  Motivated by recent experiments of successfully carving out stable carbon
atomic chains from graphene, we investigate a device structure of a carbon
chain connecting two zigzag graphene nanoribbons with highly tunable
spin-dependent transport properties. Our calculation based on the
non-equilibrium Green's function approach combined with the density functional
theory shows that the transport behavior is sensitive to the spin configuration
of the leads and the bridge position in the gap. A bridge in the middle gives
an overall good coupling except for around the Fermi energy where the leads
with anti-parallel spins create a small transport gap while the leads with
parallel spins give a finite density of states and induce an even-odd
oscillation in conductance in terms of the number of atoms in the carbon chain.
On the other hand, a bridge at the edge shows a transport behavior associated
with the spin-polarized edge states, presenting sharp pure $\alpha$-spin and
$\beta$-spin peaks beside the Fermi energy in the transmission function. This
makes it possible to realize on-chip interconnects or spintronic devices by
tuning the spin state of the leads and the bridge position.
