ISSN 1063�780X, Plasma Physics Reports, 2011, Vol. 37, No. 5, pp. 455–460. © Pleiades Publishing, Ltd., 2011.Published in Russian in Fizika Plazmy, 2011, Vol. 37, No. 5, pp. 492–497.

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1 1. INTRODUCTION

Research and development for new materials andtheir manufacturing have been accelerating during thelast years. Processing by plasma assisted techniques isused in various areas of production and manufacturing[1, 2].

The plasma state covers an energy range spanningseveral orders of magnitude, more than the otherphases together. In length scales it extends fromnanometers to kiloparsecs (1 parsec = 3.26 light�years).

In the plasma, a large number of variables affect thebalance between the competing chemical and physicalprocesses. Appropriate reliable diagnostic techniquesare necessary to provide an understanding of theplasma types and of the phenomena taking place [3, 4].

Electrostatic probes are indispensable diagnostictools for low�pressure plasmas. In such plasmas theelectrons are not at energetic equilibrium with ions orneutrals, having a much greater temperature, Te, thanthe temperature of the ions, Ti, and of the neutrals, Tn.Although the most common theoretical approachstarts with the presumption that the energy distribu�tion function is a Maxwellian one, both for electronsand ions, in low�pressure plasmas the electron energydistribution function (EEDF) for electrons is usually anon�Maxwellian one and the electron temperature hasto be regarded as the effective value (Teff) correspond�ing to the mean electron energy computed fromEEDF [5–7].

The purpose of this paper is to present a particularcomputational technique for calculating the EEDFfrom the current�voltage characteristic of a Langmuirprobe. The experimental distribution is compared to

1 The article is published in the original.

the Maxwell and Druyvesteyn theoretical distribu�tions. The electron density and temperatures that pro�vide the best fit to the experimental data are specifiedfor both distributions. The method is particularlyimportant in the case of plasmas which contain morethan one electronic population (from the viewpoint ofthe energy distribution). The computational tech�nique includes a smoothing procedure for the elimina�tion of the statistical fluctuations of the experimentaldata and is adapted in order to allow a simple, but rig�orous control for the uncertainty propagation from theprimary Langmuir probe data to the EEDF parame�ters, as long as the smoothing uncertainty is not toogreat.

2. DETERMINATION OF THE ELECTRON ENERGY DISTRIBUTION FUNCTION

The microscopic properties of plasma are fullydescribed by the distribution functions of its compo�nents [7, 8]. Let us consider the plasma produced fromgas, with a pressure situated between 0.1 and 5 Torr,without magnetic fields, and let us assume that, forthis plasma, the EEDF is isotropic, stable and homog�enous (the request for plasma homogeneity is notmandatory if movable Langmuir probes are used,making possible EEDF measurements with spatialresolution). For this plasma, the transition region ofthe probe voltage�current contains information aboutthe EEDF.

The EEDF measurements are based on theDruyvesteyn method [9], which shows that the elec�tron energy probability function (EEPF) is propor�tional to the second derivative of the probe voltage�current characteristic. The EEDF is the product of theEEPF and the square root of electron energy.

PLASMADIAGNOSTICS

Computational Technique for Plasma Parameters Determination Using Langmuir Probe Data1

C. Negreaa, b, V. Maneaa, V. Covleaa, and A. Jipaa

a Faculty of Physics, University of Bucharest, P.O. Box MG�11, RO�0771253 Bucharest–Magurele, Romaniab Institute of Space Science, P.O. Box MG�23, RO�077125, Bucharest–Magurele, Romania

Received July 22, 2010; in final form, November 11, 2010

Abstract—In the present work, we consider a new numerical method for processing the experimental infor�mation on the electron energy distribution function obtained with a Langmuir probe in a low�pressureplasma. This method offers the possibility to establish the temperature and concentration of the electrons fordifferent forms of the distribution function. Some specific difficulties of the previous methods used to do suchestimations are surpassed using the method proposed in this work.

