Paola Pietra

Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit

Abstract. 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 test examples are presented to help interpret the theory.

Numerical Discretization of Energy-Transport Models for Semiconductors with Nonparabolic Band Structure

The energy-transport models describe the flow of electrons through a semiconductor crystal, influenced by diffusive, electrical, and thermal effects. They consist of the continuity equations for the mass and the energy, coupled with Poissons equation for the electric potential. These models can be derived from the semiconductor Boltzmann equation. This paper consists of two parts. The first part concerns the modeling of the energy-transport system. The diffusion coefficients and the energy relaxation term are computed in terms of the electron density and temperature, under the assumptions of nondegenerate statistics and nonparabolic band diagrams. The equations can be rewritten in a drift-diffusion formulation which is used for the numerical discretization. In the second part, the stationary energy-transport equations are discretized using the exponential fitting mixed finite element method in one space dimension. Numerical simulations of a ballistic diode are performed.

A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schrödinger Equation

We apply Wigner transform techniques to the analysis of the Dufort--Frankel difference scheme for the Schrodinger equation and to the continuous analogue of the scheme in the case of a small (scaled) Planck constant (semiclassical regime). 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 or not caustics develop. Numerical test examples are presented to help interpret the theory and to compare the Dufort--Frankel scheme to other difference schemes for the Schr{odinger equation.

A PHASE PLANE ANALYSIS OF TRANSONIC SOLUTIONS FOR THE HYDRODYNAMIC SEMICONDUCTOR MODEL

We present an analysis of transonic solutions of the steady state 1-dimensional unipolar hydrodynamic model for semiconductors in the isoentropic case. The approach is based on construction of the orbits of the system in the electron density-electric field phase plane and on representation of discontinuous solutions of the hydrodynamic boundary value problem by a union of trajectory pieces. These pieces are related by shocks obeying jump and entropy conditions. A continuation argument in the length of the semiconductor device under consideration is applied to construct a continuum of sub- and transonic solutions, which contains at least one solution for every positive length. We also present numerical results illustrating the various possible solution profiles. For this we use a regularization of the problem, adding artificial diffusion to obtain singularly perturbed problems which are then solved numerically using continuation in the regularization parameter.

Identification of doping profiles in semiconductor devices

This paper is devoted to the identification of doping profiles in the stationary drift-diffusion equations modelling carrier and charge transport in semiconductor devices. We develop a framework for these inverse doping problems with different possible measurements and discuss mathematical properties of the inverse problem, such as the identifiability and the type of ill-posedness. In addition, we investigate scaling limits of the drift-diffusion equations, where the inverse doping problem reduces to classical (elliptic) inverse problems. As a first concrete application we consider the identification of piecewise constant doping profiles in p–n diodes. Finally, we discuss the stable solution of the inverse doping problem by regularization methods and their numerical implementation. The theoretical statements are tested in a numerical example for a p–n diode.

A Discretization Scheme for a Quasi-Hydrodynamic Semiconductor Model

A discretization scheme based on exponential fitting mixed finite elements is developed for the quasi-hydrodynamic (or nonlinear drift–diffusion) model for semiconductors. The diffusion terms are nonlinear and of degenerate type. The presented two-dimensional scheme maintains the good features already shown by the mixed finite elements methods in the discretization of the standard isothermal drift–diffusion equations (mainly, current conservation and good approximation of sharp shapes). Moreover, it deals with the possible formation of vacuum sets. Several numerical tests show the robustness of the method and illustrate the most important novelties of the model.

A Hierarchy of Diffusive Higher-Order Moment Equations for Semiconductors

A hierarchy of diffusive partial differential equations is derived by a moment method and a Chapman–Enskog expansion from the semiconductor Boltzmann equation assuming dominant collisions. The moment equations are closed by employing the entropy maximization principle of Levermore. The new hierarchy contains the well-known drift-diffusion model, the energy-transport equations, and the six-moments model of Grasser et al. It is shown that the diffusive models are of parabolic type. Two different formulations of the models are derived: a drift-diffusion formulation, allowing for a numerical decoupling, and a symmetric formulation in generalized dual-entropy variables, inspired by nonequilibrium thermodynamics. An entropy inequality (or H-theorem) follows from the latter formulation.

An Adaptive Mixed Scheme for Energy-Transport Simulations of Field-Effect Transistors

Energy-transport models are used in semiconductor simulations to account for thermal effects. The model consists of the continuity equations for the number and energy of the electrons, coupled to the Poisson equation for the electrostatic potential. The movement of the holes is modeled by drift-diffusion equations, and Shockley--Read--Hall recombination-generation processes are taken into account. The stationary equations are discretized using a mixed-hybrid finite-element method introduced by Marini and Pietra. The two-dimensional mesh is adaptively refined using an error estimator motivated by results of Hoppe and Wohlmuth. The numerical scheme is applied to the simulation of a two-dimensional double-gate MESFET and a deep submicron MOSFET device.

Weak Limits of the Quantum Hydrodynamic Model

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.

AN EFFECTIVE MASS MODEL FOR THE SIMULATION OF ULTRA-SCALED CONFINED DEVICES

In this paper, we present the derivation and the simulation of an effective mass model, describing the quantum motion of electrons in an ultra-scaled confined nanostructure. Due to the strong confinement, the crystal lattice is considered periodic only in the one-dimensional transport direction and an atomistic description of the entire cross-section is given. Using an envelope function decomposition, an effective mass approximation is obtained. It consists of a sequence of one-dimensional device-dependent Schrodinger equations, one for each energy band, in which quantities retaining the effects of the confinement and of the transversal crystal structure are inserted. In order to model a gate-all-around field effect transistor, self-consistent computations include the resolution, in the whole domain, of a Poisson equation describing a slowly varying macroscopic potential. Simulations of the electron transport in a simplified one-wall carbon nanotube are presented.