Magnetic Exchange in Metal Clusters: Mathematical Modelling and Physical Results

Magnetic Exchange in Metal Clusters: Mathematical Modelling and Physical Results



J.J.Borras-Almenar1, J.M.Clemente-Juan1, E.Coronado1, A.V.Palii1,2, B.S.Tsukerblat3

1Instituto de Ciencia Molecular,Universidad de Valencia,

46100 Burjassot (Valencia) Spain;

1,2Institute of Applied Physics of the Academy of Sciences of Moldova,

2028 Kishinev, Moldova;

3Chemistry, Ben-Gurion University of the Negev,Beer-Sheva 84105,Israel

In this article we present a new conception of the magnetic exchange and double exchange in clusters containing orbitally degenerate metal ions. The main theoretical tools for the study of polynuclear clusters (Heisenberg- Dirac- Van-Vleck model of exchange and Anderson’s model of the double exchange ) are invalid as applied to the orbitally degenerate systems. As distinguished from the situation in spin-clusters where the exchange interaction is predominating isotropic, the orbital degeneracy of the constituent metals is shown to produce new kinds of the interionic interactions resulting in an anomalously high magnetic anisotropy in these systems. Using the developed effective Hamiltonian approach combined with the technique of the irreducible tensor operators and efficient computer programs we have analyzed orbitally dependent magnetic exchange in localized systems and the double exchange in mixed-valence metal clusters. This study leads to a new concept of “anisotropic double exchange”. The energy pattern as well as the character of the magnetic anisotropy is closely related to the ground terms of the ions in crystal field, electron transfer pathways and overall symmetry of the systems being affected also by spin –orbital interaction and vibronic coupling ( Jahn-Teller and pseudo Jahn-Teller effects). Special attention is paid on the origin of the magnetic anisotropy in the face-shared binuclear unit [Ti2Cl9]3-. The theory was applied to the study of the ferromagnetic spin alignment in the heterobinuclear Cr(III)Fe(II) cyano-bridged dimer.


Finite molecular clusters of exchange coupled ions are currently important for many areas of research such as solid state chemistry, magnetism and biochemistry. They provide new possibilities in design of nanometer size magnets possessing the unusual magnetic properties of paramagnetic-like behavior and quantum tunneling of magnetization (1,2). The interplay between the electron delocalization (double exchange) and magnetic interactions play a crucial role in the properties of many mixed valence (MV) compounds of current interest in solid state materials science. Till now the the existed models of the magnetic interactions took into consideration only orbitally non-degenerate terms of the interacting ions. Anderson and Hasegawa (3) proposed the theory of the double exchange and the usually accepted model for the magnetic exchange in the low-dimensional and extended materials was based on the Heisenberg-Dirac-Van Vleck (HDVV) model (4). In our papers (5-6) we developed the theories of the magnetic exchange and the double exchange in the materials containing orbitally degenerate ions in the high-symmetric crystal fields. Hereafter we discuss the main approaches and the role of the orbital degeneracy in the problem of the exchange paying attention on the main physical manifestations.


The theory of the double exchange proposed by Anderson and Hasegawa (3) is essentially based on the assumption that the interacting metal ions (in both oxidation degrees) are orbitally non-degenerate (spin systems). The main conclusion of the theory is that the delocalization of the extra electron over two spin cores produces a linear spin dependence of the double exchange splitting resulting thus in the ferromagnetic ground state of the dimer. At the same time in these spin systems the double exchange is magnetically isotropic. However, in a variety of compounds the metal ions possess orbitally degenerate ground states. In this case the conventional theory of the double exchange proves to be inapplicable.

The electron transfer (double exchange) Hamiltonian can be presented in a conventional way as:


where creates an electron on the orbital of site A with the spin projection and annihilates an electron on the orbital of site B; are the intersite one-electron transfer integrals. In eq.(1) all relevant transfer pathways are included. We will consider only transfer processes with participation of orbitals, so denote the cubic basis related to the axes of sites. The pairs and are considered in three high-symmetric topologies: edge-shared , corner-shared , and face shared bioctahedra. Figure 1 shows the effective overlap of the d- orbitals giving rise to the most efficient transfer pathways. The T-P isomorphism (6) allows us to assign the T1(2) basis to the P-states (L=1).The calculation of the matrix of the double exchange operator gives the following result:


where is the reduced matrix element of the creation operator that

can be considered as an irreducible tensor of rank 1(m,m’=0,+1,-1) in the orbital subspace and as that of rank ½ in spin space, are the transfer parameters in the angular momentum representation (5). for , and for -ions, with for orbital triplets and for orbital singlets. One can see that


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The overlap patterns related to the most efficient transfer pathways: (a) D2h, (b) D4h, (c) D3h.


