Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

Abstract

Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be rewritten in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment.

Relativistic theory of magnetic inertia in ultrafast spin dynamics

“…Thus, a comparison can be made between the Gilbert damping parameter and the magnetic inertia parameter of a pure system. As noticed above, both the parameters are given by the magnetization susceptibility tensor, however it should be noted that the quantiy r α p β is imaginary itself, because [11],…”

“…Therefore one has to work together with those terms and cannot only perform the dynamics with an individual term. In an earlier work [11] we have shown that taking an uniform magnetic field along with the gauge A = B×r 2 will preserve the hermiticity. The essence of the uniform field lies in the assumption that the skin depth of the electromagnetic field is longer than the thickness of the thin-film samples used in experiments.…”

“…The damping tensor can be decomposed to have contributions from an isotropic Heisenberg-like, anisotropic Ising-like and Dzyaloshinskii-Moriya-like tensors. The anti-symmetric Dzyaloshinskii-Moriya contribution has been shown to lead to a chiral damping of the form M × (D × ∂M /∂t) [11]. Experimental observations of chiral damping have been reported recently [56].…”

“…Note that the dynamics only considers the local dynamics as we have not taken into account the time derivative of particle density operator (for details, see [11]). Incorporating the latter would give rise the local as well as non-local processes (i.e., spin currents) within the same footing.…”

“…For detailed calculations yet including the exchange field see Ref. [11]. The second term in the spin Hamiltonian Eq.…”