Magnetic anisotropy and orbital magnetic moment in Co films and Co/X bilayers ( X=Pd and Pt)

Abstract

Magnetocrystalline anisotropy (MCA) energy and the anisotropy of the orbital magnetic moment (AOM), relations between them, and methods of their calculation are investigated for the (001) fcc Co film and $\mathrm{Co}/X$ bilayers ($X=\mathrm{Pd}$ and Pt), using a realistic tight-binding model. The relations derived by Bruno [Phys. Rev. B 39, 865 (1989)] and van der Laan [J. Phys.: Condens. Matter 10, 3239 (1998)], including the AOM of Co only, are found to give largely incorrect mean values of the MCA energy but reproduce its oscillatory variation versus the Co thickness, with scaling factors needed for the bilayers. The presently proposed extension of the Bruno relation, with the AOM of both Co and Pd or Pt included, predicts the correct sign of the MCA energy for both bilayers and reproduces its magnitude and oscillation pattern very well, without extra scaling, for the Co/Pd bilayer, though this relation is not satisfied locally, by the MCA and AOM layer terms. A similarly extended van der Laan relation fails to reproduce the MCA energy, its sign, and the oscillation pattern, and largely overestimates its magnitude. For all investigated systems, the Co orbital moment oscillates versus the Co thickness with the 2 monolayer (ML) period for the out-of-plane direction of magnetization and the 5 ML period for the in-plane direction while the oscillations of the Pd and Pt orbital moments are more complex and similar for both magnetization directions. The exact and approximate MCA energies obtained with the force theorem and the perturbation theory (PT), respectively, are close to each other for the Co film and the Co/Pd bilayer. For the Co/Pt bilayer, only the mean value of the MCA energy is well approximated by the PT while its oscillation amplitude is overestimated a few times due to the large SOC in Pt. It is also shown that the MCA energy includes the intraband term, usually neglected, but, in fact, finite and vital for systems without the inversion symmetry as its magnitude is comparable to that of the interband term for the Co/Pd bilayer.