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Plasma ElEctronics: applications in microelectronic Device Fabrication © 2006 by Taylor & Francis Group, LLC Series in Plasma Physics Series Editor: Steve Cowley, Imperial College, UK and UCLA, USA Other books in the series: An Introduction to Inertial Confinement Fusion S Pfalzner Aspects of Anomalous Transport in Plasmas R Balescu Non-Equilibrium Air Plasmas at Atmospheric Pressure K H Becker, R J Barker and K H Schoenbach (Eds) Magnetohydrodynamic Waves in Geospace: The Theory of ULF Waves and their Interaction with Energetic Particles in the Solar-Terrestrial Environment A D M Walker Plasma Physics via Computer Simulation (paperback edition) C K Birdsall, A B Langdon Plasma Waves, Second Edition D G Swanson Microscopic Dynamics of Plasmas and Chaos Y Elskens and D Escande Plasma and Fluid Turbulence: Theory and Modelling A Yoshizawa, S-I Itoh and K Itoh The Interaction of High-Power Lasers with Plasmas S Eliezer Introduction to Dusty Plasma Physics P K Shukla and A A Mamun The Theory of Photon Acceleration J T Mendonça Laser Aided Diagnostics of Plasmas and Gases K Muraoka and M Maeda Reaction-Diffusion Problems in the Physics of Hot Plasmas H Wilhelmsson and E Lazzaro The Plasma Boundary of Magnetic Fusion Devices P C Strangeby Non-Linear Instabilities in Plasmas and Hydrodynamics S S Moiseev, V N Oraevsky and V G Pungin © 2006 by Taylor & Francis Group, LLC Series in Plasma Physics PlaSma ElEctronicS: applications in microelectronic Device Fabrication t makabe Keio University, Japan Z Petrovic´ Institute of Physics Belgrade, Serbia New York London © 2006 by Taylor & Francis Group, LLC Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-7503-0976-8 (Hardcover) International Standard Book Number-13: 978-0-7503-0976-9 (Hardcover) Library of Congress Card Number 2005056888 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Makabe, T. (Toshiaki). Plasma electronics : applications in microelectronic device fabrication / by T. Makabe and I. Petrovic p. cm. -- (Series in plasma physics) Includes bibliographical references and index. ISBN 0-7503-0976-8 1. Plasma engineering. I. Title. II. Series. TA2020.M35 2005 621.044--dc22 2005056888 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of Informa plc. http://www.crcpress.com © 2006 by Taylor & Francis Group, LLC Preface Overthepastthreedecadeslow-temperatureplasmaapplicationshavebeen extendedfromprimarilylightingtothefabricationofmicroelectronicdevices andnewmaterials,farexceedingourexpectations.Radiofrequencyplasmas rangingfrom105 Hzto109 Hz,inparticular,arenowusedtoprocessmetal- lic,semiconductor,anddielectricmaterialsforthefabricationofultralarge- scale integrated (ULSI) circuits and to deposit various kinds of functional thin films and to modify the surface properties. Without plasma-produced ions and dissociated neutral radicals for etching and deposition on wafers, microelectronicsmanufacturingforULSIcircuitwouldsimplybeunfeasible. The advent of ULSI fabrication has greatly changed how the field of plasma science is approached and understood. Low-temperature non-equilibrium plasmas are sustained mainly by electron impact ionization of a feed gas drivenbyanexternalradiofrequencypowersource.Theselow-temperature plasmas acquire characteristics intrinsic to the feed gas molecules as deter- minedbytheiruniqueelectron-collisioncross-sectionsets.Thisuniqueness meansthatplasmasmustbeunderstoodusingquantum,atomic,andmolecu- larphysics.Thedisparatetimeandspatialscalesinvolvedinlow-temperature plasma processing (submeter to nanometer and seconds to nanoseconds) makes plasma processing an inherently stiff problem. The characteristics of low-temperatureplasmascontrastmarkedlywithhighlyionizedequilibrium plasmasthatareensemblesofchargedparticleswhosebehaviorcanbeun- derstoodthroughtheirlong-rangeCoulombinteractionsandcollectiveeffects andcharacterizedbyplasmafrequency,Debyelength,andelectrontempera- ture.Thefundamentalcollisionandreactionprocessesoccurringbothingas phase and on surface in low-temperature plasmas are the bases for