Advances in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 43
Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Max-Planck-Institutfu’r Quantenoptik Garching bei Munchen Germany
Editorial Board P. R. BERMAN University of Michigan Ann Arbol; Michigan
M. GAVRILA F0.M. Instituut voor Atoom-en MolecuulJLsica Amsterdam The Netherlands M . INOKUTI Argonne National Laboratory Argonne, Illinois W. D. PHILLIPS National Institute for Standards and Technology Gaithersburg, Malyland
Founding Editor SIRDAVIDR. BATES
Supplements 1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed. 2. Cavity Quantum Electrodynamics, Paul R. Berman, Ed. 3. Cross Section Data, Mitio Inokuti, Ed.
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK
Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANK-INSTITUT FUR QUANTENOPTIK MUNICH, GERMANY
Volume 43
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1
Contents CONTRIBUTORS
. . .. . . . .. ... . . . . ... .. ... .. . . .. .. . .. .. . . . . . .. . . .. . . . .. .. . . . .. .... ... .. . . .. . . .. . . .. ..
ix
Plasma Processing of Materials and Atomic, Molecular, and Optical Physics. An Introduction Hiroshi Tanaka and Mitio Inokuti ...................................................................
111. IV. V. VI.
.....
and Collision Processes.. .... . . . . ... .. .. , . , .. .. . . ... .. . . .. . . . . .. .. . . Plasma Diagnosis and Modeling , .. , . , .. .. .. ..... . . .. . .. . . . .. .. Pertinent Topics from Atomic, Molecula ....................... Acknowledgments . .. . . . .. .. . .. .. .. ... .. . . . References ........................................................................
1 3 7 12 16 16
The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas R. Winkler I. Introduction.. . . .. .. , .. .. ... .. . . . .. .. . . . .. .. .
.......................
11. Kinetic Description of the Electrons .. , . , . . .. . . . . , , . .. .. , . , , . , . ,... . .. . . .. .. .... .. .. .. . . 111. Electron Kinetics in Time- and Space-Independent Plasmas . . . . . .. . . . . . . .. . .. . . . . . . .
IV. V. VI. VII. VIII.
Electron Kinetics in Time-Dependent Plasmas. .. . . .. .. . .. . .. .. . Electron Kinetics in Space-Dependent Plasmas.. .. , . , .. .. , .. . . . . . . . .. . . . . .. .. .. .. . . . . . Concluding Remarks . .. ... .. . . . .. . . .. . .. .. . . . . . . . .. . . . . . . . .. Acknowledgments .. .. . . . .. .. . . . . . . . . . . .. , .. . . .. . . . . . . . .. .. ... .. .. ... .. .. . .. .. . . . ..... . .. . ....................... References .. . . .. . .... .. . . . .. .. . .. . . . .. . . .. . .
20 24 32 47 61 15 16 76
Electron Collision Data for Plasma Chemistry Modeling WL. Morgan I. Dedication
.......................................................................
19 80 81 111. Sources of Data and Interpretations.. .. . . 90 n! Discussion of Data for Specific Processes and Species ............................... ... 104 V. Concluding Remarks: Journals, Databases, and the World Wide Web VI. Acknowledgements .. .. .. . .. . . . .. .. .. . . . . . .. . . . .. ... .. .. .. . .. .. .. . .. . . .. . . . . .. .. . .. . . . . . . . 107 107 VII. References ... . . .. . .. 11. Introduction., , .. , . , .... . . . .. .. ... .. . . . . . .. ..
....................... ........................
vi
CONTENTS
Electron-Molecule Collisions in Low-Temperature Plasmas: The Role of Theory Carl mnstead and fincent McKoy I. Introduction.,.............................................................................
1 11
11. Types of Cross Sections
111. Cross-Section Calculatio 1V. Methods in Current Use.. ............................................................... V. Areas for Future Progress ...............................................................
124 139
VII. References
Electron Impact Ionization of Organic Silicon Compounds Ralf Basner, Kurt Becker, Hans Deutsch, and Martin Schmidt I. Introduction.. .............................................................................
147
11. Ionization-Cross-Section Measurements 149 111. Semiempirical Calculation of Total Single Ionization Cross Sections. .............. 156 I\! Ionization Cross Sections of SiH, (x = 1 to 4) and of Selected Si-Organic
Compounds ............................................................................... V. Comparison with Ion Formation Processes and Ion Abundances in Plasmas ...... VIII. References
..........................
160 177
182
Kinetic Energy Dependence of Ion-Molecule Reactions Related to Plasma Chemistry P B. Armentrout I. Introduction. .............................................................................. 11. Experimental Methods .......... 111. Reactions with Silane (SiH,). ...........................................................
IV. V. V1. V11. VIII.
Reactions Involving Organosilanes ................................. Reactions with Silicon Tetrafluorid ............................................ Reactions with Silicon Tetrachloride (SiCI,) .......................................... ........... Reactions with Fluorocarbons (CF, and C2Fs) Miscellaneous Thermochemical Studies ...............................................
......................................................................... X. Acknowledgment.. ....................................... XI. References ................................................................................
I88 189 I95 204 207 215 219 223 225 226 226
CONTENTS
vii
Physicochemical Aspects of Atomic and Molecular Processes in Reactive Plasmas Yoshihiko Hatano I. Introduction
................................................
231
ctive Plasmas. .................. 11. Atomic and 111. Overview and Comments on Free Radical Reactions in Reactive Plas
I\! Deexcitation of Excited Rare Gas Atoms by Molecules Containing Group IV elements.................................................................................... V. Comments on Atomic and Molecular Processes in Reactive Plasmas from Physicoche ............................................................. VI. References .............................................................
235 240 240
Ion-Molecule Reactions Werner Lindinger, Armin Hansel and Zdenek Herman I. Introduction 11. Reaction Ra 111. Types of Ion-Molecule Processes ....................................................... 1V. Effect of Internal Energy and Temperature on IM Processes.. ....................... V. Concluding Remarks ..................................................................... .................... VI. Acknowledgments .... VII. References .................................................................................
243 249 253 279 288 289 289
Uses of High-Sensitivity White-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing L. I T Anderson, A.N. Goyette, and JE. Lawler ........................................................................... White-Light Absorption Spectroscopy 111. The Uses of High-Sensitivity White-Light Absorption Spectroscopy in the CVD of Diamond Films ........................................................................ n! The Uses of High-Sensitivity White-Light Absorption Spectroscopy in Other CVD Environments.. .............................................................. V. Other Uses of High-Sensitivity White-Light Absorption Spectroscopy.. ............ VI. Conclusion.. ............................................................................... VII. Acknowledgments ........................................................................ VIII. References
295 296 303 332 334 337 338 338
viii
CONTENTS
Fundamental Processes of Plasma-Surface Interactions Rainer Hippler I. Introduction 111. Scattering of Ions at Surfaces. ..........................................................
I\! Physical Sputtering.. ..................................................................... v1 Chemical Effects ......................................................................... VI. References
358 36 1 367 370
Recent Applications of Gaseous Discharges: Dusty Plasmas and Upward-Directed Lightning Ara Chutjian I. Dust in Plasma Environments ...........................................................
374
11. Elves, Red Sprites, and Blue Jets
I\! References
Opportunities and Challenges for Atomic, Molecular, and Optical Physics in Plasma Chemistry Kurt Becker, Hans Deutsch and Mitio Inokuti 1. Acknowledgement.. ...................................................................... I1 References ................................................................................ SUBJECT INDEX.. .................................................................................. CONTENTS OF VOLUMES IN THIS SERIES. ........................................................
406 406 407 4 15
Contributors
Numbers in parentheses indicate pages on which the author’s contributions begin.
L. W. ANDERSON(295), Department of Physics, University of Wisconsin, Madison, WI 53706
(187), Department of Chemistry, University of Utah, PETERBRUCEARMENTROUT Salt Lake City, UT 841 12 RALF BASNER(147), Institut f i r Niedertemperatur-Plasmaphysik, Universitaet Greifswald, D- 17487 Greifswald, Germany KURTH. BECKER(147, 399), Dept. of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030 ARACHUTJIAN (399), Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91 109 HANS DEUTSCH(147), Institut f i r Physik, Universitaet Greifswald, D-17487 Greifswald, Germany (295), Department of Physics, University of Wisconsin, Madison, A. N. GOYETTE WI 53706 ARMINHANSEL(243), Institut f i r Ionenphysik, Universitaet Innsbruck, A-6020 Innsbruck, Austria
YOSHIHIKOHATANO(23 l), Department of Chemistry, Tokyo University of Technology, Tokyo 152-0033, Japan ZDERNEK HERMAN(243), J. Heyrovsky Institute of Physical Chemistry, Academy of Science of the Czech Republic, CZ-18223 Prague, Czech Republic
RAINERHIPPLER (341), Institut fiir Physik, Universitaet Greifswald, D- 17487 Greifswald, Germany ix
X
CONTRIBUTORS
MITIOINOKUTI (l), Physics Division, Argonne National Laboratory, Argonne, IL 60439 J. E. LAWLER (295), Department of Physics, University of Wisconsin, Madison, WI 53706
WERNERLINDINGER (243), Institut fir Ionenphysik, Universitaet Innsbruck, A-6020 Innsbruck, Austria VINCENT MCKOY(1 1 l), A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, CA 91 125 W. LOWELL MORGAN (79), Kinema Research & S o h a r e , Monument, CO 80132 ARTIN IN SCHMIDT (147), Institut fir Niedertemperatur-Plasmaphysik, Universitaet Greifswald, D-17487 Greifswald, Germany
ROLFWINKLER(19), Institut flir Niedertemperatur-Plasmaphysik, Universitaet Greifswald, D- 17487 Greifswald, Germany CARL L. WINSTEAD(1 1 l), A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, CA 91 125
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
PLASMA PROCESSING OF MATERIALS AND ATOMIC, MOLECULAR, AND OPTICAL PHYSICS. AN INTRODUCTION HIROSHI TANAKA Department of Physics, Faculty of Science and Technologv, Sophia University, Tobo, Japan
MITIO INOKUTI Physics Division. Argonne National Laboratory, Argonne. Illinois
I. Introduction ...................................................................... 11. Plasma Formation and Collision Processes A. Plasma Structure and Molecular Species .................................... B. Behavior of Particles in a Plasma. ........................................... C. Chemical Reactions on a Substrate Surface ................................. 111. Plasma Diagnosis and Modeling.. ............................................... A. Plasma Monitoring ......................... B. Plasma Simulation.. ........................ IV: Pertinent Topics from Atomic, Molecular, and Optical Physics ................ A. Electron Collisions with Molecules.. ........................................ B. Transport and Reactions of Chemically Active Species. .................... C. Remarks on Theoretical Studies ............................................. V Acknowledgments.. .............................................................. VI. References. ....................................
1
3 3 3 6 7
12 12 14 15
16 16
I. Introduction Low-temperatureplasmas are generated by glow discharges of low-pressure gases and are widely used in industry for chemical-vapor deposition (CVD),plasma etching, and other treatments of solid surfaces for the manufacture of useful materials such as amorphous silicon (a-Si: H) for solar cells and of ultra largescale integrated (LSI) circuits. These examples may be viewed as a new application of atomic, molecular, and optical physics to materials science and technology (Lieberman and Lichtenberg, 1994; Bruno et al., 1995; Fujishiro et al., 1998). In the manufacture of ultra LSI circuits, for instance, it is desirable rapidly to create prescribed patterns with scales less than 0.1 pm and fine structures with an aspect ratio of over 10 on a silicon wafer with a diameter of more than 30 cm. To I
Copyright 6 2000 by Academic Press All rights of reproduction in any form reserved ISBN: 0-12-003843-9/ISSN: 1049-25OX $30.00
2
Hiroshi Tanaka and Mitio Inokuti
this end, one needs an ion beam that has a high current, is well collimated, and is spatially uniform, and a source of such a beam is a high-density plasma. Much research is being carried out to design optimal plasma sources for use under different conditions. Problems to be solved arise from the presence of a magnetic field, from the enormous variety of reacting species present, and from their interactions with vessel walls. The densities of electrons or ions in plasmas used for material processing are lo9 to 10” ~ m - The ~ . electrons are characterized by an energy distribution with a mean energy of several electron volts (corresponding to a temperature T, of some 10,000K). Most of the electrons have kinetic energies below 20 e\! Yet a modest number of them with relatively high kinetic energies are responsible for electronic excitation, dissociation, and ionization of molecules. Discharges by radiofrequency (rf) waves or microwaves are usually generated in gas at pressures of 10 to lop2torr, at which two-body collisions among electrons, ions, and molecules are decisive for the structure of discharges and for transport of particles. Thus, characteristics of a resulting plasma reflect the properties of atomic and molecular species, including the cross sections for the two-body collisions with electrons, ions, radicals, and other species and transition probabilities governing interactions with photons. The densities of neutral molecules (in the ground electronic state) are higher by orders of magnitude than the densities of electrons or ions; therefore, one often characterizes such a plasma as “weakly ionized.” The kinetic energies of ions are much lower than the kinetic energies of electrons, and correspond to a temperature Tinot much higher than the gas temperature TN;therefore, one often describes such a plasma with the modifiers “low-temperature” and “thermally non-equilibrium.” In a weakly ionized low-temperature plasma, various molecular species are abundantly generated either directly or indirectly as a consequence of electron collisions with molecules, and many of the molecular species readily react with other species. For this reason, one sometimes calls such a plasma “chemically reactive.” Applications of chemically reactive plasmas are widespread over organic and inorganic materials, in part because of the relatively low cost of generating of such plasmas. The large variety of chemically active species generated in a plasma is sometimes a disadvantage because they may initiate many reaction pathways, which may be difficult to analyze and to control. It is thus clearly important to further develop the technology of plasma processing of materials toward advanced goals, including the identification and characterization of usefbl reactions, the optimization of physical parameters for the generation of a plasma best suited for a given purpose, the easiest and surest control of plasma properties, the best economics, and the minimization of any potentially adverse impact of the technology on the human environment and health (ASET, 1998). A rational and reliable approach to these goals must be based on a full understanding of the fundamentals of plasma chemistry at the
PLASMA PROCESSING OF MATERIALS
3
molecular level, the scope of which largely belongs to atomic, molecular, and optical physics. The fundamentals are illustrated (though not exhaustively represented) by the following articles in the present volume. The crucial importance of the fundamentals of atomic, molecular, and optical physics has been seen earlier in fusion-plasma research, astrophysics, and radiation physics. Plasma chemistry is a relatively recent addition to the list of fruitful applications of atomic, molecular, and optical physics.
11. Plasma Formation and Collision Processes A. PLASMA STRUCTURE AND MOLECULAR SPECIES Figure 1 (Japan Society of Applied Physics, 1993) shows schematically the basic structure of a glow-discharge reactor using coupling with a high-frequency capacitor, and also shows the electric potential between the electrodes in the presence of a plasma. A high-frequency (13.56 MHz) electric field is applied to a gas at a pressure of lop2 to 1 torr between the electrode S, on the power-source side and the electrode S, on the earth side; this causes excitation, dissociation, and ionization of molecules by electron collisions with molecules, and leads to the formation of a self-sustaining glow-discharge plasma. The central part of positive glow, called a bulk plasma, has an electric potential Vp,which is always positive and is comparable to the first ionization potential of a molecule of the major constituent of the gas. The electrode connected to earth (or the anode) has a lower potential than the bulk plasma, and is subjected to impacts of positive ions, as discussed by Hippler (1999). The electrode on the power-source side (or the cathode) also has a lower potential than the bulk plasma, because electrons follow the high-frequency electric field, but ions do not, so that a negative load is applied to the blocking capacitor. As a consequence, a negative self-bias potential V, is established in a region near the cathode; this region does not glow and is called the plasma sheath. The glow is most intense in a region of the bulk plasma close to the sheath, showing the presence of many electrons of relatively high energies. In addition to the capacitor-coupled reactor shown in Fig. 1, there are two other classes of reactors, namely, those coupled with an inductor and those excited by microwaves. Various gases are used depending on the materia!s treated, as Table 1 (Samukawa, 1999) summarizes.
B. BEHAVIOR OF PARTICLES IN A PLASMA Figure 2 illustrates collisions and reactions of particles in a low-temperature plasma used for plasma CVD (for instance, the formation of a-Si:H films by SiH,, on the left-hand side) and for plasma etching (for instance, microprocessing
4
Hiroshi Tanaka and Mitio Inokuti
FIG. 1. Schematic diagram of a high-frequency glow-discharge reactor, and the electric potential between electrodes (Japan Society of Applied Physics, 1993). The left-hand panel shows the basic structure of the reactor. The right-hand panel shows the electric potential. TABLEI .
GASESCOMMONLY USED FOR PLASMA ETCHING Materials treated Silicon
Silicon dioxide Aluminium alloys
Classification
Molecular species
Fluorides Chlorofluorides Chlorides Bromides Fluoride/hydrogen Fluorocarbons Chlorides Chlorofluorides Bromides
CF,, SF,, NF,, SiF,, BF,, CBrF,, XeF, CCIF,, CCI,F,, CCI,F, C,CIF,, C2C12F, CCI,, SiCI,, PCI,, BCI,, CI,, HCI Br,, HBr CHF,, CF, H, CzF,, C,F,, C4Fs CCI,, BCl,, SiCI,, CI,, HCl CC12F2,CC1,F Br,, BBr,
+
FIG. 2. Collisions and reactions of particles in a low-temperature plasma. The left-hand panel shows a plasma used for chemical-vapor deposition, is., the formation of a-Si : H by a discharge of SiH, gas. The right-hand side shows etching of a silicon surface by a discharge of CF, gas.
PLASMA PROCESSING OF MATERIALS
of Si surfaces, on the right-hand side). It is useful to distinguish three temporal stages of the numerous and in general complex atomic and molecular processes occurring in such a plasma. The first stage may be called physical or initial, and includes excitation, dissociation, and ionization of molecules by electron collisions (Basner et al., 1999; Morgan, 1999). The second stage may be called physicochemical or secondary, and includes reactions of reactive species such as subexcitation electrons (electrons with kinetic energies below the first electronicexcitation threshold of the major constituent molecule), photons emitted by excited molecules, positive and negative ions (Armentrout, 1999; Lindinger, et al., 1999), excited atoms or molecules, and free radicals (Hatano, 1999) with other molecules. The third stage may be called chemical or thermal, and includes further reactions of the products of the second stage, which occur under nearly thermal-equilibrium conditions. The products of the third stage undergo diffusion, and some of them proceed to react with the surfaces of the reactor walls (Hippler, 1999). Let us discuss further collisions and reactions in chemical etching of Si and SiO, in a CF, plasma. Electron energy-loss spectra (Kuroki et al., 1992) of CF, show broad bands without vibrational structure, indicating that most of the lowlying excited states have repulsive adiabatic potential surfaces, leading to immediate dissociation. In other words, electron collisions mostly result in the dissociation of CF, into CF,, CF,, CF, and other radicals (Winters and Inokuti, 1982), which are all chemically reactive, as represented by e+CF, + C F , + F + e CF, F, e CF+F,+F+e.
+ +
The detection of neutral radicals is not straightforward in general, and continues to be a subject of current research (Sugai et al., 1995; Cosby, 1993; Mi and Bonham, 1998; Motlagh and Moore, 1998). One method uses threshold-ionization mass spectrometry (Sugai et al., 1995), and another is based on the adsorption of radicals on a tellurium surface (Motlagh and Moore, 1998). As a consequence of electron collisions, dissociative ionization also occurs; this leads to CF:, which are readily detected by mass spectrometry (Poll et al., 1992). Subsequent light emission provides information about the formation of electronically excited dissociation fragments. The cross sections and appearance potentials for various optical emissions can be studied in many ways useful for plasma diagnostics (Becker, 1994). Radicals, subexcitation electrons, and ions react predominantly with groundstate molecules of the major constituent, which are most abundant, and less frequently among themselves. Through diffusion and repeated secondary reactions,
6
Hiroshi Tanaka and Mitio Inokuti
they lead to F, FZ,and polymers C,F, (including saturated and unsaturated bonds between adjacent carbon atoms), as represented by
+
+
F CF, -+ CF3 F2 CF3 CF, -+ C2F5 F2.
+
+
Furthermore, clusters (i.e., aggregates of atoms or molecules) and particulates (i.e., small particles of solid) are also formed. The particulates thus formed may contaminate the base surface, influence plasma properties and structure, or have other undesirable consequences (Kushner, 1994). Furthermore, interactions of these particulates or of dust particles otherwise present are important to plasma properties and behavior; therefore, they are a subject of extensive current research (Chutjian, 1999). Low-temperature plasmas are also generated by the use of electron cyclotron resonance (ECR) in a gas at low pressure, lop3 to lo-' Pa, where secondary reactions are probably negligible. However, in an afterglow of an ECR plasma, secondary reactions cannot be disregarded.
c . CHEMICAL &ACTIONS
ON A
SUBSTRATE SURFACE
Reactive species formed in a plasma and unreacted molecules introduced for etching may reach the substrate surface, and some of them will be adsorbed, with the probability depending on the temperature, the electric properties, the state of chemical binding, and the structure of the surface. If the logarithm of the rate of etching of a Si02 surface depends linearly on 1 / T , the reciprocal of the temperature T of the surface, one may be justified in regarding the etching process as a chemical reaction occurring under a nearly thermal equilibrium, and may determine an (apparent) activation energy from such an Arrhenius plot. The rate of etching is greater in general when the activation energy is lower, but depends also on the free-energy change of the reaction. The reaction proceeds in general in the direction of increased binding energies. In our example, F atoms adsorbed on the Si02 surface do not react with Si atoms because the Si-F bond dissociation energy (130 kcal/mol) is smaller than the Si-0 bond dissociation energy (192 kcal/mol). However, the etching reaction proceeds because the adsorption of C,F,. on the Si02 surface results in the formation of a CO bond (with bond dissociation energy as large as 256.7kcal/mol) and also in the formation of SiO, from Si and F atoms released in the gas, as represented by Si(surface)
+ 4F +. SiF,
PLASMA PROCESSING OF MATERIALS
7
When 0, is added to the gas, electron collisions with 0, may lead to dissociation, producing chemically active 0 atoms. They react with CxF,, to produce CO, CO,, and COF,, which are released into the gas, as represented by
+
+
+
+ +
0 CF, + COF, F O+CF, + C O + 2 F COF F 0 COF + CO, F
o+c+co etc. In addition to the chemical etching described above, there are other kinds of etching processes, namely physical etching and ion-assisted etching.
111. Plasma Diagnosis and Modeling A. PLASMAMONITORING For the understanding of plasma properties and for the control of a plasma reactor, it is important to detect electrons, ions, and other active species present in a plasma and to measure their densities. To this end, various methods have been developed, including measurements of radicals by absorption spectroscopy (Anderson et al., 1999) or optical-emission spectroscopy, measurements of electron densities and electric fields by probes, and measurements of ions by mass spectrometry (Matsuda et al., 1983; Robertson et al., 1983). In particular, neutral and nonemitting radicals (for instance, radicals in the electronic ground state) are expected to be abundantly present in a nonequilibrium plasma and have become measurable recently (Sugai et al., 1995; Cosby, 1993; Mi and Bonham, 1998; Motlagh and Moore, 1998). To measure radicals containing a small number of hydrogen atoms such as SiH and CH in the electronic ground state, use has been made of spectroscopic methods such as optical-emission spectroscopy (OES) (Matsuda et al., 1983), infrared laser absorption spectroscopy (IRLAS) (Itabashi et al., 1988; Goto, 1990), the laser-induced fluorescence (LIF) method (Lee et al., 1983), and coherent anti-Stokes Raman spectroscopy (CARS) (Hata, 1989). As we discussed in Section ILC, nonemitting radicals such as SiH, and CF, are regarded as precursors of thin-film formation or chemical etching. These radicals have begun to be analyzed by laser-spectroscopic methods (for instance, absorption spectroscopy with infrared radiation from a semiconductor laser, absorption spectroscopy in an internal cavity, and resonance ionization spectrometry) and by threshold ionization methods, sometimes leading up to the determination of
8
Hiroshi Tanaka and Mitio Inokuti
formation cross sections and reaction rate constants. These methods are nondestructive and are particularly advantageous for plasma monitoring. Figure 3 illustrates an example of absorption spectroscopy using multiple reflection of infrared radiation from a semiconductor laser (Itabashi et al., 1988). Among the control parameters of a semiconductor laser operated at a few tens of Kelvins for infrared radiation of 0.1 mW and line width of 10 MHz, one tunes the wave number to a few tens of cm-' by adjusting the temperature, and to a few cm-' by adjusting the current. The use of multiple reflection increases the absorption path length and permits detection of radicals at densities as low as lo9 to 1O'O ~ m - This ~ , method has made it possible to observe infrared absorption by rotational and vibrational transitions in SiH,, SiH2, and SiH radicals and to determine their densities. The power of the method has been demonstrated in the determination of a density of 10" to 10l2cmP3of SiH, in a plasma of a SiH,-H, mixture (Itabashi et al., 1990). The SiH3 radical is the most important precursor of the formation of a-Si : H in plasma CVD. The novelty of the method consists in the use of infrared spectra, which permit high-precision measurements on polyatomic molecules and radicals at low densities, while the spectroscopy in the visible and ultraviolet regions also has been widely used for plasma monitoring. Table 2 (Japan Society of Applied Physics, 1993) shows a comparison of various methods. Progress in plasma monitoring with advanced methods of atomic, molecular, and optical physics, as exemplified above, has contributed greatly to plasmaprocessing technology, especially in improving the reproducibility of a treatment procedure.
200cm
pyrrx glass
(imcr radius o f 10 em)
I am pli f i r r
pulse generator 3Spps 0.45msec
FIG. 3. An apparatus for infrared laser absorption spectroscopy using multiple reflection (Itabashi
et al., 1988).
TABLE2. METHODS USED FOR
Method
Object of measurement
Infrared laser absorption spectroscopy (IRLAS)” Coherent anti-Stokes Raman spectroscopy (CARS)* Laser-induced fluorescence (LIF)’ Optical-emission spectroscopy (OES)d Mass spectrometry”‘ Langmuir probe
“Itabashi ef id.,1990.
DIAGNOSTICS OF SiH, PLASMAS
Neutral molecules (in the ground state) SiH,, SiH,, SiH,, SiH Neutral molecules (in the ground state) SiH,, SiH,, H,, Si,H, Neutral molecules SiH,, SiH, Si, Emitting states SiH*, Si* H2*, H*
Limit of detection
‘Hata, 1989.
Remarks
1 0 ~ - 1 0cm-, ~~
Good
High sensitivity
10” cm-,
Excellent
1Oh ~ r n - ~ I 0, cm-3
Excellent Good
Possible measurement of gas temperature Not easy to quantify Not easy to quantify
Positive or negative ions Electron density and energy
bLee et al., 1983.
Spatial resolution
dMatsuda et al., 1983.
Inapplicable Good
‘Robertson et al., 1983.
Perturbs plasma, material accumulation on probes
E?
n
CJ
50 %
6g
10
Hiroshi Tanaka and Mitio Inokuti
B. PLASMA SIMULATION Complementary to plasma monitoring, theoretical modeling of atomic and molecular processes is also valuable for the understanding and control of plasma properties (Winkler, 1999). Theoretical modeling is usually based on microscopic physical and chemical properties, such as cross sections and transition probabilities of major molecular species, and aims at deriving plasma properties concerning spatial structure and transport of chemically reactive species. Recall that, in the technological development of plasma processing, one tries to realize a desirable plasma and to control chemical reactions by adjusting macroscopic parameters, such as input electric power, pressure, temperature, speed of gas flow, composition of a gas mixture, structure and materials of electrodes and walls, and resist materials used on the base surface. Theoretical modeling provides a link between microscopic physical and chemical properties and macroscopic control parameters, and thus attempts to predict an optimal condition for a plasma with desired properties. Theoretical modeling may be canied out by using either Monte Carlo simulation or a method of transport equations. These techniques are complementary rather than competitive. For a complex problem involving a complicated geometry, for instance, Monte Carlo simulation is practically the only approach, but making the result fully trustworthy and aniving at its physical meaning require high expertise and mature judgment. Even with the modem computer, a method of transport equations is tractable only for a sufficiently simplified problem, but it offers useful insights even without a complete solution and provides a clear physical interpretation of a solution and a range of its applicability. A brief sketch of an example of modeling is as follows. A radio-frequency (rf) discharge generates a bulk plasma containing nearly the same number of positive and negative particles and a sheath containing predominantly positive ions; both of them are periodically modulated by the rf field. The electron distributionf (x, v, t ) as a function of position x, velocity v, and time t in a gas under an oscillating applied electric field F may be determined by the Boltzmann equation (Winkler, 1999)
where [af/E!t], on the right-hand side is the collision term representing the variation of the distribution resulting from all collisions and has a complex structure. Many different approaches have been adopted, depending upon how much of the spatial and geometric structure is incorporated. Here we shall look at an example of the modeling of an 0, plasma within the relaxation continuum model (Shibata et al., 1995). This model incorporates the phase shift of the electric field as a result of temporal evolution of the plasma as a relaxation phenomenon, and accounts for
11
PLASMA PROCESSING OF MATERIALS
individual electron collision processes (Itikawa et al., 1989) and also for subsequent processes. These include in particular the temporary capture of an electron of 6.5 eV by an 0, molecule, during which a part of the electron energy readily transfers to the nuclear motion, leading to vibrational excitation. The temporary negative-ion state may also decay by electron dissociative attachment, e
+ 02(x3c,-)+ o;(~H,)
-+ o-(,P)
+ o(~P),
at a high probability, a process leading to an accumulation of 0- ions in the plasma. As we pointed out in Section ILA, both of the electrodes have a negative potential compared to the center of the plasma, and therefore the negative ions are prone to be trapped in the plasma. Excited electronic states (A3ZT. C3A,, c'Z;, and B3C;) of 0, at excitation energies of 9.7 to 12.1eV are repulsive in the Franck-Condon region of the ground state, and the excitation to these states results in dissociation. e
+ 0, + o ( ~ P )+ o ( ~ P )+ e
+
-+ o ( ~ P ) o('D)
+ e,
which causes active 0 atoms to accumulate in the plasma, as seen in Fig. 4. Eventually, 0 atoms and 0- negative ions recombine to yield 0, and electrons, as represented by 0-
+ o ( ~ P I,D )
+ 0,
+ e.
According to the modeling, the electrons thus released contribute to the maintenance of the plasma. An 0, plasma is often used for ashing (removing carbon from polymers deposited on a substrate surface and reactor walls during
- 1.61
(a)
r
I
Distance ( rnm )
(b)
I
Distance ( mm )
FIG. 4. Spatial density distributions of each particle in the parallel plate 0, rf discharge at rut = n/2 for p = 0.5 Torr andf = 13.5 MHz. (a) V, = 75 V and (b) V , = I50 V.
12
Hiroshi Tanaka and Mitio Inokuti
processing). Furthermore, advanced modeling is also carried out to treat pulsed excitation, two-dimensional cases, and surface formation.
IV. Pertinent Topics from Atomic, Molecular, and Optical Physics A. ELECTRON COLLISIONS WITH MOLECULES Electron collisions with molecules initiate the first step in plasma generation, as we saw in Section ILB, and therefore represent the most fundamental topic of plasma chemistry. Let us briefly discuss cross-section data for electron collisions as used in modeling studies (Tanaka and Boesten, 1995; Christophorou et al., 1996; Christophorou et al., 1997; Christophorou and Olthoff, 1998; Morgan, 1999). One uses the notion of the cross section to express the probability of a collision of an electron with a specific molecule. Suppose that a beam of unit flux of electrons of a fixed momentum enters a gas consisting of a single chemical species at unit density. Then, the number of electrons scattered into a solid-angle element around the direction given by angle 8 measured from the direction of incidence is called the differential cross section a(@. This differential cross section can be further classified in terms of the quantum state n of the molecule left after the collision; thus, the number of electrons scattered in the same way as above and leaving the molecule in state n is called the differential cross section for the excitation to state n and is designated by a,(@. When the state of the molecule after the collision is the same as that before the collision, we call the collision elastic, and the differential cross section for this process may be written as oo(8).The integral of the differential cross section over all possible scattering angles is the (integrated) cross section qn, which is a function of the electron lunetic energy, i.e.,
s
qn = 2n ~ ~ (sin 8 Ode, )
where the factor 2n comes from integration over the azimuthal angle. The integral of the differential cross section a,(8) multiplied by 1 - cos 8 is the momentumtransfer cross section
which is a function of the electron kinetic energy and determines the mean energy transferred to the translational motion of the molecule upon elastic scattering of an electron. The integral of the cross section q,, multiplied by the electron speed v and the distributionf(x, v, t ) is the reaction rate for electron collisions.
PLASMA PROCESSING OF MATERIALS
13
Electron e n e r g y ( e V 1
FIG. 5. Cross sections (in units of cm2) of CF, for electron collisions as functions of electron energy (in eV), according to Kurachi and Nakamura (1990).
Figure 5 (Kurachi and Nakamura, 1990) presents a survey of electron collision cross sections of CF,. In addition to the momentum-transfer cross section qm, it shows the vibrational-excitation cross sections qy3 and qv4 (for two different vibrational modes), the (total) electronic-excitation cross section qe, the dissociation cross section qdn,the electron-attachment cross section qa, and the (total) ionization cross section qi. Each of the cross sections is a function of the electron kinetic energy and reflects the physics of the collision process, which is being clarified by theory. The cross sections designated as “total” can be discussed in greater detail in terms of different contributions, which are designated as “partial” cross sections. For chemical etching, polyatomic halogen-bearing molecules are often used. Electron collisions with these molecules often lead to negative ions through electron dissociative attachment, e + CF, + (CF,)* + CF,
+F
or CF,
+ F-,
which usually occurs via a temporary negative-ion state in competition with vibrational excitation. (See the region near 8 eV in Fig. 5.) The negative ions thus produced tend to be accumulated in the plasma, and play the role of scavenger of excess electrons in the plasma. They also contribute to reducing the electric charging of the base surface. Attachment of electrons of thermal energy also occurs often with halogen-containing molecules; measurements with electrons in the microelectron volt domain (Dunning, 1995) are beginning to be made. The determination of electron-collision cross sections over a broad range of kinetic energy cannot be accomplished in a single experiment, and requires the
14
Hiroshi Tanaka and Mitio Inokuti
use of many different methods that are complementary to one another (Morgan, 1999). The methods include the electron-beam method (Trajmar and McConkey, 1994), the electron swarm method (Crompton, 1994), the beam attenuation method, and the microwave-cavity method used with pulse radiolysis (Shimamori, 1995). The electron-beam method permits measurements of the cross section for each channel of excitation distinguishable within the energy resolution, its angular dependence, and its dependence on the electron lunetic energy. However, in general, arriving at absolute values of cross sections is not straightforward. The swarm method determines macroscopic parameters describing the transport of electrons in a gas, and one deduces a set of cross sections consistent with the measured parameters from an analysis of electron transport through the solution of the Boltzmann equation or Monte Carlo simulations, as discussed in Section 1II.B. Certainly the analysis provides an opportunity for a comprehensive survey of cross-section data (Christophorou et al., 1996; C h s tophorou et al., 1997; Christophorou and Olthoff, 1998). However, great care is required to make certain that all the major processes are properly accounted for. The beam attenuation method determines the grand total cross section, that is, the sum of the elastic-scattering cross section and all inelastic-scattering cross sections, absolutely and often reliably. The grand total cross section is useful as a test of a cross-section set and as an upper bound for an individual cross section. The microwave-cavity method allows one to derive electron-attachment and other cross sections at thermal energies (Shimamori, 1995).
B. TRANSPORT AND REACTIONS OF CHEMICALLY ACTIVESPECIES Electrons, ions, and other chemically reactive species, as well as molecules originally present in a plasma, diffuse, interact among themselves, and are transported throughout the reactor. Full modeling of the transport requires consideration of numerous elementary processes, including reactions of radicals or excited atoms with molecules (Hatano, 1999), ion-molecule reactions (Armentrout, 1999; Lindinger et al., 1999), electron-ion recombination, and recombination of positive and negative ions. The diffusion and transport are influenced by pressure, speed of gas flow, geometry and structure of the plasma, and reactor walls. During this stage, it is possible to control some of the processes and to help accomplish a desired goal of plasma processing. Let us consider an SiH, plasma with rare-gas (He, Ar,or Xe) additives. The metastable state (23S, at the excitation energy of 19.8eV) of He, for instance, is produced by electron collisions, although at a small cross section (of the order of lo-'' cm2), and gradually accumulates in a plasma because it has a long radiative lifetime (6 x lo5 s) and may be transported over a considerable distance without being affected by the electric field, unless it is quenched by collisions with molecules. The excitation energy stored in such a state is available for a variety of
15
PLASMA PROCESSING OF MATERIALS
chemical reactions. The total reaction rates and branching ratios for reactions of He* and Ar* (with excitation energy of 11.6 eV) with various molecules used in plasma processing have been determined by pulse-radiolysis and flowing-afterglow methods (Tsuji et al., 1989a; Tsuji et al., 1989b; Yoshida et al., 1992a; Yoshida et al., 1992b), as seen in Table 3. It has been reported that Xe* (with excitation energy of 8.23 eV) in an SiH, plasma produced a-Si : H with greatly improved resistance against photodegradation (Matsuda et al., 1991). STUDIES C. REMARKS ON THEORETICAL
Some of the reaction rates are amenable to theoretical prediction based on advanced calculations on the electronic structure in quantum chemistry. Computer codes are becoming available for the evaluation of electronic structures of the ground state and low-lying excited states of polyatomic molecules to a chemically meaningful precision. Theories of electron-molecule collisions are also being developed with the use of the R-matrix method, the Schwinger multichannel variational method, the multiple-scattering method, and other methods (Huo and Gianturco, 1995; Winstead and McKoy, 1999). One particular area where theoretical studies are especially appropriate concerns electronically excited states of molecules, which should be abundant in a chemically reactive plasma. Apart fi-om metastable states, it is in general difficult to prepare excited states of molecules, and especially of radicals, in a TABLE3. REACTIONS OF
CH, AND SM,
WITH
METASTABLE STATES AND IONS OF RARE GASES
Total rate constants, measured in 1O-Io cm3 s-' Rare-gas species Ar(3P2) Ar(,P,) He(2,S) He(2IS) Ar+(2P) He+('S)
Molecule
5.4
CH, SiH,
5.8
6.0 5.7
1.9 2.3
7.9 7.6
13.4 1.0
13.0 21.8
Partial cross sections, measured in lo-'' cm2 He(23S)
+ SiH,
+ Si* SiH* SiH SiH.: e SiH,, SiH, SiH, --$ Si* SiH* SiH e SiH.: SiH,, SiH,
+
Ar(,P2)
+
+
0.074 =0 =0 18 =0 0.27 4 4-25 0 13-94
The data are taken from Tsuji et al., 1989a, 1989b and from Yoshida et al., 1992a, 1992b.
16
Hiroshi Tanaka and Mitio Inokuti
copious and known amount suitable for measurements of their chemical reactions. Thus, theoretical studies on excited-state reactions are particularly desirable. Furthermore, with the increasing variety of molecular species used in plasma processing, it is difficult to carry out measurements on all of them; thus, one expects theoretical studies to provide knowledge about reactions that is not accessible by experiment. Even for molecules in the ground electronic state, our knowledge about cross sections is largely limited to the room-temperature condition, in which vibrational and rotational states are populated in a thermal distribution. Then, for a diatomic molecule, the ground vibrational state is predominantly populated. However, for a polyatomic molecule, normal modes with small quanta must be appreciably excited. For the full understanding of kinetics in plasma chemistry, it is important to assess the role of the internal energy of reactant molecules.
V. Acknowledgments The present work is supported by the US. Department of Energy, Office of Science, Nuclear Physics Division, under Contract No. W-3 1- 109-Eng-38.
VI. References Anderson, L. W., Goyette, A. N., and Lawler, J. E. (1999). In the present volume. Armentrout, I? B. (1999). In the present volume. ASET (Association of Super-Advanced Electronics Technologies) (Ed.) ( 1998). Digest of Papers, International Forum on Semiconductor Technology, Kyoto, March 9-10, ASET (Tokyo). Basner, R., Becker, K., Deutsch, H., and Schmidt, M. (1999). In the present volume. Becker, K. H. (1994). In H. Ehrhardt and L. A. Morgan (Eds.), Electron collisions with molecules, clusters. and surfaces (p. 127). Plenum Press (New York). Bruno, G., Capezzuto, P., and Madan, A. (Eds.) (1995). Plasma deposition of amorphous siliconbased materials. Academic Press (San Diego). Chnstophorou, L. G., Olthoff, J. K., and Rao, M. V. V S. (1996). 1 Phys. Chem. Ref: Data 25, 1341. Christophorou, L. G., Olthoff, J. K., and Rao, M. V V S. (1997). 1 Phy.s. Chem. Ref: Data 26, I . Christophorou, L. G . , and Olthoff, J. K. (1998). 1 Phys. Chem. ReJ Data 27, 1. Chutjian, A. (1999). In the present volume. Cosby, P. C. (1 993). 1 Chem. Phys. 98, 9544. Crompton, R. W. (1994). In M. Inokuti (Ed.), Advances in atomic, molecular; and opticalphysics, Vol. 33, Cross-Section Data (p. 97). Academic Press (San Diego). Dunning, F. B. (1995). 1 Phys. B 28, 1645. Fujishiro, S., Garscadden, A,, and Makabe, T. (Eds.) (1998). Papers from the International Workshop on Basic Aspects of Noneqirilibrium Plasmas Interacting with SurJaces, Shirahama, Wakayama, Japan, January 2 6 2 7 , 1997, published in . I Vac. Sci. Techno/. A 16, 215 (1998). Goto, T. (1990). Trends Chem. Phys. 1, 69. Hata, N. (1989). Report of the Electrotechnical Laboratoty 901, 75.
PLASMA PROCESSING OF MATERIALS
17
Hatano, Y. (1999). In the present volume. Hippler, R. (1999). In the present volume. Huo, W. M., and Gianturco, F. A. (Eds.) (1995). Computational methods for electron-molecule collisions. Plenum Press (New York). Itabashi, N., Kato, K., Nishiwaki, N., Goto, T., Yamada, C., and Hirota, E. (1988). Jpn. 1 Appl. Phys. 27, L1565. Itabashi, N., Nishiwaki, N., Magane, M., Naito, S., Goto, T., Matsuda, A,, Yamada, C., and Hirota, E. (1990). Jpn. 1 Appl. Phys. 29, L505. Itikawa, Y., Ichimura, A,, Onda, K., Sakimoto, K., Takayanagi, K., Hatano, Y., Hayashi, M., Nishimura, H., and Tsurubuchi, S. (1989). 1 Phys. Chem. Re$ Data 18, 23. Japan Society of Applied Physics (Ed.) (1993). Amorphous silicon. Ohm-sha (Tokyo). Kurachi, M., and Nakamura, Y. (1990). In T. Takagi (Ed.), Proceedings of the 13th Symposium on Ion Sources and Ion-Assisted Technology, Kyoto (p. 205). Kyoto University (Kyoto). Kuroki, K., Spence, D., and Dillon, M. A. (1992). 1 Chem. Phys. 96, 6318. Kushner, M. J. (Ed.) (1994). Proceedings of the NATO Advanced Research Workshop on Formation, lkansport, and Consequences of Particles in Plasmas, Chateau de Bonas, 1993, published in Plasma Sources Sci. Technol. 3, 239. Lee, H. U.,Deneufville, J. P., and Ovshinsky, S. R. (1983). 1 Non-Cryst. Solids 59/60, 671. Lieberman, M. A,, and Lichtenberg, A. J. (1994). Principles of plasma discharges and material processing. John Wiley & Sons (New York). Lindinger, W., Hansel, A,, and Herman, Z. (1999). In the present volume. Matsuda, A,, Kaga, T., Tanaka, H., and Tanaka, K. (1983). 1 Non-Cryst. Solids 59/60, 687. Matsuda, A,, Mishima, S., Hasezaki, K., Suzuki, A,, Yamasaki, Y., and McElheny, P. J. (1991). Appl. Phys. Lett. 58, 2494. Mi, L., and Bonharn, R. A. (1998). 1 Chem. Phys. 108, 1910. Morgan, W. L. (1999). In the present volume. Motlagh, S., and Moore, J. H. (1998). 1 Chem. Phys. 109, 432. Poll, H. U., Winkler, C., Margreiter, D., Gill, V, and Mark, T. D. (1992). Int. 1 Mass Specfrom. Ion Proc. 112, 1. Robertson, R., Hils, D., Chatham, H., and Gallagher, A. (1983). Appl. Phys. Lett. 43, 544. Samukawa, S. (1 999). Personal communication. Shibata, M., Nakano, N., and Makabe, T. (1995). 1 Appl. Phys. 77, 618. Shirnamori, H. (1995). In Proceedings of the International Symposium on Electron- and PhotonMolecule Collisions and Swarms, Berkeley, July (p. B-1 ). Sugai, H., Toyoda, H., Nakano, T., and Goto, M. (1995). Contrib. Plasma Phys. 35, 4; 415. Tanaka, H., and Boesten, L. (1995). In L. J. Dubti, J. B. A. Mitchell, J. W. McConkey, and C. E. Brion (Eds.), Physics of Electronic and Atomic Collisions. XIX International Conference, Whistler, Canada, July-August (p. 279). American Institute of Physics (Woodbury). Trajmar, S., and McConkey, J. W. (1994). In M. Inokuti (Ed.), Advances in atomic, molecular, and optical physics, Vol. 33, Cross-section data (p. 63). Academic Press (San Diego). Tsuji, M., Kobarai, K., Yamaguchi, S., Obase, H., Yamaguchi, K., and Nishimura, Y. (1989a). Chem. Phys. Lett. 155, 481. Tsuji, M., Kobarai, K., Yamaguchi, S., and Nishimura, Y. (1989b). Chem. Phys. Lett. 158, 470. Winkler, R. (1999). In the present volume. Winstead, C., and McKoy, V. (1999). In the present volume. Winters, H. F., and Inokuti, M. (1982). Phys. Rev. A 25, 1420. Yoshida, H., Kawamura, H., Ukai, M., Kouchi, N., and Hatano, Y. (1992a). J Chem. Phys. 96,4372. Yoshida, H., Ukai, M., Kawamura, H., Kouchi, N., and Hatano, Y. (1992b). 1 Chem. Phys. 97, 3289.
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ADVANCES IN ATOMIC MOLECULAR. AND OPTICAL PHYSICS. VOL . 43
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS OF ELECTRONS IN WEAKLY IONIZED PLASMAS R . WINKLER Institut fiir Niedertemperatur.Plasmaphysik. 17489 Greifwald. Germany
I . Introduction A . The Role B. Basic Interaction Processes of Electrons ..................................... I1. Kinetic Description of the Electrons ............................................. A . Velocity Distribution Function and Velocity Space Averages ............... B. The Boltzmann Equation of the Electrons ................................... C. Expansion of the Velocity Distribution and the Kinetic Equation .......... D. Macroscopic Properties and Macroscopic Balances of the Electrons ....... 111. Electron Kinetics in Time- and Space-Independent Plasmas.................... A . Basic Equations and Consistent Macroscopic Balances ..................... B. Some Remarks on the Calculation of the Isotropic Distribution ............ C. Examples of Distribution Functions and Macroscopic Quantities ........... D. Kinetic Treatment of Gas Mixtures .......................................... E. Inclusion of the Electron-Electron Interaction ............................... F. Remarks on Additional Aspects of the Steady-State Kinetics ............... I\! Electron Kinetics in Time-Dependent Plasmas ........... ................. A . Basic Equations for the Distribution Components .... B. Macroscopic Balance Equations and Lumped Dissipation Frequencies..... C . Some Aspects of the Numerical Solution of the Basic Equation System ... D. Temporal Relaxation of the Electrons in Time-Independent Fields ......... E . Reponse of the Electrons to Pulselike Field Disturbances .................. F. Remarks on Additional Aspects of Time-Dependent Kinetics v: Electron Kinetics in Space-Dependent Plasmas ................................. A . Basic Equations and Their Representation by the Total Energy B. The Consistent Balance Equations in Space-Dependent Plas C . Characteristic Features of the Spatial Relaxation of the Electrons .......... D. Response of the Electrons to Pulselike Field Disturbances ................. E . Remarks on Additional Aspects of Space-Dependent Kinetics ............. VI . Concluding Remarks ............................................................. ..... ..................... VII . VIII. References ........................................................................
19
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51 60 61 61 63 64 70 13 75 76 16
Copyright 0 2000 by Academic Press All rights of reproduction in any form reserved ISBN: 0-12-003843-9/ISSN: 1049-25OX $30.00
20
R. Winkler
I. Introduction A. THEROLEOF ELECTRONS AND THEIRNONEQUILIBRIUM BEHAVIOR
Weakly ionized plasmas are complex systems involving several interacting particle components. In the simplest case, they consist largely of unexcited atoms, i.e., neutral gas particles, and to a lesser extent of electrons and positive ions. Usually the main power source of the plasma is provided by an electric field acting upon the charged particles and sustaining the plasma in this way. Owing to their small mass, the electrons are incapable of losing a larger part of their kinetic energy by elastic collisions with heavy particles. Therefore, a bad energetic contact of the electrons with the heavy particle components of the plasma is established. As a consequence, under the action of the electric field, the electrons reach a mean kinetic energy that is remarkably higher than that of the heavy plasma components, and the plasma becomes an anisothermal medium. In other words, a significant portion of the electrons populate the high-energy region of their energy space. These electrons become energetically capable of overcoming the threshold energies above which inelastic electron collision processes with the atoms or molecules take place. However, in each of these inelastic collision processes, the corresponding electron loses at least the threshold energy of this process, i.e., the electron is transferred from the region of higher kinetic energies to the low-energy region. Thus, the occurrence of inelastic collisions causes an efficient depopulation of the electron energy space in the range of higher energies. As an immediate consequence, the electron population in the region of inelastic collisions markedly decreases with increasing energy. This interplay between the action of the electric field and the elastic and inelastic collision processes causes the electron component to generally reach a state far from the thermodynamic equilibrium. This nonequilibrium behavior of the electron component cannot be described using the well-developed methods of thermodynamics for equilibrium conditions. Thus, the requirement arises that the state of the electron component established in anisothermal plasmas or its temporal and spatial evolution can be described only on an appropriate microphysical basis. In principle, the microphysical study of electron behavior can follow two quite different approaches. One way consists of the formulation of an adequate electron kinetic equation and its approximate solution. The other approach uses the techniques of the particle simulation. Both approaches have particular advantages and disadvantages. For example, the electron simulation technique can be more easily applied to a more complicated geometry than can the solution approach of the kinetic equation. However, only a limited number of electrons can be treated
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
21
in any simulation. As a result, the statistics limits the resolution and thus the accuracy reached, for instance with respect to the energy distribution of the electrons in the region of higher energies. This limitation usually does not occur when solving the kinetic equation of the electrons, and the computational expenditure connected with the treatment of the kinetic equation is generally substantially lower. However, in either of these microphysical approaches, the complex interplay of the action of an electric (and possibly of an additional magnetic) field and of elastic and various inelastic electron collision processes with atoms or molecules of the gas has to be taken into account in detail. The primary purpose of the kinetic treatment of the electron component in anisothermal plasmas is the determination of its velocity distribution function or only its energy distribution. The various macroscopic properties of the electrons can then be obtained from the velocity distribution function by appropriate averages over the velocity space of the electrons. The macroscopic nonequilibrium properties of the electrons are critical to the global behavior of the plasma. Because of their large mean energy, the electrons are the only plasma component that is capable of causing inelastic collisions with atoms and molecules, thus leading to excitation, dissociation, or ionization. This is usually the basic process through which the first activation of the working gas takes place. As a result of this activation, other collision processes and chemical reactions between the activated heavy particles of the plasma are often initiated. The electron velocity distribution function depends to a large extent on the special plasma conditions considered. Among them, the lumped fiequency of various inelastic electron collision processes and the structure of the electric field acting upon the electrons must be mentioned. The spectrum of important electron collision processes is broad and includes elastic collisions as well as such inelastic collisions as excitation, dissociation, ionization, and attachment. With respect to the inelastic collision processes, remarkable differences between electron collisions with atoms and with molecules generally have to be taken into account. As detailed below, inelastic collisions with atoms are characterized by high energetic thresholds of several electron volts. However, in addition to such electronic excitation processes, the rotational and vibrational excitation of molecules in collisions with electrons occurs. Both of these inelastic processes have remarkably lower energy thresholds: some hundredths and some tenths of electron volts, respectively. Because of the low energy loss in each collision event, the impact of the rotational excitation (and also the deexcitation of rotational states) on the kinetics of the electrons is usually of less importance in the anisothermal plasmas typically maintained by an electric field. Therefore, these inelastic collisions are often neglected. Sometimes they are dealt with in a manner similar to elastic collisions.
22
R. Winkler
Furthermore, the detailed procedure ultimately used to determine the velocity distribution sensitively depends on the type of plasma and is quite different when studying the electron kinetics in steady-state, time-dependent, or space-dependent plasmas. If the electric current and thus the density of electrons and excited atoms and molecules grows in the plasma, electron collisions with excited atoms and molecules and the Coulomb interaction between the electrons become increasingly important and have to be included in the kinetic study of the electron behavior. B. BASICIINTERACTION PROCESSES OF ELECTRONS The electrons in weakly ionized plasmas generally undergo two basic impacts, namely, the action of an electric (and possibly of an additional magnetic) field and the interaction with heavy particles in binary elastic and inelastic collisions (Desloge, 1966; Shkarofsky et al., 1966; Golant et al., 1980). Because of the negative charge of the electrons, the electric field accelerates the electrons in a direction opposite to that of the electric field. If the electric field in the plasma is parallel to a fixed space direction, as is often the case, the action of the electric field causes a change in the component of the vectorial electron velocity parallel to the fixed space direction. As a consequence of the sole field action, this velocity component plays an exceptional role in causing anisotropy of the velocity distribution function. Moreover, the sole action of the electric field naturally causes a change, generally an increase, in the individual and consequently the mean electron energy. However, in addition to the action of the electric field, several types of binary collision processes between the electrons and the atoms or molecules occur in the plasma. Each collision event causes a change in the velocity direction of the colliding electron. As a result, each electron collision process leads to a pronounced, more or less isotropic scattering of the electrons and of their vectorial velocities in all space directions. Thus, the electron collision processes tend to reduce the anisotropy of the velocity distribution produced by the action of the electric field. In addition to the scattering of the electrons, the collision processes cause a change in the electron energy. Since the mean energy of the electrons is considerably larger than that of the heavy particles, the electrons generally lose energy when undergoing elastic or inelastic collisions. The velocity distribution of the electrons finally established in special plasma conditions is, essentially, the result of a complex interplay between the action of the field on the electrons and the various binary collision processes of the electrons. For a better understanding of the collisional interaction, some basic aspects of the most significant electron collision processes, i.e., the elastic and exciting
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
23
collisions, are considered in the following (Shkarofsky et al., 1966; Golant et al., 1980). The vectorial velocities of- both colliding particles before and after each collision event, denoted by V and 3,V*, respectively, satisfy the momentum and energy conservation law
z,
-.
-+
m,z+MV =m,ij*+MV*
m M “U2+-V2+Ehc 2 2
(1)
m M =2(u*)2+-(V*)2fEaf 2 2
where m, and M are the mass of the electron and the heavy particle, respectively, Ebeand Eaf mean the respective internal energy of the heavy particle before and after the collision event, and u, V , u*, V* denote the absolute values of the corresponding vectorial velocities. The energy conservation law of elastic collisions is obtained with Ear = Ebe and that of exciting collisions with Eaf
’
Ehe.
As already mentioned above, in the course of each binary interaction, the electron undergoes a change in its kinetic energy. Because the mean energy of the electrons is considerably higher than that of the heavy particles, the latter are usually considered to be at rest. Under this approximation, the main contribution to the energy losses of the electrons in elastic and exciting collisions can be deduced from the conservation laws in Eqs. (1) and (2). The energy losses are given by the two expressions
where 8 is the scattering angle related to the elastic collisions and E,, - Ehe represents the increase in the internal energy of the heavy particle in the excitation process. Expressions ( 3 ) and (4) reflect important properties of the elastic and exciting electron collision processes already mentioned in the introduction. So, expression (3) indicates that the energy loss in each elastic collision is proportional to the electron energy m,u2/2 before the collision and to the very small mass ratio m,,M. Thus, the bad energetic contact between the electrons and the heavy particles by elastic collisions becomes immediately obvious. However, according to expression (4), in each exciting collision the electron loses the energy Eaf- Ebcthat is necessary to excite the heavy particle from its lower energy level Eheto its higher level Eaf. The conservation laws in Eqs. (1) and (2), related to the elastic and exciting co!isions, represent four scalar equations connecting the two vzctorial velocities Z, V before the collision event with the corresponding ones ;*, V* after the event.
24
R. Winkler
z,
If the initial velocities ? are given, the components of the velocities i*,?*, i.e., six scalar quantities, have to be determined in order to describe the result of the binary collision. Thus, when using Eqs. (1) and ( 2 ) , two scalar quantities remain undetermined. The remaining lack of knowledge on the collision process can be eliminated by using additional information on the electron scattering process as being involved in the differential scattering cross section o ( m , v 2 / 2 ,cos 0 ) of the corresponding collision process (Desloge, 1966; Shkarofsky et al., 1966; Golant et al., 1980). The differential cross sections are the basis for determining, by appropriate averaging over the solid angle of scattering sin 0 d0 dq5, total cross sections Q(m,v2/2), as detailed below in the framework of the expansion of the kinetic equation. With respect to the binary inelastic collision processes of the electrons with the heavy particles, other important types have to be mentioned (Shkarofsky et al., 1966; Golant et al., 1980). Usually these collision processes are subdivided into conservative and nonconservative processes, i.e., with respect to the conservation or alteration of the number of electrons in the course of the collision event. Other important conservative collision processes are the dissociation of molecules and the deexcitation of excited atoms or molecules. In each dissociation process, the colliding electron loses the dissociation energy of the molecule and at least two heavy-particle fragments are formed as a result of the dissociation process. In each deexciting collision process, the colliding electron receives the excitation energy from the excited heavy particle. Thus, in deexciting collisions with an excited atom, the electron is transferred to the region of considerably larger energies by one collision only. Important nonconservative collision processes with respect to the kinetics of the electrons are ionization and attachment. While in the first process, after each collision event, two electrons result, in the second process the colliding electron is lost and a negative ion is generated by the attachment of an electron to the neutral heavy particle. In the course of an ionization event, the ionization energy has to be covered by the initially available electron energy and the remaining electron energy is distributed among both the electrons. In the case of attachment, the electron itself and its initial energy disappear from the electron component of the plasma.
11. Kinetic Description of the Electrons A. VELOCITY DISTRIBUTION FUNCTION AND VELOCITY SPACE AVERAGES The distribution of the electrons with respect to their velocity space 3 at the coordinate space position 2 and at time t is described by the velocity distribution
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
25
function F(5,2, t ) (Desloge, 1966). Then, the contribution dn(2, t ) of the velocity space interval d3 around 5 to the electron density TI(?, r) is determined by the expression dn(?, t ) = F(5, 2, t)dG with d i = dv, dv),dv:. If the velocity distribution F(3,?, t ) is known, important macroscopic properties of the electrons can be calculated by appropriate velocity space averaging over the distribution. To give some examples, the density n(?, t),the density of the mean energy urn(;, t ) and the vectorial particle current density y(2, t ) of the electrons are given by the averages (Desloge, 1966)
s
n(2, t ) = F(3,2, t ) d;
In the introduction of the velocity distribution and the examples, Eqs. ( 5 ) to (7), of macroscopic quantities, the usual normalization of the velocity distribution on the electron density has been used. In special conditions, e.g., in steady-state or in time-dependent conditions with only conservative electron collision processes included, the electron density becomes a constant and can easily be separated from the velocity distribution. Especially in these cases, a normalization of the velocity distribution on one electron is often used. B. THEBOLTZMANN EQUATION OF THE ELECTRONS The different microphysical processes-the field action and the various binary collision processes-in which the electrons are involved in a weakly ionized plasma lead to a complex redistribution of the electrons in their phase space, i.e., their combined coordinate and velocity space. According to the concept of the short-range interaction in binary electron collisions, the appropriate phase space balance equation for the electron velocity distribution F(3, 2, t ) is given by the Boltzmann equation (Desloge, 1966; Shkarofsky et al., 1966; Golant et al., 1980)
Here -eo denotes the charge of the electrons, $2, t ) is the electric field, and C"'(F) and C;T(F)are the collision integrals for elastic collisions and important conservative inelastic collisions, i.e., the Ith excitation or dissociation process of the electrons in collisions with the ground-state atoms or molecules of the gas. For simplicity in the W h e r representation, only the most essential electron
26
R. M’nkler
collision processes have been taken into account in the kinetic equation, Eq. (8). For a specific plasma, other collision processes-for example, the excitation and deexcitation of excited atoms or molecules and the ionization of and attachment to ground-state and excited atoms or molecules-may be of importance, and corresponding collision integrals will have to be added to the right side of Eq. (8). To avoid the more complex treatment of ionizing collisions in the kinetic approach, however taking the energy dissipation in these collisions approximately into account, ionization is often treated in the same way as excitation, neglecting the appearance of an additional electron after the ionization event. The statistical description of the plasma electrons by using the Boltzmann equation is based on a classical concept. The collision-free part of this equation has been derived in the frame of the classical statistics. The collision integrals for the various binary electron collision processes describe the collision events largely as a classical process. Only the properties of the colliding particles before and after each collision event are described by the coupled momentum and energy balances of the particles. However, the real evolution of the particle system in space and time during the collision process is not considered in the collision integrals. The quantum mechanical aspect of the collision events is described by the corresponding collision cross section and the change in the internal energy of the heavy particles in the energy balance equation of the colliding particles. In agreement with these properties of the Boltzmann equation, the velocity distribution F(G,2, t ) has been introduced above on a purely classical basis. The kinetic equation is very complex and covers a tremendous number of special electron kinetic problems. Consequently, there does not seem to be any chance of finding some kind of “general solution” of this equation that can later be adapted to the specific plasma conditions of interest. As a consequence, for different plasma conditions-for instance, steady-state, time-dependent, or space-dependent problemsdifferent solution approaches and numerical techniques have been developed and applied. In addition, the specific structure of the electric field acting upon the electrons is of particular importance for the establishment of a special symmetry in the velocity distribution and thus for a specific simplification of the solution approach. Therefore, the objective in the following parts of this chapter can only be to give a certain introduction to the study of the kinetics of the electrons under different plasma conditions and to illustrate some typical aspects of the kinetics.
c.
EXPANSION OF THE VELOCITY DISTRIBUTION AND THE m E T I C EQUATION
To find an approximate solution of the kinetic equation, an orthogonal expansion of the velocity distribution with respect to the direction z / u of the velocity G is commonly used in the treatment of the kinetic equation. Depending on the
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
27
structure of the electric field and on the expected inhomogeneity of the plasma, a reduced expansion with respect to one angle coordinate only or a more complex expansion with respect to both angle coordinates of 3/v is used. If the electric field and the inhomogeneity in the plasma are parallel to a fixed space direction-for example, the Zz direction of the coortjnate space-the velocity distribution becomes symmetrical around the field E(z, t ) = E(z, t)ZZ, gets the reduced dependence F ( U , u,/u, z , t ) , and can be given the expansion (Shkarofsky et al., 1966; Golant et al., 1980)
in Legendre polynomials Pn(vz/v), with U being the kinetic energy. In this expansion, the dependence of the velocity distribution F ( U , v,/v, z , t ) on the direction z / v is fixed by the Legendre polynomials Pn(u,/v). Thus, averages with respect to the angle space G/v over the velocity distribution and appropriate weight functions can be performed. For example, with d3 = v2 dv d('v/u), the angle space averages over the velocity distribution F and over the product of and F , yield according to (5) and ( 7 ) the expressions
=-(-)2 1
3
' I 2u f i ( U , z , t ) d U Z ,
me
because of the orthogonality relation of the Legendre polynomials. This means that the contributions of the interval dv of the absolute value of the velocity to the density n(z, t ) and the particle current density ?(z, t ) are, up to scalar factors, completely determined by the lowest two coefficientsfo(U, z , t ) andfi ( U ,z , t ) of the expansion in Eq. (9). Thus, the lowest coefficient fo(U, z , t ) represents the isotropic part of the velocity distribution, and all other terms of the expansion in Eq. (9) are contributions to the anisotropy of the velocity distribution. The representations in Eqs. (10) and (1 I ) additionally show that, in a strict sense, the expression U'12fo(U,z, t ) represents the energy distribution of the electrons and (1/3)(2/m,)'12Ufi ( U ,z , t ) represents the energetic distribution of their particle current density. The latter possesses a component in the Zz direction only.
28
R. Winkler
The substitution of the expansion in Eq. (9) into the kinetic equation, Eq. (8), leads after several intermediate rearrangements to an analogous expansion in Legendre polynomials of the entire kinetic equation and, because of the orthogonality of the polynomials, ultimately to a hierarchy of equations (Shkarofsky et al., 1966; Golant et al., 1980). This equation system includes the expansion coefficientsJr(U, z , t ) , and its approximate solution finally yields these coefficients and thus the velocity distribution. In deriving the equation system, it is commonly assumed with respect to the collision integrals that the atoms or molecules are at rest before the collision events. Furthermore, each collision integral is additionally expanded with respect to the mass ratio m e / M , and only the leading term with regard to m,/M of each collision integral has been taken into account in each coefficient of the Legendre polynomial expansion of the kinetic equation. This infinite system of equations has to be truncated in order to obtain a closed system and its approximate solution. It has been found in recent years that a restriction of the expansion in Eq. (9) to its lowest two terms already leads to an unexpectedly good approximation for the velocity distribution under many plasma conditions. In this so-called two-term approximation, the system
a
- - (2
au
3 U2NQ"(U)fo) M
a
e ) 1 1 z U ' 1 2-atf i
+ U aza fo
-
a
e,,E(z, t)U % f U
of two equations for the two expansion coefficientsfo(U, z , t ) andfi ( U ,z , t), i.e., the isotropic distribution and the single contribution to the distribution anisotropy according to the two-term approximation, is ultimately obtained, where N denotes the density of the ground-state atoms or molecules.
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
29
With respect to the elastic collisions and the excitation or dissociation by electron collisions considered in Eq. (8) the collision cross sections
1
@(U ) = ae’(U , cos 0)( 1 - cos 0) sin 0 dB d4
J
@,(U) = O ~ ( ~ , C O S ~s)i nC8 O d BSd~4 occur in system (12). As can be seen from Eqs. (13) and (14), these cross sections are obtained by averaging the differential cross sections ae‘(U , cos 0) and ay(U, cos 0) together with further weight factors over the solid angle of scattering. @ ( U ) is the well-known cross section for momentum transfer in elastic collisions, @ ( U ) is the total cross section, @,(U) is a further “generalized” total cross section of the Ith excitation or dissociation process, and Uf“is the corresponding excitation or dissociation energy. By these cross sections, an anisotropic scattering in elastic as well as in exciting or dissociating collisions is taken into account in Eqs. (12). To obtain a simpler structure of system (12), it is usually assumed that excitation and dissociation take place with isotropic scattering of the colliding electron. As a consequence, all “generalized” total cross sections @,( U ) become zero and the last term of the second equation of (12), involving these cross sections, disappears. Equations (12), simplified by the assumption of isotropic scattering in exciting and dissociating collisions, represent the basic equations for studying many quite different problems in electron kinetics. In particular, the additional simplification to steady-state, purely time-dependent, or purely space-dependent plasma conditions allows a detailed microphysical analysis of various electron kinetic problems related to each of these plasma conditions.
D. MACROSCOPIC PROPERTIES AND MACROSCOPIC BALANCES OF THE ELECTRONS
Because of the orthogonality of the expansion in Eq. (9), all essential macroscopic quantities of the electrons can be represented by energy space averages over the lowest two expansion coefficients, fo(U, z , t ) and fi ( U , z , t). This is similar to the averaging in Eqs. (10) and (1 l), already a consequence of the integration over the angle space ; / v .
30
R. mnkler
So, the electron density n(z, t), the mean energy density unt(z,t ) (i.e., the mean electron energy times its density), and the particle and energy current densities j(z, t ) =j&, t)& andje(z, t ) =j,,(z, t)Zz are given by the expressions
n(z, t ) =
1: 1:
u,(z, t ) =
U''2fo(U, z , t ) dU U"'2fo(U,Z, t ) dU
,
rOO
The particle and energy current densities&, t ) andje(z, t ) possess a z component only because of the rotational symmetry of the velocity distribution F ( U , vzIv, z , t ) around the direction of 2:. This is an immediate consequence of the assumption that the field action occurs only parallel to this direction. The power and momentum gain from the electric field P'(z, t ) and I f ( , , t ) are given by
e0 Iqz, t ) = -n(z, t ) -E(z, t) me
The power losses P'(z, t ) and Pp(z, t ) by elastic collisions and by the Ith excitation or dissociation process, the lumped power loss P"'(z, t ) in inelastic collisions, and the total power loss P'(z, t ) in collisions have the representation P'(z, t ) = 2
%J2/m, A4
Pfs(z, t ) = UFJ2/mc
U2NQd(U)fo(U,Z , t ) dU
1:
U Np;"(U)fo(U,Z, t ) dU
P ( Z , t) = CP$(Z,t) I
P (z, t ) = F ( Z , t ) + P'"(z, t )
(24)
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
31
Similarly, the momentum losses Z"(z, t ) and Zf5(z,t ) by elastic collisions and by the Ith excitation or dissociation process, the lumped momentum loss Z"'(z,I ) in inelastic collisions, and the total momentum loss Z'(z, t ) in collisions are given by P(Z,t )
=
1;
~
3%
U3I2NQ"(U)fi( U ,Z , t ) dU
U[U'/2NQ;"(U)- ( U - U f " ) 1 / 2 N ~ . ' l ( U ) ] f i (tU ) dU , z , (26) P ( z ,t)=
c
Zf"(z, t )
I
Z"(z, t ) = P ( z , t )
+P ( Z , t)
Finally, the mean collision frequency v y ( z , t ) and the corresponding rate coefficient kF'(z, t ) of the Ith excitation or dissociation process are represented by the averages vf'(z, t ) = J2/m,
UNQF(U)f,(U,z,t ) dU/n(z, t )
k;"(z, t ) = v;"(z,t ) / N
(29) (30)
If, for example the isotropic and anisotropic distributions fo(U, z , t ) and f , ( U , z , t ) have been determined by solving the equation system of the twoterm approximation, Eqs. (12), adapted to a specific kinetic problem, the steadystate values, the temporal evolution, or the spatial alteration of the macroscopic quantities can be calculated by appropriate energy space averaging over these distribution functions according to the corresponding representation given in Eqs. (15) to (30). Furthermore, appropriate energy space averaging over Eqs. (12), derived through two-term approximation from the Boltzmann equation, yields the consistent macroscopic balance equations of the electrons. In particular, the particle and power balance can be derived from the first equation of system (12) and the momentum balance equation, normalized on the electron mass m,, can be derived from the second equation of (12). These balance equations are
a
-/2(Z,
at
t)
+ aza j.,(z, t) = 0 -
a u,(z, t ) + a jez(z,t ) = P/(z,t ) - P ( Z , t )
-
at
-
az
(32)
All macroscopic quantities occumng in the balance equations, Eqs. (3 1) to (33), have already been introduced in Eqs. (15) to (30).
32
R. mnkler
Valuable information about the physics involved in the kinetic treatment of a specific problem can be obtained by considering the consistent macroscopic balance equations of the electrons, Eqs. (31) to (33), adapted to the specific kinetic problem. On the right side of the power and momentum balance, Eqs. (32) and (33), a difference between the corresponding gain from the electric field and the total loss in collisions occurs. Gain and loss terms arise on the right side of the particle balance equation, Eq. (3 l), too if nonconservative electron collision processes (for instance, ionization and attachment) are additionally taken into account in the kinetic equation, Eq. (8), and thus in the equation system (12).
111. Electron Kinetics in Time- and Space-Independent Plasmas Kmetic studies of plasmas in steady state represent the conventional area of electron kinetics. Such studies have been made in many atomic and molecular gases and in mixtures of such gases. In addition to the basic electron collision processes (elastic collisions and exciting and dissociating collisions with groundstate atoms and molecules), exciting and deexciting electron collision processes with excited atoms and molecules and at higher electron density, the Coulomb interaction between the electrons have been partly taken into account. These investigations are largely performed on the basis of the two-term approximation, allowing for anisotropic scattering in elastic collisions, but assuming mainly isotropic scattering in the conservative inelastic collision processes. A. BASICEQUATIONS AND CONSISTENT MACROSCOPIC BALANCES Let us briefly consider some aspects of the kinetic treatment based on the twoterm approximation. When the equation system (12) is adapted to time- and space-independent plasmas, the simplified equation system (Shkarofsky et al., 1966; Winkler et al., 1982) U*NQd(U)fO(U)]
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
33
is obtained for the determination of the isotropic and anisotropic distributions AdU) andfi(U). In this case, the electron particle balance [Eq. (3 I)] is automatically satisfied. This means that for plasmas in steady state, no restriction on the electron density exists and the density can be freely chosen. Therefore, the constant electron density n is usually separated from the velocity distribution according to F( U , u , / u ) = n k ( U , u,/u), and, consequently, in the two-term approximation, the same separation procedure is applied to the isotropic and anisotropic distribution, giving A( U ) = nJ;( U ) with i = 0 and 1. The substitution of this relation into the representation of the density given in Eq. (1 5) leads to the normalization condition
fa(
for the one-electron normalized isotropic distribution U ) . When isotropic scattering in all conservative inelastic collision processes is assumed, which means vanishing cross sections G.',( U ) , the second equation of (34) can easily be resolved with respect to the anisotropic distribution and can be used to eliminate fi ( U ) from the first equation of (34). Following this procedure and performing the separation of the density, the equation
for the normalized isotropic distribution jb( U ) is obtained. Commonly this equation and Eq. (35) are used to determine the normalized isotropic distribution. Consideration of Eq. (36) shows that various quantities of the collision processes and a few plasma parameters are involved in its coefficients and naturally have an immediate impact on its solution. With respect to the atomic data of the various collision processes, these are the momentumtransfer cross section @(U ) , the total cross sections @'( U ) , the corresponding excitation or dissociation energies U;' of the ground-state atoms or molecules, and the mass ratio m,/M. With regard to the plasma parameters, the electric field strength E and the density N of the atoms or molecules occur, but only in the form of the reduced field strength E I N . All these quantities have to be known for specific weakly ionized plasma in order to determine the isotropic distribution fo(U) by solving Eq. (36).
34
R. Wtnkler
If the isotropic distribution f a ( U ) has been obtained from Eq. (36), the normalized anisotropic distribution ( U ) can be determined from the second of Eqs. (34) according to the expression
3
With both the normalized distributions, the steady-state values of all important macroscopic quantities can be calculated, up to the electron density n as common factor, using the reprFsentations in Eqs. (15) to (30) and replacing the distributions fi ( U ,z , t ) by nfi ( U ) with i = 0 and 1 in all integrals. Furthermore, the adaptation of the consistent power and momentum balance, Eqs. (32) and (33), to plasmas in steady state leads to the equations P ' -F -n n I'_ - I" -n n
(39)
According to the power balance, Eq. (38), the mean power gain from the electric field is compensated for by the mean power loss in collisions, and this happens for any given gas and its specific atomic or molecular data and for any reduced field strength E I N . An analogous compensation occurs in the momentum balance, Eq. (39), between the mean momentum gain from the field and the mean momentum loss in collisions. It should be mentioned that the additional inclusion of nonconservative electron collision processes in the kinetic study of a plasma in steady state does not really make sense from a strict point of view. In such a case, fulfillment of the consistent electron particle balance would require the production and the loss of electrons to completely compensate for each other in any small volume of the plasma and at any time. But, for a given gas and its specific atomic data, such a requirement can not naturally be satisfied for any reduced field strength. Thus, E I N would no longer be a parameter of Eq. (36) if it was extended to nonconservative collision processes. B. SOME REMARKS ON THE CALCULATION OF THE ISOTROPIC DISTRIBUTION Equation (36) represents a linear ordinary differential equation of second order with the additional terms fa(U Uf") involving the shifted energy arguments U U,?. These terms are caused by the occurrence of the various conservative inelastic electron collision processes with corresponding energy losses Uf" > 0 in these collision events. The solution of Eq. (36) is sought on an appropriate energy range 0 5 U 5 Urno, where the upper limit Uw has to be chosen in such a way
+
+
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICENTS
35
that the solution,&(U) becomes negligibly small for energies larger than the upper limit Urn. To determine the physically desired solution of the second-order equation, two "boundary conditions" have to be imposed on the solution. The normalization condition, Eq. (35), can be used as one of these. Furthermore, an asymptotic analysis of Eq. (36) for large energies U shows that the desired solution can be isolated when the boundary condition &(Urn)= 0 is applied, neglecting additionally for U Uf"> U" the terms&((/ U,'") involving the shifted energies. Various techniques for solving Eq. (36) [or system (34)] have been developed in the past. A very efficient solution technique involves performing a discretization of the second-order differential equation, Eq. (36), at all internal points of an equidistant energy grid, adapted to the energy region 0 5 U 5 U w ,using a finite-difference approach with second-order-correct difference analogues. In this discretization, the above-mentioned distribution termsf,( U Uf")with shifted energy arguments occurring in Eq. (36) can be represented on the same energy grid by appropriate parabolic interpolation, using those discrete function values of the isotropic distribution fo( U ) that are immediate neighbors with respect to the shifted distribution values &(U U,'"). Because of their energy shift, these terms disturb the tridiagonal structure of the resultant linear equation system for the discrete function values. Furthermore, a discretization of the normalization condition, Eq. (35), on the same grid can be obtained by applying the well-known Simpson rule. This integration rule should be slightly improved to obtain a more accurate integration of the factor U'12 just above zero energy in the integral occurring in Eq. (35). The discretization of the ordinary differential equation, Eq. (36), and of the two mentioned boundary conditions leads finally to a complete linear equation system whose inhomogeneity results from the discretized normalization condition, Eq. (35). An efficient resolution of this system becomes possible if those terms obtained by the parabolic interpolation are iteratively treated in the resolution procedure. A particular advantage of this solution approach is that other collision processes and even the nonlinear Coulomb interaction between the electrons can be included and can be successfully treated after corresponding extensions of the solution technique.
+
+
+
+
C. EXAMPLES OF DISTRIBUTION FUNCTIONS AND MACROSCOPIC QUANTITIES To illustrate the behavior of electron kinetic quantities in steady-state conditions, weakly ionized plasmas in neon and molecular nitrogen are considered as typical representatives of atomic and molecular gas plasmas. The essential differences between these plasmas with respect to the electron kinetics are the energy regions where the electron collision processes in each gas
36
R. Winkler
occur and their different intensities. While in the atomic plasmas the inelastic collision processes usually occur with lower intensity in the region of higher energies, in the molecular plasmas an intensive vibrational excitation at lower energies and a pronounced electronic excitation already at medium energies generally take place. Using two different scales for the cross-section values, the important inelastic collision cross sections of Ne and N2 are shown in Fig. 1 together with the respective cross section @(u)for momentum transfer in elastic collisions, denoted by d. With respect to Ne, the individual collision cross sections have been taken from Hayashi (1996). The relevant total cross sections P ( U ) , @ ( U ) , and Q ( U ) for the respective lumped excitation of the s and p states and the ionization from the neon ground state (left) are denoted by s, p , and i. The corresponding energy losses in these inelastic processes are U' = 16.62, Up = 18.38, and U' = 21.56 eV With regard to N2 (right), a reduced set of inelastic collision processes with the ground-state molecule is used for simplicity. This set has been proposed by Phelps and Pitchford (1985) and includes the total cross sections Q ( U ) , Q'(U), Q'(U), and Q ( U ) for the respective lumped excitation of the vibrational, triplet, and singlet states and the ionization. These cross sections are denoted by v, t , s, and i. The corresponding energy losses in these processes are UL' = 1.0, U' = 7.5, U' = 13.0, and U' = 15.6eV. Despite the lumped description of some individual excitation processes by one total cross section, the characteristic features of the different collision processes and their consequences for the various electron kinetic properties are preserved to a large extent. The ionization process in each of the two gases has been taken into account. However, as already mentioned above, this process is henceforth treated as an excitation process, conserving the electron number in this inelastic collision process. Additional details on the various collision cross sections are reported for Ne by Zecca et al. (1996) and for N2 by Itikawa (1994). 2 1
0
0
5
10
15 20 U [evl
FIG. 1. Collision cross sections
25
30
for neon and
37
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
The significant differences between the cross sections for the electron collisions in the Ne and N2 plasmas in terms of their intensity and the energy region of their occurrence can be clearly seen from Fig. 1. These differences are the main reason for the very different kinetic properties of the electrons in the two plasmas, which will be illustrated in the following discussion for both steady-state plasmas. To fix the plasma parameters, involved in the kinetic equations, Eqs. (36) and (37), and in the representation of the macroscopic quantities in Eqs. (1 5) to (30), the gas density N = 3.54. cm-', i.e., a density which corresponds to 1 torr pressure at 0°C gas temperature, is henceforth used in the calculations. Equation (36) has been solved for a range of field strengths E using the respective mass ratio m,/M and the set of collision cross sections for Ne and N2 presented in Fig. 1. The resultant !nergy dependence of the isotropic and anisotropic distributions A ( U ) (left) and,f,(U) (right) in the neon plasma is shown at field strengths E between 0.2 and 10 V/cm in Fig. 2. The structural change of both distributions is the result of the competing action of the electric field and of the elastic and inelastic collisions. At field strengths lower than about 0.5V/cm, only elastic collisions with a small energy loss in each collision event take place. An increase in the field strength causes a monotone growth in the isotropic distribution at higher energies. However, if the electron population markedly overcomes the energy threshold I/" of the lowest excitation process, a large structural change in the energy dependence of the isotropic distribution becomes obvious. This is mainly the result of the growing occurrence of inelastic collisions. In each exciting collision,
E inV/cm 0.2
0.35 0.5
E in V/cm 2
_
-.
-
0.2 lo''*
-
-1 10-2,
0.35 0.5
-~
-
2
5
7
.......... 11
38
R. Winkler
the colliding electron suffers the large energy loss Us. As a consequence, this electron is backscattered into the low-energy region of the isotropic distribution, which finally leads to an efficient depopulation of the isotropic distribution at higher energies. Thus, particularly at medium electric field, a pronounced nonequilibrium behavior of the isotropic distribution or, in other terms, particularly large deviations from the course of the well-known Maxwell distribution, i.e., a straight line in the semilogarithmic plot, is usually found. Additional growth of the field increasingly causes a smoothing of the isotropic distribution by the stronger field action, reducing the nonequilibrium behavior. At lower energies, the values of the anisotropic distribution, obtained from the isotropic distribution by means of Eq. (37), are relatively small compared with those of the isotropic distribution. However, at higher energies, where the inelastic collisions occur, the anisotropic distribution approaches the corresponding isotropic distribution. Using these normalized distributions, some transport properties and the mean power losses in the different collision processes have been determined for neon by means of Eqs. (1 6) to (1 8) and (21) to (24) and have been represented as a function of the field strength in Fig. 3. The mean energy u,/n and the magnitudes of the reduced particle and energy current densities j , / n and je,/n of the electrons (left) naturally increase with the field strength because of the correlated enlargement of the field action on each electron. However, it can be seen that the growth of the mean energy is quite different from that of the current densities. The large structural change in the course of the mean energy with growing field strength around E = 0.5 V/cm is caused by the transition from the action of only elastic collisions to the increasing effect of inelastic collisions. Owing to the large energy loss in each inelastic
.
I
7;'
0
2
4
6
E [Wcm]
8
10
0
2
4 6 E [Wcm]
8
Fic;. 3. Transport properties and mean power losses of the electrons in the neon plasma.
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
39
collision event, a remarkably smaller increase in the mean energy with the field is obtained at higher field strengths. This is confirmed to a large extent by the representation of the mean power losses in the various collision processes, also given in Fig. 3 (right). It can be easily observed from the course of the loss in elastic collisions P " / n and of the total loss in collisions P / n that in the steady-state neon plasma, the dominant contributor to the power loss changes around the field strength E = 0.5 V/cm from elastic to inelastic collisions. With respect to the latter, Fig. 3 additionally shows that the excitation of the s levels represents the dominant power-loss channel among the three inelastic power losses P'ln, Pl'ln, and P'/ti in the range of field strengths considered. This behavior can be expected from the course and magnitude of the corresponding total collision cross sections for neon shown in Fig. 1. It can be further seen from Fig. 3 that the mean electron energy u,,,/n in the atomic gas plasma reaches relatively large values compared with the low mean energy of the gas atoms of about 0.03 eV. Let us now briefly consider the field dependence of the same kinetic quantities in the N2 plasma. Figure 4 illustrates the evolution of the isotropic and anisotropic distributionsj& U ) (left) andfi ( U ) (right) as calculated for field strengths between 0.1 and 100 V/cm. Because of the very intensive vibrational excitation in a narrow range around 2.5 eV, a strong decrease in the isotropic distribution in this energy region is found for all field strengths considered. However, in the gap between the vibrational and the triplet excitation, i.e., between about 4 and 7 eV, a remarkably reduced decrease in the isotropic distribution is obtained owing to the
5
FIG. 4. Isotropic and anisotropic distribution in nitrogen for various field strengths
40
R. Winklev
occurrence of only elastic collisions at these energies. Then, starting with the triplet excitation, a stronger decrease in the isotropic distribution is found at higher energies. This illustration of the evolution of the isotropic distribution in the molecular plasma and the preceding one for the atomic plasma in Fig. 2 make it obvious that the structure and magnitude of the important electron collision processes substantially determine the detailed energy dependence of the isotropic distribution, and according to relation (37), that of the anisotropic distribution too. Furthermore, a comparison of the isotropic distributions in both plasmas calculated for the same field strength generally shows that the main population of the distribution in the molecular plasma is noticeably shifted to the region of lower energies. This energy shift is mainly a consequence of the larger intensity of the inelastic collision processes in the molecular plasma and their occurrence at substantially lower energies. This statement is also reflected in the values of the mean energy u,,,/n obtained for the nitrogen plasma. Its field dependence is shown in Fig. 5 (left), together with those of the reduced current densities j z / n and jJn. Except for the highest field strengths, the mean energy in the molecular gas has values of only around 1 eV Thus, the mean energy in N2 is almost one order of magnitude smaller than that in the neon plasma at the same field strength. The evolution of the mean total power loss by collisions P r / n and by the various collision processes in the nitrogen plasma is also displayed in Fig. 5 . It can be seen (right) that at almost all field strengths considered, the mean loss by vibrational excitation P"/n is the dominant power loss channel. Only at field strengths below about 0.2 V/cm, where the power loss by elastic collisions P"/n becomes dominant, and above about 60 V/cm, where the power loss by the triplet excitation P'/n becomes dominant, is this not the case.
0.1 E [v/crn]
1
10
E [Wcrn]
FIG. 5 . Transport properties and mean power losses of the electrons in the nitrogen plasma.
100
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
0
2
4
6 E [V/cm]
8
1
0
0.1
10
1
41
1 0
E [Vlcm]
FIG. 6 . Mean collision frequencies in the neon and nitrogen plasmas.
With respect to the application of electron kinetic quantities in an extended quantitative plasma description, the mean collision frequencies v;' (or the corresponding rate coefficients k;') related to the various inelastic electron collision processes are of particular importance. According to Eq. (29), these mean collision frequencies are determined by the isotropic distribution. The evolution of the various mean collision frequencies with growing field strength is presented in Fig. 6 for the neon (left) and nitrogen (right) plasmas. Quite different evolutions with the field strength and very different magnitudes of the various mean collision frequencies can be observed in the two plasmas. At lower fields, a very sensitive increase in the three mean collision frequencies v ' , 4, and v' for the excitation of the s and p states and the ionization in the neon plasma occurs because of the sensitive dependence of these frequencies on the high-energy tail of the isotropic distribution. The same holds with respect to the mean frequencies v', v', and v' for the excitation of the triplet and singlet states and the ionization in the nitrogen plasma, but this increase happens at substantially higher field strengths. As is to be expected, the dominant collision frequency in the nitrogen plasma is the mean frequency v" of the vibrational excitation.
D. KINETIC TREATMENT OF GAS MIXTURES So far, the kinetics of the electrons in steady state has been considered for plasmas in pure gases. However, in many applications, mixtures of some gases occur, and the kinetic treatment of the electrons has to include all important electron collision processes with each mixture component. Because of the abovementioned short range of the electron-heavy particle interaction, all these
42
R. Winkler
processes are considered to occur independently of one another. As a consequence of this concept, the collision integrals related to all mixture components have to be summarized on the right side of the Boltzmann equation, Eq. (8). Then, for plasmas in steady-state, instead of Eqs. (36) and (37), the extended equations (Winkler et al., 1982)
for the determination of the isotropic and anisotropic distributions $l(U ) and f i ( U ) in the mixture plasma are obtained. In these equations, the quantities @(U), @ ( U ) , UF, m e / M , and N are replaced by the corresponding quantities e ; ' ( ~@(u), ), U;, me/Mk and Nk related to the kth mixture component. By using the same boundary condition; and the sAmesolution technique as adapted for Eq. (36), both distributions h ( U ) and f i ( U ) for a given mixture composition can be calculated from Eqs. (40) and (41) if the atomic data for all important electron collision processes occurring in the mixture are available. As a further consequence, the mean power losses P"/n and p;"/n of the onegas-component plasma, given in the representations in Eqs. (21) and (22), are replaced by the corresponding mean power losses P f l n and e / n related to the kth mixture component. All these losses are then summarized according to F / n = CkP $ / n Ck e / n to get the total mean power loss in the mixture plasma. The same holds for the various mean momentum losses and the mean collision frequencies of the electrons. To illustrate the variation in electron kinetic quantities in a mixture plasma when its composition is changed, the mixture of Ne and N2 is considered. The total gas density N , N2 of the mixture is again supposed to be 3.54 . 10I6 ~ m - ~ , i.e., the same value as in the one-gas-component plasmas. For the field strength E = 10V/cm and for neon-to-nitrogen mixture ratios beAmeen100 : 0 and 50 : 50, the alteration of the normalized isotropic distributionfo(U) is shown in Fig. 7 (left). From this figure, a pronounced variation in the isotropic distribution with increasing admixture of the molecular component can be observed. Owing to the very intensive inelastic collision processes and their distinctly lower energy
+
+
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
43
1oo neon
FIG. 7. Isotropic distribution in various Ne :N2 mixtures (left) and in pure Ne under the additional impact of the electron-electron interaction (right).
thresholds in nitrogen compared to neon, small admixtures of nitrogen (only a few percent) lead to drastic changes in the isotropic distribution and, thus, in the related macroscopic quantities of the electrons. In particular, the addition of one or some molecular components to an atomic plasma presents the most sensitive case with respect to the change of the electron kinetics. For example, addition of only a few percent of molecular gases can cause about half the power input from the electric field to be dissipated by electron collisions into the molecular admixture components. If, in a plasma with a single gas component instead of a mixture plasma of different gases, collision processes with excited atoms or molecules of the same gas are additionally taken into account, each kind of excited particle has to be treated as a mixture component in the frame of the electron kinetics. Thus, the same equations, Eqs. (40) and (41), are the basis for the study of the electron kinetics influenced additionally by elastic and conservative inelastic electron collisions with excited particles of the same gas.
E. INCLUSION OF THE ELECTRON-ELECTRON INTERACTION
With increasing density of the electrons in the plasma, in addition to the binary electron collisions with gas particles, the Coulomb interaction between the electrons becomes more and more important, and its impact on the kinetics of the electrons has to be considered. Finally, if this interaction process dominates
44
R. Winkler
the kinetics of the electrons in the plasma, the isotropic distribution approaches the well-known Maxwell distribution. Following the conventional approach (Winkler et al., 1982), the electronelectron interaction can be sufficiently described by adding a Fokker-Planck term to the kinetic equation, Eq. (36). This leads to the extended kinetic equation
(42) for the normalized isotropic distributioni(U). The quantity Y in this equation is given by the expression
and includes, in addition to the vacuum permittivity co, the electron density n and the mea? electron energy u,,,/n. The latter is determined by the average U312fo(U ) dU over the normalized isotropic distribution. For steady state, the density n is an additional parameter of Eq. (42). In principle, the mean energy u,n/n has to be self-consistently determined in the solution of Eq. (42). Because of the weak dependence of the Coulomb logarithm log A on the mean energy and on the density, this quantity is usually approximately determined and treated as a fixed value in the solution of Eq. (42). However, its consistent treatment does not present a problem. Except for the weak dependence of log A on the density, the Fokker-Planck term in Eq. (42) contains the density n as a factor. Thus, the density n controls as a parameter the impact of the electron-electron interaction in Eq. (42). Because of the term of the electron-electron interaction the extended kinetic equation, Eq. (42), is nonlinear in the isotropic distribution. However, despite this serious complication, an extended solution technique based on the same ideas as those used for the solution of Eqs. (36) and (40) can be applied. In this extended approach, the nonlinearities occurring in the Fokker-Planck term are iteratively treated, in a way similar to the treatment of the above-mentioned terms involving the shifted energy arguments. Thus, in the frame of the finite-difference approach,
Jr
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
45
in each cycle of the iterative solution technique, a linear equation system determining the discrete values of the isotropic distribution on the energy grid has to be efficiently solved. Since the coefficients of the linear system now contain the discrete distribution values contained in the preceding cycle, these coefficients also have to be iteratively treated in this extended approach. The additional impact of the electron-electron interaction on the isotropic distributionjb(U) in the neon plasma at E = 2V/cm and N = 3.54. 10l6 cm-3 is illustrated in Fig. 7 (right). These distributions have been determined by solving the extended kinetic equation, Eq. (42), including the consistent treatment of log A, for electron densities n between 3 10" and 3 . l O I 4 cmP3. The distribution formally related to the density 0 belongs to the limit without the electron-electron interaction. It can be clearly seen from this figure that with increasing electron density at unchanged field E and gas density N,a monotone reduction in the pronounced nonequilibrium behavior of the isotropic distribution occurs, and at the highest electron density considered, a Maxwell distribution is almost established. It should be additionally emphasized that a sufficiently large electron density is required in order to cause a significant impact of the electronelectron interaction. Because of the competing action of the electric field term and the electron-electron interaction in Eq. (42), the required electron density sensitively varies with the change of the field strength in the plasma. +
F. REMARKS ON ADDITIONAL ASPECTSOF THE STEADY-STATE KINETICS The preceding representation of main aspects of the steady-state kinetics of the electrons is based on the so-called two-term approximation of the electron velocity distribution and on the corresponding two-term treatment of the electron Boltzmann equation including the most important electron collision processes. The latter means that, in addition to elastic collisions, exciting, dissociating, and approximately ionizing collisions of the electrons with atoms or molecules have been taken into account, where the heavy particles have been supposed to be in their ground state and at rest. Furthermore, in the frame of this two-term approximation, it has been briefly illustrated how this approach can be extended to treat the electron kinetics in gas mixtures and to include the Coulomb interaction between the electrons in plasmas containing larger electron densities. In the same frame of the two-term approximation, some other extensions have been used. Some examples of these extensions are briefly presented in the following. The impact of a finite gas temperature has been considered (Shkarofsky et al., 1966; Winkler et al., 1990), to allow an energy transfer from the atoms or molecules back to the electrons in elastic collisions at very low electric fields. Exciting electron collisions with excited atoms or molecules have been included (Winkler et a]., 1983) in the kinetic treatment. This impact is of greater
46
R. Winklev
importance for the electron kinetics in molecular gases, since a substantial portion of the molecules of the electronic ground state can be vibrationally excited. The energy losses and collision intensities associated with the excitation of vibrationally excited molecules in electron collisions are usually different from those associated with the excitation of vibrationally unexcited molecules. The energy transfer back to the electrons by electron collisions of the second kind with excited atoms has been taken into account (Winkler et af.,1983). These deexciting collisions can have a large impact on the population of the isotropic distribution at higher energies and become important at low electric fields and when electronically excited states are sufficiently populated. The superimposed action of an electric and magnetic field has been analyzed (Winkler, 1972). If such a superposition acts upon the electrons and if the two fields are not parallel, the velocity distribution loses its symmetry around the direction of the electric field, and the expansion of the distribution in Legendre polynomials [Eq. (9)] has to be replaced by that in spherical harmonics with respect to the whole angle space Z / u of the electron velocity 'i In two-term approximation of this extended expansion, then, a vectorial anisotropic distribution is involved instead of a scalar. A kinetic treatment of this superimposed field action has revealed, for example, that owing to the additional action of the Lorentz force, the isotropic distribution now sensitively depends on the magnitude of the magnetic field and the angle between the two fields. The expansion of the velocity distribution in Legendre polynomials, Eq. (9), presents an expansion with respect to the angle coordinate u,/u. Its lowest term is the isotropic part, and all additional terms are contributions to the anisotropy of the velocity distribution. In the frame of the two-term approximation of this expansion, the angle dependence of the velocity distribution is described by P,(u,/u) and thus by a linear dependence on u=/v. Particularly with respect to the kinetic treatment of plasmas that involve intensive inelastic electron collision processes, operate at larger electric field strengths, or are characterized by distinctly anisotropic scattering in electron collision processes, there have been doubts as to whether the two-term approximation treats the kinetics of the electrons with sufficient accuracy and describes the almost convergent solution of the electron Boltzmann equation. To improve the accuracy of the solution, the conventional two-term solution approach has been extended to a multiterm approach (Winkler et af., 1984; Winkler et af., 1985a; Winkler et af., 1985b). In this case, instead of just the lowest two terms, th: first m terms with m 2 2 of the expansion (9), i.e., the expansion coefficients fo(U), . . . ,A,,- I ( U ) , are taken into account. Then, instead of Eq. (34), a system of rn ordinary differential equations with additional terms of shifted energy arguments is obtained. After is completion by appropriate boundary conditions, the system has to be solved as a boundary-value problem on an appropriate energy range 0 5 U 5 Urn'.Several
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
47
techniques have been developed to solve this system and thus to find multiterm solutions of the Boltzmann equation. Various applications of the multiterm solution approach have shown (Winkler et al., 1985b) that under usual plasma conditions, the largest corrections of the two-term solution by the corresponding multiterm results are obtained in molecular plasmas like N2, CO, and COz at medium field strength. However, it should be emphasized that the substantial behavior of the two lowest expansion coefficients U ) and .fi ( U ) and of all related macroscopic quantities is already found by using the two-term approximation. The main corrections of the two-term results are already obtained by the fourterm approximation, and the convergent solution is reached by a six- to ten-term approximation. The corrections, for example, of the isotropic distribution f o ( U ) and the first contribution ,fi ( U ) to the distribution anisotropy by the multiterm treatment start in the molecular plasmas just above the range of intensive vibrational excitation and at medium electron energies reach maximal corrections of up to about a factor of 2. The largest corrections occur in the tail of the distributions and finally approach about one order at the highest energies. It has been generally found (Winkler et al., 1985b) that larger corrections of the two-term solution by the convergent multiterm solution result if the lumped intensity of the inelastic collision processes, characterized by C, U), is large and becomes comparable with the intensity of the elastic collisions, characterized by @(U), and if this happens in a substantial part of the relevant energy range 0 5 u 5 U".
as(
IV. Electron Kinetics in Time-Dependent Plasmas According to the relevant power and momentum balance, Eqs. (38) and (39), the electron kinetics in steady-state plasmas is characterized by the conditions that at any instant the power and the momentum input from the electric field are dissipated by elastic and inelastic electron collisions into the translational and internal energy of the gas particles. This instantaneous complete compensation of the respective gain from the field and the loss in collisions usually does not occur in time-dependent plasmas, and often the collisional dissipation follows with a more or less large delay-for example, the temporally varying action of a timedependent field. Thus, the temporal response of the electrons to certain disturbances in the initial value of their velocity distribution or to rapid changes of the electric field becomes more complicated, and the study of kinetic problems related to time-dependent plasmas naturally becomes more complex and sophisticated. Despite this extended interplay between the action of the binary electron collisions and the action of the electric field, the electron kinetics in time-
48
R. Winkler
dependent plasmas can, in many cases, also be treated with good accuracy on the basis of the time-dependent two-term approximation. A. BASICEQUATIONS FOR THE DISTRIBUTION COMPONENTS When specifying the kinetic treatment to purely time-dependent plasmas with isotropic scattering in the conservative inelastic collision processes, from system (12) the simplified system (Wilhelm and Wmkler, 1979; Winkler and Wuttke, 1992; Loffhagen and Winkler, 1994; Winkler, 1993; Winkler et al., 1995) R)li2U'/'
-A, a at
a at
- e,E(t)
-(-A) a u
au a au
a
2U2NQd(U)fo]
3
- fi - eoE(t)U -fo
+ U k @ ( U ) + CI NQ$(U)
1
fi
=0
is obtained in the frame of the two-term approximation. It describes the temporal evolution of the isotropic and anisotropic distributionsf,(U, t) andfi ( U ,I ) . Thus, in a strict sense, even in two-term approximation, a system of two partial differential equations of first order with the additional terms fo(U U,?, t ) of shifted energy arguments remains to be solved. To obtain a simpler struture of this mathematical problem, the system has often been reduced in the past by neglecting the first term in the second of Eqs. (44), i.e., the derivative offi(U, t) with respect to time (Wilhelm and Winkler, 1979; Winkler and Wuttke, 1992; Loffhagen and Winkler, 1994). When this additional approximation is accepted, the anisotropic distribution can be eliminated by means of the second of Eqs. (44), and finally a partial differential equation of second order with additional terms of shifted energy arguments is obtained. Some remarks about the validity limits of this approximation will be made below in connection with the presentation of some results. However, in recent years, techniques for solving system (44) numerically without additional reductions or simplifications have been developed (Winkler el al., 1995; Winkler, 1993). This modem approach is used as the basis of the following explanations concerning the time-dependent two-term treatment. The system of partial differential equations of first order, Eqs. (44), usually has to be treated as an initial-boundary-value problem on an appropriate energy region 0 5 U 5 U" and for times t 2 0, where the time represents the evolution direction of the kinetic problem. Initial values for each of the distributionsf,( U , t ) and .fi ( U , t), suitable for the problem under consideration, have to be fixed, for example at t = 0. Appropriate boundary conditions for the system are given by the requirementsfo(U 2 Urn,t ) = 0 andfi(0, t ) = 0.
