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Spin-Fluctuation Theory of Cluster Magnetism

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Metal Clusters and Nanoalloys

Part of the book series: Nanostructure Science and Technology ((NST))

Abstract

Recent developments in the theory of the magnetic properties of transition-metal clusters at finite temperatures are reviewed. After a brief introductory overview of the experimental situation, we present the theoretical framework, which is based on a Hubbard–Stratonovich functional-integral formulation of the corresponding canonical and grand-canonical equilibrium partition functions. The theory is applied to Fe, Co, and Ni clusters by using the static approximation to the functional integration and by solving the underlying single-particle electronic structure by means of a real-space recursive expansion of the local Greens functions. First, we formally recover the T = 0 limit of the theory and with it the most significant ground-state properties of small magnetic clusters. Second, we consider the low-temperature limit of the local spin-fluctuation energies ΔF l (ξ) at different atoms l, as a function of the local exchange fields ξ. Representative results for the size, structural, and local-environment dependence of ΔF l (ξ) in Fe N and Ni N clusters are discussed. The interplay between fluctuations of the module and of the relative orientation of the local magnetic moments is analyzed. The strong dependence of the spin-excitation spectrum on the local atomic environment and on interatomic bond-length relaxations is demonstrated. Third, we derive a simple relation between the low- and high-temperature values of the cluster magnetization per atom, which takes into account the effects of short-range magnetic order (SRMO). Using this relation, and by comparison with experiment, an important degree of SRMO in the clusters is inferred even at relatively large T. Finally, the temperature dependence of the magnetic properties of Fe N clusters having for N < 25 atoms is discussed. To this aim, the statistical averages in the static approximation are performed rigorously by using a parallel-tempering Monte Carlo sampling of all exchange-field configurations \(({\xi }_{1},\ldots {\xi }_{N})\). In this way, the interplay between different local environments, the possibility of SRMO, and the size-dependent electronic structure are taken into account on the same footing. Representative results for the average magnetic moment per atom, the local magnetizations, and nearest neighbor spin-correlation functions are shown. A remarkable dependence on size and structure is revealed, which reflects the importance of the electronic structure to the cluster spin excitations. The correlation between local atomic environment and finite T magnetism is analyzed in some detail by means of the spin-correlation functions. The role of bond-length relaxations on the temperature-dependent properties is quantified. An interpretation of our electronic results in terms of Ising or Heisenberg models of localized magnetism implies a strong dependence of the effective interatomic exchange couplings on size and local coordination number, which defies straightforward transferability and easy generalizations. Finally, we conclude with a glimpse into future research directions.

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Notes

  1. 1.

    As usual, μB = eℏ ∕ 2mc = 5. 79 ×10 − 5 eV/T stands for the Bohr magneton.

  2. 2.

    For the sake of clarity, a hat (ˆ) is used to distinguish operators from numbers.

  3. 3.

    Recent applications of the mean-field approach to TM clusters and nanostructures at T = 0 may be found, for example, in [29] and [61–63], and in references therein.

  4. 4.

    As in an N-step random walk the root mean square average of the total magnetic moment per atom of a cluster having N uncorrelated local moments of size μ0 is \({\mu }_{0}/\sqrt{N}\).

  5. 5.

    From Fig. 5.10 it is possible to infer the distance dependence of cluster “Curie” T C(N). Despite changes as a function of size and structure, T C(N) always remains of the same order of magnitude as in the bulk. The actual value of T C is the result of an interplay between two effects directly related to the reduction of the coordination numbers: the enhancement of local magnetic moments and the reduction of the number of NN couplings. For some clusters, this may lead to incidental compensations and to values of T C(N) close to the bulk one. However, in most cases one of the contributions dominates over the other (see Fig. 5.10).

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Acknowledgements

This work was supported in part by CONACyT-Mexico (grant No. 62292), by the Deutsche Forschungsgemeinschaft, and by the Mexico-Germany exchange program PROALMEX. The authors (RGA and JDD) would like to acknowledge the kind hospitality and support of the University of Kassel.

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Authors and Affiliations

  1. Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Coahuila, CU Camporredondo, 25000, Saltillo, Mexico

    R. Garibay-Alonso

  2. Instituto de Física, Universidad Autónoma de San Luis Potosí, Alvaro Obregón 64, 78000, San Luis Potosí, Mexico

    J. Dorantes-Dávila

  3. Institut für Theoretische Physik, Universität Kassel, Heinrich Plett Straße 40, 34132, Kassel, Germany

    G. M. Pastor

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  1. R. Garibay-Alonso
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  3. G. M. Pastor
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Correspondence to G. M. Pastor .

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Garibay-Alonso, R., Dorantes-Dávila, J., Pastor, G.M. (2013). Spin-Fluctuation Theory of Cluster Magnetism. In: Metal Clusters and Nanoalloys. Nanostructure Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3643-0_5

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