Implementation of the relativistic effects on spin in first-principles electronic structure and magnetic property calculations

An extended single Hamiltonian, which can properly treat the couplings between the two spin states, has been applied to study the relativistic effects including the spin–orbit coupling inside atoms, the coupling with the electric field in the interstitial region, and the spin–orbit torque in ferromagnetic Fe, Co, and Ni metals. The calculated results show that the spin–orbit coupling is strong enough to cause enhancement of the minority-spin d electrons in these metals, which reduces the spin magnetic moment of the Co and Ni metals by 0.3764 μ_B and 0.0619 μ_B, respectively. The z-component of the d-orbital orbital magnetic moments obtained are 0.0375 μ_B, 0.1323 μ_B, and 0.0628 μ_B, respectively, for the Fe, Co, and Ni metals. The calculated dependence of the Landau–Lifshitz damping coefficient, λ, on the temperature has a similar trend as that of the experimental one in low temperatures. This study suggests that the low-temperature Lorentzian broadening is not due to a temperature sensitive mechanism, but may be due to energy transfer or resonances with magnetic field induced degrees of freedom. The observed drop of λ with the increase of temperature is found to be due to the increase of the chemical potential relative to the Fermi level.