PHYSICSOFPLASMAS VOLUME8,NUMBER6 JUNE2001 Langmuir probe analysis for high density plasmas Francis F. Chena) ElectricalEngineeringDepartment,UniversityofCalifornia,LosAngeles,California90095-1594 ~Received 14 December 2000; accepted 28 February 2001! High-density, radio-frequency plasmas used in semiconductor processing have progressed to densities n>531011cm23, where the methods used to interpret Langmuir probe characteristics in low-density(109–11cm23) plasmareactorsarenolongervalid.Thoughtheoryandcomputationsfor arbitrarily dense collisionless plasmas exist, they are difficult to apply in real time. A new parametrizationanditerationschemeisgivenwhichpermitsrapidanalysisofLangmuirprobedata using these theories. However, at high n, measured ion saturation curves are shown which do not agree in shape with the ‘‘correct’’ theory, yielding anomalously high values of n. The discrepancy with independent measures of n, which can exceed a factor of 2, is believed to be caused by charge-exchangecollisionswelloutsidethesheath.Probedesignsforavoidingthisdiscrepancyare suggested. © 2001 American Institute of Physics. @DOI:10.1063/1.1368874# I. BACKGROUND perature T , the OML current to a cylindrical probe is given i by F G A majority of the critical steps in the fabrication of a ccoapmapcuittievrecdhiispchnaorwgeisnvuoselvdefpolrasthmeaseprporcoecsessisnegs.aTrheegsrtaadnudaalrldy I5Apjr A2px1/21ex~12erf~x1/2!! being replaced by so-called high density plasmas, particu- larly inductively coupled plasmas ~ICPs! and helicon wave 2 A plasmas, which are both driven by radio-frequency power. !ApjrAp 11x, ~1! x@1 Thesesourcesarecapableofincreasingtheplasmadensityn fromthe109–11cm23 rangetotheorderof531012cm23. In wherex[2eVp/KTi, Vp istheprobevoltage,Ap theprobe thelowdensityregime,itiscommonpracticeintheindustry area,and jr therandomthermalioncurrent.AsTi!0,theTi to use the orbital motion limited ~OML! theory of ion col- dependencesofxand jr cancel,andafinitelimitingvalueof lection. This theory can be applied successfully well outside the OML current exists: S D its intended range, but its error is greatly enhanced at high A 2 ueV u 1/2 densities. Although a suitable theory exists, it is normalized I ! A ne p . ~2! p p M in such a way that the result must be known before the cal- Ti!0 culation is begun. In this paper we present a method for At the opposite extreme of dense plasmas and thin sheaths, parametrizing the theoretical curves so that fast, real-time ions enter the sheath with the so-called Bohm velocity analysis of probe curves at any density can be made with modern computers. This paper will treat only cylindrical n 5~KT /M!1/2, ~3! B e probes, since spherical ones are impractical. Except at the so that the saturation ion current is end, collisions will be neglected because in high density plasmas the sheaths are much thinner than the mean free I’aneA n , ~4! p B path. Attention will be focused on saturation ion currents, independently of V , since the sheath adds very little to the which present the most difficult problems. p probe radius R . Here, an5n is the ion density at the The OML theory of ion collection was developed by p s sheath edge, with a’1/2. The exact value of adepends on Mott-SmithandLangmuir,1whofoundthattheioncurrentto the conditions in the presheath, which can cause the ‘‘satu- a negatively biased probe is independent of the shape of the ration’’ current to increase with V , even for a plane probe. plasma potential V(r) as long as the current is limited only p Since the presheath thickness is generally >R , there is no bytheangularmomentumoftheorbitingions.Thisrequired p simple way to treat a plane probe theoretically. eitherthearbitraryassumptionofa‘‘sheathedge’’s,beyond Between 1926 and 1957 many probe papers appeared, whichtheionenergydistributionwasMaxwellian,oraV(r) butalloftheminvolvedthearbitraryassumptionofasheath varying so slowly that no ‘‘absorption radius’’ inside of edge,sincecomputersdidnotexisttohandlethedisparityin which all ions are drawn in exists between the probe and scale length between the sheath region and the quasineutral infinity.Thisconditionisneversatisfiedevenatmodestden- plasma region. In 1957 Allen, Boyd, and Reynolds ~ABR!2 sities. For s!‘ and a Maxwellian ion distribution at tem- derived a relatively simple differential equation which could be solved to give V(r) for all r without division into sheath, a!Electronicmail:[email protected] presheath, and plasma regions. However, this theory was 1070-664X/2001/8(6)/3029/13/$18.00 3029 ©2001AmericanInstituteofPhysics Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp 3030 Phys.Plasmas,Vol.8,No.6,June2001 FrancisF.Chen only for spherical probes and only for T 50, so that ions i moved radially into the probe, there was no orbital motion, and the absorption radius was at infinity. Chen3 later ex- tendedtheT 50 calculationstocylindricalprobes.Forfinite i T , ions with small angular momentum J would strike the i probe and be collected, while those with large J would miss the probe and contribute twice to the ion density at any ra- dius r which they reached. Thus, the density used to