Quasiclassical kinetic equation for charge-carrier transport in a semiconductor

Abstract

Starting with the Uehling–Uhlenbeck quasiclassical generalized Boltzmann equation, a kinetic equation is derived for the distribution function f(ε,t) for charge carriers of energy ε and effective mass m in a semiconductor in the presence of an electric field E. Charge-carrier scattering with optical and acoustic phonons through strain and polar interactions are taken into account. Introducing a Legendre-polynomial expansion of the distribution function and assuming spherical constant energy surfaces, the following equation emerges in the limit of large charge carrier-to-phonon energy: (∂f0/∂t)+(eE/3) ((2ε/m))1/2 ((∂/∂ε) +(1/ε))(f1(ε,0)e−t/τ−T̂eE ((2ε/m))1/2 (∂/∂ε) f0)=(P−aQf0) (∂f0/∂ε) +(F−aGf0) (∂2f0/∂ε2)+Cf0 (1−af0). Collision integrals P,Q,F,G, and C, as well as the collision frequency τ, are evaluated explicitly. The quantum parameter a is 1 in the degenerate domain and 0 in the classical domain. The first two Legendre components of f(ε,t) are written f0 and f1, respectively, and T̂ is a time-dependent operator. Working in steady state and passing to the classical limit it is found that acoustic-strain interactions dominate over all others. Leading terms in this expansion give a new quasiclassical equation for the distribution of charge carriers of a semiconductor in an electric field. Further reducing this equation in the limits of equipartion and mu/p≪1, where p is carrier momentum and u is acoustic phonon velocity, gives a generalized Druyvesteyn equation which incorporates exclusion effects.