Interval Computations of Ferromagnetic Resonance Conditions in Exchange-Biased Bilayers

Abstract

The main purpose of this paper is to promote the use of interval calculus in physics. As an example we use the system consisting of two thin films, one ferromagnetic and another one antiferromagnetic, deposited one atop of the other. We successfully and accurately simulate the positions of resonance fields of such a system, as seen in ferromagnetic resonance experiment. Interval calculations have revealed the presence of 1, 2 and sometimes even 4 distinct equilibrium configurations of the system, all corresponding to the same resonance field, when the field has a component antiparallel to that of cooling field, while only 1 such position when it points in the opposite direction. In both cases only a single resonance line is observed. As an added value we show that the exchange-biased system is in the metastable state, out of true thermodynamical equilibrium. Resonance conditionsWe use the model for ferromagnetic/antiferromagnetic (FM/AF) bilayers introduced by Hu et al. [1]. The relevant part (we neglect the Zeeman energy of the AF component) of the free energy per unit area of a ferro-/antiferromagnetic bilayer can be written asis the direction perpendicular to the sample plane, u = (1, 0, 0) is the direction of the magnetic field during cooling (an easy, in-plane direction for FM layer magnetization, M FM ), K in and K out are respective uniaxial anisotropy constants for FM component, σ is the domain wall energy density in the AF layer, J 1 and J 2 are the bilinear and biquadratic exchange constants at the interface, respectively. γ is the gyromagnetic ratio and t is thickness of a ferromagnetic part.In spherical coordinates the ferromagnetic resonance (FMR) condition iswhere the angles θ and ϕ describe the orientation of magnetization of the FM layer (M FM ) at resonance and all the derivatives have to be evaluated at the equilibrium position(s) of the entire system, that is of both vectors:ω is the fixed frequency of the microwave radiation, here 9.248 GHz. Our measurements were performed with external field located in the plane perpendicular to the sample plane but containing the in--plane easy axis. Other details concerning sample preparation, experiment and values of all obtained relevant parameters were already published in [2] and [3]. The nature of exchange bias effect is still poorly understood [4,5] and the goal of our investigation was to shed some light on this phenomenon. Classical simulations of the FMR spectraTo calculate the resonance field for a given orientation of an external field, one has to follow a rather tedious procedure. For each magnitude of the external field an equilibrium position(s) of the system has (have) to be found first. This step alone is a challenging task, requiring to solve a system of 4 highly non-linear equations, ∇E = 0, and determining which solutions correspond to the free energy minima. Then, using Eq. (2), the resonance frequency (frequencies) is (are) determined. The field(s), at which so found resonance frequency is equal to the one used in experiment, is (are)...