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We study the charge and spin transport under high electric field (up toseveral kV/cm) on the surface of topological insulator Bi$_2$Se$_3$, where theelectron-surface optical phonon scattering dominates except at very lowtemperature. Due to the spin mixing of conduction and valence bands, theelectric field not only accelerates electrons in each band, but also leads tointer-band precession. In the presence of the electric field, electrons cantransfer from the valence band to the conduction one via the inter-bandprecession and inter-band electron-phonon scattering. The electron density ineach band varies with the electric field linearly when the electric field isstrong. Due to the spin-momentum locking, a transverse spin polarization, withthe magnitude proportional to the momentum scattering time, is induced by theelectric field. The induced spin polarization depends on the electric fieldlinearly when the latter is small. Moreover, its magnitude is inverselyproportional to the temperature and is insensitive to the electron density athigh temperature. Our investigation also reveals that due to the large relativestatic dielectric constant, the Coulomb scattering is too weak to establish adrifted Fermi distribution with a unified hot-electron temperature in thesteady state under the electric field. After turning off the electric field inthe steady state, the hot carriers cool down in a time scale of energyrelaxation which is very long (of the order of 100-1000 ps) while the spinpolarization relaxes in a time scale of momentum scattering which is quiteshort (of the order of 0.01-0.1 ps).

Hot-carrier transport and spin relaxation on the surface of topological insulator