Patricio Farrell

Dilara Abdel, Zeina Amer, Daniel Fritsch, Yiannis Hadjimichael

Imke Weitkamp



The Leibniz group NUMSEMIC develops and numerically solves nonlinear PDE models. These models are often inspired by charge transport in innovative semiconductor devices. In particular, applications include perovskite solar cells, memristors, nanowires, quantum wells, lasers as well as doping reconstruction. To translate these applications into mathematical models, we rely on nonlinear drift-diffusion, hyperelastic material models, inverse PDE problems, localized landscape theory and atomistic coupling. Our methodologies include physics-preserving finite volume methods, data-driven techniques as well as meshfree methods.

Mathematical research topics

  • Modeling with and numerical solution of nonlinear systems of partial differential equations
  • Nonlinear drift-diffusion models, hyperelastastic materials, inverse PDE problems, localized landscape theory and atomistic coupling
  • Physics preserving finite volume methods on Voronoi meshes
  • Charge transport in semiconductors
  • Preconditioners and anisotropic meshing strategies
  • High dimensional meshfree approximation
  • Data-driven techniques for ill-posed inverse problems


  • 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

  • Nanowires: Nanowires have many potential applications, for example they may be used to build even smaller MOS transistors. 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 combining charge transport with continuum mechanics are needed to explain the cause.

    Cooperations: RG1, RG3, Paul-Drude-Institut

  • Memristors: The von Neumann architecture is far from ideal for AI applications due to its unacceptably high energy consumption. Memristors help to emulate the extremely efficient computing power of human brains. We develop complex charge transport models which incorporate mobile point defects and Schottky barrier lowering to theoretically understand the shape and asymmetries of the hysteresis curves observed in experiments.

    Cooperations: TU Ilmenau

  • 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)

  • Imaging techniques: We model several semiconductor-based imaging techniques to predict fluctuations in doping profiles such as the laser beam induced current (LBIC) or the lateral photovoltage scanning (LPS) method. Mathematically, we need to solve an inverse problem which we achieve via machine learning techniques.

    Cooperations: RG3, Institut für Kristallzüchtung, University of Florence, SISSA (Trieste, Italy)

  • 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. Our group extends the van Roosbroeck model to incorporate additional physical effects (heterostructures, heat transport and light emission). In particular, we couple our charge transport model with a Helmholtz problem.

    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. Additionally, we solve inverse problems arising semiconductor-based imaging techniques via machine learning techniques.

    Cooperations: WG DOC, University of Florence, SISSA