Plasma Diagnostics 75

Fig. 1. Emission of ionized argon lightat various z positions in a helicon dis-charge.

PRINCIPLES OF PLASMA PROCESSINGCourse Notes: Prof. F.F. ChenPART A7: PLASMA DIAGNOSTICS

X. INTRODUCTIONDiagnostics and sensors are both measurement

methods, but they have different connotations. Diagnos-tic equipment is used in the laboratory on research de-vices and therefore can be a large, expensive, and one-of-a-kind type of instrument. Sensors, on the other hand,are used in production and therefore have to be simple,small, unobtrusive, and foolproof. For instance, end-point detectors, which signal the end of an etching stepby detecting a spectral line characteristic of the underly-ing layer, are so important that they are continually beingimproved. Practical sensors are few in number but con-stitute a large subject which we cannot cover here. Welimit the discussion to laboratory equipment used tomeasure plasma properties in processing tools.

Diagnostics for determining such quantities as n,KTe, Vs, etc. that we have taken for granted so far can beremote or local. Remote methods do not require inser-tion of an object into the plasma, but they do require atleast one window for access. Local diagnostics measurethe plasma properties at one point in the plasma by in-sertion of a probe of one type or another there. Remotemethods depend on some sort of radiation, so the win-dow has to be made of a material that is transparent tothe wavelength being used. Sometimes quartz or sap-phire windows are needed. The plasma can put a coatingon the window after a while and change the transmissionthrough it. Probes, on the other hand, have to withstandbombardment by the plasma particles and the resultingcoating or heating; yet, they have to be small enough soas not to change the properties being measured.

XI. REMOTE DIAGNOSTICS1. Optical spectroscopy

One common remote diagnostic is optical emis-sion spectroscopy (OES), which is the optical part of themore general treatment of radiation covered in Part B. InOES, visible light is usually collected by a lens and fo-cused onto the slit of a spectrometer. The detector can bea photodiode, a photomultiplier, or an optical multichan-nel analyzer (OMA). With a photodiode, interferencefilters are used to isolate a particular spectral line. Opti-cal radiation can also be used to image a plasma in thelight of a particular spectral line using an interference

Part A776

Fig. 2. Schematic of a local OESprobe.

Fig. 3 Example of data on opticalemission vs. z.

filter and a sensitive CCD camera. Fig. 1 shows theemission from ionized argon recorded with a narrow-band filter for the 488 nm line of Ar+. A photomultiplier can see only one part of thespectrum at a time, but it is the most sensitive detectorfor faint signals. An OMA records an entire range ofwavelengths on a CCD (charge-coupled detector) and isthe convenient for scans of a single line or for recordingan entire spectrum.

By comparing the intensities of different spectrallines, one can determine not only the atomic species pre-sent but also the electron temperature, density, and theionization fraction. The relative intensities of two lineswith different excitation thresholds can yield KTe. Therelative intensities of an ion line and a neutral line can beused to estimate the ionization fraction. In principle, linebroadening contains a large amount of information, butonly for hot, highly ionized plasmas. For instance, Dop-pler broadening yields the velocity of the emitting ion oratom. Stark broadening or pressure broadening givesinformation on density. This is because, at high densi-ties, collisions interrupt the emission of radiation, andhence the line cannot contain a single frequency. Inplasma processing, the most useful and well developedtechnique is actinometry. In this method, a known con-centration of an impurity is introduced, and the intensi-ties of two neighboring spectral lines, one from theknown gas and one from the sample, are compared.Since both species are bombarded by the same electrondistribution and the concentration of the actinometer isknow, the density of the sample can be calculated.

Though most optical methods average over a raypath in the plasma, a more local measurement of lightemission can be made with a probe containing a smalllens coupled to an optical fiber. Such a probe is shownin Fig. 2, and data from it in Fig. 3.. The lens collectslight preferentially from a small focal spot just in front ofit. The Ar+ light collected by it is localized under theantenna if B0 = 0, as would be expected in ICP operation.

2. Microwave interferometryAnother useful remote diagnostic is microwave

interferometry. A beam of microwave radiation islaunched by a horn antenna into a plasma through a win-dow. According to Eq. (A5-1), these waves can propa-gate in the plasma if > p. From (A5-1) it is easilyseen that the phase velocity in the plasma is

Plasma Diagnostics 77

Fig. 4. Schematic of a microwaveinterferometer (Chen, p. 91).

2 2 1/ 2(1 / )p

ck

=

. (1)

This is faster than the velocity of light, but it is quite allright for phase velocity to be > c as long as the groupvelocity is < c. The microwave beam therefore has alonger wavelength inside the plasma than in air. Thepresence of the plasma therefore changes the phase of themicrowave signal, a change which increases with thedensity of the plasma. The standard setup is shown inFig. 4. The microwave beam from a generator is splitinto two parts, one going through the plasma and theother going through a waveguide toward the detector,where the two beams are recombined By adjusting thereference signal with an attenuator and phase shifter, thetwo signals can be made to cancel each other, so that thedetector shows zero signal when there is no plasma. Ifthe plasma density is increased slowly, the signal goingthrough the plasma will have undergone fewer oscilla-tions, and this phase shift will cause the nulled detectorto give a finite dc signal output. If the plasma densityreaches a value such that the wave loses exactly onewavelength, the detector will again return to zero; thesignal is shifted by one fringe. By counting the numberof fringes either on the way up to maximum density or onthe way down, one obtains a measure of the average den-sity traversed by the microwave beam. Though this il-lustrates the principle of interferometry, it is not normallydone this way. First, is usually chosen so that p / isa small number; then, the phase shift is linearly propor-tional to n. Second, the entire reference leg can be re-placed by a mirror on the opposite side of the plasma toreflect the beam back into the launching horn. The beamthen travels twice through the plasma and suffers twicethe phase shift. Besides increasing the sensitivity, thismethod obviates phase shifts in the reference leg due tosmall changes in room temperature, which change thelength of the waveguide. If the plasma is always on, it isdifficult to set the initial null of the detector. There arevarious ways to get around this which we need not ex-plain here. Modern network analyzers can do most ofthese calibrations automatically, but the principle of op-eration is always the same.

