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

One of the cornerstone phenomena in spintronics is spin pumping by dynamical magnetization which is steadily precessing (around, e.g., the z-axis) with frequency ω₀, due to absorption of microwaves of frequency ω₀ under the resonance conditions, in the absence of any bias voltage. The two-decades-old “standard model” of this effect, based on the scattering theory of quantum transport attuned to the problem of adiabatic pumping of spin or charge, predicts that component I Sz of spin current vector (I_Sx , I_Sz , I_Sz ) α ω₀ is time-independent while I_Sx(t) and I I_Sy(t) harmonically oscillate in time with single frequency ω₀; whereas pumped charge current is zero I ≡ 0 in the same adiabatic α ω₀ limit. Here we employ more general time-dependent quantum transport formalism to predict unforeseen-by-standard-model features of spin pumping—precessing localized magnetic moments (LMMs) within ferromagnetic metal (FM) or antiferromagnetic metal (AFM) with spin-orbit (SO) coupling in the precessing region, which could be either intrinsic or proximity induced, will pump spin I Sα(t) and charge I(t) currents all four of which harmonically oscillate, at sufficiently long times for transient currents to die away, at both even and odd integer multiples nω₀ of the driving frequency ω₀. Such high-harmonics are cut off at nmax ≤ 8, in our one-dimensional model with the Rashba SO coupling, with possibility to increase nmax by increasing the strength of the SO coupling.