Fluid models of gas discharges require the input of transport coefficients and rate coefficients that depend on the electron energy distribution function. Such coefficients are usually calculated from collision cross-section data by solving the electron Boltzmann equation (BE). In this paper we present a new user-friendly BE solver developed especially for this purpose, freely available under the name BOLSIG+, which is more general and easier to use than most other BE solvers available. The solver provides steady-state solutions of the BE for electrons in a uniform electric field, using the classical two-term expansion, and is able to account for different growth models, quasi-stationary and oscillating fields, electron–neutral collisions and electron–electron collisions. We show that for the approximations we use, the BE takes the form of a convection-diffusion continuity-equation with a non-local source term in energy space. To solve this equation we use an exponential scheme commonly used for convection-diffusion problems. The calculated electron transport coefficients and rate coefficients are defined so as to ensure maximum consistency with the fluid equations. We discuss how these coefficients are best used in fluid models and illustrate the influence of some essential parameters and approximations.

https://www3.nd.edu/~sst/teaching/AME60637/reading/2005_PSST_Hagelaar_Pitchford_bolsig.pdf

Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models

This paper presents theoretical calculations of the large-signal admittance and efficiency achievable in a silicon p-n-v-ns Read IMPATT diode. A simplified theory is employed to obtain a starting design. This design is then modified to achieve higher efficiency operation as specific device limitations are reached in large-signal (computer) operation. Self-consistent numerical solutions are obtained for equations describing carrier transport, carrier generation, and space-charge balance. The solutions describe the evolution in time of the diode and its associated resonant circuit. Detailed solutions are presented of the hole and electron concentrations, electric field, and terminal current and voltage at various points in time during a cycle of oscillation. Large-signal values of the diode's negative conductance, susceptance, average voltage, and power-generating efficiency are presented as a function of oscillation amplitude for a fixed average current density. For the structure studied, the largest microwave power-generating efficiency (18 percent at 9.6 GHz) has been obtained at a current density of 200 A/cm2, but efficiencies near 10 percent were obtained over a range of current density from 100 to 1000 A/cm2.

Large-signal analysis of a silicon Read diode oscillator

http://www.inf.ufes.br/~luciac/comcie/JFNK-Knoll.pdf

Jacobian-free Newton-Krylov methods: a survey of approaches and applications

We have developed a numerical device model that consistently describes the current-voltage characteristics of polymer:fullerene bulk heterojunction solar cells. Bimolecular recombination and a temperature- and field-dependent generation mechanism of free charges are incorporated. It is demonstrated that in poly[2-methoxy-5-(3('),7(')-dimethyloctyloxy)-p-phenylene vinylene]- (OC1C10-PPV-) and [6,6]-phenyl C-61-butyric acid methyl ester- (PCBM-) (1:4 wt. %) based solar cells space-charge effects only play a minor role, leading to a relatively constant electric field in the device. Furthermore, at short-circuit conditions only 7% of all free carriers are lost due to bimolecular recombination. The model predicts that an increased hole mobility together with a reduction of the acceptor strength of 0.5 eV will lead to a maximum attainable efficiency of 5.5% in the PPV/PCBM-based solar cells.

Device model for the operation of polymer/fullerene bulk heterojunction solar cells

A self-consistent iterative scheme for the numerical calculation of dc potentials and currents in a one-dimensional transistor model is presented. Boundary conditions are applied only at points representing contacts. Input data are: doping profile, parameters governing excess carrier recombination, parameters describing the dependence of mobility on doping and on electric field, applied emitter and collector voltages, and a trial solution for the electrostatic potential. The major limitation of the present approach results from use of Boltzmann rather than Fermi statistics. Convergence of the iteration scheme is good for low and moderate injection levels.

A self-consistent iterative scheme for one-dimensional steady state transistor calculations

A general small-signal theory of the avalanche noise in IMPATT diodes is presented. The theory is applicable to structures of arbitrary doping profile and uses realistic ( \alpha \neq \beta in Si) ionization coefficients. The theory accounts in a self-consistent manner for space-charge feedback effects in the avalanche and drift regions. Two single-diffused n-p diodes of identical doping profile, one of germanium and the other of silicon, are analyzed in detail. For description of the noise of the diodes as small-signal amplifiers the noise measure M is used. Values for M of 20 dB are obtained in germanium from effects in the depletion region only, i.e., when parasitic end region resistance is neglected. Inclusion of an assumed parasitic end resistance of one ohm for a diode of area 10-4cm2produces the following noise measure at an input power of 5×104W/cm2, and at optimum frequency: germanium 25 dB, silicon 31 dB. For comparison, a noise figure of 30 dB has been reported [1] for a germanium structure of the same doping profile as used in the calculations. Measurements of silicon diodes of the same doping profile are not available, but typically silicon diodes give 6-8 dB higher noise figures than germanium diodes of comparable doping profile.

H.K. Gummel

Papers

Large-signal analysis of a silicon Read diode oscillator

A self-consistent iterative scheme for one-dimensional steady state transistor calculations

A small-signal theory of avalanche noise in IMPATT diodes

Design of Read diode oscillators

Avalanche-region characterization of Read diodes