+
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
49
B. MACROSCOPIC BALANCE EQUATIONS AND LUMPED DISSIPATION FREQUENCIES For time-dependent plasmas, the macroscopic balance equations in Eqs. (3 1) to (33) take the simplified form (Wilhelm and Winkler, 1979; Winkler and Wuttke, 1992; Winkler, 1993) d zn(t) =0
(45)
d u,,l(t)= P'(t) - P'(t) dt
-
d
-j:(t) = I / ( [ )- Z C ( t ) dt
(47)
and the total power and momentum loss in collisions, Eqs. (24) and (28), can be rewritten into the representation v , ( U)U3izf;(U , t ) dU
(48)
where the energy-dependent lumped frequencies v,( U ) and v,( U ) for power and momentum dissipation in collisions have been introduced by the expressions v,(U) =
+ CI NQ;"(U)
~
U
+ 1N Q ; ' ( U )
v,,,(U)= J2/m,U'i2 N Q 1 ( U )
I
Since, again, in time-dependent conditions, only conservative collisions are considered for simplicity, the consistent particle balance, Eq. (45), simply says that the electron density n is time-independent. Because of the linear dependence of all terms of the basic system (44) on the isotropic or anisotropic distribution, instead of the latter, the one-electron normalized distributions fo( U , t ) / n and fi ( U , t ) / n can immediately be introduced into this system. If nonconservative collisions are also considered, the electron density n(t) becomes time-dependent, and such a normalization will no longer be possible. As can be seen from the power and momentum balance, Eqs. (46) and (47), a temporal evolution of the mean energy density ulgI(t) and/or of the particle current density j J t ) is initiated if the instantaneous compensation of the respective gain from the field and the corresponding total loss in collisions is disturbed for any reason. Generally, by collisional dissipation, the electron component tries to reduce these disturbances and to again establish the compensated state in both
50
R. Winkler
balances. The rapidity of the collisional dissipation of power and momentum ultimately determines whether the compensation in both balances and thus the establishment of the steady state occurs almost immediately or with a noticeable temporal delay after the occurrence of a disturbance. The representations of the total power loss in collisions F ( t )and of the mean energy density u,,(t), Eqs. (48) and (16) clearly indicate that the rapidity of the dissipation of the kinetic energy per volume unit U3I2f,(U,t ) dU contained in the energy interval dU is determined by the lumped energy dissipation frequency v,(U). In an analogous manner, the representations of the total momentum loss I'(t) in collisions and of the particle current density jl(t), Eqs. (49), (17), and ( l l ) , show that the rapidity of the dissipation of the contribution (1/3)Jz/m,Ufi ( U , t ) dU of the energy interval dU to the particle current density (or, in other terms, of the contribution of dU to the momentum of the electrons per volume unit) is determined by the lumped momentum dissipation frequency Vn1( U ) . These energy-dependent dissipation frequencies are very important in characterizing the rapidity of the response of the electron component in different regions of the energy space to disturbances-for example, of the established steady state. These dissipation frequencies can be determined when the atomic data for the important electron collision processes and the gas density are known. To give an example, Fig. 8 represents both these dissipation frequencies for neon and molecular nitrogen at the gas density N = 3.54 . 10l6 ~ m - calculated ~, by means of the atomic data for these gases given in Fig. 1. It can be observed from this figure that in both gases, the momentum dissipation frequency exceeds the energy dissipation frequency by at least one order of magnitude. This means that the momentum dissipation occurs much faster than the energy dissipation. 10"
I
I "
FIG. 8. Lumped
I
/
' ' ~ " ' ' ' 1 ' '
' 1
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
51
The energy dissipation frequency has a complicated energy dependence in both gases. According to relation (3), the energy loss in an elastic collision event is proportional to the mass ratio m e / M , and thus very small compared with that in an inelastic collision event. As an immediate consequence, the lumped energy dissipation frequency assumes very small values in those energy regions where only elastic collisions occur, but considerably larger values at those energies where inelastic collisions happen. Thus, the efficiency of the collisional energy dissipation substantially depends on the electron population and its temporal evolution in different parts of the energy region.
c. SOME ASPECTSOF THE NUMERICALSOLUTION OF THE BASICEQUATION SYSTEM The numerical solution of the initial-boundary-value problem based on the equation system (44) can be performed (Winkler et al., 1995) by applying a finite-difference method to an equidistant grid in energy U and time t. The discrete form of the equation system (44) is obtained using, on the rectangular grid, second-order-correct centered difference analogues for both distributions fo(U, t ) / n andfi ( U , t ) / n and their partial derivatives of first order. In this discretization approach, in order to describe the termsf,(U U F , t ) / n with shifted energy arguments on the same grid, each of these function values is first represented at unchanged energy U U,'.' on the two neighboring grid lines of the time grid related to the centered discretization point by using the corresponding second-order-correct analogue with respect to the time. Second, these function values with shifted energy argument are represented on the energy grid by appropriate parabolic interpolations with respect to the energy at fixed time. An analogous discretization of the above-mentioned initial and boundary conditions leads finally to a complete linear equation system for the simultaneous determination of the discrete values of the isotropic and anisotropic distribution at all energy points U, of the range 0 p U p Urn and at a new time step if all discrete values of both distributions at the preceding time step t, are already known. Thus, starting from the initial values of both distributions and advancing from one time step to the next, the temporal evolution over the entire energy range of the normalized isotropic and anisotropic distribution & ( U , t ) / n and fi ( U , t)/n can be calculated, and, by performing the corresponding energy space averages over the two distributions according to the representations in Eqs. (1 5) to (30), the temporal evolution of the various macroscopic properties of the electrons can be determined. An efficient resolution of the resulting linear equation system for each time step becomes possible if all discretized terms related to the distribution fo(U + U;', t)/n with shifted energy arguments are iteratively treated in the
+
+
52
R. Winkler
resolution procedure. Then, in each cycle of the iteration process, a reduced form of the well-known bitridiagonal linear equation system has to be solved. For such a system, a fast algorithm is available in the literature that makes an efficient resolution of the system possible.
D. TEMPORAL RELAXATION OF THE ELECTRONS IN TIME-INDEPENDENT FIELDS To illustrate the relaxation of the electrons (Wilhelm and Winkler, 1979), the temporal evolution of their velocity distribution under the action of a timeindependent field and of the important electron collision processes has been calculated by solving system (44). The solution procedure started at I = 0 from a Gaussian distribution as the initial value of the isotropic distribution and from a vanishing anisotropic distribution, i.e., from
with the center energy U,, the energy width U,,, and the factor c used to normalize the initial value of the isotropic distribution on one electron according to U’/*fO(U,0) d U / n = 1. In the following, some results concerning the relaxation in neon and in nitrogen are presented and discussed. They have been determined again for N = 3.54. 10l6 cmP3 and with fixed energy width U, = 2 eV at the initial value of the isotropic distribution. The results for neon have been calculated for the two field strengths E = 0.2 and 10V/cm and for the center energy U, = 15 e\! Figure 9 shows the temporal evolution of the isotropic distribution f o ( U , t ) / n at the fields 0.2 V/cm (left) and 10 V/cm (right) on a logarithmic time scale. As just detailed, the evolution starts at t = 0 from an isotropic distribution with a
Jr
FIG. 9. Temporal evolution of the isotropic distribution in neon.
53
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
single peak around 15 eV, and the course of the relaxation finally leads to the establishment of the corresponding steady-state distributions. Because of the logarithmic time scale, the representation starts somewhat after the beginning of the relaxation process. At these times, a distribution peak at low energies is created by the backscattering of electrons that have undergone inelastic collisions at the very beginning of the relaxation. During the relaxation process, quite different evolutions occur, with strong depletion of the electron population at higher energies under low field action (left) and strong enhancement of the electron population at higher energies under the action of a larger field (right), and very different steady-state distributions are finally established. With respect to the total relaxation time of the isotropic distribution, i.e., the time needed to reach steady state, quite different times-of about lop4s (left) and about lop's (rightFare found. This already makes it clear that the relaxation time for the establishment of the steady state in a time-dependent electric field can differ by several orders of magnitude and drastically depends on the field strength and thus on the collisional energy dissipation efficiency of the electrons. The pronounced variation of this efficiency can also be expected from the strong field dependence of the total energy loss in collisions P c / n given for the steady-state neon plasma in Fig. 3. The establishment of the steady state and the corresponding relaxation times can be well evaluated when the behavior of the gain from the field and the total loss in collisions in the power and momentum balance are considered during the relaxation process. For the same conditions as those considered in Fig. 9, the gain-to-loss ratios are displayed in Fig. 10. The ratio Pf(t)/P"(t)is presented at the top of the figure, and the ratio I f ( r ) / Z c ( t ) at the bottom. Because of the vanishing initial value used
.
- 102
power
neon, E=lOV/cm
power
. E m CD
l o i 1 10-1
10-
1 0 8 ' 1 0 lo6 lo5 1 0 t [sl
10lo
10-8
t [SI
FIG. 10. Temporal evolution of the power and momentum gain-to-loss ratios in neon.
10-7
54
R. Mnkler
for the anisotropic distribution, the initial values for the power input !"(t) and the total momentum loss F ( t ) vanishes, too. Thus, the power and momentum ratios formally start from zero and infinity, respectively, i.e., from large disturbances in the power and momentum balance. The representation of the power gain-to-loss ratio shows that this quantity undergoes a quite different evolution at the two field strengths, involving orders of magnitude changes until the value 1-i.e., the complete compensation of power gain and loss-is established. Furthermore, this representation clearly shows that the compensation occurs just at those quite different relaxation times deduced above from the establishment of both isotropic distributions in Fig. 9. However, the representation of the momentum gain-to-loss ratio at the bottom of Fig. 10 makes it obvious that the evolution of this quantity toward the value 1 takes place much faster. The almost complete compensation of the momentum gain and loss is reached much earlier, at about lop9 s, and this happens nearly independent of the field strength. This evolution is largely in agreement with the magnitude and the weak energy dependence of the lumped frequency v,,,(U) for momentum dissipation by collisions in neon presented above in Fig. 8. Consideration of the consistent power and momentum balance, Eqs. (46) and (47), clearly shows that the establishment of the steady state simultaneously requires the complete compensation of the respective gain and loss terms and its continuation with growing time in both these balances. This means that the relaxation process and the corresponding relaxation time are mainly determined by the considerably slower establishment of the power balance. Let us now analyze the channels by which the power gain from the electric field is dissipated by the various collision processes during the course of relaxation and the manner in which these channels determine the quite different relaxation times found in the neon plasma at the two field strengths. These losses are illustrated in Fig. 11 for the same plasma conditions as those considered in Fig. 9. Consideration of Fig. 1 1 immediately shows that at the low field (left), after an initial relaxation phase, the mean power loss by elastic collisions P"/n becomes the dominant energy loss channel. This means as was already seen from Fig. 9 (left), that the linal establishment and compensation of the power balance occur only by elastic collisions in the region of low electron energies. However, as can be seen from Fig. 8, for neon, the lumped frequency v,( U ) for energy dissipation in collisions has very small values at lower energies, which makes the large relaxation time in neon at this field strength immediately understandable. To some extent an opposite behavior becomes obvious at the higher field strength in Fig. 11 (right). During the entire relaxation process, the mean power loss by inelastic collisions P'"/n dominates by orders of magnitude. This means that the energy dissipation mainly occurs in the energy range above the energy threshold U' of the lowest inelastic collision process and thus with a power
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
55
FIG. 11. Temporal evolution of the mean power gain P'/n, the mean power losses P'yn and P"Yn in elastic and inelastic collisions, and the power loss P / n in all collisions.
dissipation frequency v,(U) orders of magnitude larger, as can be seen from Fig. 8. Furthermore, Fig. 12 illustrates the temporal evolution of the mean energy u,,,(t)/nand the reduced particle current densityj,(t)/n in the neon plasma for the same relaxation processes as are considered in Fig. 9. The mean energy shows a monotone decrease from its initial value u,(O)/n x U,, determined by the Gaussian distribution [Eqs. (52)], to its respective steady-state value in the time-independent field. The magnitude of the reduced electron current density increases in a nonmonotone manner from its initial value of zero to its steadystate value in the respective field. The results obtained for the two relaxation processes presented in Figs. 9 through 12 belong to one and the same initial value of the isotropic distribution. Let us now briefly consider the impact of different initial values on the relaxation time. For the same two field strengths considered so far, but for two center 16-J
r6
16
14:
neon, E=lOV/crn
z : I
FIG. 12. Temporal evolution of the mean energy and reduced particle current density.
56
R. Wnkler 10'
I E, U, in V/cm and eV
7
4
,
,
/
I
,
,
lo-"
,
,
,
10'O
'""10-9' t [sl
1o 8
" " "
'
FIG. 13. Power and momentum gain-to-loss ratios in neon at different center energies
energies, namely U, = 10 and 15 eV at E = 0.2 V/cm and U,. = 15 and 20 eV at E = 10 V/cm, the temporal evolution of the power and momentum gain-to-loss ratos is shown in Fig. 13. The two courses of the momentum gain-to-loss ratio obtained with U, = 15 eV agree in the frame of the chosen representation. Figure 13 makes it obvious that different initial values for the isotropic distribution and thus different initial phases in the temporal relaxation process are of less importance for the entire duration of the relaxation process. The course of the temporal relaxation in other gases under the action of a timeindependent electric field is, to a certain extent, similar to that found in neon. Figure 14 presents some results of the relaxation in molecular nitrogen at E = 10V/cm starting from the same initial value of the isotropic distribution used in Figs. 9 through 12. However, from Fig. 14 (left), an evolution of the
1$1'
"'l&lO'
'''I+' t
""l'o-8' " ' l ( j . 7
[sl
FIG. 14. Evolution of the isotropic distribution and of the gain-to-loss ratios in nitrogen
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
57
isotropic distribution, that is rather different from that in neon, given in Fig. 9 (right), can be observed. Starting with the center energy U,. = 15 e\! the main part of the electrons undergoes an intensive backscattering to lower energies in the initial relaxation phase. This backscattering is mainly caused by the overlapping action of the triplet and singlet excitation at the beginning, and later by the triplet excitation. In the course of the complex action of the field and the various collision processes, the steady-state distribution, populated at very low energies only, is finally established in a relaxation time of some lop7s. Despite this relaxation time, the almost complete compensation in the momentum balance becomes established very early in the relaxation process, at some s.
E. RESPONSE OF THE ELECTRONS TO PULSELIKE FIELDDISTURBANCES Results concerning the temporal relaxation of the electrons toward steady state in time-independent electric fields are very helpful in interpreting and understanding the more complex behavior of the electrons in time-dependent electric fields (Wilhelm and Winkler, 1979) or the electron response to pulselike field disturbances (Winkler and Wuttke, 1992; Lomagen and Winkler, 1994). Now, in time-dependent fields E(t), the relation between the typical time tE characterizing a noticeable temporal field alteration and the relaxation time t,.(E) needed to establish the steady state at each instantaneous field strength E(t) of the entire spectrum of the field values occurring in the course of the field disturbance becomes very important. If the inequality tE >> t,.(E) holds at each instant of the field alteration, the steady state related to the instantaneous field E can be established at each instant of the slowly changing field. However, in the opposite limit tE << t,.(E), the electrons follow the alteration of the field with large delay, and at each instant their state remains far from the steady state related to the instantaneous field. An intermediate state between both these limits is illustrated by the following results. The temporal field transition starts from the steady state, goes through a disturbance, and again reaches the undisturbed field. This field pulse is described by E(t) = (E,, - Eud)[1 sin (2xt/tp,, - n / 2 ) ] / 2 for 0 p t p t,,d and by E(t) = Eud for t 2 tpd,where ELrd,E,, and tpl, denote the undisturbed field, the peak value of the field, and the pulse duration. Figure 15 shows in a logarithmic time scale the two field pulses with the same pulse duration of s as used in the calculation. The isotropic and anisotropic distributions obtained from the solution of the steady-state kinetic equation, Eq. (36), related to the undisturbed field are used as initial values for both distributions in the time-dependent treatment of the electron response to the respective field disturbance. Figure 16 illustrates for neon the evolution of the isotropic distribution up to the establishment of the steady state in the undisturbed field for the field pulses of Fig. 15. If the field substantially
+
+
58
R. Winkler
t [SI
FIG. 15. Two field disturbances applied to the neon plasma
increases during the field pulse, as happens in Fig. 16 (left), a strong enhancement of the isotropic distribution at higher energies results, and the undisturbed state is reached with a considerable delay behind the end of the field pulse. In the opposite case of the field variation (right), a strong diminution of the isotropic distribution at higher energies occurs; however, the undisturbed state is reached immediately at the end of the field pulse. The detailed course of the relaxation can be better understood if the evolution of the corresponding power and momentum gain-to-loss ratios for the two pulselike disturbances is considered. These ratios are shown in Fig. 17. In both transitions connecting steady states, large deviations between gain and loss of about one order of magnitude occur in the power balance, while deviations between gain and loss of only about 1 percent are obtained in the momentum balance. The latter is an immediate consequence of the rapid momentum dissipation by collisions in neon, as illustrated by v,(U) in Fig. 8, compared with the chosen duration of the field pulse. When the field of the pulse increases from its low value of 0.5 V/cm (left), the relaxation time related to the establishment of steady states at these low field values amounts to some 10-5 s. Therefore, with the field increase, there is a large delay in the power dissipation, and a large power gain-to-loss ratio rapidly become established. However, when the field in
FIG. 16. Response of the isotropic distribution to the two temporal field disturbances
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS 10'4
/')
j
neon
m
59
I
neon, E=lW0.5-10V/cm
power
10"
'."-
I
momentum
I
1.02
I
! ib'
0.98 ,
I
,
' '
t
Is1
'io6
I
FIG. 17. Power and momentum gain-to-loss ratios for the pulselike field disturbances
the pulse approaches its maximum value of 10 V/cm, the relaxation time related to the establishment of steady states at these larger field values decreases to some lop7 s. Thus, in the immediate neighborhood of the field maximum, this very short relaxation time leads to the establishment of almost steady states related to the instantaneous field values. If the field in the pulse then decreases to its undisturbed value of 0.5 V/cm, an increasingly larger relaxation time controls the temporal evolution, and again large deviations between gain and loss occur in the power balance. When the field pulse is switched off at t = lop6 s, a relaxation process in the undisturbed field follows, and the steady state is finally reached with the large relaxation time of some s. Almost the reverse evolution can be observed for the other field disturbance (right). The response of the electrons is controlled at the beginning by the short relaxation time of some lop7 s, at around half the pulse duration by the long relaxation time of some lop5s, and close to the end of the pulse again by the short relaxation time. As a result, large deviations between gain and loss in the power balance slowly arise at the beginning, reach their maximum at about half the pulse duration, and vanish almost without any delay at the end of the field pulse. This interpretation of the electron response to the field disturbances is largely confirmed by the temporal course of the power gain from the field P.'/n and power losses P'yn, Pin/n,and P / n in elastic, inelastic, and all collisions given in Fig. 18. For example, if the field pulse starts and ends with the low field (left), at the beginning of the pulse and in the later relaxation phase the power gain is almost compensated for by the power loss in elastic collisions, and this leads to the large relaxation time at these periods. However, around the pulse maximum, the power
60
R. Winkler neon,E=0.5+ IO-+O.SV/crn
...
.
FIG. 18. Temporal evolution of the power gain and losses in both field disturbances.
gain is compensated for predominantly by the loss in inelastic collisions, and the evolution in this period is controlled by the short relaxation time.
F. REMARKS ON ADDITIONAL ASPECTSOF TIME-DEPENDENT KINETICS The preceding examples concerning the time-dependent kinetics of the electrons have illustrated the complexity of the temporal relaxation of the electron component toward steady state in a time-independent electric field and the broad range of relaxation times involved in such relaxation processes. Furthermore, it has been demonstrated how these relaxation times control the response of electron kinetic quantities to single pulselike temporal disturbances of the electric field being superimposed to established steady states. In the same frame of the time-dependent two-term approximation, kinetic problems in quite different time-dependent fields have been dealt with (Wilhelm and Winkler, 1979; Winkler, 1993). So, for example, the temporal evolution of the kinetic properties initiated by jumplike and continuous field transitions between steady states, by pulselike field alterations, or by decaying electric fields and the periodic evolution (Winkler, 1993) caused by the action of highfrequency electric fields in a wide range of field frequencies should be mentioned in this context. In these time-dependent lunetic studies, a variety of electron collision processes similar to those treated in the steady-state kinetics has been treated. In addition to these processes, nonconservative electron collision processes, such as ionization and attachment, and even the nonlinear electron-electron interaction have been taken into account. Besides the various types of electron collisions, other electron generation and destruction processes, such as the chemo-ionization in collisions between excited heavy particles in decaying plasmas or the injection of beamlike electrons into plasma, have been included as particle sources or sinks
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
61
in the electron Boltzmann equation. Owing to the consideration of nonconservative electron collisions and of other electron generation and destruction processes, the electron density becomes a time-dependent quantity and has to be included in the study of the temporal evolution of the isotropic and anisotropic distribution. To enhance the accuracy of electron kinetic quantities obtained using the timedependent two-term approximation, similar to the improvements achieved in the steady-state kinetics, the two-term solution approach could be extended by developing a time-dependent multiterm solution approach (Lofkagen et al., 1996; Lofkagen and Winkler, 1996) for the electron Boltzmann equation. In the case of an m-term approximation with m 2 2, instead of the system (44) containing two partial differential equations, an extended system of m partial differential equations is obtained from the Boltzmann equation. This system has to be solved numerically as an initial-boundary-value problem using an approach analogous to that briefly described above for system (44). The application of this multiterm approach to some atomic and molecular gases has shown that improvements in the two-term results comparable to those found by the multiterm treatment of the electron kinetics in steady-state plasmas are obtained during the temporal evolution.
V. Electron Kinetics in Space-Dependent Plasmas As in time-dependent plasmas, the electron component in space-dependent plasmas is usually characterized by the property that its local power and momentum gain from an electric field cannot be dissipated at almost the same space position by elastic and inelastic electron collisions into translational and/or internal energy of the gas particles. Thus, the spatial response of the electrons to, for instance, a disturbance of their velocity distribution or a spatial change of the electric field in the plasma is of a complex nature (Tsendin, 1995; Kolobov and Godyak, 1995; Winkler et al., 1996; Winkler et al., 1997). As shown below, the way to understand such a response is usually very different from that followed in the explanation of the corresponding temporal behavior of the electrons. Despite the complicated character of the electron response, in many cases the electron lunetics of inhomogeneous plasmas can be analyzed with good accuracy by using the space-dependent two-term approximation including the impact of all important binary electron collision processes with the gas particles and the action of a possibly space-dependent electric field. A. BASICEQUATIONS AND THEIRREPRESENTATION BY THE TOTALENERGY When specifying the kinetic treatment based on the two-term approximation to purely space-dependent plasmas with isotropic scattering in the important
62
R. mmkler
conservative inelastic collision processes, from system (12) the simplified system (Winkler et al., 1996; Winkler et al., 1997)
is obtained. This system describes the evolution of the isotropic and anisotropic distributions f o ( U , z ) and f i ( U ,z ) in the coordinate and energy space of the electrons. A consideration of the second equation of this system shows that, in principle, this equation can easily be resolved with respect to the anisotropic distribution and the result can be used to eliminate this distribution from the first equation of system (53). Then a second-order partial differential equation for the isotropic distribution with additional terms fo(U U r , z ) of the shifted energy arguments U U, is obtained. A particular disadvantage is that this equation contains cross-derivative terms, which causes its direct numerical treatment to become heavy and less stable. This disadvantage can be avoided when system (53) is converted into its standard form. A suitable representation of system (53) for its numerical solution is obtained when the kinetic energy U is replaced by the total energy c according to t = U W(z), where W ( z )= e, JiE(Z) d5 is the potential energy of the electrons in the electric field. This transformation and the elimination of the anisotropic distribution, with the hrther abbreviation U,, = t - W(z), leads finally to the parabolic equation (Sigeneger and Winkler, 1977a; Sigeneger and Winkler, 1997b; Sigeneger and Winkler, 1996)
+
+
+
for the transformed isotropic distribution &(c, z ) =fo(U,,, z), including the additional terms A(t U , ,z ) with the shifted energy arguments t U,Y. Furthermore, the equation
+
+
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
63
fi
z) =
for the determination of the transformed anisotropic distribution
fi ( Ucz,z) is obtained.
(6,
As detailed below, the parabolic equation, Eq. (54) describes the evolution of the isotropic distribution and has to be solved as an initial-boundary-value problem on a nonrectangular solution region whose boundaries are partly determined by the spatial course of the electric field and thus by the specific kinetic problem considered. The parabolic problem has to be completed by appropriate initial and boundary conditions, which are briefly described below. Equation (54) makes it obvious that the natural evolution direction of the problem is that of the total energy t. This means that according to the conventional treatment of parabolic problems, the solution has to be sought by advancing step by step in the total energy direction and solving Eq. (54) for each discrete total energy over the entire range of the relevant coordinate space.
B. THECONSISTENT BALANCE EQUATIONS IN SPACE-DEPENDENT PLASMAS In purely space-dependent plasmas, the consistent particle, power, and momentum balance equations of the electrons, Eqs. (31) to (33), have the simplified representation (Sigeneger and Winkler, 1997a; Sigeneger and Winkler, 1996) d dz
- j,(z) = 0
(56)
d j&) = P'(z) - F(z) dz
(57)
-
where all macroscopic quantities occurring in these balances are given by the expressions in Eqs. (15) to (30) when their integral representations are specified for purely space-dependent conditions. Since, for simplicity, only conservative inelastic collision processes are considered in the basic kinetic equations, Eqs. (53), the consistent particle balance, Eq. (56), says that the particle current density of the electrons j , is a conservative quantity and becomes independent of the z coordinate. Furthermore, the power and momentum balance, Eqs. (57) and (58), show that an incomplete compensation of the respective gain from the electric field and the corresponding loss in collisions enforces the occurrence of sources or sinks in these balances. This leads, according to the power balance, Eq. (57), to a spatial evolution of the energy current densityj,,(z) and, according to the momentum balance, Eq. (58), to a spatial evolution of the mean energy density u,(z) of the electrons. Thus, the magnitude of these sources or sinks compared, for example, with the respective
64
R. Winkler
gain from the electric field makes an assessment of the degree of the nonlocal or nonhydrodynamic behavior of the electron component possible. This will be considered in greater detail below. In contrast to the situation in plasmas in steady state or in time-dependent plasmas, the electron density n(z) in space-dependent plasmas always depends on the z coordinate, and this happens already if only conservative inelastic collisions are considered. As an immediate consequence, it no longer makes sense to separate the density from the isotropic and anisotropic distribution of the electrons.
c. CHARACTERISTIC FEATURES OF THE SPATIAL RELAXATION OF THE ELECTRONS Basic aspects of the spatial relaxation of the electrons in collision-dominated plasmas can be revealed when the evolution of the electrons whose velocity distribution has been disturbed at a certain space position is studied under the action of a space-independent electric field (Sigeneger and Winkler, 1997a; Sigeneger and Winkler, 1997b). Sufficiently far from this position in the field acceleration direction of the electrons, a uniform state finally becomes established. Such relaxation problems can be analyzed on the basis of the parabolic equation for the isotropic distribution, Eq. (54), when the initial-boundary-value problem is adopted to the relaxation model. To enforce a disturbance in the velocity distribution, the anisotropic distributionf,(t, z ) has been fixed at the position z = ,O by an appropriate boundary value; i.e., the boundary condition f,(6, z = 0) = f i (t) is applied. For the boundary value, the Gaussianlike distribution
with center energy U, and energy width U, is used. The factor c serves to normalize the boundary value of the anisotropic distribution. According to Eqs. (1 1) and (17) and because U = t at z = 0, this boundary value describes the energetic distribution of the electron particle current density at z = 0 and determines the space-independent value of j , . If negative values are chosen for the uniform electric field E, the electron acceleration occurs in the direction z > 0 and the potential energy W ( z ) linearly decreases with growing z. For this case, the relevant solution region of the spatial relaxation problem in the ( 6 , z ) space is illustrated in Fig. 19jlefi) by the enclosed area. On its lefi boundary at z = 0, the boundary value f i ( t ) ,just detailed, is imposed, and its lower boundary t = W ( z ) at z > 0 corresponds to zero kinetic energy.
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
65
-5 1 -12 8
-08 N
0.6
-
T12
-P l . s -
08
306
04 0
10
20
30
40
50
z [cml FIG. 19. Solution region of the relaxation problem (left) and spatial relaxation of density and energy current density in neon (right).
The evolution direction of (54) is given by the total energy t Its numerical solution starts at a sufficiently large energy em and is continued down to negative total energies E . Appropriate initial and boundary conditions related to z > 0, used when solving Eq. (54) as an initial-boundary-value problem, are additionally given in this figure as a function of z. The numerical solution of Eq. (54) as an initial-boundary-value problem, specified to the spatial relaxation problem in uniform electric fields, can be obtained (Sigeneger and Winkler, 1996) by using a finite-difference approach according to the well-known Crank Nicholson scheme for parabolic equations. Based on this approach, the spatial relaxation of the electron component has been studied in plasmas of neon and molecular nitrogen and for some electric field strengths, again using the gas density N = 3.54 . 10l6 cm-3 and appiying the parameter values U,. = 5 eV and U,,, = 2 eV to fix the boundary value f i ( U ) according to Eq. (59). The spatial evolution of the density n(z)/n(oo)and the energy current density jez(z)/je2(oo), normalized on their respective values in the established uniform state at large z, is shown in Fig. 19 (right) for a neon plasma at the field E = -5V/cm. The figure clearly illustrates that the boundary value ( U ) for the anisotropic distribution initiates a weakly damped, spatially periodic relaxation of the density and energy current density of the electrons, and that the corresponding relaxation length becomes very large and takes about 100 cm at this field. This periodic relaxation behavior is in substantial contrast to the largely monotone evolution of all important electron kinetic quantities in the temporal relaxation process shown above.
66
R. Winkler
FIG. 20. Spatial relaxation of the isotropic distribution at two field strengths.
For the two field strengths E = - 1 and -5 V/cm, Fig. 20 illustrates the initial part of the spatial relaxation of the normalized isotropic distribution fo(U,z)/n(co) in neon together with the respective boundary value of the normalized anisotropic distribution that initiates the relaxation process. The figure makes it obvious that the relaxation behavior of the isotropic distribution is very different at the two field strengths. While at the higher field value, a weakly damped, distinctly periodic evolution with a short period length dominates, the spatial relaxation at the lower field occurs in a less pronounced way and with a substantially larger period length. This means that the detailed relaxation course of the isotropic distribution is controlled to a larger extent by the field strength acting upon the plasma. These results indicate that the more or less damped periodic evolution in the relaxation process is a typical feature of the spatial electron relaxation at medium electric field strengths. This means that despite the continuous power dissipation by collisions, a damped periodic response of the electrons occurs during the relaxation process of the electrons. The basic processes involved in this periodic response are the following: Driven in the positive z direction by the uniform electric field, the electrons gain potential energy from the electric field, and the electron population at higher energies monotonously increases. If a remarkable number of electrons overcomes the energy threshold of the lowest excitation process, these electrons undergo inelastic collisions. As a result of these events, the electrons lose the excitation energy, i.e., with respect to neon, the excitation energy U s of the lumped s states, and are backscattered to the region of low energies, where they again undergo the field action and gain potential energy from the field. The continuous repetition of this basic sequence leads to the periodic response of the electrons in the relaxation process. If this basic mechanism dominates, the inherent period length 1 satisfies the simple relation e,EA = U’.
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
67
Let us now consider the spatial relaxation process in terms of the local compensation of the gain and loss in the power and momentum balance, Eqs. (57) and (58). Figure 21 shows the spatial evolution of the respective gain-to-loss ratios of the power (top) and momentum (bottom) balances at the same two field strengths, - 1 V/cm (left) and -5 V/cm (right). The figure makes it obvious that the deviations from the local compensation of gain and loss are generally large and extend over a large spatial range. This is an immediate reflection of large relaxation lengths and of a pronouncedly nonlocal behavior of the electron component. A comparison of the ratios related to the power balance with those of the momentum balance shows that the deviations from the local compensation are generally somewhat more pronounced in the power balance. In contrast to the temporal relaxation, in the spatial relaxation process the establishment of the uniform state occurs, at fixed field strength, on the same scale in both balances. This means that the coupling between the isotropic and anisotropic distributions is much closer in the spatial than in the temporal relaxation process. This point can be W h e r analyzed by considering the spatial evolution of the individual contributions to the power and momentum balances, Eqs. (57) and (58). These contributions, all normalized on the respective gain terms P f ( w ) and Z1(00) in the established uniform state at large z, are illustrated in Fig. 22 for the relaxation process in neon at the field strength -5 V/cm. The representation of the power balance (left) shows that, in particular, the normalized power loss by inelastic collisions P'"(z)/Pf(00) oscillates with a large amplitude around the normalized field term P f ( z ) / P f0(0).The large deviations between these terms are compensated for to a large extent by correspondingly large oscillations around zero of the normalized source term d / d z [ j e , ( z ) ] / P f ( w of ) the power balance (dotted-dashed curve) containing the spatial derivative of the energy current density je,(z). E = -1 V/cm
power
. ! -
0
~~
momentum
'a
- 1
05 0
10
20 z
30
[cml
40
50
0
10
20
30 [cml
40
FIG 21 Gain-to-loss ratio of the power and momentum balances for two field strengths.
50
68
R. Winkler
The behavior of the individual terms in the momentum balance (right) is similar to that in the power balance. Now the normalized momentum loss in elastic collisions Ze'(z)/Zf(co) oscillates around the oscillating momentum gain Zf(z)/Z'(co), and the somewhat lesser deviations between these quantities are compensated for to a large extent by the normalized source term (d/dz) [(2/3m,)u,(z)]/Zf(co)of the momentum balance (dotted-dashed curve) containing the spatial derivative of the mean energy density um(z). The relaxation of the isotropic distribution, shown for two field strengths in Fig. 20, and the corresponding evolution of the gain-to-loss ratios of the power and momentum balance, displayed in Fig. 21, indicate that the relaxation length sensitively depends on the field strength. In order to reveal the dependence of the relaxation length on the magnitude of the uniform field, the relaxation process in neon has been studied for several field strengths, keeping all other parameters unchanged. Figure 23 shows the spatial relaxation course of the normalized density n(z)/n(co) (left) and of the mean energy u,(z)/n(z) (right) of the electrons at field values between -0.5 and -15 V/cm. From both figures, a
50 0
50 0
FIG. 23. Spatial relaxation of density and mean energy at various field strengths.
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
69
drastic change in the relaxation behavior and the involved relaxation length can be immediately seen. This change is caused by the alteration of the dominant relaxation mechanism with increasing field strength. As mentioned above, at medium field values, i.e., for neon around about -2 V/cm, the gain in potential energy and the energy loss in the lowest excitation process, i.e., the excitation of the lumped s states, dominate. These are the two basic processes that cause a weakly damped periodic relaxation of the electrons and unexpectedly large relaxation lengths. In the limit that only these two power transfer channels are active, an undamped, purely periodic spatial evolution without any spatial relaxation is excited. The occurrence of additional power loss channels causes a damping of the spatial evolution to take place. Thus, with decreasing field, the power loss in elastic collisions monotonously grows and finally becomes the dominant power loss process. In that case, the basic mechanism for the excitation of the periodic response is increasingly lost and a strongly damped, aperiodic relaxation process with a remarkably reduced relaxation length occurs. However, when the field strength is increased above the range of medium field values, because of the excitation of higher electronic states and finally because of the ionization, additional power loss channels become active and of importance comparable to that of the lowest possible loss channel. As a consequence, the electrons undergo different inelastic collision processes and are backscattered into the region of low energies with considerably different energy losses. This backscattering by different energy losses causes a strong increase in the damping and a marked diminution of the relaxation length. Furthermore, the increase in the field strength leads to a monotone decrease in the period length, since a comparably large gain in potential energy is reached in a lesser distance. So far, examples concerning spatial relaxation in the neon plasma have been presented and discussed. In order to briefly illustrate the impact of the kind of gas and thus of the specific atomic data, the relaxation behavior for the same conditions with respect to the gas density, the field strength, and the parameters of the boundary value of the anisotropic distribution has been studied in neon and molecular nitrogen. Using a logarithmic scale, Fig. 24 shows the relaxation behavior of the isotropic distribution in both gases at E = -10V/cm. The comparison makes it immediately obvious that a quite different spatial evolution takes place. While in neon a weakly damped periodic evolution occurs, a strongly damped periodic relaxation with a rather small relaxation length is found in nitrogen at somewhat higher energies. One main reason for the strongly damped relaxation process in nitrogen is the intensive overlap in the initial relaxation phase of the lowest two power loss channels leading to the vibrational and the triplet excitation. These two loss channels correspond to rather different energy losses in both types of excitation, and thus to a quite different backscattering of the electrons. This is a typical property of the molecular gases compared with the
70
R. Winkler
FIG. 24. Spatial relaxation of the isotropic distribution in neon and nitrogen.
atomic ones. After a distance of about 2 cm, the uniform state has already become established in the isotropic distribution in nitrogen, while about a 20-fold distance is required in the neon plasma.
D. RESPONSE OF THE ELECTRONS TO PULSELIKE FIELDDISTURBANCES The study of the spatial electron relaxation in uniform fields has demonstrated its complexity and its sensitive dependence on the field strength. In these relaxation studies, a local Pisturbance at z = 0 has been initiated by the choice of the boundary value f; ( U ) for the anisotropic distribution according to Eq. (59), and the succeeding spatial relaxation in a uniform electric field has been analyzed. Now, the response of the electron component to a spatially limited disturbance of the electric field is considered (Sigeneger and Winkler, 1996) as an example of the inhomogeneous electron kinetics acted upon by a space-dependent electric field. Sufficiently far from the field disturbance region, uniform states in and opposite to the acceleration direction of the electrons by the field should occur. If it is assumed that the field direction inside and outside the field disturbance region remains unchanged, the response of the electrons can be studied by using a kinetic approach similar to that as applied above for the relaxation studies in unifonn fields. Unlike the case with these relaxation problems, in the current case the isotropic distribution related to the homogeneous state in the undisturbed electric field is used sufficiently far from the field disturbance region as a boundary value for the isotropic distribution&(€,z ) when solving the parabolic equation, Eq. (54). Furthermore, since the electric field E(z) now becomes space- dependent in the field disturbance region, the potential energy W ( z ) no longer linearly decreases with z and the relevant solution area in the (6, z ) space becomes somewhat more complicated than that illustrated in Fig. 19.
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
71
Applying an analytical description similar to that for the temporal field pulse, the response of the electrons to a spatial field pulse has been calculated for neon by solving the parabolic equation, Eq. (54). Figure 25 shows the corresponding behavior of the isotropic distribution using a linear (left) and a logarithmic (right) scale. This allows illustration in more detail of the response in the low-energy range and in the high-energy tail of the distribution. In addition, the field pulse is shown at the top, and the thick lines on the distribution surface mark the spatial region where the field pulse is superimposed. At z = 0, the homogeneous isotropic distribution related to the undisturbed field has been taken as the boundary value for the isotropic distribution, and this homogeneous state remains established over a short distance. As can be seen from the field course, the pronounced diminution of the field in the pulse region between z = 5 and 15 cm causes, relative to the isotropic distribution, a strong enhancement in the low-energy region and a drastic reduction in the tail. In the field acceleration direction, behind the field disturbance region, a weakly damped periodic relaxation process in the distribution occurs. The overall response of the isotropic distribution of the electrons to the spatial field pulse can be qualitatively understood in the following manner. Although the spatial evolution starts from a homogeneous state in the undisturbed field, the spatially limited field pulse initiates a disturbance in the kinetics of the electrons and thus in the isotropic distribution too. As is known from the preceding study of the spatial relaxation, large relaxation lengths compared with the extent of the field disturbance region correspond to all field values covered by the field pulse. Therefore, during the action of the field pulse, no remarkable relaxation toward the establishment of the homogeneous state in the undisturbed field becomes possible, and a large relaxation distance in the undisturbed field is required in addition for the establishment of the undisturbed isotropic distribution.
FIG. 25. Response of the isotropic distribution to a spatially limited field pulse.
72
R. Mnkler
This qualitative interpretation is c o n h e d by considering the corresponding disturbances initiated by the field pulse in the consistent power and momentum balances, Eqs. (57) and (58). For the same field disturbance given in Fig. 25, the spatial evolution of the power (left) and momentum (right) gain-to-loss ratios is shown in Fig. 26 by the full curves. In this case, the field pulse acts in the region from z = 5 to 15 cm. It can be observed that starting with or even somewhat before the field pulse, the power and momentum ratios deviate increasingly from their value in the undisturbed state, reach their largest deviation with some spatial delay with respect to the position of maximal field disturbance at about z = 10 cm, and end up with a weakly damped, spatially periodic behavior at about the end of the field disturbance. In Fig. 26, the gain-to-loss ratios for an additional field disturbance are displayed by the dashed curves. These ratios have the same field pulse shape seen in Fig. 25; however, the field pulse now extends over twice the distance, i.e., over the region from z = 5 to 25 cm. Despite the remarkable enlargement of the field disturbance range, a very similar disturbance in both ratios and a comparable relaxation course behind the field pulse is established in both cases because of the generally large relaxation length at medium field strength. To illustrate the response of the electrons from the macroscopic point of view in more detail, Fig. 27 shows for the field disturbance given in Fig. 25 the spatial behavior of the individual terms of the power balance, Eq. (57) (left), normalized on the power gain P f ( 0 ) in the undisturbed state, and the spatial evolution of the reduced density n(z)/n(O)and mean energy u,(z)/n(z) (right). The evolution of the normalized power balance terms indicates that the power loss in inelastic collisions follows to a remarkable extent the alteration of the power gain from the field pulse. However, toward the end of the field pulse, the inelastic loss responds with a certain overshooting and ends up with a weakly damped oscillation around the undisturbed power gain at z > 15 cm. The remaining local deviations between the power gain and the inelastic loss are largely compensated for according to the
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
0
z [cml
10
20
73
30
z Icml
FIG. 27. Response of the power balance terms and of the density and mean energy
power balance, Eq. (57),by an alternating source and sink (dotted-dashed curve) containing the spatial derivative of the energy current density jeZ(z). Figure 27 (right) shows the corresponding response of the reduced density and mean energy of the electrons. It can be observed that the field pulse causes a large alteration, in particular, of the electron density. With regard to the field acceleration direction of the electrons, both responses start somewhat before the field pulse.
E.
REMARKS ON
ADDITIONAL ASPECTSOF SPACE-DEPENDENT KINETICS
On the basis of the space-dependent two-term approximation, including elastic and conservative inelastic electron collision processes, substantial aspects of the inhomogeneous electron kinetics, such as the spatial relaxation behavior in uniform electric fields and the response of the electron component to spatially limited pulselike field disturbances, have been demonstrated and the complex mechanism of spatial electron relaxation has been briefly explained. In these cases, starting from a specific choice of the boundary condition for the velocity distribution, the succeeding spatial evolution of the electrons in the field acceleration direction up to their establishment of a steady state has been studied. By using the same approach and an adequate choice of the boundary condition, the spatial evolution of the electron kinetic quantities can be analyzed in all those space-dependent electric fields that do not reverse their direction with growing z . In such studies, nonconservative inelastic electron collisions can also be included and will cause, in accordance with the consistent particle balance, Eq. (56), a spatial evolution of the particle current densityj,(z) also. In particular, by this approach, the kinetics of the electrons in moving and standing striations, occurring at low discharge currents in dc column plasmas, can be investigated (Sigeneger et al., 1998). Under the action of the highly modulated, spatially periodic electric field of the striations, an undamped, spatially periodic evolution of all electron kinetic properties is established.
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R. Khkler
Furthermore, the space-dependent two-term solution approach could be extended to a multiterm solution approach (Petrov and Winkler, 1997) to the space-dependent kinetic equation. By this extension, it becomes possible to accurately describe the spatial evolution of the electron component under conditions of larger anisotropy in the velocity distribution of the electrons. By using, for example, a 10-term approximation, even the kinetic treatment of the cathode region of dc discharges could be performed (Petrov and Winkler, 1998). This region is characterized by the occurrence of a very large electric field strength close to the cathode and by an extreme spatial decrease of the field in the succeeding cathode fall region. All applications and extensions of the space-dependent two-term approximations mentioned above are related to the condition that the electric field acts parallel to a fixed space direction and that the inhomogeneity concerning the electron component takes place parallel to the same space direction. In this case, a symmetry of the electron velocity distribution around this direction is established and its expansion in Legendre polynomials, Eq. (9), is adequate to describe the velocity distribution by a two-term approximation or, with greater accuracy, by a multiterm approximation. A quite different situation with respect to the spatial behavior of the electron component arises if the inhomogeneity of the plasma is caused by a plasma confinement. The electron kinetics established in the radial direction of the positive column of dc glow discharges, which usually operates in a cylindrical discharge tube with an isolated wall, presents a representative example (Kortshagen, 1995; Uhrlandt and Winkler, 1996; Pfau et al., 1996; Alves et al., 1997) of such a condition. From the point of the electron kinetics, this space-dependent problem is somewhat more complex than those considered above. The electric field in the axially homogeneous column plasma consists of a superposition of the constant axial electric field and of the radially varying radial space charge-field, i.e., k ( r ) = E,.(r)& Ez&. Thus, the direction of the total electric field E(r) is different from the radial direction in which the inhomogeneity of the plasma column occurs. Therefore, the expansion of the velocity distribution in Legendre polynomials can no longer be used and has to be replaced by an expansion in spherical harmonics (Uhrlandt and Winkler, 1996). Because of the cylindncal symmetry of the total electric field in the column plasma, in two-term approximation the expansion of the velocity distribution can be represented by the expression f ( U , G / u , r ) = ( 2 ~ ) ~ ’ ( r n ~ ~ ) ~ ’ ~ [ rf )o+f,(U, ( U , r)u,./u + L ( U , r)uZ/u].This expansion includes, in addition to the isotropic distribution f o ( U , v), a radial component f,( U , r ) and an axial component f , ( U , r) of the vectorial anisotropic part of the velocity distribution. In particular, this radial distribution component allows the particle and energy current density of the electrons in the radial direction to be described and thus reveals significant aspects of the electron confinement by the radial
+
THE BOLTZMANN EQUATION AND TRANSPORT COEFFICIENTS
75
space charge field. When this expansion is substituted into the appropriate version of the electron Boltzmann equation, a system of three partial differential equations for the three scalar expansion coefficients fo(U, Y), &(U, Y), and f , ( U , r ) is obtained. The elimination of both anisotropic distributions and the replacement of the kinetic by the total energy leads finally to a partial differential equation of elliptic type for the transformed isotropic distribution. This elliptic equation has been solved as a boundary-value problem (Uhrlandt and Winkler, 1996; Pfau et al., 1996; Winkler el al., 1997) to find the actual radial alteration of the isotropic distribution caused by the space charge confinement.
VI. Concluding Remarks By the preceding representations, an attempt has been made to give, on the basis of the electron Boltzmann equation, an introduction to the kinetic treatment of the electron component in steady-state, time-dependent, and space-dependent plasmas and to illustrate by selected examples the large variety of electron kinetics in anisothermal weakly ionized plasmas. In this representation, particular emphasis has been placed on a uniform basis for the electron kinetics under different plasma conditions. The main points in this context concern the consistent treatment of the isotropic and anisotropic contributions to the velocity distribution, of the relations between these contributions and the various macroscopic properties of the electrons (such as transport properties, collisional energy- and momentum-transfer rates and rate coefficients), and of the macroscopic particle, power, and momentum balances. Furthermore, special attention has been paid to presenting the basic equations for the kinetic treatment, briefly explaining their mathematical structure, giving some hints as to appropriate boundary and/or initial conditions, and indicating main aspects of a suitable solution approach. Such complete studies of electron kinetic problems allow the essential nonequilibrium properties of the electron component to be revealed and a deeper understanding of the interplay between the various microphysical processes involved in the kinetics of the electrons to be gained. In particular, this point has been illustrated by some examples concerning the temporal and spatial relaxation of the electrons and the electron response to temporal and spatial pulselike disturbances of the electric field. Future aspects of the study of electron kinetics based upon the electron Boltzmann equation certainly involve its extension to spatially two- and even three-dimensional kinetic problems, to coupled space- and time-dependent problems, to more complex field structures, and to more sophisticated boundary conditions. First attempts in these directions have already been undertaken in the literature (Meijer, 1991; Yand and Wu, 1996) or are on the way.
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R. Winkler
Another important point, closely connected with electron kinetics, concerns the self-consistent treatment of electron kinetics, of the particle and/or power balance equations for heavy particles (such as different ions and excited atoms or molecules), and of the Maxwell equations (or a reduced version such as the Poisson equation or appropriate electric circuit equations) to obtain a more complete description of all important plasma components as well as of the internal electric field. This self-consistent treatment is usually tricky and is based on an iterative approach to the solution of the various types of equations involved (Lomagen and Winkler, 1994; Uhrlandt and Winkler, 1996; Yang and Wu, 1996). To integrate the electron kinetic equation in such an approach adequately, a very effective solution procedure for this equation is of particular importance, although remarkable progress with respect to the speed of computation has been achieved in recent years.
VII. Acknowledgments I would like to thank my colleagues Dr. Detlef Loffhagen and Florian Sigeneger for their kind assistance in preparing the figures of this paper.
VIII. References Alves, L. L., Gousset, G., and Ferreira, C. M. ( 1 997). Phy.s. Rev. E 55, 1. Desloge, E. A. (1966). Statistical physics. Holt, Rinehart and Winston (New York), p. 273. Golant, V E., Zhilinsky, A. P., and Sakharov, I. E. (1980). Fundamentals ofplasma physics. John Wiley & Sons (New York), pp. 16, 108, 181. Hayashi, M. (1996). Europh.y.sics Conference Abstracts 20E, Part A, 13; private communications. Itikawa, Y. (1994). In M. lnokuti (Ed.). Cross Section Data, AAMOP Series (vol. 33, p. 253) Academic Press (Boston). Kolobov, V I., and Godyak, V A. (1995). IEEE Trans. Plasma Sci. 23, 503. Kortshagen, U. (1995). Plasma Sources Sci. Technol. 4, 172. Loithagen, D., and Winkler, R. (1994). 1 Comput. Phys. I 12, 91. Loithagen, D., Winkler, R., and Braglia, G. L. (1996). Plasma Chem. Plasma Process. 16, 287. Loithagen, D.. and Winkler, R. (1996). Plasma Sources Sci. Technol. 5. 710. Meijer, I? M. (1991). The Electron Dynamics of RF Discharges. Dissertation, Rijks-University, Utrecht. Petrov, G., and Winkler, R. (1997). 1 Phys. D: Appl. Phjx 30, 53. Petrov, G., and Winkler, R. (1998). Plasma Chem. Plasma Process. 18, 113. Pfau, S., Rohmann, J., Uhrlandt, D., and Winkler, R. (1996). Contrib. Plasma Phys. 36, 449. Phelps, A. V, and Pitchford, L. C. (1985). JILA Injwm. Center Rep. 26, 1. Shkarofsky, 1. P., Johnston, T. W., and Bachynski, M. I? (1966). The particle kinetics of plasmas. Addison-Wesley (Reading, Mass.), pp. 46, 70, 119, 161, 243. Sigeneger, F., and Winkler, R. (1996). Contrih. Plasma Phys. 36, 55 1 . Sigencger, F., and Winkler, R. (1997a). PIasmu Chem. Plasma Process. 17, 1.
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Sigeneger, F., and Winkler, R. (l997b). Plasma Chem. Plasma Process. 17, 281. Sigeneger, F., Golubovskii, Yu. B., Porokhova, 1. A,, and Winkler, R. (1998). Plasma Chem. Plasma Process. 18, 153. Tsendin, L. D. (1995). Plasma Sources Sci. Technol. 4, 200. Uhrlandt, D., and Winkler, R. (1996). 1 Phys. D: Appl. Phys. 29, 115. Wilhelm, J., and Winkler, R. (1979). Journal de Physique, Colloque C7 40, 251. Winkler, R. (1972). Contrib. Plasma Phys. 12, 317. Winkler, R. (1993). In C. M. Ferreira and M. Moisan (Eds.). Microwave discharges-fundamentals and applications, NATO AS1 Series, Series B (vol. 302, p. 339) Plenum Press (New York). Winkler, R. B., Wilhelm, J., and Winkler, R. (1982). Contrib. Plasma Phys. 22, 401. Winkler, R. B., Wilhelm, J., and Winkler, R. (1983). Ann. Ph?/sik (Leipz.) 40, 90, 119. Winkler, R., Braglia, G. L., Hess, A., and Wilhelm, J. (1984). Contrib. Plasma Phys. 24, 657. Winkler, R., Braglia. G. L., H a s , A,, and Wilhelm, J. (1985a). Contrib. Plasma Phys. 25, 351. Winkler, R., Braglia, G . L., and Wilhelm, J. (1985b). XVH-th ICPIG, Budapest, Invited Paper, p. 22. Winkler, R., Wilhelm, J., Braglia, G. L., and Diligenti, M. (1990). N Nuovo Cimento 12D, 975. Winkler, R., and Wuttke, M. W. (1992). Appl. Phys. B 54, 1. Winkler, R., Braglia, G. L., and Wilhelm, J. (1995). Contrib. Plasma Phys. 35, 179. Winkler, R., Sigeneger, F., and Uhrlandt, D. (1996). Pure & Appl. Chem. 68, 1065. Winkler, R., Pewov, G., Sigeneger, F., and Uhrlandt, D. (1997). Plasma Sources Sci. Technol. 6, 118. Yang, Yun, and Wu, Hanming (1996). 1 Appl. Phys. 80, 3699. Zecca, A,, Karwasz, G.P., and Brusa, R. S. (1996). Rivista del Nuovo Cimento 19, I .
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING WL. MORGAN Kinema Research & Softare, L.L.C. Monument, CO I. Dedication.. . . .. . . . . . . . . . . . . . . .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . , . , . , , 11. Introduction . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . , . , . 111. Sources of Data and lnterpretations .. . . . . . . . . . . , , . , . , . , . , .. , . , . .. , . .. , . , . , . . . , . . A. Transport Coefficients . . . . .. . . . . . .. .. . . . . . .. .. . . . . . . . , , .. , . , . , . B. Obtaining Cross Sections from Electron Transport Data.. . . _ _. .., . , , . , , , . .. C. The Traditional Approach in Gas Discharge Modeling . .. . . .. . . . . . . . . . . . . . . D. The Roles of Beam Data, ab Initio Cross-Section Calculations, and Swarm Data. .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . , . . .. E. Verification, Validation, and Confirmation of Numerical Plasma Chemistry Models . . ... . . . .. . . . . . . . . . .. ... . . .. .. . . . . . . . .. .. . .. . . . . . . . . . . .. . . .. IV. Discussion of Data for Specific Processes and Species.. . . . .. . . . . . . . . . . . . . . . . . . A. Cross-Section Sets.. .. . . . . . . . . . . . . . . . . . . .. . . .. .. . . . .. .. . . . . . . . . . . . . . . . . . . . . . .. B. Elastic and Momentum-Transfer Cross Sections . .. . . . .. . . . . . . . . . . . . . . . . .. .. C. Electron Impact Ionization and Dissociation . .. ................ 1. Ionization . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . a. Detachment: Ionization of Negative Ions 2. Electronic Excitation and Dissociation. . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . , . D. Attachment. . . . . . . . . .... . . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . 1. Dissociative Attachment and Vibrational Excitation. . . . . .. . . . . . . . . . . . . . . . 2. Transient Attachment and Long-Lived Negative Ions .. . . . . .. . . . . . . . . . . . . E. Recombination ...... 1. Dissociative ...... 2. Collisional Radiative Recombination . . . . . . .. . . . . . . . .. .. . . .. . . . . . . . . . . . . .. 3. Recombination at High Gas Pressures: Inelastic Enhancement of Dissociative Recombination . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. , . , . , .. . , . , . . . V Concluding Remarks: Journals, Databases, and the World Wide Web . . . . . . . . . A. Issues Concerning the Dissemination of Data.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . B. Useful Web Sites .... .................... VI. Acknowledgements.. . . . . . . . . . .. . . . . . . . .. . . . . . . . .. .. . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . VII. References. . . . . . . . . .. . . .. . .. . . . . . . . .. . . . . . . . . . .. . . . . , . . .. . . . . . . . . . . . . . . . . . . . . . . . ..
79 80 81 82 85 86 86 87 90 90 90 91 92 94 94 97 97 99 101 101 103 103 104 104 105 107 107
I. Dedication I dedicate this article to the late Professor Sir David R. Bates, F.R.S. (19161994), who was the founding editor of the Advances in Atomic and Molecular Physics series and my collaborator and mentor for the last dozen years of his life. David performed the original theoretical work on many of the atomic and 79
Copyright !cj2000 by Acadcrnic Press All rights of reproduction in any form reserved ISBN: 0-12-003843-9/ISSN: 1049-25OX $30.00
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FVL. Morgan
molecular processes that are directly involved in the plasma chemistries being discussed in this book. This list includes radiative recombination, dissociative recombination, collisional radiative recombination, bimolecular and termolecular ion-ion neutralization, and ion-molecule collisions. He was an early enthusiast for the use of digital computers in computing cross sections and rate coefficients for atomic and molecular collisions and in modeling such processes in ionized gases.
11. Introduction The electron collision processes that occur in most laboratory plasmas are listed in Table 1. The 300-K neutral gas pressures in plasma applications (excluding fusion plasmas) range from about 1 mtorr for the inductively coupled plasmas used in microelectronics processing applications to 1 to 10 torr for lighting applications to an atmosphere or more for excimer lasers or corona discharge processing of materials. The relative importance in a particular plasma of the various processes listed in Table 1 will depend upon the species of atoms or molecules present, the neutral gas density and temperature, the average electron energy, the distribution function, and the fractional ionization. A major obstacle to accurate numerical modeling of plasma chemistry has been the lack of reliable electron collision cross-section data. Electron collisions drive the entire processing plasma chemistry and, hence, are among the most important and critical processes that we need to consider. In addition to cross sections for elastic scattering and momentum transfer, to which the electrical and thermal conductivities are directly related, it is necessary to consider all important TABLE 1.
ELECTRON COLLISION Collision type Elastic/momentum transfer Rotational excitation Vibrational excitation Electronic excitation/dissociation Ionization Attachment Dissociative recombination Termolecular recombination Termolecular attachment
PROCESSES IN PLASMAS
Comments Average energy loss equals approximately 2m/M Usually included in elastic because of small energy loss Cross sections often large; typically 0.1-0.25 eV energy loss Metastable excitation and dissociation are important to plasma chemistry May be dissociative Usually dissociative; negative ions are very important in plasma chemistry Often has a large rate coefficient and is an important electron loss process Collisional rahative recombination; important at high pressure May be important at high pressure
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING
81
electron production and loss channels for ionization, attachment, and recombination, as well as inelastic electron energy-loss processes. Because the molecular gases most often used in plasma processing tend to be readily dissociated by electron collisions, densities of atomic and radical dissociation products are likely to be relatively large in high-density, low-pressure processing plasmas, and so electron collision cross sections for molecular and atomic targets are required. For many of these processes, few measurements are available and the body of knowledge is very limited. Theoretical calculations have helped to fill some of the gaps in the database, but for some processes we must rely on estimates that come from either approximate models or analogies to other systems for which data are available.
111. Sources of Data and Interpretations This section consists mostly of a discussion of how we are to interpret and reconcile the electron collision cross sections that we obtain from our three primary, but disparate, sources of such data and how these interpretations are related to the modeling and simulation for which we are assembling the basic collision data in the first place. All our electron impact cross-section data arise from three sources: 1. Electron beam measurements (Christophorou, 1984), in which the textbook scattering experiment is performed whereby an electron beam is passed through a very low density gas a n 4 we hope, either single scattering events are recorded as functions of scattering angle and incident electron energy or the excited or charged scattering products are observed. 2. Ab initio quantum theoretical calculations (Huo and Gianturco, 1995; Winstead and McKoy, 1999), in which Schrodinger’s wave equation is solved in some approximation appropriate to the scattering problem and processes. 3. Electron swarm measurements (Huxley and Crompton, 1974), in which a burst of electrons is observed to drift along an electric field applied to a low-density gas and various transport coefficients, such as the drift velocity, transverse or longitudinal diffusion coefficients, attachment or ionization coefficients, and so on, are measured as functions of the applied electric field divided by the pressure or the gas number density (i.e., E / p or E I N ) ; collision cross sections, which are related to the transport coefficients through Boltzmann’s transport equation (Morgan, 1979; Morgan and Penetrante, 1990), can be extracted by a process of inversion.
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R L . Morgan
Beam collision measurements represent the ideal for us in terms of potential quality of data, but they are the most scarce in terms of quantity. Many early beam measurements were of relative cross sections; nevertheless, they are useful when used in conjunction with calculations or swarm measurements. Ab initio calculations of electron impact cross sections for complex molecules, as discussed by Winstead and McKoy (1999), have become very sophisticated but require enormous computational resources for large molecules. The third technique has been in use for some three decades. There is a very large body of literature reporting on measurements and interpretations of electron transport or swarm coefficients in many of the same gases in which we are currently interested. This is an excellent technique, as I describe below, for estimating cross sections when no other data are available. Whereas the beam and ab initio approaches directly yield cross sections from which we compute the transport coefficients that are used in modeling, Cross sections ==+calculated transport coefficients in the swarm approach, cross sections are themselves derived from measured transport coefficients (usually in carefully controlled experiments), as described by the sequence below: Measured transport coefficients =+cross sections Cross sections ==+calculated transport coefficients On the one hand, the cross sections that are derived from swarm data cannot be expected to possess the accuracy and detailed structure of good beam measurements or ab initio calculations, but, on the other hand, they naturally produce (if the procedure is carried out well) cross-section sets that accurately reproduce the macroscopic observables that are relevant to real plasmas. Such quantities are drift velocities or mobilities, which are directly connected with the power deposition in a discharge plasma, diffusion coefficients, and attachment and ionization coefficients, which are intimately related to the ionization balance of a plasma. These are the quantities that are used directly in most plasma models and that are measured in laboratory plasmas. A. TRANSPORT COEFFICIENTS
The object of modeling plasma chemistry is to relate the external variables, such as discharge power or current, applied voltage, gas composition, pressure, temperature, discharge geometry, etc., which one can adjust in a laboratory device, to properties that are of interest, such as electron and ion densities, densities of excited or radical species, plasma radiation characteristics, etc. The electron impact processes in an ionized gas drive the plasma chemistry. In
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING
83
modeling plasma chemistry, we use our knowledge of the microscopic physics of electron collisions with atoms and molecules to construct macroscopic transport coefficients, which are then used in rate equations and transport equations to predict discharge properties and species densities (Bukowski et al., 1996). The combined continuity and momentum and the energy fluid equations describing the transport of electron density and average energy in a plasma having gas density N,,, electron density N,, and electron temperature T , are
where E is the electric field vector in the plasma and Po is an external electron heating term, such as the period averaged inductive heating due to RF inductive coils external to the plasma (Jaeger et al., 1995). Here re is the electron flux vector and Q, is the heat flux vector, which are given, respectively, by
re= - D V N ~- - EIse
(3)
and 5
Q, = -r,kT, 2
- A,V(kT,)
(4)
where 0' is the electrical conductivity, D is the diffusion coefficient, and A, = $ W e D is the thermal conductivity. R,, the rate of gain or loss of electrons due to ionization, attachment, or recombination, is merely the product of the appropriate rate coefficient (defined below), the electron density, and a molecular density. E, is the rate of energy gain or loss and is equal to the exo- or endothermicity of the reaction times the reaction rate. The definitions of the transport coefficients in terms of the microscopic electron collision cross sections follow. The current density in a plasma is given by 1 j, = crE = eN,p,E = -eN,v, = - - eN, 3
'
(5)
The Greek letter u is used here to denote electrical conductivity. It is also used with the subscripts rn, e, or i to denote momentum-transfer and elastic and inelastic electron collision cross sections, respectively. These are the standard notations.
84
KL. Morgan
where vd is the drift velocity and&(&)is the electron energy distribution function, which has units of eVP3l2and is normalized according to
Since it is usually in the direction of the applied electric field, the drift velocity is usually denoted as the scalar vd. a,(e) is known as the momentum-transfer cross section and is defined by o,(E)
= 2 n / : w ( l
-cosO)sinOdO
(7)
where do,(&, O)/dB is the differential cross section for elastic scattering. The momentum-transfer cross section is also known in transport theory as the diffusion cross section. For a uniform differential cross section, i.e., do,(&, @/do = constant, the elastic and momentum-transfer cross sections are equal, i.e., om(&)= o,(e). When the differential cross section is strongly peaked in the forward direction, om(&)< o,(e), and when it is peaked in the backward direction, om(&)> o,(~). Other transport coefficients are the transverse diffusion coefficient
and the rate coefficients for inelastic collisions
Derived quantities often seen in the plasma literature are 1. The characteristic energy DIP,,
where p, = v d / E is the electron mobility, which has units of energy and is equal to the electron temperature T, when the electrons have a Maxwell-Boltzmann distribution &(E) = (2/&)exp(-s/kTe); D / p e is a measure of how nonMaxwell-Boltzmann & ( E ) is. 2. The ionization and attachment coejicients ci = ki,,N,,/vd and q = k,, N n / v d ,which have units of cm-' and represent the increase or decrease in electron density per centimeter resulting from, respectively, ionization and attachment; these quantities are directly measurable in a drift tube.
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B. OBTAINING CROSS SECTIONS FROM ELECTRON TRANSPORT DATA The process of obtaining electron collision cross sections from electron swarm data involves inserting cross-section models in the collisional terms of Boltzmann’s equation, calculating the distribution function and, hence, the swarm coefficients (Morgan, 1979; Morgan and Penetrante, 1990; Winkler, 1999), altering the model cross sections, and then iterating until an acceptable match between measured and computed coefficients is found. These techniques were developed by Phelps and various collaborators in the 1960s (see Phelps, 1968; Shkarofsky et al., 1966; and Huxley and Crompton, 1974 for reviews of the methodology). One can also use numerical optimization techniques to manipulate the model cross sections (Taniguchi et al., 1987; Morgan, 1991a; Morgan, 1991b; Morgan, 1993). The so-called two-term expansion of Boltzmann’s equation for transport of electrons in a dc electric field is typically used for swarm analysis. If one neglects the spatial and temporal dependence of the distribution function f ( r , v, t ) in Boltzmann’s equation and expressesf = f ( v ) as the first two terms of a spherical harmonic expansion
f(v) = M u )
V
+ U . f ,( u )
(10)
-
the following scalar equation for the energy distribution function f , ( ~ ) (where E = mu2/2) in a gas having temperature Tg and density N,,, and under the influence of a dc electric field E, is obtained:
-(-)*-(- -) 1 eE 3 N,,
+
d E df, dE urn dE
c I
[(E
+
&I)
GI(&
+i{?+)~’[f,(.,+kT
-
dE
+
El)
M E+
El)
- E C,(E)
h(41 = 0
The transport equation for electrons is usually written in terms of electron energy because it is the energy loss E, that is quantized in inelastic collisions between electrons and atoms or molecules. Here it is assumed that the populations of the excited levels, labeled by i, are small enough that superelastic collisions and transitions among excited states are unimportant. If that is not the case, there is an additional sum over excited states having terms [ ( E - E , ) CJ,(E - E , ) & ( E - E , ) E o , ( E ) ~ , ( E ) ] ,which are multiplied by the fractional populations of the excited states. The electron collision cross sections involved are the momentum-transfer cross section G,(E) and the set of inelastic cross sections G , ( E ) for transitions from the ground state to the various excited states i. Note that the electric field and gas density enter only through the term containing EIN,,. This equation does a remarkably good job of describing the transport of electrons under the influence of an electric field in most gases. There
+
86
WL. Morgan
are, of course, more sophisticated treatments that go beyond the simple two-term spherical harmonic expansion. See the article by Winkler in this book for hrther details on Boltzmann’s equation and its solution.
c. THE TRADITIONAL APPROACH IN GAS DISCHARGE MODELING Most applications of swarm-derived cross sections in gas discharge modeling use what amounts to an effective momentum-transfer cross section. This is obtained by fitting calculated and measured electron swarm data and may include sizable inelastic contributions (Pitchford and Phelps, 1982). For this reason, the cross sections that are derived from swarm data may not agree with those that are separately measured in beam devices or computed using ab initio quantum techniques. The general wisdom is that this approach works well because the cross sections are derived by fitting exactly the same coefficients, i.e., drift, diffusion, ionization, attachment, etc., as are used in fluid models of discharge plasmas. This leads to a self-consistency that has worked well over the years. D. THEROLESOF BEAMDATAAB INIT10 CROSS-SECTION CALCULATIONS, AND SWARM DATA Modern high-vacuum beam measurement techniques and modern ab initio multichannel quantum calculations performed on supercomputers can provide very accurate cross sections for low-energy elastic and inelastic collisions. Often, however, when such data are assembled into a model for a molecule and transport calculations are performed, the agreement with measured transport, i.e., swarm, coefficients is poor. This leaves us somewhat dubious about the value of modeling a processing reactor using very detailed and correct cross sections if the model does not reproduce very accurately the plasma measurables in a well-defined, well-controlled swarm experiment. An easily conceivable example of this is one where we assemble a detailed model using what we consider to be accurate cross sections from disparate sources and find that the computed Townsend ionization coefficient CI differs from that measured in a drift tube by an order of magnitude or more. There are two reasons for such disagreements. First, the individual and independent errors (both in magnitude and in energy dependence) in the separate cross sections from different sources conspire to produce a possibly sizable overall error. Second, including all known cross sections does not necessarily mean including all possible collision processes. This is somewhat analogous to the missing matter problem in cosmology: All that we know may be only a fraction of what there is. This is where swarm analysis can make a very important contribution. By their nature, swarm-derived cross sections include all possible processes, either explicitly as individual cross sections or implicitly contained
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 87
within other cross sections. This is another reason why swarm-derived cross sections often differ from beam measurements and calculations. The best procedure for dealing with the potential problem of a collection of cross sections producing erroneous plasma transport coefficients is to 1. Assemble the most complete models that we can, using data from the sources discussed above. 2. Perform swarm calculations for conditions appropriate to transport measurements when such data are available. 3. Systematically renormalize the cross sections in order to reproduce the measured transport coefficients.
E. VERIFICATION, VALIDATION, AND CONFIRMATION OF NUMERICAL PLASMA CHEMISTRY MODELS Anyone who performs plasma chemistry modeling recognizes that cross sections and rate coefficients for an individual process are rarely used in isolation; rather, they are generally used as part of a model involving a number of other processes. An example of this can be seen in the schematic model for C2F6 plasma chemistry shown in Fig. 1. While not much is known about C2F6 plasma
FIG. 1. Schematic diagram of a plasma chemistry model for C2F,.
88
WL. Morgan
chemistry, this is a reasonable guess as to what the kinetic scheme might look like, although there other possibilities as well. Key parts of the model are the cross-section sets for C2F6 and for the radicals that are formed when it is dissociated by electron impact. The first step in constructing a plasma chemistry model in which one might have some confidence is to assemble a set of electron collision cross sections that can be shown to yield transport coefficients that are in agreement with swarm measurements. Such a cross-section set was assembled for C2F6by Hayashi and Niwa (1987) and is shown in Fig. 2. The origin of the various cross sections is described by Hayashi and Niwa as follows: 1. Momentum transfer om: The cross section was taken from those for CH4, C2H6, and CF4 and improved by analysis of swarm data. 2. Vibrational excitation or:Born theoretical cross sections were used as the starting point and then were improved by swarm analysis. 3. Dissociation o d n : The measured total dissociation cross section a d of Winters and Inokuti (1982) was used, and rsdn was determined from Cd,, = o d - of. 4. Electronic excitation oe:The two electronic excitation cross sections were determined by analysis of swarm data. 5 . Ionization oi:The cross section measured by Beran and Kevan (1969) was scaled by a factor of 0.79 based on swarm analysis. 6. Dissociative attachment a*:The cross section measured by Harland and Franklin (1974) was scaled by a factor of 2.2 1 and was made to coincide at 4eV with the cross section derived by Hunter and Christophorou (1984). This swarm analysis made use of the attachment and ionization coefficients and the dnft velocity. Generally, however, the best swarm-derived cross sections are obtained when the transverse diffusion coefficient or the characteristic energy D / p , (equal to T, for a Maxwell-Boltzmann distribution function) is used as well as the drift velocity. ud and D sample different parts of the electron energy distribution function & ( E ) and have different sensitivities to the momentumtransfer cross section om(&)in different energy ranges. Hence the most accurate cross sections are obtained when both of these transport quantities are available. Hayashi and Niwa made use of Hunter’s D / p e data, but, unfortunately, the measurements were performed in a mixture of 90% CH4 and 10% C2F6. Since the cross sections for methane are not very accurate, the uncertainty in the cross sections derived from the characteristic energy measured in this mixture can be expected to be large. Picasso said that “art is the lie that helps us see the truth.” The same might be said of models and modeling. Oreskes et al. (1 994) have discussed the concepts of verification, validation, and confirmation as they apply to numerical models of
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 89 100
h
10
N€! c
z
Y
#
I
rn u)
3
0
0.1
0.01
0.01
0.1
1
-
Momntum transfer Vibrational excitation
--o --c
Dissociation
o
10
100
Energy (eV)
*--
-o . Electronicexcitation --+ .
+ ionization A
0
Dissociative attachment vatretch; Takagi eta/. (19%)
FIG. 2. Cross-section set for CzFh assembled by Hayashi and Niwa (1987); the vibrational excitation cross sections measured by Takagi et al. (1994) are also shown.
physical phenomena. They point out that verification, a demonstration of truth, is not possible with numerical models. Confirmation, the framing of empirical observations as deductive consequences of a general theory, is not applicable to modeling. The authors regard a claim that a model has been verified because its results match observed data as being the logical fallacy of “confirming the consequent.” The best that one can achieve is validation, or establishment of legitimacy, of models. That is, the model does not contain known flaws and is internally consistent, but it is not necessarily a representation of reality. The C2F6cross-section set shown in Fig. 2 provides very good agreement with measured swarm data. Despite this, there are more recent data of higher quality than those used to construct this cross-section set that, as shown below, differ substantially from these cross sections. Using these more recent data, one can construct (Morgan, 1998) a new C2F6cross-section set that is different from that shown in Fig. 2 but that also agrees well with the swarm measurements. Hence, we must, in dealing with models and modeling, be aware of and live with the
90
KL. Morgan
uncertainty and lack of uniqueness of our models. This, however, should not hinder us in gaining valuable insights by constructing and using models.
IV. Discussion of Data for Specific Processes and Species A. CROSS-SECTION SETS Table 2 lists a number of atoms and molecules for which cross-section sets have been compiled or detailed cross-section reviews have been written. In Table 2, unless otherwise noted, the cross-section sets have been evaluated by comparison with electron transport data.
B. ELASTIC AND MOMENTUM-TRANSFER CROSS SECTIONS Figure 3 shows the momentum-transfer cross section for C2F6 derived from swarm data by Hayashi and Niwa (1987) and the recent beam measurements by Takagi et al. (1994). Clearly the differences are substantial. Very recent unpublished calculations by Winstead and McKoy (1997) show much more structure than the measurements but agree quite well (see McKoy et al., 1998 for a comparison of differential cross sections) with the Takagi et al. results, especially for the momentum-transfer cross section. These large differences in the momentum-transfer cross section imply, of course, that when constructing a cross-section set that will agree with the measured transport coefficients, one must modify other cross sections to bring things back into agreement. An excellent example of the uncertainty that can be found in elastic or momentum-transfer cross sections is shown in the review of CF4 cross sections by Christophorou et al. (1996). Elastic cross sections are difficult to measure using beam techniques below about 1 or 2 eV. At energies greater than 2 eV, most beam measurements in CF4 are in good agreement, and theory also agrees well with these results. Below 2 eV, however, there are no absolute beam measurements, the theoretical results show no Ramsauer minimum, and the swarm-derived cross sections show a deep Ramsauer minimum but differ by nearly two orders of magnitude in the value of the cross section at the minimum. Recent ab initio calculations by Isaacs et al. (1998a) lend, for the first time, theoretical support for the Ramsauer minimum in CF4 elastic scattering. Generally, when one is constructing a cross-section set, one finds it necessary to make use of swarm results at low energies and measurements and theory at intermediate and high energies because the swarm-derived cross sections become more uncertain as inelastic channels open up.
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 9 1 CROSS-SECTION SETS
TABLE 2. AVAILABLE FOR VARIOUS ATOMSAND MOLECULES
Species
Reference Phelps and Pitchford (1985); Hayashi (1987); Itikawa ei al. (1986)" Phelps (1985); Itikawa et a/. (1989)" Buckman and Phelps (1985); Tawara et a/. (1990)" Land (1978) Lowke et al. (1973); Hayashi (1990) Hayashi (1987); Yousfi and Benabdessadok (1 996) Hayashi (1990) Hayashi and Niwa (1987) Hayashi (1 987) Hayashi ( 1990); Yousfi and Benabdessadok (1996) Rockwood ( 1973); Hayashi ( 1990) Laher and Gilmore (1990)".'; Itikawa and Ichimura (1990)".h Morgan (1 9 9 q b Morgan and Tischenko (1997) Hayashi (1990) Hayashi (1990) Hayashi (1987) Morgan (1 992b)" Morgan (1992b)"; Penin et al. ( 1996) Hayashi (1987) Morgan (1992a); Morgan et al. (1998) Morgan (1992a); Morgan (1998) Morgan (1992a) Phelps and Van Brunt (1988) Morgan (1992b)d; Bordage et al. (1996); Christophorou et al. (1996)" Hayashi and Niwa (1987); Christophorou and Olthoff (1998a)" Christophomu et al. (1997a)" Christophorou et al. (1997b)" Nagpal ei al. (1995); Nagpal and Garscadden (1996)'; Morgan (1998) Nagpal and Garscadden (1994)c; Morgan (1998)c Morgan (1998) Morgan (1998) Hayashi (1998) Jeon and Nakamura (1998) "
Comprehensive review without swarm analysis.
'No swarm analysis because of lack of swarm data. Partial results. of previous work.
" Review
c. ELECTRONIMPACT IONIZATION AND DISSOCIATION Dissociation, attachment, and ionization are among the most important inelastic electron collision processes that one needs to be able to treat in plasma chemistry modeling. The dissociation cross section is very difficult to measure directly but
92
EL. Morgan
40
t Elastic cross section - Takagi ef a/. (1994) Momentum transfer cross section i b d - -0- Momentum transfer cross section - Hayashi and.Niwa (1987) ~
01 1
10
100
Energy (eV) FIG. 3. C2F, elastic and momentum-transfer cross sections.
can often be inferred from electron swarm data. The attachment rate, from which a cross section can be deduced, is easy to measure in a swarm apparatus. Ionization cross sections are relatively easy to measure using beam techniques. As discussed below, one can make use of beam-measured ionization cross sections and swarm-measured ionization coefficients in order to extract valuable information about the electronic excitation and dissociation cross sections.
1. Ionization
Perhaps the first comprehensive set of measurements, which is still widely used, of the ionization cross sections for a large number of gases, H2,
Dz,He, Kr, CO, N20, CH4, Hz, Ne, N2, Ar, COZ, Xe, SF6, C A
was performed by Rapp and Englander-Golden (1965). Beran and Kevan (1969) measured the ionization cross sections at 70eV (and at 35 and 20eV in some cases) of 62 gases. These were mostly organic and halogenated organic molecules. Since then, Freund and his collaborators (Freund, 1987; Wetzel et a f . ,1987; Hayes et al., 1987; Shul et al., 1988; Hayes e t a f . ,1988; Shul et al., 1989; Freund et a f . , 1990a; Freund et a f . , 1990b; Shul et a f . , 1990) have performed ionization cross-section measurements on many gases, as have Becker and his collaborators (see Becker, 1999, and references contained therein; Becker, 1994; Becker and Tarnovsky, 1995). Unlike their predecessors, Freund’s and Becker’s groups have
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 93
measured ionization cross sections of radical fragments as well as the parent molecule. The trend in recent years has been not only to provide total ionization crosssection data but to measure the partial cross sections for dissociative ionization. An example is shown in Fig. 4 for C2F6.These are important data for modeling in that they show that the dominant ionization product is CF;. Figure 4 also shows the ionization cross section, which is nearly indistinguishable from the measured total cross sections, computed by Nishimura et al. (1998) using the binary encounter Bethe (BEB) technique (Kim and Rudd, 1994; see also Winstead and McKoy, 1999). This technique, which is relatively straightforward, provides generally accurate total ionization cross sections. Kim and his collaborators (Kim and Rudd, 1994; Hwang et al., 1996; Kim et al., 1997a; Kim et al., 1997b; Ali et al., 1997; Nishimura et al., 1998) have used this theory to compute the ionization cross sections for a large number of molecular species: C2F6, C2H2, C2H4, C2H69 C3FX3 C3H8, C6H6 CF4, CH, CH2, CH3, cH4 co, c02, c o s , cs, cs2 Ge2H6, GeH, GeH2, GeH3, GeH4 H, H2, H20, He, He+, Li++ N2, N20, NH3, NO, NO2, Ne, 0 2 , 0 3 , s2, SF6, so2 Si(CH3)4, Si2H6, SiF, SiF2, SiF3, SiH, SiH2
10
-?
/bid -CF'
-a
ibrd -Total
0.01
0
20
40
60
Energy (eV) FIG. 4. Total and partial cross sections for ionization of CzF6.
80
100
94
KL. Morgan
These cross sections are all presented on his Web site, which is mentioned at the end of this article. The BEB technique does well on ionization cross sections of some of these radicals that have been measured leading us to expect that it may provide reasonable estimates of the total ionization cross sections of radical species for which we have no measurements. a. Detachment: Ionization of Negative Ions. A process that can be important in electronegative plasmas of high fractional ionization is collisional detachment of negative ions. The detachment cross sections for H-, C-, 0-, and F- were systematically measured by Peart et al. (1970, 1979a, 1979b, 1979c) in the 1970s. More recent measurements (Vejby-Chnstensen et al., 1996) on 0- and D- using an ion storage ring have investigated the near-threshold behavior and the possible existence of resonances, which were not found. Pindzola (1996) has performed distorted wave calculations on detachment of 0- and D- and has gotten reasonable agreement with the storage ring measurements. He did find however, that his results are very sensitive to the choice of polarization potential. Fortunately, useful simplifications can be found. Robinson (1965) describes a modification of the classical impulse approximation using Slater’s rules to estimate the average kinetic energy of the bound electron. Esaulov (1980) notes that a plot of IJ(E)E:versus E I E ~where , E, is the electron affinity of the neutral atom, provides an approximate universal curve for the detachment cross section. 2. Electronic Excitation and Dissociation
One of the most important features of using plasmas for processing is the copious quantity of radical fragments created by molecular dissociation. The ability to predict the densities of radical species in a plasma is one of the major goals of modeling. Because large molecules have a very great number of electronic excitation channels and many repulsive potential curves, once an electron excites an electronic transition, there are many paths available that lead to dissociation. For big molecules, one can expect that most electronic excitation will lead to dissociation. Measurement of the cross sections for electron impact dissociation of molecules into neutral ground-state fragments is, unfortunately, difficult. There have been relatively few such measurements published in all the electron collision literature. Although cross sections for dissociative excitation of molecules are relatively easy to measure, such processes often have much higher excitation thresholds and smaller cross sections than dissociation into ground-state fragments. The difficulty of neutral dissociation cross-section measurements can be seen in the Christophorou et al. (1996) review of CF4, where the neutral dissociation cross-section measurements span an order of magnitude in peak value, have very
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING
95
different energy dependences, and fall well below the cross section deduced from swarm data. Similarly, in their review of CHF3 cross sections, Christophorou et al. (1997a) estimate a total neutral dissociation cross section by subtracting the measured total ionization cross section from the total dissociation cross section (i.e., including dissociative ionization) measured by Winters and Inokuti (1982) and find that it is much larger than and has a completely different energy dependence from the neutral dissociation measurements published by Goto et al. (1994) and by Sugai et al. (1 995). Although such a subtraction potentially has quite large errors associated with it, the striking differences are not due to such errors. Modern sophisticated electron-molecule collision theory, like that described by Winstead and McKoy in this volume, can be very usefd in providing insight into the magnitude and energy dependence of dissociation cross sections. Such calculations have been performed, for example, for NF3 by Rescigno (1995) and represent some of the best data that we have for that molecule. Similarly, McKoy et al. (1998) have computed a dissociation cross section for CHF3 and have obtained a result that is much more like that deduced by Christophorou et al. (1997a) in the review mentioned above than like the cross sections measured by Goto et al. (1 994) and by Sugai et al. (1 995). Perhaps the best means of obtaining a realistic value for a composite dissociation cross section is to make use of swarm data and ionization crosssection data. The procedure works as follows. One assembles a set of low-energy cross sections and adjusts it to agree with drift velocities, characteristic energies, and attachment coefficients, if the attachment cross section is large. Then one postulates an electronic excitation-dissociation cross section having the correct threshold and energy loss. One can either use a model cross section and adjust the magnitude or adjust the energy dependence and magnitude of the dissociation cross section in order to match the ionization coefficient. This is very effective if one has good ionization cross sections because the ionization coefficient 1
1
cx - (T,(E)J;)(E) E ud
~
dE/
~
depends not only on the ionization cross section (presumably well known) but on the distribution function f0(~),which is, via Boltzmann’s equation, a function of the cross sections at energies less than E . This is how Hayashi and Niwa (1987) arrived at the dissociation cross sections for C2F6 shown in Fig. 5. The utility of such techniques can be seen in Fig. 6, where I have plotted various measured and swarm-derived cross sections for electron impact dissociation of molecular nitrogen. Winters (1966) and Cosby (1993) have measured the cross section for dissociation of N2 in electron collisions. Winters measured the total dissociation cross sections [as we have seen previously in the Winters and Inokuti (1 982) measurements of dissociation of C2FB] and subtracted the
JKL, Morgan
96
Winters and- lnokuti (1982)
/=-- '
Ionization -Total Cross Section
$--* Hayashi and Niwa (1987)
10
100
Energy (eV) FIG. 5 . Total dissociation cross section (Winters and Inokuti, 1982; Hayashi and Niwa, 1987), total ionization cross section, and swarm-derived dissociation cross sections for C2F6.
- Cosby (?992)1 rewmmended - Phelps and Pitchford (1985) - sum of singlet states - . 0.75 x Phelps 8 Pitchford cross section 1.
0
50
100
Energy (eV) FIG. 6 . Dissociation cross sections for N2,
150
200
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 97
measured total ionization cross section. Cosby used a crossed beam apparatus with microchannel plate detection to measure the dissociation cross section directly. Also shown in Fig. 6 along with Cosby’s recommended dissociation cross section is the total cross section for the sum of the singlet excitations used by Phelps and Pitchford (1 985) in their cross-section set. This last multiplied by a scale factor of 0.75 is also plotted in Fig. 6. In addition to the CF4 and C2F6 cross sections already mentioned, Winters and Inokuti (1 982) measured total dissociation cross sections for CHF, and C3F8. Similarly, in addition to Nz, Cosby and Helm (1992) measured dissociation cross sections for 0 2 , CO, CO2, Cl2, and NOz. Motlagh and Moore (1998) recently measured absolute total dissociation cross sections for the production of CF,, CH3, CzF5,CH2F, and CHF2 radicals from CH4, CF4, CHF,, CH2F2,CH3F, C2F6, and C3F8 for electron impact energies between 10 and 500eV When good measurements of dissociative ionization cross sections are available, total dissociation cross-section measurements, such as those of Motlagh and Moore, can be invaluable. D. ATTACHMENT Formation of negative ions is often very important in processing plasmas. It is very common in plasmas containing halogenated molecules. Common examples are C12, F2, NF3, BCl,, HCl, HBr, HF, and cyclo-C4F8 as well as oxygencontaining molecules such as O2 itself, H20, SO2, and N 2 0 . A terrific amount of work in measuring the attachment rates and cross sections for a very large number of molecules was done at the Oak Ridge National Laboratory over a 25-year period beginning in the mid-1960s. This work through the early 1980s has been summarized in a book by Christophorou (1984). Smith and Spanel (1994) and Chutjian et al. (1 996) have reviewed recent advances in attachment theory and experimental techniques. I . Dissociative Attachment and Vibrational Excitation A number of molecular species, such as NF3, HC1, HBr, HF, and N 2 0 , exhibit strong dependence of attachment rate on temperature. This comes about because the dissociative attachment cross section increases with increasing vibrational quantum number (see Christophorou et al., 1994). This can be seen in the temperature dependence of the N 2 0 dissociative attachment cross section shown in Fig. 7a, and its effect on the attachment rate coefficient can be seen in Fig. 7b. The attachment cross section (Christophorou et al., 1971), where the products are NO 0 - , is very temperature-dependent (Chantry, 1969), as shown in Fig. 7b, which means that it is very sensitive to the degree of vibrational excitation. In a plasma, one does not need an elevated gas temperature to populate the molecular
+
98
WL. Morgan
vibrational levels. In N20, for example, the lower vibrational levels have energies of 0.073, 0.159, and 0.276eV with statistical weights of 2, 1, and 1, respectively. They would be expected to be populated at a fairly high vibrational temperature in a plasma as a result of electron collisions. At atmospheric gas pressures, the vibrational levels may be depopulated because of vibrational-to-translational energy transfer processes. At pressures on the order of torr, the vibrational temperature can be elevated as a result of anharmonic pumping. At the very low millitorr pressures found in inductively coupled processing plasmas, the vibrational temperatures may be expected to be on the order of the electron temperature. As we have seen, in some gases this can lead to very large dissociative attachment rates. Christophorou and Stockdale (1968) published a diagram that is sometimes usefit1 in assessing whether or not a given molecule might be expected to have a
J
lWO 100
I 0
2
4
6
8
I
Electron Temperature (ev)
FIG. 7. (a)Temperature dependence of effective dissociative attachment cross section in N20; (h) temperatwe dependence of dissociative attachment rate coefficient.
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING
105
.
a I
99
.I
,.
0.01
0.1
1
10
Energy of Cross-Section Maximum (ev)
FIG. 8. Correlation between the peak values of dissociative attachmentcross sections and energies of the cross-section maximum.
significant dissociative attachment cross section. Figure 8 plots the peak value of the dissociative attachment versus the energy of the cross-section maximum for a large number of molecules. The curve turns down above about 5 eV as a result of the competing channels of electronic excitation. Using thermochemical data and electron affinities, one can use this correlation diagram to very roughly estimate the possible significance of dissociative attachment in a particular molecule.
2. Transient Attachment and Long-Lived Negative Ions Some molecules, such as BC13 and c-C4Fs, are able to attach an electron and form a long-lived negative-ion complex (Christophorou, 1978). The interpretation of electron attachment data is made difficult in such gases because the measured attachment coefficient q / N becomes a function of pressure as well as of the size of the experimental apparatus. In the following I will describe attachment issues in BC13. There have been four quantitative measurements of the attachment cross section or the attachment rate coefficient in BC13 since 1959: 1. Buchel’nikova (1959) performed a direct measurement of the electron attachment cross section for energies between 0 and 1 eV using an electron beam apparatus.
100
KL. Morgan
2. Stockdale et al. (1972) performed a measurement of the attachment rate coefficient by thermal electrons in an N2/BC13 mixture. 3. Petrovic et al. (1990) measured the rate coefficients for low-energy electron attachment to BC13, as functions of E / N , electric field divided by gas number density, in an electric discharge in an N2/BC13 mixture. 4. Tav et al. (1998) very recently measured the attachment rate of electrons in an N2/BCI3 mixture and derived an attachment cross section. The Stockdale et al. rate coefficient measurements were performed at gas pressures in the range from 5 to 15 torr, and those of Petrovic et al. over the range 100 to 400 torr. The rate coefficient was observed to be independent of pressure in both measurements. The Tav et al. measurements were performed at 700 ton. The possible plasma chemical processes involving BC13 in these measurements are
1. 2. 3. 4.
+
e BCl, + BCl,* BCl,* -+ BCI, e BCl,* N, .+ BCl, N, e BCl, + BC1, C1-
+
+
+
+
+
Here BC1;* is the metastable negative ion state that eventually autodetaches or is stabilized by a collision with a molecule of the background gas. The electron affinity of BC13 is 0.33 eV, which is some 13 kT at 300 K. The energy threshold for the fourth dissociative attachment process is about 1 eV. The relative cross section for this process has recently been measured by Jiao et al. (1997). Looking at the correlation curve shown in Fig. 8, we would expect the dissociative attachment cross section to be relatively small. Tav et al. estimate the cross section to be smaller than about 5 x lo-'* cm2, which is the smallest cross section that can be detected with their apparatus. Figure 9 shows the attachment cross section that I derived (Morgan, 1997) fiom Petrovic's data along with that derived by Tav et al. from their data, that measured by Buchel'nikova, and the raw data from a relative measurement by Olthoff (1985). Based on his experimental geometry, Olthoff estimated the autodetachment lifetime of BC1;* to be about 60 ps. Since Buchel'nikova performed her measurements nearly 40 years ago, before the advent of modem high-vacuum techniques, her measurements are suspect. Christophorou and Stockdale (1968) have noted that there are large discrepancies between her measured cross sections for a wide variety of molecules and more recent measurements. Figure 9 also shows the BC13 momentum-transfer cross section recently computed by Isaacs et al. (1998b), which shows a sharp temporary negative-ion resonance at 0.25 eV. The attachment cross section for the formation of BCI,* has several consequences for plasma chemistry as well as for the interpretation of the other swarm
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 101
0 Morgan (1997) -0- Buchel'nikova (1959) Olthoff (1985) raw data; relative 0 eta/. (1998) lsaacs eta/. (1998)
+Tav
-
-
0.1
0.0
0.4
0.2
0.6
0.8
1.o
Energy (eV) FIG. 9. BC13 attachment and momentum-transfer cross sections.
measurements and derived cross sections. With regard to the plasma chemistry of plasma processing discharges, the cross section and rate coefficient are large enough that, even without the formation of C1- and Cl;, we can expect a reasonable negative ion density. The 60 ,us autodetachment lifetime of BCl;* is long enough for it to be stabilized to BCl; by collisions with neutral atoms and molecules even at a pressures of several milliton: Because the attachment cross section is large, overlaps the BC13 vibrational excitation thresholds, and lies in the energy range of the Ramsauer minimum of Ar, it and the quite large momentumtransfer cross section may be expected to play a crucial role in the derivation of BCl, inelastic cross sections from drif? velocities measured in Ar/BC13 gas mixtures as has been done by Mosteller et al. (1993) and Nagpal and Garscadden (1 994).
E. RECOMBINATION I. Dissociative Recombination
In most molecular gases, dissociative recombination (Bates, 1950; Bardsley, 1968; Biondi, 1976; Biondi, 1982) AB++e-+
A+B
102
WL. Morgan
is the dominant volumetric electron loss process. The dissociative recombination cross section is a product of two factors: Od,.(E) = 6,.(E)S where 6, a 1 / is ~ the capture cross section and S is the survival factor, which is related to the curve-crossing probability. The rate coefficient at thermal energies, except in those rare cases for which there are no accessible crossings of the ionic and predissociating repulsive neutral curves, is typically in the range lop7 to cm3/s with a dependence on electron temperature of T;'I2. The dissociative recombination rate coefficients have been measured for a number of molecular ions, although few of the ions are species that we expect to find in the processing discharges of interest here. Dissociative recombination is a curve-crossing process and, typically, has large rate coefficients, on the order of lo-' to 4 x lop7 cm3/s (or even much greater for molecular clusters) at 300K. The following table (Mitchell, 1990) lists the rate coefficients for species that might be of interest in plasma processing. Species H ~ (= o 0) N:(u = 0) 0:
co+ co:
H2Of OH' NO+ NH; NH; NH+ CH,f CH,f CH; CH+ C2H+ C2H,+ C2H,f C3G c3Hf C4H,+ C2Hf
k,. (cm3/s) 1.6 x 10-8(300/T)0.43 3.6 x 10-X(300/T)".42 1.95 x 10-7(300/T)0.7 1.0 x 10-7(300/T)".46 3.8 x 10-7 3.15 x 10-7(300/T)0.s 3.75 x 10-8(300/T)0.S 4.3 x 10-7(300/T)0.37 3.1 x 10-7(300/T)0.s 3.0 x 10-7(300/T)0.s 4.3 x 10-8(300/T)0.5 3.5 x 10-7(300/T)0.S 3.5 x 10-7(300/T)0.s 2.5 x 10-7(300/T)0.s 1.5 x 10-7(300/T)0.42 2.7 x 10-7(300/T)0.s 2.7 x 10-7(300/T)0.s 4.5 x 10-7(300/T)0.s 1.0 10-7 3.5 10-7 6.2 10-7 7.4 x 10-7
Such rate coefficients are usually measured in thermal systems, where the electron temperature is equal to the gas temperature. In modeling plasmas, one
ELECTRON COLLISION DATA FOR PLASMA CHEMISTRY MODELING 103
typically uses the electron “temperature” T , = 2(6)/3, instead of the gas temperature T. Vibrational excitation can affect the overall rate coefficient and the temperature dependence. For example, when all vibrational levels are included the dissociative recombination rate coefficients for H2 and N2 are 2.3 x 10-7(300/T)o.4and 1.8 x 10-7(300/T)”.39,respectively, i.e., a number of times larger than the rate for the lowest ( u = 0) vibrational level.
