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Hole spin relaxation in bilayer WSe$_2$

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. ....