MODELLING AND SIMULATION OF LASER CHEMICAL PROCESSING (LCP) FOR THE MANUFACTURING OF SILICON SOLAR CELLS DISSERTATION zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.) an der Universität Konstanz Fachbereich Physik vorgelegt von Andreas Fell Fraunhofer Institut für Solare Energiesysteme Freiburg im Breisgau 2010 i Table of contents Table of contents........................................................................................................................i 1 Introduction.......................................................................................................................1 1.1 Motivation ..................................................................................................................1 1.2 LCP – status of research .............................................................................................3 1.3 Literature review........................................................................................................7 1.4 Aims and scopes .........................................................................................................7 2 Mathematical modelling ...................................................................................................9 2.1 Overview of physical effects in LCP..........................................................................9 2.2 Optics........................................................................................................................11 2.2.1 Complex refraction index of silicon .....................................................................12 2.2.2 Absorption ............................................................................................................13 2.2.3 Reflection .............................................................................................................15 2.3 Thermodynamics ......................................................................................................19 2.3.1 Heat transport.......................................................................................................19 2.3.2 Phase change solid – liquid ..................................................................................21 2.3.3 Evaporation...........................................................................................................23 2.3.4 Species transport...................................................................................................25 2.4 Fluid dynamics.........................................................................................................27 2.4.1 Basic fluid flow ....................................................................................................28 2.4.2 Multiphase flow....................................................................................................28 2.4.3 Melting and solidification.....................................................................................29 2.5 Chemistry .................................................................................................................30 2.5.1 Photochemical radical generation.........................................................................30 2.5.2 Reactions ..............................................................................................................31 3 Fundamentals of the simulation methods .....................................................................32 3.1 PDE solving by finite differences.............................................................................32 i ii TABLE OF CONTENTS 3.1.1 Spatial discretization............................................................................................ 33 3.1.2 Time integration................................................................................................... 36 3.1.3 Alternating direction explicit (ADE) method ...................................................... 38 3.2 Fluent ....................................................................................................................... 39 3.2.1 Solver basics........................................................................................................ 40 3.2.2 User defined code – UDS and UDF..................................................................... 41 3.2.3 Multiphase flow ................................................................................................... 42 4 Implementation............................................................................................................... 43 4.1 Heating and radical generation in the liquid jet ....................................................... 43 4.2 LCPSim: optics and thermodynamics at the reaction spot ...................................... 47 4.2.1 Solving of the heat transport equation ................................................................. 48 4.2.2 Implementation of the ADE method.................................................................... 51 4.2.3 Treatment of the surface geometry ...................................................................... 55 4.2.4 Intensity distribution, surface reflections and raytracing..................................... 56 4.2.5 Semi analytical heat sources ................................................................................ 61 4.2.6 Surface recession due to evaporation................................................................... 61 4.2.7 Diffusion of impurity atoms in silicon melt......................................................... 64 4.2.8 Multilayer implementation................................................................................... 66 4.2.9 Grid generation and adaptation............................................................................ 67 4.2.10 User settings and program structure................................................................. 71 4.2.11 Verification ...................................................................................................... 73 4.3 Adaption of Fluent ................................................................................................... 78 4.3.1 Separate temperature fields.................................................................................. 78 4.3.2 Melting / solidification and density change ......................................................... 79 4.3.3 Free surface heat transfer ..................................................................................... 81 4.3.4 Verification .......................................................................................................... 83 4.4 Coupling of LCPSim and Fluent.............................................................................. 87 4.4.1 Coupling algorithm.............................................................................................. 88 TABLE OF CONTENTS iii 4.4.2 Verification...........................................................................................................90 5 Simulation results............................................................................................................92 5.1 Heating and radical generation in the liquid jet........................................................92 5.2 Influence of basic parameters in silicon laser processing.........................................95 5.3 Dry laser ablation......................................................................................................99 5.3.1 Silicon...................................................................................................................99 5.3.2 Silicon nitride layer ............................................................................................101 5.4 Ablation of silicon by LCP.....................................................................................102 5.4.1 Evaporation.........................................................................................................102 5.4.2 Melt expulsion by the liquid jet..........................................................................104 5.4.3 Cooling effect.....................................................................................................109 5.5 Dry laser doping .....................................................................................................111 5.6 LCP doping.............................................................................................................112 5.6.1 Simulations with flat top profile.........................................................................113 5.6.2 Influence of inhomogeneous intensity profile....................................................115 6 Summary........................................................................................................................122 7 Deutschsprachige Zusammenfassung..........................................................................127 A Program code and settings ...........................................................................................132 A.1 LCPSim: start.m.....................................................................................................132 A.2 LCPSim: gridadapt_2d.m.......................................................................................135 A.3 LCPSim: ade_2d.m.................................................................................................139 A.4 LCPSim: sims.m.....................................................................................................142