  Excitation of surface plasmon waves in extrinsic graphene is studied using a
full-wave electromagnetic field solver as analysis engine. Particular emphasis
is placed on the role played by spatial dispersion due to the finite size of
the two-dimensional material at the micro-scale. A simple instructive set up is
considered where the near field of a wire antenna is held at sub-micrometric
distance from a disk-shaped graphene patch. The key-input of the simulation is
the graphene conductivity tensor at terahertz frequencies, being modeled by the
Boltzmann transport equation for the valence and conduction electrons at the
Dirac points~(where a linear wave-vector dependence of the band energies is
assumed). The conductivity equation is worked out in different levels of
approximations, based on the relaxation time ansatz with an additional
constraint for particle number conservation. Both drift and diffusion currents
are shown to significantly contribute to the spatially dispersive anisotropic
features of micro-scale graphene. More generally, spatial dispersion effects
are predicted to influence not only plasmon propagation free of external
sources, but also typical scanning probe microscopy configurations. The paper
set the focus on plasmon excitation phenomena induced by near field probes,
being a central issue for the design of optical devices and photonic circuits.
