  We propose, for the first time, a valley Seebeck effect in gate tunable
zigzag graphene nanoribbons as a result of the interplay between thermal
gradient and valleytronics. A pure valley current is further generated by the
thermal gradient as well as the external bias. In a broad temperature range,
the pure valley current is found to be linearly dependent on the temperature
gradient while it increases with the increasing temperature of one lead for a
fixed thermal gradient. A valley field effect transistor (FET) driven by the
temperature gradient is proposed that can turn on and off the pure valley
current by gate voltage. The threshold gate voltage and on valley current are
proportional to the temperature gradient. When the system switches on at
positive gate voltage, the pure valley current is nearly independent of gate
voltage. The valley transconductance is up to 30 {\mu}S if we take Ampere as
the unit of the valley current. This valley FET may find potential application
in future valleytronics and valley caloritronics.
