Phase analysis of quantum well states in metallic multilayers: Periodicity and thickness dependence of the oscillatory magnetic coupling

A phase accumulation approach has been used to model Cu metallic quantum well (MQW) states in the fcc $M/\mathrm{C}\mathrm{u}/\mathrm{f}\mathrm{c}\mathrm{c}$ $M\left(100\right)\mathrm{}$ systems, where $M=\mathrm{Fe},$ Co, or Ni. The electronic states at $k_{\parallel }=0.00\mathrm{}{\AA }^{-1}$ and at $k_{\parallel }=0.98\mathrm{}{\AA }^{-1}$ along the $\overline{\Gamma X}$ direction—the locations of the belly and the neck of the Cu Fermi surface, respectively—are considered. MQW states crossing the Fermi energy ${(E}_F)$ at these points in the two-dimensional Brillouin zone are associated with the long- and short-period oscillatory magnetic coupling observed in these systems. The model predicts that states cross $E_F$ at the belly with a period of 5.7 Cu ML and a neck with a period of 2.6 Cu ML, in excellent agreement with the observed crossings in photoemission and inverse photoemission, as well as the oscillation periods seen in magnetic measurements. The fact that the period with which MQW states cross the Cu Fermi energy is solely a property of the spacer layer follows naturally from application of the model. Furthermore, the $E_F$ crossing of a given state near the neck is found to occur at successively greater thicknesses when the ferromagnetic (FM) layer is changed from Fe to Co to Ni. This systematic shift is consistent with what is seen in magnetic measurements and can be directly related to the energy of projected minority spin band gaps in the FM layer. The model gives an intuitive picture of how the minority spin band structure of the FM layer affects the electronic states of the nonmagnetic layer, showing why the thickness shift of a Fermi energy crossing depends on the location of projected band gaps of the FM layer.