  We report on the dynamics of electron spins in n-doped CdTe layers that
differs significantly from the expected response derived from the studies
dedicated to electron spin relaxation in n-GaAs. At zero magnetic field, the
electron spin noise spectra exhibit a two-peak structure - a zero-frequency
line and a satellite - that we attribute to the electron spin precession in a
frozen random nuclear spin distribution. This implies a surprisingly long
electron spin correlation time whatever the doping level, even above the Mott
transition. Using spatiotemporal spin noise spectroscopy, we demonstrate that
the observation of a satellite in the spin noise spectra and a fast spin
diffusion are mutually exclusive. This is consistent with a shortening of the
electron spin correlation time due to hopping between donors. We interpret our
data via a model assuming that the low temperature spin relaxation is due to
hopping between donors in presence of hyperfine and anisotropic exchange
interactions. Most of our results can be interpreted in this framework. First,
a transition from inhomogeneous to homogeneous broadening of the spin noise
peaks and the disappearance of the satellite are observed when the hopping rate
becomes larger than the Larmor period induced by the local nuclear fields. In
the regime of homogeneous broadening the ratio between the spin diffusion
constant and the spin relaxation rate has a value in good agreement with the
Dresselhaus constant. In the regime of inhomogeneous broadening, most of the
samples exhibit a broadening consistent with the distribution of local nuclear
fields. We obtain a new estimate of the hyperfine constants in CdTe and a value
of 0.10 Tesla for the maximum nuclear field. Finally, our study also reveals a
puzzle as our samples behave as if the active donor concentration was reduced
by several orders of magnitudes with respect to the nominal values.
