Numerical methods are used to determine the Electron Energy Distribution Function (EEDF) from I(V) probe characteristics, which are measured using a cylindrical Langmuir probe in the case of weakly ionized plasmas. This task becomes difficult when measurement is complicated by the presence of an external magnetic field or in high pressure plasma because of collision between electrons and heavy particles within the sheath formed around the probe tip. In this case, the electron current must be calculated using the Swift law instead of the Langmuir law. The numerical methods consist of determining the derivative functions of the I(V) probe characteristics in the case of a noisy signal and correcting the EEDF taking into account the electron diffusion coefficient within the sheath formed around the probe collector. Algorithms are given to detail the methods step by step, which can be used to write homemade codes. The methods are tested in the case of different plasma reactors described in the literature, such as microwave plasma and rf (radio-frequency) and dc (direct current) plasma reactors working at different pressures with or without magnetic field. The results show the effect of pressure or magnetic field on the I(V) probe characteristics because of the change in the electron diffusion coefficient.

Langmuir probe diagnostics provide useful information about plasma parameters (electron energy or density, ion density, plasma potential or floating potential, etc.) in both basic research and plasma processing.1–5 This simple method is widely used to investigate the plasma under different experimental conditions (low and high pressure plasma, thermal plasma, or magnetized plasma). The probe is biased and immersed into the plasma. It is used to measure the current–voltage characteristics of the surrounding plasma in the vicinity of the probe collector. Unfortunately the theoretical analysis of the I(V) probe characteristics is generally complicated and depends on the experimental conditions and on the type of probe used during the experiment. The I(V) probe characteristic can be drastically impacted by the probe collector geometry or by the sheath formed around the collector because of the diffusion of charged particles, collision processes within this sheath, external effects such as a magnetic field used to confine the plasma. Theories have been developed taking into account different plasma configurations to determine reliable results,6–10 depending on models and hypotheses, but they are available in specific cases only.

The determination of electron energy distributions in plasma is a crucial aspect in plasma physics. This distribution influences the characteristics and the behavior of the plasma, which in turn impacts its applications in material processing, plasma-based propulsion, and semiconductor manufacturing.11–13 However, the interpretation of the data acquired from the probe cannot be challenged without numerical methods. In this article, we will discuss how numerical methods can be used to accurately determine the electron energy distribution from Langmuir probe measurements, in different plasmas.

In order to simplify the study, we consider the case of a cylindrical probe collector, which is the most frequent geometry used in weakly ionized plasma. The case of totally ionized plasma is not considered in this article since other phenomena such as the effect of the ion viscosity on the collected current, which complicate the interpretation, should be taken into account.14 

The basic theory, developed by Druyvesteyn,15 shows the dependence of the I(V) probe characteristics on the Electron Energy Distribution Function (EEDF) within the plasma surrounding the probe. In the following part, details of the theories used to determine the EEDF are given, with the different hypotheses used to elaborate the laws, and consequently, these laws can be used under specific conditions only.

Considering a cylindrical probe negatively biased at a voltage (−VP) (in reference to the plasma potential) and looking at the electron current impinging on the probe collector, the electrons with velocity v contained in the elementary volume of plasma dV and impinging on the elementary collector surface δAp in 1 s are contained in the hemispherical volume centered on δAp and of radius v (see Fig. 1).

FIG. 1.

Electrons with velocity v and contained in dV are impinging on the elementary collector surface δAp.

FIG. 1.

Electrons with velocity v and contained in dV are impinging on the elementary collector surface δAp.

Close modal
The elementary volume dV in cylindrical coordinates is given by
dV=vdθvsinθdɸdv.

Considering the Electron Velocity Distribution Function f(v), the number of electrons contained in dV with a velocity ranging from v and v + dv is dne = f(v)dV. Electrons of velocity v impinging on δAp are contained in the solid angle dΩ=δApcosθv2, and their number is given by dne=f(v)dVdΩ4π. It corresponds to the elementary current intensity die = evdne. Hence, the total current collected on δAp is ie=vminɸ=02πθ=0θmaxdieθ,ɸ,vdθdɸdv.

In the following part, values of ie, e, and Vp are arbitrarily considered positive in the different equations.

The electrons impinging on δAp are collected if the kinetic energy Ee=12mevcosθ2eVP and θmax=2eVPmev2.

The minimum velocity value vmin is obtained when θ = 0 and is given by vmin=2eVPmev21/2.

Hence, the total current collected on δAp is
ieVp=eδAP4vminvf(v)12eVPmev2dv.
(1)
This equation is named the Langmuir law, which is obtained considering the electron velocity distribution function.15 It is
ieVp=geVPεe1/2g(εe)1eVPεedεe=geVPεe1/2εeeVPfεedεe,
(2)
with g=2meeδAP4. When we consider the Electron Energy Distribution Function g(εe), gεe=12meεe1/2fv.
Considering the function F(x, y) and the integral u=φ(x)f(x)F(x,y)dy, the first derivative function is given by
ux=Fx,fxfxFx,φxφx+φxfxFx,yxdy.
(3)
In the case of the electron current intensity of the probe, assuming u = ie, x = VP, and y = v, the first derivative function vs Vp of the Langmuir law is given by
ieVP=e2δAP2mevmin1vfvdv.
(4)
In addition, the second derivative function is
2ieVP2=e2δAP4me1VPfvmin.
(5)
Assuming the electron energy distribution function, the second derivative becomes
2ieVP2=14e5/22meδAPVP1/2fεe,
(6)
with εe = eVP.