2. Collisional Radiative Recombination At high gas pressures, the dominant volumetric electron loss process is often what is known as collisional radiative recombination. This is a multistep process whereby an electron is captured by an ion and collisionally stabilized into a Rydberg state by either a neutral or a charged third body. Further collisions on average drive electrons to lower levels, where, at some particular level called the bottleneck, radiative rates exceed collision rates and electrons drop down to a stable ground state. A number of authors, beginning with Bates, Kingston, and McWhirter (1 962), have modeled this process in detail, using manifold levels, cross sections, and radiative rates. For many practical purposes, the process can be simplified into a single formula for the net recombination rate as a function of gas density, electron density, and electron temperature (Flannery, 1996): e-
+ A+ + e-
-+
A
+ e- : k = 2 x
+M
-+
A
+M : k =
e- + A +
cmh/s
where p is the reduced mass in amu of the molecular collision partners and T, is the electron temperature in eV 3. Recombination at High Gas Pressures: Inelastic Enhancement of Dissociative Recombination It was found experimentally by Warman et al. (1979) and by Armstrong et al. (1982) that in some molecular gases at pressures ranging from 100 ton to several atmospheres, the dissociative recombination rate coefficient is enhanced by termolecular effects. This was simulated for electrons in HzO, NH3, and COz by Morgan and Bardsley (1983) and by Morgan (1 984a, 1984b) using numerical Monte Carlo techniques. The effect comes about because of the large low-energy inelastic collision cross sections in these gases. H 2 0 and NH3 are both strong dipoles having very large rotational excitation cross sections, and C 0 2 has a large vibrational excitation cross section at 0.083 eV 2: 3 kT. When an electron has a collision with one of these molecules while in the vicinity of a molecular ion, it
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can lose a significant enough amount of energy to enhance its probability of recombining dissociatively with the molecular ion. This process has been called collisional dissociative recombination by Bates (1980, 1981). At present there is no simple general theory of the process.
V. Concluding Remarks: Journals, Databases, and the World Wide Web A. ISSUESCONCERNING THE DISSEMINATION OF DATA As can be seen from the bibliography, most data discussed in this article are available either in the standard physics and chemistry journals or in conference proceedings, which may be more difficult to obtain than are journals. A number of the works cited in this article are at present unpublished but sooner or later will appear in the literature. Comprehensive review papers are rare in this field and there is no single data center or repository that covers the field. There once was such a data center at the Joint Institute for Laboratory Astrophysics (JILA) in Boulder, Colorado, but it closed in the mid-1990s after more than 25 years of operation. At the time of its closing, the JILA Data Center had a computerized database and library of papers on microfiche that comprised more than 22,000 entries. Operating such a data center is a formidable enterprise that, it appears, few organizations are willing to support. Fortunately there were a number of reports and bibliographies that were published by the JILA Data Center that are still very useful today, although they are becoming increasingly difficult to obtain as time goes on. It appears that the World Wide Web is fast becoming a kind of global database that is readily accessible from anywhere in the world by anyone having an account on the Internet. In a panel discussion at a recent conference on atomic and molecular data, Smith et al. (1998) pointed out that the “classic database, a book with limited availability containing data critically evaluated by experts, is gradually becoming obsolete.” As a forum for dissemination of atomic and molecular data, this trend of posting data on the Web is not without controversy. There are frequent discussions -at conferences [see, for example, Mohr and Wiese (1998)l and in journals and trade magazines-of the use and possible misuse of the Web. Indeed, in a mixed state of optimism and pessimism, Heller (1996) entitled one such discussion “Chemistry on the Internet -the Road to Everywhere and Nowhere.” The two major concerns about the Web as a universal database seem to be 1. The lack of standards and review and assessment by experts in the field 2. The often transient nature of Web sites, which makes true archiving and future referencing of data problematical
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Despite these valid concerns, the Web will continue to increase in importance as the medium of choice for both formal and informal dissemination, exchange, and even publication of data. People have always exchanged data informally and have always been aware that, even when using data from refereed publications, they must beware of bad data. Publication of data in a refereed journal is no guarantee that the data are of high quality or even correct. The stories are legion of papers that accumulate a large number of citations because they have significant errors or are wrong. That Web sites posting data may be short-lived is, in many ways, no different from publishing data in proceedings or, for that matter, even books. Indeed, I have cited in this article a number of sources of data that, frankly, may be difficult for some readers to obtain.
B. USEFULWEB SITES There are a number of sponsored data centers in the world that deal, in varying degrees, with the kinds of atomic and molecular data that would be of interest to the users of this book. They all have found the Web to be the dissemination medium of choice. In addition, there are a number of informal sites that I list here as well. Despite the potential deficiencies referred to above, the Web has become the dominant medium for data dissemination and exchange by scientists worldwide, and this article cannot be complete without a listing of some of the important data sites on the Web. Even if the URLs (universal resource locators) of the sites change or disappear altogether, I have listed the institutions responsible for the sites. Just as people interested in the numerical solution of Boltzmann’s equation have been able to locate me 20 years after JILA Report No. 19 (Morgan, 1979) appeared, despite the moving around that I have done in that time, the authors of these Web sites may be locatable for some time to come.
1. “Databases for Atomic and Plasma Physics,” maintained by Yuri Ralchenko, Weizmann Institute of Science:
http://plasma-gate.weizmann.ac.il/ This is a list of hyperlinks to some three dozen atomic physics and plasma physics databases or data collections worldwide that reside on the Web. 2. The Oak Ridge National Laboratory (ORNL) Controlled Fusion Atomic Data Center: http ://www-cfadc.phy. ornl .gov This database is primarily oriented toward multiply charged species (Schultz et al., 1998).
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3. NIFS Database, National Institute for Fusion Science, Toki, Japan:
http://dbshino.nifs.ac.jp This database contains a large amount of atomic and molecular data in a number of different categories (Murakami et al., 1998). 4. GAPHYOR Data Center, UniversitC Paris-Sud, Orsay, France: http ://gaphyor. 1pgp.u-psud.fr This database has been under development for nearly two decades and contains about 500,000 entries on the properties and reactions of atoms, molecules, and neutral or ionized plasmas (Delcroix et al., 1998). 5 . The NIST Databases, National Institute of Standards and Technology, Gaithersburg, MD:
http://physics.nist.gov/PhysRefData/contents.html NIST has numerous databases on its Web site (see Dragoset et al., 1998), including the ionization cross sections of Y.-K. Kim and his collaborators (Kim et al., 1998) referred to previously:
http://physics.nist.gov/PhysRetData/Ionization/Xsection.html and, very recently, the complete tables of molecular constants of diatomic molecules. The NIST Web site
http://www.eeel.nist.gov/81 1/refdata (Christophorou and Olthoff, 1998b) contains the numerical data from the critical reviews of electron impact cross sections for CF4, CHF3, CC12F2,and CzF6 by Christophorou et al. (1996, 1997a, 1997b, 1998a), which have been mentioned in this article. 6. The Atomic and Molecular Data Information System (AMDIS) of the International Atomic Energy Agency in Vienna: http :/ /www. iaea.org/programmes/ amdis This is primarily for fusion plasma research, but it includes much related to plasma chemistry as well. 7. Finally, many tables of cross-section data discussed in this article can be found on the Kinema Research & Software Web site: http :/ /www.kinema.com/kinema/.
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VI. Acknowledgements This work was partially supported by SEMATECH, Inc., through a contract from the California Institute of Technology. The research on electron attachment to BC13 was performed while I was a visitor in the Institute for Theoretical Atomic and Molecular Physics (ITAMP) at the Harvard-Smithsonian Center for Astrophysics.
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ADVANCES IN ATOMIC, MOLECULAR. A N D OPTICAL PIfYSICS, VOL 43
ELECTRON-MOLECULE COLLISIONS IN LOW-TEMPERATURE PLASMAS The Role of Theory CARL WINSTEAD AND VINCENT MCKOY A. A. Noyes Laboratory of Chemical Physics. California Institute of Technology, Pasadena, California 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . , . . . . , , . . . , , , . , , . . . . . . , . 11. Types of Cross Sections.. . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . _ . . . . A. Elastic Scattering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . B. Momentum Transfer . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Vibrational Excitation. ................ D. Attachment. .. . . . . , . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. , . . . . , . . E. Electronic Excitation. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . F. Ionization. . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 111. Cross-Section Calculations at Low Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Physical Considerations. . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . .. . . .. . B. Implications for Choice of Method.. . . . . . . . . . _ _.. . , . . . . . . . . . . . . . . . . . . _ _.._ _ . IV Methods in Current Use .. . . . . . ........... A. Kohn Variational Method., . . . . , . , . . . . , , . . . , , . , . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . , B. Schwinger Multichannel Method. . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. .. . . . . . . . . C. R-Matrix Method . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . D. Ionization: Binary-Encounter-Bethe Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Illustrative Examples.. . ............. .. .. .. .. . 1. Elastic Scattering by . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2. Electron Impact Excitation of CzH4... . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . 3. Electron Impact Ionization of SF6.... . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . V Areas for Future Progress.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .
A. Treatment of Electronic Excitati B. Reactive Scatte ' ..... .............. VI. Acknowledgement ..... .............. VII. References .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction This chapter discusses the contribution that theoretical methods can make to a knowledge of electron-molecule collision behavior, and thereby to an understanding of low-temperature plasmas. Its aim is to survey both the relevant problems and the methods that have been developed to treat those problems. 111
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Without delving deeply into the workings of any specific method, we will try to convey both the capabilities and the limitations of present theoretical approaches, and to point out directions for future progress. If one hopes to develop detailed, predictive models of plasmas, microscopic information such as electron-molecule collision probabilities clearly is needed. But why obtain that information from theory? The short answer is that experimental data are often absent and-given the difficulty of the measurements and the paucity of research groups conducting them-in many cases are likely to remain so indefinitely. A longer answer would add that, as both theoretical methods and computer hardware improve, theory is, at least in some areas, becoming competitive with experiment in terms of accuracy and time to solution. To set the stage for the subsequent discussion, let us briefly recall some features of the electron-molecule collision problem as it arises in low-temperature, nonequilibrium plasmas. Most salient is that the electron kinetic energies are low, with the energy distribution often peaking at a few eV. Thus electrons in the higher-energy tail of the distribution may be responsible for important inelastic processes, such as electronic excitation and ionization of molecules, whose thresholds often lie above 10 eV, but very little of the plasma chemistry will be driven by truly high-energy electrons-that is, electrons whose energies greatly exceed the average lunetic energies of molecular valence electrons. This simple fact has profound consequences for both theory and experiment, because in both instances it is far more difficult to work with low-energy electrons. In the case of theory, many simplifying approximations that can be applied at high energy are excluded, and it becomes necessary to employ a many-body approach that treats the projectile electron and the electrons belonging to the molecule on an equal footing, with a proper accounting being made for the indistinguishability of electrons. As we will see later, the low-energy electron-molecule collision problem is far from hopeless; certain simplifications can usually be made without seriously impairing accuracy. What remains to be solved, however, is still formidable-a version of Schrodinger’s equation for the motion of several (perhaps several dozen) electrons. As this is a second-order partial differential equation with three spatial degrees of freedom per electron, direct integration is completely out of the question. The goal of theory, then, is to develop practical methods of approximation that allow one to extract reliable collision information. At the quantum mechanical level, collision information takes the form of a scattering amplitude ,f, a complex number whose square modulus If l2 is proportional to the collision cross section 6,which is a measure of the cotision probability. Specifically, for an electron incident with initial momentum Ak,, the cross section-for a collision that leads to an electron departing with final momentum A$ is
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The quantum mechanical cross section has units of area and bears the same relation to the collision probability as does the geometrical cross-sectional area of a classical object. AHowever,where a geometrical cross section _wo_ulddepend only on the directions k, and kf of the initial . and final momenta, a(k,,k,) depends also on the magnitudes lk,I = k, and Ik, I = k f , in general strongly so. Equivalently, one can say that the quantum mechanical cross section depends on the incident and final kinetic energies El = ti'k:/2m, and E, = hzk:/2m,. In part, this energy dependence arises because the molecule acts less like a solid object than like a potential field affecting the electron's motion. Another part arises from the wavelike character of the quantum mechanical collision partners and is thus analogous to the wavelength (or energy) dependence of, for example, the Rayleigh cross section for the scattering of light by dust. However, an additional type of energy dependence arises from the possibility of exciting (or deexciting) internal motions of the molecule, with a corresponding decrease (or increase) in E, relative to El. Because these internal motions are quantized, with only certain energy levels allowed, the electron kinetic energy difference E, - E l can take on only certain values, and we can, in fact, often deduce which internal motions were involved froin a measurement of this difference. For this reason, and because the individual probabilities for such inelastic collisions are of inherent interest, one generally speaks not of a single collision cross section but of a whole set of cross sections, each associated with a different process. In the following section, we will describe some of the cross sections that are most important in low-temperature plasmas. We will then turn to methods for computing such cross sections, describing the methods that are in most common use for low-energy electron-molecule collisions and giving a few examples of their application. A concluding section discusses areas where hrther progress is needed.
11. Types of Cross Sections A. ELASTICSCATTERING The elastic cross section is the cross section for scattering without loss (or gain) of total kinetic energy. It is of primary importance simply because it is, at low impact energies, the largest cross section by far, often an order of magnitude larger than all inelastic cross sections combined. Although the concept of elastic scattering is easy to grasp, in practice the term elastic is used in several qualified senses, particularly in connection with theoretical studies. Because it is important to be aware of these shades of meaning, we devote this section to a discussion of them. Strictly defined, elastic scattering leaves the sum of the electron and molecular kinetic energies the same after the collision as before, and therefore leaves the
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molecule in the same rotational, vibrational, and electronic energy levels where it was found initially. True elastic cross sections of this kind have been measured for only a very few molecules, such as CH4 (e.g., Muller et al., 1985), where the rotational spacings are large. The elastic cross sections most commonly available from experiment are vibrationally elastic, meaning that the molecule remains in its initial (at the temperatures of interest, usually ground) vibrational state, but neither the initial nor the final rotational state is specified. The vibrationally elastic cross section is therefore a sum over initial rotational states (with appropriate weights, often those given by a thermal distribution) and an average over final rotational states. In an analogous way, one can define the electronically elastic cross section, which combines the cross sections for vibrationally elastic and inelastic processes associated with a single electronic level. Theoretical cross sections are almost always obtained in the $xed-nuclei approximation: The nuclei are treated as charge centers fixed in space, and the electronic motion problem (for both the impinging electron and the electrons belonging to the target molecule) is solved in the field generated by the nuclei. If, in solving this problem, we specify that the initial and final electronic states of the molecule are to be the same, we obtain a sort of elastic cross section-the fixednuclei electronically elastic cross section-but, clearly, some additional reasoning is needed to make a connection with any of the experimentally determined “elastic” cross sections. That such a connection is possible was demonstrated explicitly by Chase (1956), who noted that, provided that the collision takes place in a time that is brief compared to the periods of any internal nuclear motions (rotations or vibrations), the combined electronic-nuclear motion problem can be factored, to a good approximation, into separate electronic and nuclear problems. This adiabatic approximation is the analogue, for scattering problems, of the familiar Born-Oppenheimer approximation that justifies a fixed-nuclei treatment of bound-state problems. Letting Tlf represent the transition amplitude (the quantity whose square mod+ determines the cross section) from rovibronic state Y,@, i)to state Y,@,R), where and R are, respectively, electronic and nuclear coordinates, the adiabatic approximation amounts to setting
with E,,($ being rovibrational wavefunctions and $;,,@;j) being electronic wavefunctions determined at fixed nuclear coordinates 4. Associated with $;,,,(;; i)is a fixed-nuclei electronic transition amplitude Tjf(R)in terms of which
Rotational and vibrational motion are _almost always treated separately within the adiabatic approximation; that is, S ( R ) is fac_tored further into_ a rotational component Qk) and a vibrational component x(Q), where k and Q are orienta-
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tional and vibrational coordinates, the justification again being different time scales for the two hnds of motion. Moreover, under the conditions that prevail in most experiments and in plasmas, one is almost always justified in taking the classical limit (e.g., Shimamura, 1984), in which vibrationally elastic (rotationally unresolved) cross sections can be obtained simply by averaging the cross section over all possible molecular orientations, rather than summing over rotational transitions. Vibrationally elastic and inelastic amplitudes are then evaluated as
where uif are the initial and final vibrational states. Implicit in the classical limit is the idea that the initial rotational state of the target is irrelevant, and therefore that the rotationally unresolved cross section is independent of the distribution of initial rotational states and hence of the gas temperature. For nonpolar molecules, this result holds even if the classical limit is not taken, but rather the sum over final rotational states is carried out explicitly (Shimamura, 1984); however, for molecules having a dipole moment, the cross section does depend somewhat on the initial state (Shimamura, 1990). Because the fixed-nuclei amplitude must be computed multiple times in order to evaluate the preceding expression, obtaining vibrationally resolved cross sections is quite expensive and usually forgone. Instead, the elastic cross section reported in most theoretical papers is computed at the equilibrium nuclear geometry alone. It is explicitly or implicitly assumed that the resulting cross section is comparable in some way to the experimental elastic cross section, but what, if anything, justifies this assumption, and to which elastic cross sectionvibrationally elastic or vibrationally summed-is the fixed-nuclei elastic cross section comparable? First, note that in the common case of small molecules composed of light nuclei at moderate temperatures, most molecules will be in their ground vibrational sgtes, which we denote by xo. Because xo is peakzd at the equilibrium geometry Qeq,if we assume that q, is weakly dependent on Q, we can make a Taylor expansion:
Truncated at the first-derivative terms and inserted in Eq. (4), this expansion gives
and
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where in both cases the neglected terms vanish in the harm_onicapproximation. Thus we can justify equating the fixed-nuclei cross section at Qeq with the vibrationally elastic cross section, but only by making the additional (and somewhat circular) assumption that the scattering process is, insensitive to nuclear motion. On the other hand, we could choose to view Tif(Qeq)as a single-point quadrature for the integrals of Eq. (4); in this view, the appropriate comparison is to the vibrationally summed cross section, even though the approximation to any given term in the sum (except possibly the first) is likely to be poor. Indeed, the_ inescapable conclusion is that electronically elastic cross sections computed at Qeqcannot be directly, reliably, and unambiguously compared to measured elastic cross sections unless vibrational excitation is weak; otherwise, the computed elastic cross section is perhaps best viewed as a first approximation to the vibrationally summed cross section from the ground vibrational state ui = 0. It should also be added that the population of excited vibrational states, though typically small at or near room temperature, is not necessarily negligible, and that therefore cross sections for v, > 0 may be required. In contrast to the rotational case, the vibrationally summed cross section may be expected to depend strongly on the initial state; although relations among the summed cross sections for different ui can be derived (Shimamura, 1992), the expressions are rather complicated and the assumptions rather restrictive. In general, it appears that it will be necessary to accept the necessity of carrying out multiple fixed-nuclei calculations if fully satisfactory results are to be derived. B. MOMENTUM TRANSFER Although elastic scattering is generally the dominant process in low-energy electron-molecule collisions, not all elastic collisions are equally important in understanding transport phenomena in plasmas. Clearly, an electron that is elastically scattered through a tiny angle is not much different from an unscattered electron. The momentum-transfer cross section oMT is an angle-integrated cross section that reflects the relatively greater importance of large-angle scattering by including a weighting factor (1 - cos 0), where 0 is the scattering angle:
oMT= 2x
1;:
(O)( 1 - cos 0) sin 0 dO
-
in terms of the differential cross section da/dR. The momentum-transfer cross section has various uses. For example, the rate at which a unit flux of electrons of velocity u and mass m transfers energy through elastic collisions to target molecules of mass M and density p can be shown through a simple kinematic v, to the momentum-transfer cross argument to be 2 ( m / M ) c T ~ ~ E pproportional section. Because m / M is very small, many collisions are required in order to transfer a significant amount of energy to the heavy particles; one requirement for
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creating a low-temperature plasma is that the density p be sufficiently low to suppress such heating. In plasmas of this type, the primary significance of gMT is as a measure of the power deposition from an applied oscillating electric field into the electron energy distribution, for it is the collisions with large momentum change that transform directed velocity acquired from the field into random, thermal motion (Cecchi, 1990). The momentum-transfer cross section is easily calculated for any of the various “elastic” cross sections discussed in the preceding section. A momentum-transfer cross section can also be defined analogously for inelastic processes, although the concept is most useful in the elastic case. C. VIBRATIONAL EXCITATION
Our discussion of elastic scattering in Section 1I.A has already led us into the subject of vibrational excitation. Here we would like to consider vibrational excitation in somewhat more detail, looking in particular at conditions where vibrational excitation is most important. A thorough treatment of the theoretical aspects of the subject may be found in the review by Herzenberg (1984). The adiabatic approximation used in Section 1I.A to separate vibrational and electronic motion relied on the assumption that the collision duration was much shorter than the period of any nuclear motion. While this is almost always an excellent assumption when applied to rotation, it not infrequently breaks down in the case of vibration. Since typical vibrational energies are on the order of 0.1 e y we can anticipate that nonadiabatic effects may be important when the electron kinetic energy is significantly less than 1 eV. Note that such slow electrons occur not only for low incident energies but also just above the threshold for any inelastic process. Similarly, nonadiabatic effects may be expected in the presence of resonances (temporary anions) whose width is significantly less than 1 eV, since the resonance lifetime may then be a large fraction of a vibrational period. While the breakdown of the adiabatic approximation is not a prerequisite for vibrational excitation (indeed, we gave an adiabatic expression for the vibrationalexcitation amplitude in Section ILA), there is nonetheless a strong correlation between large interaction times and vibrational excitation. Thus large vibrationalexcitation cross sections, of the same order as the vibrationally elastic cross section, tend to be found at low impact energies or in association with resonances. Vibrational excitation by electron impact is also favored when the transition is allowed by infrared selection rules and has a large oscillator strength (Herzenberg, 1984). Crudely speaking, the time-dependent electric field due to the passing electron induces such transitions in the same manner as the oscillating field associated with infrared radiation. Because this mechanism operates at long range, most of the excitation of such optically allowed transitions takes place via weak collisions, and a perturbative approach (the first Born approximation)
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usually suffices for computing the cross section. The resulting cross sections peak close to threshold and thereafter decrease monotonically; however, because the decrease is only as (energy)-’, the cross sections for such dipole-allowed excitations can remain significant to fairly high energies.
D. ATTACHMENT Attachment processes (Christophorou et al., 1984) are those in which the incident electron is permanently captured, so that there is no outgoing free electron present in the final state. In potential scattering, an electron may be captured temporarily in a resonance, but (if we neglect radiative capture) it will ultimately depart again; thus, any resonance has a nonzero width and finite lifetime. Ordinarily the same is true in electron-molecule scattering: The temporary anions formed by collisions necessarily have energies above the redetachment limit, and thus (again barring radiative capture, stabilization via collision with a third body, etc.) will eventually decay to the neutral molecule plus an electron. Under some circumstances, the resonance lifetime may be very long. In particular, if there is a large conformational change in passing from the neutral molecule to the anion, and if there are many low-frequency vibrational modes among which the excess energy can be dispersed, then the time before the original conformation is recovered and the electron ejected may be very long; and, of course, the greater this lifetime, the greater the probability that the anion will be permanently stabilized by giving up some of its vibrational energy in a collision with another molecule or the chamber wall. In such cases, the distinction between a temporary and a permanent negative ion may be immaterial. Apart from such special cases, however, the negative ions that result from electron collisions will be dissociation fragments of the original molecule. Dissociative attachment, the process e-+AB+A-+B
(9)
can occur when the electron’s kinetic energy lies above the dissociation limit for the anion; letting E be the electron kinetic energy and D,4B-the dissociation energy of the A-B bond in the anion, the requirement is E > DAB-,or, in terms of quantities that are more readily available from the literature, E > DAB- EA,, where DAB is the A-B bond strength of the neutral molecule and EA,4 is the electron affinity of A. Dissociative attachment is usually observed in association with resonances in the elastic cross section, because the presence of a resonance enhances the probability that the nuclei will reach a configuration at which AB- is more stable than AB e- before the electron departs. Likewise, dissociative attachment cross sections typically increase rapidly with temperature, because the initial nuclear kinetic energy of vibrationally excited molecules decreases the kinetic energy that must be transferred from electronic to nuclear motion to achieve dissociation.
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Even with resonant or vibrational enhancement, cross sections for dissociative attachment tend to be small-at least one and generally two orders of magnitude smaller than the elastic cross section. Dissociative attachment is nonetheless an important process. At energies below the threshold for electronic excitation, dissociative attachment is generally the only mechanism for dissociation that proceeds with a detectable cross section; moreover, dissociation by electronic excitation generally leads to neutral or positively charged fragments, so that dissociative attachment may be the principal mechanism for forming negative ions in many cases. E. ELECTRONIC EXCITATION Electronic excitation (Trajmar and Cartwright, 1984; Allan, 1989) is among the most important inelastic phenomena occurring as a result of electron-molecule collisions in the 0 to 50-eV energy range, both because the cross sections tend to be large (within an order of magnitude of the elastic cross section) and because the consequences-transformation of a significant amount of energy from kinetic energy of the projectile electron to internal energy of the molecule-are dramatic. The large mismatch between electron and nuclear masses precludes direct collisional energy transfer; thus, as we have already discussed, vibrationalexcitation and dissociative-attachment cross sections tend to be small unless they are enhanced by an indirect mechanism such as a resonance. However, in electronic excitation, the initial energy transfer is between electrons, and a large direct cross section is therefore possible, while resonant enhancement is not precluded. Once a molecule has been placed in an excited electronic state by a collision, several fates are open to it. The tamest is to radiate its energy away and return to the ground state. However, low-energy electron impact excitation is not constrained by optical selection rules, so the excited state may well be metastable, in which case it may survive to collide and react with another molecule. Moreover, many excited states prepared by electron impact will be dissociative, either because the excited-state potential surface has no minimum or because the vertical (i.e., without change of nuclear positions) transition brought about by the collision reaches a region of the upper surface that lies above the dissociation limit. The fragments produced by such dissociative excitations include atoms, neutral radicals, stable molecules, and occasionally ions (via AB* -+ A + B-). Because the number of electronic excitation processes possible for any given molecule is not merely large but infinite, some selection principles are clearly needed to identify those that are most important for study. Conservation of total spin for the electron-molecule system leads to the selection rule for the molecular electronic states AS = 0, f l ; thus, in the case-by far the most common--of a singlet ground state, only singlet and triplet excited states are accessible. Beyond
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that, the most important processes are, obviously, those with the largest cross sections and those that lead to some desired result-which, in the plasma processing context, is usually dissociation. Chemical and physical principles can inform a choice of excitation processes for study. For example, excitations that remove an electron from a bonding orbital and/or promote an electron to an antibonding orbital are likely to contribute to dissociation. Electronic-structure calculations exploring the excited-state potential-energy surface may be used in this connection both to confirm (or refute) the dissociative character of the excitation and to identify possible products. In selecting transitions that are likely to have large cross sections, several factors enter: Those transitions with the lowest thresholds tend to have the largest cross sections, transitions into valencelike orbitals are generally stronger (and more prone to resonant enhancement) than transitions to Rydberg orbitals, and optically allowed transitions with large oscillator strengths tend to have large electron impact cross sections. In the last case, the mechanism is the same as that already mentioned in connection with vibrational excitation in Section 1I.C. Armed with these heuristics and some spectroscopic or computational information about the singlet and triplet excitedstate manifolds, one has a reasonable hope of identifying the most important excitation processes.
F. IONIZATION Like electron impact excitation, electron impact ionization is important in plasmas as a source of reactive species, including both the parent ion (if it is stable) and various cationic fragments that may be formed by dissociation following ionization. In a sense, electron impact ionization is merely the extension of electron impact excitation out of the discrete spectrum and into the continuum. A correct qualitative conclusion would be that ionization cross sections, like discrete excitation cross sections, should be fairly large. However, there are important differences. Fewer constraints apply to the ionization process than to discrete excitation. When two free electrons are present, they may divide up the available energy and momentum in an infinite distribution of ways. Moreover, ionization is always allowed by optical selection rules, and, as we have discussed, optically allowed transitions tend to have enhanced electron impact cross sections. As a result of these considerations, the ionization cross section tends to be somewhat larger than the summed discrete excitation cross section, unless the latter is enhanced by resonances. The maximum value of the ionization cross section also tends to occur at a higher energy than the maximum in the discrete excitation cross section, in part because spin-changing excitations and resonant effects that can make large contributions to the latter are restricted to the low-energy region.
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111. Cross-Section Calculations at Low Energy A. PHYSICAL CONSIDERATIONS Having outlined the types of electron-molecule cross sections that are relevant in low-temperature plasmas, we turn to the calculation of such cross sections. Before examining particular computational techniques, it is instructive to consider the constraints that physical conditions and accuracy requirements place on the choice of calculational method. In this way, we shall better understand both the motivation behind the design of the methods in use today and the strengths and weaknesses of those methods. To begin with an obvious but critical point, molecules are not spherical. Spherical symmetry is assumed early on in most textbook discussions of scattering theory, but only to simplify the discussion-nothing fundamental is different in the nonspherical case. However, the low symmetry produces great technical as well as pedagogical complications. A scattering process is specified by the behavior of the collision partners at great distances; in the electronmolecule case, because we can think of the center of mass as fixed in the molecule, the initial and final conditions are specified by the asymptotic motion of the electron alone. Natural quantum numbers to specify this free-electron m_otion are the initial linear momentum with which it appr2aches the molecule, tlk,, and the linear momentum long after the collision, hkj. Alternatively, we could specify the initial energy k2kf/2m and angular momentum quantum numbers ( l , ,m,) and the final energy and angular momentum, A2k,?/2m and ( l , , m f ) . In either case, the natural coordinate system for thinking of the asymptotic motion is a spherical polar system with its origin at the molecule. For a spherical scatterer, that coordinate system would remain appropriate even at small distances, and it clearly would be convenient if the same coordinates could be employed everywhere. A related and much greater convenience follows from observing that the total electronic angular momentum will be conserved in scattering from a spherical center. With all quantities expressed in polar coordinates, imposition of the angular momentum conservation requirement could then be used to remove the angular coordinates of all the electrons from the problem completely. It is incomparably easier to solve an N-dimensional electronic structure problem built up from functions of a single variable (describing the radial motion of individual electrons) than to solve the 3Ndimensional problem built up from functions of three variables that results in the molecular case. As we have just implied, solutions to the many-electron scattering problem, like solutions to the many-electron bound-state problems of quantum chemistry, are obtained in terms of products of one-electron functions, subject to constraints of spin, exchange antisymmetry (the Pauli principle), and possibly spatial (point
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group) symmetry. Because spherical symmetry is broken, one must either persist in using basis functions adapted to spherical symmetry, at the expense of slow convergence, or, if one uses a more appropriate basis set to represent the wavefunction near the molecule, connection must somehow be made to eigenstates of angular or linear momentum at large distances. The specific computational methods that we will examine in Section IV illustrate different approaches to this problem. A minor additional complication is that an average over molecular orientations (or an explicit consideration of the initial and final rotational wavefunctions-see Section 1I.A) must be taken before computed quantities are compared to measured cross sections. Scattering is easiest to think about in a one-particle picture, where a projectile is scattered by a field of force associated with some potential. If both collision partners are structureless, we can always reduce the problem to this form by removing the center-of-mass motion. When one of the collision partners is composite and possesses internal modes that may be excited by the collision, this appealing potential-scattering picture no longer applies, although it remains a useful conceptual tool and limiting case. The identity of particles adds a further level of complication to electron-molecule (or electron-atom) scattering: Because electrons are indistinguishable, the overall electronic wavefunction for the electron-molecule system is antisymmetric under exchange of any two electrons’ coordinates. Exchange strongly affects both elastic and inelastic scattering of low-energy electrons and must be taken into account if reliable results are to be obtained. Thus, rather than solving for a one-electron wavefunction describing the scattered electron, we must solve for a many-electron wavefunction that treats all electrons on an equal footing. There is further reason why a many-electron treatment is required in order to obtain accurate results. As mentioned earlier, an adiabatic approximation, wherein the nuclei are considered fixed in space while the electronic portion of the problem is solved, can usually be applied to low-energy electron-molecule scattering without significant error, because the nuclei move sufficiently slowly compared to the projectile electron. If the same were true of the electrons belonging to the molecule, one could employ a fixed target electronic wavefunction that would, together with the nuclear charges, generate a static charge distribution defining a one-electron potential-scattering problem. At the cost of making the potential nonlocal, exchange effects can be incorporated into this oneelectron problem, resulting in what is called the static-exchange approximation. In fact, however, the valence electrons of molecules have velocities comparable to that of the low-energy projectile, and they therefore respond dynamically to its presence, necessitating a true many-particle treatment of the collision. As would be expected, this response, known as polarization, is most significant under circumstances where the collision time is long-that is, for collisions at the lowest impact energies and in the presence of resonances. As a rule of thumb, taking
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polarization effects into account is necessary in order to obtain qualitatively correct cross sections below 5 eV impact energy and leads to noticeably improved results at energies up to about 20eV B. IMPLICATIONS FOR CHOICE OF METHOD
To summarize the results of the preceding section, low-energy electron-molecule collision calculations must take into account the nonspherical nature of molecules, the exchange and polarization interactions between the projectile and target electrons, and the possibility that the molecule may be excited by the collision to a higher electronic state, with corresponding loss of energy by the scattered electron. Depending on circumstances, it may also be necessary to take into account vibrational excitation and/or to allow for nonadiabatic effects, i.e., the coupling of electronic and nuclear motions, as discussed in Sections 1I.A and 1I.C. Treatment of dissociative attachment or electron impact ionization is still more difficult because the problem becomes one of reactive scattering, in which the particles produced by the collision are not the same as those that were present before the collision. Indeed, we will not take up the detailed treatment of either dissociative attachment or electron impact ionization in this article, although we will describe later a simple but remarkably successful model that gives estimates of the ionization cross section. In many ways, the accurate treatment of low-energy electron-molecule scattering problems closely resembles the accurate treatment of bound-state molecular electronic structure, which is the subject matter of computational quantum chemistry. Many aspects of the physics are comparable, as are the computational difficulties that they engender. In fact, we will see shortly that many of the methods of quantum chemistry can be borrowed for or adapted to the scattering problem. However, it is important to bear in mind that there are major differences as well. The most obvious difference-that at least one electron is not permanently bound to the molecule-implies that it may be necessary to employ a representation of the wavefunction extending over all space. In contrast, conventional quantum chemistry exploits representations that are localized in the vicinity of the molecule. At the very least, any localized representation of a scattering wavefunction will have to be carefully designed and justified. A further serious technical complication is the loss of a variational principle for the energy. Traditional quantum chemical methods are founded on the Rayleigh-Ritz variational principle for the Hamiltonian or energy operator, or on a combination of variational and perturbative techniques. This variational principle states that, of all possible wavefunctions satisfying the correct boundary conditions, the true ground-state wavefunction gives the lowest energy; thus, given a set of possible approximate wavefunctions, the optimal approximate wavefunction is that whose energy is lowest. In a scattering problem, however, the energy is not quantized but
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is, rather, a continuously variable parameter that is specified in advance as part of the problem definition. For this reason, the highly developed and sophisticated variational methods used by quantum chemists to obtain accurate electronic wavefunctions cannot be applied to electron-molecule scattering; other methods of approximation tailored to the scattering problem must be designed and implemented. In the next section, we will outline those methods that have been most extensively applied to the sorts of polyatomic molecules associated with plasma processing, and we will give examples of the application of those methods to specific problems.
IV. Methods in Current Use In this section, we look at several methods in current use for calculating electronmolecule collision cross sections relevant to low-temperature plasmas. For the most part, we will avoid technical details, which in any case can readily be found elsewhere (Huo and Gianturco, 1995; Winstead and McKoy, 1996), although we will attempt to describe enough of the implementation to bring out the advantages, disadvantages, and limitations of each method. We will conclude with illustrative examples in which different methods are applied to the same elastic and inelastic electron--molecule collision problems. The fist three methods we will discuss are based on variational principlesnot minimum principles for the energy, but stationary principles for the scattering amplitude or some related quantity. While these methods are the most elaborate and computationally demanding, they are also potentially the most flexible and the most accurate, in that they make the fewest simplifications and approximations. More approximate methods are also in use, and descriptions can be found elsewhere (e.g., Huo and Gianturco, 1995). Because of its extraordinary utility, we will also briefly consider the method of Kim and Rudd (Kim and Rudd, 1994; Hwang et al., 1996) for obtaining electron impact ionization cross sections, which is based on a very simple model of the electron-target interaction. A. KOHNVARIATIONAL METHOD
The Kohn variational principle (Kohn, 1948) can be formulated in a number of closely related ways (e.g., Nesbet, 1980; Rudge, 1990). Their common feature is that the variational expression involves the Hamiltonian operator H together with an operator describing the scattering: either the so-called T operator that is most closely connected to the scattering amplitude or the related reactance ( K ) or scattering ( S ) operators. The Kohn expression is so contrived that the portion of the expression that depends on H-a matrix element between two approximate wavefunctions-approaches zero quadratically as those wavefunctions approach
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the exact scattering wavefunctions, while the remaining portion approaches the exact T-matrix (or S- or K-matrix) element. This is a statement of the variational property: The error in the Kohn expression is second-order with respect to the error in the wavefunctions. Like other variational methods, the Kohn expression is implemented numerically by restricting the possible wavefunctions to some specific functional form dependent on a set of parameters. Mathematical techniques are then applied to locate a stationary point, i.e., a combination of parameter values for which all derivatives of Kohn expression, considered as a function of the parameters, are zero. In large-scale applications such as electron-molecule scattering, the only practical functional form is a linear expansion, II/ = x i x , where f; are known functions and the coefficients xi are the variational parameters. The stationarity requirement then leads to a system of linear equations that can be solved efficiently by the techniques of linear algrebra. Note that, because only a stationary (rather than a minimal or maximal) point is located, there is no easy way to compare the quality of two different approximate solutions, nor is one assured, as one is in a minimization or maximization problem, that enlarging the basis set (the set ofJ;) will lead to a result that is at least as good as the previous approximation. Nonetheless, variational principles remain powerful techniques for scattering problems, in that, by eliminating the need to solve Schrodinger’s equation directly, they permit the inclusion among the basis functions of elaborate many-particle wavefunctions that can provide an accurate and detailed representation of both the isolated target molecule and the electron-molecule interaction. Of particular importance is the ability to include functions representing many different electronic configurations (i.e., sets of orbital occupations), which is important to the description of both polarization and electron impact excitation. The principal advantage of the Kohn method is that it involves only matrix elements of the Hamiltonian and of the electron-molecule interaction. Because the latter is itself a component of the Hamiltonian, it follows that the Kohn expression can be applied to any situation for which one can evaluate the Hamiltonian matrix elements. As mentioned earlier, bound-state electronicstructure studies have been advanced to a high degree of sophistication; since all such studies are based on techniques involving the Hamiltonian, it would seem at first glance that the Kohn method could be implemented virtually without additional effort or computational cost. Indeed, when a basis set is introduced and the Hamiltonian operator thereby transformed into a Hamiltonian matrix, the largest block of that matrix can be, and is (e.g., Rescigno et al., 1995a), obtained by conventional electronic-structure methods, using basis functions localized in space. Now, however, we encounter the complications catalogued in Section 111. First of all, there is the matter of boundary conditions: For correct results to be obtained from the Kohn method, the approximate wavefunction must behave appropriately at large distances from the molecule-namely, as an electronic
xi
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wavefunction for the target molecule times the wavefunction of an incident or scattered electron with the proper kinetic energy and direction of propagation. To provide this form, new basis functions must be introduced. Asymptotically, these additional functions are products of electronic states of the molecule with angular momentum eigenstates for a free electron satisfying ingoing- or outgoing-wave boundary conditions; at short range, the ingoing-wave functions must be regularized, i.e., prevented from becoming infinite at the origin (Rescigno et af., 1995a). The Hamiltonian matrix elements involving these new functions are not available from standard quantum chemistry programs, and procedures for evaluating them by quadrature must therefore be developed as part of the implementation of the Kohn method (McCurdy and Rescigno, 1989; Rescigno et af.,1995a). In evaluating these matrix elements, the most severe difficulties are encountered in the treatment of exchange interactions, which give rise to sixdimensional integrals whose quadrature would be prohibitively expensive. A further approximation is thus introduced to deal with exchange interactions (Rescigno and McCurdy, 1988). The Kohn method has been successfully applied to a number of small molecules (for a recent review, see Rescigno et af., 1995b). To date, most applications have been to linear molecules or to hydndes such as CH4 and C2H6. For such molecules, the expansion of the nonlocalized portion of the wavefunction in terms of eigenstates of angular momentum is either partially separable (because angular momentum about the nuclear axis is conserved in linear molecules) or rapidly convergent (because the presence of symmetrically arranged hydrogens does not strongly perturb the charge density from spherical or linear symmetry). More recently, studies of nonhydride polyatomics such as NF3 (Rescigno, 1995) and CF4 (Isaacs et af.,1998) using the Kohn method have been reported. Studies of molecules in which the departure from spherical symmetry is still greater can be expected to be progressively more demanding, though how much so remains to be seen.
B. SCHWINGER MULTICHANNEL METHOD Our own approach to the low-energy electron-molecule scattering problem is referred to as the Schwinger multichannel, or SMC, method. Like the Kohn method, the SMC method employs a variational technique to obtain an approximation to the scattering amplitude. The origin of the SMC method is in a variational principle put forward by Schwinger (1947) for the solution of problems in potential scattering. Modifications to the original Schwinger form that take into account complexities of the many-particle problem (Takatsuka and McKoy, 1981; Takatsuka and McKoy, 1984) facilitate its application to electronmolecule collisions.
ELECTRON-MOLECULE COLLISIONS
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Unlike the Kohn principle, the Schwinger principle involves matrix elements of a fairly complicated operator, the interaction-free Green’s function for the electron-molecule system, as well as matrix elements of the electron-molecule interaction itself. The result is a method that is formally of “higher order” than the Kohn variational principle (Takatsuka et al., 198l), meaning that as a general rule (but not always) it will produce a better approximate wavefunction, and therefore a better approximation to the scattering amplitude, from a given basis set. Of course, this potential advantage must be set off against the cost of evaluating a more difficult class of matrix elements. Perhaps the principal practical advantage of the Schwinger variational expression is that all matrix elements arising in it involve the electron-molecule interaction. (The Green’s function does not occur alone but in a product involving the interaction.) Because the electron-molecule interaction vanishes at large distances, the matrix elements that enter the Schwinger expression are insensitive to the asymptotic behavior of the basis functions; in particular, localized basis functions that vanish asymptotically can be employed. This is in sharp contrast to the Kohn method, where wavefunctions extending over all space are needed in order to impose the correct boundary conditions (in the Schwinger method, the boundary conditions are incorporated through the Green’s function). Although the SMC modification of the Schwinger expression adds terms that do not involve the interaction and would therefore appear to require long-range basis functions, in fact the contributions of these terms can be shown to cancel asymptotically (Winstead and McKoy, 1993). The SMC method thus preserves the valuable property of requiring only short-range basis functions for its implementation. In practice, the short-range functions used in both the Kohn and SMC methods are taken to be the so-called Cartesian Gaussians used in bound-state quantum chemistry, because these functions provide the best balance between an efficient description of the wavefunction and ease of evaluation of the necessary matrix elements. From these Gaussian functions, a flexible variational basis set of manyelectron basis functions can be constructed. Even though short-range functions may be used exclusively in the representation of the wavefunctions, free-electron wavefunctions extending over all space do occur in the definition of the SMC expression, and mixed matrix elements involving both these functions and Cartesian Gaussians must thus be evaluated. However, when the free-electron functions are chosen to be plane waves, exp(ik . ;), the resulting integrals, in contrast to those arising in the implementation of the Kohn method, have a convergent closed form. In fact, the expressions that arise are variants of those that are used to evaluate matrix elements involving Gaussians alone, and the same efficient algorithms may be used to evaluate them. Moreover, by using a representation of the Green’s function in term: of plane waves and carrying out a numerical quadrature in the wave vector k , we can reduce the Green’s function term in the SMC expression to the same sort of
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Carl Winstead and Vincent McKoy
integrals and can therefore evaluate it to any required accuracy (Lima et al., 1989). However, evaluation of the Green’s function matrix elements remains the most time-consuming task, and therefore the greatest obstacle to the application of Schwinger-type methods. Because the SMC method is dominated by procedures that are highly structured and repetitive-namely, the evaluation of large numbers of basic integrals and the manipulation of those integrals to construct matrix elements occurring in the variational expression-it is very amenable to parallelization. We have recently discussed the parallelization of the SMC method and the performance achieved in some detail (Winstead and McKoy, 1996). Although constructing an efficient distributed-memory parallel implementation has required considerable human effort, much of the progress that we have achieved in applying the SMC method to polyatomic molecules used in plasma processing of materials has been due to the resulting capability to distribute both the work and the memory requirements of large problems over hundreds of processors. C. R-MATRIXMETHOD The R-matrix method (Kapur and Peierls, 1938; Wigner and Eisenbud, 1947) is a variational technique that relies on the explicit partitioning of space into an inner collision region and an asymptotic region. The collision problem within the finite inner volume can be treated much like a bound-state problem, by constructing and diagonalizing a Hamiltonian matrix. The scattering character of the problem is reflected in the boundary conditions imposed on the surface that separates the inner and outer regions. These boundary conditions are conveniently incorporated via an extra term in the Hamiltonian, known as a Bloch operator (Bloch, 1957), which renders the inner-region Hamiltonian Hermitian. Diagonalizing this Hamiltonian, which is independent of the collision energy, effectively solves the collision problem in the inner region. Scattering information at a given collision energy is then extracted by constructing an inner-region solution in terms of the eigenfunctions of the Hamiltonian and imposing continuity conditions at the boundary surface to obtain the wavefunction in the outer region. If the boundary surface lies entirely beyond the range of the potential, the scattering amplitudes may be evaluated immediately; otherwise, it may be necessary to propagate the solutions numerically to larger distances before the asymptotic form can be extracted. If, in the latter case, the boundary surface is chosen to be sufficiently distant that the surviving terms in the potential are simple in form (e.g., the leading one or two electrostatic multipole moments), then the propagation into the asymptotic region will not pose a significant computational challenge. Despite having intuitive appeal and certain computational advantages, the Rmatrix method has to date seen very limited application (Nestmann et al., 1994;
ELECTRON-MOLECULE COLLISIONS
129
Sarpal et al., 1994; Sarpal et al., 1998; Beyer et al., 1997) to nonlinear molecules. As discussed, e.g., by Schneider ( 1 995), one obstacle to its wider use is that the highly developed variational methods for bound states that one would like to adapt to the inner-region problem employ basis functions whose matrix elements are easy to compute over an infinite volume, but not at all easy to compute over a finite volume. Somewhat ironically, then, the explicit introduction of a collision region where the strong interactions occur alleviates some computational difficulties, only to introduce new ones. In the cases of atoms and linear molecules, the necessary finite-volume matrix elements are readily obtainable by one- or two-dimensional numerical quadrature, and the R-matrix method has been extensively and successfully applied to such problems. In the molecular case, one might resort to three-dimensional quadrature, but a more palatable alternative is to develop procedures (Nestmann et al., 199 1 ; Pfingst et al., 1994; Morgan et al., 1997) for obtaining the finite-volume matrix elements as the difference between the infinite-volume matrix elements and the contributions to those matrix elements from outside the scattering region (the latter being evaluated in some efficient manner). Because only a few nonlinear molecules-methane, ozone, and cyclopropane-have been studied to date, and because no studies of electron impact excitation of such molecules have yet been made, it is too early to assess the utility of current implementations of the R-matrix method for the study of general polyatomic molecules. In particular, it remains to be seen whether, in addition to the usual scaling problems that render ab initio calculations by all methods rapidly more expensive as the number of heavy atoms increases, there are complications peculiar to the the R-matrix method or its implementation that will arise when the molecular symmetry is low and the number of heavy atoms large. D. IONIZATION:BINARY-ENCOUNTER-BETHE MODEL Detailed treatment of the electron impact ionization of molecules is extremely difficult. To all of the complications that exist for inelastic electron-molecule scattering is added the complication of having two unbound electrons and a positive ion, all interacting strongly via Coulomb potentials, present in the final state. The technical difficulties engendered by this physical situation have up to now largely precluded first-principles studies of ionization of molecules by lowenergy electron impact, although steps in this direction are being made (McCurdy and Rescigno, 1997; McCurdy et al., 1997). At high energies, on the other hand, one can expect that the collision physics will simplify considerably; with the projectile moving much faster than the molecular electrons, correlation and identity of particles (i.e., polarization and exchange effects) become less important, and the potential energy of interaction
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Carl Winstead and Vincent McKoy
becomes a smaller fraction of the total energy of the projectile. As already mentioned in Section II.F, electron impact ionization is always allowed under optical selection rules; thus, as other mechanisms become rapidly less important with increasing energy, the relatively long-range interaction between the timedependent electric field associated with the passing electron and the transition dipole moment for ionization becomes the major factor determining the ionization cross section. As a result, the high-energy ionization cross section can be computed accurately from a simple expression, due originally to Bethe (1930), which requires as input only the oscillator strength distribution (that is to say, the photoionization cross section). Not unexpectedly, the Bethe model does not predict ionization cross sections at low impact energies well. Various attempts (e.g., Margreiter et al., 1990, and references therein) have therefore been made to combine the Bethe expansion with a simple model that is appropriate at low energies, such as the binaryencounter model (Vriens, 1969), a semiclassical treatment of inelastic scattering whose ingredients are the Mott cross section for a painvise electron collision, the kinetic energy distributions for the electrons of the molecule, and the electronic binding energies (ionization potentials). Despite its simplicity, the binaryencounter model often works reasonably well for ionization (less well for discrete excitations), and it has better qualitative behavior in the near-threshold region than the Bethe approximation, although its asymptotic behavior is incorrect. The most successful effort to merge the binary-encounter theory with the Bethe model has been that of Kim and Rudd (1994). Their binary-encounter-Bethe (BEB) model behaves correctly at both low and high energies and has a remarkably simple form, expressible in a single equation:
o(T) =
t+u+l
In this expression, which applies to an individual molecular orbital, T is the incident electron energy and t and u are normalized energies, t = T/B and u = U/B, with B the binding energy of the orbital and U the average electron kinetic energy for the orbital; S is given by S = 4.naiNR2/B2,with N the orbital occupation number, a. the Bohr radius, and R the Rydberg constant. The total ionization cross section is obtained by summing Eq. (10) over all orbitals for which T exceeds B . It is notable that the photoionization cross sections does not appear in the BEB expression, although it is required in the Bethe model; instead, a simple functional form has been assumed (Kim and Rudd, 1994). A more elaborate model that does incorporate the photoionization cross section, referred to as the binary-encounter-dipole (BED) model, was also proposed by Kim and Rudd (1994). The BED model is suitable for calculation not only of the total ionization cross section but of the singly differential cross section, i.e., the cross section for a
ELECTRON-MOLECULE COLLISIONS
131
given impact energy and a given asymptotic kinetic energy of the ejected electron. In the few cases where accurate experimental data are available for comparison, the BED model appears to be quite successful; however, the necessity for first obtaining an extensive and accurate set of partial photoionization cross sections for the molecule in question, either from theory or by experiment, limits its applicability. In contrast, one can, using standard electronic structure programs and no more than a modest-priced personal computer, readily compute Hartree-Fock orbitals, and thus the orbital kinetic and binding energies U and B, for almost any molecule likely to be found in the gas phase. The BEB model can therefore be applied to any molecule or radical of interest to plasma processing. Kim, Rudd, and coworkers (Kim and Rudd, 1994; Hwang et al., 1996; Kim et al., 1997; Ali et al., 1997; Ali et al., 1998) have tested the BEB model on a variety of molecules for which experimental data exist, with generally excellent results. In some cases, they obtained improved results when the HartreeFock binding energy of the outermost orbital was replaced by the measured vertical first ionization potential. In a study of the halomethane series CHfi (A' = H, F, C1, Br, I), Harland and coworkers (Vallance et al., 1997) recently found that the BEB model did not work as well for molecules containing heavier atoms as it did for CH4 and CH3F. More extensive comparisons between the model and experimental results will be necessary to determine whether this observation holds true in general.
E. ILLUSTRATIVE EXAMPLES In this section, we briefly present a few examples that illustrate recent applications of some of the theoretical methods discussed above. The first example is electronically elastic scattering by nitrous oxide, N20;the second is electronically inelastic scattering by ethylene, C2H4. These examples have been chosen not because of any special relevance to plasma processing, but because it is possible in these cases to compare theoretical results obtained by different methods both to each other and to experimental data. We conclude the section with an example application of the BEB model to electron impact ionization of SF6.
I . Elastic Scattering by N 2 0
Low-energy elastic electron scattering by nitrous oxide has been the subject of several recent theoretical studies. In addition, multiple experimental determinations have been made of the integral and differential elastic cross sections as well as of the total scattering cross section, and these measurements are in generally good mutual agreement. Moreover, electron collisions with N 2 0 at low energies are marked by dramatic resonance and dipole-potential effects that pose a
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Carl Mnstead and Vincent McKoy
challenge to theoretical methods and illustrate some the limitations imposed by simplifying approximations. For all these reasons, then, N 2 0 makes an instructive example. Recent high-level theoretical studies of electron scattering by N 2 0 include the calculations of Sarpal et al. (1996a, 1996b) and of Morgan et al. (1997) using versions of the R-matrix method and the calculation of Winstead and McKoy (1998) using the SMC method. These three calculations were carried out at similar levels of approximation; in particular, each incorporated polarization effects, and each was carried out in the fixed-nuclei approximation. In addition, Sarpal et al. considered the effect of including correlation in the description of the target molecule's electronic wavefunction, and both Sarpal et al. and Morgan et al. incorporated long-range scattering by the dipolar potential. Somewhat more approximate calculations neglecting polarization effects were also reported recently by Michelin et al. (1996a, 1996b) and by da Costa and Bettega (1998). In Fig. 1, we show these theoretical integral cross sections together with selected experimental data (Szmytkowski et al., 1984; Johnstone and Newell, 1993). Clearly, the most remarkable feature of the cross section at these energies is the sharp resonance peak near 2.2eY which the calculations determine to be a sharp resonance or quasi-bound state of 'II symmetry, arising from the temporary trapping of the projectile electron in a virtual valence orbital of 71 type. Each of the three calculations that includes polarization is reasonably successful in reproducing the resonance feature; however, there are noticeable differences in detail among the calculations and between the calculations and experiment. Specifically, the resonance maxima in the calculated cross section are larger than the maximum in the measured total cross section, the resonance positions obtained by Sarpal et al. and by Morgan et al. are somewhat different from the observed resonance position, and the resonances calculated by Morgan et al. and by Winstead and McKoy are narrower than the observed feature. The first and third of these observations find a common explanation in the calculations' omission of nuclear vibrational motion. Although the precise effects of including vibration can be determined only by carrying out the appropriate calculation, the general effect is easily predicted: a broadening of the resonance peak with a corresponding diminution of its maximum value. Within the fixednuclei approximation, such broadening arises from simple averaging over nuclear positions, because resonances energies are typically quite sensitive to molecular geometry. It is thus surprising, and perhaps coincidental, that Sarpal et al. obtained approximately the correct width, especially given that they overestimated the maximum value. Otherwise similar calculations omitting target correlation (Sarpal et al., 1996a; Sarpal et al., 1996b) agree better with the other two calculations. In any event, taken together, these calculations demonstrate the necessity of including nuclear motion if detailed qualitative agreement with experiment is required.
133
ELECTRON-MOLECULE COLLISIONS
0.0
2.5
5.0
7.5
10.0
12.5
Electron Energy (eV) FIG. 1. Integral cross sections for elastic scattering of electrons by NzO. Short dashes, R-matrix calculation of Sarpal et al. (1996a, I996b); long dashes, R-matrix calculation of Morgan et al. (1 997); solid line, SMC calculation of Winstead and McKoy (1998); squares, measurements of Johnstone and Newell (1 993). The total scattering cross section measured by Szmytkowski et al. (1984) is also shown for comparison (crosses)
The disagreement among the calculations on the energy of the resonance arises from the difficulty of determining an appropriate representation of polarization effects in the underlying ab initio calculations. In both the R-matrix and SMC methods (and in the Kohn method as well), polarization is accounted for by virtual excitations into closed electronic channels. In other words, terms that have the form of a scattering electron in the presence of an electronically excited target molecule are included in the trial space of the variational method, even though the corresponding excitation processes (channels) have thresholds lying above the resonance energy, and the channels are thus "closed". These virtual excitations account for the dynamic correlation between the motions of the target electrons and the motion of the projectile by allowing the charge density of the target to relax from its isolated-molecule form. When an uncorrelated or only partially correlated description of the isolated-molecule wavefunction is used, however (as in virtually all electron-molecule collision calculations to date), fully representing the dynamic correlation within the collision complex-i.e., polarization-without inadvertently adding some degree of correlation to the description of the target molecule's wavefunction has proved difficult. (Recall that the overall wavefunc-
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Carl mnstead and Vincent McKoy
tion must be fully antisymmetric in the coordinates of all electrons.) Such overcorrelation tends to pull the calculated resonance position below the experimental position. In obtaining the SMC results shown in Fig. 1, we introduced (Winstead and McKoy, 1998) a new approach for determining the closed-channel configurations to include in such calculations that appears to be highly successful in avoiding overcorrelation; that we obtained a better resonance position than the other calculations is almost certainly attributable to the use of this approach rather than to any other factor. Differential elastic cross sections at 8 and 10 eV are shown in Fig. 2. While the agreement between calculation (Morgan et al., 1997; Winstead and McKoy, 1998) and experiment (Johnstone and Newell, 1993) is remarkably good at 10 eV, it is less so at 8 eV. Near 8 eV there is a second shape resonance, considerably broader than the 211 resonance discussed above, which is attributable to *Z symmetry of the overall wavefunction. An incomplete representation of the effects of this resonance, the neglect of nuclear motion, or some other factor may be responsible for the failure of either calculation to reproduce the observed form of the differential cross section, and further study will be required to pin down the origin of the discrepancy, which is particularly disappointing in light of the excellent agreement seen in the integral cross sections (Fig. 1) at the same energy. In sum, these studies of N 2 0 illustrate that current ab initio techniques can be quite successfbl in some respects, up to and including the semiquantitative prediction as well as the interpretation of low-energy elastic electron scattering phenomena; at the same time, both expected and unexpected departures in detail from the observed cross sections are possible at the levels of approximation that are currently common. Where similar calculations are used to make predictions in the absence of experimental data, these limitations must be understood. 2. Electron Impact Excitation of C,H4
Low-energy elastic electron scattering by ethylene (C2H4) is also marked by a prominent shape resonance which, as in NzO, can be associated with a n-type virtual orbital, in this case the antibonding n* orbital conjugate to the n orbital forming one part of the carbon-carbon double bond. Such low-lying virtual valence orbitals affect not only elastic scattering but inelastic scattering as well, for they give rise to low-lying electronically excited states that often have large electron impact excitation cross sections. In the case of ethylene, a sufficiently energetic projectile electron can promote one of the target's electrons from the n occupied orbital to this n* orbital, thereby producing either the a 3B,ustate or the A ' B l l ,state, depending on whether the overall spin coupling is triplet or singlet. Excitation cross sections for the triplet state have been computed both using the SMC method (Sun et al., 1992) and using the Kohn method (Rescigno and Schneider, 1992), while Allan (1994) has measured the differential cross section
135
ELECTRON-MOLECULE COLLISIONS L
-? "E0 2.5 2
2.0
I
0 4
-1.5
G
/
0
z! 1.0 0
/
Q,
m
0.5
m 0
uL 0.0 L 0
30
80
90
120
150
180
Scattering Angle (deg)
0 0
30
60
90
120
160
0
Scattering Angle (deg) FIG. 2. Differential cross sections for elastic scattering of electrons by N 2 0 at (a) 8 eV and (b) 1OeV Dashed line, R-matrix calculation (Morgan eta[., 1997); solid line, SMC calculation (Winstead and McKoy, 1998); squares, measured values (Johnstone and Newell, 1993).
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Carl Winstead and Vincent McKoy
over a range of angles at selected energies as well as the differential cross section at a fixed angle of 90" as a function of energy (see also Love and Jordan, 1995). Thus the a 3B,ustate of ethylene is one of the very few excitation processes in a polyatomic molecule for which high-level theoretical calculations can be compared to each other and to experimental data. In Fig. 3, the integral cross sections obtained by Kohn and SMC methods are shown. Clearly there are differences in detail, but the overall qualitative and quantitative similarity is striking. In Fig. 4, which shows the calculated differential cross sections at 15 eV impact energy along with the measured cross section at 14.18 eV, it may be observed that the qualitative agreement extends to the comparison with experiment. Not only do the calculations agree well with each other, but they also reproduce the pronounced backscattering observed experimentally. As observed by Allan (1994), the measured and calculated cross sections differ in magnitude by an almost uniform factor of approximately two at all energies and angles. Both calculations severely restricted the number of channels explicitly considered; the SMC calculation included only the electronic ground state and the a 3 B l , ,state, while the Kohn calculation also included the A ' B l r ,state (as a closed channel, below its own threshold). It is natural to suspect that the apparent overestimation of the cross section may be a shortcoming of such a few-channel approximation; however, further calculations and measurements, ideally covering a variety of excitation processes and molecules, will be needed to clarify this point.
3. Electron Impact Ionization of SF,
As discussed earlier (Section N D ) , the BEB model provides a simple but generally effective means of approximating molecular electron impact ionization cross sections. As an illustration of the BEB model, we show in Fig. 5 the ionization cross section for SF6 as obtained from Eq. (10) and the orbital binding and kinetic energies of Table 1, which were computed on a personal computer using the freely available program GAMESS (Schmidt et al., 1993). Also shown in the figure is the measured cross section of Rapp and Englander-Golden (1965). Considering the simplicity of the BEB model, the agreement with experiment is quite satisfactory, the principal difference being that the BEB result is about 15% smaller at its peak, which also occurs at somewhat lower energy. As was discussed earlier, similar (and often better) agreement is observed for many other molecules, although there are cases in which agreement appears to be poor (Vallance et al., 1997). On the whole, however, the BEB model combines wide applicability with a fairly high degree of reliability and is thus of great usefulness given the current absence of higher-level alternatives that are generally applicable to polyatomic molecules.
137
ELECTRON-MOLECULE COLLISIONS 1.5 n
"E0 W 4
I
0 1.0 r3
W
c: 0
3 0
~
Q)
0.5
m m 0
k
u 15
10
5
0
20
Impact Energy (eV) FIG. 3. Integral cross sections for electron impact excitation of the ii 3L?B,,(7')state of CzH4. Solid line, SMC calculation (Sun et a/., 1992); dashed line, Kohn calculation (Schneider et al., 1992).
"
0
30
60
90
120
150
180
Scattering Angle (deg) FIG. 4. Differential cross sections for electron impact excitation of the ii 3Bl,(Z') state of C2H4. Solid line, SMC calculation (Sun et al., 1992) at 15 eV impact energy; dashed line, Kohn calculation (Schneider eta/., 1992) at 15 eV; squares, measured values of Allan (1994) at 14.18e\! multiplied by a factor of 2.
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Carl WTnstead and Encent McKoy
10
100
Impact Energy ( e V )
1000
FIG. 5. Total cross section for electron impact ionization of SF6. Solid line, as calculated from Eq. (10) and the values in Table 1; squares, measured values (Rapp and Englander-Golden, 1965). TABLE 1 ORBITAL PARAMETERS USED IN BEB CALCULATION ON
SF6
Orbital
occupancy
Binding energy (ev)
Kinetic energy (ev)
2 4 6 2 2 6 2 6 4 2 6 6 6 6 6 4
2519.21 718.75 718.75 718.75 259.93 196.84 52.71 48.49 46.42 31.84 26.50 23.41 2 1.08 20.70 19.80 19.30
3297.17 1013.38 1013.44 1013.47 509.81 479.11 84.44 100.41 111.10 104.31 84.35 77.33 91.24 91.96 99.12 89.35
ELECTRON-MOLECULE COLLISIONS
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V. Areas for Future Progress A. TREATMENT OF ELECTRONIC EXCITATION Although the first-principles methods that we have described-the Kohn, SMC, and R-matrix methods-are all capable of treating electron impact excitation, and indeed have been employed to calculate a variety of excitation cross sections, the present state of this field of endeavor can hardly be considered satisfactory: The calculations are too difficult, and their results are too approximate. To a large extent, the origin of these interrelated shortcomings is understood, and a brief consideration of the factors limiting our ability to perform accurate calculations of excitation cross sections points out areas where further development will be rewarded. Though we focus here on the problem of electronic excitation because it is of such critical importance to the generation of reactive species in plasmas, this is not to say that the challenges remaining in the accurate treatment of vibrational excitation, even of diatomic molecules, are insignificant (see, e.g., Sun et al., 1995; Morrison and Sun, 1995; Morgan, 1995). It is important, first of all, to recognize that obtaining accurate descriptions of electronically excited states for molecules with more than two or three heavy atoms can in itself be a challenging problem, and one can hardly expect to obtain a quantitatively reliable transition amplitude without a good description of the state into which the transition occurs! Given the unfavorable scaling with molecular size of high-accuracy methods for computing excited states, we are likely to be confined to semiquantitative descriptions of larger polyatomics for some time. For smaller polyatomics, with up to perhaps a half dozen heavy atoms, it should generally be possible, given adequate computer resources, to obtain reasonably accurate results for at least the lowest few excited states by employing, for example, the configuration-interaction (CI) method (Shavitt, 1977) or the related multiconfiguration self-consistent field (MCSCF) method (Wahl and Das, 1977). The latter approach, when based on state-averaged electron densities, appears to be a promising route to obtaining a balanced and fairly compact description of several states at once. However, one must in either case be prepared to adapt the subsequent scattering calculation to the more sophisticated and complicated description of the target states. Both the Kohn and the R-matrix methods are presently capable of employing CI target wavehctions, and a similar capability is being implemented within the SMC method. Because in each method a larger (N 1)-particle configuration space for the target-plus-electron system is built up from the N-particle target configuration space, the size of the latter-and hence the accuracy of the description of the target states-will be limited by the necessity of keeping the scattering calculation manageable. Even when we have implemented such sophisticated methods for dealing with excited states in our electron-molecule scattering studies, however, the accuracy
+
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Carl Winstead and Vincent McKoy
of our calculations may still be limited by other factors. One of these is the high density of excited states in polyatomic molecules and the concomitant likelihood of strong interchannel coupling. Almost all electron impact excitation calculations on molecules to date have been carried out in a few-channel approximation, in which only the ground state and a handful of excited states (from as few as one up to about ten) are included in the calculation. The few-channel approximation works best when the thresholds of the excited states are well separated in energy and when this energetic isolation combines with factors such as the state symmetry to make the collision-induced interaction with states excluded from the calculation weak. Although such isolated states are found in polyatomic molecules-for example, the n* states of alkenes such as CzH4(Section IYE.2)they are the exception rather than the rule. For many molecules of interest in plasma processing applications, notably the hydrofluoro- and perfluoroalkanes used in etching, numerous excited states lie within a few eV of the first excitation threshold. Ideally, all states that are near each other in energy and might couple strongly would be included in the scattering calculation. In practice, it is not yet possible to carry out such extensive calculations, at least on a routine basis, and the consequences of employing a few-channel description instead have not yet been thoroughly investigated. A few benchmark calculations employing a large number of channels would be of immediate value, but in the long run, it seems likely that more elaborate methods, possibly allowing an implicit or collective representation of large numbers of channels that are not explicitly represented, may be needed. Such approaches, if developed, might also prove useful in the treatment of Rydberg excitations, where coupling not only to a manifold of discrete states but also to the ionization continuum appears to be important (Gil et al., 1994). It is worth mentioning separately that improvements in the treatment of electron impact excitation of molecules along the lines discussed above are likely to entail more than just increasing the size and complexity of the calculations, for increasing the size and complexity of the calculations will almost certainly aggravate numerical problems. Three examples may suffice. First, linear dependence within the one-particle basis is a more severe problem for larger molecules and more diffhe excited states, and it is exacerbated in collision calculations by the need to include functions describing the unbound electron in the vicinity of the molecule. Second, as target wavefunctions grow more sophisticated and additional channels are included, it will be increasingly necessary to account properly for the identity of electrons and to avoid overcompleteness in the ( N + 1)-particle space, which can give rise to spurious resonance effects in the cross section (Friedman, 1967; Feshbach and Friedman, 1968; Feshbach, 1992); current ad hoc methods (e.g., Lengsfield and Rescigno, 1991) for controlling such resonances are unlikely to perform satisfactorily in extended multiconfiguration calculations. Third, it may be necessary to design or
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redesign the computational approach for efficient execution on large-scale parallel machines in order to obtain sufficient computational resources (not only raw speed but also memory and disk space) for the most demanding multichannel calculations on polyatomic targets. B. REACTIVESCATTERING The greatest challenges facing the theoretical study of electron impact on molecules can be drawn together under the label reactive scattering, by which we mean collision processes whose final state consists of something other than the original collision partners (an electron and a neutral molecule). Dissociation into neutral fragments, dissociative attachment, and electron impact ionization (including dissociative ionization) all fall into this category, and for none of these is theory presently in a satisfactory state. Dissociative attachment can be divided into resonant and nonresonant cases. The resonant case is fairly amenable to theoretical treatment (Bardsley et al., 1964; O’Malley, 1966). In that case, the dissociation process can often be well modeled semiclassically in terms of the lifetime of the temporary anion and the survival probability for it to move from the geometry at which attachment occurs to the point beyond which the anion is more stable than the neutral. While a fully detailed theoretical treatment can be complex (O’Malley, 1966), the minimal ingredients to form a useful estimate of the cross section are an anion potential energy surface and a resonance lifetime or width, each of which can be computed in a fairly straightforward manner. Except at very low electron energies, cross sections both for formation of vibrationally excited resonances and for nonresonant dissociative attachment are vanishingly small; on the other hand, attachment processes for electrons at thermal energies can have extremely large cross sections (Christophorou et al., 1984). Even at these low energies, attachment may lead to dissociation if one of the fragments has a large enough electron affinity to compensate for the breaking of a chemical bond. Accurate theoretical treatment of dissociative attachment through vibrationally excited resonances or through nonresonant mechanisms poses considerably greater difficulties than shape-resonant attachment, in that an explicit coupling between nuclear and electronic motion becomes necessary. Moreover, the dissociative attachment cross section can be expected to vary markedly with the initial vibrational state of the molecule, an expectation verified by observations of a temperature dependence in some cases (Christophorou et al., 1984). There is both great need and much room for progress in this area. In the case of electron impact ionization, the difficulties are mostly a result of the altered boundary conditions in the final state, which make computational procedures developed to treat elastic scattering and discrete excitations inapplicable. Besides this technical issue, there is also the need to consider the interaction
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of the departing electrons through the long-range Coulomb force not only with the residual ion but with each other. McCurdy and coworkers (McCurdy and Rescigno, 1997; McCurdy et al., 1997) have recently discussed an approach to electron impact ionization that is adapted to molecular problems, but wider application will be necessary before its usefulness can be assessed. In the meantime, the simple yet often highly successful BEB model of Kim and Rudd, which was described in Section N D , remains the best means of computing electron impact ionization cross sections for molecules. From a practical point of view, neutral dissociation and dissociative ionization are perhaps the most important reactive collision processes. The whole purpose of the plasma is, after all, the generation of reactive fragments, and the surface chemistry is determined in great part by which reactive species are generated, and in what proportions. Especially in the case of dissociation to neutral fragments, theoretical methods have the potential to make a great contribution, because the experimental determination of accurate neutral-dissociationcross sections is very difficult. A principal problem is the quantitative detection of radicals that are produced in their ground electronic state and thus cannot be observed by emission spectroscopy. Detection of such radicals by mass spectroscopy is greatly complicated by the need to discriminate against ions produced by dissociative ionization of the parent gas, either within the main collision volume or within the ionization region of the spectrometer. Quantitative mass spectrometry has also been impeded by a dearth of absolute ionization cross sections for radicals, although the measurements of Becker and coworkers (e.g., Tarnovsky et al., 1993; Tarnovsky et al., 1994) are addressing this need. Indeed, although measurements of cross sections for dissociation to neutral products have been reported for a few gases relevant to plasma etching (Nakano and Sugai, 1992; Goto et al., 1994; Sugai et al., 1995; Toyoda et al., 1997), more recent results (Mi and Bonham, 1998; Motlagh and Moore, 1998) call those measurements into question. From a computational point of view, it can be argued that neutral dissociation and dissociative ionization are not true reactive scattering processes, in that, in almost all cases, the electron impact and nuclear motion components of the process occur on such different time scales that they can be decoupled and treated as separate events. That is, electron impact first promotes the nuclei vertically to a higher (excited or ionized) potential energy surface, then the nuclei move apart on that surface (possibly making one or more nonadiabatic crossings to new surfaces). The electron collision problem is thus solvable in the fixed-nuclei approximation, and the nuclear dynamics problem requires no “memory” of how the initial conditions were generated. However, even if the latter problem is thus simplified, it remains formidable-and it cannot be ignored, for in plasma modeling applications, one is likely to be more interested in the production cross sections for various dissociation fragments than in the underlying electron
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impact excitation and ionization cross sections. Of course, the problem of treating nuclear dynamics lies outside the scope of electron-molecule collisions, and it requires distinct expertise and computational approaches. Future developments in this area are thus likely to involve collaboration between researchers with complementary capabilities and interests.
VI. Acknowledgments Funding by Sematech, Inc., and by the U.S. Department of Energy, Office of Basic Energy Sciences, is gratefully acknowledged.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS RALF BASNER AND MARTIN SCHMIDT Institut fu’r Niedertemperatur Plasmaphysik Greifwald, Greifwald, Germany
KURT BECKER Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ
HANS DEUTSCH Institut f i r Physik, Ernst-Moritz-Amdt Universitat Greifwald, Greifwald, Germany I. Introduction.. .................. 11. Ionization-Cross-Section Meas
........
nts .........................................
A. General Remarks. ............................................................. B. The Fast-Neutral-Beam Apparatus ........................................... C. The High-Resolution Double-Focusing Mass Spectrometer ................. 111. Semiempirical Calculation of Total Single Ionization Cross Sections .......... 1V Ionization Cross Sections of SiH (x = 1 to 4) and of Selected Si-Organic Compounds .................................... A. Silane and Its Radicals ............... B. Tetramethylsilane [Si(CH,),] ......... C. Tetraethoxysilane [Si(O-CH2-CH3)4......................................... D. Hexamethyldisiloxane [(CH,),-Si-&Si-CH3),] ............................ V Comparison with Ion Formation Processes and Ion Abundances in Plasmas.. . VI. summary.. ........................................................................ VII. Acknowledgments ............................. VIII. References ........................................................................
147 149 149 151 153 156 160 160 168 170 172 177 181 182 182
I. Introduction Nonthennal low-temperature plasmas used in plasma processing and other plasma-assisted applications are mainly composed of hot electrons (average electron energy 0.5 to 5eV) and cold ions and neutral gas molecules, which both have energies corresponding to temperatures in the range 300 (room temperature) to 600K. In a stationary (steady-state) plasma, the ion loss rate equals the ion production rate. Electron impact ionization of the neutral heavy particles in the plasma (atoms and molecules in their ground state or in excited states) is an important ion-formation process. In many plasmas it is the dominant ion-formation process. The electron impact ionization of molecules produces the 147
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parent molecular ion and, by dissociative ionization and dissociation, fragment ions and neutral products such as radicals and smaller neutral stable molecules. Thus, the dissociative ionization is not only important for the charge carrier production, but also an essential step in initiating plasma chemical reactions. In the plasma volume, the ions may undergo a variety of ion-molecule reactions. However, the ions are even more important in the sheath regions of the plasma near the wall of the plasma reactor or near the electrodes, where there is a large potential drop. In this region, the ions are the main carrier of lunetic and potential energy, which may be transferred to the surface, where, in addition to recombination, other processes such as sputtering, chemical reactions, etc., may be triggered. The efficiency of the ionization process for a given electron energy and a given target is determined by the ionization cross section of that particular target. The partial ionization cross sections describe the formation of the various fragment ions, and the total ionization cross section is obtained as the sum of all partial ionization cross sections. The ion-formation rate in a plasma is determined by the rate coefficient multiplied by the electron and neutral gas density. The rate coefficient for this two-body collision is given by the integral over the product of the ionization cross section a(u),the velocity of the electrons v (the velocity of the heavy molecules is neglected), and their velocity distribution f ( u ) (no and n, denote the number density of the target molecules and the electrons, respectively):
R
s
= none a(u)uf(u)du
Ionization cross sections have been determined with high accuracy (to better than 10%) for simple atomic gases like the noble gases, some metallic vapors such as Hg, and atmospheric gases. The database on ionization cross sections for most of the feed gases used in plasma chemical applications and, in particular, in plasma processing is very limited. Since there is an ever-increasing multitude of plasma-assisted processes using a growing number of different precursor molecules, the need for ionization-cross-section data for molecules and radicals continues to grow at a rapid pace. Organic silicon compounds (silicon organics) are an important class of precursors relevant to plasma chemistry, particularly for plasma-assisted thin-film deposition applications. A slight, often minimal, variation of the discharge conditions of a deposition plasma containing Si-organic compounds can have a profound impact on the properties of the deposited film, e.g., it can change the film properties from those of an inorganic film to those of an organic film. The fabrication of microelectronic and other semiconductor devices relies on the plasma-assisted deposition of SiO,xfilms using admixtures of Si-organic compounds and 0,. The use of “mild” discharge conditions (i.e., low
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 149
ion energies) leads to the formation of Si-organic polymeric films with the wellknown and desirable properties of the silicones. This article describes recent advances in the experimental determination of electron impact ionization cross sections for silane (SiH,); its radicals, SiH,v (x = 1 to 3); and the Si-organic molecules tetramethylsilane (TMS), Si(CH& tetraethoxysilane (TEOS), Si(O-CH,-CH3),; and hexamethyldisiloxane (HMDSO), (CH,),-Si-O-Si-(CH,)3, which is one of the simplest siloxane compounds. These are “model” substances, and the results obtained for these species may be used in efforts to predict the ionization properties of other, more complex Si-organic molecules. The ionization cross sections of the stable compounds were measured using a high-resolution double-focusing mass spectrometer. The cross-section data for the radicals were obtained in a fast-neutralbeam apparatus. In the following sections of this article, we describe the principles of ionization cross-section measurements, including a brief description of the fast-beam apparatus and the high-resolution double-focusing mass spectrometer employed in the present studies. A comprehensive review of semiempirical calculations of total ionization cross sections is given. Comparisons between these calculated cross sections and the experimental results are presented. The decomposition of the various molecules in a low-temperature plasma is discussed on the basis of the measured ionization-cross-section data, and comparisons are made with the results of in situ plasma diagnostics studies using mass spectrometric techniques.
11. Ionization-Cross-Section Measurements We review the general concept of ionization-cross-section measurements and describe briefly the two experimental techniques employed in the particular ionization-cross-sectionmeasurements discussed in this article. A comprehensive review of electron impact ionization is given in the book by Mark and Dunn (1985). A. GENERALREMARKS
The ionization of a molecule by controlled electron impact under well-defined single-collision conditions is most commonly described in terms of an ionization cross section. The experimental determination of ionization cross sections requires in principle an arrangement consisting of a vacuum chamber in which collisions occur between a well-characterized beam of electrons and the target gas under study at a reduced pressure (ion source). The reduced pressure is one of several experimental conditions that have to be employed in order to ensure welldefined single-collision conditions in the ion source. The target gas is most
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commonly admitted into the ion source region as a static gas target or as a welldefined molecular beam. The resulting product ions are extracted from the ion source and detected by a suitable ion detector. The number of positive ions produced in the ion source, the ion current I+, is given by the following expression:
I f = I;N,.cJ.L
(2)
where I, refers to the electron current, No denotes the neutral gas density, L specifies the length of the path of the electron beam in the target gas, and the quantity o is the ionization cross section. The ionization cross section CJ depends on the relative velocity of the colliding electrons and the target molecules (or on the energy of the electrons relative to the energy of the target gas molecules). In electron collisions, the relative collision energy in the center-of-mass system is to a very high approximation identical to the electron energy in the laboratory system, since the electrons have a much smaller mass than the target molecules. The formation of a positive ion from a molecule requires a minimum collision energy, a threshold for ionization or an appearance energy for the formation of that ion. Typical appearance energies for the formation of a singly positively charged parent ion for most molecules are in the range of 5 to 15 eV; appearance energies for multiply charged parent ions and for fragment ions can be significantly higher. The determination of the total ionization cross section of a molecule requires in principle a careful measurement of all quantities in Eq. (2). Partial ionization cross sections can be obtained if the detection of the ion current is restricted to a particular product ion. In the latter case, a mass selective device-e.g., a mass spectrometer-has to be employed. The most commonly used mass spectrometers for this purpose are magnetic, radio-frequency, quadrupole, and time-of-flight mass spectrometers. In all cases, the detection sensitivity of the instrument may vary with the mass of the detected ions and must be known accurately. Another experimental parameter that has to be known accurately is the electron beam current. The beam current either has to be kept constant with sufficient stability during the course of a cross-section measurement or has to be measured as the energy of the ionizing electron beam is varied, with the recorded ion signal then being normalized to the varying beam current. Typical impact energies in ionization-cross-section measurements range from about 5 up to 100 or 200eV, or in some cases even up to 1 or several key In ionization-crosssection measurements relevant to plasma physical and chemical applications, it is sufficient to operate the electron beam with an energy width of about 0.5 eV [full width at half maximum (FWHM)], which can be obtained easily by using a tungsten filament cathode or an indirectly heated oxide-coated cathode as the source of the electrons. The density of the target gas and the length over which the electron beam interacts with the target gas are quantities that are not easy to
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 151
measure in most ionization cross-section apparati. Therefore, it has become common practice to utilize normalization procedures. This typically involves gases of well-known ionization cross sections such as Ar or Kr and requires the simultaneous measurement of the known ionization cross section and the unknown cross section for the target under study under essentially identical experimental conditions. The reliability of the mass spectrometric technique for absolute measurements of parent ionization cross sections and also for fragment ionization cross sections was demonstrated by Mark and coworkers (Stephan et al., 1985; Leiter et al., 1989; Margreiter et al., 1990; Poll et al., 1992). These authors pointed out that discrimination effects and the loss of energetic fragment ions must be accounted for in order to obtain reliable dissociative ionization cross sections. Excess kinetic energies of several electron volts per ion can be observed frequently for light fragment ions produced by dissociative ionization of heavy parent molecules via states with higher potential energies as a consequence of momentum conservation (Mark, 1984). Excess kinetic energies of the ions can influence ionization-crosssection measurements primarily in two ways: (1) in the ion source because of a lower extraction efficiency for energetic ions and (2) in the ion transport optics, which is typically less efficient in transporting energetic ions from the ion source through the mass analyzer to the ion detector. Practical solutions of this ion discrimination problem involve extensive ion trajectory modeling in conjunction with in situ experimental studies such as ion-beam sweeps across the entrance slit of the mass spectrometer in two dimensions to determine the shape of the ion beam for each product ion. If necessary, modifications have to be made into the ion source, the ion extraction stage, and the ion transport system in order to minimize and/or to quantify discrimination effects. The fast-beam technique has been found particularly useful in the study of energetic fragment ions, since it employs an “open” system with no limiting apertures between the ion source and the ion detector (see discussion below). A new approach for minimizing the effects of ion discrimination involves the use of position-sensitive detectors in conjunction with an “open” time-of-flight mass spectrometer as described, for example, by Straub et al. (1995). B. THEFAST-NEUTRAL-BEAM APPARATUS A detailed description of the fast-beam apparatus and of the experimental procedure employed in the determination of absolute partial ionization cross sections has been given in previous publications (Wetzel et al., 1987; Freund et al., 1990; Tamovsky and Becker, 1992; Tamovsky and Becker, 1993). For the measurements of the cross sections of silane radicals, a dc discharge biased at typically 2 to 3 kV through SiD, served as the primary ion source. Deuterated rather than protonated target species were used in these studies to facilitate a
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better separation of the various product ions from a given parent [ionization cross sections are insensitive to isotope effects (Miirk and Egger, 1977; Mark, et al., 1977; Basner et al., 1995a) to a very high degree of approximation]. The primary ions were mass selected in a Wien filter, and a fraction of them were neutralized by near-resonant charge transfer in a charge-transfer cell filled with an appropriately chosen gas for resonant or near-resonant charge transfer. Efficient charge transfer is often not critically dependent on an exact match of the ionization energies of the charge-transfer partners. The residual ions were removed from the target gas beam by electrostatic deflection, and most species in Rydberg states were quenched in a region of high electric field. The neutral beam was subsequently crossed at right angles by a well-characterized electron beam (5 to 200 eV beam energy, 0.5 eV FWHM energy spread, 0.03 to 0.4mA beam current). The product ions were focused in the entrance plane of an electrostatic hemispherical analyzer, which separates ions of different charge-to-mass ratios (i.e., parent ions from fragment ions and singly from multiply charged ions). The ions leaving the analyzer were detected by a channel electron multiplier (CEM). A schematic diagram of the fast-beam apparatus is shown in Fig. 1. The neutral beam density in the interaction region can be determined from a measurement of the energy deposited by the fast neutral beam into a pyroelectric crystal whose response is first calibrated by a well-characterized ion beam, as discussed by Wetzel et al. (1987). As an alternative, the well-established Kr or Ar absolute ionization cross sections (known to better than 5%) can be used to calibrate the pyroelectric crystal. The calibrated detector is then used to determine the flux of the neutral target beam in absolute terms. This procedure avoids the frequent and prolonged exposure of the delicate pyroelectric crystal to fairly intense ion beams (Freund et al., 1990; Tarnovsky and Becker, 1992).
FIG. 1. Schematic d i a p m of the fast-neutral-beam apparatus.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 153
It was established for each target that all fragment ions (except for D+; see discussion below) with an excess kinetic energy of less than 2.5 eV per fragment ion are collected and detected with 100% efficiency using a combination of in situ experimental studies and ion trajectory modeling calculations (Tarnovsky et al., 1993). Furthermore, careful threshold studies revealed little evidence of the presence of excited target species (vibrationally excited species, metastables, and species in high-lying Rydberg states) in the incident neutral SiD,T(x = 1 to 3) beams. These experimental checks are necessary for any target studied using the fast-beam technique in order to ensure that the measured cross sections are free from systematic uncertainties to the maximum extent possible (Wetzel et al., 1987; Freund et al., 1990; Tarnovsky and Becker, 1992; Tarnovsky and Becker, 1993). In addition, experimental checks pertaining specifically to hydrogen- and deuterium-containing targets were carried out as described in detail in the literature, as discussed by Tarnovsky et al. (1996a). Little evidence of the presence of D+ fragment ions from the dissociative ionization of all three SiD, targets was found. On the other hand, ion trajectory modeling calculations using a standard software package (SIMION 1992, 1996) suggested significant losses of D+ fragment ions from SiD, for excess kinetic energies as low as 0.5eV per fragment ion. This rendered it difficult, if not impossible, to determine reliable absolute partial D+ ionization cross sections for any of the SiD, targets using this experimental approach. C. THE HIGH-RESOLUTION DOUBLEFOCUSING MASSSPECTROMETER The ionization cross-section measurements for SiH, and for the stable Si-organic molecules were carried out in a high-resolution ( m / A m = 40,000) doublefocusing sector field mass spectrometer (MCH 1310) of E-H configuration with a Nier-type electron impact ion source, as described by Tarnovsky et al. (1994) and Basner et al. (1997a). A schematic diagram of the apparatus is shown in Fig. 2. The ion source was operated at a gas pressure in the range 0.1 to 1 mPa compared to a background pressure of 0.001 mPa. The pressures were measured by a spinning-rotor viscosity gauge. The gases (SiH, and Ar) or liquids (the Si organics) were placed in a reservoir connected to the ionization chamber via a UHV high-precision leak valve. The vapor pressure at room temperature of the liquid Si organics is sufficiently high to produce a target gas pressure in the ion source in the mPa regime. The electron gun was operated with a stabilized electron beam current of 10pA emitted from a directly heated tungsten band cathode. The impact energy was vaned from 5 up to 100 eV The energy spread of the electron beam, which is collimated by a weak longitudinal magnetic field of 200 G, was about 0.5 eV (FWHM). The temperature of the ion source could be varied between 100 and 200°C. The ions were extracted from the ionization region by a penetrating electric field. The acceleration voltage between the ion
154
R. Basner; M. Schmidt, K. Becker; and H. Deutsch
i
i C
FIG. 2. Schematic diagram of the high-resolution double-focusing mass spectrometer. The insert shows an enlarged view of the ion extraction optics for the high-extraction-efficiency mode. P,pusher electrode; C, collision chamber; B, electron beam (z mrection); S , , collision chamber exit slit; S,, penetrating field extraction slit; S3, grounded slit; S4 and S,, deflector electrodes.
source and the entrance slit of the mass spectrometer was 5 kV The ion repeller potential was kept at the potential of the ionization chamber. Argon, whch was used as reference gas, was always added to the target gas for calibration purposes. The ion efficiency curves (relative ionization cross sections) for Ar and the target gas under study in a well-defined mixture were measured simultaneously in an effort to ensure identical operating conditions for the detection of the ions from the target under study and the Ar ions. The measured relative partial ionization cross sections were put on an absolute scale by normalizing the total single ionization-cross-section curve to the total Ar ionization-cross-section curve with a value of 2.77 x cm2 at 70 eV (Rapp and Englander-Golden, 1965). Earlier measurements of the NF, parent and fragment ionization cross sections in our apparatus (Tarnovsky et al., 1994) revealed a significant loss of energetic fragment ions caused by the known discrimination effects. As a consequence, the ion optics for the extraction, acceleration, and deflection of the ions was rebuilt based on ion-trajectory simulations in conjunction with in situ experimental studies in an effort to minimize and/or quantify the discrimination against energetic fragment ions. The modified mass spectrometer could thus be operated either in a high-mass-resolution mode (with significant discrimination effects present) or in a high-extraction-efficiency mode (by partially sacrificing the highmass-resolution capability). Figure 3 shows the ion-extraction efficiencies in the two modes of operation as a function of the excess kinetic energy of the fragment ion as obtained from ion-trajectory simulations. The curves demonstrate that significant ion losses occur in the high mass resolution mode for excess energies as low as 0.1 eV and that about 50% of all ions are lost for an excess energy of
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 155 I
0
'
I
2
'
I
4
'
I
'
6
I
8
'
I
10
Excess kinetic energy [ eV ] FIG. 3. Calculated ion extraction efficiency of the ion source as a function of excess kinetic energy per fragment ion for the high-mass-resolution mode (diamonds, +) and the high-extractionefficiency mode (circles, O), using the SIMION ion trajectory modeling code (1992, 1996).
1 eV per fragment ion. By contrast, there are essentially no ion losses for excess energies of up to 3 eV per fragment ion when the instrument is operated in the high-extraction-efficiency mode. Recent ionization-cross-section measurements for SO, (Basner et al., 1995a) carried out in our modified mass spectrometer as well as in the fast-neutral-beam apparatus, which is less sensitive to discrimination effects, confirm experimentally that the mass spectrometer operated in the high-extraction-efficiencymode can detect energetic fragment ions efficiently as long as the excess kmetic energy is not too large [e.g., less than about 1 eV per fragment in the case of SOf, S+, and O+ from SO,, as discussed by Basner et al. (1995a)l. For all measurements reported here, the data acquisition procedure included a sweep of the ion beam across the entrance slit of the mass spectrometer, and absolute cross sections were obtained by integrating the ion signal over the horizontal ion beam profile (see Fig. 4). Mark and coworkers (Leiter et al., 1989; Poll et al., 1992) demonstrated that the determination of reliable ionization cross sections for energetic fragment ions requires in general a sweep of the ion beam horizontally and vertically across the entrance slit of the mass spectrometer and an integration of the ion signal over the measured two-dimensional beam profile. These authors also pointed out that the vertical sweep is important only for very energetic fragment ions and that reliable cross sections for less energetic fragment ions can be obtained by integrating over the horizontal profile alone. The high-resolution mass spectrometer used here allows us to obtain horizontal beam profiles but no vertical beam profiles. As discussed below, the only
156
R. Basnec M. Schmidt, K. Becker, and H. Deutsch
0.0
I -1000
-750
-500
-250
0
250
500
750
1000
Deflection voltage [ V ] FIG. 4. Normalized ion currents of the extracted H' (squares, W), H2+ (diamonds, +), Si+ (triangles, A),and SiH,' (circles, 0 )fragment ion beams produced by dissociative electron impact ionization of silane, SiH,, as a function of the horizontal deflection voltage.
fragment ions studied here for which a vertical integration could result in a noticeable change in the cross section are the CH,+ ions from TMS, HMDSO, and TEOS and the H+ and H2+ ions from SiH,. In both the fast-beam apparatus and the double-focusing mass spectrometer, absolute cross sections can be determined with uncertainties of f 15% for the parent ionization cross sections and f 18% for the dissociative ionization cross sections. These error margins include statistical and all known systematic uncertainties and are typical for ionization-cross-section measurements carried out with this apparatus (Tarnovsky and Becker, 1992; Tarnovsky and Becker, 1993).
111. Semiempirical Calculation of Total Single Ionization Cross Sections Up until about 5 years ago, the calculation of absolute electron impact ionization cross sections for molecules relied largely on empirical and semiempirical methods and on simplistic additivity rules because of the complexity of more rigorous calculations for these processes and these targets (Younger and Mark, 1985). In the past 5 years, several new developments have emerged:
1. Several modifications of conventional additivity rules (Otvos and Stevenson, 1956; Fitch and Sauter, 1983; Deutsch and Schmidt, 1985)
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 157
that attempt to account for molecular bonding in different ways (Bobeldijk et al., 1994, Deutsch et al., 1997). 2. The Deutsch-Mark (DM) formalism, which combines a Gryzinski-type energy dependence of the cross section with quantum mechanically calculated molecular structure information (Deutsch et al., 1993; Deutsch et al., 1994) 3. A binary-encounter-Bethe (BEB) theory (Kim and Rudd, 1994; Hwang et al., 1996; Kim et al., 1997) 4. A theory based on the calculation of the maximum in the electron impact ionization cross section as a function of the electron-molecule approach geometry and subsequent averaging over all orientations (Vallence et al., 1996; Harland and Vallence, 1997) In this article we will compare total single ionization cross sections for the Sicontaining compounds calculated using the modified additivity rule of Deutsch et al. (1997, 1998a, 1998b) and, where available, with the BEB method of Kim and coworkers (Kim and Rudd, 1994; Hwang et al., 1996; Kim et al., 1997). The BEB theory of Kim and collaborators (Hwang et al., 1996; Kim et al., 1997) is a simpler version of the more rigorous binary-encounter-dipole (BED) theory, which was first developed for the calculation of atomic ionization cross sections by Kim and Rudd (1994). The BED model combines the binary-encounter theory and the Bethe theory and expresses the total single-ionization cross section in terms of the binding energies of the occupied orbitals, the average kinetic energy of the bound electrons in their orbitals, the electron occupation number, and the continuum dipole oscillator strength. The total single ionization cross section is then obtained by summing over all occupied orbitals. The simpler BEB model eliminates the need to know the continuum dipole oscillator strength, which is known only for the simplest molecules, by approximating this quantity by a simplified expression (Hwang et al., 1996; Kim et al., 1997). The modified additivity rule introduced by Deutsch et al. (1997) attempted to account for the effects of molecular bonding by introducing empirically determined weighting factors that depend on the atomic orbital radii and the electron occupation numbers of the various atomic orbitals. A detailed comparison with existing molecular ionization cross section data for molecules of the form AB, suggested the following explicit form of the ionization cross section o+(AB,) of such a molecule:
Here rA,r5 and tA,tBrefer to, respectively, the radii and the effective number of electrons of atoms A and B, and o+(X)denotes the total single electron impact
158
R. Basner, M. Schmidt, K. Becker, and H. Deutsch
ionization cross section of the atoms X.The weighting factorsf, and f E are given by
The exponents a and p are explicitly dependent on rA,r,, we have
tA,and t,. Specifically,
where the functions g , and g2 are shown in Fig. 5 and the arguments are given by
The two curves in Fig. 5 were obtained empirically from a fitting procedure using a few benchmark cross sections as discussed by Deutsch et al. (1997, 1998a). A special case arises for hydrides where the second atom has a radius smaller than the radius of the H atom, (e.g., H,O, OH, and HF) and in cases where the radii of both atoms are smaller than the radius of the H atom (NO, N20, and NO,). The ionization cross section for these molecules is determined by geometric effects
Parameter a', p',
f,6*
FIG. 5. Functional dependence of the exponents a , 8, y, and 6 on a* , p*, y * , and 6* (see text for further details).
