  Transport and gain properties of a resonant-phonon terahertz quantum cascade
laser are calculated using nonequilibrium Green's functions. Impurity
scattering is shown to be responsible for contrasting nonlinear effects in the
transport and the gain properties. For typical doping concentrations, the
current density is found to be weakly sensitive to the impurity scattering
strength. In contrast, the calculated gain is found to be very sensitive to the
impurity scattering strength. This difference is attributed to the strong
momentum dependence of the long-range coupling to charged impurities.
Small-momentum impurity scattering is shown to be responsible for an incoherent
regime of resonant tunneling processes. These new insights into the crucial
role of impurity scattering open a new route of improvement of terahertz
quantum cascade lasers by engineering of the doping profile.
