Magnetic relaxation in cyanide based single molecule magnets
Introduction
The most fascinating developments of the last decade in the field of molecule-based magnetism involve the discovery and characterization of single molecule magnets (SMMs) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. The majority of known SMMs contains ions with orbitally non-degenerate ground states. For SMMs of this type the energy barrier for magnetization reversal appears as a result of the combination of the large spin S of the ground state and a significant negative zero-field splitting parameter DS. In an attempt to increase both the energy barrier for magnetization reversal and the lifetime of magnetization researchers have turned to new types of SMMs [11], [12], [13] that contain ions with unquenched orbital angular momenta in the ground state. Recently, we demonstrated [14], [15], [16] that in the family of new cyano-bridged pentanuclear Mn(III)2Mn(II)3 clusters representing SMMs of this kind, the first-order single ion anisotropy and the anisotropy of exchange interaction are responsible for the formation of the barrier for reversal of magnetization. As for the lifetime of magnetization for both types of SMMs containing orbitally degenerate and orbitally non-degenerate ions the state of theory is very scarce. However, the unusually slow magnetic relaxation and the resultant magnetic bistability are of primary importance for the SMM behavior. As a rule the relaxation of single molecule magnets is described with the aid of the Arrhenius law:and the height of the barrier for magnetization reversal U is obtained from the temperature dependence of the relaxation time. Meanwhile, from the theory of non-radiative transitions it is well known that the activation law for the relaxation time is only valid for extremely high temperatures [17] at which the SMM properties of all known such type clusters disappear. On the other hand, the commonly accepted phenomenological description of the relaxation time of magnetization in SMMs does not answer the question what is the mechanism of magnetic relaxation. To our knowledge the only attempt to understand this mechanism was made in the paper of Sessoli and coauthors [18]. However, this examination concerns SMMs with orbitally non-degenerate ions and only contains an approximate estimate of the relaxation time. For SMMs containing ions with unquenched orbital angular momenta a microscopic model of magnetic relaxation has never been discussed. At the same time in view of practical applications the problem of relaxation in SMMs necessitates special attention. Actually, the relaxation of magnetization in existing magnetic clusters with SMM properties is still very fast. For instance, for the Mn12Ac cluster having a barrier of about 61 K the relaxation time at T = 2 K is about 1.4 months [18]. Meanwhile, a relaxation time acceptable for applications should be at least 4.7 × 108 s = 15 years at room temperature. The aim of the present work is the elaboration of the model of magnetic relaxation in the cyano-bridged pentanuclear Mn(III)2Mn(II)3 clusters and elucidation of the conditions for the observation of long relaxation times in SMMs containing ions with unquenched orbital angular momenta.
Section snippets
Electronic Hamiltonian of the Mn5–cyanide cluster
The model for the interpretation of the observed temperature dependence of the dc magnetic susceptibility has been considered in detail in our papers [14], [15], so here we briefly describe only the Hamiltonian of the system. The molecular structure of the Mn5–cyanide cluster is shown in Fig. 1. The metal skeleton represents a trigonal bipyramid containing two Mn(III) ions (1 and 2) in the apical positions and three Mn(II) ions (3, 4, and 5) in the equatorial positions. Each Mn(III) ion is in
Barrier for reversal of magnetization
The sign of the local magnetic anisotropy is determined by the sign of the trigonal component of the crystal field. Providing relatively strong positive trigonal field (Δ > 0), each Mn(III) behaves as a spin s = 1 ion with the quenched (to a second order) orbital angular momentum. So in this case the Mn5–cyanide cluster can be considered as a spin-system containing two spins s1 = s2 = 1 and three spins s3 = s4 = s5 = 5/2. Since the exchange interaction is antiferromagnetic the spin of the ground state of the
Hamiltonian of electron–phonon interaction
The evaluation of the probabilities of non-radiative transitions that facilitate magnetic relaxation requires a model of electron–phonon interaction Hamiltonian and phonon dispersion law. For transition metal ions being an example of a small-radius center this Hamiltonian is based on the ligand-field theory. We employ a quasi-molecular (cluster) model that considers the impurity center as a complex formed by the central ion and the adjacent ions of the lattice. The displacements of the ligands
Probabilities of one-phonon transitions. Cascade responsible for relaxation in the Mn5–cyanide clusters
Since the spectrum of low-lying electronic states of the Mn5 cyanide cluster contains very close in energy levels (Fig. 2a), we assume that direct one-phonon transitions between these levels are responsible for the relaxation of magnetization. This effect takes place in the presence of resonance frequencies in the phonon spectrum. The probability of one-phonon p → s transition per second is expressed as in [25]here is the
Time and temperature dependence of the populations of the MJ-states
The time and temperature dependence of the populations of the cluster states was obtained from the solution of the set of master equations. For each state there were taken into account all transitions populating and depopulating this state. The probabilities of these transitions have been calculated with aid of the expressions above listed. When solving the set of master equations it was assumed that a very small constant magnetic field is applied to the system so as at the initial moment
Acknowledgments
The research described in this publication was made possible in part by Award No. MOC2-2611-CH-04 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF) and Award No. MTFP-04-07 Follow-On-Award of the Moldovan Research and Development Association (MRDA) and CRDF. Financial support of the Supreme Council for Science and Technological Development of Moldova is also appreciated. KRD gratefully acknowledges the support of the Department
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