solve Poisson’s equation for V(r) depends on the current I, which is unknown. This difficult problem was solved by Bernstein and Rabinowitz ~BR!4 in 1959, but only for monoenergetic ions. The angular momentum forms an effective potential barrier for the ions, and those with sufficient energy E to overcome the barrier are collected. Thus, the constants of FIG.1. Curvesofionsaturationcurrentforincreasingplasmadensity~from motion E, J determine the fate of each velocity class. In Ref.13!.Theabscissaisproportionaltoioncurrent,andtheordinatetothe addition,V(r) mayhavealocalminimuminwhichionscan probepotential.ThedashedlinehastheslopeofOMLtheory:I2}V. betrappedinclosedorbits.Fortunatelytherehasneverbeen, to our knowledge, any indication of the existence of such a population of collisionally trapped ions. A simpler method, valid only for highly negative probes, was given by Lam,5 II. PARAMETRIZATION OF LAFRAMBOISE CURVES who took advantage of the disparity in scale lengths at vari- In 1965, Chen13 showed that the apparent linearity of ousradii.Usingboundarylayertechniquesfromaerodynam- current–voltage (I2–V) curves of ion current was fortuitous ics, he derived a graphical method for computing ion cur- and unrelated to the OML formula of Eq. ~2!. For instance, rents. With modern computers, however, this method is no this dependence is found in ABR theory, which has no or- longer useful. Computations based on the ABR, BR, and bital motions, and also for spherical probes, for which OML Lam theories were given by Chen.3,6 Experimental verifica- theory would predict a linear I–V dependence. Figure 1, tion of the BR results was done by Chen etal.7 copied from that paper, shows ABR curves of ion current TheBRcomputationswereextendedtoMaxwelliandis- overalargerangeofj , wherej ~or,simply,j!istheratio tributionsinthedissertationofLaframboise.8Sinceeachve- p p of probe radius to Debye length locityclass(E,J) haditsownidiosyncrasies,andtherewere convergence problems in the solution of the integral equa- j [j[R /l , l [~eKT /ne2!1/2. ~5! tions, these calculations were difficult and nontrivial. Unfor- p p D D 0 e tunately, only the cases b[T /T 50, 0.5, and 1 were i e treated; if bhad been taken to be 0.1, the results could have The slope of the curves at low jp ~low density! is indeed been used forthwith, without the nonuniform convergence consistent with linear I2–V, but the curves bend at high jp problems in the case T 50. When T !0, one might expect ~high density!, approaching true ion saturation with constant i i the BR–Laframboise ~BRL! results to reduce to the ABR I. Figure 2 shows the I–V curves of Laframboise8 for Ti results, but they do so only for spheres, not for cylinders. 50 and various values of j. Since they cannot be easily Thereasonisthatasr!‘ whileT !0,theangularmomen- recalculated, it is these curves that we wish to represent by i tum J takes the indeterminate form ‘30, which is zero for analytic functions for arbitrary values of jand Vp. Follow- spheresbutfiniteforcylinders.Byasymptoticanalysisofthe ing Ref. 8, we use the following normalizations: governing equations, Laframboise showed that this limit de- pendsonwhetherV(r) variesfasterorslowerthan1/r2. For cylinders,itvariesmoreslowly,causingJtobefiniteevenif T 50. Consequently, the ABR theory cannot be used for i cylindrical probes; we must use the Laframboise curves or the BR results, which are only slightly different from each other for b!1. Further computations of this type were given later by VirmontandGodard,9butonlyforsphericalprobes.Numer- ousextensionsofcollisionlessprobetheoryhavebeenmade; for instance, to collisional plasmas by Cohen,10 to flowing plasma by Chung etal.,11 and to magnetized plasmas by Stangeby.12 However, the collisionless theories worked out in the 1960s are still state-of-the-art and are appropriate for high density, low pressure plasmas. These results, however, are normalized to units that depend on the variables to be FIG.2. Laframboisecurvesofnormalizedioncurrentvsnormalizedprobe determined and are therefore difficult to apply to the experi- potential for T50. The curves are labeled by the value of j5R /l . i p p D ment. ~FromRef.8,Fig.40.! Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp Phys.Plasmas,Vol.8,No.6,June2001 Langmuirprobeanalysisforhighdensityplasmas 3031 FIG.4. ExampleoffittingLaframboisedata~points!withafour-parameter function ~line!. The radius of the points is ’2%. Points for which exact values were available are shown by the large squares. The orbital motion limitisshownbythedashedline. functions A(j), B(j)... are found so that the parameters FIG.3. Laframboisecurvesforioncurrentatlowvaluesofj~fromRef.8, ABCD can be evaluated for arbitrary j. In stage 1, the Fig.43!.Eachcurveisforconstanth,andjisplottedontheabscissafrom curves of Fig. 2 were carefully digitized by direct measure- righttoleft. ment.Forlowvaluesofhandjweusedtheexpandedgraph in Ref. 8, shown in Fig. 3. In addition to these graphs, Laframboise8 gave the numerical values of the points which e~V 2V ! h[h[2 p s , were actually computed. Since the curves were no doubt in- p KT e terpolated by a draftsman, in