  We theoretically investigate the longitudinal spin Seebeck effect, in which
the spin current is injected from a ferromagnet into an attached nonmagnetic
metal in a direction parallel to the temperature gradient. Using the fact that
the phonon heat current flows intensely into the attached nonmagnetic metal in
this particular configuration, we show that the sign of the spin injection
signal in the longitudinal spin Seebeck effect can be opposite to that in the
conventional transverse spin Seebeck effect when the electron-phonon
interaction in the nonmagnetic metal is sufficiently large. Our linear-response
approach can explain the sign reversal of the spin injection signal recently
observed in the longitudinal spin Seebeck effect.
