  We perform a comparative \textit{ab-initio} study of Mn-doped Germanium
semiconductor using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation
functional, DFT+$U$ and Heyd-Scuseria-Ernzerhof hybrid functional (HSE). We
show that the HSE functional is able to correctly account for the relevant
ground state properties of the host matrix as well as of Mn-doped
semiconductor. Although the DFT+$U$ and the HSE description are very similar,
some differences still remain. In particular, the half-metallicity is lost
using DFT+$U$ when a suitable $U$ value, tuned to recover the photoemission
spectra, is employed. For comparison, we also discuss the case of Mn in
Silicon.
