Spin configurations in hard/soft coupled bilayer systems: Transitions from rigid magnet to exchange-spring

Abstract

We investigate equilibrium properties of an exchange-spring magnetic system constituted of a soft layer (e.g. Fe) of a given thickness on top of a hard magnetic layer (e.g. FePt). The magnetization profile M (z) as a function of the atomic position ranging from the bottom of the hard layer to the top of the soft layer is obtained in two cases with regard to the hard layer: i) in the case of a rigid interface (the FePt layer is a single layer), the profile is obtained analytically as the exact solution of a sine-Gordon equationwith Cauchy's boundary conditions. Additional numerical simulations also confirm this result. Asymptotic expressions of M (z) show a linear behavior near the bottom and the top of the soft layer. In addition, a critical value of the number of atomic planes in the soft layer, that is necessary for the onset of spin deviations, is obtained in terms of the anisotropy and exchange coupling between the adjacent plane in the soft layer. ii) in the case of a relaxed interface (the FePt layer is a multilayer), the magnetization profile is obtained numerically for various Fe and FePt films thicknesses and applied field.

Spin configuration of magnetic multi-layers: effect of exchange, dipolar and Dzyalozhinski–Moriya interactions

“…14 we present different magnetic profiles for typical values of the EI (λ EI = 1.44), DDI (λ DDI /λ EI ≈ 10 −2 ) and DMI (λ DMI /λ EI ≈ 0.1), as can be found in Refs. [35,38,39] is two orders of magnitude weaker and thereby its contribution to the alignment of the magnetic moments at the interface is very small. Nonetheless, it tends to align the interface along the z axis and this effect propagates throughout all sub-layers by the in-plane exchange interaction, causing a global decrease in the MP.…”

“…The exact solution for the MP has been obtained for the system with RI, in the absence of the external magnetic field [38]. For definiteness, the SMS-HMS lies on the xy plane and the out-ofplane anisotropy in the HMS slab points in the z direction.…”

“…Moreover, the angle deviation with respect to the z axis θ i − θ i+1 of two adjacent magnetic moments s i , s i+1 is supposed to be small. For convenience, we denote ξ i ≡ θ i − θ e , where θ e is the equilibrium polar angle of the multilayer system as defined in [38]. For typical materials d s 1, thus θ e 0.…”

“…In our case, since the DDI vector is parallel to the HMS anisotropy (along the z axis), a strong enough interaction aligns the magnetic moments at the interface (and all the magnetic moments in the HMS) in the direction θ = 0, thus driving the system into an effective RI state. As such, the asymptote can be found by calculating ξ s L using the analytical expressions [38], i.e. Eqs.…”

“…In such a case one can write down the (finite-difference) equation for ξ from the energy minimization. The solution of this equation can be written as [38] …”