under- standingtheirbehaviorandexploitingthemforpracticalapplications. In the emerging nanotechnology era, device design, reliability, and the design of integrated plasma processes for their fabrication are tightly cou- pled. Being able to predict feature profile evolution under the influence of spatio-temporallyvaryingplasmasisindispensablefortheprogressofnano- technology.Predictionofplasmadamageanditsmitigationarealsocrucial prediction&mitigationare2processandwillbeperformedthroughaseries of vertically integrated numerical modeling and simulations ranging from the reactorscaletothosecognizantofdeviceelementssubtendingtheplasma. Motivatedbytheimportantroleofplasmasintechnologyandtheneedfor simulationstounderstandtheassociatedcomplexprocesses,weemphasize © 2006 by Taylor & Francis Group, LLC in this book academic fusion among atomic and molecular physics, surface physics, the Boltzmann transport theory, electromagnetic theory, and com- putationalscienceasplasmaelectronics.Wedothistodescribeandpredict thespaceandtimecharacteristicsoflow-temperatureplasmasandassociated processingintrinsictospecificfeedgases.Anunderlyingthemethroughthis work is computer-aided plasma analysis and synthesis, with emphasis on computationalalgorithmsandtechniques. ThisbookisbasedonaseriesoflecturespresentedatKeioUniversityas part of its graduate program. The university’s interest in the subject matter and feedback were essential parts of developing this text. We believe that the book is well suited as an instrument for self-instruction through its topical exercisesandproblemsarrangedineachchapter. It is a pleasure to acknowledge our debt to David Graves and Robert Robson, who have helped in a variety of ways during the long period of our research life composing plasma electronics. Finally, we are indebted to T.Yagisawaforfigurepreparationandhisattentiontodetail. ToshiakiMakabeandZoranLj.Petrovic KeioUniversity © 2006 by Taylor & Francis Group, LLC Contents 1 Introduction..........................................................1 1.1 PlasmaandItsClassification.....................................1 1.2 ApplicationofLow-TemperaturePlasma.........................2 1.3 AcademicFusion................................................3 References............................................................4 2 PhenomenologicalDescriptionoftheCharged ParticleTransport....................................................5 2.1 TransportinReal(Configuration)Space..........................5 2.1.1 MomentumBalanceofElectrons..........................6 2.1.2 EnergyBalanceofElectrons...............................7 2.2 TransportinVelocitySpace.....................................10 2.2.1 ElectronVelocityDistributionand SwarmParameters.......................................10 2.2.2 IonVelocityDistributionandMeanEnergy...............18 2.3 ThermalEquilibriumandItsGoverningRelations...............20 2.3.1 BoltzmannDistributioninRealSpace....................21 2.3.2 MaxwellDistributioninVelocitySpace...................22 References...........................................................25 3 MacroscopicPlasmaCharacteristics.................................27 3.1 Introduction....................................................27 3.2 Quasi-Neutrality................................................27 3.3 Charge-SeparationinPlasmas...................................28 3.3.1 SpatialScaleofCharge-Separation.......................28 3.3.2 TimeScaleforCharge-Separation........................29 3.4 PlasmaShielding...............................................30 3.4.1 DebyeShielding.........................................30 3.4.2 MetalProbeinaPlasma..................................31 3.5 ParticleDiffusion...............................................34 3.5.1 AmbipolarDiffusion.....................................34 3.5.2 SpatialandTimeScaleofDiffusion.......................35 3.6 BohmSheathCriterion..........................................37 3.6.1 BohmVelocity...........................................37 3.6.2 FloatingPotential........................................38 