Solution of the multivalley Boltzmann transport equations in Si and GaAs based on the time scales of hydrodynamic equations

Abstract

The previously developed hydrokinetic transport theory is used to arrive at a multivalley transport model for the electron distribution function evolving at the energy relaxation scale. The hydrokinetic distribution described by hydrodynamic parameters, including the density, mean energy, and average velocity, is introduced to approximate the kinetic distribution. The developed multivalley hydrokinetic model, together with the Monte Carlo method, is applied to study nonequilibrium energy and momentum distribution functions of electrons in n-type Si 〈100〉 and GaAs. It is shown that the hydrokinetic concept can be used to characterize extreme nonequilibrium phenomena of the distribution and transport parameters in terms of the relaxation scales of hydrodynamic parameters. The study suggests that evolution of the distribution is strongly influenced by energy relaxation. It is also found that in ultrafast transient situations the influence of velocity relaxation on the distribution function is more pronounced if the ratio τε /τm is larger, where τε and τm are energy and momentum relaxation times, respectively. In general, similar influences of energy and momentum dependences also show in the relaxation times. In Si at room temperature, the ratio is near or below 10 at low or medium field, and the distribution, which is subjected to a rapid change in field, weakly depends on the velocity relaxation. In the Γ valley of GaAs, although the ratio is not larger than that in Si, effects of velocity relaxation are considerably stronger due to much more pronounced velocity overshoot. The hydrokinetic distribution at the energy relaxation scale therefore provides a good description for electrons in Si in extreme nonequilibrium situations, but not in GaAs during the strong overshoot/undershoot interval. In the L valleys the ratio is much larger than 10 at low or medium fields. Consequently, The L-valley distribution function subjected to a drastically increasing field from a low value is also strongly influenced by velocity relaxation even though no overshoot is observed.