Equations (5) and (6) are the Druyvesteyn Equations used to determine the Electron Velocity Distribution Function (EVDF) or the Electron Energy Distribution Function (EEDF) from the electron current intensity collected by a negatively biased probe of area δAP.15 

Details of the theory given above show that both the Langmuir law and the Druyvesteyn law are available only in the case where electrons are not disturbed in the sheath formed around the probe. The energy of electrons collected by the probe depends on the initial electron energy and on the biased voltage only. Hence, these equations can be used when the disturbance of the plasma due to the probe can be neglected, i.e., when the electron diffusion rate is high enough to cope with the drain of electrons from the plasma to the probe surface. If the diffusion coefficient is too low, the measured distribution function will be underestimated compared to the real distribution function of the undisturbed plasma. The disturbance is greater when the gas pressure and the probe area increase and as the electron velocity decreases. This effect is also observed in magnetized plasma because of the electron gyration within the magnetic field. In all these cases, corrections of the Langmuir and Druyvesteyn equations are necessary, taking into account the electron diffusion within the sheath formed around the probe. This has been done by Swift.16 

The electron flow dΦ(εe) of energy ranging from εe and εe + e does not only depend on the biased voltage but also on the electron diffusion through the sheath surrounding the probe collector. Assuming a radial diffusion through the sheath, the electron flow is given by
dΦεe=Dεenεer=Dεefεedεer,
where r, f(εe), and D(εe) are the sheath radius, the EEDF, and the electron diffusion coefficient, respectively. Consequently, the EEDF change is due to the diffusion through the sheath and is fεef0εedεe=diεer0rdreDεeS, where diεe=eSdΦεe and r0 is the sheath thickness.

From this equation, it can be seen that the larger the diffusion coefficient, the lower is the change in the EEDF.

Under these conditions, Langmuir law (2) becomes
die=gεe1/2εeeVPf0εe+dieεer0rdreDεeSdεe;
then,
die=gεe1/2εeeVPf0εedεe1+gεeeVPεe1/2rr0dreDεeS.
The total current collected by the probe of radius rp biased at the potential Vp with a sheath radius rs is now16 
ieVP=geVPεe1/2εeeVPf0εedεe1+εeeVPεeΨεe,
(7)
where Ψ(εe) is called the diffusion parameter and
Ψεe=gεe1/2rPrsdrDεeeS.
(8)

Equation (7) is the modified form of the Langmuir law (2) proposed by Swift,16 taking into account the diffusion coefficient of the electron through the sheath. When the electron diffusion through the sheath can be neglected, ψ(εe) = 0, and consequently, ie(VP) is given by Eq. (2). The Swift law [Eq. (7)] has been used by different authors6–10 to study the effect of magnetic fields or moderate-collisional plasma on I(V) probe characteristics. These authors have developed models assuming hypotheses and considering the product of the electron velocity and the diffusion coefficient, ve(εe)D(εe), constant within the sheath thickness.8 Consequently, the diffusion parameter Ψ(εe, V) = Ψ(εe). It does not depend on the probe bias voltage. For a probe tip of radius rp and length l, the Druyvesteyn method is reasonably applicable if λe>0.75rplnπl4rp, and if λe<rp7lnπl4rp, the EEDF can be calculated using the first derivative function of the electron current die/dV.6,8 Hence, it is necessary to adjust the probe radius to satisfy the corresponding validity conditions. By this way, authors consider a mean value for the electron mean free path all over the EEDF. To prevent error due to these assumptions and approximations, the following part gives a numerical method that does not need any additional hypotheses.

Using Eq. (7) and the same derivation method already used Eq. (3), the partial derivative functions ieVPVP and 2ieVPVP2 are (in a first approximation) given by
ieVPVP=geVPeεe1/2f0εedεe1+εeeVPεeΨεe2
(9)
and
2ieVPVP2=Cεe1/2f0εeCeVP2εe3/2f0εeΨεedεeεe+εeeVPΨεe3.
(10)
Then calculations using Eqs. (7), (9), and (10) give
A=ieVPεe=ieVPVPVPεe=gεe1/2εeeVPf0εe1+εeeVPεeΨεe
and
B=ieVPVPεe=2ieVPVP2VPεe=geεe1/2f0εe1+εeeVPεeΨεe2.
Hence, the value of Ψ(εe) can be calculated using the ratio A/B,17,18
Ψεe=eεeeVPieVPVP2ieVPVP2+1εeεeeVP.
(11)

It is worth noting that in Eq. (10), because diedVp=diedVapp0 and d2iedVp2=d2iedVapp20, the ratio die/dVpd2ie/dVp20.

The diffusion parameter is expected to be positive. Otherwise, the electron flow should increase when it reaches the collector. The condition Ψεe0 is fulfilled only if die/dVpd2ie/dVp2εeeVpe.

