  We investigate the spin-dependent thermoelectric effect of graphene flakes
with magnetic edges in the ballistic regime. Employing static, respectively,
dynamic mean-field theory we first show that magnetism appears at the zigzag
edges for a window of Coulomb interactions that increases significantly with
increasing flake size. We then use the Landauer formalism in the framework of
the non-equilibrium Green's function method to calculate the spin and charge
currents in magnetic hexagonal graphene flakes by varying the temperature of
the junction for different flake sizes. While in non-magnetic gated graphene
the temperature gradient drives a charge current, we observe a significant spin
current for hexagonal graphene flakes with magnetic zigzag edges. Specifically,
we show that in the "meta" configuration of a hexagonal flake subject to weak
Coulomb interactions, a pure spin current can be driven just by a temperature
gradient in a temperature range that is promising for device applications.
Bigger flakes are found to yield a bigger window of Coulomb interactions where
such spin currents are induced by the magnetic zigzag edges, and larger values
of the current.
