In this paper, we discuss a number of problems found in the literature related to experimental measurements of rf discharge electrical and plasma parameters with different electromagnetic probes. Incorrect evaluations of discharge power and the inaccurate measurement of basic plasma parameters with electrical (Langmuir), magnetic (B-dot), and microwave probes are among the troubling issues found in many recent publications on rf plasma. The purpose of this review is to show the origination of errors and ways to their mitigation based on the three-decade development of contemporary rf discharge diagnostics.

## I. INTRODUCTION

“The supreme judge of any physics theory is the experiment. Without experiments theoreticians turn sour”—Lev Landau

Experiment is the criterion of truth. Functionality and scaling found in experiments allow us to enhance the engineering process in the manufacturing of plasma-based equipment for many branches of today's technology. Experiments also allow us to verify theoretical and computational models and frequently stimulate modeling of the results found in experiments. Therefore, the credibility of data obtained in the experiment is of paramount importance for science and technology.

Unfortunately, not all experiments are a criterion of truth. In our opinion, many experiments with rf plasma from the last few decades suffer from negligence of some basic requirements for gas discharge diagnostics. Specifically, the negligence of vacuum hygienics, incorrect measurement of discharge power, and plasma parameters with different types of probes are the most frequent examples resulting in distorted and, sometimes, merely erroneous data.

An example of unsatisfactory experiment is ignorance of vacuum hygienics that requires degassing the discharge vessel and achieving the ultimate vacuum pressure being, at list, one-two orders of magnitude lower than the working gas pressure. Otherwise, in absence of the powerful working gas flow, due to gas desorption from discharge vessel caused by its heating and ion bombardment of the vessel and electrodes, the working gas becomes contaminated.

But the biggest concern *in rf discharge diagnostics is negligence of limitations of diagnostics used and incorrect measurement of discharge power.* The results of such negligence are significant errors in the inferred rf discharge parameters such as the discharge power P_{d}, plasma density n, electron temperature Т_{е}, rf electric field E, and current density J, reaching some time up to an order of magnitude.

In this Tutorial, we discuss sources of errors in experimental measurements of rf discharge electrical and plasma parameters. Among them are incorrect evaluation of discharge power (in Sec. II), errors in measurement of the basic plasma parameters and electron energy distribution function (EEDF) with Langmuir probes (LPs) (in Sec. III), and errors in measurements with magnetic and microwave (MW) probes (in Sec. IV). For all these cases, we show known resolutions of the problem and some examples of contemporary rf discharge diagnostics where these problems were resolved. The paper is finalized with concluding remarks.

## II. POWER MEASUREMENT IN RF DISCHARGE

Discharge power (together with driving frequency, plasma geometry, gas composition, and its pressure) is one of the discharge parameters that define the discharge state. In the majority of published experiments with rf plasma, rf discharge power is measured by an inline rf power meter placed between the rf power source (rf generator or amplifier) and the matching network (matcher). Measured this way, rf power (that is the transmitted power consumed from the rf power source P_{g}) usually is considered the discharge power and frequently used in simulations as the input parameter for calculation of plasma parameters and rates of plasma-chemical processes.

The distribution of the rf power flowing from rf power source to rf discharge is shown in Fig. 1. As seen in Fig. 1, the rf source transmitted power P_{g} = P_{f} − P_{r} goes to the matcher and then (in case of inductively coupled plasma, ICP) goes to the rf coupler (inductive coil, antenna) and only then to the rf discharge. Here, P_{f} and P_{r} are the forward and reflected powers measured with the inline power meter. Due to power loss in the matcher P_{m} and that in the coupler P_{c}, the real rf power delivered to rf discharge P_{d} is less (and sometimes, much less) and (due to plasma nonlinearity) is not proportional to the measured power P_{g}. The power loss in the matcher can reach up to 90% for capacitive coupled plasma (CCP) and considerably less (but not negligible) for ICP, while the power loss in the antenna (coupler coil) may be a considerable part of the total measured power P_{g}.

Therefore, the real discharge power absorbed by discharge is P_{d} = P_{g} − P_{m} − P_{c}, and for its realistic evaluation, the knowledge of the power loss in the matcher and coupler is needed.

The delivered to rf discharge power P_{d} splits to the power of electron heating P_{e} and that of ion acceleration P_{i} in the sheaths near the chamber wall and electrodes. A simple and reliable method for measurement of P_{m}, P_{c}, and P_{d} has been developed in our lab at Sylvania three decades ago and since has been used by few rf plasma groups over the world, but is ignored by the majority of published works on rf plasma sources.

ICPs operate on the principle of a conventional ac transformer, with the primary multi- or single-turn coil (named interchangeably in shop jargon—an antenna, an induction coil, a coupler coil, or a coupler) and a single-turn secondary comprising a toroidal rf current. In a conventional transformer, with a ferromagnetic core, the coupling coefficient k between the primary and the secondary windings is close to unity, k ≈ 1, and ratios of voltage V and current I of primary to secondary circuit are V_{1}/V_{2} ≈ I_{2}/I_{1} ≈ N_{1}/N_{2} ≈ N. In the case of ICP, this coupling is often essentially less than unity, and the voltage and current induced in the plasma cannot be simply defined by the turn ratio N, resulting in V_{2} ≡ V_{p} < V_{1}/N and I_{2} ≡ I_{p} < I_{1}N. Here, N is the number of turns in the ICP coupler coil; V_{p} and I_{p} are the plasma induced rf voltage and current, respectively. There, voltage and current are rms values.

Loose coupling between the inductor coil and the plasma and the coil loss limit power transfer efficiency η_{g} = P_{d}/P_{g} of conventional ICP operating in the ICP mode when P_{d} = P_{p}. Here, P_{p} is the discharge power deposited into the plasma, P_{g} is the power consumed from rf generator and transmitted to the matcher, P_{g} = P_{m} + P_{c} + P_{p}, and P_{c} is the power loss in the coupler coil. Therefore, the rf power deposited into plasma P_{p} is always less than the power measured at the inductor coil terminals and is smaller than P_{g}. Frequently, the power transfer efficiency is defined as η_{c} = P_{p}/(P_{g} − P_{m}) = (1 + P_{c}/P_{d})^{−1}accounting only for coupler loss, η_{c} > η_{g} = P_{d}/P_{g}. Note that a properly designed ICP matcher does not need an inductor, and due to the high Q-factor of the matcher capacitors, $ Q C\u226b Q L=\omega L / ( R 0+ R p)$, the loss in the capacitors is negligibly small compared to that in the antenna inductor. Here, R_{0} and R_{p} are the Ohmic rf resistance of the set on the chamber inductor coil (thus, accounting for hardware losses) not loaded with plasma and the plasma resistance transformed to the primary coil circuit.

The equivalent circuit of the coupler coil loaded with plasma is shown in Fig. 2(a), while two matching networks are shown in Figs. 2(b) and 2(c). The matcher in Fig. 2(b) made of variable rf air or vacuum capacitors has negligible power loss, since the loss in the coupler coil exceeds loss in such capacitors by nearly or more than two orders of magnitude. Losses in conventional L, T, or Pi matching networks having inductors should always be taken into account.

It has been shown on the basis of ICP transformer model^{1} (followed from the Maxwell equations) that *P _{m}, P_{c}, and P_{d} can be found from the lump sum parameters of the ICP primary circuit shown in Fig. 2(a) measured with and without plasma*. According to this model, the equivalent circuit of the coupler loaded with plasma can be represented by series connected inductance L, resistors R

_{0}and R

_{p}, corresponding to the coupler (induction coil) rf resistance R

_{0}, and that transformed to the primary circuit by plasma R

_{p}. The analytical expressions for L, R

_{0}, and R

_{p}can be found in Ref. 1. The inductance L is resultant of the antenna coil inductance without plasma L

_{0}affected by induced currents in plasma and in nearby conductors (like ICP chamber and rf shield). The resistance R

_{0}corresponds to antenna coil resistance affected by nearby conductors, but without plasma. The values of L

_{0}and R

_{0}without plasma can be measured with an L-Q meter or by measuring the inductor voltage V

_{0}(t) and current I

_{0}(t) without plasma resulting in R

_{0}= ωL

_{0}/Q

_{0 }= Re(V

_{0}/I

_{0}) measured near resonance condition. Here, ω is the angular rf frequency and Q

_{0}is the coupler Q-factor.

*It is essential that measurements of L*In this case, the measured values of L

_{0}, Q_{0}, V_{0}, and I_{0}were performed on already installed coupler on the chamber with all auxiliary metal and dielectric parts._{0}and R

_{0}automatically account for hardware effect on coupler impedance. Note, that L

_{0}< L′

_{0}and R

_{0}> R′

_{0}, where L′

_{0}and R′

_{0}are values measured in stay alone antenna inductor. Note that, due to complicated shape of a real experimental device, a reliable calculation of diamagnetic effect and rf losses in nearby supplemental ICP parts is problematic.

According to Ref. 1, for evaluation of P_{p}, P_{m}, and P_{c}, the coupler coil current I_{c} and the power delivered to the matcher P_{g} or to coupler P_{t} = P_{p} + P_{c} have to be measured with and without plasma. Then, the power absorbed by the plasma P_{p}, the power lost in the matcher P_{m} and in antenna coil P_{c} can be found.

In shown in Fig. 2(b), the antenna coupler is connected to the power source via tuning-matching network (matcher) that performs tuning function for compensation of coupler reactive impedance, i.e., to resonate it with the driving frequency ω and transforming function of its Ohmic impedance to the coaxial cable characteristic resistance of 50 Ω. This cable resistance provides optimal (without overloading) performance of the power source usually having its output resistance R_{g} essentially less than 50 Ω. This “mismatching” is purposely done to increase the power source efficiency; otherwise, with R_{g} = 50 Ω, the additional power loss equal to P_{g} would dissipate in the power source.

_{1}and C

_{2}is shown in Fig. 2(b). At certain values of C

_{1}and C

_{2}, one can resonate the coupler coil to the driving frequency and to match it to the generator. Precise tuning and matching can be achieved with multi-step variation of capacitance C

_{1}and C

_{2}. Resonating of the coupler inductance can be done by adjusting the driving frequency ω, while having C

_{1}and C

_{2}combination fixed near the optimal preadjusted values. The resonance and matching condition of this circuit are satisfied when

For evaluating the antenna coil resistance R_{0}, the coil current I_{0} has to be measured in the point a, or better, at the grounded end of the coil a′, thus reducing a possible capacitive coupling between the coil rf voltage and current sensor.

Due to cross interaction of the tuning and transforming adjustment of the matcher in Fig. 2(b), the tuning to resonance and matching to minimize reflected power require multi-step adjustments of variable capacitors C_{1} and C_{2}. The tuning and matching can be made practically independently with matching circuit shown in Fig. 2(c).^{2} Here, the tuning is mainly enabled by adjusting capacitor C = (Lω^{2})^{−1}, and transforming is mainly enabled with a step-down transformer and a variable capacitor C_{3}. For ideal matching at resonated ICP antenna, the output resistance of the transformer is equal to R_{0} + R_{p}. More details of such matching networks can be found in the original work^{2} and in Refs. 3–5.

_{g}= η

_{c}. By measuring without plasma (R

_{p}= 0) the inline power P

_{0}= R

_{0}I

_{0}

^{2}at point b and inductor current I

_{0}at the point a, and those with plasma when P = (R

_{0}+ R

_{p})I

^{2}, one can obtain

**V**(t) ⋅

**I**(t),

Another advantage of this method is there is no need for exact tuning and matching when the phase shift φ between the voltage and current waveforms is not too large (say, $cos \varphi \u22730.5$) and reflected power P_{r} is less than transmitted power P_{t}, $ P r \u2272 P t$. Otherwise, at large detuning and mismatching, when $cos\phi \u226a1$, the limited resolution of the measurement scope leads to a significant error in evaluation of the measured power. This error is due to near orthogonal vectors of rf voltage and current of the coupler coil. In typical ICP, the coupler loaded with plasma has a Q-factor $ Q \u227310$ and a power factor $cos\phi \u2248 Q \u2212 1\u226a1$. Then, the relative error in power measurement, *ΔP/P = ΔφQ*, where *Δφ* is the absolute phase error in radian. For example, at Q = 20 and *Δφ* = 1°^{ }= 0.0174 rad, the power measurement error ΔP/P can reach 35%.

