The Wigner–Boltzmann Monte Carlo method applied to electron transport in the presence of a single dopant

The Wigner–Boltzmann model is a partial integro-differential equation which describes the time dependent dynamics of quantum mechanical phenomena including the effects of lattice vibration as a second-order approximation. Recently a Monte Carlo technique exploiting the concept of signed particles has been developed for its ballistic counterpart, in one and two-dimensional space. In this work, we introduce an extension to the Wigner–Boltzmann model in three-dimensional geometries adapted for the treatment of the scattering term. As an application, we study the dynamics of an electron wave packet in proximity of a Coulombic potential in the presence of absorbing boundary conditions. This mimics the presence of a dopant atom buried in a semiconductor substrate. By using this method, one can observe how the lattice temperature eventually affects the dynamics of the wave packet.