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Modelling and Diagnostics of Low Pressure Plasma Discharges

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Modelling and Diagnostics of Low Pressure Plasma Discharges Lehrstuhl fur¨ Technische Elektrophysik Modelling and Diagnostics of Low Pressure Plasma Discharges Peter Scheubert Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Elektrotechnik und Informationstechnik der Technischen Universitat¨ Munchen¨ zur Erlangung des akademischen Grades eines Doktors-Ingenieur (Dr.-Ing.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr.-Ing. habil. A. W. Koch Prufer¨ der Dissertation: 1. Univ.-Prof. Dr. rer. nat. G. Wachutka 2. Univ.-Prof. Dr. rer. nat R. P. Brinkmann 3. Priv.-Doz. Dr.-Ing., Dr.-Ing. habil. P. Awakowicz Die Dissertation wurde am 17:10:2001 bei der Technischen Universitat¨ Munchen¨ eingereicht und durch die Fakultat¨ fur¨ Elektrotechnik und Informationstechnik am 26:02:2002 angenommen. to all carrots growing on this planet Contents Zusammenfassung 1 Summary 3 Acknowledgements 4 1 Introduction 6 1.1 Application of low pressure, low temperature plasmas . 6 1.2 Modelling of low pressure plasmas . 7 1.3 Diagnostics . 9 1.4 An introductory approach to plasma models . 11 1.4.1 A simple Monte Carlo model . 11 1.4.2 Results . 13 1.4.3 Discussion . 16 2 Theory 19 2.1 Hydrodynamic models . 19 2.1.1 The Boltzmann equation and its moments . 20 2.1.2 Conservation of mass . 22 2.1.3 Conservation of momentum . 23 2.1.4 Conservation of energy . 24 2.2 Application of hydrodynamic models for low pressure plasmas . 25 2.2.1 General properties of low pressure plasma . 26 2.2.2 Conservation of mass . 27 2.2.3 Conservation of momentum . 28 2.2.4 Conservation of energy . 31 2.3 Energy transfer to the plasma . 32 2.3.1 Heating of the discharge . 32 2.3.2 Electrodynamic model . 34 2.3.3 RF-fields versus electrostatic fields . 37 I 3 Implementation 39 3.1 Application of hydrodynamic models . 39 3.2 Discretisation . 41 3.2.1 Semiconductor models . 41 3.2.2 Finite box schemes . 42 3.2.3 Upwind schemes . 44 3.3 Boundary conditions . 48 3.3.1 Electrodynamic model, Poisson equation . 49 3.3.2 A “simple” test problem . 49 3.3.3 Transport equations for ions . 50 3.3.4 Transport equations for electrons . 51 3.3.5 Boundary conditions for finite volume schemes . 52 4 Input data 55 4.1 Impact ionisation . 55 4.2 Ion momentum loss . 59 4.3 Elastic electron-neutral collisions . 60 4.3.1 Electric RF-conductivity . 61 4.3.2 Electron momentum loss . 65 4.3.3 Heat conduction . 67 5 1-D Results 69 5.1 Static solutions of the hydrodynamic equations . 69 5.1.1 Drift diffusion approximation versus momentum conservation equation 69 5.1.2 The eigenvalue of the electron temperature . 71 5.1.3 The most simple demonstration example . 72 5.1.4 Influence of average electron density . 74 5.1.5 Influence of neutral gas pressure . 75 5.1.6 Sensitivity analysis . 79 5.2 Dynamic solutions of the hydrodynamic equations . 81 5.2.1 Comparison of drift diffusion and two-moment model . 83 5.2.2 Asymmetric discharges, self bias . 85 5.2.3 Self Excited Electron Resonance Spectroscopy (SEERS) . 87 6 2-D Results 92 6.1 Antenna design . 93 6.1.1 Influence of coil geometry . 93 6.1.2 Experimental validation . 96 6.1.3 Skin effect . 96 6.2 Influence of neutral gas pressure . 98 6.2.1 Pressure variation in a planar chamber . 98 6.3 Influence of the reactor geometry . 99 6.3.1 Influence of aspect ratio . 100 II 6.3.2 Experimental validation . 101 6.4 Input power and discharge efficiency . 103 6.5 A design study . 108 6.5.1 A model problem . 108 6.5.2 General considerations . 109 6.5.3 An optimised chamber geometry . 109 6.6 Rules of thumb for ICP design . 112 6.6.1 Low pressure versus high pressure . 112 6.6.2 Antenna geometry . 113 6.6.3 Discharge geometry . 113 7 Conclusion 116 A Tables 117 III Zusammenfassung Niederdruck-Plasmaverfahren sind zum unverzichtbaren Bestandteil moderner Hochtechnolo- gieprozesse geworden. Neben einer Vielzahl innovativer Anwendungen im Bereich Ober- flachenbehandlung¨ bzw. -veredelung sowie Beleuchtungstechnik ist vor allem der Einsatz von Niederdruckentladungen in der Fertigung von Halbleitern bzw. Flachbildschirmen von wirtschaftlicher Bedeutung. Typischerweise sind Plasmaverfahren bei der Herstellung einer modernen integrierten Schaltung an bis zu 50 Teilprozeßschritten beteiligt. Die zunehmende Integrationsdichte und die stetig steigenden Anforderungen an die Ausbeute der Einzelschritte erfordern in zunehmendem Maße ein grundlegendes Verstandnis¨ der komplexen physikalischen sowie plasmachemischen Prozesse. Die vorliegende Arbeit gibt einen Uberblick,¨ wie die in Niederdruckplasmen ablaufenden Transportprozesse mittels geeigneter mathematischer Modelle verstanden werden konnen.