground state with high |M_J| and small mixing between the +M_J and −M_J components in order to reduce the relaxation through quantum tunnelling of magnetization. The most trivial condition to get a high-M_J ground-state doublet is to have a large J values which occur for the second half of the lanthanide series, with the Tb(III), Dy(III), Ho(III), Er(III) and Tm(III) complexes being the best choices.

2. The second condition for having high |M_J| ground state and the magnetization reversal barrier is the high symmetry of the complex. This can be achieved for complexes with a pseudoaxial symmetry like D_4d , C_5h , D_6d , or for any symmetry of order 7 or higher. The most suitable situation occurs when the second-order uniaxial anisotropy determined by the parameter A20〈r2〉α (also known as D) dominates. For example, comparing two high-symmetric octacoordinated complexes, one with the antiprismatic D_4d symmetry and another one with the cubic symmetry (A20〈r2〉=0), we could see that the O_h geometry is unfavorable to exhibit a large barrier for the magnetization reversal. In contrast, the systems with D_4d symmetry have strong uniaxial anisotropy (either positive or negative), which is provided by many SIMs.

3. In most cases the off-diagonal terms A44〈r4〉βO^44, A64〈r6〉γO^64 etc., allows quantum tunnelling of magnetization and they also effectively reduce the barrier. In some cases, however, such terms cannot lead to the quantum tunnelling in the ground state and thus they do not preclude from SIM behavior of the complex. Such situation is exemplified by the Ho(III) complex of D_2d symmetry. The calculated splitting diagram for the J = 8 ground state of Ho(III) in this environment evidences that the two components of the ground-state doublet are composed by the following M_J values: (+7, +3, −1, −5) and (−7, −3, +1,+5). It is seen that although these two functions are formed by an extensive mixture of different M_J states, they cannot participate in tunnelling because there is no overlap between these two sets of M_J values. Another way to avoid the destructive effect of quantum tunnelling is to use lanthanide ions with non-integer J – values [e.g. Dy(III) or Er(III) ions]. In this case the quantum tunnelling is forbidden by the Kramers theorem and becomes allowed (but rather weak) only through the hyperfine coupling of the 4f – electrons with nuclei possessing non-integer spin values, so that the total electron-nuclear angular momentum of the complex becomes integer.

Although the above discussed point change model of the CF is of great help in establishing the criteria for rational design of SIMs based on 4f-ions, strong precaution should be made if one is aimed to apply this approach to the description of the dc magnetic properties and especially to the analysis of the relaxation behaviour of SIMs. Indeed, this approach does not take into account the covalence effects. In order to avoid these limitations of the point charge model more sophisticated approaches are required, such as, for example, the exchange charge model of CF [72], [73], [74], [75] or ab initio calculations [76], [77].

In summary, one can say that the main requirement to obtain good SIMs based on lanthanide complexes is to have strongly axial CF. In view of this it has been recently suggested to consider Dy complexes with low coordination numbers (1 and 2) as good candidates to obtain SIMs with unprecedentedly high barriers (see examples in Ref. [76]), but such complexes are often not too stable. Fortunately, there is a possibility to obtain almost perfect axial CF with coordination number that is higher than one or two, when such CF is created by two negatively charged axial ligands, while the remaining equatorial ligands are nearly neutral. This has recently led to a series of dysprosium complexes [Dy(OPCy₃)₂(H₂O)₅]³⁺ (Cy = cyclohexyl) [77], [Dy(OP^tBu(NHⁱPr₂)₂)₂(H₂O)₅]³⁺ [78], [Dy(BIPM)₂]⁻, (BIPM = {C(PPh₂NSiMe₃)₂}²⁻) [79] and [Dy(bbpen)Br] (bbpen = N,N′-bis(2-hydroxybenzyl)-N,N′-bis(2-methylpyridyl)ethylenediamine) [80], for which very high effective barriers of U_eff = 543, 735, 813 and 1025 K, respectively, have been found. Finally, quite recently the complex [Dy(O^tBu)₂(py)₅][BPh₄] exhibiting the record barrier U_eff ≈ 1755 K (1220 cm⁻¹) has been reported [81]. This is a pentagonal bipyramidal complex of the type [DyX₂L₅]⁺, where L are the neutral and X are the anionic donors which induce almost perfect axial CF.