The phase shift that the plasma causes can becalculated as follows. If k0 = /c is the propagation con-stant in air and k1 is that in the plasma, we have

Part A778

0

2

4

6

8

0 1 2 3 4 5x (mm)

Inte

rfero

met

er s

igna

lVacuumPlasma

Fig. 5. Fringe shifts as the path lengthis changed. The lines are analytic fitsthrough the points.

Fig. 6. Fringe patterns views alongthe axis can show the shape of the

plasma [Heald and Wharton, 1978].

0 1( )k k dx = , (2)where k1 is given by Eq. (1) as

1/ 2

1 0 1c

nk kn

=

. (3)

Here we have replaced p2/2 by n / nc , where nc is thecritical density defined by

22

0

cn em

= . (4)

The phase shift is then1/ 2

0( )1 1c

n xk dxn

=

. (5)We see that the phase shift measures only the line inte-gral of the density, not the local density. If is highenough that n radians, (6)where is the average density over the path length L.In the reflection method, the integral (or L) must be dou-bled. Fig. 5 gives an example of the interferometer out-put in the double-pass method as the mirror is moved tochange the path length. The fringe shift is clearly seen,but it is also evident that the waveform has been dis-torted. This is because the microwave generator did notgive a pure signal, and its harmonics at higher suf-fered a different phase shift. By fitting the curves to sinewaves and their harmonics and adjusting the relativephases, one can recover the phase shift of the fundamen-tal and thus get the density. Fig. 6 shows an end view ofa dense plasma, in which the path length was so long thatmany fringes are seen, revealing the shape of the plasma.

Microwave interferometry is useful for calibrat-ing Langmuir probes. With a probe, one can measure thedensity profile across a radius or diameter of the plasma,but the absolute value of the density may not be knownaccurately. By using the measured density profile tocompute the integral in Eq. (6), one can find the absolutedensity by measuring the microwave phase shift. Theerrors in this method come from the fact that the plasmais not perfectly planar, and the microwave beam is not

Plasma Diagnostics 79

Fig. 7. Perpendicular alignment ofinjection laser and collection optics[Scime et al., Plasma Sources Sci.Technol. 7, 186 (1998)].

Fig. 8. LIF data on Ti parallel (solidpoints) and perpendicular (openpoints) to B0, showing anomalouslyhigh KTi [Kline et al., Phys. Rev. Lett.88, 195002 (2002]).

perfectly parallel. Refraction can cause part of the beamto miss the collector, and reflections from the chamberwalls can cause spurious waves.

3. Laser Induced Fluorescence (LIF)This diagnostic is both non-invasive and local

because it uses intersecting beam paths. Furthermore, itis the only way to measure Ti without using a large en-ergy analyzer. One laser, tuned to a particular transition,is used to raise ions to an excited state along one paththrough the plasma. The excited ions fluoresce, givingoff light at another frequency, and this light is collectedby a lens focused to one part of the path, providing thelocalization. Doppler broadening of the line yields theion velocity spread in a particular direction. The equip-ment is large, expensive, and difficult to set up, so that itis available in a relatively few laboratories. LIF istreated in more detail in Part B. Figure 7 shows a typicalLIF setup, and Fig. 8 shows data taken in a heliconplasma.

XII. LANGMUIR PROBES1. Construction and circuit

A Langmuir probe is small conductor that can beintroduced into a plasma to collect ion or electron cur-rents that flow to it in response to different voltages. Thecurrent vs. voltage trace, called the I-V characteristic,can be analyzed to reveal information about n, Te, Vs(space potential), and even the distribution function fe(v),but not the ion temperature. Since the probe is immersedin a harsh environment, special techniques are used toprotect it from the plasma and vice versa, and to ensurethat the circuitry gives the correct I V values. Theprobe tip is made of a high-temperature material, usuallya tungsten rod or wire 0.11 mm in diameter. The rod isthreaded into a thin ceramic tube, usually alumina, to in-sulate it from the plasma except for a short length of ex-posed tip, about 210 mm long. These materials can beexposed to low-temperature laboratory plasmas withoutmelting or excessive sputtering. To avoid disturbing theplasma, the ceramic tube should be as thin as possible,preferably < 1 mm in diameter but usually several timesthis. The probe tip should extend out of the end of thetube without touching it, so that it would not be in elec-trical contact with any conducting coating that may de-posit onto the insulator. The assembly is encased in avacuum jacket, which could be a stainless steel or glasstube 1/4 in outside diameter (OD). It is preferable tomake the vacuum seal at the outside end of the probe as-

Part A780

Fig. 9. A carbon probe tip assemblywith RF compensation circuitry [Suditand Chen, Plasma Sources Sci.Technol. 4, 162 (1994)].

vPROBE

R

(a)

PROBE

R

v

(b)

Fig. 10. Two basic configurations forthe probe circuit.

sembly rather than at the end immersed in the plasma,which can cause a leak. Only the ceramic part of thehousing should be allowed to enter the plasma. Somecommercial Langmuir probes use a rather thick metaltube to support the probe tip assembly, and this canmodify the plasma characteristics unless the density isvery low. In dense plasmas the probe cannot withstandthe heat unless the plasma is pulsed or the probe is me-chanically moved in and out of the plasma in less than asecond. When collecting ion current, the probe can beeroded by sputtering, thus changing its collection area.This can be minimized by using carbon as the tip mate-rial. Ordinary pencil lead, 0.3mm in diameter works welland can be supported by a hypodermic needle inside theceramic shield. One implementation of a probe tip as-sembly is shown in Fig. 9.