The measurement with the matcher shown in Fig. 2(c) has been used in measuring the ICP discharge power.^{1,3–5} The matcher in Fig. 2(c) has a definite advantage compared to other known schemes; they are (a) deep decoupling between tuning and matching functions, (b) ability to separately measure η_{g} (in the point b) and η_{c} (in the point a), and (c) low matcher disturbance caused by the oscilloscope rf probe due to low matcher impedance in the points of measurement.

An example of ICP matcher circuit with incorporated voltage and current sensors is shown in Fig. 3.^{3} This combined matcher-sensor can operate at frequencies 13.56, 6.78, and 3.39 MHz and allows for measurement of input and output rf voltages and currents to find the corresponding powers. The matcher has a homemade wideband transmission-line transformer^{6} with very low (<1%) loss, able to operate without forced cooling at $ P g \u22720.7 kW$. Two calibrated capacitive voltage dividers and commercial wideband current transformer for current measurement were fed to a two-channel rf vector voltmeter having phase resolution of 0.1°.

The experimental chamber with antenna block is shown in Fig. 4. A five-turn antenna coil shown in Fig. 5 was shielded from plasma with a low loss Faraday shield (shown in Fig. 5) and enclosed in a metal cap having diameter equal to that of the chamber to prevent electromagnetic interference with nearby electronics. Due to Faraday shield, rf plasma potential was considerably lower than T_{e}/e that excluded capacitive discharge mode and allowed for reliable measurement with Langmuir and magnetic probes. Some results of rf measurement in this ICP are presented in Figs. 6 and 7 for discharge power P_{d} = P_{p} = 100 W and three frequencies in the range of argon pressure between 0.3 and 100 mTorr.

Contrary to ICP, the discharge current in CCP is closed through rf electrode sheaths, resulting in capacitive CCP impedance. Therefore, a compensating inductance has to be included in the CCP matcher. A typical CCP matcher [shown in Fig. 8(a)] consists of the inductor L (having its impedance **Z**_{L} = R_{0} + jωL) connected to capacitors C_{1} and C_{2} and (via a large dc blocking capacitor $ C d\u226b C 1+ C 2$) to CCP having its impedance **Z**_{d} = R_{p} − j(ωC_{sh})^{−1}. Here, R_{0} is the inductor's rf Ohmic resistance, C_{sh} is the resultant capacitance of the both rf sheaths, and R_{p} is the resultant of all discharge resistances associated with variety rf power dissipation processes (collisional and stochastic electron heating and ion acceleration in the rf sheaths). Compensation of capacitive impedance (i.e., resonating) and impedance transforming of the circuit in Fig. 8(a) are achieved by adjusting variable capacitors C_{1} and C_{2}, similar to the matching circuit in Fig. 2(b).

The induction coil current I is measured at point a, while the power P is measured at point b. Both P and I measurements have to be performed with and without plasma as described for ICP. The measurements can be done (1) with an inline power meter or (2) with measuring of voltage **V** and current **I**, as P = IV cos φ using vector voltmeter having high phase resolution, or (at the presence of considerable harmonic level in the voltage and/or current wave forms) by integration of **V**(t) ⋅ **I**(t) product, according to formula (3).

A detailed study of the electrical characteristics in symmetrical CCP at 13.56 MHz has been performed in Refs. 7–12, using a symmetric (push–pull) matcher with compensation of stray capacitance of rf electrodes to ground. rf plasma potential V_{p} in symmetrically driven CCP is much less than that in asymmetrically driven CCP. Therefore, the radial rf current proportional to V_{p} is negligible comparing to axial discharge current I. Also, experimental data obtained in symmetric CCPs are more convenient for comparison with 1D and 2D theory and simulation, while plasma probe diagnostics are easier and more reliable than those in asymmetric CCP.

The wave forms of the discharge voltage V(t), current I(t), and plasma rf potential V_{p}(t) are shown in Fig. 9, and discharge power P_{d} measured with different methods accounting for matcher power loss is shown in Fig. 10.^{7,8}

Note that measurement of rf plasma potential with a scope rf probe (SRP) is not a trivial task, since due to very low Langmuir probe (LP) capacitance to the plasma. The connection of SRP to LP, or to rf electrodes, can considerably detune and disbalance the matcher, thus to change the value of the discharge voltage and current. Therefore, it is advised to use rf probes with minimal input capacitance (1/10 and 1/100 probes) and do not disconnect rf probe during rf discharge tuning and operation.

A severe error can occur when one tries to find rf plasma potential by measurement of the rf potential with a floating Langmuir probe using the scope rf probe. Measured this way, the rf potential can be an order of magnitude (and more) less than the real plasma rf potential. The problem is that the input impedance of the rf probe $ Z 0\u2248 ( \omega C 0 ) \u2212 1\u226a Z pr\u2248 ( \omega C pr ) \u2212 1$, while for measurement with reasonable accuracy, the opposite inequality, $ Z 0\u226b Z pr$ or $ C 0\u226a C pr$, is a must. Here, C_{0} is the rf probe input capacitance and C_{pr} is the Langmuir probe capacitance to plasma defined by the probe sheath. In practice, C_{pr} is about 3 pF and 15 pF for 1/100 and 1/10 rf probes, while the input capacitance of a simple 1/1 scope rf probe is (50–150) pF that makes such probes unsuitable for rf measurement. Methods for accurate measurement of rf plasma potential have been considered in Refs. 12 and 13 and evaluation of the power lost in CCP matcher in Ref. 11.

An example of discharge power measured in argon CCP driven at 13.56 MHz is given in Fig. 11 together with independently measured power of ion acceleration P_{i} in rf electrode sheaths.^{9,10} It is known (see Fig. 1) that absorbed by CCP, discharge power P_{d} splits between the power absorbed by electrons P_{e} and that directly (not through electron temperature, like ambipolar field and floating potential) absorbed by ions P_{i} in the rf sheaths dc field due to rf field rectification, P_{d }= P_{e} + P_{i}. It has been shown^{9,10,14} that at relatively large discharge voltage and current, when the rf voltage drop across the rf sheaths prevails the voltage drop across the plasma, but is not large enough for CCP to transit to the γ-mode, the relations between found in experiment CCP parameters closely satisfy to scaling: $ P e\u221d n\u221d I\u221d V$, while $ P i\u221d n 2\u221d I 2\u221d V 2$. This gives an opportunity to separately evaluate P_{e} and P_{i} from the measured in experiment CCP power scaling as it shown in Fig. 11. Note that relative power loss in CCP matchers P_{m}/P_{d}, usually, considerably exceeds that in ICP matchers and ICP antennas, and can reach an order of magnitude.^{15,16} That means that the measurement of CCP power as one at the matcher input P_{g} = P_{m} + P_{d} can overestimate the real absorbed by the CCP power P_{d} up to order of magnitude.

The described above experiments with CCPs were performed three decades ago using symmetric matching network conceptually similar to that shown in Fig. 8(a). As we mentioned before, the network concept with a step-down rf transformer developed for ICP,^{2} shown in Fig. 2(c) and for CCP shown in Fig. 8(b), has many advantages compared to the traditional concept shown in Figs. 2(b) and 8(a). Similarly, to ICP, the matcher circuit shown in Fig. 8(b) is much more robust and convenient in application to CCP than one in Fig. 8(a). The measurement procedure with the circuit shown in Fig. 8(b) is the same as that shown in Fig. 2(c) for ICP. In both these matcher circuits,

Measurements of rf voltage and current are located in the matcher points having low (up to zero at exact resonance) phase shift between voltage and current and having low (reference to ground) impedance and rf voltage that minimizes the matcher disturbance by scope probes and effect of stray capacitance on the current sensor. It also does not need an expensive high voltage rf scope probe. Note that standard 1/100 scope probes are unsuitable for measurement of high rf voltage.

Contrary to the traditional matcher circuits shown in Figs. 2(b) and 8(a), the matcher circuits shown in Figs. 2(c) and 8(b) enable practically undependable tuning and transforming functions.

There is no need in these matchers for exact tuning and matching. For accurate power measurement, it is quite sufficient that cos φ > 0.5 and reflected power P

_{r}be less than forward power P_{f}. Here, $\phi $ is the phase shift between the voltage and current in the point of measurement. Usually, the phase shift $\phi $ in such measurement is much less than the phase shift ϕ in ICP antenna (or at CCP rf electrode). In case of ideal matching, at resonance, $\phi =0$, cos φ = 1, and the power at this point P = VI.

## III. LANGMUIR PROBE DIAGNOSTICS IN RF DISCHARGES

### A. General consideration

Any type of plasma probe diagnostics implies an inferring of local plasma parameters not distorted by the probe presence. The Langmuir probe is a rare case, where plasma disturbance by probe is accounted by theory, provided the basic requirements for classical Langmuir probe application are satisfied.^{13}

The electrical (Langmuir) probe introduced by Langmuir^{17} has been the major plasma diagnostics tool for a century. It was mostly by means of the electrical probe and the plasma spectroscopy that contemporary knowledge of the gas discharge plasmas has been achieved. Langmuir probes have also been extensively used for diagnostics in industrial plasma devices operated at relatively low gas pressure. Basics of the electrical probe theory and the technique covering various aspects of Langmuir probes [including measurement of the electron energy distribution function, EEDF, F(ɛ)] are given Refs. 13 and 17–29.

The commonly used probe techniques are the classical Langmuir method where plasma parameters are inferred from the electron part of the probe I/V characteristic, assortment of methods utilizing the ion part of the probe characteristic (IPPC), and measurements of the electron energy distribution function EEDF based on differentiation of the probe characteristic. These techniques differ in complexity as well as in the capability to reveal detailed and accurate information about plasma electrons.

Simplicity of the Langmuir probe concept creates perception of the probe measurement being a straightforward and predictable procedure. In fact, as mentioned by a renowned plasma diagnostics expert,^{26} “There is no plasma diagnostics method other than probe diagnostics where the danger of incorrect measurements and erroneous interpretation of results are so great.” This notion made a half century ago stays true (even in larger extend) today with more complex plasmas requiring sophisticated probe measurement techniques.

Interpretation of the probe measurements can be intricate and confusing. Therefore, it is not surprisingly that erroneous results and incorrect applications of Langmuir probes are common in the literature. The errors mainly arise from the poor design of probe experiment, incorrect probe construction, probe circuit design, probe contamination, and lack of awareness of possible error sources. Sometimes, the applicability and limitations of probe theories used for the evaluation of plasma parameters are ignored in spite of these issues have been frequently considered in periodical literature and probe reviews. At the same time, there has been significant progress in the probe experiment design effectively addressing the aforementioned problems, and professional probe measurement examples made by different author groups can be found in periodic literature, mainly in relation to gas discharge basic science, gas discharge lamps, and rf plasmas typical for plasma processing reactors.

A probe immersed in plasma inevitably causes a local plasma disturbance creating probe sheath and presheath that form the electron and ion currents to the probe; nevertheless, undistorted by the probe plasma parameters can be inferred from the probe I/V characteristic. Generally, a probe has to be small enough to neglect plasma perturbations beyond of those accounted by the probe theories.

Sometimes, to obtain a probe characteristic looking similarly to those shown in textbooks (ones with sharp transition to electron saturation current), beginners use too large probes causing considerable perturbation in the plasma ionization and electron energy balances and discharge current redistribution, resulting in erroneous plasma parameters obtained with such probes.

Here, we consider three major Langmuir probe diagnostics: (a) the classical Langmuir procedure where the plasma density and the electron temperature are inferred from the electron part of the probe characteristic, (b) the probe diagnostics based on the ion part of the probe characteristic according to various theories of orbital and radial ion motion around the probe, and (c) the probe diagnostics of electron energy distribution function (EEDF) according to the Druyvesteyn formulation. All these methods are considered for commonly used cylindrical probes, in frame of existing collisionless probe theories in the absence of the magnetic field. Methods of probe diagnostics in collisional, anisotropic, and magnetized plasmas are described in Refs. 19–24.

