Plasma Sources Sci. Technol. 9 Introduction To Gas Discharges
Plasma Sources Sci. Technol. 9 (2000) 517–527. Printed in the UKPII: S0963-0252(00)15592-6Introduction to gas dischargesN St J BraithwaiteThe Open University, Oxford Research Unit, Foxcombe Hall, Boars Hill, Oxford OX1 5HR,UKReceived 1 December 1999, in final form 13 June 2000Abstract. This is a tutorial article. An introductory discussion of direct current gasdischarges is presented. Beginning with basic ideas from kinetic theory, gas dischargeplasmas are described in terms of phenomena observed in the laboratory. Various models areintroduced to account for electrical breakdown, plasma boundaries and the longitudinal andtransverse structure of discharges.1. IntroductionThis article is intended to set the scene for more detaileddiscussions about the physics of laboratory plasmas. Thereare many types of plasma source such as those basedon discharges created by direct current (dc), capacitivelycoupled radiofrequency (rf) (RIE, PECVD), inductivelycoupled rf (ICP, TCP, Helicon) and microwaves (ECR,Surfatron). The technological applications of plasmasformed in these sources are numerous and include thin filmdeposition, semiconductor processing, materials treatments(modification of surface physics and surface chemistry,sterilization), lamps, light sources and displays, thick filmdeposition, waste treatment and materials analysis.In section 2 some basic results from kinetic theory arerecalled. The nature of the plasma state and of laboratoryplasmas in particular is described in section 3. Sections 4and 5 examine the production and loss of charged particles,including the breakdown phase of a gas discharge. Insection 6 the spatial structure of a steady state, self-sustainingdischarge is discussed, for medium and low pressure. A finalbrief section illustrates the flow of energy in a low-pressure,laboratory plasma. radius of argon atom, rAr 1.5 10 10 m;Boltzmann’s constant, k 1.38 10 23 J K 1 ;mass of argon atom, MAr 40 1.67 10 27 kg;mass of electron, m 9.11 10 31 kg;electronic charge, 1.60 10 19 C.2.1. Moving neutrals (in a thermalized group)At room temperature the average speed of an argon atom(atomic mass 40) is 3.5 102 m s 1 , 2kTv̄ .(2)MAt a pressure of 1 Torr (130 Pa) the random flux of argonatoms at room temperature is 2 1025 atoms m 2 s 1 , nc̄kT n.(3)42π MAt a pressure of 1 Torr (130 Pa) the distance travelled byan argon atom between collisions will be on average about0.11 mm,1(4)λ 4σ ng2. Just particles2σ πrAr.The kinetic description of a stationary gas considers largepopulations of gas atoms with a range (or distribution) ofspeeds moving in all directions. The distributions of speedand velocity are characterised by a mean energy, ε , whichis linked by the form of the distribution to the kinetictemperature, kT . Pressure, p, in a gas is a measure of thedensity of thermal energy associated with the number of gasatoms per unit volume, ng . Thus,p ng kT(1)(SI units: p is in Pa when ng is in m 3 and kT is inJ ; one pascal is equivalent to a joule per cubic metre).Similar concepts apply to charged particle populations. Inthe following some more useful rabbits are pulled out of thehat of kinetic theory. Use will be made of the following data:0963-0252/00/040517 11 30.00 2000 IOP Publishing Ltd(5)The frequency of collisions between gas atoms at roomtemperature is 3.5 106 s 1 ,ν v̄λ(6)A flow of neutrals arises when there are gradients in pressure(i.e. gradients in density and/or temperature).2.2. Moving chargesThe introduction of charge changes a few things. First, chargeis a source of electric field through which every charge exertsforces on any other charges in inverse square proportionto their separation (r 2 ). Second, an externally imposedelectric field will apply forces to any charges that enter it.517
N St J BraithwaiteThird, charges that move relative to magnetic fields alsoexperience forces. The Lorentz force conveniently combinesthe electric and magnetic effects for a chargeF q(E v B )(7)in which: q is the quantity of charge in coulombs; E is electricfield in volts per metre; v is velocity in metres per second; here implies the vector product; B is the flux density inteslas.If work is done on a charge it gains energy1mvd22 eV(8)where vd is a drift velocity. The energy eV can be gainedby moving a given distance, x1 , in the direction of an electricfield: x1eV eE dx.(9)0Elastic collisions randomize particle motions leading to amean thermal speed as equation (2).2.3. Elastic and inelastic collisionsBy far the most common encounter in gases is betweenpairs of particles (binary collisions). When particles interact(collide) momentum and energy must be conserved. Thereare three clear classes of event.(i) Elastic: momentum is redistributed between particlesand the total kinetic energy remains unchanged, e.g. ef ast Aslow elessf ast Aless slow .