There are two basic ways to apply a voltage V tothe probe and measure the current I that it draws from theplasma, and each has its disadvantages. In Fig. 10a, theprobe lead, taken through a vacuum feedthru, is con-nected to a battery or a variable voltage source (bias sup-ply) and then to a termination resistor R to ground. Tomeasure the probe current, the voltage across R is re-corded or displayed on an oscilloscope. This arrange-ment has the advantage that the measuring resistor isgrounded and therefore not subject to spurious pickup.Since the resistor is usually 10-1000, typically 50,this is not a serious problem anyway. The disadvantageis that the bias supply is floating. If this is a small bat-tery, it cannot easily be varied. If it is a large electronicsupply, the capacitance to ground will be so large that acsignals will be short-circuited to ground, and the probecannot be expected to have good frequency response.The bias supply can also act as an antenna to pick up rfnoise. To avoid this, one can ground the bias supply andput the measuring resistor on the hot side, as shown inFig. 10b. This is usually done if the bias supply gener-ates a sweep voltage. However, the voltage across Rnow has to be measured with a differential amplifier orsome other floating device; or, it can be optoelectroni-cally transmitted to a grounded circuit. The probe volt-age Vp should be measured on the ground side of R so asnot to load the probe with another stray capacitance.

To measure plasma potential with a Langmuirprobe, one can terminate the probe in a high impedance,such as the 1 M input resistance of the oscilloscope.This is called a floating probe. A lower R, like 100K,can be used to suppress pickup. The minimum value of

Plasma Diagnostics 81

I

V

Ion saturation

Electron saturation

Transition

"Knee"

Floating potential

(a)

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-100 -50 0 50V

I (A)

(b)

Fig. 11. (a) An idealized I V char-acteristic showing its various parts;(b) a real I V curve from an ICP.

R has to be high enough that the IR drop through it doeschange the measured voltage. A rough rule of thumb isthat IsatR should be much greater than TeV, or R >>TeV/Isat, where Isat is the ion saturation current definedbelow. The voltage measured is not the plasma potentialbut the floating potential, also defined below. The largevalue of R means that good frequency response is diffi-cult to achieve because of the RC time constant of straycapacitances. One can improve the frequency responsewith capacitance neutralization techniques, but eventhen it is hard to make a floating probe respond to RFfrequencies.

2. The electron characteristicA Langmuir I-V trace is usually displayed upside

down, so that electron current into the probe is in the +ydirection. The curve, resembling that in Fig. 11a, hasfive distinct parts. The point at which the curve crossesthe V axis is called the floating potential Vf. To the leftof this the probe draws ion current, and the curve soonflattens out to a more or less constant value called the ionsaturation current Isat. To the right of Vf, electron currentis drawn, and the I-V curve goes into an exponential part,or transition region, as the Coulomb barrier is lowered toallow slower electrons in the Maxwellian distribution topenetrate it. At the space potential Vs, the curve takes asharp turn, called the knee, and saturates at the electronsaturation current Ies. Actual I V curves in RF ormagnetized plasmas usually have an indistinct knee, asshown in Fig. 11b.

The exponential part of the I V curve, whenplotted semi-logarithmically vs. the probe voltage Vp,should be a straight line if the electrons are Maxwellian:

exp[ ( ) / )]e es p s eI I e V V KT= , (7)

where, from Eq. (A4-2), 1/ 2

v / 42

ees e e

KTI eAn en Am

= =

, (8)

A being the exposed area of the probe tip. Eq. (7) showsthat the slope of the (ln I)Vp curve is exactly 1/TeV andis a good measure of the electron temperature. As longas the electrons are Maxwellian and are repelled by theprobe, the EEDF at a potential V < 0 is proportional to

2 2( v ) / | |/ ( v / 2 )(v) e e ee e em eV KT eV KT m KTf + = . (9)

Part A782

0.1

1.0

10.0

100.0

0 5 10 15 20V

I e (m

A)

MaxwellianModified dataRaw Data

Fig. 12. A semilog plot of electroncurrent from an I V curve in an ICP.

We see that f(v) is still Maxwellian at the same Te; onlythe density is decreased by exp(e|V|/KTe). Thus, theslope of the semilog curve is independent of probe areaor shape and independent of collisions, since thesemerely preserve the Maxwellian distribution. However,before Ie can be obtained from I, one has to subtract theion current Ii. This can be done approximately by draw-ing a straight line through Isat and extrapolating it to theelectron region. One can estimate the ion contribution more accu-rately by using one of the theories of ion collection dis-cussed below, but refinements to this small correction areusually not necessary, and they affect only the high-energy tail of the electron distribution. One easy itera-tion is to change the magnitude of the Isat correction untilthe ln I plot is linear over as large a voltage range as pos-sible. Fig. 12 shows a measured electron characteristicand a straight-line fit to it. The ion current was calcu-lated from a theoretical fit to Isat and added back to I toget Ie. The uncorrected points are also shown; they havea smaller region of linearity.

3. Electron saturationSince Isat is cs because of the Bohm sheath crite-

rion, Ies, given by Eq. (8), should be (M/m) times aslarge as Isat. In low-pressure, unmagnetized discharges,this is indeed true, and the knee of the curve is sharp andis a good measure of Vs. For very high positive voltages,Ies increases as the sheath expands, the shape of the curvedepending on the shape of the probe tip. However, ef-fects such as collisions and magnetic fields will lower themagnitude of Ies and round off the knee so that Vs is hardto determine. In particular, magnetic fields strongenough to make the electron Larmor radius smaller thanthe probe radius will limit Ies to only 10-20 times Isat be-cause the probe depletes the field lines that it intercepts,and further electrons can be collected only if they diffuseacross the B-field. The knee, now indistinct, indicates aspace potential, but only that in the depleted tube of fieldlines, not Vs in the main plasma. In this case, the I Vcurve is exponential only over a range of a few KTeabove the floating potential and therefore samples onlythe electrons in the tail of the Maxwellian. One mightthink that measurement of Ies would give information onthe electron density, but this is possible only at low den-sities and pressures, where the mean free path is verylong. Otherwise, the current collected by the probe is solarge that it drains the plasma and changes its equilibrium

Plasma Diagnostics 83properties. It is better to measure n by collecting ions,which would give the same information, since plasmasare quasineutral. More importantly, one should avoidcollecting saturation electron current for more than a fewmilliseconds at a time, because the probe can be dam-aged.

