Ferdinand Schürrer

A direct multigroup-WENO solver for the 2D non-stationary Boltzmann-Poisson system for GaAs devices: GaAs-MESFET

We propose a direct solver to the non-stationary Boltzmann-Poisson system for simulating the electron transport in two-dimensional GaAs devices. The GaAs conduction band is approximated by a two-valley model. All of the important scattering mechanisms are taken into account. Our numerical scheme consists of the combination of the multigroup approach to deal with the dependence of the electron distribution functions on the three-dimensional electron wave vectors and a high-order WENO reconstruction procedure for treating their spatial dependences. The electric field is determined self-consistently from the Poisson equation. Numerical results are presented for a GaAs-MESFET. We display electron distribution functions as well as several macroscopic quantities and compare them to those of Monte Carlo simulations. In addition, we study the influence of the used discretization on the obtained results.

A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes

We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of the phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Greens functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi-exact agreement of the steady state Wigner and Greens functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations.

Energy conservation and H-theorem in the scalar nonlinear Boltzmann equation and in its multigroup representation

Abstract A multigroup representation of the scalar nonlinear Boltzmann equation is given. In analogy to the properties of a well-formulated scattering kernel of the Boltzmann equation, the symmetries of the scattering coefficients, governing energy conservation and the H -theorem in the multigroup model, are discussed and illustrated by two examples.

On the relaxation of single hard‐sphere gases

The nonlinear Boltzmann equation is solved to examine the Maxwellization of a spatially uniform hard‐sphere gas using the Laguerre moment method. The computations are carried out for two different classes of initial conditions. Emphasis is layed on the characteristic times for the relaxation of the distribution function toward the equilibrium. As a result, in the thermal energy range the relaxation takes place within few mean collision times, regardless of the initial state. Depending on whether the high‐energy tail is initially overpopulated or underpopulated the relaxation time in this part of the spectrum is a function of the molecular velocity via 1/v or log v, respectively. The solutions are compared with those of the linearized Boltzmann equation.

A General Form for Global Dynamical Data Models for Three-Dimensional Systems

The information contained in a scalar time series and its numerical derivatives is used to construct a global model for the underlying dynamical system, using a model transformation presented previously. Here, however, we analytically determine the most general form for the transformed model in the case of a three-dimensional model ansatz. We then test this method by reconstructing global models for known chaotic dynamical systems, and comparing correlation and topological measures of the re-constructed and original systems. We also do a preliminary investigation of a real data set consisting of the sunspot number over the last 200 years.

Multigroup solutions of the nonlinear Boltzmann equation

The nonlinear Boltzmann equation is solved numerically to examine the Maxwellization of spatially homogeneous gases, using the multigroup method. By applying the Krook–Wu scattering model, an exact solution of the Boltzmann equation (BKW mode) is reproduced with high accuracy. The numerical code is also used for hard‐sphere molecules. Initial distributions are a Maxwellian with tail cutoff and distributions composed of two δ peaks. For the latter class, a strong transient overpopulation of the distribution function is observed, which may amount to several orders of magnitude.

On the relaxation of binary hard-sphere gases

A spatially homogeneous and isotropic binary gas mixture consisting of hard spheres of different mass is considered. Using the multigroup method, the Boltzmann equation describing this system is solved numerically to examine the Maxwellization of the individual components. In one class of examples, a gas with one very dilute component is considered for various values of the ratio of the particle masses. By this means, the range of validity of some simplifications frequently used in transport theory, such as the ‘heat bath approximation’ which allows a linearization, or the Rayleigh and Lorentz gas approximations are examined. The second example concerns a gas mixture with equal number densities but very different particle masses. Taking δ-peaks as initial distributions, we find that the relaxation of this system is divided into three stages: (i) The relaxation of the distribution function of the light component towards a Maxwellian distribution; (ii) The relaxation of the heavy component; (iii) Convergence of the temperatures of the two subsystems.

A Multigroup Approach to the Coupled Electron‐Phonon Boltzmann Equations in InP

Abstract We propose a very general multigroup formalism to the Boltzmann transport equations governing the transient transport regime in polar semiconductors. The electrons, which are described via a three‐valley model, and the longitudinal optical phonons are treated in a dynamic way. Hence, these multigroup equations yield reliable results in the cases of high electron densities and high electric‐field strengths. Computations are performed for InP by taking into account all of the relevant scattering mechanisms. We investigate the coupled electron‐phonon system in response to a uniform external electric field. The results are compared to experimental and theoretical data.

The nonlinear Boltzmann equation including internal degrees of freedom

A spatially homogeneous and isotropic gas, consisting of particles with internal degrees of freedom is considered. The corresponding extended Boltzmann equations are transformed into a system of scalar kinetic equations. The elastic and inelastic collisions are described by means of a generalized Krook–Wu model. The principle of microscopic reversibility is taken into account. A closed analytic form has been found for the resulting scalar scattering kernels. Their symmetries are sufficient to prove energy conservation and an H theorem.

Scalar nonlinear Boltzmann equation for gas-mixtures

A scalar form of the nonlinear Boltzmann equation for a spatially homogeneous and isotropic gas-mixture is given. Integrations are carried out explicitly on the assumption that the particles scatter like rigid spheres of different mass. A closed analytic representation has been found for the resulting scalar hard-sphere scattering kernel, which reveals two basic symmetries that govern energy conservation and anH-theorem.