(ii) Inelastic: momentum is redistributed between particlesbut a fraction of the initial kinetic energy is transferredto internal energy in one or more of the particles (i.e.excited states or ions are formed), e.g. A ef ast A eslower eslower A e . (iii) Superelastic: a third class also needs to be anticipated—here there is more kinetic energy after the collision.Momentum is conserved and internal energy in theparticles entering into a collision is transferred intokinetic energy, e.g. Aslow Bslow Af aster Bf aster .Detailed analysis of binary collisions reveals some usefulgeneral points. Lighter particles (m) cannot lose much energy elasticallyto heavier particles (M)—at best a fraction 2m/M,nevertheless, substantial changes in momentum occur(cf tennis). A moving particle on striking elastically a stationaryone of equal mass head-on can transfer all of its kineticenergy (cf billiards). Lighter particles can lose virtually all their kinetic energythrough inelastic collisions with heavier objects (cfsandblasting). Equal mass particles can lose no more than half theirkinetic energy inelastically on collision (cf ion impactionization). Binary collisions in which at least one particle is chargedmay be dominated by long-range Coulomb forces.518Figure 1. Calculation of mean free path.2.4. Cross section and mean free path—elastic collisionsA thorough study of collisions considering scattering anglesand impact parameters will be covered in a companion article.Here a simpler ‘zeroth order’ approach is presented to accountfor the more general features.Consider an electron moving through a number ofstationary argon atoms; see figure 1. Simple elasticencounters only will be dealt with here so the target atomsare viewed as hard spheres and the electron is assumed tobehave as a point mass; effects arising from its charge are notincluded. The question to be addressed is how far the electroncan be expected to penetrate before having a collision.The number of target (argon) atoms in a cuboid xyz2is ng xyz. Each atom presents a cross section, π rAr σ,obscuring the electron path. Viewed through the face xy thetotal area blocked by atoms will be ng xyzσ . When the cuboidextends as far as one mean free path (λ), virtually the entireface xy will be obscured so(ng xyλ)σ xyλ 1ng σ(10)and the electron is very likely to have a collision(cf equation (4) which has a factor of four to account for thesize of the projectile atom). From equation (6), the frequencyof encounters can be estimated for projectiles, electrons inthis case, with a mean speed v̄v v̄ng σ.(11)The scaling of mean free path and frequency of collisions withpressure, through ng , and energy, through v̄, are consistentwith intuition.In practice cross sections are not actually independentof energy, even for elastic collisions. High-energy electronsspeed past so quickly that the chances of interacting with theouter shell electrons on an atom are reduced. Also, at very lowenergy, quantum mechanics may prevail, taking an electronaround an atom with a marked inability to interact overa narrow range of energy—the so-called Ramsauer effect.Figure 2 illustrates these features for argon, showing alsoinelastic cross sections. The simple hard sphere cross sectionis about 3 10 20 m2 .3. Characterizing plasmas3.1. Similarities and differencesAll plasmas have a number of features in common. Forinstance, they are composed of equal amounts of positive
Introduction to gas dischargesFigure 4. Confined (laboratory) plasmas.Figure 2. Cross sections for argon (approx./schematic).ionized gas. Nowadays, continuous electrical dischargesare often used to create a dynamic steady state in whichthere is a balance between production (sources) and loss(sinks). An overflowing water tank maintains a dynamicsteady state as water flows out at the same rate as it flows in.If the production and loss processes are physically separated,as often is the case for laboratory plasmas that are non(thermal-) equilibrium, then energy invested in productionis effectively transported to the place of loss and a steadyflow of energy must be supplied to sustain the steady state.Laboratory plasmas are formed when gas is ionizedby driving an electric current through it, or by shiningelectromagnetic radiation into it. Generally, these means ofplasma formation give energy directly to the free electronsin the plasma. Electron–atom collisions then liberate moreelectrons and to some degree heat the gas. The electrons endup quite a bit hotter than the ions, since the electrons carrythe electrical current or absorb the electromagnetic energy,but the presence of the vessel means there is neither time norspace for thermal equilibrium.Figure 3. A 2D map of parameter space (density-energy).4. Origins of species in plasmasand negative charge carried by particles that are arrangedwithout any local ordering, being free to move. One wayto