collecting the data set we gave S D I I KT 21/2 I ~6! extra weight to those points for which exact values were i5 i5 i e [ i , known.Anexampleofadatasetandthefunctionalfitusing I enA 2pM nJ 0 p r Eq. ~8! is shown in Fig. 4 on a log–log plot. The scatter in whereV andV aretheprobeandspace~plasma!potentials, thepointsarisesfromerrorsinreadingFigs.2and3because p s I is the ion current to a cylindrical probe with area A of the finite width of the lines. With four parameters i p 52pR L, and J is a random ion current per unit density (ABCD) tobevariedintheleastsquaresfit,amultiplicityof p r ~evaluated at T !. Equation ~6! is invariant to the system of solutionscouldbeobtaineddependingonthestartingvalues. e units,butitisconvenienttoexpressI andeinmks,withthe WefirstfittheslopesBandDtotheleftandrightportionsof i other quantities in cgs. Note that hdepends on V and T , thecurve,respectively,andthenadjustedthevaluesofAand s e andionn,allquantitiesthatarenotknownuntiltheanalysis C to get an overall fit. Only then were all four parameters is complete. varied to get the final least squares minimization. Steinbru¨chel etal.14,15 and Mausbach16 have param- The entire data set and the corresponding least squares etrized these curves with a two-parameter function of the fits are shown in Fig. 5. All available values of jare listed, form but to avoid clutter some values are not plotted. The curve i5AhB, ~7! for j50 agrees with the OML limit given by Eq. ~2!. Close examination of Fig. 5 will show that the slope changes dis- but it is clear that the bend in the I–V curves in Fig. 1 for large jcannot be represented by so simple a function. In- stead, we have used the following four-parameter fitting function: 1 1 1 5 1 , ~8! i4 ~AhB!4 ~ChD!4 where the parameters ABCD are functions of j. The first term on the right in Eq. ~8! is dominant for small h, giving an approximate i2}hdependence, while the second term dominates at large h, where the slope is smaller. The ratio C/A determines where the bend in the curve occurs, and the exponent4affectsthesharpnessofthebend.Fortunatelythe same exponent could be used for all curves. The parametrization proceeds in two stages. In stage 1, FIG. 5. The Laframboise data set ~points! and analytic fits ~lines! for all values of ABCD are found which give good fits to the availablevaluesofj.Thecurvesareinthesameorderasinthelegend,but curves of Fig. 2 for the available values of j. In stage 2, forclaritysomecurvesarenotdrawn. Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp 3032 Phys.Plasmas,Vol.8,No.6,June2001 FrancisF.Chen TABLEI. FittingparametersusedforthecurvesofFig.5. TABLEII. CoefficientsforcalculatingABCD(j) forj.3. j A B C D a b c d f g 0 1.585 0.451 1.218 0.526 A 1.142 19.027 3.000 1.433 4.164 0.252 1 1.453 0.477 1.233 0.517 B 0.530 0.970 3.000 1.110 2.120 0.350 2 1.445 0.494 1.224 0.514 C 0.000 1.000 3.000 1.950 1.270 0.035 2.5 1.412 0.531 1.255 0.486 D 0.000 2.650 2.960 0.376 1.940 0.234 3 1.142 0.541 2.146 0.316 4 1.433 0.646 1.306 0.378 5 1.513 0.670 1.244 0.351 10 1.473 0.635 1.166 0.257 20 1.384 0.622 1.104 0.181 attempts to smooth over them yielded poor results. More 50 1.203 0.544 1.095 0.091 insightintothisbehaviorcanbeseenfromFig.3,whereitis 100 1.181 0.532 1.067 0.055 seen that the curves change discontinuously to a horizontal linenearj53.ThereasonforthisisthatforcylinderstheBR theory converges poorly for small j, yielding currents larger continuously at j53. This value of jseparates the region than the OML limit. Laframboise argues that the ion current ~j,3! in which the OML limit is approximately valid from cannot exceed this limit because, when a thin sheath is the region ~j.3! where it is not. The physical meaning is formed, it shields the plasma from the probe potential, and that,forj>3,theformationofanabsorptionradiusbeginsto ions cannot be drawn in from large distances. For lack of a limittheprobecurrent.Exceptforthepointh50,whichhas better procedure, he arbitrarily cuts off the ion current when little experimental value, the fitting error over the entire it reached the OML value. This limit is not observed in ex- range of hand jis less than ’3%, and in most cases less periment. Laframboise8 shows data by Sonin17 which follow than 1%–2%. The values of the parameters ABCD used in the extrapolation of the curves of Fig. 3 without an OML Fig.5areshowninTableI;asexplainedabove,thisisbyno cutoff. This is physically reasonable, since any small colli- means a unique set of values. sionfarfromtheprobecanchangetheangularmomentumof In stage 2, we attempt to express the parameters ABCD anincomingion.TheobvioussolutionistoignoretheOML asanalyticfunctionsofj.ThevaluesinTableIareplottedin limit and use the extrapolation of the curves of Fig. 3. Un- Fig. 6~a!. We see that all the curves have a discontinuity at fortunately, the parameters ABCD(j) then behave even j53,exceptfortheOMLexponentB.Thesejumpsarereal; more erratically than in Fig. 6~a!, and we were unable to fit them to smooth functions. The fitting of ABCD(j) to analytic functions was car- ried out in two steps. In step 1, the values for j,3 were ignored,andtheoriginofthecurveswasshiftedtoj53.The following functional forms were used: A,B,D~j!5a1b~j2c!dexp@2f~j2c!g#, ~9! C5a1bexp@2cln~j2d!