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.

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A detailed investigation of the scaling relations recently proposed by [J. d'Albuquerque e Castro, D. Altbir, J. C. Retamal, and P. Vargas, Phys. Rev. Lett. 88, 237202 (2002)] to study the magnetic properties of nanoparticles is presented. Analytical expressions for the total energy of three characteristic internal configurations of the particles are obtained, in terms of which the behavior of the magnetic phase diagram for those particles upon scaling of the exchange interaction is discussed. The exponent $\eta $ in scaling relations is shown to be dependent on the geometry of the vortex core, and results for specific cases are presented.Comment: 6 pages, 4 figure

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

d'Albuquerque e Castro et al. Reply: The first point in the Comment [1] regards the scaling relations we have obtained for the magnetic phase diagram (MPD) of cylindrically shaped clusters. The diagram is based on the relative stability of three internal configurations of the clusters, namely, ferromagnetic in plane, ferromagnetic out of plane, and vortex. It consists of regions in the D H plane (D diameter of the basis, and H height of the cylinders), within which one of the three configurations has the lowest total energy E tot . The latter is given as the sum of the dipolar, exchange, and crystalline anisotropy terms. The boundary between any two regions in the MPD, say, ferromagnetic (F) and vortex (V), is determined by the equationThe main result in our Letter is the finding that the MPD for large clusters and full strength of the exchange coupling J can be obtained from those corresponding to smaller systems and a weaker value of J, with all the other parameters kept fixed, just by scaling the D and H axes. In other words, we have found that the two relations E F tot D; H; xJ E V tot D; H; xJ and E F tot D=x ; H=x ; J E V tot D=x ; H=x ; J hold, with 0:55 (within 1% error) and x < 1. Guslienko and Novosad disagree with the value we have obtained for . They think it should be exactly equal to 0.5, although they do not provide any proof for this claim. Instead, they present arguments which they understand would make their claim plausible. Their arguments go as follows.They keep just the exchange and the dipolar interactions and consider two configurations, namely, ferromagnetic (any of the two) and the vortex one. For the first one, they approximate the dipolar term by that of a continuous distribution of magnetic moments, which they call W m . For the vortex configuration, they consider just the exchange energy (relative to that of the ferromagnetic configuration), which they label W ex . By comparing the expressions in the two equations, Guslienko and Novosad concluded that it would be ''natural to express all dot sizes in dimensionless form in units of L ex 2A=M 2 s