Electron charge and spin delocalization revealed in the optically probed longitudinal and transverse spin dynamics in $n$-GaAs

The evolution of the electron spin dynamics as consequence of carrier delocalization in $n$-type GaAs is investigated by the recently developed extended pump-probe Kerr/Faraday rotation spectroscopy. We find that isolated electrons localized on donors demonstrate a prominent difference between the longitudinal and transverse spin relaxation rates in magnetic field, which is almost absent in the metallic phase. The inhomogeneous transverse dephasing time $T_2^{\ast }$ of the spin ensemble strongly increases upon electron delocalization as a result of motional narrowing that can be induced by increasing either the donor concentration or the temperature. An unexpected relation between $T_2^{\ast }$ and the longitudinal spin relaxation time $T_1$ is found, namely that their product is about constant, as explained by the magnetic field effect on the spin diffusion. We observe a two-stage longitudinal spin relaxation which suggests the establishment of spin temperature in the system of exchange-coupled donor-bound electrons.