Nonlinear Electrical Transport in Semiconductors : a Nonequilibrium Thermodynamics Approach

The electrical transport problems in semiconductors are nonequilibrium phenomena and most conveniently described by the hydrodynamic equations (or balance equations) that involve the directly measurable, macroscopic variables of the system [1–8]. Such nonequilibrium state-variables are the carrier density, drift velocity, and temperature in the conventional hydrodynamic description [1–4] and also momentum and heat fluxes are included in the generalized hydrodynamic equations approach [5–8]. Due to the many-particle nature, the observables are defined microscopically as the statistical average of the relevant operators over the nonequilibrium occupancy, the single-particle distribution function. In the linear response regime (or near equilibrium) where the external perturbation is weak, there exists the firm foundation between the desired macroscopic quantities and the underlying microscopic physics. For instance, the Ohmic conductivity can be calculated by either solution to the Boltzmann equation or the Kubo formula, equivalently [9]. However, in the high-field regime there remain the fundamental difficulties in understanding the nonlinear transport properties. First of all, it is not a priori obvious what are the relevant quantities to be specified, since physics depends on space and time scales under consideration. Also, it is not known how to comply with the macroscopic law of thermodynamics under general nonequilibrium situation, in particular the second law of thermodynamics. On the other hand, the modern miniaturization trend of semiconductor devices demands a better understanding of this long-standing problems ever since. In this paper, we attempt to tackle this nonlinear transport problem in semiconductors while giving much effort to constructing thermodynamic consistencies in the theory. In solids, the electrical current density j is most mi-