  We study the spin dynamics in the presence of impurity and electron-electron
(e-e) scattering in a III-V semiconductor quantum well with arbitrary
spin-orbit coupling (SOC) strength and symmetry at finite temperature. We
derive the coupled spin-charge dynamic equations in the presence of inelastic
scattering and provide a new formalism that describes the spin relaxation and
dynamics in both the weak and the strong SOC regime in a unified way. In the
weak SOC regime, as expected, our theory reproduces all previous
zero-temperature results, most of which have focused on impurity-scattering
induced spin-charge dynamics. In the regime where the strength of the Rashba
and linear Dresselhaus SOC match, known as the SU(2) symmetry point,
experiments have observed the spin-helix mode with a large spin-lifetime whose
unexplained non-monotonic temperature dependence peaks at around 75 K. As a key
test of our theory, we are able to naturally explain quantitatively this
non-monotonic dependence and show that it arises as a competition between the
Dyakonov-Perel mechanism, suppressed at the SU(2) point, and the Elliott-Yafet
mechanism. In the strong SOC regime, we show that our theory directly
reproduces the only previous known analytical result at the SU(2) symmetry
point in the ballistic regime. It also explains, as we have shown previously,
the rise of damped oscillating dynamics when the electron scattering time is
larger than half of the spin precession time due to the SOC. Hence, we provide
a unified theory of the spin-dynamics in two dimensional electron gases in the
full phase diagram experimentally accessible.
