A terahertz quantum cascade laser design that combines a wide gain bandwidth, large photon-driven transport and good high-temperature characteristics is presented. It relies on a diagonal transition between a bound state and doublet of states tunnel coupled to the upper state of a phonon extraction stage. The high optical efficiency of this design enables the observation of photon-driven transport over a wide current density range. The relative tolerance of the design to small variations in the barrier thicknesses made it suitable for testing different growth techniques and materials. In particular, we compared the performances of devices grown using molecular-beam epitaxy with those achieved using organometallic chemical vapor deposition. The low-threshold current density and the high slope efficiency makes this device an attractive active region for the development of single-mode quantum cascade lasers based on third-order-distributed feedback structures. Single-mode, high power was achieved with good continuous and pulsed wave operation.