  A quantum dot formed in a suspended carbon nanotube exposed to an external
magnetic field is predicted to act as a thermoelectric unipolar spin battery
which generates pure spin current. The built-in spin flip mechanism is a
consequence of the spin-vibration interaction resulting from the interplay
between the intrinsic spin-orbit coupling and the vibrational modes of the
suspended carbon nanotube. On the other hand, utilizing thermoelectric effect,
the temperature difference between the electron and the thermal bath to which
the vibrational modes are coupled provides the driving force. We find that both
magnitude and direction of the generated pure spin current are dependent on the
strength of spin-vibration interaction, the sublevel configuration in dot, the
temperatures of electron and thermal bath, and the tunneling rate between the
dot and the pole. Moreover, in the linear response regime, the kinetic
coefficient is non-monotonic in the temperature $T$ and it reaches its maximum
when $k_BT$ is about one phonon energy. The existence of a strong intradot
Coulomb interaction is irrelevant for our spin battery, provided that
high-order cotunneling processes are suppressed.
