  We study the electron transport through a graphene nanoribbon-superconductor
junction. Both zigzag and armchair edge graphene nanoribbons are considered,
and the effects of the magnetic field and disorder on the transport property
are investigated. By using the tight-binding model and the non-equilibrium
Green's function method, the expressions of the current, conductance, normal
tunneling coefficient, and Andreev reflection coefficient are obtained. For a
clean system and at zero magnetic field, the linear conductance increases
approximatively in a linear fashion with the on-site energy. In the presence of
a magnetic field and a moderate disorder, the linear conductance exhibits
plateau structures for both armchair and zigzag edges. The plateau values
increase with the width of the graphene ribbon. With a wide sample width, a
saturated plateau value of $|\nu|e^2/h$ emerges at the filling factor $\nu$.
For a small filling factor, the conductance can reach the saturated value at a
small width, but for a high filling factor, it requires to have a quite wide
sample width to reach the saturated value. In particular, the Andreev
reflection coefficient is always at 0.5 after reaching the saturated value,
independent of any system parameters. In addition, we also consider the finite
bias case, in which the Andreev reflection coefficient and normal tunneling
coefficient are studied.
