Mesoscopic mechanism of exchange interaction in magnetic multilayers

We discuss a mesoscopic mechanism of exchange interaction in ferromagnetnormal metal-ferromagnet multilayers. We show that in the case when the metal’s thickness is larger than the electron mean free path, the relative orientation of magnetizations in the ferromagnets is perpendicular. The exchange energy between ferromagnets decays with the metal thickness as a power law. Suggested PACS index category: 05.20-y, 82.20-w Typeset using REVTEX 1 Both the experiment and the theory of ferromagnet-normal metal-ferromagnet multilayers have attracted a lot of attention . An example of such a structure consisting of two ferromagnetic films separated by a nonferromagnetic metallic film is shown in Fig.1. In the case when the metal thickness L is much smaller than the electron scattering mean free path l the sign of the exchange interaction energy between the ferromagnet’s magnetizations oscillates as a functions of L with a period of order of the Fermi wave length. As a result the magnetic structure of the system oscillates between ferromagnetic and antiferromagnetic orientations of the ferromagnet’s magnetizations . The explanation of this phenomenon is based on the fact that the interlayer exchange energy is due to Ruderman-Kittel interaction between electron spins in different ferromagnets. In the case of low temperatures and at |r − r| ≫ l the exchange energy between two localized spins < J(r, r) > averaged over the scattering potential configurations decays exponentially with |r− r| . Here r and r are coordinates of spins and brakets <> stand for averaging over realizations of the scattering potential in the metal and the ferromagnets. Recent experiments on ferromagnet-metal-ferromagnet multilayers [8] imply, however, that the exchange energy between the ferromagnets does not decay exponentially at L ≫ l and that the equilibrium relative orientation of the ferromagnet’s magnetizations is perpendicular independently of L. Phenomenologically, this situation can be described by an effective energy per unit area E = −J(m1 ·m 0 2) +B[(m 0 1 ·m 0 2) 2 − 1] (1) in the case 2B ≫ |J|. Here m1 and m 0 2 are, averaged over the film’s volumes, unit vectors parallel to magnetizations of the ferromagnetic films. Indices 1,2 indicate the first and the second ferromagnetic film respectively, J and B(θ) are bilinear and biquadratic coupling coefficients. In general, B(θ) is a smooth function of the angle θ between m1 and m 0 2. In this paper we discuss a theory of this phenomenon. It has been shown in [9−11] that the exponential decays of the average < J(r, r) > is connected to the fact that it has a random sign at large L. The modulus of the exchange interaction decay with L as a power 2 law. We can introduce a local exchange energy J(ρ) between the ferromagnets as an average of J(r, r) over a ferromagnet’s surface area of order of L. Here ρ is coordinate along the films. We assume that J(ρ) is small enough and spatial dependence of the magnetizations on the scale of order of L can be neglected. According to Slonczewski , biquadratic term proportional to B in Eq.1 can originate from the existence of spatial fluctuations of the sign of exchange interaction J(ρ) = 〈J〉+ δJ(ρ) along the layers. The fluctuations of J(ρ) cause fluctuations of directions of magnetizations. Energy associated with spatial fluctuations in the magnetization’s directions can be represented as E(J(ρ),mi(ρ)) = − ∫ dρJ(ρ)(m1(ρ) ·m2(ρ)) +αd ∫ dρ [ ∂m1(ρ) ∂ρ · ∂m1(ρ) ∂ρ + ∂m2(ρ) ∂ρ · ∂m2(ρ) ∂ρ ] (2) where the first term corresponds to the interfilms exchange energy, the second term is associated with the gradients of magnetizations inside the films, d is the ferromagnetic film’s thickness and α is a coefficient characterizing the exchange energy value in the ferromagnets. In the case when δJ(ρ) ≫ 〈J〉 and J(ρ) has a random sign, the energy E(J,mi(ρ)) has a minimum at a sample specific realization mi(ρ) = m 0 i + δmi(ρ; [δJ ]) with m 0 1⊥m 0 2 [12,14] and B ≡ B0 αd G(θ) (3) G = ∫ dρ〈δJ(ρ)δJ(0)〉 (4) Here B0 is a number of order unity . Let us consider the case when J(ρ) has random sign due to mesoscopic fluctuations of Ruderman-Kittel oscillations inside the metal . We assume that the ferromagnetic film’s thickness d > Ls = √