  The origin of anomalous resistivity peak and accompanied sign reversal of
Hall resistivity of ZrTe$_5$ has been under debate for a long time. Although
various theoretical models have been proposed to account for these intriguing
transport properties, a systematic study from first principles view is still
lacking. In this work, we present a first principles calculation combined with
Boltzmann transport theory to investigate the transport properties in
narrow-gap semiconductors at different temperatures and doping densities within
the relaxation time approximation. Regarding the sensitive
temperature-dependent chemical potential and relaxation time of semiconductors,
we take proper approximation to simulate these two variables, and then
comprehensively study the transport properties of ZrTe$_5$ both in the absence
and presence of an applied magnetic field. Without introducing topological
phases and correlation interactions, we qualitatively reproduced crucial
features observed in experiments, including zero-field resistivity anomaly,
nonlinear Hall resistivity with sign reversal, and non-saturating
magnetoresistance at high temperatures. Our calculation allows a systematic
interpretation of the observed properties in terms of multi-carrier and Fermi
surface geometry. Our method can be extended to other narrow-gap semiconductors
and further pave the way to explore interesting and novel transport properties
of this field.