4. Space potentialThe time-honored way to obtain the space poten-

tial (or plasma potential) is to draw straight lines throughthe I V curve in the transition and electron saturationregions and call the crossing point Vs, Ies. This does notwork well if Ies region is curved. As seen in Fig. 11b, agood knee is not always obtained even in an ICP with B0= 0. In that case, there are two methods one can use.The first is to measure Vf and calculate it from Eq. (A4-4), regarding the probe as a wall. The second is to takethe point where Ie starts to deviate from exponentialgrowth; that is, where ( )eI V is maximum or ( )eI V iszero. If ( )eI V has a distinct maximum, a reasonablevalue for Vs is obtained, but it would be dangerous toequate the current there to Ies. That is because, accord-ing to Eq. (7), Ies depends exponentially on the assumedvalue of Vs.

5. Ion saturation current1

a) Plane probes. The measurement of Isat is thesimplest and best way to determine n. At densities aboveabout 1011 cm-3, the sheath around a negatively biasedprobe is so thin that the area of the sheath edge is essen-tially the same as the area of the probe tip itself. The ioncurrent is then just that necessary to satisfy the Bohmsheath criterion:

1/ 20.5 ( / )sat eI eAn KT M= , (10)

where the factor 0.5 represents ns/n. This value is onlyapproximate; when probes are calibrated against otherdiagnostics, such as microwave interferometry, a factorof 0.6-0.7 has been found to be more accurate. Note thatEq. (10) predicts a constant Isat, which can happen onlyfor flat probes in which the sheath area cannot expand asthe probe is made more and more negative. In practice,Isat usually has a slope to it. This is because the ion cur-rent has to come from a disturbed volume of plasma (thepresheath) where the ion distribution changes from iso-

1 For detailed references, see F.F. Chen, Electric Probes, in "Plasma Diagnostic Techniques", ed. by R.H.Huddlestone and S.L. Leonard (Academic Press, New York, 1965), Chap. 4, pp. 113-200.

Part A784

-0.03

0.00

0.03

-160 -140 -120 -100 -80 -60 -40 -20 0 20V

I (m

A)

BRL theoryLinear fitBohm current

p = 20

Te = 3 eV, n = 4 x 1012 cm-3

Vf Vs

= 0.74

Fig. 13. Illustrating the extrapolationof Ii back to the floating potential toget Isat. In this case, the Bohmcoefficient 0.5 in Eq. (10) has to bereplaced by 0.74 to get the rightdensity.

p

a

Vo

Rp

Fig. 14. Definition of impactparameter p.

tropic to unidirectional. If the probe is a disk of radius R,say, the disturbed volume may have a size comparable toR, and would increase as the |Vp| increases. In that case,one can extrapolate Ii back to Vf to get a better measureof Isat before the expansion of the presheath. This is il-lustrated in Fig. 13. Better saturation with a plane probecan be obtained by using a guard ring, a flat washer-shaped disk surrounding the probe but not touching it. Itis biased at the same potential as the probe to keep thefields planar as Vp is varied. The current to the guardring is disregarded. A section of the chamber wall can beisolated to be used as a plane probe with a large guardring.

b) Cylindrical probesi) OML theory. As the negative bias on a probe

is increased to draw Ii, the sheath on cylindrical andspherical probes expands, and Ii does not saturate. For-tunately, the sheath fields fall off rapidly away from theprobe so that exact solutions for Ii(Vp) can be found. Weconsider cylindrical probes here because spherical onesare impractical to make, though the theory for them con-verges better. The simplest theory is the orbital-motion-limited (OML) theory of Langmuir.

Consider ions coming to the attracting probe frominfinity in one direction with velocity v0 and various im-pact parameters p. The plasma potential V is 0 at andis negative everywhere, varying gently toward the nega-tive probe potential Vp . Conservation of energy and an-gular momentum give

2 20 0

0

a aa

mv mv eV eVpv av

= +

= (11)

where eV < 0 and a is the distance of closest approach tothe probe of radius Rp . Solving, we obtain

1/ 22 2

00 0 0

1 , 1a a aaV v Vmv mv p a aV v V

= + = = +

.(12)

If a Rp, the ion is collected; thus, the effective proberadius is p(Rp). For monoenergetic particles, the flux to a

2 F.F. Chen, Phys. Plasmas 8, 3029 (2001).3 F.F. Chen, J.D. Evans, and D. Arnush, Phys. Plasmas 9, 1449 (2002)4 I.D. Sudit and F.F. Chen, RF compensated probes for high-density discharges, Plasma Sources Sci. Technol. 3,162 (1994).5 N. Hershkowitz, How Langmuir Probes Work, in Plasma Diagnostics, Vol. 1, Ed. by O. Auciello and D.L. Flamm(Acad. Press, N.Y., 1994), Chap. 3, p. 113.

Plasma Diagnostics 85probe of length L is therefore

1/ 202 (1 / )p a rR L V V = + , (13)

where r is the random flux of ions of that energy.Langmuir then extended this result to energy distribu-tions which were Maxwellian at some large distance r = sfrom the probe, where s is the sheath edge. The ran-dom flux r is then given by the usual formula

1/ 2

2i

rKTn

M =

. (14)

With Ap defined as the probe area, integrating over allvelocities yields the cumbersome expression

erf( ) [1 erf ( ) ]p rsA ea

= + +

, (15)

where 2

2 2/ , ,p i paeV KT a R

s a

=

.

Fortunately, there are small factors. In the limit s >> a,when OML theory applies, if at all, we have

Part A786tion radius exist, and OML theory is inapplicable.Nonetheless, the I2 V dependence of Isat is oftenobserved and is mistakenly taken as evidence of orbitalmotion.

ii) ABR theory. To do a proper sheath theory,one has to solve Poissons equation for the potential V(r)everywhere from the probe surface to r = . Allen,Boyd, and Reynolds (ABR) simplified the problem byassuming ab initio that Ti = 0, so that there are no orbitalmotions at all: the ions are all drawn radially into theprobe. Originally, the ABR theory was only for sphericalprobes, but it was later extended to cylindrical probes byChen1, as follows. Assume that the probe is centered at r= 0 and that the ions start at rest from r = , where V = 0.Poissons equation in cylindrical coordinates is

( ) /00

1 , eeV KTe i eV er n n n n e

r r r = =

. (17)

To electrons are assumed to be Maxwellian. To find ni,let I be the total ion flux per unit length collected by theprobe. By current continuity, the flux per unit length atany radius r is

( )/ 2 , where 2 /i i in v I r v eV M = = = . (18)

Thus, -1/22

2i i

I eVnv r M = =

. (19)

Poissons equation can then be written

-1/2/

00

1 22

eeV KTV e I eVr n er r r r M

= (20)

Defining1/ 2

, ese

eV KTcKT M

, (21)

we can write this as

( )-1/20

0

212

e

s

KT e Ir n ee r r r r c

=

(22)

or

( )-1/202 00

212

e

s

KT Ir er r r r n cn e

= . (23)

Plasma Diagnostics 87

Fig. 15. ABR curves for ().

