Nonequilibrium free energy, H theorem and self-sustained oscillations for Boltzmann-BGK descriptions of semiconductor superlattices

Abstract

Semiconductor superlattices (SL) may be described by a Boltzmann-Poisson kinetic equation with a Bhatnagar-Gross-Krook (BGK) collision term which preserves charge, but not momentum or energy. Under appropriate boundary and voltage bias conditions, these equations exhibit time-periodic oscillations of the current caused by repeated nucleation and motion of charge dipole waves. Despite this clear nonequilibrium behavior, if we `close' the system by attaching insulated contacts to the superlattice and keeping its voltage bias to zero volts, we can prove the H theorem, namely that a free energy $\Phi (t)$ of the kinetic equations is a Lyapunov functional ($\Phi \ge 0$, $d\Phi /dt\le 0$). Numerical simulations confirm that the free energy decays to its equilibrium value for a closed SL, whereas for an `open' SL under appropriate dc voltage bias and contact conductivity $\Phi (t)$ oscillates in time with the same frequency as the current self-sustained oscillations.

15 pages, 3 figures, minor revision of latex file

References in corpus (3)

• Generalized drift-diffusion model for miniband superlattices
• Nonlinear electron and spin transport in semiconductor superlattices
• Self-sustained current oscillations in the kinetic theory of semiconductor superlattices