  We present a microscopic theory for the description of the bias-controlled
operation of an exciton-polariton-based heterostructure, in particular, the
polariton laser. Combining together the Poisson equations for the scalar
electric potential and Fermi quasi-energies of electrons and holes in a
semiconductor heterostructure, the Boltzmann equation for the incoherent
excitonic reservoir and the Gross-Pitaevskii equation for the exciton-polariton
mean field, we simulate the dynamics of the system minimising the number of
free parameters and for the first time build a theoretical threshold
characteristics: number of particles vs applied bias. This approach, which also
accounts for the nonlinear (exciton-exciton) interaction, particle lifetime,
and which can, in principle, account for any relaxation mechanisms for the
carriers of charge inside the heterostructure or polariton loss, allows to
completely describe modern experiments on polariton transport and model new
devices.
