  The D0 experiment has reported a direct search for a charged Higgs boson
produced by $q \bar q$ annihilation and decaying to $t \bar b$ final state, in
the $180 \leq M_{H^+} \leq 300$ GeV mass range. The analysis has lead to upper
limits on the production cross section in the framework of the two Higgs
doublet model types I, II and III. We compare the predictions of two different
scenarios in the framework of the two Higgs doublet type III to the cross
section limits reported by D0 collaboration, and we obtain constraints on the
charged Higgs mass, for the case when the charged Higgs mass is bigger than the
top quark mass. Also, searches for the charged Higgs boson with a mass smaller
than top quark mass are considered, we discuss the possible limits on the
charged Higgs boson mass obtained from measurements of the ratio
$R_\sigma=\sigma_{t \bar t}^{l+jets}/\sigma_{t \bar t}^{dileptons}$ within the
two Higgs doublet model type III.
