Quantum theory of thermal noises for steady-state hot-electron transport under a strong electric field

Current fluctuation and thermal noises of an electron system biased at finite electric field E and steady current I are studied. Owing to the separability of the motion of the center of mass of electrons from the relative motions of electrons for a parabolic band, we have derived and solved a Langevin-type equation for the fluctuating center-of-mass velocity operator δV^. The formulas for the current fluctuation spectrum ${\Gamma }_{\Delta I}$(ω), the voltage fluctuation spectrum ${\Gamma }_{\Delta V}$(ω), the diffusion coefficient $D_s$(t), and zero-frequency noise temperature $T_n$ have been derived microscopically. In particular, we obtain a generalized fluctuation-dissipation relation for the fluctuating force operator at a finite electric field and in nonequilibrium steady state. Analytical and numerical calculations of these quantities have been carried out for both degenerate and nondegenerate electron systems. While our results for a degenerate electron system may provide a better understanding of the quantum noise at extremely low lattice temperature, the essential features of our calculated diffusion coefficient and noise temperature for a nondegenerate electron system are in good agreement with those of Monte Carlo simulation.