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 159
alone, which is accomplished by setting the factors containing the ratios of the electron numbers equal to 1, which leads to
+
o+(AB,) = [ ( n & / ( n n r i ) ~ " a + ( ~[)( n n 4 / ( n r $ ) l P n a + ( ~ )
(7)
where the exponents c1 and B are now also determined solely by the ratios of the atomic radii; i.e., the factors containing the effective electron numbers are set equal to unity. The above-described modified additivity rule can be extended (Deutsch et al., 1998a; Deutsch et al., 1998b) to the calculation of electron impact ionization cross sections for molecules of the form AXBY,A,B,C,, and A,B,C,D,. The corresponding expressions in their simplest form (factors of n that cancel have been omitted for simplicity) are a+(AXBY)= [(.;>/(.~)l*[X5A/(xrA+ Y4e)lxa+(4
+ [oi~~)/(x~;)l"lVre/(x5A +Y5A11YO+@> O+(A,B."C,) = [oi + z ) / x I " [ ( r 2 / ( 4+ 1 . 2 c ) I " [ X 4 A / o i t B + z t c ) l x o + ( 4 + [(x + z ) / Y l ' w > / ( r ;+ r;>l"re/(xrA + ztc)lva+(B)
+ [(x+r)/zl'"(1.2c)/(r;+ r31"z5c/(x4'4 +Yte)lzo+(C)
(8)
(9)
and
+ t + 4/PlX[(r;)/(4 + 4 + &I" x [PtA/(&i + t t c + utn)ba+(A>
0 + ( A P ~ , C , 4=) Ks
+ [@ + t + u)/s]"(r3/(r; + r; + r31"
+ t t c + urD>ls~+(B) + [o, + s + 4 / j I Y x [<.$>/(d + 4 + &IY[ttc/@t,4+ S t B + uto>lta+(C) + Lo, + s + t ) / u I " ( r W d + 4+ .$>I6 x [ 4 d @ t A + sre + t t c > l u ~ + ( ~ ) x
[&/@
(10
The exponents c1 and P for the molecules AxBJ,are determined in a fashion similar to that used for the case of the molecules AB,. The exponents a, P, and y for the molecules A,B& are obtained from Fig. 5 using the functions
fz= g,(x*),P = g*(P*>>Y = &(Y*) where the exponents a*, p*, and y* are given by
+ .c)l[rA/(te + 4 C ) l P* = c(re/(rA + rc)I[te/(t.4+ tc>l Y* = [(rc/(l;l+ Q ) " C / ( t A + tell a* = [ ( Y A / ( %
(1 1)
( 12a)
(12b) (12c)
160
R. Basner; M. Schmidt, K. Becker; and H. Deutsch
The exponents a, p, y, and 6 for the molecules A,B,C,D, are obtained from Fig. 5 using the fimctions = g,(a*>,
P = g2(P*),
Y = gz(Y*),
6 = g2(6*)
(13)
where the arguments are given by a* = [ T A / ( y B
+ TC + T D ) l [ l A / ( l B + l C + 501
(14a)
p*
+ rC +
( 14b)
= [TB/(rA
Y* = [TC/(yA
+ + (D)] + rB + r D ) l [ t C / ( t A + 58 + lo)] + r B + r C ) l [ t D / ( t A f ( B + 5C)I T D ) I [ ~ B / ( ~ A (C
(1 4 4
(1 4 4 6* = LTD/(TA We note that the factors containing the effective electron numbers in the expression for a* and p* are set equal to unity for the molecules A,B,,, as in the case of the molecules AB,, if one or both atoms have radii smaller than the radius of the H atom. In the case of the molecules A,B,vCz and A,B,C,D,, the same applies, if the radii of the atoms are smaller than or equal to the radius of the H atom.
IV. Ionization Cross Sections of SiH, (x = 1 to 4) and of Selected Si-Organic Compounds In this chapter we present a summary of ionization cross-section results (absolute partial and total ionization cross sections and appearance energies) for SiH,, for the SiH, (x = 1 to 3) radicals, and for three selected Si-organic compounds. A. SILANE AND ITS RADICALS Silane, SiH,, is used for plasma-assisted thin-film deposition of amorphous silicon (Turban et al., 1980; Robertson et al., 1983; Doyle et al., 1990; Tochikubo et al., 1990), silicon nitride (Konuma, 1992) , SiO, (Tissier et al., 1991), and (Ti,Si)N (He et al., 1995). Complete sets of electron collision cross sections for S M , for the modeling of silane-containing plasmas have been proposed by several authors (Morgan, 1992; Nagpal and Garscadden, 1994; Kim and Ikegawa, 1996; Penin el al., 1996). Electron impact ionization cross sections of silane (Chatham et al., 1984; Haaland, 1990; Krishnakumar and Srivastava, 1995; Basner et al., 1997a) and of its radicals, SiH, (x = 1 to 3) (Tarnovsky et al., 1996b), have been studied by several authors. Absolute partial SiH4 ionization cross sections were reported by Chatham et af. (1984) and Krishnakumar and Srivastava (1995), who used quadrupole mass spectrometric methods. Basner et al. (1997a) employed the double-focusing mass spectrometer described above.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 161
Haaland (1990) used a Fourier-transform mass spectrometer (FTMS) and estimated the cross-section values for the Si-containing ions by scaling the published values of Chatham et al. (1984) and Morrison and Traeger (1973) to his absolute values at 50 eV: The resulting cross sections of Haaland (1990) are lower by a factor of 2 to 3 compared to the data reported by the other groups. The partial ionization cross sections reported by the other groups show differences in the absolute values of about 30% for SiH,+ and less for the other Si-containing fragment ions. Significant differences, however, were reported by the various groups for the H+ and H,+ partial ionization cross sections. Haaland (1990) was the only author who did not report any data for H+ and H,+ fragment ions. Here we present a detailed summary of the data measured with the doublefocusing mass spectrometer. The mass spectrum of silane was measured with the mass spectrometer in the high-extraction efficiency mode at an electron impact energy of 70 eV: The measured spectrum agrees well with the spectrum given in the standard Eight Peak Index (1974) and in other mass spectrometric databases, with the exception of the H+ and H,+ ions. The reasons for the discrepancy for these two ions are (1) the low acceptance of conventional mass spectrometers (e.g., quadrupole mass spectrometers) for nonthermal ions with a significant amount of excess kinetic energy and ( 2 ) the possibility of an additional production mechanism of hydrogen from silane in the ion source by pyrolytic decomposition of silane at the hot filament of the electron gun. Doubly charged ions were detected at m / z = 14, 14.5, 15, and 15.5, but with very low intensities of 0.3% (Si2+), 0.9% (SiH2+), 0.6% (SiHi+), and 0.06% (SiH:+) of the intensity of the SiH2+ signal, which is the most intense peak in the spectrum (base peak). No evidence of the presence of SiH,+ ions was found. We note that the mass resolution of the mass spectrometer even in the high-extraction-efficiency mode was sufficient to separate the small ion signal arising from the ionization of background N, from that of 2*Si+. The measured appearance energies and cross sections (at 70 eV) are presented in Table I. The values measured by our group are in good agreement with the data from other authors and with tabulated data of thermochemical and ion formation energies. The comparatively low appearance energies of SiH2+ and Si+ may be related to a breakup of the parent silane molecule in which a stable H, molecule is removed, followed by a molecular rearrangement of the residual ion in the case of SiH,+. Selected partial ionization cross sections are shown in Fig. 6 together with data of Chatham et al. (1984) and Krishnakumar and Srivastava (1995). The partial cross section (see also Table I) for each silicon-containing fragment ion was obtained by adding the various isotope contributions. The total single ionization cross section of SiH, is shown in Fig. 7. Also shown are the data of Chatham et al. (1984), Krishnakumar and Srivastava (1995), and Haaland (1 990). The experimental results are compared with calculations using the modified additivity rule discussed earlier and with the result of the BEB model (Ali et al.,
162
R. Basner. M. Schmidt, K. Becker. and H. Deutsch TABLEI
MEASURED APPEARANCE ENERGIES
AND PARTIAL IONIZATION CROSS SECTIONS AT 70 eV IMPACT ENERGY FOR THE VARIOUS FRAGMENT IONIZATION IONS PRODUCED BY DISSOCIATIVE OF SlLANE
m/z
Ion
AE (eV)
Cross section at 70eV (10-16cm2)
31 30 29 28 2
SiH: SiH: SiH+ Sit
12.2f0.5 11.6f0.6 15.1f0.5 13.6f0.5 24.3 f 1.O 24.5 f0.6 Total
1.67 2.18 0.64 0.59 0.035 0.28 5.40
1
0
H$ H+ SiH,
20
40
60
80
100
Electron energy [ eV ]
FIG. 6. Absolute partial SiH3+ and SiH2+ and Sic ionization cross sections as a function of electron energy. The squares represent the data of Basner et al. (1997a) for, respectively, SiH,+ (filled squares) and SiH2+ (open squares); circles refer to the data of Chatham et al. (1984) for, respectively, SiH3+ (filled circles) and SiH2+ (open circles); the triangles refer to the data of Krishnakumar and Srivastava (1995) for, respectively, SiH,+ (filled triangles) and SiH2+(open triangles); the Si+ data are also shown (+, Basner et al., 1997a; *, Chatham et al., 1984; and x, Krishnakumar and Srivastava, 1995).
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 163
0
20
40
60
80
100
Electron energy [ eV ]
FIG. 7. Absolute total single SiH, ionization cross section as a function of electron energy. The various symbols refer to the following data: triangles (A), Basner et al., (1997a); circles (O),Chatham et al. (1984); squares (W), Krishnakumar and Srivastava (1995); diamonds (+), Haaland (1990); the dashed line represents a calculation using the modified additivity rule (see text for details), and the dotted line refers to the BEB calculation of Kim and coworkers (Ali et al., 1997).
1997). There is excellent agreement between the calculations based on the modified additivity rule and the BEB model, and the experimental results of Chatham et al. (1984) and those of Basner et al. (1997a). Up to 30 eV, there is also good agreement with the data of Krishnakumar and Srivastava (1995). However, their cross section declines more rapidly with increasing impact energy. The cross section of Haaland (1990) is significantly smaller in the entire energy range. Table I also lists the partial ionization cross sections measured by Basner et al. (1997a) for the various silane fragment ions. A detailed study of the formation of H2+ ions by Basner et al. (1997a) showed that H,+ in the ion source is formed (1) by electron impact dissociative ionization of SiH4 with an appearance energy of 24.5 eV and a significant amount of excess kinetic energy and (2) by electron impact ionization of H2 formed by pyrolytic decomposition of SiH4 at the hot filament with an appearance energy of 15.4eV and no excess energy. The rate coefficient for electron impact ionization depends very sensitively on the ionization energy, as is demonstrated in Fig. 8. We calculated the ratio of the rate coefficients for total ionization of silane for different electron temperatures using the total ionization cross sections of Chatham et al. (1984) and Krishnakumar and Srivastava (1995). The primary difference between the two crosssection data sets at low energies is a 0.6-eV difference in the measured ionization energy [with Chatham et al. (1984) reporting the lower value]. As expected, the
164
R. Basnev, M. Schmidt, K. Beckev, and H. Deutsch 26
\.
2 42 [
04
06
08
10
12
'.4
16
18
20
22
Electron Temperature [ eV 3
FIG. 8. Ratio of the rate coefficient for total ionization of silane for different electron temperatures using the total ionization cross sections of Krishnakumar and Snvastava (1995) (labeled kKnahna,)and Chatham et al. (1984) (labeled kChatllam).
rate coefficient based on the lower ionization energy is always larger. It is, however, interesting to see how rapidly the difference between the two rate coefficients increases with decreasing electron temperature; it reaches a factor of 2 for mean electron energies below about 0.7 eV even though the 0.6 eV energy difference is comparatively small. For each of the three free radicals SiD, (x = 1 to 3), relative partial parent ionization cross sections were measured in the fast-beam apparatus from threshold to 200 e y followed by a measurement of the relative partial cross sections for the corresponding fragment ions. The measurements were limited to singly charged ions, since cross sections for the formation of doubly charged ions were found to be at or below the detection sensitivity of the fast-beam apparatus (i.e., peak cross sections were below 0.05 x 10-I6cm2). The parent ionization cross sections were then put on an absolute scale by normalization to the wellknown Kr or Ar benchmark cross sections, as discussed before. All dissociative ionization cross sections were subsequently normalized to the parent ionization cross section for a given target. In all cases, careful threshold studies were carried out to check for the presence of excited species in the incident neutral beam and to determine the appearance energies for the various product ions. This is particularly crucial for dissociative ionization processes, since the appearance energy when compared to thermochemical and spectroscopic data for the formation of a particular fragment ion provides information about the (minimum) excess kmetic energy with which the fragment ion is formed.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 165
Figure 9 and Table I1 show a summary of the measured partial cross sections for the SiD, (x = 1 to 3) free radicals as well as of their ionization and appearance energies. In all cases, only two channels for the formation of singly charged ions were found to have appreciable cross sections. The peak cross sections for all other singly charged ions and for all multiply charged ions were found to be less than 0.1 x 10-''crn2. Figure 9 (top) shows the absolute cross sections for the formation of SiD3+ and SiD,+ ions from the SiD, free radical from threshold to 200eX Both curves represent the result of a single data run. We found cross sections at 70eV of 3.68f0.50 x 10-'6cm2 (SiD3+) and 1.13f 0.21 x 10p'6cm2 (SiD2+). The measured appearance energy for the SiD3+ parent ion of 8.0f0.5eV is very close to the known 8.14-eV ionization energy of SiD, in its vibrational ground state (Lias et al., 1988, Wagman et al. 1982; Chase et al., 1985). We found no evidence of an extended curvature in the near-threshold region or of a significant shift of the measured appearance energy to lower values. This indicates that the vibrational excitation of the SiD, radicals in the target beam is negligible and that there is no appreciable contamination of the target beam due to the presence of metastable SiD, radicals or SiD, radicals in long-lived Rydberg states. The measured appearance energy of the SiD2+ fragment ions from SiD, of 11.3f0.7eV is only marginally higher than the thermochemical minimum energy required for the formation of this fragment ion (Chase et al., 1985; Wagman et al., 1982; Lias et al., 1988; Herzberg, 1950; Handbook of Chemistry and Physics, 1985). This indicates that the SiD,+ fragment ions are formed with little excess kinetic energy. Very similar results were obtained for ionization of the SiD, and SiD free radicals. These results are shown in Fig. 9 for SiD, (center) and for SiD (bottom). The absolute cross-section curves for the formation of SiD2+ and SiD+ ions from SiD, represent the average of several data runs. Cross sections at 70eV of 3.75f0.55 x 10-'6cm2 (SiD,+) and 1.27f0.23 x 10-"cm2 (SiD+) were found. The measured appearance energies of the SiD,+ parent ions of 8 . 5 f 0 . 5 e V is close to the known SiD, ionization energy of 8.92eV in its vibrational ground state (Chase et al., 1985; Wagman et al., 1982; Lias et al, 1988). Threshold studies revealed properties of the SiD, target beam (and of the SiD target beam, see below) similar to those of the SiD, target beam (see discussion above). The measured appearance energy of the SiD+ fragment ions from SiD, of 12.2 f0.7 eV is higher than the thermochemical minimum energy required for the formation of this fragment ion by less than 1 eV, assuming a Si-D bond dissociation energy of about 3 eV (Chase et al., 1985; Wagman et al., 1982; Lias et al., 1988; Herzberg, 1950; Handbook of Chemistry and Physics, 1985). This indicates that the SiD+ fragment ions are formed with little excess kinetic energy. The absolute cross-section curves for the formation of SiD+ and Si+ ions from the SiD free radical (Fig. 9, bottom) represent individual data runs. Cross sections at 70 eVof 3.706 0.55 x lo-'' cm2 (SiD+) and 1.25 f0.22 x cm2
166
R. Basner. M. Schmidt, K. Becker. and H. Deutsch
FIG. 9. Top: Absolute cross sections for the formation of the SiD,+ parent ions (full circles, 0 ) and the SiD2+ fragment ions (full squares, W) from SiD, as a function of electron energy. Center: Absolute cross sections for the formation of the SiD,+ parent ions (full circles, 0 ) and the SiDf fragment ions (full squares, W) from SiD, as a function of electron energy. Bottom: Absolute cross sections for the formation of the SiD+ parent ions (full circles, 0 ) and the Si+ fragment ions (full squares, U) from SiD as a function of electron energy.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 167 TABLE11
MEASURED APPEARANCE ENERGIES AND PARTIAL IONIZATION CROSS SECTIONS AT 70EV FOR THE VARIOUS PARENT AND FRAGMENT IONSPRODUCED BY ELECTRON IMPACT IONIZATION AND DISSOCIATIVE IONIZATION OF THE SID, ( X = 1 to 3) FREERADICALS ~p
m/z
Ion/parent
34
SiD, k/SiD, SiD,' /SiD, SiD, SiD,+/SiD2 SiD+/SiD, SiD, SiD+/SiD Si+/SiD SiD
32 32 30
30 28
AE (eV) 0 . 8 f 0.5 1 1.3 f0.7 Total 8.5 f 0.5 12.2f0.7
Total 7 . 6 f 0.5 11.3f 0.7 Total
Cross section at 70eV (1O-"cm2) 3.68 1.13 4.81 3.75 1.27 5.02 3.70 1.25 4.95
(Si+) were found. The measured appearance energies of the SiD+ parent ions of 7.6f0.5eV is close to the well-known ionization of 7.89eV of SiD in its vibrational ground state (Chase et al., 1985; Wagman et al., 1982; Lias et al., 1988). The measured appearance energy of the Si+ fragment ions from SiD of 1 1.3 f0.7 eV is higher than the thermochemical minimum energy required for the formation of this fragment ion by less than 1 eV (Chase et al., 1985; Wagman et al., 1982; Lias et al., 1988; Herzberg, 1950; Handbook of Chemistry and Physics, 1985). This indicates that the Si+ fragment ions are formed with little excess kinetic energy. The cross-section values at 70eV for all ions are summarized in Table I1 together with the measured appearance energies for easier reference. In summary, four observations should be noted: (1) the parent ionization cross section for all three targets SiD, (x = 1 to 3) has essentially the same maximum value of 3.7 x cm2 at 70 eV, (2) for all three targets, parent ionization is the dominant process, and the most prominent dissociative ionization channel is the one in which one D atom is removed, i.e., SiD, + SiD,_, 'D; (3) the cross section for the formation of the dominant fragment ion also has essentially the same value of about 1.2 x lo-'' cm2 (at 70eV) for all three targets; and (4) the dominant fragment ions are formed with little excess kinetic energy. There are some notable similarities between the present SiD, cross sections and the cross section data obtained previously for CD, (Tarnovsky et al., 1996a) and SiF, (Hayes et al., 1989a; Hayes et al., 1989b; Shul et al., 1989). Similar to what we found for SiD,, the ionization of the CD, radicals was also dominated by
168
R. Basner, M. Schmidt, K. Beckec and H. Deutsch
parent ionization and the parent ionization cross section had essentially the same value for all CD, targets. Furthermore, dissociative ionization of the CD, radicals was also dominated by a single channel, which involved the removal of a D atom. However, the present SiD, cross sections are typically larger than the corresponding CD, cross section by more than a factor of 2. On the other hand, the previously measured SiF, ionization cross sections showed maximum values comparable to and in some cases even larger than the present SiD, cross sections. However, dissociative ionization channels were found to be much more important for SiF,r than for SiD,. The presence of strong dissociative ionization channels appears to be characteristic for all fluorine-bearing molecules and radicals. A comparison of the experimentally determined total single S B r ionization cross sections with calculated cross sections using the modified additivity rule discussed earlier shows that there is overall good agreement in terms of the absolute values, but that the calculated cross sections reach their maximum at a somewhat lower energy, around 50eY and decline more rapidly toward higher impact energies compared to the experimental cross sections (Tarnovsky et al., 1996b). B. TETRAMETHYLSILANE [Si(CH,),] Tetramethylsilane (TMS) is one of the simplest Si-organic compounds. It is frequently used as a precursor in the plasma-assisted chemical vapor deposition of polymers (Tajama and Yamamoto, 1987; Favia et al., 1992) and of SiN and Sic films (Peter et al., 1993). TMS is also observed as a reaction by-product in processing plasmas containing more complex Si-organic monomers (Schmidt et al., 1994). Mass spectrometric databases (see, e.g., the Eight Peak Index, 1974) contain some information on the mass spectral cracking pattern of TMS at 70 eV impact energy. Electron impact ionization cross sections for TMS under controlled single collisions have been measured by McGinnis et al. (1995) using a FTMS technique and by Basner et al. 1996) using their double-focusing sector-field mass spectrometer. A detailed comparison of the cross-sections of McGinnis et al. (1995) and Basner et al. (1996) reveals good agreement in terms of the cross-section shapes, but rather poor agreement in terms of the absolute cross section values for most of the intense ion peaks in the TMS mass spectrum. The values of McGinnis et al. (1995) are generally significantly smaller than those given by Basner et al. (1996). The TMS mass spectrum measured with the double-focusing mass spectrometer in the high-extraction-efficiency mode agrees well with the mass spectrum found in the Eight Peak Index (1974). The mass spectrum obtained in the high-mass-resolution mode differs only for the mass-tocharge ratio m l z = 15, which corresponds to the CH,' ion. In the highextraction-efficiency mode, the intensity of this peak is three times higher than in the high-mass-resolution mode. This indicates that the CH,+ ions are formed with a significant amount of excess kinetic energy, but it also means that all other
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 169
ions are formed with essentially near-thermal energies. There was no evidence for the formation of doubly charged ions. Appearance energies for the various ions from TMS have been measured in previous electron impact and photo-ionization experiments (Potzinger and Lampe, 1970; Distefano, 1970). The ionization and appearance energies obtained by Basner et al. (1996) are summarized in Table 111, which also shows a proposed TABLEI11 MEASURED APPEARANCE ENERGIES AND PARTIAL IONIZATION CROSS SECTIONS AT 70 eV FOR VARIOUS FRAGMENT IONSPRODUCED BY DISSOCIATIVE ELECTRON IMPACT IONIZATION OF TMS. THEFRAGMENT IONS ARE ARRANGED ACCORDING TO THEIRDECOMPOSITION ROUTE,H LOSS BY CH DECAY, REMOVAL OF A COMPLETE METHYL GROUP,AND H TRANSFER FROM CH, GROUPS,RESPECTIVELY H loss by CH, decay mlz
57 55 53 42
Ion SiC2H5+ SiC,H3+ SiC2H+ SiCH2+
AE (eV)
Cross section at 70eV (10-I6cm2)
17.6f0.4 20.8 f0.4 25.1 f 0.4 22.7% 0.4
0.13 0.28 0.27 0.35
Removal of complete methyl groups Ion
AE (eV)
mlz
88 73 58 43 28 15
Si(CH,),+ Si(CH,),+ Si(CH,)2+ Si(CH3)+ Si+ CH:
Cross section at 70eV (10-"cm2) 0.16 10.74 0.25 1.57 0.49 0.96
9.9f0.4 10.1f0.3 17.6410.4 20.1 f0.3 21.2k 0.4 23.450.6
H transfer from CH, groups mlz
Ion
AE (eV)
Cross section at 70eV (10-'6cm2)
74 59 45 44 31 29
HSi(CH,),+ HSi(CH,),+ HzSi(CH,)+ HSi(CH,)+ H, Si+ HSi+ Si(CH,),
10.4f0.4 15.6f0.4 13.8f0.3 17.2f0.4 18.9f 0.4 21.23~0.3 Total
0.93 0.25 1.33 0.41 0.33 0.74 19.19
170
R. Basner, M. Schmidt, K. Becker, and H. Deutsch
complete decomposition scheme for the TMS molecule following electron impact and the cross sections at 70 eV The appearance energies in Table I11 are in good agreement with other values (Potzinger and Lampe, 1970; Distefano, 1970) within the stated uncertainty of f0.5 eV of the data of Basner et al. (1996). The decomposition scheme of the TMS molecule shown in Table I11 indicates that the formation of the Si-containing fragment ions proceeds mainly via three different mechanisms: 1. The removal of a complete methyl group 2. The removal of complete methyl groups and additional H atoms (perhaps connected with the formation of a stable CH, molecule as a reaction by-product) 3. The removal of a CH, or CH group, with one or two H atoms remaining with the ion Ion formation process 3 requires a lower energy than processes 1 and 2. A possible explanation for the exception at m / z = 73 and 74 could be the remarkably small energy difference between the ionization energies of the parent ion and the most abundant fragment ion at m / z = 73. The small 0.2-eV difference in the appearance energies of the two ions is most likely due to the lower energy of the planar Si(CH,),+ ion compared to the tetragonal Si(CH3),+ ion (McGinnis et al., 1995). The total and selected partial electron impact ionization cross sections of TMS are shown in Fig. 10. Also shown in Fig. 10 are the calculated total single TMS ionization cross sections from the modified additivity rule and from the BEB model of Kim and coworkers (Ali et al., 1997). There is reasonably good agreement between the two calculated cross sections and between the calculated cross sections and the measured cross section of Basner et al. (1996) (at least for impact energies below about 80 eV). The cross section of McGinnis et al. (1995) is considerably smaller than the two calculated cross sections and the measured cross section of Basner et al. (1996). C. TETRAETHOXYSILANE [Si(O-CH,-CH,),] Tetraethoxysilane (TEOS) is used in plasma-assisted thin-film deposition techniques for the formation of SiO, films and also for plasma polymerization (Fracassi et al., 1992; Raupp et al., 1992; Pai et al., 1992; Ray et al., 1992; Foest et al., 1998). The distribution of fragment ions formed in collision processes with electrons of 70 eV can be found in standard mass spectrometric databases such as the Eight Peak Index (1974). Electron impact ionization cross sections for the most intensive fragment ions have been measured by Holtgrave et al. (1993) in the energy range from threshold to 50eV using Fourier transform mass spectrometry. The mass spectrum of TEOS as found in standard mass spectrometric
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 171
0
20
40
60
80
100
Electron energy [ eV ]
FIG. 10. Absolute ionization cross sections of TMS as a function of electron energy. The squares (D) and circles (0)refer to the total ionization cross sections of McGinnis et a/. (1995) and Basner et a/. (1996), respectively. Also shown are the calculated cross section of a m and coworkers (Ah et al., 1997) (dotted line) and a calculated cross section using the modified additivity rule (dashed line). The partial cross section for the most abundant fragment ion ( m / z = 73) from Basner et a/. (1996) is indicated by the triangles (A).
databases (1974) shows the molecular ion and many fragment ions with appreciable intensities. The ionization cross-section measurements carried out with the double-focusing mass spectrometer (Basner et al., 1999) operated in the high-extraction-efficiency extraction mode showed a mass spectrum that agrees well with the mass spectrum of the Eight Peak Index (1974) for ions with a mass number ( m / z )higher than 89. The intensities of the ions with smaller m / z values appear with a somewhat higher intensity in our measurement. The ion spectrum produced by 70-eV electron impact on TEOS, limited to those ions with relative intensities higher than 1% of the most abundant ion, is presented in Table IV together with the ions' appearance energies, their relative intensities, and their ionization cross sections at 70eV For clarity of presentation, only ions for the dominant isotope 28Siare included. The ions are listed in the order of decreasing 0 content and decreasing mass number. The ionization energy of the TEOS molecule is 7.7kO.3eV The most intense peak (base peak) was found at m / z = 193, corresponding to the fragment ion Si0,C,HT3, which is the result of the loss of a CH, group from the TEOS molecule. We found a high probability for the formation of ions for which the accompanying neutral partner may be expressed by the sum formula 0,,C,H2,+,. Thus, the fragment ions at m / z = 179, 163, 149, 135 and 119 are readily identified with the loss of C2H5, OC,H,, OC2H5 CH,, OC2H5 C2H4, and OC2H5 OC2H4 neutral fragments, respectively. It is impossible to identify unambiguously the accompanying neutral
+
+
+
172
R. Basner, M. Schmidt, K. Becker, and H. Deutsch
fragments formed in a dissociative ionization process when one detects only the ions, as in our experiment. However, some valuable information can be extracted even without detection of the neutrals. For instance, (1) measurements in the high-mass-resolution mode showed that two kinds of ions with different appearance energies contribute to the ion signal detected at mlz = 105, 103, 91, 89, 75, 61, 45, and 29, (2) pure Si+ could not be detected, and (3) the smallest Sicontaining ion is SOH+. It is also obvious that the observed CO’ ion is the result of the thermal decomposition of the TEOS molecule at the hot cathode surface, followed by CO formation. This was verified by a measurement of the appearance energy of CO+, which agreed well with the ionization energy of the CO molecule and which gave no indication of even the slightest amount of excess kinetic energy. Many smaller fragment ions, such as those with mlz = 2,15,26,27,28,29 (C2H:), 43, 45 (SOH’ and C2HSO+),62, and 63, are formed with significant excess kinetic energy as shown by the broadening of the corresponding ion beams. The electron impact ionization cross section measurements by Holtgrave et al. (1993) were limited to the 20 ions with the highest abundances and covered the electron energy range from threshold to 50 eV. Their results are largely similar to those obtained with the double-focusing mass spectrometer except for a few observations. Their ionization and appearance energies are generally higher than the values measured in the double-focusing mass spectrometer. Holtgrave et al. (1993) obtained their values from a fitting procedure which they applied in order to describe their cross sections by an analytical expression. The higher values for the formation of the COH,+ and C2HSO+ may be caused by the pyrolytical decomposition of the TEOS molecule followed by the formation of C,HSOH, as discussed by Voronkov et al. (1978). The appreciable difference in the total ionization cross sections at 70 eV [2.69 x lo-” cm2 by Holtgrave et al. (1993) vs. 3.8 x cm2 by Basner et al. (1999)l may be understood by the fact that Holtgrave et al. (1993) limited their investigation to the 20 most abundant ions. If we use the same 20 ions and determine their combined ionization cross section, we arrive at a value of 2.8 x lo-” cm2, which is in excellent agreement with the value reported by Holtgrave et al. (1993). The total and a representative set of partial ionization cross sections are presented in Fig. 11. We also show the result of a cross-section calculation using the modified additivity rule (see before). The calculated cross section is somewhat lower than the experimental value, particularly at higher impact energies. This might be due to the contribution of ion pair formation processes to the measured cross section at higher energies that are not included in the calculation. D. HEXAMETHYLDISILOXANE [(CH,),-Si-0-Si-(CH,),] Hexamethyldisiloxane (HMDSO) is one of the simplest siloxane compounds; it has been successfully used in plasma-assisted thin-film deposition applications
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 173 TABLEIV MEASURED APPEARANCE ENERGIES AND PARTIAL IONIZATION CROSS SECTIONS AT 70eV FOR THE VARIOUS PARENT AND FRAGMENT IONS PRODUCED BY ELECTRON IMPACT IONIZATION AND DISSOCIATIVE IONIZATION OF TEOS. mlz
Ion
AE (eV)
208 207 193 179 177 165 151 137 123 163 149 147 135 133 121 107 105 103 93 91 89 79 119 118 117 105 103 91 90 89 77 76 75 63 62 61 75 74 73 61 47
Si0,C8H2,+ SiO4C8Hly+ SiO4C,HI,+ SiO4C6HlS+ Si04C6H13+ Si04C,H, 3 + Si04C4HI Si04C3H9+ Si04C2H7+ sio3 C6H Is f Si03C, H,,+ SiO,C,H, Si03C4H, Si03C4Hy+ Si0,C3Hy+ Si03C2H,+ Si03C2Hs+ SiO,C,H,+ Si03CHs+ Si03CH3+ Si03CH+ SiO, H,+ Si02C4H,I + SiO, C4HI ,+ Si02C4H9+ SiO,C,H,+ Si02C4H,+ SiO, C, H, + Si02C2H6+ Si02C2H5+ Si02CHS+ SiO,CH4+ SiOC,H,+ Si02H3+ SiO2H2+ SiOzH+ Si0,C2H,+ SiOC,H6+ SiOC,H,+ SiOCH5+ SiOH,+
7.2f0.3 8.7f0.4 8.4f0.3 8 . 2 f 0.3 13.3& 0.5 10.13Z0.5 10.4rt 0.4 12.5 3Z 0.5 13.1+0.6 12.23~ 0.3 11.4 f 0.3 14.1 f 0.8 11.5f0.7 15.9f0.7 13.5f0.8 15.3f 0.4 21.4f 0.8 25.5f 1.0 16.9f0.6 25.1 rt 0.8 18.4rt0.8 19.3+0.5 15.2f0.6 16.6f 1.0 16.8f 1.0 16.6f0.8 19.2 f 1 .O 17.1 f 0 . 7 19.8f0.8 19.4f0.8 18.8f 0.6 22.8f0.8 24.4f0.8 21.8f0.5 23.4f0.7 29.7+ 1.2 23.1 f 0.8 15.8f 0.8 21.8 f 0.5 24.1 f0.9 25.4f 1.0
,+ ,
+
,+
Cross sechon at 70 eV (1 0-%m2) 1.63 0.48 5.72 1.39 0.069 0.42 0.17 0.12 0.17 3.20 4.49 0.12 1.07 0.17 0.3 1 0.67 0.20 0.03 0.22 0.13 0.13 2.82 1.73 0.093 0.092 0.29 0.12 0.56 0.15 0.13 0.22 0.048 0.014 1.52 0.68 0.053 0.067 0.048 0.124 0.034 0.051 (continued)
174
R. Basner, M. Schmidt, K. Becker, and H. Deutsch TABLE n! (continued) Ion
m/ z
45 45 43 31 29 29 28 27 26 15 14 2
ws
26.5+0.9 11.1 Zk 0.9 14.0Zk0.7 13.11!10.7 15.1 Zk 0.7 22.1f0.7 10.8f0.7 16.3& 0.3 13.63~0.3 15.3k0.5 17.3 k 0.5 l5.9* 0.8 Total
SOH+ C2H50+ C*H,O+ CH, O+ CH O+ C2H5+ C2H4+
C2H,+ C2H2+ CH, CH2' H*+ Si(OCH,CH,), +
I
Cross section at 70eV (I0-lhcm2)
AE (eV)
.
'
'
l
~
0.79 0.13 0.35 0.071 0.29 2.51 0.55 1.56 0.27 0.71 0.092 0.11 37.2
~
~
l
-
l
~
,
,
~
3
40
-
-
a
'
30
,... . . .....'
-
I
C
.-+ al
20 v,
2 0
-
-
-
E
.B10 -
-
d
-
.c
c
-
0
0
m--x-m-m--m--m--*--m--m
I
0
20
*
I
,
I
40
,
I
I
I
.
I
60
I
I
80
,
,
,
I
,
100
Electron energy [ eV ]
FIG. 11. Absolute ionization cross section of TEOS as a function of electron energy. The circles (0)refer to the total ionization cross section of Basner et ul. (1999), and the dotted line represents the total ionization cross section of Holtgrave et al. (1993). The inverted triangles (V)denote the cross section obtained by Basner et al. (1999) for only those ions that were measured by Holtgrave et al. (1993). Also shown is a calculated cross section using the modified additivity rule (dashed line). Several partial ionization cross sections from Basner et al. (1999) are also shown: diamonds (e), m / z = 193; crosses (x), m / z = 149; triangles (A),m / z = 163; and stars (*), m / z = 208.
~
,
.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 175
(Sarmadi et al., 1995; Sawada et al., 1995). The suitability of HMDSO-based polymers formed by plasma techniques for corrosion protection has been known for a long time (Benz, 1987), as have the unique optical properties of such films (Poll et al., 1993). Electron impact ionization data for HMDSO in the literature include the mass spectral cracking pattern in standard mass spectrometric databases (see, e.g., the Eight Peak Index, 1974), appearance energies for some fragment ions (Dibeler et al., 1953; Borossay and Szepes, 1971), and previous mass spectrometric measurements of ionization cross sections from threshold up to 50 eV by Seefeldt et al. (1985). The most recent ionization-cross-section data are those of Basner et al. (1998, 1999), which include data for ions with low intensity and range in energy up to 100 eV The observed mass spectrum reported by Basner et al. (1998, 1999) agrees in its general features with the published data in the standard mass spectrometric data bases. However, the Eight Peak Index (1974) contains 10 different mass spectra with distinct differences. Table V lists the various ions, their relative intensities at 70 eV impact energy, their appearance energies, and the total and partial cross sections at 70eV (Basner et al., 1998; Basner et al., 1999; Foest, 1998). The ionization energy of the HMDSO molecule is 8 . 8 f 1.3 eV The ion spectrum produced by 70-eV electron impact on HMDSO is characterized by a dominant signal at m / z = 147, which results from the dissociative ionization of TABLEV
MEASURED APPEARANCE ENERGIES AND PARTIAL IONIZATION CROSS SECTIONS AT 70 eV FOR THE VARIOUS PARENT AND FRAGMENT IONS PRODUCED BY ELECTRON IMPACT IONIZATION AND DISSOCIATIVE IONIZATIONOF HMDSO mlz
162 147 133 131 73 73 66 59 52 45 45 43 15
Ion Si,OC,H,,+ SizOCSH,st Si,OC,H,,+ Si20C4Hll+ SiC,H,+ Si,OH+ Si20C4H12++ SiC2H,iSi,OC,H,++ SiCH,+ SOH+ SiCH,+ CH,+ Si, O W 3Ih
AE (eV)
Cross section at 70 eV ( 10-%m2)
8 . 8 f 1.3 9 . 6 f 0.5 14.8f 0.9 15.8f0.7 16.3 f0.6 25.3f 1.5 26.8f 0.6 22.0f 0.6 32.6% 0.8 21.4f0.7 21.4f 1.4 28.4f 0.7 14.7f 0.8 Total
0.017 16.7 0.24 0.64 1.42 0.26 1.49 0.96 0.3 1 0.84 0.1 1 0.38 0.35 25.5
176
R. Basnec M. Schmidt, K. Becker, and H. Deutsch
the parent molecule and the removal of a CH, radical from the HMDSO molecule. The molecular ion (mlz = 162) is found in the mass spectrum with a very small intensity (0.1% of the intensity of the m l z = 147 base peak). The ion at m / z = 73, with a relative intensity of 9.3%, results from the breaking of one of the Si-0 bonds in the center of the molecule. It is noteworthy that we found evidence of the presence of doubly charged ions (mlz = 66, 52) with relatively high intensities, whereas the corresponding singly charged ions with the same mass were not observed. A comparison of the spectra measured in the high-massresolution mode and in the high-extraction-efficiency mode shows the same relative intensities for all ions with the exception of m l z = 15, CH,+. As was the case for the other two Si-organic compounds, TMS and TEOS, the CH3+ ion from HMDSO is formed with an excess kinetic energy of 3 eV or more. This was verified by studies of the corresponding horizontal ion beam profile and by measurements of the neutral gas component in a plasma using a plasma monitor (Foest, 1998). The appearance energies of the most abundant ion at m / z = 147 reported by Foest, et al. (1998) agree well with the values given by Dibeler et al. (1953). Partial ionization cross sections for the formation of the ion at m l z = 147, for the CH; ion and the total single HMDSO ionization cross section are presented in Fig. 12. The cross section for the dominant ion ( m l z = 147) reaches a
- t
FIG. 12. Absolute ionization cross section of HMDSO as a function of electron energy. The squares (m) denote the total ionization cross section of Basner et a/. (1999); the open circles (0)refer to the data of Seefeldt et a/. (1985). A calculated cross section based on the modified additivity rule is shown as the diamonds (+). Also shown are two partial ionization cross section for the fragment m / z = 147 (full circles, 0 )and the CH3+ ion at m / z = 15 (triangles, V).
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 177
maximum near 25 eV For the other fragment ions, the maximum in the ionization cross section is shifted to higher energies. The shape of the CH,’ ionization cross is different from all the other cross-section curves. The CH3+ cross-section curve reaches a first plateau at an electron energy of about 40 eV, and the cross section subsequently increases again up to an energy of 100eV This indicates the presence of at least two channels leading to the formation of CH;, with the second, more prominent channel having a minimum energy of about 50eV The excess kinetic energy of the CH,’ ion causes the measured partial ionization cross section for this ion to be a lower limit of the “true” cross section as a result of discrimination effects (see discussion before). However, the impact of this on the total HMDSO ionization cross section is small. The partial cross sections for the ions measured in the earlier study of Seefeldt et al. (1985) ( m / z = 66, 73, 131, and 147) were found to be larger values in the recent study of Basner et al. (1998, 1999) by factors ranging from 1.5 to 5. A possible explanation could be the fact that Seefeldt et al. (1985) had to rely on an indirect measurement of the gas pressure in the ion source, which was perhaps more susceptible to systematic errors. The experimentally determined HMDSO total single ionization cross section is compared with the result of a calculation using the modified additivity rule discussed before. The level of agreement between the experimentally determined and the calculated cross sections is better than 16% at all energies, which constitutes satisfactory agreement for such a complex target molecule. cm2) is The total HMDSO ionization cross section at 70eV (2.2 x marginally higher than the TMS cross section (1.9 x cm2) and significantly smaller than the TEOS cross section (3.8 x cm2).
V. Comparison with Ion Formation Processes and Ion Abundances in Plasmas Electron impact ionization of the parent molecule is only one of several important ion formation processes in nonthermal plasmas. Secondary processes such as electron impact ionization of neutral fragments produced by dissociation of the parent molecule and ion-molecule reactions are other mechanisms contributing to the formation of plasma ions. It is interesting to compare ion abundances in a realistic plasma with the ion abundances predicted from electron impact ionization cross sections measured under single-collision conditions. Although mass spectrometry of plasma ions is a known and well-developed diagnostic method (Osher, 1965; Drawin, 1968; Schmidt et al., 1999), its application to plasmas for thin-film deposition is not very common. The main reasons are deleterious effects of insulating deposits on the ion collection orifice (which connects the mass spectrometer to the plasma) and on the ion transfer optics, which render it
178
R. Basnel: M. Schmidt, K. Becker, and H. Deutsch
difficult to maintain a stable ion collection efficiency. Nonetheless, plasma mass spectrometric studies have been carried out in silane discharges (Yamamoto et al., 1997; Haller, 1980; Howling et al., 1994), as well as in TMS-, TEOS-, and HMDSO-containing plasmas (Peter et al., 1993; Schmidt et al., 1994; Basner et al., 1995b; Basner et al., 1997b; Foest et al., 1997; Foest et al., 1994; Wrobel et al., 1983). We present a brief summary of the key points of these experiments in this chapter. Ion-molecule reactions were found to play an important role in the ion formation processes in silane plasmas. For instance, a plasma produced in an ECR discharge in silane 0, = 0.16 to 0.8 Pa) (Yamamoto et al., 1997) contains all the Si-containing ions that one would expect on the basis of measured electron impact ionization cross sections for silane, but their relative intensities as measured by plasma mass spectrometry are very different from what one would expect on the basis of the partial ionization cross sections. SiH; is the dominant ion in the plasma (even though SiH2+ has a larger partial ionization cross section), and the intensities of the other SiH,+ (x = 0 to 2) ions are one order of magnitude smaller. This has been explained by Yamamoto et al. (1997) on the basis of ion-molecule reactions between the Si-containing ions and the silane molecules that result in H transfer from the molecule to the positive ion. This process eventually transforms a large fraction of the SiH,+ (x = 0 to 2) ions into SiH; ions. By contrast, in rf' discharges with a silane pressure of more than 2Pa, cluster ions were found to be dominant (Haller, 1980). The mass spectrometric observation of positive and especially of negative cluster ions is a very interesting finding, because the formation of these ions represents the fist step in a reaction chain leading to the formation of dust particulates in the plasma volume under plasma conditions that are routinely used in the deposition of highquality a-Si : H films (Howling et al., 1994). Plasmas containing Si-organic parent molecules show a pronounced decline in the concentration of the parent molecule after the discharge has been ignited. The main loss channel for the Si-containing ions is the thin-film formation on the electrodes and on the walls of the discharge reactor. At the same time, the concentration of stable light reaction products such as H,, CH,, and C,H, increases. As an example, Fig. 13 shows the temporal behavior of the concentration of TMS molecules and of the concentrations of some stable reaction products (H,, CH,, and several small hydrocarbons) after ignition of the discharge. The formation of stable Si compounds is another possible process. For instance, octamethyltrisiloxane, (CH3),-Si-O-Si(CH3),-O-Si(CH3)3, and tetramethysilane, Si(CH3),, were observed in HMDSO plasmas (Wrobel et al., 1983; Jurani et al., 1994; Charles et al., 1992). Stable secondary products may be formed by gas-phase reactions initiated, for example, by electron impact-induced dissociation processes and by dissociative ionization. Other reaction pathways are also possible. It is known that there is a significant concentration of CO in TEOS
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 179
v
010’ v)
8
5
c-(
loo
lo-’
lo-* 0
50
100
150
200
250
300
350
Process time [ s 3 FIG. 13. Time dependence of the intensity of various ions (upper diagram) and neutrals (lower diagram) in a TMS-containing rf dmharge before and after ignition of the plasma.
plasmas (Foest, 1998, Kickel et al., 1992). The CO may be the result of the interaction of the plasma with the deposited film, a notion supported by the fact that the electron impact ionization of TEOS under single-collision conditions does not lead to the formation of CO with an appreciable cross section (Basner et al., 1999). Thermal decomposition is an unlikely process because of the absence of hot surfaces in the rf-discharge vessel. The result of the complex plasma chemical reactions in plasmas containing Siorganic compounds is gas mixtures with relatively high concentrations of H2 and CO (in the case of TEOS) and smaller concentrations of lighter hydrocarbons as well as selected Si-organic compounds. The various neutral constituents are ionized primarily by electron impact. A detailed mass spectrometric investigation of the ion abundances (Foest et al., 1997) shows the presence of essentially all fragment ions of the parent molecules that were observed by electron impact ionization under single-collision conditions in addition to the ions of the lighter neutrals. Figure 14 shows the results of mass spectrometric studies of the ion components of Ar-TMS, Ar-TEOS, and Ar-HMDSO plasmas obtained in a rfdischarge reactor in comparison with calculated relative ion formation rates using the measured partial ionization cross sections and a Maxwellian energy distribu-
180
R. Basnel; M. Schmidt, K . Beckel; and H.Deutsch
tion of the plasma electrons corresponding to an average energy of 3 eV. For all smaller ions, the calculated relative ion formation rates are much lower than the measured ion intensities. More realistic electron energy distribution functions are characterized by a loss of some of the more energetic plasma electrons and consequently result in lower formation rates for the lighter ions with higher appearance energies. Therefore, the difference between the calculated rates and the measured ion currents would be greater than those shown in Fig. 14. Thus, the results displayed in Fig. 14 demonstrate the important role of secondary processes in the formation of the various ions in realistic deposition plasmas using gas mixtures that contain Si-organic compounds. Secondary processes of relevance include dissociation into neutral particles, ionization of dissociation products, and ion-molecule reactions. The importance of ion-molecule reactions is demonArH+ strated by, for example, the observation of protonated ions such as H3+, (Fig. 13), and COH+.A typical ion spectrum of an Ar-TEOS rf discharge is presented in Fig. 15 (Foest, 1998; Basner et al., 1997). The measured current of the COH+ ions is nearly the same as that of the Ar+ and ArH+ ions. The H3+and CH3+currents are one order of magnitude smaller, and the TEOS ions with mlz = 208, 193, and 149 have relative intensities that are two orders of
FIG. 14. Comparison of mass spectrometrically obtained ion abundances in TMS-, TEOS-, and HMDSO-containing rf discharges (open bars) with calculated ion formation rates using measured partial ionization cross sections and a Maxwellian energy distribution of the plasma electrons corresponding to an average energy of 3 eV (full bars).
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 181 50
25 I
.
,
,
l
100
75 .
,
.
I
.
.
.
I
125 I.
.
I
25
20
'
:-I 0.25
25
50
75
Power [ W
100
125
1
FIG. 15. Relative intensities of various ions in an Ar-TEOS rf discharge as a function of the nominal rf power.
magnitude smaller. The concentration of the TEOS ions decreases with increasing power, while the light product ions increase with increasing power (and decreasing concentration of TEOS).
VI. Summary Recent studies of the formation of positive ions following electron impact on silane (SiH,), the SiH, (x = 1 to 3) free radicals, and the Si-organic molecules TMS, TEOS, and HMDSO have been summarized in this article. The studies include the experimental determination of absolute partial ionization cross sections under well-defined single-collision conditions, the calculation of total single ionization cross sections, studies of the excess kinetic energy of the fragment ions produced by dissociative ionization, and plasma mass spectrometric studies. The main features of the decomposition of the silane molecule and the subsequent secondary plasma chemical reactions involving the SiH, free radicals in a silane plasma can be understood on the basis of the ionization data obtained under single-collision conditions combined with information regarding ion-molecule reactions (Kickel et al., 1992). The ionization cross-section data for
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R. Basner, M. Schmidt, K. Becker, and H. Deutsch
SiH, (x = 1 to 4) are a crucial component of a broader ionization cross-section database for halogen- and hydrogen-containing molecules and free radicals of interest to plasma chemical applications. Other species in this database include CH,, CF,, SiF,, SiH, (x = 1 to 4), NH,r, and NF, (x = 1 to 3). Ionization cross sections and plasma mass spectrometry involving TMS, TEOS, and HMDSO provide a rudimentary understanding of the decomposition of these prototypical Si-organic species in realistic deposition plasmas. In all cases, main decomposition routes proceed via the loss of one or more methyl, CH,, groups. The dominant ion in all cases is the one in which a single CH, group is removed. The molecular ion is always present, but only in the case of TEOS with appreciable intensity. Doubly charged ions were found only for HMDSO. The important role of secondary ion-molecule reactions, particularly in mechanisms leading to the formation and destruction of low mass-to-charge ( m / z ) species in the plasma, has been demonstrated.
VII. Acknowledgments One of us (KB) would like to acknowledge support from the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy and from the U.S. National Science Foundation.
VIII. References Ali, M. A,, Kim, Y.-K., Hwang, W., Weinberger, N. M., and Rudd, M. E. (1997). 1 Chem. Phys. 106, 9602. Basner, R., Schmidt, M., Deutsch, H., Tamovsky, V;, Levin, A,, and Becker, K. (1995a). 1 Chem. Phys. 103, 211. Basner, R., Foest, R., Schmidt, M., Kurunczi, P., Becker, K., and Deutsch, H. (1995b). In K. Becker, W. E. Cam, and E. E. Kunhardt (Eds.), Proceedings Xnr ICPIG, Hoboken, U.S.A. p. 4-31, Basner, R., Foest, R., Schmidt, M., Sigeneger, F., Kurunczi, l?, Becker, K., and Deutsch, H. ( 1 996). Int. 1 Muss Spectrom. Ion Proc. 153, 65. Basner, R., Schmidt, M., Tamovsky, V;, Becker, K., and Deutsch, H. (1997a). Int. 1 Muss Spec. Ion Proc. 171, 83. Basner, R., Foest, R., Schmidt, M., Hempel, F., and Becker, K. (1997b). In M. C. Bordage (Ed.), Proceedings XXIII ICPIG. Toulouse, France, p. IV-196. Basner, R., Foest, R., Schmidt, M., Becker, K., and Deutsch, H. (1998). Int. 1 Muss Spectrom. Ion Proc. 176, 245. Basner, R., Foest, R., Schmidt, M., and Becker, K. (1999). Advunc. Muss Specfrom., (on CD-ROM). B e n , G. (1987). Bosch Techn. B m , 219, 1. Bobeldijk, M., van der Zande, W. J., and Kistemaker, P. G. (1994). Chem. Phys. 179, 125. Borossay, J., and Szepes, L. (1971). Adv in Muss Specfrom. 5 , 700. Charles, C., Garcia, P., Grolleau, B.. and Turban, G . (1992). 1 Vuc. Sci. Technol. A 10, 1407.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 183 Chase, M. W., Jr., Davis, K. A,, Downey, J. R., Frurip, D. J., McDonald, R. A,, and Syverud, A. N. (1985). 1 Phys. Chem. Ref Data 14, 1. Chatham, H., Hils, D., Robertson, T., and Gallagher, A. (1984). 1 Chem. Phys. 91, 1770. Deutsch, H., and Schmidt, M. (1985). Conk Plusma Phys. 25, 475. Deutsch, H., Cornelissen, C., Cespiva, L., Boacic-Koutecky, V, Margreiter, D., and Mirk, T. D. (1993). Int. J: Mass. Spectrom. Ion Process. 129, 43. Deutsch, H., Mark, T. D., Tarnovsky, V, Becker, K., Cornelissen, C., Cespiva, L.. and BonacicKoutecky. V (1994). Int. 1 Mass. Spectrom. Ion Process. 137, 77. Deutsch, H., Becker, K., and Miirk, T. D. (1997). Int. 1 Mass. Spectrom. Ion Process. 167/168, 503. Deutsch, H., Becker, K., and Mark, T. D. (1998a). Contr. SASP, Going/Kitzbuhl, Austria, 4/46. Deutsch, H., Becker, K., Basner, R., Schmidt, M., and Mirk, T. D. (1998b).1 Phys. Chem. 102, 8819. Dibeler. V H., Mohler, V. L., and Reese, R. M. (1953). 1 Chem. Phys. 21, 180. Distefano, G. (1970). Inorg. Chem. 9, 1919. Doyle, J. R., Dougthy, D. A,, and Gallagher, A. (1990). 1 Appl. Phys. 68, 4375. Drawin, H.W. (1968). In W. Lochte-Holtgreven (Ed.), Plasma diagnostics. Amsterdam. Eight peak index qf mass spectra, 2nd edn, Mass Spectrometry Data Center, Aldermaston. Favia, P., Lamendola., R., and d’Agostino, R. (1992). Plasma Sources Sci. Technol. I , 59. Fitch, W. L., and Sauter, A. D. (1983). Analyt. Chem. 55, 832. Foest, R. (1998). Ph.D. thesis, Emst Moritz Arndt University, Greifswald, Germany. Unpublished. Foest, R., Schmidt, M., Hannemann, M., and Basner, R. (1994). In L. G. Chnstophorou and D. R. James (Eds.) (New York) Gaseous Dielectrics VII (p. 335). Foest, R., Basner, R.,,Schmidt, M., Kurunczi, P., and Becker, K. (1997). In J. F. P. Conrads and G. Babucke (Eds.), Proceedings of the 12th international conference on gas discharges and their applications. (Vol. I, p. 547). Foest, R., Basner, R., Schmidt, M.. Hempel, F., and Becker. K. (1998). In L. G. Christophorou and J. K. Olthoff (Eds.), Proceedings of the Vlll international symposium on gaseous dielectrics. Plenum Press (New York). Fracassi, F., d’Agostino, R., and Favia, P. (1992). 1 Electrochem. Sac. 139/9, 2636. Freund, R. S., Wetzel, R. C., Shul, R. J., and Hayes, T. R. (1990). Phys. Rev. A 41, 3575. Haaland, P. (1990). Chem. Phys. Lett. 170, 146. Haller, 1. (1980). Appl. Phys. Lett. 37, 282. Handbook ofchemistry and Physics, 65th ed., R. C. Weast, M. J. Astle and W. H. Beyer (Eds.), CRC Press (Boca Raton). (1985). Harland, P. W., and Vallence, C. (1997). Int. 1 Mass. Spectrom. Ion Proc. 171, 173. Hayes, T. R., Wetzel, R. C., Biaocchi, F. A., and Freund, R. S. (1989a). 1 Chem. Phys. 88, 823. Hayes, T.R., Shul, R. J., Biaocchi, R. A,, Wetzel, R. C., and Freund R. S. (1989b). 1 Chem. Phys. 89, 4035. He, J.-W., Bai, C.-D., Xu, K.-W., and Hu, N.-S., (1995). Surf Coat. Technol. 74-75, 387. Herzberg, G. (1950). Molecular spectra and molecirlar structure (vols. 1 and Ill). Van Nostrand Reinhold, New York. Holtgrave, J., Riehl, K., Abner, D., and Haaland, P. D. (1993). Chem. Phys. Letters 215, 548. Howling, A.A., Sansonnens, L., Dorier, J.-L., and Hollenstein. C. (1994). 1 Appl. Phys. 75, 1340. Hwang, W., Kim, Y. K., and Rudd, M. E. (1996). J Chem. Phys. 104, 2965. Jurani, R., Lamendola, R., d’Agostino, R., and Tmovec, J. (1994). In P. Lukac (Ed.), Contributed Papers. 10th. Symposium on Elementary Processes and Chemical Reactions in Low Temperature Plusma, Stara Lesna. Slovakia, (p. 28). Comenius University (Bratislava). Kickel, B. L., Griffin, J. B., and Armentrout, P. B. (1992). Z. Phys. D 24. 101. Kim, K.-S., and Ikegawa, M. (1996). Plusmu Sources Sci. Technol. 5, 31 1. Kim, Y.-K. and Rudd, M. E. (1994). Phys. Rev. A 50, 3954.
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Kim, Y.-K., Hwang, W., Weinberger, N. M., Ah, M. A., and Rudd, M. E. (1997). 1 Chem. Phys. 106, 1026. Konuma, M. (1992). Film deposition by plasma techniques. Springer-Verlag (Berlin). Krishnakumar, E., and Srivastava, S. K. (1995). Contrib. Plasma Phys. 35, 395. Leiter, K., Schreier, P., Walder, G., Mirk, T. D. (1989). Int. 1 Mass. Spectrom. Ion Proc. 87, 209. Lias, S. G., Bartmess, J. E., Liebrnan, J. F., Holmes, J. L., Levine, R. D., and Mallard, W. G. (1988). 1 Phys. Chem. Ref: Data 17, 1. Margreiter, D., Waldner, G., Deutsch, H., Poll, H. U., WinMer, C., Stephan, K., and M&k, T. D. (1990). Int. 1 Mass. Spectrom. Ion Proc. 100, 143. Mirk, T. D., and Egger, F. (1977). 1 Chem. Phys. 67, 2629. Mirk, T. D., Egger, F., and Cheret, M. (1977). 1 Chem. Phys. 67, 3795. Mirk, T. D. (1984). In L. G. Chnstophorou (Ed.), Electron-molecule interactions and their applications (Vol. 1). Academic Press (Orlando). Miirk, T. D., and Dunn, G. H. (1985). (Eds.) (1985). Electron impact ionization. Springer-Verlag (Vienna). McGinnis, S., Riehl, K., and Haaland, P. D. (1995). Chem. Phys. Lett. 232, 99. Morgan, W. L. (1992). Plasma Chem. Plasma Proc. 12, 477. Morrison, J. P., and Traeger, J. C. (1973). Int. 1 Mass Spechom. Ion Proc.11,289. Nagpal, R., and Garscadden, A. (1994). In L. G. Christophorou and D. R. James (Eds.), Gaseous dielechics VII, @. 39) Plenum (New York). Osher, J. E. (1965). In R. H. Huddlestone and S. L. Leonard (Eds.) Plasmadiagnostic Techniques. Academic Press (New York). Otvos, J. W, and Stevenson, D. P. (1956). 1 Americ. Chem. Soc. 78, 546. Pai, C. S., Miner, J. F., and Foo, P. D. (1992). 1 Elecmchem. Soc. 139/3, 850. Penin, J., Lervy, O., and Bordage, M. C. (1996). Conrr Plasma Phys. 36, 3. Peter, S., Pintaske, R., Hecht, G., and Richter, F. (1993). Surf: Coat. Technol. 59, 97. Poll, H. U., Wmkler, C., Grill, V, Margreiter, D.. and Mirk, T. D. (1992). Int. 1 Mass. Spectrom. Ion Proc. 112, 1. Poll, H. U., Meichsner, J., Arzt, M.. Friedrich, M., Rochotzki, R., and Kreyig, E. (1993). Surf: Coat. Techn. 59, 365. Potzinger, P.,and Lampe, F. W. (1970). 1 Phys. Chem. 74, 719. Rapp, D., and Englander-Golden, P. (1965). 1 Chem. Phys. 43, 5. Raupp, B. G., Cale, T. S. and Hey, H. (1992). 1 Vac. Sci. Technol. B 10, 37. Ray, S. K., Maiti, C. K., Lahiri, S. K., and Chakrabati, N. B. (1992). 1 Vac. Sci. Techno/.B 10, 1139. Robertson, R., Hils, D., Chatham, H., and Gallagher, A. (1983). Appl. Phys. Lett. 43, 544. Sarmadi, A. M., Ying, T. H., and Denes, F. (1995). Euv. Polym. J. 31/9, 847. Sawada, Y., Ogawa, S., and Kogoma, M. (1995). 1 Phys. D: Appl. Phys. 28, 1661. Schmidt, M., Foest, R., Basner, R., and Hannemann, M. (1994). Acta Phys. Uniu Comenianae 35, 217. Schrmdt, M., Foest, R., and Basner, R. (1998). 1 De Physuqye IV 8, 231. Seefeldt, R., Moller, W., and Schmidt, M. (1985). Z. Phys. Chem. (Leipzig) 266, 797. Shul, R.J., Hayes, T. R., Wetzel, R. C., Biaocch, F. A., and Freund, R. S. (1989). 1 Chem. Phys. 89, 4042. SIMION (1992). Version 5.0, ldaho National Engineering Laboratory, EG&E Idaho Inc., ldaho Falls, ID. SIMION (1996). Version 6.0 (3-D), Energy Science and Technology Software Center. Stephan, K., Deutsch, H., and Mirk, T. D. (1985). 1 Chem. Phys. 83, 5712. Straub, H. C., Renault, P., Lindsay, B. G., Smith, K. A., and Stebbings, R. F. (1995). Phys. Rev. A 52, 1115. Tajama, I., and Yamamoto, M. (1978). JT Polym. Sci.:Part A Polym. Chem. 25, 1737.
ELECTRON IMPACT IONIZATION OF ORGANIC SILICON COMPOUNDS 185 Tarnovsky, V, and Becker, K. (1992). Z. Phys. D 22, 603. Tarnovsky, V, and Becker, K. (1993). 1 Chem. Phys. 98, 7868. Tamovsky, V, Kurunczi, P., Rogozhnikov, D., and Becker, K. (1993). Inr. 1 Mass Spectrom. Ion Proc. 128, 181. Tarnovsky, V, Levin, A., Becker, K., Basner, R., and Schrmdt, M. (1994). Int. 1 Mass Spec Ion Proc. 133, 175. Tamovsky, V, Levin, A,, Deutsch, H., and Becker, K. (1996a). 1 Phys. B 29, 139. Tarnovsky, V, Deutsch, H., and Becker, K. (1996b). 1 Chem. Phys. 105, 63 15. Tissier, A,, Khallaayoune, J., Gerodolle, A,, and Huizing, B. (1991). 1 Physique IV 1, C 2 4 3 7 . Tochikubo, F., Suzuki, A,, Kakuta, S., Terazono, Y., and Makabe, T. (1990). 1 Appl. Phys. 68, 5532. Turban, G., Catherine, Y., and Grolleau, B. (1980). Thin Solid Films,67, 309. Vallence, C., Harland, P. W., and MacLagan, R. G. A. R. (1996). 1 Phys. Chem. 100, 15021. Voronkov, M. G., Mileshkevich, V P., and Yuzhelevskii, Yu. A. (1978). The siloxane band, Consultants Bureau (New York). Wagman, D. D., Evans, W. H., Parker, V B., Schumm, R. H., Halow, I., Bailey, S. M., Chuney, K. L., and Nutall, R. L. (1982). 1 Phys. Chem. ReJ Duta 11, 1. Wetzel, R.C., Biaocchi, F. A,, Hayes, T. R., and Freund, R. S. (1987). Phys. Rev. A 35, 559 (1987). Wrobel, A.M., Kryszewski, M., and Gazicki, M. (1983). 1 Mucromol. Sci.-Chem. A20, 583. Yamamoto, Y., Suganuma, S., Ito, M., Hori, M., and Goto, T. (1997). 1 Appl. Phys. 36, 4664. Younger, S.M., and M h k , T. D. (1985). In T. D. Mirk and G. H. Dunn (Eds), Electron impact ionization Springer-Verlag (Vienna)
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ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL 41
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS RELATED TO PLASMA CHEMISTRY P B. ARMENTROUT Chemishy Department. University of Utah. Salt Luke City. UT
I . Introduction ....................................................................... I1. Experimental Methods ............................................................ A . General .............................................. B. Advantages ....................................................................
................................................... 2. Surface Ionization Source ................................................. 3. Flow-Tube Ion Source ........................................... D. Methods of Analysis .......................................................... 1. Exothermic Reactions ...................... 2 . Endothermic Reactio 3. Relationship between Thresholds and Bond Energies .................... 4 . Conversion of Cross Sections to Rate Constants ......................... 111. Reactions with Silane (SiH, ) ..................................................... A . Rare Gases (He+, Net, Ar+, &+,Xe') .....................
....................................... C . Si+ ........... ......................... D. Transition Metal Ions ......................................................... E . Thermochemistry of Silicon Hydrides ....................................... 1. SiH and SiH: .......................................................... 2 . SiH' and SiH, ............................................................. 3 . SiH, and SiH: .............. 4 . Si2HT Species ............................................................. IV Reactions Involving Organosilanes .............................................. A . Si+ + CH,, C,H, ............................................................. B . Si+ H,SiCH, ............................................................... C. Thermochemistry of Organosilanes.................................. V Reactions with Silicon Tetrafluoride (SiF,) ...................................... .......................... A . Rare Gases (He+, Ne+, Ar+, Kr+) B. O+, O z , N+, and N$ ......................................................... C. Si+ ............................................................................ D. Thermochemistry of Silicon Fluorides ....................................... v1. Reactions with Silicon Tetrachloride (SiC1, ) ....................................
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D. Thermochemistry of Silicon Chlorides ....................................... 187
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Copyright 2000 by Acadcmic Press All nghts of reproduction in any form reserved ISBN . 0-12.003843.9/ISSN . 1049-25OX $30.00
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I? B. Armentrout VII. Reactions with Fluorocarbons (CF, and C,F,) .................................. A. Rare Gases (He+, Nef, Ar+) CF, ......................................... B. Oc, 0: + CF,, C,F, ......................................................... C. Themnochemistry of Fluorocarbons.. ..... VIII. Miscellaneous Thennochemical Studies.. ........................................ A. 0: CH,. .................................................................... B. Sulfur Fluorides.. .............................................. IX. Conclusions ....................................................................... X. Acknowledgments X1. References.. .......................................................................
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I. Introduction An important tool in the fabrication of microelectronic devices is the use of
plasmas to etch and deposit silicon, silicon oxide, silicon carbide, silicon nitride, and other semiconductor material layers. During etching, highly reactive radicals and ions present in the plasma bombard the surface and volatilize components of the surface. Deposition of silicon dielectric films involves polymerization of silicon-containing radicals and ions. The detailed role of gas-phase ion-molecule reactions in these processes is as yet unclear. Although radical-molecule reactions dominate the chemistry at the surface, gas-phase ion-molecule reactions are a substantial source of neutral radicals that can rival their production via electron impact dissociation (DeJoseph et al., 1984; Haller, 1983). Further, high plasma densities (ionization fractions of 10-4 to are typical in low-pressure plasma etching reactors, so that ion chemistry becomes even more important. Thus, it seems clear that ion-molecule reactions must be included in accurate modeling of the plasma environment (Kushner, 1992), a conclusion substantiated by the recent report Database Needs f o r Modeling and Simulation of Plasma Processing (Database, 1996). One of the difficulties faced in such modeling is that fundamental information relevant to the plasma environment is often not available. For example, Chatham and Gallagher (1985) have modeled ion-molecule reactions in a silane dc discharge-specifically, the reactions of SiH; SiH, (n = 0 to 4 t a n d calculated the distribution of ion species containing more than one Si atom. These reactions have been experimentally investigated, but most only at thermal energies (Mandich et al., 1988; Mandich et al., 1990; Mandich and Reents, 1989; Mandich and Reents, 1991; Reents and Mandich, 1990; Reents and Mandich, 1992). Chatham and Gallagher (1985) note that the accuracy of their calculations is most constrained by the simplifying assumption that the ion-molecule reaction rate constants are independent of collision energy. Although this can be true for efficient exothermic processes (see Section II.D.4), we have shown that this is not a valid assumption for many ion-molecule reactions, and a number of good
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examples are provided throughout this article. Kinetic energy-dependent rate constants (or cross sections) are crucial to understanding plasma reactors because ion energies are usually near thermal in the bulk plasma but can range from tens to hundreds of electron volts in the sheaths. Another critical need identified in Database Needs for Modeling and Simulation of Plasma Processing (Database, 1996) is the measurement of thermodynamic data for species of interest in plasmas (radicals and ions). Such data provide benchmarks for comparison with calculated potential energy surfaces, allow energetically unfavorable reaction pathways to be identified, and supply information necessary to estimate unknown reaction rates by transition state theory. Such thermodynamic information is a critical tool in understanding deposition and etching processes and in evaluating the optimum conditions for plasma reactors ( h i s et al., 1992). With these needs in mind, we have studied a number of ion-molecule reactions related to plasma deposition and etching of silicon over a broad range of kinetic energies using guided ion-beam mass spectrometry. The basic information provided is absolute reaction cross sections as a function of kinetic energy. In most studies, this information is interpreted to determine the mechanisms of the reactions and the relevant thermochemistry of the reactive ionic and neutral species that are of potential importance in plasma environments. In many cases, auxiliary studies expressly designed to measure thermodynamic information have been performed to augment our other studies.
11. Experimental Methods A. GENERAL
The technique we have used in our studies is guided ion-beam tandem mass spectrometry (Ervin and Armentrout, 1985). Ions are produced by one of several methods described below. The ions are then focused into a beam, mass analyzed, and decelerated to a desired lunetic energy. The ion beam is injected into an rf octopole beam guide (Teloy and Gerlich, 1974; Gerlich, 1992), which acts as an ion trap. The benefits of the octopole trap are discussed below. The octopole passes through a reaction cell containing a neutral reactant gas maintained at sufficiently low pressure to ensure single-collision conditions. (Pressure-dependence studies easily isolate contributions from multiple-collision processes.) The product and unreacted ions drift out of the gas chamber to the end of the octopole, where they are focused into a quadrupole mass filter for mass separation and detected by using a secondary electron scintillation detector and standard ion pulse counting techniques. Data collection is under computer control, which allows extensive signal averaging. The instrument incorporates extensive differ-
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ential pumping, which allows precise control over the conditions in the critically important reaction region and is one feature distinguishing our apparatus from similar commercial triple quadrupole instruments. Laboratory (lab) ion energies are converted to energies in the center-of-mass (CM) frame E, the energy actually available for chemical reactions. This conversion utilizes the stationary target approximation E = E,,,m/(m M), where m and M are the masses of the neutral and ionic reactants, respectively. Motion of the neutral reactant is explicitly considered in the analysis of the data. At each CM energy, the intensities of transmitted reactant I,. and product I!, ions are converted to absolute reaction cross sections for each product channel a,](E), using the formulae, I,. = (I,. XI,) exp(-na,,,l) and a,, = atOt(I,,/XZp), where n is the number density of the neutral reactant gas and 1 is the length of the collision cell, 8.26 cm (Ervin and Armentrout, 1985). These cross sections have absolute uncertainties of f20%, up to f50% for some charge-transfer reactions. Relative cross-section values are accurate to about f5%.
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B. ADVANTAGES There are several advantages to using guided ion beam techniques to study ionmolecule reactions. The first is that the octopole ion-beam guide allows for highly efficient product collection. Because of the large number of rods, the trapping field is more homogeneous and effective than in comparable quadrupole devices. This enables us to accurately measure absolute reaction cross sections (Ervin and Armentrout, 1985; Burley et al., 1987a) even for inefficient channels (one in every lo6 collisions). A second benefit of the octopole is that the absolute kinetic energy scale can be accurately determined (Ervin and Annentrout, 1985; Burley el al., 1987a). This is possible because the octopole can act as a highly efficient retarding energy analyzer, and efficient collection of ions does not require an extraction potential. Such potentials, which are frequently used in other beam experiments, distort low collision energies. Use of the octopole, on the other hand, permits very well defked kinetic energies as low as 0.05eV (lab). (The homogeneity of the octopole field is an important consideration in achieving these low energies and one that quadrupole fields cannot match.) The maximum ion kinetic energy in our instrument is 1 keV (lab); thus, measurements can be made at all energies found in plasma environments. Another advantage of these experimental techniques is that each reaction can be studied under single-collision conditions. Thus true microscopic reaction rates can be determined, rather than just the composite rates of depletion and formation of a given species, as in studies using plasma reactors.
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c. ION SOURCES In our work, ions may be generated by using several types of ion sources. This versatility permits the production of atomic and polyatomic ions with controlled internal excitation. This is important for two reasons. First, ions produced in a plasma environment have varying degrees of vibrational and electronic excitation. Such excitation can have a significant effect on reactivity, as illustrated below. Second, accurate derivation of thermochemistry from the kinetic energy dependence of the reaction cross sections requires ions with well-characterized internal energies. This is most easily achieved by creating ions in their vibrational and electronic ground states. 1. Electron Impact Source
One method of ion production is a standard electron impact (EI) source with variable electron energy. This source produces ions having a distribution of internal states such that higher electron energies increase the probability of producing electronically excited ions. Only for ions with no low-lying excited states, such as the rare gas ions, can a ground-electronic-state beam be produced with the EI source. Even in such cases, electron energies below the appearance energy of higher-energy electronic states must be used, and a distribution of the spin-orbit levels of the ground-state beam is generated. For rare gas ions, this is generally thought to produce a statistical distribution of the 'P,,, ground and 2P,,2excited spin-orbit levels. In our work, this has been tested experimentally in the cases of Ar+ (Dalleska and Armentrout, unpublished work), Kr+ (Ervin and Armentrout, 1986), and Xe+ (Ervin and Armentrout, 1989). 2. Surface Ionization Source
Another simple means of creating ions is a surface ionization source. This works effectively for species having low ionization energies, which in this work include atomic silicon and atomic transition metals. Typically, a rhenium filament resistively heated to about 2200K is used. Silane or the vapors of a transition metal complex or salt are directed at the filament, where decomposition and ionization occur. It is generally believed that the electronic state distribution of the ions formed is in equilibrium at the filament temperature. This generally produces ground-state ions, e.g., exclusively Si+(*P), with a distribution of spinorbit levels associated with the filament temperature. 3. Flow-Tube Ion Source Because of the importance of controlling the internal energy of the ions, the ion source is a critical feature of our instrument when used for the study of polyatomic ions. The use of flow-tube technology ensures that the ion energy is well defined. Our flow tube (Schultz and Armentrout, 1991a) can be varied in
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length (25,50,75, or 100 cm) and is pumped by a 600-cfin roots blower. The flow tube is followed by a 13-cm-long region that is held to about torr by an unbaffled 6-in diffusion pump. Ion lenses in this region have an open construction in order to maximize pumping speed. Ions exit through an aperture into the first chamber of the guided ion-beam apparatus (held below torr by a baffled 6-in diffusion pump). For a flow tube that is 1 m in length, the ions undergo 1O5 collisions with the carrier gas (typically He or Ar) at a pressure of 0.5 ton. For the internal modes of polyatomic ions, Lineberger and coworkers have shown that these collisions serve to thermalize the ions, at worst to 1000 K, but more typically to 300 K (Leopold et al., 1985; Leopold et al., 1986b; Leopold et al., 1987; Leopold and Lineberger, 1986a). Some limitations on the degree of collisional cooling do exist. For vibrational excitation, Ferguson has shown that vibrational quenching of small diatomic ions is very inefficient in He (Ferguson, 1986). The use of argon as a bath gas or the addition of a more efficient quencher to the flow can alleviate such problems. For example, we have shown that we can effectively quench excited vibrational states of 0; and NZ, primarily by self-quenching collisions with 0, and N, (Weber et al., 1993; Schultz and Armentrout, 1991b), and the excited ,PI/, spin-orbit level of Xe+ is quenched by methane (Ervin and Armentrout, 1989). In other cases, charge-transfer reactions can be used to produce state-specific beams, e.g., ionization of Kr by CO+ to form Kr+(2P3,2) (Ervin and Armentrout, 1986). Diagnostic charge-transfer reactions with argon, in the case of N; (Schultz and Armentrout, 1991b); methane, in the case of Krf, Xe+, and 0; (Ervin and Armentrout, 1986; Ervin and Armentrout, 1989; Fisher and Armentrout, 1991a); and nitrogen, in the case of O+ (Burley et al., 1987b), can be performed to ensure that these ions are thermalized. Our data for the covalently bound species, e.g., SiF; (n = 1 to 4) (Fisher et al., 1993), are consistent with thermalized ions, and studies of more weakly bound species, such as N i (Schultz and Armentrout, 1992a), Fe(C0): (n = 1 to 5) (Schultz et al., 1992b), Cr(C0); (n = 1 to 6) (Khan et al., 1993), and H30+(H20),,(n = 1 to 5) (Dalleska et al., 1993) have shown that the internal energy of these ions is adequately described by a Maxwell-Boltzmann distribution of rotational and vibrational states corresponding to 298 K.
- -
D. METHODSOF ANALYSIS I . Exothermic Reactions
The collision cross section for ion-molecule reactions at low energies is predicted by the Langevin-Gioumousis-Stevenson (LGS) model (Gioumousis and Stevenson, 1958), o,,,(~)
= ze(2cc/~)I/*
(1)
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 193
where e is the electron charge, c1 is the polarizability of the neutral reactant, and E is the relative kinetic energy of the reactants. Many exothermic reaction cross sections follow this type of energy dependence, although deviations are commonly seen (Armentrout, 1987). 2. Endothermic Reactions
The kinetic energy dependence of the cross sections for endothermic reactions can be analyzed using the empirical formula (Armentrout, 1992)
where a. and n are adjustable parameters, E is the relative kinetic energy, and Eo is the threshold energy. The sum is over all rovibrational and electronic states of the reactants having energies Ei and populations g,, where Cg, = 1. Before comparison with the data, this model is convoluted with the kinetic energy distributions of the reactants, as described elsewhere (Chantry, 1971; Lifshitz et al., 1978; Ervin and Armentrout, 1985). To obtain accurate thermodynamic information, it is necessary to pay attention to several details of the analysis. It is critical to explicitly include the internal energy distribution of the reactants in order to obtain accurate thermochemistry. In addition, in collision-induced dissociation studies, the neutral reactant of choice is generally Xe, for reasons described elsewhere (Loh et al., 1989; Schultz el al., 1992a; Khan et al., 1993). The effects of multiple ion-neutral collisions must be considered, and we generally accomplish this by extrapolating our data to zero-neutral-pressure, rigorously single-collision conditions (Hales et al., 1990; Schultz et al., 1992b). Using these methods, the thermodynamic thresholds of ion-molecule processes can be determined, in favorable cases to within f0.05 eV (lab). A number of studies have demonstrated that these methods can provide accurate thermochemistry on a variety of transient species (Armentrout, 1992; Dalleska et al., 1993; Dalleska et al., 1994; Rodgers and Armentrout, 1997). 3. Relationship between Thresholds and Bond Energies
Converting the thresholds measured using the methods described above to thermodynamic information is straightforward if there are no reverse activation barriers to the reaction under investigation. Reverse activation barriers are often absent in ion-molecule processes because of the long-range ion-induced dipole or higher-order interactions. Activation barriers can occur when the reaction path is complex (Chen et al., 1994; Haynes et al., 1995) or when there are spin or orbital angular momentum restrictions (Armentrout, 1987; Armentrout, 1992). Our studies suggest that the measured threshold E, often corresponds to the
194
F? B. Armentrout
asymptotic energy difference between reactants and products, although this is ideally verified by multiple measurements using several chemical reactions. Two general types of reactions can be considered: collision-induced dissociation (CID) processes, reaction (3),
AX:
+ Rg +. AX:-,
+X
+ Rg
(3)
or rearrangement reactions such as reaction (4), A+
+ BC
AB+
+c
(4)
In the first case, the threshold equals the bond energy, D(M:-, -X).In the second case, the threshold equals the differences in the product and reactant heats of formation such that D(Af - B) = D(BC) - E,. When the threshold analysis is performed using Eq. (2), all sources of reactant energy are included, such that the bond energies so determined correspond to thermodynamic values at OK (Dalleska et al., 1993; Armentrout and Kickel, 1996). Conversion to 298-K values can be achieved using standard thermodynamic functions. In this work, 298-K heats of formation for ions are reported using the thermal electron convention. Values from the literature that use the stationary electron convention should be increased by 0.064 eV for comparison to these values. In much of our early work, however, the internal energy of the reactants was not always included in the threshold analysis, and it was generally assumed that the thresholds corresponded to 298 K values. It is now believed that such values are incorrect, although by only a small amount. Subsequent work has shown that the best means of accurately correcting these early studies is to adjust the measured thresholds to 0-K values (Armentrout and Kickel, 1996). In many cases, this is achieved approximately by adding the rotational energy of the reactants to the reported thresholds. Such corrections have been performed for a number of the systems reported in this review. 4. Conversion of Cross Sections to Rate Constants
Cross sections can be converted to phenomenological rate constants by using the formula k((E)) = VOW)
(5)
+
where v, = (2E/p)’I2, p = m M / ( m M ) is the reduced mass of the reactants, and m and M are masses defined above. The rate constants are a function of the mean relative energy of the reactants ( E ) = E (3/2)ykbT, where y = M / (rn M ) and T is the temperature of the reactant gas (300 K). In the limit that E + 0, k((E)) approaches the “bulk” thermal rate constant for the temperature T’ = yT. In most cases, the room-temperature rate constant can be estimated by examining the behavior of k((E)) at the lowest interaction energies. In many of
+
+
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 195
the papers reviewed below, near-thermal rate constants derived in this manner were tabulated and compared with literature values. Interested readers should see the original papers for such information. For higher energies, absolute rate constants as a function of temperature, k( T ) , can be easily obtained by averaging k(E) over a Maxwell-Boltzmann distribution of relative energies (Ervin and Armentrout, 1985), although internal degrees of freedom are not well characterized by this translational temperature. the rate constant kLGs is Note that when o(E) is well represented by aLGs(E), independent of energy and temperature. Hence, comparison of the energy dependence of measured cross-sections with the LGS collision limit provides a means of quickly assessing whether thermal rate constants can be accurately extrapolated to higher-energy conditions. The data contained in this report belie this commonly used assumption, even in cases where the cross sections are at low energies. modeled accurately by cLGS
111. Reactions with Silane (Sill,) A. RAREGASES(He+, Ne+, Ar+,Kr+, Xef) Of the many studies included in this review, that concerning the dissociative charge-transfer reactions of silane with the rare gas ions (He+, Ne+, Ar+,Kr+, and Xe+) is one of the most interesting (Fisher and Armentrout, 1990b). Results for statistical distributions of the spin-orbit states of Ne+, Ar+,Kr+ and Xe+ along with state-specific data for the 'P3/2 ground spin-orbit states of Kr+ and Xe+ were included. Figure 1 shows the results obtained for He+('SlI2). The total at low cross section is comparable to the calculated collision cross section oLGS energies and a hard sphere estimate at higher energies (>5eV). The products observed in the He system are SiH; (n = 0 to 3), but the smaller ions, Si+ and SiH+, dominate by over an order of magnitude. The Ne+ system shows comparable results, although the total efficiency of the reaction is down by about a factor of 3. Further, SiH+ is slightly favored over Si+ at all energies, and no SM; is observed. Figure 2 shows the results obtained for Kr+ formed by electron impact at 23 eV and thus having a near statistical distribution of 'P3/2 and 'PI/, states (Ervin and Armentrout, 1986). State-specific results for Kr+(2P3/2) differ only slightly from those shown. Again the total cross section mimics the behavior of the calculated collision cross section, but the products now favor larger molecular ions, SiH: and SiHl, with only minor amounts of Si+ and SiH+ observed. Results for Xe+ (both state-specific 'P3/2 and a statistical distribution of 2P3/2 and 'P1/, states) show efficient reaction at all energies. No Si+ and SiH+ products are formed, and yields of SiH3f and SiHl are comparable.
196
P B. Armentrout ENERGY (eV, Lab)
I
10-11 10-1
'
100 I
"
"
'
101
"
I
100
I
I 101
ENERGY (eV, CMI
FIG. 1. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of He+(2S,,2)with SiH,. The solid line shows the total cross section. The dashed line shows the collision cross section given by the maximum of either the ion-induced dipole (LGS) or the hard sphere cross section. Reprinted with permission from Fisher and Armentrout (1990b). Copyright 1990, American Institute of Physics.
The difference in the behavior of these four ions is easily rationalized. The ionization energies (IEs) of He and Ne fall in a region where resonant ionization preferentially removes an electron from the 3a, orbital of SiH,. Photoionization studies indicate that ionization from this orbital yields Sif and SiHt products almost exclusively (Cooper et al., 1990). In contrast, the IEs of Kr and Xe are resonant with the lower-energy 2t, orbital of SiH,. Photoionization from this orbital results in less dissociation, preferentially forming SiH; and SiHr. Thus, the dominant products observed are consistent with the photoionization results for removal of an electron from the two distinct valence molecular orbitals of silane. Given the observations made for the reactions of Het, Ne+, Krt, and Xe+ with silane, our results for reaction of silane with Ar+ (formed by electron impact ionization at 25 eV), shown in Fig. 3, are surprising. It can be seen that there is a dramatic dependence of the total cross section on kinetic energy. The reaction cross section at low energies is about two orders of magnitude smaller than the calculated collision cross section and declines more rapidly with increasing kinetic energy. The cross section reaches a minimum near 2eV before rising sharply at higher energies. Further, the favored products differ sharply in the low-
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS ENERGY (eV.
197
iabi
FIG. 2. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of Kr+(’P) (in a statistical distribution of spin-orbit states) with SiH,. The solid line shows the total cross section. The dashed line shows the collision cross section given by the maximum of either the ion-induced dipole (LGS) or the hard sphere cross section. Reprinted with permission from Fisher and Armentrout (1990b). Copyright 1990, American Institute of Physics.
and high-energy regions of the cross section. Below 2eV, SiH: dominates the products, with smaller amounts of SiHr and SiH+. Above 2 eV, SiH+ and Si+ are the dominant products. On the basis of our observations for the other four rare gas systems, the explanation for these results seems straightforward-namely, below 2eV, ionization occurs by removing an electron from the 2t, orbital, although inefficiently because there is not good Franck-Condon overlap between the IEs of Ar and the 2T, state of SiH,f. Above 2 eV, ionization is preferentially from the 3a, orbital. This hypothesis seems confirmed by the observation that the difference between the adiabatic IEs of Ar and the 3a1 orbital of SiH, is 2.2 eV, as indicated by the arrow in Fig. 3. Although this explanation for the behavior of the Ar+ + SiH, system seems secure, it should be realized that it involves a very interesting coupling of translational to electronic (T-E) energy in the charge-transfer process. Thus, it corresponds to a breakdown in the Born-Oppenheimer approximation. To our knowledge, this type of coupling had not been described in the literature previous to our work. Further, it is observed in a number of other systems, as noted below, although the consequences are rarely as dramatic as those of Fig. 3.
198
I? B. Armentrout ENEf7GY ieV. Lab)
ENERGY
lev, CMi
FIG. 3. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of Ar+(*P) (in a statistical distribution of spin-orbit states) with SiH,. The solid line shows the total cross section. The dashed line shows the collision cross section divided by 10, given by the maximum of either the ion-induced dipole (LGS) or the hard sphere cross section. The arrow indicates the energy level (2.2 eV) where the 3a, state of SiH, becomes accessible. Reprinted with permission from Fisher and Armentrout ( I 990h). Copyright 1990, American Institute of Physics.
B.
o+,o;, N’,
AND
N;
We have also investigated the reactions of silane with more reactive ions (Kickel et al., 1992). Oq4S), Or(,lIg, u = O), Nf(3P), and N;(’C:, u = 0) all react with silane by dissociative charge transfer to form SiH; (n = 0 to 3). The overall reactivities of the O+, OT, and Nf systems show little dependence on kinetic energy, but for the case of N;, the reaction probability and product distribution rely heavily on the kinetic energy of the system. The results for N: are similar to those observed for the Ar+ system, although the changes in behavior with energy are less sharp, presumably as consequence of the increased number of states (vibrational and rotational) available to the molecular species. These results can be understood by comparison with the rare gas ion study (Fisher and Armentrout, 1990b) and can again be explained in terms of vertical ionization from the 1t, and 3a, bands of SiH,. Specifically, IE(0,) is similar to IE(Xe), while IE(0) and IE(N) are most comparable to IE(Kr), and IE(N,) % IE(Ar). Hence, dissociative charge transfer from O,: O+, N+, and N : yields results directly parallel with
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS
199
those for the relevant rare gas ions. The correspondence of the Ar and N, results provides additional reinforcement for our interpretation of the T-E coupling observed in the rare gas charge-transfer processes. Other products observed in these systems include S O H + (a major product), SiO+, SOH:, and SOH; in the reactions of O+ with silane. Reactions of 0; with silane formed these same products, and again SOH+ was produced in good yield. The 0,' reaction also yields SiO,H; ( n = 0 to 3), where both SiO: and Si02H+have-sizable cross sections. Atomic N+ reacts with silane to form SiNH,f (n = 0 to 2 ) , where both SiNH+ and SiNH: are abundant products. In all cases, these SiOH;, SiO,H;, and SiNH; product ions are formed in exothermic reactions, so that only limits on the thermochemistry of these species can be determined.
C. Sis In addition to providing fimdamental data pertinent to the modeling efforts of Chatham and Gallagher (1985), our study of the reaction of atomic silicon ions with silane (Boo and Armentrout, 1987) was very informative regarding the thermochemistry of silicon hydride radicals and cations. The atomic silicon ions were produced by surface ionization such that only the 2P ground state is present. Carefkl attention to the isotopes of silicon was required to accurately assess the cross sections for individual chemical channels. Proper identification was aided by performing these experiments with deuterium-labeled silane (SiD,) as well. Results for reaction of 28Si+with SiH, are shown in Fig. 4. Two processes are observed at thermal energies. The dominant product is Si,H:, but about 10% of the total reactivity involves an exchange of the two silicon atoms, i.e., Si+* SiH, + Sit Si*H4, a process that is readily observed by the production of the minor isotopes (29 and 30) of atomic silicon ions (low-energy region in Fig. 4a). Subsequent work by Mandich et al. (1988) confirms these observations. Theoretical studies of Raghavachari (1 988a) show that these reactions occur by insertion of Si+ into an Si-H bond of silane to form a H, SiSiH+ intermediate. Hydrogen migration yields the symmetric H,SiSiH: species, which is the global minimum for this system. Clearly, reversible hydrogen atom migrations can reform the reactants with and without exchange of the silicon atoms. In addition, the H3SiSiH+ intermediate can dehydrogenate to form the Si,H: product, which has two low-energy isomers. The ground state has the two hydrogen atoms bridging the silicon-silicon bond, Si(H),Si+. Lying 0.30 eV higher in energy is an isomer in which both hydrogen atoms are bound to the same silicon atom, H,SiSi+ (Raghavachari, 1988a). Linear acetylenelike structures are not local minima for the Si,Hl species. Interestingly, Raghavachari's calculations indicate that it is the higher-energy H,SiSi+ isomer that is
+
+
200
P B. Armentrout ENERGY (eV, Lab)
.-
1
10-2 10''
,.
,
",.;:
--
,
A
100
, , ,
,,,I 10'
ENERGY (eV. CM
ENERGY lev. Lab)
ENERGY lev. CU)
FIG. 4. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of "Si+ with SiH,. Part a shows SiH: cross sections, and part b shows Si,H: cross sections. The solid lines show dLGSand the total cross section for all products in parts a and b. In some cases, the cross sections shown, which are for specific masses, can be attributed to distinct chemical species in different regions of the kinetic energy spectrum. Reprinted with permission from Boo and Armentrout (1987). Copyright 1987, American Chemical Society.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 20 1
most readily formed by reaction of Si+ with SiH, because there is a lower barrier to its formation than there is for the ground-state Si(H2)Si+ isomer. At higher energies, several additional reaction pathways open. These are shown in reactions (6) to (1 1). These reactions are endothermic in all cases, and their cross sections can be analyzed to provide thermodynamic information regarding the products. A particularly interesting aspect of reactions (6) to (8) is that both ionic and radical silicon hydrides are formed such that coupled information about the cations and neutrals can be obtained from these data. This is discussed in more detail in Section 1II.E. Si+
+ SiH,
-+
SiH'
+ SiH,
(6)
+=
S i H l + SiH,
(7)
+ SiH -+ Si: + 2H, -+ Si,H+ + H, + H Si,H; + H .+ SiH;
(8)
(9) (10)
(1 1)
-+
D. TRANSITION METALIONS We have also examined the kinetic energy dependence of the reactions of several transition metal ions ( M + )with silane. These studies include all of the group 3 metals (Sc+, Y+, La+, and Lu+) (Kickel and Armentrout, 1995b) and the remaining first-row transition metal ions, Ti+, V+, and Crf (Kickel and Armentrout, 1994); Fe', Co+, and Ni+ (Kickel and Armentrout, 1995a); and Mn+, Cu+, and Zn+ (Kickel and Armentrout, 1995~).In several cases, the reactivity of these metals as a function of their electronic state was examined. A wide range of products are observed, including MSiH; (n = 0 to 3) and ions formed by hydrogen (MH+ SiH, and MH,f SiH,) and hydnde (MH SiH;) transfer. In most cases, the energy dependence of the cross sections can be analyzed in detail to give thermodynamic information regarding bonds between transition metal species and silicon hydrides. The periodic trends in the reactivity and thermochemistry are understood reasonably well. A detailed description of the results of these studies is beyond the scope of this review.
+
+
+
E. THERMOCHEMISTRY OF SILICON HYDRIDES
Thennochemistry for silicon hydride species is derived from either atomic silicon or silane reactions. However, Grev and Schaefer (1992) have found that the
202
I? B. Armentrout TABLE I
THERMOCHEMISTRY (IN ev) OF Species SM SiH2 SiH, SiH, SiHC SiH: SiH:
AfH; (this work)" 4.03 f 0.07c 2.93 f 0.07f 2.20f0.09f 11.84f0.06C.' 11.98f 0.OW 10.32f0.08f
SILICON
HYDRIDES AT OK"
A,H$ (literature) 3.89f O.Osd.e 2.85 f 0.07g 2.14~k0.03~ 0 . 4 6 f 0.02d 1 l.80f 0.05' 12.00f0.06k 10.28f0.03
AfHt (theory)h 3.87 2.84 2.13 11.81 11.99 10.26
Values reported here differ from those in the original citations as discussed in the text, Sections II.D.3 and 1II.E. bGrevand Schaefer, 1992. C Bet al., ~ 1990. ~ dChase et al., 1985. eBerkowitzet al., 1987. /Boo and Armentrout, 1987. gFrey et al., 1986. Seetula et al., 1991. 'Elkind and Armentrout, 1984. 'Boo and Armentrout, 1991. kAfHt(SiH,) + IE(SiH,) from footnote e. 'AfH,0(SiH3)+ IE(SiH,) from Johnson et al. (1989).
commonly used 0-K heats of formation for Si and SiH,, 4.62f0.08 and 0.455 f0.02 eV, respectively (Chase et al., 1985), are incompatible with each other theoretically. They calculate Af H,O(Si)=4.69 f0.02 eV given the experimental heat of formation for silane. In subsequent experimental investigation of silicon fluoride thermochemistry (Fisher et al., 1993), we have concluded that this discrepancy is real and derived a value for the heat of formation of Si' at 0 K of 12.80, f0.03 e\! Combined with the precisely measured IE(Si) = 8.15172f 0.00003 eV, this gives AfH;(Si) =4.65 f0.03 eV, which falls in between the JANAF value (Chase et al., 1985) and the theoretical value. In calculations throughout this manuscript, we adopt our values for the atomic heats of formation. As noted above, analysis of the kinetic energy dependence of reactions (6) to (8) allows the sums of the product heats of formation to be determined. To derive heats of formation for specific silicon hydride radicals and ions, additional information must be employed. Several additional studies have provided information on SiH, SiH+, and SiH: that can be combined with the results for reactions (6) to (8) to provide a complete set of data. In most cases, the thresholds determined in the original work are adjusted to 0-K values, as discussed in Section II.D.3. Conversion between 0- and 298-K values uses the thermodynamic information in Boo and Armentrout (1987). These results are listed in Table 1 and reviewed in the following sections. 1. SiH and SiH:
We have measured the heat of formation of SiH by analysis of the reaction, CH,f (Boo et al., 1990). The revised value obtained, Si+ CH, +SiH
+
+
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 203
4.03 f0.07 eV, is somewhat high but, given the experimental errors, compares reasonably with the generally accepted value of 3.89f 0.09 eV (Chase et al., 1985; Berkowitz et al., 1987), which agrees with theory (Table I). It seems likely that our value is slightly elevated because the reaction to form SiH + CH: competes directly with the much more favorable and less endothermic process Si+ CH, +SiH' CH, (see Section N A ) . When the heat of formation of SiH, 3.89eV, is combined with the thermochemistry from reaction (8), a corrected E,, value of 0.95 f0.05 eV, we obtain A,Ht(SiH,f) = 10.32f 0.09 eV This is comparable to the 10.28f0.03 eV value obtained when A,H,O(SiH,) =2.14f0.03 eV (Seetula et al., 1991) is combined with the ionization energy of SiH,, 8.135f0.005 eV (Johnson et al., 1989).
+
+
2. SiH+ and SiH, We have measured the bond energy of Si+-H by examining the reactions of Si+ H2 and CH, (Elkind and Armentrout, 1984; Boo et al., 1990) to form SiH+ + H, CH,. These studies provide Do(Si+-H) = 3.23 f0.04 and 3.17f0.06 eV, respectively, in good agreement with each other and with spectroscopic values of 3.2040.08 and 3.22f0.03 eV (Douglas and Lutz, 1970; Carlson et al., 1980). Our average bond energy, 3.20f 0.05 eV, can be converted to the heat of formation for SiH' listed in Table I. This table also shows that our value agrees well with experimental values from the JANAF tables (which are derived from the results of Douglas and Lutz) and from the photoionization study of Berkowitz et al. (1987), and with the theoretical value (Grev and Schaefer, 1992). In our work, we combine A, H,O(SiH+)= 1 1.84 eV with the thermochemistry from reaction (6), a corrected Eo value of 0.78f0.05eV, to give A,H,O(SiH,) = 2.20 f0.09 eV This is higher than the best literature value available at the time of our study, 2.074~0.05 eV (Walsh, 1981), but subsequent studies have determined a value of 2.14*0.03 eV (Seetula et al., 1991), and theory predicts 2.13 eV (Grev and Schaefer, 1992).
+
3. SiH, and SiHt A revised analysis of the kinetic energy dependence of reaction (7) yields AfH; (SiH2) A,H,O(SiH;) = 15.02 f 0.06 eV, which also equals 2 ArH,O (SiH,) IE(SiH,). Using IE(SiH,) = 9.15 f 0.02 eV (Berkowitz et al., 1987; Berkowitz, 1989), we therefore obtain ArH,O (SiH,) = 2.93 f0.07 eV and AfH,O (SiH:) = 12.09f0.07eV The former value is in reasonable agreement with that recommended by Walsh (Frey et al., 1986), 2.85 f 0.07 eV, and with theory, 2.84 eV (Grev and Schaefer, 1992). The latter value agrees with that measured in the 0.08 eV (Boo and Armentrout, reactions of Si+ with C2H6 and C2D6, 1 1.982~
+
+
204
l? B. Armentrout
1991), with photoionization values, 11.99f 0.03 eV (Berkowitz et al., 1987), and with theory, 11.99eV (Grev and Schaefer, 1992). 4. Si,H$ Species
Analysis of the cross sections for reactions (9) to (1 1) also provides information regarding several Si,H; species. These include the OK heats of formation for Si:, 14.28+0.09eV; Si2H+, 13.28h0.07eV; and Si,H;, 11.59f0.09eV As discussed in our original work, it is possible that these values are upper limits. The observation that formation of Si,H; is exothermic establishes that its heat of formation is less than 13.26 eV.
