  Being a true two-dimensional crystal, graphene has special properties. In
particular, a point-like defect in graphene may have effects in the long range.
This peculiarity questions the validity of using a supercell geometry in an
attempt to explore the properties of an isolated defect. Still, this approach
is often used in ab-initio electronic structure calculations, for instance. How
does this approach converge with the size of the supercell is generally not
tackled for the obvious reason of keeping the computational load to an
affordable level. The present paper addresses the problem of substitutional
nitrogen doping of graphene. DFT calculations have been performed for 9x9 and
10x10 supercells. Although these calculations correspond to N concentrations
that differ by about 10%, the local densities of states on and around the
defects are found to depend significantly on the supercell size. Fitting the
DFT results by a tight-binding Hamiltonian makes it possible to explore the
effects of a random distribution of the substitutional N atoms, in the case of
finite concentrations, and to approach the case of an isolated impurity when
the concentration vanishes. The tight-binding Hamiltonian is used to calculate
the STM image of graphene around an isolated N atom. STM images are also
calculated for graphene doped with 0.5 % concentration of nitrogen. The results
are discussed in the light of recent experimental data and the conclusions of
the calculations are extended to other point defects in graphene.
