  The relaxation process of electron spin in systems of electrons interacting
with piezoelectric deformation phonons that are mediated through spin-orbit
interactions was interpreted from a microscopic point of view using the formula
for the electron spin relaxation times derived by a projection-reduction
method. The electron spin relaxation times in two polymorphic structures of GaN
were calculated. The piezoelectric material constant for the wurtzite structure
obtained by a comparison with a previously reported experimental result was
P_pe=1.5 x 10^29 eV/m. The temperature and magnetic field dependence of the
relaxation times for both wurtzite and zinc-blende structures were similar, but
the relaxation times in zinc-blende GaN were smaller and decreased more rapidly
with increasing temperature and magnetic field than that in wurtzite GaN. This
study also showed that the electron spin relaxation for wurtzite GaN at low
density could be explained by the Elliot-Yafet process but not for zinc-blende
GaN in the metallic regime.
