Spin ladder

Zener appears to have been the first to consider this problem to some depth in his theoretical work on ferromagnetic crystals of the type l.a. Ca.MnO, (Zener 1951). For x = 0 one has LainMnin03 but for x 0 some of the Mn will be 4+, and so we have the structure I. a / a Mn (Mn CL in which some Mn-Mn pairs will be mixed valence, that is, MiF Mnj) or MnJ) Mn . Mnm is 3d4 (S = 2) and MnIV is 3d3 (S = 3/2), and Zener proposed that the excess electron (also called itinerant electron or Zener electron) on Mn111 can travel to the MnIV via a doubly-occupied p-orbital of 

Schematically, the double exchange interaction can be written in the form of a spin Hamiltonian operator (working on the system spin AB —for short, S) as 

And the combination of Heisenberg superexchange plus Zener double exchange results in zero-field energy levels in terms of the system spin S 

The first signs of this spin richness are seen in the magnetism of the trinuclear iron-sulfur cluster [3Fe-4S], a distorted cube of alternating Fe and S comers from which one Fe has been removed. In the fully oxidized state, [3Fe-4S]1+ all three iron 

FIGURE 11.7 Spin ladders for the dinuclear FeIII-Fe11 cluster. The two metal ions are subject to superexchange (7) and double exchange (B) with J B,J B, or J = B/4. The three ladders are not normalized to the same energy scale. 

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The two iron ions of the Rieske cluster are antiferromagnetically coupled therefore, the ground state has a spin S = while excited states of the spin ladder S = I, i, I, and, are at energies -3J, 8J,... 

Concerning molecule-based magnets, the first spin-ladder was synthesized (/ -EPYNN)[Ni (dmit)2] (/ -EPYNN = / -7V-ethylpyridinium o-nitronyl nitroxide). Within the crystal lattice, the radical cation /7-EPYNN units are arranged in ID chains with ferromagnetic interactions. The chains of [Ni(dmit)2] moieties in the ladder formation exhibit coexistent antiferromagnetic interactions.1031,1032... 

A deeper analysis [80] on this Cu - Dy chain showed that NN and NNN interactions are efficient above Tc and that inter spin-ladder ferromagnetic interaction is observed in this family of compounds [79-82]. Attempts were made to use the 3d tectons with other lanthanide precursors, such as hexanuclear lanthanide clusters. Regular chains were obtained with Dy-Cu ferromagnetic interaction but their close packing prevents observation of SCM behaviour [83]. 

Radicals have been known for many years to form organic paramagnetic materials with numerous magnetic properties (ferro- or ferri-magnetism, spin Peierls transition, spin frustration, spin ladder systems) (see [51-60] for verdazyl radicals, [61-68] for thiazyl radicals, [69] for nitronyl nitroxide and [70-78] for Tempo radicals) (Fig. 6). When they are in their cationic form, they are valuable candidates for an association with the M(dmit)2 systems they will then provide the magnetic properties thanks to their free electron(s), whereas the M(dmit)2 moieties will provide the electrical properties. 

In contrast, within (p-EPYNN)[Ni(dmit)2], first synthesized in 1996 [79], it has been proven that spin-ladder chains of the Ni(dmit)2 moiety coexist with the ferromagnetic one-dimensional chain of the p-EPYNN radical cation. Spin-ladders are of interest because of their potential applications in the area of quantum magnets and because it has been predicted that holes doped into even-leg ladders may pair and possibly superconduct [90-92]. 

Very often, however, these dyads are not isolated in the solid state and interact with neighboring ones at least along one preferential direction. In this case, we can distinguish two important situations the alternated spin chain and the spin ladder. As shown in Scheme 5, the alternated spin chain is characterized with two different magnetic interactions, noted / and aJ with 0 < a < 1. Note that if a = 0, one recovers the singlet-triplet behavior while if a = 1, we are in the presence of a uniform spin chain. The spin ladder is also characterized by two J values, noted J// and J in the following. 

There is a fundamental difference between the uniform spin chain on one hand and the alternated spin chain or spin ladder on the other. Indeed, in the latter cases, the ground state is the singlet state and the susceptibility thus goes to zero at the lowest temperatures, with an activated part of the susceptibility between 0 and T (Xmax)- On the other hand, as shown by Bonner and Fisher [65], in the uniform spin... 

Scheme 5 Models of the spin chain (left) and the spin ladder (right). The black dots represent the spin carriers...

Table 1 Magnetic characteristics of the spin ladders in [Cp2Mo(dmid)][TCNQF4] and [Cp2W (dmid)][TCNQF4] ...

Finally, one other example of a spin ladder has been reported within these extensive series of [Cp2M(dt)]+ salts, in the AsFfi salt of [Cp2W(dsit)]+ [34]. We will see below that the other AsFfi (Sects. 3.3.1 and 3.3.2) salts of [Cp2Mo... 

We have seen above several examples where [Cp2M(dt)]+ (M = Mo, W) complexes organize in the solid state into low dimensional structures, leading to characteristic magnetic behaviors such as spin chains (eventually alternated) or spin ladders. The extensive use in later years of dithiolene ligands such as dmit or dddt was aimed at... 

Dinuclear clusters ferrous site distortion, 38 175 spin ladder, 38 182-183 Dinuclear cobalt complex, 45 291-293 Dinuclear complexes osmium, electrochemistry, 37 321-323 quadruply bridged, 40 187-235 axial ligand substitution properties, 40 232-234... 

Spin ladders, 38 180-183 Spin-lattice relaxation, 13 205 iron-sulfur centers, 47 486-487 dinuclear clusters, 47 435-436 HiPlP, 47 447... 

The fully ionic solids (region I) afforded band insulators, 1 1 Mott insulators with ground states of antiferromagnets (E b(21) and F b(22) in Fig. 1) or spin-Peierls systems, ferroelectrics, ferromagnets, spin-ladders, and nonlinear transport materials (switching and memory). 

The RVA, presented here, is a variational method invented recently in the context of spin-ladders [22], It is a method very closely related to the DMRG scheme with the main differences coming from the fact that only one state is retained as the best candidate for the ground state in the RVA on the contrary to the DMRG which considered much more states [21], Most often, the results obtained with it are less accurate than the DMRG ones, but it is much easier to get a physical insight into the problem the... 

A spin ladder is an array of coupled spin chains. The horizontal chains are called the legs, the vertical ones, rungs. In the case of spin one-half antiferromagnet spin-ladders, these systems show a. remarkable behaviour in function of the number of leg there is a gap in the excitation spectrum of even-leg ladders and, on the contrary, no gap in the excitation spectrum of odd-leg ladders. In terms of correlation lengths, this means that there is short (long) -range spin correlation in even (odd) -legladder (see [24] for a review). 

With this classification of states we may write the following generalized Dimer-RVB state for a two-leg spin ladder,... 

Fig. 27 Magnetic heat capacity for PhBABI for 7 < 100 K showing variation with external magnetic field (left) zero-field magnetic heat capacity showing fits (right) to ID AFM chain, 2D AFM square planar, 2D AFM square planar bilayer, singlet-triplet spin pairing (ST), and spin ladder models.

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