Quantum-well theory of the exchange coupling in magnetic multilayers with application to Co/Cu/Co(001)

Two parallel calculations of the exchange coupling in a Co/Cu/Co(001) trilayer, both using the same realistic $s,\hspace{0.5em}p$, 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 (QW) 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 (FS) 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 $T=0\mathrm{}$ K following the usual 1/$N^2$ dependence. The amplitude of the calculated coupling $\approx 1.2\hspace{0.5em}{\mathrm{m}\mathrm{J}/\mathrm{m}}^2$ 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.