begin a classification is to quantify: (i) the density of thecharge carriers (so many per cubic metre); and (ii) the thermalenergy of electrons in particular. Figure 3 shows this kind oftwo-dimensional (2D) classification of plasmas.Even within a small region of the density–energyparameter space there are innumerable distinctly differentplasmas. For example, a plasma containing electrons andargon ions is distinguishable from one containing negativeions and positive ions formed from fragments of a moleculeof sulphur hexafluoride.3.2. Equilibrium and steady statesThe plasma state can be realised as a thermal equilibriumphase of matter beyond the conditions of gases. However, justas amorphous solids appear to be steady-state (though non(thermal)-equilibrium) solids, so too laboratory-generatedplasmas often exist as steady, non-(thermal)-equilibriumstates. It must be noted also that the lifetime of individualparticles in laboratory plasmas may only be a small fractionof a second, so the steady state is a kind of dynamic (butnon-thermal) equilibrium.3.3. Laboratory plasmas—sustaining the steady stateEarly laboratory experiments in which capacitors weredischarged through a gas provided transient sources ofA laboratory gas discharge is confined; that is it has physicalboundaries—see figure 4. Charged particles are created andlost both within the volume and at the confining surfacesto varying degrees. Not only does this provide a finerclassification of plasmas but also it hints at the possiblescaling issues for plasma sources.Table 1 lists various reactions that can take place at asurface exposed to a plasma. The first two concern etchingand deposition processes that are in turn enhanced by thearrival of energy brought by other particles. Tables 2 and 3 listphenomena that take place in the gas phase, where particlesare ionized, some molecular gases are broken up and othersagglomerate (oligomerize). The last process is the first stagein the formation of particulate matter in plasmas.4.1. Surface production of electronsThe lowest energy required to remove an electron from asolid is the so-called work function (eφ). Lesser quantitiesof energy simply will not do the job (quantum mechanics).The energy can arrive in various forms: thermal (phonons,kT ), photons (hν), internal potential energy of atoms andions (eV , eVi ), kinetic energy ( 21 mv 2 , 21 Mv 2 ).(i) The flux of electrons from thermionic emission showsa strong temperature dependence with a weaker butsignificant dependence on materials. The theoreticalexpression (attributed to Richardson and Dushman) isj AT 2 exp( eφ/kT )(A m 2 s 1 ).(12)519
N St J BraithwaiteTable 1. Surface reactions.ReactionsDescriptionEvidenceAB Csolid A BCvapourAB A Bsolide A AA AA A e (from surface)A (fast) A e (from nsecondary emissionsecondary emissionmaterial erosionthin film formationmajor loss processAuger electronsAuger electronsTable 2. Gas phase reactions involving electrons.ReactionsDescriptionEvidencee A A e e A A e e e A A e e A e A hνe A A e e e AB A B e A e B e A Be A B A Belastic p ionizationfragmentationdissociative ionizationdissociative attachmentvolume recombinationthermal electronsconductivitylight emissionionization efficiencyresidual gas analysisplasma decay and steady-stateTable 3. Gas phase reactions involving ions and neutrals.ReactionsA B B A ‘resonant’ for B AA B B A A B A B e A B A B e A B A B e A BC A B Ce A B A BA B AB A B AB DescriptionEvidencecharge exchangeelastic scatteringexcitationionizationPenning ionizationfragmentation/dissociationvolume recombinationoligomerizationoligomerizationion energy spectraion energy spectraionization efficiencyionization efficiencyionization efficiencyresidual gas analysisplasma decayion mass spectraresidual gas analysisTable 4 gives some data on thermionic emission forvarious materials. The work function is sensitive tocrystal orientation and purity. Richardson’s theorypredicts a value for the constant A that is larger thangenerally observed.Photons are effective removers of electrons provided thathν eφthe excess energy being in the kinetic energy of emittedelectrons. Data in table 4 indicate that ultraviolet (UV)radiation will give rise to photoemission whereas visibleradiation ( 3 eV) will not (λ (in µm) 1.2/E (in eV)).Very energetic photons (e.g. x-rays) release inner shellelectrons from the atoms and the spectrum of emittedelectrons gives a signature characteristic of the material(cf the analytical technique ‘XPS’).