N. Ben Abdallah

On a hierarchy of macroscopic models for semiconductors

This paper shows that various models of electron transport in semiconductors that have been previously proposed in the literature can be connected one with each other by the diffusion approximation methodology. We first investigate the diffusion limit of the semiconductor Boltzmann equation towards the so‐called ‘‘spherical harmonic expansion model,’’ under the assumption of dominant elastic scattering. Then, this model is again connected, either to the energy‐transport model or to a ‘‘periodic spherical harmonic expansion model’’ through a diffusion approximation, respectively making electron–electron or phonon scattering large. We provide the mathematical background which makes the Hilbert expansions associated with these various diffusion limits rigorous.

An energy-transport model for semiconductors derived from the Boltzmann equation

An energy-transport model is rigorously derived from the Boltzmann transport equation of semiconductors under the hypothesis that the energy gain or loss of the electrons by the phonon collisions is weak. Retaining at leading order electron-electron collisions and elastic collisions (i.e., impurity scattering and the “elastic part” of phonon collisions), a rigorous diffusion limit of the Boltzmann equation can be carried over, which leads to a set of diffusion equations for the electron density and temperature. The derivation is given in both the degenerate and nondegenerate cases.

Coupling one-dimensional time-dependent classical and quantum transport models

A transient model for one-dimensional charge transport in an open quantum system is proposed. In the semiclassical limit, it reduces to the inflow boundary value problem for the classical transport equation. On this basis, the coupling of classical and quantum transport models through an interface is investigated. Suitable interface conditions are derived through asymptotic formulas involving the quantum reflection–transmission coefficients and time delays.

Combining statistical and expert evidence using belief functions: Application to centennial sea level estimation taking into account climate change

Estimation of extreme sea levels for high return periods is of prime importance in hydrological design and flood risk assessment. Common practice consists of inferring design levels from historical observations and assuming the distribution of extreme values to be stationary. However, in recent years, there has been a growing awareness of the necessity to integrate the effects of climate change in environmental analysis. In this paper, we present a methodology based on belief functions to combine statistical judgements with expert evidence in order to predict the future centennial sea level at a particular location, taking into account climate change. Likelihood-based belief functions derived from statistical observations are combined with random intervals encoding expert assessments of the 21st century sea level rise. Monte Carlo simulations allow us to compute belief and plausibility degrees for various hypotheses about the design parameter.

High field approximations of the spherical harmonics expansion model for semiconductors

Abstract. We present an asymptotic analysis (with the scaled mean free path as small parameter) of the spherical-harmonics expansion (SHE-) model for semiconductors in the case of a large electric field. The Hilbert and Chapman-Enskog expansions are performed and the dependence of macroscopic parameter-functions such as the mobility and the diffusivity on the details of the considered elastic and inelastic scattering processes are investigated.¶For example, we verify so called velocity-saturation mobility models, so far obtained by heuristic considerations, by means of an asymptotic analysis for certain scattering processes.

An accelerated algorithm for 2D simulations of the quantum ballistic transport in nanoscale MOSFETs

An accelerated algorithm for the resolution of the coupled Schrodinger/Poisson system, with open boundary conditions, is presented. This method improves the sub-band decomposition method (SDM) introduced in [N. Ben Abdallah, E. Polizzi, Subband decomposition approach for the simulation of quantum electron transport in nanostructures, J. Comput. Phys. 202 (1) (2005) 150-180]. The principal feature of the here presented model consists in an inexpensive and fast resolution of the Schrodinger equation in the transport direction, due to the application of WKB techniques. Oscillating WKB basis elements are constructed and replace the piecewise polynomial interpolation functions used in SDM. This procedure is shown to reduce considerably the computational time, while keeping a good accuracy.

Diffusive transport of partially quantized particles: Existence, uniqueness and long time behaviour

A self-consistent model for charged particles, accounting for quantum con- finement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a three dimensional Poisson equation with the particle density as the source term. This density is the product of a two dimensional surface density and that of a one dimensional mixed quan- tum state. The surface density is the solution of a drift-diffusion equation with an effective surface potential deduced from the fully three dimensional one and which involves the diagonalization of a one dimensional Schr¨ odinger operator. The overall problem is viewed as a two dimensional drift-diffusion equation coupled to a Schr¨odinger-Poisson system. The latter is proven to be well posed by a convex minimization technique. A relative entropy and an a priori

A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs

L^2

On the Convergence of the Boltzmann Equation for Semiconductors Toward the Energy Transport Model

estimate provide enough bounds to prove existence and uniqueness of a global in time solution. In the case of thermodynamic equilibrium bound- ary data, a unique stationary solution is proven to exist. The relative entropy allows to prove the convergence of the transient solution towards it as time grows to infinity. Finally, the low order approximation of the relative entropy is used to prove that this convergence is exponential in time.

Space lateral transfer and negative differential conductance regimes in quantum waveguide junctions

We model a nanoMOSFET by a mesoscopic, time-dependent, coupled quantum-classical system based on a sub-band decomposition and a simple scattering operator. We first compute the sub-band decomposition and electrostatic force field described by a Schrodinger-Poisson coupled system solved by a Newton-Raphson iteration using the eigenvalue/eigenfunction decomposition. The transport in the classical direction for each sub-band modeled by semiclassical Boltzmann-type equations is solved by conservative semi-lagrangian characteristic-based methods. Numerical results are shown for both the thermodynamical equilibrium and time-dependent simulations in typical nowadays nanoMOSFETs.