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In order to study spin-wave excitations of itinerant ferromagnets a relativistic first-principles method based on the adiabatic approach is presented. The derivatives of the free energy up to second order with respect of the polar and azimuthal angles are derived within the framework of the magnetic force theorem and the fully relativistic Korringa-Kohn-Rostoker method. Exchange and spin-orbit coupling are thus incorporated on equal footing in the Hamiltonian. Furthermore, a detailed comparison to classical spin Hamiltonians is given and it is shown that the magnetocrystalline anisotropy energy contains contributions from both the on-site anisotropy and the off-site exchange coupling terms. The method is applied to an Fe monolayer on Cu͑001͒ and Au͑001͒ surfaces and for a Co monolayer on Cu͑001͒. The calculations provide with the gap at zero wave number due to the spin-orbit coupling and uniaxial anisotropy energies in good agreement with the results of the band energy difference method. It is pointed out that the terms in the spin-wave Hamiltonian related to the mixed partial derivatives of the free energy, absent within a nonrelativistic description, introduce an asymmetry in the magnon spectrum with respect to two in-plane easy axes. Moreover, in the case of an in-plane magnetized system the long-wavelength magnons are elliptically polarized due to the difference of the second-order uniaxial and fourth-order in-plane magnetic anisotropy.

Effective pair exchange interactions between Mn atoms in III-V and group-IV diluted magnetic semiconductors are determined from a two-step first-principles procedure. In the first step, the self-consistent electronic structure of a system is calculated for a collinear spin structure at zero temperature with the substitutional disorder treated within the framework of the coherent-potential approximation. The effective exchange pair interactions are then obtained in a second step by mapping the total energies associated with rotations of magnetic moments onto an effective classical Heisenberg Hamiltonian using the magnetic force theorem and one-electron Green functions. The formalism is applied to Ga 1Ϫx Mn x As alloys with and without As antisites, and to Ge 1Ϫx Mn x alloys recently studied experimentally. A detailed study of the behavior of pair exchange interactions as a function of the distance between magnetic atoms as well as a function of the concentrations of the magnetic atoms and compensating defects is presented. We have found that due to disorder and the half-metallic character of the system the pair exchange interactions are exponentially damped with increasing distance between the Mn atoms. The exchange interactions between Mn atoms are ferromagnetic for distances larger than the ones corresponding to the averaged nearest-neighbor Mn-Mn distance. The pair exchange interactions are also reduced with increasing concentrations of the Mn atoms and As antisites. As a simple application of the calculated exchange interactions we present mean-field estimates of Curie temperatures.

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