### B. Classical Langmuir probe

The main assumptions for the classical Langmuir probe theory and applicability is an isotropic Maxwellian EEDF and the absence of electron collisions in the area of plasma perturbation caused by the probe. Although, due to discharge current, the EEDF in gas discharge plasma is always anisotropic, and the anisotropy is usually small enough to affect Langmuir probe measurement even at rather low gas pressure.

It is known for a long time that in gas discharge plasma, electrons are not in equilibrium within their own ensemble, which results in a significant departure of the electron energy distribution function F(ɛ) from the equilibrium Maxwellian distribution. The main reason for this is the absence of a thermodynamic equilibrium between direct and reverse processes in electron kinetics of gas discharge plasma at low gas pressure. For example, electron–ion creation and gas excitation are due to the impact of fast electrons in the plasma volume, while electron and radiation loss are due to plasma particle and radiation escape to the wall. An EEDF depletion (due to inelastic collisions), in its high-energy part (at ɛ > ɛ*), or EEDF tail enhancement (due to selective electron heating) is the main reason for non-Maxwellian EEDF in gas discharge plasma. Here, ɛ* is the excitation energy.

Although, in general, Coulomb electron–electron collisions with frequency $ \nu ee\u221d n \epsilon \u2212 3 / 2$ tend to support a Maxwellian EEDF, the collision rate is usually too small, in order to bring the bulk of low energy electrons to an equilibrium with the high-energy electrons responsible for inelastic collisions. Therefore, the rates of electron inelastic collisions calculated from classical Langmuir probe data are, usually, in strong disagreement with those found by other independent methods.

When we have no idea what is a real EEDF, we assume it to be a Maxwellian (what else?). Indeed, in gas discharge plasma at low gas pressure (when Langmuir probe is applicable), the EEDF is never Maxwellian in the full range of the relevant electron specter covering elastic (ɛ < ɛ*) and inelastic (ɛ > ɛ*) energy range. Here, ɛ^{i} is the ionization energy and ɛ*is the excitation energy.

_{p}(V) of a small electrode (probe) immersed into plasma. In accordance with the Langmuir probe theory, the undistorted by probe local plasma parameters, the electron temperature T

_{e}and the plasma density n, can be inferred from the probe characteristic I

_{p}(V). The probe I/V characteristic is given by the sum of the electron and ion components,

_{p}is the probe current, I

_{e0}is the electron saturation current at the plasma potential (V = V

_{s}= 0), I

_{e0}= enS

_{p}(T

_{e}/2

*π*m)

^{1/2}, e and m are, respectively, the electron charge and mass, T

_{e}is the electron temperature in the units of energy (eV), S

_{p}is the probe collecting area, I

_{i}is the ion current on the probe, and V is the probe potential referenced to the plasma potential.

An exemplary probe characteristic I_{p}(V) in linear scale is shown in Fig. 12(a). The probe characteristic consists of three zones, the ion attracting part at V < V_{f}, the electron repelling part at V < V_{s }= 0, and the electron attracting or electron saturation part at V > V_{s}. Here, V_{f} is the floating probe potentials. Each part of the I_{p}(V) carries some information about the plasma parameters; however, reliability and accuracy of the plasma parameters inferred from different parts of the probe characteristic are substantially different.

In the classical Langmuir procedure, the plasma parameters (T_{e} and n) are found from the electron repelling part of the probe characteristic. It starts by extrapolating of the ion current I_{i}(V) from its high negative potential (where the probe electron current is negligible) to the unidentified yet plasma potential. Fortunately, the plasma parameters inferred with the Langmuir probe procedure are not sensitive to the accuracy of the ion current extrapolation (since at V > V_{f}, $ I i\u226a I e$) and linear extrapolation of I_{i}(V) is sufficient for obtaining the electron part of the probe characteristic I_{e}(V) = I_{p}(V) − I_{i}(V). On the other hand, at V < V_{f}, I_{e}(V) obtained from such ion current extrapolation comes out uncertain because the ion extrapolation impact becomes critical. This issue is discussed later in the paper.

_{e}(V)] shown on Fig. 12(b) is presented in many textbooks and review papers. The linear segment of the ln[I

_{e}(V)] defines the electron temperature T

_{e}, and the asymptotic crossing point defines the plasma potential, V = V

_{s }= 0 and the electron saturation current I

_{e0}= I

_{e}(V = V

_{s}). Then, the electron temperature T

_{e}and the plasma density n are found according to the well-known formulas

However, in practice, for a properly designed small probe, there is no clearly displayed saturation of the electron current at V > V_{s}, and a distinctive brake point on the ln[I_{e}(V)] curve; the transition from the electron repelling to the electron attracting area is rather smooth. An experimental probe characteristic looks like one shown in Fig. 12 only when the probe is large and/or plasma density is high, so that the probe sheath thickness s is very small compared to the probe radius a_{p}.

The desire to obtain textbook-like probe characteristics with a visible saturation and a sharp bend leads some beginners to use excessively large probes. Such probes create plasma disturbances, which are difficult to account for, and may cause probe characteristic distortion due to final probe circuit resistance resulting in significant inaccuracy in inferred plasma parameters.

Smooth transition of the probe current to saturation area creates uncertainty for pinpointing the plasma potential on the I_{e}(V) curve and, therefore, for the plasma density inferred from the probe characteristic. This is illustrated in Fig. 13 for the probe characteristics I_{p}(V) and their first derivatives I′_{p} (V) measured in the CCP at 13.56 MHz in the benchmark argon gas at 0.03 and 0.3 Torr.^{28}

As seen in Figs. 13(a) and 13(b), there is no distinct bend of the probe characteristics at the true plasma potential located at maximum of dI_{p}/dV. Presentation of the electron parts of the probe characteristics (EPPC) in a semi-log scale ln[I_{e}(V)] shown in Fig. 14 allows to find the plasma potential in accordance with the Langmuir procedure V_{sL} (at the point of the asymptotic crossing). The plasma potential for p = 0.03 Torr found this way is somewhat higher than the true plasma potential V_{s} found at d^{2}I_{p}(V)/dV^{2} = 0 and marked in Fig. 14 by a horizontal arrow. According to Fig. 14, at 0.03 Torr (and that is typical for plasmas having a Maxwellian distribution for low energy electrons), the true plasma potential V_{s} falls closer to the inflection point V_{si} rather than to the asymptotic crossing point V_{sL}.

A much larger error in determining the plasma potential occurs at p = 0.3 Torr. In this case, the value of V_{sL} is noticeably lower than the true plasma potential V_{s}, while at the inflection point V_{si} is even further from V_{s}. Such a wide gap in determining the plasma potentials lead to an order of magnitude error in finding the electron saturation currents I_{e0} and, correspondingly, in the calculated plasma density. Note that I_{p}(V) in the electron saturation area (V > V_{s}) is not an exponent and the slope of the line interpolating ln[I_{e}(V)] would depend on applied voltage span going beyond the plasma potential, which adds uncertainty to finding the plasma potential V_{sL} at the asymptotic crossing point.

The case of p = 0.03 Torr shown in Fig. 14 corresponds to a non-Maxwellian, two-temperature structure of the EPPC with the temperature of cold electrons T_{ec} = 0.73 eV and the temperature of hot electrons T_{eh} = 4.2 eV and the Langmuir procedure yields the plasma density n_{L} = 5.9 × 10^{9} cm^{−3}. The corresponding values from the electron energy probability function (EEPF), f(ɛ) = F(ɛ)ɛ^{−1/2} shown in Fig. 15, are T_{ec} = 0.50 eV and T_{eh} = 3.4 eV, with the effective electron temperature T_{eff} = 2/3 áeñ = 0.67 eV, and plasma density n = 4.4 × 10^{9} cm^{−3}. Here, áeñ is the average over EEDF mean electron energy. In this case, the error of the plasma density calculated from the EPPC following Langmuir procedure is 34%.

In case of 0.3 Torr, this error becomes even more significant. The differences between the true plasma potential V_{s} found at d^{2}I_{p}(V)/dV^{2} = 0 and those found at the intersection point V_{sL} and at the inflection point V_{si} are seen in Fig. 14. The electron saturation current I_{eo} found at V_{sL} and V_{si}, and calculated from these I_{eo} plasma densities, are lower than the true values of I_{e0}, by factors of 2.6 and 14.

The electron temperature found according to the Langmuir procedure from the linear part of ln[I_{e}(V)] for 0.3 Torr (shown in Fig. 14), T_{eL} = 1.37 eV, while the effective electron temperature found from the shown in Fig. 15 EEPF T_{eff} = 3.4 eV and T_{eh} = 0.71 eV. These numbers and those for 0.03 Torr demonstrate how misleading can be values of slow and fast electron temperatures found from the EPPC for non-Maxwellian plasmas by following the Langmuir routine.

The main reason for the so large disparity in inferred values of the n and T_{e} at 0.3 Torr is a non-Maxwellian, Druyvesteyn-like EEPF. This type of distribution is typical in dc and rf discharges in Ramsauer gases (Ar, Kr, Xe) at $ \omega 2\u226a \nu en 2$ and/or at relatively low plasma density when electron–electron collision frequency ν_{ee }∼ nT_{e}^{−3/2} is not high enough to Maxwellize the electron energy distribution even for low energy electrons. Here, ω is the angular rf frequency and ν_{en} is the electron-neutral collision frequency.

A non-Maxwellian EEDF in both the elastic (ɛ < ɛ*) and inelastic (ɛ > ɛ*) energy ranges is typical for gas discharge plasmas. The EEDF in the elastic energy range may get close to the Maxwellian distribution [dependent on ν_{en}(ɛ) function, ω/ν_{en} ratio, and the plasma density], while in the inelastic energy range (ɛ < ɛ*), mainly, due to inelastic (excitation and ionization) electron collisions with atoms), the EEDF inevitably diverts from the Maxwellian distribution of low energy electrons in the elastic energy range.

In the vicinity of the floating potential V_{f}, only hot electrons with energy $\epsilon > e V f \u2273 \epsilon \u2217$ reach the probe and the EPPC (inferred as I_{e }= I_{p} − I_{i}) gives extremely inaccurate presentation of the electron temperature (energy distribution) of the hot electrons T_{eh} due to uncertainty in the ion current approximation.

When the plasma density is high (roughly, at $ n \u2273 10 11 c m \u2212 3$), the EEDF in the elastic energy range gets close to the Maxwellian distribution, and the plasma parameters found from the Langmuir routine and the EEDF do not differ significantly. However, the electron distribution temperature of fast electrons T_{eh} may be lower than T_{eL} (due to inelastic processes and lack of e–e collisions) or higher than T_{eL} (due to stochastic electron heating, preferably of hot electrons, in low pressure CCP and ICP).

In conclusion of this part, let us state the main problems breaking the validity and thus limiting the application of the classic Langmuir probe diagnostics:

Non-Maxwellian EEDF;

Uncertainty in the plasma potential evaluation;

Arbitrariness in the ion current approximation.

These problems are intrinsic to the Langmuir procedure and they limit its applicability. They all can be avoided by differentiation of the probe characteristic obtaining the EEDF in a wide range of electron energy, giving more complete information about plasma electrons than any other diagnostic method.

It would be helpful to reiterate some important requirements and limitations for correct Langmuir probe experiment given in numerous books and reviews, but frequently neglected:

Small probe: a

_{p}⋅ ln(πl/4a_{p}), b, $ \lambda D\u226a \lambda e$ [a_{p}/λ_{e}< 10 is not enough, since, usually, l/a_{p}≈ 10^{2}and ln(πl/4a_{p}) ≈ 4]. Here, a_{p}and b are the probe tip and the adjacent probe holder radius, respectively, l is the probe tip length, λ_{D}is the electron Debye radius, and λ_{e}is the electron mean free path.Negligible probe circuit resistance, R

_{c}≤ 10^{−2}T_{e}/I_{e0}; otherwise, the probe sheath voltage would be less than the voltage applied to the probe resulting in distorted probe characteristic. Probe and chamber surface contamination, Ohmic resistance of the rf filter inductors, and insufficient ion current to the metal chamber to conduct electron probe current all contribute to the probe circuit resistance R_{c}.The absence of rf and/or low frequency time variable voltage V

_{pt}(t) across the probe sheath. Due to nonlinear impedance of the probe sheath, the presence of a large plasma time variable plasma potential V_{p}(t) leads to significant distortion of the measured probe characteristic. The criterion negligible for the time variable plasma potential effect is $ V pt( t) \u2272 T el / 3$.

