The generation of a spin-polarized electron current (spin current) in a normal metal without a bias voltage is one of central issues in realizing spintronics applications.
A standard model used in analyzing spin current generation, or so-called spin pumping, consists of a ferromagnet attached to a normal metal lead, where the spin current is induced into the metal by a precession of magnetization in the ferromagnet. Focusing on the "adiabaticity", which is quantified using a comparison between the frequency and a relaxation rate of the relevant system, we investigate the role of nonadiabaticity in the spin pumping. For the purpose, we consider a minimum model consisting of a magnetic quantum dot attached to a normal metal lead and obtain the dependence of the spin current on the frequency of the precession using full counting statistics. This evaluation shows that the steady-state population of the quantum dot remains unchanged by the precession owing to a rotational symmetry about the axis of precession. This implies that in the adiabatic limit the spin current is absent and that spin pumping is entirely a nonadiabatic effect. We also find that the nonadiabatic spin current depends linearly on the frequency in the low-frequency regime and exhibits an oscillation in the high-frequency regime. The oscillation points to an enhancement of spin pumping by tuning the frequency of precession.