
Coworker:  A. Glitzky, A. Mielke, M. Liero, M. Thomas 
Cooperation: 
Institut für
Helmholtz Zentrum für Materialien und Energie, Group SE1
SiliziumPhotovoltaik
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 realworld processes, Project D22 
Scope
Photovoltaic cells are built from layers of different materials. In thinfilm solar cells, where rough interfaces are used for light trapping, the interfaces have a strong impact on the functionality of the device. Nanoscaletreatment 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 (aSi:H/cSi) are investigated by our partners at the HelmholtzZentrum Berlin für Materialien und Energie (HZB). The key issues for enhancing the efficiency are the reduction of recombination losses at the aSi:H/cSi interface and the improvement of the chargecarrier transport over the heterointerface.
The bulk equations are driftdiffusion models for the charge carriers coupled with ODEs for immobile defects, which may absorb electrons or holes. The light, generating electronhole 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 nonvanishing state density in the aSi:H mobility gap provide complications, e.g. tunneling of electrons from cSi layers into defect states with energy levels inside the band gap of aSi 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 reactiondiffusion systems and for energydriftdiffusion systems, Nonlinearity 24 (2011) pp.13291346,
WIAS Preprint 1485 (2010) 
A. Glitzky
Uniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems,
accepted for publication in Math. Nachr.
WIAS Preprint 1443 (2009) 
A. Glitzky ,
K. Gärtner
Energy estimates for continuous and discretized electroreactiondiffusion systems,
Nonlinear Analalysis 70 (2009) pp. 788805,
WIAS Preprint 1222 (2007) 
A. Glitzky ,
K. Gärtner
Existence of bounded steady state solutions to spinpolarized driftdiffusion systems,
SIAM J. Math. Anal. 41 (2010) pp. 24892513,
WIAS Preprint 1357 (2008) 
A. Mielke
Weakconvergence methods for Hamiltonian multiscale problems,
Discr. Cont. Dynam. Systems Series A 20 (2008) pp. 5379,

Poster
Modeling of electronic properties of interfaces in solar cells  Talk A. Mielke, A. Glitzky: Modeling of electronic properties of interfaces in solar cells
Papers

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 reactiondiffusion effects in bulk and interfaces,
submitted for publication in ZAMP
WIAS Preprint 1603 (2011)
Talks
 A. Mielke: Gradient structures for electroreactiondiffusion systems with applications in photovoltaics, First Interdisciplinary Workshop of the GermanRussian Interdisciplinary Science Center (GRISC) "Structure and Dynamics of Matter", October 1820, 2010, Freie Universität Berlin and HelmholtzZentrum Berlin für Materialien und Energie, October 19, 2010
 A. Glitzky: Analysis of electronic models for solar cells, WIASDay, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, February 21, 2011
 M. Liero: MATHEONProject D22: Modeling of Electronic Properties of Interfaces in Solar Cells, PVcomBTreffen 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 reactiondiffusion equations, Annual Meeting GAMM, April 1821, 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 1821, 2011, Graz, Austria, April 20, 2011
 M. Liero: Solarzellen und Mathematik, 16. Berliner Tag der Mathematik, BeuthHochschule 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
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