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

TL;DR

This paper investigates the nonequilibrium dynamics of semiconductor superlattices modeled by a Boltzmann-BGK equation, proving a free energy decay in closed systems and linking oscillations to self-sustained current behavior in open systems.

Contribution

It establishes a Lyapunov functional for the Boltzmann-BGK model of superlattices and connects free energy oscillations to self-sustained current oscillations.

Findings

Free energy decays to equilibrium in closed superlattices.

In open superlattices, free energy oscillates with current.

Numerical simulations confirm theoretical predictions.

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 ($\Phi\geq 0$, $d\Phi/dt\leq 0$). 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.