Inadequacy of the asymptotic approximation for the interlayer coupling in Fe/Ag/Fe and Fe/Au/Fe (001) trilayers

Abstract

We discuss the basic assumptions underlying the stationary phase approximation to the interlayer exchange coupling between magnetic materials across nonmagnetic spacers. We show that, for temperatures and spacer thicknesses of interest, it cannot be applied to systems such as Fe/Ag/Fe and Fe/Au/Fe ͑001͒ trilayers. Such inadequacy results from particular confinement effects caused by the Fe layers in the Ag and Au extremal Fermi-surface ''neck'' states. This is of special relevance to the interpretation of experimental data. ͓S0163-1829͑99͒04738-4͔The bilinear exchange coupling J between magnetic materials across nonmagnetic metallic spacers oscillates with decaying amplitude between positive ͑ferromagnetic͒ and negative ͑antiferromagnetic͒ values, as the spacer layer thickness N is increased. For systems with crystalline layers and sharp interfaces, J can be calculated at any temperature T from an expression of the form 1-5where f (E) is the Fermi-Dirac distribution function, k ជ ͉͉ is a wave vector parallel to the layers, F(E,k ជ ͉͉ ,N) is given in terms of the electronic Green functions of the system, and the sum over k ជ ͉͉ is restricted to the two-dimensional Brillouin zone ͑BZ͒. The prominent features of the oscillatory behavior of J, i.e., period, phase, and amplitude, are related to the electronic structure of the multilayered system. While numerical evaluations of Eq. ͑1͒ do not highlight this relationship, simple approximations, currently referred to as the stationary phase ͑SP͒ method, allow semianalytical expressions for J to be derived. They express the coupling asymptotically ͑i.e., for large values of N) as a sum of contributions coming from extremal wave vectors of the spacer Fermi surface ͑FS͒ in the direction perpendicular to the layers. 6,7,4 The oscillation period of each contribution is determined by the spacer FS, whereas the amplitude and phase are regulated also by the degree of confinement of the carriers within the spacer layer.The stationary phase method has been successfully applied to analyze both experimental data and results of full numerical calculations of J in several systems, such as Co/ Cu/Co ͑001͒ and Fe/Cu/Fe ͑001͒ multilayers. 2,4,[7][8][9] Good agreement between total-energy calculations and the SP approximation were obtained in these systems. However, as we show here, there are cases in which this method is inadequate for spacer thicknesses and temperatures of experimental interest. This is precisely what happens in Fe/Ag/Fe ͑001͒ and Fe/Au/Fe ͑001͒ trilayers. To understand why this is so, it is instructive to review the basic assumptions of the method.The SP method relies on the fact that, for fixed E and k ជ ͉͉ , the function F in the integrand of Eq. ͑1͒ oscillates with N. Thus, F can be expanded in a Fourier series in N, whose coefficients c s and corresponding wave vector k Ќ are functions of E and k ជ ͉͉ . The following approximations are then introduced to allow an analytical evaluation of the integrals over these variables. The first one is based on the fact that t...

Origin of the temperature dependence of interlayer exchange coupling in metallic trilayers

“…This gives rise to an additional temperature dependence of the IEC since the energy interval of interest around the Fermi energy increases with temperature 4,5 . The same may in principle apply to the norm of r σ 6 .…”

“…(ii) interface contribution The argument φ σ of the complex reflection coefficients r σ = |r σ |e iφσ at the spacer/magnet interface may be highly energy dependent. This gives rise to an additional temperature dependence of the IEC since the energy interval of interest around the Fermi energy increases with temperature 4,5 . The same may in principle apply to the norm of r σ 6 .…”

Engineering the interlayer exchange coupling in magnetic trilayers

“…It is well known that the carriers located at the belly of the NM Fermi surface induce the long-period oscillation (~10–18 Å or 5–9 monolayers (MLs)) 3 16 from the QWSs crossing. However, the carriers at the neck of the NM Fermi surface are related to the short-period oscillation (~4–6 Å or 2–3 MLs) 16 17 18 19 , but the microscopic mechanism is not clear up to now.…”

Switching off the magnetic exchange coupling by quantum resonances

“…( 3), on which Eq. ( 6) is based, is only an approximation and the higher-order terms are not rigorously negligible for trilayers with noble-metal spacer [20].…”