  The optical spectra of two dimensional (2D) materials exhibit sharp
absorption peaks that are commonly identified with exciton and trions (or
charged excitons). In this paper, we show that excitons and trions in doped 2D
materials can be described by two coupled Schrodinger-like equations - one
two-body equation for excitons and another four-body equation for trions. In
electron doped 2D materials, a bound trion state is identified with a four-body
bound state of an exciton and an excited conduction band electron-hole pair. In
doped 2D materials, the exciton and the trions states are the not the
eigenstates of the full Hamiltonian and their respective Schrodinger equations
are coupled due to Coulomb interactions. The strength of this coupling
increases with the doping density. Solutions of these two coupled equations can
quantitatively explain all the prominent features experimentally observed in
the optical absorption spectra of 2D materials including the observation of two
prominent absorption peaks and the variation of their energy splittings and
spectral shapes and strengths with the electron density. The optical
conductivity obtained in our work satisfies the optical conductivity sum rule
exactly. A superposition of exciton and trion states can be used to construct a
solution of the two coupled Schrodinger equations and this solution resembles
the variational exciton-polaron state, thereby establishing the relationship
between our approach and Fermi polaron physics.