Field induced single-ion magnets with positive axial and strong rhombic anisotropy

In all examples so far discussed the presence of a considerable magnetization reversal barrier formed by uniaxial easy-axis anisotropy represents the key prerequisite for the creation of SMM. Recently a slow magnetic relaxation has been also discovered in some complexes of Kramers ions with dominant easy-plane magnetic anisotropy and strong rhombic anisotropy. Such unusual behavior takes place only in the presence of external magnetic dc field and so these complexes are often termed “field induced SIMs”. The majority of field induced SIMs represent the high-spin Co(II) complexes [61], [62], [63], [64], [65], [66], [67], [68]. In this Section we will briefly discuss one such example representing the recently reported Et₄N[Co^II(hfac)₃] (hfac = hexafluoroacetylacetonate) complex [68], whose structure is shown in Fig. 18a. The nearest ligand surrounding of the Co(II) ion formed by six oxygen atoms exhibits strong axial and rhombic distortions.

Fig. 18:

Molecular structure of Et₄N[Co^II(hfac)₃] (C – gray, F – light green) (a), and frequency dependences of the out of phase ac susceptibility measured at dc field B = 0.1 T and temperatures ranging from 1.8 to 3.3 K with increment of 0.1 K (b).

Dynamic ac magnetic susceptibility measurements revealed no frequency dependence of the in-phase (χ′M) and out-of-phase (χ″M) signals in the absence of an applied dc field. However, when the ac measurements were performed in the presence of a small external field of 0.1 T the complex showed typical SMM behaviour (Fig. 18b).

The magnetic properties of the Co(II) complex are often analyzed based on the following Griffith Hamiltonian that explicitly takes into account the unquenched orbital angular momentum of the Co(II) ion [82], [83], [84]:

This Hamiltonian operates within the basis of the ground octahedral ⁴T_1g – term of the Co(II) ion which represents the mixture of ⁴T_1g – terms arising from the terms ⁴F (ground) and ⁴P (excited) of the free Co(II) ion by the cubic component of the CF. The ⁴T_1g – term can be associated with the fictitious orbital angular momentum L = 1 and the spin S = 3/2. Factor 3/2 in SOC operator [first term in eq. (22)] and in Zeeman operator [last term in eq. (22)] appears due to the fact that the matrix of the orbital angular momentum operator L^defined in the ⁴T_1g (⁴F) basis differs by this factor from the matrix of L^defined in pure atomic ⁴P – basis. The orbital reduction factor κ describes both the covalence effect and the admixture of the excited ⁴T_1g(⁴P) term to the ground term ⁴T_1g (⁴F) by the cubic CF. The second term in eq. (22) describes the splitting of the ⁴T_1g – term into orbital singlet ⁴A_2g (M_L = 0) and the orbital doublet ⁴E_g(M_L = ±1) caused by the axial (tetragonal) distortion of the octahedron. Finally, the third term in eq. (22) is responsible for the further splitting of the tetragonal orbital doublet induced by the rhombic distortion of the octahedral surrounding of the Co(II) ion.

The sign of the axial CF parameter Δ_ax plays crucial role in the magnetic behavior of the Co(II) complex since it determines the sign of the magnetic anisotropy of the system. Thus, providing Δ_ax < 0 the ground term of the axially distorted complex proves to be an orbital doublet, and the system exhibits negative magnetic anisotropy χ_⊥−χ_|| < 0 which corresponds to the existence of an easy axis of the magnetization. In contrast, for positive Δ_ax the anisotropy is also positive which means that the system possesses an easy plane of magnetization. Note, however, that in the presence of strong rhombic CF the terms “easy axis” and “easy plane” are only of conditional character.

The main problem in fitting the dc magnetic data for the powder sample is that they are only slightly sensitive to the change of the sign of Δ_ax. An additional complication arises from the fact that the dc magnetic data prove to be almost independent of Δ_rh. As a result, the usage of the Hamiltonian, eq. (22), in which Δ_ax and Δ_rh are regarded as fitting parameters can give a multitude sets of the best fit parameters. Therefore, without additional independent information about the sign of the parameter Δ_ax, neither the adequacy of a model nor the correctness of these parameters can be tested. From this point of view the usage in addition to the dc magnetic measurement of complementary spectroscopic techniques, like EPR providing a direct access to the sign of the magnetic anisotropy (incorporated in the principal values of g-tensor for the ground Kramers doublet of the Co(II) ion) has been shown to be quite useful as well as the quantum-chemical evaluation of the parameters Δ_ax and Δ_rh.