#1f~12glnj!. Thus,eachparameterA,B,C,orD,wasfittedusingsixother parametersabcdfg, whichweshallcallcoefficientstoavoid confusion. Possible values for these are given in Table II. This is by no means a uniquely optimized set; we simply show that a set exists which can be used to reproduce the Laframboisecurvesaccurately.Figure6~a!showstheresult- ing curves of ABCD(j), as analytic fits to the points in that figure. With these smoothed parameters, the calculated data points of Fig. 5 can be fitted within 3% down to j53. Instep2,wesacrificeaccuracyinordertofittheparam- eters ABCD(j) for all known values, j , of j. The param- j j etersABCD arechosennottogivethebestfittothedatabut togiveareasonablefitwhilevaryingmoresmoothlyasfunc- tions of j. Since C becomes large for j,3, the second term in Eq. ~8! is negligible for small j. We therefore choose a function C(j) which fits the pointsj.3 in Fig. 6 and which diverges rapidly for j,3. The function D(j) is then imma- terialforj,3andneedstobefittedonlyforlargej.Having chosen C(j) and D(j), we then fix C and D at their FIG.6. TheparametersA,B,C,andDforavailablevaluesofjandanalytic smoothed values while optimizing A and B. This results in a fitstothemfor:~a!j.3and~b!allj. new set of parameters ABCD, given in Table III, which are Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp Phys.Plasmas,Vol.8,No.6,June2001 Langmuirprobeanalysisforhighdensityplasmas 3033 TABLEIII. DegradedparametersforfittingthecurvesofFig.5. j A B C D 0.1 1.141 0.496 1.218 0.526 1 1.199 0.477 1.233 0.517 2 1.194 0.515 1.224 0.514 2.5 1.234 0.537 1.255 0.486 3 1.415 0.522 1.294 0.316 4 1.568 0.446 1.306 0.378 5 1.546 0.422 1.244 0.351 10 1.426 0.583 1.166 0.257 20 1.415 0.654 1.104 0.181 50 1.255 0.583 1.095 0.091 100 1.130 0.486 1.067 0.055 to be fitted with new functions A(j) and B(j). The new set of functions, involving new coefficients abcdf, is as fol- lows: 1 A5a1 , 1 1 2 bjc dln~j/f! B,D5a1bjcexp~2djf!, ~10! C5a1bj2c. TableIVgivesthenewcoefficients,andFig.6~b!showsthe points and fitting curves of step 2, valid for all j. In spite of thefactthatthepointsinFig.6~b!arestillerraticandthefits FIG.7. ParametricfitstotheLaframboisedatafor:~a!j55and~b!j520. The points have a radius of 2%. The solid line ~fit 1! shows the curve poor, the smooth curves in this figure yield I–V curves optimizedforj>3;thedashedline~fit2!,thecurveoptimizedforallj. agreeing with those in Fig. 2 to within ’5% for 0,j,100. Thelargestdiscrepanciesoccurforj’4–5,wherebothterms in Eq. ~8! are significant. Again, we emphasize that the pa- gas discharges T rarely exceeds 0.1 eV, the correction for i rametersofTableIIIarebynomeansoptimized.Bydegrad- finite T is entirely negligible except for h,1. i ing the stage 1 fit using other local minima in the least- squares process, it may be possible to make the points III. ANALYSIS PROCEDURE ABCD(j) lie on smoother curves that can be fitted more j readily. To obtain accurate values of n, T , and V ~but not T ! e s i Comparison of the two parametrizations is shown for fromtheI–V characteristic,themostdifficulttaskistosepa- two sample values of jin Figs. 7~a! and 7~b! on log–log rate the ion current I from the electron current I , and vice i e plots. Although it is not easy to see on this scale, the step 1 versa, in the region near the floating potential, where both coefficients give closer fits than the step 2 coefficients. The contribute to the total current I. Now that we have accurate largest discrepancies are for h,1, where the ion current is ioncurvesforsmallvaluesofh,wecansubtractI fromI , i e only a small contribution to the total current anyway. Note that the fits of Fig. 5 are better than those of Fig. 7 because the parameters ABCD in Fig. 5 were not calculated from analytic functions. Finally, we consider the effect of finite ion temperature. Figure 8 shows the Laframboise calculations8 for i versus h at j510 for various ratios T /T . Since in partially ionized i e TABLEIV. CoefficientsforcalculatingABCD(j) forallj. a b c d f A 1.12 0.00034 6.87 0.145 110 B 0.50 0.008 1.50 0.180 0.80 C 1.07 0.95 1.01 — — D 0.05 1.54 0.30 1.135 0.370 FIG. 8. Laframboise data for the variation of ion I–V curves with ion temperature,atj510. Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp 3034 Phys.Plasmas,Vol.8,No.6,June2001 FrancisF.Chen FIG.9. SampleI–Vcurvetobeanalyzed~I .0here!.Thedataweretaken e with an 0.15 mm diam, 1 cm long probe in a 900 W, 2 MHz ICP in 20 mTorr of argon. The density was of order 431011cm23, and KT was FIG. 10. Initial determination of V and T from derivative of the I–V e s e ’2eV. curve.Ioncurrentispositivehere. and then I from I , in an iterative procedure. We shall il- e i B. Step 2 lustrate this technique using data from an inductively coupled discharge in 10 mTorr of argon, taken with an rf- The data are smoothed to remove digital noise, and compensated probe of radius 0.075 mm and length 1 cm dI/dV is computed; the maximum of this gives a first esti- ~Hiden ESPion®!. A similar iterative scheme is used by mate of the space potential V . This estimate will not affect s Hopkins and Graham.18 the final determination of V , which is found from the ion s Except in negative-ion plasmas, it is inadvisable to at- and electron fits and the condition of quasineutrality. If the tempt to obtain the electron density n from the saturation electron distribution is Maxwellian, the ratio I/(dI/dV) e electron current, although this has been done successfully in yields a first estimate of T : e quiescent, field-free plasmas of very low density. There are I KT several reasons for this. The space potential V , at which n I }exp@e~V 2V !