Energy diagram for the 3T1()-4A2() MV dimers: (a) D2h, D3h; and (b) D4h. A short notation is used.


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the matrix of the double exchange is diagonal with respect to the spin quantum numbers . Thematrix element of the double exchange proves to be proportional to (this dependence is contained in the 6j-symbol in eq.(2)) as well as in the case of non-degenerate ions (spin systems). The matrix elements of the double exchange for orbitally degenerate dimers depend also on the orbital quantum numbers . This dependence produces a magnetic anisotropy in the orbital subsystem. The character of this anisotropy is closely related to the set of transfer integrals in eq.(2), reflecting both the point symmetry of the dimer and the specific choice of physically significant transfer pathways. Strong magnetic anisotropy of the degenerate double exchange systems is to be considered as the main physical consequence of this dependence.

Figure 2 shows the energy splitting for a pair with different overall symmetries. Providing symmetries (Fig.2a) the energy pattern involves three pairs (+ and -) of levels with S=1/2, 3/2, 5/2; the energies are . All these levels correspond to . The spectrum contains also one highly degenerate level at . This level comprises states with all S values, each belonging to . In the case of symmetry (Fig.2b) we meet the reverse situation. The state with involves all S-values and corresponds to , while all the states with the energies possess . One can see that the system exhibit strong magnetic anisotropy with the C4 easy axis of magnetization meanwhile systems are also anisotropic but possess only Van Vleck-type paramagnetism in the ground state. More complicated cases of the “anisotropic double exchange”
are considered in refs (6,7).


Suppression of the magnetic anisotropy by the PKS vibration as

illustrated by the singlet-triplet pair in the D4h system.

The vibronic interaction (pseudo Jahn-Teller effect (8,9)) in MV compounds is usually strong coupled to the electronic motion. In order to illustrate (at least qualitatively) the influence of the vibronic interaction we employ the Piepho-Krauzs-Schatz (PKS) model (9,10) dealing with the so called out-of-phase mode q ( this mode is constructed from the breathing modes of the octahedral subunits). The main effect of the vibronic interaction is illustrated in Fig.3 where the adiabatic potentials of a singlet-triplet pair are depicted. The vibronic interaction is operative within the sets of states with a given full spin, Fig.3 shows the selected S=5/2 levels. One can see that the gap t between the levels with S=5/2 and ML =0 and |ML|=1 is strongly decreased in a deep minima of the lower sheet of the adiabatic potential. Thus the effect of the localization of the extra electron is accompanied by the reduction of the anisotropy induced by the double exchange in the orbitally degenerate system.

In the case of orbital degeneracy of the constituent ions, the isotropic spin-Hamiltonian of the magnetic exchange HDVV model becomes invalid even as a zeroth order approximation. In our recent papers (11,12) we proposed a new approach to the problem of the kinetic exchange between orbitally degenerate multielectron transition metal ions. Our consideration takes into account explicitly complex energy spectrum of charge transfer crystal field states exhibited by the Tanabe-Sugano diagrams. Taking advantage from the symmetry arguments we have deduced the effective exchange Hamiltonian in its general form for an arbitrary overall symmetry of the dimer taking into account all relevant electron transfer pathways. The effective Hamiltonian was constructed in terms of spin-operators and standard orbital operators (cubic irreducible tensors). All parameters of the Hamiltonian incorporate physical characteristics of the magnetic ions in their crystal surroundings. In fact, they are expressed in terms of the relevant (in a given overall symmetry) transfer integrals and crystal field and Racah parameters for the constituent ions.

Along with the isotropic spin-spin interactions the effective Hamiltonian in the case of orbital degeneracy contains terms like (orbital matrices) and mixed terms like containing both types of operators. All these operators can be expressed in terms of the irreducible tensor operators acting in the orbital and spin subspaces. Then the effective Hamiltonian can be represented as a linear combination of the irreducible products. The last step of the mathematical procedure involves decoupling of these products and the calculation of the eigenvectors and energy levels (11). Symmetry properties of the effective Hamiltonian are studied in ref (12).

The results can be illustrated by the application of the developed approach to the binuclear unit [Ti2Cl9] 3 in Cs3Ti2Cl9 that represents a face shared cluster with D3h overall symmetry. Figure 1c shows the most important electron transfer pathway . The energy levels are obtained as the functions of the ratio , where is associated with the e-orbitals in the trigonal symmetry. The model also takes into account local trigonal crystal field (parameter ) and also spin-orbital coupling. Figure 4 shows that the calculated magnetic susceptibility is in a good agreement with the experimental data (13). In the agreement with the experimental data the system exhibits the magnetic anisotropy arising from the orbitally dependent exchange interaction.


