Exchange coupling in magnetic multilayers: effect of partial confinement of carriers

The exchange coupling J(l) between magnetic layers across a non-magnetic spacer is observed to oscillate as a function of the spacer thickness l. In an earlier work a theory of the oscillatory exchange was proposed which shows that the oscillation periods are characteristic of the spacer. The theory relied on the assumption of an infinitely large exchange splitting in the magnetic layers, which leads to complete confinement of magnetic carriers of one spin in the ferromagnetic configuration of the sandwich. While this may be valid for strong ferromagnets such as Co or Ni, the complete confinement model is not a realistic approximation for iron which has holes in its majority-spin band. The theory is now generalized to the case of partial confinement of carriers in the spacer appropriate to a sandwich with weakly ferromagnetic layers. An exactly solvable hole-gas model of the coupling as well as numerical tight-binding results are presented which demonstrate that the oscillation period is unaffected but the amplitude and phase depend critically on the degree of confinement. Asymptotic expansions for J(l) valid at finite temperature and for an arbitrary single tight-binding band are also obtained. They show that the period and temperature dependence of the oscillations are directly related to the properties of the spacer Fermi surface but the amplitude and phase depend also on the exchange splitting in the ferromagnetic layers.