/KT #, e 5 e, ~12! s e e p s e dI /dV e is determined, is usually found from the inflection point of e p theI–V curveorfromthe‘‘knee’’atwhichtheextrapolated which is T in eV. These curves are shown in Fig. 10, with e linesofthesaturationandtransitionregionscross.Thispoint T read directly from the right-hand scale. From the mini- e is ill defined, and n depends exponentially on the choice of mum in the dI/dV curve, the space potential is seen to be e V . Magnetic fields and collisions can move the knee of the ’13 V. Since the ion contribution to I has not yet been s e curve. Radio-frequency fluctuations can distort this particu- subtracted, the apparent T varies with V . Taking a poten- e p larlynonlinearpartofthecurve.Drawinglargeelectroncur- tial ’2T more negative than V , we estimate T to be e s e rentstotheprobecanalsodepletetheplasmaordamagethe about 2.1 eV in this example. probe. If n and n were to differ by as much as 0.1%, Pois- i e son’s equation shows that d2V /dx2 would be on the order s of 200 V/cm2, which is impossible outside of a sheath. C. Step 3 Therefore, apart from low density (n,331010cm23), rf- Aroughestimateofplasmadensityncanbefoundfrom free plasmas, n is in principle best determined from the ion e I (V ): saturation current, assuming quasineutrality; and any dis- e s agreement between ni and ne18,19 simply shows the error in I~Vs!’Ie~Vs!5neS~KTe/2pm!1/2. ~13! measuring ne. In the following procedure, we do not use This yields a density of ’1.731011cm23. A second esti- electron saturation except for an initial estimate of n. How- matecanbefoundfromtheBohmformulaofEq.~4!applied ever, it will be seen that there are still problems with the tothecurrentatthemostnegativeprobepotentialmeasured. theories of ion collection. Since the sheath has expanded at that potential, ashould be given a value @1/2, perhaps 2. This yields a density of ’6 A. Step 1 31011cm23. These estimates may differ considerably, but they are needed only to provide an order of magnitude. Thecurrenttotheendareaoftheprobeissubtractedout by dividing the measured probe current I by an aspect ratio factor A , defined by r D. Step 4 S52pR L, A 5~S1pR2!/S5~11R /2L!. ~11! p r p p Having working values of V , T , and n, we can now s e Dependingonthevalueofj,theendoftheprobecancollect evaluatel , j,andJ fromEqs.~5!and~6!.Theparameters D r from a hemispherical sheath or a plane one. Since the cor- ABCD(j) canthenbeevaluatedusingEq.~9!ifj@3,orEq. rectionissmall,wehavesimplyassumedasmallincreasein ~10! if jis less than 3 or close to 3, together with Tables II the cylindrical area. The entire I–V curve in this example is andIV,respectively.Equation~10!canalwaysbeusedbutis shown in Fig. 9. somewhat less accurate. Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp Phys.Plasmas,Vol.8,No.6,June2001 Langmuirprobeanalysisforhighdensityplasmas 3035 FIG. 11. Square of saturation ion current vs probe voltage as measured FIG.12. Semilogplotofelectroncurrentasoriginallymeasured~diamonds! ~dots!andascomputedafteroptimization~smoothline!. andaftersubtractionofioncomponent~circles!.ThelineisafittoaMax- welliandistribution. E. Step 5 H. Step 8 The theoretical ion current can now be found as a func- A least squares fit of the I data is made by varying T e e tion of V from Eq. ~6! and Ve, yielding the line shown in Fig. 12. The resulting p s valueofVe isthespacepotentialrequiredbyquasineutrality, I 5nJ i, ~14! s i r regardlessofwherethe‘‘knee’’oftheI–V curveislocated. withigivenbyEq.~8!asafunctionofh.Usingtheprelimi- ForeachtrialvalueofT orVe, jandhwillchange,andthe e s nary values of V and T , we can convert I (h) into I (V ) ion curve has to be recalculated. The I data are then also s e i i p e using Eq. ~6!. The measured and calculated curves of I2 are recalculated because the ion subtraction has changed. De- i shown in Fig. 11. The reason that we plot I2–V rather than spite the complexity, these least square fits of a 500-point i I –V is that measured I2–V curves tend to be linear over a datasetrequirelessthan15sona400MHzcomputerusing i i larger range of densities than one would expect from this an uncompiled spreadsheet program. Since T has changed, e theory. the ion curve in Fig. 11 will no longer fit the data. I. Step 9 F. Step 6 Steps 4–8 are repeated until consistent values of n, T , e squaTrehsefivtaltuoesthoefdnataa,ndavVoisdianrge tthheenreagdijounstendeafrorVsa.lTeahset V5si5,.1a5n3d1V0se11acrme2o3b,taiTnee5d.2I.0n9tehVis,exVasem5p1le4,.9theeV,resaunltdisVnsi valueofncontrolsthemagnitudeofthecurve,andthevalue 