c speed of light in vacuum (= 2.99792458•10 8 ms-l ) c specific heat JEc shift energy for conduction band edge JE shift energy for valence band edge v df field enhancement factor ( absolute permittivity ( 0 permittivity constant in vacuum (= 8.854187818•10-12 Asv-lm-l )Notation The Solution of Sparse Systems of Linear Equations Direct Methods Ordering Methods Relaxation Methods Alternating Direction Methods Strongly Implicit Methods Convergence Acceleration of Iterative Methods References A Glimpse on Results Breakdown Phenomena in MOSFETs The Rate Effect in Thyristors References

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Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The classical transport picture is associated with Boltzmann-like particles that evolve in the phase-space defined by the wave vector and real space coordinates. The evolution consists of consecutive processes of drift over Newton trajectories and scattering by phonons. In the quantum regions, a convenient description of the transport is given by the Wigner-function formalism. The latter retains most of the basic classical notions, particularly, the concepts for phase-space and distribution function, which provide the physical averages. In this work we show that the analogy between classical and Wigner transport pictures can be even closer. A particle model is associated with the Wigner-quantum transport. Particles are associated with a sign and thus become positive and negative. The sign is the only property of the particles related to the quantum information. All other aspects of their behavior resemble Boltzmann-like particles. The sign is taken into account in the evaluation of the physical averages. The sign has a physical meaning because positive and negative particles that meet in the phase space annihilate one another. The Wigner and Boltzmann transport pictures are explained in a unified way by the processes drift, scattering, generation, and recombination of positive and negative particles. The model ensures a seamless transition between the classical and quantum regions. A stochastic method is derived and applied to simulation of resonant-tunneling diodes. Our analysis shows that the method is useful if the physical quantities do not vary over several orders of magnitude inside a device.

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