These limitations have been analyzed in detail in Ref. 13 and will be considered later in connection with EEDF measurement where these limitations have a dramatic effect on the accuracy of EEDF measurement.

### C. Ion part of the probe characteristic

_{ei}is found from the IPPC in the vicinity of the floating potential, V

_{f}. For a single probe,

The plasma density n_{i} is found from the ion saturation current at a large negative potential according to one of the few ion current theories. Two basic collisionless theories are used today for inferring the plasma density from the IPPC.^{18–22} One of them, the radial motion theory (RMT), which accounts only for the radial ion motion to the probe, was proposed by Allen *et al*.^{30} and modified by Chen^{31} for the practical cylindrical probe. The RMT is also frequently referred to as the ABRC theory, named after its authors. Another one, the orbital motion theory (OMT), which accounts for the orbital ion motion, was proposed by Mott-Smith and Langmuir^{32} and later refined by Bernstein and Rabinowitz^{33} and Laframboise.^{34} The refined OMT is frequently referred to as the BRL theory.

Both RMT and OMT assume the absence of ion-neutral collisions in the orbital zone (r < R_{o}) and/or the probe sheath area (a_{p} < r < a_{p} + s); they also assume Maxwellian electron and ion energy distributions. There is no clear boundary between the applicability areas of each theory. Applicability of the OMT usually implies $ a p/ \lambda D\u226a1$, while the RMT applicability suggests a final ratio a_{p}/λ_{D} > 1. Here, a_{p} is the probe radius, λ_{D} is the electron Debye length, R_{o} is the radius of orbital zone, and s is the sheath width.

While some publications affirm a plausible agreement of the plasma densities n_{i} found from the IPPC with the densities found from the EPPC using Langmuir procedure and other independent diagnostics, special studies on this subject^{27,28,35,36} pointed toward substantial differences between them. These studies demonstrated vast difference of n_{i} values found through OMT and RMT compared to the n calculated from the EPPC using the Langmuir procedure and from the measured EEDF.

For example, in Ref. 35, the plasma density found in the positive column of the dc discharge in helium at 40 mTorr and the discharge current 0.2 A, from the EEDF, from the orbital theory for electrons, from the OMT and the RMT for ions are, respectively, related as 1.0, 0.85, 9.0, and 0.25 (36 times difference in n_{i} values!). Similar patterns of significant differences between the plasma densities were found for helium and nitrogen over the range of discharge currents and gas pressures.^{35}

Significant disagreements between plasma densities inferred from the OMT for ions and the EEPF were found in a CCP at argon pressure 0.3 and 0.03 Torr.^{28} The plasma density at p = 0.03 Torr found from the OMT was 2.5 times larger than from the EEPF and 3.3 times larger than for p = 0.3 Torr. On the other hand, calculation of the ion saturation currents according to the RMT^{30,31} using plasma densities found from EEPF at 0.03 Torr showed 6.2 times larger current than found in the experiment.^{28} This corresponds to underestimated plasma density inferred from the RMT similarly to that in Ref. 35. The ion saturation current calculated according to the RMT for 0.3 Torr showed 0.72 of the measured current, although, at p = 0.3 Torr, the probe sheath is strongly collisional $( \lambda i\u226a a p+ s< R 0)$ making both the RMT and OMT not applicable.

The ion current I_{sim} to a cylindrical probe in argon plasma at 1, 10, and 100 mTorr has been calculated by Iza and Lee^{36} using particle-in-cell simulation. These gas pressures correspond to near-collisionless, weakly/moderate, and highly collisional ion motion in the probe vicinity. The simulations assumed a Maxwellian EEDF and fixed plasma density n_{0} and electron temperature.

Plasma densities calculated according to the radial ABRC, orbital BRL, and Tichy *et al*.^{37} (accounting for ion collisions) theories using the I_{sim} values have shown dramatic discrepancy with the plasma density n_{0} set in the simulation. Thus, at 1, 10, and 100 mTorr, the BRL theory, gave n_{i}/n_{0} ≈ 3, 4, and 2, while the ABR theory gave n_{i}/n_{0} ≈ 0.3, 0.45, and 0.14, respectively. Smaller discrepancies were found at elevated gas pressures (p > 0.1 Torr) using the Tichy theory:^{37} n_{i}/n_{0} ≈ 2, 0.75, and 0.9.

Comparison of plasma densities found from the EEDF measurement, cutoff, and hairpin (HP) microwave probes, and from IPPC using OML theory has been performed in argon ICP between 7 and 22 mTorr.^{38} The corresponding ratios for plasma densities were found as 1.0, 1.1, 1.45, and about 3, which, again, show considerable difference between the plasma density found from the ion part of the probe characteristic and those found from three others electrical and microwave probe methods. Close values for plasma densities found from EEDF and cutoff probe [by measuring the plasma frequency ω_{pe} = e(4πn/m)^{1/2}] are not fortuitous, as both methods are based on theories with minimal and realistic assumptions. Indeed, both methods (within their applicability, i.e., when a_{p} ⋅ ln(πl/4a_{p}), b, λ_{D} ≪ λ_{e}, and ν^{2}_{en} ≪ ω^{2}_{pe}) *do not depend on the probe geometry* (if the probe surface is not concave) and particular shape of isotropic EEDF.^{38} The results of above studies are presented in Table I, where the plasma densities n obtained by different methods are related to those, n_{0}, found as appropriate integrals of the measured EEDFs.^{13}

Source . | Gas/pressure . | Langmuir EPPC . | Ion OMT . | Ion RMT . | Cutoff probe . | Hairpin probe . |
---|---|---|---|---|---|---|

Ref. 28 | Ar, 30 mTorr | 1.34 | 2.5 | 0.16 | — | — |

Ref. 28 | Ar, 0.3 Torr | 0.38/0.07 | 3.3 | 1.4 | — | — |

Ref. 35 | He, 40 mTorr | 0.85 | 9 | 0.25 | — | — |

Ref. 38 | Ar, 7–22 mTorr | — | 2.6–3.25 | — | 1.1 | 1.5 |

Ref. 36 | Ar, 1 mTorr | — | 3 | 0.3 | — | — |

Ref. 36 | Ar, 10 mTorr | — | 4 | 0.45 | — | — |

Ref. 36 | Ar, 0.1 mTorr | — | 2 | 0.14 | — | — |

Source . | Gas/pressure . | Langmuir EPPC . | Ion OMT . | Ion RMT . | Cutoff probe . | Hairpin probe . |
---|---|---|---|---|---|---|

Ref. 28 | Ar, 30 mTorr | 1.34 | 2.5 | 0.16 | — | — |

Ref. 28 | Ar, 0.3 Torr | 0.38/0.07 | 3.3 | 1.4 | — | — |

Ref. 35 | He, 40 mTorr | 0.85 | 9 | 0.25 | — | — |

Ref. 38 | Ar, 7–22 mTorr | — | 2.6–3.25 | — | 1.1 | 1.5 |

Ref. 36 | Ar, 1 mTorr | — | 3 | 0.3 | — | — |

Ref. 36 | Ar, 10 mTorr | — | 4 | 0.45 | — | — |

Ref. 36 | Ar, 0.1 mTorr | — | 2 | 0.14 | — | — |

Plasma densities ratios n/n_{0} calculated according to OMT and RMT to those found in Ref. 38 through particle-in-cell simulation are also included in this table. As seen from Table I, the plasma density values inferred from the IPPC using the OMT and RMT significantly diverge (often above an order of magnitude). Both IPPC and OMT data are consistently different (up to an order of magnitude) from the corresponding values obtained from EPPC and microwave probes. Ironically, some less discrepancy is observed for relatively high gas pressure when collisionless OMT and RMT are not applicable at all.

Table I shows that plasma densities found from the EPPC are in plausible agreement with those found with the cutoff probe and EEDF (except of argon CCP at p.0.3 Torr, when the EEDF is strongly non-Maxwellian in the elastic energy range). In the last case, using the true plasma potential found through the EPPC differentiation results in even larger error (by the factor of 0.07). Apparently, the classical Langmuir procedure is not applicable for such plasmas.

Large diversion between the plasma density obtained from the IPPC and the EPPC is attributed to many unrealistic assumptions put into IPPC theories.^{27} The most frequently mentioned reason for the discrepancy is ion-neutral collisions. Indeed, both the EPPC and the IPPC theories assume a collisionless motion of the charged particles in the area perturbed by the probe, in the space charged sheath and presheath, and in the orbital zone; in abbreviated form, this means $ \lambda i\u226b( a p+ s )$, R_{o}, where λ_{i} is the ion mean free path, s is the probe sheath thickness, and R_{o} is the radius of the orbital zone.

Let us roughly estimate what is the maximal gas pressure p_{max} at which the collisionless condition $ \lambda i\u226b R o$ is satisfied. For the cylindrical OMT, i.e., for the BRL theory, the radius of the orbital zone given in Ref. 39 is R_{o} = a_{p}(−eV/T_{i})^{1/2}; thus, the absence of ion collisions is equivalent to $ \lambda i\u226b a p ( eV / T i ) 1 / 2$, where T_{i} is the ion temperature. For the cylindrical RMT, i.e., for the ABR/Chen theory at a highly negative biased probe, the sheath thickness may be approximated as s ≈ 10λ_{D}, and the collisionless condition is equivalent to $ \lambda i\u226b10 \lambda D$.

Assuming here and later the sign “ $\u226b$” staying for the factor of 10, and selecting typical plasma parameters n = 1 × 10^{11} cm^{−3}, T_{e} = 3 eV, T_{i} = 0.03 eV, the probe radius a_{p} = 5 × 10^{−3} cm, the ion mean free path for argon λ_{i }= (p/300) cm, and the probe voltage V = −60 V, we obtain for applicability of OMT the maximal pressure p_{max} = 1.5 mTorr, and for RMT p_{max} = 7.4 mTorr. If n = 1 × 10^{10} cm^{−3}, RMT can be applied up to p_{max} = 2.3 mTorr. Such values of $ p max \u2272(1\u22122) mTorr$ correspond to the low end of the gas pressure range used for the plasma processing and basic research applications.

It would be interesting to evaluate and compare with IPPC applicability the value of p_{max} for applicability of the Langmuir and Druyvesteyn procedures, which, correspondingly, involve I_{e}(V) and d^{2}I_{e}(V)/dV^{2}. There, the effect of electron-neutral collisions leads to depletion of plasma density around the probe beyond of that accounted by the collisionless Langmuir probe theory and, thus, to reduction of the electron current to the probe at the probe potential close to the plasma potential V_{s} where $ s\u21920$. This plasma density depletion in the probe presheath is caused by decreasing of electron diffusion to the probe at growing gas pressure, see review^{13,24} and cited original works there.

The collisionless condition for applicability of the EPPC can be written as a_{p} ⋅ ln(πl/4a_{p}), b, λ_{D} ≪ λ_{e}. Taking the parameter values selected in the previous estimate and assuming for argon λ_{e} = (1.5 × 10^{−2}/p) cm, we find p_{max} = 740 mTorr. Thus, the gas pressure range where the IPPC can be applicable is two orders of magnitude smaller than that for the EPPC. This is another reason to be cautious when choosing the ion part of the probe characteristic for the plasma diagnostics.

There were many attempts to mend IPPC theories by accounting for ion collisions,^{37,39–41} but they have not resulted in any established reliable methods for practical applications. Obstacles in refining of IPPC methods, to our mind, come from the issues not addressed in IPPC original theories. Let us discuss these deficiencies.