517.1eV. of V its slope. The result is the thin line in Fig. 11. The s discrepancy between theory and experiment near V is ex- J. Step 10 s pected, since the contribution of electrons has not yet been Theelectroncurrentcannowbesubtractedfromtheion subtracted. In general, the value of V given by this fit s data to give the true ion current. This can be done in two ([Vi) will differ from that required to fit the electron cur- s ways. If the ion-corrected electron current is subtracted, the rent([Ve); thiswillbediscussedlater.Notethatwedonot s result is shown in Fig. 13. If the theoretical electron current vary Te at this step. From Eq. ~6!, we see that i2 and hboth ~assumingaMaxwelliandistribution!issubtracted,theresult vary as T21, so that a fit of i2(h) is independent of T . e e Unfortunately, i depends weakly on T through the value of e j, and iteration is necessary. G. Step 7 Next, the calculated ion current is subtracted from the probe current to obtain the electron current. We do not sub- tractthemeasuredioncurrent,sincethatcontainsanelectron component.Figure12showsasemilogarithmicplotofI , as e compared with the raw data. In this case, subtracting the ion current has greatly improved the linearity of this curve. The solid line is a graph of the equation FIG.13. Linearplotofsaturationioncurrentaftersubtractionofcorrected I 5neS~KT /2pm!1/2exp@e~V 2Ve!/KT # ~15! electron current. The line is the theoretical fit. The points at the right are e e p s e electroncurrentswhichappearbecauseofthemismatchbetweenthespace using the value of n determined by the ion fit. potentialsassumedfortheionandelectronfits. Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp 3036 Phys.Plasmas,Vol.8,No.6,June2001 FrancisF.Chen TABLEV. ParametersforfittingtheABRcurves. TABLEVI. CoefficientsforcalculatingABCD(j) forABRtheory. A B C D a b c d 0.5 0.314 0.493 0.314 0.493 A 0.864 1.500 0.269 2.050 1 0.846 0.488 1.033 0.360 B 0.479 20.030 20.010 — 1.5 1.511 0.468 2.152 0.283 C 1.008 1.700 0.336 2.050 2.5 3.637 0.409 4.329 0.298 D 0.384 20.150 0.013 — 4 6.677 0.423 10.63 0.200 6 14.79 0.367 21.67 0.140 10 21.75 0.430 55.68 0.158 30 291.5 0.205 417.1 0.039 Ref. 13, including Fig. 1 of the present paper. The param- 50 796.2 0.167 1113 0.008 etersABCD(j) forthebestfittotheABRdataareshownin 70 1691 0.122 2110 20.030 TableV.Theseparametersarefittedwiththefollowingfunc- tions of j: A,C5ajb1cjd, ~not shown! has a discontinuity near Vs because of the dis- ~18! crepancy between Ve and Vi. Non-Maxwellian electrons B,D5a1blnj1c~lnj!2. s s would cause these curves to differ even more. The fitting coefficients abcd are shown in Table VI. Some disagreement between Ve and Vi can be expected s s The OML, BRL, and ABR theories predict different because: ~a! the ion current for h’0 cannot be calculated magnitudes and shapes for the ion saturation curves. This accurately, being sensitive to small perturbations and poor can be seen in Fig. 14, which compares the three theories convergence;~b!itisfitleastwellbyouranalyticfunctions; with the same data used for the example of Figs. 9–13. It is and~c!itisfurthermoresensitivetoiontemperature~cf.Fig. seen that the OML curve fits the linear I2–V dependence of 8!. However, it is surprising that the discrepancy is so large the data almost exactly. The deviation of the data from this and is not always of the same sign. This problem has not line at the highest voltages is in the ‘‘wrong’’ direction and been encountered before, because no attempts had been is probably caused by secondary electron emission. The madetoevaluateI nearh50accurately.Anotherpossibility i ABRcurvefitsthedatawellatlowvoltagesbutshowssome is that ion trapping in closed orbits is occurring at low po- saturationathighvoltages.TheBRLcurveshowsmoresatu- tentials. However, Laframboise8 points out that with cylin- ration and does not follow an I2–V dependence at all. As drical probes trapped ions can escape by moving parallel to showninthefigure,eachtheoryrequiresadifferentvalueof the axis. As is evident from the erratic behavior of Fig. 13 densityinordertofitthedata.Ingeneral,theBRLandOML near Vs, it is difficult to separate Ii and Ie near the space theories agree at small values of j, as they should, since potential, and this may affect the apparent electron distribu- Laframboise8 forced them to do so as explained in Sec. II. tion function at low energies. For large j, the data follow the OML curve much more closely than the BRL curve, even though the OML theory is IV. ANALYSIS WITH OTHER PROBE THEORIES not expected to be accurate for j.3. Indeed, the BRL fit is sopooratlargejthatthereisconsiderablelatitudeinchoos- It is apparent from our sample case ~Fig. 11! that the ing the combination of n and Vi that gives the best fit. In BRL theory diverges from the experimental points at large s practice, we chose a combination that also straightens the negative voltages. As will be seen in Sec. VI, this discrep- lnI –V curve as much as possible. The ABR theory in gen- ancy vanishes at low densities but becomes large at high e eral fits the shape of the data curves better than does the densities.Forthisreason,weneedtotesttheaccuracyofthe BRLtheory;but,asweshallsee,theresultingvaluesofnare other available theories. Ion currents predicted by the OML theory are simply given by Eq. ~2!. The curves of the ABR theory, however, require parametrization. The procedure is essentiallythesameasthatinSec.II,andthedetailswillbe omitted.SincethebendintheABRcurvesisoppositetothat intheBRLcurves~Fig.5!,thefittingfunctionofEq.