No abstract

The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schrödinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Schödinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

An improved closure relation - based on the entropy principle - is implemented in a Hydrodynamic model for electron transport. Steady-state electron transport in the “benchmark” n+ - n - n+ submicron silicon diode is simulated and the quality of the model is assessed by comparison with Monte Carlo results.

The effects of viscosity, previously neglected in electronic device stimulations, are studied using a non‐standard hydrodynamic model, following Anile and Pennisi. Results are compared with those of Gardner.

Hydrodynamical models are suitable to describe carrier transport in submicron semiconductor devices. These models have the form of nonlinear systems of hyperbolic conservation laws with source terms, coupled with Poisson's equation. In this article we examine the suitability of a high resolution centered numerical scheme for the solution of the hyperbolic part of these extended models, in one space dimension. Because of the lack of physically significant exact analytical solutions, the method is assessed against a benchmark for the system of compressible, unsteady Euler equations with source terms, which has an exact solution; the latter is shown to be nearly identical to the numerical one. The method is then used to solve the extended hydrodynamical model (EM) based on the maximum entropy closure recently introduced by Anile, Romano, and Russo, simulating a ballistic diode n + − n − n + , which models a metal oxide semiconductor field effect transistor (MOSFET) channel. Results are presented for the reduced-and full-equation EM formulation at steady state, for an initially discontinuous electron density at the junctions. Transient results show the evolution of highly nonlinear waves emanating from the neighborhood of the junctions. Introduction.Enhanced functional integration in modern electron devices requires an accurate modeling of energy transport in semiconductors in order to describe high-field phenomena such as hot-electron propagation, impact ionization, and heat generation in the bulk material. Furthermore, when using compound semiconductors for high frequency applications, usually one deals with multivalley band structures and in these cases the transfer of carriers from one valley to the other must also be modeled. The standard drift-diffusion models cannot cope with highfield phenomena because they do not comprise energy as a dynamical variable. Also they do not incorporate dynamical transfer of carriers from one valley to the other and this renders them ill-suited for simulating time dependent high frequency phenomena. Therefore, generalizations of the drift-diffusion equations have been sought which would incorporate energy as a dynamical variable and which also could treat time dependent high frequency phenomena. Because of their mathematical similarity to the equations of compressible fluid flow, these models are called hydrodynamical models. Semiconductor hydrodynamical models are obtained from the infinite hierarchy of moment equations of the semiclassical Boltzmann transport equation (BTE) by a suitable truncation procedure. This requires making suitable assumptions on (i) choosing the appropriate moments, (ii) closing the hierarchy of moment equations by finding appropriate expressions for the N + 1 order moment in terms of the previous