As mentioned above, the electron temperature, specifically, the electron energy distribution EEDF is the key factor defining the shape of the IPPC. All IPPC theories make *a priori* assumption of a Maxwellian EEDF that is not the case for gas discharge plasmas. The high-energy tail of the EEPF usually has its distribution temperature T_{eh} = [dlnf(ɛ)/dɛ]^{−1} different from that for the main body of electrons T_{el}; that is true even for very dense plasmas, where the main body of thermal electrons in the elastic energy range (ɛ < ɛ*) is Maxwellian. The tail distribution temperature T_{eh} can be lower than T_{el} due to electron inelastic collisions at ɛ > ɛ*, or higher than T_{el}, due to selective heating of high-energy electrons occurring in the low pressure CCP^{29} and ICP^{42} in the regime of stochastic electron heating.

Transition from T_{el} to T_{eh} in the EEPF of gas discharge plasmas takes place around the transition energy ɛ_{tr} defined by processes of the EEPF tail depletion by electron cooling due to excitation, ionization, and electron escape to the wall, and enrichment by electron rf heating and electron energy redistribution via e–e collisions. Except for the very low gas pressure, the transition energy ɛ_{tr} is close to ɛ*, and this energy can be lower or higher than the energy corresponding to the probe floating potential V_{f}. Since in IPPC routine, the measured electron temperature T_{ei} is defined at the floating potential V_{f}, the T_{ei} value appears to be close to the bulk electron temperature T_{el} when ɛ_{tr} < |eV_{f}|, and T_{ei} is close to the tail electron temperature T_{el} when ɛ_{tr} > |eV_{f}|. Which case occurs in particular experiment is not known in advance and can be found only after EEDF measurement.

Another source of uncertainty is the difference between values of T_{el} and T_{eh} found from the probe characteristic and those found from the EEPF measurement. Therefore, one cannot be sure of using correct value of the electron temperature into the IPPC routine unless the real (rather than assumed) EEPF is known *a priori*.

The floating potential V_{f} (at which electron current to the probe equal to ion current) is defined by the EEDF tail corresponding to electrons having high enough energy to reach the floating probe (ɛ > |eV_{f}|). If T_{eh} < T_{el} (depleted tail of the EEPF), the value of |V_{f}| is lower, while if T_{eh} > T_{el} (enhanced tail of the EEPF), |V_{f}| is higher than the value predicted by using T_{el}.

^{43}that in the case of a non-Maxwellian EEDF, the plasma potential in the presheath, the ion current to the probe, and the Debye length are defined neither by T

_{el}, or T

_{eh}, nor by the electron effective temperature T

_{eff}= 2/3〈ɛ〉, but by the electron screening temperature T

_{es},

Screening temperature T_{es} is weighted by low energy electrons, and for EEPFs shown in Fig. 15, T_{es} is close to T_{el} < T_{eff} for bi-Maxwellian EEPFs, while T_{es} > T_{eff} for the Druyvesteyn-like distribution. In case of a Maxwellian EEDF, all electron temperatures are equal: T_{es} = T_{el} = T_{eh} = T_{eff }= T_{e}. The departure from a Maxwellian EEDF could be accounted for in IPPC theories by replacing T_{e} with T_{es}, but that cannot be done without knowledge of the EEDF.

More effects are unaccounted for in the existing IPPC theories and their evaluation may require additional assumptions about floating potential, ion temperature, and EEDF. Let us consider just a few of them. The ion ambipolar drift in the probe vicinity can make the probe ion current significantly different from its value calculated under no-drift assumption postulated in theories. In bounded gas discharge plasmas with $ T e\u226b T i$, the ion drift velocity v_{i} may considerably exceed the ion thermal velocity v_{iT}, in the discharge volume (but its center) since v_{iT} ≈ (T_{i}/M)^{1/2} < v_{i} < v_{s} = (T_{es}/M)^{1/2}. Although there are theories to account for the unidirectional ion motion in plasma,^{20} we are not aware of applying them to IPPC analyses of gas discharge plasmas. In any case, that would require further assumptions about the ambipolar field and ion drift velocity and its direction reference to the probe orientation.

The deformation of the probe sheath or the orbital zone at high negative probe potential, from cylindrical (at low potential) to ellipsoidal (and even spherical at high potential), may significantly distort the ion motion around the probe, thus affecting the ion current to the probe. Such a deformation takes place at high negative probe potential, low plasma density, and small ion temperature. According to the OMT theory, the ion current dependence is I_{i}(V)_{ }∼ V^{1/2} for a cylindrical orbital zone and I_{i}(V)_{ }∼ V for a spherical zone. The orbital zone around cylindrical probe approaches a spherical shape when its size becomes comparable to the probe half-length, l/2*.* For typical plasma parameters and the probe radius (n = 10^{11} cm^{−3}, T_{e} = 3 eV, T_{i} = 0.03 eV, a_{p} = 5 × 10^{−3} cm), the orbital zone radius, R_{0} = 1.7 mm is comparable with l/2 of a practical probe length, l = (3–6) mm. A similar effect would show up at the radial ion motion at low plasma density and large negative probe potential, when the probe sheath thickness s ≈ 10 λ_{D} is comparable to l/2*.* Not taking these deformations into account is a serious drawback in applying present IPPC theories to real experiments. We believe that together with non-Maxwellian EEDF and ion collisions, the above-mentioned effects non-accounted in IPPC theories are the main reason for notorious inaccuracy in the plasma parameters inferred with using ion current theories.

We do not accept “surprising validity of OML theory,”^{44} based on the fact that the ion current to the cylindrical probe can be well fitted to $ I i( V)\u221d V 1 / 2$ dependence by varying two unknown values: the plasma potential and the plasma density. This “validity” has not been proven with independent plasma density measurements in the same plasma. As Langmuir and many others have pointed out, the dependence $ I i( V)\u221d V 1 / 2$ is not a sufficient condition to ensure validity of OML theory.

Our understanding is that none of the existing theories for the ion part of the probe current can consistently yield accurate plasma parameters and make dependable tools for plasma diagnostics. An exception could be the RMT applied to a very dense plasma when $ \lambda D< a p\u226a \lambda i$, with a Maxwellian EEDF up to the electron energy exceeding the floating probe potential, but the very existence of such a condition has to be confirmed by EEDF measurements or established through a strong theoretical argument.

Applying old techniques, which use the EPPC and the IPPC for processing of the probe characteristic obtained in the non-equilibrium plasma, may lead to significant errors (up to an order of magnitude, see Table I) in calculating the plasma basic parameters. The existing classical theories for the EPPC and IPPC have many assumptions that are usually not held in real experiments. The absence of Maxwellian EEDF, sensitivity of IPPC to weak ion collisions, ion drift, and deformation of ion collection symmetry (cylindrical to spherical) at high negative probe voltage are main reasons for making these techniques unsuitable for contemporary probe diagnostics of highly non-equilibrium gas discharge plasma.

Therefore, we recommend the plasma parameter probe diagnostics by acquisition of the full probe characteristic followed by its differentiation (to get EEPF and then plasma parameters calculation as appropriate integrals of the measured EEDF) as the most informative and reliable way for contemporary Langmuir probe diagnostics.

### D. EEDF measurement at low gas pressure

The electron velocity distribution function (EVDF), **F**_{v}(**r**,**v**,t), is the most complete characteristic of plasma electrons. Here, **r** and **v** are coordinate and velocity vectors, respectively. Having the EVDF, one can find the plasma basic parameters, transport, and chemical reaction coefficients as corresponding integrals of the EVDF. Usually, except the case when the ratio of the electric field to the gas density, E/N is very high, electrons in the bounded gas discharge plasma exhibit relatively small anisotropy, and their energy distribution can be quite accurately represented by the sum of an isotropic EVDF, F_{v0}(r,v,t), and a small anisotropic part, **F**_{v1}(**r**,**v**,t). For practical purposes, since $ F v 0\u226b| F v 1|$, the plasma parameters, electron transport, and reaction coefficients are defined only by F_{v0}(**r**,v,t) or by the electron energy distribution function (EEDF) F_{ɛ}(**r**,ɛ,t).

^{45}has shown that in isotropic plasmas (within limitation of the Langmuir probe theory and at V < 0), the EEDF can be found by differentiation of the probe characteristic according to

As recently published in a prestigious magazine, the attempt to measure EEPF in gas discharge plasma using just first derivative assuming $ f(\epsilon )\u221d d I e / dV$ because of using a flat probe is erroneous. The expression $ f(\epsilon )\u221d d I e / dV$ is correct only for mono-directional EEPF but not for isotropic EEPF in gas discharge plasma.

As has been showed in Ref. 13, the contribution of the ion current to d^{2}I_{p}/dV^{2} is negligible, unless the dynamic range of d^{2}I_{p}/dV^{2} measurement exceeds 5–6 orders of magnitude that is an essentially more than realistic range of EEPF measurement. Due to deficiency of ion current theories, the subtraction of the ion current from probe characteristic prior to its differentiation results in an additional error in the measured EEPF.^{13}

Still, making accurate measurements and differentiation of the probe characteristic did not turn out to be a trivial task. It took over three decades after Druyvesteyn published his formula to demonstrate the first experimental results and two more decades to make EEPF measurement a standard procedure for diagnostics of non-equilibrium gas discharge plasmas.

The basics of EEDF measurements and various techniques for differentiation, smoothing, and processing of the probe I/V characteristics as well as the analysis of distortions in EEDF measurement and remedies for their mitigation can be found in Refs. 13, 23, and 27 and in the references given there. Here, we shortly discuss the sources of errors and ways of their mitigation, earlier reviewed in our works.^{13,23,27}

EEPF measurements yield meaningful results only when they contain accurate information about electrons, which reside in elastic energy range ɛ < ɛ* forming the main body of the electron part of the probe characteristic. The EEPF in the energy interval between ɛ = 0 and ɛ = 〈ɛ〉 = 3T_{e}/2 contains the majority of electrons in the EEPF; they are responsible for the electron density number and rates of electron transport processes. Distortion in this part of EEPF significantly affects the accuracy of corresponding data found from the measured EEPF.

The high-energy tail of the EEPF defines inelastic processes of electron collisions with atoms (such as excitation and ionization) and fast electron escape to the wall. Due to the limited dynamic resolution of EEDF, the measurement (that is the ratio of the maximal EEPF signal to its minimal one exceeding the noise) is 60–80 db (3–4 orders of magnitude) for a good experiment. Acceptable EEPF data with sufficient energy resolution require that the energy gap between the zero point and the peak of the second derivative of the probe characteristic d^{2}I_{p}/dV^{2} to not exceed (0.3–0.5)T_{e} and that the high-energy tail beyond the inelastic threshold is not masked by noise.

These requirements call for an EEPF measurement arrangement with combination of a high dynamic range (to resolve the EEPF tail) and high-energy resolution (to avoid loss of low energy electrons in the measured EEPF). Note that for plasmas with low electron temperature $( T e\u226a\epsilon \u2217 )$ that are typical for discharges at a relatively high pΛ product, or in afterglow plasma, the limited EEPF dynamic resolution prevents to obtain reliable EEPF measurement in the inelastic energy range (ɛ > ɛ*). Here, p is the gas pressure and Λ is the plasma characteristic size. For example, at a Maxwellian EEPF, the dynamic range of measurement 3–4 orders of magnitude corresponds to the maximal resolved electron energy ɛ_{max} = (6.9–9.2)T_{e} that prevents accurate EEPF measurements in their inelastic energy range (ɛ > ɛ*) for electron temperatures T_{e} < ɛ*/(6.9–9.2).

Many published EEPF data obtained in laboratory experiments and in commercial reactors (using homemade and commercial probe instruments) have heavily distorted (and sometimes absent) EEPFs in the range of low energy electrons at $ \epsilon \u2272 \u27e8 \epsilon \u27e9$ that comprise the majority of electron population. Therefore, calculated from such distorted EEPF plasma density n, electron temperature T_{e} and rates of plasma transport processes are in essential errors with real values. For this reason, some manufacturers of commercial probe stations, instead of using expression (9), recommend to find plasma density using electron saturation current according to the Langmuir procedure (valid only for Maxwellian EEPF). The basis for such an “ostrich” approach is that distorted and undistorted probe characteristics looks very similar, contrary to their second derivatives, where distortion is seen at a glance. In a well-done experiment with Maxwellian EEPF in the elastic energy range, the plasma density found from the probe characteristic via I_{e0} should be very close to that found by integration of the measured EEPF.

