  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 non-Markovian effect
on spin pumping to describe spin current generation driven by arbitrary
precession frequency of magnetization 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 nano-spintronic devices show that we need to include
non-Markovian effects. In contrast to the one-way-only nature of the spin
current generation under the Markovian dynamics, we find that the non-Markovian
dynamics exhibits a temporal backflow of spin, call spin backflow for brevity.
We capture the phenomenon by introducing its quantifier, and show that the
backflow significantly reduces the amount of spin current when the frequency
exceeds the relaxation rate. This prevents 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.