These results show that the diffusion parameter can be calculated directly using the ratio of the first to the second derivative functions of the experimental Ie(VP) probe characteristic without any additional model. However, the value of Ψ(εe) depends on the noise measured with the experimental signal during the data acquisition, and it is necessary to determine accurate values of the derivative functions. The larger the signal/noise ratio, the better the determination of Ψ(εe).

The next part of this article presents numerical methods to determine the EEDF from the I(V) probe characteristics considering the general case of the electron current given by Swift Eq. (7).

The EEDF can be calculated using the values of the first and second derivatives of the electron current as a function of the applied potential (in reference to the plasma potential). Different methods can be used to determine the derivative functions. The easiest should be to measure the change in ie vs applied voltage Vapp. This method was used by Medicus,19 but it is generally not efficient because of the low signal/noise ratio. “Analog differentiation” methods can also be used.20,21 However, additional equipment such as active differentiators composed of cascading operational amplifiers is needed, which generally decreases the apparatus response time necessary for a high energy resolution.22 The second derivative function of the probe measurement is not a direct value, but it results in a convolution product between the “true” second derivative and the instrumental (or apparatus) functions. It can have an effect on the results as a shift of the plasma potential position or a broadening of the derivative function shape. A correction is necessary, which can be done with the knowledge of the instrumental function only.8,23 Numerical methods can also be used to smooth data by suppressing the noise of the recorded signals. These methods do not need additional electronic equipment. The data smoothing method proposed by Hayden24 or the optimal Wiener filtering method25 considers the convolution product between the signal and the apparatus function of the probe. The peculiarity of this convolution product is that the noise spectrum is statistically uncorrelated with the instrumental function. However, these methods need the knowledge of the apparatus function of the setup, which is also generally unknown. In the case of a Langmuir probe, it depends on the sheath formed around the probe, which changes with the biased voltage. Hence, these numerical methods that involve the knowledge of the apparatus function are also not easy to use.

Other numerical methods based on polynomial interpolation, such as the Lagrange polynomial interpolation, the cubic spline interpolation method,25 or the Savitzky–Golay algorithm,26 are more suitable to smooth the I(V) probe characteristic. These methods do not depend on apparatus functions. Moreover, the results can be improved by gradually changing the polynomial parameters, and any change in the curve (shift of the plasma potential value or change in the derivative function shape) can be detected. Hence, any error due to the smoothing effect can be prevented.

The following part presents two efficient numerical methods based on polynomial interpolations and Fourier transforms. We compare these methods in the case of different experiments proposed in the literature.

The method was first used to determine the second derivative of the probe characteristic by Fujita et al..27 This method is derived from the Savitzky–Golay algorithm and is based on the least squares principle to fit the experimental data with a polynomial curve and on the differential operation by convolution with weighting functions. The step h=xixi1 between two successive points xi,yi must be constant. It is assumed that the smoothed value y(j) in the vicinity of a point i can be estimated by a quadratic polynomial yj=a2ji2+a1ji+a0, where j = −m + i, …, −1 + i, i, i + 1, ….i + m. m is a suitable integer selected as the fitting width.

The coefficients a0,1,2 are determined considering that the squared error I between the estimated value y(j) and the measured value y0(j) is minimized,
I=j=m+ij=m+iy0jyj2andIak=0,1,2=0.
This equation system is resolved using the Kramer determinant,
ak=0,1,2=1wk=0,1,2j=mj=mwk=0,1,2jy0j+i,
with
w0=134m212m+3,
w1=13mm+12m+1,
w2=130mm+12m+14m21
and
w0j=3mm+115j2,
w1(j)=j,
w2(j)=3j2mm+1.

Consequently, the fitted value yi=a0, the first derivative yi=yii=a1h, and the second derivative yi=2yii2=2a2h2. This method consists of considering Taylor development of the second order for each experimental point f(x):

fx=fx0+hfx+h22fx++R(h), where h is the step between two points (ji), which must be constant, and R(h) is the rest obtained after minimization of squared error I. The smaller the step between the two points, the more precise are the a0, a1, and a2 values.

Figure 2 shows the SDM algorithm.

FIG. 2.

Algorithm for the SDM method.

FIG. 2.

Algorithm for the SDM method.

Close modal

This method has been tested to determine the second derivative function of the I(V) probe characteristics under different experimental conditions. Figure 3 shows the second derivative function of the I(V) probe characteristic measured in argon microwave plasma working at a frequency of 2.45 GHz, a pressure of 2 Torr, and an incident power of 100 W. The results are obtained using different m values corresponding to a fitting width of 2m + 1 values. It can be seen that the noise decreases with increasing m value, and good results are obtained for m larger than 10. However, there appears a broadening of the curve, with increasing m producing a slight shift of the potential plasma value determined at the zero-crossing point with the V axis when the second derivative becomes negative (indicated by the vertical arrow). The plasma potential value shifts from 13.7 V for m = 5 to 14.4 V for m = 20. This is the main problem observed with this method. This shift decreases with increasing density of points in the data array. Hence, the m value should be gradually increased to control any shifting or broadening effect and to check the results of integrating the second derivative and compare the results to the initial experimental I(V) curve. Thus, any shift in the plasma potential or broadening of the second derivative will be detected.

FIG. 3.

Second derivative function calculated using the SDM method in the case of the I(V) probe characteristic measured in argon microwave plasma working at a frequency of 2.45 GHz, a pressure of 2 Torr, and an incident power of 100W.