Fig. 16. VI curves derived from ().

The Debye length appears on the left-hand side as thenatural length for this equation. We therefore normalizer to D by defining a new variable :

1/ 20

20

, eDD

r KTn e

. (24)

Eq. (23) now becomes

( )

( )

( )

-1/2

0

-1/2

01/ 22 1/20

0 0

0 0

1 22

1 22

22

2 2

s

D s

e e

e

I er n c

I en c

n eI M en KT KT

eI M eKT n

=

=

=

=

(25)

Defining1/2

0 02 2e

eI MJKT n

, (26)

we arrive at the ABR equation for cylindrical probes:

-J e

= . (27)

For each assumed value of J (normalized probe current),this equation can be integrated from = to any arbi-trarily small . The point on the curve where = p (theprobe radius) gives the probe potential p for that valueof J. By computing a family of curves for different J(Fig. 15), one can obtain a J p curve for a probe ofradius p by cross-plotting (Fig 16). Of course, both Jand p depend on the unknown density n0, which one istrying to determine from the measured current Ii. (KTe issupposed to be known from the electron characteristic.)The extraction of n0 from these universal curves is atrivial matter for a computer. In the graphs the quantityJp is plotted, since that is independent of n0. Note thatfor small values of p, I2 varies linearly with Vp, as inOML theory, but for entirely different reasons, sincethere is no orbiting here.

Part A788

Fig. 17. Definition of absorptionradius.

Fig. 18. Effective potential seen byions with angular momentum J.

iii) BRL theory. The first probe theory whichaccounted for both sheath formation and orbital motionswas published by Bernstein and Rabinowitz (BR), whoassumed an isotropic distribution of ions of a single en-ergy Ei. This was further refined by Laframboise (L),who extended the calculations to a Maxwellian ion dis-tribution at temperature Ti. The BRL treatment is con-siderably more complicated than the ABR theory. InABR, all ions strike the probe, so the flux at any radiusdepends on the conditions at infinity, regardless of theprobe radius. That is why there is a set of universalcurves. In BRL theory, however, the probe radius mustbe specified beforehand, since those ions that orbit theprobe will contribute twice to the ion density at anygiven radius r, while those that are collected contributeonly once. The ion density must be known before Pois-sons equation can be solved, and clearly this depends onthe presence of the probe. There is an absorption ra-dius (Fig. 17), depending on J, inside of which all ionsare collected. Bernstein solved the problem by express-ing the ion distribution in terms of energy E and angularmomentum J instead of vr and v. Ions with a given Jsee an effective potential barrier between them and theprobe. They must have enough energy to surmount thisbarrier before they can be collected. In Fig. 18, the low-est curve is for ions with J = 0; these simply fall into theprobe. Ions with finite J see a potential hill. With suffi-cient energy, they can climb the hill and fall to the probeon the other side. The dashed line through the maximashows the absorption radius for various values of J. The computation tricky and tedious. It turns outthat KTi makes little difference if Ti/Te < 0.1 or so, as itusually is. Laframboises extension to a Maxwellian iondistribution is not normally necessary; nonetheless, La-framboise gives the most complete results. Fig. 19shows an example of ion saturation curves from the BRLtheory. One sees that for large probes (Rp/D >>1) theion current saturates well, since the sheath is thin. Forsmall Rp/D, Ii grows with increasing Vp as the sheath ra-dius increases.

One might think that the ABR result would berecovered if takes Ti = 0 or Ei = 0 in the BRL computa-tion. However, this happens only for spherical probes.For cylindrical probes, there is a problem of nonuniformconvergence. Since the angular momentum is Mvr, for r ions with zero thermal velocity have J = (M)(0)(),an indeterminate form. The correct treatment is to calcu-late the probe current for Ti > 0 and then take the limit Ti

Plasma Diagnostics 89

Fig. 19. Laframboise curves for Ii Vcharacteristics in dimensionless units,in the limit of cold ions. Each curve isfor a different ratio Rp/D.

0

1

2

3

4

5

6

200 400 600 800 100Prf (W )

Den

sity

(101

1 cm

-3)

BRLmW aveBRL*ABRABR

Fig. 20. Comparison of n measuredwith microwaves with probes usingtwo different probe theories.

0.0

0.5

1.0

1.5

2.0

2.5

-70 -60 -50 -40 -30 -20 -10 0 10V

I (m

A)4/

3

Fig. 21. Extrapolation to get Ii at Vf.

0

1

2

3

4

5

6

300 450 600 750 900Prf (W)

n (1

011 c

m-3

)

Ne(MW)Ni(CL)Ni(ABR)Ne,sat

10 mTorr

Fig. 22. Comparison of microwaveand probe densities using the floatingpotential method (CL), ABR theory,and Ie,sat.

0, as BRL have done. The BRL predictions have beenborne out in experiments in fully ionized plasmas, butnot in partially ionized ones.

iv) Comparison among theories. It is not rea-sonable to reproduce the ABR or BRL computationseach time one makes a probe measurement. Chen2 hassolved this problem by parametrizing the ABR and BRLcurves so that the Ii V curve can be easily be created forany value of Rp/D. One can then compute the plasmadensity from the probe data using the ABR and BRLtheories and compare with the density measured with mi-crowave interferometry. Such a comparison is shown inFig. 20. One sees that the ABR theory predicts too low adensity because orbiting is neglected, and therefore thepredicted current is too high and the measured current isidentified with a lower density. Conversely, BRL theorypredicts too high a density because it assumes more or-biting than actually occurs, so that the measured currentis identified with a high density. This effect occurs inpartially ionized plasmas because the ions suffer charge-exchange collisions far from the probe, outside thesheath, thus losing their angular momentum. The BRLtheory assumes that the ions retain their angular mo-mentum all the way in from infinity. One might expectthe real density to lie in between, and indeed, it agreesquite well with the geometric mean of the BRL and ABRdensities.

