Design of nano structures for applications in photovoltaics

A key role for the development of new efficient technologies of thin film solar cells is played by the mathematical modelling and numerical simulation of complex growth processes of the films, their resulting material properties and finally, their opto- electronic properties. One focus of investigations concerns the derivation of simplified model equations, that enable parameter studies and optimisation of material properties, such as for growth processes of thin Si-films, or coating of a-Silicon substrates with photoaktiv polymer films.

Semiconductor simulation in photovoltaics

In cooperation with Helmholtz Centre Berlin for Materials and Energy we performed 2D and 3D simulations for a novel concept in thin film photovoltaics by using the WIAS-TeSCA software. The equipment of a standard device with electrically conducting point contacts embedded in an appropriate passivation layer shows a beneficial effect on the solar cell performance. In the present model study, especially, the effect of the point contact radius and the interfacial defect density have been investigated.



  • H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Science, Springfield, 2017, IV+933 pages, (Collection Published).
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  • B. Wagner, B. Rech, A. Münch, V. Mehrmann, eds., Proceedings of the Workshop Mathematics in Industry: Technologies of Thin Film Solar Cells, WIAS, Berlin, 2010, 68 pages, (Collection Published).

  Articles in Refereed Journals

  • S. Bergmann, D.A. Barragan-Yani, E. Flegel, K. Albe, B. Wagner, Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model, , 25 (2017) pp. 065015, DOI 10.1088/1361-651X/aa7862 .
    We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel-Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.

  • M. Dziwnik, A. Münch, B. Wagner, An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit, Nonlinearity, 30 (2017) pp. 1465--1496.
    We propose a phase field model for solid state dewetting in form of a Cahn-Hilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the well-known Young's equation.

  • M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thin-film phase separation, SIAM Journal on Applied Mathematics, 75 (2015) pp. 38--60.
    In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

  • A. Glitzky, A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 64 (2013) pp. 29--52.
    We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.

  • A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 479--499.
    We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (X,Φ,K) defining the evolution $dot U$= - K(U) DΦ(U). Here Φ is the driving functional, while the Onsager operator K(U) is symmetric and positive semidefinite. If the inverse G=K-1 exists, the triple (X,Φ,G) defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential Ψ*(U, Ξ)= ½ ⟨Ξ K(U) Ξ⟩. Then, the two functionals Φ and Ψ* can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.

  • A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM Journal on Mathematical Analysis, 44 (2012) pp. 3874--3900.
    We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generation-recombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level.

  • A. Glitzky, Analysis of electronic models for solar cells including energy resolved defect densities, Mathematical Methods in the Applied Sciences, 34 (2011) pp. 1980--1998.
    We introduce an electronic model for solar cells including energy resolved defect densities. The resulting drift-diffusion model corresponds to a generalized van Roosbroeck system with additional source terms coupled with ODEs containing space and energy as parameters for all defect densities. The system has to be considered in heterostructures and with mixed boundary conditions from device simulation. We give a weak formulation of the problem. If the boundary data and the sources are compatible with thermodynamic equilibrium the free energy along solutions decays monotonously. In other cases it may be increasing, but we estimate its growth. We establish boundedness and uniqueness results and prove the existence of a weak solution. This is done by considering a regularized problem, showing its solvability and the boundedness of its solutions independent of the regularization level.

  • A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011) pp. 1329--1346.
    In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reaction-diffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature.

  • N. Allsop, R. Nürnberg, M.Ch. Lux-Steiner, Th. Schedel-Niedrig, Three-dimensional simulations of a thin film heterojunction solar cell with a point contact/defect passivation structure at the heterointerface, Applied Physics Letters, 95 (2009) pp. 122108/1--122108/3.

  • H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of drift-diffusion equations for semiconductor devices: The 2D case, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009) pp. 1584--1605.
    We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.

  • J.A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg, A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube, Note di Matematica, 28 (2008) pp. 185--201.
    We construct two bi-Lipschitz, volume preserving maps from Euclidean space onto itself which map the unit ball onto a cylinder and onto a cube, respectively. Moreover, we characterize invariant sets of these mappings.

  • R. Haller-Dintelmann, H.-Chr. Kaiser, J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de Mathématiques Pures et Appliquées, 89 (2008) pp. 25--48.

  • J. Elschner, H.-Chr. Kaiser, J. Rehberg, G. Schmidt, $W^1,q$ regularity results for elliptic transmission problems on heterogeneous polyhedra, Mathematical Models & Methods in Applied Sciences, 17 (2007) pp. 593--615.

  • H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two dimensional domains, NoDEA. Nonlinear Differential Equations and Applications, 13 (2006) pp. 287-310.

  Contributions to Collected Editions

  • V. Mehrmann, A. Mielke, F. Schmidt, D -- Electronic and photonic devices, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 229--232.

  • A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2 -- Mathematics for optoelectronic devices, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243--256.

  • H. Gajewski, H.-Chr. Kaiser, H. Langmach, R. Nürnberg, R.H. Richter, Mathematical modelling and numerical simulation of semiconductor detectors, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 355--364.

