The research activities of the FWF Project P21685 "Wigner-Boltzmann Particle Simulations" focus on the phase space description of the electron behavior in nanometer semiconductor structures. The Wigner-Boltzmann equation provides a relevant physical model for a variety of transport conditions characterizing these structures. It is defined by two operators, which, as implied by the name, impose quantum-coherent or scattering dominated evolution. While the former is manifested by oscillations in the solution due to quantum superpositions, the second strives towards classical equilibrium causing decoherence and irreversibility. Which of these regimes prevail depends on the physical scales, which is theoretically investigated.
A relation called scaling theorem is derived, which estimates how physical scales determine the choice between classical or quantum transport regimes. It is shown that two mechanisms affect the purely quantum evolution, causing in parallel decoherence of the electron evolution. The first one could be expected from the linearity of the operator formulation of the problem: an increase of the phonon coupling is equivalent to a relative decrease of the involved energy scales. The second mechanism is associated with a reduced parameter giving rise to a decrease of the effect of higher order derivatives of the Wigner potential. It is an aspect of the scaling theorem, which elucidates the heuristic picture of a "scattering-induced reduction of the coherence length", where electrons "carry" the information about the electric potential during their free flight. Without scattering the flight lasts forever, so that all spatial points are correlated. Alternatively the distance between the correlated points decreases with the increase of the scattering rates, as they give rise to shorter flights.
The scaling theorem determines classes of physical problems with equivalent numerical aspects. Processes with very different initial conditions, momenta, electron-phonon coupling, phonon energies, and local evolution time may have equivalent evolutions provided that these physical quantities are properly scaled