References...........................................................39 © 2006 by Taylor & Francis Group, LLC 4 ElementaryProcessesinGasPhaseandonSurfaces.................41 4.1 ParticlesandWaves.............................................42 4.1.1 ParticleRepresentationinClassicaland QuantumMechanics.....................................42 4.1.2 LocallyIsolatedParticleGroupandWavePackets........44 4.2 CollisionsandCrossSections...................................46 4.2.1 ConservationLawsinCollisions.........................47 4.2.2 DefinitionofCollisionCrossSections.....................49 4.2.3 TheDistributionofFreePaths............................53 4.2.4 RepresentationofCollisionsinLaboratoryand CMReferenceFrames....................................54 4.3 ClassicalCollisionTheory.......................................57 4.3.1 ScatteringinClassicalMechanics.........................58 4.3.2 ConditionsfortheApplicabilityoftheClassical ScatteringTheory........................................64 4.4 QuantumTheoryofScattering..................................64 4.4.1 DifferentialScatteringCrossSectionσ(θ).................66 4.4.2 ModifiedEffectiveRangeTheoryin ElectronScattering.......................................71 4.5 CollisionsbetweenElectronsandNeutral Atoms/Molecules..............................................72 4.5.1 ResonantScattering......................................73 4.6 Electron–AtomCollisions.......................................75 4.6.1 EnergyLevelsofAtoms..................................75 4.6.2 Electron–AtomScatteringCrossSections.................77 4.7 Electron–MoleculeCollisions...................................81 4.7.1 Rotational,Vibrational,andElectronicEnergy LevelsofMolecules......................................81 4.7.2 RotationalExcitation.....................................83 4.7.2.1 RotationalEnergyLevels........................83 4.7.2.2 RotationalExcitationCrossSections.............84 4.7.3 VibrationalExcitation....................................87 4.7.3.1 VibrationalEnergyLevels.......................87 4.7.3.2 VibrationalCrossSections.......................88 4.7.4 ElectronicExcitationandDissociation....................89 4.7.4.1 ElectronicStatesofMolecules....................89 4.7.4.2 CrossSectionsforElectronicExcitation ofMolecules....................................91 4.7.5 ElectronCollisionswithExcitedAtoms andMolecules...........................................92 4.8 NonconservativeCollisionsofElectronswithAtoms andMolecules..................................................95 4.8.1 Electron-InducedIonization..............................95 4.8.2 ElectronAttachment.....................................97 4.8.2.1 DissociativeElectronAttachment................99 4.8.2.2 NondissociativeElectronAttachment...........100 © 2006 by Taylor & Francis Group, LLC 4.8.2.3 IonPairFormation............................101 4.8.2.4 ElectronAttachmenttoExcitedMolecules.....101 4.8.2.5 RateCoefficientsforAttachment..............102 4.8.3 Electron–IonandIon–IonRecombination..............103 4.8.4 Electron–IonandElectron–ElectronCollisions..........105 4.9 HeavyParticleCollisions.....................................105 4.9.1 Ion–MoleculeCollisions...............................106 4.9.1.1 ChargeTransfer,Elastic,andInelastic ScatteringofIons ............................. 106 4.9.1.2 Ion–MoleculeReactions.......................109 4.9.2 CollisionsofFastNeutrals.............................110 4.9.3 CollisionsofExcitedParticles..........................111 4.9.3.1 Chemi-IonizationandPenningIonization.....112 4.9.4 CollisionsofSlowNeutralsandRateCoefficients......116 4.9.4.1 QuenchingandTransportofExcitedStates....116 4.9.4.2 KineticsofRotationaland VibrationalLevels.............................118 4.10 PhotonsinIonizedGases.....................................119 4.10.1 EmissionandAbsorptionofLineRadiation...........119 4.10.2 ResonantRadiationTrapping.........................122 4.11 ElementaryProcessesatSurfaces.............................123 4.11.1 EnergyLevelsofElectronsinSolids...................124 4.11.2 EmissionofElectronsfromSurfaces...................127 4.11.2.1 Photo-Emission.............................127 4.11.2.2 ThermionicEmission ....................... 