FIG. 3.

Second derivative function calculated using the SDM method in the case of the I(V) probe characteristic measured in argon microwave plasma working at a frequency of 2.45 GHz, a pressure of 2 Torr, and an incident power of 100W.

Close modal

This second numerical method can be compared to the “analog method” of Kortshagen and Schlüter,20 but it does not involve additional apparatus. Hence, there is no need of the knowledge of the apparatus function of new equipment. The principle of the method consists in using a Fourier transformation to isolate harmonic components, which are obtained based on the original experimental I(V) probe characteristic by the simulation of the effect of a sinusoidal superimposed perturbation on the measured signal. This method is efficient in the case of noisy signals.28–31 

Let us consider the part of the experimental I(V) probe characteristic corresponding to the negatively biased probe with respect to the plasma potential. In this case, electrons are repulsed and only electrons of kinetic energy larger than eVp (Vp is the biased voltage) are collected.

Suppose that a sinusoidal component ut=u0sin(ωt) is added to the applied voltage, V(t) = VP + u(t).

Assuming the I(VP) curve as infinitely derivable, the Taylor expansion gives
IVP+u(t)=IVP+i=1nutii!IiVP+ut+utnεut,
where Ii[Vp + u(t)] is the ith derivative of I[Vp + u(t)] with respect to t and u0nε[u(t)] is the rest, limnu0nεut0.
Considering the third order expansion of the previous equation, we obtain
IvP+ut=IVP+14ut2I2vP+ut+utIvP+ut+324ut3I3vP+ut×sinωt14ut2I2vP+utcos2ωt124ut3I3vP+utsin3ωt+ut3εut.

This equation shows that the second derivative appears in the term related to the second harmonic 2ω. Isolating these components, it is possible to obtain a signal whose amplitude is proportional to the second derivative of the electron current I(Vp).

The experimental I(Vp) probe characteristics are a data array [(x1, y1), …, (xn, yn)], corresponding to n acquisition points. We simulate for each of these points (xi, yi) using a sinusoidal signal ut=u0sin(ωt) with w = 2πf, which is added to the applied voltage xi = VP. We obtain for each point a new array containing ns points [(xi + u(t)], [Ip(xi + u(t)]. The value of the current intensity Ip[xi + u(t)] can be deduced from the experimental I(Vp) curve using a Lagrange polynomial interpolation of degree (m − 1) through m consecutive points (xk, yk) of the initial experimental data array for x = xi + u0 sin(ωt) ranging from xk=1 to xk=m, i.e., xi+uo<xm and xiuo>x1, with xn > xn−1,25 which gives
IPx=k=1mykikmxxiikmxkxi.
In our program, we use m = 4.

Hence, at each point (xi, yi) of the initial data array, a new data array containing ns points corresponding to the simulated probe characteristic is determined, which would be obtained when a sinusoidal superimposed component is added to the initial applied potential (Vp). This new signal consists of multiple harmonic components, and one of them is the second harmonic, which is proportional to cos(2wt) = cos(4πft). Using the direct Fourier transform, this component can be isolated in the frequency domain. Using the inverse Fourier transform and dividing it by 14u02cos4πft, we obtain ns points corresponding to xi=VP,yi=2IPVP2. This procedure must be done for any (xi, yi) points of the initial experimental data array, and the output is xi=VP,yi=1nsns2IPVP2. The second derivative values calculated for the ns points are close to each other, and the final value used is the mean value of these second derivative values calculated over the ns points.

It is worth noting that in this calculation, the frequency f must be between ffNy,fNy (where fNY is the Nyquist frequency) to prevent any “aliasing effect.”25 The Nyquist frequency is given by fNy=12Δ with Δ = tech, where tech is the arbitrary value of the time between two consecutive points [(xi + u(t)], [Ip(xi + u(t)] of the simulated new array.25 

Contrary to the previous method, this one does not need a constant interval between two successive points. Figure 4 shows the SHC algorithm.

FIG. 4.

Algorithm for the SHC method.

FIG. 4.

Algorithm for the SHC method.

Close modal

Figure 5 shows the second derivative function calculated using this method for the same case previously seen for the SDM method on the same I(V) probe characteristic obtained in argon microwave plasma working at a frequency of 2.45 GHz, a pressure of 2 Torr, and an incident power of 100 W. Measurements are performed using the following parameters: sampling time tech = 10−6 s, ns = 250 points, and frequency f = 242187 Hz (the Nyquist frequency is 0.5 MHz). Different uo values are used and compared to the results previously obtained using the SDM method for m = 20. No broadening of the curve and shift in the plasma potential are observed with increasing uo. It is worth noting that by increasing uo, we can filter the noise contained in the data array. uo has a filtering effect on the noise.

FIG. 5.

Second derivative function calculated using the SHC method in the case of the I(V) probe characteristic measured in argon microwave plasma working at a frequency of 2.45 GHz, a pressure of 2 Torr, and an incident power of 100 W. Comparison with results obtained using the SDM method (m = 20).

FIG. 5.

Second derivative function calculated using the SHC method in the case of the I(V) probe characteristic measured in argon microwave plasma working at a frequency of 2.45 GHz, a pressure of 2 Torr, and an incident power of 100 W. Comparison with results obtained using the SDM method (m = 20).

