Quantum Tansport
 

The transport theory of quantum particles

This is nowadays an interesting and fast-developing field of mathematical physics, not only because of its intrinsic mathematical interest, but also in view of the remarkable applications to nanoelectronic devices modeling. The recent progresses in semiconductor engineering, in fact, make it possible to build ultra-small semiconductor structures in which the quantum effecs on the charge carriers not only are no more negligible but, on the contrary, constitute the basis of  the devices functioning. An accurate mathematical description of the quantum transport in such quantum devices becomes necessary in  order to perform numerical simulation, which allow to reduce laboratory time and expenses, to simulate peculiar working conditions and to optimize device parameters.
 

Wigner equation

The research interest of our group is mainly focused on the Wigner description of quantum transport. The Wigner formulation of Quantum Mechanics has some features that akes it a desirable alternative to the Schroedinger formulation to several respects:

it presents striking formal analogies with the classical theory of kinetic equations, suggesting the quantum analogues of Boltzmann and Vlasov equations;

it allows to investigate the semi-classical limit of a quantum system in a particularly clear fashion;

it allows to set up boundary value problems for open systems, simulating the interface with ohmic contacts.

However, these properties have to be taken very carefully, since the analogy with the kinetic theory is seriously restricted by the fact that the Wigner functions (Wigen transform of wave functions) are not  densities in phase-space as they may take negative values.
The main topics of our investigation are the following.

Mathematical  properties of the Wigner or Wigner-Poisson equations in bounded domains with semiclassical inflow boundary condidtions

Boundary value problems for the Wigner equation

Generalized Wigner functions for electron-phonon interactions

Multi-band Wigner functions.