  A novel lasing scheme for terahertz quantum cascade lasers, based on
consecutive phonon-photon-phonon emissions per module, is proposed and
experimentally demonstrated. The charge transport of the proposed structure is
modeled using a rate equation formalism. An optimization code based on a
genetic algorithm was developed to find a four-well design in the
$\mathrm{GaAs/Al_{0.25}Ga_{0.75}As}$ material system that maximizes the product
of population inversion and oscillator strength at 150 K. The fabricated
devices using Au double-metal waveguides show lasing at 3.2 THz up to 138 K.
The electrical characteristics display no sign of differential resistance drop
at lasing threshold, which suggests - thanks to the rate equation model - a
slow depopulation rate of the lower lasing state, a hypothesis confirmed by
non-equilibrium Green's function calculations.
