Magnetic relaxation in cyanide based single molecule magnets

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Abstract

The present contribution is aimed at the elaboration of the model of magnetic relaxation in the cyano-bridged pentanuclear Mn(III)2Mn(II)3 cluster that belongs to a new family of single molecule magnets (SMM) containing ions with unquenched orbital angular momenta. We proceed from the energy pattern of the cluster formed by the trigonal component of the crystal field acting on the ground-state cubic terms 4T1(t24) of the Mn(III)-ions, spin–orbital interaction and Heisenberg exchange between Mn(II) and Mn(III) ions. The ground state of the cluster possesses the total angular momentum projection ∣MJ = 15/2, while the energies of the excited states increase with decreasing ∣MJ∣ values, thus giving rise to a barrier for the reversal of magnetization. The monophonon transitions between the states ∣MJ〉 and ∣MJ ± 1〉, ∣MJ ± 2〉 induced by electron–vibrational interaction are shown to be allowed. The rates of all possible transitions between the states with 1/2 < MJ < 15/2 are calculated in the temperature range 0.1 K < T < 3 K. With the purpose of calculation of the temperature dependence of the relaxation time of magnetization we solve the set of master equations for the populations nMJ(t) of the ∣MJ〉 states of the Mn(III)2Mn(II)3 clusters. The relaxation time is shown to diminish from 1012s to 10 s with decrease in temperature from 1 K to 3 K for the cluster [Mn(III)(CN)6]2[Mn(II)(tmphen)2]3 (tmphen = 3,4,7,8-tetramethyl-1,10-phenanthroline) with the trigonal crystal field parameter Δ = −251 cm−1. The obtained values of the relaxation time are in qualitative agreement with the temperature dependence of the ac susceptibilities observed for this SMM. In order to reveal the possibility of enhancing the relaxation time of magnetization in the family of clusters Mn(III)2Mn(II)3 we vary the trigonal crystal field parameter ∣Δ∣(Δ < 0) and demonstrate that increase in ∣Δ∣ leads to a considerable growth of the relaxation time.

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:τ(T)=τ0expUkTand 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]Wps=2π2[p]κνs|s|Vκν|p|2n¯(ωκν)n¯(ωκν)+1δωκν-|Δsp|here n¯(ωκν) is the

Time and temperature dependence of the populations of the MJ-states

The time and temperature dependence of the populations nMJ 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

References (26)

  • B.Z. Malkin

    Crystal field and electron–phonon interaction in rare-earth ionic paramagnets

  • E. Clementi et al.

    At. Data Nucl. Data Tables

    (1974)
  • D. Gatteschi et al.

    Angew. Chem. Int. Ed.

    (2003)
  • D.D. Awschalom et al.

    Phys. Today

    (1995)
  • R. Sessoli et al.

    J. Am. Chem. Soc.

    (1993)
  • R. Sessoli et al.

    Nature

    (1993)
  • H.J. Eppley et al.

    J. Am. Chem. Soc.

    (1995)
  • S.M.J. Aubin et al.

    Inorg. Chem.

    (1999)
  • M. Soler et al.

    Chem. Commun.

    (2000)
  • A. Müller et al.

    Chem. Phys. Chem.

    (2001)
  • A.J. Tasiopoulos et al.

    Angew. Chem. Int. Ed.

    (2004)
  • J.J. Sokol et al.

    J. Am. Chem. Soc.

    (2002)
  • H.J. Choi et al.

    Inorg. Chem.

    (2004)
  • Cited by (0)

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