DOI: 10.1134/S1063780X11050096

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The Druyvesteyn method is in principle wellunderstood, but the particular aspects related to itsnumerical implementation are subject of increasedinterest due to the recent technical advances in theautomated acquisition and manipulation of the Lang�muir probe data. The signals extracted from theplasma environment contain noise determined by thestatistical character of the involved processes. Theexperimental devices themselves can contribute to thenoise of the data acquisition systems, due to insuffi�ciently filtered high�tension generators or due to elec�tromagnetic interferences. It is thus necessary to use inthe routines for the data manipulation some methodsby which to eliminate this noise from the experimentaldata.

The solution is to include a smoothing procedure inthe numerical calculation of the second derivative ofthe probe current. We start in our application from themethod proposed by Savitzky and Golay [10]. In thismethod, the current–voltage characteristic of theLangmuir probe is fitted with a polynomial aroundeach voltage value and the second derivative is calcu�lated analytically from the expression of the polyno�mial after the values of the fit parameters are deter�mined.

We use a slight variation from the method originallyproposed in [10] by using a first degree polynomial forthe local fit of the experimental data. This requiresapplying the procedure twice, because a first degreepolynomial only contains information on the firstderivative. This variation was chosen because theintermediate obtainment of the first derivative allows asimple determination of the plasma potential, Vp, asthe value of the voltage for which the first derivative ofthe experimental data is maximal, reducing the uncer�tainty on the value of Vp to the uncertainty of the probevoltage measurement, which is usually as low as 0.1 V.

The smoothing depth is determined by the numberof experimental points around each voltage value forwhich the fit is performed. An advantage of our doublestep procedure is that the smoothing depth requiredfor obtaining the same level of smoothness of the sec�ond derivative is considerably smaller, which meansthat a certain value of the second derivative is obtainedon the basis of fewer experimental points. This isimportant in the case of plasmas with two electronicgroups, in which the smoothing depth required forobserving the low concentration group is large. We alsomention that the smoothing depth does not influencethe position of the maximum of the first derivative andso, the determined value of the plasma potential.

Once the second derivative of the voltage�currentcharacteristic is obtained, the EEDF is calculated bymeans of the Druyvesteyn formula:

(1)2

3 2

8( ) ,e e

p

m d IF E E

e A dV=

where Ap is the probe surface, while e and me representthe electron charge and mass.

3. DETERMINATION OF THE ELECTRON CONCENTRATION AND EFFECTIVE TEMPERATURE. CASE OF A SINGLE

ELECTRONIC GROUP

The obtainment of the EEDF allows the determi�nation of the electron concentration and effectivetemperature. This is done by assuming a certain func�tional form of the distribution function, having asparameters the electron concentration and tempera�ture, and by fitting with it the experimental EEDF.Usually more than one form of the distribution func�tion is possible and all forms must be successivelytested, choosing the one which best minimizes theχ2 variable.

The method will be presented in this section for thecase of plasmas with a single electronic group, forwhich alternative, more direct methods are possible.In the following section, the method will be adaptedand applied for the case of plasmas with two electronicgroups, for which it is the only method capable ofoffering, for each of the two groups, the concentrationand effective temperature.

For fitting the experimental EEDF we use theGauss–Newton nonlinear fitting algorithm [10]. Inthe following we will give a brief description of themethod, applied to the case of a two�parameter fit�ting function. Let us denote the fitting function byf(E; n, T), where E is the function argument (theenergy), while n and T are the fitting parameters, theelectronic concentration and temperature, respec�tively. We will perform the linearization of the functionf by means of a first�order Taylor expansion aroundsome probable values of the concentration and tem�perature, say n0 and T0:

(2)

With the linear (approximate) form of the function,we perform a two�parameter least squares fit of theEEDF, from which we obtain a new set of values forthe parameters n and T. The next step is to reinitializen0 and T0 to the newly obtained values of the parame�ters and to perform a new linearization around thesevalues, followed by a new linear regression. The con�vergence of the method is good and requires few itera�tions until stable values of n0 and T0 are obtained, giv�ing the electron concentration and effective tempera�ture. The only real difficulty of the method is itssensitivity to the proper initialization of the fit param�eters, which is not a real impediment in the case of atwo�parameter fit.

