  We investigate the hole spin relaxation due to the Rashba spin-orbit coupling
induced by an external perpendicular electric field in bilayer WSe$_2$. The
Rashba spin-orbit coupling coefficients in bilayer WSe$_2$ are constructed from
the corresponding monolayer ones. In contrast to monolayer WSe$_2$, the
out-of-plane component of the bilayer Rashba spin-orbit coupling acts as a
Zeeman-like field with opposite directions but identical values in the two
valleys. For in-plane spins, this Zeeman-like field, together with the
intervalley hole-phonon scattering, opens an intervalley spin relaxation
channel, which is found to dominate the in-plane spin relaxation in bilayer
WSe$_2$ even at low temperature. For out-of-plane spins, this Zeeman-like field
is superimposed by the identical Hartree-Fock effective magnetic fields in the
two valleys, and hence different total effective magnetic fields between two
valleys are obtained. Owing to the large difference of the total fields at
large spin polarization, different out-of-plane spin relaxation times in the
two valleys are obtained when the intervalley hole-phonon scattering is weak at
low temperature and low hole density. This difference in the spin relaxation
times can be suppressed by enhancing the intervalley hole-phonon scattering
through increasing temperature or hole density. Moreover, at large spin
polarization and low temperature, due to the weak intravalley hole-phonon
scattering but relatively strong hole-hole Coulomb scattering, the fast spin
precessions are found to result in a quasi hot-hole Fermi distribution
characterized by an effective hot-hole temperature larger than the temperature,
which also enhances the intervalley scattering. During this process, it is
interesting to discover that the initially equal hole densities in the two
valleys are broken in the temporal evolution, and a valley polarization is
built up. ....
