  We study the nonequilibrium Seebeck spin transport across metal-magnetic
insulator interfaces. The conjugate-converted thermal-spin transport is
assisted by the exchange interaction at the interface, between conduction
electrons in the metal lead and localized spins in the insulating magnet lead.
We predict the rectification and negative differential spin Seebeck effect and
resolve their microscopic mechanism, as a consequence of the
strongly-fluctuated electronic density of states in the metal lead. The
rectification of spin Peltier effect is also discussed. The phenomena predicted
here are relevant for designing efficient spin/magnon diode and transistor,
which could play crucial roles in controlling energy and information in
functional devices.