( ) ( )0 0 0 0

0 0

0 0, ,

( ; , ) ( ; , )

( ) ( ) .n T n T

f E n T f E n T

f fn n T T

n T

=

∂ ∂+ − + −

∂ ∂

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COMPUTATIONAL TECHNIQUE FOR PLASMA PARAMETERS 457

Because the fitting method performs a linearregression of the experimental EEDF, which, in itsturn, is determined by means of local linear regres�sions, the experimental uncertainties can be easilypropagated from the initial voltage–current charac�teristic to the electron concentration and effectivetemperature. One problem in this case is the uncer�tainty introduced by the smoothing process itself,which is difficult to estimate. In the case of plasmaswith a single electronic group, the smoothing depthrequired for the obtainment of good quality data is rel�atively low, so we can neglect the uncertainty deter�mined by the smoothing process.

In Fig. 1 we present the results of applying ourmethod to plasma with a single electronic group. TheGauss–Newton fitting algorithm was used, assumingfor the distribution function either a Maxwellian or aDruyvesteyn form, the results being represented withcontinuous lines light and dark, respectively. Theexperimental EEDF is plotted with full circles. Theascending region of the EEDF was not taken into con�sideration, because this region is poorly fitted by any ofthe presumed functions that could theoreticallydescribe the experimental distribution. The pointswhich were included in the fit represent the decreasingregion of the EEDF.

It is apparent from Fig. 1 that the Maxwellian dis�tribution offers a better agreement with the experi�mental EEDF. Table 1 contains the results of the fitwith the two presumed forms of the distribution func�tion, along with the associated uncertainties.

Even if in this particular case the Maxwell distribu�tion offers a better description of the experimentalEEDF than the Druyvesteyn distribution, this shouldnot be interpreted as a signature of thermodynamicalequilibrium. In this sense, the temperature deter�mined by the Maxwell fit should still be regarded as aneffective temperature.

The electron concentration and effective tempera�ture can be obtained directly from the Langmuir probeexperimental data, by analyzing the electronic regionof the voltage�current characteristic [3]. For our set ofexperimental data, the obtained value of the electronictemperature is 30900 K, with a dispersion of 200 K,while the value of the electronic concentration is 9.1 ×1015 m–3, with a dispersion of 2.5 × 1015 m–3. One cannotice that the values obtained from the fitting proce�dure performed on the EEDF are in agreement withthe ones obtained directly from the Langmuir probecurrent�voltage characteristic, the effective tempera�ture having in the latter case a lower dispersion, due tothe fact that it was calculated in a more direct manner.This is the reason for which in the case of plasmas witha single electronic group, the direct method of deter�mining the electron temperature is preferable, due toits lower uncertainty. Our method becomes important

in the case of plasmas with multiple electronic groups,for which the direct method cannot provide theparameters characterizing each group.

4. DETERMINATION OF THE ELECTRON CONCENTRATION AND EFFECTIVE

TEMPERATURE. CASE OF TWO ELECTRONIC GROUPS

The data presented in this section correspond tothe current�voltage characteristics of plasmasobtained in a reflex plasma reactor [7]. The dischargemechanism, in this case, is the origin of three groupsof electrons within the bulk plasma region, i.e., thecold group of electrons (ultimate electrons or bulkelectrons), the hot group of electrons (secondary elec�trons), and the first group of electrons (primary elec�trons). The cold group of electrons is dominant and,its number density is almost two orders of magnitudehigher than the number density of the hot group. Theleast numerous are the primary electrons. We couldobtain information on the temperature, number den�sity and EEDF (EEPF) for both the cold and hotgroups of electrons. No measurements have beenmade on the primary electrons. However, since we canmeasure the plasma potential and the anode voltagedrop, we can establish the energy of primary electronswhen they enter the bulk plasma region [7].

