Solar cell
Modeling of Electronic Properties of Interfaces in Solar Cells

<--  Principle of solar cells: photons generate electron-hole pairs, electrons/holes move to the contacts. The aim consists in minimizing the recombination losses.

Coworker: A. Glitzky, A. Mielke, M. Liero, M. Thomas
Cooperation: Institut für Helmholtz Zentrum für Materialien und Energie, Group SE1 Silizium-Photovoltaik
PVcomB ( Photovoltaics Competence Center Berlin)
ODERSUN AG Berlin - Frankfurt (Oder) - London
Period: June 2010 - May 2014
Support: DFG Research Center MATHEON Mathematics for key technologies: Modelling, simulation, and optimization of real-world processes, Project D22


Photovoltaic cells are built from layers of different materials. In thin-film solar cells, where rough interfaces are used for light trapping, the interfaces have a strong impact on the functionality of the device. Nanoscale-treatment of interfaces like doping near the interface or deposition of atoms into the interface is used to tune the electronic properties. Solar cells built from layers of amorphous and crystalline silicon (a-Si:H/c-Si) are investigated by our partners at the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB). The key issues for enhancing the efficiency are the reduction of recombination losses at the a-Si:H/c-Si interface and the improvement of the charge-carrier transport over the heterointerface.
The bulk equations are drift-diffusion models for the charge carriers coupled with ODEs for immobile defects, which may absorb electrons or holes. The light, generating electron-hole pairs, is treated as a given source term. While the equations in the bulk are well established, the modeling of the kinetics of defects on transition layers and at interfaces is a topic of current research in the physics community. The band offsets at the interface and the non-vanishing state density in the a-Si:H mobility gap provide complications, e.g. tunneling of electrons from c-Si layers into defect states with energy levels inside the band gap of a-Si layers.
So far, there is no mathematical theory for special interface conditions. For heterogeneous materials always the classical interface conditions are used, like continuity of fluxes and chemical potentials.
The aim of the project is to find adequate models for the electronic properties of solar cells including interface kinetics, to investigate their analytical properties, to derive suitable numerical approximation schemes, and to provide simulation results.

Preparatory work

  • A. Mielke
    A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity 24 (2011) pp.1329-1346,
    WIAS Preprint 1485 (2010)
  • A. Glitzky
    Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems,
    accepted for publication in Math. Nachr.
    WIAS Preprint 1443 (2009)
  • A. Glitzky , K. Gärtner
    Energy estimates for continuous and discretized electro-reaction-diffusion systems,
    Nonlinear Analalysis 70 (2009) pp. 788-805,
    WIAS Preprint 1222 (2007)
  • A. Glitzky , K. Gärtner
    Existence of bounded steady state solutions to spin-polarized drift-diffusion systems,
    SIAM J. Math. Anal. 41 (2010) pp. 2489-2513,
    WIAS Preprint 1357 (2008)
  • A. Mielke
    Weak-convergence methods for Hamiltonian multiscale problems,
    Discr. Cont. Dynam. Systems Series A 20 (2008) pp. 53-79,
  • Poster
    Modeling of electronic properties of interfaces in solar cells
  • Talk A. Mielke, A. Glitzky: Modeling of electronic properties of interfaces in solar cells


  • A. Glitzky
    Analysis of electronic models for solar cells including energy resolved defect densities,
    to appear in Math. Methods Appl. Sci.
    WIAS Preprint 1524 (2010)
  • A. Glitzky, A. Mielke
    A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces,
    submitted for publication in ZAMP
    WIAS Preprint 1603 (2011)


  • A. Mielke: Gradient structures for electro-reaction-diffusion systems with applications in photovoltaics, First Interdisciplinary Workshop of the German-Russian Interdisciplinary Science Center (G-RISC) "Structure and Dynamics of Matter", October 18-20, 2010, Freie Universität Berlin and Helmholtz-Zentrum Berlin für Materialien und Energie, October 19, 2010
  • A. Glitzky: Analysis of electronic models for solar cells, WIAS-Day, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, February 21, 2011
  • M. Liero: MATHEON-Project D22: Modeling of Electronic Properties of Interfaces in Solar Cells, PVcomB-Treffen TU Berlin Berlin, March 11, 2011
  • A. Mielke: 0 durch 0 oder Grenzschichten für Photovoltaik, Vortragsreihe "MathInside" an der Urania, Berlin March 22, 2011
  • M. Liero: Derivation of effective interface conditions for reaction-diffusion equations, Annual Meeting GAMM, April 18-21, 2011, Graz, Austria, April 19, 2011
  • A. Glitzky: Analysis of electronic models for solar cells including energy resolved defect densities, 82nd Annual Meeting GAMM, April 18-21, 2011, Graz, Austria, April 20, 2011
  • M. Liero: Solarzellen und Mathematik, 16. Berliner Tag der Mathematik, Beuth-Hochschule Berlin, Mai 7, 2011
  • A. Glitzky: An electronic model for solar cells including active interfaces, Workshop "Mathematical Modelling of Organic Photovoltaic Devices" Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK, June 9, 2011