  • U. Bandelow, H. Gajewski, H.-Chr. Kaiser, Modeling combined effects of carrier injection, photon dynamics and heating in Strained Multi-Quantum-Well Laser, in: Physics and Simulation of Optoelectronic Devices VIII, R.H. Binder, P. Blood, M. Osinski, eds., 3944 of Proceedings of SPIE, SPIE, Bellingham, WA, 2000, pp. 301--310.

  • H.-Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of Schrödinger-Poisson systems into the drift-diffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 1328--1333.

  Talks, Poster

  • M. Hintermüller, Nonsmooth structures in PDE constrained optimization, Optimization Seminar, Chinese Academy of Sciences, State Key Laboratory of Scientific and Engineering Computing, Beijing, China, June 6, 2017.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, Salzburg Mathematics Colloquium, Universität Salzburg, Fachbereich Mathematik, Austria, June 9, 2016.

  • M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinite-dimensional Systems, March 14 - 18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

  • A. Glitzky, Drift-diffusion models for heterostructures in photovoltaics, 8th European Conference on Elliptic and Parabolic Problems, Minisymposium ``Qualitative Properties of Nonlinear Elliptic and Parabolic Equations'', May 26 - 30, 2014, Universität Zürich, Institut für Mathematik, organized in Gaeta, Italy, May 27, 2014.

  • M. Liero, On gradient structures and geodesic convexity for reaction-diffusion systems, Research Seminar, Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, April 17, 2013.

  • M. Liero, On gradient structures for drift-reaction-diffusion systems and Markov chains, Analysis Seminar, University of Bath, Mathematical Sciences, UK, November 21, 2013.

  • A. Glitzky, Continuous and finite volume discretized reaction-diffusion systems in heterostructures, 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.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, Workshop ``Material Theory'', December 16 - 20, 2013, Mathematisches Forschungsinstitut Oberwolfach, December 17, 2013.

  • A. Mielke, Using gradient structures for modeling semiconductors, Eindhoven University of Technology, Institute for Complex Molecular Systems, Netherlands, February 21, 2013.

  • TH. Koprucki, A. Glitzky, A. Fischer, Electronic and thermal effects in organic semiconductors, Organic Photovoltaics Workshop, Oxford University, Mathematical Institute, UK, April 2, 2012.

  • M. Liero, Interfaces in reaction-diffusion systems, Seminar ``Dünne Schichten'', Technische Universität Berlin, Institut für Mathematik, February 9, 2012.

  • M. Liero, Interfaces in solar cells, 5th Annual Meeting Photonic Devices, February 23, 2012, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, February 24, 2012.

  • M. Liero, WIAS-TeSCA simulations in photovoltaics for a point contact concept of heterojunction thin film solar cells, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24 - 28, 2012, WIAS Berlin, September 25, 2012.

  • A. Glitzky, An electronic model for solar cells taking into account active interfaces, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24 - 28, 2012, WIAS Berlin, September 27, 2012.

  • A. Mielke, Multidimensional modeling and simulation of optoelectronic devices, Challenge Workshop ``Modeling, Simulation and Optimisation Tools'', September 24 - 26, 2012, Technische Universität Berlin, September 24, 2012.

  • A. Mielke, Using gradient structures for modeling semiconductors, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24 - 28, 2012, WIAS Berlin, September 24, 2012.

  • M. Liero, Derivation of effective interface conditions for reaction-diffusion equations, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Applied Analysis, April 18 - 21, 2011, Technische Universität Graz, Austria, April 19, 2011.

  • A. Glitzky, An electronic model for solar cells including active interfaces, Workshop ``Mathematical Modelling of Organic Photovoltaic Devices'', University of Cambridge, Department of Applied Mathematics and Theoretical Physics, UK, June 9, 2011.

  • A. Glitzky, Analysis of electronic models for solar cells including energy resolved defect densities, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Applied Analysis, April 18 - 21, 2011, Technische Universität Graz, Austria, April 20, 2011.

  • A. Mielke, Thermodynamical modeling of bulk-interface interaction in reaction-diffusion systems, Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), June 20 - 23, 2011, Gargnano (Brescia), Italy, June 20, 2011.

  • A. Mielke, Mathematical approaches to thermodynamic modeling, Autumn School on Mathematical Principles for and Advances in Continuum Mechanics, November 7 - 12, 2011, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

  • 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. Mielke, Gradient structures for reaction-diffusion systems and semiconductor equations, 81th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2010), Session on Applied Analysis, March 22 - 26, 2010, Universität Karlsruhe, March 24, 2010.

  • A. Mielke, Gradient structures for reaction-diffusion systems and semiconductor models with interface dynamics, International Conference on Evolution Equations, October 11 - 14, 2010, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, October 12, 2010.

  • A. Mielke, The GENERIC formulation for dissipative temperature-dependent materials, International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2010), August 30 - September 2, 2010, Technische Universität Berlin, Institut für Mechanik, Berlin, September 1, 2010.