  We present self-consistent calculations of electron transport in graphene
nanoconstrictions within the Hartree approximation. We consider suspended
armchair ribbons with V-shaped constrictions having perfect armchair or zigzag
edges as well as mesoscopically smooth but atomically stepped constrictions
with cosine profiles. Our calculations are based on a tight-binding model of
the graphene and account for electron-electron interactions in both the
constriction and the semi-infinite leads explicitly. We find that electron
interactions result in (i) Electrons accumulating along edges of the uniform
ribbon and along zigzag and cosine constriction edges but not along armchair
constriction edges. (ii) The first subband showing almost perfect transmittance
due to localization at the uniform graphene boundary except at low energies for
the cases of zigzag and cosine constrictions where Bloch stop-bands form in
related periodic structures. (iii) The second subband being almost perfectly
blocked by the constriction. (iv) Electron interactions favor intra-subband
scattering while the non-interacting electron theory predicts predominance of
inter-subband scattering. (v) Conductance quantization for the first few
conductance steps being more pronounced for armchair constrictions but less so
for zigzag constrictions. (vi) A much more prominent 2e^2/h conductance plateau
for the cosine constriction than is found in the absence of electron
interactions. Possible implications for recent experiments are briefly
discussed.
