**Abstract**

In designing the terahertz quantum cascade lasers, electron injection manner indeed plays a significant role to achieve the population inversion. The resonant tunneling process is commonly employed for this injection process but waste more than 50% fraction of populations out of the active region owing to resonance alignment, and the injection efficiency is obviously degraded due to thermal incoherence. An alternative approach is to consider the phonon-assisted injection process that basically contributes to most of the populations to the upper lasing level. However, this manner is still not realized in experiments if a short-period design only containing two quantum wells is used. In this work, it is found in this design that the population inversion is indeed well improved; however, the optical gain is inherently low even at a low temperature. Those two opposite trends are ascribed to a strong parasitic absorption overlapping the gain. The magnitude of this overlap is closely related to the lasing frequency, where frequencies below 3 THz suffer from fewer effects.

**Keywords:** intersubband transition, terahertz, quantum cascade lasers, parasitic channels, optical gain