Close modal

Concerning the rest of the Taylor expansion depending on (uo)n, this one can always converge to 0 by changing the coordinate system to obtain uo < 1. With the numerical method, it is always possible to define a new coordinate system for the I(V) probe characteristic after the data acquisition, so the second derivative can be calculated using a uo value lower than 1. This means that the same results are obtained using uo > 1 and uo/α < 1 for the second derivative values. In the second case, the X-axis coordinates (V values) are divided by α and the Y-axis coordinates (second derivative values) are multiplied by α2. Thus, the condition on the rest of the Taylor expansion limnuo/αnεut0 is fulfilled.

The filtering effect on the noise obtained with different uo values can be seen in Fig. 6. The figure shows the spectral density measured in the frequency domain in the case of I(V) probe characteristic measured in the plasma “bubble.”32 The rate used to apply the bias voltage is 165.74 V/s (6.033 ms/V). This probe characteristic will be considered later in the text (paragraph V, application case C).

FIG. 6.

Spectral density measured on the second derivative obtained using the SHC method for different uo values. Calculations are performed using the I(V) probe characteristic performed in a plasma bubble using a grid bias voltage equal to V = −1 V.32 

FIG. 6.

Spectral density measured on the second derivative obtained using the SHC method for different uo values. Calculations are performed using the I(V) probe characteristic performed in a plasma bubble using a grid bias voltage equal to V = −1 V.32 

Close modal

It can be seen that the large values of the frequency oscillations disappear with increasing uo. For uo = 0.01, oscillations are observed up to about 24 kHz; for uo = 0.02, oscillations are observed up to about 13kHz; and for uo = 0.5, only very low frequency remains (<2.5 kHz).

Regarding both methods (SDM or SHC), the best way to determine smoothed data is to gradually increase the parameter values (m and uo corresponding to SDM and SHC methods, respectively) to prevent any distortion or shift in the second derivative function and then to integrate the second derivative function and compare the results to the experimental I(V) characteristic. Keeping in mind that the lower the step between two experimental points the more accurate the results are, it is worth noting that results obtained by one of the two previous methods (SDM or SHC) can be improved by fitting these results using the SDM method. In this case, the fitted second derivative value corresponds to parameter ao (see details previously given).

In the following part, the SHC method is used to determine the EEDF from I(V) probe characteristics measured in different plasma proposed in the literature. The effect of electron diffusion through the sheath formed around the probe collector is considered.

As previously shown, the Druyvesteyn equation is available under collisionless conditions only. In this case, calculations using the Langmuir law [Eq. (2)] fit the experimental I(V) probe characteristic. Otherwise, the Swift law [Eq. (7)] is required. In these conditions, the EEDF is obtained taking into account the diffusion parameter by means of 7 steps:

  1. The first and second derivative functions ieVV and 2ieVV2 of the experimental Ie(V) characteristic are calculated using one of the previous numerical methods, and the Druyvesteyn Eq. (6) is used to calculate f0εe, which corresponds to Ψεe=0 in Eq. (7).

  2. Ψεe is calculated using Eq. (11) for each V value of the data array.

  3. Using Eq. (10) and the previous values of Ψεe and f0εe, the new values of 2ieVV2 are calculated. Then steps (2) and (3) are repeated for all V values of the experimental data array Ie(V).

  4. The values of 2ieVV2 calculated are compared to the experimental values (exact values) previously calculated in step (1). The difference 2ieεeV2exp2ieεeV2EQ(10) is due to the effect of the electron diffusion through the sheath.

  5. The EEDF f0εe is incremented using the difference between the two second derivatives,
    εe1/2fεe=εe1/2f0εe+2ieεeV2exp2ieεeV2EQ(10).
    (12)
  6. The new values of 2ieVV2 are calculated using Eq. (10) and the new value of the EEDF previously calculated Eq. (12). Again, 2ieεeV2exp and 2ieεeV2EQ(10) values are compared, and the increase in the EEDF using Eq. (12), by changing f0εe to fεe, is still realized as long as the difference between both second derivative functions remains larger than an arbitrary limit. This happens generally after two or three iterations.

  7. Finally, integration of the last EEDF, corresponding to the best fit 2ieεeV2EQ(10) versus V using Eq. (7), is done to check the good agreement between the calculated I(V) probe characteristic and the experimental values.

We have tested the method on different I(V) probe characteristics obtained in different plasma reactors and different experimental conditions: microwave plasma, without and with confining magnetic field, plasma “bubble,” and rf plasma with confining magnetic field. All the I(V) probe characteristics that we use have been published in the literature, except the first one, which has been measured in our own reactor. The results are given and discussed in the following part.

Different points must be checked before data treatment to obtain reliable I(V) probe characteristics using Langmuir probes. One of the phenomena becomes important for the correct interpretation of the results: the influence of the collision of charged particles with neutrons in the space charged sheath surrounding the probe.33 Collision effects are important in plasma with a gas pressure larger than several 100 Pa. When the gas pressure is too high, a very large probe radius provides distortions of the second derivative, and the position of the plasma potential determined at the zero crossing point of the second derivative function can be changed. According to David et al.,33 the determination of the plasma potential using the second derivative can be done using a probe of radius rp and length Lp if the condition S=keke+1ke+lnLprp>1, where ke = λe/rp.33 

According to Godyak et al.,22 reliable measurements by means of Langmuir probes must have a good energy resolution. This one is defined by the energy interval Δεe between the zero crossing point of the second derivative function and the maximum of the EEDF. Information concerning low energy electrons can be lost if the energy resolution is too low (depending on the data acquisition system). The criterion for acceptable energy resolution is Δεe < Te. In practice, Δεe/Te < [0.3–0.5] corresponds to a good resolution.