IV. Reactions Involving Organosilanes A. Si+
+ CH,,
C2H,
Several of our studies were designed to investigate the reactions of atomic ground-state silicon ions with hydrocarbon gases that are potentially present in plasma systems. The fist of these involved reactions with methane (Boo et al., 1990). The results, illustrated in Fig. 5, show that only endothermic processes are ENERGY (eV. Lob)
10. 0
20, 0
ENE?GY
30.0
rev. CU)
FIG. 5 . The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of 28Si+with CH,. The solid line shows the total cross section for all products. The arrow indicates the H X H , bond energy at 4.5eV Reprinted with permission from Boo et al. (1990). Copyright 1990, American Chemical Society.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS
205
observed. SiH+ and SiH,C+ are the major ionic products, with smaller amounts of CH:, SiCH+, and SiCH:. The energy dependence of the SiH3C+ cross section shows strong evidence that there are two major pathways for forming this species. At low energies, Si+-CH, is formed, while the higher-energy form (beginning near 2 eV) could be HSiCHi or a triplet state of SiCH:. This cross section reaches a maximum at an energy of 4.5 eV, corresponding to the H-CH, bond energy. This behavior indicates that this product decomposes to Si+ CH,, thereby reforming the reactant ion. Two features in the SiCHl cross section are also observed. These can be explained by neutral products of H, (lower threshold process) and 2H (beginning near 5 eV). All observed products are consistent with a reaction that occurs by insertion of Si+ into a C-H bond to form a HSiCH: intermediate. A comparable study on the reactions of Si+(,P) with ethane (do, l,l,l-d,, and d6) was later conducted (Boo and Armentrout, 1991). At thermal energies, exothermic formation of Si+-CH, accounts for 90% of all reactivity. Exothermic dehydrogenation to form SiC,H: and demethanation to form SiCH; are observed, but these processes are inefficient. When CH,CD, is employed as the reactant neutral, all products are observed to incorporate hydrogen and deuterium atoms in near-statistical distributions at low energies, e.g., SiCHD,f and SiCH,D+ have equal cross sections that are about 6 times larger than those for SiCH: and SiCDt. This suggests that the reaction occurs by insertion of Si+ into a C-H, C-D, or C-C bond to form the primary HSi+-CH2CD3, CH,CD,SiD+, and CH,-Si+-CD3 intermediates, respectively, which can rapidly interconvert through a cyclic HDSi(CH2CD2)+ intermediate.
+
B. Si+ + H,SiCH,
In reactions of Si+(,P) with methylsilane, SiH,CH, (Kickel et al., 1992), the major ionic products formed at thermal energies in exothermic reactions are SiCH: and Si,HCH:, and, above l e y SiH,CH:, which is formed in an endothermic process. A number of other minor products (SiH+, SiH:, SiCHi, SiHCH:, Si,CH:, Si,CH:, and Si,H2CH3f) are also observed. Labeling experiments involving 30 Si+ provide additional mechanistic information. The general mechanistic details of this system can be understood by extending the potential energy surfaces calculated by Raghavachari (1 988b). His calculations indicate that the structures of the major products are silicon-methyl cation for SiCHl and Si+-SiH-CH, for Si,HCH:, with a HSi-Si+-CH, structure lying 0.36eV higher. As for the reaction of Si+ with silane, the reaction of Si+ with methylsilane proceeds primarily by insertion of Si+ into a silicon-hydrogen bond to form HSi+-SiH,CH,, followed by rapid and reversible hydrogen migrations to form H, Si+-SiHCH, and H, Si+-SiCH3. These various intermediates decompose by cleavage of the silicon-silicon bond (to form the
206
l? B. Armentrout
+
+
SiH SiH2CH3f and H,Si SiCH: major product channels and several minor products), by dehydrogenation (to form the Si2HCH3f major product), or by H atom loss (to form a minor Si,H,CH,f product in an exothermic process). A minor reaction pathway involves insertion of Si+ into the Si-C bond of methylsilane.
c. THERMOCHEMISTRY OF ORGANOSILANES The studies discussed in Section IV involve the formation of a number of organosilane cations and neutrals. Analysis of these cross sections using the methods outlined above permits the extraction of thermochemistry for many of these species. After correcting for the internal energy of the reactants (Section III.D.3), we obtain the results summarized in Table 11. The value for SiCH+ is the average of three determinations from reactions of Si+ with CH,, C2H2,and C2H4 (Boo et al., 1990). The value for SiCHt is determined from reaction of Si+ with C,H, (Boo et al., 1990). Reaction with methane gave a value for SiCH: that was 0.5 eV higher and was therefore discounted. Reaction of Si+ with ethane provides
TABLE I1
THERMOCHEMISTRY (IN ev) OF ORGANOSILANE COMPOUNDS CATIONS~ Species SiCH+ SiCHt SiCH: SiC,H; SiC,H: SiC,H: SiCH, SiHCH: SiH,CH: SiH,CH, SiH,(CH3)2 Si2CH: Si,CH; Si,HCH: Si,H,CH:
A,% (this work) 14.73f0.24h l2.39& 0.13' 10.07&0.07h 11.06f0.10' < 12.1Oh 10.13f 0.05' 3 . 5 f 0.3' < 10.92&0.14" 9.28f 0.14' <1.86f0.09' -0.74f 0. 17d 13.1050.08' 11.36f 0.06" <12.68' 10.8f0.3"
A,H,,, (literature)
AND
A, H298 (theory)
>9.69,' < I 1.25'
3.15,' 3.2' 10.5, 10.8f0.1" 8.93 f0.1" 1.3, -1.0f 0.1
3.23 f0.0W
1.44f 0.04"
11.86'
"Values reported here differ from those in the original citations as discussed in the text, Sections II.D.3 and IVC. Ion heats of formation use the thermal electron convention. Boo et ul., 1990. ' Wlodek et ul. 1991. Kickel el a/., 1992. Shin et a/., 1990. 'Estimated in Walsh, 1981. "Allendorf and Melius, 1992. "Shin and Beauchamp, 1989. ' Raghavachari, 1988b.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS
207
an upper limit for A,HH,"(SiCH:) of 10.55eV, in good agreement with the value obtained from reaction with methane, 10.07f0.07 eV. Other reactions in the ethane system provide the thermochemistry for the SiC,H; species listed. The remaining thermochemistry comes from our work on reactions of Sit with methylsilane. The heat of formation for SiH,(CH,), listed is derived by combining our results with photoionization measurements of Shin et al. (1990). Values from the literature are also shown in Table 11, although these values are uncorrected from the reported 298-K values, as the required molecular constants are not available.
V. Reactions with Silicon Tetrafluoride (SiF,) A. RARE GASES(He+, Ne+, Ar+,Kr+) He+, Net, Ar+ (Weber and Armentrout, 1989a), Art(2P3/2,1,2),and Kr+('P,/,) (Kickel et al., 1993) react with SiF, exclusively by dissociative charge-transfer reactions. Results for Ar+ in a statistical distribution of spin-orbit states are shown in Fig. 6. All SiF; (n = 0 to 3) products are observed in all systems,
0.0
ILG
3.0
20.0
1
'
1
'
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40.0
1
'
1
'
60.0
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'
i
' -
_ _ _ _ _ _ _ _ _ _ _: %Ol
=tot
.
...................'".....; .................... .....--.. ......... ?-?
.-.W.WWWW..WWWIWWW....W.
0
.
c z
, SiF4+
2
lOG5
w !
t
*
**,.,
:
c
) I
10.0
t+.'+++*+*+%'**tttw+(+,;
f' t'
; ' ' '
-
+ (0'
:
s 10-1
C. 0
SiF2
t
2
10-2
* . . . L
*.***
<
Ar'
-
' '
11 1 1 . 3 1
S:F4
!SiF.' 'r I*
1
'
1 ' '
20.0
' '
I ' 30.0
40.0
50.0
208
I! B. Armentrout
except for Sif in the reactions with Ar+ and Kr+. SiF: is observed only in the
Ar+ system. At h g h energies, the dominant products are SiF; in the Ar and Kr systems, and SiF+ in the Ne and He systems. There is some evidence in the Ne system for production of an excited state of SiF; measured to lie 5.7 eV above the ground state. In the Ne+ and Ar+ reactions, the thresholds measured for the various channels are near those calculated using literature thermochemistry. In contrast, large reaction barriers are observed in the He+ system. This can be attributed to the very different IEs of He and SiF, and was visualized by a detailed examination of how the potential energy surfaces involved vary as the IEs of the rare gases change. In our early work on the Ar+ SiF, system (Weber and Armentrout, 1989a), it was established that charge transfer (CT) to form SiF: was near resonant, so that appreciable amounts of this ion are formed at thermal energies. Features in the CT cross section were speculatively assigned to the different spin-orbit states of Ar+. Later state-specificwork (Kickel et al., 1993) showed that this assignment was incorrect, as both spin-orbit states showed the same complex behavior with kinetic energy (Fig. 7). Instead, the unusual shape of this cross section is attributed to an exothermic and adiabatic electron-transfer process at the lowest
+
ENERGY
10-1
(eV.
Lob) 100
1'
mi
8
10'
'"
0 " z
2 c 100 II)
cn a U n
10-1
FIG. 7. The variation of product cross sections with translational energy in the laboratory kame (upper scale) and the center-of-mass frame (lower scale) for the reaction of Ar'. with SiF,. Closed symbols represent data taken with Ar+ generated in the EI ion source (statistical distribution of spinorbit states) and open symbols show data for the FT ion source (largely 'P,,,). The dashed line shows the threshold analysis of the FT data using Eq. 2. The lower full line shows this model convoluted over the reactant energy distributions. The upper full line shows this convoluted model and includes a contribution from the 2P,,, state. Reprinted with permission from Kickel et al. (1993). Copyright 1993, American Chemical Society.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 209
energies. The efficiency of this process depends on the lifetime of the transient [Ar. SiF,]+ complex, which rapidly decreases with increasing kinetic energy. At slightly higher kinetic energies, the charge-transfer efficiency increases as the system reaches resonance (maximum Franck-Condon overlap) with the vertical IE of SiF,, which is displaced from the adiabatic IE by about 1.2 eV (see Section VD).
B. O+, O l , N+,and N : As shown in Fig. 8, reaction of atomic O+ with SiF, (Fisher and Armentrout, 1991d) yields dissociative charge-transfer products as the major species; however, two additional reactions are particularly interesting. Two strong features are observed in the cross section for SiF:, and the latter has an onset characteristic of the dissociative charge-transfer reaction (12) when A = 0.
A'
+ SiF,
+ SiFT + F + A +. SIFT +AF + SiF: + A
Hence, the lower-energy feature must correspond to reaction (13 ) , in which the OF radical is formed. Analysis of this cross section yields a threshold that ENERGY
lev.
Lab)
3
c
t
10-I l o - '
UL
ENERGY lev.
cnl
FIG. 8. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of O q 4 S ) with SiF, The solid line shows the total cross section. The dashed line shows the collision cross section given by the maximum of either the ion-induced dipole (LGS) or the hard sphere cross section. Arrows indicate the literature thresholds for process (14) at 1.57f0.03 eV, process (12) at 0.36f0.10eV, and process (13) at 2.64f0.04eV. Reprinted with permission from Fisher and Armentrout (1991d). Copyright 1991, Elsevier Science Publishers.
210
P B. Armentrout
corresponds to a 0-K heat of formation for OF of 1.O 1 f0.14 eV, comparable to the literature value of 1.13f0.43 eV (Chase et al., 1985). Reaction (14) is endothermic, so that the threshold provides the difference in IEs of 0 and SiF,, as discussed below. Results for the reaction of 0; with SiF, are shown in Fig. 9 (Fisher and Armentrout, 1991d). Figure 9a shows results for ions formed in the flow-tube source, while Fig. 9b shows data obtained when 0; is formed by electron impact at 50 eV Dissociative charge transfer is the dominant process observed, but minor amounts of SiOF,f (n = 1 to 3) products are formed in this system. Clearly, the results differ greatly for the two ion sources and indicate that reactions of excited 0; ions with SiF, are more efficient than reactions of ground-state ions. This is explained by noting that the metastable first excited state of 0; is near resonance with the *TI ground state of SiFt, while the Ol(2n,) ground state is over 4 eV off resonance. Llke Of, Nf reacts with SiF, by reactions (12) to (14). In this case, however, reaction (13) is exothermic, whereas reaction (12) is endothermic. The SiFi ion formed in the former reaction decomposes at higher energies to form SiFi. Analysis of the threshold for SiF; formation yields a threshold that can be combined with literature thermochemistry to provide a heat of formation for the NF radical, AfH; = 2.75 f0.17 eV This value is in good agreement with the less precise literature value of 2.58f0.34eV (Chase et al., 1985). As with O+, reaction (14) is endothermic when A = N, so that the threshold provides the difference in IEs between N and SiF,. Reaction of N;(,C;, u = 0) with SiF, is very similar to that for Ar+,again because Ar and N, have very similar IEs. One interesting difference is the efficient production of SiF,Nl at thermal energies in an exothermic reaction. This product can be thought of as a complex where N;(2Cl) has substituted for F(,P) in SiF,, or, more realistically, as a complex of the closed shell molecules, SiF: and N,. SiF,N; is actually more stable than SiFt (which is most realistically thought of as a complex of SiFi and F) because N, has a higher polarizability (1.74 A3) (Rothe and Bernstein, 1959) than fluorine atoms (0.557 A3) (Miller and Bederson, 1977). C. Sis Like the reactions of silicon ions with silane, the reactions of silicon ions with silicon tetrafluoride can provide information about both the ions and neutral silicon fluoride species (Weber and Armentrout, 1988). Reactions (15) to (17), analogous to processes (6) to (8), are the only reactions observed. Si'
+ SiF,
+ SiF+ + SiF, + SiF; + SiF;
+ SiF, + SiF
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 2 11 ENERGY lev. Labl
0.0
I
20 .o
40.0
I
I
60.0 I
I
ENERGY lev, CM)
ENERGY lev, Labl 10-1
loo
lo1
1021
000000
wB
ENERGY le V, CMI
FIG. 9. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass h m e (lower scale) for the reaction of SiF, with O:(2H,, u = 0) (part a) and 0: formed by electron impact at 50eV @art b). The solid lines show the total cross sections. The dashed line in part b shows the collision cross section, the maximum of oLGSor the hard sphere cross section. Arrows in part (I indicate the thresholds for formation of SiF: in its X, B, and D states. Reprinted with permission from Fisher and Armentrout (1991d). Copyright 1991, Elsevier Science Publishers.
212
I? B. Armentrout
No Si,F; species, formed in reactions analogous to processes (9) to (1 l), were observed, despite a careful search. As Fig. 10 shows, formation of SiF+ SiF, dominates the product spectrum and is only slightly endothermic. Formation of SiF: SiF and SiF; SiF, have much higher thresholds, and the cross section for the SiF: ion exhibits two features indicating that SiF F is probably formed at higher energies. The thermodynamic interpretation of these results in our paper was hampered by inaccurate information in the literature. On the basis of the thermochemistry listed in Table 111, we calculate that reactions (15) to (17) have endothermicities of 0.14 f0.07, 1.46f0.09, and 2.09f 0.10 eV at 0 K. This compares with measurements of 0.14f0.05, 2.39f0.12, and 2.52f0.14eV made in our work (corrected for the internal energy of the reactants). Clearly the dominant process, reaction ( 15), begins at its thermodynamic limit, but reactions (16) and (17) do not. Reaction (17) competes directly with the much more favorable reaction (1S), and this may reduce its probability at threshold enough that it is difficult to measure the onset accurately. Reaction (16) requires a severe rearrangement (F atom transfer) to form products, and so competition with
+
+
+
+
ENERGY (eV. Lab) 100
101
1
ENERGY (e V. M
FIG. 10. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for reaction of Si+ with SiF,. The first feature in the SiF: cross section corresponds to SlFZneutral products, while the second feature corresponds to SiF F neutral products. The arrow marked E,, indicates the thermodynamic threshold for the Si F. The arrows at 6.4, 9.1, and 6.0eV (top to bottom) charge-transfer process to form SiF: show the thermodynamic thresholds for the dissociative processes that form Si+ F SiF,, SiFz F SiF, and SiF+ F SiF, respectively. Reprinted with permission from Weber and Armentrout (1988). Copyright 1988, American Institute of Physics.
+
+ +
+ +
+ +
+ +
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 2 13 TABLE 111
THERMOCHEMISTRY (IN ev) OF
SILICON RADICALS AT OK
Species Si SiF SiF, SiF, SiF4 Si+ SiFt SiFi SiF: SiF:
FLUORIDE IONSAND
A[H; (this work)
A,H; (theory)"
A,H; (JANAF)*
4.66f 0.03 -0.5 1 f0.09 -6.60f 0.06 -10.30f 0.05
4.67 -0.63, -0.64 -6.54, -6.62 -10.32, -10.41
4.62f 0.08 -0.23f0.13 -6.08f 0.13 -11.22f0.17 - 16.68f 0.01 12.77f0.04
12.81f0.03 6.57f 0.05 4.19f 0.05 - 1.27f0.03 - 1.37f 0.06
12.81 6.73, 4.19, -1.33, -1.35,
6.60 4.20 -1.35 -1.38
"The first values (uncertainties of 0.08eV) listed are from Ricca and Bauschlicher, 1998. The second values (uncertainties of 0.08 eV for neutrals and 0.17eV for ions) are from lgnacio and Schlegel, 1990a and 1990b. 'Chase el al., 1985.
reaction (15 ) may again restrict the probability of this process. This indicates that the fluorine atoms are not particularly mobile, in contrast to the hydrogen atoms in the Si+ SiH, system.
+
D. THERMOCHEMISTRY OF SILICON FLUORIDES Our work (Kickel et al., 1993) on the reactions of Nf(3P), N2+(2Xl), A I - + ( ~ P ,,2) ~ , ~ ,and Kr+(2P3,2) with SiF, was designed primarily to determine the thermochemistry for SiF: and SiF: with high accuracy. This work is recent enough that the original citations already include the internal energy of the reactions, and so no changes to the reported thermochemistry are required. From analysis of the charge-transfer reactions in the Of and N+ systems, the adiabatic IE of SiF, is determined to be 15.292rO.OSeV The values obtained in the two systems are in excellent agreement with each other. Because SiF, undergoes a large Jahn-Teller distortion upon ionization, the Franckxondon factors coupling the ground states of the neutral and the cation are very small. Hence, measurement of its adiabatic IE by photoionization methods has been difficult. The vertical IE is 16.46f0.04eV (Bull et al., 1970), but reported values for the adiabatic LE vary widely, ranging from about 15 to 16 eV (see the review of the literature in Kickel et al., 1993). The advantage of using charge-transfer reactions to measure the adiabatic IE is that nuclear motion must occur during the time scale of a charge-transfer reaction, thereby allowing the SiF: ion time to relax to its ground-state geometry during the ionization process.
2 14
I? B. Armentrout
Dissociative charge-transfer reactions of Ar+, Kr+, and N; to form SiF: are all endothermic. Analyses of the energy dependence of these cross sections yield thermochemistry in good agreement with one another. On the basis of these results, we determined an average heat of formation for SiFl at OK as - 1.27f0.03 eV. This agrees nicely with literature values, which are generally less precise (Kickel et al., 1993). Additional information regarding other SiF: cations and SiF, radicals was obtained by a follow-up study on the collisioninduced dissociation (CID) and charge-transfer reactions of SiF: (n = 1 to 4) (Fisher et al., 1993). An example is shown by the CID reaction of SiF: with Xe in Fig. 11, where it can be seen that sequential losses of fluorine atoms are the dominant reactions, along with a small amount of a charge-transfer process to form Xef. Analysis of such energy-dependent CID cross sections using Eq. (2) (as illustrated in Fig. 11) yields the following O-K bond-dissociation energies (BDEs): Do(SiF:-F)=0.85f0.16ey D,(SiF;-F)= 6.29fO.lOeV, Do(SiF+-F) = 3.18 f0.04 e y and Do(Si+-F) = 7.04f 0.06 eV. The ionization energies, IE(SiF,) = 10.84f0.13 eV and IE(SiF,) = 9.03 f0.05 e y were also measured through analysis of endothermic charge-transfer reactions. Combined with the studies discussed above (Weber and Armentrout, 1988; Kickel et al., 1993), these results allowed us to derive heats of formation for the silicon fluoride cations and neutrals, Table 111, that provide a self-consistent set of thermochemENERGY 0
.o
10 0
(eV.
Lab) 20 .o
N -
u 100
LD
I
0
i
2 H 0 k W U v) v)
10-1
v)
0
u U
0 .o
5 .o ENERGY
10 .o feV. CMI
15 .O
FIG. 1 1 . The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for reaction of SiF: with Xe. The dashed line shows the threshold analysis of the SiF: cross section using Eq. (2). The full line shows this model convoluted over the reactant energy distributions. Reprinted with permission from Fisher et al. (1993). Copyright 1993, American Chemical Society.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 2 15
ical data for the silicon fluoride species. The thermochemical values determined here are considerably different from available literature values, as exemplified by the values listed from the JANAF tables (Chase et al., 1985), but are in good agreement with theory (Ignacio and Schlegel, 1990a; Ignacio and Schlegel, 1990b; Ho and Melius, 1990; Darling and Schlegel, 1993). The agreement with very recent high level ab initio calculations (Ricca and Bauschlicher, 1998) is particularly gratifying.
VI. Reactions with Silicon Tetrachloride (SiC14) A. RAREGASES(He+, Ne+, Ar+) The only reactions of SiCl, with He+, Ne+, and Ar+ are dissociative chargetransfer processes (Fisher and Armentrout, 1991b). The example of the Arf system (statistical distribution of spin-orbit states) is shown in Fig. 12. The total reactivity in all three systems is fairly high, with total cross sections that are comparable to the collision cross section at all energies. All SiCl; (n = 0 to 4)
ENERGY
lev, CMl
FIG. 12. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for reaction of Arf(2P) (in a statistical distribution of spin-orbit states) with SiC1,. The solid line shows the total cross section. The dashed line shows the collision cross section, the maximum of uLos and the hard sphere cross section. Reprinted with permission from Fisher and Armentrout ( 1991b). Copyright 1991, American Chemical Society.
216
P B. Armentrout
species are observed for all three systems, except that SiClz is not seen in the Ne+ system. The major product observed in the He+ and Ne+ systems is SiCl+, with smaller amounts of SiCll and SiCl; and much smaller amounts of Si+ and SiClz. These processes decline with increasing energy in all cases, indicating that the reactions are exothermic. At thermal energies in the Ar+ reactions, the dominant product is SiCl:, with small amounts of SiCl:. Formation of SiCl;, SiCl+, and Si+ is endothermic, but they reach appreciable cross sections at elevated kinetic energies. The thresholds for the former two products agree nicely with known thermodynamic values and can be used to derive thermochemical information (see Section V1.D). Note that the SiCl+ cross section (Fig. 12) has two features that correspond to the difference in energy between C12 C1 and 3C1 neutral products. The observed threshold for Si+ lies well above the thermodynamic limit.
+
B. 0;
Reaction of SiCl, with 0; in its ground electronic and vibrational state is also very efficient, having a cross section comparable to the collision cross section at all energies (Fisher and Armentrout, 1991b). The dominant product at thermal energies is Sic$, with small amounts of SiOCli also formed. The observation of exothermic formation of the latter product indicates that AfHo(OSiCll) t4.21 eV At slightly higher kinetic energies, SiCll is formed in an efficient, endothermic process that can be used to provide thermodynamic information, as discussed below. Above about 4 eV, several additional reaction channels open. These include SiCl;, SiCl+, and SiOCl; (n = 0 to 2). Formation of the former product occurs at energies above the thermodynamic limit, but SiCV is formed promptly at the calculated threshold. Analysis of the thresholds for SiOClf and SiOCl; leads to heats of formation of 8.41 f0.17 and 7.16f0.13 eV, respectively. This requires knowledge of the appropriate neutral products, which were assigned as OC1 2C1 and OC1 C1, respectively, on the basis of an analysis of the sequential bond energies in the SiOCl$ species.
+
+
C. Si+ Like the reaction of Si+ with SiF,, the reaction of Si+ with SiCl, yields no SiCl,-, species of products containing two silicon atoms, but only the SiCl; reactions (18) to (20) (Weber and Armentrout, 1989b):
+
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 2 17
Reactions (18) to (20) are much more favorable energetically than the analogous reactions in the silicon tetrafluoride system. As shown in Fig. 13, exothermic production of SiCP SiC1, is the dominant process and occurs on nearly every Sic1 and SiCll SiC1, production are both found to be collision. SiCl; slightly endothermic. At higher energies, the dissociative channels, SiCli C1+ Sic1 and SiCP C1+ SiCl,, respectively, account for the higher-energy features in these two cross sections. Isotopic labeling studies with ,OSi+ indicate that the SiCV SiC1, and SiClt Sic1 product channels are coupled to each other and are produced by a direct mechanism involving a simple chlorine atom transfer. As expected, SiCll has the silicon isotope completely scrambled, indicating that it is formed through an intimate collision involving a symmetric intermediate. These mechanisms were interpreted in terms of molecular orbital correlations. The cross-section behavior and proposed mechanisms are consistent with those in our analogous study of the Si+ SiF, reaction (Weber and Armentrout, 1988).
+
+
+
+
+
+
+
+
ENERGY (eV. Lab)
100
101
ENERGY (el!
I
CM)
FIG. 13. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-massframe (lower scale) for reaction of Si+ with SiCl,. The solid line shows the total cross section, and the dotted line shows this reduced by 30% (the lower error limit in the magnitude). The dashed line shows the LGS collision cross section. The first feature in the SiCl: cross section corresponds to SiCI, neutral products, while the second feature corresponds to SiCl + CI neutral products. Reprinted with permission from Weber and Armentrout (1989b). Copyright 1989, American Chemical Society.
218
P B. Armentrout
D. THERMOCHE~~ISTRY OF SILICON CHLORIDES Molecular information is not available to properly convert heats of formation of the silicon chloride radicals and ions between 0 and 298 K; hence, all values will be discussed in terms of 298-K thermochemistry. The errors in the experimental values encompass any differences that might arise. The results fkom the present work and from the literature are summarized in Table n! In our study of the reactions of Ar+ SiCl,, thresholds for the formation of SiCP and SiClt provide heats of formation of 8.844~ 0.11 and 8.02 f0.1 1 eV, respectively. These compare favorably with the theoretical calculations of Darling and Schlegel (1993), who find 8.92 f0.18 and 8.08f0.18 eV, respectively. Reaction of 0; with SiC1, yields SiCll in an exothermic reaction, consistent with the relative ionization energies, IE(02) = 12.071eV (Lias et al., 1988) and IE(SiC1,) = 11.79f 0.01 eV (Bassett and Lloyd, 1971). SiCl, also reacts with 0; in a slightly endothermic process to form SiClt, but the products can conceivably be O2 C1 (as we originally assumed) or O,Cl, as pointed out by Darling and Schlegel(l993). Therefore, analysis of the threshold for this system leads to two alternative values for AfH&(SiCl,f), 4.33 f0.07 and 4.50f0.07 eY The latter value is in reasonable agreement with the theoretical value of 4.67 f0.18 eV and with the recommended value of Weber and Armentrout (1989b), 4.42 k 0.13 eV, which was based largely on several appearance energy measurements. We take the latter of our two alternative values to be most definitive. These ionic heats of formation can now be combined with the thresholds measured for reactions (19) and (20) to provide neutral heats of formation. In
+
+
TABLE IV
THERMOCHEMISTRY (lN ev) OF
SILICON CHLORIDE AT 298 Ka
Species Sic1 SiCl, SiC1, SiC1, SiCl+ Sicl:
sicl: sicl,+
A, Hg,8 (this work)b
IONS AND
RADICALS
AfH&, (theory)’
ArH;98 (JANAF)~
1.8f0.2 -1.58zt0.15 (-2.80
1 . 6 2 f 0.08 -1.66% 0.08 -3.32f 0.08
2.05 f0.07 -1.75 f0.03 -4.05f0.17 - 3 . 4 7 ~ t0.09e -6.87f0.01
8 . 8 4 f 0.1 1 8 . 0 2 f 0.1 1 4 . 5 0 f 0.07 (5.27
8.92 f0.18 8.08f0.18 4 . 6 7 f 0.18
a Ion heats of formation use
4 . 9 9 f o.od
the thermal electron convention. Values reported here differ from those in the original citations as discussed in the text, Sections II.D.3 and V1.D. ‘Best estimates from Darling and Schlegel, 1993. Chase et al., 1985. ‘Walsh, 1983.fLias et al., 1988.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 2 19
contrast to the fluoride system, where the thresholds of the two analogous reactions, processes (16) and (17), do not occur at the thermodynamic limit, the chloride system appears to provide reasonable thermochemistry. We obtain thresholds of 0.4 f0.1 and 0.3 f0.2 eV, compared with theoretically calculated results of 0.41 and 0.27eV (Darling and Schlegel, 1993). Combining these thresholds with our heats of formation for SiClr and SiCl;, we obtain heats of formation for the neutral products, SiCl, and SiC1, of - 1.58f0.15 and 1.8 f0.2 eV, respectively. These can be compared to theoretical values of - 1.66f 0.08 and 1.62f0.08 eV, respectively (Darling and Schlegel, 1993). The best literature values are - 1.75f0.03 eV (Chase et al., 1985) for SiC1, and 1 . 6 f 0.4eV (Walsh, 1983) and 2.05 f 0.07 eV (Chase et al., 1985) for SiC1. The observation that reaction (18) is exothermic allows us to determine an upper limit for the heat of formation of SiCl,, AfH&,, < - 2.80eV This agrees with theory and experiment (Table IV),although more experimental work is clearly needed to define the thermochemistry of this species.
VII. Reactions with Fluorocarbons (CF, and C,F,)
+
A. RARE GASES(Hef, Ne+, Ar+) CF, In several studies, we have examined reactions yielding ionized fluorocarbons. In one, dissociative charge-transfer reactions of CF, with He+, Ne+, and Ar+ were studied from thermal to 50eV (Fisher et al., 1990a). As shown in Figs. 14 and 15, only CF,' (n = 1 to 3) products are observed in the reactions of Ne+ and Ar+, produced under conditions that should yield a statistical distribution of 2P3/2and ,P,/,spin-orbit states. Clearly, the reaction with Ar+ is quite efficient, occurring at the collision rate at all energies examined. This is because charge transfer is near resonant: IE(Ar) = 15.755 eV, while photoionization appearance energies (AEs) for CF: are 15.56 (Cook and Ching, 1965), 15.52 (Noutary, 1968), and d 15.35 eV (Walter et al., 1969). As for most tetrahedral molecules, ionization of CF, has a large Jahn-Teller distortion, so that CF: is very difficult to produce and the AE of CF: is difficult to measure. This is discussed further below. Smaller product ions, CF; and CF+, are observed at higher energies, but their thresholds do not correspond to the thermodynamic values. Rather, they correlate directly with the onset for ionization of CF, to the C2T2and D2A, states of the ion, respectively. This correlation is shown more clearly in the reaction of CF, with Ne+, Fig. 15. Here, the efficiency of the reaction at thermal energies is down about an order of magnitude because the IE of Ne is not resonant with any electronic bands in CF,. Even though the reaction Ne+ CF, + CF: 2 F Ne, is exothermic, the CF; cross section shows an apparent onset for
+
+
+
220
I! B. Armentrout ENERGY CeV. Lob)
A
t
B
I1 ENERGY (e V. CM)
FIG. 14. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of CF, with Ar+(2P) (in a statistical distribution of spin-orbit states). Cross sections for CF: and CF+ have been multiplied by a factor of 10. The dashed line shows the collision cross section, the maximum of oLGSand the hard sphere cross section. Arrows indicate the thresholds for formation of CFZ in its X, A, B, C, and D states. Reprinted with permission kom Fisher et al. (1990a). Copyright 1990, American Institute of Physics.
efficient production that correlates with the C state of CF;. Llkewise, the onset for CF+ is well above the thermodynamic limit of 2.26 eV, but close to the origin of the D state of CF;. The reaction of He+ with CF, shows a very strong dependence on lunetic energy, yielding CF: and CF,f in nearly equal yields at low energies and then primarily CF; and CF+ at elevated kinetic energies. Small amounts of C+ and F+ are seen at high kinetic energies for the He+ reactant, and these appear to correlate with the onset of the E2T2state of CF,. Reaction rates at thermal energies for these reactions determined in our work compare reasonably well with previous measurements.
B. o+,0;
-4- CF4, C2F6
The reactions of O+ and 0;with CF, and C2F6 have also been examined (Fisher and Armentrout, 1991~).In both systems, the predominant ions formed correspond to dissociative charge-transfer reactions, with small amounts of FCO+ and F2CO+ being formed. As for the rare gas system, the thresholds and shapes of the
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 22 1
a -tD,
I
I
$ , I
dissociative charge-transfer cross sections are explained in terms of vertical ionization to various electronic states of CF; and C,F$. Although charge transfer is endothermic for both ionic reactants with CF,, atomic O+ reacts efficiently at thermal energies by fluoride transfer to form CFT OF. For C2F6, charge transfer from O+ is nearly resonant at thermal energies, so that the total reaction cross section follows uLGSat low energies (below 1 eV). CFT is the dominant product in this energy region, but C2F$ OF is also formed in abundance. Formation of C2Ff 0 F, endothermic by 1.13eV, becomes the dominant reaction above 1 eV From the 0; CF, and O+ C2F, systems, AfH&(F2CO+) = 6.30f 0.08 eV is determined, a value that compares well with the literature, 6.37f 0.15 eV (Lias et al., 1988).
+
+ +
+ +
+
c. THENOCHEMISTRY OF FLUOROCARBONS Our study of the reactions of rare gas ions with CF, made it clear that the heat of formation of CF: was not well established. Because of the severe Jahn-Teller distortion upon ionization, measured photoionization thresholds for formation of
222
P B. Armentrout
CF: from CF, are generally too high, and even photoelectron-photoion coincidence experiments provide only upper limits (Brehm et al., 1974; Powis, 1980). We therefore performed a series of experiments designed to determine this heat of formation (Fisher and Armentrout, 1990d). In contrast to a literature report (Babcock and Streit, 198l), we demonstrated that there can be no fluoride transfer equilibrium between CF,f and SFf at thermal energies by examining both the forward and reverse reactions (21):
This conclusion agrees with that of Sieck and Ausloos (1990). In our work, we measured the threshold for formation of CF: from the reaction of ground state Kr+(2P,,2) with CF, to establish AfHo(CF$). This value agreed well with a less precise value derived from collision-induced dissociation of CF: by Xe. We reported a threshold of 0.24 f0.07 eV, which can be corrected to a 0-K threshold of 0.28eV after including the internal energy of the reactants. This threshold corresponds to AE,(CF$/CF,) = 14.28f 0.07 eV and AfH,"(CF,f)= 3.87f 0.07 eV This agrees well with a value, AE,,,(CF,f/CF,) = 14.2 f0.1 eV, derived from the observation of an efficient reaction (22) at thermal energies (Tichy et al., 1987):
HC1'
+ CF, + CF$ + HF + C1
(22)
Further, it has seemingly been c o n h e d by photoionization studies of CF,Br, which find AfHl(CF,f) = 3.76f 0.05 eV (Clay et al., 1994). However, a more recent photoionization study examined several compounds to determine values of AfH,"(CF;) = 4.25 f0.04 and AE, = 14.67%0.04 eV (Asher and Ruscic, 1997). In light of this work, we have reexamined our Kr+ CF, data and find that a 0-K threshold as high as 0.53f0.12eV is consistent with our data. This threshold corresponds to AfH;(CF,f)=4.12f0.12 and AE, = 14.53f0.12ey which is then in reasonable agreement with the values of Asher and Ruscic, but unfortunately not definitive. Ultimately, the difficulty in resolving these disparate values is that measured thresholds (reaction or photoionization) can be too high if the onset for CF: production is inefficient (for instance, because of FranckCondon factors or competition with other channels). The higher values for AfHl(CFi) are not consistent with the observation of an efficient reaction (22) at thermal energies (Hansel et al., 1998), which may indicate that the higher AE and AfH" values are still technically upper limits.
+
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 223
VIII. Miscellaneous Thermochemical Studies A. 0:
+ CH,
Several studies were aimed at determining the thermochemistry of additional species of potential importance in plasma systems. One of these (Fisher and Armentrout, 1990c) involved studying the reaction of O;(,II,, u = 0) with CH,. Analysis of the kinetic energy dependence of the slightly endothermic reactions (23) and (24),
0;
+ CH, + 0,H + CH: + O2 + CH,f
provided measurements of the heat of formation of the hydroperoxyl radical, AfH&(H02) = 0.16 f0.05 eV, and the ionization energy of methane, IE(CH,) = 12.54f0.07 eV: These values are in reasonable agreement with literature thermochemistry, as discussed in detail in our work. The former value can be combined with other heats of formation in the literature to yield the bond energies, D,,,(H-OO) = 2.09 f0.05 eV and D,,,(H-OOH) = 3.83 f0.05 eV: A more comprehensive study (Fisher and Armentrout, 1991a) of this reaction system found three previously reported reaction products, those from reactions (23) and (24) and CH200H+ H, formed in the only exothermic process. Several minor products, CH,O,f, H30+, and CO:, are also observed at higher kinetic energies. Reactions of excited 0: ions (formed by electron impact) were also examined and shown to react more efficiently than ground-state ions. The thermochemistry and potential energy surfaces for this reaction were discussed in detail, as were the effects of vibrational, electronic, and translational energy on the reaction system. This provided complementary information to the thorough studies of this reaction system using selected ion flow tubes (Van Doren et al., 1986; Barlow et al., 1986). In our work, a 298-K heat of formation for CH,O,f of 8.74 f0.07 eV was measured and tentatively assigned to the methyne hydroperoxy ion structure, HC+-0-OH.
+
B. SULFURFLUORIDES Finally, a comprehensive study of the thermochemistry of sulfur fluoride cations and neutrals was performed (Fisher et al., 1992). Endothermic charge-transfer reactions of several of the SF: ions were examined. Analyses of these cross sections yielded the ionization energies of SF from reactions with Xe; of SF, from reactions with CH,I, NO, and C,H,Br; of SF, from reactions with Xe and 0,; and of SF, from reactions with NO, C6H,CF,, and CH31. IE(SF,) was not measured in our work, as a precise photoelectron value (10.08 f0.05 eV) was already available in the literature (DeLeeuw et al., 1978). Some of our values
224
F! B. Armentrout TABLE V IONIZATION
Species SF SF2 SF3 SF4 SFS
ENERGIES(IN eV) OF
SULFUR
FLUORIDE RADICALS
This worku
JANAF~
Theory
10.16f 0.17
10.09f0.1 10.29f0.3 9.24f0.7 12.154=0.3 11.14f0.37
10.13,‘ 10.31; 10.22e 10.15,‘ 10.07,d 10.15‘ 8.36,’ 8.27; 8.24‘ 11.90,’ 11.85; 11.90‘ 9.71,’ 9.63: 9.52‘
8.18zk 0.07 11.69f0.06 9.60f0.05
“Fisher et al., 1992. hChaseet al. 1985. ‘Irikura, 1995. Uncertainties are 0.16-0.20eV dCheung et al., 1995. ‘Bauschlicher and Ricca, 1998.
disagree with literature thermochemistry as exemplified by values listed in the They are in reasonable JANAF tables (Chase et al., 1985), given in Table I? agreement with subsequent ab initio studies (Irikura, 1995; Cheung et al., 1995; Bauschlicher and Ricca, 1998). We also examined the energy dependence of the cross sections for CID of SF,f (n = 1 to 5) with Xe (Fisher et al., 1992). Analysis of these cross sections yielded the 0-K bond-dissociation energies (BDEs) listed in Table VI. Experimental values in the literature (Chase et al., 1985; Lias et al., 1988) prior to our study had large error bars, but subsequent photoionization studies by Ng and coworkers (Cheung et al., 1995) probably provide the most accurate heats of formation for SFt, SF:, and SF: that are presently available. The heat of formation for SF,f has been measured by the same group using a sophisticated photoelectron-photoion coincidence experiment (Evans et al., 1997). BDEs calculated using these values and heats of formation for S+ and SF+ from TABLE V1
BONDDISSOCIATION ENERGIES (IN ev) OF SULFUR FLUORDE CATIONS AT
OK
Species
Do (this work)”
Do(exp.)b
D” (Theory)
S+-F SF+-F SFi-F SF:-F SFi-F
3.56f0.05 4.17k 0.10 4.54 It 0.08 0.36f0.05 4.60f0.10
3.79f0.17 3.87f0.22 4.26f0.16 0.50f0.14 3.99f0.14
3.72,’ 3.80d 3.94,’ 3.86d 4.22,” 4.15: 4.16‘ 0.49,’ 0.58,d 0.56‘ 3.97,’ 3.85; 3.87‘
“Fisher et al., 1992. bCalculated from 0-K heats of formation for S+ and SF+ in Chase et al., 1985; for SF: ( n = 2 to 4) in Cheung et al., 1995; and for SF: in Evans et al., 1997. ‘Bauschlicher and Ricca, 1998. dlrikura, 1995. Uncertainties are 0 . 0 W . 0 8 e V ‘Chueng etal., 1995.
KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 225
Chase et al. (1985) are given in Table VI. These agree nicely with theoretical values (Irikura, 1995; Cheung et al., 1995; Bauschlicher and Ricca, 1998). Overall, our BDEs for SF+, SF:, SF:, and SF,f agree with the literature values within or just outside the combined experimental errors. Nevertheless, the agreement between our values and the literature is somewhat disappointing and deserves some speculative comment. For SF:, SF; and SF;, the BDEs determined by CID are slightly large. CID thresholds can exceed the thermodynamic BDEs for strongly bound species simply because the probability that all of the collision energy is transferred from kinetic energy into the reaction coordinate leading to dissociation becomes increasingly small as the threshold increases (a so-called kinetic shift). We have shown that this can lead to CID cross sections that are difficult to model accurately in the threshold region and thus yield elevated thresholds (Aristov and Armentrout, 1986; Sievers et al., 1996). Clearly, such changes in CID thresholds need not be systematic problems; if they were, CID results for the strongly bound SiF; species would also be in error, but they are not (see Section YD). For SF+, our BDE is lower than the other values, but in this system, there was evidence for excited SFf species (Fisher et al., 1992), which may have been incompletely quenched, leading to a low threshold. In the case of SF:, it is possible that this weakly bound species was incompletely thermalized in the flow-tube source, a conclusion consistent with the slightly lower IE (by 0.16 to 0.2 1 eV) measured in the same experiment (Table V). The SF,f molecule is a special case where our measured CID threshold disagrees with the best experimental and theoretical values by a large amount (Table VI). The dissociation behavior of this species is particularly complicated because the lowest-energydissociation product, SF;, rapidly dissociates further to SF; at slightly higher kinetic energies. This complex behavior makes analysis of the cross section more difficult. Another factor that may be influential was pointed out by Irikura (1995), who calculated that there is a substantial energy (0.47eV) required for geometry relaxation of the SF: fragment. Because the threshold for dissociation is at fairly high kinetic energies, such a large relaxation energy may make it difficult to observe the thermodynamic threshold in this system.
IX. Conclusions Guided ion-beam mass spectrometry has proven its ability to measure absolute reaction cross sections over a wide range of kinetic energies. The use of the octopole ion trap allows the energy range between thermal and several to hundreds of electron volts to be bridged with no loss in collection efficiency. Such kinetic energy4ependent information is critical to accurate modeling of ion-molecule reactions of importance in plasma chemistry. While the assumption
226
l? B. Armentrout
that ion-molecule rate constants are independent of energy is often used (because no kinetic energy-dependent information is available), we find this is rarely the case. Even for reactions that follow the LGS collision limit at low energies (e.g., those illustrated in Figs. 1,2,4,9b, 12, 13, and 14), the cross sections often reach a total cross section that follows a hard sphere collision limit at elevated energies, so such that the associated rate constant will increase with increasing energy (as E l l 2 ) at high energies. In other cases, the reactions are endothermic and cannot follow oLGS (e.g., data in Figs. 5 , 8,9a, 10, and 11). In still others, unusual kinetic energy behavior of exothermic reactions leads to cross sections that deviate sharply from the LGS limit (e.g., Figs. 3, 7, and 15). Unfortunately for modelers, there are no simple rules that predict the behavior of ion-molecule reactions at elevated kinetic energies. These must generally be measured on a case-by-case basis. Thermochemistry for silicon hydride, silicon fluoride, silicon chloride, organosilane, and sulfur fluoride radicals and cations obtained using guided ion-beam methods is reviewed here. In general, we find that our results provide accurate information, although the precision of the method does not compare with that of spectroscopic and photoionization methods. In some cases, these methods provide access to thermochemistry for species that are difficult to measure in other ways, e.g., the ionization energies of tetrahedral molecules that undergo large geometry changes upon ionization. In such cases, accurate adiabatic onsets can be measured because there is sufficient time during reaction for the required nuclear motion. In other cases, collisional processes (especially those at high kinetic energies) can occur impulsively, so that energy transfer is incomplete. This can lead to a breakdown in the Bomappenheimer approximation, which leads to nonintuitive dependence on the hnetic energy (e.g., Fig. 3).
X. Acknowledgment I thank my coworkers on this project, Drs. Bong Hyun Boo, Mary Ellen Weber, Ellen R. Fisher, and Bernice L. Kickel, for their substantial contributions to the studies described here. Primary funding for this work was supplied by the Air Force Wright Aeronautical Laboratories, and partial support was obtained from the National Science Foundation.
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KINETIC ENERGY DEPENDENCE OF ION-MOLECULE REACTIONS 229 Reents, W. D., and Mandich, M. L. (1990). 1 Chem. Phys. 93, 3270. Reents, W. D., and Mandich, M. L. (1992). 1 Chem. Phys. 96, 4429. Ricca, A,, and Bauschlicher, C. W., Jr. (1998). 1 Chem. Phys. 102, 876. Rodgers, M. T., and Armentrout, I? B. (1997). 1 Phys. Chem. A 101, 1238. Rothe, E. R., and Bemstein, R. B. (1959). 1 Chem. Phys. 31, 1619. Schultz, R. H., and Armentrout, F! B. (1991a). Int. 1 Muss Spechorn. Ion Pmc. 107, 29. Schultz, R. H., and Armentrout, I? B. (1991b). Chem. Phys. Lett. 179,429. Schultz, R. H., and Armentrout, I? B. (1992a). 1 Chem. Phys. 96, 1046. Schultz, R. H., Crellin, K. C., and Armentrout, I? B. (1992b). 1 Am. Chem. Sac. 113, 8590. Seetula, J. A,, Feng, Y., Gutman, D., Seakins, I? W., and Pilling, M. J. (1991). 1 Phys. Chem. 95, 1658. Shin, S. K., and Beauchamp, J. L. (1989). 1 Am. Chem. Sac. 111, 900. Shin, S. K., Corderman, R. R., and Beauchamp, J. L. (1990). Int. Muss Spechorn. Ion Proc. 10 1,257. Sieck, L. W., and Ausloos, F! J. (1990). 1 Chem. Phys. 93, 8374. Sievers, M. R., Chen, Y.-M., and Armentrout, I? B. (1996). J Chem. Phys. 105, 6322. Teloy, E., and Gerlich, D. (1974). Chem. Phys. 4, 417. Tichy, M., Javahery, G., Twiddy, N. D., and Ferguson, E. E. (1987). Int. 1 Muss Spechom. Ion Proc. 79, 231. Van Doren, J. M., Barlow, S.E., Depuy, C. H., Bierbaum, V M., Dotan, I., and Ferguson, E. E. (1986). 1 Phys. Chem. 90, 2772. Walsh, R. (1981). Acc. Chem. Res. 14, 246. Walsh, R. (1983). 1 Chem. Sac. Famday Duns. 1 79, 2233. Walter, T. A,, Lifshitz, C., Chupka, W. A., and Berkowitz, J. (1 969). 1 Chem. Phys. 5 1, 353 1. Weber, M. E., and Armentrout, F! B. (1988). 1 Chem. Phys. 88, 6898. Weber, M. E., and Armentrout, F! B. (1989a). 1 Chem. Phys. 90, 2213. Weber, M. E., and Armentrout, F! B. (1989b). 1 Phys. Chem. 93, 1596. Weber, M. E., Dalleska, N. F., Tjelta, B. L., Fisher, E. R., and Armentrout, P. B. (1993). 1 Chem. Phys. 98, 7855. Wlodek, S., Fox, A., and Bohme, D. K. (1991). 1 Am. Chem. SOC.113,4461.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
PHYSICOCHEMICAL ASPECTS OF ATOMIC AND MOLECULAR PROCESSES IN REACTIVE PLASMAS YOSHIHIKO HATANO Department of Chernistvy. Tokyo Institute of Technology, Meguro-ku. Tokyo 152-8551. Japan
I. Introduction 11. Atomic and Molecular Processes in Reactive Plasmas.......................... 111. Overview and Comments on Free Radical Reactions in Reactive Plasmas IV Deexcitation of Excited Rare Gas Atoms by Molecules Containing Group IV Elements ............................................................... V Comments on Atomic and Molecular Processes in Reactive Plasmas from Physicochemical Viewpoints ..................................................... VI. References . . .................... ........ ........
23 I 232 233 235 240 240
I. Introduction A brief survey is given of physicochemical aspects of atomic and molecular processes that are of great importance in reactive plasmas. The processes are composed of the interaction of molecules, in most cases polyatomic molecules, with reactive species such as electrons, ions (both positive and negative), free radicals, and excited atoms and molecules. Topics are chosen from recent studies of some elementary processes in reactive plasmas. Some comments are also given on future problems that call for more work in reactive-plasma research from the viewpoints of physicochemical studies of gas-phase reaction dynamics and kinetics, such as radiation chemistry and photochemistry. Reactive plasmas are generally characterized as plasmas in which component polyatomic molecules have an important role. Information and ideas, as well as experimental techniques in physical chemistry, particularly in reaction dynamics and kinetics studies, are greatly needed to control the essential features of atomic and molecular processes in reactive plasmas and thus to obtain desired products of reactive-plasma processing such as chemical-vapor deposition (CVD) and etching (Hatano, 1991). Atomic and molecular processes may be classified into the following three groups: 1. Atomic and molecular processes in a bulk plasma 23 1
Copyright 0') 2000 by Acadcmic Press All rights of reproduction in any form reservcd ISBN: 0- I2-003843-9/ISSN: 1049-25OX $30.00
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2. Atomic and molecular processes in the region of plasma-surface interactions 3. Atomic and molecular processes in solids It is generally accepted that all three groups are of great importance in the control of reactive plasmas (Hatano, 1991). This article treats only the first group.
11. Atomic and Molecular Processes in Reactive Plasmas The primary activation of parent molecules in reactive plasmas is through the collision of molecules with electrons in a wide energy range. Molecules thus receive energies from electrons and form reactive species such as excited or ionized states of molecules, free radicals, and electrons of low energies. These species interact with each other or with stable molecules. The succession of events in atomic and molecular processes that follow the primary activation is summarized in Table I (Hatano, 1991; Tanaka et al., 1996). In analyzing these processes, workers in a reactive-plasma research field should understand, at least TABLE I
ATOMICAND MOLECULAR PROCESSES IN AB+AB+ + e+AB** +AB* AB**+AB+ e+A+B AB++A’ +B ABf + AB or S+Products AB+ + e- +AB* A B f + S---+F’roducts e- S+Se - +nAB-+e; AB*+A + B +AB + BA +AB+hv AB* S+AB S* AB* +AB+(AEl)* U+A, +C+D A AB+A2B +A,+B
+
+
+
+
+
REACTIVE PLASMAS
Direct ionization Superexcitation] (Direct excitation) Excitation Autoionization Dissociation Ion dissociation Ion-molecule reaction Electron-ion recombination Ion-ion recombination Electron attachment Solvation Dissociation Internal conversion and intersystem crossing Isomerization Fluorescense Energy transfer Excimer formation Radical recombination Disproportionation Addition Abstraction
Source: (Hatano, 1991 ; Tanaka et a[., 1996).
ASPECTS OF ATOMIC AND MOLECULAR PROCESSES
233
in general terms but hopefully comprehensively, the present status of the knowledge of atomic-collision research and elementary reaction dynamics. Following are a brief description of and related comments on the atomic and molecular processes in reactive plasmas listed in Table I. Molecules AB in collisions with electrons distributed over a wide range of their energy, which is characterized with an electron temperature or a mean energy, are directly ionized and excited into superexcited states (Hatano, 1999) above their first ionization potentials and excited states below them. Superexcited states AB** may be autoionized or dissociated to neutral fragments, i.e., free radicals or stable product molecules. Electronically excited states AB* may also be dissociated to neutral fragments. Parent ions directly formed via direct ionization or indirectly formed via autoionization are dissociated to fragment ions. It should be noted that free radicals are formed simultaneously in the dissociation of the parent ion. Absolute cross sections as a function of the electron-molecule collision energy are needed, therefore, for both ionization and dissociation. Formed ions are quickly converted to other ions via ion-molecule reactions, whose reaction rates are dependent on the pressure of molecules. It should again be noted that free radicals are also formed in ion-molecule reactions and that the collision energy dependence of these reactions has not been fully understood. In some cases, negative ions are produced in electron-molecule collisions. Electrons with characteristic energies are selectively captured to form negative ions. It is generally accepted that large neutral clusters and larger aggregates of molecular products such as dust can capture electrons at large cross sections (Hatano, 1986). In some cases, particularly in a gas system of polar molecules AB, free electrons or ions are solvated with AB-dipoles to form solvated electrons and ions, respectively (Hatano, 1986). Such species would have an important role in reactive plasmas. The recombination of positive ions with electrons or negative ions may also contribute to formation of free radicals. In reactive plasmas consisting of the mixture of a rare gas with an additive host gas of polyatomic molecules, the collisional energy transfer from an excited rare gas atom to a constituent polyatomic molecule to form ions and free radicals is of great importance.
111. Overview of and Comments on Free Radical Reactions in Reactive Plasmas In the bottom part of Table I, the reactions of free radicals in reactive plasmas are summarized. As described in the preceding sections, there are various reactive or transient species in reactive plasmas. It is generally accepted, however, that free
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Y: Hatano
radicals are the most important species leading to desirable products of reactiveplasma processing (Matsuda and Hata, 1989). Precursor states or processes that contribute to the formation of free radicals are also important. In this section, therefore, an overview of and comments on free radical reactions are given from a physicochemical viewpoint. Free radicals produced from precursor states or processes described in the preceding section decay through one of the following reactions (Hatano, 1991; Tanaka et al., 1996): 1. 2. 3. 4.
Recombination with other free radicals to form stable molecules Disproportionation to two stable molecules Addition to or insertion to stable molecules to form larger molecules Abstraction of atoms or free radicals from stable molecules 5 . Reaching the surface in a reactive-plasma system and reacting with the surface or the so-called dangling chemical bonds to finally attach to the surface
Ions in a bulk plasma can be accelerated with the electric potential of a plasma sheath near a solid surface so that they reach the surface, quickly neutralize, and finally attach to the surface. It is experimentally difficult to discriminate this mechanism from free radical mechanism 5, the deposit of reactive species near the surface. To clarify the roles of free ra&cals in reactive plasmas, it is necessary to measure the products in free radical reactions 1 though 5, which have not been klly understood in the physicochemical studies. Free radicals in SiH, plasmas have been measured using newly developed infrared diode laser absorption spectroscopy (Itabashi et al., 1990; Kono et al., 1993). Their densities are, in brief, [SiH,] :[SiH,] : [SiH] : [Si] = loL2: loLo: 1O'O : lo9 cm-,, i.e., SiH3 radicals are the most abundant, and their density is 2 orders of magnitude smaller than the density of SiH, itself. This result has been explained as follows (Tanaka et al., 1994; Nomura et al., 1995; Itabashi et al., 1989). The reactivity of SiH, with SiH, is much less than the reactivity of the other free radicals in the present system. In particular, SiH, reacts easily with SiH, to form larger free radicals, which may contribute to the formation of clusters and dust. The effective lifetime of SiH, is, therefore, exceptionally long compared with those of the other free radicals, and SiH, can reach predominantly a solid surface and produce amorphous silicon. It is accepted, therefore, that SiH, is the most desirable free radical for deposit of amorphous silicon. In SiH, plasmas, the effect of rare gases as additives has been examined to see expected changes in the distribution of the free radical density among SiH, where x = 0 to 3. In fact, the quality of the amorphous silicon produced is clearly dependent on added rare gases (Matsuda et al., 1991). The obtained results have
ASPECTS OF ATOMIC AND MOLECULAR PROCESSES
235
been discussed in terms of the differences in the excitation energy of the metastable state of rare gases.
IV. Deexcitation of Excited Rare Gas Atoms by Molecules Containing Group IV Elements Binary mixtures of rare gases with polyatomic molecules have frequently been used for gases in reactive plasmas, in which energy-transfer processes between excited rare gas atoms and molecules are considered to be very important. There were, however, very few cross-section data reported on molecules of Si or Ge hydrides, in comparison with those on hydrocarbons (Ukai and Hatano, 1991). Cross sections for the deexcitation of excited rare gas atoms by molecules containing group IVelements and also branching ratios of the product formation have been measured using, respectively, a pulse radiolysis method combined with time-resolved spectroscopy and a flowing afterglow method combined with optical-emission or laser-induced fluorescence (LIF) spectroscopy (Hatano, 1991; Tanaka et al., 1996; Yoshida et al., 1991; Yoshida el al., 1992a; Yoshida et a1.,1992b; Yoshida et a1.,1993; Tsuji et al., 1989a; Tsuji et a1.,1989b; Tsuji et al., 1990; Sekiya et al., 1987; Sekiya et al., 1989; Sekiya and Nishimura, 1990; Balamuta et al., 1983; Bolden et al., 1970). The following results were obtained. The deexcitation cross-sections are dependent on electronic states of excited rare gas atoms and target molecules. They are compared in detail with related theoretical results to find some regularities of cross-section values in correlation with fundamental parameters of target molecules, from which unknown crosssection values for any molecules can be estimated with enough accuracy. Availability of this estimation will be of great importance in finding new candidates of host molecules in reactive-plasma research. Deexcitation cross sections of He(23S), He(2'S), and He(2'P) by CH,, SiH,, GeH,, C,H,, Si,H,, CF,, SiF,, and SiCl, have been measured at a mean collisional energy corresponding to room temperature to understand general features of the cross-section values dependent on the electronic states of excited helium atoms and the target molecules containing group IV elements and to understand the deexcitation mechanism. In the deexcitation of the metastable atoms, He(23S) and He(2'S), the magnitude of the cross sections suggests a short-range electron exchange interaction. The cross section values oM for He(23S) reacting with molecule M are compared, as shown in Fig. 1, with the semiempirical formula o<
Ni.IP-l',
(1)
where Ni and IP are the number of electrons in M to be ionized in the He(23S)-M collision and the lowest ionization potential of M , respectively.
236
I: Hatano
= t
b
FIG. 1. Relation of deexcitation cross sections u M with Eq. (1) for He(23S) (Yoshida et al., 1992b).
It has been generally considered that a major part of the deexcitation processes of He(23S), He(2'S), and He(2'P) in collisions with molecules at the mean collisional energy corresponding to the room temperature is understandable in terms of Penning ionization, because the excitation energy of these excited states is much higher than the ionization potential of almost all the molecules. Optical emissions in a UV-visible region from Si* atoms in several excited electronic states produced in the He(23S) SiH, collision are observed at a total emission cross section of 0.081 A2. Minor emissions from other excited fragments, i.e., H*, SiH*, SiHt and SiH,*,are also observed in this collision. An LIF detection is applied to non-emitting fragments, H, Si, SiH, SiH,, and SiH3, and the obtained results indicate that such fragments make only a minor contribution to the product formation. It is thus considered that the cross section of 0.081 A2 represents the total cross section for producing the emitted neutral fragments, or possibly the total cross section for the neutral dissociation of SiH,. The deexcitation cross section of He(23S) by SiH, is, however, determined to be 18 A2, as shown in Fig. 1, and this value is over 200 times as large as that of the optical emission cross section. It is therefore concluded, as expected, that a major part of the deexcitation processes in He(23S) SiH, collisions should be processes other than the neutral fragmentation, i.e., Penning ionization, which means the formation of e-(x I4). In the present deexcitation processes, associative ionization SiH; may not be important. A similar discussion has also been presented on He(23S) GeH, and He(23S) CH, collisions; it is summarized in Table 11. The excitation energies of He(23S),He(2's), and He(2'P) are much larger than the ionization potentials of CH,, SiH, and GeH,, whereas those of the lowest
+
+
+
+
+
ASPECTS OF ATOMIC AND MOLECULAR PROCESSES
237
TABLE I1
DEEXCITATION CROSS SECTIONS OF He(23S)BY CH,, SiH, OR
GeH,
IN COMPARISON WITH THE RESPECTIVE CROSS SECTIONS FOR PRODUCTS FORMATION (IN UNITS OF A*)
Total deexcitation cross sections (uM) He(2'S)
Product formation cross sections
+ CH, 5CH;(~ 4 4) C* CH* CH
12 -
0.05 1 -
-
He(2'S)
+ GeH,
SiH* SiH SiH,, SiH, GeH:(x 5 4) Ge* GeH* GeH GeH, , GeH,
-
0.081 0 0 0 -
0.44 0 -
-
Source: (Yoshida et al., 1991; Yoshida et al., 1992b; Bolden et al., 1970; Tsuji et al., 1989b).
excited states of Ar, i.e., Ar(3P2), Ar(3P,), Ar(3Po), and Ar(IP1), are slightly lower than or comparable with the ionization potentials of these molecules summarized in Fig. 2. It is presumed that Penning ionization is not a major deexcitation of these argon excited states and that other energy-transfer processes are more important. In the measurements of the deexcitation cross-sections of Ar(3P2), Ar(3P,), AI(~P~),and Ar('Pl) by CH,, SiH,, and GeH,, the crosssection values obtained are dependent on the electronic states of both excited argon atoms and target molecules. The magnitude of these values is discussed in terms of energy-transfer mechanisms or interactions responsible for the deexcitation transition. In the deexcitation of the metastable atoms, AI-(~P,)and Ar(3P0), the magnitude of the cross sections indicates an adiabatic transition from the interaction potentials for Ar* - M(M = CH,, SiH,, and GeH,) to those for Ar - M* due to a long-range potential curve crossing, while the deexcitation of the resonant atoms, Ar(3PI)and Ar('P1), occurs by resonant energy transfer due to a long-range dipole-dipole interaction. Absolute cross sections for both the total deexcitation of AI-(~P,,,)and the product formation are summarized in Table
III.
238
I: Hatano
T
0
- - _ - _ He
Ar
GeH,
SiH,
CH,
FIG. 2. Energy level diagram of He, Ar, CH,, SIH,, and GeH,. Excitation energies of He* [He* = He(2,S), He(2'S), and He(2'P)I and Ar*[Ar* = Ar(,P2), Ar(,P,), Ar(,PO), and Ar(lP,)] and the first adiabatic (a) and vertical (v) ionization potentials of CH,, SiH,, and GeH, are shown (Yoshida et al., 1992a).
TABLE 111
DEEXCITATION CROSS SECTIONS OF /w(~P,,,) BY CH,, SiH, , OR GeH, IN COMPARISON WITH THE RESPECTIVE CROSS (IN UNITS OF A2) SECTIONS FOR PRODUCTS FORMATION Total deexcitation cross sections (oM)
+
Ar(3P2,0) CH,
+
Ar(3P2,0) SiH,
+
-
AI-(~P~,")GeH,
o*,=
I0 I
c -101
CH:(x I4) C* CH*(A) CH(W CH2 CH, SiHT(X5 4) Si* SiH*(A) SiH(X) SiH, SiH, GeH:(X I 4) GeH* GeH*(A) GeH(X) GeHz GeH,
Product formation cross sections 0
0 4.1
... ... 0.27 4.0 4.6
...
... 1.7 0.03 0
... ...
Source: (Yoshda et al., 1992a; Tsuji et al., 1989a; Tsuji et al., 1990; Sekiya et al., 1987; Sekiya et al., 1989; Sekiya and Nishimura, 1990; Balamuta et al., 1983).
ASPECTS OF ATOMIC AND MOLECULAR PROCESSES
239
In the deexcitation of the metastable atoms, the smaller cross sections for CH, in comparison with those for SiH, and GeH, are due to the larger vertical ionization potential for CH4 than for SiH, and GeH,. It is understandable that SiH, and GeH, have almost the same cross section values since these two molecules have an almost equal density of Rydberg states as a result of the close vertical ionization potentials of the molecules. Table 111shows that ionic products are almost negligible in these collisions and that optically emissive products as well as the ground-state free radicals RH(X) as products, where R is C, Si, or Ge, are minor in terms of product branching ratios. It is concluded, therefore, that larger free radicals, RH, and RH,, are of great importance in the product branching ratios. The excitation energies of the lowest excited states of Ne, i.e., Ne(3P2), Ne(3Pl),Ne(,P,), and Ne('P,), are intermediate between those of He and Ar, as shown in Fig. 3. Deexcitation cross sections of Ne(3P,), Ne(,P,), and Ne(3Po) by CH,, SiH,, GeH,, CF,, and SiF, have been measured at mean collisional energy corresponding to room temperature. The cross-section values are also dependent on the electronic states of excited neon atoms and the target molecules. A small difference in the cross-section values between metastable atoms, Ne(3P2) and Ne(,P,), and a resonant atom, Ne(3P,), indicates that Penning ionization by Ne(3P,) is mainly governed by an electron exchange interaction rather than by a dipole-dipole interaction. This conclusion is based on a relatively small oscillator strength value of the Ne('S,)-Ne(3P,) transition as a result of a small spin-orbit coupling in Ne(,P,). From this viewpoint, the cross section for the collisional deexcitation of Ne('P,) by a variety of molecules have been recently measured
*Ol
....
19
10-
. 2 17t
Ne'
-
16- 3Po 16.72 3Po 16.72 15- 3Po 16.72
14-
-
13.
13.60
"i
16.21 16'45
-
12'30 11.98
11
- -
OLNe
CH4
-
-
SiH4 GeH4 CF4
-
SiF4
FIG. 3. Energy level diagram of Ne, CH,, SiH,, GeH,, CF,, and SiF,. Excitation energies of Ne*[Ne* = Ne(3P,), Ne('P,), and Ne(3Po)]and the first and second vertical ionization potentials of the molecules are shown (Yoshida et a/., 1993).
240
!I Hatano
and compared systematically with those for Ne(3PI),to confirm the conclusion on the mechanism of the deexcitation of Ne(3Pl). The cross section values for the Ne(' PI) deexcitation are well elucidated by the dipole-dipole interaction between Ne('P,) and M . The product formation in the collisions of the lowest excited states of Ne with molecules has not been studied yet. Further systematic study of the Ne, Kr, Xe-A4 systems like that done for the He, Ar-M systems is greatly needed.
V. Comments on Atomic and Molecular Processes in Reactive Plasmas from Physicochemical Viewpoints Future perspectives and comments on atomic and molecular processes in reactive plasmas from physicochemical viewpoints are given as follows (Hatano, 1991). 1. Absolute cross sections should be measured for electron impact dissociative excitation of molecules, leading in particular to non-luminescent fragments. 2. Products and their branching ratios should be measured for free radical reactions. 3. More attention should be paid to precursors and precursor processes that form free radicals. 4. A rate-determining process for the formation of important free radicals should be identified in the analysis of complex mechanisms of reactive plasmas. 5. Theories for each process should be studied to estimate cross-section values that have not been obtained experimentally. 6 . A universal expression of product yields like the quantum yield in photochemistry and the G value in radiation chemistry should be defined in reactive-plasma research. 7. Experimental techniques that are frequently used in physicochemical studies should be applied to reactive plasmas. These techniques are, for example, the use of deuterated compounds, the analysis of stable products in the gas phase, and the use of matrix isolation or trapping of reactive species at low temperatures combined with electron-spin resonance (ESR) or optical spectroscopy.
VI. References Balamuta, J., Golde, M. F., and Ho, Y. S. (1983). 1 Chem. Phys. 79, 2822. Bolden, R. C., Hemsworth, R. S., Shaw, M. J., and Twiddy, N. D. (1970). 1 Phys. B. 3 , 61.
ASPECTS OF ATOMIC AND MOLECULAR PROCESSES
24 1
Hatano, Y. (1986). In D. C. Lorents, W. E. Meyerhof, and J. R. Peterson [Eds.], Electronic and atomic collisions, p. 153, Elsevier (Amsterdam). Hatano, Y. (1991). In T. Goto [Ed.], Proceedings of the International Seminar on Reactive Plasmas, Nagoya University, p. 341. This paper summarizes physicochemical aspects of atomic and molecular processes in reactive plasmas which have been discussed in detail in the joint research project entitled “Control of Reactive Plasmas” (R. Itatani, head) as supported by Grant-in-Aid for Scientific Research on Priority Areas, Ministry of Education, Science, and Culture, Japan. Hatano, Y. (1999) Phys. Reports., 313, 109. Itabashi, N., Kato, K, Nishiwaki, N., Goto, T., Yamada, C., and Hirota, E. (1989). Jpn. 1 Appl. Phys. 28, L325. Itabashi, N., Nishiwaki, N., Magane, M., Naito, S., Goto, T., Matsuda, A., Yamada, C., and Hirota, E. (1990). Jpn. 1 Appl. Phys. 29, L505. Kono, A,, Koike, N., Okuda, K., and Goto, T. (1993). Jpn. 1 Appl. Phys. 32, L543. Matsuda, A,, and Hata, N. (1989). In K. Tanaka [Ed.], Glow-discharge hydrogenated amorphous silicon, Chap 2, KTK Scientific (Tokyo). Matsuda, A,, Mashima, S., Hasezaki, K., Suzuki, A,, Yamazaki, S., and McElheny, F? J. (1991). Appl. Phys. Lett. 58, 2494. Nomura, H., Akimoto, A., Kono, A,, and Goto, T. (1995). 1 Phys. D:Appl. Phys. 28, 1977. Sekiya, H., Hirayama, T., and Nishimura, Y. (1987). Chem. Phys. Lett, 138, 597. Seluya, H., Obase, H., and Nishimura, Y. (1989). 1 Chem. Soc. Jpn. 1989, 1210 (in Japanese). Sekiya, H., and Nishimura, Y. (1990). Chem. Phys. Lett. 171, 291. Tanaka, H., Boesten, L., and Hatano, Y. (1996). Oyobutsuri (AppLPhys). 65, 568 (in Japanese). Tanaka, T., Hiramatsu, M., Nawata, M., Kono, A,, and Goto, T. (1994). 1 Phys. D: Appl. Phys. 27, 1660. Tsuji, M., Kobarai, K., Yamaguchi, S., Obase, H., Yamaguchi, Y., and Nishimura, Y. (1989a). Chem. Phys. Lett. 155, 481. Tsuji, M., Kobarai, K., Yamaguchi, S., and Nishimura, Y. (1989b). Chem. Phys. Lett. 158, 470. Tsuji, M., Kobarai, K., and Nishimura, Y. (1990). 1 Chem. Phys. 93, 3133. Ukai, M., and Hatano, Y. (1991). In R. W. Crompton, M. Hayashi, D. E. Boyd, and T. Makabe [Eds.], Gaseous electronics and its applications, p. 5 1., KTK Scientific (Tokyo). Yoshida, H., Morishima, Y., Ukai, M., Shinsaka, K., Kouchi, N., and Hatano, Y. (1991). Chem. Phys. Lett. 176, 173. Yoshida, H., Kawamura, H., Ukai, M., Kouchi, N., and Hatano, Y. (1992a). 1 Chem. Phys. 96,4372. Yoshida, H., Ukai, M., Kawamura, H., Kouchi, N., and Hatano, Y. (1992b). 1 Chem. Phys. 97,3289. Yoshida, H., Kitajima, M., Kawamura, H., Hidaka, K., Ukai, M., Kouchi, N., and Hatano, Y. (1993). 1 Chem. Phys. 98, 6190.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
ION-MOLECULE REACTIONS WERNER LINDINGER AND ARMIN HANSEL Institut f i r Ionenphysik Universitaet Innsbruck Innsbruck, Austria
ZDENEK HERMAN 1 Heyrovsv Institute of Physical Chemistry. Academy of Sciences of the Czech Republic, Prague, Czech Republic
I. Introduction. . . . .. .. . . . . . . . ... .. . . . . . .. . . .. . .. .. .. .. . . . .. .. .. . .. .. . . ... . . .. . . . . . . ..
11. Reaction Rate Constants of Ion-Molecule Reactions . . . . .. .. . . . . . .. .. .. . . .. .. ..
A. Ion-Induced Dipole Reactions B. Ion-Permanent Dipole Interacti C. Ion Dipolelnduced Dipole Interaction . . . . .. .. . . . .. .. . .. . . .... .. . . . . . . .. . . . . III. Types of Ion-Molecule Processes. ... . .. . . .. . . . .. ... . . .. . . . .. . . .. .. .. .. . . . .. .. .. . A. Chemical Reactions of Ions .. . . . .. . . .... . . . .. .. . .. . . . .. .. . . ... . . .. . ... .. .. . . . 1. Reactions with Rearrangements of Bonds 2. Proton-Transfer Reactions . . .. . . . . . .. . . .. . . . .. . . . . .. . .. . . , .. . . .. .. . . .. . . .. . 3. Isomerization Reactions . . . . .. .. ... .. .. .. . . . .. .. . .. .. . . . .. .. .. .. .. .. .. .. .. . 4. Switching Reactions . .. . . . .. ... .. . . .. . . . . . . . .. . .. . . .. .. . . . .. .. . . . . . . .. . . .. . B. Charge Transfer from Single Charged Ions.. . .. . .. .. . . ... .. .. .. . . . . . . ... . . _ . C. Reactions of Multiply Charged Ions 1. Charge Transfer Involving Mu1 2. Chemical Reactions of Doubly D. Vibration Deexcitation and Excitation of Molecular Ions . .. . . .. .. .. .. .. .. .. 1. Repulsive Interaction . . . . .. . . . . . . . . . . . . . . . . .. . . ... . . . . . .. .. . . . . . . .. .. . . .. . . .......... 2. Ion-Induced Dipole Interaction. .. . . . E. Association Reactions . . . . .. .. . . ... . . . . . .. . . .. . .. .. .. ... .. . . ... . ... . ... .. . . .. . IV Effect of Internal Energy and Temperature on IM Processes . . .. . ... . . . . .. .. .. . 1. Spin-Orbit States of Rare Gas Ions. .. . . . . .. . . . .. .. . . . .. .. . . . .. .. . . . . . . .. .. .. . 2. Electronic Excitation of Reactant Ions . .. . . .. . .. .. . . . .. . . . . . .. . . .. .. .. .. .. .. . 3. Vibrational Excitation of Reactant Ions
V Concluding Remarks VI. Acknowledgments . . . . . . . . . . ... . . . .. .. .. .. ... .. ..... .. .. . . . .. . . . . .. . . . .. . . . . . . . . . . VII. References .. . . . .. . . . . . .. . . . . . .. .. . .. .. .. . . . .. . . .. . .. .. .. .. . .. . . . . . . . .. .. .. .. .. .. ..
243 249 249 250 25 1 253 253 253 255 256 259 262 261 267 271 212 272 214 211 219 28 1 282 283 286 288 289 289
I. Introduction A chapter on ion-molecule reactions (IMR) within a volume on fundamentals of plasma chemistry quite naturally addresses the question of the fUrther fate of ions We dedicate this chapter to Professor Eldon E. Ferguson in appreciation of his pioneering work in the field of ion-molecule reactions. 243
Copyright 0 2000 by Academic Press All rights of reproduction in any form reserved ISBN: 0-12-003843-9/ISSN: 3049-25OX $30.00
244
K LindingeK A . Hansel, Z. Herman
in a plasma as a result of reactions with neutrals present. Primary ions usually are created by electron impact ionization, nondissociatively, as in the case of rare gases, e.g., argon, h + e -+ h + + 2 e
(1)
provided that the energy of the ionizing electron is higher than the ionization potential of argon, I€'(&) = 15.76 eV, or both, nondissociatively [Eq. (2a)l and dissociatively [Eq. (2b)], as in the case of many molecules, e.g., O,,
provided that the energy of the ionizing electron is higher than the appearance potential of 0' being formed from 0, AP(O+ from 0,) = 18.99 eV. Cross sections for electron impact ionization increase from their onset at electron energies equal to the ionization potential of the respective neutral to a maximum of typically cm2 at electron energies of 70 to 120eV, then decreasing toward higher electron energies. Reviews on electron impact ionization have been published by Mark (1986), Bottcher (1985), and Becker (1998), and this volume contains several chapters on this subject. But what about the further fate of the ions? The concentration of any specific type of ion [X+]within a plasma is governed by its production, loss, and diffusion processes. Production is due to electron impact ionization, but in many cases it is also due to reactions of precursor ions with neutrals, resulting in ions X + . Also, photoionization can be a substantial ion production process. Losses of ions X' occur as a result of ion+lectron and ion-ion recombination, respectively (Bates, 1985), and of ion-molecule reactions, the subject of this chapter. Thus the density [X+] at a specific location in the plasma is governed by
d[X+]/dt= D,A[Xf]
+ k,[ef]N + vp - vI - cr[X+][e,]
(3)
Here D,A[X+] is a diffusion term, [ef] denotes the density of fast electrons with energies high enough for production of ions X+ by electron impact ionization from neutrals N with a rate constant k,, ef+N
-+
X++Y+2e
(4)
whereas [e,] denotes the density of slow plasma electrons, which usually is much higher than [e,-1.These slow plasma electrons recombine with ions X+ with the recombination constant cr under the emission of a photon hv, e , + X + -+ X + h v
(5)
when X+ is an atomic ion, or dissociatively, e,+X+
-+
Y+Z
(6)
245
ION-MOLECULE REACTIONS
when X + is a molecular ion. In the first case, values of CI are of the order of lo-" cm3 . S - I , whereas the second case (dissociative recombination) is much faster, with a being of the order of lo-' cm3 s-'. vp and vl are production and loss terms, respectively, representing reactions leading to the production of X + ions from precursor ions P+ reacting, e.g., via charge transfer with neutrals X ,
p + + x +. x + + p
(7)
and resulting in losses of X + ions as a result of reactions with other neutrals M present in the plasma, again in the simplest way by charge transfer,
X'+M
+.
M++X
(8)
Equation (3) represents cases where neither electrical nor magnetic fields as well as photoionization play a significant role. If the plasma is in a steady state, then d[X+]/dt= 0, transforming Eq. (3) into a steady-state equation that has been successfully used to describe the negative glow of hollow-cathode discharges in argon and nitrogen, respectively, with traces of H20, which are the simplest cases of plasmas in steady state (Lindinger, 1973; Howorka et al., 1974). Because of their simplicity, these negative-glow plasmas are quite instructive, as the following example will show. Figure 1 represents radial density profiles of the main ions Ar+,ArH+, H20+, and H,O+ observed by mass spectrometric analysis in a hollow-cathode discharge of 2 cm diameter and 3 cm length at a pressure of 0.34 torr (Arwith 0.15% H20) and a discharge current of 3 mA. These main ions are created in the following processes:
Ar+:Ar + ef ArH': A r ' +H20
+ +
H30+: ArH+ H 2 0 H30f: H20+ H 2 0 H 2 0 + : Ar+ + H 2 0
+.
Ar+ +2e
k,
(94
-+
ArH' +OH H30+ Ar H30+ +OH H20++Ar
k, k2 k3 k4
(9b) (9c) (94 (94
+.
+. +.
+
The main loss of ions, other than through diffusion, is due to recombination of H30+: H30f: H 3 0 f + e ,
-+
products
a
(9f)
For the following considerations, we should note that the cylindrical negative glow extending along the axis of the discharge represents a field-free region at a potential of about 300V above the cathode wall potential. Formation of Ar+ by reaction (9a) is due almost exclusively to the collisions of fast electrons [ef] (Ekin% 50 eV) from the dark space with Ar atoms. These fast electrons originate from secondary electron emission after impact from ions diffusing out of the negative glow and then traversing the cathode fall region, thereby gaining lunetic
246
W Lindingec A. Hansel, Z. Herman
FIG. 1. Radial density profiles of the main ions Ar+,ArHf, H20+, and H 3 0 f and the sum of all ions Z x;' observed by mass spectrometric analysis in a hollow-cathode discharge of 2 cm diameter and 3 cm length at a pressure of 0.34 tom (Arwith 0.15% H20) and a discharge current of 3 mA (Lindinger, 1973).
energy prior to impact on the cathode wall, as indicated in Fig. 2. In view of the low contribution of water vapor (only 0.15%) to the Ar discharge gas, the concentrations of the secondary ions ArH+, HzO+ and H30+, as shown in Fig. 1, are quite substantial, indicating that the conversion of ions through reactions (9a) through (9e) is extremely fast, with rate constants having values of the order of cm3 . s-'. This was confirmed by treating the data of Fig. 1 using the respective steady-state equations [Eq. (3)] for the above ions. The modified Eq. (3), taking into account the main processes (9a), (9b), and (9e) governing the density of Ar+, reads,
d[Ar+]/dt = 0 = D,A[Ar+]
+ k,[e,][Ar]
- k,[Ar+][H20] - k4[Ar+][H20] (10)
and the one for ArH+, making use of (9b) and (Sc), has the form
d[ArH+]/dt = 0 = D,A[ArH+]
+ k,[Arf][H20] - k2[ATH+][H20]
(11)
ION-MOLECULE REACTIONS
247
RADIUS [cml
FIG. 2. Section through a cylindncal hollow cathode and potential distribution. Va, anode potential; Vp, space potential; Vk,cathode potential; KFR, cathode fall region (Howorka et al., 1974).
By using the same set of rate constants for the steady-state equations of all main ions, consistency was achieved in that d[X+]/dt= 0 was obtained at all radial distances (Lindinger, 1973). In Eq. (1 0), the values for the diffusion term as well as the values for the reaction terms are known from the measured profiles of the Ar+ ions; thus, the term k,[Ar][ef]can be calculated, so that the radial dependence of the density of the fast electrons [ef] was obtained-f the order of lo5 cmP3, increasing from the central axis of the negative glow (r = 0) toward its edge at about r = 4 111111. From the steady-state equation of H30+, d [ H 3 0 f ] / d t= 0 = D,A[H,O+]
+ k2[ArHf][H20]+ k3[H20+][H20]
- m30+l[e,l
(14
where the radial density of the slow plasma electrons [e,J was taken as equal to the radial density of the sum of all main ions (see Fig. l), a recombination constant u = 5.5 x lo-' cm3 . s-' was obtained, which is quite reasonable in view of the enhanced electron temperature (T, E 0.1 eV) in the negative glow. Using an assumed temperature dependence u T;0.5, the above value scales to
-
248
W Lindingec A . Hansel, Z. Herman
-
c( = 1.1 x 10-6 cm3 s-' at room temperature, which is in good agreement with values obtained by other authors (Biondi et al., 1971; Herd et al., 1980). Consistency was achieved so easily in the above case because the negative glow of the hollow-cathode plasma is field-free, the reactions involved are fast (on every collision), and the plasma temperature is uniform and quite low. In general, however, a plasma is considerably more complicated than the case of a hollowcathode plasma and the situation represented by Eq. (3). Ions produced by electron impact usually are not all in their ground state, but have a distribution of vibrational (in the case of molecular ions) and even electronic states, and reaction rate constants and products of a reaction often depend on the excited state of the ion. Furthermore, collisions of excited ions with neutral gases can lead to deexcitation of the ions to lower excited states or to the ground state; or when electric fields are present, ground-state ions may even become vibrationally excited as a result of collisions with the neutral gases in the plasma. Besides binary reactions as indicated in Eq. (3), association reactions or ternary reactions can also occur, especially when the pressure of the plasma is high enough (typically above a torr), but dissociation of ions will also be observed as a result of collisions with neutrals. All these basic processes should be considered when modeling of a more complicated plasma than the above example of the negative glow of a hollow-cathode discharge is done, and it is the aim of this chapter to discuss the fundamentals of the main types of ion-molecule reactions; however, we will not include actual cases of plasma modeling. We will stick to the general term, ion-molecule reactions (IMR), although we also include excitation and deexcitation processes, which are not reactive in the sense that they do not change the identity of the ions, but rather change their state of excitation. In the past, a variety of reviews on IMR have been written (Ferguson, 1968; Ferguson, 1992; Ferguson et al., 1969; Lindinger, 1984; Lindinger, 1986; Lindinger and Smith, 1983; Adams and Smith 1983), and in several of these, various swarm techniques that have been applied for measuring reaction rate constants were described in great detail; therefore, we refer to these publications for experimental details. We will start with the question of which values of rate constants we can expect for IMR, after which the various types of IMR will be discussed. Here again, we will mention the well-known facts about IMR that were described in detail in earlier reviews only briefly and will mainly concentrate on new results obtained since then. In further sections, we will deal with the influence of vibrational and rotational energy as well as temperature on ionneutral interactions, and we also will show examples of thermochemical data obtained from swarm-type experiments. We will strictly limit ourselves to positive ions and will mainly, but not exclusively, present data obtained by swarm experiments with the ability to change the relative kinetic energy between the reactants. As we will see later, it is this energy dependence of the rate constants and reaction channels that yields detailed information on the reaction
ION-MOLECULE REACTIONS
249
mechanisms involved. For the many reactions investigated at room temperature only, we refer to the compilations of Ikezoe et al. (1987) and Anicich (1993), and for the field of cluster reactions we want to draw attention to the review of Castleman and Wei (1994).
11. Reaction Rate Constants of Ion-Molecule Reactions A. ION-INDUCED DIPOLEINTERACTION
Reactions of ions A+ (with A+ denoting atomic or molecular ions) with neutrals B proceeds with reaction rate constants k, defined by
d[A+l = -k[A+][B] dt where d[A+]/dtrepresents the change in the ion density as a function of time in a given volume as a result of reactive collisions with B, which is present with a density [B] The dimension of k is cubic centimeters per second (in binary collisions), and its relation to the cross section ~ ( v is )
J
k = a(v)f(v)v dv wheref(v) is the velocity distribution function. Thus, in a first approximation,we may write
k
(15)
E (OV)
How large, then, can values of k be for specific reactions? Each neutral atom or molecule has a polarizability a, which means that an electric dipole is induced when the neutral particle is put into an electric field. Whenever an ion approaches a neutral (molecule or atom), its Coulomb field induces a dipole in this neutral, which results in an attractive force. This leads to the formation of an ion-neutral collision complex when the impact parameter is below a critical value, as has been shown by Gioumousis and Stevenson (1958). The rate constant for formation of such complexes is independent of temperature and has the value 1/ 2
kL = 2rce($) where c1 is the polarizability of the neutral collision partner and m, is the reduced mass of the ion and the neutral. kL is called the Langevin limiting value and can
250
W Lindingec A. Hansel, Z. Herman
be seen as a capture rate constant; that is, the value indicates the rate at which the reactants are captured in spiraling orbits. B. ION-PERMANENT DPOLE INTERACTION In cases where neutral reactants already possess a permanent dipole moment pD, the capture rate constant is larger than kL. Su and Bowers (1979) have derived the expression
which is called the average dipole orientation (ADO)limit kADO.The dipole locking constant c, which depends on the ratio pD/d/’, can be considered qualitatively as cos (0), where (0) is the average orientation angle of the dipole. Values of c are listed by Su and Bowers (1979). The rotational motion of the molecule is hindered by the presence of permanent dipoles and, in general, in systems having strongly anisotropic potentials. Thus a variety of more complex theories have been developed to account for not only permanent dipole but also quadrupole moments (Troe, 1985). A new computational technique involving a combination of adiabatic capture and centrifugal sudden approximations (ACCSA) was applied by Clary (1984); this theory predicts sharply increasing rate constants as the temperature decreases. Parameterization of the ion-polar molecule collision rate constant by trajectory calculations was done by Su and Chesnavich (1982), leading to the temperature-dependent expression k,(T) = kL kcap,where
-
I0.4767~+ 0.6200 kcap=
(x
+ 0.5090)2 +0.9754 10.526
x>2 x~ 2
with x = 1/&
= pd/(2~kgT)‘/’
Values of k, are often similar to those of often the two differ by less than 10%. How fast do IMR actually proceed in comparison with kL and k,, respectively? The above discussion infers that kL and kc represent upper limits for rate constants of actual IMR. Indeed, experimentally measured rate constants hardly ever exceed these values, and in cases where substantially higher values have been reported in the literature, careful reexamination has shown that the reactions proceed with rate constants k 5 kL or k,, respectively.Exceptions are reactions of ions with a substantial dipole moment, as will be shown below.
ION-MOLECULE REACTIONS
25 1
It should be stressed that in the case of exoergic proton-transfer reactions involving small reactants neutrals (masses up to 100 dalton), measured values of k invariably agree with k, to within a few percent. When rate constants are needed for plasma model calculations, it is recommended that calculated values k, be used unless there are very reliable experimental data available. Also, hydrogen abstraction reactions often proceed with rate constants k, when this reaction channel is the only one occurring, and even a large group of charge-transfer processes proceed at the collisional rate (usually, when large Franck-Condon factors exist for the transition from the ground state of the neutral into the resonant ionic state).
c. ION DIPOLE-INDUCED DIPOLEINTERACTION Larger rate constants than k, have been observed and confirmed for reactions of ArH: (Rakshit, 1982; Smith et al., 1992). Investigations by Praxmarer et al. (1994, 1996) including reactions of ArHt (and ArDt) with nonpolar neutrals such as CH,, C,Hs, C4HI0,N2, 02,and CO (the permanent dipole moment of CO is very small, only 0.11 D) showed rate constants that exceed kL by approximately 20%, while the experimental values obtained simultaneously for reactions of H; with CO, CH,, and N, were equal to kL within 5% (see Table I). These results clearly indicate that the interaction between the permanent ion dipole and the induced dipole of the molecules is the predominant cause of the increase of the measured rate constant to above kL in the case of nonpolar neutrals. The experimental results are in agreement with calculations of the capture rate constant based on an ion-neutral interaction potential of the form
which extends the effective potential (on which the Langevin model is based) by the term -ptOnol/R6.Here pionis the permanent dipole moment of the ion (which is >8D, defined with respect to the origin at the center of mass of the ArHT ion) (Bogey et al., 1987; Hobza et al., 1993). This extra term should reflect the effect of dipole-induced dipole interaction on the rate constant. The calculated capture rate constants kL,IDincluding the extra term lead to values above kL,in agreement with experimental results. In the case of polar neutrals, an additional interaction between the ion dipole and the neutral permanent dipole may also be of significant influence. The experimental results from Praxmarer et al. (1994) on the reactivity of ArH: with polar molecules such as H 2 0 and NH, are in excellent agreement with recent calculations from Clary (1995) using a rotationally adiabative capture theory. There it is found that dipoldipole, dipole-induced dipole, and dispersion terms
TABLE I RATE CONSTANTS IN UNITS OF
cm3 . S-' FOR THE REACTIONS kH: AND H: WITH THE NONPOLAR GASESNo, N,, Xe; 0, AND D,; AND THE POLAR GASESH,O, NH,,SO,, CH31, AND C,H,I
co, CH,, a,AND
klk, Reactant
k (300 K) this work
klk,
Smith et al., 1992
k,
(300K)
(80 K)
1.02 0.90 1.02 1.24 0.85 1.15
0.77 0.75 0.82 1.10 0.69 0.83
1.32 1.17 1.24 1.13 1.17 1.33
2.50 2.56 2.00 2.13 2.25
2.01 1.88 1.50 1.61 1.80
0.60 0.42
0.68 1.09
1.23 1.24 1.36 1.33 1.33 1.25 1.30 0.88 0.38
1.27 1.16 1.14
k this work
k,
klk,
Product
0 0.11 0 0 0
1.94 1.74 2.03 2.40 1.04 2.49
1.95 1.89 2.03 2.35 2.17 2.75
0.99 0.92 1.00 1.02 0.48 0.91
NOH+ N2H+ COH+ CH: KrH+ XeH+
1.85 1.47 1.63 1.62 1.91
4.81 4.39 4.93 5.39 5.86
4.50 4.14 4.48 5.36 6.10
0 0
0.65 0.58
1.59
Product
fl(A3)
P @)
NOH+ N2H+ COH+ CH; KrH; XeH+ XeH:?
1.70 1.74 1.95 2.59 2.48 4.04
0.15
H, O+
1.45 2.26 3.72 7.97 10.0
1.19
NH: S02H+ CH41+ C,H,I+
1.18
AQH+ ArH, D+ ArD2H+
1.58 0.79
1.79
0.97 1.07 1.06 1.10 1.01 0.96 1.04 0.36 0.36
H30+
NH: S02H+ CH41f C2H$ 02H+ H, D+ D, HC
ION-MOLECULE REACTIONS
253
produce an enhancement (about 18%) over the rate constant calculated with only the ion-dipole and ion-induced dipole terms in the potential energy surface.
111. Qpes of Ion-Molecule Processes A. CHEMICAL REACTIONS OF IONS
I . Reactions with Rearrangements of Bonds Whenever complicated rearrangements of bonds occur during a reaction, it is ldcely that the lifetime of the ion-neutral collision complex at room temperature is not long enough to allow for a reaction on every collision, i.e., the reaction probability is smaller than unity, and thus the reaction rate constant is smaller than the collisional limiting value k,. The rate constants of these reactions usually show a negative temperature (and also energy) dependence. A decrease in temperature (or energy) causes an increase in the complex lifetime and therefore an increase in reaction probability. In the absence of potential barriers in the reaction path, the rate constants of these reactions have a tendency to increase (and even approach the value k, in some cases) when the temperature reaches values of only a few kelvin (Le Garrec et al., 1997). At high enough temperatures or energies, new reaction channels will open up that do not require rearrangements of bonds (such as charge transfer, proton transfer, etc.), and thus the rate constants will show a positive temperature (energy) dependence above that point, so that the overall shape of the rate constant as dependent on temperature (or energy) has a pronounced minimum at a certain temperature (energy). We want to demonstrate such a case by discussing the reaction between 0; and CH,, which is one of the most extensively studied reactions (Ferguson, 1988). Investigations of this process have been carried out using selected ion-flow dnft tube (SIFDT) and variable-temperature selected ion-flow tube (VT-SIFT) methods (Van Doren et al., 1986; Smith et al., 1978; Dotan et al., 1978; Durup-Ferguson et al., 1984; Adams et al., 1985; Alge et al., 1981; Viggiano el al., 1990b) in order to obtain the temperature dependence of the rate constant and to determine the products of the reaction. SIFDTexperiments have been used (Albritton et al., 1979; Lindinger and Smith, 1983) to measure the reaction rate constant as dependent on the mean center-of-mass energy KE,, , and to study the influence of different buffer gases on the rate constant for this reaction. Measurements of the reaction rate constant involving CRESU experiments (Rowe et al., 1984) have been performed down to temperatures as low as 20K, and guided ion-beam studies have been used to study the cross section for this reaction up to an energy of lOeV (Fisher and
254
u(
Lindinger, A . Hansel, Z. Herman
Armentrout, 1991). Despite the existence of several strongly exoergic reaction channels in the reaction,
02f
+ CH,
+ +
+ CH30: H -+ H30+ HCO
+ CH,O + HCO'H + H,O + CH30+ + OH -+ CH; + HO, + CH: + O2 + H2f
+ 23 kcal . mol-' + 113 kcal . mol-I + 53 kcal . mol-' + 71 kcal . mol-' + 78 kcal . mol-'
(204 (20b) (20c) (204 (20e)
- 5.5 kcal . mol-'
(20f)
- 13.8 kcal . mol-'
(20g)
at thermal and subthermal energies this reaction proceeds only via channel (20a), and channels (20b) through (20e) have not been observed. In several studies the influence of internal excitation of the reactants was investigated. The influence of electronic excitation of 0; was investigated by Lindinger et al. (1979); that of vibrational excitation of the 02f ion was investigated by Durup-Ferguson et al. (1984), Alge et al. (1981), Albritton et al. (1979) and Lindinger and Smith (1983); and the influence of rotational excitation of CH, on the reaction was observed by Viggiano et al. (1990b), who extended earlier work of Adams et al. (1985). It was established that reaction (20) with ground state 0; ions involves a sequence of three successive steps occurring within the collision complex and proceeds via a double minimum-potential surface with a large intermediate barrier (Ferguson, 1988; Van Doren et al., 1986; Barlow et al., 1986). The structure of the C H 3 0 t product ion of reaction (20a) has been determined in several studies. Van Doren et al. (1986) carried out a detailed study of the reactivity of the different isomers of the CH30; ion with a large number of neutrals and concluded that the product of reaction (20a) is the methylene hydroperoxy cation, CH,OOH+. Kirchner et al. (1989) used high-energy collision-induced dissociation (CID) and came to the same conclusion as Van Doren et al. (1986), namely, that the product ion is CH200H+ and not protonated formic acid. Figure 3 shows the temperature (and energy) dependence of the overall rate constant for the reaction of 0: with CH, as obtained in a SIFDT experiment by Viggiano et al. (1990~)and Miller et al. (1994). The strong increase in the rate constant at elevated temperature is caused by vibrational and rotational excitation of 0; and CH,, all of which are driving the reaction. The increase of k toward higher transitional energies above 0.1 eV is correlated with the appearance of the slightly endoergic charge-transfer and dissociative chargetransfer channels, respectively, neither of which requires long-lived complexes. The strong increase toward low temperatures is a consequence of the increase in complex lifetime, as discussed above.
255
ION-MOLECULE REACTIONS
4 translational energy
.-
0
-9 u
g*=*x.
5
7
,*-.
a,
SIFT 170 K SlFT298K a SlFT430K o SIFT 545 K fl x
'
0,'
+
CH,
1 o-'*
4
products
- 0
- FDT 300 K
There are many reactions that show this characteristic behavior of reactivities and thus rate constants: limited by collision complex liftetime at low temperatures or energies and increasing toward higher temperatures as a result of the appearance of direct endoergic channels. A few examples of this kind are the reactions of CH: with NH, (Glosik et al., 1998); S+ with CH,, C2H2,and C,H,, respectively (Zakouril et al., 1995); Si+ with HCI (Glosik et al., 1995); N+ and N,: respectively, with HCl (Glosik et al., 1993); and O+ with NO (Le Garrec et al., 1997). 2. Proton-Transfer Reactions
Proton-transfer reactions are the most common ionic processes occurring in many technical and natural plasmas. They are dominant in nearly all plasmas containing hydrocarbons or other hydrogen-bearing compounds (Goodings et al., 1979). The presence of even small amounts of these compounds leads to the conversion of atomic ions into protonated species. Chemical ionization mass spectrometry (Harrison, 1992), in its wide variety of methods, uses proton-transfer processes for soft ionization of the neutral trace compounds which are thus detected; and the recently developed proton-transfer reaction mass spectrometry (PTR-MS), which allows for on-line monitoring of complex mixtures of nearly any of the existing volatile organic compounds, is solely based on PTR from H 3 0 + and NH,f ions, respectively (Hansel et al., 1995; Hansel et al., 1998c; Lindinger et al., 1998a, Lindinger et al., 1998b).
