High-harmonic generation in spin and charge current pumping at ferromagnetic or antiferromagnetic resonance in the presence of spin-orbit coupling

One of the primary mechanisms used in spintronics is the transfer of angular momentum by means of pure spin current pumping, which can occur due to the magnetization dynamics produced by stationary precession of the localized magnetic moment under ferromagnetic or antiferromagnetic resonance and in the absence of any applied voltage. This is a well-known phenomenon that occurs due to the absorption of low-power microwaves of frequency $omega_0$ under resonance conditions. Consequently, as predicted by the literature, the spin current along the precession axis is proportional to $omega _0$, while the x and y components oscillate harmonically at the resonance frequency. Additionally, the charge current can only be zero when there is no inversion symmetry; This opens the opportunity to manipulate and produce THz frequencies through the Antiferromagnetic resonance phenomena. We reexamine the spin pumping problem through two exact general formalisms: a time-dependent non-equilibrium Green's functions (TDNEGF) and the Floquet formalism combined with the stationary Non equilibrium Green's function (NEGF) formalism, thereby predicting unforeseen spin pumping features in a system of localized moments precessing within a Ferromagnetic (FM) or Antiferromagnetic (AFM) resonance, in which the conduction electrons are subjected to the effects of spin-orbit coupling. It was observed that both spin and charge current components oscillate in integer multiples of the characteristic frequency. Besides that, it has been found that the frequency cutoff increases following the magnitude of the spin-orbit parameter, reaching a value of ($N_{max}approx 11$) in a FM or AFM system in one dimension and even larger values ($N_{max}approx 25$) in a 2D system defined on a honeycomb lattice. In this study, we utilize the Floquet NEGF formalism, integrating the Floquet scattering matrix with the established TDNEGF calculations, and validating the time-dependent calculations of the NEGF.