Spin backflow: A non-Markovian effect on spin pumping

The miniaturization of spintronic devices, specifically, nanoscale devices employing spintronics, has attracted intensive attention from a scientific as well as engineering perspective. In this paper, we study a non-Markovian effect on spin pumping to describe spin current generation driven by the magnetization of arbitrary precession frequency in a quantum dot attached to an electron lead. Although the Markovian approximation can be used when driving is sufficiently slow compared with relaxation times in electron tunneling, recent developments in nanospintronic devices show that we need to include non-Markovian effects. In contrast to the one-way-only nature of the spin current generation under Markovian dynamics, we find that non-Markovian dynamics exhibit a temporal backflow of spin, called spin backflow for brevity. We capture the phenomenon by introducing its quantifier, and show that the backflow reduces the amount of spin current significantly when the frequency exceeds the relaxation rate. This prevents an unphysical divergence of the spin current in the high-frequency limit that occurs under the Markovian approximation. We believe our analysis provides an understanding of the spin pumping particularly in regard to producing a more efficient spin current generation over shorter time scales by going beyond the conventional Markovian approximation.