  We investigate the electronic, thermal, and optical characteristics of
graphene like SiC$_x$ structure using model calculations based on density
functional theory. The change in the energy bandgap can be tuned by the Si
atomic configuration, rather than the dopants ratio. The effects of the
concentration of the Si atoms and the shape of supercell are kept constant, and
only the interaction effects of two Si atoms are studied by varying their
positions. If the Si atoms are at the same sublattice positions, a maximum
bandgap is obtained leading to an increased Seebeck coefficient and figure of
merit. A deviation in the Wiedemann-Franz ratio is also found, and a maximum
value of the Lorenz number is thus discovered. Furthermore, a significant red
shift of the first peak of the imaginary part of the dielectric function
towards the visible range of the electromagnetic radiation is observed. On the
other hand, if the Si atoms are located at different sublattice positions, a
small bandgap is seen because the symmetry of sublattice remains almost
unchanged. Consequently, the Seebeck coefficient and the dielectric function
are only slightly changed compared to pristine graphene. In addition, the
electron energy loss function is suppressed in Si-doped graphene. These unique
variations of the thermal and the optical properties of Si-doped graphene are
of importance to understand experiments relevant to optoelectronic
applications.