Treating the charge-exchange collisions rigor-ously in the presheath would be an immense problem,but recently Chen et al.3 have found an even easier, for-tuitous, way to estimate the plasma density in ICPs andother processing discharges. The method relies on find-ing the ion current at floating potential Vf by extrapolat-ing on a graph of Ii4/3 vs. Vp, as shown in Fig. 21. Thepower 4/3 is chosen because it usually leads to a straightline graph. At Vp = Vf, let the sheath thickness d be givenby the Child-Langmuir formula of Eq. (A4-7) with V0 =Vf. The sheath area is then A = 2(Rp+d)L. If the ionsenter the sheath at velocity cs with density ns = n0cs, theion current is Ii = n0eAcs, and n can easily be calculatedfrom the extrapolated value of Ii(Vf). Note that if Rp

Part A790

Fig. 23. EEDF curves obtained with aLangmuir probe in a TCP discharge[Godyak et al. J. Appl. Phys. 85,3081, (1999)].

is not always this good. The OML theory (not shown)also fits poorly. Though this is a fast and easy method tointerpret Isat curves, it is hard to justify because the CLformula of Eq. (A4-7) applies to planes, not cylinders,and the Debye sheath thickness has been neglected, aswell as orbiting and collisions. This simple-minded ap-proach apparently works because the neglected effectscancel one another. From the preceding discussion, it isclear that the rigorous theories, ABR and BRL, can errby a factor of 2 or more in the value of n in partiallyionized plasmas. There are heuristic methods, but thesemay not work in all conditions. It is difficult for Lang-muir probes to give a value of n accurate to better than1020%; fortunately, such accuracy is not often required.

6. Distribution functionsSince the ion current is insensitive to Ti, Lang-

muir probes cannot measure ion temperature, and cer-tainly not the ion velocity distribution. However, carefulmeasurement of the transition region of the I V char-acteristic can reveal the electron distribution if it is iso-tropic. If the probe surface is a plane perpendicular to x,the electron flux entering the sheath depends only on thex component of velocity, vx. For instance, the Maxwelldistribution for vx is

2 2 21(v ) exp( v / v ), v 2 /vM x x th th eth

f KT m

=

.

(28)

The coefficient normalizes f(v) so that its integral over allvxs is unity. If f(v) is not Maxwellian, it will have an-other form and another coefficient in front. The electroncurrent that can get over the Coulomb barrier and becollected by the probe will therefore be

min

2min

vv (v ) v , v ( )e x x x s p pI eAn f d m e V V eV

= = =

(29)

where vmin is the minimum energy of an electron that canreach the probe, and Vs = 0 by definition. Taking the de-rivative and simplifying, we find

min

min

v

v =v

vv (v )

vv (v )x

e xx x p

p p p

xx x

p

dI ddeAn f dVdV dV dV

deAn fdV

=

=

Plasma Diagnostics 91

Fig. 24. An I V curve of a bi-Maxwellian EEDF.

-20 -10 0 10 20 30Vp - Vs

Ele

ctro

n cu

rrent

(a)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-20 -15 -10 -5 0 5 10eV/KTe

Ele

ctro

n C

urre

nt 0 5 10 15

Vrf V)

Te = 3 eVHelium

(b)

Fig. 25 . (a) The center curve is thecorrect I V curve. The dashed onesare displaced by 5V, representingchanges in Vs. At the vertical lines,the average Ie between the displacedcurves is shown by the dot. The linethrough the dots is the time-averagedI V curve that would be observed,differing greatly from the correctcurve. (b) Computed I V curves forsinusoidal Vs oscillations of variousamplitudes.

min

2min

v vv (v ) (v )

vx

ex x

p x

dI e neeAn f A fdV m m=

= = ,(30)

so that f(vx) can be found from the first derivative of the I V curve. If the probe is not flat, however, one has totake the three-dimensional distribution g(v) = 4v2f(v),where v is the absolute value |v| of the velocity, and takeinto account the various angles if incidence. Withoutgoing into the details, we then find, surprisingly, that f(v)is proportional to the second derivative of the I Vcurve:

2

2 (v)e

p

d I fdV

(31)

This result is valid for any convex probe shape as long asthe distribution is isotropic, and for any anisotropic dis-tribution if the probe is spherical. To differentiate I Vdata twice will yield noisy results unless a good deal ofsmoothing is employed. Alternatively, one can dither theprobe voltage by modulating it at a low frequency, andthe signal at the dither frequency will be proportional tothe first derivative. In that case, only one further deriva-tive has to be taken to get f(v). Figure 23 is an exampleof non-Maxwellian f(v)s obtained by double differentia-tion with digital filtering.

In special cases where the EEDF consists of twoMaxwellians with well separated temperatures, the twoKTes can be obtained by straight-line fits on the semilogI V curve without complicated analysis. An example ofthis is shown in Fig. 24.

7. RF compensationLangmuir probes used in RF plasma sources are

subject to RF pickup which can greatly distort the I Vcharacteristic and give erroneous results. ECR sourceswhich operate in the microwave regime do not have thistrouble because the frequency is so high that it is com-pletely decoupled from the circuitry, and the measuredcurrents are the same as in a DC discharge. However, inRF plasmas, the space potential can fluctuate is such away that the circuitry responds incorrectly. The problemis that the I V characteristic is nonlinear. The V isactually the potential difference Vp Vs , where Vp is aDC potential applied to the probe, and Vs is a potentialthat can fluctuate at the RF frequency and its harmonics.If one displaces the I V curve horizontally back and

Part A792

Fig. 26. Design of a dogleg probe.

