  In this article, we use a many-body approach to study the absorption spectra
of electron-doped two-dimensional semiconductors. Optical absorption is modeled
by a many-body scattering Hamiltonian which describes an exciton immersed in a
Fermi sea. The interaction between electron and exciton is approximated by an
effective scattering potential, and optical spectra are calculated by solving
for the exciton Green's function. From this approach, a trion state can be
assigned as a bound state of an electron-exciton scattering process, and the
doping-dependent phenomena observed in the spectra can be attributed to several
many-body effects induced by the interaction with the Fermi sea. While the
many-body scattering Hamiltonian can not solved exactly, we reduce the problem
to two limiting solvable situations. The first approach approximates the full
many-body problem by a simple scattering process between the electron and the
exciton, with a self-energy obtained by solving a Bethe-Salpeter equation
(BSE). An alternate approach assumes an infinite mass for the exciton, such
that the many-body scattering Hamiltonian reduces to a Mahan-Nozieres-De
Dominicis (MND) model. The exciton Green's function can then be solved
numerically exactly by a determinantal formulation, and the optical spectra
show signatures of the Fermi-edge singularity at high doping densities. The
full doping dependence and temperature dependence of the exciton and trion
lineshapes are simulated via these two approximate approaches, with the results
compared to each other and to experimental expectations.
