Quantum oscillations of the spin density in magnetic multilayers

Abstract

An asymptotic stationary phase formula is derived for the oscillatory spin density induced in a nonmagnetic spacer sandwiched between two semi-infinite ferromagnets. It gives an explicit dependence for the polarization on the spacer layer thickness and on the distance from the ferromagnet-spacer interface. Both dependences are shown to oscillate with the same periods as the exchange coupling between the ferromagnetic layers. The magnitude of the polarization is governed by the degree of confinement of carriers in the spacer quantum well and by the curvature of the spacer Fermi surface. The formula is applied to a Co/Cu/Co ~001! trilayer described by tight-binding bands fitted to an ab initio band structure. Its validity is tested against a fully numerical calculation using the same band structure. As in the case of the oscillatory exchange coupling, the induced polarization is dominated by the contribution of the Cu Fermi surface neck extrema leading to a short period oscillation of 2.6 atomic planes. An interesting non‐Ruderman-Kittel-Kasuya-Yosida initial decay of the induced polarization is discussed. @S0163-1829~99!06309-2# Oscillatory exchange coupling between two ferromagnetic layers separated by a nonmagnetic metallic spacer layer 1 is one of the rare manifestations of quantum interference effects in a metallic system structured on a nanometer length scale. It was proposed early on by Edwards and co-workers 2,3 that the oscillatory exchange coupling is due to size quantization of the energy of electrons confined in a quantum well by the spin-dependent potentials of the magnetic layers. For large spacer layer thicknesses ( ’10 atomic planes!, an analytic formula for the coupling has been derived. 2‐4 The asymptotic formula is based on the stationary phase approximation and is referred to as analytic quantum well ~AQW! formula. It allows us to determine the oscillation periods, amplitudes and rate of decay of the coupling unambiguously and relate them to the details of the band structures of the ferromagnets and nonmagnetic spacer. Applications of the AQW theory 2‐4 using a realistic band structure 5‐7 lead to oscillation periods and amplitudes of the coupling that are in agreement with numerical total energy calculations 6,8‐12 and also with experimental results for multilayers with noble metal spacers. 13 The photoemission experiments of Ortega and Himpsel 14,15 and Segovia et al. 16 confirm the energy quantization of electrons confined in a nonmagnetic spacer layer. Although it is assumed implicitly that quantum confinement of electrons leads also to oscillations of the spin density in the spacer layer, the spin density has never been analyzed analytically. The induced moment in a nonmagnetic spacer has been calculated numerically. 17‐19 A very small moment