This topic is currently not treated at the WIAS
One option to for semiconductor device manufacturing is optical lithography. In this process, a pattern defined by a photomask is transferred by light into a photosensitive film (photoresist) by light. The photoresist processing steps for an exemplary pattern transferred into silicon dioxide on the top of a silicon wafer are sketched in the following picture. Zoom Image: © 2007 T. Schnattinger
 Exposure:
During this step, the molecules of the Photo Acid Generator (PAG) hit by photons split off acid molecules. The obtained acid concentration is the input for the next process step.  PostExposure Bake (PEB):
Photogenerated acid acts as a catalyst for splitting polymer bonds. After a given time t _{ PEB}, the process step yields the concentration of the still existing polymer bonds, also called inhibitor concentration. The socalled metamodel, describes the the post exposure bake by a system of reaction diffusion equations:  fast acid base neutralization: A + B > X _{ AB},
 acid catalyzed inhibitor deprotection: A + M > A + X _{ M},
 diffusion of acid and base.
 Development:
During this step the deprotected resist regions are dissolved.  Etch:
Etching then creates structures in the blank parts of the substrate surface.
Highlights
  Therefore, the research network MD^{3} ”Material Development for Double exposure and Double patterning” funded by the Seventh EU Framework Programme (FP7), a number of process options for the 32 nm technological node have been investigated with respect to their technological and economical feasibility. 
Publications
Articles in Refereed Journals

A. Fiebach, A. Glitzky, A. Linke, Convergence of an implicit Voronoi finite volume method for reactiondiffusion problems, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 141174.
Abstract
We investigate the convergence of an implicit Voronoi finite volume method for reaction diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, meshindependent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a longterm simulation of the MichaelisMentenHenri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities. 
A. Fiebach, A. Glitzky, A. Linke, Uniform global bounds for solutions of an implicit Voronoi finite volume method for reactiondiffusion problems, Numerische Mathematik, 128 (2014), pp. 3172.
Abstract
We consider discretizations for reactiondiffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, meshindependent global upper and lower $L^infty$ bounds for the chemical potentials. These bounds provide the main step for a convergence analysis for the full discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete GagliardoNirenberg inequalities. For the proof of the GagliardoNirenberg inequalities we exploit that the discrete Voronoi finite volume gradient norm in $2d$ coincides with the gradient norm of continuous piecewise linear finite elements. 
J. Fuhrmann, A. Fiebach, A. Erdmann, P. Trefonas, Acid diffusion effects between resists in freezing processes used for contact hole patterning, Microelectronic Engineering, 87 (2010), pp. 951954.
Abstract
Double patterning following an litho?lithoetch scheme is a possible option to create structure widths below the nominal resolution of optical light with current exposure technology. Interactions between the first and second resist layers may influence the final structure created. In this paper, we perform a model based investigation of the possible consequences of the diffusion of photogenerated acid from the second resist to the first one. As a consequence, less acid is available for the deprotection reaction, and we observe a tendency to an increase of the CD values of the primary structure. We attempt to explain observed footing effects in contact holes by this effect. 
J. Fuhrmann, A. Erdmann, Ch.R. Szmanda, A. Fiebach, M. Uhle, A model of selflimiting residual acid diffusion for pattern doubling, Microelectronic Engineering, 86 (2009), pp. 792795.
Contributions to Collected Editions

A. Erdmann, F. Shao, J. Fuhrmann, A. Fiebach, G.P. Patsis, P. Trefonas, Modeling of double patterning interactions in lithocurelithoetch (LCLE) processes, in: Optical Microlithography XXIII, Part I, M.V. Dusa, W. Conley, eds., 7640 of Proceedings of SPIE, SPIE, 2010, pp. 76400B/176400B/12.
Abstract
This paper uses advanced modeling techniques to explore interactions between the two lithography processes in a lithocure etch process and to qualify their impact on the final resist profiles and process performance. Specifically, wafer topography effects due to different optical properties of involved photoresist materials, linewidth variations in the second lithography step due to partial deprotection of imperfectly cured resist, and acid/quencher diffusion effects between resist materials are investigated. The paper highlights the results of the simulation work package of the European MD3 project. 
J. Fuhrmann, A. Fiebach, G.P. Patsis, Macroscopic and stochastic modeling approaches to pattern doubling by acid catalyzed crosslinking, in: Advances in Resist Materials and Processing Technology XXVII, Part I, R.D. Allen, M.H. Somervell, eds., 7639 of Proceedings of SPIE, SPIE, 2010, pp. 76392I/176392I/9.
Abstract
Pattern doubling by crosslinking of a spacer triggered by residual acid diffusion from a previously developed primary structure into the spacer is a possible option to create structure widths below the nominal resolution of the optical process. An advantage of such a process step would be the selfalignment to the primary structure, which would render a second exposure step unnecessary. Using macroscopic and stochastic modeling approaches, we demonstrate that it may be possible to control the width of the secondary structure created by crosslinking by the amount of quencher base added.
Talks, Poster

A. Fiebach, A. Glitzky, K. Gärtner, A. Linke, Voronoi finite volume methods for reactiondiffusion systems, MoMaS Multiphase Seminar Days  Journées MoMaS Multiphasiques, BuressurYvette, France, October 7  9, 2013.

A. Fiebach, A. Glitzky, A. Linke, Voronoi finitevolume methods for reactiondiffusionsystems, 33. Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo 2012), Universität Rostock, Institut für Mathematik, May 4  5, 2012.

A. Fiebach, Problems in double patterning: resist footing, The 16th European Conference on Mathematics for Industry (ECMI 2010), July 26  30, 2010, Bergische Universität Wuppertal, July 27, 2010.

J. Fuhrmann, A. Fiebach, G.P. Patsis, Macroscopic and stochastic modeling approaches to pattern doubling by acid catalyzed crosslinking, Workshop ``SPIE Advanced Lithography'', San Jose, California, USA, February 21  25, 2010.

K. Gärtner, Charge explosion simulations, 7th Meeting of the Detector Advisory Committee for the European XFEL, European XDAC, Hamburg, May 26, 2010.

K. Gärtner, Plasma effects  Status of simulations (10'), 8th Meeting of the Detector Advisory Committee for the European XFEL, European XDAC, Hamburg, November 15, 2010.

A. Fiebach, J. Fuhrmann, Numerical issues for reactiondiffusion processes in semiconductor photoresists, Workshop ``Evolution Equations, Related Topics and Applications'', München, September 7  11, 2009.

J. Fuhrmann, A. Fiebach, A. Erdmann, P. Trefonas, Acid diffusion effects between resists in freezing processes used for contact hole patterning, 35th International Conference on Micro and Nano Engineering (MNE 2009), Ghent, Belgium, September 29  30, 2009.

J. Fuhrmann, Modeling and simulation of post exposure bake processes in double patterning, 7th Fraunhofer IISB Lithography Simulation Workshop, September 25  27, 2009, Hersbruck, September 26, 2009.

J. Fuhrmann, Reactiondiffusion processes in polymer resists, Workshop ``Evolution Equations, Related Topics and Applications'', September 7  11, 2009, HelmholzZentrum München, September 8, 2009.

A. Fiebach, PEsimulation: Benchmarking of numerical solvers for 2D and 3D geometries, 6th Fraunhofer IISB Lithography Simulation Workshop, September 19  20, 2008, Athen, Greece, September 19, 2008.

A. Fiebach, J. Fuhrmann, Some reaction diffusion problems in semiconductor device fabrication, Workshop on PDE Approximations in Fast Reaction  Slow Diffusion Scenarios, Leiden, Netherlands, November 10  14, 2008.