~8!has been modified, and the normalized ion current i is replaced by Jj , which is independent of n p Jj 5I R ~e/KT !2~2MKT !1/2 p i p e e 5@~AhB!41~ChD!4#1/4. ~16! Here I is the ion flux, not current, per cm length, and cgs i units are used. In practical units, Eq. ~16! can be written I ~mA!50.327~Jj !T ~n /A!1/2L/j , ~17! i p eV 11 p wheren isninunitsof1011cm23, Aistheatomicnumber FIG.14. Comparisonofionsaturationdatawiththeshapesofcurvespre- 11 dictedbythreeprobetheories.Thedensitiesrequiredforthefitsareshown ofthegas,andLtheprobelengthincm.Thecomputeddata inthelegend.Thevalueofjforthisexampleis5.0fortheBRLdensityand tobefittedweretakenfromtheoriginalcurvesreproducedin 4.1fortheOMLdensity. Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp Phys.Plasmas,Vol.8,No.6,June2001 Langmuirprobeanalysisforhighdensityplasmas 3037 FIG.16. RatiosofdensitiesobtainedwiththeBRL,OML,andABRtheo- riesfortestcaseswithvaryingvaluesofj~computedwiththeBRLdensity!. Thestarredpointslyingoffthelinewerecalculatedassuminganincorrect proberadius. underestimating the density. Since it accounts for an absorp- tion radius, which the OML theory does not, it could have predicted a lower ion current and higher density than does the OML theory; but apparently the effect of angular mo- mentum is larger, and the ABR density is always lower than theOMLdensity.AlthoughtheBRLandOMLtheoriesboth incorporate angular momentum, the OML formula neglects FIG.15. Comparisonofdensityandtemperaturevaluesobtainedfromdif- the formation of an absorption radius and allows the probe ferentmethodsofprobeanalysisforsamplecaseswithj5~a!1.1,~b!3.1, potential to attract ions from large distances, thus overesti- ~c!4.6,and~d!9.1. mating the current and underestimating the density. How- ever, the BRL theory does not necessarily give the right an- swer, because it does not fit the shape of the ion I–V too low to be realistic. When the OML theory is applicable, the values of Ve and Vi agree well, but for large jthese characteristic as well as the OML theory does. s s A comparison of the three theories for our test cases is valuesdivergebecauseoftheinaccuracyofthetheoriesnear showninFig.16,whichplotstheBRL/OMLandABR/OML the plasma potential. Usually the OML fit requires too large a value of Vi, while the BRL and ABR theories require too density ratios as functions of j. The BRL/OML ratio ap- small a values of Vi. proachesunityasj!0,asexpected,andincreasesmonotoni- s callywithj.Asjincreasesandthesheathbecomesthin,the probepotentialisshieldedfromdistantpointsintheplasma, V. COMPARISON OF THEORIES WITH EXPERIMENT aneffectneglectedinOMLtheory.Hence,OMLpredictstoo Wehaveanalyzedsome15probecurvestakenbyEvans large a current and too small a density, the error increasing and Zawalsky20 with various probe radii and plasma condi- with j. To test the sensitivity to j, we analyzed two I–V tions in an ICP, using a Hiden ESPion® probe system. Ex- curves assuming an incorrect value of R , and hence of j. p perimentaldetailsareoutsidethescopeofthispaperandwill These cases are shown by the two points marked with a star be given in a separate paper. The values of KT are insensi- ~*! and lie well off the trend line of the other points. The e tive to the method of analysis used, but the densities n can ABR/OML ratio is usually below unity, for reasons stated vary by a factor of 3 or more from theory to theory, leading above. to a large uncertainty in the interpretation of the data. The WebelievethatBRLtheoryisalsoinaccuratebecauseit discrepancy, however, follows a trend that can be discerned applies to strictly collisionless plasmas. In the presence of in the four cases chosen for illustration in Fig. 15. The bar charge-exchange collisions in a gas like argon, the angular marked ‘‘Hiden’’ is the density given by the ESPion® soft- momentumofanincomingionfarfromtheprobecaneasily ware package and differs from the OML result only because be destroyed by a collision, after which the ion will be ac- the value of KT involved a slightly different fit to the data. celeratedradiallybytheprobe’selectricfield.Enoughangu- e The values of KT , shown by the connected points, are al- larmomentumremains,however,forthecurrenttobelower e most the same for all methods, but the values of n differ by thanthatpredictedbyABRtheory.Ourpreviousexperimen- an amount which increases with j5R /l . The BRL for- tal check of BRL theory7 was done in a thermionic p D malism consistently yields density values larger than the Q-machine,whichwasfullyionizedandgavelargevaluesof OMLtheory,whiletheABRdensitiesarealwayslowerthan jbecauseofthelowtemperature.There,theioncurrentwas the OML density. well saturated and fitted the shape of the BRL curves. In a TheABRtheory,whichneglectsangularmomentumand