## 7. Quantum semiconductor simulations

The enormous progress of the computer and telecommunication industry relies heavily on the development of modern semiconductor materials and the miniaturization of semiconductor devices. The numerical simulation of new materials and structures is of paramount importance in industrial applications, in order to reduce costly experiments and to accelerate production cycles. The challenge is not only to provide accurate and computationally cheap models of the physical processes, but also to develop efficient numerical techniques. The group of Ansgar Juengel, who joined the Faculty of Mathematics and Geoinformation towards the end of 2006, is working on the mathematical modeling of quantum structures, such as laser diodes or resonant tunneling diodes, and their numerical simulation using Schroedinger or quantum fluid equations.

For quantum structures, in which collisional effects cannot be neglected, the use of macroscopic quantum models may be reasonable. The main achievement of Juengel’s group was the derivation of new quantum fluid model hierarchies and their numerical approximation. It turns out that viscous effects in such models yield stable numerical schemes but may influence strongly the coherency of the quantum states. Therefore, hybrid models were developed, which couple Schroedinger models in ballistic regions with quantum fluid models (for instance, density-gradient equations or quantum hydrodynamics) in collisional regions.

The fluid models are discretized using finite-difference or finite-element approximations, whereas the Schr¨odinger equation (for instance, for quantum wire applications) is approximated by pseudo-spectral methods.

This viewpoint may be unified by employing Wigner equation models which allow for the inclusion of quantum collisional effects. A comparison of the various viewpoints (quantum fluid, Schroedinger, Wigner) is on-going work with the group of Arnold. The development of efficient simulation codes for modern quantum structures, such as like quantum wires and quantum dots, requires new insight in the underlying quantum physics. Therefore, a cooperation with the group of Held is planned to model and simulate quantum dots.