  We revisit the problem of the decoherence and relaxation of a central spin
coupled to a bath of conduction electrons. We consider both metallic and
semiconducting baths to study the effect of a gap in the bath density of states
(DOS) on the time evolution of the density matrix of the central spin. We use
two weak coupling approximation schemes to study the decoherence. At low
temperatures, though the temperature dependence of the decoherence rate in the
case of a metallic bath is the same irrespective of the details of the bath,
the same is not true for the semiconducting bath. We also calculate the
relaxation and decoherence rates as a function of external magnetic fields
applied both on the central spin and the bath. We find that in the presence of
the gap, there exists a certain regime of fields, for which surprisingly, the
metallic bath has lower rates of relaxation and decoherence than the
semiconducting bath.
