High-field balance equations for electronic transport in weakly nonuniform systems

Balance equations for the spatial distribution of particle number density n(R), drift velocity v(R), and internal energy u(R) [or electron temperature $T_e$(R)] in high-field electronic transport of a weakly nonuniform electron-impurity-phonon system are derived using a model in which the dynamic interactions between different fluid elements are removed by a self-consistent mean-field approximation and thermodynamic forces are neglected. The resistive forces due to impurity and phonon scattering and the electron energy-loss rate to the phonon system have expressions similar to those in uniform case except that now the drift velocity and electron temperature are replaced by the corresponding spatially variable field parameters. Being a form of first-order space-time differential equation, the balance equations determine v(R) and $T_e$(R) as well as the total electric potential if the initial and boundary conditions are given; thus they can easily be applied to the discussion of high electric field transport in weakly nonuniform systems.