Another important point concerning the probe measurements is the dynamic range, i.e., the ratio between the maximum value of the EEDF and the minimum value. According to Godyak et al.,22 a good dynamic range corresponds to 3–4 orders of magnitude. This corresponds to εemax ranging [7–9]Te. In the case of a Maxwellian electron energy distribution function, this corresponds to a ratio fεe0fεedεe, ranging from 4 × 10−4 to 3 × 10−3 assuming kTe = 1 eV. This condition seems excessive, and a dynamic range corresponding to 2–3 orders of magnitude could also be appropriate and more easily accessible.

Small energy resolution and a good dynamic range in measurements with control of probe contamination give confidence in data acquisition.22 

Figure 7 shows the electron current ie vs V = VpVapp [Vp is the plasma potential (here it is 14 V) and Vapp is the applied voltage], and measurements are performed in the expansion of a microwave argon plasma working at a frequency of 2.45 GHz, an incident power of 100 W, and a pressure of 2 Torr. The Langmuir probe is a cylindrical single probe (tungsten wire of 5 mm length and 0.1 mm diameter) located in the plasma expansion. The ion current collected at saturation is determined at large values of the negative biased voltage. The collected electron current is calculated by removing the ion current from the total current. In this case, collision has no effect on the plasma potential of the probe characteristic because the ratio S ranges from 2.1 to 7.2, considering λe ranges from 2 10−4 to 5 10−4 m. The energy resolution corresponds to Δεe/Te = 0.07.

FIG. 7.

Electron current and first and second derivative functions versus V = VpVapp in the case of argon microwave plasma working at a frequency of 2.45 GHz, an incident power of 100W, and a pressure of 2Torr.

FIG. 7.

Electron current and first and second derivative functions versus V = VpVapp in the case of argon microwave plasma working at a frequency of 2.45 GHz, an incident power of 100W, and a pressure of 2Torr.

Close modal

The figure also shows the first and second derivative functions calculated using the SHC method.

Figure 8 shows the change in Ψεe versus εe for different values of Vapp. The values are calculated using Eq. (11). It can be seen that the diffusion parameter decreases with increasing electron energy and converges to 0 at large electron energy values. The low energy electrons are mainly affected by the disturbance due to electron collision in the sheath.

FIG. 8.

Diffusion parameters versus (εe − eV) for different values of Vapp; measurements are performed under the same experimental values that have been mentioned in Fig. 7 .

FIG. 8.

Diffusion parameters versus (εe − eV) for different values of Vapp; measurements are performed under the same experimental values that have been mentioned in Fig. 7 .

Close modal

Under these conditions and using the process previously described, the real EEDF corresponding to the Swift law can be calculated. Figure 9 compares the EEDF calculated using the Druyvesteyn equation [Eq. (6)] and the corrected one calculated using [Eq. (10)]. The change between both the EEDFs is mainly observed at low electron energy. Figure 10 compares the experimental ie(V) probe characteristic obtained first with values calculated using the Swift law [Eq. (7)], taking into account the electron diffusion parameter, and second with values calculated using the Langmuir law [Eq. (2)], which corresponds to Ψεe=0. The results show that better agreement is obtained when the electron diffusion parameter is taken into account. Nevertheless, this effect is not important in the present case because the diffusion coefficient does not change drastically at low pressure. Under these conditions, the error bar corresponding to the calculated first and second derivative functions of I(V) can have an important effect on the result.

FIG. 9.

Comparison of the EEDF calculated using the Druyvesteyn Eq. (6) [i.e., assuming ψ(εe) = 0] and using Eq. (10).

FIG. 9.

Comparison of the EEDF calculated using the Druyvesteyn Eq. (6) [i.e., assuming ψ(εe) = 0] and using Eq. (10).

Close modal
FIG. 10.

Comparison of the I(V) experimental values with results of integration of the second derivative calculated using Eqs (2) and (7).

FIG. 10.

Comparison of the I(V) experimental values with results of integration of the second derivative calculated using Eqs (2) and (7).

Close modal

The method has also been tested in the case of magnetized plasma. Figure 11 gives the results obtained in magnetized plasma sustained in hydrogen. The probe characteristic has been measured by Cortazar et al.34 in a microwave plasma working at 2.45 GHz with a power equal to 1500 W and sustained in hydrogen at 0.19 Pa. The Langmuir probe is a cylindrical single probe (tungsten wire of 6 mm length and 0.5 mm diameter) located in the middle of the reactor along the Z-axis of the cylindrical plasma chamber. It is parallel to the magnetic field Bz, whose intensity is equal to 97 mT. A schematic of the experimental setup is shown in Ref. 34.

