Giovanni Frosali

Conditions for runaway phenomena in the kinetic theory of particle swarms

The velocity distribution of a spatially uniform diluted guest population of charged particles moving within a host medium under the influence of a D. C. electric field is studied. A simplified one‐dimensional Boltzmann model of the Kac type is adopted. Necessary conditions and sufficient conditions are established for the existence, uniqueness, and attractivity of a stationary non‐negative distribution corresponding to a specified concentration level. Conditions for the onset of the runaway process are established.

Wigner Approach to the Two-Band Kane Model for a Tunneling Diode

Abstract In this article we study a simple model for describing the electron behavior in a heterogeneous material characterized by two energy bands. The model, the so-called Kane system, is composed of two Schrödinger-like equations coupled by a k · P term which describes the interband tunneling. We give a physical justification of the Kane system and reformulate it in terms of Wigner functions. The well-posedness of the corresponding Wigner system is investigated in a suitable Hilbert space.

RUNAWAY PARTICLES FOR A BOLTZMANN-LIKE TRANSPORT EQUATION

In this paper we consider the linear Boltzmann equation which describes the evolution of one species of charged particles, electrons or ions, in a single component neutral gas. Collisions among charged particles are neglected and the acceleration field is dependent on time. We investigate a necessary condition on the behavior of both collision frequency and acceleration field so that the time-dependent solution relaxes towards a steady-state. Under suitable conditions on the acceleration field, an asymptotic behavior similar to a travelling wave in velocity space is pointed out. Various examples with different behaviors of the acceleration field for large times are given, in the simple context of the BGK model.

Inelastic scattering models in transport theory and their small mean free path analysis

In this paper we perform an asymptotic analysis of a singularly perturbed linear Boltzmann equation with inelastic scattering operator in the Lorentz gas limit, when the parameter corresponding to the mean free path of particles is small. The physical model allows for two-level field particles (ground state and excited state), so that scattering test particles trigger either excitation or de-excitation processes, with corresponding loss or gain of kinetic energy. After examining the main properties of the collision mechanism, the compressed Chapman–Enskog expansion procedure is applied to find the asymptotic equation when the collisions are dominant. A peculiarity of this inelastic process is that the collision operator has an infinite dimensional null-space. On the hydrodynamic level this is reflected in the small mean free path approximation being rather a family of diffusion equations than a single equation, as is the case in classical transport theory. Also the appropriate hydrodynamic quantity turns out to be different from the standard macroscopic density. Copyright

Scattering theory relevant to the linear transport of particle swarms

The long-time behavior of the velocity distribution of a spatially uniform diluted guest population of charged particles moving within a host medium under the influence of a D.C. electric field is studied within the framework of scattering theory. We prove the existence of wave and scattering operators for a simplified one-dimensional model of the linearized Boltzmann equation. The theory is applied to the study of the long-term behavior of electrons and the occurrence of traveling waves in runaway processes.

ASYMPTOTIC ANALYSIS FOR A PARTICLE TRANSPORT EQUATION WITH INELASTIC SCATTERING IN EXTENDED KINETIC THEORY

In this paper we perform the asymptotic analysis for a linear transport equation for test particles in an absorbing and inelastically scattering background, when the excited species can be considered as non-participating. This model is derived in the frame of extended kinetic theory and rescaled with the Knudsen number ∊. After examining the main properties of the collision model and of the scattering operator in the case with an infinite interval of energy as well as the case with a finite interval, the modified (compressed) Chapman–Enskog expansion procedure is applied to find the asymptotic equation for small mean free path. A specific feature of this model is that the collision operator has an infinite-dimensional null-space. The main result is that in the small mean free path approximation on level we obtain a free molecular flow for a suitable hydrodynamic quantity, rather than the diffusion which is typical for linear transport problems.

A quantum kinetic approach for modeling a two‐band resonant tunneling diode

We present a mathematical study of a two‐band quantum kinetic transport model. The multiband model, derived in the envelope function theory, is designed to describe the dynamics in semiconductor devices when the interband conduction‐valence transition cannot be neglected. The Wigner formulation consists of a four‐by‐four system, containing two effective mass Wigner equations (one for the electron in conduction band and one for the valence band) coupled by pseudo‐differential operators arising from the electric field in the semiconductor. The existence and uniqueness of a solution to the initial value problem are proved in a L 2‐setting for sufficiently regular electric potentials. An extension of the single band splitting‐scheme algorithm is presented to solve the one‐dimensional system for a bounded domain. Finally, we show some numerical results concerning the simulation of an interband resonant diode.

Multiband quantum transport models for semiconductor devices

The modeling of semiconductor devices, which is a very active and intense field of research, has to keep up with the speed at which the fabrication technology proceeds; the devices of the last generations have become increasingly smaller, reaching a size so small that quantum effects dominate their behaviour. Quantum effects such as resonant tunneling and other size-quantized effects cannot be described by classical or semiclassical theories; they need a full quantum description [Fre90, JAC92, KKFR89, MRS90, RBJ91, RBJ92]. A very important feature, which has appeared in the devices of the last generation and which requires a full quantum treatment, is the presence of the interband current: a contribution to the total current which arises from transitions between the conduction and the valence band states. Resonant interband tunneling diodes (RITDs) are examples of semiconductor devices which exploit this phenomenon; they are of big importance in nanotechnology for their applications to high-speed and miniaturized systems [YSDX91, SX89]. In the band diagram structure of these diodes there is a small region where the valence band edge lies above the conduction band edge (valence quantum well), making interband resonance possible.

Kinetic and Hydrodynamic Models for Multi-Band Quantum Transport in Crystals

This chapter is devoted to the derivation of k⋅p multi-band quantum transport models, in both the pure-state and mixed-state cases. The first part of the chapter deals with pure-states. Transport models are derived from the crystal periodic Hamiltonian by assuming that the lattice constant is small, so that an effective multi-band Schrodinger equation can be written for the envelopes of the wave functions of the charge carriers. Two principal approaches are presented here: one is based on the Wannier-Slater envelope functions and the other on the Luttinger-Kohn envelope functions. The concept of Wannier functions is then generalized, in order to study the dynamics of carriers in crystals with varying composition (heterostructures). Some of the most common approximations, like the single band, mini-bands and semi-classical transport, are derived as a limit of multi-band models. In the second part of the chapter, the mixed-state (i.e. statistical) case is considered. In particular, the phase-space point of view, based on Wigner function, is adopted, which provides a quasi-classical description of the quantum dynamics. After a theoretical introduction to the Wigner-Weyl theory, a two-band phase-space transport model is developed, as an example of application of the Wigner formalism to the k⋅p framework. The third part of the chapter is devoted to quantum-fluid models, which are formulated in terms of a finite number of macroscopic moments of the Wigner function. For mixed-states, the maximum-entropy closure of the moment equations is discussed in general terms. Then, details are given on the multi-band case, where “multi-band” is to be understood in the wider sense of “multi-component wave function”, including therefore the case of particles with spin or spin-like degrees of freedom. Three instances of such systems, namely the two-band k⋅p model, the Rashba spin-orbit system and the graphene sheet, are examined.

Neutron transport operators in C and L2 spaces

We study the stationary neutron transport equation (in its integral form) in plane and spherical symmetry. The investigation is carried out in both L2 and C spaces, by means of standard methods of functional analysis. The equivalence between the eigenvalue problems in the two spaces is proved, and the space C seems often to be more appropriate than L2 for investigating the main properties of the eigenfunctions. Results of functional analysis are applied in establishing some properties of the eigenvalues and the eigenfunctions. Moreover, the continuous dependence of neutron flux on optical and spatial parameters is shown.