  We investigate the spin-pumping-induced electric current on the surface of a
three-dimensional topological insulator hybridized with a ferromagnet, namely,
ferromagnetic topological insulator. In order to do this, we establish the
microscopic formalism and construct the perturbation theory using a Keldysh
Green's function approach. We analyze how this electric current is generated by
an exchange interaction and an external ac magnetic field, which is the driving
force of ferromagnetic resonance as well as the spin pumping. The mechanism is
as follows. First, the ferromagnetic resonance is driven and a zero-momentum
magnon emerges. It is the fluctuation from the saturation magnetization
pointing parallel to the precession axis of the ferromagnetic resonance. After
then, the spin pumping is generated with the zero-momentum magnon being the
carrier of spin. The zero-momentum magnon and the topological insulator surface
state couples through the exchange interaction and the spin carried by the
magnon is transferred to it. Owing to the spin-momentum locking, the
transferred spin is converted into the momentum of topological insulator
surface state leading to the generation of electric current flowing
perpendicular to the precession axis of the ferromagnetic resonance. It is
quadratic in the amplitude of external ac magnetic field whereas it is linear
to the strength of the exchange interaction. The associated electric voltage is
described by the spectrum of zero-momentum magnon. The non-linearity of
spin-pumping-induced electric current in the ac magnetic field as well as the
linearity in the exchange-interaction strength reflects that the surface of
ferromagnetic topological insulator has a high-performing functionality of
generating the electric charge current by magnetic controlling.
