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Heavy charged Higgs bosons ($H^\pm$) of a Type II 2-Higgs Doublet Model(2HDM) can be detected at future electron-positron Linear Colliders (LCs) evenwhen their mass is larger than half the collider energy. The single Higgs mode$e^+e^-\to t\bar b H^- + ~{\rm{c.c.}} \to 4b +{\rm{j}}{\rm{j}} + \ell +p_T^{\rm{miss}}$ (where j represents a jet and with $\ell=e,\mu$) contributesto extend the discovery reach of $H^\pm$ states into the mass region$M_{H^\pm}\gsim \sqrt s/2$, where the well studied pair production channel$e^+e^-\to H^-H^+$ is no longer available. With a technique that allows one toreconstruct the neutrino four-momentum in the decay $t\to b W^+\to b\ell^+\nu$, one can suppress the initially overwhelming main irreduciblebackground due to $e^+e^-\to t\bar t b\bar b$ (via a gluon splitting into$b\bar b$ pairs) to a negligible level. However, for currently foreseenluminosities, one can establish a statistically significant $H^\pm$ signal onlyover a rather limited mass region, of 20 GeV or so, beyond $M_{H^\pm}\approx\sqrt s/2$, for very large or very small values of $\tan\beta$ and providedhigh $b$-tagging efficiency can be achieved.Comment: 18 pages, latex, 8 figures (version to appear in EPJC

Detection of heavy charged Higgs bosonsin $e^ + e^-\to t\bar b H^-$ production at future Linear Colliders