¨ In gleichem Maße wird auf eine Validierung der theoretischen Daten Wert gelegt. Neben allge- meinen Betrachtungen zur Gultigk¨ eit von hydrodynamischen Beschreibungen liegt der Schwer- punkt bei der Beschreibung moderner induktiver Plasmaquellen, wie sie vermehrt in der Halb- leiterfertigung eingesetzt werden. Die Modellrechnungen werden vielfach mit experimentellen Daten verglichen. Es wird nachgewiesen, daß hydrodynamische Modelle in der Lage sind, mit großer Genauigkeit Elektronendichteverteilungen sowie Teilchenflusse¨ vorherzusagen. Neben einer ausfuhrlichen¨ und vergleichenden Diskussion verschiedener Modellsysteme wird im eindi- mensionalen Fall eine Empfindlichkeitsanalyse durchgefuhrt,¨ die den Einfluß der zugrundelie- genden Eingabedaten der Modelle diskutiert. Große Aufmerksamkeit wird hierbei der Frage der Gultigk¨ eit vereinfachter Modelle gewidmet, d.h. in welchen Bereichen evtl. vereinfachte, und deshalb schnellere und stabilere Modelle eingesetzt werden konnen.¨ Die gesamte Verof¨ fentlichung gliedert sich in drei Teile. Um einen qualitativen Vergleich der verschieden komplexen Modellansatze¨ zu ermoglichen,¨ wird zunachst¨ die allgemeine Klasse hydrodynamischer Modellsysteme aus der Boltzmann-Gleichung hergeleitet. Anhand einzelner Vereinfachungen gelangt man zu Erhaltungsgleichungen fur¨ Masse, Impuls und Energie. Die Frage einer numerischen Losung,¨ ebenso wie die Wahl geeigneter Randbedingungen, wird mit Hinblick auf bereits vorliegende Arbeiten anderer Autoren ausfuhrlich¨ diskutiert. Im zweiten Teil werden fur¨ eindimensionale Testprobleme die verschiedenen denkbaren Modellansatze¨ qualitativ verglichen und die Gultigk¨ eitsbereiche sowie Empfindlichkeiten gegenuber¨ den ver- wendeten Eingabedaten analysiert. Im letzten Teil schließlich werden zweidimensionale in- duktive Entladungen behandelt. Theoretische Ergebnisse werden mit experimentellen Daten, gewonnen aus Sondenmessungen, verglichen und bestatigen¨ in einem weiten Parameterbereich 1 die Anwendbarkeit hydrodynamischer Transportmodelle. Die abschließenden Abschnitte sind der Thematik Designregeln gewidmet. Exemplarisch wird dargestellt, wie mittels geeigneter optimierter Geometrie der Entladungskammer ein optimales Prozeßergebnis erzielt werden kann. Die wesentlichen physikalischen Grundmechanismen, die bei dem Entwurf von Entladungen ber”ucksichtigt werden sollten und eine Reihe von “Faustregeln” die Kammerdesign ohne auf- wendige Modellrechnungen ermoglichen,¨ werden diskutiert. 2 Summary The presented thesis deals with various aspects of models for low pressure discharges. As well the theoretical background of hydrodynamic plasma models as the comparison of calculated data with experimental results are discussed. In the first part of this work hydrodynamic conservation equations are derived from the Boltzmann equation. Main focus is the application to low pressure plasmas. Different model systems for electrons and ions are presented. The question in which case simplified models can be applied, is treated as well as numerical aspects and algorithms for obtaining a solution are discussed. In a second part hydrodynamic models were used for performing one-dimensional simula- tions. The sensitivity of the model in dependence of input parameters like momentum exchange frequencies was analysed. As well different model systems were compared qualitatively in or- der to get an estimate for the error introduced by using simple and numerical more stable model systems. In the third part hydrodynamic models were applied to simulate different kinds of low pres- sure discharges. Main focus were inductively driven plasmas like they are used in semiconduc- tor fabrication as high density plasma source. Calculated data for this kind of discharge were compared with experimental values obtained from Langmuir probe measurements. Theory and experiment show very good agreement. In a final part the scaling laws and geometry dependence derived from hydrodynamic models were used to assemble a set of general reactor design rules. Examples were presented, how reactor performance can be optimised by careful choice of geometry parameters. Also general dependencies on external parameters like discharge pressure were discussed. 3 Acknowledgements For giving me the opportunity to perform this work as a member of the Institute for Physics of Electrotechnology, I would like to thank Univ.-Prof. Dr. rer. nat. Gerhard Wachutka. In numerous discussions he gave me support, pointing me especially on the analogies, showing the close relationship between semiconductor device simulation and modelling of low temperature plasma processes. Also, the excellent computer network and the various software tools being present at his institute were an essential prerequisite which made this work possible. I am greatly indebted to Univ.-Prof. Dr. rer. nat. Ralf-Peter Brinkmann for essentially supporting me in critical moments of my work. By knowing the right numerical method for the right problem, he saved me more than once
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