256
W LindingeK A . Hansel, Z. Herman
Any neutral particle, atom or molecule, possesses a proton affinity (PA), and if PA( Y) is sufficiently larger PA(X), then the reaction XH++Y-+YH'+X
(21)
is exothermic and proceeds on every collision, i.e., with a rate constant close to k,. For each PTR, the common relation -RTlnK = AH - TAS
(22)
holds, and the process in forward and reverse direction XH+ + Y
% YH+X
(23)
kr
has the equilibrium constant K = kf/kr. At equilibrium, the net rate of change is zero. Hence, at equilibrium, d[XH+]/dt= -$[XH+][YI
+ k r [ Y H + ] [ 4= 0
(24)
and therefore [m+I[Xl/[XH+I[YI= kf/kr = K
(25)
K can be obtained experimentally by measuring the ratio [YH+] [XHf] as dependent on the density [Yl while the density [XIis kept constant. One thus obtains linear increases in the ratio, the slope of which is a direct measure of K. Values of K obtained at different temperatures are then plotted in the form of van't Hoff diagrams, from which both A H and AS are obtained. AH is by definition the difference in the PA. Another approach is to measure the forward and reverse rate constants, the ratio of which yields K and APA (= AH) as described above. In this way, many values of PAS have been obtained using temperature variable flowing afterglow and similar techniques (Walder and Franklin, 1980; Adams et al., 1989; Fehsenfeld et al., 1976), and a compilation of data was published by Lias et al. (1988). 3. Isomerization Reactions As the complexity (atomicity) of the ions increases, a greater number of structural isomers are possible. Theoretical calculations are sometimes available for small molecular ions that indicate which structural isomers have minima on the potential energy surface. Experimentally, in SIFT and SIFDT studies, the existence of two or more structural isomers at the same molecular weight is often manifest by curvature on ion decay plots as a result of the isomers' different reactivities with specific monitor gases. Then one has to determine if the different reactivity is due to the presence of excited states of a given structural isomer or to
ION-MOLECULE REACTIONS
257
the presence of other structural isomers or even some mixture of excited ions and structural isomers. Theoretical calculations predict the existence of two structural isomers HCO+/COH+ as a result of minima on the potential energy surface. Experimentally, Freeman et al. (1987) observed in their SIFT study two ion-molecule reactions that produce the two isomers HOC+ and HCO+ with different branching ratios. These two reactions are C+ H,O (branching ratio HOC+/HCO+ = 5) and CO+ H, (branching ratio HOC+/HCO+ x 1) (Freeman et al., 1987). A technique based on the very different proton affinity of CO at 0 compared to CO at C was used to distinguish between the two isomers. In addition, the reaction with H, was observed to also isomerize HOC+ into the more stable HCO+ cation. Further examples are the structural isomers of C,H; and C3H;. The linear and cyclic structures of both the C,H; and C,H; ions are formed in electron impact ionization from methylacetylen (CH,CCH) (Smith and Adams, 1987), n-butane, or propane (Hansel et al., 1989). The linear and cyclic isomers of C,H; and C3H: are distinguished by either their different association rates with CO or their different reactivity with C2H2.In all cases, the linear (or open-chain) structural isomers l-C,H;, l-C,H;) are more reactive then the respective cyclic forms (C-C~H;, c-C~H;). A hrther demonstration of the power of the SIFDT technique in this type of study is the determination of the thermochemistry of HNC+ (and HNC) and the determination of the mechanism by which HCN+ isomerizes into HNC+ in collisions with CO,. The two isomeric cations HCN+ and HNC+ are distinguished by the use of monitor gases SF,, CF,, O,, and Xe, which react differently with the two isomers. For the determination of the heat of formation of HNC+, it is necessary to ensure that the mass 27+ ions are in the lowest energy state (also vibrationally deexcited). This can be done by quenching HNC+ ions in collisions with the He buffer gas at elevated kinetic energies prior to the energy-dependent determinations of the reaction rate constants with CO and Xe. (As discussed in Section III.D, the quenching probability increases with increasing collision energy, if He is used as the quencher. Obviously molecules are expected to have a higher quenching efficiency than rare gases even at room-temperature collisions, but the HCN+ and CNH+ cations are very reactive with molecules, leaving only rare gases as possible quenchers.) The first reaction considered
+
+
is endothermic at room temperature and is promoted by kinetic energy. The rate constant as a function of relative kinetic energy gives an Arrhenius plot yielding for the heat of formation of HNC+ the value 322.4kcal .mol-', in good agreement with theory. The heat of formation of the neutral HNC results from
258
K Lindinger, A . Hansel, Z. Herman
measurements of the ionization energy (IE) of HNC obtained from the slightly endothermic charge-transfer reaction
JXN~?,=,,
+ Xe
+=
Xe+
+ HNC
(27)
The Arrhenius plot gives a value IE(HNC) = 12.04 eV f 0.01 eV, and this in turn yields for the heat of formation of the neutral HNC the value 45 f 1 kcal . mol-’ (Hansel et al., 1998a). Recently it has been proposed by Petrie et al. (1990) that the reaction HCN+
+co(co,)
+
+ HNC+ c o ( c o , )
(28)
occurs and that the reaction mechanism is a two-step proton transfer, NCH’
+ CO + (NC . . .H+-CO)
(294
followed by (CN . . . Hf-CO)
+ CNH’
+ CO
(29b)
and similarly with C02, each of the two proton-transfer reactions being exothermic. The rotation of the neutral CN & the complex, implied in order for the proton to leave the C atom and return to the N atom, is facilitated by the large C=N dipole moment, 1.45 Debye, with the N atom being the negative end of the dipole. The argument was made by Petrie et al. (1 990) that the validity of this two-step model (called forth-and-back proton transfer) could be established by looking at the kinetic energy dependence of Eqs. (29), the expectation being that the initial proton transfer, Eq. (29a), is highly efficient and quite insensitive to relative KE,, up to several eV, a generalization that is discussed in Section III.A.l. However, the occurrence of Eq. (29b) depends on the lifetime of the reactant complex and hence will obviously decrease as this lifetime decreases with complex internal energy content, i.e., relative collision energy. This follows from all unimolecular reaction theory, at any level of sophistication. Figure 4 shows that this expectation is clearly met. At low KE,, i0.05 eV, essentially all of the reaction product, >95%, is the isomerization product HNCf. Above relative kinetic energies of 0.1 eV, the isomerization product abundance falls, to essentially zero at KE,, = 0.6 eV. The isomerization product is replaced by HCO:, i.e., reaction (29a), as expected, but also by simple charge transfer to produce CO;. The charge transfer is endothermic by the differences in ionization potential of C 0 2 and HCN, 13.77 - 13.60 = 0.17 eV, has its onset at the threshold, and then somewhat exceeds the simple proton tranfer to produce HC0,f. The conclusion was made by Hansel et al. (1998b) that the proposed “forth-and-back’’ model for the isomerization reaction is supported. Presumably such an isomerization reaction will occur efficiently, not only for CO and CO,, but for other molecules M whose proton affinity is such as to make the analogues of both Eqs. (29a) and (29b) exothermic, i.e., such that PA(CN at C) < PA(M) < PA(CN at N) or 124.5 < PA(M) < 148.8 kcal/mol. [PA(CO at C) = 141.4;
ION-MOLECULE REACTIONS
259
100%
E
.-
80%
zl
P
2 .V
60%
%-
2
s
40%
.3 -me
20% 0%
KE [eV]
FIG. 4. Relative product ion distribution of HCN+ (Hansel et al., 1998b).
+ CO, + products, as a function of KE,,
PA(C0,) = 128.5 kcal/mol.] However many molecules A4 with proton affinities in this range undergo other, hence preemptive, reactions with HCN+. 4. Switching Reactions
The product channels observed in the reactions of dimer and dimerlike ions A.B+ with neutrals C, A.B++C -+
C++A+B + A.C+ B + A++B+C
+
where A , B, and C represent atoms or small molecules, follow a pattern that is nearly exclusively controlled by the energy involved in these processes. Following earlier work of Shul et al. (1987a, 1987b), Giles et al. (1989), and Adams et al. (1980), where reactions of this type were examined at room temperature, Praxmarer et al. (1993) systematically investigated the energy dependencies of the overall reaction rate constants and the branching ratios of Kr2f reactions with 11 neutral reactants, using a selected ion-flow drift tube (see Table 11). At high enough exoergicities, AH< - 0.8 eV [always with respect to the charge-transfer channel (30a)], charge transfer and dissociate charge transfer are the only channels observed at all energies (from thermal energy up to ca. 1 e y KE,,) and the rate constants are independent of KE,,. At exoergicities between -0.8 and x 0 eV, both charge transfer and switching are taking place, but as the example of the reaction with H 2 0 shows (see Fig. 5), charge transfer takes over at elevated KE,, at the expense of the switching process. Because of the shortening of the lifetime of the collision complex (Kr: - H20), the probability for switching decreases, so that the fast charge transfer via Franck-Condon transition
TABLE II RATE CONSTANTS k AND PRODUCT ION DISTRIBUTIONS OF Kr: WITH 18 NEUTRAL REACTANTS AT THERMAL ENERGIES, AND ALSO THE IONIZATION ENERGIES OF THE REACTANTS, THE THERMAL CAPTURE RATE CONSTANTS k,, AND THE EXOERGICITY FOR THE RESPECTIVE CHARGE-TRANSFER REACTIONS USING THE “ADIABATIC” RE(Q:) = 12.85 eV
Ion
Reactant
Kr:
NO
NO2 m 3
H2 S c2H4
C3H8
Ionization energy (eV) 9.25 9.78 10.15 10.47 10.51 10.95
11.18 11.41 11.51
0 2
12.06
3
1
i;:
?3 Production distribution NO+ NO:
m:
H2S+ C2H: C2H:(3O%) C2Ht(25%) C3H:(5%) C-,H;(30%) C,H,f( 10%)
cos+ c2H: C2H:(35%) C2Ht(5%) C, H6+(60%) 0: 0:
k
(10-10 cm3/s) 0.1 6.2 16 8.2 7.9 8.1
kc cm3/s)
6.4 7.4 20.1 12.4 9.9 10.1
Exoergicity (ev)
Reference
2
-3.62 -3.09 -2.72 -2.4 -2.36 -1.92
Thls work This work Giles et al., 1989 Giles e? nl., 1989 Giles et aL, 1989 Giles et al.,1989
$
0%
a 5
$
‘ 3 .a
8.5 9.2 9
9.8 9 9.8
-1.69 -1.46 -1.36
Giles e? d., 1989 Giles et nl., 1989 Giles et nZ., 1989
6.2 4.3
5.7
-0.81
This work Giles e? nZ., 1989
-
4
N
;;
ION-MOLECULE REACTIONS
m w ?P
22
z"4
26 1
262
W Lindingel; A . Hansel, Z. Herman
E,
(ev)
FIG. 5 . Rate constants as a function of E,, for the reaction Kr: et al., 1993).
+ H,O + products (Praxmarer
becomes dominant. When charge transfer becomes endothermic, only switching is observed at room temperature; however, at elevated KE,,, charge transfer is also observed, increasing as KE,, becomes larger. This observed pattern of charge transfer and switching processes is consistent with the vertical-transition model (Franck-Condon principle) as discussed by Bearman et al. (1976), who interpreted the cross sections for ionic excitation in low-energy charge-transfer collision between He; and some diatomic neutrals. In analogy to that, in the cases of Kr; reactions, it is not the total recombination energy RE(Kri) = 12.85 eV that is available, but only the “effective” recombination energy Re,&$) = 11.91 eV, which is determined, as shown in Fig. 6, by the vertical transition from Kr2f to the repulsive state of b - K r at the equilibrium distance Ro(Kr:). As in the case of the proton-transfer reactions mentioned earlier, the study of the energy dependencies of switching reactions yields a wealth of thermodynamic data-e.g., in the case of the reactions of Krl investigated by Praxmarer et al. (1993), binding energies for Kr.COf and &.Cot of 1.07 f0.08 eV and 0.82 f0.03 e v respectively, and that for (CO,); of 0.65 f0.03 eV were obtained. B. CHARGE TRANSFER FROM SINGLY CHARGED IONS Information on rate constants of charge-transfer reactions of singly charged atomic and molecular ions in the thermal and slightly hyperthermal range has
ION-MOLECULE REACTIONS
263
W
61
R. FIG. 6 . Calculated potential curves for the dimer ion Kr: and the van der Waals molecule Kr,. Indicated are the equilibrium distances, R,(Kr;) = 2.79A and R,(Kr2) = 4.0065 A, and the dissociation energy, DE(Kr:) = 1.15 eV. The well depth DE(Kr2) = 0.024 eV. The recombination energy of Kri, RE(Kr:) = 12.85 eV. The value RE,,(Kr;) = 11.91 f0.23 eV indicates the effective energy available if a K r l ion accepts an electron in a long-range electron jump, as discussed in the text (Franck-Condon transition). Erep= 0.94 eV is the difference between RE and RE,, (Praxmarer et al., 1993).
been conveniently obtained from SIFT and SIFDT studies. In this section, we will treat some aspects of the charge-transfer (electron-exchange) reactions and the dependence of the rate constants on the kinetic energy of the reactants. In collisions with molecular targets, the products may be formed in either nondissociative charge-transfer processes (3 1) or dissociative processes (32): A++BC + A+BC+
(31)
A++BC + A + B + + C
(32)
The recombination energy of atomic projectiles A+ is equal to the ionization energy of the particle from its ground (or excited) neutral state. In the case of simple molecular reactant ions, the effective recombination energy is close to the energy gained in vertical transitions between the particular state of the molecular ionic and neutral species; it is close to the adiabatic ionization energy if the minima of the respective potential energy curves lie above each other, but it may be considerably smaller if the positions of the minima differ appreciably. The special case of charge transfer involving dimer molecular ions A4; (rare gases Ari, Kr:, and also N t ) was already discussed in Section III.A.4. Here the maximum “adiabatic” recombination energy is not available to its full extent, as
264
W Lindinger, A. Hansel, Z. Herman
the neutral products in a vertical transition end on the repulsive M-M potential energy curve above the nominal energy limit (Fig. 6). Thus the effective recombination energy was found to be somewhat lower (by approximately 1 eV) than the nominal recombination energy (Praxmarer et al., 1993; Praxmarer et al., 1998). Considerable insight into the mexhanism of a charge-transfer process provides the translational energy dependence of the charge-transfer reaction rate as obtained from SIFDT experiments. Basically, two groups of reactions can be distinguished (Lindinger, 1986): (1) Reactions that are fast at room temperature and do not show a translational energy dependence up to KE,, of a few eV; In these cases, large Franck-Condon factors (FCF), i.e., FCF > lop5, usually exist between the ground state of the neutral reactant and its resonant ionic state (resonant with respect to resonant charge transfer). (2) Reactions with rate constants considerably smaller than k, at thermal energy that show strong energy dependence, which often reflects an access to endoergic channels at elevated KE,, values. For these processes, the FCF for resonant charge transfer is usually significantly smaller than and with that the probability for charge transfer per collision is smaller than unity. An example of the first type is the reaction Ar+
+
H, (Fig. 7), where the rate constant is large and practically independent of KE,,. The recombination energy of Ar+ can be deposited into H, in a resonant way. Direct evidence for it was provided by scattering experiments (Hierl et al., 1977): The translational energy distribution of the charge-transfer products showed that the product HZ was formed, in large impact parameter collisions, in a resonant way, preferentially in v = 1 in collisions with AI-+(~P,/,)and in u = 2 in collisions with Ar+(,P,/,). Examples of the second type are the reactions Ar+ 0, and Ar+ CO in Fig. 7 or the reactions Ne+ N,, Ne+ CO,, and Ne+ CO (Fig. 8). All of them are slow at room temperature, although they are nominally strongly exothermic processes, but in all cases the FCFs for resonant charge transfer are well below lo-’. The room-temperature values of the rate constants for the reactions of Ar+ with 0, and CO are 5 x lo-” cm3 . s-’ and respectively, both decreasing with increasing KE,, because 4 x lo-“ cm3 . SKI, of the declining collision complex lifetimes as KE,, increases. In both cases, a strong increase in k is observed when KE,, reaches high enough values so that the sum of the energies RE(Ar+) KE,, is sufficient to allow for population of the (a411,) of 0: state (at KE,, above 0.35 ev) and the A state of CO+ (at KE,, higher than 0.9 eV), respectively, in a “resonant charge-transfer” process. Both these states are connected to the respective neutral ground states by large FCFs, and that resonant charge transfer indeed occurs has been proven for the Ar+-02
+
+
+
+ +
+
265
ION-MOLECULE REACTIONS I
I
, 1 1 1 1 1 ,
I
A r t + H2
A r * + CO
I . , .
i
110-11
Arc+ 0,
..
A * A I
003
I
I I I I
I
1
I
I 1 1 1 1 1 1
01
1
3
KECmleVI
FIG. 7. Energy dependence of the rate constant for the reaction of Ar+ with H,, 0, as a function of KE,, (Dotan and Lindinger, 1982).
system in a SIFDT experiment using a monitor reaction method (Lindinger et al., 198 1).
+
In the latter two cases (Ne+ N, and CO, respectively), the increase in k is accompanied by the opening of dissociative channels (N+ or C+ formation, respectively). As the endoergicity inferred from the results matches for the energy defect to form N$(C2C:) with a strongly predissociative channel to N + N , the interpretation is that the missing energy to open this charge-transfer channel is again supplied from the translational energy of the reactants, analogously, as in the Ar+ O2 and Ar+ CO system. The conversion of translational energy into internal energy to bridge an energy defect of a charge-transfer process was also confumed by the results of beam-scattering experiments on CH: and CH,f formation in Kr+ CH, collisions (Herman and Friedrich, 1995): While largeimpact-parameter collisions lead to the forward scattering of the reaction product formed in resonant charge transfer via a long-distance electron jump, smallimpact-parameter collisions give rise to backward-scattered product formed in inelastic momentum-transfer, intimate collisions in which translational energy transformed into internal energy may open new, endoergic channels of the charge-transfer process.
+
+
+
+
266
W Lindinger, A . Hansel, Z. Herman
t
.
(AEERYSTWYTIil %
:. 5.
(INNSEHUCKI Ne++ CO,
‘*lo mm
1 r10-’2 0.03
0.1
xE,,
(eVl
10 20
FIG. 8. Energy dependence of the rate constants for charge transfer of Ne+ with NZ,C 0 2 and CO (Lindinger and Smith, 1983).
SIFDT studies on charge transfer in collisions with polyatomic molecules (Praxmarer at al., 1998) provide further information on reaction rates of dissociative and nondissociative processes. The rate constants for charge transfer between simple atomic and molecular projectiles (Ar+, Kr+, Xe+, N,: CO+, Ar;, etc.) with ethane, propane, and butane were found to be always large and close to the capture rate constant. The relative abundance of the reaction products fit the breakdown patterns of the particular molecular ions well, if plotted at molecular ion internal energies that correspond to the recombination energies of the projectiles. Figure 9 shows the results for charge transfer to the propane molecule. Thus it appears that the charge transfer at thermal and slightly hyperthermal energies occurs as a resonant process of depositing the recombination energy into the quasi continuum of excited states of the polyatomic molecular ion. An increase in the fragment ion yields at increased collision energies was found to be due to collision-induced excitations of the product ions to their dissociation limits in collisions with the helium buffer gas atoms rather
ION-MOLECULE REACTIONS
267
FIG. 9. Breakdown pattern for propane: QET calculation, solid line; measurments: (e),C3H:; (+), C,H:; ( x ) , C,H;. Symbols are present charge-exchange results from projectile ions with (effective) recombination energies RE,&ri) = 1 1.9 e y 12.9ey RE,,dArzf)=13.7eV, RE(Krf)= 14.00eV, RE(CO+)= RE(Xe+)= 12.13ey RE,,I(N:)= 14.01 e y RE(N:)= 15.6eV, RE(&+)=15.82eV (Praxmarer et al., 1998).
(A),C,H,+; (U), C3H:; (V),C,Ht;
than to inelastic charge transfer from hyperthermal projectile ions. The latter mechanism is obviously also present, but the collision-induced excitation mechanism prevails.
c. REACTIONS OF MULTIPLY CHARGED IONS Reactions in collisions of multiply charged ions with atoms and molecules are of particular importance in highly energized gaseous systems like plasmas. Chargetransfer processes (Neuschafer et al., 1979; Stori et d.,1979; Peska et al., 1979; Smith et al., 1980; Lindinger, 1983; Herman, 1996) between doubly-charged ions and atoms (molecules) from thermal energies up to collision energies of many keV have been studied. More recently, bond-forming chemical reactions of doubly charged ions have been observed and described (Weisshaar, 1993; Price et al., 1994). The thermal data are usually obtained from SIFT and SIFDT experiments (Spears et al., 1972; Johnsen and Biondi, 1978; Howorka, 1977; Stori et al., 1979; Smith et al., 1980; Lindinger, 1983) and offer information on rate constants even for state-selected reactant (electronic) states, while beam data using product “translational energy spectroscopy” or product chemiluminescence measurements provide information on state-to-state processes.
1. Charge Transfer Involving Multiply Charged Ions Charge-transfer reactions differ from those of single charged ions in that the interaction usually results in the formation of two singly charged ions that repel
268
W Lindinger, A. Hansel, Z. Herman
each other along a Coulomb repulsive potential, and thus they possess a rather high relative translational energy, usually 3 to 5 eV. The process can be described generally as A + + + B +. A f + B f
(33)
where A and B are atoms or molecules. For a collision between two atomic species, the interaction between the reactants can be described by a relatively flat ion-induced dipole interaction term combined with a repulsive term at small internuclear separations; the interaction between the products is primarily determined by a Coulomb repulsion term. The two terms cross at a acute angle at an internuclear separation R, = 14.4/AE (for R, in A, AE in eV, neglecting the polarization interaction term); the crossings are well localized, and in most cases the probability of electron transfer can be described within the Landau-Zener formalism. The situation may be qualitatively exemplified by Fig. 10: If the crossing occurs at very large internuclear separations (small Ah‘), the two terms cross adiabatically and the transition probability is very small (A); if they cross at very small internuclear separations (large AE),the terms are adiabatically split, and the probability of the system’s ending on the product potential energy curve is small again (C). It is only if the single-passage probability through the crossing point is p x 0.5 that the system has a strong chance of ending on the product potential energy curve (B). This leads to the “reaction window” concept of the rate constant (cross section), which was developed by Spears et al. (1972): The
FIG. 10. Schematics of potential energy curves for the reaction A++ + E + A+ + E + . For discussion, see text; only case (B) leads to charge transfer of a sizable cross-section. RW = reaction window (Herman, 1996).
269
ION-MOLECULE REACTIONS
charge transfer occurs with a high probability if the crossings occur within 2.5 to 5.5A (depending on collision energy), or for reaction exoergicities of 2.6 to 5.8eV (see RW in Fig. 10). Thus, charge-transfer reactions involving doubly charged ions of type (33) represent a category of thermochemically driven reactions, where the rate constant depends on the heat of the reaction. The applicability of this “reaction window” concept was shown for a variety of thermal atomic ion-atom systems (Spears et al., 1972; Smith et al., 1980; Smith and Adams, 1980; Lindinger and Smith, 1983, Lindinger, 1983). It also applies well for populations of electronic states in atomic doubly charged ion-molecule ( F h i k et al., 1993; Herman, 1996) and molecular doubly charged ion-atom or ion-molecule charge-transfer processes (Ehbrecht et al., 1996), as shown in beam experiments, The values of thermal rate constants for rare gases from swarm experiments (Smith and Adams, 1980; Smith et al., 1980; Lindinger and Smith, 1983, Lindinger, 1983) fit the reaction window concept quite well (Fig. ll), with the highest values reaching lop9cm3 . s-’ if there is a final state of the products available for which the crossing occurs at about 4 A. For reactions deviating from that, the rate constants drop steeply. The measured rate constant for the state-tostate reaction
+
AI-++(~P) He(’S) + Ar+(’P,)
+ He+(’S)
(34)
1
FIG. 1 1. Rate constants of various reactions between doubly charged rare gas ions and neutrals as a function of R, (Smith and Adams, 1980).
270
W! Lindinger, A . Hansel, Z. Herman
was well reproduced by theoretical calculations based on the Landau-Zener formalism (Friedrich et al., 1986). In fact, the value of the rate constant can be used to eliminate the reactant and product ion reactive states. Thus, for the reaction of Hg++ with Ar and Kr, the results of the flow drift experiments showed (Hansel et al., 1992), on the basis of the curve-crossing reaction window concept combined with the core-conservation argument, that in the reaction with Ar, excited Hg++(3D) react fast and the ground-state Hg++(lS) represent a slowreacting species, while in reactions with Kr, just the opposite is the case: The fastreacting species are the ground-state Hg++('S) ions. This conclusion was in agreement with the results of beam experiments in which products were identified by translational energy measurements (Hansel et al., 1992). In reactions with molecular targets, dissociative processes are often observed. This is the case of the Xe++('D2) reaction with oxygen, where both molecular and atomic oxygen ions are formed as products (Adams et al., 1979):
+ 0, + 0; + Xe+ Xef+('D2) + 0, + 0' + 0 + Xe' Xe++('D,)
(354
On the other hand, in the reaction of Ar++(3P)with methane, only nondissociated CHZ was observed as a product; this was interpreted as formation of the other product Ar+ in the excited Ar+*('S,) state, which takes up a substantial amount of the available energy and makes the exoergicity of the process too small to make the dissociative charge transfer possible (Smith and Adams, 1980). Such a formation of an excited projectile h a 1 state has been observed in other reactions, too, notably in collisions of He++ with some molecular targets (NO, NH,, H,S) in which the reaction window concept directs the process to formation of the excited He+*(2P)state and the ground state of the molecular product ion, thus making determination of its vibrational energy distribution possible (Farnik et al., 1993; Herman, 1996). Information on vibrational and rotational state distribution in the molecular products of charge-transfer processes (33) comes from beam-scattering (Herman, 1996) and beam spectroscopic (Ehbrecht et al., 1996) experiments: In nondissociative reactions, at higher collision energies, the product vibrational state distribution is very close to that expected from the overlaps of the FranckCondon factors of the particular electronic states involved at higher collision energies, and at low collision energies, it is still close to it. The rotational temperature of the molecular products in reactions CO++ with a variety of gases (Ehbrecht et al., 1996) was found to be quite low (400 to 800 K) and close to the temperature of the ion source. Thus, charge-transfer reactions (33) presumably tend to produce rotationally cold products. For reactant ions of higher recombination energy, double charge transfer was observed of the type A + + + B -+ A + B + +
(36)
ION-MOLECULE REACTIONS
27 1
+
This appears to be the dominant charge-transfer process in Ar++ N, collisions (Neuschfer et al., 1979) and in many reactions of Ne'+ (Smith and Adams, 1980). In collisions of this ion with Kr and Xe, both single-charge transfer [reaction (33)] and double-charge transfer [reaction (36)] were observed, and the rate constants were found to be of the order of lo-' cm3 . s-I.
2. Chemical Reactions of Doubly Charged Ions The occurrence of chemical reactions of doubly charged ions was briefly mentioned in flow-tube studies of Ca++ and Mg++ interactions with simple molecules (Spears et al., 1972). Several chemical reactions of transition metal doubly charged ions (Ti++, Ta++) in collisions with hydrogen and simple hydrocarbons, leading to singly charged chemical products, have been reported (Tonkyn and Weishaar, 1986; Ranasighe et al., 1991; Weishaar, 1993). More recently, bond-forming chemical reactions of molecular dictations have been observed (Price et al., 1994). They are usually accompanied by competitive charge-transfer processes and may be of a nondissociative type, as, for example, CF;++D, CF:++D,
+ CF2Df+D+ + CFl+DZ
(374 (37b)
+
or of a dissociative type, as, for example, in the system CO:+ D,, where both nondissociative (Co,', C02D+) and dissociative (CO+, COD, O+) products of both bond-forming and charge-transfer reactions were observed (Price et al., 1994). A beam-scattering study of reactions (37a) and (37b) showed that the dynamics is governed by the Coulomb repulsion between the products, which recoil with large kinetic energy that peaks at about 6 eV for reaction (37a) and at about 4eV for reaction (37b) (DolejSek et al., 1995). In conclusion, one should mention that chemical reactions of doubly charged ions leading to doubly charged ions and neutral reaction products have also been detected, i.e., processes
A+++BC + AB+++C.
(38)
One might take as examples of the processes observed the following reaction (Ranasighe et al., 1991), Ta++
+ CH,
-+ TaCH;'
+ H,
(39)
and an analogous reaction with Zr+.+,or the recently reported very interesting dication chemical reaction (Tosi et al., 1998): Ar+++N2
-+
ArN+++N
(40)
272
K Lindinger, A . Hansel, 2. Herman
D. VIBRATIONAL DEEXCITATION AND EXCITATION OF MOLECULAR IONS Already by 1925, Pierce (1925) had discovered that the dispersion of sound was caused by excitation and deexcitation of small neutral molecules. Since then, the measurement of vibrational quenching and excitation of neutral molecules has been an active area of research. By contrast, the vibrational relaxation of molecular ions is a more recent field of research. It was not until the late 1970s that quenching rate constants for ion-neutral pairs were reported by Huber et al. (1977); Kim and Dunbar (1979); and Jasinslu and Browman (1980). The acquisition of systematic quenching rate constant data for molecular ions colliding with neutrals has only occurred following the development of suitable measurement technology in the form of flowing afterglows and selected ion-flow drift tubes (SIFDT) (Howorka et al., 1980; Lindinger and Smith, 1983). Most of this research was inspired and conducted by Eldon Ferguson, who also has presented several reviews on this subject (Ferguson, 1984; Ferguson, 1986).
1. Repulsive Interaction
As long as repulsive interaction is dominant, there is no fundamental difference in the theoretical description between the quenching of ions and the quenching of neutral molecules. This problem was solved fist by Landau and Teller (1936), leading to the famous Landau-Teller equation for the vibrational quenching rate constant.
where p is the reduced mass, o the vibrational frequency, k the Boltzmann constant, T the temperature, and 1 the range parameter for the repulsive exponential interaction V(r) exp(-r/l). 1 has a typical value of -0.2A. This has been derived in different ways and is discussed in detail by Cheng et af. (1970). It is essentially the classical expression of the adiabatic criterion for energy transfer, intergrated over a Maxwellian velocity distribution. The most favorable case investigated so far for ion vibrational quenching to be dominated by repulsive interactions is for quenching of N2f(u) by He, the atom of smallest polarizability. Figure 12 shows results obtained in Innsbruck (Kriegel et al., 1988; Kriegel et al., 1989) on the vibrational excitation of N:(u = 0) and quenching of N,'(u = 1) as a function of KE,,, the only case so far where both have been measured. Figure 13 shows a Landau-Teller plot of the kq from Fig. 12. Above 0.25 eV, the plot is very linear, and the range parameter (slope) deduced is 0.22 A, in excellent agreement with a subsequent quantum calculation by Miller et al. 1988). The deviation at KE,, < 0.25 eV, showing an enhancement in kq, is due to impurities that are present in the helium buffer gas and lead
-
-
ION-MOLECULE REACTIONS
273
FIG. 12. Measured values of the excitation rate constant k, and of the quenching rate constant kq for the N:(u)-He system as a function of E / N m e g e l et al., 1988).
FIG. 13. Landau-Teller plot of In kq vs. (KE,,J'I3 of kq (open symbols) (Knegel et al., 1988, 1989).
(solid line) as compared to measured values
274
K Lindinger, A . Hansel, Z. Herman
to erroneously h g h rate constants at low E / N (long reaction time), as was proved by refined measurements by Kato et al. (1995). Smaller values for kq at low E / N and thus low KE,, are also expected from theoretical calculations performed by Zenevich et al. (1992).
2. Ion-Induced Dipole Interaction There is always some attraction between molecules and even between atoms, and therefore the role of the attractive interaction becomes significant at sufficiently low temperature, yielding a characteristic positive deviation of kq from the Landau-Teller plot. In the case of neutrals for which are are strong electrostatic attractive potentials, this leads to a minimum for kq versus T , as has been observed for the very polar hydrogen halides (Yardley, 1980). The repulsive interaction, dominant for KE,, > D, (D, is the well depth in the interaction potential), gives the strong increase in kq with KE,, described by the LandauTeller plot [Eq. (41)], whereas long-range domination of the quenching (at KE,, < 0,) always yields a decrease in kq with increasing KE,,, causing a minimum in the function of kq as dependent on KE,, at values KE,, 2 D,. In the case of ions, the attractive electrostatic potential almost always dominates vibrational quenching at thermal energies, because of the chargeinduced dipole force for nonpolar molecules and the even stronger charge4ipole interaction for polar molecules. The case of the above-mentioned N:(u)-He system is an exception, as D, = 0.017 eV is so small that the minimum of kq falls below room temperature, and thus only the increase due to repulsive forces is observed in experiments at elevated energies. For the quenching of NO+(u) in collisions with many different neutrals, the expected decline of kq with KE,, increasing from room temperature up to a few tenths of an eV has been observed (Federer et al., 1985), and higher vibrational states generally are quenched faster than lower ones (Pogrebynya et al., 1993; Lindinger, 1987; Hansel et al., 1999). An example of these findings is presented in Fig. 14, showing the rate constants kq for quenching of NO+ ( u = 4) and NOf(u = l), respectively, with CH,. The quenching of O l ( u ) by 14 different gases (Bohnnger et al., 1983) and of NO+(u) by 17 gases (Federer et al., 1985) shows for both sets of measurements a strong correlation between the magnitude of the quenching rate constants and the polarizabilities of the neutral quenchers. Higher polarizabilities lead to deeper well depths in the ion-neutral interaction potentials and therefore to longer lifetimes of the ion-neutral collision complexes, thus increasing the probability for quenching of the ionic vibration. On the basis of these data, Ferguson (1984, 1986) developed a vibrational relaxation model assuming that colliding ion-neutral pairs form a transient
ION-MOLECULE REACTIONS 5E-IU
- IE-IU "&'V)
I
I
I
I
I
275
I
,
7 4.': -
0
I'
A
9
0
,'O A
I
,'O
A
-- - _ _ _ ---.
,A'
A$
I
A AMA.
k43 rxp
0 klOexp A A k l O e x p (Richlrret81. I W B ) LI.I) llicurel ~
_
IE-ll
k43 lhcorcl ' a
*
.
~ I
'
0.1
1
complex under the influence of the electrostatic attractive potential of an ioninduced dipole interaction,
A B f + C + [AB+.C]*
(42)
with a complex lifetime z and a rate constant for complex formation k, being the collisional rate constant. When ABf(u) is vibrationally excited, the transient complex [ABf.C]* may either decay unimolecularly back to the reactants or, alternatively, undergo vibrational predissociation into ABf(u' < u) C, with a rate constant kup,
+
ABf(v)
+c A k"
kAB+(U' < u) + c
[AB+(U).C]*
(43)
The quenching rate constant kq for the overall process AB+(U)
+ c + AB+(V' < v ) + c
(44)
276
W Lindinger: A . Hansel, Z. Herman
is given by
which reduces to kq = k,k,,/k, = k,k,t when k,, >> kvp, which is the case for small quenching rate constants, i.e. k9 << k,. Conversely, kq x k, when kup>> k,,. Information about the complex lifetime t and the rate constant for vibrational predissociation kup can be obtained from the study of three-body association reactions by Ferguson (1984, 1986),
AB+ + C + M +. A B + . C + M
(46)
where the transient complex [AB+.c]* is stabilized by a third body of M ,
[AB+.c]* + M -+ A B + . C + M
(47)
before it can decay unimolecularly (Bates, 1979; Herbst, 1980). The stabilization rate constant k, can usually be assumed to be equal to the collisional rate constant k, (in the case of M = He, k, x k,/4). For low-pressure conditions, it can be shown that k3 = k,k,z (Ferguson, 1984). Under the assumption that t is approximately the same for [AB+(u = O).c]* and [ABf(u # O).c]*, k9 and k3 are related by
k9 = kupk3/k,
(48)
when k3 << k,. This assumption receives justification from the data in Fig. 15, which show a strong correlation between k3 and k9. Values of kupobtained from
10-9
-
Y
k3(crn6s-1) FIG. 15. Correlation between vibrational quenching and three-body association for O:(u) and NO+(u) with some neutrals (Ferguson, 1986).
ION-MOLECULE REACTIONS
277
k3 and k4 are within one order of magnitude, lop9 s-' 2 k, 2 lo-'' s-I, while the corresponding values of k4 and k3 vary over three orders of magnitude. Thus, from known values of kq, k3 can be estimated to within an order of magnitude, and vice versa. Recent observations by Hawley and Smith (1991) of fast quenching rate constants at very low temperatures (below 5K) support the above-described concept of Ferguson (1986). E. ASSOCIATION REACTIONS
In association reactions, an intermediate from a binary reactive collision is stabilized by a third body, which removes the excess energy: A + + B C + M + ABC++M
(49)
The mean lifetime of small (4to 5 atoms) ion-molecule reaction intermediates at room-temperature collisions is typically of the order of lo-'' s. Therefore, association reactions involving three-body stabilization may be expected to play a minor role in hot, low-density plasmas. However, their importance increases with decreasing temperature and increasing plasma density. The mean lifetime of an intermediate also increases with the complexity of the system. Its decomposition may be expected to follow the unimolecular reaction kinetics, and the value of the mean lifetime (reciprocal of the unimolecular decomposition rate constant) increases steeply with the number of internal degrees of freedom in the system, even in the simplest Kassel form, hni= 1/T = v(E - E,/E)"-', where E is the total energy content of the intermediate, E , its dissociation energy, v a critical frequency factor, and s the number of degrees of internal energy in the system. Some polyatomic ions, such as CH:, were found to form very long-lived complexes (T > lop6s) even at room temperature. Three-body association rate constants for CH: with a variety for diatomic and triatomic molecules, as measured by the SIFT technique (Adams and Smith, 1981), are given in Fig. 16. They vary from partner to partner (from about cm6 . s-I), but to they correlate strongly with the binding energies of the constituents of the complex. They show also a strong temperature dependence T-", with u in the range 2.7 to 4.4. The association rate of CH: with ammonia in He buffer was investigated in a SIFDT experiment in the KE,, energy range 0.047 to 0.2 eV (Saxer et al., 1987) and found to be in effective competition with two bimolecular reactions of CH2NHi and NH; formation. The association rate constant k3 declined from to cm6 . s-' as a function of KE,, (Fig. 17) in the collision energy range investigated. The lifetimes T~ of the respective complexes (CH: .NH3)* were estimated to be to lo-' s.
278
W Lindinger, A. Hansel, Z. Herman
FIG. 16. Temperature dependence of rate constants for the association of CH: neutrals (Adams and Smith, 1981).
with various
FIG. 17. Association Rate constant k3 and collision complex lifetime td for the association of CH: with NH3 as a function of KE,,,.f represents stabilization efficiency in helium, which was found to be 0.6 (Saxer et al., 1987).
279
ION-MOLECULE REACTIONS
The SIFDT techniques was used also to investigate the competition between association and bimolecular reactions in the system SF,f H2S (Zangerle et al., 1993). Analysis of the reaction kinetic data showed that the rate constant for ternary association, at a temperature close to room temperature, was 3.0 x lop2' cm6 s-I, and that the reaction product SF,HF+ was formed by a slow dissociation of the associate (SFf.H2S), k- = 3.0 x lo-'' cm3 . s-'. The associate was also found to react further with H,S,
+
+ H2S +. H,Sf + SF, + HS ( k = 1.4 x lop9 cm3 - s-I). Data on dissociation
SFf.H2S
(50)
in a fast reaction of the associate ion in collisions with Ar and comparison of them with results on dissociation of other associate ions were used to estimate the dissociation energy of (SFf.H2S)as about 2.2 f 0.5 eV. A method developed to predict the rates of ion-molecule association reactions (Olmstead et al., 1976) was based on a quick randomization of energy in the collision complex and on treating the backward decomposition of the collision complex by an application of the RRKh4 theory. The method was successfully applied to predict both the pressure dependence and the temperature dependence of the association rates of proton-bound dimers of ammonia, methylamine, and dimethylamine.
IV. Effect of Internal Energy and Temperature on IM Processes The effects of energy on ion-molecule rate processes have been investigated by a variety of methods. The influence of reactant translational energy, as studied by SIFDT (selected ion-flow dnft-tube) techniques in swarm experiments, by ICR, and by beam and other single-collision techniques, is reviewed in other parts of this chapter or of this book. In this section, we will concentrate specifically on the influence of reactant internal energy on ion-molecule reactions. There are basically two sources of data that address this problem: 0
Variable-temperature swarm methods provide data on ion-molecule reactions over a wide range of temperatures and reveal the influence of vibrational and rotational energy of the reactants on reactivity. Ion reactants can be prepared in specific states in flow tubes, and their reactivity can be investigated; in combination with state-to-state studies by other (single-collision) methods (Ng, 1988), they offer valuable information on how the specific electronic and vibrational excitation of the reactants influences reaction rates.
Pioneering studies using variable-temperature flowing afterglow tubes (Lindinger et al., 1974) provided data in the temperature range 80 to 900K; present
280
FV Lindinger, A . Hansel, Z. Herman
instrumentation (Hierl et al., 1996) enables high-temperature studies at 300 to 1300 K. The influence of both internal and translational energy on reactivity can nowadays be studied conveniently by the variable-temperature selected ion-flow drift tube (VT-SIFDT) technique (Adams et al., 1985) over the temperature range 85 to 550 K (Viggiano and Moms, 1996) and from 300 to 1800 K (Dotan et al., 1997). Examples of temperature-dependent reactions are O+ N,, O+ 0, (Hierl et al., 1997), N+ 02,and N t 0, (Dotan et al., 1997), which are especially important in the earth's ionosphere. In case of the exothermic reaction
+
+
+
+
0 + + N 2 + NO++N
(51)
the rate constant increases above l000K by almost an order of magnitude (Fig. 18). The data are in agreement with an earlier interpretation, obtained with discharge-excited N2 (Schmeltekopf et al., 1968), that vibrational excitation of the neutral reaction partner N2(u = 2) (present in amounts of a few percent) increases the rate constant by a factor of 40 in comparison with N,(u = 0). The effect of translational energy (diamonds in Fig. 18) does not seem to be nearly as big. The data imply the existence of a barrier along the reaction coordinate as a result of crossing of potential energy surfaces. In an analogous study of the charge-transfer reaction N;+02
(52)
+ 0t+N2
an increase in both rotational and translational energy up to about 0.3 eV causes a decrease in the rate constant, but further increase has only a minor effect. However, an increase in the vibrational energy of the neutral reactant 0, has a r
.
.
b,
3 lcr".
z--
.
.
,
.
Present NOAAO NOAA(KEonly)
0 W
.
. . , . . . . , . . . . 0 * +N,
I predicted
'ffl
.I
i
C
I 8
6
c
8
4
10"
e
NO'+ N
--f
'8
4
-
8
1043.
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l&l'-
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U
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.
.
,
.
'
. . . . '
L
,
'
-
,
'
*
-
.
,
ION-MOLECULE REACTIONS
28 1
large effect, namely for O,(u = 2); this was connected with the possibility of opening a new reaction channel-the formation of Ot(a411,). The VT-SIFDT technique provides a powefil tool for studying the influence of rotational and vibrational excitation of the reaction partners (which are in thermal equilibriium equilibrium with the buffer gas) on their reactivity, separated from the influence of the translational ion energy which only ions gain from the drift field. A recent extensive review (Viggiano and Morris, 1996) summarizes the results of numerous studies of these effects on rate constants and branching ratios. It appears that rotational and translational energy have a similar effect on driving endothermic reactions. For exothermic reactions, large effects were found only if one or both reactants had a large rotational constant; this suggests that a change from low to moderate J affects reactivity, while a change from moderate to large J has little effect. Vibrational energy effects on the rate of chemical reactions were found to vary greatly. In some cases, very strong effects of vibrational energy in charge-transfer reactions may be connected with available resonances and large Franck-Condon factors.
1. Spin-Orbit States of Rare Gas Ions The first group of reactions concerns reactions of rare gas ions in specific spinorbit states. In SIFT and FDT studies, the influence of the two spin-orbit states (,P,/, and 2 P , j z )of Kr+ and Xe+ ions in reactions with various molecules was investigated (Adams et al., 1980; Jones et al., 1982). In charge-transfer reactions with several simple molecules, reaction rate constants of the ground state Kr+(’P,/J were found to be larger by about an order of magnitude (CO, 0,, N,O, COS); in other reactions (CH,, CO,, H,, NH,),they were about the same as the rates for the excited (3P,i2) state. The difference in reactivity of Ar+ spin-orbit states could not be investigated in SIFT experiments, presumably because of rapid quenching of the upper state by the buffer gas. However, a large amount of data on AI-+(~P,/,) and Ar+(’P,,,) comes from single-collision experiments, in which state-selected ions were prepared by photoionization and product ions were measured in coincidence with the threshold photoelectron [threshold electron-seconary ion coincidence (TESICO)] meethod and its variations (Koyano and Tanaka, 1980). The authors studied both charge transfer and chemical reactions of the state-selected argon ions. The chemical reaction with hydrogen,
+
Ar+(2P3/2,2 P , i 2 ) H, + ArH’
+H
(53)
282
ctl Lindinger, A. Hansel, 2. Herman
was found to proceed about 1.5 faster with the upper 1/2 state than with the ground state. Avery big effect was found for the charge-transfer process (Tanaka, et al., 1981a):
where the reaction with the 1/2 state was found to be about an order of magnitude faster than the reaction with the 3/2 state. On the other hand, in an analogous reaction with D,, the rates of both processes were just about equal. The explanation is in the defects in resonance between the levels of the reactant [Ar+(2P3/2,2P,/2) H,(u = 0) or D,(v = O)] and product states [Ar H:(u) or D:(u)]; the charge-transfer process is faster, if the energy defect is smaller. These conclusions are confirmed by scattering experiments (see Section 1II.B) and by measurements of rates of the backward process in dependence on the vibrational energy of the molecular reactant (see further on). An interesting case of internal energy effects in the charge-transfer reaction Ar+ N, and in the reverse process, N$(u) Ar, will be discussed separately.
+
+
+
+
2. Electronic Excitation of Reactant Ions Many examples of differences in the reaction rates for ground-state and electronically excited (metastable) states of reactant ions come from SIFT and SIFDT studies. In reactions of simple ions, reactions of ground and excited O+ ions are of interest: While the reaction of the ground-state ion,
+
O+(4S) N, + NO' + N
(55)
is a slow process, excited ions O+(,D,'P) react fast in charge transfer (predominant) (56a) and chemical reaction (56b) (Smith and Adams, 1980): O+(,D, ,P)
+ N,
+ N:
+0
-+ N O + + N Metastable oxygen ions 02(a4n,) react fast with atoms or molecules (Ar,N,, CO, H2) in reactions that are endothermic with ground-state oxygen ions; the respective rate constants do not depend much on translational energy. An interesting case is the charge-transfer reaction of electronically excited NO+(a3X+)with Ar The reaction is endothermic by 0.09 eV, and its rate constant increases with translational energy from the thermal value of 3 x lo-" cm3 . s-' to a value of about 9 x cm3 . s-' at KE,, x 3 eV in a way typical of slightly endothermic reactions (Dotan et al., 1979).
ION-MOLECULE REACTIONS
283
3. Vibrational Excitation of Reactant Ions
A wealth of data exists on the effects of vibrational energy of molecular reactants on reaction rates. Detailed data come from methods by which vibrationally stateselected ions could be prepared. In swarm experiments, an important step forward was the development of a new method in which the SIFT techniques were combined with laser-inducedfluorescence (LIF) detection for monitoring the ion vibrational states (Kato et al., 1993). In this way, both the vibrational states of the reactant ion and the vibrational states of some reaction products could be detected, and the influence of vibrational energy on reaction rates of thermal ions (where the translational-tovibrational energy transfer is negligible) could be studied. The method was primarily used to study reactions of N l ( u = 0 to 4) with Ar (which will be discussed separately), N, and 0, (Kato et al., 1993), Kr (Kato et al., 1996), H, (de Grouw et al., 1999, and HCl (Krishnamurthy et al., 1997). In the reaction N ; ( ~ = O t o 4 ) + K r + NZ+Kr+
(57)
a dramatic increase in the reaction rate constant of almost two orders of magitude was observed when going from u = 0 to u = 4 (Fig. 19). The results were interpreted by a model that assumes that only reactions in close energy resonance with the product states N,(u = 3) Krf(2P,,2) occur and that the transition probabilities are strongly influenced by Franck-Condon factors between potential
+
FIG. 19. Rate constants for the reaction N:(o) + Kr --f N, + Kr as a h c t i o n of the vibrational quantum number u. Langevin rate constant kL = 8.1 x lo-'' cm' . s-' (dashed line). The prediction is based on Franck-Condon factors for energy-resonant transitions (solid line), whereas the dotted line is based on Franck-Condon factors assuming a relative shift of 0.02 8, of the vibrational wave functions of N: and N, (Kato et aL, 1997).
284
W Lindinger, A . Hansel, Z. Herman
energy curves of the molecular products, including a perturbation during a close approach of the (N2-K.r)+ pair (dotted line in Fig. 19). In the reactions with H, and HCl, no special effect of internal excitation of N:(u = 0 to 4) was observed. A negative dependence of the reaction rate constant on internal energy of exothermic reactions
and NH,f(u)
+ NH,
-+
NH:
+ NH,
(59)
was well described by a simple model based on statistical RRKM calculations with constraints (inactive vibrations, steric hindrance) (Uitenvaal et al., 1995). However, trajectory calculations on reaction (58) indicate that the integral rate is a very complicated interplay of the particular state-to-state processes of very different cross sections (Eaker and Schatz, 1985; see also Ng, 1988). Single-collision beam experiments using state-selected reactants (the TESICO method and its variants, as mentioned above) provide detailed data mostly for hyperthermal collision energies. In reactions of hydrogen ions with rare gases, the chemical reactions H$(u)
+ He
-+
HeH'
+H
(60)
H;(u)
+ Ne
-+
NeH'
+H
(61)
and
are endothermic with ground-state reactants, by 0.8 eV and 0.6 eV for reactions (60) and (6 l), respectively. In agreement with pioneering photoionization studies (Chupka, 1975, and references cited therein), vibrational excitation of the molecular reactant in reaction (60) to states at or over the endothermicity bamer ( u ? 3) increased the cross-section dramatically; reactant vibrational energy was found to have a much larger effect on overcoming the barrier than reactant translational energy (Baer et al., 1986); experimental results were supported by theoretical calculations. The results for reaction (6 1) were similar, but the difference in the effect of vibrational vs. translational energy was not so pronounced (Herman and Koyano, 1987). On the other hand, a rather small effect of vibrational excitation was found for the exothermic chemical reaction Hl(u)+Ar
-+
ArH++H
(62)
The rate showed a tendency to increase by about 40% when the excitation of the molecular ion reactant increased from u = 0 to u = 4 (Tanaka et al., 1981b). In studies of other exothermic ion-molecule chemical reactions in which more than three atoms are involved, the effect of vibrational excitation has been
285
ION-MOLECULE REACTIONS
generally negative. This holds for reactions (58) and (59) and also for a more complicated reaction (Herman et al., 1986):
+ CH,
CH:(v)
+.
CH,f
+ CH,
(63)
In all these exothermic reactions, the rate had a tendency to decrease with increasing internal excitation, but the effect was by no means large. A very different picture emerges from studies of vibrational energy effects on simple charge-transfer reactions. A prime example is the reaction (Tanaka et al., 1981b)
At all collision energies, there was a strong tendency to promote the reaction with H : (v = 2) (Fig. 20). The results were interpreted as preferential formation of the AI-+(~P,,~) state, which is in closest energy resonance with Hi(u = 2), and they tie in nicely with state-selected studies of the reverse process [see reaction (54) above]. Considerable attention, both experimentally (Cole et al., 1984) and theoretically (Lee and DePristo, 1984), was given to the charge-transfer reaction
a process competing with chemical reaction (58). Here, the relative rate was found to decrease almost linearly from 1.0 at v = 0 to about 0.3 at v = 10 (see also Ng, 1988).
- A r e * H z : d2
+
1
I
Ar+
I
+
H2 : cr, 1
h
5
B \
01
-.-
0
1 v
2 of
3 H;
L
I
0 v
o f 'H:
1 V
+
Of
2
3
L
Hi
FIG. 20. State-selected cross sections for the reaction Hl(u) Ar + products as a function of the vibrational quantum number u, obtained at different collision energies Ec,m(Tanaka et al., 1981b).
286
E! Lindinger, A . Hansel, Z. Herman
4. Internal Energy Effects on Charge Transfer in the System (Ar
+ N2)+
A special discussion is needed of the influence of internal energy on the chargetransfer process
Nl(u)
+ Ar
-+ Ar' + N 2
(66)
and the reverse process
Practically all existing methods used in studies of ion-molecule reactions-the swarm methods SIFT, SIFDT, and SIFT-LIF; the single-collision methods with state-selected reactants; and guided beam and beam scattering-as well as numerous theoretical investigations concentrated on this particular system. As a result, reactions (66) and (67) are among the most thoroughly state-to-stateinvestigated ion-molecule processes. As in other parts of this review, we cannot provide a full review of the numerous studies of this system, but will concentrate on the main issues; complete references can be found in the quoted original papers. The first impetus to investigate this system in more detail came from a SIFDT study of reactions (66) and (67) (Lindinger et al., 1981) that summarized the previous findings and showed a dramatic increase in the rate constant for reaction (67) with translational energy (Fig. 21), suggesting that the product molecular ion was formed in a slightly endothermic process in the N,f(X, u > 0) state. In the reverse charge-transfer process, reaction (66), the rate constant was fast for vibrationally excited N;, but it showed a strong energy dependence for
FIG. 21. Energy dependence of the rate constant for the charge-transfer reaction Arf + N 2 . (Lindinger et al., 1981).
ION-MOLECULE REACTIONS
FIG. 22. Energy dependence of u # 0) Ar (open symbols) and
N:(X,
+
287
the rate constants for the charge-transfer reactions N:(X, u = 0) Ar (solid symbols) (Lindinger et al., 1981).
+
N;(u = 0), reflecting the endothermicity of this reaction path (Fig. 22). In a VTSIFDT investigation of reaction (66) at elevated temperatures, the effect of rotational energy on the reaction rate was found to be comparable to the effect of translational energy (Viggiano et al., 1990a). A SIFT-LIF study (Kato et al., 1996) confirmed a steep increase in the rate constant of reaction (66) with vibrational excitation of N$ at thermal collision energies (from 0 . 0 9 ~ cm3 . s-' for u = 0 to 4.0 x lo-'' cm3 . s-' for u = 1, and then with little change up to u = 4). Photoionization methods made it possible to prepare state-selected ions for single-collision studies of the cross section of reaction (66) on both vibrational quantum number and translational energy (Kato et al., 1982b; Govers et al., 1984; Liao et al., 1986). The results showed that the reaction with N;(u = 0) remained an order of magnitude smaller than that with u = 1 up to relative collision energies of about 40 eV; thereafter it gradually increased, and above 140eV the two cross sections were about the same (Fig. 23). In an ingenious extension of the guided-beam experiment (Liao et al., 1985), the ion product Arf of reaction (66) was allowed to react hrther with hydrogen; from the difference in its reactivity, one could infer that most of the product ions were in the ground Arf('P,/J state; the fraction of the Ar+(2P,/2)increased from practically zero at the collision energy of 5 eV to about 20% above 40 eV Information on the reactivity of the spin-orbit states of Ar+ in reaction (67) was obtained by the TESICO method (Kato et al., 1982a): The ratio of cross
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K Lindingec A . Hansel, Z. Herman
V
V
V'
V'
\ 260 eV
O00
I
V'
2
0
I
v'
2
0
I
"l
7
320eV 0
I i 2
+
FIG. 23. Relative total cross sections of uL,,/bd=, for the charge-transfer reaction N:(u') Ar as a function of the vibrational quantum number u' obtained at different collision energies Ec,m,(from Liao ef al., 1986).
sections CT(1/2)/a(3/2) was about 0.6. Beam-scattering experiments (Friedrich et al., 1984; Rockwood et al., 1985) showed that the Nt(v = 1) state was primarily produced in a slightly endothermic process (0.092 eV) rather than the ground state, for which the reaction is exothermic by 0.179 eV Higher vibrationally excited states of the product molecular ion could be populated in intimate, momentum-exchange collisions at elevated collision energies. Thresholds for production of the particular vibrationally (and at elevated energies also electronically) excited states were identified in guided-beam experiments carried out over a wide relative collision energy range 0.1 to 90 eV (Tosi, 1992). Experimental results have been confirmed and rationalized by theoretical calculations based on curve-crossing arguments (Parlant and Gislason, Parlant and Gislason, 1986; 1987; Nikitin et al., 1987; Clary and Sonnenfroh, 1989).
V. Concluding Remarks In this chapter, we have been mainly concerned with the energy dependence of ion-neutral processes ranging from chemical reactions of ions to charge-transfer
ION-MOLECULE REACTIONS
289
processes, association reactions, and finally internal excitation and deexcitation of molecular ions in swarm-type environments, as they yield a wealth of general information on the mechanisms involved, such as complex formation, direct processes of particle exchange, and Franck-Condon transitions in chargeexchange reactions. We did not summarize as many data as possible, but rather have chosen characteristic examples of reactions in order to extract generalizations for the different types of ion-neutral processes. There is also a large amount of data on reactions investigated at room temperature; these data can be found in compilations mentioned throughout the text of this chapter. Because of space limitations, we have limited ourselves to reactions of positive ions and have omitted processes like collisional dissociation which, so far as multiple collisions are concerned are strongly related to vibrational excitation and deexcitation, and we have also omitted isotopic exchange processes, which show many similarities to proton-transfer reactions, isomerization processes, and switching reactions. We have dealt with the reactions of small ions and molecules only, but there is also an enormous amount of data on reactions of large molecules, which follow different patterns from the ones described here. We have briefly mentioned the use of data on IMR in plasma modeling and for the understanding of interstellar molecular synthesis as well as of ionospheric chemistry, and we also want to point out the applications of IMR in various methods of chemical ionization. The most recent one, developed in our laboratory, namely proton-transfer reaction mass spectrometry (PTR-MS), allows for on-line monitoring of volatile organic compounds at levels as low as a few parts per trillion and is therefore applicable for environmental, food, and medical research involving investigations of fast metabolic and enzymatic processes.
MI. Acknowledgement This work was supported by the “Fonds zur Forderung der Wissenschaftlichen Forschung,” Project P-12429 and P-12022 and by the grant no. 20319710351 of the Grant Agency of the Czech Republic. We want to thank Professor Eldon E. Ferguson for many helpful discussions.
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USES OF HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY IN CHEMICAL YAPOR DEPOSITION AND PLASMA PROCESSING L. W ANDERSON A.N. GOYETTE, AND J E . LAWLER Department of Physics, University of Wisconsin, Mudison, WI I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . .. , . , . , .. . , . , , 11. High-Sensitivity White-Light Absorption Spectroscopy.. . . . . . . . . . . . . . . . . . . . . . . . 111. The Uses of High-Sensitivity White-Light Absorption Spectroscopy in the CVD of Diamond Films. _ _.. . . . . . . . . . . . _ _.. . . . . _.. . . . . _ _ _ _.,. . . . , . , . , . , A. Measurement of CH, Radical Densities.. . . . . . . . . . . . . . . .. . . B. Measurement of CH Radical Densities and [H]/[H,] Ratios.. . . . . . . . . .. . . . . C. Measurements of C,H, Densities . . .. .
I\!
V:
VI. VII. VIII.
D. Detection of Other Species during the ........................... E. Measurement of C, Radical Densities F. Spectroscopic Temperature Determinat The Uses of High-Sensitivity White-Light Absorption Spectroscopy in Other CVD Environments . . . . . . . . , . . , . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . A. CVD of GaAs.. . . . . .. .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . B. CVD of Silicon.. . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . Other Uses of High-Sensitivity White-Light A. Etching Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . B. Argon Ra uency Plasmas.. . . . . . . . . . . . . . . . . . . . .. .. . .. . . .. . . . . . . . . . . . . . Conclusion.. .................................................... Acknowledgments . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . .. .. . . . . . .. . . . . . . . . . . . . . . . . . References . . . , . . . . , . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . .. . . . .. . . . . . . .. . . . . . . .. .
295 296 303 304 313 319 323 325 328 332 332 333 334 334 336 331 338 338
I. Introduction This paper reviews the uses of high-sensitivity white-light absorption spectroscopy in studying the vapor phase during the chemical vapor deposition (CVD) of various materials and in processing plasmas. This paper discusses the following: (1) techniques for high-sensitivity white-light absorption spectroscopy and their limitations, (2) the use of high-sensitivity white-light absorption spectroscopy in 295
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the CVD of diamond films, (3) the use of high-sensitivity white-light absorption spectroscopy in the CVD of GaAs and Si films, and (4)other applications of high-sensitivity white-light absorption spectroscopy. Although this diagnostic has been used for the study of the gas-phase composition during the CVD of several materials, it has been most extensively applied to the study of the CVD of diamond films. It is therefore natural that we focus our discussion on its use in the CVD of these films. Absorption spectroscopy is a method for determining the absolute column density for the various atoms, radicals, and molecules in the vapor phase. There are relatively few methods for determining gas species densities in CVD systems that do not perturb the species densities. High-sensitivity white-light absorption spectroscopy is one such method. It is reliable, is easy to use, and yields absolute measurements. In addition, as discussed in this paper, it can be used with radiating systems that have bright backgrounds such as glow discharges. Other noninvasive methods such as laser-induced fluorescence, coherent anti-stokes Raman spectroscopy (CARS), and two- or three-photon photoionization are all more difficult to use and present serious difficulties for absolute calibration. Laser-induced fluorescence, while very sensitive, often does not work at high total gas densities, where excited levels are quenched, or for predissociating excited levels. Both coherent anti-stokes Raman spectroscopy and two- and threephoton photoionization are nonlinear techniques that are difficult to calibrate absolutely. Two- and three-photon photoionization is very difficult to use in a glow discharge. Other methods using a sampling probe such as mass spectrometry can perturb the gas species.
11. High-Sensitivity White-Light Absorption Spectroscopy Figure 1 shows a schematic diagram of the apparatus used by Menningen et al. (1995a) for high-sensitivity white-light absorption spectroscopy studies of the CVD of diamond films in a hollow-cathode CVD reactor. The continuum light source is typically an ultrastable high-pressure Xe or D, arc lamp. The 1 mm x 2mm arc of a Xe lamp is imaged at the center of the CVD reactor and is reimaged onto the entrance slit of a spectrometer with unity magnification. Ultraviolet-grade fused silica lenses and windows are used so that light is transmitted at wavelengths from 190nm to the infrared. A multilayer dielectric filter with a bandpass centered near the spectral range of interest is typically placed in front of the entrance slit of the spectrometer. The dielectric filter helps reduce scattered light in the spectrometer. Either a photodiode array or a charge-coupled device (CCD) array, placed at the output plane of the spectrometer, is used as the detector. This discussion concentrates on the use of the diode array, with some comments on the use of a
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Speclrcmeler
Narrowband liller
I
1
I-water
coded mbe
Gas Row
FIG. 1. Schematic diagram of the hollow-cathode glow-discharge CVD and high-sensitivity white-light absorption spectroscopy apparatus. (Reprinted with permission from Menningen et a/., 1995a, Conirih. Plasma Phys. 35, 359, 1995 Wiley-VCH, Inc.)
CCD array at the end of the section. Typical photodiode arrays from Princeton Instruments and other manufacturers have individual diodes with a center-tocenter spacing width of 25 pm and a height of 2 mm. This height matches the height of the 1-to-I image of the arc in a Xe lamp. The center-to-center spacing of the photodiodes and the dispersion of the spectrometer determine the spectral resolution of the system. The use of a diode array rather than a sequentially scanned single-channel detector such as a photomultiplier has two major advantages. First, because all channels are detected simultaneously, data are obtained rapidly, leading to high signal-to-noise ratios provided that the experiment is limited by photon statistics. Second, fluctuations in the intensity of the light source appear in all channels in nearly the same way. This means that fluctuations in the intensity of the light source do not prevent the detection of weak absorption features, as they would if a sequentially scanned single-channel detector were used. A typical photodiode array has 5 12, 1024, or 2048 diodes that conduct current when they are illuminated. The quantum efficiency of the photodiodes typically ranges from 0.38 at a wavelength of 275 nm to 0.74 at 575 nm. The photodiode current discharges a charged capacitor that is connected in parallel with the photodiode. The charge required to recharge the capacitor is measured during the readout. The charge required for each diode is converted into a digital signal. The capacitor across a photodiode in a typical array is completely discharged, and hence the signal “saturates” when about 1.2 x 10’ photoelectrons have passed through the photodiode. The readout noise of each photodiode in the array is 1800 photoelectrons. One “count” of the 16-bit digitizer for the photodiodes is equivalent to 1800 photoelectrons, so that the readout noise is equal to fone count. Poisson statistical noise exceeds 1800 photoelectrons once the charge carried by the photocurrent exceeds 3% of saturation. The signal-to-noise ratio is about 1O4 for a single near saturated readout. Nevertheless, by digitally storing
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and summing repeated readouts of a spectrum, one can obtain signal-to-noise ratios of lo5 to lo6. The arc lamp used as the white-light source combined with a typical 0.5-m-focal-length spectrometer yields enough photons per second at wavelengths from 200nm to the near infrared to significantly exceed the dark current of 10 counts per diode per second at the operating temperature of the diode array, which is -40°C. Of course, precise electronic gating is essential in order to obtain a high signal-to-noise ratio. The gating signals are provided by the readout circuitry of the diode array. The diode arrays from Princeton Instruments have two different amplifiers, one for the readout of the odd diodes and the other for the even diodes. If the two amplifiers do not have the same gain, then an absorption spectrum will show alternating higher and lower absorbances for consecutive diodes. Slow drifts in the ratio of the gain of the amplifiers for the even and odd diodes results in alternating higher and lower signals for alternate diodes in every spectrum. This is overcome by normalizing the sum of the even diodes to be equal to the sum of the odd diodes for each of the repeated readouts before storing the readout digitally. This results in greatly increased signal-to-noise ratios and permits one to measure very small absorbances. It should be pointed out that it is not necessary to use an intensified array to measure small absorbances. Intensified arrays have an important advantage for certain low-light-level experiments, particularly in astronomy. Dark noise can be overcome even with very low light levels by using an intensifier. The disadvantages of an intensified array in a typical absorption experiment include extra expense, vulnerability to damage from “room” light levels, a reduced saturation fluence, noise from the intensifier, a reduced linearity, and a reduced spectral resolving power. Both glow discharges and hot-filament CVD reactors have been used for the growth of diamond films. We first analyze the use of a diode array in highsensitivity white-light absorption spectroscopy with a glow-discharge CVD reactor. The analysis follows that of Menningen et al. (1995a). The transmittance of a glow-discharge system is obtained as follows: Four spectra are recorded, all with the same integration time. The first spectrum is taken with the light from the arc lamp blocked off and with the glow discharge on. This spectrum is cr,E D, where aE is a constant of proportionality, E is the number of photons in the integration time that result from the light emitted by the glow discharge, and D is the number of electrons that pass through a diode as a result of dark current during the integration time. The dark current is due to thermal electrons rather than to photoelectrons. Second, a spectrum is taken with the light from the arc lamp passing through the CVD reactor and with the glow discharge on. This spectrum is given by cc,,(L - B E ) D, where aLE is a constant of proportionality and L - B E is the number of electrons due to the photons from the arc lamp during the integration time, L, minus the photons absorbed in the CVD reactor during the integration time, B, plus the photons due to the emission from
+
+
+ +
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the glow discharge during the integration time, E. Third, a spectrum is taken with the light from the arc lamp passing through the CVD reactor but with the glow discharge in the reactor off. This spectrum is given by ELL 0, where aL is a constant of proportionality. Finally, the fourth spectrum is taken with the light from the arc lamp blocked off and the glow discharge turned off. This spectrum records the number of electrons during the integration time due to dark current, D. After the four spectra are taken, the quantity
+
is calculated. The transmittance of the glow discharge is given by
T=-
L-B L
The quantity T* differs from T because the response of the diode array is not a perfectly linear function of the number of photons incident on the diodes, so that aLE,aL,and aE are not the same constants of proportionality. It can be shown that
Solving for T , one obtains
From this equation, it is easily seen that if aL, uLE, and aE were all identical, then T would be equal to T*. The absorbance of the glow discharge is given by
If we define A* = 1 - T*, then it can be shown that
The readout from the diode array directly yields T*, from which one obtains A*. The difference between A and A* is due to the nonlinearity of the response of the diode array, i.e., it is due to slight differences between aL and a L E . In an experimental measurement, B is usually much less than L (typically B 5 10p3L),and E is between B and L. Menningen et al. (1995a) have measured the constant of proportionality a as a function of the percent of saturation of the diodes in the array. The constant a decreases slowly as the percent of saturation increases (decreasing by about 2% as the percent saturation increases from 0 to 95%), and then decreases rapidly when the percent of saturation reaches 96%.
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Figure 2 shows measurements of tl from Menningen et al. (1995a). From their data, it is seen that if the saturation due to L is about 80% and if EIL = lop2, then
(5) 2:
1 - 2 x lop4, so that
Thus it is possible to measure an absorbance of lop2 with an accuracy of 10% without any correction for the nonlinearity of the response of the diode array. In order to be a little more quantitative, we use the data from Menningen et al. (1995a), in which tlLL is 65% of saturation and a,E is 0.1% of saturation. From their measurements of tl, we find that t l L / a L E= 1.000021, that (aLE- ctL)/ctLE = -2.1 x lo-', and that (aLE- ctE)/tlLE = -1.127 x lop2. Thus
1.02
1.01 1.oo ld
0.99
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0.98
-4
1.0
4
0.8 0.6< -
0
1 7
20
7 1
40 60 80 48 Saturation
- .-d
100
FIG. 2. Plot of the relative proportionality constant ct as a function of the saturation (65,536 counts). Each data point represents at least lo8 photoelectrons; thus the uncertainty due to Poisson The data were accumulated using a very stable incandescent lamp and various statistics is accurately gated integration times. Plot ( a ) is the result of 42 tests of 5 different diodes across the array. In each test, the number of counts for a given exposure was compared with the number of counts recorded for a reference exposure corresponding to approximately 50% of saturation. The a value drops suddenly beyond 96% saturation. Plot (b) depicts the same data as (a) on an expanded scale in order to show the rapid dropoff of a as the percent of saturation exceeds 96%. (Reprinted with permission from Menningen et al., 1995a, Contrib. Plasma Phys. 35, 359, 0 1995 Wiley-VCH, Inc.)
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30 1
In this situation, it is possible to measure an absorbance of 4 x lop4 with an accuracy of about 10% without making any correction for the nonlinear response of the diode array. Of course, if one measures the US, then with corrections the error in the absorbance can be due only to Poisson statistics. The difference between A and A* due to the nonlinearity of the diode array increases as EIL increases. The quantity EIL, which is the ratio of the spectral radiance of the glow emission to the spectral radiance of the arc lamp, sets a limitation on the measurement of the absorbance unless corrections are made for variations in a. Menningen et al. (1 995a) note that for molecular spectra, E I L is typically to lop3, and high sensitivity in the measurement of the absorbance is easily achieved. Wamsley et al. (1993) have noted that for strong atomic transitions in the visible or ultraviolet, E I L may be near unity, i.e., the spectral radiance at the wavelength of the line emission may be comparable to that of the arc lamp. In / L ) be on the order of a few this situation, the quantity [(aE- M L E ) / ~ L E ] ( Emay percent, so that one must measure the quantities aE and aLEand determine the correction if one desires to obtain an absorbance with an accuracy greater than a few percent. If one uses a synchrotron as the white-light source, then the value of L is about 1000 times greater than that for the arc lamp, and so EIL is very small even for strong atomic transitions and an absorbance can be measured with a small uncertainty without having to correct for the nonlinearity of the array. If the glow-discharge emission has a spectrum that is broad compared to the absorption feature, then even if the emission is relatively strong, the baseline of the divided spectrum is affected, but the absorption feature is not obscured, since all the diodes are affected in approximately the same manner. Thus the absorption due to CH, at 216nm is not obscured by the broad, relatively featureless absorption spectrum due to other hydrocarbon species that covers the region near 2 16 nm. This is discussed in detail in Section 111. A. We have discussed the problems associated with the nonlinear response of the diode array when it is used with a glow discharge or other radiating CVD reactor. The problems associated with the nonlinearity of the diode array when it is used with a nonradiating (thermally assisted) CVD reactor are almost nonexistent. The hot filament in a hot-filament CVD reactor emits essentially no radiation in the region near 200nrn. In this situation, EIL = 0 and uL = aLE,so that A = A* and the absorbance can be measured without any significant correction. Thus, for nonradiating CVD reactors, the nonlinearity of the diode array does not limit the measurement of the absorbance due to the gas-phase molecules or radicals. It should be noted that our analysis assumes that the absorbance is small, L M L - B, so that the value of CY is not changed due to the absorption. The use of a CCD array for measurement of the absorbance is similar to the use of a diode array. We briefly discuss the important differences. Charge-coupled device arrays have far lower levels of dark noise than diode arrays and proportionally lower levels of readout noise. A research-grade CCD array
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typically has a readout noise of 5 to 10 photoelectrons per pixel, whereas a diode array typically has a readout noise of 1500 to 2000 photoelectrons per diode. Bergeson et al. (1995) have used a back-illuminated, boron-doped CCD array, which has useful quantum efficiencies in the UV and VUV, in their highsensitivity absorption spectroscopy experiment on Fe+ ions in a glow discharge. Their experiment used a storage ring as a continuum source in the W a n d VW, and achieved spectral resolving powers of 300,000 with a 3-m-focal-length vacuum echelle spectrometer. The combination of VLJV wavelengths down to 150 nm and very high spectral resolving powers yielded low fluences on this array, hence the CCD array was advantageous. The CCD arrays are in general two-dimensional arrays, but this is not a disadvantage, since a column of pixels can easily be summed during readout. Bergeson et al. (1995) found a simple solution to the CCD gating problem recognized by Wamsley et al. (1993). A CCD array “rasters” the stored photoelectrons across the array during readout. This creates the possibility of “smearing” the spectrum and makes precise gating difficult. Wamsley et al. (1993) used an image intensifier to provide precise electronic gating of the exposures required for digital subtraction. For reasons described earlier, intensifiers are undesirable. Bergeson et al. (1 995) achieved the precise gating of various exposures by using a frame transfer from an unmasked to a masked part of the CCD array. This gating technique avoids all of the problems associated with intensified arrays. Because of the combination of a storage ring, a 3-m echelle spectrograph, and a state-of-the-art VW sensitive array, Bergeson et al. (1 995) achieved atom/ion detection limits of 3 x 10’ cm-2 in their experiment. White-light absorption spectroscopy is especially valuable for determining the absolute column density or the line integral of the density along the line of sight of the absorbing atoms, molecules, or free radicals in a chemical vapor deposition reactor. The intensity Z(v) of a light beam at a frequency v after it passes through a deposition reactor is given by I(v) = Zo exp { - J n(x)adx}, where Zo is the intensity of the beam incident on the reactor; n(x) is the density of the absorbing atoms, molecules, or free radicals as a function of the position x in the reactor; o is the optical absorption cross section at the frequency v; and dx is the element of the path length along the beam. The intensity of the light as it exits the reactor is obtained by evaluating the integral from the entrance to the output of the reactor. The absorption cross section is a rapidly varying function of the frequency, and hence the output intensity also depends on the frequency. The expression for Z(v) is I ( v ) S 1,(1 - Jn(x)adx) for situations where Jn(x)adx << 1. We use the notation J n(x)o dx = iioL, where ii is the average density along the optical path. The absorbance can be rewritten as
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The equivalent width of the feature is defined as W, = 1-d.1 Thus for the situation where iioL
1
<< 1,
s
W, = iioLdv = .nr,cJ;,iiL where r, is the classical radius of the electron, c is the speed of light, and&;, is the absorption oscillator strength of the transition. It is possible to calculate ii if one measures the equivalent width W,,. In this case, it is not necessary to know the spectral line shape function to obtain ii. It is also possible to obtain ii from a measurement of [I, - I(v)]/Ioat a particular value of v, such as the value of v at line center, if one knows the spectral line shape function. Both methods have been used to obtain fi. For example, in order to obtain ii for CH or Cz, measurements of W,, have been used, whereas in order to obtain ii for CH,, measurements of [I,- I(v)]/Ioat line center have been used. If iioL is not small compared to unity, then one must use a curve of growth analysis to obtain ii. In this paper, we discuss the determination of ii for several atoms, molecules, and free radicals in CVD reactors and processing plasmas.
111. The Uses of High-Sensitivity White-Light Absorption Spectroscopy in the CVD of Diamond Films As stated in the introduction, high-sensitivity white-light absorption spectroscopy has been used more extensively in studying the gas-phase composition during the CVD of diamond films than in other applications of CVD. In this section we review the use of high-sensitivity white-light absorption for the measurement of i for the various free radicals and molecules present in the CVD of diamond under different conditions. Diamond is transparent from the infrared to the ultraviolet. It is the hardest known material. It has the highest known thermal conductivity (at 300 K) of any material, but it is an electrical insulator. It also has the highest known bulk modulus and the lowest known compressibility. Synthetically grown diamond comes close to or matches natural diamond in these properties. Also, diamond is relatively inert chemically, making it useful in corrosive or reactive environments. Diamond has the same crystalline structure as Si, and thus can be doped for applications as a semiconductor. The extremely large band gap of diamond (5.5 eV) will make a diamond semiconductor diode or transistor tolerant of high temperatures and very resistant to thermal runaway. A diamond semiconductor is relatively insensitive to damage by ionizing radiation, so that it might be possible
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L. K Anderson, A.N. Goyette, and J E . Lawler
to use a diamond semiconductor in high-radiation conditions such as in a nuclear reactor. It may be possible to make an extremely short-wavelength diode laser using a diamond semiconductor. Because of the extraordinary physical properties of diamond and because of its possible use as a very-wide-band-gap semiconductor, the CVD of diamond is of great interest. There are a number of current or prospective applications for CVDgrown diamond, including uses as thermal conductors for heat transfer, scanning tunneling microscope tips, and x-ray windows, and in high-fidelity speakers. Studies of diamond CVD processes are also motivated by fundamental scientific considerations. Diamond growth occurs in nature and in traditional industrial processes under high-pressure and high-temperature conditions. A detailed understanding at a microscopic level is thus an important scientific goal of low-pressure-growth studies. The CVD of diamond can be carried out in a variety of ways. It is necessary to have a gaseous source of carbon and an activation mechanism. For example, diamond can be grown using a feed gas mixture of CH, (about 0.5% by volume) or other hydrocarbon species and H, or using fullerenes and Ar with a small fraction of H,. The activation mechanisms include the use of a hot filament, a dc glow discharge, a microwave discharge, an arcjet, and a flame. The chemical composition of the vapor phase is quite different for hydrogen-rich and hydrogendeficient feed gas mixtures. The chemical composition of the gas phase for a given mixture and pressure is relatively independent of activation mechanism. This section reviews the use of high-sensitivity white-light absorption spectroscopy to determine the vapor-phase densities of free radicals and molecules for various feed gases and activation mechanisms in these environments. High-sensitivity white-light absorption spectroscopy has been extensively used to study quantitatively gas-phase densities of free radicals and molecules during the CVD of diamond films. This technique has been used to characterize the deposition environments during the CVD of diamond films using hot filaments, dc glow discharges, microwave discharges, arcjets, and flames for gas activation. The densities of a number of different gas-phase molecules and free radicals have been determined from spectral features ranging from just above the vacuum ultraviolet cutoff through the visible. In this section, we discuss the results obtained using high-sensitivity white-light absorption spectroscopy by species. For each species, different activation methods and different input gases are compared in order to examine similarities and differences in the gas-phase chemistry during the CVD of diamond. OF CH, RADICAL DENSITIES A. MEASUREMENT
High-sensitivity white-light absorption spectroscopy was used in the detection of CH, and C,H, in the CVD of diamond by Childs et al. (1 992) in a hot-filament
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reactor used for the CVD of diamond films. The methyl radical, CH,, is thought to be a key precursor to nucleation and growth of diamond films in hydrogen-rich CVD environments. Measuring the absolute density of CH, in a CVD reactor is particularly challenging. As with many other polyatomic free radicals, the excited electronic levels of CH, all dissociate. Consequently, laser-induced fluorescence (LIF) measurements of CH3 are not possible. Prior to the work of Childs et al. (1992), the CH, radical had been detected using resonantly enhanced multiphoton ionization (REMPI) (Celii et al., 1989), infrared diode laser absorption spectroscopy (Celii et al., 1988), and near-threshold ionization mass spectrometry (Toyoda et al., 1988). Compared to these techniques, high-sensitivity absorption spectroscopy, even deep in the ultraviolet, is very simple. It is noninvasive, linear, and very sensitive, and it provides absolute densities without difficult calibration. REMPI is a nonlinear technique and hence is difficult to calibrate. Infrared diode laser absorption spectroscopy is difficult because it probes transitions with absorption cross sections that are orders of magnitude smaller than those of the electronic transitions probed by high-sensitivity white-light absorption spectroscopy in the ultraviolet and visible. As a result, infrared diode laser absorption spectroscopy is a much less sensitive technique. Near-threshold ionization mass spectrometry involves extracting and analyzing the gas from the CVD system, with the possibility that the sampling affects the gas composition. REMPI and near-ionization mass spectrometry are both difficult to use in discharges in which the gas is partially ionized. A schematic diagram of the reactor used by Toyoda et al. (1994) is shown in Fig. 3. They used eight hot tungsten filaments in parallel and a gas manifold to spread the input gas flow uniformly across the filaments. The 0.25-mm-diameter tungsten filaments were 1.9 cm long, 1.6 cm apart, and located 1.5 cm from a Si (100) substrate. The filaments operated at 2600 K. The substrate was 9.5 cm long and 1.3 cm wide and was resistively heated to 1100 K. The feed gas contained 1% CH4 diluted in H, at a total pressure of 20 torr and had a flow rate of 100 sccm. Diamond growth was confirmed both by observing the diamond crystals using scanning electron microscopy and by detecting in the Raman spectrum of the films at the 1332 cm-’ line characteristic of diamond. The high-resolution absorption spectrum of the B(,A‘,) t X(,A’,’) transition in CH, (Herzberg and Shoosmith, 1956) shows two main features at 215.7nm and 216.3 nm but does not show any significant rotational structure as a result of the rapid dissociation of the CH, excited electronic level. Childs et al. (1992) were able to resolve the two features in the CH, absorption spectrum with a spectral resolution of 0.12 nm, which was about 3 times the theoretical resolution limit of 0.04nm for their instrument. For most measurements, however, Childs et al. (1992) used the spectrometer at a lower spectral resolving power with the input slits opened up and with the averaging of neighboring pixels, since the width of the CH, absorption feature at 216.3 nm is 1.2nm. Although the two features of the CH,
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L. K! Anderson, A N Goyette, and J E . Lawler
FIG. 3. Schematic diagram of the hot-filament CVD and high-sensitivity whte-light absorption spectroscopy apparatus. (Reprinted with permission from Toyoda et al., 1994, J Appl. Phys., 75,3 142 8 1994 American Institute of Physics.)
absorption spectrum near 2 16nm are not resolved, the sensitivity of the system is increased significantly, as is shown in Fig. 4. The spectrum shown in Fig. 4 has a comparatively narrow CH, absorption feature near 216nm superposed on a broadband absorption. The value of i L is obtained by dividing the peak value for the absorbance by the absorption cross section. The geometry produces a welldefmed value of L of 12.7 cm. The value of i was determined with an accuracy of f20%. The largest source of uncertainty was due to the broad absorption, which makes the baseline uncertain. By estimating the smallest signal that could be detected with the noise and on the sloping baseline, they reported a detection limit of 2 x 10" CH, radicals/cm3. 1.ooo
0.982
! .
190
260
2iO
220
1
230
Wavelength (nm) FIG. 4. Typical CH, absorption profiles for ( a ) 2.0% CH, in H, with a spectral limit of resolution of 0.12 nm,and (b) 4.0% CH, in H, with a spectral limit of resolution of 0.76nm. Both spectra were taken with a filament temperature of 2600 K. (Reprinted from K. L. Menningen et al., Chem. Phys. Lett., 204, 573, 0 1993, with permission from Elsevier Science.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
307
Childs et al. (1992) measured the CH, density as a function of the fraction of CH, in the input gas and found that the CH, radicals increased linearly from 0 in pure H2 to about 3 x l o i 3 CH, radicals/cm3 at 4% CH, in the input feed gas and were nearly constant at 3 x lo’, CH, radicals/cm’ as the CH, fraction in the input gas increased from 4 to 7%. Diamond growth occurs only for CH, fractions less than 2%, in the input gas. For CH, fractions greater than 2%, other forms of carbon are grown rather than diamond films. It is clear that something other than low CH, density causes the diamond growth to cease for input CH, fractions above 2%. These results of Childs et al. (1992) on CH, density were very similar to the results obtained by Hsu (1991) using near-threshold ionization mass spectrometry. Following the work of Childs et al. (1992), high-sensitivity white-light absorption spectroscopy was used for further studies of the CH, density in diamond CVD reactors. Menningen et al. (1993) also studied CH3 production in a hot-filament CVD system. They measured the CH, density as a function of the filament temperature for a feed gas with different fractions of CH, diluted in either He or H,. Their apparatus was very similar to that of Childs et al. (1992) except that a single long filament was used. The filament was kept under tension so that it did not sag when heated. The filament I - V characteristics were stabilized by a carbonization treatment in which the filament was heated to a temperature of 2700K in 7% CH, for 1 h. Figure 4 shows a typical absorption profile of the B(,A’,) t X(2A’,’) transition of CH, as a function of wavelength from Menningen et al. (1993). Menningen et al. (1993) also saw absorption due to acetylene, C2H2,at wavelengths to the blue of the CH, features. Their results, which are shown in Fig. 5, were presented as an Arrhenius plot of the CH, density as a function of 1/T, where T is the filament temperature. The filament temperature was measured using a single-color optical pyrometer. The CH, density reached a maximum near T = 2500 K for a feed gas with 2% CH, in H,, T = 2400K for 1% CH, in H,, and 2300K for 0.5% CH, in H,. At higher filament temperatures, the CH, density decreased, probably because of hydrocarbon filament poisoning, which reduced the ability of the hot filament to dissociate H,. For filament temperatures less than 2700 K and for a feed gas of CH, diluted in He, the CH, density was an order of magnitude lower than when CH, diluted in H, is used. The conclusion was that most of the CH, was produced by dissociation of H, on the filament, forming atomic hydrogen, followed by abstraction reactions of the H with the CH, rather than by thermal dissociation of CH, on the filament. Thus the abstraction reaction H CH, t, H, CH, is the primary source of CH, radicals. From the Arrhenius plot for temperatures less than 2300K, it was found that the activation energy for CH, production ranges from 1.67 eV for 4% CH, in H, to 1.82 eV for 2% CH, in H,. Finally, they studied the production of CH, when tert-butyl peroxide [(CH,),-COOC-(CH,), , abbreviated as TBP] replaced CH, as the
+
+
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L. W Anderson, A.N. Goyette, and J E . Lawler
-
?
10
'
4
L
tl
**
A A
** o * A
0 1.0% CH4 in Hg A 0.5% CHI in Hg
4.0
4.5
1/T
5.0
(lo-*
5.5
E
0
K-l)
FIG. 5. Arrhenius plot of CH, concentration versus filament temperature for (A) 0.5%. (0)1 .O%, and (*) 2.0% CH, in H, and (*) 2.0% CH, in He. Activation energies were obtained from data for filament temperatures below 2300 K. (Reprinted from K. L. Menningen et al., Chem. Phys. Lett., 204, 573, (01993, with permission from Elsevier Science.)
carbon precursor in the feed gas. It was thought that the large number of CH, radicals attached to TBP might produce a significant fraction of CH, by thermal dissociation on the filament, but this was not the case. The CH, densities for TBP diluted in H, were comparable to the CH, densities for CH, diluted in H,. As was the case with CH, the CH, density was much higher for about 1% TBP diluted in H, than for TBP diluted in He. Thus, for TBP, abstraction reactions were the primary source of CH, for temperatures below 2500K. The effective activation energy for the production of CH, with TBP was measured to be 1.69eV/ molecule. The data of Celii and Butler (1991) yielded activation energies nearly a factor of 2 smaller than those reported by Menningen et al. (1993). Measurements of the CH, density with a feed gas of C,H, diluted in H, have been conducted by Toyoda et al. (1994) in a hot-filament CVD reactor. The CH, density as a function of the fraction of C,H, in the input gas is more strongly affected by poisoning of the hot filament than for CH, diluted in H,. For a filament temperature of 2260K, the CH, density increased from near zero for a very low C,H, fraction in the input gas to about 2 x lo', cmP3 for a C,H, fraction of 0.5%, and then, decreased probably because of filament poisoning, to near zero for a C2H2fraction of 1.5%. They also found that at the same filament temperature, when the hot filament was not poisoned, the CH, density using C,H, in the input gas was about the same as the CH, density using the same carbon mole fraction of CH, in the input gas. The only difference was that the hot filament became poisoned at a lower carbon mole fraction for C2H2than for CH,.
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
309
The effect of filament poisoning also appears in measurements of Toyoda et al. (1994) on the time dependence of the CH, density after C,H, begins flowing into the CVD reactor chamber. For an input mole fraction of 0.5% C,H, in the feed gas, the CH, density increased monotonically to its steady-state value of 2 x lo', cmP3. In contrast, at higher mole fractions of C,H,, the time-resolved CH, dependence was quite different. For example, for a 2% mole fraction of C2H2,the CH, density increased to approximately 3 x lo', cmP3 after the C,H, had been flowing for 5 rnin and then decreased to nearly zero after 25 min as the hot filament became poisoned. The mechanism for CH, production when the feed gas was composed of C2H2diluted in H, probably included the dissociation of H, into atomic hydrogen at the hot filament, followed by reactions that add hydrogen to C,H, until a C2H5 radical results. The reaction C2H5 H ++2CH3 then results in the production of CH,. Clearly, if poisoning eliminates the active surface area of the hot filament where H, is dissociated, then CH, is not produced. It is also possible that some C,H, is dissociated on the hot filament, followed by the reactions that produce CH,. Again, if poisoning eliminates the active surface area of the hot filament where C,H, is dissociated, then CH, is not produced. Toyoda et al. (1 994) concluded that filament poisoning is important at low filament temperatures and/or high input fractions of C,H,. In a related experiment, Menningen et al. (1 994) studied the effect of substrate pretreatment on the time evolution of CH, in a hot-filament CVD reactor in which the feed gas was 5% CH, in H, for the first 30 min followed by 0.5% CH, in H2 for about an hour. In this situation, the CH, density rapidly increased to about 3 x lo', cmP3 and then slowly decreased to about 2.5 x lo', cmP3 after 30 min. Then, when the CH, fraction in the input feed gas was decreased to 0.5%, the CH, density fell rapidly to about 6 x lo', cm-,, where it remained. In another experiment, the input gas was 3% C,H, in H, for 30min followed by 0.36% C2H2 in H, for about an hour. Under these conditions, the CH, density rose rapidly to 2 x lo', cmP3 when the C,H, flow began. It then fell to about 2 x 10'2cm-3 within 10min. When the C,H, concentration was decreased after the first 30min the CH, density rose rapidly to 1.8 x lo', cmP3 in about 5 min and then decreased to a value of 8 x 10I2cm-, after another 5 or 10 min. These experiments demonstrated clearly that the hot filament is poisoned in the pretreatment process when C,H, is used. They also showed that the surface poisoning of the filament is somewhat reversible. Bulk carbonization of the filament tends to be much more permanent. In addition to studies using CH, or C,H, diluted in H,, the CH, density has been measured and diamond growth studied for other hydrocarbons in the feed gas. Menningen et al. (1995b) have carried out diamond growth in a hot-filament CVD reactor with a feed gas composition of CH,, C,H,, C2H4, or C,H, diluted in H,, and also with CH4/02 diluted in H,. They concluded that the identity of the carbon precursor in the feed gas does not affect the growth of the diamond
+
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L. W Anderson, A.N. Goyette, and JE. Lawler
film or the CH, density. Only the input carbon mole fraction affected the diamond growth or the CH, density. The chemical reactions set in motion by the atomic hydrogen produced by dissociation of H, at the hot filament produce a wide variety of carbon-containing free radicals and molecules. Similar gas-phase CH3 densities, diamond growth rates, and diamond film morphologies were observed using similar carbon mole fractions of CH,, C2H,, C2H4, or C3H8.It is clear that gas-phase reactions involving CH,, CH,, CH,, CH, and H proceed quickly enough to reach partial equilibrium, thereby completely scrambling the identity of the original carbon precursor. No particular carbon precursor provided a significant advantage for CH, production or for diamond growth. This is an especially important result. Addition of 0, to the feed gas mixture was also investigated. The role of 0, in the CH,/02 mixture is interesting. It was found that the presence of only a few percent by volume of 0, suppressed the CH, density as a result of gas-phase reactions of 0, with the CH,. Typically the CH, densities were reduced by a factor of 2 or 3 as a result of reactions with 0,. The 0, also reacted with hydrocarbons on the filament to suppress filament poisoning. In addition, the density of CH, in both dc glow discharges and microwave discharges has been examined. Childs et al. (1994a) have used high-sensitivity white-light absorption spectroscopy to obtain the first measurements of the CH, density in a hollow-cathode glow-discharge CVD reactor. The detection of whitelight absorption in a glow discharge is challenging, since the discharge may have very intense emission at the same wavelength as the absorption feature of interest. As discussed in Section I1 of this paper, the detection of the white-light absorption is straightforward provided that the spectral radiance of the lamp is greater than or equal to the spectral radiance of the discharge at the wavelength of the absorption feature. The hollow cathode was made of a 3-mm-diameter Ta tube and was located 1.5 cm below the Si (100) substrate upon which the diamond film was grown. Childs et al. (1994a) measured the CH, column densities 1.2 cm above the hollow cathode. They estimated the path length from the area on the Si (100) substrate that was affected by the discharge in order to extract CH, densities. They found that the density of CH, was about 2.7 x lo', cmP3 for 1% CH, diluted in H, at 30 ton: This is comparable to the density of CH, in a hot-filament CVD reactor under similar conditions of feed gas pressure, flow, and composition. They also found that the CH, density decreased by about 30% from the hollow cathode to the substrate. Childs et al. (1994a) also measured the temperature distribution and the CH, density for the hollow-cathode CVD reactor. Menningen et al. (1995a) followed up the work of Childs et al. (1994a) with more complete measurements of the CH, density in the same hollow-cathode glow-discharge CVD reactor. Their apparatus is shown schematically in Fig. 1. They obtained absolute CH, column densities from the transmittance as follows: The transmittance is given by T = Z/Zo = exp (-%L), so that one obtains the column density by dividing the natural logarithm of the
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
3 11
minimum transmittance by the peak absorption cross section. The peak absorption cross section for CH, has been measured by Glanzer et al. (1977) and Moller et al. (1986). A least-squares fit to the high-resolution data of Glanzer et al. (1977) yields a(T) = 3.88 x exp(-T/969.6)cm2, where T is the neutral gas temperature in kelvin. The gas temperature was obtained from rotational temperature analysis of H, as described in Section 1II.F of this paper. Childs et al. (1994a) also took into account that the CH3 absorption feature is superposed on broad background absorption partially due to other hydrocarbon species. The CH, column density for a discharge current of 1 A rises from 0 at 0% CH, in the feed gas to about 2.6 x lo1, cm-* at 1.5% CH, in the feed gas and was constant from that level up to 2.5% CH, in the feed gas. At a discharge current of about 0.7A, the column density is about 1.6 x lo', cm-, at 1.5% CH, in the feed gas. Menningen et al. (1 995a) also found that the CH, density in a hollow-cathode CVD reactor was about the same as in a hot-filament CVD reactor. As shown in Fig. 6, the CH, density in a hollow-cathode reactor was large even at high enough CH, fractions in the feed gas that diamond cannot be grown. It is clear that the CH, density is not the limitation that prevents diamond from being grown for CH, fractions higher than about 2% in the feed gas in a hollow-cathode CVD reactor. Erickson et al. (1996) have measured the CH, column density for a microwave CVD reactor with a feed gas of CH4 diluted in H2. A schematic diagram of their apparatus is shown in Fig. 7. The temperature of the discharge was determined to
0
1
2
3
Input CH, mole fraction (%) FIG. 6 . Absolute CH, column density measured 12 mm above the cathode as a function of input CH, mole fraction. The solid symbols are recorded for a discharge current of l.OA, and the open symbols correspond to a discharge current of 0.65 to 0.74A. The solid and dashed lines are to guide the eye. (Reprinted with permission from Menningen et al., 1995a, Contrib. Plasma Phys. 35,359, 0 1995 Wiley-VCH, Inc.)
312
L.
Anderson, A.N. Goyette, and JE. Lawler
FIG. 7. Schematic diagram of the microwave discharge CVD apparatus. (Reprinted with permission from Erickson et al., 1996, Plasma Sources Sci. Technol.,5, 761, 0 1996 IOP Publishing, Ltd.)
be about 1200K from the rotational analysis of the G'Z; -+BIZ: emission band of H,. They measured the transmittance, and, using the peak CH, absorption cross section at 1200K and a path length estimated to be about 6 cm, they obtained a CH, density. The CH, density was measured as a function of the CH, fraction in the feed gas. The results of their measurement are shown in Fig. 8. The CH3 density is 6 x lo', cm-3 at 1% CH, in H,. These results are similar to the results for a feed gas of CH4 diluted in H, in both hot-filament and hollow-cathode CVD reactors.
FIG. 8. CH, density versus input CH, mole fraction in a microwave CVD reactor. (Reprinted with permission from Erickson et al., 1996, Plasma Sources Sci. Technol., 5, 761, 0 1996 1OP Publishing, Ltd.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
3 13
Loh and Capelli (1997) have measured the CH, density in a supersonic arcjet during the growth of diamond films. The arcjet used CH, diluted in H2 for the feed gas. The spectroscopy system used was similar to that used by Childs et al. (1992) except that a CCD array rather than a diode array was used for the detector. They measured the transmittance near 216.3 nm and obtained the column density by dividing the logarithm of the transmittance by the peak absorption cross section. The column density was obtained as a function of the CH, fraction in the feed gas, and as a function of the process pressure. At 2% CH, in the feed gas, they found that the CH, column density was about 4 x lo’, cmP2. The CH, densities were used to obtain a “sticking” coefficient under diamond growth conditions. for CH, that was on the order of This section has discussed the use of high-sensitivity white-light absorption spectroscopy to measure the CH, density in various CVD reactors used for diamond film growth. It is clear that this technique is useful for detecting low densities of free radicals that do not radiate. The CH, radical is thought to play a key role in the CVD of diamond. Using high-sensitivity white-light absorption spectroscopy, it has been demonstrated that the CH, density is relatively independent of the particular carbon precursor in the feed gas. The CH, density is not the primary limitation on the diamond film growth rate with a “carbonrich” feed gas mixture, since the CH, density increases as the carbon content in the feed gas increases up to a carbon content much above the range of carbon contents for which diamond growth occurs.
