The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections These quantum corrections are $O( {\hbar ^2 } )$ The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms These quantum terms allow particles to tunnel through potential barriers and to build up in potential wellsThe three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind methodSimulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v

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The quantum hydrodynamic model for semiconductor devices

A semiclassical two-temperature model for ultrafast laser heating

Basic Semiconductor Physics.- Microscopic Semi-Classical Models.- Derivation of Macroscopic Equations.- Collisionless Models.- Scattering Models.- Macroscopic Semi-Classical Models.- Drift-Diffusion Equations.- Energy-Transport Equations.- Spherical Harmonics Expansion Equations.- Diffusive Higher-Order Moment Equations.- Hydrodynamic Equations.- Microscopic Quantum Models.- The Schr#x00F6 dinger Equation.- The Wigner Equation.- Macroscopic Quantum Models.- Quantum Drift-Diffusion Equations.- Quantum Diffusive Higher-Order Moment Equations.- Quantum Hydrodynamic Equations.

/pdf/transport-equations-for-semiconductors-24lyaczrd5.pdf

Transport Equations for Semiconductors

Since Stratton published his famous paper four decades ago, various transport models have been proposed which account for the average carrier energy or temperature in one way or another. The need for such transport models arose because the traditionally used drift-diffusion model cannot capture nonlocal effects which gained increasing importance in modern miniaturized semiconductor devices. In the derivation of these models from Boltzmann's transport equation, several assumptions have to be made in order to obtain a tractable equation set. Although these assumptions may differ significantly, the resulting final models show various similarities, which has frequently led to confusion. We give a detailed review on this subject, highlighting the differences and similarities between the models, and we shed some light on the critical issues associated with higher order transport models.

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A review of hydrodynamic and energy-transport models for semiconductor device simulation

This paper shows that various models of electron transport in semiconductors that have been previously proposed in the literature can be connected one with each other by the diffusion approximation methodology We first investigate the diffusion limit of the semiconductor Boltzmann equation towards the so‐called ‘‘spherical harmonic expansion model,’’ under the assumption of dominant elastic scattering Then, this model is again connected, either to the energy‐transport model or to a ‘‘periodic spherical harmonic expansion model’’ through a diffusion approximation, respectively making electron–electron or phonon scattering large We provide the mathematical background which makes the Hilbert expansions associated with these various diffusion limits rigorous

On a hierarchy of macroscopic models for semiconductors

Electron dynamics in silicon is investigated by means of improved momentum- and energy-balance equations including particle diffusion and heat flux. The resulting system of partial differential equations is numerically solved in a variety of field configurations including strong discontinuities, in order to enhance velocity overshoot effects. It is found that diffusion, usually neglected in previous studies, plays a major role, and considerably modifies the features of the velocity vs distance curve, leading to an increase of the carrier drift velocity in the low-field region, i.e. before experiencing the effect of the strong field. In addition, it is found that, in order to take full advantage of velocity overshoot effects in MOSFET's, a structure must be designed having the strongest possible field at the source-end of the channel, where carrier density is controlled by the gate.

G. Baccarani

Papers

An investigation of steady-state velocity overshoot in silicon