Two parallel calculations of the exchange coupling in a Co/Cu/Co͑001͒ trilayer, both using the same realistic s, p, and d tight-binding bands with parameters determined from the ab initio band structures of bulk Cu and Co, are reported. The coupling is first calculated within the framework of the quantum-well ͑QW͒ formalism in which the periodic behavior of the spectral density is exploited to derive an analytic formula for the coupling valid for large spacer thicknesses. On the other hand, an alternative expression for the coupling, referred to as cleavage formula, is derived that allows accurate and efficient numerical evaluation of the coupling. An analytic approximation to this expression, valid in the asymptotic region of large spacer thickness, is also obtained. These two approaches are discussed in relation to other existing theoretical formulations of the coupling. The numerical results for the coupling obtained from the cleavage formula are first compared with the analytical QW calculation. The agreement between the two calculations is impressive and entirely justifies the analytical QW approach. The numerical calculation fully confirms the result of the QW formalism that, for trilayers with thick Co layers, the short-period oscillation due to the minority electrons from the vicinity of the Cu Fermi-surface ͑FS͒ necks is dominant, the contribution of the long-period oscillation being negligible. This is shown, in the analytical QW formalism, to be due to the existence of bound states for the minority-spin electrons at the Cu FS necks in the ferromagnetic configuration. The dominant short-period oscillation has been confirmed by spin-polarized scanning electron microscopy and observed directly in the most recent photoemission experiments. The full confinement of the minority electrons at the neck of the Cu FS also leads to a strong temperature dependence of the short-period oscillation and an initial decay of the coupling with spacer thickness N that is much slower than predicted by the usual 1/N 2 law. For the electrons at the belly of the Cu FS, the confinement is weak in both spin channels and the long-period oscillation hardly changes between zero and room temperatures. In addition, the belly contribution to the coupling decreases at Tϭ0 K following the usual 1/N 2 dependence. The amplitude of the calculated coupling Ϸ1.2 mJ/m 2 at the first antiferromagnetic peak of Cu is only a factor of 3 larger than the observed coupling strength. Finally, the coupling for 2 ML of Co embedded in Cu has also been evaluated from the cleavage formula. A large initial coupling strength ͑3.4 mJ/ m 2 ) and comparable contributions from the short-and long-oscillation periods are obtained. This is in complete agreement with theoretical results reported by other groups.

Oscillations of the giant magnetoresistance (GMR) with nonmagnetic spacer layer thickness are predicted in the current-perpendicular-to-plane (CPP) geometry. The methods of the quantum-well theory of the oscillatory exchange coupling are applied to the Kubo formula to derive general selection rules for the GMR oscillation periods. The selection rules are illustrated for single-orbital tight-binding and parabolic band models. They predict that the CPP GMR oscillates not only with the expected Fermi-surface period but also with additional periods determined by the potential steps between the magnetic and nonmagnetic layers.

Using a torque formula, oscillations of the exchange coupling between two magnetic layers embedded in a nonmagnetic metal are calculated from the spin current across the structure. It is shown that each component of the spin current for a fixed value of the wave vector parallel to the layers exhibits quasiperiodic oscillations as a function of the ferromagnet thickness. Ideas of the theory of quasicrystals are used to derive a general

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The results of a rigorous quantum calculation of the current-perpendicular-to-plane giant magnetoresistance ͑CPP GMR͒ of a Co/Cu/Co͑001͒ trilayer without impurity scattering are reported. The conductances per spin in the ferromagnetic ͑FM͒ and antiferromagnetic ͑AF͒ configurations of the magnetic layers are computed from the Kubo formula. The electronic structure of the Cu and Co layers is described by fully realistic s, p,d tight-binding bands fitted to ab initio band structures of Cu and ferromagnetic fcc Co. Depending on Co thickness, the CPP GMR ratio can be as high as 90%. The whole calculated effect is due solely to quantum reflections of electrons from perfectly flat Co/Cu interfaces. The CPP GMR ratio is found to oscillate both with Co and Cu thickness, the respective oscillation amplitudes being 12 and 6 % of the average GMR. The resistances in each spin channel per unit cross-sectional area of the trilayer range from 3 to 7 f⍀ and oscillate with an amplitude ϳ 0.5 f⍀. An analytic asymptotic formula for resistance oscillations originating from the Cu Fermi surface is applied to analyze the numerical results. It is found that the resistance oscillations for majority electrons in the FM configuration have periods dominated by the extremal radii of the Cu Fermi surface. These are the same periods as observed in the oscillatory exchange coupling. However, the amplitude of resistance oscillations with the Fermi-surface periods is negligibly small for the minority electrons in the FM configuration and for electrons of either spin orientation in the AF configuration. The resistance oscillations of these electrons are dominated instead by periods determined by cutoffs of the conductance due to a mismatch between the Co and Cu bands across the Co/Cu interfaces. ͓S0163-1829͑97͒05121-7͔