  The gapless energy band spectra make the structures based on graphene and
graphene bilayers with the population inversion created by optical or injection
pumping to be promising media for the interband terahertz (THz) lasing.
However, a strong intraband absorption at THz frequencies still poses a
challenge for efficient THz lasing. In this paper, we show that in the pumped
graphene bilayer structures, the indirect interband radiative transitions
accompanied by scattering of carriers caused by disorder can provide a
substantial negative contribution to the THz conductivity (together with the
direct interband transitions).
  In the graphene bilayer structures on high-$\kappa$ substrates with point
charged defects, these transitions almost fully compensate the losses due to
the intraband (Drude) absorption. We also demonstrate that the indirect
interband contribution to the THz conductivity in a graphene bilayer with the
extended defects (such as the charged impurity clusters, surface corrugation,
and nanoholes) can surpass by several times the fundamental limit associated
with the direct interband transitions and the Drude conductivity. These
predictions can affect the strategy of the graphene-based THz laser
implementation.