The EEPFs in many published measurements have very limited dynamic range (sometimes, less than 1–2 orders of magnitude) that is not enough to analyze the EEPF tail responsible for excitation and ionization processes. In this case, the measured EEPF has no advantage compared to probe characteristic.

In the measurement of the probe I/V characteristic, it is important to realize that due to error augmentation inherent to differentiation procedure, even small errors, which are tolerable (actually invisible) in classic Langmuir probe diagnostics, can result in enormous distortion in the inferred EEPF. Deterioration effects of probe contamination and changing of the probe work function, probe circuit resistance and electron collisions, and low- and high-frequency noise are practically invisible on the measured probe I/V characteristic, but are usually clearly seen in the resultant EEPF that allows, in a glance, to see the problem. Therefore, to obtain a reliable EEPF, special attention must be paid to the accuracy of the probe measurements themselves and to the correct application and to limitation requirements for the traditional Langmuir probe method.

Comprehensive studies of EEPFs in capacitive^{46} and inductive discharges^{25} that revealed some new electron kinetics effects^{29,46–48} have been performed in argon rf plasmas with a real-time display of the EEPF and the plasma basic parameters T_{e} and n. The examples of measurements of EEPFs in argon CCP^{29} and ICP^{25} in a wide range of gas pressure are shown in Figs. 16 and 17, respectively.

The real-time display (t = 0.1–1 s) of the probe characteristic and its derivative $ d 2 I p / d V 2\u221d f(\epsilon )$ used in the measurements shown in Figs. 16 and 17 is a convenient way to detect problems and try to address them at the moment of probe measurements. *The main problem in EEPF measurement is a large gap between zero and maximum of EEPF,* $ \delta \epsilon \u2273 T e $ *and flattering of EEPF at low electron energy that witnesses inadequate design of the probe experiment associated with the probe contamination, change of the probe work function, distortions by the low and rf frequency noise, and a stray impedance in the probe circuit, leading to EEPF distortion in its low energy range*.^{23,25,27}

The majority of EEPFs found in literature measured at a similar discharge condition differ from those in Figs. 16 and 17 by the inability to resolve $( \delta \epsilon \u2273 T e)$ a low energy peak typical in CCP and ICP at low gas pressure (seen in Figs. 16 and 17) and by a low dynamic range of measurement preventing to detect high-energy electrons for evaluation of rates of excitation, ionization, and electron escape to the wall.

EEPFs measured in argon ICPs at a similar condition with four different commercial probe instruments, a, b, c, and d^{27} are shown in Figs. 18–21. Presented in Fig. 18(a), EEPFs measured at low rf power demonstrate a Maxwellian EEPF in elastic energy range (ɛ < ɛ* = 12.7 eV); then, at ɛ > ɛ*, EEPFs fall faster than that at (ɛ < ɛ*). Such an EEPF shape is typical for argon plasma at relatively high plasma density when the electron–electron collision frequency $ \nu ee\u221d n \epsilon \u2212 3 / 2$ is high enough to Maxwellize the EEPF in the elastic energy range. Seen in Fig. 18(a), the energy interval having Maxwellian EEPF is growing with the discharge power P_{d}, and at P_{d} = 2 kW when plasma density reaches 10^{12} cm^{−3}, the EEPF is very close to Maxwellian distribution up to ɛ = 22 eV > ɛ^{i }= 15.7 eV.

Very different EEPFs in the same plasma were obtained with other instruments shown in Fig. 18(b), where EEPFs at a low electron energy are close to Maxwellian distributions only at low rf power, but evolve into Druyvesteyn-like distribution at elevated power, i.e., quite opposite to that seen in Fig. 18(a). Moreover, the EEPF in Fig. 18(b) demonstrates much less maximal resolved electron energies ɛ_{max} than those in Fig. 18(a). The EEDF Druyvesteynization with the increase in plasma density contradicts to gas discharge physics that requires the opposite trend seen in Fig. 8(a), since the electron–electron collisions that Maxwellize EEDF grow with the discharge power $( \nu ee\u221d n\u221d P d)$.

Nonetheless, the Druyvesteynization effect is a common case in the majority of published EEPFs measured in dense plasmas, as shown in Fig. 20 demonstrating quite a different shape of EEPF than that in Fig. 17 at a similar plasma condition: convex in Fig. 20 and concave in Fig. 17. The measured Druyvesteyn EEPF in high density helicon plasma even has been declared in PRL publication as a new feature of helicon plasma.

To legitimate the Druyvesteynization effect, the “parameterized Druyvesteyn distribution,” called ν-distribution, has been used in some papers for description of apparently distorted in their low energy part EEPFs. The essence of such an approximation is to fit distorted experimental EEPFs to the function $ g(\epsilon )\u221d exp ( \epsilon / \xi ) \nu $, which coincide with Maxwellian at ν = 1 and with Druyvesteyn at ν = 2 distribution. Such an EEPF presentation has no relevance and does not give any insight into plasma kinetics processes in gas discharge plasma.

It has been shown^{13,25} that the *Druyvesteynization effect is result of unaccounted stray probe circuit resistance* R_{0} *mainly due to probe and metal chamber contamination, chamber sheath, and rf filter resistances. The violation of basic requirement for Langmuir diagnostics:* [a_{p} ⋅ ln(πl/4a_{p}), b, $ \lambda D\u226a \lambda e$], *the presence of time variable voltage in the probe sheath, change in probe work function during due to probe heating by time dependent electron current, and magnetic field are other reasons for EEPF distortion at low electron energy* frequently found in the literature (see Refs. 13, 25, and 27 with analysis of EEPF distortions and remedies for their mitigation).

Note that, according to the classical Langmuir probe theory, the position of maximum ɛ_{m} and zero crossing of EEPF should coincide. However, in practice, there is always some gap δ_{ɛ} between them. *When δ _{ɛ} is not too large and the EEPF looks as a Maxwellian distribution at electron energy* ɛ > ɛ

_{m}corresponding to the maximum in f(ɛ), extrapolation of the measured EEPF from its Maxwellian part to the zero electron energy at d

^{2}I

_{p}/dV

^{2}= 0 may recover true EEPFs at low electron energy. This corrected EEPF should be used with the integration of the EEPF for evaluation of the effective electron temperature T

_{e}= 2/3 〈ɛ〉 and plasma density n, according to expression (9). Found in this way, the electron temperature occurs somewhat lower and the plasma density larger than those without extrapolation; yet, plasma density appears to be very close to that found from the electron saturation current I

_{e0}.

In support of this recommendation, note that if the EEPF is Maxwellian in a large electron energy interval, T_{el} < ɛ < ɛ*, it should be Maxwellian at ɛ < T_{e}, since the Maxwellization is stronger at lower electron energy, since ν_{ee} is proportional to ɛ^{−3/2}.

Another comparison of EEPF obtained with the different (than b and c) commercial probe instruments is shown in Fig. 21. Contrary to previous cases, the Druyvesteynization effect is absent and there is a plausible agreement between the EEPFs measured in the same argon ICP with instruments a and d, although the instrument d has some less energy resolution (larger δ_{ɛ}) and more noise reducing dynamic range of measured EEPF.

In the case shown in Fig. 18(b), the main reason for Druyvesteynization is the voltage drop δV across the probe circuit resistance, ΔV = I_{p}R_{0} resulting in a reduction of probe voltage across the probe sheath and thus in reduction of $ d 2 I p / d V 2\u221d f(\epsilon )$ at large probe current. The EEPF distortion is very sensitive to the probe circuit parameter $\rho = R 0/ R p min= e R 0 I e 0/ T e\u221d n / T e$, where I_{e0} is the electron saturation current to the probe. As seen in Figs. 18(b) and 19(b), the Druyvesteynization effect in these experiments is observed only for high plasma density $ n\u221d f (\epsilon )$, when f(ɛ) > 2 ⋅ 10^{10} (eV)^{−3/2} cm^{−3}.

Calculation^{25} of the circuit resistance effect for Maxwellian EEPF shown in Fig. 22 suggests that in order to have less than 3% error in EEPF measurement near the plasma potential, the requirement ρ ≤ 10^{−2} is mandatory. This is not easy to satisfy in experiment with high density plasma, but there is way to compensate ΔV = I_{p}R_{0} by properly designing the probe measurement arrangement^{13,25} as it has been done for results presented in Figs. 16, 17, 18(a), and 19(a).

The principle for compensation of ΔV due to final probe circuit resistance is schematically shown in Fig. 23. Here, the plasma floating potential (reference to ground) of a reference probe (RP) (placed in the vicinity of the measurement probe, MP) is fed to an operational amplifier that adds the voltage drop ΔV to the probe measuring circuit. In other words, the operational amplifier generates the additional voltage equal to δV and applies it to the measurement probe circuit, thus compensating the ΔV effect.

The circuit shown in Fig. 23 compensates for not only probe voltage drop due to final probe circuit resistance but also for any low frequency noise generated between the reference probe and the grounded chamber wall, provided that the operational amplifier has large enough frequency and phase bandwidth, for example, (0.1–1.0) MHz that is quite enough to compensate low frequency noise. Unfortunately, it is problematic to use such a circuit for probe rf compensation at rf discharge driving frequency, typically for 1356 MHz and its few harmonics.

The compensation of the final circuit resistance considered here is limited by the amplifier's maximal output voltage V_{max} > I_{p}R_{0} and by the ion current to the discharge chamber (I_{ich} > I_{p}). The second limitation is met at relatively high gas pressure when the ratio of plasma density ratio at the plasma boundary near the wall n_{ch} to that at the plasma center n_{0}, n_{ich}/n_{0} = h is too small. At the lowest gas pressure corresponding to the collisionless ion motion in the chamber, h ≈ 0.2–0.3. At elevated gas pressure p corresponding to the regime of variable ion mobility, $ h\u221d p \u2212 1 / 2$ and $ h\u221d p \u2212 1$ at constant ion mobility. For example, EEPF measurements in argon ICP^{25} (D = 20 cm and H = 10 cm) were limited to p = 0.3 Torr; at higher pressure, the ion current to metal chamber was not enough to conduct electron saturation current to the probe.

Note, that using a reference probe RP does not compensate for the voltage drop on the external part of the probe circuit resistance like resistance of rf filter and wires. However, this external resistance can be readily measured and accounted for in probe characteristic processing or compensated with an analog circuit known as a gyrator, which is a current-to-voltage converter with a negative input resistance equal to the external circuit resistance.^{29,46}

^{13,29,46}Nonetheless, this issue has not been properly addressed in many published experiments performed with rf plasmas.

*Simply putting some filter (tuned on the discharge driving frequency and its harmonics) into the probe circuit is not sufficient to remove the probe distortion associated with rf oscillation of the plasma potential*. Indeed, to design and make an adequate filter, one should have knowledge of the plasma rf potential V

_{prf}reference to ground and the probe impedance reference to plasma Z

_{pr}. It has been shown

^{49}that to avoid a noticeable distortion in EEDF measurement, the rms rf voltage across the probe sheath V

_{shrf}should be essentially less than the electron temperature expressed in volts, namely, V

_{shrf}< (0.3–0.5)T

_{e}/e. Therefore, the filter impedance Z

_{f}(for fundamental and harmonic frequencies) should satisfy the following relationship:

Relation (10) is practically impossible to satisfy when Z_{pr} is defined solely by the probe sheath impedance, usually corresponding to a reactance of the probe sheath capacitance of less than 1 pF. Therefore, a shunting electrode is placed in the probe vicinity or in the area that is rf equipotential with the rf plasma potential near the measuring probe. The shunting electrode is connected to the probe with a capacitor whose capacitance is much larger than the probe sheath capacitance but is small enough to not introduce a noticeable stray displacement current into the probe measurement circuit. With the shunting electrode, the impedance of the LP referenced to plasma can be reduced by 1–2 orders of magnitude being enough with adequate filter impedance to satisfy inequality (10) for fundamental and its harmonics having large rf potential. One can find a more detailed consideration and recommendation on rf compensation of LP in Ref. 13.

