Modeling of Electronic Properties of Interfaces in Solar Cells

Coworker	Annegret Glitzky, Alexander Mielke, Matthias Liero, Reiner Nürnberg
Cooperation	HZB, Institute for Silicon Photovoltaics (B. Rech) HZB, Institute for Heterogeneous Material Systems (M. Ch. Lux-Steiner) PVcomB (Photovoltaics Competence Center Berlin) together with Matheon-projects C10 (B. Wagner) and D23 (F. Schmidt) D. Lockau (ZIB and HZB) 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

Scope

Solar cells consist of layers of different semiconducting materials. In thin-film solar cells 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. Our partners at the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB) are investigating solar cell concepts containing layers of amorphous and crystalline silicon (a-Si:H/c-Si). 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 capture and release electrons or holes. The light that generates the electron-hole pairs, is treated as a given source term in our modelling, while it is calculated in project D23. 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 was no mathematical theory for special interface conditions. For heterogeneous materials always the classical interface conditions are used, like the continuity of normal fluxes and chemical potentials.

The aim of this project is to find reasonable hybrid models for electronic properties of solar cells coupling partial differential equations in the bulk with reaction-diffusion systems on the interface via suitable linear and nonlinear transmission conditions. These models are then studied regarding their analytical properties. Finally, the derivation of suitable numerical approximation schemes is necessary to provide simulation results.

Matheon Poster: Modeling of electronic properties of interfaces in solar cells
Matheon Talk: Modeling of electronic properties of interfaces in solar cells (January 2010, A. Mielke and A. Glitzky)

Hole current in heterostructured solar cell

Simulation using WIAS-TeSCA: Hole current for a structured interface in a heterostructured solar cell.

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

Publications

A. Glitzky
Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems,
Math. Nachr. 284 17-18 pp. 2159-2174, (2011)
WIAS Preprint 1443 (2009)
A. Mielke
A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,
Nonlinearity 24 pp.1329-1346, (2011)
WIAS Preprint 1485 (2010)
A. Glitzky
Analysis of electronic models for solar cells including energy resolved defect densities,
Math. Methods Appl. Sci. 34, pp. 1980-1998, (2011)
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
Thermomechanical modeling of energy-reaction-diffusion systems including bulk-interface interactions
to appear in DCDS-S
WIAS Preprint 1661 (2011)
A. Glitzky
An electronic model for solar cells including active interfaces and energy resolved defect densities
submitted for publication in SIAM J. Math. Analysis
WIAS Preprint 1663 (2011)
M. Liero
Passing from bulk to bulk/surface evolution in the Allen-Cahn equation
submitted for publication in NoDEA
WIAS Preprint 1676 (2012)

Proceedings

A. Glitzky
On the analysis of an electronic model for solar cells including energy resolved defect densities,
PAMM, Proceedings in Applied Mathematics and Mechanics, Special Issue: GAMM Annual Meeting 2011 - Graz. pages Wiley-VCH Verlag, Weinheim, pp. 675-676, (DOI: 10.1002/pamm.2011110327), (2011)
A. Glitzky and J. A. Griepentrog
On discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations,
In J. Fort, J. Fürst, J. Halama, R. Herbin, and F. Hubert, editors, Finite volumes for complex applications VI: Problems and perspectives, pp. 533-541. Springer Proceedings in Mathematics 4, Heidelberg, (2011)
M. Liero
Derivation of effective interface conditions for reaction-diffusion equations,
PAMM, Proceedings in Applied Mathematics and Mechanics, Special Issue: GAMM Annual Meeting 2011 - Graz. pages Wiley-VCH Verlag, Weinheim, pp. 677-678 (DOI: 10.1002/pamm.201110328), (2011)

Talks

A. Mielke
Gradient structures for reactions-diffusion systems and semiconductor models with interface dynamics
International Conference on Evolution Equations, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, (2010)
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)
A. Mielke
Thermodynamical modeling of bulk-interface interaction in reaction diffusion systems,
Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), Gargnano (Brescia), (2011)

Publications related to the project

A. Glitzky, K. Gärtner
Energy estimates for continuous and discretized electro-reaction-diffusion systems,
Nonlinear Analalysis 70 pp. 788-805, (2009)
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 pp. 2489-2513, (2010)
WIAS Preprint 1357 (2008)
A. Mielke
Weak-convergence methods for Hamiltonian multiscale problems,
Discr. Cont. Dynam. Systems Series A 20 pp. 53-79, (2008)
WIAS Preprint 1357 (2008)

Last Update: 24.01.2012