128 4.11.2.3 Field-InducedEmission.....................129 4.11.2.4 PotentialEjectionofElectronsfromSurfaces byIonsandExcitedAtoms..................132 4.11.3 EmissionofIonsandNeutralsfromSurfaces..........135 4.11.3.1 SurfaceNeutralization......................136 4.11.3.2 SurfaceIonization...........................136 4.11.4 Adsorption...........................................138 References..........................................................140 5 TheBoltzmannEquationandTransportEquationsof ChargedParticles..................................................143 5.1 Introduction..................................................143 5.2 TheBoltzmannEquation.....................................143 5.2.1 TransportinPhaseSpaceandDerivationofthe BoltzmannEquation...................................144 5.3 TransportCoefficients........................................146 5.4 TheTransportEquation.......................................150 5.4.1 ConservationofNumberDensity......................151 5.4.2 ConservationofMomentum...........................151 5.4.3 ConservationofEnergy................................153 5.5 CollisionTermintheBoltzmannEquation....................153 © 2006 by Taylor & Francis Group, LLC 5.5.1 CollisionIntegral.......................................153 5.5.2 CollisionIntegralbetweenanElectronanda GasMolecule...........................................154 5.5.2.1 ElasticCollisionTerm J ......................155 elas 5.5.2.2 ExcitationCollisionTerm J ...................158 ex 5.5.2.3 IonizationCollisionTerm J ...................158 ion 5.5.2.4 ElectronAttachmentCollisionTerm J ........ 159 att 5.6 BoltzmannEquationforElectrons..............................160 5.6.1 SphericalHarmonicsandTheirProperties ..............160 5.6.2 VelocityDistributionofElectrons.......................163 5.6.2.1 VelocityDistributionunderUniform NumberDensity:g0............................164 5.6.2.2 VelocityDistributionProportionalto ∇ n(r,t):g1.....................................165 r 5.6.3 ElectronTransportParameters..........................170 References..........................................................174 6 GeneralPropertiesofChargedParticleTransportinGases.........175 6.1 Introduction...................................................175 6.2 ElectronTransportinDC-ElectricFields........................175 6.2.1 ElectronDriftVelocity..................................175 6.2.2 DiffusionCoefficients...................................178 6.2.3 MeanEnergyofElectrons...............................180 6.2.4 Excitation,Ionization,andElectron AttachmentRates.......................................181 6.3 ElectronTransportinRadio-FrequencyElectricFields..........184 6.3.1 RelaxationTimeConstants..............................184 6.3.2 EffectiveFieldApproximation..........................189 6.3.3 ExpansionProcedure...................................192 6.3.4 DirectNumericalProcedure............................195 6.3.5 Time-VaryingSwarmParameters.......................199 6.4 IonTransportinDC-ElectricFields.............................201 References..........................................................203 7 ModelingofNonequilibrium(Low-Temperature)Plasmas.........205 7.1 Introduction...................................................205 7.2 ContinuumModels............................................207 7.2.1 GoverningEquationsofaContinuumModel............207 7.2.2 LocalFieldApproximation(LFA).......................210 7.2.3 Quasi-ThermalEquilibrium(QTE)Model...............211 7.2.4 RelaxationContinuum(RCT)Model....................212 7.2.5 PhaseSpaceKineticModel..............................214 7.3 ParticleModels................................................216 7.3.1 MonteCarloSimulations(MCSs) ....................... 216 7.3.2 Particle-in-Cell(PIC)andParticle-in-Cell/MonteCarlo Simulation(PIC/MCS)Models.........................219 © 2006 by Taylor & Francis Group, LLC