To minimize global and local plasma perturbance, a telescopic probe with a very thin (0.17 mm OD) adjacent probe holder [shown in Fig. 24(b)] has been used for ICPs.^{4,25} In the both cases, ICP was arranged to be free of capacitive coupling (with rf plasma potential essentially less than electron temperature, $ V prf\u226a T e / e$).

A multi-frequency rf compensated probe used for measurement in commercial plasma reactors is shown in Fig. 24(c). This probe consists of the measuring P_{1} and reference probe P_{2}. The second probe has a twofold purpose: (a) to compensate the probe circuit resistance and to remove low frequency noise, and (b) to increase rf coupling between the measurement probe and plasma.^{13} A similar double probe shown in Fig. 25 had been used for EEPF measurement in CCP operating at 13.56 MHz.^{46}

Let us consider in more detail the design and operation of probe measurement in a symmetric CCP at 13.56 MHz. A circuit for EEPF measurement is shown in Fig. 26.^{46} The reference (P_{2}) probe potential, passing an rf filter F and a voltage divider (10:1), is fed to the input of the probe driver OP_{2} with a gain of G = 10. The measuring probe span voltage is applied to the rf electrodes of CCP, whereas the measuring probe P_{1} is ground via a current-to-voltage converter OP_{1} having a negative input resistance equal to the dc resistance of the rf filter.

The parameters of the probe ramp voltage are shown in Fig. 26. To maintain a clean probe surface, the probe is biased with a high negative voltage 50 V referenced to its floating potential between the probe–voltage ramps. Note that using a very high voltage for probe cleaning leads to probe erosion and change of its collecting surface. The output voltage applied to rf electrodes consists of the input probe ramp voltage minus the voltage on the reference probe, thus compensating the effect of final probe circuit resistance and low frequency noise. It also automatically sets the center of probe voltage ramp to be equal to the floating potential of the reference probe. This feature prevents the measuring probe overloading with excessive negative or positive probe voltage leading to the probe destruction.

The plasma potential spectrum (needed for rf filter design) measured in this symmetric CCP is shown in Fig. 27 for asymmetric (with one electrode being grounded) and symmetric CCP drive. In the last case, the value of V_{prf} at fundamental frequency is two orders of magnitude less than that at asymmetric drive, while for the second harmonic, V_{prf} ≈ 7 V in both cases. At ideal symmetry, the fundamental and all odd harmonics should be absent.

For symmetrical drive setting, all, except the second harmonic, have their potential essentially lower than T_{e}/e; therefore, the filtering needs to be done only for the second harmonic, 27.12 MHz. The corresponding filter circuit (Fig. 28) assembled with sub-miniature inductors, resistors, a capacitor, and a mini connector to be inserted into probe cavity of the probe is shown in Fig. 25. The inductors were chosen to have their self-resonance frequency close to the second harmonic. One inductor resonating near 13.56 MHz is set just for case of non-ideal symmetry. The impedance of the self-resonating coil inductor Z_{c} = ωLQ = (ωC)^{−1}Q, where L is the coil inductance, C is the coil self-capacitance, and Q is the coil Q-factor. The inductor resonating with its self-capacitance provides a maximal filter resistance.

The calculation of probe shunting capacitance C_{ch} (consisting of the capacitance of the sheath around the reference probe in series with coupling capacitor of 30 pF) resulted in about 12–16 pF that was much larger than the measuring probe capacitance. The probe impedance reference to plasma at this capacitance for 27.12 MHz was no more than 500 Ω, which was 330 times less than the filter impedance at this frequency shown in Fig. 29. That means that the rf voltage at the second harmonic across the measuring probe sheath should be less than tens of mV.

In practice, due to the impossibility to find sub-miniature inductors with self-resonance exactly fitting to 13.56, 27.12, and 40.68 MHz, and due to the shift of the resonance frequency by stray capacitance and temperature change, the real impedance of the inserted probe filter may differ from that measured outside the probe construction. For this reason, the filter bandwidth Δω = ω/Q should be large enough (to accommodate frequency shift). The EEPFs measured with the above described probe arrangement is shown in Fig. 16.

In design of the probe measurement circuit, it is essential to minimize the measuring probe capacitive coupling (capacitance) to ground by placing the rf filter as close as possible to the probe, as it shown in Figs. 25 and 26. An attempt to use the probe (and its wiring) capacitance to resonate a choke placed outside the discharge chamber makes problematic to build a multi-frequency filter.

A special problem may occur at using L-C filters considered above in inductive plasmas when a strong rf magnetic field (that maintain the ICP) can induce a large rf voltage on multi-turn filter inductors. At such a scenario, instead of minimizing the probe sheath rf voltage, the filter inductors would generate an additional rf voltage there. A proper orientation and thorough rf shielding of the rf filter are required to avoid such an effect. For this reason, the building of ICP without antenna capacitive coupling providing negligible rf plasma potential is the best way to mitigate probe rf distortions. That approach has been used to obtain high-quality EEPFs (shown in Fig. 17) without any rf filter.

At the end of this section, let us demonstrate examples of high-quality EEPF measurement made in laboratory and industrial plasmas. The common features of these measurements are small energy difference δ_{ɛ} between the zero and maximum of EEPFs, well resolved low energy part of EEPFs, and large dynamic range of EEPF measurements that allow us to detect a high-energy EEDF tail.

EEPF measurements^{25} in argon ICP at p = 10 mTorr in a wide range of discharge power P_{d} is shown in Fig. 30. At low $ P d\u221d n$ in the elastic energy range (ɛ < ɛ*), one observes a low energy peak typical for ICP in the regime of anomalous skin effect. With an increase in the discharge power, due to the growing electron–electron collision frequency $ \nu ee\u221d n \epsilon \u2212 3 / 2$, the EEPF in the elastic energy range evolves into Maxwellian distribution.

EEPF Maxwellization with growing plasma density is a very common feature of gas discharge plasmas. With the increase in plasma density, the Maxwellization starts from electrons of lowest energy, when involving electrons with higher energy. As clearly seen in Fig. 30, at a large discharge power (P_{d} = 200 and 400 W), the EEPF is Maxwellian at ɛ < ɛ* and then starting at ɛ ≈ ɛ* declines due to electron inelastic collision and fast electron escape to the wall. Since the electron–electron collision frequency sharply drops with electron energy, the ν_{ee} collisions do not affect EEPF shape at ɛ > ɛ*, and as seen in Fig. 30, the electron distribution temperature $ T ed\u221d [ df ( \epsilon ) / d \epsilon ] \u2212 1$ (EEPF slopes) remain the same. At a considerably higher discharge power and plasma density, a Maxwellian EEPF can be observed in a wide range of electron energy covering the tail of fast electrons with energy ɛ > ɛ* and ɛ^{i}, as seen in Fig. 18(a) for P_{d} = 2 kW corresponding to plasma density n ≈ 10^{12} cm^{−3}.

EEPFs measured in different ICP arrangements with different chamber sizes, rf powers, rf frequencies, and probe instruments are shown in Fig. 31.^{50} They demonstrate the low energy peak at low gas pressure typical for ICP in a regime of anomalous skin effect.^{25} The low energy peak is absent in the majority of published EEPFs at similar conditions but was recovered in many EEPF simulations.

Note that the measurements in Fig. 31 were performed at 3, 20, and 50 mTorr, and P_{g} = 50 W that is the power consumed from rf generator, while the measurements in Fig. 30 were performed at 10 mTorr and the discharge power P_{d} between 6.25 and 400 W.

EEDF measurements in chemically active processing plasmas may present a serious challenge due to rapid probe and chamber wall contamination with reaction materials, and large plasma rf potential at driving frequency and its harmonics. The probe coating affecting probe measurements may occur in less than few milliseconds, and the probe cleaning before measurements may not be enough. But probes may be cleaned in intervals between successive fast data acquisitions needed to reduce noise and to avoid effects of probe work function change distorting the probe characteristic.^{23} The choice and effectiveness of a particular cleaning method depends on the actual deposition rate and may require a bit of experimentation while looking at the real-time EEPF display. Monitoring the probe characteristics and its derivatives makes possible immediate recognition of distortions.

Different techniques for mitigating problems mentioned above give opportunities for high-quality EEDF measurements in harsh environments of plasma processing reactors. Examples of such measurements are given Figs. 32 and 33. Figure 32 demonstrates EEPFs measured in a two-inductor ICP reactor operated with Ar, O_{2}, HBr, and HBr + O_{2} at 15 mTorr and 2 MHz.^{51} Similarly, to argon data, the measurements in reactive gases demonstrate the baseline quality (with small values of δ_{ɛ} < T_{e}/2 and dynamic range of EEDFs more than three orders of magnitude) in the reactive gas environment giving confidence in these data.

Similarly, trustworthy EEPFs are shown in Fig. 33, and they were measured in an ICP reactor operating in pure Ar gas and in Ar–H_{2} mixture.^{52} The high-energy resolution (δ_{ɛ} ≈ 0.1T_{e}) and dynamic range of 3–4 orders of magnitude are evidence of good EEPF measurement accuracy. Such data can be used with full confidence for plasma density, electron temperature, and reaction rate calculations as appropriated integrals of EEDF.

## IV. MAGNETIC AND MICROWAVE PROBE DIAGNOSTICS

Magnetic (B-dot) probes have been used for a long time in the measurement of time varying magnetic field in inductive and pulsed discharges.^{53} Magnetic probe, or B-dot probe, measurement is based on a simple idea of measurement of a rf voltage U induced on a small coil or loop by the time varying magnetic field. From magnetic probe measurement, the magnitude and the phase of the rf electric field and the current density distribution over the plasma volume can be determined.^{54,55} In turn, these data allow one to find the spatial distribution of the plasma conductivity and rf power deposition.^{56,57} Together with the plasma density and the EEDF, measured with a Langmuir probe, the magnetic probe diagnostics permits one to evaluate the effective electron collision frequency^{58,59} and the electron drift oscillatory velocity and, thereby, to estimate the role of the stochastic heating and the nonlinear electrodynamic effects (due to rf magnetic field) in ICP electron kinetics.^{61–63}

**E**,

**B**, and

**J**are vectors of the electric field, the magnetic induction, and the current density, respectively, and

*μ*

_{0}is the permeability of free space. The displacement current is neglected in (11) since it is on the order of ω

^{2}/ω

_{pe}

^{2}, which is always negligibly small in an ICP. A detailed theory of magnetic probe measurement and practical hints has been reviewed in Ref. 53.

_{θ}, one component of rf current J = J

_{θ}, and two components of the magnetic induction, radial B

_{r}and axial B

_{z}. Thus, from (11) it follows,

The integration constants in Eq. (12) are defined by the boundary conditions for a particular discharge configuration. For a typical ICP experiment with a metal chamber having radius R and length H, these boundary conditions are E_{θ} = 0 at r = R and at z = H. *Note that integration and differentiation of equation* (12) *have to be executed in both magnitude and phase domains*.

Markedly, there are two ways to infer the electric field: from the axial component of the magnetic field B_{z} or from the radial component B_{r}. This feature of a two-dimensional ICP is beneficial in B-dot diagnostics since it provides an additional option for measuring the azimuthal electric field.

_{r}and B

_{z}components). This can be done either with two differently oriented probes or with a single rotatable probe. In either case, there are two B-dot signals generated by B

_{r}and by B

_{z}components,

The phases in (13) ϕ = ϕ(r, z) are the same for the two signals and U_{0r,z }= SNωB_{r,z} is the voltage measured by B-dot probe with its axis parallel to one of magnetic field components. Here, S is the probe loop area and N is the number of winding turns.