B. MEASUREMENT OF CH
RADICAL
DENSITIES AND [H]/[H2] RATIOS
Measurements of the density of a number of free radicals and molecules other than CH, have been made using high-sensitivity white-light absorption spectroscopy. Childs et al. (1992) made the first high-sensitivity white-light absorption spectroscopy observation of the CH free radical in a glow discharge. They obtained the CH column density as a function of position between the hollow cathode and the Si substrate. The apparatus used was identical to that used by Childs et al. (1994a) to study CH, except that the system was used to measure white-light absorption near 432 nm or 3 14 nm in order to study the A 2 A t X 2 n (0,O) or C2C+ t X211 (0,O) bands of CH, respectively, with a spectral resolution of 0.12 nm. Figure 9 shows the emission and absorption spectra near 432 nm due to the A 2 A t X 2 H (0,O) band, in which both the R and Q branches are observed. They found that the CH column density decreased from about 2.5 x 10l2cm-2 at 1 mm above the hollow cathode to 0.5 x 10l2cmP2 at 5 mm above the hollow cathode. They also measured the CH, column density and found that the CH, column density decreased more slowly as a function of position above the hollow cathode than did the CH column density. As shown in Fig. 9(a), absorbances as
3 14
L. FT Anderson, A.N. Goyette, and J E. Lawler
1
.-
I
1 .GQOOo
0.99995
0.99990
r 4
0.99985 0.4 h
0.2 0.2
t
F
0.0 420
425
430
435
440
Wavelength (nm) FIG. 9. ( a ) Typical absorption spectrum of the A 2 A t X211 (0,O) band of CH, taken at 2 mm from the cathode with 1.O% CH, in H, used as the feed gas. (b) Typical line emission spectrum from the discharge under the same conditions as (a). The strong line at 434nm is Hy. (Reprinted with permission from Menningen et al., 1995%Contrib. Plasma Phys. 3 5 , 3 5 9 , o 1995 Wiley-VCH, Inc.)
small as a few parts per million were detected. Absorbances as small as a few parts per million have also been measured in a hot-filament CVD reactor. This is remarkable sensitivity in the presence of a discharge emitting radiation at the wavelengths of the absorption band. Clearly, high-sensitivity white-light absorption spectroscopy will have numerous important applications because of this extraordinary sensitivity. Childs et al. (1994b) have measured both the CH, and CH column density in a hot-filament diamond film CVD reactor. They measured the CH column density using the CH absorption spectra due to the C2Z+ t X 2 n (0,O) band near 3 14nm, and the CH, column density from the 2 16nm feature as described in the previous section. They obtained CH, and CH densities by dividing the column density by the path length, which was taken to be the same as the filament length. The CH column density was obtained by measuring the equivalent width of the Q bandhead over a wavelength interval containing a known set of rotational lines (Moore and Broida, 1959). The column density was determined using the gas temperature, the measured equivalent width, and the known band oscillator strength. They took the gas temperature as varying linearly from 1200K at the substrate to 500 degrees below the filament temperature just above the filament. Their results were relatively insensitive to the gas temperature. The CH density is relatively insensitive to the temperature because many rotational lines in the Q branch are included, and the CH, is relatively insensitive to the temperature because the peak CH, absorption cross section changes by only about 0.1% per degree.
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
3 15
Childs et al. (1994b) used their measurements of the CH and CH, densities to obtain the hydrogen dissociation ratio, [H]/[H,]. This can be obtained because the reactions CH,
+ H * CH + H,
CH3 + H
* CH2 +H2
are very fast. The equilibration times for the reactions are 10 and 20ps, respectively, for the experimental conditions. During 20ps, CH, CH,, or CH, diffuse only about 1 mm, so the [H]/[H,] ratio is an average over about 1 mm. The model of Goodman and Gavillet (1990, 1991) showed that the abstraction reactions are in equilibrium in the gas phase under typical diamond CVD conditions to within about 1 mm of surfaces. Childs et al. (1994b) calculated the equilibrium constant keq for the reaction CH,
+ 2H + CH + 2H2
from the tabulated thermodynamic data and the gas temperature. The dissociation ratio is given by
The CH, density can be calculated from the measured CH, density and the H ++CH, H, is dissociation ratio [H]/[H,]. The equilibration time for CH, about 300ps, so that the CH, density is obtained with about a 4-mm spatial resolution. The 4-mm spatial resolution is determined by the diffusion length of CH, in 300ps. Figure 10 shows the results of Childs et al. (1994b) for the densities of CH,, CH,, and CH and the dissociation ratio [H]/[H,] as functions of distance from the filament for a feed gas of 1% CH, diluted in H,. The CH and CH, densities have absolute uncertainties of about 35 and 50%, respectively, and relative uncertainties of about 15 and 20%, respectively. Childs et al. (1994b) also gave determinations of CH,, CH,, and CH densities and the dissociation ratio [H]/[H,] as functions of the input CH, fraction and the filament temperature. An interesting result was the clear demonstration that input fractions of CH, greater than 1.5% result in a severe reduction of the dissociation ratio [H]/[H,]. The reason for this is probably filament poisoning. As discussed in the previous section, filament poisoning is probably the primary reason why diamond growth does not occur for higher CH, fractions. The CH, densities do not decrease for CH, input fractions greater than 1.5%. The reduction of atomic hydrogen, which preferentially etches the graphitic phases, is probably why diamond growth does not occur for higher CH, input fractions.
+
+
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L. K Anderson, A.N. Goyette, and 1E.Lawler 10
10
15
10
10
1
10
'3
10
12
;
10
11
2
10
10
-
5 .4
.2
N
X
710
-l
c
X
10 10
-z
-3
0
z X
10 Distance from filament (mm)
FIG. 10. [H]/[H,] ratio (0)and CH4 density (0) derived from the measured CH, density (A) and CH density (X) as a function of distance from the filament for an input CH, mole fraction of 1 .O% and a filament temperature of 2500 K. (Reprinted from M. A. Childs et al., 1994b, Phys. Lelf. A , 194, 1 19, 0 1994, with permission from Elsevier Science.)
Menningen et al. (1995b) have studied the effects of oxygen on the CH density by adding a small amount of 0, to a feed gas composed of a hydrocarbon diluted in H, in a hot-filament CVD reactor. They found that at higher filament temperatures, the CH, and CH densities are lower when 0, is added to the feed gas than when there is no 0, present. They proposed that this is due to rapid reactions that form CO and thereby tie up some of the carbon. Menningen et al. (1995a) have studied the CH and CH, densities in a hollowcathode dc-discharge CVD reactor. They observed the white-light absorption spectra of CH using either the A2A t X 2 n (0,O) band near 432nm or the C2Zi t X 2 n (0,O) band near 314nm as a function of the position with a spectral resolution of 0.04nm. Figure 9 shows a typical spectrum of the A 2 A t X211 (0,O) band. Also shown is the CH emission from the discharge. It is remarkable that the very small absorption can be detected in the presence of the emission due to the same band. The equivalent width W,, of the Q branch bandhead was obtained by integrating each spectrum over a spectral interval containing a known number of rotational lines. The band oscillator strength is the sum of the oscillator strengths for each rotational line in the Q branch weighted by the Boltmann statistical fraction of the CH molecules in the lower rotational level. For a discharge current of 1 A and a feed gas composed of 1% CH, diluted in the H,, the CH column density decreased from about 2.5 x 10" cm-' at a location 2 mm above the hollow cathode, to about 2 x 10" cm-, 7 mm above the hollow cathode, as shown in Fig. 11. Absolute densities of CH were determined using an Abel inversion of a fit to the spatial dependence of the column density.
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
3 17
Distaace from cathode (mm)
FIG. 11. Spatial profile of the measured CH column densities for a feed gas of 1.O% mole fraction of CH, in H, as measured by (0)the A 2 A c X211 (0,O)band and (0) the C2Zf c X211 (0,O)band of CH. (Reprinted with permission from Menningen et al., 1995a, Contrib. Plasma Phys. 35, 359, 0 1995 Wiley-VCH, Inc.)
Figure 12 shows the CH density as a function of the position between the hollow cathode and the Si substrate. Menningen et al. (1995a) compared the spatial distributions of the CH, and CH densities in hot-filament and hollow-cathode glow-discharge CVD reactors. The results of their comparison are shown in Fig. 13. Both the CH and CH, densities are slightly higher in the hollow-cathode than in the hot-filament reactor. The CH, density decreases gradually between the hollow-cathode/hot-filament and the substrate. In contrast, the CH density decreases rapidly as a function of the distance from the hollow cathode or hot filament. The CH, and CH densities were used to obtain the hydrogen dissociation ratios [H]/[H,] as a function of the position between the hollow-cathode/hot-filament and the substrate. The results
Distance fmm cathode (m)
FIG. 12. Spatial profile of the absolute CH density resulting from an Abel inversion of the measured CH column densities. The column densities were smoothed prior to the Abel inversion. (Reprinted with permission from Menningen et al., 1995a, Contrib. Plasma P h p . 35, 359, 0 1995 Wiley-VCH, Inc.)
318
L. kV Anderson, A.N. Goyette, and J E . Lawler T7
Distance from cathode (mm)
FIG. 13. Comparison of CH, and CH densities as a function of position (distance from the cathode or filament) in the hollow-cathode (open symbols) and hot-filament (closed symbols) CVD systems. The CH3 densities in the hollow-cathode system were obtained by dividing the measured column densities by an estimated path length of 2.5 cm. (Reprinted with permission from Menningen et al., 1995a, Contrib. Plasma Phys. 35, 359, 6 1995 Wiley-VCH, Inc.)
are shown in Fig. 14. The [H]/[H,] ratio was about 0.08 at a position 1 mm above the hollow cathode and decreased rapidly in a hollow-cathode CVD reactor. In a hot-filament CVD reactor, the [H]/[H2] ratio was 0.02 at a position 2 mm above the filament and then decreased gradually, being about 0.0 15 at 9 mm from the filament. Erickson et al. (1996) tried to detect CH absorption in a microwave CVD reactor using CH4 diluted in H, as the feed gas. They were unable to detect the CH absorption. Using the detection limit of their system, they placed an upper limit of 10" ~ r n -on~ the CH density. From this and their measured CH, density, they placed an upper bound on the hydrogen dissociation ratio [H]/[H2] of 0.008.
0.08
0.00
1
=
-
0
3
6
9
1 2 1 5
Distance (m)
FIG. 14. Derived [H]/[H2] ratio in the hollow-cathode (open squares) and hot-filament (filled squares) systems as a function of distance from the cathode or filament. (Reprinted with permission from Menningen et al., 1995a, Contrib. Plasma Phys. 35, 359, 0 1995 Wiley-VCH, Inc.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
3 19
High-sensitivity white-light absorption spectroscopy has been used to measure the CH densities in various CVD reactors. These measurements show remarkable sensitivity. The CH and CH, densities have been used to obtain the dissociation ratio [H]/[H2].
OF C2H2 DENSITIES C. MEASUREMENT
The acetylene molecule, C2H2,was detected using high-sensitivity white-light absorption spectroscopy in the first experiments using the technique with a diamond hot-filament CVD reactor (Childs et al., 1992; Menningen et al., 1993; Toyoda et af., 1994). The ultraviolet absorption spectrum of C2H2was detected simultaneously with that of CH,. The ultraviolet absorption spectrum of C2H2 appears most prominently as a series of fluctuations in the transmittance spectrum in the region between 190 and 205 nm. An example of such a transmittance spectrum is shown in Fig. 15. To determine the C2H2density directly from these absorption features requires careful and detailed analysis, since these features contain complex unresolved rotational and vibrational structures. In addition, the structure of the transmittance spectrum changes significantly as a function of both the substrate temperature and the filament temperature. At higher temperatures, higher vibrational levels of the molecule become populated, giving rise to the appearance of additional subbands and creating an extremely complicated and, for these experiments, unresolved vibrational structure. For the experiments of Toyoda et al. (1994), neither the absorption cross section nor the detailed gas kinetic temperature distribution in the CVD reactor was known. The temperature
FIG. 15. Typical absorption profile with the CH, absorption at 216nm and the C,H, absorption at 194nm. (Reprinted with permission from Toyoda et al., 1994, 1 Appl. Phys., 75, 3142, 0 1994 American Institute of Physics.)
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L. J Y Anderson, A.N. Goyette. and 1E.Lawler
distribution in the CVD reactor is especially important because C2H2is a stable molecule and may exist throughout the reactor, not only in the regions where the reactive chemistry takes place. This is in contrast to the situation for a free radical such as CH,, which reacts quickly before it can diffuse from the region where it is produced and thus exists only in the reactive regions of the chamber. Toyoda et al. (1994) calibrated the C,H, absorption feature at 194nm in the following manner. They assumed that for high C2H2 concentrations, the hot filament was completely poisoned, so that no atomic hydrogen was produced by dissociation at the filament. In this situation, reactions between atomic hydrogen and C,H, do not occur, and the C2H2mole fraction is determined by the input fraction of C,H2 and the total pressure in the CVD reactor. They tested this assumption by measuring the absorbance at 194 nm for different filament temperatures while holding the substrate temperature constant as the input mole fraction of C2H2 was changed under conditions for which the filament was poisoned. At a constant hot-filament temperature, the magnitude of the absorbance at 194 nm was found to be directly proportional to the input C,H2 mole fraction. Their results are shown in Fig. 16. The linearity of the absorbance as a function of the C,H2 mole fraction confirmed that the filament was poisoned. By extrapolation to lower absorbances, they were able to calibrate the dip in the absorbance at 194 nm to yield the C,H, mole fraction for given hot-filament and substrate temperatures for situations in which the filament is not poisoned. From Fig. 17, Toyoda et al. (1 994) determined the absorbance per mole fraction of the 194 nm feature as a function of the filament temperature.
Percentage Cz H2 FIG. 16. Plot of the C,H, absorbance at 194nm versus input C2H2 mole fraction for filament 1925 K, ( + ) 2149K, (0)2329K, and (A)2515K.(Reprinted with permission temperatures of (a) from Toyoda et al., 1994,J Appl. Phys., 75, 3142, 0 1994 American Institute of Physics.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
32 1
z ! z0.2 L
z
I 1800
n 40.0
-
,
8
,
-
2000
1
I
1
I
2200
.
I
,
.
2400
. .21
Filament Temperature (K
10
FIG. 17. Absorbance per mole fraction of C,H, for the absorption feature at 194nm versus filament temperature. (Reprinted with permission from Toyoda et al., 1994,J Appl. fhys., 75, 3142, 6 1994 American Institute of Physics.)
If significant temperature gradients exist, thermal diffusion can cause the gas mixture to separate, so that it has a different composition at different locations. Toyoda et al. (1994) analyzed the possible separation of the gas in their hotfilament CVD reactor. They found that if their system were static, then significant thermal separation would occur. This was not the case for their reactor, however, and the flow of fresh feed gas was sufficient so that the C2H2 concentration differed by no more than 10% throughout their CVD reactor despite the presence of large thermal gradients in the CVD reactor. Toyoda et al. (1994) made several interesting observations based on their C,H, measurements. They found that for a feed gas of C2H2diluted in H,, there was a detectable amount of CH, only when the C,H, was consumed, that is, when the C,H, mole fraction was reduced from its value in the feed gas. The C,H, is consumed in chemical reactions driven by atomic hydrogen when the filament is not poisoned. They found that the total carbon content in CH, is much less than that lost from C2H2 and that much of the C,H, must be converted into species other than CH,, such as C,H,, CH,, or other hydrocarbon species. They also studied the time-resolved evolution of C,H, in their diamond CVD reactor, the results of which are shown in Fig. 18. The C2H2concentration increases up to approximately the concentration of C,H, in the feed gas if the C2H2 mole fraction in the feed gas is sufficient to poison the hot filament. This was the situation for a C2H2fraction in the feed gas greater than 1.5%. For input fractions less than this, the hot filament is not poisoned, and the C2H2 concentration remains less than 0.5% at all times as a result of the reactions with atomic hydrogen. In order to understand the data shown in Fig. 18, it is necessary to
322
L. FV Anderson, A.N. Goyette, and LE. Lawler
N
3:1.0 0
0.01
0
-
a
1
.
.
8
I
I
20 40 Time (minute)
.
.
I
60
FIG. 18. Evolution over time of the C2H2mole fraction for (0)2.0%, (0)1.5%, and (A) 1.0% C2H, input mole fraction. The C2H, mole fraction was below detection limits for 0.5% C,H2 input mole fraction. The apparent overestimation of the C2H, mole fraction at some times arose from random fluctuations over the long period during which each series of measurements was taken. (Reprinted with permission from Toyoda et al., 1994,J Appl. Phys., 75, 3142, 0 1994 American Institute of Physics.)
know that the l / e time for filling the chamber with C,H, when the hot filament is poisoned is about 30min. Thus, even at high input mole fractions, the C,H, density changes on a time scale of about 30min. The poisoning of the filament has been studied in detail by Toyoda et al. (1 994). The resistivity of the tungsten filament increases monotonically with time as the hot filament undergoes carbonization after the flow of C,H, in the feed gas begins. The input power (current times voltage) to the hot filament decreases, partially because of the reduced power load to dissociate H, as the filament becomes poisoned and partially because the more permanent carbonization alters the bulk resistance of the hot filament. They observed that the filament temperatures measured with one- and two-color optical pyrometers differed significantly. From this, they concluded that both the temperature and the emissivity of the hot filament changed as the filament was poisoned. They observed that as C2H2 flowed into the CVD reactor at a concentration high enough to poison the filament, bright patches formed on the filament and then spread to cover the entire filament as it became poisoned. They speculated that the carbon was removed from the filament by reactions with atomic hydrogen. After a patch became poisoned, it was unable to dissociate H,. This reduced the atomic hydrogen concentration near that patch, allowing hydrocarbons to accumulate in nearby regions of the filament. This resulted in a chain reaction whereby the poisoning of the filament spread over the entire filament and may
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
323
explain why patches were formed that spread over the entire filament rather than the entire filament uniformly changing its brightness. In related research, Menningen et al. (1994) studied the effects of a pretreatment of the substrate by exposure to a high concentration of a hydrocarbon diluted in H, before hot-filament diamond CVD. A pretreatment was found to enhance nucleation density and film uniformity in microwave reactors by Stoner et al. (1992), Bou et al. (1992), Barnes and Wu (1993) and Wolter et al. (1993). Menningen et al. (1994) carried out two sets of experiments. One used a 5% mole fraction of CH, in H, flowed for 30 min followed by 7.5 h of diamond growth with a feed gas of 0.5% CH, in H,. The other used 3% C,H2 in H, flowed for 30min followed by diamond growth for 7.5 h with a feed gas of 0.36% C,H, in H,. In both of these experiments, no significant effects on nucleation density or film uniformity as a result of the pretreatment were observed. Thus the effects of a pretreatment of the substrate are very different in a hot-filament CVD reactor and a microwave CVD reactor. Menningen et al. (1994) carried out a number of time-resolved species density measurements as part of their substrate pretreatment studies. They observed that when 3% C2H2diluted in H2 was used as the feed gas, the CH, density increased quickly to about 2 x lo1, cmP3, followed by a rapid decline to about 2 x 10I2cmP3 during the pretreatment. When the C,H, concentration was reduced to 0.36%, the CH, density abruptly increased to about 1.8 x lo', cm-, and then quickly fell to about 8 x lo', cm-,, where it remained for the duration of the diamond growth. The C,H, mole fraction increased from about 0.02 to 0.035 during the pretreatment and then fell to about 0.0075 during the diamond growth. Interesting time-resolved results were also found when CH, was used in the pretreatment. During the pretreatment the CH, density was about , it remained for the 2.8 x loi3cm-, and fell to about 7 x 10I2~ m - ~where duration of the growth period. In this section, we have reviewed the use of high-sensitivity white-light absorption spectroscopy to study the C,H, density during diamond growth in hot-filament CVD. Many interesting effects due to filament poisoning have been observed. D. DETECTION Of OTHER SPECIES DURING THE CVD
OF
DIAMOND
Several other atoms, molecules, and free radicals have been detected in diamondgrowth CVD reactors using high-sensitivity white-light absorption spectroscopy. During the studies of diamond CVD with a hot-filament reactor, Menningen et al. (1995b) observed several impurities. Absorption due to Zn at 213.9nm (4s2 ' S o -+ 4s4p ' P I )was observed. The Zn was assumed to have evaporated from Znplated washers used on the hot-filament assembly. The density of Zn was determined to be 4.7 x lo9 cm-3 using the known oscillator strength for the
324
L. K Anderson, A.N. Goyette, and J E . Lawler
transition. Also observed by Menningen et al. (1995a) was In at 304.0nm (5p 2Pl/2+ 5d 2D3/2). The reactor had previously been used for an experiment with In. Although carefully cleaned, the reactor had apparently retained enough In to be observed with high-sensitivity white-light absorption spectroscopy. The observed In density was about 7 x 10' ~ m - ~ . In a hollow-cathode glow-discharge CVD reactor, Menningen et al. (1995a) observed several impurities using high-sensitivity white-light absorption spectroscopy. The hollow cathode was made from Ta. Some Ta is sputtered from the hollow cathode. Several Ta I absorption lines were observed. The Ta density was determined to be about 5 x 1010cm-3,using absorption from the 5d36s2 'F,(, level at 340.8nm into a J = 5 / 2 level. The upper level of the transition IS unassigned. Using x-ray backscattering analysis, they found that the diamond films grown contained Ta impurities. Another impurity discovered in the hollowcathode discharge was NH. The NH was detected using absorption from the A 2 H t X 2 C - (0,O) band at 336.1 nm. They determined the NH density to be 1.8 x 10" cmP3. No nitrogen impurity was detected by x-ray backscattering in the diamond films grown. It is clear that high-sensitivity white-light absorption spectroscopy is useful for detecting low levels of impurities in CVD reactors. This may become important for process control in the use of CVD reactors for manufacturing materials. Welter and Menningen (1997) have detected both CN and OH in an oxyacetylene flame used for atmospheric diamond growth. Figure 19 shows a typical absorption spectrum from the OH radical. Both species were primarily found in the outer parts of the flame, rather than in the flame feather, where diamond is grown. They measured column densities as a function of distance from the substrate. Column densities of 2 x 10l2cm-2 to 2 x 1013cm-2 for CN OH absorption spectrum
On A 'I?+!X
'q (24)band
0.9091
m
MU
MI
m
m.3
ZBO
z7o
W h n g h (nm)
FIG. 19. Typical absorption spectrum from the A 2 Z + t X 2 n (2,O) vibrational band of the OH radical. (Reprinted with permission from Welter and Menningen, 1997, J: Appl. Phys., 82, 1900, (c) 1997 American Institute of Physics.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
325
and 1 x 10l6cm-, to 8 x 10l6 cmP2 for OH were measured under their experimental conditions.
E. MEASUREMENT OF c2 RADICAL DENSITIES Among the species that have been investigated using high-sensitivity white-light absorption spectroscopy in the CVD of diamond is the C, molecule. In conventional hot-filament CVD or plasma-enhanced CVD, in which the feed gas consists of a small fraction of hydrocarbon gas-no more than a few percent-diluted in molecular hydrogen, the reactive chemistry is dominated by atomic hydrogen. In these environments, the presence of C2 has been detected using optical-emission spectroscopy but not in absorption spectroscopy. Under these circumstances, Muranaka et al. (1990) has suggested that the presence of C, may be responsible for amorphous carbon deposition. The presence of the C2 molecule has also been detected in CVD environments that utilize very different, hydrogen-deficientgas chemistries. These novel diamond CVD environments use a microwave plasma with a small fraction of either hydrocarbon gas or fullerene vapor diluted in argon, with only a few percent of molecular hydrogen added to the feed gas (Gruen et al., 1994a; Gruen et al., 1994b). In these gas chemistries, C2 has been detected using optical-emission spectroscopy, but rather than its presence indicating graphitization and “poor” diamond quality, the films grown have very high quality, nanocrystalline diamond content. In fact, Gruen et al. (1995) proposed that C, was the primary diamond growth species in these environments. The structure and properties of these nanocrystalline diamond films differ significantly from those of the microcrystalline films grown in the hydrogen-rich conventional gas chemistry. In addition to their very high quality, these nanocrystalline diamond films are extremely smooth, with average surface roughness of only tens of nanometers. The C2 molecule has also been observed using laser-induced fluorescence in diamond CVD using oxyacetylene torches by Matsui et al. (1989) and using optical-emission spectroscopy in arcjets by Stalder and Sharpless (1990). Welter and Menningen (1997) used high-sensitivity whitelight absorption spectroscopy to determine C, column densities in an oxyacetylene torch used for diamond CVD. Quantitative analysis of C, densities using high-sensitivity white-light absorption spectroscopy has been done in microwave CVD environments by Goyette et al. (1998c). Goyette et al. (1998a) also used this method to determine C, densities in nonconventional, hydrogen-deficient microwave CVD environments used for nanocrystalline diamond growth. The high-sensitivity white-light absorption spectroscopy of hydrogen-deficient microwave CVD used for nanocrystalline diamond growth done by Goyette et al. (1998a) used the (0,O) vibrational band of the d311 t a3II electronic transition of C2, known as the Swan system. Figure 20 shows a typical absorption spectrum from this band. The bandhead for the (0,O) Swan band is located at 5 16.5nm. The
326
L. W Anderson, A.N. Goyette, and 1E.Lawler
0 . 9 9 9 2 1 . .
,
,
.
,
,
,
,
*
2 ; ; ~ 3 ; ~ ~ 3 ~ ; ; n
n
n
n
n
n
n
n
n
n
n
m
(m)
FIG. 20. Typical absorption spectrum from the d 3 n , c a3n,(0,O) Swan band of C,. (Reprinted with permission from Goyette et al., 1998c, J Vac. Sci. Technol. A , 16, 337, 0 1998 American Vacuum Society.)
structure of the band is such that rotational lines in the P branch accumulate near the bandhead, and the R-branch rotational lines extend through the tail of the band. Since the bandhead is such a prominent feature of this band but is still optically thin under these conditions, the equivalent width of the bandhead was integrated to give the C2 densities, rather than integrating across the entire band. The bandhead contains rotational lines originating from nearly 30 rotational levels. The C, densities in these nanocrystalline diamond CVD environments were determined for both hydrocarbon and fullerene precursors under the variation of several processing parameters. For chemistries employing hydrocarbon precursors, the C , density for typical nanocrystalline diamond-growth conditions was approximately 3 x 10" cmP3. These typical conditions consisted of a feed gas ratio of Ar : H, : CH, = 97 : 2 : 1, a pressure of 100 ton, a microwave power of 800 W, and a substrate temperature of 800°C. The C2 density was observed to vary most dramatically with chamber pressure and substrate temperature, increasing approximately as the cube of the chamber pressure and decreasing approximately exponentially with increasing substrate temperature. Microwave power had almost no effect on the C, density. Increased hydrogen fractions in the feed gas resulted in a decrease in C, density, as a result of increased C, destruction rates from chemical reactions converting C, into various hydrocarbon species. Increasing the carbon fraction in the feed gas increased the C, density initially. Because the carbon fraction in the feed gas could not be varied independently of the total hydrogen fraction in the feed gas, the total hydrogen fraction in the feed gas increased with the carbon fraction in the feed gas. This increase in the total hydrogen fraction in the feed gas resulted in increased reaction rates for the chemical reactions that convert C, into hydrocarbon species. Above a specific critical total hydrogen fraction, the rates
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
327
of these reactions became sufficient to balance the increased C, production rate. Above the critical total hydrogen fraction, the C, density decreased. For chemistries employing fullerene precursors, the C2 density for diamondgrowth conditions also was approximately 3 x 10" cmP3. The fullerenes were sublimated from a fullerene-containing soot and delivered to the plasma with an Ar carrier gas. The vapor pressure of C,, as a function of its temperature is well characterized, and changing the temperature of the fullerene-containing soot from which the fullerenes sublimated regulated the partial pressure of C,, delivered to the plasma. What is remarkable is that typical nanocrystalline diamond-growth conditions for plasmas employing fullerene precursors contain a significantly smaller total carbon content than plasmas using hydrocarbon precursors. The partial pressure of C,, introduced into the plasma is typically a few mtorr. Although each molecule contains 60 carbon atoms, the ratio of the C, density to the total density of carbon introduced into the plasma was determined to be a factor of 12 greater for fullerene precursors than for hydrocarbon precursors. Thus the production efficiency of C, from fullerene precursors is more than an order of magnitude greater than from hydrocarbon precursors. As with plasmas using hydrocarbon precursors, the C, density in fullerene-containing plasmas varied with changes in total chamber pressure, hydrogen fraction in the feed gas, but only slightly with microwave power. The C2 density was found to increase with the temperature of the fullerene-containing soot, but not as rapidly, however, as the equilibrium vapor pressure of C,,. This indicated that the partial pressure of C,, delivered to the plasma was somewhat less than the vapor pressure of C,, in the sublimator. Thus the production efficiency calculated for C, from fullerene precursors was in fact a lower-bound estimate, based upon the assumption that the equilibrium partial pressure of fbllerenes in the sublimator was delivered to the plasma. Welter and Menningen (1997) used the (0,O) vibrational band of the d311 t a311 Swan system in order to calculate the C2 column density in their oxyacetylene torch. Unlike Goyette et al. (1998a), they integrated the equivalent width across the entire vibrational band. They found that the C, radical was produced in the inner cone of the flame and had elevated densities in the acetylene feather. In fact, the extent of the feather increases as the C,H, : 0, ratio in the torch feed gas is increased, and the column density of C, correspondingly increases. Klein-Douwel et al. (1995), and Matsui et al. (1989) have observed a correlation between the Cz distribution in the flame and the local growth rate and diamond quality. The measured column densities of Welter and Menningen (1997) were on the order of lOI3 ern-,. It is important to note that the flame of an oxyacetylene torch is far from homogeneous and has very distinct regions in the line of sight. Therefore, the column density is a line-of-sight average across a region in which large species gradients occur. This is in contrast
328
L. K Anderson, A.N. Coyette, and J E . Lawler
to the other types of CVD discussed in this paper, in which the gas-phase composition is relatively homogeneous along the line of sight. High-sensitivity white-light absorption spectroscopy has also been used in the case of C2 as a method of providing calibration for optical-emission spectroscopy. Emission measurements alone are not suitable for quantitative analysis of groundstate densities for a variety of reasons. The populations of the radiating levels depend upon both radiative and nonradiative excitation and deexcitation rates and provide limited information on ground-state densities. Nevertheless, Goyette et al. (1998b) measured the dependence of C2 emission intensity from the (0,O) Swan band on the population of the lower level of that transition using highsensitivity white-light absorption spectroscopy. They found that there was a linear correlation between the observed emission intensities and the observed C2 lowerlevel densities across a range of C2 density spanning two orders of magnitude. This is a very useful result for trend studies. Since the emission intensity of this band was found to be directly proportional to the absolute density for C2, this extends the sensitivity with which the density of that species can be estimated.
F. SPECTROSCOPIC TEMPERATURE DETERMINATION High-sensitivity white-light absorption spectroscopy has also been used in the determination of gas kinetic temperature in diamond CVD environments. The gas kinetic temperature is an important parameter in these reactive environments, since the rates and endpoints of chemical reactions are temperature-dependent. In addition, knowledge of the gas kinetic temperature is necessary in order to calculate column or absolute densities of species from their measured absorbances. Absorption cross sections are temperature-dependent, and the relative populations of the various levels represented in the integrated equivalent width are also temperature-dependent. Determination of gas kinetic temperatures in reactive discharges is therefore important. Invasive methods for measuring the temperature, such as the insertion of a thermocouple into the discharge, affect the discharge, and are influenced by such things as the heat deposited on the thermocouple as a result of recombination on the surface of the thermocouple. Thus a noninvasive, in situ method is required for reactive plasmas. Various spectroscopic methods have been used to determine gas kinetic temperatures: Doppler broadening of atomic hydrogen lines, coherent anti-stokes Raman spectroscopy, and optical-emission spectroscopy are a few (Gicquel et al., 1996). Each method has its limitations. Doppler broadening measurements require high spectral resolution and precise knowledge of all linebroadening mechanisms; coherent anti-stokes Raman spectroscopy requires a complex laser system; and optical-emission spectroscopy measures the population distribution of excited, radiating energy levels, which may or may not be in equilibrium with the gas kinetic temperature.
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
329
The energy separations between rotational levels in a diatomic molecule in a given vibrational and electronic state are typically small compared with the thermal translational energy. Nearly all gas kinetic collisions produce a change in the rotational quantum number, whereas collisions producing a change in the vibrational and electronic quantum numbers occur much less frequently. Consequently, the relative rotational population distribution in a sufficiently long-lived vibrational state has a Boltzmann distribution with a rotational temperature that reflects the gas kinetic temperature. The relative emission or absorption intensity is then I x Sryexp (-ElkT,,) where S,?,, is the Honl-London line strength, E is the energy of the upper or lower level of the transition for emission or absorption, respectively, k is Boltzmann’s constant, and Trotis the rotational temperature. Most often, rotational temperatures are derived from molecular emission, thus probing the rotational distribution of an excited electronic level. This requires that collisions equilibrate the rotational levels in a time that is short compared to the lifetime of the excited level. The excited-level lifetime may be primarily determined by radiation or by other effects, such as collisional quenching of the excited level. If rotational temperatures are derived from molecular absorption, the rotational population distribution of a low-lying level such as the ground level or a metastable level is measured. Long-lived, low-lying levels are likely to have rotational levels that are in equilibrium as a result of collisions. This, however, requires sufficient signalto-noise ratios in the absorption spectra to accurately determine the relative absorption of individual rotational lines. In this use, the multielement detection is especially beneficial in accumulating high photon statistics and eliminating effects due to lamp drift. Goyette et al. (1 998a) used high-sensitivity absorption spectroscopy in order to calculate rotational temperatures from the (0,O) vibrational band of the d311 t a311 Swan band of C,. They had sufficient spectral resolution in order to resolve individual R-branch rotational lines in the tail of this band. Highangular-momentum quantum number rotational lines from the P branch of this band overlap the lower-angular-momentum quantum number rotational lines from the R branch. Consequently, rotational lines from the R branch with high rotational quantum numbers were chosen for analysis with a Boltzmann plot of the natural logarithm of the absorption versus the lower rotational level energy. If the population distribution is Boltzmann, this plot yields a straight line whose slope is inversely proportional to the rotational temperature of this vibrational level. Analysis of these particular rotational lines in absorption did yield linear Boltzmann plots with well-defined rotational temperatures near 1300°C. The total energy separation of lower levels among all the rotational lines used in the
330
L. W Anderson, A.N. Goyette, and J E . Lawler
analysis was relatively small in comparison to the energy separation of lower levels among all the rotational lines in this band, however. In order to give confidence that these rotational temperatures were indicative of the rotational population distribution of the entire vibrational band, another method was used to calculate a rotational temperature from unresolved lines of the entire band. Goyette et al. (1998a) divided the experimental band spectra into bins, and calculated the integrated absorption of each bin. Each bin contained several unresolved rotational lines. Theoretical line intensities for the rotational lines present in each bin were calculated and summed. Both experimental and theoretical bin intensities were normalized to the total integrated band intensity. These bin intensities were then compared with the integrated absorption from theoretical spectra for various rotational temperatures. The rotational temperatures for the experimental spectra were determined from a least-squares fit that involved iterating until the overall difference between experimental and theoretical integrated absorption was minimized. This method of spectral synthesis does not require that individual rotational lines be resolved. Rotational temperatures determined in this manner from the (0,O) vibrational band of the d311 t a 3 n Swan band of C, agreed within 50°C with those determined from Boltzmann plots. Booth et al. (1998) determined both vibrational and rotational temperatures from high-resolution absorption spectra from the CF, radical in low-pressure capacitively coupled rf plasmas. These plasmas used C,F, as the feed gas. Experimental band profiles were compared with simulated band profiles in order to determine a rotational temperature. Figure 21 shows an experimentally obtained CF, spectrum and a theoretically simulated spectrum with a rotational temperature of 400 K. The absorption was measured out of the X (O,O,O) level of CF,.
Warelonom (nm)
FIG. 21. High-resolution specburn of the A (0,5,0)t X (O,O,O) vibrational band of CF, near 252 nm. Also shown is a theoretical spectrum of the same band simulated with a rotational of 400 K. (From Booth ef al., Plasma Sources Sci. Technol., in press.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
33 1
Optical-emission spectroscopy using multielement detection may also be used to determine rotational temperatures of diatomic molecules. As with absorption, multichannel detection for emission has an advantage over sequentially scanned detection in that high photon statistics are accumulated rapidly. However, as mentioned previously, care must be taken when interpreting emission spectra for temperature analysis. The population distribution of excited, radiating energy levels is probed. Depending on the lifetime of the levels, the population distribution of the excited levels may or may not have time to equilibrate with the gas kinetic temperature. The population of these levels depends upon both excitation rates and radiative and nonradiative deexcitation rates into and out of these levels. In their studies of gas-phase species in various diamond CVD systems, Childs et al. (1994a), Menningen et a f . (1995a), and Erickson et al. (1996) used rotational temperatures determined from the R branch of the G'C; + B'C: (0,O) vibrational band of H,. As a result of the feed gas composition, the H, molecule was present in abundance. Furthermore, because H, is a very light molecule, its low moment of inertia results in comparatively large energy separations between rotational lines. Thus, high spectral resolution is not necessary in order to resolve individual rotational lines from this molecule and compare their relative intensities. In order to reconfirm whether the spectroscopically determined rotational temperature from this band of H, was indicative of the gas kinetic temperature, Goyette et al. (1996) conducted a calibration experiment. They initiated a weak dc discharge and enclosed it in a resistively heated furnace. The gas kinetic temperature of the discharge was then predominantly determined by the temperature of the furnace, with slight additional gas heating due to Joule heating of the discharge, which was calculated knowing the power deposition into the discharge. Their analysis included a detailed calculation of the radial temperature variation. They were able to place an upper bound on the gas kinetic temperature in the discharge. The furnace had a very small opening from which discharge emission could escape. The emission was focused onto the entrance slit of a spectrometer, and spectroscopic and gas kinetic temperatures were compared. They found that the rotational temperatures determined from G'C; + B'C: (0,O) R-branch emission did indeed correlate with the gas kinetic temperature under diamond-growth conditions. For pressures above 20 torr and gas kinetic temperatures above 700°C, they found that the rotational and gas kinetic temperatures agreed. In addition, they investigated another commonly used band of H,, the d 3 n , +. a 3 C l (0,O) Fulcher band. Rotational lines from the R branch of this band were used for analysis; although a Boltzmann plot of these lines did yield a straight line, the rotational temperature determined from the slope of this line was independent of the gas kinetic temperature. Thus the Fulcher band is unsuited for use in measuring the gas kinetic temperature. This example illustrates the difficulty of accurate gas kinetic temperature measurement
332
L. W Anderson, A.N. Goyette, and J E . Lawler
in a discharge and the caution that must be taken in interpreting rotational temperatures determined by optical-emission spectroscopy as indicators of gas kinetic temperature. Goyette et al. (1998d) performed a similar calibration experiment to test whether rotational temperatures determined from the second positive system, C311 + B311, of N, and the first negative system, B2E + X 2 C , of N i are suited for use in measuring gas temperature. Nitrogen is used in various CVD processes and is present as an impurity species. For the second positive system of N,, vibrational bands from the Av = 2 and Av = 3 series were investigated. They found that the rotational temperatures determined from the (2,O) vibrational band were within the bounds for the gas kinetic temperature for pressures above 0.5 torr. Rotational temperatures determined from the (3,l) and (4,l) vibrational bands of this system did not correlate with the gas kinetic temperature. For the fist negative system of N l , the rotational temperatures determined from the (0,O) vibrational band were linearly correlated with the gas kinetic temperature, but were consistently 1 5 0 T higher than the upper bound placed upon the gas kinetic temperature. High-sensitivity white-light absorption spectroscopy has several advantages in the spectroscopic determination of gas kinetic temperatures. The method is noninvasive, in situ, and relatively simple. Multichannel detection allows high signal-to-noise ratios even in absorption, as well as accurate determination of relative line intensities without difficulties due to lamp drift. Additionally, the determination of relative rotational population distributions in ground or metastable electronic levels of diatomic molecules alleviates the concerns associated with the accuracy of rotational temperature analysis using optical-emission spectroscopy.
IV. The Uses of High-Sensitivity White-Light Absorption Spectroscopy in Other CVD Environments A. CVD
OF
GaAs
Killeen (1992) has used high-sensitivity white-light absorption spectroscopy in the ultraviolet for in situ studies of gas-phase species during the CVD of GaAs. Killeen used a diode array at the exit plane of a 0.25-m spectrometer. The continuum radiation sources used were a D, lamp, a tungsten halogen lamp, and a hollow-cathode lamp. Measured absorbances less than 1Op3 were achieved using a modest integration time of 0.5 s. Killeen (1992) used a horizontal-flow metalorganic CVD reactor with trimethylgallium (TMGa) and H, as the feed gases. When the graphite susceptor was left at room temperature, only absorption from TMGa was observed. When
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
333
the susceptor was inductively heated to 700"C, absorbances from not only TMGa but also other species were detected. Absorption features from atomic gallium, GaH, and GaCH, were identified. The GaCH, molecule had been postulated as a gas-phase intermediate in the CVD of GaAs, but had not been previously detected. Several other unidentified absorption features were also observed. Relative concentrations of atomic Ga, GaH, GaCH,, and TMGa were obtained as functions of temperature in an isothermal cell. A sharp decrease in the TMGa concentration occurred between 450 and 500°C. Conversely, the GaCH, concentration increased significantly in the same temperature range. The atomic gallium and GaH concentrations increased sharply between 500 and 600°C. When arsine, ASH,, was added to the flowing cell, dramatic changes in the gallium fragments occurred. When the ratio of arsine to TMGa in the feed gas was 0.12, atomic gallium absorption was not observed and the GaH and GaCH, concentrations were reduced. These changes in the concentration of the intermediate atomic gallium, GaH, and GaCH, were believed to be the result of the suppression of chemical reactions producing these intermediates by the abstraction of hydrogen from ASH, by CH, radicals. Detection of absorption from ASH, A2Al --+ X2Bl in the region from 400 to 450 nm supported this hypothesis. Similarly, subhydnde species of arsenic had been postulated to exist in these types of CVD environments, but none had been detected until the work of Killeen (1992). From the work of Killeen (1 992), high-sensitivity white-light absorption spectroscopy has important applications in the metalorganic CVD of GaAs.
B. CVD OF SILICON Toyoda et al. (1 995) have used high-sensitivity white-light absorption spectroscopy in the ultraviolet using a photodiode array to study the concentration of silyl radicals, SiH,, in an rf discharge with a feed gas of SiH, diluted in H,. The 3
(...,...,...,...
(4
P ;2
il $
0 FIG. 22. SiH, radical density and relative particulate density as a function of RF power for 20 Pa and 12ccm of a 50% SiH4/5O% H, mixture. (Reprinted with permission from Toyoda et al., 1995, Jpn. 1 Appl. Phys., 34, L448, 0 1995.)
334
L. u( Anderson, A.N. Goyette, and J E . Lawler
SiH, radical has a broad absorption feature between 205 and 240nm. The absorbance from this band was monitored, and from this, the relative SiH, concentration was determined as a function of the rf power. The production of dust particles in this type of discharge was also monitored by measuring the transmittance at wavelengths longer than 250 nm. Unfortunately, particulate scattering in the wavelength range of the SiH, absorption affected the background measurements, making an accurate determination of the SiH, absorbance difficult. Figure 22 displays the relative S M , and particulate densities determined by Toyoda et al. (1995). It is clear that high-sensitivity white-light absorption spectroscopy is of use in SiH, discharges for CVD of Si.
V. Other Uses of High-Sensitivity White-Light Absorption Spectroscopy A. ETCHINGPLASMAS Booth et al. (1998) have used high-sensitivity white-light absorption spectroscopy to measure the absolute densities of CF, in low-pressure capacitively coupled rf plasmas using C F , C,F,, and SF, as feed gases. Other reactive intermediate species detected in these plasmas include CF, SiF,, S,, and AlF. They used a high-pressure Xe arc lamp as a source of broadband U V light. A fused silica prism was used to select light from the Xe arc in the spectral region of interest. The absorption features Booth et al. (1998) investigated were located between 2 10 and 280 nm. Peak transmittances of dielectric-layer interference filters this deep in the ultraviolet typically do not exceed 20%. A fused silica prism has a considerably higher transmittance at these wavelengths, and photon statistics may be accumulated more rapidly than in experiments using an interference filter. This is especially important for wavelengths below 300 nm, as transmissive optical filters become very lossy, making photon statistics considerably more difficult to accumulate than in the visible. Booth et al. (1998) used a UV-enhanced CCD array to detect the dispersed light from their spectrometer. Fractional absorptions of were routinely detectable with their system. Figure 23 shows a typical transmittance spectrum showing the bands of the A (0, v', 0) t X (O,O,O) series of CF,. The A (0, u', 0) t X (0,1,0) series was also detected. Booth et al. (1998) used the A (0,6,0) t X (O,O,O) vibrational band located at 249nm in order to calculate the CF, density. They used the absorption cross section for this band measured by Sharpe et al. (1987). Booth et al. (1998) observed that the CF, density was considerably higher in a C,F, plasma at lOOmtorr and 100 W rf power than in a CF, plasma at the same
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY 1
s
..-(I)
335
.oo
0.99
E
(I)
c
0.98
0.97
220
240
230
260
250
270
280
(nm) FIG. 23. A typical transmittance spectrum showing the bands of the A (O,u’,O)t X (O,O,O) series of CF7, Weak absorption from the A ( O , u ’ , O ) c X (0,1,0) series of CF, is also present. (Booth et al., 1998, Plasma Sources Sci. Technol., in press.)
pressure and power. They measured a CF2 density of 4.3 x 1013radicals/cm3 in a C,F, plasma and only 4.5 x 10I2 radicals/cm3 in a CF, plasma. The CF, radical is an important species associated with the polymerization, selectivity, and wall passivation mechanisms in oxide etching processes. Because a Si wafer was placed in these reactive plasmas using fluorinated gases, SiF, absorption was detected near 220nm. Figure 24 shows a typical absorption spectrum from this
C
.-0
(ID
.;0.999
-
(ID
c
e
I-
0.998 -
0.997 2;o’
v - -
9
. ’ 215
.
5
- .220 ’ -
I
II1 V-1
. .
- 225 ’ . . - . ’ 230
wavelength (nm) FIG. 24. A typical transmittance spectrum showing the bands of the A ( O , u ’ , O ) c X (O,O,O) series of SiF2 Weak absorption from the A ( O , u ’ , O ) c X (0,1,0) series of SiF, and vibrational bands from the CF radical are also present. (Booth et a/., 1998, Plasma Sources Sci. Technol., in press.)
336
L. W Anderson, A.N. Goyette, and J E . Lawler
intermediate species. In this wavelength region, absorption from the CF radical was also detected. Booth et al. (1998) also detected the S, molecule in SF, plasmas operated at 50 mtorr and 100 W. B. ARGON RADIO-FREQUENCY PLASMAS Wendt et al. (1998) have used high-sensitivity white-light absorption spectroscopy to determine the density of Ar metastables in an Ar inductively coupled RF plasma. A schematic of their apparatus is shown in Fig. 25. Their apparatus differs from those previously discussed in this paper in that it is a double-pass, rather than a single-pass, experiment. White light from a high-pressure Xe arc lamp was directed along the axis of the cylindrical discharge. Windows at each end of the discharge allowed optical access along the entire length of the plasma. A planar mirror located at the far window opposite the lamp enabled a second pass through the plasma. A mechanical chopper was placed in front of the arc lamp in order to selectively block light from the Xe arc. The readout of the diode array was gated in order to obtain spectra of light from the arc lamp combined with the plasma emission and of the plasma emission alone. The subtraction of these two spectra allowed the absorption features to be isolated from the plasma emission. Light from the lamp in the absence of the plasma was divided into this in order to obtain an absorption spectrum. Wendt et al. (1998) determined the complete steady state Ar metastable density. The total Ar metastable density was measured as a function of pressure
Photo Diode
I ' 'i Xenon Lam
h,
1 5 0 ham
\ Chdpper Wheel
Banipass Filter
ICP Plasma Chamber
U
FIG. 25. Double-pass absorption spectroscopy system used in the detection of Ar metastable densities. (From Wendt et al., 1998.)
HIGH-SENSITIVITY WHITE-LIGHT ABSORPTION SPECTROSCOPY
337
and of discharge power. Steady-state densities did not vary significantly in these experiments, with values for the total Ar metastable densities ranging between 6.5 x 10"' atoms/cm3 and 8.5 x 10'' atoms/cm3 as the pressure was vaned between 5 and 50 mtorr and the discharge power from 50 to 400 W. As the pressure was increased from 5 to 20mtorr, the Ar metastable density increased slightly. As the pressure was increased from 20 to 50mtorr, the Ar metastable density decreased. This behavior was observed for all discharge powers investigated. Wendt et al. (1 998) qualitatively explained this pressure dependence as the combined effects of increasing electron density and decreasing electron temperature with increasing pressure. The metastable production rate is proportional to the electron density, which increases with pressure; however, the metastable production rate also decreases as the electron temperature decreases with increasing pressure. They inferred that at lower pressures, the rise in the electron density, and consequently the Ar metastable production, supersedes the decrease in metastable production associated with the reduction in electron temperature with increasing pressure. Thus the Ar metastable density initially increases with increasing pressure. At higher pressures, the decrease in metastable production associated with the reduction in electron temperature with increasing pressure prevails.
VI. Conclusion In this paper, we have reviewed the uses of high-sensitivity white-light absorption spectroscopy in the study of gas-phase species in the chemical-vapor deposition of various materials and in processing plasmas. High-sensitivity white-light absorption spectroscopy is a remarkably powerfd diagnostic of such environments because it is nonperturbative, is an in situ technique, yields direct quantitative information regarding species densities, and is extremely sensitive even in radiating environments. Densities of stable molecules, free radicals, atoms, and molecular and atomic ions can all be determined with this technique. Furthermore, high-sensitivity white-light absorption spectroscopy is inexpensive, very reliable, and straightforward to implement. Applications of high-sensitivity white-light absorption spectroscopy highlighted in this review have included the determination of absolute steady-state densities of several gas-phase species including reactive intermediates, spectroscopic gas kinetic temperature determination, and impurity detection and quantification in processing environments. High-sensitivity white-light absorption spectroscopy is a fairly recent technique, with high sensitivities in radiating environments becoming realizable with the advent of photodiode arrays and research-grade CCD arrays. It holds much promise for continued use, both in the
338
L. K Anderson, A.N. Goyette, and LE. Lawler
diverse set of applications and environments discussed in this review and in numerous potential applications not yet explored.
VII. Acknowledgments The U.S. Army Research Office under grant DAAH-04-96-1-0413 supported this work. The authors wish to thank M. Schmidt and S. McVay for their assistance in typing this manuscript.
VIII. References Barnes, P. D., and Wu, R. L. C. (1993). Appl. Phys. Lett. 62, 37. Bergeson, S. D., Mullman, K. L., and Lawler, J. E. (1995). Ashophys. J. 464, 1050. Booth, J. l?, Cunge, G., Neuilly, F., and Sadeghl, N. (1998). Plasma Sources Sci. Technol. 7, 432. Bou, P., Vaudenbulche, L., Herbin, R., and Hilton, F. (1992). 1 Mater Res. 7 , 257. Celii, F. G., and Butler, J. E. (1991). New Diamond Sci. Technol. 1, 201. Celii, F. G., Pehrsson, P. E., Wang, H. T., and Butler, J. E. (1988). Appl. Phys. Lett. 52, 2043. Celii, F. G., and Butler, J. E. (1989). Appl. Phys. Lett 54, 1031. Celii, F. G., Pehrsson, l? E., Wang, H. T., Nelson, H. H., and Butler, J. E. (1989). AIP Conference Proceedings 191, 747. Childs, M. A,, Menningen, K. L., Chevako, P., Spellmeyer, N. W., Anderson, L. W., and Lawler, J. E. (1992). Phys. Lett. A 171, 87. Childs, M. A,, Menningen, K. L., Toyoda, H., Anderson, L. W., and Lawler, J. E. (1994a). Europhys. Lett. 25, 729. Childs, M. A,, Menningen, K. L., Toyoda, H., Ueda, Y., Anderson, L. W., and Lawler, J. E. (1994b). Phys. Lett. A 194, 119. Erickson, C. J., Childs, M. A., Anderson, L. W., and Lawler, J. E. (1995). 1 Mater Res. 10, 1108. Erickson, C. J., Jameson, W. B., Watts-Cain, J., Menningen, K. L., Childs, M. A., Anderson, L. W., and Lawler, J. E. (1996). Plasma Sources Sci. Technol. 5, 761. Firchow, S. J., and Menningen, K. L. (1997). Bull. Am. Phys. SOC. 42, 1760. Gicquel, A,, Hassouni, K., Breton, Y., Chenevier, M., and Cubertafon, J. C. (1996). Diam. Rel. Muter 5, 366. Glinzer, K., Quack, M., and Troe, J. (1977). Proceedings of the 16th Symposium (International) on Combustion @. 949). The Combustion Institute (Pittsburgh). Goodwin, D. G., and Gavillet, G. G. (1990). 1 Appl. Phys. 68, 6393. Goodwin, D. G., and Gavillet, G. G. (1991). Proceedings of the 2nd International Conference on New Diamond Science and Technology, @. 335). R. Meissier, J. T. Glass, J. E. Butler, and R. Roy (Eds.), Materials Research Society (Pittsburgh). Goyette, A. N., Jameson, W. B., Anderson, L. W., and Lawler, J. E. (1996). 1 Phys. D 29, 1197. Goyette, A. N., Lawler, J. E., Anderson, L. W., Gruen, D. M., McCauley, T. G., Zhou, D., and Krauss, A. R. (1998a). 1 Phys. D 31, 1975. Goyette, A. N., Lawler, J. E., Anderson, L. W., Gruen, D. M., McCauley, T. G., Zhou, D., and Krauss, A. R. (1998b). Plasma Sources Sci. Technol. 7, 149-153. Goyette, A. N., Matsuda, Y.,Anderson, L. W., and Lawler, J. E. (1998~).1 Vac. Sci. Technol. A 16, 337.
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Goyette, A. N., Peck, J. R., Matsuda, Y., Anderson, L. W., and Lawler, J. E. (1998d). 1 Phys. D 31, 1556. Gruen, D. M., Liu, S., Krauss, A. R., and Pan, X. (1994a). Appl. Phys. Lett. 64, 1502. Gruen, D. M., Liu, S., Krauss, A. R., and Pan, X. (1994b). 1 Appl. Phys. 75, 1758. Gruen, D. M., Zuiker, C., Krauss, A. R., and Pan, X. (1995). 1 Yac. Sci. Technol. A 13, 1628. Herzberg, G., and Shoosmith, J. (1956). Can. 1 Phys. 24, 523. Hsu, W. L. (1991). Appl. Phys. Lett. 59, 1427. Killeen, K. P. (1992). Appl. Phys. Left. 61, 1864. Klein-Douwel, R. J. H., Spaanjaars, J. J. L., and ter Meulen, J. (1995). 1 Appl. Phys. 78, 2086. Loh, M. H., and Capelli, M. A. (1997). Appl. Phys. Lert. 70, 1052. Matsui, Y., Yuuki, A,, Sahara, M., and Hirose, Y. (1989). Jpn. 1 Appl. Phys. 28, 1718. Menningen, K. L., Childs, M. A,, Chevako, F!, Toyoda, H., Anderson, L. W., and Lawler, J. E. (1993). Chem. Phys. Lett. 204, 573. Menningen, K. L., Childs, M. A,, Toyoda, H., Anderson, L. W., and Lawler, J. E. (1994). 1 Mate,: Res. 9, 915. Menningen, K. L, Childs, M. A,, Toyoda, H., Ueda, Y., Anderson, L. W., and Lawler, J. E. (1995a). Contrib. Plusma Phys. 35, 359. Menningen, K. L., Erickson, C. J., Childs, M. A,, Anderson, L. W., and Lawler, J. E. (1995b). 1 Mate,: Res. 10, 1108. Moller, W., Mozyhukhin, E., and Wagner, H. Gg. (1986). Be,: Bunsenges. Phys. Chem. 90, 854. Moore, C. E., and Broida, H. P. (1959). 1 Res. Natl. Bur Stand. Sect. A 63, 19. Muranaka, Y., Yamashita, H., Sato, K., and Miyadera, H. (1990). 1 Appl. Phys. 67, 6247. Sharpe, S., Hartnett, B., Sethi, D. S., and Sethi, H. S. (1987). 1 Photochem. 38, 1. Stalder, K. R., and Sharpless, R. L. (1990). 1 Appl. Phys. 68, 6187. Stoner, B. R., Williams, B. E., Wolter, S. D., Nishimura, K., and Glass, J. T. (1992). J Muter Res. 7, 257. Toyoda, H., Kojima, H., and Sugui, H. (1988). App. Phys. Left. 54, 2043. Toyoda, H., Childs, M. A,, Menningen, K. L., Anderson, L. W., and Lawler, J. E. (1994). 1 Appl. Phys. 75, 3142. Toyoda, H., Goto, M., Kitagawa, M., and Hirao, H. (1995). Jpn. 1 Appl. Phys. 34, L448. Wamsley, R. C., Mitsuhasi, K., and Lawler, J. E. (1993). Rm. Sci. Instrum. 64, 45. Welter, R. D., and Menningen, K. L. (1997). 1 Appl. Phys. 82, 1900. Wendt, A. E., Beale, D. F., Hitchon, W. N. G., Keiter, E., Kolobov, V,Mahoney, L., Pierre, A. A,, and Stittsworth, J. (1998). U. Kortshagen (Ed.), Electron Kinetics and Applications of Glow Discharges. NATO AS1 Series, in press. Wolter, S. D., Stoner, B. R., Glass, J. T., Ellis, F! J., Buhaenko, D. S., Jenkins, C. E., and Southworth, I? (1993). Appl. Phys. Lett. 62, 1215.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS MINER HIPPLER Institut fur Physik, Ernst-Moritz-Arndt-UniversitatGreifswald, DomstraJe 1 Oa, Greifswald. Germany I. Introduction.. . . . .. .. . . . . . .. . . ... .. . . . .. .. . . ... . . .. .. . . . . . .. . . . . . . . . . .. . . .. . .. . . .. . 11. Theoretical Considerations . . . .. . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . .. . . . . . .. . . . . .
Binary Collision Model.. . .. . . . . . . . .. . . . .. .. . . .. . . . .. . . . .. . . . . . . .. .. . .. .. . . .. . 1. Scattering Angle and Energy Transfer . _... _.. . . . . . . .. . . . .. . .. . . . .. .. . . .. . 2. Stopping Power . . . .. . . . a. Nuclear Stopping Power.. . . . . .. . . . . .. . . . .. . . . . . .. .. . . . . . . . . . .. . . .. .. .. . b. Electronic Stopping Power. 3. Sputtering Yield.. .. .. . . . . . .. .. . . . .. . . . . . .. . . . . . .. .. .. . .. .... .. . . . . . . . .. . . . . 4. Computer Simulations Based on the Binary Collision Model.. .. . . . . . . . . Molecular Dynamics Model.. .. .. .. . . . ... . .. . . . . . .. . . . .. .. . .. . . .. .. .. . . . . . . . . Scattering Potentials ... . . .. . . . .. .. . . . .. .... .. . .. .. .. . . . . . .. . .. . . .. .. .. . . .. . . . . .......... 1. Repulsive Potentials. .. .. a. Screened Coulomb Potentials. . . . . .. . . . . .. . . . . . . . . . . . . .. . . .. . . . . ... . . . . . b. Born-Mayer Potential .. . . .. . . . . . . .. .. .. .. . . . . . . . . . . . . .. . . .. . . .. . . . . . . . . 2. Attractive Potentials. . . . . . . . . . . .. . . . . . .. . . .. . . . . . . . . . . . . .. .. . . .. .. .. . . . . . . . . a. Embedded Atom Potential. .. .. . . . .. . . .. .. . . . . . .. . . . .. .. . . .. . . .. . . . . . . . . 111. Scattering of Ions at Surfaces.. .. . . . .. .. . . . .. . . . . . . A. Implantation., . , .. . , , ., . , . , , . , . , . , , . . . . . , .. . . . . . . . .. . . . .. . . . . . .. . . .. . . . . .. .. .. . B. Backscattering.. . . .. . .. . . .. . . . .. . . . . . . . . . . . . . . . . . . I\! Physical Sputtering ............................................................... A. Projectile Energy Dependence. . . . . . .. . ... . . . .. . . . .. .. .. . .. .... . . . . . . . . . . . . .. . B. Angular Dependence.. .. . .. . . . . . . . .. .. . . . .. .. . . . . . .. . . . . . .. .. . . . . .. . . . . . . . .. . .
.................................... .......................................................
341 343 343 344 345 346 348 349 351 352 353 353 353 354 355 358 358 358 359 361 361 362 364 367 367 370
I. Introduction The interaction between plasma and surrounding walls plays an important role in almost all kinds of plasmas, including low-temperature plasmas for technical applications and high-temperature plasmas for fusion research. Many of the underlying fundamental processes are only partly examined, and details of the interactions of the plasma particles with solid surfaces (substrate, walls) are very often unknown. The main constituents of a plasma, e.g., electrons, ions, neutrals, 341
Copyright (02000 by Academic Press All rights of reproduction in any form reserved. ISBN: 0-12-003843-9/ISSN: 1049-250X $30.00
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Rainer Hippler
radicals, and metastables, all interact differently with a solid, giving rise to a large variety of different effects. Interest arises because of the fundamental importance of such processes and because of the technological applications for which these processes are important. For example, plasma etching due to heavy particle impact on surfaces plays an important role in the manufacturing of solid-state devices and computers. On the other hand, the erosion of walls surrounding the plasma by energetic particle impact is one of the major problems in fusion devices, giving rise to an unwanted cooling of the fusion plasma. In this chapter we shall review the basic mechanisms of the interaction of sufficiently energetic atoms or ions with solid surfaces. Among the various interaction processes are scattering of the impinging atomic particle from the surface, deposition on and implantation into the surface, and modification and erosion of the surface (Fig. 1). The contents of the present article are not intended to give a full coverage of the plasma-surface interactions; rather, they describe some of the major mechanisms and provide a few illustrative examples of the various processes.
Secondarv products
FIG. 1. The interaction of an incident atomic particle with kinetic energy E, and charge q with a solid surface (see text).
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 343
11. Theoretical Considerations The basic theoretical concepts describing the interaction of a sufficiently massive and energetic particle with a surface are the binary collision (BC) model and the molecular or classical dynamics (MD) model. A. BINARYCOLLISION MODEL The binary collision model is applicable as long as classical trajectories are justified and quantum mechanical trajectory effects are negligible. The interaction of the impinging particle with the surface is considered through individual binary collisions with the atoms that constitute the surface of the solid, and only one binary collision event at each instant is considered. The total interaction of the ion with the atoms that constitute the solid and the interactions among the target atoms then fall into a sequence of binary collisions. The binary collisions themselves may be treated fully classically or quantum mechanically or within the semiclassical approximation. In most cases of practical relevance to plasma physical applications, a classical description suffices. In that case, a relatively simple and straightfonvard relationship exists between the scattering angle, the impact parameter: the lunetics of the collision and that is complicated only by the frequently not so well known interaction potential between the colliding partners. The interaction of an ion or atom with another atom leads to an exchange of momentum and energy. Conservation of momentum and energy requires that the total momentum and the total energy be the same before and after each collision; i.e., for a target atom initially at rest, (1) Eo = EI + E2 + QiOss where Eo is the kinetic energy of the projectile prior to the collision, El and E2 are the kinetic energies of the projectile and the target atom, respectively, after the collision, and Qloss is the inelastic energy transfer. Similarly, conservation of momentum requires
mlGo = mlGl
+ m2G2
(2) where m, and m2 are the mass of the projectile and the target atom, respectively, Go is the projectile velocity prior to the collision, and GI and are the projectile and target atom velocity, respectively, after the collision. Equation (2) may be rewritten as
s2
m, uo = m , u1 cos 8, 0 = m, u, sin O1
+ m2u2cos 8, + m2v2sin 8,
(3)
where O1 and 8, are the angles of the scattered projectile and the recoiling target atom, respectively, with respect to the incident direction.
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Rainer Hippler
1. Scattering Angle and Energy Transfer
For one collision partner at rest prior to the collision, the kinematics are depicted in Fig. 2. The exact amounts of transferred momentum and energy depend on the details of the collision, for example, impact parameter b and scattering angle Bs. In the laboratory system, the energies E, and E2 of the projectile and the target atom, respectively, after the collision are given as
where A = m2/m1 is the mass ratio and f 2 = 1 - (1 + A - ’ ) Q / E o . Simpler relations are obtained in the center-of-mass system. The center-of-mass scattering angle Ocm relates to the scattering angles el and O2 as tanel = tane -
-
For the kinetic energies, we obtain
FIG.2. Binary collision kinematics.
Af sinO,,
1
+ Af cos Ocm sin Ocm
I
-fCOSeCm
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 345
The maximum energy transferred from the projectile to a target atom EmaX(Q) is obtained for Ocm = 90":
and equals
for Q = 0 (f = 1). The scattering angles depend not only on the collision parameters-for example, the impact parameter &but also on the interaction potential V(r). Particularly at low relative velocities of the colliding partners, a proper choice of the interaction potential may become crucial. In the center-of-mass system, the scattering angle Ocm may be obtained as 00
dr
Ocm = x - 2b s,o
(9)
r2 J1 - V(r)/Ecm- (b/r),
where R, is the minimum distance during the collision, obtained from 1 - V(R,)/Ecm- (b/&J2 = 0
(10)
and where
is the center-of-mass energy. 2. Stopping Power
Projectile ions impinging on the surface that hit the surface without being reflected or backscattered are either deposited onto the surface or implanted into the solid. The depth distribution of the implanted projectile ions is governed by the energy loss of the projectile inside the substrate. The specific energy loss per distance or stoppingpower S is expressed as dE S(E) = dx It is related to a stopping cross section S,,(E) as
S(E) = NS,(E)
where N is the atom density in the solid. The concept of a stopping is based on several approximations, e.g., the continuous-slowing-down approximation, and
346
Rainer Hippler
straight-line trajectories (e.g., ICRU, 1993; Inokuti, 1996). The maximum distance R, of the implanted ions is obtained from
1;
S(E) dx = Eo
where E, is the kinetic energy of the incident projectile, E = E(x) is the kinetic energy of the projectile inside the solid, and x is the distance from the surface. Again, Eq. (14) holds only in the limit of the continuous-slowing-down approximation, which becomes questionable for energy losses AE comparable to E,, which is the case at low kinetic energies. To the total stopping power both nuclear S, and electronic S, stopping contribute.
a. Nuclear Stopping Power. The nuclear stopping power results from the direct interaction (scattering) of the two colliding nuclei with each other. This interaction leads to a relatively large momentum and energy transfer from the projectile to the substrate atoms, and, thereby, to a large displacement of the involved atoms. The nuclear stopping power may be calculated from the cross section for Rutherford scattering, i.e., the scattering of two unscreened nuclei with projectile charge Z,e and target atom charge Z2e, as (Sigmund, 1981) ml
da(E,E,) = n-(Z m2
I
dE 'I E2Eo
for0 5 E 5 Em,
Z e2
(15)
Equation 15 holds for an unscreened Coulomb potential and hence is justified for sufficiently large projectile energies, i.e., for t 2 1. Here the reduced energy t is defined as
where E J a ) = ZlZ2$/a, and a is the so-called screening length (see below). At lower projectile energies ( E i l), the colliding nuclei penetrate less deeply into each other's Coulomb field, and the screening of the repulsive nuclear potential by surrounding electrons becomes important. Assuming an interatomic potential V(R) of the type
V(R)0:R-lJrn the energy loss cross section may be expressed as (Sigmund, 1981)
do(E,EO) S C,
dE E;El+"'
~
for 0 5 E IEm,,
(17)
FUNDAMENTAL PROCESSES OF PLASM-SURFACE INTERACTIONS 347
where (19) and 1, is a dimensionless factor that varies slowly from 1, = 24 at low energies where m = 0 to 1, = at high energies where rn = 1 (Sigmund, 1981). The stopping cross section S,,(E,) is obtained from
+
and y = 4mlm2/(ml m2)2. As a function of incident energy, the stopping cross section thus rises approximately linearly from low energies (m = 0) and, afier reaching a maximum, decreases with E i l at large energies where m = 1 (Fig. 3). In compact form, the stopping cross section S,(E,) may be expressed as (Sigmund, 1981)
where the so-called reduced stopping power ~ ~ ( is6 a) universal function largely independent of projectile and target mass and charge. For the reduced nuclear
0,001
0.01
0,l
1
10
Reduced energy E
FIG 3. Reduced nuclear stopping power vs. reduced energy 6 calculated for three different interaction potentials (solid lines; after Sigmund, 1981). Also shown are the predictions from Eq. (23) (open circles).
348
Rainer Hippler
stopping power s,(c), (Eckstein, 1991)
an approximate analytical expression was given by 3.441,/71n(t +2.718)
”(‘)
=1
+ 6 . 3 5 5 4 + ~ ( 6 . 8 8 2 -4 1.708)
where t is defined by Eq. (1 6). The reduced nuclear stopping power calculated for three different interaction potentials is shown in Fig. 3. It is quite apparent that particularly at low reduced energies, significant deviations may arise for different choices of the interaction potential. Also shown is the reduced stopping power calculated with the help of Eq. (23). b. Electronic stopping power. The electronic stopping power results from the interaction of the impinging ion or atom with the electrons inside the solid. The electronic stopping cross section may be calculated from quantum mechanical theory. Bethe’s (1930) result is based on the first Born approximation and may be expressed as (Inokuti, 1971; Paul et al., 1994)
where T = Eo x m,/m,, E , is the incident energy, m, and mp are the electron and projectile mass, respectively, z is the projectile charge, Z is the number of target atom electrons, a. is the Bohr radius, R is the Rydberg energy, and In is the mean excitation energy. Although Eq. (24) is based on atomic scattering theory, it has, particularly for large incident velocities, certain merits in describing the interaction of fast particles with solids as well. Figure 4 displays the stopping cross section (in units of eV.cm2) for protons incident on solid (amorphous) carbon and for a mean excitation energy I, = 78 eV calculated with the help of Eq. (24). In comparison with other estimates for the stopping cross section (Kaneko, 1993; Ziegler et al., 1985), Eq. (24) appears reasonable for large energies T >> I,, which are, however, of little relevance to low-temperature plasma physics. The apparent failure at low energies largely reflects the use of a mean excitation Zn that is a reasonable approximation at high energies but ignores the electronic shell structure of atoms and solids. Particularly in solid materials, typical excitation energies (e.g., electron-hole pair production, excitation of plasmons) can be as small as a few eV; such small energy losses are thus not properly accounted for if a mean excitation energy is used. To account for this deficiency, so-called shell corrections as well as other corrections have been introduced. Better agreement is thus expected by theories that additionally take the shell structure or, alternatively, the dielectric response of the electrons inside a solid into account (e.g., Egerton, 1996; Paul et al., 1994, and references therein).
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 349
incident energy (keV)
FIG.4. The electronic stopping cross section vs. incident energy for protons on amorphous carbon: Solid line: Bethe theory [Eq. (24)]; dashed line: Lindhard and Scharff(1961) [Eq. (25)]; 0: Kaneko (1993); and A: Ziegler et al. (1985).
An analytical formula based on the dielectric response model that is applicable at low particle velocities was given by Lindhard and Scharff (1961):
The results from Eq. (25) are also shown in Fig. 4. Whde electronic stopping may yield to a significant excitation including ionization of electrons, thereby causing an additional stopping of fast-moving projectiles, the momentum and energy transfer to the substrate nuclei nevertheless remain relatively small. The major contribution to the sputtering yield therefore stems from the nuclear stopping, and contributions from electronic stopping are generally weak and comparatively small. Exceptions may occur at very high projectile energies where the nuclear stopping is small, and with highly charged ions at low velocities because of the large potential energy then carried by the projectile (potential sputtering, e.g., Sporn et al., 1997). 3. Sputtering Yield
Sputtering, unlike (thermal) evaporation, is a collision process by which an atom formerly bound to a solid becomes liberated; the energy needed to liberate the atoms is provided by binary collisions either with the projectile (direct knock-on) or with other target atoms that received their kinetic energy through a sequence of
350
Rainer Hippler
collisions (collision cascade) originating from the projectile. The number of atoms involved in such a collision cascade may vary widely, depending, for example, on the kinetic energy of the projectile and on the projectile/target atom combination. The sputtering yield Y(Eo)is defined as the number of sputtered atoms per incident projectile ion, number of sputtered atoms y(Eo)= number of projectile ions
(26)
It is a function of the projectile energy Eo and also depends on collision parameters like projectile mass ml, target atom mass m2, projectile and target nuclear charge 2 , e and Z2e, where e is the elementary charge, and the angle of ion incidence einc. Within the collision cascade model, the sputtering yield is calculated from the energy deposited in a certain depth of the target or substrate, which is related to the stopping power Sfl(E,),and from the transport of the moving substrate atoms to the surface. The sputtering yield Y may be expressed as (Bohdansky, 1984) where Q is the so-called yield factor, g ( & / E t h ) factor that accounts for threshold effects, dE0IEth)
=
- (Eth/E0)2’31
is a semiempirical correction
(l - Eth/EO)2
(28)
where the threshold energy Eth,which is a function of the mass ratio m2/ml (Bohdansky, 1984; Eckstein et al., 1991), may be calculated from an analytical expression that was obtained from a fit to experimental and theoretical sputtering yield data (Eckstein et al., 1983),
5 UO = 7(E2-o.54+
1.12
0.15k)
Following Sigmund (1 98 l), the yield factor Q can be expressed as O1
Q = 0.01 1 7 5 ~ 0x ~UO
(30)
where U, is the surface binding energy of the atoms and a. is the Bohr radius. In the derivation of Eq. (30), Sigmund assumed a screened Born-Mayer potential with a 2-independent screening constant, unlike the case in Eq. (43) (see below). The dimensionless factor 01 = a(E0, ml/m2, Oinc) depends weakly on the incident energy Eo but shows a pronounced dependence on the mass ratio m2/m1 and on Oinc, the angle of incidence relative to the surface normal. Figure 5 shows the dependence of 01 as a function of the mass ratio m2/ml as calculated by Sigmund (1981) and according to Bohdansky ( e g , Eckstein et al., 1983). Sigmund’s
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 35 1 1
'
'
'
.
,
I
.
,
I
,
.
,
.
.
.
,
,
, " ' , T
1.5 -
/
FfSigrnundJ Bohdansky
1I
iI
1.0 -
a
II
I
I
I
-
/ I
,
II
0,s
~
/ /
/
I
I
/
/&-
--\\.
0,O'
'
"',,..'
'"'....'
'"',,,.L
calculation shows a rather steep dependence of a on m 2 / m l ,while Bohdansky's result, which is close to a % 0.17, is in better agreement with experiment. The reason for this apparent deficiency of Sigmund's calculation is an overestimation of the energy deposited inside the solid by light ions. 4. Computer Simulations Based on the Binary Collision Model
The binary collision approximation (BCA) model was the first to be used in computer simulations of ion-solid interactions (Bredov et al., 1958). The usefulness of computer simulations was further demonstrated by Robinson and Oen (1963) during their discovery of the channeling effect. Computer simulations based on the BCA model in essence fall into two categories, those that assume a crystalline structure of the solid and those that, as in calculations based on the TRIM code, assume a randomized or structureless target. In simulations assuming a crystalline structure, the collision sequence is deterministic once the impact point of the projectile at the surface and its direction into the solid are given. A list of target atom positions is then required; this can be constructed as in standard solid-state theory with the help of three primitive translation vectors, starting from a basis of one or more atoms. A major and elaborate task in such simulations is, hence, the calculation of a list of next neighbors for each collision sequence and the search procedure to find the next collision partner. These problems are to a large extent circumvented by simulations that employ a randomly structured solid target. Now the next collision partner may be found
352
Rainer Hippler
by a random selection process, which is why these simulations are sometimes called Monte Car10 programs. A rather popular code is the TRIM (transport of ions in matter) program (Biersack and Haggmark, 1980), which comes in various versions and modifications, such as TRIM.SP (Moller and Eckstein, 1984) and TRIDYN (Biersack and Eckstein, 1984). A discussion of the various simulation programs may be found in Eckstein's (1991) book. B. MOLECULAR DYNAMICS MODEL
In the classical (CD) or molecular dynamics (MD) model, the movement of all atoms inside a solid is studied as a function of time. The model thus takes the interaction with all neighboring atoms into account. The starting point is Newton's equation for the motion of a single atom i, which is governed by the interaction forces from all target atoms in the neighborhood,
-
d2Zi(t) mi -= C Fii = F,(Zi(t)) dt2 j=l +
where N is the number of atoms taken into consideration. The forces pv can be calculated from the interaction potentials assumed in the calculations. Once all forces interacting on atom i have been computed, Eq. (31) may be integrated numerically to yield the new position Zi(t At). It is apparent that a good algorithm is required in order to achieve good computational accuracy and sufficient computational speed simultaneously. A common algorithm is the socalled Verlet (1967) algorithm, but other schemes are in use as well (Eckstein, 1991; Frenkel and Smit, 1996). The Verlet algorithm is based on a Taylor expansion of Zi(t At),
+
+
F.(t)
Zi(t+ At) = Zi(t) + ;i(t) At + 2m,
1 d3Zi(t)
+ 7 7(At)3+ 0
~ ) (32) ~ )
and of Zi(t - At), Z,(t
gi(t)
1 d3Zi(t) (At)3 O((Atl4) 3 dt3
+
- At) = ZJt) - ;i(t) At + -(At)2 - -2mi
(33)
+
Summing these two equations, one obtains the new position Zi(t At) from the former two positions Zi(t)and Zi(t - At),
Zi(t+ At) = 2SEi(t) - Zi(t - At) +
k(t) mi
(At)*
(34)
which is accurate to within U((At)4). Obviously, making the time step At sufficiently small enhances the computational accuracy at the expense of long
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 353
computing times. Hence, the choice of the step size is a compromise between these two limitations; typical time steps are in the range of several fs, and several hundred time steps may be required for a full calculation. The molecular dynamics approach has become quite popular for scattering at low projectile velocities, where the interaction can be limited to rather few atoms and where small cell sizes (several hundred to a few thousand) may suffice. C. SCATTERING POTENTIALS There are various choices for the interaction potential V ( r )between projectile and target atom or between two colliding target atoms to be found in the literature. In most cases, the interaction potentials are only approximately known, however. 1. Repulsive Potentials
The interaction between two colliding atoms, particularly for small internuclear separations, is dominated by the Coulomb force between the two positively charged nuclei, which is repulsive by nature. Surrounding electrons modify and partly screen the repulsive Coulomb potential, particularly at large internuclear distances, and therefore realistic potentials drop off more quickly than pure Coulomb potentials. a. Screened Coulomb Potentials. A simple Ansatz is to use screened Coulomb potentials
where,f;.(r/a) is a screening hnction and a is a (typical) screening length. The simplest screened Coulomb (Bohr) potential uses a single exponential for the screening function, = exp(-r/a)
(36)
For the screening parameter, Firsov (1958) proposed aF =
0.8853~~ 112 213
(2y2+z,
(37)
while Lindhard et al. (1 968) used
where a. is the Bohr radius. This choice of screening function ignores the shell structure of realistic atoms, which are composed of many electron shells. A more
354
Rainer Hippler
realistic Ansatz, therefore, is a screened Coulomb potential that (partly) take the shell structure into account. The first one to come to mind is the Moliere potential, which is obtained using Eq. (35) with a modified screening functionf,, f d r ) = 0.35 exp(-0.3r/aF)
+ 0.55 exp(-1.2r/aF) + O.lOexp(-6.0r/aF) (39)
Another more realistic Ansatz is the Ziegler-Biersack-Littmark which the screening functionfZBLis given by
potential, for
+ 0.2802 exp(-0.4O29r/azBL) + 0.5099 exp(-0.9423r/azBL) + 0.1818 exp(-3.2r/azBL)
j&.(r) = 0.02817 exp(-0.2016r/az,,)
(40)
with the screening parameter uZBL given by
Figure 6 compares these screening functions with one another. As a major difference, we note that the Bohr potential drops off rather quickly, while the other screening functions show a significantly weaker dependence on the internuclear separation. b. Born-Mayer Potential. A repulsive potential that is frequently used in the literature largely because of its simplicity is the Born-Mayer potential (Born and
lntemuclear distance (a) FIG.6. A comparison of the screening functions V(r)/Vc(r)for the Coulomb (dash-dotted line), Bohr (solid line), Molitre (dashed line), and Ziegler-Biersack-Lindhard (dotted line) potentials.
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 355
Mayer, 1932), V(r)= CBM~ x P ( - ~ / ~ B M )
(42)
For the energy parameter CBMand the screening length uBM, Andersen and Sigmund (1 965) proposed CBM
= 52(Z,Z2)1’4eV
and
uBM
= 0.219
8,
(43)
2. Attractive Potentials
So far we have discussed interaction potentials that are purely repulsive and, hence, do not take into account the fact that realistic interactions, particularly for atomic combinations that form stable molecules or solids, may have to become attractive for large internuclear separations. Such potentials exhibit a potential well D at a certain internuclear distance r,,. A frequently applied potential was proposed by Morse (1929), V ( r ) = Dexp[-2aM(r - rO)]- 2Dexp[-ccM(r - ro)]
(44)
where ccM is a constant that determines the slope of the potential and, hence, the zero point. Also in use is another attractive potential introduced by Lennard and Jones (1924). A popular version of the Lennard-Jones potential is the so-called 6-12 potential, V(r)= 2 D c ) 6 - D c ) 1 2
(45)
where the rP6dependence is due to the dipoldipole interaction’s leading to van der Waals forces that govern the binding of van der Waals complexes but not necessarily that of other compounds. No physical justification is given for the second r-12 term except that it has to drop off more quickly than the first term to yield a (partly) attractive potential. In Fig. 7 the purely repulsive Molihre potential is compared with the (partly) attractive Morse potential. It is to be noted from this comparison that the Morse (like the Lennard-Jones) potential, while providing a realistic description of the attractive part of the interaction, becomes insufficient at low internuclear separations where the purely repulsive potentials are more adequate. While more sophisticated potentials calculated, for example, by employing the Dirac-FockSlater (DFS) method (Eckstein eta!., 1992), have recently become available, such potentials are generally more complicated and are available only in numerical form, and are thus not very handy for the calculations of interest here. For Si-Si collisions, the interaction potential, being repulsive for small and attractive for
356
Rainer Hippler
Internuclear distance (a,)
FIG. 7. V(r)/Vc(r)vs. internuclear distance for the Morse potential (solid line) and the Molikre potential (dashed line) of Ni-Ni. VJr) is the Coulomb potential.
Internuclear distance (a,)
FIG. 8. A comparison of the Si-Si screening function calculated within the Dirac-Fock-Slater model (Eckstein et al., 1992) and for the Mokre potential.
large internuclear distances, may be approximated by Eq. (35) with a screening fimction given by
1
(
fi,$T(r/UF)= [l - 0.005713(r/aF)2]0.35 exp -0.28-
-0.002327(&)
exp[ -o.I~(;
:F)
- 5.757)’]
1
(46)
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 357
Figure 8 compares the Dirac-Fock-Slater potential for Si-Si collisions with the corresponding Molibre potential. While for low internuclear separations the two potentials are almost indistinguishable from each other, significant differences occur for medium and large internuclear separations, where both potentials are weak. Nevertheless, trajectories calculated with these two potentials differ significantly at large impact parameters, where the DFS potential becomes attractive (Fig. 9).
n
s 6)
9
.3
n
Distance (A) FIG. 9. Calculated trajectories for Si-Si collisions (from Eckstein, 1991).
358
Rainer Hippler
a. Embedded Atom Potential. In the embedded atom model, the energy Ui of each atom inside a metal is calculated from the energy needed to embed the atom, at a given locality, in the local electron density provided by the surrounding atoms (Daw and Baskes, 1983),
ui = +(pi)
+ -1 c K] 2 i
(47)
where K, describes the core-core repulsion between atoms i and j . The electron density pi may be approximated by a superposition of atomic densities $Jrij) that depend on the interatomic distance between atoms i and j ,
where
with effective charges Zi(r) and Z,(r). The embedding function F is chosen to fit the bulk properties of the solid. A rather simple form is F ( p ) IX but various other and more accurate choices for the embedding function have been proposed; for a discussion, see, e.g., Frenkel and Smit (1996).
111. Scattering of Ions at Surfaces A. IMPLANTATION
The interaction of energetic (fast) ions or atoms with surfaces leads to the deposition of the incident atomic or molecular particles on the surface and to implantation into the solid. The range of the incident projectiles in the solids is largely determined by the incident kinetic energy. Figure 10 shows calculated trajectories for 10-keV H+ ions at normal incidence on solid carbon and solid gold. Compared to that for the gold target, the projected range is larger in carbon while the lateral spread is smaller, which is due to the smaller stopping power of low-Z atoms. Naturally, the projected range inside the solid increases with incident projectile energy. Figure 11 shows experimental results for the mean projected range of He+ ions in silicon in comparison with theoretical results obtained from analytical theory and from TRIM calculations (Eckstein, 1991). In general, the agreement between experiment and theoretical predictions is rather good.
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 359
Lateral spread (nm)
Lateral spread (nm) FIG. 10. Projected trajectories for 10-keV H+ ions on carbon and gold (100 trajectories each) calculated with the TRIM code (after Eckstein, 1991).
B. BACKSCATTERING Particularly at low incident energies, the incident ion has a fair chance to escape deposition or implantation and be backscattered instead. The particle reflection coefficient R is defined as the ratio of the number of reflected relative to the number of incident particles (ions),
R(Eo,einc)
=
number of reflected particles number of incident particles
(50)
The calculated particle reflection coefficients for nickel with H+ and Ne' ion bombardment versus incident energy using the TFUDYN code (Biersack and Eckstein, 1984) are displayed in Fig. 12. The reflection coefficient is rather large
360
Rainer Hippler
Incident energy (eV)
FIG. 11. Mean projected range vs. incident energy for He' incident on silicon. The experimental results ( 0 ) are compared with mean ranges obtained from analyt~caltheory (solid line) and from TRIM calculations (dashed line) (after Eckstein, 1991).
at low incident energies, amounting to about 50% at 100 eV for incident H+ and to about 30% for Ne' ions. The reflection coefficient decreases monotonically toward larger energies (Eckstein, 1991). This behavior reflects the fact that the average scattering angle decreases with incident energy and increasing projectile mass.
ion energy (eV) FIG. 12. Reflection coefficient of nickel for H+ and Ne+ ion impact vs. incident energy (after Eckstein, 1991).
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 36 1
IV. Physical Sputtering A. PROJECTILE ENERGYDEPENDENCE Measured sputtering yields for nickel by H+, Ne', and Ni+ ion bombardment versus incident energy are displayed in Fig. 13. Also shown are theoretical calculations based on TRIDYN code (Biersack and Eckstein, 1984) and on the Bohdansky (1984) formula. It is rather obvious that the sputtering yields for the heavier Ne' and Ni' projectiles are quite different, both in magnitude and with respect to the position of the maximum, from those with Hf impact (after Eckstein, 1991). There are several mechanisms that contribute to the total sputtering yield. According to Eckstein (1 99 1), one may distinguish between primary knock-on atoms (PKA), which receive kinetic energy and momentum directly from the projectile, and secondary knock-on atoms (SKA), which receive energy and momentum from other fast-moving target atoms. A further distinction arises from the actual direction of the projectile at the time of the interaction, i.e., whether the projectile is moving inward brojectile-in) or outward @rojectile-out)with respect to the target surface. The relative importance of the four contributions as a function of incident energy is displayed in Fig. 14. At low incident energies, the projectile-out contribution dominates, largely because there is little energy to transfer and the chance of the projectile becoming backscattered is relatively
102r
'
"""'
'
102
"'''''1
' """"j
' " " " 1
103
. , . . 104
,
105
Ion energy (eV) FIG. 13. Sputtering yield of nickel for H+ and Ne' ion impact vs. incident energy (after Eckstein, 1991).
3 62
Rainer Hippler
Incident energy (eV) FIG. 14. Contribution of the various types of sputtering processes (see text) to the total sputtering yield vs. incident energy for Ne+-Ni collisions S. 178 (after Eckstein, 1991).
large. Here, primary knock-on is by far the most important process. At higher energies, the fraction of backscattered projectiles decreases with E i 2 , while the total energy being transferred to the target atoms increases. This increases the number of projectile-in events at the expense of the projectile-out processes. Now, secondary knock-on through the formation of a collision cascade dominates the sputtering event.
B. ANGULAR DEPENDENCE
The pronounced angle-of-incidence dependence of the ct parameter is displayed in Fig. 15. Since the energy that is deposited by the projectile in a certain depth from the surface increases with decreasing angles of incidence eincroughly in proportion to c0s-I 0,,,, a similar increase in the sputtering yield Y may be expected. The experimental data (Roth et al., 1979) vaguely follow this prediction for small angles but differ significantly at large angles of incidence. Here the sputtering yield goes through a maximum at 0%:, after which it decreases again. Better agreement is obtained with more detailed calculations based on the collision cascade model (Sigmund, 1981) or with TRIM calculations. According
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE TNTERACTIONS 363
4- Trim
30
0
60
90
Angle of ion incidence (deg) FIG. 15. The parameter a vs. angle of incidence Q,,,.
to Yamamura et al. (1983, 1984), the angular dependence of CI may be expressed as
The following empirical expressions for f and q have been given by Yamamura e t a l . (1983, 1984):
f = (0.94 -
where IZ is the atom density of the target material (in atoms per A3). Tabulated values off and q may be also found in Eckstein et al. (199 1). According to these formulae, the maximum of the sputtering yield is found for
after which the sputtering yield decreases.
364
Ruiner Hippler
c. ENERGYDISTRIBUTION OF SPUTTERED PARTICLES Within the collision cascade model, the energy distribution N(E) of sputtered atoms is proportional to
where E.s is the kinetic energy of the sputtered target atoms. In the derivation of Eq. (55), a potential of the type V ( R ) 0: R-‘’’” [Eq. (17)] has been assumed. For m = 0, Eq. (55) refers to Thompson’s (1 968) formula. At low energies, threshold effects due to the possible maximum energy transfer need to be taken into account (Betz and Wien, 1994),
where En,,, is given by Eq. (7) or (8). The energy distribution of sputtered target atoms are frequently determined by optical means. In that case, the Doppler shift and Doppler broadening of emitted spectral lines of excited atoms are investigated (e.g., Betz and Wien, 1994). Figure 16 shows the velocity distribution of sputtered Fe atoms in the ground state and in the metastable state following 10keV Ar’ bombardment of iron (Schweer and Bay, 1982). Also shown is the velocity distribution predicted by Thompson’s formula [Eq. (55) with m = 01. Taking the surface binding energy as a fit parameter, the predicted velocity distributions are in reasonable agreement with experiment for the both groundstate and metastable atoms. Nevertheless, the velocity distributions of groundstate and metastable atoms differ significantly from each other. The observed shift toward larger velocities of the metastable-atom distribution may be caused by deexcitation mechanisms that prevent excited or metastable atoms from escaping intact from the surface if their velocity is too slow. Among the possible mechanisms, we mention resonance tunneling of electrons from a sputtered excited atom back to the surface (Hagstrum, 1954; Ghose and Hippler, 1998). It should be mentioned here that resonance tunneling may be prevented by oxidation of a metal or semiconductor surface, leading to the formation of a band gap in the solid. For example, Fig. 17 displays line-broadening measurements that were obtained for a clean and oxygen-covered aluminum surface. While a rather narrow line shape largely determined by the spectral resolution of the optical spectrometer is observed for the oxygen-covered A1 surface, a much broader line shape is obtained for sputtered Al atoms emerging from a clean Al surface. The resonance tunneling model may be incorporated into Thompson’s formula by adding an exponential escape factor depending on the velocity
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 365
Velocity (km/s) FIG. 16. Velocity distribution of sputtered iron atoms in the ground state ( 0 ) and in the metastable a5F5state (0) following 10-keV Ar' bombardment of Fe (Schweer and Bay, 1982). The solid and dashed lines are corresponding fits according to Eq. (57) (see text).
component uI perpendicular to the surface, allowing those atoms to escape from the surface without being ionized only if their velocity is sufficiently fast:
where A and a are constants that depend on the transition probability for such a nonradiative transition and the interaction distance. Another striking feature observed in such experiments is the very large Doppler broadening of spectral lines from excited A12+ ions. In that case, the broadening of the ionic line is found to be almost double that of the neutral atomic line. Moreover, when the ionic line profile is fitted to Eq. (57), with U, taken as a free parameter, one obtains a rather high value of Uo = 500 eV (Reinke et al., 1991; Hippler and Reinke, 1992). These observations imply that an appreciable fraction of the excited A12+ ions has kinetic energies of more than 1 key This is considerably larger than the kinetic energies with which the excited neutral A1 atoms leave the surface and may be taken as evidence of different production mechanisms for excited A1 atoms and A12+ ions. Presumably, the A12+ions are produced in closer and, hence, more violent encounters than the neutral atoms, which are believed to result from the sputtering cascade.
366
Rainer Hippler
1.o
s $ .-m 0
0.8
0.6
c
-
v
0.4
U J
i7j
0.2 0.0
309.0
309.2
309.4
Wavelength (nm) 1.o h
U
$ 0.8
.m
0
0.6
c
-
W
0.4
.-0 in 0.2 0.0 452.4
452.6
452.8
453.b
453.2
Wavelength (nm) FIG. 17. Line shape of (a) the 3s23d -+ 3s23ptransition (1 = 3092.7 A) in neutral Al and ( b )the 3d +3p transition (d = 4529 A) in ionic A12+following 300-keV Ar' bombardment of clean ( 0 ) and an oxidized (A) aluminum target. The solid line represents a reference line measured with a hollowcathode lamp. In (a), dashed lines are corresponding calculations based on the resonance tunnel model (after Reinke et al., 1991).
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 367
V. Chemical Effects A. CHEMICAL SPUTTERING AND PLASMA ETCHING
Unlike physical sputtering, where the sputtered particles receive sufficient kinetic energy to overcome the surface binding energy via collisions with other energetic particles, surface erosion by chemical effects relies on chemical reactions by which the bonds of surface atoms with their neighbors are broken. Chemical effects are, hence, rather sensitive to the chemical affinity (reactivity) of the agent-surface combination and may show a pronounced temperature dependence. Because of this, a general picture of the chemical erosion process is much more difficult to achieve, and no simple picture has emerged yet. However, as a general remark, chemical sputtering is considered a multistep process, finally leading to the formation of a volatile molecule that escapes into the gaseous phase. Since the formation of these molecules may be exothermic, no minimum energy transfer is required for chemical sputtering. Figure 18 shows the sputtering yield of carbon by low-energy C', O', and Ne' ions. Whereas the sputtering yield for C+ and Nef ions has a threshold at about 100 eV incident energy, the threshold energy appears to be much lower or even zero for 0' ion bombardment. While C+ and Ne+ ions are subject to physical sputtering, 0' ions that form stable molecules with C (CO, CO,) additionally give rise to chemical sputtering, particularly at low energies, where physical sputtering is weak. Hence, this example also demonstrates the selectivity of chemical sputtering and its dependence on a particular agent-substrate combination. As a further significance of chemical sputtering, we
1.2
- l,o c
0
'= ln
k
+ d .
0,a
m
v
z .-
sr
0,6
ol
E
0.4
5a 0,2
0.0
I
0,1
1
10
Energy (keV) FIG.18. Sputtering yield of carbon by low-energy C', (after Roth et al., 1979).
O ' , and Ne+ ions versus incident energy
368
Rainer Hippler
000'
'
300
'
'
400
'
' 500
'
I
600
700
800
I
Temperature (K)
FIG.19. Temperature dependence of the sputtering yield of silicon by H+ ions and its correlation with the formation of SiHt ions detected in a mass spectrometer (after Roth et al., 1979).
mention the pronounced temperature dependence of the sputtering yield that is frequently observed. For example, Fig. 19 shows the measured temperature dependence of the sputtering yield of Si by 300-eV H+ ions (Roth et al., 1979) as well as the measured signal from the detection of the SiH, fragment ion SiH; in a mass spectrometer. The measurements indicate that chemical formation of volatile SiH, molecules is the main erosion process. Further evidence for chemical sputtering is inferred from measurements of the sputtered particles' velocity. While the kinetic energy distribution of particles ejected by physical sputtering is well described by Eq. (9, the energy distribution of atoms resulting from chemical sputtering should be controlled by thermal desorption, which follows
Figure 20 displays the energy distribution of SiF, and SiF, molecules sputtered from Si by 5-keV Ari ions while the Si surface was subject to a XeF, gas flux of 7 x 10l6 molecules/cm2 (Roth et al., 1979). The occurrence of volatile SiF, molecules indicates chemical sputtering. Moreover, the measured kinetic energy dependence of the sputtered molecules is significantly different, i.e., shifted to lower energies, from what is expected for pure physical sputtering. Nevertheless, at the higher kinetic energies (above about 1 eV), the results follow an E-' dependence, which is consistent with the physical sputtering picture provided that a surface binding energy as low as 0.3 eV, and, hence, significantly lower than the regular binding energy, is assumed. Toward lower energies there are indications for additional thermal desorption consistent with chemical sputtering. Chemical and chemically enhanced sputtering have important applications, for example, in dry etching in computer chip production or plasma cleaning. To achieve the small structures required for the most advanced computer technol-
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 369
Energy (eV) FIG. 20. Energy distribution of molecules sputtered from Si by 5-keV Ar+ ions while XeFz is allowed to the reactor (after Roth et a/., 1979).
ogies, a high directionality together with a good etching rate is required. Figure 2 1 displays results from Cobum (1994) that compare chemical etching of silicon by XeF, molecules, sputtering of Si by 450-eV Ar+ ions, and the combined effect, the last resembling the situation in a real plasma, where chemically active molecules and energetic ions simultaneously interact with surfaces. The comparI
'
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XeF2 gas only,
90
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,
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,
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Time (sec) FIG.21 Etch rate ofpolycrystalline silicon under the influence of XeF, gas and 450-eV Art ion impact and under the combined influence (after Cobum, 1994).
370
Rainer Hippler
ison shows that neither chemical etching by XeF, nor physical sputtering by Ar+ ions provides high erosion rates, in contrast to the combined interaction of XeF, and energetic Ar' ions, which gives x10 times larger rates with the simultaneous benefit of a high directionality. This synergetic effect most probably has to do with the activation of chemical reaction by the Ar' impact, which enables the formation of volatile SiF4 molecules.
VI. References Andersen, H. H., and Sigmund, P. (1965). Nucl. Ins& Meth. 38, 238. Bethe, H. (1930). Ann. Physik 5, 325. Betz, G., and Wien, K. (1994). Inf. 1 Mass Speck Ion Proc. 140, 1. Biersack, J. P., and Eckstein, W. (1984). Appl. Phys. 34, 73. Biersack, J. P., and Haggmark, L. G . (1980). Nucl. Instr: Meth. 174, 257. Bissessur, V, and Tsong, I. S . T. (1990). Nucl. Instr: Meth. B52 129. Bohdansky, J. (1984). Nucl. Insir. Meth. Phys. Res. B2, 587. Born, M., and Mayer, J. E. (1932). Z. Physik 75, 1. Bredov, M. M., Lang, I. G., and Okuneva, M. N. (1958). Sov. Phys.-Tech. Phys. 3, 228. Cobum, J. W. (1994). Appl. Phys. A 59, 451. Daw, M. S., and Baskes, M. I. (1983). Phys. Rev. Leffers 50, 1285. Daw, M. S., and Baskes, M. 1. (1984). Phys. Rev B 29, 6443. Eckstein, W. (1991). Computer simulations of ion-solid interactions, Springer Series in Materials Science 10. Springer-Verlag (Heidelberg). Eckstein, W., Garcia-Rosales, G., Roth, J., and Ottenberger, W. (1983). Sputtering data, Report IPP 9/82. Max-Planck-Institut f i r Plasmaphysik (Garching). Eckstein, W., Bohdansky, J., and Roth, J. (1991). In Atomic andplasma-material interaction data for fusion, Vol. 1, Nuclear fusion, @. 51). International Atomic Energy Agency (Vienna). Eckstein, W., Hackel, S., Heinemann, D., and Fricke, B. (1992). Z. Physik D 24, 171. Egerton, R. F. (1996). Electron energy-loss spectroscopy in the electron microscope. Plenum (New York). Firsov, 0. B. (1958). Sov-Phys. JETP 6, 534. Frenkel, D., and Smit, B. (1996). Understanding molecular simulation. Academic Press (San Diego). Ghose, D., and Hippler, R. (1998). In D. R. Vij (Ed.), Luminescence of solids @. 189). Plenum (New York). Hagstrum, H. D. (1954). Phys. Rev. 96, 336. Hippler, R., and Reinke, S. (1992). Nucl. Instr. Mefh. B68, 413. Inokuti, M. (1971). Rev Mod. Phys. 43, 297. Inokuti, M. (1996). Int. 1 Quantum Chem. 57, 173. International Commission on Radiation Units and Measurements (1993). ICRU Report No. 49. ICRU (Bethesda, MD). Kaneko, T. (1993). At. Data Nucl. Data Tabl. 53, 271. Lennard, J. E., and Jones, 1. (1924). Proc. Roy SOC.A 106, 441, 463. Lindhard, J., and ScharfT, V M. (1961). Phys. Rev 124, 128. Lindhard, J., Nielsen, V , and ScharfT, M. (1968). Mut.-Phys. Med. K. Dan. Vidensk. Selsk. 36, 10. Moller, W., and Eckstein, W. (1984). Nucl. Insir. Mefh. B2, 814. Morse, P. M. (1929). Phys. Rev. 34, 57.
