  Using the self-consistent tight-binding Bogoliubov-de Gennes formalism we
have studied the effect of nearest neighbor spin-singlet bond (SB) correlations
on Josephson coupling and proximity effect in graphene SNS Josephson junctions
with conventional s-wave superconducting contacts. Despite the s-wave
superconducting state in the contacts, the SB pairing state inside the junction
has d-wave symmetry and clean, sharp interface junctions resemble a
'bulk-meets-bulk' situation with very little interaction between the two
different superconducting states. In fact, due to a finite-size suppression of
the superconducting state, a stronger SB coupling constant than in the bulk is
needed in order to achieve SB pairing in a junction. For both short clean
zigzag and armchair junctions a d-wave state that has a zero Josephson coupling
to the s-wave state is chosen and therefore the Josephson current decreases
when a SB pairing state develops in these junctions. In more realistic
junctions, with smoother doping profiles and atomic scale disorder at the
interfaces, it is possible to achieve some coupling between the contact s-wave
state and the SB d-wave states. In addition, by breaking the appropriate
lattice symmetry at the interface in order to induce another d-wave state, a
non-zero Josephson coupling can be achieved which leads to a substantial
increase in the Josephson current. We also report on the LDOS of the junctions
and on a lack of zero energy states at interfaces despite the unconventional
order parameters, which we attribute to the near degeneracy of the two d-wave
solutions and their mixing at a general interface.