Fig. 27. I V curves taken with andwithout an auxiliary electrode.

forth around a center value V0, the average current Imeasured will not be I(V0), since I varies exponentially inthe transition region and also changes slope rapidly as itenters the ion and electron saturation regions. The effectof this is to make the I V curve wider, leading to afalsely high value of Te and shifting the floating potentialVf to a more negative value. This is illustrated in Fig. 25.

Several methods are available to correct for this.One is to tap off a sinusoidal RF signal from the powersupply and mix this with the probe signal with variablephase and amplitude. When the resultant I V curvegives the lowest value of Te, one has probably simulatedthe Vs oscillations. This method has the disadvantagethat the Vs oscillations can contain more than one har-monic. A second method is to measure the Vs oscilla-tions with another probe or section of the wall which isfloating, and add that signal to the probe current signalwith variable phase and amplitude. The problem withthis method is that the Vs fluctuations are generally notthe same everywhere. A third method is to isolate theprobe tip from the rest of the circuit with an RF choke(inductor), so that the probe tip is floating at RF frequen-cies but is fixed at the DC probe bias at low frequencies.The problem is that the probe tip does not draw enoughcurrent to fill the stray capacitances that connect it toground at RF frequencies. One way is to place a largeslug of metal inside the insulator between the probe tipand the chokes. This metal slug has a large area andtherefore picks up enough charge from the Vs oscillationsto drive the probe tip to follow them. However, we havefound4 that the best way is to use an external floatingelectrode, which could be a few turns of wire around theprobe insulator, and connect it through a capacitor to apoint between the probe tip and the chokes. The chargecollected by this comparatively large probe is then suf-ficient to drive the probe tip so that Vp - Vs remains con-stant. Note that this auxiliary electrode supplies only theRF voltage; the dc part is still supplied by the externalpower supply. The design of the chokes is also critical:they must have high enough Q to present a resonantlyhigh impedance at both the fundamental and the secondharmonic of the RF frequency. This is the reason thereare two pairs of chokes in Fig. 9. One pair is resonant at, and the other at 2. Two chokes are used in series toincrease the Q. A compromise has to be made betweenhigh Q and small physical size of the chokes. Figure 26shows a dogleg design, which permits scans in two di-rections. Figure 27 shows an I V curve taken with andwithout the auxiliary electrode, showing that the chokes

Plasma Diagnostics 93

VR

Fig. 28. A double probe.

VR

Fig. 29. A hot probe.

themselves are usually insufficient. Without proper RF compensation, Langmuirprobe data in RF discharges can give spurious data on Te,Vf, and f(v). However, if one needs to find only theplasma density, the probe can be biased so that V neverleaves the ion saturation region, which is linear enoughthat the average Isat will be the correct value.

8. Double probes and hot probes5

When Vs fluctuates slowly, one can use themethod of double probes, in which two identical probesare inserted into the plasma in close proximity, and thecurrent from one to the other is measured as a function ofthe voltage difference between them. The I V charac-teristic is then symmetrical and limited to the region be-tween the Isats on each probe. If the probe array floatsup and down with the RF oscillations, the I V curveshould not be distorted. This method does not work wellin RF plasmas because it is almost impossible to makethe whole two-probe system float at RF frequencies be-cause of the large stray capacitance to ground. Even ifboth tips are RF compensated, the RF impedances mustbe identical. Hot probes are small filaments that can be heatedto emit electrons. These electrons, which have very lowenergies corresponding to the KT of the filament, cannotleave the probe as long as Vp Vs is positive. As soon asVp Vs goes negative, however, the thermionic currentleaves the probe, and the probe current is dominated bythis rather than by the ion current. Where the I V curvecrosses the x axis, therefore, is a good measure of Vs.The voltage applied to the filament to heat it can beeliminated by turning it off and taking the probe data be-fore the filament cools. One can also heat the probe bybombarding it with ions at a very large negative Vp, andthen switching this voltage off before the measurement.In general it is tricky to make hot probes small enough.For further information on these techniques and on be-havior of Langmuir probes in RF plasmas, the reader isreferred to the chapter by Hershkowitz5.

XIII. OTHER LOCAL DIAGNOSTICS1. Magnetic probes

a) Principle of operation. Fluctuating RF mag-netic fields inside the plasma can be measured with amagnetic probe, which is a small coil of wire, perhaps 2mm in diameter, covered with glass or ceramic so as toprotect it from direct exposure to charged particles.

Part A794

Fig. 30. A magnetic probe with abalun transformer.

When the coil is placed in a time-varying magnetic fieldB, an electric field is induced along the wire accordingto Faradays Law:

/d dt = E B . (32)

Integrating this over the surface enclosed by the coil withthe help of Stokes theorem to convert the surface inte-gral to a line integral, we obtain

( ) inddS dS d V = = B E E! ! " . (33)Here the line integral is along the wire in the coil and is the magnetic flux through the coil, which is BA,where A is the area of the coil. The induced voltage Vindis measured by a high-impedance device like an oscillo-scope. If there are N turns in the coil, the voltage will beN times higher; hence,

indV NAB= ! (34)

The dot indicates the time-derivative and is the origin ofthe name B-dot probe. The minus sign indicates thatthe induced electric field is in the opposite direction tothat obtained when the right-hand rule is applied to B.For a sinusoidal signal, B-dot is proportional to B, sothat the probe is more sensitive to higher frequencies. Toobtain B from the measured Vind, one can use a simpleintegrator consisting of a resistor and a capacitor toground to obtain

1indB V dtNA

= . (35)One only has to be sure that the RC time constant of theintegrator is much longer than the period of the signal.

b) Construction. The probe itself can be assimple as ten turns of thin wire wound on a core ma-chined out of boron nitride. The coil can be placed in-side a ceramic tube or a closed glass tube. Such a tube isnecessarily larger than a Langmuir probe shaft and maydisturb the plasma downstream from the source. If theaxis of the coil is parallel to the tube, the component of Bparallel to the shaft will be measured. If the coil axis isperpendicular to the shaft, one can change from Br to Bmeasurement by rotating the shaft by 90. Sometimesthree coils are mounted in the same shaft to measure allthree B components at the same time.

Plasma Diagnostics 95

V

s

G1 G2 G3

C

Fig. 31. A gridded energy analyzer.

