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Semiconductor superlattices (SL) may be described by a Boltzmann-Poissonkinetic equation with a Bhatnagar-Gross-Krook (BGK) collision term whichpreserves charge, but not momentum or energy. Under appropriate boundary andvoltage bias conditions, these equations exhibit time-periodic oscillations ofthe current caused by repeated nucleation and motion of charge dipole waves.Despite this clear nonequilibrium behavior, if we `close' the system byattaching insulated contacts to the superlattice and keeping its voltage biasto zero volts, we can prove the H theorem, namely that a free energy $\Phi(t)$of the kinetic equations is a Lyapunov functional ($\Phi\geq 0$, $d\Phi/dt\leq0$). Numerical simulations confirm that the free energy decays to itsequilibrium value for a closed SL, whereas for an `open' SL under appropriatedc voltage bias and contact conductivity $\Phi(t)$ oscillates in time with thesame frequency as the current self-sustained oscillations.

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