  In this paper, the production of the charged Higgs pair associated with the
$Z^0$ boson is analyzed in the minimal extension of the standard model the
so-called two-Higgs-doublet model (2HDM). The process $e^+e^- \rightarrow
H^+H^-Z^0$ is calculated at the tree level including all the possible diagrams
in 2HDM. The numerical analysis is performed in consideration of the current
experimental constraints and various scenarios for the free parameters of the
model. The results are presented as a function of center-of-mass energy, the
charged Higgs mass ($m_{H^\pm}$), and the ratio of the vacuum expectation
values ($t_\beta$). The unpolarized cross section, taking into account the
results in the flavor physics, gets up to $0.278\text{ fb}$ for
$m_{H^\pm}=175\text{ GeV}$, and it declines with decreasing $m_{H^\pm}$ in
Type-I. However, it gets down to $0.073\text{ fb}$ for $m_{H^\pm}=500\text{
GeV}$ in Type-II. Further, the calculation is also carried out in the
non-alignment scenario and low-$m_{h^0}$ scenario. The effect of the polarized
incoming $e^+$ and $e^-$ beams shows that the cross section is enhanced by a
factor up to 2.5 at P(+0.60,-0.80) polarization configuration. Decay channels
of the charged Higgs, possible final states of the process, and some
differential distributions belong to the charged Higgs and $Z^0$ boson are
examined for each scenario. The analysis shows that some channels have higher
branching ratio such as $H^+\rightarrow t \bar{b}$, $H^+\rightarrow W^+h^0$,
and $H^+\rightarrow W^+H^0$. These decay channels are essential for the charged
Higgs searches in the lepton colliders regarding the scenarios interested. The
detection of the charged Higgs is a powerful sign for the extended scalar
sectors, and the results show the potential of a future lepton collider.
