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Wigner-Poisson and nonlocal drift-diffusion model equations for semiconductor superlattices
A Wigner-Poisson kinetic equation describing charge transport in doped
semiconductor superlattices is proposed. Electrons are supposed to occupy the
lowest miniband, exchange of lateral momentum is ignored and the
electron-electron interaction is treated in the Hartree approximation. There
are elastic collisions with impurities and inelastic collisions with phonons,
imperfections, etc. The latter are described by a modified BGK
(Bhatnagar-Gross-Krook) collision model that allows for energy dissipation
while yielding charge continuity. In the hyperbolic limit, nonlocal
drift-diffusion equations are derived systematically from the kinetic
Wigner-Poisson-BGK system by means of the Chapman-Enskog method. The
nonlocality of the original quantum kinetic model equations implies that the
derived drift-diffusion equations contain spatial averages over one or more
superlattice periods. Numerical solutions of the latter equations show
self-sustained oscillations of the current through a voltage biased
superlattice, in agreement with known experiments.Comment: 20 pages, 1 figure, published as M3AS 15, 1253 (2005) with
correction
Nonequilibrium free energy, H theorem and self-sustained oscillations for Boltzmann-BGK descriptions of semiconductor superlattices
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
of the kinetic equations is a Lyapunov functional (, ). 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 oscillates in time with the
same frequency as the current self-sustained oscillations.Comment: 15 pages, 3 figures, minor revision of latex fil
Self-sustained current oscillations in the kinetic theory of semiconductor superlattices
We present the first numerical solutions of a kinetic theory description of
self-sustained current oscillations in n-doped semiconductor superlattices. The
governing equation is a single-miniband Boltzmann-Poisson transport equation
with a BGK (Bhatnagar-Gross-Krook) collision term. Appropriate boundary
conditions for the distribution function describe electron injection in the
contact regions. These conditions seamlessly become Ohm's law at the injecting
contact and the zero charge boundary condition at the receiving contact when
integrated over the wave vector. The time-dependent model is numerically solved
for the distribution function by using the deterministic Weighted Particle
Method. Numerical simulations are used to ascertain the convergence of the
method. The numerical results confirm the validity of the Chapman-Enskog
perturbation method used previously to derive generalized drift-diffusion
equations for high electric fields because they agree very well with numerical
solutions thereof.Comment: 26 pages, 16 figures, to appear in J. Comput. Phy
Convergence of a Finite Volume Scheme for a Corrosion Model
In this paper, we study the numerical approximation of a system of partial
dif-ferential equations describing the corrosion of an iron based alloy in a
nuclear waste repository. In particular, we are interested in the convergence
of a numerical scheme consisting in an implicit Euler scheme in time and a
Scharfetter-Gummel finite volume scheme in space
Two mini-band model for self-sustained oscillations of the current through resonant tunneling semiconductor superlattices
A two miniband model for electron transport in semiconductor superlattices
that includes scattering and interminiband tunnelling is proposed. The model is
formulated in terms of Wigner functions in a basis spanned by Pauli matrices,
includes electron-electron scattering in the Hartree approximation and modified
Bhatnagar-Gross-Krook collision tems. For strong applied fields, balance
equations for the electric field and the miniband populations are derived using
a Chapman-Enskog perturbation technique. These equations are then solved
numerically for a dc voltage biased superlattice. Results include
self-sustained current oscillations due to repeated nucleation of electric
field pulses at the injecting contact region and their motion towards the
collector. Numerical reconstruction of the Wigner functions shows that the
miniband with higher energy is empty during most of the oscillation period: it
becomes populated only when the local electric field (corresponding to the
passing pulse) is sufficiently large to trigger resonant tunneling.Comment: 26 pages, 3 figures, to appear in Phys. Rev.
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