Development of discontinuous Galerkin methods applied to the Liouville-von-Neumann equation

• Applicant: Professor Dr.-Ing. Dirk Schulz
• Subject Area: Electronic Semiconductors, Components and Circuits, Integrated Systems, Sensor Technology, Theoretical Electrical Engineering
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 446321675

The investigation of the transport behavior within nanoelectronic and nanophotonic devices demands a high accuracy of the numerical methods used to describe realistic scenarios considering coherent or incoherent effects. Based on the Liouville-von-Neumann equation in center mass coordinates, the stationary and transient transport can be determined by applying a finite volume method. By the introduction of a complex absorbing potential, the open structure of devices can be taken into account, and by choosing an appropriate basis in the approximation, a representation in the phase space can be realized to incorporate scattering mechanisms. These advantages allow a more flexible implementation of transient algorithms compared to non-equilibrium Green's functions approaches and avoid the disadvantages of conventional numerical methods solving the Wigner-Transport-Equation based on the application of discrete Fourier transforms. The discontinuous Galerkin method as a numerical method has been used and developed in recent years, because of its high approximation order and numerical efficiency in many fields of research. However, this methodology has not yet been applied to the Liouville-von-Neumann equation. With this methodology, however, in addition to a further increase in efficiency, a significantly improved description of a large number of interaction effects would be possible in comparison to existing methods, for which a significant reduction in the numerical dispersion is essential to avoid phase errors. The aim of the project is therefore the development of a sophisticated discontinuous Galerkin method, to solve the Liouville-von-Neumann equation in such a way that it can be extended to other important applications such as spintronics or phonon transport.

• DFG Programme: Research Grants