(ii) An important source of energy for electron emissionfrom surfaces at plasma boundaries is in the internalenergy of particles such as ions and excited states(especially the metastable ones). In the case of positiveions, recombination on a surface releases to that surfacean amount of energy equivalent to that invested in theion’s production, namely eVi , the ionization energy. Ifthe total ion energy exceeds twice the work functionof the surface 21 Mv 2 eVi 2eφ then in additionto neutralization a secondary electron may be released.520Escape from the material is still subject to a statisticalfactor. Table 5 shows ionization energy data for a rangeof gases; comparison with work function data in table 4shows that the threshold condition above is usually meteven without taking account of the kinetic energy ofincident ions. Neutralization of 10 or 100 eV ionsin practice seems to release secondary electrons withalmost equal effectiveness. Typical values are givenin table 6, in which γi records the effectiveness ofelectron release being the average number of electronsper incident ion. High energy electron impact (100s ofeV) is also effective in releasing electrons.(iii) Electric fields of around 108 V m 1 are enough to pullelectrons directly out of solids in measurable quantities.This kind of field is most easily achieved around sharpfeatures.4.2. Surface production of positive ionsBombardment by a stream of high-energy particles can leadto the formation of positive ions on or near surfaces inquantities useful for analytical purposes (cf SIMS). Hotsurfaces (2000 K) of high work function materials will ionizelow ionization energy atoms, some ions of the hot surfacematerial would also be produced. At lower temperatures(1200 K) metal ions can be derived thermionically fromsalts coated on filaments. None of these mechanisms is
Introduction to gas dischargesTable 4. Thermionic emission data. The theoretical value of A is1.2 106 .Materialφ (eV)A (A m 2 K 2 )Electrons(m 2 s 1 ) at 2000 KWBaOThO2Al2 O3CuSi4.53.42.93.84.44.96.0 1052.5 1041.6 1021.4 1046.5 105—8.2 10191.7 10212.0 10209.5 10191.4 1020—Table 5. Excitation (V ) and ionization (Vi ) energies of variousatoms and molecules.GasV (eV) VI (eV) Ionization 4NO2 —N2 O —H2 412.5917.319.213.8H2 H2 e H2 H H e H H kinetic energy e H H kinetic energy e H H N2 N2 e N N e O2 O 2 e O O e Ar Ar e He He e CO CO e C O e C O e CO e CO2 CO 2 e CO O e CO O e C O O e NO NO e O N e O N e NO2 NO 2 e NO O e N 2 O N 2 O e N2 O e NO N e NO N e H 2 O H 2 O e HO H e HO H e HCl HCl Table 6. Secondary electron emission coefficient (γi ) for impactof slow positive ions ( keV)—indicative values only.SurfaceAlCuSiSiWWIon ArAr Ar He Ar He γi0.120.060.020.170.100.26particularly useful for filling a large vessel with ions at theconcentrations encountered in low and medium pressure gasdischarges around 1015 –1017 m 3 (cf figure 3)—that requiresvolume processes.4.3. Volume ionization (simultaneous electron and ionproduction)Table 5 lists the energies necessary to remove an electronfrom a range of gas atoms.(i) Hot gas will ionize itself (effectively through kineticenergy of the random motion of particles). Since 1 eVis equivalent to a kinetic temperature of 11 000 K, onewould not expect much thermal ionization below a fewthousand kelvin. This is the regime of high-pressure arcsand thermal equilibrium,A B( ) A B( ) e .(ii) Matter absorbs photons when their energy exactlymatches a transition of electrons between energy levels.That means that photo-excitation of gas atoms is aresonant process (being associated with a transitionfrom one atomic level to another). Transitions betweenthe lower excitation levels are generally in the UV(see table 4) but some of the transitions among thehigher lying levels correspond with the visible. Sinceionization energies tend to be two or three times thevalues of work functions, photo-ionization is alwaysassociated with deep UV radiation, except when itinvolves interactions between photons and neutralsalready excited to (temporary) higher states.A hν A e A hν A e .(iii) Long-lived excited states (metastables) can be importantsources of ionization among species where the ionizationenergy of species A is less than the excitation energy ofspecies BA B A B e .This is known as the Penning effect.(iv) The major volume ionization process arises from highenergy impacts between electrons and atoms. So, anelectron with kinetic energy greater than eVi for aparticular species can ionize that species, ef ast A A eslow eslow.The ionization event is a statistical process that issummarized in terms of how far an electron travels onaverage before it takes part in an ionizing collision (forelectrons below the ionization energy this is essentiallyan infinite distance). This is called the mean free pathfor ionization, λi .The way to describe the probability of an ionizingcollision is as for elastic processes. Suppose the numberdensity of ga
The kinetic description of a stationary gas considers large populations of gas atoms with a range (or distribution) of speeds moving in all directions. The distributions of speed and velocity are characterised by a mean energy, ε, which is linked by the form of the distribution to the kinetic
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