Modeling of Electronic Properties of Interfaces in Solar Cells
Coworker  Annegret Glitzky, Alexander Mielke, Matthias Liero, Reiner Nürnberg 
Cooperation 

Period  June 2010  May 2014 
Support 
DFG Research Center MATHEON Mathematics for key technologies: Modelling, simulation, and optimization of realworld processes, Project D22 
Scope
Solar cells consist of layers of different semiconducting materials, e.g. amorphous silicone (aSi:H) and crystalline silicone (cSi). In thinfilm solar cells 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. Our partners at the HelmholtzZentrum Berlin für Materialien und Energie (HZB) are investigating solar cell concepts containing layers of amorphous and crystalline silicon (aSi:H/cSi). 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 capture and release electrons or holes. The light that generates the electronhole pairs, is treated as a given source term in our modeling, 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 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 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 reactiondiffusion 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.
In September 2012 the International Workshop Mathematics for Semiconductor Heterostructures  Modeling, Analysis, and Numerics organized by F. Nier, H.Ch. Kaiser, K. Gärtner, and A. Glitzky took place at WIAS Berlin.
Publications

Feel the heat: Nonlinear electrothermal feedback in Organic LEDs
Advanced Functional Materials, online: DOI: 10.1002/adfm.201303066, 2014.
WIAS Preprint 1839, 2013. 
Convergence of an implicit Voronoi finite volume method for reactiondiffusion problems
WIAS Preprint 1827 2013. 
Uniform global bounds for solutions of an implicit Voronoi finite volume method for reactiondiffusion problems
Numer. Math., online: DOI 10.1007/s0021101406046, 2014.
WIAS Preprint 1718, 2012 
Selfheating, bistability, and thermal switching in organic semiconductors
Phys. Rev. Lett., 110 pp. 126601/1126601/5, 2013. 
Gradient structures and geodesic convexity for reactiondiffusion systems
Phil. Trans. R. Soc. A, 371 (2013) pp. 20120346/120120346/28
WIAS Preprint 1701, 2012 
Selfheating effects in organic semiconductor crossbar structures with small active area
Organic Electronics, 13, pp. 24612468, 2012
WIAS Preprint 1693, 2012 
Passing from bulk to bulk/surface evolution in the AllenCahn equation
NoDEA, Volume 20, Issue 3 , pp. 919942, 2013. Online First
WIAS Preprint 1676, 2012 
An electronic model for solar cells including active interfaces and energy resolved defect densities
SIAM J. Math. Anal., 44 pp. 38743900, 2012
WIAS Preprint 1663, 2011 
Thermomechanical modeling of energyreactiondiffusion systems, including bulkinterface interactions
Discrete Contin. Dyn. Syst.S, 6(2) pp. 479499, (2013).
WIAS Preprint 1661, 2011 
A gradient structure for systems coupling reactiondiffusion effects in bulk and interfaces,
Z. Angew. Math. Phys. 64, pp. 2952, (2013).
WIAS Preprint 1603, 2011 
Analysis of electronic models for solar cells including energy resolved defect densities,
Math. Methods Appl. Sci. 34, pp. 19801998, 2011
WIAS Preprint 1524, 2010 
A gradient structure for reactiondiffusion systems and for energydriftdiffusion systems,
Nonlinearity 24 pp. 13291346, 2011
WIAS Preprint 1485, 2010 
Uniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems,
Math. Nachr. 284 1718 pp. 21592174, 2011
WIAS Preprint 1443, 2009
Proceedings

Electrothermal modeling of organic semiconductors describing negative differential resistance induced by selfheating
in: Proceedings of the 13th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD 2013, 1922 August 2013, J. PIPREK, L. CHROSTOWSKI, eds., IEEE Conference Publications Management Group, Piscataway, NJ, USA, 2013, pp. 7778.
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 WileyVCH Verlag, Weinheim, pp. 675676, (DOI: 10.1002/pamm.2011110327), 2011
On discrete SobolevPoincaré 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. 533541. Springer Proceedings in Mathematics 4, Heidelberg, 2011 
Derivation of effective interface conditions for reactiondiffusion equations,
PAMM, Proceedings in Applied Mathematics and Mechanics, Special Issue: GAMM Annual Meeting 2011  Graz. pages WileyVCH Verlag, Weinheim, pp. 677678 (DOI: 10.1002/pamm.201110328), 2011
Talks

