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 effects in spintronics is spin pumping by dynamical magnetization that is steadily precessing (around, e.g., the z-axis) with frequency ω₀, due to absorption of low-power microwaves of frequency ω₀ under the resonance conditions and in the absence of any applied bias voltage. The two-decades-old "standard model" of this effect, based on the scattering theory of quantum transport, predicts that component I_Sz of spin current vector (I_Sx,I_Sy,I_Sz)α ω₀ is time-independent while I_Sx(t) and I_Sy(t) oscillate harmonically 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 features of spin pumping -- precessing localized magnetic moments within ferromagnetic metal (FM) or antiferromagnetic metal (AFM), whose conduction electrons are exposed to spin-orbit coupling (SOC) of either intrinsic or proximity origin, will pump both spin I_Sα(t) and charge I(t) currents. All four of these functions harmonically oscillate in time at both even an odd integer multiples nω0 of the driving frequency ω₀. Such high-harmonics are cut off at nmax, with possibility to tune n_max≤4 in the chosen for demonstration one-dimensional FM or AFM models with the Rashba SOC by increasing its strength. Finally, we conjecture that two-dimensional magnetic materials offer the optimal setting for experimentally confirming high harmonics in spin pumping as their magnetic ordering at finite temperature crucially relies on magnetic anisotropy originating from strong intrinsic SOC. To demonstrate this, we compute high harmonics for honeycomb lattice of doubly proximitized graphene with both magnetic ordering (due assumed magnetic overlayer) and SOC (due to assumed transition metal dichalcogenide underlayer).