We investigate regularizing dispersive effects for various classical equations, e.g., the
Schrödinger and Dirac equations. After Wigner transform, these dispersive estimates are
reduced to moment lemmas for kinetic equations. They yield new regularization results for
the Schrödinger equation (valid up to the semiclassical limit) and the Dirac equation.

I. Gasser

P. A. Markowich

B. Perthame

Dispersion Lemmas Revisited

A numerical study of the dispersive limit of the quantum hydrodynamic equations for
semiconductors is presented. The solution may develop high frequency oscillations
when the scaled Planck constant is small. Numerical evidence is given of the fact that in
such cases the solution does not converge to the solution of the formal limit equations.

Paola Pietra

Carsten Pohl

Weak Limits of the Quantum Hydrodynamic Model

We apply Wigner-transform techniques to the analysis of difference methods for
Schrödinger-type equations in the case of a small Planck constant. In this way we are
able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence
for average values of observables as the Planck constant tends to zero. The
theory developed in this paper is not based on local and global error estimates and does
not depend on whether caustics develop or not.Numerical examples are presented to help interpret the theory.

Peter A. Markowich

Semiclassical Analysis of Discretizations
of Schrödinger-type Equations

A linear (given potential) steady-state Wigner equation is considered in conjunction
with inflow boundary conditions and relaxation-time terms. A brief review of the use of
inflow conditions in the classical case is also discussed. An analytic expansion of
solutions is shown and a recursion relation derived for the given problem under certain
regularity assumptions on the given inflow data. The uniqueness of the physical current
of the solutions is shown and a brief discussion of the lack of charge conservation
associated with the relaxation-time is given.

Horst Lange

Bruce Toomire

P. F. Zweifel

Inflow Boundary Conditions in Quantum Transport Theory

In this paper, we perform a mathematical study of a semiconductor superlattice. Since thethickness of the layers is very small, quantization plays an important role. The modellingis therefore given by a Schr&#246;dinger equation with a periodic potential. The scaled latticethickness is denoted by a small parameter &#8455; which is of the same order of magnitude asthe Planck constant. When this parameter tends to zero, i.e., the semi-classical limit, weobtain classical transport of the charge carriers described by a Vlasov equation.

Philippe  Bechouche

Semi-classical Limit in a Semiconductor Superlattice

Recent experiments on the formation of crystalline CO2 from a newly discovered binary
phase consisting of CO2 and C2H2 at 90° K fail to be adequately simulated by Avrami
equations. The purpose of this note is to develop an alternative to the Avrami model
which can make accurate predictions for these experiments. The new model uses empirical
approximations to the distribution densities of the volumes of three-dimensional
Voronoi cells defined by Poisson-generated crystallization kernels (nuclei). Inside each
Voronoi cell, the growth of the crystal is assumed to be linear in diameter (i.e., cubic in
volume) until the cell is filled by the CO2 crystals and the C2H2 (thought of as a waste
product). The cumulative growth curve is computed by averaging these individual growth
curves with respect to the distribution density of the volumes of the Voronoi cells. Agreement
with the experiments is excellent.

Terry Gough

Reinhard Illner

Modeling Crystallization Dynamics when the Avrami Model Fails

Carl L. Gardner

Guest Editorial

We consider the motion of a single electron under phonon scattering caused by a crystal
lattice. Starting from the Fröhlich Hamiltonian in the second quantization formalism
we derive a kinetic transport model by using the Wigner transformation. Under the
assumption of small electron-phonon interaction we derive asymptotically the operator
representing electron-phonon scattering in the Wigner equation. We then consider some
scaling limits and finally we give the connection of our result to the well known Barker-Ferry equation.

Florian Frommlet

Christian Ringhofer

A Wignerfunction Approach to Phonon Scattering

The “smooth” quantum hydrodynamic (QHD) model is derived specifically to handle
in a mathematically rigorous way the discontinuities in the classical potential energy
which occur at heterojunction barriers in quantum semiconductor devices. Smooth QHD
model simulations of the resonant tunneling diode are presented which exhibit enhanced
negative differential resistance when compared with simulations using the original O(ħ2)
QHD model. In addition, smooth QHD simulations of a classical electron shock wave are
presented which agree with classical hydrodynamic model simulations and which do not
exhibit the spurious dispersive oscillations of the O(ħ2) QHD model.

Theory and Simulation of the Smooth Quantum
Hydrodynamic Model

In this work we discuss relativistic corrections for the description of charge carriers in a
quantum mechanical framework. The fundamental equation is the Dirac equation
which takes into account also the electron's spin. However, this equation intrinsically
also incorporates positrons which play no role in applications in solid state physics. We
give a rigorous derivation of the Pauli equation describing electrons in a first order
approximation of the Dirac equation in the limit of infinite velocity of light. We deal
with time-dependent electromagnetic potentials where no rigorous results have been
given before. Our approach is based on the use of appropriate projection operators for
the electron and the positron component of the spinor which are better suited than the
widely used simple splitting into ‘upper (large)’ and ‘lower (small) component’. We also
systematically derive corrections at second order in 1/c where we essentially recover the
results of the Foldy-Wouthuysen approach. However, due to the non-static problem,
differences occur in the term which couples the electric field with the spin.

Norbert J. Mauser

Rigorous Derivation of the Pauli Equation With Time-dependent
Electromagnetic Field

About this journal

Journal profile

Meet the editorial board

Editor spotlight

Submit your proposal

Do you think there is an emerging area of research that really needs to be highlighted? Or an existing research area that has been overlooked or would benefit from deeper investigation? Raise the profile of a research area by leading a Special Issue.

Special Issues

title

https://images.hindawi.com/journals/vlsi/vlsi.banner.jpg

twitter:card

twitter:site

twitter:title

twitter:url

twitter:image

VLSI Design | Hindawi

VLSI Design

This Issue is now open for submissions.
Papers are published upon acceptance, regardless of the Special Issue publication date.

VLSI Design

International Workshop on Quantum Theory

International Workshop on Quantum Theory

Articles