The aim of this group is to develop and analyze physics preserving numerical techniques for innovative semiconductor devices. Few discoveries have shaped our modern society like semiconductors have. They are the backbone of every electronic device. A world without semiconductors is one without television, pacemakers, satellites, cell phones, air planes and computers - and by extension a world without the Internet, social media and online communication. New semiconductor techniques, materials and devices innovate established technologies. Among them are low-cost perovskites for next-generation solar cells, resource-efficient nanowires as well as accurate lasers for self-driving cars.
Mathematical research topics
Numerical solution and simulation of nonlinear systems of partial differential equations
Finite volume methods on Voronoi meshes
Modified Scharfetter-Gummel techniques for nonlinear diffusion
Drift-diffusion equations
Charge transport in semiconductors, in particular coupling them to atomistic models as well as stress and strain
Physics preserving numerical techniques
Preconditioners and anisotropic meshing strategies
High dimensional meshfree approximation for surrogate models
Applications
Perovskites: About ten years ago engineers showed for the first time that low-cost perovskites could be used to convert
sunlight into electricity. Since then their efficiency has greatly improved, giving hope to replace or modify (via tandem solar cells) less efficient yet widely-used silicon-based solar cells soon. Simulating perovskite solar cells is extremely challenging due to stiffness: Apart from electrons and holes a third ionic species has to be considered which moves about twelve orders of magnitude more slowly. This means that different time scales are present in the model which leads to numerical difficulties.
Cooperations: RG1, RG3, Helmholtz Zentrum Berlin, University of Oxford, Inria Lille/Universty of Lille
Crystal growth: The lateral photovoltage scanning (LPS) method helps to reconstruct tiny fluctuations within a semiconductors
crystal such as silicon in a nondestructive way. This knowledge is important because it provides insight into how the temperature field is
distributed during the growth of a semiconductor crystal. Given that silicon melts at 1687K, it is impossible to measure
the temperature directly. Therefore, the crystal growth community tries to infer the temperature distribution post
mortem from these fluctuations, so called striations, within the crystal. We model, analyze and simulate the LPS method.
Cooperations: RG3, Institut für Kristallzüchtung, University of Florence, SISSA (Trieste, Italy)
Nanowires: Nanowires have the potential to greatly
reduce the amount of bulk silicon needed in established solar cells or MOS transistors thus enabling cheaper and more resource-efficient solar cells. Useful electronic properties of these thin wires can be controlled via elastic strain. For example, bending nanowires changes the band gap. However, deformation-related, piezoelectric, and in particular flexoelectric contributions create a complicated potential landscape which is poorly understood and leads to unexpectedly slow charge carrier transport. Careful simulations are needed to explain the cause.
Cooperations: RG1, RG3, Paul-Drude-Institut
Quantum wells: We model and simulate random alloy fluctuations in band edge profiles within a full device. To achieve this, we combine random atomic fluctuations in band edges with macroscale drift diffusion processes. The spatially randomly varying band edges are implemented in ddfermi. Quantum effects are taken into account via localization landscape theory (LLT).
Cooperations: RG1, Tyndall National Institute (Cork, Ireland)
Source: Pang Kakit (CC BY-SA 3.0)
Lasers: Semiconductor lasers are needed in many areas: For example,
semiconductor-based LiDAR (light detection and ranging) sensors improve autonomous driving as they
are accurate, comparatively small and thus mass market friendly. Moreover, high precision lasers are needed in quantum
metrology and quantum computing. Since building new laser prototypes is costly, it is important to understand for all
these cases how a new setup works before production. Thus simulations of semiconductor lasers will not only
provide scientific insights but also help to reduce development costs. To achieve this, the group will extend the van
Roosbroeck model to incorporate more than two charge-carrier species and include additional physical effects
(heterostructures, heat transport and light emission).
Cooperations: RG1, RG2, RG3, Ferdinand-Braun-Institut
Neural networks/surrogate models: The core of machine learning algorithms consists of a (usually high-dimensional) optimization problem. To find a
minimizer within such complex structures it is often beneficial to resort to surrogate models, which will be minimized
instead of the original problem. Due to the curse of dimensionality it is often not feasible to build meshes. For this
reason meshfree methods help to efficiently build surrogate models for high-dimensional problems. In particular, we will
focus on positive definite kernels within an efficient multilevel residual correction scheme. Additionally, we solve inverse problems arising semiconductor-based imaging techniques via machine learning techniques.
Cooperations: WG DOC, University of Florence, SISSA
The Leibniz group is organizing a mini symposium at the SIAM Conference on Mathematical Aspects of Materials Science 2021 with the title Modelling and simulation of charge transport in perovskites. Dilara Abdel won a SIAM MS student travel award for attending the conference to give a talk.
