  We report a combined theoretical and experimental study of electrical
transport in weakly-coupled doped superlattices. Our calculations exhibit
negative differential conductivity at sufficiently high electric fields for all
dopings. In low-doped samples the presence of impurity bands modifies the
current-voltage characteristics substantially and we find two different current
peaks whose relative height is changing with the electron temperature. These
findings can explain the observation of different peaks in the current-voltage
characteristics with and without external THz irradiation in low-doped samples.
  From our microscopic transport model we obtain quantitative agreement with
the experimental current-voltage characteristics without using any fitting
parameters. Both our experimental data and our theory show that absolute
negative conductance persists over a wide range of frequencies of the
free-electron laser source.