The difficult part is to take the signal out throughthe probe shaft without engendering too much RFpickup. One way is to use a very thin rigid coax, whichis then connected to the scope with a 50- cable. Thecoil in this case can be a single turn formed from thecenter conductor looped around and soldered to the con-ducting shield. If the shaft has to traverse a long paththrough the plasma, a better way is to use a multi-turncoil to increase the signal voltage, and then bring the twoends of the coils through the shaft with a twisted pair ofwires. Outside the plasma, the wires are connected to abalun (balanced-to-unbalanced) 1-to-1 transformer sothat the signal can be carried to the scope with an unbal-anced line. Such a probe is shown in Fig. 30. The trans-former can also have a turns ratio that amplifies the sig-nal voltage. With magnetic probes there is always thedanger of capacitive pickup through the insulators. Onecan check this by rotating the probe 180. The magneticsignal should be the same in magnitude but shifted 180in phase, while the capacitive signal would be the samein both orientations. Whether or not the probe and leadsshould be shielded with slotted conductors is a matter ofexperimentation; the shield can help or actually make thepickup worse.

2. Energy analyzersGridded energy analyzers are used to obtain bet-

ter data for ion and electron energy distributions than canbe obtained with Langmuir probes. However, these in-struments are necessarily largeat least 1 cubic centi-meter in volumeand will disturb the plasma down-stream from them. A standard gridded analyzer has fourgrids: 1) a grounded or floating outer grid to isolate theanalyzer from the plasma, 2) a grid with positive ornegative potential to repel the unwanted species, 3) asolid collector with variable potential connected to thecurrent measuring device, and 4) a suppressor grid infront of the collector to repel secondary electrons. InFig. 31, s is the sheath edge. Grid G1, whether floatingor grounded, will be negative with respect to the plasma,and therefore will repel all electrons except the most en-ergetic ones. One cannot bias this grid positively, sinceit will then draw so much electron current that the plasmawill be disturbed. It is sometime omitted in order to allowslower electrons to enter the analyzer. Grid G1 alsoserves to attenuate the flux of plasma into the analyzer sothat the Debye length is not so short there that subse-quent grid wires will be shielded out. In the space be-

Part A796

Fig. 32. Energy analyzer with onlyone grid and a collector

co p p er fo il

Fig. 33. Construction of an RF cur-rent probe.

hind Grid G1, there will be a distribution of ions whichhave been accelerated by the sheath but which still hasthe original relative energy distribution (unless it hasbeen degraded by scattering off the grid wires). Theseare neutralized by electrons that have also come throughG1. These electrons also have the original relative en-ergy distribution, but they all have been decelerated bythe sheath. Grid G2 is set positive to repel ions andnegative to repel electrons. For example, to obtain fi(v),we would set G2 sufficiently negative (V2) to repel all theelectrons. The ions will then be further accelerated to-ward the collector. This collecting plate C, at Vc, wouldcollect all the ion current if it were at the same potentialas V2. By biasing it more and more positive relative toV2, only the most energetic ions would be collected. Thecurve of I vs. Vc would then give fi(v) when it is differ-entiated. When ions strike the collector, secondary elec-trons can be emitted, and these will be accelerated awayfrom the collector by the field between C and G2, leadingto a false enhancement of the apparent ion current. Toprevent this, Grid G3 is fixed at a small negative poten-tial (about 2V) relative to C) so that these electrons areturned back. Variations to this standard configuration arealso possible.

In a plasma with RF fluctuations, energy analyz-ers would suffer from nonlinear averaging, just as Lang-muir probes do. Because of their large size, and there-fore stray capacitance, it would not be practical to drivethe grids of an energy analyzer to follow changes inplasma potential at the RF frequency. However, one candesign the circuitry to be fast enough to follow the RFand then record the oscillations in collected current as afunction of time during each RF cycle. By selecting datafrom the same RF phase to perform the analysis, one can,in principle, obtain the true energy distribution. Thistechnique cannot be used for Langmuir probes, becausethe currents there are so small that the required frequencyresponse cannot be obtained. RF-sensitive energy ana-lyzers have been made successfully by at least twogroups; one such analyzer is shown in Fig. 32.

3. RF current probeCurrent probes, sometimes called Rogowski coils,

are coils of wire wound on a toroidal coil form shapedlike a Life Saver. Figure 33 shows such a coil. Currentpassing through the hole induces a magnetic field in theazimuthal direction, and this, in turn, induces a voltage inthe turns of wire. The current driven through the wire is

Return loop

Faraday shield

Plasma Diagnostics 97

Fig. 34. Schematic of a POP[Shirakawa and Sugai, Jpn. J. Appl.Phys. 32, 5129 (1993)].

Fig. 35. Peak at p (at the right) movesas Prf is increased [ibid.].

then measured in an external circuit. The coil must take areturn loop the long way around the torus to cancel theB-dot pickup that is induced by B-fields that thread thehole. Current probes are usually large and can be boughtas attachments to an oscilloscope, but these are unsuit-able for insertion into a plasma. The probe shown here isnot only small (~1 cm diam) but is also made for RF fre-quencies. It is covered with a Faraday shield to reduceelectrostatic pickup, and the windings are carefully cali-brated so that the B-dot and E-dot signals are small com-pared with the J-dot signal. An example of a J-dotmeasurement was shown in Fig A6-17.

4. Plasma oscillation probeWhen used in a plasma processing reactor,

Langmuir probes tend to get covered with insulatingcoatings so that they can no longer properly measure dccurrent. A plasma oscillation probe avoids this by meas-uring only ac signals, which can pass capacitivelythrough the coatings. A filament, like a hot probe (Fig.34), is heated to emission and biased to ~100V nega-tively to send an electron beam into the plasma. Such abeam excites plasma waves near p. These high-frequency oscillations are picked up by a probe and ob-served on a spectrum analyzer. If a peak in the responsecan be detected (Fig. 35), it will likely be near = p,and this gives the plasma density. Various spurious ef-fects, such as multiple peaks or surface waves, can causethe resonant to differ from p, but when the signal isclear, a good estimate of n can be obtained.