To describe the carrier transport processes in novel nanoelectronic devices, an accurate description in the nanometer scale has to
be used. Accordingly, in this regime, the effects of quantum mechanics have to be considered as well. The Wigner formulation of
quantum mechanics provides a convenient phase space formulation of quantum theory.
The main difficulties in the implementation of a deterministic approach to solving the Wigner equation arise due to the huge amount of memory
consumption. This issue is solved by stochastic approaches, that are, however, offset by their computational time requirements.
Deterministic methods also show difficulties in the discretization of the diffusion term in the differential equation because of rapid variations
of the Wigner function in the phase-space. Unfortunately the commonly used higher-order schemes show very different
The developed approach uses an integral formulation of the Wigner equation of a single fundamental wave packet. In this method
the differentiation can be avoided.
We consider the evolution of an initial condition described by the superposition (in phase-space) of particular fundamental packages.
To calculate the distribution at desired time steps, all "fundamental evolutions" of the Wigner equation have to be summated.
The Wigner equation has to be solved for each packet, which only depends on the initial location (in phase-space)
of the packet. As the particular calculations are independent from each other, this method is well suited for parallelization.
The calculations have been parallelized by Message Passing Interface over several nodes and OpenMP on each node and show excellent scalability.
With this parallelization it was also possible to eliminate another drawback of a sequential procedure.
For the calculation of one particular solution of the Wigner equation a convenient
history of previous distributions has to be stored. As the amount of stored data rises rapidly with resolution and
simulation time, the applicability of the sequential method becomes limited. As each fundamental packet only has to know its own history,
the overall history can be split into the history of the tasks on the particular nodes.
Further improvements of memory consumption, and consequentially of the resolution, can be achieved by an alternative simulation technique.
Originally the solutions for each time step arise sequentially. At end time the history of each distribution in phase-space of all discrete
packets is stored.
If the separate results are summed at end time, only the distribution history for the currently running simulations has to be stored.
This will be the focus of future investigations.