Figure 11 shows the corrected EEDF calculated taking into account the Swift law; using Eq. (10), it is compared to the EEDF calculated using the Druyvesteyn equation [Eq. (6)]. Figure 12 compares the experimental electron current values with those calculated using the Swift law (7) and using the Langmuir law (2). It can be seen that the best fit with experimental values is obtained after correction using the Swift law. The magnetic field acts on the electron motion and consequently on their radial diffusion coefficient component, which depends on the Larmor radius.35 The radial diffusion coefficient is lower when the probe is oriented along the magnetic field than perpendicularly and at low electron energy. Consequently, the electron energy within the sheath around the probe collector is changed because it depends on the applied voltage and also on the electron diffusion coefficient.

FIG. 11.

Comparison between the EEDF calculated using the Druyvesteyn equation [Eq. (6)] and corrected using equation [Eq. (10)] in the case of microwave plasma working at a frequency of 2.45 GHz, at an incident power equal to 1500 W, and sustained in hydrogen at a pressure of 0.19 Pa. The Langmuir probe is located in the middle of the reactor along the cylindrical reactor axis and is parallel to the magnetic field. The applied magnetic field intensity is equal to 97 mT.

FIG. 11.

Comparison between the EEDF calculated using the Druyvesteyn equation [Eq. (6)] and corrected using equation [Eq. (10)] in the case of microwave plasma working at a frequency of 2.45 GHz, at an incident power equal to 1500 W, and sustained in hydrogen at a pressure of 0.19 Pa. The Langmuir probe is located in the middle of the reactor along the cylindrical reactor axis and is parallel to the magnetic field. The applied magnetic field intensity is equal to 97 mT.

Close modal
FIG. 12.

Comparison between the experimental I(V) probe characteristic, the I(V) probe characteristic calculated using the Langmuir law (no electron diffusion through the sheath), and the probe characteristic calculated considering the Swift law according the method previously described in the text.

FIG. 12.

Comparison between the experimental I(V) probe characteristic, the I(V) probe characteristic calculated using the Langmuir law (no electron diffusion through the sheath), and the probe characteristic calculated considering the Swift law according the method previously described in the text.

Close modal

The method has also been tested in a plasma “bubble” using the results obtained by Stenzel and Urrutia.32 The Ie(V) probe characteristic is measured in a plasma produced by a filamentary cathode (Vdischarge = 30 V and Idischarge = 100 mA) in argon at 5 × 10−4 m Torr. A magnetic field of 2 × 10−3T is applied, and a spherical grid of coarse mesh (0.25 mm) around the plasma bubble is biased at 1 V. A schematic of the experimental setup is shown in Ref. 32. As done previously, Figure 13 we compare the EEDF calculated using Eq. (6) and the one corrected in the case of Eq. (10). Figure 14 compares the experimental Ie(V) probe characteristics with the one calculated first using the Langmuir law and second calculated using the Swift law. The magnetic field and the pressure gas are too low to have a significant effect on the Ie(V) probe characteristic, and because of numerical errors on the values of the first and second derivative due to the signal noise, the best agreement between experiment and theory is obtained using the Langmuir law.

FIG. 13.

Comparison of the EEDF calculated for a plasma “bubble” using data given by Stenzel and Urrutia32 using the Druyvesteyn Eq. (6) and corrected in the case of the Swift Eq. (10).

FIG. 13.

Comparison of the EEDF calculated for a plasma “bubble” using data given by Stenzel and Urrutia32 using the Druyvesteyn Eq. (6) and corrected in the case of the Swift Eq. (10).

Close modal
FIG. 14.

Comparison of the I(V) curve calculated using the Langmuir law [Eq. (2)] and using the Swift law [Eq. (7)] with experimental data.

FIG. 14.

Comparison of the I(V) curve calculated using the Langmuir law [Eq. (2)] and using the Swift law [Eq. (7)] with experimental data.

Close modal

The method has also been tested for measurements performed by means of a Langmuir probe (rp = 25 μm and lp = 3 mm) by Calderelli et al.,20 in an RF plasma reactor working at a frequency of 27.12 MHz, at a power of 1 kW, and in argon at pressure ranging from 0.51 × 10−3 to 1 × 10−3 Torr. In this article, the authors compare different methods to determine the second derivative function of the I(V) characteristic. We have tested the SHC method on the I(V) curve used in Ref. 20 and obtained the results at 200 W using a low magnetic field intensity equal to 0.03 T. Calculations have been performed to determine the second derivative function for different values of u0 ranging from 0.9 to 4 (u0 is the adjustable parameter of the SHC method). Results vs (VpVapp) are compared and shown in Fig. 15. It can be seen that the second derivative is strongly noisy for low values of u0, up to u0 = 3, and is well defined for larger values. It is defined only on one decade for u0 = 0.9 and over two decades for u0 = 4. For u0 larger than 3, only 1% of the signal intensity cannot be correctly treated. The plasma potential Vp is equal to 67.0, 66.5, and 65.4 V for u0 = 0.9, 3, and 4, respectively. Integrating the second derivative function vs V and comparing it to the experimental I(V) curve, it can be seen that the best fit is obtained for u0 = 4 (see Fig. 16).

FIG. 15.

Second derivative of the probe characteristic versus VpVapp, calculated using the SHC method with u0 = 0.9, 2, 3, and 4. I(V) probe characteristics have been measured by A. Calderelli et al.20 

FIG. 15.

