  We study theoretically the effect of the fermion and boson densities on the
superconductivity transition critical temperature $(T_c)$ of a two dimensional
electron gas (2DEG), where superconductivity is mediated by a Bose-Einstein
condensate of exciton-polaritons. The critical temperature is found to increase
with the boson density, but surprisingly it decreases with the 2DEG density
increase. This makes doped semiconductor structures with shallow Fermi energies
better adapted for observation of the exciton-induced superconductivity than
metallic layers. For the realistic GaAs-based microcavities containing-doped
and neutral quantum wells we estimate $T_c$ as close to 50K. Superconductivity
is suppressed by magnetic fields of the order of 4T due to the Fermi surface
renormalisation.