_{0}= (

*μ*

_{0}SNω)

^{−1}[(dU

_{0}r/dz − dU

_{0z}/dr)

^{2}+ (U

_{0z}dϕ/dr − U

_{0r}dϕ/dz)

^{2})]

^{1/2}and tan α = /(U

_{0z}dϕ/dr − U

_{0r}dϕ/dz)/(dU

_{0}r/dz − dU

_{0z}/dr)

_{θ}, found from two different components of magnetic field E

_{θ}= E

_{0}sin (ωt + ϕ

_{E}): from B

_{z}and B

_{r}components,

*Having measured the magnitude and phase of* E_{θ} *and* J_{θ}*, one can readily find the local values of the plasma conductivity, effective electron collision frequency, absorbed power density, and electron oscillatory velocity*.^{53}

B-dot measurements are usually made with a small coil or loop encapsulated in a dielectric shell as shown in Fig. 34(a). Used for a half century, such B-dot probes prone to significant local plasma disturbance around the probe that distorts rf current path and corresponding magnetic field in the probe vicinity. As a result, the local rf electric field and current inferred from such measurement would be distorted.

The effect of plasma density depletion around a spherical probe^{64} and similar calculation for a cylindrical probe^{65} are shown in Figs. 35 and 36. The calculations^{64,65} were based on the fluid model accounting for ion inertia and nonlinear ion friction force for neutral plasma separated by a probe sheath from the probe as shown in Fig. 37.

The normalized plasma density n/n_{0} is presented as a function of the normalized distance x from the plasma–probe sheath interface (x = r/ρ) for different collisionality parameters β = ρ/l_{i} shown in Figs. 35 and 36. Here, n_{0} is the undisturbed plasma density, ρ = a + s is the radius of the plasma–sheath interface, and λ_{i} is the ion mean free path, where a is the probe radius and s is the sheath thickness. The plasma density depletion seen in Figs. 35 and 36 stretches on the length of few sheath radius ρ, and the depletion raises with the ion collisionality β = ρ/l_{i}. Also, the plasma depletion for a cylindrical probe is larger and stretches deeper into plasma than that for a spherical probe. Plasma density depletion for large distance from the cylindrical probe and the relative plasma density at the plasma–sheath interface h = n_{ρ}/n_{0}, forming electron and ion current to the probe, are shown in Fig. 36 for different collisionality.

*It is essential that the area of the plasma depletion caused by magnetic or microwave probe coincides with the plasma area of main probe interaction with plasma currents (for B-dot probe) and microwave field (for MW probe).* Therefore, evaluation of the plasma rf electric field and current density with the conventional encapsulated B-dot probe shown in Fig. 34(a) should be in error. For the same reason, evaluation of plasma density with some published microwave probes expected to be underestimated.

*Note that widely used for plasma diagnostics the expression for ion current to negatively biased probe, I _{i }= 0.6Sen_{0}*v

*$\beta =\rho / \lambda I\u226a1$. Here S = 2πρl is the probe collecting area, l is the probe length, ρ is the sheath radius, and v*

_{s}, as seen in Fig. 36, is valid only at_{s}is the ion sound speed (Bohm velocity).

It is worth to note that a recently proposed 3D magnetic probe consisting of three electrostatically shielded loops oriented perpendicularly to each other around the common center forming a small cage can be just partially transparent for plasma and rf current. The plasma density inside such a small cage would be considerably reduced and rf current would avoid this probe, resulting in erroneous data obtain with this 3D B-dot probe. Moreover, it is meaningless to use a 3D magnetic probe in axially symmetric ICP since the two components of magnetic field are enough to recover the third component using the Maxwell equations.

Note that in some works, a plausible agreement shown between plasma densities obtained from EEPF measured with Langmuir probe and from microwave probe is erroneous and is a consequence of underestimated plasma density obtained with Langmuir probe due to the effect of Druyvesteynization (considered in the Sec. III) and due to plasma density depletion around the microwave probe.

As any probe inserted in plasma, a magnetic probe (as well a microwave probe) disturbs plasma parameters around the probe. Contrary to the Langmuir–Druyvesteyn theory (within the area of its applicability), the theories of magnetic and microwave probes do not account for plasma disturbance around the probe.

Insertion of incapsulated B-dot probe obstructs and deviates the rf current path around the probe as shown in Fig. 38. To resolve the problem of the plasma density depletion and rf current obstruction, a transparent 2D magnetic probe shown in Fig. 34(b) has been proposed and used in Refs. 53–56.

B-dot measurement along the axis of argon ICP driven at 6.78 MHz in the experimental chamber shown in Fig. 39 performed with conventional encapsulated and current transparent B-dot probes are shown in Fig. 40. The magnitude and phase of the azimuthal rf electric field E_{θ}(z) in Fig. 40 were measured with two different transparent D-dot probes made of thin wire. The one shown in Fig. 34(b) is a 2D probe, where the spatial derivative dB_{z}/dr is measured by the probe single loop moving in the radial direction, while the axial derivative dB_{r}/dz is proportional to the difference of magnetic fluxes sensed by two parts of the eight-shaped double loop.

The second transparent probe is a single-turn large loop with its radius equal to the radial position r_{1} of the first probe. Provided the loop impedance is much lower than that of rf voltage measuring circuit, the induced on the loop end voltage V_{l }= 2πr_{1}E_{θ} and E_{θ} = V_{1}/2πr_{1}, which is in excellent agreement (in magnitude and phase) with E_{θ} obtained from the first probe measurement according to expression (12). Coincidence of these measurements proves the validity of the 2D transparent probe. Another proof of the transparent probe validity has been given in Ref. 66 by measuring the radial distribution of the axial current density J_{z}(r) in cylindrical ICP at a wide range of argon pressure and comparing its integral with independently measured discharge current.

Very different magnitude and phase of the azimuthal electric field axial distributions E_{θ}(z) seen in Fig. 40 was obtained with the conventional encapsulated and transparent B-dot probes. A similar huge disagreement for the current density distribution J_{θ}(z) (in magnitude and phase) obtained with conventional and transparent D-dot probes is seen in Fig. 40. From Fig. 40, the disagreement between measurements with transparent and encapsulated B-dot probes may reach up to an order of magnitude.

The encapsulated B-dot probe shows the presence of some rf current flowing in the probe location, although rf discharge current cannot penetrate into the encapsulated probe. The probe shows a current that does not exist, and there is no proof that this probe shows plasma current that would exist without probe insertion.

A variety of plasma kinetic and electrodynamic effects in ICP at low gas pressure have been revealed with the transparent B-dot probe: rf field phase bifurcation and non-monotonous rf field decay in plasma, collisionless and negative rf power absorption, second harmonic currents flowing normally to the main discharge current, and ponderomotive effects on plasma density distribution.^{54–60} Later, all these effects were recovered in ICP simulations, but many of them could not be found in our experiments with the conventional encapsulated probes on the same ICP arrangement.

A variety of potential problems must be addressed to obtain reliable data with B-dot measurements. The most common problem is associated with a relatively large rf plasma potential referenced to ground due to capacitive coupling between plasma and the induction coil. Because of stray capacitance between the plasma and the probe winding, there is common mode rf interference on the B-dot probe. Probe electrostatic screening, a center-taped probe winding, and a center-tapped balun transformer are common means of reducing common mode interference. The use of a grounded electrostatic screen between the rf coil and plasma can reduce the plasma rf potential and the common mode leakage to magnetic probe.

The B-dot should be loaded to transformer or transmission cable having impedance, which is much larger than the probe impedance; otherwise, the B-dot signal would be reduced and the rf magnetic field in the surrounding plasma may be disturbed by the probe rf current. Proper electrostatic and magnetic shielding of connecting wires and grounding of the center-tapped point are important issues in B-dot probe design. To avoid the jamming effect, magnetic and microwave probes should be rf compensated at ICP driving frequency.

Oscilloscopes that are commonly used in B-dot measurements are not the best instruments due to their poor sensitivity and phase resolution. Acceptable sensitivity and phase resolution can be achieved with phase-meters or vector-voltmeters having a high sensitivity and signal averaging features. It is evident that the B-dot probe should be as small as possible to minimize the convolution effect and attain the maximum spatial resolution.

The microwave (MW) probe diagnostics proposed in Refs. 67–69 are based on the resonance response in the absorption or reflection spectrum of some electrodynamic structure (probe) immersed into a plasma. Depending on the probe structure and particular resonance mode, the probe resonance frequency ω_{r} is some modeled function of the plasma frequency ω_{pe} (corresponding to the local plasma density) and the electron temperature T_{e}; ω_{r} = ω_{r}(ω_{pe}, s). It is believed that apart from the plasma density, the electron temperature and electron collision frequency ν_{en} could be inferred from analysis of the probe resonance curve, but there are many reasons for doubt in reliable measurement of electron temperature and electron collision frequency with existing microwave probes.

In spite of over half century history and attempts to refine of MW probes, they did not become a routine plasma diagnostic tool. Some problems with theoretical and experimental aspects of the plasma parameter measurement with microwave probes have been considered in Ref. 70 and shortly boil down to the following:

Unaccounted effect of plasma density depletion around the probe where the main interaction of MW field occurs. This leads to the underestimation of the plasma density inferred from MW probe measurement.

Uncertainty in the MW probe sheath capacitance that depends not only on distorted nearby plasma density yet is extremely sensitive to the shape of EEDF through the probe dc floating potential and Debye length, which is defined by the electron screening temperature T

_{es}[see formula (7)].Effect of uncompensated rf plasma potential leading to modulation of the sheath around grounded microwave or magnetic probe at rf plasma source. That results in the augmentation of the MW probe sheath that affects the inferred plasma density value and appearance of false widening of the resonance curve leading to overestimate the inferred value of the electron collision frequency.

A practical diagnostic probe should be based on the simple basic principles with minimum questionable assumptions and minimal plasma disturbance. In this respect, the cutoff^{68} and hairpin (HP)^{69} microwave probes seems to have more chances than others to become a real plasma diagnostic tool. Contrary to the rests of old and recently proposed MW probe modifications, the cutoff and hairpin probes are less sensitive to the probe sheath that strongly depends on *a priori* unknown EEDF. They are also less sensitive to the assumed sheath model and to the rf plasma potential when used in rf plasmas.

The sheath and plasma density depletion effects are expected to be negligible for cutoff and hairpin probes when $ a+ s=\rho \u226a c / | \epsilon p |$, where c is the distance between transmitting and receiving probe pins of the cutoff probe and between two hands of the hairpin probe, and ɛ_{p} is the relative plasma permittivity. For the hairpin probe operated at (ω > ω_{pe}), |ɛ_{p}|_{,} ≈ 1, while for the cutoff probe operated at ω = ω_{pe} and $ ( \nu eff / \omega pe ) 2\u226a1$, |ɛ_{p}| ≈ (ν_{eff}/ω_{pe})^{2}, where ω_{pe} is the plasma frequency and ν_{eff} is the effective electron collision frequency. Since |ɛ_{p}| for the cutoff probe is less than that for the hairpin probe, the inequality $\rho \u226a c / | e p$ is easy to satisfy for cutoff probes. Nonetheless, above inequality is not satisfied in some published works where the distance between probe pins c is comparable with the probe sheath radius ρ.

It is not a trivial task to evaluate |ɛ_{p}| at ω = ω_{pe} in essentially multi-dimension space between the probe pins, since due to Landau dumping at ω ≈ ω_{pe}, the plasma Q-factor Q_{p }= ω/n_{eff} can be considerably less than ω_{ep}/ν_{en}. Therefore, experimental and computational studies of the cutoff and hairpin probe geometry effects are desirable to find a quantitative criterion for accurate measurement with these probes.

## V. CONCLUDING REMARKS

Motivation to write this review was the author's desire to help an experimentalist to avoid typical drawbacks found in some experimental works in the field of low temperature plasma. The most common drawback of these papers (and their reviewers) was the ignorance of limitations for old classic diagnostics, and awareness of known contemporary diagnostics methods for correct measurement of discharge power, basic plasma parameters and electromagnetic fields, and rf current density in rf plasma using Langmuir, magnetic, and microwave probes. The analysis of errors origination and ways to their mitigation with examples of correct high-quality measurement performed by many different authors, hopefully, help experimentalists to attain mastership in contemporary rf discharge diagnostics.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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