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

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

2010-11-12

arxiv.org/abs/1011.3007v2

J. Stat. Mech. (2011) P01018

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

15 pages, 3 figures, minor revision of latex file

Scientific paper

10.1088/1742-5468/2011/01/P01018

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.

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