The values Δ_ax = 428.29 cm⁻¹, Δ_rh = 90.34 cm⁻¹ obtained through the quantum chemical calculations [68] for the Et₄N[Co^II(hfac)₃] complex indicate the presence of non-uniaxial magnetic anisotropy with strong positive axial and significant rhombic contributions. These values of the CF parameters as well as the free-ion value λ = −180 cm⁻¹of the SOC parameter for the Co(II) ion have been used in Ref. [68] in both the evaluation of the dc magnetic properties and the analysis of EPR spectra. The only parameter has been considered as an adjustable one, namely, the orbital reduction factor κ. The best fit value of this parameter is κ = 0.72. It is remarkable that with the only fitting parameter one can reproduce quite well both the dc magnetic data and the effective g-tensor derived from the EPR spectra. Thus the calculated values g_ZZ ≈ 2.51, g_XX ≈ 4.01, g_YY ≈ 5.32 of the principal values of g-tensor for the ground Kramers doublet prove to be quite close to those (g_ZZ = 2.502, g_XX = 4.251, g_YY = 5.467) obtained from simulation of EPR spectra.

The energy levels of the Co(II) ion calculated as functions of the parameter Δ_ax for found values Δ_rh = 90.34 cm⁻¹, κ = 0.72 and λ = −180 cm⁻¹ are shown in Fig. 19. Such kind of plot represents a generalization of the well-known Griffith diagram [82] to the case of tri-axial symmetry. The vertical section marked by dashed red line corresponds to the found axial CF value Δ_ax = 428.29 cm⁻¹ and shows the evaluated energy spectrum consisting of six Kramers doublets. The two low-lying doublets arise from the spin-orbital splitting of the tetragonal ⁴A_2g -term (this splitting can be approximately described by the ZFS spin Hamiltonian DS^Z2 with positive D value), and the upper four doublets appear as a result of the spin-orbital splitting of the ⁴E_g-term. The first excited doublet is separated from the ground one by the energy gap of around 178 cm⁻¹, and the second excited doublet lies ≈ 463 cm⁻¹ above the ground one.

Fig. 19:

Dependences of the energy levels of the Co(II) ion on the parameter Δ_ax calculated with κ = 0.72, Δ_rh = 90.34 cm⁻¹ and λ = −180 cm⁻¹. Vertical section (red dashed line) corresponds to the value Δ_ax = 428.29 cm⁻¹. The central part of the plot shown by dashed lines corresponds to the area of the forbidden values of Δ_ax for which |Δ_ax| < 3Δ_rh|.

Let us briefly discuss the most ambiguous question concerning the origin of slow magnetic relaxation in Kramers ions exhibiting easy plane anisotropy. Several explanations of such slow relaxation have been recently proposed. One explanation is based on the assumption that this relaxation is a result of strong rhombic anisotropy, with the effective barrier for spin reversal being determined by the parameter E through the approximate relation U_eff ≈ 2| E |. This estimation sometimes gives the values of U_eff which are close to those found from the Arrhenius plot [61].

For other systems such estimation does not provide correct values of U_eff and so it has been proposed [85] that relaxation between the levels M_S = −1/2 and M_S = +1/2 is slowed by a phonon bottleneck and involves the Orbach process through the first excited doublet with M_S = ±3/2. In the framework of this picture the effective barrier is approximated by the energy gap between the ground (±1/2) and excited (±3/2) doublets, that is U_eff ≈ (D² + 3E²)^1/2. In some cases U_eff found in this way proves to be close to the barrier extracted from the Arrhenius plot but for other systems such explanation fails. Thus for the above considered complex Et₄N[Co^II(hfac)₃] the Arrhenius plot shows the presence of the barrier U_eff ≈ 19.5 cm⁻¹ [68] that is one order of magnitude smaller than the energy gap ( ≈ 178 cm⁻¹) between the ground and first excited Kramers doublets. So one should exclude the Orbach process from the consideration and consider another possibility to reproduce the temperature dependence of the relaxation time by taking into account one-phonon direct processes (dominating in the low temperature region) in combination with the two-phonon Raman processes (important at higher temperatures).

The most comprehensive and non-equivocal explanation of the origin of slow magnetic relaxation in Kramers complexes with an easy plane anisotropy has been given in Ref. [62]. It has been shown that the mentioned slow relaxation is a general consequence of time-reversal symmetry (van Vleck cancellation) that hinders direct spin–phonon transitions between the sublevels of the ground Kramers doublet. The hyperfine interaction with the I = 7/2 nuclear spin breaks time reversal symmetry and opens channels for direct spin–phonon relaxation. The relaxation between the Kramers-conjugate states can also occur through Orbach and Raman processes albeit, as we have seen, the Orbach process is often irrelevant because of too large energy gap between the ground and excited Kramers doublets. At the same time the hyperfine interaction gives rise to a quantum tunneling that masks the relaxation phenomenon at zero dc field.

To summarize, one can say that the three main prerequisites to get slow relaxation in such kind of complexes are [62]: (i) half-integer spin, (ii) strong magnetic anisotropy, (iii) minimized hyperfine interaction, that is, the usage of Kramers ions having stable isotopes with zero nuclear spin.