Dynamic consequences of exchange enhanced anisotropy in ferromagnet/antiferromagnet bilayers

– The phenomenon of exchange anisotropy is well known in terms of static magnetization properties such as enhanced coercivity and magnetization loop shifts. These effects are primarily associated with effective anisotropies introduced into a ferromagnet by exchange coupling with a strongly anisotropic antiferromagnet. These effective anisotropies can be understood as manifestations of a more fundamental exchange-induced susceptibility. We show that a consequence of this view is that a class of unusual dynamic effects associated with the exchange susceptibility should also exist. The effects become apparent near the ordering temperature of the antiferromagnet and affect domain wall velocities, domain wall resonances, and precessional switching of the ferromagnet. Exchange anisotropy is a term coined to describe the enhancement of magnetic anisotropies in a ferromagnet through contact with an anisotropic antiferromagnet [1]. Suitably field cooled, it is possible to prepare a ferromagnet/antiferromagnet structure in such a way as to observe a variety of static and quasi-static magnetic properties associated with exchange anisotropy [2, 3]. One of the most well-known phenomena is the shift of magnetization loops called exchange bias. Enhanced coercivity is also observed, and can be distinct from the bias shift [4]. Both the bias and coercivity require a strong contact exchange interaction between the ferromagnet and antiferromagnet. The common way to understand exchange anisotropy is as an effective anisotropy originating in the antiferromagnet. An antiferromagnet with strong anisotropy will affect the magnetic properties of an adjacent ferromagnet if the interface spins of the two materials are correlated through exchange interactions. This gives rise to an effective magnetic anisotropy in the ferromagnet that can have unusual symmetry properties upon reversal of the ferromagnet magnetization. Typically these properties can be understood by identifying reversible and irreversible magnetization processes in the antiferromagnet [5–8]. In this letter we argue that this concept of an effective magnetic anisotropy in the ferromagnet due to its exchange coupling to the antiferromagnet is also valid for a number of c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10553-8 R. L. Stamps et al.: Dynamic consequences of exchange enhanced etc. 513 interesting dynamic problems for which there is a clear separation of time and length scales. To be more specific, we argue that the degrees of freedom of the antiferromagnet can be integrated out leading to an effective Hamiltonian for the ferromagnet modified by interface energies which can be expressed by a susceptibility tensor. These are quite general results which are a generalisation of our previous work [9] to dynamical problems. The conditions under which such an approach is valid is a separation of time and length scales in the sense that the antiferromagnet is in (local) thermal equilibrium responding to the slow dynamics of the ferromagnet. Such a situation is met in a number of important problems, ranging from basic research to applications. Examples are domain wall dynamics, ferromagnetic resonance or switching of single-domain ferromagnetic particles, to name just a few. The key idea is to realize that the ferromagnet is only affected by exchange coupling across the interface to a magnetic moment somehow induced at the interface of the antiferromagnet. The magnitude of the moment is determined by the exchange coupling across the interface and any applied external magnetic fields [10]. The corresponding effective field acting on the ferromagnet can be defined locally at each lattice site. At site i along the interface on an atomistic level it is given by hα(i) = − ∂ ∂Sα(i) Ho + Jintσα(i), (1) where σ is the nearest neighbour spin across the interface on the antiferromagnet side,Ho is the Hamiltonian representing all other energies affecting spins within the ferromagnet including the external applied field and magnetic anisotropies and α denotes Cartesian components. This effective field determines the dynamics of the ferromagnet. If this dynamics is slow enough and if the fields acting on the antiferromagnet vary slowly in space the antiferromagnet stays in (local) thermal equilibrium so that a thermal average restricted to the antiferromagnet can be performed in eq. (1) resulting in an effective field acting on the ferromagnet given by h̃α(i) = − ∂ ∂Sα(i) Ho + Jintmα(i), (2) where mα(i) is the thermally averaged interface magnetization on the antiferromagnet (AFM) side. This effective field can be used, for instance, in the Landau-Lifshitz-Gilbert equations for the spins in the ferromagnet determining their dynamics under the assumption of slow dynamics as stated above. To proceed we assume linear response to be valid, in which case the reversible part of the antiferromagnet interface magnetization is given by mα(i) = Jintχ (1) αβ Sβ(i) + μoχ (2) αβ Hβ , (3) where summation over double appearing indices is understood. The first term describes the response to the interlayer exchange coupling Jint, and the second term is the response to an external applied magnetic field H. A third term of the form mirr(i) has to be added to eq. (3) for a disordered antiferromagnet representing contributions from irreversible magnetic moments pinned at the interface. This will be discussed later. The susceptibilities χ αβ and χ (2) αβ in general differ because the exchange interaction only affects the AFM interface layer while the external field is applied to all AFM layers. These susceptibilities represent the response of the AFM interface layer to the effective fields. The linear response as written in eq. (3) is a general form valid for situations in which the local effective fields vary slowly on atomic length scales and for which the time scales in the 514 EUROPHYSICS LETTERS FM layer and the AFM layer are separated in the sense that processes in the FM layer are slow as compared to those in the AFM layer, which itself remains locally in thermal equilibrium. An important insight can be gained by examining the effective free energy of the FM layer corresponding to eq. (2) in which the AFM degrees of freedom are integrated out