  We study spin-dependent transport in a suspended carbon nanotube quantum dot
in contact with two ferromagnetic leads and with the dot's spin coupled to the
flexural mechanical modes. The spin-vibration interaction induces spin-flip
processes between the two energy levels of the dot. This interaction arises
from the spin-orbit coupling or a magnetic field gradient. The inelastic
vibration-assisted spin flips give rise to a mechanical damping and, for an
applied bias voltage, to a steady nonequilibrium occupation of the harmonic
oscillator. We analyze these effects as function of the energy-level separation
of the dot and the magnetic polarization of the leads. Depending on the
magnetic configuration and the bias-voltage polarity, we can strongly cool a
single mode or pump energy into it. In the latter case, we find that within our
approximation, the system approaches eventually a regime of mechanical
instability. Furthermore, owing to the sensitivity of the electron transport to
the spin orientation, we find signatures of the nanomechanical motion in the
current-voltage characteristic. Hence, the vibrational state can be read out in
transport measurements.
