The Semiclassical Approach

When we talk about semiclassical approach then we talk about a well-defined group of approximations. Let us briefly review these last ones in the following list.

1. Quantum size effects. First of all, the dimensions of the device simulated have to be such that the envelope wavelength of the carriers (in our case electrons or holes) are negligible with respect to the characteristic length of the device. In that case, the particles can be described by wave-packets well-localised in the phase-space. In this case, we can consider the particles as "biliard balls".

2. Slow Physical Phenomena. This means that the phenomena simulated in GNU Archimedes, i.e. the dynamics of electrons or holes, are significantly slow with respect to the dynamics of the electric and/or magnetic field. This means that we work in a physical context in which it is justified to use electrostatic instead of full set of Maxwell's equations. Then we simulate only the Poisson equation, neglecting the potential retardation effects and the coupling with the photons.

3. The effective mass approximation. It is well-known, both from the quantum theory of matter and from physical experiments, that a particle moving in a periodic potential, as the potential experienced by a particle in a lattice, can be described as a free particle with a mass lightly smaller than the original one. Then if an electron move in a semiconductor lattice, its mass will be smaller by a well-defined factor. This is the approximation we will adopt in GNU Archimedes, in order to take into account the effects of the lattice on the particles.

4. The scattering events. The scattering are considered as semiclassical, i.e. they are obtained from quantum theory of scatterings, but the scattering events are considered instantaneous, uncorrelated and localised in space and time.

5. The Many Body effects. In our simulations, we neglect the Pauli principle, i.e. all the particles in the simulation have interation with each other. Probably, the Pauli principle will be taken into account in a next version of GNU Archimedes, even if it seems, from experiments, that in the real world and for enough diluted doping concentrations, the electrons don't interact with other electrons, i.e. there are no collisions between electrons.