Magnetite thin films: A simulational approach

doi:10.1016/j.physb.2006.05.268

Physica B: Condensed Matter

Volume 384, Issues 1–2, 1 October 2006, Pages 224-226

https://doi.org/10.1016/j.physb.2006.05.268 Get rights and content

Abstract

In the present work the study of the magnetic properties of magnetite thin films is addressed by means of the Monte Carlo method and the Ising model. We simulate L×L×d magnetite thin films (d being the film thickness and L the transversal linear dimension) with periodic boundary conditions along transversal directions and free boundary conditions along d direction. In our model, both the three-dimensional inverse spinel structure and the interactions scheme involving tetrahedral and octahedral sites have been considered in a realistic way. Results reveal a power-law dependence of the critical temperature with the film thickness accordingly by an exponent $ν = 0.81$ and ruled out by finite-size scaling theory. Estimates for the critical exponents of the magnetization and the specific heat are finally presented and discussed.

Introduction

Parallel to the development of films growth techniques, a great interest on thin-film magnetism has recently arisen. Several works deal with properties of Ising-type films [1] and more recently the interest has been focused on iron oxides like maghemite [2]. However, to our knowledge, simulations of magnetite (Fe₃O₄) thin films have not been reported. In consequence and due to the technological importance of these systems at nanometric scale, we stress on the magnetic properties and critical behavior of magnetite thin films using the Monte Carlo method and the Ising model.

Section snippets

Model and simulation

As is well known, magnetic crystallizes in the Fd3m inverse spinel structure, with 8 Fe³⁺ ions located in tetrahedral sites plus (8 Fe³⁺ and 8 Fe²⁺) ions distributed into octahedral sites, per unit cell. In our model, magnetic ions Fe_A³⁺, Fe_B³⁺, and Fe_B²⁺ are represented by Ising spins where labels A and B refer to tetrahedral and octahedral sites respectively, while oxygen ions are consider as non magnetic. The Ising Hamiltonian describing our system reads as follows: $H = - \sum_{〈 i, j 〉} J_{ij} ε_{i} ε_{j} σ_{i} σ_{j} .$ The

Results and discussion

Fig. 1 shows the temperature dependence of the specific heat per spin c for different film thicknesses. Hence, a well-defined lambda-type behavior is observed in agreement with a thermal-driven ferrimagnetic to paramagnetic phase transition occurring at some T_C where c tends to diverge. Furthermore, as the film thickness decreases, T_C becomes smaller. Such reduction is ascribed to the breaking of symmetry at the surface having a smaller coordination number and consequently a lower density of

Conclusion

From our results, a remarkable reduction of T_C as the film thickness becomes smaller is concluded. This fact is attributable to the breaking of symmetry at the surface and consequently to a lower density of magnetic bonds. Such dependence is ruled out by finite-size scaling theory and well characterized by an exponent $ν = 0.81$ . Compared to what happens in nanoparticles, results suggest a different class of universality depending on topological features.

Acknowledgements

This work was supported by Univ. de Antioquia Mediana Cuantía GICM 2005, Project SIU24-1-28 and COLCIENCIAS Project no. 1115-05-17603. Simulations were performed on the Hercules cluster (University of Antioquia) http://urania.udea.edu.co/facom/index.

References (5)

D.P. Landau et al.
Phys. Rev. B
(1990)
K. Binder et al.
Phys. Rev. Lett.
(1995)
J.W. Tucker
J. Magn. Magn. Mater.
(2000)
Y. Laosiriataworn et al.
Phys. Rev. B
(2004)
O. Iglesias et al.
J. Magn. Magn. Mater.
(1999)
O. Iglesias et al.
J. Magn. Magn. Mater.
(2000)
O. Iglesias et al.
Phys. Rev. B
(2001)

There are more references available in the full text version of this article.

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Magnetite thin films: A simulational approach

Abstract

Introduction

Section snippets

Model and simulation

Results and discussion

Conclusion

Acknowledgements

Phys. Rev. B

Phys. Rev. Lett.

J. Magn. Magn. Mater.

Phys. Rev. B

J. Magn. Magn. Mater.

J. Magn. Magn. Mater.

Phys. Rev. B