Second derivative of the probe characteristic versus VpVapp, calculated using the SHC method with u0 = 0.9, 2, 3, and 4. I(V) probe characteristics have been measured by A. Calderelli et al.20 

Close modal
FIG. 16.

Comparison between the experimental I(V) probe characteristic measured by Calderelli et al.20 with the results obtained by integration of the second derivative measured using the SHC method for u0 = 0.9, 3, and 4.

FIG. 16.

Comparison between the experimental I(V) probe characteristic measured by Calderelli et al.20 with the results obtained by integration of the second derivative measured using the SHC method for u0 = 0.9, 3, and 4.

Close modal

Calderelli et al.20 tested the same method (named the ac superimposed method in Ref. 20) using u0 < 1. The results shown in Fig. 6(b) in Ref. 20 are strongly noisy because of the low value of u0. These are similar to ours obtained for u0 = 0.9. It is worth noting that the shoulder observed at (VpVapp) ranging from 20 to 25 eV, which is observed in Ref. 20 using the Savitzky–Golay method, the Gaussian filter, the b spline, and the Blackman window methods and in the present work (see Fig. 15 for u0 = 4), disappeared in Ref. 20 when using the “analog” method, probably because of a smoothing effect.

In Ref. 20, the authors measured a plasma potential value of Vp = 66.95 V using the same method, Vp = 67, 14 V using the Savitzky–Golay method, and Vp = 68.08 V using the “analog” method.

Figure 17 compares the experimental values of the Ie(V) probe characteristic first with the values calculated using the Langmuir law and second using the Swift law after correction of the EEDF using Eq. (10). It can be seen that in the present case, the two laws give similar results. The change due to the correction of the EEDF taking into account the effect of electron diffusion parameter is not important. This is due to the low pressure and low intensity value of the magnetic field (0.03 T) used to confine the plasma and due to the error bars on the electron current value and consequently on the first and second derivative functions.

FIG. 17.

Comparison of the I(V) experimental values measured by Calderelli et al.20 with the values calculated first using the Langmuir law [Eq. (2)] and second using the Swift law [Eq. (7)].

FIG. 17.

Comparison of the I(V) experimental values measured by Calderelli et al.20 with the values calculated first using the Langmuir law [Eq. (2)] and second using the Swift law [Eq. (7)].

Close modal

This article reports on numerical methods used to determine the derivative functions and the EEDF from the I(V) probe characteristics measured using a cylindrical Langmuir probe. The effect of electron diffusion within the sheath formed around the probe is taken into account. The electron diffusion coefficient depends on the plasma pressure (e-neutral collision) and also on the magnetic field, which can be used to confine the plasma (electron gyration in the field). In these cases, the electron current collected by the probe is not representative of the EEDF within the plasma bulk, and some corrections are necessary.

After a theoretical review of the electron current collected by a negatively biased probe (retardation conditions), numerical methods are used to determine the second derivative function of the I(V) probe characteristic and to correct the EEDF, taking into account the electron diffusion within the probe sheath. These methods are applied to various cases reported in the literature or to our own experiments carried out at the laboratory and corresponding to different plasma conditions (microwave, radio frequency, and direct-current plasmas with or without confining magnetic field). Results show that when a low magnetic field intensity (2 or 30 mT) is applied along an axis parallel to the probe, no greater change is observed on the I(V) probe characteristic. The change is of the order of the measurement errors due to the signal noise. However, for a larger magnetic field value (97 mT), a significant change is obtained compared to the I(V) probe measured without magnetic field, so the Druyvesteyn and Langmuir equations cannot be used because of the effect of the low electron diffusion through the sheath (Swift law). Results show that the noise observed on the curve when Ie is calculated using the Swift law depends strongly on the error due to the calculation of the ratio die/dVpd2ie/dVp2.

Regardless of the theory used to measure the EEDF (with or without correction due to the electron diffusion parameter), the electrostatic probes can be used to study the plasma only if the disturbed volume by the probe has a characteristic length (rd) much smaller than the electron energy relaxation length λε, which is the characteristic length of the undisturbed EEDF formation and depends on the electron collision cross sections with any other species present in the plasma.6,11,36 Hence, as long as rd ≪ λε, the EEDF measured by the probe corresponds to the undisturbed EEDF within the plasma. For larger rd values, the effect of collision processes within the sheath drastically changes the EEDF. Hence, measurements are not representative of the EEDF within the plasma bulk.

The probe disturbed length rd is of the order of the probe sheath radius (probe radius + sheath thickness) only in the case of a collisionless regime in the sheath. This condition is generally not fulfilled, especially because of ions of mean free path being much lower than the electron mean free path. Hence, for a cylindrical probe of length L and radius rp, we use rdrPlnLrP.

The authors would like to express their deep gratitude to O.D. Cortázar, A. Megía-Macías, R.L. Stenzel, J.M. Urrutia, and F. Filleul for their contributions in providing the I(V) data files recorded in different plasma reactors and for information and discussions concerning these experiments. The accessibility to these data has granted the opportunity to test these numerical methods under different plasma conditions.

The authors have no conflicts to disclose.

J. L. Jauberteau: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal). I. Jauberteau: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal).

Data are available if they are requested.

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