Magnetic properties of 3D bimetallic cyano-bridged systems {M(II)3[Cr(III)(CN)6]2}.13H2O (M=Co, Fe,Ni) attract considerable interest as molecule based magnets with promising application and represent interesting systems for testing different models of the exchange interactions. The early papers devoted to these kind of bimetallic systems were based on the theoretical model of the potential exchange interaction dealing with the orbitally nondegenerate localized orbitals (4). This model predicts short range ferromagnetic interaction through a strict orthogonality of the magnetic orbitals in a bimetallic system. Using the strategy based on this theoretical background M.Verdaguer and coworkers performed a rational design of high Tc 3D ferromagnetic and ferrimagnetic cyanide- bridged systems (14). During last decade this successful route allowed to raise Tc from 5.5K in Prussian blues to 315K (14) creating thus the first room-temperature molecule based magnet.

The further progress in this area led to the revealing of the ferromagnetic properties of two bimetallic cyanide-bridged systems


Magnetic behavior of the [Ti2Cl2]-3 unit: comparison with the theoretical curve (solid line) calculated at te/ta=-0.154, ta=52028 cm-1, =-320 cm-1, =155 cm-1 and orbital reduction k=0.71. (Inset) Temperature dependence of the degree of the anisotropy, compared with the theoretical curve (solid line).

with (Tc=16K and 23K). This observation turned out to be in a sharp contrast to the orthogonality rules in the localized model of potential exchange and motivated the development of the model of kinetic exchange that takes into account the effect of electron delocalization. In our paper (15) we have extended the Anderson’s theory of the double exchange to the case of a heterobimetallic pair. The delocalization was supposed to occur over two non-equivalent spin cores comprising nondegenerate orbitals. Second order effect of the double exchange is proved to give rise to a ferromagnetic kinetic exchange in the system under consideration. Although the proposed orbital model (10) provided an explanation of the ferromagnetic spin alignment the problem of the kinetic exchange in the named bimetallic compounds is far from being completed and requires more general consideration with due account of the orbital degeneracy of the ground terms of Fe(II) and Co(II) ions.

In the orbital angular momentum representation the kinetic exchange Hamiltonian consists of two commuting parts arising from the second order processes connecting the ground state with the high spin and low spin CT states:


The full Hamiltonian proves to be isotropic in spin subspace and axially symmetric in orbital subspace, so that and are the good quantum numbers.

One can easily find two sets of sublevels corresponding to and :



As one can see from Eqs. (4) the energy pattern of Cr(III)Fe(II) contains two superimposed groups of the energy levels (with and ) each obeying the Lande’s rule with the total spin of the pair taking the values .

One can see that the Heisenberg-like energy pattern with is antiferromagnetic due to the mixing with the low spin CT states while the levels with form the ferromagnetic pattern appearing as a result of the competition between the contributions of high and low spin CT states. These results are in agreement with the theory of superexchange proposed by Anderson (16) and Goodenough and Kanamori (see (17) and refs. therein). It is to be underlined that along with the full spin quantum number the energy levels are enumerated by the quantum numbers . The dependence on reflects the magnetic anisotropy of the whole system arising from the non-vanishing orbital angular momentum of ion in its ground state. It is remarkable that the first order axial orbital magnetic contribution exists only in one of two subsets of the levels, namely in the ferromagnetic pattern for which . This contribution is axial with the axis of the bioctahedron being the easy axis of magnetization. On the contrary, the antiferromagnetic pattern is expected to possess only the second order orbital magnetism (Van Vleck paramagnetism). One might say that although energy subsets obey the Lande’s rule they can not be considered as the Heisenberg schemes due to strong orbital contributions to the magnetic moments.

Regarding the spin alignnement it is to be inquired whether the overall effect of the kinetic exchange is ferro- or antiferromagnetic. The lowest energy level with |ML|=1 belongs to the maximum spin S=7/2 and one finds that ; the lowest level with ML=0 is that possessing the minimum spin for which Comparing these energies one finds the inequality that assures the ferromagnetic ground state:

As distinguished from the finite clusters, the treatment of the extended systems requires some approximate approaches (for example, mean field approximation) Nevertheless, our conclusion about the nature of the ferromagnetic ground state of the bimetallic Cr(III)Fe(II) compounds undoubtedly remains valid for 3D-systems. Some results essentially based on the point symmetry are applicable to the pair of ions only (axial magnetic anisotropy). At the same time it is to be noted that the orbitally dependent (anisotropic) interactions of the effective Hamiltonian are of primary importance for both dimers and 3D-systems. In the last case they are responsible for the cooperative phenomena in solids.


Financial support of INTAS is highly appreciated (Grant INTAS 00-0651). Authors indebted to the Supreme Council on Science and Technological Development of Moldova for the financial support (grant 111). We thank Oleg Reu for his kind help in the computer artwork.


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