0 5 10 15 20E, eV

16.0

15.5

14.5

13.0

log(F(E))

Neonp = 1.3 mbarV = 350 VI = 25 mA

15.0

14.0

13.5

Fig. 1. Experimental EEDF (full circles) of a plasma witha single electronic group fitted by a Maxwellian (light line)and by a Druyvesteyn (dark line) function.

Table 1

Distribution function n, m–3σn, m–3 Teff, K σT, K

Maxwellian 1.1 × 1016 1.3 × 1014 29600 500

Druyvesteyn 7.7 × 1015 7.1 × 1013 31900 300

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In the case of plasma with two electronic groups,we use an adapted variant of the procedure describedin the previous section. The function f will this timehave four parameters, two concentrations and twoeffective temperatures: f(E; n1, T1, n2, T2). In this case,we will perform the first order Taylor expansion of thefunction with respect to all four parameters:

(3)

where we have denoted by the index 0 the fact that thepartial derivatives are calculated at the point (n01, T01,n02, T02).

From this point forward, the iterative procedure issimilar to the one already described in the previoussection. The four parameter function is chosen as thesum of two independent distribution functions, the

1 1 2 2 01 01 02 02

1 01 1 011 10 0

2 02 2 022 20 0

( ; , , , ) ( ; , , , )

( ) ( )

( ) ( ) ,

f E n T n T f E n T n T

f fn n T T

n T

f fn n T T

n T

=

⎛ ⎞ ⎛ ⎞∂ ∂+ − + −⎜ ⎟ ⎜ ⎟

∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂+ − + −⎜ ⎟ ⎜ ⎟

∂ ∂⎝ ⎠ ⎝ ⎠

forms of which are again determined by multiple trials,in which all possible combinations are tested (forexample, between a Maxwellian and a Druyvesteyndistribution function):

(4)

The main problem of this four�parameter fittingalgorithm is its great sensitivity to the initialization ofthe fit parameters, which complicates the obtainmentof a fully convergent fit on a set of noisy experimentaldata.

The stability of the method can be increased if,instead of using a single global distribution function,written as the sum of two independent distributionfunctions, one uses the two functions to separately fitthe electronic groups of the EEDF. At the first step,one performs a Gauss–Newton nonlinear fit, with thefunction f1, of the EEDF region corresponding to theelectronic group with the higher concentration. Theobtained fitting curve is subtracted from the experi�mental EEDF, followed by a Gauss–Newton nonlin�ear fit, with the function f2, of the values obtained

1 1 2 2 1 1 1 2 2 2( ; , , , ) ( ; , ) ( ; , ).f E n T n T f E n T f E n T= +

17

0 5 10 15 20E, eV

0 5 10 15 20

16

15

14

13

log

(F(E

))

Nitrogenp = 0.1 mbarV = 366 VI = 100 mА

Fig. 2. Experimental EEDF (full circles) of a plasma with two electronic groups fitted by two Druyvesteyn (left) and by two Max�wellian (right) functions.

17

0 5 10 15 20E, eV

0 5 10 15 20

16

15

14

13

log

(F(E

))

Nitrogenp = 0.08 mbarV = 374 VI = 50 mA

Fig. 3. Experimental EEDF (full circles) of a plasma with two electronic groups fitted by two Druyvesteyn (left) and by two Max�wellian (right) functions.

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COMPUTATIONAL TECHNIQUE FOR PLASMA PARAMETERS 459

(after the subtraction of f1) in the region correspondingto the electronic group with the lower concentration.This second fitting curve is subtracted from the initialexperimental EEDF, and, in a new step, the values ofthe EEDF (after the subtraction of f2), in the region ofthe electronic group of higher concentration, are againfitted independently with the function f1. The proce�dure is repeated until one obtains stable values of thefour parameters. The successive subtraction of the fit�ting curves from the experimental EEDF has the pur�pose of eliminating the contribution of one of the elec�tronic groups, when the region corresponding to theother electronic group is fitted.

