One of the cornerstone effects in spintronics is spin pumping by dynamical magnetization that is steadily precessing (around, for example, the
We report results of largescale groundstate density matrix renormalization group (DMRG) calculations on t
 Award ID(s):
 2110041
 NSFPAR ID:
 10307360
 Publisher / Repository:
 Proceedings of the National Academy of Sciences
 Date Published:
 Journal Name:
 Proceedings of the National Academy of Sciences
 Volume:
 118
 Issue:
 44
 ISSN:
 00278424
 Page Range / eLocation ID:
 Article No. e2109978118
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract z axis) with frequencyω _{0}due to absorption of lowpower microwaves of frequencyω _{0}under the resonance conditions and in the absence of any applied bias voltage. The twodecadesold ‘standard model’ of this effect, based on the scattering theory of adiabatic quantum pumping, predicts that component of spin current vector ${I}^{{S}_{z}}$ is timeindependent while $({I}^{{S}_{x}}(t),{I}^{{S}_{y}}(t),{I}^{{S}_{z}})\propto {\omega}_{0}$ and ${I}^{{S}_{x}}(t)$ oscillate harmonically in time with a single frequency ${I}^{{S}_{y}}(t)$ω _{0}whereas pumped charge current is zero in the same adiabatic $I\equiv 0$ limit. Here we employ more general approaches than the ‘standard model’, namely the timedependent nonequilibrium Green’s function (NEGF) and the Floquet NEGF, to predict unforeseen features of spin pumping: namely precessing localized magnetic moments within a 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 $\propto {\omega}_{0}$ and charge ${I}^{{S}_{\alpha}}(t)$I (t ) currents. All four of these functions harmonically oscillate in time at both even and odd integer multiples of the driving frequency $N{\omega}_{0}$ω _{0}. The cutoff order of such high harmonics increases with SOC strength, reaching in the onedimensional FM or AFM models chosen for demonstration. A higher cutoff ${N}_{\mathrm{m}\mathrm{a}\mathrm{x}}\simeq 11$ can be achieved in realistic twodimensional (2D) FM models defined on a honeycomb lattice, and we provide a prescription of how to realize them using 2D magnets and their heterostructures. ${N}_{\mathrm{m}\mathrm{a}\mathrm{x}}\simeq 25$ 
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