FUNDAMENTAL PROCESSES OF PLASMA-SURFACE INTERACTIONS 37 1 Paul, H., Berger, M. J., and Bichsel, H. ( I 994). In Atomic and molecular data needed in radiotherapy. IAEA-TECDOC (chap. 7). International Atomic Energy Agency (Vienna). Reinke, S., Rahmann, D., and Hippler, R. (1991). Vacuum 42 807. Robinson, M. T., and Oen, 0. S. (1963). Phys. Rev. 132, 2385. Roth, J. (1983). In R. Behnsch (Ed.), Sputtering by Particle Bombardment II (p. 91). Springer (Heidelberg). Roth, J., Bohdansky, J., and Ottenberger, W. (1979). IPP-Report 9/26, Garching. Schweer, B., and Bay, H. L. (1982). Appl. Phys. A 29, 53. Sigmund, P (1981). In R. Behrisch (Ed.), Sputtering by Particle Bombardment I, (p. 9). Topics in Applied Physics 47, Springer-Verlag (Heidelberg). Smith R. (Ed.) (1997). Atomic and ion collisions in solids and at surfaces Cambridge University Press (Cambridge). Spom, M., Libiseller, G., Neidhart, T., Schrnid, M., Aumayr, F., Winter, H. P, Varag, P., Grether, M., Niemann, D., and Stolterfoht, N. (1997). Phys. Rev Letters 79, 945. Thompson, M . W. (1968). Phil. Mag. 18, 377. Verlet, L. (1967). Phys. Rev. 159, 98. Yamamura, Y. (1984). Rad. Effects 80, 57. Yamamura, Y., Itikawa, Y., and Itoh, N. In Angular dependence of sputtering yields of monoatomic solids. Rep. IPPJ-AM-26. Nagoya University, Institute of Plasma Physics. Ziegler, J. F., Biersack, J. P., and Limnark, U. (1985). The stopping and ranges of ions in matter (vol. 1). Plenum (New York).
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
RECENT APPLICATIONS OF GASEOUS DISCHARGES: DUSTY PLASMAS AND UPWARD-DIRECTED LIGHTNING AM CHUTJIAN Jet Propulsion Laboratory and California Institute of Technology, Pasadena, CA
............................. I. Dust in Plasma Environments s ....................................... A. Dust in Plasma-Processing B. Suspension, Alignment, and Ordering........................................ C. Dust in Astronomical Plas 11. Elves, Red Sprites, and Blue Jets ................................................ 111. Acknowledgments I\! References ........................................................................
3 74 375 379 381 386 395 395
Intriguing phenomena have recently been observed in gaseous discharges that involve dust particles; and in discharges which, when produced in the ionosphere, lead to formation of upward-directed lightning. These phenomena abound both here on earth and in astronomical objects. Dusty plasmas are found in laboratory RF generators such as those used for microelectronics manufacture; in our planetary system as the stuff of rings and comets; and in planetary nebulae as nucleation sites for simple and complex molecules. Lightning occurs within planetary atmospheres (Yair et al., 1998; Strangeway, 1995) as well as in the earth’s atmosphere and ionosphere. Lightning-accompanied upward-directed ionospheric discharges-so-called elves, red sprites, and blue jets-have recently been detected in the earth’s atmosphere by shuttle, aircraft, and ground observatories. The role of lightning and dust in the nebular gas and during the formation of our own solar system has been studied (Horhnyi et al., 1995). While these environments are seemingly different, they display a commonality because of the plasma. Electron attachment and ionization play key roles, their rates determined by species cross sections and by the electron-energy distribution function in the plasma (Chutjian et al., 1996). Negative-ion formation and surface charging lead to the growth, trapping, and crystallization of dust. In a thunderstorm cloud, the charge separation leading to lightning, the lightning electro373
Copyright 63 2000 by Academic Press All rights of reproduction in any form reserved. ISBN: 0-12-003843-9/ISSN: 1049-25OX $30.00
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magnetic pulse itself, and the quasistatic electric fields establish an electron energy distribution function that depends upon the electric field distribution, atmospheric composition, conductivity, and altitude. Boltunann transport equations are used to describe the electron and ion behaviors in both phenomena (Gurevich, 1978; Drallos and Wadehra, 1989).
I. Dust in Plasma Environments Dust is found in a broad range of environments, including combustion chambers (Podfilipski and Jarosinsla, 1998), plasma discharges (Rosenberg, 1996), arcjet thrusters (Tiliakos et al., 1998), comets, planetary rings (Horhnyi, 1996), planetary nebulae (Horanyi et al., 1995), stellar atmospheres (Dalgarno et al., 1997), and the interstellar medium (Allamandola and Tielens, 1989). Trapped dust affects the operation of semiconductor and micromachined devices either by altering the purity of the final etched surface or deposited layer, or by introducing grit into the micromachined structure (Selwyn et al., 1998; Choi and Kushner, 1994). Dust provides an additional high-area surface for heterogeneous chemical synthesis. It serves as a third body for chemical-bond formation involving neutral and/or ionized reactants and absorbs part of the energy of bond formation, with another part remaining as kinetic, rotational, vibrational, or electronic excitation available to the product species. Two specific environments will be discussed. Negative ions play a role in the operation of weakly ionized plasmas on earth (Garscadden, 1994). They also play a role in molecule formation in the interstellar medium, especially with H and H,. The interactions can be complex, eventually leading to the formation of complex species such as polyaromatic hydrocarbons (PAHs) from streaming and ejected neutrals and ions of stellar winds, quasars, and supernovae (Biham et al., 1998). In both home-based and astronomical dusty plasmas, grains will in general be negatively charged. This follows from the requirement that the net fluxes of electrons and ions to a grain surface be zero. To see this, let us denote the ion and electron masses and temperatures, respectively, as mi, Ti and me, T,. Let 4, be the potential of the grain surface and e the magnitude of electron charge. Assuming small, spherical grains (of diameter A smaller than the Debye length A) and the absence of secondary electrons, the condition is 1 - e4,/kT, = (mi/m,)”2 exp(e+,/kT,) For a simple case of protons and for Ti= T, = T, one has the result that dS= -2.51kT/e. Physically, plasma electrons generally travel faster than ions; hence their flux is greater, and the grain has to charge to a negative potential in order to repel enough electrons to maintain equality with the positive ion flux. In the more general case where electrons can be released from the grain via
RECENT APPLICATIONS OF GASEOUS DISCHARGES
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photoionization, ion sputtering, or secondaries the grain may be either positively or negatively charged. The sign of charge can also depend on the local temperatures and fluxes in a spatially distributed plasma, so that one can have regimes of alternating surface charge with an associated dipole or higher multipole moment.
A. DUSTIN PLASMA-PROCESSING DISCHARGES Discharges in methane (CH,), silane (SiH,), and germane (GeH,) are routinely used in plasma processing to deposit amorphous hydrogenated silicon-germanium films (a-SiGe : H) and amorphous hydrogenated carbon films (a-C : H). These films have wide application in the manufacture of low-band-gap solar cells, linear image sensors, thin-film transistors for liquid crystal displays, and other microelectronic circuit components. The plasmas are rich in positive and negative ions (Howling et al., 1993a), clusters (Howling et al., 1993b), radicals (Kae-Nune et al., 1994; Watanabe et al., 1996a; Kawasaki et al., 1997; Kawasaki et al., 1998a), and dust particles (Watanabe et al., 1996b; Kawasaki et al., 1998b; Kawasaki et al., 1998~).Use of threshold ionization mass spectrometry has given information on total surface recombination probabilities (Kae-Nune et al., 1996; Perrin et al., 1998). The radical species have been identified, and their density has been measured in absorption, using a hollow-cathode lamp or an Ar ion laser (intracavity) (Watanabe et al., 1996b; Kawasaki et al., 1998c), and in emission (Kawasaki et al., 1998b; Kae-Nune et al., 1996). As examples, the absorptions X I A l + BIB, in SiH, and 3p2 ID2 + 3p4s'P; in Si and the emissions A 2 A + X217 in SiH (Kawasaki et al., 1997) and 4p5s'P: --+ 4p2 ID2 in Ge (Kawasaki et al., 1998d) have been used to monitor the spatial density profiles of a plasma and surface recombination (Shiratani et al., 1998). A radio-frequency (RF) plasma reactor, consisting of circular capacitive plates, is combined with one or two polarized lasers to measure the Rayleigh light scattering from the generated particles. Several experimental arrangements may be found, such as in Fig. 1 of Watanabe et al. (1996b) and in Fig. 1 of Kawasaki et al. (1998~). Details of production of Si particles with diameters in the range 10 to 100nm are given in Shiratani et al. (1996a). The physics and chemistry of the plasma are determined by a range of atomic and molecular collision phenomena. These include elastic and superelastic electron collisions, ionization, two- and three-body electron attachment, dissociative excitation and ionization, positive ion-molecule collisions, negative ionmolecule collisions, metastable-state collisions, electron and ion recombination, and ion-ion neutralization. Surfaces play a role through secondary electron emission, heterogeneous reactions, and generation of ground-state, vibrationally excited H, through recombination (Orient and Chutjian, 1999; Gough et af., 1996; Schermann et al., 1994; Hiskes, 1990). A detailed discussion of these phenomena, including cross sections, rate constants, transport data, and method
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of solution to Boltzmann’s equation for the SiH,-H, discharge, may be found in Perrin et al. (1996) and Leroy et al. (1998). The spatial and temporal growth of particles with diameters in the range 1 to lOOnm is an intriguing phenomenon. A number of recent studies have been carried out on SiH, and GeH, discharge plasmas. Given here are just a few results using laser light scattering (LLS) methods, combined with analysis of the spatial and temporal evolution of the LLS intensity (Shiratani and Watanabe, 1998). Shown in Fig. 1 is one apparatus used to measure the LLS intensity and evolution. The reactor consists of two plates (1OOmm diameter) separated by 45 mm. The bottom plate is grounded, and the RF power (about 3 to 30 MHz and 0.05 to 1.O W/cm2 power density) is impressed on the top electrode. Pure SiH, gas (Shiratani and Watanabe, 1998), or mixtures of SiH, and GeH, in Ar (Kawasaki et al., 1998a) have been used as the source gas. The light source is an argon-ion laser with a beam that is sometimes expanded into a rectangular cross section for increased scattering volume. Scattered radiation is detected perpendicular to the directed beam using a scanning monochromator tuned to the incident Ar-ion wavelength (488 nm) or to one of the characteristic emission lines in the atom or radical. For particles of diffusion coefficient D and density n p , the density gradient within the reactor in a direction x normal to the electrode planes is described by the diffusion equation (Shiratani and Watanabe, 1998)
RF Power Supply
x% ,Aperture
Computer
FIG.1. Experimental arrangement of the radio-frequency reactor (R)and Ar-ion laser to measure laser light scattering from suspended dust particles in various reactant mixtures (Shiratani and Watanahe, 1998).
RECENT APPLICATIONS OF GASEOUS DISCHARGES
377
Let N g , Tg, mg, and dg denote the density, temperature, mass, and diameter of the support gas; dsi and msi the density and mass of the Si atoms, n the number of atoms per Si dust particle, and k the Boltrmann constant. The particle diameter A (and hence its mass from the known species density) is given by
The time evolution of the plasma decay after the RF is turned off is measured at various positions x in the reactor, from which the second derivative $np/ax2 in Eq. (2) and hence A in Eq. (3) are evaluated. Representative examples of the spatial distribution of Ar and Ge emissions for a GeH,-Ar plasma are shown in Fig. 2. The time evolution of the particle density and size for an SiH,-Ar plasma can be seen in Fig. 3 of Watanabe et al. (1996b). These measurements are typically made through polarized LLS measurements (Kawasaki et al., 1998d).
A -
0 0
t
GND
10
20 30 Position (mm)
40
t
RF
FIG 2. Spatial distributions of intensities of Ar, Ge, and GeH, emissions, along with total particle amount, in a GeH, (lO%bAr RF discharge (6.5MHz, 80 W, 13Pa) (Kawasaki et al., 1998a).
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FIG.3. Micrographs and representations of (a)hexagonal, ( b )BCC, and ( c )FCC crystal lattices in an 0,-SiH, RF reactor (Chu and Lin, 1994). Shaded areas in the sketch are planes normal to the optical viewing axis of the system. Blurred images are from particles in the slightly out-of-focus third plane.
Here, the intensities Ill and ZL of laser light polarized parallel and perpendicular to an observation plane are measured. Combining these measurements with the absolute total laser light scattering intensity and Mie scattering theory (requiring the particle-size distribution and the complex refractive index), one obtains the size and density of the particles (Shiratani et al., 1996b; Matsuoka et al., 1998). In another photoemission method, laser light is used to photodetach electrons in negatively charged particles and to photoionize neutral particles. The increase in electron density is related to the electron affinity and the ionization potential of the target, which in turn are related to target size (Wood, 1981; Liu et al., 1986). Results have been presented for targets of 1 to 100 Si atoms (Fukuzawa et al., 1996). The spatial and temporal properties of particle growth in silane and germane plasmas have several elements in common. These can be summarized as follows:
RECENT APPLICATIONS OF GASEOUS DISCHARGES 0
0
0
0
379
Particles are trapped nucleate, and grow at the plasma sheath boundary near the powered electrode. Radicals and atoms (GeH,, Ge, SiH, etc.) have maximum concentration at the boundary. Short-lived radicals that have high reactivity and production rate are the principal contributors to the initial growth of particles. The starting particles are negatively charged or neutral, and can be electrostatically suspended and even ordered in the plasma.
B. SUSPENSION, ALIGNMENT, AND ORDERING One of the more fascinating aspects of dusty plasmas is the phenomenon of electrostatic suspension, alignment, and crystal ordering. The forces acting on a dust grain are the resultant of the neutral drag, ion drag, electrostatic, gravitational, pressure-gradient, thermophoretic, and Stokes forces (Garscadden, 1994; Perrin et al., 1994). In the one-component plasma model, one views a particle of charge q immersed in a “sea” of the opposite charge density. The coupling constant of the particle is defined as the ratio of the Coulomb attraction to the thermal (disruptive) energy. If the negatively charged species has an average interparticle distance d and temperature Tj (K), then the coupling constant can be expressed (in SI units) as (Ikezi, 1986; Thomas et al., 1994; Chu and Lin, 1994) 4, r =4nt,kTjd
(4)
If the ratio r is of the order unity, the plasma is said to be strongly coupled. It can then exhibit unusual phenomena such as the formation of liquid and solid structures (Chu and Lin, 1994). If the ratio r exceeds about 170, a Coulomb lattice can be formed (Dubin, 1990). In general, ordered structures are favored under conditions of large particle capacitance or large negative charge (strong Coulomb attraction), low gas temperature (only gentle collisions), and low gas density (low positive-ion density, and hence small Debye screening). Examples of hexagonal, BCC, and FCC crystal structures have been reported by Chu and Lin (1994). Use was made of a 14-MHz reactor (1 W) and a 0.2-tom mixture of 0, and SiH,. Results are presented in Fig. 3. From microimages and Langmuir probe measurements of T,, q, and N,, this plasma corresponds to a coupling parameter M 200, in the regime (r 3 170) for a Coulomb cloud to be formed. Increasing the RF power resulted in a “melting” of the structures as a result of the reduced Debye screening. Dust grains of TiO, have also been suspended by gravitational plus electrostatic forces in a higher-power (30 W) 13.56-MHz discharge (Melzer et al., 1994). Formation of dust grains of diameter 20 pm with intergrain spacing d = 880 pm were observed. By measuring the response of the cloud motion to a
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modulated dc bias, the total charge on the particle was determined. To see this, one calculates the force balance on the suspended charges (Melzer et af., 1994). If one uses x to denote the position of the particle in the sheath above the powered electrode, p for the damping constant, q and m for the particle charge and mass, respectively, E(x) for the time-averaged electric field, g for the gravitational constant, and FCxt(t)for the force arising from the modulated bias on the powered electrode, one has the following equation of motion: m3
+ mbx = qE(x) - mg + FeXt(t)
(5)
A damped particle motion results from a balance between the electrostatic force, gravity, and a superimposed "dither" given by Fext(t).The field E(x) at the particle will depend upon its location, as obtained from a solution to the onedimensional Poisson's equation describing the ion space charge and the sheath at the powered electrode. Hence the total electrostatic force qE(x) is positiondependent. Solving Eq. (S), one obtains for the time-averaged particle's position (SI units) x = xo
+ A R ( o ) sin(wt + 4)
(6)
where xo is the equilibrium position of the particle and the second term in Eq. (6) is the response of the oscillator's amplitude A , frequency response R(w), and phase 4 to the external modulation of the sheath boundary. The relation between the total charge q and both A and oo,the resonance frequency of the system, is given by Melzer et af. (1 994). Hence q can be obtained by measuring either A or wo, or both. Results from measurements of both are shown in Fig. 4. Agreement 1
/
1
I
1
I
.? 200000 2 c
t
150000
-E & 100000
6
-
50000
I 0
a Calculated from on2 A Calculated from A -
20
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30
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I
40
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1
50
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60
I
1
70
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80
Discharge Voltage (Vpp)
FIG.4. Measured charge on suspended TiO, particles of 18 pm diameter as a function of peak-topeak RF voltage. Analysis follows from Eq. ( 5 ) , where measurements are made from the oscillation amplitude factor (A) or from the resonance frequency (a) (Melzer et a l , 1994).
RECENT APPLICATIONS OF GASEOUS DISCHARGES
381
between the two approaches is good, and both lead to total charges of about q = -2 x lo5 e on an 18-pm-diameter TiO, particle. Results have recently been reported in which h e carbon particles couple to generate BCC and FCC three-dimensional crystals, close to one another in space or temporally (Takahashi et al., 1998; Hayashi and Sawai, 1998). Melting was induced by raising the reactor pressure, and hence increasing the Debye shielding. Using dielectric (glass) microspheres of 1 to 50 pm diameter, Snyder et al. (1998) have observed strings of glass beads aligned along the electric field direction. The strings in the plasma can be set in motion independently of one another. The bonding is very likely through an induced bead-to-bead dipole moment. The combined electrical, gravitational, and ion drag forces keep the particles at rest, with the electric field inducing a dipole moment on the resident particle charge (Lee et al., 1997). It would be of interest to repeat these measurements with metallic spheres to give a different type of charge polarization in the dielectric. C. DUSTIN ASTRONOMICAL PLASMAS Dust is present throughout the universe. It is in the interstellar medium as ejecta from novas and supernovas, and in protostellar and protoplanetary accretion disks. Its presence is integral to the chemical and physical evolution of the protostellar disk at each stage (Williams and Hartquist, 1999). For example, violent supernova shocks sputter and destroy dust, altering chemical abundances, cooling rates, and strength of emission lines (Shaviv et al., 1999; Tielens and Charney, 1997). Dust acts as a shield against stellar photodissociation wavelengths, allowing complex molecular species to build up within cool interstellar clouds. A dramatic photo (Fig. 5) taken with the Wide Field Planetary Camera on the Hubble Space Telescope shows a “star nursery,” consisting of a weak plasma and dust, in which particles coagulate through charging and gravitational attraction to form new stars and planets. In planetary magnetospheres, material ejected from a moon of Saturn by ion impact can lead to brilliant and complex ring structures (Whipple, 1981; Whipple et al., 1985; Goertz, 1989; Hartquist et af., 1992; Hartquist et al., 1997; Horbnyi, 1993). Dust is also a prominent part of comets, and recent Rosat satellite results have shown that submicron grains from the comet surface can scatter solar x-ray radiation (Owens et al., 1998). Several reviewers have discussed the fimdamental role played by dust in the physical processes occurring in planetary magnetospheres, leading to tori at Jupiter and Saturn and to planetary rings at Jupiter, Saturn, Uranus, and Neptune via sputtering of surfaces (Johnson, 1994; Johnson, 1996; Johnson et al., 1998). Plasma-dust interactions occur in the cometary atmosphere via a combination of solar radiation pressure, plasma drag (inelastic ion-surface collisions), and electromagnetic forces (Whipple, 1981; Whipple et al., 1985; Hartquist et al.,
3 82
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FIG.5 . Hubble Space Telescope photograph of a star nursery in the Eagle Nebula. The pillarlike structures are columns of gas and dust, within which new stars have recently been formed. The column at the left is 1 light-year in length. Small globules of denser gas and dust (EGGS, or evaporating gaseous globules) are buried within the lower-density pillars. Red shows emission from S+ ions, green from neutral H, and blue from 02+. Photo credit: J. Hester and P. Scowen (Arizona State University) and NASA.
1992). Dust and ice grains play yet another role in comets and in the interstellar medium by providing a surface for chemical reaction to form complex molecular species, including polyaromatic hydrocarbons (Joblin et al., 1995), and hydrogen atom recombination to form H, (Biham et al., 1998; Vidali et al., 1998; Pirronello et al., 1997a; Pirronello et al., 1997b). The equation of motion of a dust grain of mass m and position x (with a velocity u = X) in the interstellar medium or (taken here) in a stellar region is given by (Goertz, 1989)
mX = q(E + u x B ) + Fg +Fd
7CA2
- -Prad
4
(7)
The first term on the right-hand side is the Lorentz force on the charged grain, where B and E are the local magnetic field and motion-induced electric field; the
RECENT APPLICATIONS OF GASEOUS DISCHARGES
383
second term (F,) is the gravitational force, directed toward the star; the third term (Fd)is taken as the sum of the total plasma drag on the grain; and the last term is the stellar radiation pressure on the grain (of diameter A), directed away from the star (hence the negative sign relative to Fg). This pressure will depend on the star’s luminosity, the star-grain distance, and the grain albedo. The Fd term is usually taken as the sum of a particle drag (in which billiard-ball and sticking collisions between the corotating planetary plasma and the grain induce orbital changes) and a Coulomb collision term arising from ion-ion elastic scattering (Northrop et al., 1989; Horanyi et al., 1997). As pointed out by Goertz (1989), for small grains the ratio R of electromagnetic force to gravitational force for a surface potential 4,yis R = 2.5 x lOP54, << 1 (for a l-pm-radius grain orbiting Saturn at a distance of two planetary radii). Hence these particles are primarily gravitationally bound. The grains are subject to azimuthal drag forces in the magnetosphere arising from collisions with the corotating plasma or, for cometary grains, with the incident solar wind. Effects of changing surface charge due to changes in the ambient plasma density and T, as well as changing B and E, will also affect the grain’s trajectory. A detailed three-dimensional fieldsand-trajectories solution (much as one carries out in the design of charged particle optics, with space charge) is probably needed for any detailed study of grain distribution (Horanyi and Mendis, 1987; Horanyi, 1993). The rate of interstellar formation of H, could not be explained by gas-phase collisions of H atoms because of a low collision frequency for three-body relaxation collisions. One was then led to consider the role of dust grains in interstellar medium molecular synthesis, including the synthesis of H, (Gould and Salpeter, 1963; Hollenbach and Salpeter, 1970; Hollenbach and Salpeter, 1971). In the Langmuir-Hinshelwood model of recombination (Zangwill, 1988; Levinson et al., 1997), the mechanism of recombination involves the steps of: (1) physisorptive or chemisorptive accommodation, (2) migration of the adsorbed Hatoms to one another, (3) chemical reaction in which the bond energy is now available for grain heating and vibration-rotation-translational (u, J , T ) excitation of the products, and (4) possible ejection of the products from the surface in one or more u, J , T states. In the other limiting case, described by the Eley-Rideal model, reaction occurs with one surface-accommodated species; the impinging partner may react with a single “hit”, or may bounce several times and react during any one of these trajectories (Rettner, 1994). The product species may again be ejected from the surface with some translational energy from the bond formation. Details of the recombination kinetics are given by Levinson et al. (1 997), with extensions to superthermal collisions of H and H+ on grains. Considerable work is directed at the study of H-atom recombination relevant to RF plasma discharges (Kae-Nune et al., 1996; Perrin et al., 1998). Use is made of temperature-programmed desorption (TPD) (Zangwill, 1988; Bruch et al., 1997) in conjunction with threshold ionization mass spectrometry. The surfaces studied
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Ara Chutjian
have been stainless steel, a-Si:H, and oxidized Si. A recent example of laboratory work with an interstellar-relevant target (olivine, a mixture of Fe,SiO, and Mg,SiO,) at interstellar temperatures (5 to 18 K) uses fast atomic beams of H and D atoms to study the surface recombination to give HD (Biham et al., 1998; Vidali et al., 1998; Pirronello et al., 1997a; Pirronello et al., 199713). A schematic diagram of this apparatus is given in Fig. 6. Thermal energy beams of H and D are generated in two separate beam lines using separate RF dissociation sources. After supersonic expansion, collimation, and three stages of differential pumping, the beams impinge on a surface at ultrahigh vacuum. The D, isotope is chosen to eliminate interference from H,, which is present (1) as outgassed molecules from the ultrahigh-vacuum stainless steel vessel walls and (2) as undissociated H, in the incident beam. The HD and D, production as a function of various parameters such as surface temperature, beam intensities, and exposure time is monitored with a pumped, rotatable quadrupole mass analyzer. Corrections are made for various backgrounds of HD and D, resulting from vacuum-wall recombination, ambient (unpumped) density, and undissociated H, and D, in the incident beams. An example of a TPD curve of HD from an olivine surface is given in Fig. 7 (Pirronello et al., 1997a). At the lowest coverage, the desorption kinetics are second-order; i.e., the H and D adsorbed atoms migrate Main Chamber
Hydrogen Beam Line
View Port
Hydrogen Source
\
Deuterium Source
Deuterium Beam Line FIG. 6 . Schematic diagram of a two-beam apparatus to study H, formation on grains. Separate beams of H and D atoms are produced in an W source, with about 70 to 85% dissociation of the feed H, and D, gases. The collimated, differentially pumped thermal-energy beams of H and D atoms are brought to the ultrahigh-vacuum scattering chamber, where they are adsorbed onto a grain sample (olivine, pyrolitic graphite, etc.). The HD and D, produced on the surface are desorbed using temperature-programmed desorption (TPD) and are detected by the quadrupole mass selector (QMS) (Vidali et al., 1998).
RECENT APPLICATIONS OF GASEOUS DISCHARGES
I
..-.
I
0.0
6
I
8
I 10
I
12
385
I
14
I
16
18
Temperature (K)
FIG. 7. Typical temperature-programmed desorption curve for HD from an olivine surface as a hnction of surface temperature. Curves are for exposure times to the H and D beams of (bottom to top) 0.07, 0.1, 0.25, and 0.55min (Pirronello et al., 1997a).
toward each other and react as the surface warms, then desorb. Recent work has also been reported on recombination using amorphous carbon samples (Pirronello et al., 1999). Based on these results, a modified expression for the recombination rate of surface adsorbed H atoms has been proposed (Biham et al., 1998; Pirronello et al., 1997a). The recombination rate R (cm3/s) is
where SHis the sticking fraction of H atoms on the grain surface, nH and vH are the density and velocity of the gas-phase H atoms, A is the average cross-sectional area of the grain, T ~ ,is the residence time of the atoms on the grain, ng is the density of grains, I? is the average number of sites between two adsorbed H atoms, and yf is the probability of bonding upon encounter. The factor v f ( T , a, 6 E ) is the hopping rate of adatoms resulting from surface thermal kinetic energy and barrier tunneling, with an associated total barrier height a and width 6E. Difficulties arise in accounting for the observed steep temperature dependence of the recombination efficiency in expressions such as Eq. ( 8 ) (Pirronello et al., 1997a; Pirronello et al., 1997b): The diffusion of H and D at the lowest Ts is slow, recombination depends strongly on surface Tin the range 6 to 15 K, and the recombination is thermally activated. The sharp experimental behavior calls for an additional temperature dependence in y’, or a high adatom surface density such that random walk is restricted. Further clarification of the mechanisms (random walk, barrier heights) influencing the T dependence is needed.
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Much of the characterized experimental work has been relevant to the diffuse interstellar medium, where H atoms are cooler. In addition, investigation of sticking coefficients for superthermal H and D atoms (energy range 0.1 to 10 eV, say) recombining on olivine grains would shed light on dust effects in shockheated parts of the interstellar medium or in the vicinity of supernova remnants or starburst galaxies where energetic atoms are produced via shock heating, and by charge exchange of fast outward-streaming ions with the neutral interstellar medium. At these energies, fast atoms can penetrate the bulk of the grain (Levinson et al., 1997), and a TPD analysis may show the effects of bulk diffusion to the surface, followed by recombination and evaporation.
11. Elves, Red Sprites, and Blue Jets There is a large anecdotal history of observers’ having seen above giant thunderstorms continuous darts of light, upward-rising flames, luminous trails, and discharges with shapes of inverted tree roots (Boeck et al., 1998). These phenomena have undoubtedly been present from time immemorial, but the rapid and transient nature of the discharges made their recording and quantification (and hence credibility) nearly impossible until this century. A number of recent ground and aircraft campaigns (Boeck et al., 1998; Sentman et al., 1995; Sentman and Wescott, 1995) have provided further details on the interesting effects, some of which were predicted earlier (Wilson, 1956). The emissions, with their characteristic colors and spatial extent, arise through vertically and horizontally directed electric fields from lightning-driven electromagnetic pulses (Rowland et al., 1995a; Rowland et al., 1995b; Fernsler and Rowland, 1996; Rowland 1998; Pasko et al., 1995; Pasko et al., 1997; Ma et al., 1998; Wescott et al., 1998). The optical emissions have been termed elves, red sprites, and blue jets. The characteristic colors are due to a predominance of various emission-band systems in N, and N.: These bands are excited by the moderate-energy electron energy distribution function generated in the lightning. This electron energy distribution function usually peaks at about 1 to 2eV, yet contains a tail of sufficient energy to ionize N, (energy threshold of 15.58 eV). A schematic diagram of the ionospheric emissions structures is given in Fig. 8 (Pasko et al., 1997; Rowland, 1998). Elves are an extended airglow appearing at altitudes between h = 70 and 90km, with a horizontal extent of several hundred km. Red sprites range from h = 55 to 80km, with a width of about 20 to 30km. Slender tendnls extend downward from about 55 km. Blue jets are born from the thunderstorm anvil top at 15 to 18 km,and propagate up to about 40 km altitude. These plasma structures are visible for times of the order 0.1, 10, and 300 ms for the elves, red sprites, and blue jets, respectively (Rowland, 1998). In addition, discharges called blue starters have been reported (Wescott et al., 1996). These
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387
FIG. 8. Schematic of various phenomena (top) and mechanisms (bottom) of lightning-ionospheric interactions as a function of altitude. Phenomena can give rise to alterations (A) in optical emissions (Ahw), atmospheric temperature ( A T ) , and electron density (AN,) detected as scattering of VLF signals. Other abbreviations are cloud-to-ground discharges (CG), quasistatic electric (QSE) fields and heating, electromagnetic pulse (EMP), and spacecraft (s/c) (Pasko et al., 1997).
are distinguished from blue jets by their lower terminal altitude, about h = 25 km vs 40km for jets. It has been suggested that blue starters may also be wannabe blue jets (“proto blue jets”) in which the ambient electric field E or cloud charge Q is insufficient to sustain ionization breakdown up to h = 40 km (Wescott et al., 1996; Sukhorukov and Stubbe, 1998). The red color of sprites results from a predominance of the first positive (1P) emission system in NZ,the B 311g(u’) + A ’Z: (u”). Blue elves are made up of the second positive (2P) system, C 311u(u’) -+ B 3ns(0”). Vibrational quenching of the B state (red) is
388
Ara Chutjian TABLE 1.
FREQUENTLY OBSERVED
Species
Transition
MOLECULAR TRANSITIONS AND PROCESSES UPWARD-DIRECTED DISCHARGES
RELEVANT TO
System/process description First positive ( 1P) (red) Second positive (2P) (blue) Vibrational energy transfer to pump 4.26-pm radiation in C 0 2 First negative ( I N ) (blue) Meinel (red) Dissociative attachment First negative (1N) (red) 630.205-nm emission (red) 557.889-nm emission (green) Vibrational emission of 4.26-bm radiation
efficient for altitudes below about 50 km; hence the blue color is dominant in jets. A summary of the more commonly observed emission systems and processes is given in Table I. Three scenarios for the production of sprites have been presented (Pasko et al., 1995; Pasko et al., 1997; Rowland, 1998; Morrill et al., 1998). In each case, electric fields accelerate electrons. An electron energy distribution function is established by electron-neutral atmospheric collisions. This distribution function is then used to compute the excitation rates of the various atomic and molecular components of the atmosphere. The scenarios are: (1) appearance above a thundercloud after a lightning discharge of a large quasistatic electric field that propagates upward into the ionosphere, (2) excitation of species by an additional upward-propagating electromagnetic pulse accompanying a large lightning discharge, and (3) possible excitation by accelerated runaway electrons produced via incident cosmic-ray ionization of the atmosphere within a thunderstorm cloud, with its strong electric fields (Fishman et al., 1994; Taranenko and RousselDuprk, 1996). This last mechanism, runaway air breakdown, can also give rise to blue jet excitation. It may be responsible for the y-ray flashes that have been observed by the BATSE instrument on the Compton Gamma Ray Observatory (Fishman et al., 1994). In all the high-altitude phenomena, the energy drivers are the electric fields associated with the cloud charge separation and the lightning. The transient electric and magnetic fields have three sources. There is the field due to the initial charge distribution in the cloud (no discharge yet); an electromagnetic pulse from the propagation of the time-varying return stroke(s); and the quasistatic electric field arising from the redistribution of charge after the return strokes, which propagates upward into the ionosphere. The time scale of the changes is approximately seconds or longer for establishing the initial charge distribution, to 1OP4 s for the electromagnetic pulse, and to 10-3 s for the redistribution and the
RECENT APPLICATIONS OF GASEOUS DISCHARGES
389
quasistatic electric field. The field is “quasistatic” in the sense that it exists for a time that is longer than that determined by the conductivity r~ at that altitude (see also description and Fig. 9 following). This arises from a decrease in the electron component of the conductivity as a result of many electron-gas collisions; to generation of new boundary charges at the lower altitudes as a result of cloud motion and updrafts; and to a persistence of thundercloud and polarization charges above the thundercloud (Pasko et al., 1997). The quasistatic field is also slowly varying ( 1o - ~s) relative to electron collision frequencies (< I o - ~s); hence a Boltzmann-equation description of the electron energy distribution function is valid. Effects of the earth’s magnetic field, and the weaker magnetic field associated with the slow variation of the quasistatic electric field, are usually neglected. The atmosphere is collision-dominated, so that an electron experiences many disruptive collisions in a pitch length. The neutral atmosphere is affected via electron-neutral collisions in which electrons of a time-varying distribution function excite the most abundant species N, and 0,. Excitations here include direct collisional (electron) excitation to the many electronic states of N, lying below about 15 eV above the ground state, ionization excitation to states of N; between 16 and 19 eV, and direct vibrational excitation of N, X(u”) followed by 0 - u transfer to CO,, to give rise to 4.26-pm infrared radiation [CO, (001) + (OOO)] transition (Table I). An alternative, or possibly parallel, heating mechanism can be generation of high-energy electrons in the runaway regime, followed by acceleration to MeV energies with further secondary electron generation (Taranenko and Roussel-DuprC, 1996). The effects of atmospheric absorption by 02,O,, and H,O, as well as effects of excited-state quenching, will determine the spectral characteristics of the sprites and jets at the point of observation (Morrill et al., 1998). A three-dimensional time-dependent model has been developed by Ma et al. (1998) to solve for the electric field, charge density, and Maxwell current. The last quantity is influenced primarily by the spatially anisotropic tensor conductivity in the ionosphere. Maxwell’s equations for the transient fields propagating into the ionosphere are reduced to two equations. If p is the charge density (C/m3), co the dielectric constant (C2/N . m2), p o the permeability (N/A2), r~ the conductivity tensor (S/m), and E the electric field (v/m), then the wave equation in SI units becomes
The continuity of electric current may be derived from the Maxwell’s equation for V x H , in the form
390
Ara Chutjian rad/s SeC
FIG 9. Atmospheric electrical conductivity profiles u for three latitudes and times. The relaxation 5, is given by co/u, and o = I/?, (Ma ei al.. 1998).
time
where p, is the source charge density distribution carried by the return stroke. Equations (9) and (10) show the strong dependence of E on the ionospheric conductivity profile. One example of the profile o vs altitude, at three conditions of latitude and time, is given in Fig. 9. The upward-directed electric field accelerates the ambient thermal energy electrons of mean energy t = 1.5 kT to a new distribution function that depends upon the local E field and neutral composition and density. The connection between the spatial E field and the electron energy distribution function is made through solution of either the Boltzmann equation (Pitchford et al., 1981; Phelps and Pitchford, 1985) or the derived Fokker-Planck equation (Milikh et al., 1998a). In either case, a full database of cross sections for electron-molecule (N2, 0,) excitation, ionization (both direct and dissociative), and attachment (for 0,) is needed for reliable solutions. Electron-ion and electron-atom (N, 0) scattering are usually neglected because of the small product of electron and ion or atom densities. The Boltzmann equation has been applied to the E-field disturbed ionosphere to calculate the observed photon emission rates. For an electron with velocity and space coordinates v and r, respectively, the distribution function f(r, v, t ) gives the particle density in the configuration space (r, v) at a particular time t. That is, the average number of particles in the configuration-space element or “pixel” dr . dv at time t is just f(r, v, t ) dr dv. Following the treatment of Gurevich (1978) [see also Taranenko et al. (1993a, 1993b) for further development of Eq.
This Page Intentionally Left Blank
RECENT APPLICATIONS OF GASEOUS DISCHARGES
391
(1 l), Pitchford et al. (198 l), and Phelps and Pitchford (1985)], one may write the Boltzmann equation for the electron distribution fbnctionf(r, v, t ) as
where E and B are the local electric and magnetic fields. The term So, the Boltzmann collision integral, describes the change in the electron energy distribution function as a result of elastic and inelastic electron-plasma collisions. It includes effects (where channels are open) of rotational (SfPt), vibrational (qib), and electronic (Sy')excitation; dissociation ( S F ) ; ionization (both direct and dissociative) (S;), dissociative ionization (S$); dissociative excitation (S?); electron attachment (both direct and dissociative) (S:'); and electron-ion recomEach process requires its own accurate cross section as a function bination (F). of species and electron energy. Hence a large and accurate atomic and molecular physics database is needed for reliable modeling. Results of the solutions to Eq. (1 1) for h = 90 km are shown in Fig. 10 for the assumed electron and neutral densities (Taranenko et al., 1993a). The initial condition was an electric field E = 10 V/m applied for 20 ps starting at t = 0 s. As the electric field propagates upward from the cloud discharge, it interacts with the ambient atmosphere to set up the electron densities and distribution function, such as those shown in Fig. 10. The heated electrons interact with the ambient gas (mainly N2 and 0,) to produce characteristic emission bands in elves, sprites, and blue jets. The principal bands are given in Table 1. A moderate resolution spectrum of one red sprite is shown in Fig. 11. Prominent are bands in the B + A (1P) system. There may be traces of the N l A -+ X (Meinel) system, but the signal-to-background should be improved. The Meinel system is interesting in that its presence gives some indication of the highest electron energies present in the discharge. Thus, ionic features observed by Armstrong et al. (1995) and not observed by Mende et al. (1995) could be due to problems of resolution, aerosol scattering, and low signal-to-background, but also to the electron energy distribution function in some more energetic sprites (larger E field) having a higher-energy tail (Taranenko et al., 1992). Future airborne campaigns should include a carefbl study of the Meinel system as a marker of electron energy. A detailed study of the N, emission rates has been carried out by Mom11 et al. (1998). In this study, the quasistatic electric field model (Pasko et al., 1997) was used to calculate the electric fields, and the solution to Boltanann's equation was used to calculate the electron energy distribution function as a function of altitude. Results for excitation of seven triplet states of N, are shown in Fig. 12 at h = 65 and 75 km. The temporal duration of the excitations may be understood in terms of the faster relaxation @gher conductivity; see Fig. 9) of the E field at the higher altitude.
3 92
Ara Chutjian Neutral Density ( ~ m ' ~ )
.-c0 'C
w
i
60 10-3 10-1 101
103 105 10.1 Electron Density ( ~ r n - ~ )
101 Energy (eV)
100
102
10-11 10-13 10-15 10-17 10-19 10-7 10-6 10-5
Time (s)
10-4
Time (s)
FIG.10. Ionospheric characteristics and results from Boltzmann-equation calculations (Taranenko et al., 1993a). Upper leff: Profiles o f ambient electron and neutral densities. Electron densities are shown for (a)nighttime, (b) intermediate, and (c) daytime. At an altitude of 90 km, shown are (upper right) electron energy distribution functions at the indicated times after start of the E field, (lower left) normalized electron current density and average electron energy, and (lower right) total gaseous attachment and ionization rates as a function of time after discharge.
5500
6000
7000 7500 Wavelength (A)
6500
8000
851 D
FIG. 11. Synthetic sprite spectrum (Hampton et al., 1996) with a computed fit (heavier line) using the emission systems indicated. Response-corrected sprite spectrum recorded at the Mt. Evans observatory in Colorado is shown with lighter/broken lines (Green et al., 1996).
RECENT APPLICATIONS OF GASEOUS DISCHARGES
A-
W-
C-
5-
0-
D -.
10
20
393
30
Time (ms)
x
r n 4 L l/r
/ I
\\
\\\\
r
" 4 10-2 0
1
2
3
4
5
Time (ms)
FIG. 12. Production rate of seven triplet states of N, as a function of time after initiation of the quasistatic field. Results are shown at altitudes of ( a ) 65 km and ( b ) 75 km (Mom11 et al., 1998). The notation refers to the states A 'Z:, B 'ng,W 'Au, B' 'Xu, C 311u. D 'X:, and E 'Cp'.
Less spectral information is available on elves. Shuttle observations were first made with broadband cameras with a response peaking at 440 nm (Boeck et al., 1992). Broadband filter/photometer measurements have been reported by Fukunishi et al. (1 996). Taranenko et al. (1992) calculated optical-emission intensities from lines of N, N,,N;, 0, O,, and 0;. Estimated emission intensities were in the range 10' to lo9 R. [The rayleigh (R) is the emission rate in all directions from a column of unit cross section along the line of sight. The unit is defined as 1 R = lo6 photons/cm2 . s.] The duration of the emissions is in the range 50 to 350 ms after termination of the quasistatic electric field. Considering the relevant excitation and quenching rates, 0, lines would be the most intense emitter as viewed from the shuttle or other spacecraft (Taranenko et al., 1992). Moderate-to
394
Ara Chutjian
high-resolution optical spectra would be a needed component of any future elves campaign. Given the calculations of Milikh et al. (1998b), such a low-earth-orbit campaign should also include an mfrared-measurement capability at 4.26 pm. Optical emissions for ground-state, vibrationally excited N2(d’) (“vibrons”) are dipole forbidden. Emission at 4.26pm arises from CO, via degenerate energy transfer between vibrons and u”J” levels of CO, to give rise to the dipole-allowed emission CO, (001) + (000) at 4.26pm. The lifetime of this emission is Tlife = 2.5 ms. A Fokker-Planck approximation to the Boltzmann equation is solved for the electron energy distribution function. The distribution function is then used to calculate excitation rates of N, (d’), the collisional transfer rates to CO,, and the 4.26-pm photon escape rates from the optically thick atmosphere. An outline of the collision particles is given in Fig. 13. Relevant processes, starting with N2*, are N2* deexcitation by atomic 0 (k,) and 0, (k3);vibron excitation by 02*(Q; vibron transfer to CO, (k,)and back transfer by C02* to N, (ki); absorption by CO, of a previously emitted 4.26-pm photon, and radiation of a 4.26-pm photon to the atmosphere with lifetime zlife; and 02* removal by quenching reactions with 0 (k,) and H,O (k4).The excited H,O can radiate its vibrational energy by dipole transition (Kumer and James, 1974). The excitation coefficients for the various N2(d’) levels and for C02(O01) are shown in Fig. 14 as a function of the electron quiver energy (total electron energy, including linear translation as well as cyclotron energy). These rates are computed by averaging experimental electron impact excitation cross sections for N2(u”) and CO2(O01) over the calculated distribution function. Interestingly, an electromagnetic pulse of millisecond duration will produce a delay in the infrared emission of as long as 200 s. Sources of the delay are the time for N, and kl
-
-
k2 0
Np’
COP
kg 0
02
Absorption
-
FIG. 1 3. Schematic representation of the reaction pathways for vibrons, N,(d‘ = l), produced by lightning discharges in the upper atmosphere (Milikh et a!., 1998b).
RECENT APPLICATIONS OF GASEOUS DISCHARGES
395
Electron Quiver Energy (eV)
FIG.14. Excitation rate coefficient of vibrational levels Y" in N, (solid lines), the effective rate for all levels (dotted line), and the excitation rate coefficient for CO,(OO1) (broken line) (Milikh ei al., 1998b).
C 0 2 to find one another and transfer energy, and the time for entrapped infiared photons, starting at about 85 km,to diffuse upward through the ionosphere.
111. Acknowledgments This work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, and was supported through the National Science Foundation under agreement with the National Aeronautics and Space Administration.
IV. References Allamandola, L. J., and Tielens, A. G.G.M. (Eds.) (1989). Interstellar dust. Kluwer (Dordrecht). Armstrong, R. A,, Shorter, J., Lyons, W. A,, Jeong, L., and Blumberg, W. A. M. (1995). Eos, Pansactions American Geophysical Union Fall Meeting, Abstract A41D-5. Biham, O., Furman, I., Katz, N., Pirronello, V, and Vidali, G. (1998). Mon. Not. Royal Astron. SOC. 296, 869-872. Boeck, W. L., Vaughan, 0. H., Jr., Blakeslee, R., Vonnegut, B., and Brook, M. (1992). Geophys. Res. Lett. 19, 99-102. Boeck, W. L., Vaughan, 0. H., Jr., Blakeslee, R. J., Vonnegut, B., and Brook, M. (1998). 1 Atm. SolarTerresfrialPhys. 60, 669477. Bruch, L. W., Cole, M. W., and Zaremba, E. (1997). Phy,sical adsorption: forces and phenomena. Clarendon (Oxford). Choi, S . J., and Kushner, M. J. (1994). IEEE Trans. Plasma Science 22, 138-150. Chu, J. H., and Lin, I. (1994). Phys. Rev Left. 72, 40094012. Chutjian, A., Garscadden, A., and Wadehra, J. (1996). Physics Reports 264, 393472. Dalgamo, A,, Stancil, P. C., and Lepp, S. (1997). Asfrophys. Space Sci. 251, 375-383.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 43
OPPORTUNITIES AND CHALLENGES FOR ATOMIC, MOLECULAR, AND OPTICAL PHYSICS IN PLASMA CHEMISTRY KURT BECKER Department of Physics and Engineering Physics, Stwens Institute of Technology, Hoboken, NJ
HANS DEUTSCH Institut f i r Physik, Emst-Moritz-Amdt Universitat Greifswald, Greifswald, Germany
MITIO INOKUTI Argonne National Laboratory, Physics Division, Argonne, IL
In the final chapter of this volume on fundamentals of plasma chemistry, we would like to point out future opportunities for the atomic, molecular, and optical (Ah40) physics community to contribute to a better microscopic understanding of the plasma-chemical processes in various plasma-assisted materials-processing applications as well as in other applications. We will address both gas-phase (volume) processes and plasma-surface/wall processes. We will further attempt to identify some of the important challenges for the AM0 physics community. By opportunities we mean problems that can be addressed through the application of established AM0 physics methods and techniques, and by challenges we mean problems where the development and application of new AM0 physics methods, methodologies, and techniques will be needed. Chemical reactions initiated in gas discharges and plasmas, in particular in low-temperature, nonequilibrium plasmas, have become indispensable for the advancement of many key technologies in the past 10-15 years (see, e.g., Becker et al., 1992; Garscadden, 1992). The plasma-assisted etching of microstructures and the deposition of high-quality thin films with well-defined properties have become crucial steps in the fabrication of microelectronic devices with typical feature sizes of less than 0.5 pm. The manufacture of state-of-the-art microchips now involves hundreds of process steps, most of them serial, to yield circuits with millions of discrete elements and interconnections in an area of a single square centimeter (Garscadden, 1992). Each step is a physical-chemical interaction that must be controlled. More than one-third of the process steps rely on plasma 399
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technology. Plasma-assisted surface modifications for the preparation and conditioning of biocompatible surfaces or the deposition of hard and wear-resistant coatings, for example, have become a widely utilized technology in diverse applications. The plasma environments that are typically used in these applications are collision-dominated and are characterized by average electron energies of 0.5 to 5 eV, neutral gas “temperatures” corresponding to 0.025 to 0.05 eV, and charge carrier densities of lo-* to cmP3. In such plasmas, highly excited species are readily formed and can drive “high-temperature chemistry” without the deleterious effects of real high ambient temperatures on the processed materials. The value added by these plasma-assisted or plasma-initiated chemical reactions transforms, for example, a silicon wafer costing a few dollars to microelectronic devices costing a few hundred dollars. It is also remarkable that by and large the plasma processing community has advanced by empirical methods, with little help from the plasma physics and AM0 physics communities. As the complexity of plasma-based processes increases and the costs for new microelectronic fabrication lines for advanced devices escalate to hundreds of millions,, if not billions, of dollars, the need and desire for the development of a scientific understanding of the underlying fundamental processes and plasmachemical pathways becomes inevitable. Basic AM0 physics research relevant to technologies based on plasma chemistry has a dual role to play: (1) It has to explain observed phenomena, and (2) it should be able to predict the impact of new conditions. All plasma-chemical processes incorporate two different, yet inherently related regimes: (1) volume or gas-phase processes and (2) surface or wall processes. Ideally, the two regimes would be separate, and one could focus on understanding the gas-phase processes independent from the surface/wall processes. However, in many applications, the volatile reaction products that result from the interaction of the gas-phase species generated in the plasma with the surface during the etching, deposition, or surface modification process or with the wall surrounding the plasma reactor backstream into the bulk plasma and crucially determine the steady-state properties of the plasma. Thus, the coupling between the gas-phase processes and the surface/wall processes adds a significant complication to any attempt to understand, model, and predict the plasma properties, the key plasma chemical reaction pathways, and the dominant plasma-surface/wall reactions on a microscopic scale. This is schematically illustrated in Fig. 1 for a remote plasma processing reactor (i.e., a reactor in which the generation of the reactive species and the actual processing are spatially separated). By and large, AM0 physics has done a better job in advancing our microscopic understanding of the gas-phase processes in low-temperature plasmas than it has in elucidating the surface/wall processes. The increased level of activity in efforts to provide a more scientific underpinning for plasma-chemical processes,
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FIG. 1. Schematic diagram of the interaction of gas-phase (volume) processes and surface processes in a remote plasma processing reactor.
particularly processes in low-temperature plasmas, was stimulated by the 1991 National Research Council (NRC) Report on Plasma Processing of Materials: Scientijic Opportunities and Technological Challenges (NRC, 1991). Despite the recent surge in Ah40 physics research activities related to plasma-chemical applications, much of the work has been carried out with insufficient awareness of the needs, challenges, and real problems of the user community. As a consequence, low-temperature plasma technology today remains largely a discipline in which technological advances routinely outpace the fundamental understanding at an atomistic level. Missing data for many key processes, uncertainty as to the exact nature of the key processes, and serious gaps in the database and/or questionable reliability of existing data for many processes have prevented the fill exploitation and utilization of advanced modeling codes, computerassisted design (CAD) tools, and plasma diagnostics techniques. This was noted in the 1996 follow-up NRC report, Modeling, Simulation, and Database Needs in Plasma Processing (NRC 1996). Opportunities for the A M 0 community to advance the understanding and modeling of gas-phase (volume) processes relating to plasma chemistry lie in 1. The application of existing and established experimental methods used in fundamental studies, such as electron scattering from ground-state atomic hydrogen, the rare gases, and other simple (nonreactive and easyto-handle) species, to targets of interest in plasma-chemical applications, which include reactive and corrosive species or difficult-to-produce fragments and transient species 2. The application of theoretical approaches to complex targets and processes with many final states, such as dissociation and ionization, and the further utilization of the ever-increasing computational power of massively parallel computers and supercomputers
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3. The refinement of plasma diagnostics methods that will improve reliable real-time, in-situ, nonintrusive determinations of species concentrations with high temporal and spatial resolution as well as other characteristic plasma parameters (e.g., electron, ion, and neutral temperatures, plasma potentials, etc.) 4. The critical evaluation of existing data for pure gases and gas mixtures and their compilation into complete input data sets for modeling codes
Challenges for the AM0 community lie in 1. The development of new techniques to study collision processes that have so far received little attention, but are considered crucial to a better understanding of the basic plasma-chemical pathways, such as the dissociation of molecules into neutral ground-state fragments by electron or ion impact or the vibrational excitation of complex molecules. 2. The development of techniques to prepare well-characterized targets of species in excited states (long-lived metastable atoms and molecules, electronically excited atoms and molecules, and vibrationally excited, “hot” molecules) for electron and ion collisions, including attachment and dissociative attachment studies and spectroscopic measurements. 3. The development and implementation of novel plasma diagnostics methods to facilitate faster and more reliable real-time, in-situ, and nonintrusive determinations of species concentrations with high temporal and spatial resolution as well as the measurement of other characteristic plasma parameters such as electron, neutral, and ion temperatures and plasma potentials. An example of such novel diagnostics methods is the use of a millimeter-wave Mach-Zehnder interferometer for the measurement of the effective plasma frequency and the effective frequency for momentum transfer in a high-pressure ac discharge plasma (Amorer and Kunhardt, 1999). 4. The development, exploration, and application of semiempirical crosssection calculation schemes, the derivation of scaling laws, and the establishment of common trends and tendencies of collisional and spectroscopic data for species of similar structure and/or configuration (e.g., along isoelectronic sequences, along rows and columns of the periodic table, for molecules of similar molecular or electronic structure, for molecules belonging to families such as the hydrocarbons or the fluorocarbons). The last of those items (4), in conjunction with the need to provide a select number of benchmark measurements and calculations that can test the reliability of experimental techniques and theoretical methods, acknowledges the fact that it is unrealistic to expect that measurements and calculations can or even should be
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carried out for all imaginable targets given the large number of possible targets of interest. This is particularly true in view of the rapidity with which certain gases and chemistries fall in and out of favor from an application viewpoint. An additional motivation for the fourth challenge above is the recognition that rigorous A M 0 physics methods and approaches find themselves in direct competition with quite sophisticated statistical engineering approaches developed by the user community. These methods, which are called response surface methodologies (Garscadden, 1992), analyze the sensitivity of a certain process outcome to changes in the process control parameters. In an ideal world, response surface methodologies [such as the Taguchi method (Garscadden, 1992)] and rigorous A M 0 physics methods should be synergistic. Basic A M 0 physics research should provide the framework and the guidance for the development of new chemistries and approaches, whereas the fine-tuning of a process will always have to incorporate some reliance on empiricism and prior art. No plasma - with perhaps the exception of some astrophysical plasmas can exist without a surrounding wall. This "wall" can be one or more electrodes that confine the plasma, or it can be the insulating or conducting walls of the vacuum enclosure in which the plasma is sustained, or it can be the "processed" surface in plasma processing applications such as etching, deposition, and surface modifications. There has to be a "transition" region between the bulk plasma and the surface/wall. This boundary region is called the sheath and its main purpose is to facilitate a transition from the electrical potential in the bulk plasma to the electrical potential at the surface/wall. The processes in the sheath are complex and are determined by the transport of species from the plasma to the surface/wall and by a possible (partial) backstreaming of species from the surface/wall into the bulk plasma. The key processes at the surface/wall include absorption
FIG. 2. The various components involved in the understanding of plasma-surface/wall interactions and their mutual dependence and interaction.
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and desorption, recombination, sputtering, and chemical reactions and surface processes such as diffusion, condensation, and nucleation - to name just the most important reactions. Figure 2 shows schematically the various components involved in the understanding of plasma-surface/wall interactions and illustrates their mutual dependence and interaction. The relevant species that play a role in plasma-surface/wall interactions are similar to the important gas-phase constituents and include ions, ground-state neutrals, metastables, free radicals, clusters, etc. Although the modeling of plasma-surface/wall interactions is a rather active field, most of the modeling is descriptive rather than predictive. Moreover, descriptions and models tend either to analyze the reaction products (electrons and ions and neutrals, some of which may be excited) leaving the surface after the collision of an incident ion or a neutral species with the surface, or to focus on analysis of the subsequent reactions on the surface. It is rare for any single model or description to address both aspects. There are substantial hurdles that render it difficult to achieve a detailed microscopic understanding of plasma-surface/wall processes. Foremost among these are missing data for the relevant surface/wall processes and problems in identifying the relevant surface/wall processes. Other problems arise in the description of the electron kinetics in the sheath because of the need to incorporate spatial aspects. Reliable and reproducible in-situ and realtime measurements and diagnostics of the surface/wall processes in the highly inhomogeneous plasma sheath region are also problematic. The application of the tools, techniques, methods, and expertise of A M 0 physics to the many unanswered questions posed by the processes at the interface of the plasma with the surrounding walls or at the surface in etching and deposition applications (plasma-surface/wall interactions) will provide a fertile ground for opportunities and challenges in “applied” AM0 physics. The coupling between gas-phase (volume) processes and the surface/wall processes (see Fig. 1) and the need to study surface/wall processes with realistic, “dirty” surfaces (rather than with the clean, ultrahigh-vacuum-prepared, well-characterized surfaces that are used in conventional surface science experiments) provide numerous opportunities and challenges for future experimental and theoretical A M 0 physics research. Perhaps the most obvious opportunity for A M 0 physics lies in the experimental determination and calculation of missing fundamental collision and spectroscopic data relating to surface/wall processes. In addition to sputtering, such studies should focus on 1. Ion-surface collisions, particularly studies involving well-characterized,
monoenergetic, mass-selected beams of low-energy ions (a few eV to 500 eV) incident on “real” surfaces 2. Studies of collisions involving well-characterized cluster ions with surfaces
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3. Studies of collisions of well-characterized beams of neutrals with surfaces/walls. In these applications, an extensive analysis of the composition of the reaction products, such as electrons and positively and negatively charged ions and neutrals, some of which may be in excited states, is desirable in order to obtain a microscopic understanding of the surface collision process. Whereas it is comparatively easy to analyze the charged reaction products of ion/neutral surface collisions, it is much more challenging to identify the nature and kinetic energy of the neutral reaction products, which are, however, the majority of the reaction products of such a collision. Spectroscopic and other optical techniques (such as laser-induced fluorescence studies, resonant and nonresonant laserinduced ionization, and absorption spectroscopy) might lend themselves to probing the resulting neutrals. The results of such experiments, in turn, will stimulate new studies of electron and ion collisions and spectroscopic studies with calibrated fluxes of the key desorbed species of a particular surface/wall process, notwithstanding that it might be challenging to prepare well-characterized targets of these species for such measurements. Whereas ion - surface/wall collisions are widely studied, much less work is being done in the area of neutral - surface/wall processes. Studies of the response of the surface/wall with high spatial and temporal resolution to the impact of an incident ion and/or neutral represent another fertile area for future research, not just for the AM0 physics community, but also for the surface science community at large. Combinations of techniques inherent to AM0 physics and to surface and materials science, such as the various techniques for surface structure analysis and growth analysis of deposits and the numerous powerful microscopy techniques (electron microscopy, scanning tunneling microscopy, atomic force microscopy) will be needed in order to elucidate the myriad of possible surface/wall processes triggered by the impact of an incoming ion or a neutral species (see Yan et al., 1997; Meyer et al., 1997; and references therein). Obviously, this chapter barely scratches the surface of the many opportunities and challenges that plasma-initiated chemical processes present to the AM0 physics community. The community has only begun to explore the vast possibilities offered by the many unresolved science issues relating to a better understanding of the fundamental plasma-chemical processes in many applications. The opportunities and challenges to AM0 experimentalists, theorists, and computational physicists include, but are not limited to, the application of existing experimental techniques and the development of new methods and approaches to the study of collisional and spectroscopic properties of reactive species, excited targets, complex molecules and free radicals, and multicomponent gas mixtures, the application and development of theoretical approaches to complex targets and processes with many final states; the development and
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establishment of scaling laws and semiempirical approaches, and the efficient utilization of more powerful and faster computers. Above all, the utilization of existing methods and the development of new approaches to detailed studies of surface/wall processes will provide solutions to many open questions and help establish a scientific underpinning for many technologically important application-oriented plasma-chemical processes.
Acknowledgment One of us (MI) acknowledges support of this work by the U.S. Department of Energy, Office of Science, Nuclear Physics Division under Contract No. W-3 1109-Eng-38.
References Amorer, L. E., (1999) Ph.D. thesis, Stevens Institute of Technology. Unpublished. Amorer, L. E., Kunhardt, E. E. (1999) Private communication and to be published. Becker, K., Bonham, R. A,, Lin, C. C., (1992). Z. Phys. D 24, 95. Garscadden, A. (1992). Z. Phys. D 24, 97. Meyer, F. W., Yan, Q., van Emmichoven, l? Z., Hughes, I. G., Spierings, G. (1997). Nucl. Instrum. Methods Phys. Res. B 125, 138. NRC (1991). Plasma processing of materials: scientific opportunities and technological challenges. National Academy Press (Washgton). NRC (1 996). Modeling, simulation, and database needs in plasma processing, National Academy Press (Washington, DC). Yan, Q., Burgdorfer, J., Meyer, F. W. (1997). Phys. Rev. B 56, 1589 (1997)
Index Binary inelastic collisions, 24 Blue elves, in dusty plasmas, 387 Blue jets, in dusty plasmas, 373, 386-387 Blue starters, in dusty plasmas, 386-387 Boltzmann equation two-term expansion of, 85 weakly ionized plasmas, 25-26 Bond energy, 193 Bonds, chemical reactions with rearrangement Of, 253-255 Born-Mayer potential, 354-355 Born-Oppenheimer approximation, 1 14 Bulk plasma, 3
A
Adiabatic capture and centrifhgal sudden approximations (ACCSA), 250 Aluminum alloys, plasma etching, 4 Amorphous hydrogenated carbon films, plasma processing for deposition, 375 Angular dependence, 362-363 Ansatz, 353, 354 Argon ions reaction with carbon tetrafluoride, 2 19-220 reaction with hydrogen, 264 reaction with nitrogen, 264 reaction with oxygen, 264 reaction with silane, 195-198 reaction with silicon tetrachloride, 2 15-218 reaction with silicon tetrafluoride, 207-209 Argon radio-frequency plasmas, highsensitivity white-light absorption spectroscopy, 336-337 Association reactions, 277-279 Astronomical plasmas, dust in, 38 1-386 Atomic and Molecular Data Information System (AMDIS), plasma physics web site, 106 Atomic and molecular processes, 232-240, 375 classification, 23 1-232 Attachment, low-temperature plasma, 118-119 Attachment coefficient, 84, 88 Attachment cross section, 97-101 Attractive potentials, 355-358
C Carbon tetrafluoride, reaction with rare gas ions, 2 19-220 CARS. See Coherent anti-Stokes Raman spectroscopy Characteristic energy, 84 Charge-coupled device (CCD) array, 296,302 Charge transfer from singly charged ions, 262-267 internal energy and, 286-288 multiply charged ions, 267-271 Chemical etching, 7. See also Plasma etching halogen-bearing molecules, 13 Chemically reactive plasmas, applications of,
B Backscattering, 359-360 Binary collision approximation (BCA) model, 35 1-352 Binary collision model, plasma-surface interactions, 343-352 Binary encounter Bethe (BEB) technique, 93, 129-131, 136, 157 Binary encounter dipole (BED) model, 13G131 407
Chemical sputtering, 367-370 Chemical vapor deposition (CVD), 3, 4 high-sensitivity white-light absorption spectroscopy diamond films, 298, 303-332 GaAs, 332-333 silicon, 333-334 Coherent anti-Stokes Raman spectroscopy (CARS), plasma monitoring, 7, 9 Collisional radiative recombination, 103 Collision cascade, 350, 362, 364 Collision cross section, 112 Collisions. See Electron collisions; Electronmolecule collisions; Ion-molecule collisions
408
INDEX
Configuration-interaction (CI) method, 139 Controlled Fusion Atomic Data Center (Oak Ridge), plasma physics web site, 105 Cross sections. See also individual cross sections conversion to rate constants, 194-195 Current density, 83 CVD. See Chemical vapor deposition
D Deexcitation cross sections, 235-240 Deutsch-Mark (DM) formalism, 157 Diamond, properties, 303-304 Diamond films, chemical vapor deposition (CVD), 298, 303-332 Dirac-Fock-Slater (DFS) potential, 355, 357 Direct knock-on, 349 Dissipation kequencies, 49, 50 Dissociation cross section, 9 4 9 7 Dissociative attachment, 97-99 low-temperature plasma, I 18 Dissociative recombination, 101-103 Distribution functions, steady-state plasma, 35-41 Doubly charged ions, chemical reactions of, 271-272 Dusty plasmas, 373-395 astronomical plasmas, 381-386 plasma-processing discharges, 375-379 suspension, alignment and ordering, 379-381 upward-directed lightning, 373, 386-395
E Elastic cross sections, 90-91 Elastic and exciting collisions, 22-24 Elastic scattering by nitrous oxide, 131-134 low-temperature plasma, 113-1 16 Elastic scattering cross section, lowtemperature plasma, 113-1 16 Electron beam current, I50 Electron beam measurements, 8 1 Electron collisions, 80 elastic and inelastic, 21
plasma modeling with collision data, 80-106 Electron cyclotron resonance (ECR), 6 Electron density, 30 plasmas, 2 Electron-electron interaction, 4 3 4 5 Electronic excitation low-temperature plasma, 1 19- 120, 139 of reactant ions, 282 Electronic excitation cross section, 9 4 9 7 Electronic stopping power, 348-359 Electron impact excitation, of ethylene, 134136 Electron impact ionization, 120, 147-148 of sulfur hexafluoride, 136138 Electron kinetics, 20-29 Bolztmann equation, 25-26 space-dependent plasmas, 6 1-75 steady-state plasmas, 3 2 4 7 time-dependent plasmas, 4 7 4 1 Electron-molecule collisions, 3, 5-6 low-temperature plasma, 11 1-143 in plasma formation, 3, 12-14 Electrons nonequilibrium behavior, 20-22 pulselike field disturbances space-dependent plasmas, 70-73 in time-dependent fields, 5 7 4 0 spatial relaxation, 6 4 7 0 temporal relaxation of, in time-dependent fields, 52-57 transport equation, 85 in weakly ionized plasmas, 22-24 Electron swarm data, 8 1, 86-88 Eley-Rideal model, 383 Elves, in dusty plasmas, 373, 386, 396 Embedded atom potential, 358 Endothermic ion-molecule reactions, 193 Energy current density, 30 Energy distribution, of sputtered particles, 364367 Energy transfer, plasma-surface interactions, 344345 Equation of motion, of dust grain, 382 Etching plasmas, 334335 Ethylene, electron impact excitation of, 134-136 Excited states, 15-16 Exothermic ion-molecule reactions, 192-193
409
INDEX F Fast-neutral-beam apparatus, 15 1-153 Field disturbances of electrons space-dependent plasmas, 70-73 in time-dependent fields, 57-60 Flow-tube technology, 191-192 Fluorocarbons ion-molecule reactions, 2 19-22 1 thermochemistry, 22 1-222 Free radical reactions, in reactive plasma, 233-235 Free radicals, in plasma, 7
G GaAs, chemical vapor deposition (CVD), 332-333 GAMESS (software), I36 GAPHYOR Data Center. plasma physics web site, 106 Gas discharge modeling, 86 Glow-discharge reactor, 3, 4 Group III metals, reaction with silane, 201
H Heat flux vector, 83 Helium ion reactions with carbon tetrafluoride, 2 19-220 with silane, 195-1 98 with silicon tetrachloride, 2 15-2 18 with silicon tetrafluoride, 207-209 Hexamethyldisiloxane (HMDSO), ionization cross section, 172-177 High-resolution double focusing mass spectrometer, 153-1 56 High-sensitivity white-light absorption spectroscopy, 295-296, 337 apparatus, 296-298 argon radio-frequency plasmas, 335-337 calculations, 298-303 chemical vapor deposition (CVD) diamond films, 298, 303-332 GaAs, 332-333 silicon, 333-334 etching plasmas, 334-335
Hydrocarbon gases, reactions with silicon ions, 204 Hydrogen, interstellar formation, 383-385 Hydrogenated carbon films, plasma processing for deposition, 375 Hydrogen atom recombination, 383-385 Hydroperoxyl radical, thermochemistry, 223
I Infrared laser absorption spectroscopy (IRLAS), plasma monitoring, 7-9 Internal energy, charge transfer process, 286-288 International Atomic Energy Agency, AMDIS plasma physics web site, 106 Ion current, 150 Ion dipole-induced dipole interaction, rate constants, 251-253 Ion-induced dipole interactions, 249-250, 274-277 Ionization binary encounter Bethe (BEB) model, 129-131 low-temperature plasma, 120 in plasma gases, 92-94 Ionization coefficient, 84, 88, 95 Ionization cross section, 92, 148-149 hexamethyldisiloxane (HMDSO), 172-1 77 measurement fast-neutral-beam apparatus, 15 1-1 53 high-resolution double focusing mass spectrometer, 153-156 measurement of, 149-1 56 silane, 160-1 68 tetraethoxysilane (TEOS), 170-172 total single ionization cross section, semiempirical calculation of, 156-1 60 Ion-molecule collisions, 3-6 Ion-molecule reactions, 188-1 89, 243-249, 288-289 energy, effect of on, 279-288 experimental methods, 189-195 fluorocarbons, 2 19-222 organosilanes, 204-207 reaction rate constants, 244-253 silane, 178, 195-204 silicon tetrachloride, 2 15-2 19 silicon tetrafluoride, 207-2 15
410
INDEX
Ion-molecule reactions (cont.) types association reactions, 277-279 charge transfer, 262-271 isomerization, 256-259 molecular ions, 272-277 multiply charged ions, 267-27 1 proton transfer, 255-256 rearrangement of bonds, 253-255 switching reactions, 259-262 vibration deexcitation and excitation, 272-277 Ionospheric discharges, lightningaccompanied upward-directed, 373, 386-395 Ion-permanent dipole interaction, rate constants, 25&25 1 Ions, scattering at surfaces, 358-360 Ion-surface collisions, 404 IRLAS. See Infrared laser absorption spectroscopy Isomerization reactions, rate constants, 256-259 Isotropic distribution, steady-state plasma, 34-35
K Kinema Research & Software, web site, 106 Kinetic equation, 26-28 Kinetic equations space-dependent plasma, 6 1-64 steady-state plasma, 32-34 time-dependent plasma, 48-52 Kinetics, electrons in plasma. See Electron kinetics Kohn expression, 124-125 Kohn variational method, 124-126, 139 Krypton ion reactions with silane, 195-198 with silicon tetrafluoride, 207-209
L Landau-Teller equation, 272 Langevin-Gioumousis-Stevenson(LGS) model, 192-193 Langmuir-Hinshelwood model, 383 Laser-induced fluorescence (LIF), 7, 9, 283
Laser light scattering (LLS) methods, 376 Lennard-Jones potential, 355 Lightning-accompanied upward-directed ionospheric discharges, 373, 386-395 Long-lived negative ions, 99-101 Low-temperature plasma, 147 collisions and reactions of, 3-5 cross sections attachment, I 18-1 I9 calculations, 121-131 elastic scattering, 1 13- 1 16, 131 electron excitation, 119-120, 134-138 ionization, 120 momentum transfer, 1 16- 1 17 vibrational excitation, 117-1 18 defined, 2 electron-molecule collisions, 1 1 1-143 formation of, 6 Lumped dissipation frequencies, 49, 50 Lumped power loss, 30
M Mass spectrometer, high-resolution double focusing mass spectrometer, 153-156 Mass spectrometry, proton-transfer reaction mass spectrometry (PTR-MS), 255 MCSCF method. See Multiconfiguration selfconsistent field (MCSCF) method Mean collision frequency, 3 1 Mean energy density, 30 Methane reactions with oxygen reactive ions, 223 with rare gas metastable states and ions, 15 Methyne hydroperoxy ion, 223 Modeling, plasma. See Plasma modeling Modified additivity rule, 157, 159 Molecular dynamics model, plasma-surface interactions, 3 52-3 53 Molikre potential, 355 Momentum gain, 30 Momentum losses, 31 Momentum transfer, low-temperature plasma, 116-117 Momentum-transfer cross section, 84, 90 low-temperature plasma, 1 16 Monte Carlo simulation, plasma modeling, 10 Morse potential, 355 Multiconfiguration self-consistent field (MCSCF) method, 139
INDEX Multiply charged ions, charge transfer, 267-27 1 N
National Institute for Fusion Science (NIFS), plasma physics web site, 106 National Institute of Standards and Technology (NIST), plasma physics web site, 106 Negative ions, long-lived, 99-101 Neon ion reactions with carbon monoxide, 265 with carbon tetrafluoride, 21 9-220 with nitrogen, 265 with s h e , 195-198 with silicon tetrachloride, 21 5-2 18 with silicon tetrafluoride, 207-209 Nitrogen reactive ion reactions with silane, 198-199 with silicon tetrafluoride, 209-21 0 Nitrous oxide, elastic scattering by, 131- 134 Nuclear stopping power, 346 0
Oak Ridge National Laboratory (ORNL), plasma physics web site, 105 Organic silicon compounds, electron impact ionization of, 161-182 Organosilanes ion-molecule reactions, 204207 thermochemistry, 20&207 Oxygen ions, metastable, 282 Oxygen reactive ion reactions with fluorocarbons, 220-221 with methane, 223 with silane, 198-199 with silicon tetrachloride, 2 16 with silicon tetrafluoride, 209-2 10
P Particle current density, 30 Photodiode array, 297-298 Physical sputtering, 361-367 PKA. See Primary knock-on atoms Plasma chemically active species, transport and
41 1
reactions, 14- 1 5 chemical reactions on substrate surface, 6-7 collision processes, 3 4 , 80 dust in, 373-395 electron density of, 2 electron kinetics, 20-29 Boltzmann equation, 25-26 space-dependent plasmas, 6 1-75 steady-state plasmas, 3 2 4 7 time-dependent plasmas, 47-6 1 electron-molecule collisions. See Electronmolecule collisions electron nonequilibrium behavior, 20-22 formation, 3, 12 ion-molecule collisions. See Ion-molecule collisions simulations. See Plasma modeling Plasma chemistry. future opportunities, 399406 Plasma etching, 4, 188, 367-370 chemical reactions on substrate surface, &7 high-sensitivity white-light absorption spectroscopy, 334-335 of silicon and silicon dioxide, 4-6 Plasma modeling, 10-1 2 electron collision data and, 80-106 Plasma monitoring, 7-9 Plasma-processing discharges, dust in, 375-379 Plasma-surface interactions, 34 1-342 binary collision model, 343-352 molecular dynamics model, 352-353 scattering potentials, 353-358 Power gain, 30 Power losses, 30 Primary hock-on atoms (PKA), 361 Projectile energy dependence, 361-362 “Proto blue jets,” 387 Proton-transfer reaction mass spectrometry (PTR-MS), 255 Proton-transfer reactions, 255-256 Pulselike field disturbances of electrons space-dependent plasmas, 7C73 in time-dependent fields, 57-60
R Radio-frequency (RF) plasma reactor, 375-3 76
412
INDEX
Rare gas atoms, excited, deexcitation by molecules with Group IV elements, 235-240 Rare gas ion reactions with fluorocarbons, 219-220 with methane, 15 with silane, 15, 195-204 with silicon tetrachloride, 21 5-2 18 with silicon tetrafluoride, 207-209 spin-orbit states of, 281-282 Rate constants, 244253 association reactions, 277-279 charge transfer reactions, 262-27 I ion dipole-induced dipole interaction, 25 1-253 ion-induced dipole interactions, 249-250 ion-permanent dipole interaction, 250-25 1 isomerization, 256-259 molecular ion reactions, 272-277 multiply charged ion reactions, 267-27 1 proton transfer reactions, 255-256 rearrangement of bonds, 253-255 switching reactions, 259-262 vibration deexcitation and excitation, 272-277 Reaction rate constants. See Rate constants Reactive plasma atomic and molecular processes, 232-240 characterization, 23 1 free radical reactions in, 233-235 Reactive scattering, 141-143 Recombination interstellar hydrogen atom recombination, 383-385 in plasma gases, 101-104 Recombination rate, 385 Red sprites, in dusty plasmas, 373, 386 Reduced stopping power, 347 Reflection coefficients, 359-360 Repulsive interactions, 272-274 Repulsive potentials, 353 Response surface methodologies, 403 R-matrix method, 128-129, 139
S Scattering. See also Elastic scattering; Reactive scattering of ions at surfaces, 358-360 many-electron treatment, 121-23
Scattering angle, plasma-surface interactions, 344-345 Scattering potentials, plasma-surface interactions, 353-358 Schwinger multichannel (SMC) method, 126-128, 139 Screened Coulomb potentials, 353 Secondary knock-on atoms (SKA), 361 Silane ionization cross section, 160-1 68 ion-molecule reactions, 178 with nitrogen ions, 198-199 with oxygen reactive ions, 198-1 99 with rare gas ions, 195-198 silicon hydride thermochemistry, 201-204 with silicon ions, 199-201 with transition metal ions, 201 plasma, 7-9 reactions with rare gas metastable states and ions, 15 Silicon chemical vapor deposition (CVD), 333-334 plasma etching, 4, 6 Silicon chlorides, thermochemistry, 2 18-2 I9 Silicon compounds, organic electron impact ionization of, 161-1 82 hexamethyldisiloxane, 172-1 77 silane, 160-168 tetraethoxysilane, 170-1 72 tetramethylsilane, 168-1 70 Silicon dioxide, plasma etching, 4, 6 Silicon fluorides, thermochemistry, 2 13-21 5 Silicon-germanium films, plasma processing for deposition, 375 Silicon hydrides, thermochemistry, 20 1-204 Silicon ion reactions with hydrocarbon gases, 204 with silane, 199-201 with silicon tetrachloride, 2 16-2 17 with silicon tetrafluoride, 2 10-2 13 Silicon-silicon collisions, scattering potentials, 3 55-3 57 Silicon tetrachloride ion-molecule reactions with oxygen reactive ions, 2 1 6 2 17 with rare earth ions, 2 15-2 I6 with silicon ions, 216-217 thermochemishy of silicon chlorides, 2 18-2 19
413
INDEX Silicon tetrafluoride ion-molecule reactions with rare gas ions, 207-209 with reactive nitrogen ions, 209-2 10 with reactive oxygen ions, 209-210 with silicon ions, 21&213 thermochemistry of silicon fluorides, 2 13-215 Simulations, binary collision model, 35 1-352 Singly charged ions, charge transfer from, 262-267 6-12 potential, 355 SKA. See Secondary knock-on atoms SMC method. See Schwinger multichannel (SMC) method Space-dependent plasma, electron kinetics, 61-75 Spatial relaxation, of electrons, 64-70 Spin-orbit states, rare gas ions, 281-282 Sprites, in dusty plasmas, 386, 388-392 Sputtered particles, energy distribution, 364367 Sputtering yield, 349-35 1 Steady-state plasma, 147 electron kinetics, 3 2 4 7 gas mixtures, 4 1 4 3 pure gases, 3 2 4 1 Stopping cross section, 345 Stopping power, 345-349 Structural isomers, 256-259 Sulfur fluorides, thermochemistry, 223-225 Sulfur hexafluoride, electron impact ionization of, 136-138 Surfaces, 375, 403 chemical sputtering, 367-370 physical sputtering, 36 1-367 plasma etching, 367-370 plasma-surface interactions, 341-342 binary collision model, 343-352 molecular dynamics model, 352-353 scattering potentials, 353-358 scattering of ions backscattering, 359-3 60 implantation, 358-359 Switching reactions, 259-262
T Temporal relaxation, of electrons, in timedependent fields, 52-57
Tetraethoxysilane (TEOS), ionization cross section, 17&172 Tetramethylsilane (TMS), ionization cross section, 168-170 Thermochemistry fluorocarbons, 221-222 hydroperoxyl radical, 223 organosilanes, 206-207 silicon chlorides, 2 18-2 19 silicon fluorides, 2 13-2 15 silicon hydrides, 201-204 sulfur fluorides, 223-225 Threshold energy, 193 Thunderstorms, upward-rising lightening, 373, 386-395 Time-dependent plasma, electron kinetics, 47-6 1 Total momentum loss, 3 1 Total power loss, 30 Total single ionization cross section, semiempirical calculation of, 156-1 60 Transient attachment, 99-101 Transition metal ions, reaction with silane, 20 1 Transport, in plasmas, 14-15 Transport coefficients, 82-86 Transverse diffusion coefficient, 84 Trimethylgallium (TMGa), CVD feed bas, 332
U Ultra large-scale integrated (LSI) circuits, manufacture, 1-2 Universite Paris-Sud, plasma physics web site, 106 Upward-directed lightning, in dusty plasmas, 386-395
V
Velocity distribution, expansion of, 26 Velocity distribution function, 2 1, 22, 24-25 Velocity space averages, 2 4 2 5 Verlet algorithm, 352 Vibrational excitation, 97-99 low-temperature plasma, 1 17-1 18 of reactant ions, 283-285
414
INDEX
Vibrational quenching rate constant, LandauTeller equation, 272 Vibrons, 384
Web sites, 105-106 Weizmann Institute of Science, plasma physics web site, 105 World Wide Web, as universal database, 104
W
Weakly ionized plasma deiined, 2 electron kinetics, 20-29 Boltzmann equation, 25-26 space-dependent plasmas, 6 1-75 steady-state plasmas, 32-47 time-dependent plasmas, 4 7 4 I electrons in, 22-24
X Xenon ions, reaction with silane, 195-198
Z
Ziegler-Biersack-Littmarkpotential, 354
Contents of Volumes in This Serial Volume 1
Volume 3
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. 7: Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and 1 I! Toennies High-Intensity and High-Energy Molecular Beams, 1 B. Anderson, R. I! Andres. and 1 B. Fen
The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy o f Stored Ions I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C.Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Srickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Bedder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood Volume 4
Volume 2
H. S. W. Massey-A E. H. S. Burhop
The Calculation of van der Waals Interactions, A. Dalgarno and R D. Davison Thermal Diffusion in Gases, E. A. Mason, R. 1 Munn, and Francis 1 Smith Spectroscopy in the Vacuum Ultraviolet, R R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and P Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, B 1 De Heer Mass Spectrometry o f Free Radicals, S. N. Foner
Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, E! A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holr and B. L. Moiselwitsch Resonances in Electron Scattering by Atoms and Molecules, I? G. Burke Relativistic h e r Shell Ionizations, C. B. 0. Mohr Recent Measurements on Charge Transfer, 1 B. Hasted Measurements of Electron Excitation Functions, D. u! 0.Heddle and R. G. R Keesing 415
Sixtieth Buthday Tribute,
416
CONTENTS OF VOLUMES IN THIS SERIAL
Some New Experimental Methods in Collisions Physics, R. I? Stebbings Atomic Collision Processes in Gaseous Nebulae, M. 1 Seaton Collisions in the Ionosphere, A. Dalgamo The Direct Study of Ionization in Space, R. L. F Boyd
Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, I? C. Fehsenfeld, and A. L. Schmeltekopj Experiments with Merging Beams, Roy. H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H.G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R. 1 S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s,sIU pq, C. D. H. Chisholm. A. Dalgamo, and E. R. Innes Relativistic 2-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Volume 6 Dissociative Recombination, 1 N. Bardsley and M. A. Biondi Analysis o f the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S.Kaujnan The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and ‘I: R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston
Volume 7 Physics of the Hydrogen Master, C. Audoin, 1 I? Schermann. and I? Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, 1 C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, 1 Gerratt Diabatic States of Molecules-QuasiStationary Electronic States, Thomas I? O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H.S.Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. 1 Greenfield
Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y Chen and Augustine C. Chen Photoionization with Molecular Beams, R.B. Cairns, Ualstead Harrison. and R. I. Schoen The Auger Effect, E. H. S. Burhop and W N. Asaad
Volume 9 Correlation in Excited States of Atoms, A. W Weiss The Calculation o f Electron-Atom Excitation Cross Sections, M.R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka
CONTENTS OF VOLUMES IN THIS SERIAL The Differential Cross Section of Low-Energy Electron-Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy
417
Topics on Multiphoton Processes in Atoms, l? Lambropoulos Optical Pumping of Molecules, M. Brayer, G. Goudedard, 1 C. Lehmunn, and 1 ViguC Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, WiIhelm Raith Ion Chemishy in the De Region, George C. Reid
Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Annstrong, JK and Serge Feneuille The First Born Approximation, K. L. Bell and A. E. Kingston Photoelectron Spectroscopy, R C. Price Dye Lasers in Atomic Spectroscopy, W Lunge, 1 Luther and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley 7: Huntress, Ju.
Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M. 1 Seaton The R-Matrix Theory of Atomic Process, l? G. Burke and R D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M. E Golde and B. A. Thrush
Volume 12 Nonadiabatic Transitions between Ionic and Co-valent states, R.K. Janev Recent Progress in the theory of Atomic Isotope Shift, 1 Bauche and R.J-Champeau
Volume 13 Atomic and Molecular Polarizabilihes-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. V Hertel and R Stall Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and 1 Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, R B. Somerville
Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webstel; Michael 1 Jamieson, and Ronald E. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and Two Electron Atoms, Richard Marrus and Peter 1 Mohr Semiclassical Effects in Heavy-Particle Collisions, M S.Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in IonAtom Collisions, S. % Bobashev Rydberg Atoms, S.A. Edelstein -und ? F Gallagher
418
CONTENTS OF VOLUMES IN THIS SERIAL
UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree
Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N. Fortson and L. Rlets
Volume 15 Negative Ions, H. S.W Massey Atomic Physics from Atmospheric and Astro-physical Studies, A. Dalgarno Collisions o f Highly Excited Atoms, R. E Strebbings Theoretical Aspects of Positron Collisions in Gases, 1 W Humberston Experimental Aspects of Positron Collisions in Gases, Z C. Grijith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisions at Low Energies, 1 5.Hasted The Theory o f Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization E. H. S.Burhop Excitation of Atoms by Electron Impact, D. W 0. Heddel Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions. I? G. Burke
Volume 16 Atomic Hartree-Fock Theory, M. Cohen and R. I? McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren Sources of Polarized Electrons, R. 1 Celotta and D. Z Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S.Swain Spectroscopy of Laser-Produced Plasmas, M. H. Key and R. 1 Hutcheon
Volume 17 Collective Effects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D. S. F Crothers Atomic Rydberg States, Serve Feneuille and Pierre Jacquinot Superfluorescence, M. F H. Schuurmans, Q.H. E Vrehen, D.Polder, and H. M. Gibbs Applications o f Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G. Payne, C. H. Chen. G. S.Hurst, and G. W Foltz Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun,I? L. Dufton and A. E. Kingston
Volume 18 Theory of Electron-Atomic Scattering in a Radiation Field, Leonard Rosenberg PositronGas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, 1 Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S.Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, iV Anderson and S.E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W Norcross and L. A. Collins
CONTENTS OF VOLUMES IN THIS SERIAL Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. R R Drake
Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K. Janev Interactions of Simple Ion-Atom Systems, 1 T Park High-Resolution Spectroscopy of Stored Ions, D. 1 Wineland, Wayne M. Itano, and R. S. Van Dyck, Jr Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K . Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatonic Molecules and Its Applications, i? JenE The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization o f Atomic and Molecular Photoelectrons, N. A. Cherepkov
Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T D. Mark and A. R Castleman, J r Nuclear Reaction Effects on Atomic Inner-Shell Ionization, R E. Meyerhofand J - F Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I. L. Sobel'man and A. R Enogradov Radiative Properties of Rydberg State, in Resonant Cavities, S.Haroche and 1 M. Ralmond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, 1 A. C. Gallas, G. Leuchs. H. Walther and H. Figger
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Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P 0'Brien. Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory o f Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-1 Chu Scattering in Strong MAgnetic fields, M. R. C. McDowell and M. Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More
Volume 22 Positronium-Its Formation and Interaction with Simple Systems, 1 R Humberston Experimental Aspects of Positron and Psitronium Physics, T C. GrrfJith Double Excited States, lncluding New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Pearl Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S.Swain
Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian I? Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan
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CONTENTS OF VOLUMES IN THIS SERIAL
Transition Arrays in the Spectra of Ionized Atoms, 1 Baz1che.C. Bauche-Amour, and M. Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, E. 1 Wuilleumier, D. L. Ederer; and 1 L. Picqut
Volume 24 The Selected Ion Flow Tube (SIDT): Studies of lon-Neutral Reactions, D.Smith and N. G. Adams Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. 1 Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. 1 Knize, Z. Wu, and K Happer Correlations in Electron-Atom Scattering, A. Crowe
Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor. Alexander Dalgamo: Contributions to Atomic and Molecular Physics Neal Lane Alexander Dalgamo: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgamo: Contributions to Astrophysics, David A. Wlliams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He+-He Collisions at KeV Energies, R. E Stehhings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu Model-Potential Methods, G. Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen
Schwinger Vanational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H. G. Reid Electron lmpact Excitation, R. 1 K Henry and A. E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H . Bransden and D.F! Dewangan Relativistic Random-Phase Approximation, I$! R. Johnson Relativistic Sturmian and finite Basis Set Methods in Atomic Physics, G. K E Drake and S.I? Goldman Dissociation Dynamics of Polyatomic Molecules, 7: Uzer Photodissociation Processes in Diatonic Molecules of Astrophysical Interest, Kate F! Kirby and Ewine E van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black
Volume 26 Comparison of Positrons and Electron Scattering by Gases, Walker E. Kauppib and Talbert S. Stein Electron Capture at Relativistic Energies, B. L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions-A Close-Coupling Treatment, Mineo Kimura and Neal E Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, ! I Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner; FranGoise Masnou-Sweeuws. and Annick Giusti-Suzor On the Decay of I8’Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Larry Spruch
CONTENTS OF VOLUMES IN THIS SERIAL Progress in Low Pressure Mercury-Rare Gas Discharge Research, 1 Maya and R. Lagushenko
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Cooperative Effects in Atomic Physics, J I! Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, 1 H. McGuire
Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in ElectronAtom Collisions, Joachim Kessler Electron-Atom Scattering, I. E. McCarthy and E. Weigold Electron-Atom Ionization, I. E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, Y I. Lengyel and M. I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule
Volume 28 The Theory of Fast Ion-Atom Collisions, 1 S. Briggs and 1 H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubaiv Cavity Quantum, Electrodynamics, E. A. Hinds
Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Meastable Levels Using Optical and Laser Techniques, Chung C. Lin and L. W Anderson Cross Sections for Direct Multiphoton Ionionization of Atoms, M Y Ammosov, N B. Delone, M. Yu, Ivanov, I. I. Bondar: and A. Y Masalov Collision-Induced Coherences in Optical Physics, G. S. Aganval Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski
Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electroin Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and 1 C. Nickel The Dissociative Ionization of Simple, Molecules by Fast Ions, Colin 1 Latimer Theory of Collisions between Laser Cooled Atoms, 19 S.Julienne, A. M. Smith, and K. Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in IonAtom Collisions, Derrick S. F Crothers and Louis 1 Dubd
Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. FT F Drake Spectroscopy of Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Fiiren, and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Michkle Lamoureux
Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K. L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H. Bransden and C. 1 Nobel Electron-Atom Scattering Theory and Calculations, I! G. Burke
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CONTENTS OF VOLUMES IN THIS SERIAL
Terrestrial and Extraterrestrial H,+, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K. Dolder Quantum Defect Theory and Analysis of HighPrecision Helium Term Energies, G. R F Drake Electron-Ion and Ion-Ion Recombination Processes, M. R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H. B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I. I? Grant The Chemistty of Stellar Environments, D. A. Howe, 1 M. C. Rawlings. and D. A. Williams Positron and Positronium Scattering at Low Energies, 1 R Humberston How Perfect are Complete Atomic Collision Experiments? H. Kleinpoppen and H. Handy Adiabatic Expansions and nonadiabatic Effects, R. McCarroN and D.S. F Crothers Electron Capture to the Contiuum, B. L. Moiseiwitsch How Opaque Is a Star? M. 1 Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing AfterglowLangmuir Technique, David Smith and Patrik s p a n s Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain Atoms in Cavitites and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, 1 F Williams a n d 1 B. Wang
Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A. R. Filippelli. Chun C. Lin, L. R Andersen. and 1 W McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S.Trajmar and 1 R McConkey
Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R. R Crompton Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytrc Representation of Cross-Section Data, Mitio Inokuti. Mineo Kimura, M. A. Dillon, Isao Shimamura Electron collisions with N,, 0, and 0: What We Do and Do Not Know, Yukikazu Itikuwa Need for Cross Sections in Fusion Plasma Research, Hugh I? Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto, and M. Cacciatore Guide for Users of Data Resources, Jean R Gallagher Guide to Bibliographies, Books, Reviews, and Compendia of Data on Atomic Collisions, E. R McDaniel and E. 1 Mansky Volume 34 Atom Interferometry, C. S. Adams. 0.Carnal, and 1 Mlynek Optical Tests o f Quantum Mechanics, R. Y Chio, I? G. Kwiat. and A. M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andrease Buchleitner Measurements of Collisions between LaserCooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasma, 1E. Lawler and D. A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N . A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E. C. Montenegro, R E . Meyerhof and J: H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D. L. Moores and K. 1 Reed
CONTENTS OF VOLUMES IN THIS SERIAL Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates Volume 35 Laser Manipulation of Atoms, K. Sengstock and W Erimer Advances in Ultracold Collisions: Experiment and Theory, 1 Weiner Ionization Dynamics in Strong Laser Fields, L. R DiMauro and I! Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U Buck Femtosecond Spectroscopy of Molecules and Clusters, E Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. T Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W R. Johnson, D.R. Plante, and 1 Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H. 0. Everitt and R C. De Lucia Volume 36 Complete Experiments in Electron-Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat Stimulated Rayleigh Resonances and RecoilInduced Effects, 1-Y Courtois and G. Giynbep Precision Laser Spectroscopy Using AcoustoOptic Modulators, W A. Van Wijingaarden Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan I? Dowling and Julio Gea-Banacloche Optical Lattices, I! S. Jessen and I. H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert R Krause and Sheldon Datz
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Evaporative Cooling of Trapped Atoms, Worfgang Ketterle and N. 1 Van Druten Nonclassical States o f Motion in Ion Traps, 1 I. Cirac, A. S. Parkins. R. Blatt, and I! Zoller The Physics of Highly-Charged HEavy Ions Revealed by Storage/Cooler Rings, I? H. Mokler and Th. Stohlker Volume 38 Electronic Wavepackets, Robert R. Jones and L. D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D. G. Thompson Optical and Magneto-Optical Spectroscopyof Point Defects in Condensed Helium, Serguei 1. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G. M. Lankhuijzen and L. D. Noordam Studies o f Negative Ions in Storage Rings, L. H. Andersen, E Andersen, and I? Hvelplund Single-Molecule Spectroscopy and Quantum Optics in Solids, W E. Moerner, R. M. Dickson and D. 1 Norris Volume 39 Author and Subject Cumulative Index Volumes 1-38
Author Index Subject Index Appendix: Tables of Contents of Volumes 1-38 and Supplements Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W King Storage Ring Laser Spectroscopy, Thomas U Kuhl Laser Cooling of Solids, Carl E. Mungan and Emothy R. Gosnell Optical Pattern Formation, L. A. Lugiato, M. Brambilla. and A. Gaiti
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CONTENTS OF VOLUMES IN THIS SERIAL
Volume 41
Quantum Communication with Entangled
Photons, Hamld Weinfirrter lko-Photon Entanglement and Quantum Reallity, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Sali?res, Ann L'Huiller, Philippe Antoine, and Maciej Lewenstein Atom Optics in quantized Light Fields, Matthias Freybvurger, Alois M.Herjbmmer, Daniel S.Krd, Erwin Mayr, and Wolfgang f? Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplificationby Optical Pumping, UlfJanicke and Martin Alkens
Volume 42 Fundamental Tests of Quantum Mechanics, Edward S.Fry and Thomas Walther WaveParticle Duality in an Atom Interferometer, Stephen D&r and Gerhard Rempe Atom Holography, Fuji0 Shimizu Optical Dipole Traps for Neutral Atoms, Rudolf Grimm, Matthias Weidemiillez and Yurii B. Ovchinnikov Formation of Cold (T 5 1K) Molecules, J L Bahns. f? L. Gould, and K C. Shoalley High-Intensity Laser-Atom Physics, C. 1 Joachain, M. Don; and A? J Kylstra Coherent Control o f Atomic, Molecular, and Electronic Processes, Mosha Shapiro and Paul Brumer Resonant Nonlinear Optics in Phase Coherent Media, M. D. Lukin, f? Hemme,: and M. 0.Scully The Characterizationof Liquid and Solid Surfaces with Metastable Helium Atoms, H. M o p e r
Volume 43 Plasma Processing of Materials and Atomic, Molecular, and Optical Physics. An Introduction, Himshi Tanaka and Mitio Inohti The Boltzmann Equation and Transport Coefficientsof Electrons in Weakly Ionized Plasmas, R. Mnkler Electron Collision Data for Plasma Chemistry Modeling, WL. Morgan Electron-Molecule Collisions in LowTemperature Plasmas: The Role of Theory, Carl winstead and Vincent McKoy Electron Impact Ionization of Organic Silicon Compounds, RalfBasne,: Martin Schmidt, Hans Deutsch and Kurt Becker Kinetic Energy Dependence of Ion-Molecule Reactions Related to Plasma Chemistry, R B. Armentrout Physicochemical Aspects of Atomic and Molecular Processes in Reactive Plasmas, Yoshihiko Hatano Ion-Molecule Reactions, Werner Lindinger, Armin Hansel and Zdenek Herman Uses of High-SensitivityWhite-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing, L. K Anderson, A N Goyette, and JE. Lawler Fundamental Processes of Plasma-Surface Interactions, Rainer Hippler Recent Applications of Gaseous Discharges: Dusty Plasmas and Upward-Directed Lightning, Ara Chutjian Opportunities and Challenges for Atomic, Molecular, and Optical Physics in Plasma Chemistry, Kurt Becker, Hans Deutsch and Mitio Inohti
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