This modified method is more stable and convergeswithout any difficulties, at the expense of requiring anincreased number of iterations. Figures 2 and 3present the results of applying our numerical algo�rithm to Langmuir probe data from plasmas obtainedin a reflex plasma reactor [7], for which two electronicgroups are usually obtained. In both cases, all fourpossible combinations of Maxwell and Druyvesteyndistribution functions were tested, but only the resultsfor the Maxwell–Maxwell and Druyvesteyn–Druyvesteyn combinations are presented. The bestagreement between the fitting curve and the experi�mental EEDF was obtained when two Druyvesteynforms were used for the two fitting functions, f1 and f2.Again, only the descending region of the EEDF wasfitted.

Tables 2 and 3 contain the obtained values of the fitparameters for the EEDF represented in Figs. 2 and 3,respectively. In this case, the dispersions of the calcu�lated parameters are not presented, because their val�ues are underestimated. This is due to the fact that onerequires a very deep smoothing in order to observe thelow concentration electronic group, case in which theuncertainty determined by the smoothing processitself is no longer negligible. Unfortunately, to our bestknowledge, there is no way to properly estimate thisuncertainty, which is in our case the main error source.One would expect the dispersions of the electron con�centrations and effective temperatures to be, in thiscase too, of the order of magnitude presented inTable 1.

The departure of the concentration values pre�sented in Tables 1–3 from the real concentrations ofthe analyzed electronic groups is not only influencedby the statistical uncertainties, but also by the approx�imate reproduction of the experimental data by the fit�

ting functions. This latter source of uncertainty ishowever very difficult to quantify. We could expect,even in the absence of complete uncertainty informa�tion, that the values in Tables 1–3 are good estimatesof the corresponding concentrations, representing thebest obtainable result in the case of EEDF in whichmultiple electronic groups are detected. We also men�tion that the second derivative of the probe currentitself may produce an incorrect description of theEEDF at low energy (due to electron sink) and at highenergy (due to influence of ion current). The effect ofthis departure is somewhat avoided by the fact that notall EEDF points are taken into account in the fit, butonly the points corresponding to intermediate energyvalues.

5. CONCLUSIONS

A numerical method for processing the plasmaLangmuir probe experimental data was presented andapplied for obtaining the EEDF and for determiningthe electron concentration and effective temperature.The method uses a variant of the Savitzky–Golaymethod [10] for calculating the second derivative ofthe experimental data, and the Gauss–Newton non�linear fitting algorithm for determining the plasmaelectronic parameters.

The numerical method is tested on the case ofplasma with a single electronic group and thenextended to the case of plasmas with multiple elec�tronic groups, for which the electron concentrationand effective temperature of each group can only beobtained by the analysis of the experimental EEDF. Amodified version of the Gauss–Newton algorithm isproposed in this latter case, in order to improve thestability of the algorithm convergence.

The method is applied for the analysis of experi�mental Langmuir probe data from plasmas obtained ina reflex plasma reactor [7], for which two electronicgroups are usually observed. The result of our analysisis the determination of the electron concentration andeffective temperature for each group and also of themost probable functional form of the EEDF.

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Table 2

Distribution function n1, m–3 Teff1, K n2, m–3 Teff2, K

Druyv–Druyv 7.6 × 1016 12800 3.2 × 1015 37000

Maxw–Maxw 4.7 × 1017 6500 5.3 × 1015 30300

Table 3

Distributionfunction n1, m–3 Teff1, K n2, m–3 Teff2, K

Druyv–Druyv 7.5 × 1016 12800 1.2 × 1015 54400

Maxw–Maxw 3.4 × 1017 6800 1.7 × 1015 48000

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NEGREA et al.

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