Nonlinear electrothermal feedback in organic semiconductors
Organic Photovoltaics Workshop 2013, December 10  11, 2013, University of Oxford, Mathematical Insitute, UK, December 10, 2013. 
Continuous and finite volume discretized reactiondiffusion systems in heterostructures
Workshop "Asymptotic Behaviour of Systems of PDE Arising in Physics and Biology: Theoretical and Numerical Points of View", November 6  8, 2013, Lille 1 University  Science and Technology, France, November 6, 2013. 
On gradient structures for reversible Markov chains and the passage to Wasserstein gradient flows
ERC Workshop on Energy/EntropyDriven Systems and Applications October 9  11, 2013, WIAS Berlin, October 9, 2013. 
Electrothermal modeling of organic semiconductors describing negative differential resistance induced by selfheating
13th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2013), August 19  22, 2013, Vancouver, Canada, August 20, 2013. 
Gradient structures and dissipation distances for reactiondiffusion systems
Workshop ``Material Theory'', December 16  20, 2013, Mathematisches Forschungsinstitut Oberwolfach, December 17, 2013 
Gradient structures and uniform global decay for reactiondiffusion systems
Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, April 25, 2013 
Analysis, modeling, and simulation of semiconductor devices
Kolloquium Simulation Technology, Universität Stuttgart, SRC Simulation Technology, May 14, 2013 
Using gradient structures for modeling semiconductors
Eindhoven University of Technology, Institute for Complex Molecular Systems, The Netherlands, February 21, 2013 
On gradient flows and reactiondiffusion systems
Institutskolloquium, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, December 3, 2012 
Multidimensional modeling and simulation of optoelectronic devices
Challenge Workshop "Modeling, Simulation and Optimisation Tools", September 24  26, 2012, Technische Universität Berlin, September 24, 2012 
An electronic model for solar cells taking into account active interfaces
International Workshop "Mathematics for Semiconductor Heterostructures: Modeling, Analysis, and Numerics", Berlin, 2012 
WIASTeSCA simulations in photovoltaics for a point contact concept of heterojunction thin film solar cells
International Workshop "Mathematics for Semiconductor Heterostructures: Modeling, Analysis, and Numerics", Berlin, 2012 
Using gradient structures for modeling semiconductors
International Workshop "Mathematics for Semiconductor Heterostructures: Modeling, Analysis, and Numerics", Berlin, 2012 
Electronic and thermal effects in organic semiconductors
Organic Photovoltaics Workshop, Oxford University, Mathematical Institute, UK, 2012 
Variational methods for evolution
Matheon Multiscale Workshop, Technische Universität Berlin, Institut für Mathematik, April 20, 2012. 
Interfaces in solar cells
5th Annual Meeting Photonic Devices, February 23, 2012, KonradZuseZentrum für Informationstechnik, Berlin, February 24, 2012. 
Interface conditions in reactiondiffusion equations
Seminar "Dünne Schichten", Technische Universität Berlin, Institut für Mathematik, February 2, 2012. 
Dissipative quantum mechanics: Geometry meets thermodynamics
Symposium "Recent Trends in Dynamical Systems", dedicated to Jürgen Scheurle's 60th birthday, January 11  14, 2012, Technische Universität München, Zentrum Mathematik, January 11, 2012 
Thermodynamical modeling of bulkinterface interaction in reaction diffusion systems
Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), Gargnano (Brescia), 2011 
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 
Solarzellen und Mathematik,
16. Berliner Tag der Mathematik, BeuthHochschule Berlin, Mai 7, 2011 
Derivation of effective interface conditions for reactiondiffusion equations,
Annual Meeting GAMM, April 1821, 2011, Graz, Austria, April 19, 2011 
Analysis of electronic models for solar cells including energy resolved defect densities,
82nd Annual Meeting GAMM, April 1821, 2011, Graz, Austria, April 20, 2011 
0 durch 0 oder Grenzschichten für Photovoltaik,
Vortragsreihe "MathInside" an der Urania, Berlin March 22, 2011 
MATHEONProject D22: Modeling of Electronic Properties of Interfaces in Solar Cells,
PVcomBTreffen TU Berlin Berlin, March 11, 2011 
Analysis of electronic models for solar cells,
WIASDay, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, February 21, 2011 
Gradient structures for reactionsdiffusion systems and semiconductor models with interface dynamics
International Conference on Evolution Equations, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, 2010 
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
Poster

Voronoi finite volume methods for reactiondiffusion systems
MoMaS Multiphase Seminar Days  Journées MoMaS Multiphasiques, BuressurYvette, France, October 7  9, 2013.
Voronoi finitevolume methods for reactiondiffusionsystems
33. Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo 2012), 2012.
Mathematics and simulation of the charge transport in semiconductor sensors
1. NanotechnologieWorkshop der LeibnizGemeinschaft (LeibnizNano), Berlin, January 30  31, 2012.
Publications related to the project

Energy estimates for continuous and discretized electroreactiondiffusion systems,
Nonlinear Analysis 70 pp. 788805, 2009
WIAS Preprint 1222 2007
, K. Gärtner 
Existence of bounded steady state solutions to spinpolarized driftdiffusion systems,
SIAM J. Math. Anal. 41 pp. 24892513, 2010
WIAS Preprint 1357 2008
, K. Gärtner 
Weakconvergence methods for Hamiltonian multiscale problems,
Discr. Cont. Dynam. Systems Series A 20 pp. 5379, 2008
WIAS Preprint 1357 2008  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)
Last Update: 22.04.2014