Magnetization reversal via a Stoner–Wohlfarth model with bi-dimensional angular distribution of easy axis

https://doi.org/10.1016/j.jmmm.2015.07.035Get rights and content

Highlights

  • Magnetic texture effects are included in the Stoner–Wohlfarth problem.

  • Step-like, Gaussian-like and user defined angular EADs are discussed.

  • The magnetic texture is obtained from the overall magnetization reversal.

  • Results beyond the OR method can be provided for complex systems.

Abstract

A numerical extension of the simple Stoner–Wohlfarth model to the case of bi-dimensional angular distributions of easy axis is provided. The results are particularized in case of step-like, Gaussian-like and user defined distributions. In spite of its simplicity, the model can be applied to magnetically textured thin films and multilayers with in-plane magnetic anisotropy, independently on the texture source. Exemplifications are provided for a simple ferromagnetic textured FeCo film as well as for a FeMn/FeCo/Cu/FeCo spin valve structure.

Introduction

There are more than 60 years since E.C Stoner and E.P Wohlfarth published their model about the magnetization reversal of single-domain ferromagnets [1], [2]. Despite its simplicity, the Stoner–Wohlfarth (S–W) model is still of actual interest while it provides results in good concordance with experimental observations for many system of technological impact. That is due to the fact that the presently developing nanotechnology deals frequently with assemblies of magnetic nanoentities respecting the condition of magnetic monodomains [3] (to be mentioned here biomedical and catalytic applications of magnetic nanoparticles [4], [5] or applications related to magnetic recording or sensoristics [6], [7], [8], [9] involving patterned thin films, multilayers and nanowires). The model has the great advantage that, due to its simplicity, analytical expressions can be derived in particular cases transparent to intuitive physical explanations. Moreover, the case of magnetic assemblies can be analyzed via usual statistical means of non-interacting entities with further extension to the perturbation effect of possible interactions. The most evident example is the case of magnetic monodomain nanoparticle assemblies which behavior can be easily explained starting from the S–W model of a nanoparticle [3], [10]. Very interesting is that the S–W model with related corrections can also provide in a first approximation the description of more complex effects connected to the interfacial interactions leading to exchange-spring and exchange bias phenomena, which are of large technological impact in our days [3], [9], [11], [12], [13], [14]. This is the reason for a permanent developing of the model in different versions, taking into account different additional energy contributions or suitable averaging processes. Excellent reviews of the model in respect to different applications and present achievements were provided, among others, by Radu and Zabel [15] and Tannous and Giewaltowski [16]. At this point it is to mention that only numerical solutions are suitable for a general treatment of the S–W problem, especially in case of magnetic assemblies (even in interaction), but however the involved numerical analysis is more simple, efficient and transparent as compared to the case of complex micromagnetic simulations, which on the other hand may take into account additional microstructural aspects and specific interactions among components.

As the main hypothesis of the model is the fact that all the local spins (magnetic moments) of the magnetic entity are oriented in the same preferred direction and are rotating coherently under an applied magnetic field. Hence, just one rotating representative macrospin is associated to the magnetic entity, which means that the exchange energy is infinite with respect to other magnetic energy terms and can be considered as a constant in the energy expression. As it will be shown in the next section, the magnetization reversal of the macrospin is depending on the direction of the applied field with respect to a preferred direction, which is called easy axis (EA) of magnetization. A quite realistic case of assembly of magnetic entities is the one involving an angular easy axis distribution (EAD), which can cover the limits from a unique direction (Dirac type angular distribution) to randomly distributed directions in the whole space. An angular distribution centered along a given direction reflects the case of a higher/lower magnetic texture depending on the distribution width. There are different possibilities for obtaining experimentally appropriate information about the EAD by using either Mössbauer spectroscopy [17], [18] and specific magnetometry techniques, as for example, via Orientation Ratio (OR) measurements or Flanders and Shtrikman’s principle [19], [20]. However, all these methods present quit strong limitations [20] and except the last method, make apriori assumptions about the type of the EAD function. While the magnetic texture effects are clearly reflected in the shape of the hysteresis loop of the system, one may assume that the reciprocal approach of using suitably collected hysteresis loops in order to get an as much as complete information about the distribution of easy axis might be also very effectively. For example, Kronmuller et al. have already shown and carefully analyzed different effects induced by magnetic texture and Gaussian EADs related to grain orientation on nucleation and coercive fields, in case of oriented sintered Nd–Fe–B magnets [21], [22], [23].

In this respect, the present work deals with a general numerical solution of the S–W model for bi-dimensional angular distribution of EAD. The model can be extended to different types of EADs and is consistent with the case of bi-dimensional magnetic systems, as for example thin films and multilayers presenting in plane magnetic anisotropy. In order to respect the model hypothesis, the real magnetic structures should not allow the formation of magnetic domains, fulfilling therefore conditions related to either specific thicknesses (assuring also the in plane anisotropy) or a specific island-like morphology. It is worth to mention that such a tri-dimensional type of growth (Volmer–Weber growth mode) leading to the formation of uniform nanogranular films with small oriented island-like morphologies with lateral size of a few tenths of nanometers and behaving as magnetic single domains were usually reported in case of thin films and multilayers obtained by either sputtering or thermo-ionic arc methods [24], [25], [26]. Moreover, there is a growing interest in deposition of nanoparticle-assembled thin films by femtosecond pulsed laser deposition in vacuum [27], [28] which in certain conditions can be alternative physical supports for the described model.

Section snippets

Algorithm

The starting point of the algorithm is based on the simple S–W model of a magnetic monodomain bidimensional entity, which also assumes: (i) an in plane applied field, H, (ii) an enhanced in plane shape anisotropy in order to allow the in plane reversal process and (iii) an in plane uniaxial anisotropy (anisotropy constant KF). To note that such a simplified model infers a specific physical system consisting of well-shaped bidimensional magnetic grains (e.g. circular, with in plane uniaxial

Results and discussions

An exemplification on how this theoretical approach can be used in order to determine the spin configuration in thin films and multilayers will be provided. Simple ferromagnetic FeCo films and a spin valve like structure sandwiching a Cu layer between two ferromagnetic FeCo layers (a free one and the other coupled to an antiferromagnetic Fe–Mn layer) have been obtained via the thermo-ionic vacuum arc method, as reported in [25]. The two considered systems are labeled by S1 and S2, having the

Conclusions

The S–W model is a simple tool for investigation of magnetic monodomain systems with uniaxial anisotropy. However, in many experimental situations of technological interest, it has been observed that even the systems which satisfy the above prerequisites or more complex systems consisting of S–W subsystems cannot be satisfactory analyzed in the frame of the simplest uniaxial S–W model. A new numerical approach has been considered for such situations, in case of bi-dimensional systems with in

Acknowledgments

The financial support of the Romanian National Authority for Scientific Research through the Core Programme PN 45N and Capacities E11/2014 is highly acknowledged.

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