  The spin-relaxation time for electrons in the impurity band of semiconductors
with wurtzite crystal structure is determined. The effective Dresselhaus
spin-orbit interaction Hamiltonian is taken as the source of the spin
relaxation at low temperature and for doping densities corresponding to the
metallic side of the metal-insulator transition. The spin-flip hopping matrix
elements between impurity states are calculated and used to set up a
tight-binding Hamiltonian that incorporates the symmetries of wurtzite
semiconductors. The spin-relaxation time is obtained from a semiclassical model
of spin diffusion, as well as from a microscopic self-consistent diagrammatic
theory of spin and charge diffusion in doped semiconductors. Estimates are
provided for particularly important materials. The theoretical spin-relaxation
times compare favorably with the corresponding low-temperature measurements in
GaN and ZnO. For InN and AlN, we predict that tuning of the spin-orbit coupling
constant induced by an external potential leads to a potentially dramatic
increase of the spin-relaxation time related to the mechanism under study.
