Theory of exchange coupling in magnetic multilayers

IBM Journal of Research and Development, Jan 1998 by Jones, Barbara A

Two layers of magnetic materials separated by a nonmagnetic "spacer" layer display an exchange coupling between the separated magnetic layers, as detailed by the papers on experimental aspects of magnetic multilayers in this issue. Here we describe from a personal perspective the current theoretical understanding of exchange coupling in magnetic multilayers. The understanding by this point is quite good, and involves contributions from a number of authors, of which we review a key subset. The organization of the paper is as follows. After an introduction to RKKY coupling, the next section discusses the effects of a realistic band structure (with emphasis on the work of Herman et al., Bruno and Chappert, and Stiles). The following section, which introduces quantum wells, covers the work of Mathon et al., Bruno et al., and this author. Finally, we include a discussion of the predictability of amplitude, period, and phase, including the effects of disorder.

Introduction

The theory of exchange coupling in magnetic multilayers has its basis in the interactions found between two magnetic impurities in a metal. Taking the simplest case of spin-1/2 (and avoiding issues of crystal electric fields and spin-orbit coupling), hybridization between the s-p conduction electrons of the host metal and the d- (or f-) electrons of the magnetic impurity produces an effective on-site exchange coupling at the impurity site. For s-p/d hybridization, the sign of the interaction is typically antiferromagnetic, as the conduction electrons attempt to screen the spin of the impurity in their midst. Rather than forming a negative spin-1/2 at the impurity site, however, the electrons instead spin-polarize in concentric rings around the impurity. The source of the rings of alternating polarization is that a true deltafunction in space would require, in Fourier kappa-space, all the kappa-vectors from 0 to infinity to be equally weighted, viz., Delta(r) chi int^sup chi^^^sub -chi^ dk e^sup ikr^ x 1. Since the host is a metal, there are, however, kappa-vectors only from 0 to the Fermi wave vector. The system thus cannot form a localized screening of the impurity spin, but does the closest alternative possibility, which results in an opposite alignment close in which overscreens the impurity, followed by a parallel alignment further away which overcompensates in the opposite direction, and so on with decreasing amplitude out to infinity. This is shown in Figure 1.

The periodicity of the alternations of polarization is lambda^sub F^, where Lambda^sub F^ is pi divided by a characteristic wave vector of the Fermi surface. For the simple case of a free-electron band structure, there is only one wave vector, the Fermi wave vector of order 1 Angstrom^sup -1^ , and lambda^sub F^ is around 3 Angstrom. Thus, within distances of 3 Angstrom from the impurity, the conduction electrons are aligned antiferromagnetically with the impurity, from 3 to 6 Angstrom ferromagnetically, and so on. The amplitude decay goes as 1/r^sup 3^ for three dimensions, and varies with the geometry. Such oscillations in spin polarization have been seen experimentally [1].

Interactions between two impurities arise when they are close enough to have appreciable overlap of their oscillatory screening polarizations. If a second impurity is within the spin-down region of the first impurity's conduction electron polarization, it is favored to point oppositely to the polarization, and thus ferromagnetically with the first impurity. If the second impurity is further away, in the ring of spin-up, the second impurity prefers to point down, or antiferromagnetically with the first impurity. Thus, there is an interaction between two impurities induced by the spin-polarized conduction electrons with which they interact. The interaction takes the same distance dependence as the spin polarization, but is of opposite sign. This interaction of magnetic spins mediated by conduction electrons is designated as the RKKY interaction, after Ruderman, Kittel, Kasuya, and Yoshida [2], who independently discovered it. A typical RKKY plot for a free-electron conduction band is illustrated by Figure 1. Of course, Equation (3) is just a perturbation expression, and the first-order term at that. Sometimes Equation (3) is cited as the "RKKY interaction," but this is incorrect. The RKKY interaction is the exchange coupling which arises between two (or more) spins interacting via spin polarization of conduction electrons-whether J be small or large, perturbative or not, and summed to all orders. In the cases when J is small, however, the exchange interaction calculated according to the prescription above is a good approximation.

For magnetic atoms in layers, the physical interaction arises from both the interfacial and the bulk magnetic atoms interacting with the conduction electrons in which they are immersed. The interactions between each magnetic spin and all other atoms must be superposed. Those at the interfaces have the strongest effect, both because they are closest to the other layer (1/r^sup 3^ effect, which when integrated over a layer becomes 1/r^sup 2^) and because they are more immersed in the spin sea of the spacer conduction electrons. For a rougher interface, there is even more mixing of magnetic and nonmagnetic atoms. Because of the presumed translational symmetry of the system in directions parallel to the layers, the interaction sum between each magnetic atom and all other magnetic atoms reduces to the sum along a line within one layer of interactions with all other atoms. A further simplification is typically achieved by assuming no spin-flip: that each magnetic moment is ferromagnetically aligned with every other atom in the same layer. Sums of interactions within a magnetic layer are thus avoided. Yet a further simplification is to assume, rather than discrete spins, a continuous constant spin density within each layer. The sums over the second magnetic layer thus become integrals.