partiallyionizedplasma,however,collisionswelloutsidethe ion orbiting, overestimates the ion current to the probe, thus sheathcanchangetheangularmomentumdistributionsothat Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp 3038 Phys.Plasmas,Vol.8,No.6,June2001 FrancisF.Chen determined densities analyzed with the BRL and ABR theo- ries. It is seen that there is reasonable agreement among all threemethodsatlow P ~smallj!,butthatathigh P ~large rf rf j!theBRLdensityistoohigh,andtheABRdensitytoolow compared with the microwave determination. However, the geometric mean of the BRL and ABR densities ~shown by the dotted line and starred points! agrees well with the mi- crowave measurement and has the proper slope. An exact treatment of the problem for partially ionized plasmas would require adding a collision term to the Bernstein–Rabinowitz formalism, but even then the results would depend on the pressure and species of the gas and would not be expressible in terms of universal curves. We FIG. 17. Data from a high-density helicon plasma compared with three have attempted to devise a hybrid technique, using the ABR theoretical curves. The inset shows the values of n and T obtained from e theoryforradiiaboveacriticalradiusR , toaccountforthe these fits. The 2 kW discharge was in 8 mTorr of argon with a 600 G c shielding of potentials by thin sheaths and the finite-sheath- magneticfield,andtheprobewas0.3mmindiameter. thickness OML theory for r,R to account for ion orbits in c thecollisionlessregion.Forinstance,onecouldchooseR to c the ion curves more closely follow the I2–V dependence be the mean free path, but this is usually well outside the predicted by the OML theory and, to a lesser extent, the sheath. Alternatively, one could choose R to be several De- c ABR theory. bye lengths larger than R , or to be the radius at which h p Well saturated ion curves can also be obtained in a par- 5h51. The ABR current would be calculated for a probe c tially ionized plasma by using a large probe at high density. with R 5R and h51, thus neglecting the ions’ angular p c p Figure 17 shows data from a helicon plasma in the n momenta in the exterior region. The OML current can then 51013cm23 range21attainingavalueofj556.Althoughnot be calculated for a sheath edge at r5R and a Maxwellian c shown, the BRL/OML and ABR/OML density ratios fall on ion distribution there with T <T . Setting the two currents i e an extrapolation of the trend line in Fig. 16. The straight equal to each other should yield the value of R or h, c c I2–V line corresponding to the OML theory is clearly inap- whichever is the unknown. The process is repeated for each propriate, since it crosses the axis at 154 V, and an unrea- probe potential V . The density assumed initially would be p sonably low density has to be assumed to achieve the small adjustedandtheprocessrepeateduntilthecurveagreeswith slope of the curve. The ABR theory, however, fits the data the data. Unfortunately, our attempts to carry out this proce- quite well with a reasonable density, while the BRL curve dure failed. has a lower slope than the data. In these fits, Vi has been s adjusted so that the electron current is Maxwellian after the ion current has been subtracted. Near the floating potential, VI. SUMMARY AND CONCLUSIONS the data points in Fig. 17 fall below the theory because they have not yet been corrected for the electron contribution. Theproperuseofprobetheoryforhighdensity,partially Tocomparetheproberesultswithindependentmeasure- and fully ionized plasmas is treated in this paper, which in- ments of density, we have obtained preliminary data by corporates several distinct research results: Evans and Zawalsky20 using microwave interferometry. Fig- ~1! A double parametrization technique has been devel- ure 18 shows these density measurements as a function of rf oped to facilitate the use of the Laframboise and ABR com- power P in a commercial ICP, compared with probe- putationalresults.Thisalgorithmpermitsrapidgenerationof rf theoretical I–V curves for arbitrary values of l /R , lead- D p ing to real-time analysis of probe data with fast portable computers. ~2! An iteration scheme is described which uses the pa- rametrized curves to separate the ion and electron currents collected by the probe. Although this separation fails be- tween the floating and space potentials because of inaccura- ciesinthetheory,thismethodyieldsmoreaccuratedetermi- nations of density and electron temperature than previously possible. ~3! Comparison with experiment reveals a dilemma: The OML ~or ABR! theory fits the shape of the ion satura- tion curves better than the BRL theory in regimes where the OML ~or ABR! theory should be inapplicable. The correct density,asdeterminedbymicrowaveinterferometry,liesbe- FIG.18. ValuesofdensityvsjasobtainedfromtheBRLandABRtheo- ries,ascomparedwithmicrowavemeasurements~l!.Thestars~*!arethe tween those given by the BRL and OML ~or ABR! theories. geometricmeanbetweentheBRLandABRdensities. We surmise that the cause of the failure of the collisionless Downloaded 13 Jun 2001 to 128.97.88.10. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp