Reliability of double probe measurements in nanodusty plasmas

Nonthermal plasmas are attractive sources for nanoparticles synthesis, however, their plasma properties are notoriously difficult to assess due to the chemically reactive environment and high nanoparticle concentrations. Here, we are using a floating double probe to measure the plasma properties of a nanoparticle-forming argon:silane plasma. We demonstrate good stability of current–voltage characteristics over several minutes of operation. However, unexpectedly larger electron temperatures are measured with increasing the silane mole fraction. To test the validity of these results, we developed a zero-dimensional global model to investigate the effect of the presence of nanoparticles on the plasma properties. Using this model, we show that increasing particle concentration leads to an increasing electronegativity of the plasma, causing an increase of the reduced electric field. However, this causes only a moderate increase in mean electron energy, in contrast to the much larger increase measured by the double probe. We argue that these large electron temperatures are based on the fact that a double probe measures an ‘apparent’ electron temperature, which is defined by the negative inverse slope of the logarithm of the electron energy probability function (EEPF) at an energy corresponding to the probe’s floating potential. As the silane mole fraction is increased, the plasma becomes more electronegative and the probe’s floating potential moves closer to the plasma potential. Combined with the strong non-Maxwellian EEPF, this leads to the large apparent electron temperatures obtained by the probe. Thus, the apparent electron temperatures measured with the double probe do not follow the trends in mean electron energy.


Introduction
Nanoparticles have many potential applications, including, light emission devices [1,2], biomedical imaging [3,4], electronic sensors [5], and in renewable energy [6,7]. Nonthermal plasmas (plasmas with higher electron than gas * Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. temperature) are particularly attractive sources for nanoparticle synthesis for materials that require high temperatures to achieve crystalline phases such as silicon [6,[8][9][10]. Plasma synthesis provides narrow and well-defined particle size distributions due to particle charging, reduces losses to the reactor walls, and provides selective particle heating to temperatures exceeding the gas temperature often by hundreds of Kelvin [11].
To better understand the mechanisms of nanoparticle formation in nonthermal plasmas, the need for experimentally validating plasma models increases [12,13]. Langmuir probes are widely used to assess basic plasma properties such as electron and ion densities and electron temperature [14]. However, in dusty plasmas, the use of traditional single Langmuir probes is complicated by the fact that chemical films or nanoparticles can be deposited on the probe surface [15]. Hence, in dusty plasmas often additional measures are required to protect Langmuir probes from contamination that distort the probe characteristics [16][17][18][19].
Double probes are significantly more resistant to contaminations than single probes because they essentially only analyze the ion current. Going back to the 1950s [20], double probes rely on two probes with the same surface area (symmetrical double probe) being placed in the plasma and current being measured by applying a voltage difference between the two probes. Using the appropriate theory, ion densities and electron temperatures can be derived from the currentvoltage characteristic [21]. For measurements in dusty plasmas excited with radiofrequency (RF), double probes have many advantages over traditional Langmuir probes: (i) since only the current in-between the two probes is measured, RF compensation is not necessary [22] and the electron temperature measurement is only slightly affected by the plasma potential oscillations [23]; (ii) since only the small ion current is drawn by the probes, the influence on the plasma is minimized (iii) every component that is part of the measurement is floating, thus, issues arising from improper grounding are circumvented [24]; (iv) most importantly, negatively charged dust particles are repelled by the negatively biased probe tips, preventing contamination of the probe [25]. However, since the floating double probe only measures the ion current, interpretation of the probe current-voltage (I-V) characteristic can be more challenging than for a traditional Langmuir probe, where the plasma parameters are usually obtained from the electron current instead.
First, to obtain the ion density from the double probe measurement, many researchers simply assume the probe current to be equal to the charge density at the sheath edge times the Bohm speed [22,24,26,27]. This, however, neglects the fact that ions with sufficient angular momentum can actually miss the probe tip, which is often small compared to the sheath thickness. This orbital motion of ions in the sheath around the probe is included in the original Langmuir probe theory developed by Mott-Smith and Langmuir, which is today often referred to as orbital motion limit (OML) [28]. The BRL theory works well at correctly include the potential variations inside the probe sheath [29,30]. Their theory, often referred to as BRL, works well at low pressure, where we may assume a collision less probe sheath. However, for pressures on the order of 1 Torr or 100 Pa, as commonly used for nanoparticle synthesis [8], collisional probe theory must be used to determine the ion density from the ion current of a double probe [15,[31][32][33][34].
In this paper, we apply the modified Talbot and Chou theory proposed by Tichý et al [33,35]. In 2004, this theory was validated using PIC simulations over a wide range of collisionality [36]. In the present work, we also provide some experimental validation of the theory, as will be discussed later on.
Second, the electron temperature is typically obtained from the slope of the I-V characteristic at zero voltage difference between the two probe tips [22,24,27,37]. However, rather than measuring the whole electron energy distribution function (EEDF), double probes only collect high energy electrons, i.e. the electrons whose energy is larger than the floating potential [24]. Thus, the shape of the EEDF has to be carefully considered when interpreting the results obtained from a double probe measurement [38]. Additionally, in RF discharges, stray capacitance can cause the probe potential to be lower than the floating potential, thus further increasing the complexity of the interpretation [35].
In this paper, ion densities and electron temperatures are measured using a double probe in an argon-silane (Ar+SiH 4 ) plasma for silicon nanoparticle synthesis. A surprisingly high electron temperature is found when increasing the silane mole fraction in the plasma. A simple global model of the plasma is set-up to explain these unexpected results.

Experimental setup
Double probe measurements were conducted in a lowpressure, laminar-flow, capacitively coupled plasma reactor that was used to synthesize silicon nanocrystals. A schematic of this system is shown in figure 1. A low-pressure discharge was generated by the application of 10 W radio frequency (13.56 MHz) signal to a set of two parallel copper ring electrodes, approximately 5 mm in width and 25 mm in diameter, with 20 mm spacing. The rings were placed around a pyrex glass tube with a diameter of 25 mm. Prior to the experiments, the reactor was pumped (Cit-Alcatel Annecy 2033) to a base pressure of about 30 mTorr. A silane-argon gas mixture with 95% argon and 5% silane was further mixed with pure argon to vary the silane concentration. The silane-argon mixture was inserted at flow rates between 0 and 12 sccm, while the argon flow rate remained constant at 30 sccm. The gas was then inserted into the discharge channel from the top. During the experiments, the chamber was continuously pumped, resulting in pressures between 1.4 Torr and 1.9 Torr (190 Pa and 250 Pa), depending on the gas flow rate. Silicon nanocrystals were generated in steady state and left the reactor through an orifice at the reactor exit. The double probe was fixed on top of a linear feedthrough mechanism, enabling us to measure plasma properties specially resolved.
The double probe tips were manufactured of 0.075 mm thick tungsten wire of which a length of 10 mm was exposed to the plasma. Around the wire, a 25 mm long glass capillary was placed, with an outer diameter of 0.2 mm. This capillary was supported by a 1.6 mm outer diameter two-hole ceramic tube. I-V characteristics were obtained by a source meter (Keithley SMU 2450) connected to the double probe.

Probe theory
From the I-V characteristic measured by the source meter, ion density and electron temperature were obtained. For the ion density, we applied the modified Talbot and Chou theory proposed by Tichý et al [39][40][41] as explained in detail by Bose et al [15]. The theory proposes a modification of the collected ion current, considering ion current increases due to loss of angular momentum inside the probe sheath as well as ion current reductions due to decreased ion mobility. The collisional theory yields two parameters which are multiplied with the collisionless ion current predicted by BRL. The collisionless BRL current was here calculated as described by Bose et al [15]. The ratio of electron to ion temperature used in the calculation was set to T e /T i = 100.
To obtain ion density and electron temperature, the measured probe current I as a function of applied probe voltage difference V d was fitted to the theoretical double probe I-V characteristics for a symmetrical double probe [42]: with the ion saturation current I i0 , the electron temperature T e and the Boltzmann constant k B . B describes the slop of the I-V characteristics in the ion current dominated region due to sheath expansion. Fitting equation (1) to the measurements directly yields T e from the slope around V d = 0.
Obtaining the ion density from the fit is more challenging, since the potentials of each of the probe tips (V pr ) relative to the plasma potential (V p ), V pr -V p , is unknown because we only know the potential difference between the probe tips. Since the ion current increases with the potential difference between probe tip and plasma potential, we cannot easily use the measured current magnitude to obtain the ion density. However, the collisional probe theory employed here predicts an almost linear relationship between probe tip potential and ion current. The slope of this linear relationship is an unambiguous function of the ion density. Thus, we may use the slope of the ion current increase with V d to obtain the ion density, instead of it is magnitude.
To this end, theoretical ion currents I ion were pre-calculated as a function of probe tip potential over a large range of ion densities. From this, a look-up table of the fitted slopes d(I ion )/d(V pr -V p ) and ion densities was created. The ion density was then determined by matching the measured slope B to the theoretical d(I ion )/d(V pr -V p ).
It should be noted that the electron temperature is used in the evaluation of the ion currents onto the probe so that any inaccuracy in the determination of T e will introduce an error in the calculation of the ion density. Our analysis shows that an electron temperature variation of about 1 eV will change the determined ion density by about 10%-20%.
Further uncertainty in the determined ion density arises from the depletion of electrons due to the presence of nanoparticles inside the plasma and the corresponding high degree of electronegativity, which was neglected in the evaluation of the probe measurements. This simplification would be valid in the case of collisionless ion currents and a thick probe sheath (OML regime) [43]. However, the modified Talbot and Chou theory employed here relies at least partly on the Bohm velocity with which ions enter the probe sheath, which gets modified in the presence of dust particles [44,45]. We expect this effect to introduce roughly 20% additional uncertainty to the evaluated ion density.

Probe diagnostic validation
Before performing the actual measurements, we will first attempt to validate the collisional probe theory by Tichy et al under our conditions. First, three different double probes with different tip diameters were manufactured to perform validation measurements: 0.075 mm, 0.125 mm and 0.38 mm. Then, electron temperature T e and ion density n i were measured with these different probe diameters under five different pressure conditions: 0.2, 0.5 1, 1.5, and 2 Torr. This validation experiment was performed in a different reactor, with a parallel plate configuration (GEC reference cell) [46]. From the measured electron temperature and ion density, two dimensionless numbers can be calculated [41]: (i) The Knudsen number for ions, K i , defined as: and (ii) the Debye number D λ : Here, R is the probe tip radius, λ i the mean free path for ion-neutral collisions, λ D the Debye length, e the elementary charge, k B the Boltzmann constant, n e the electron density and ε 0 the vacuum permittivity.
From this definition of K i and D λ , a linear relationship can be obtained by taking the logarithm: .
(2) From this equation, it becomes apparent that, when plotted on a log-log scale, we may expect a linear relationship between K i and D λ , with an offset depending on the gas pressure and electron density. This offset, however, should be the same for all three probe diameters. Thus, on a log-log plot of D λ over K i , the values for a given pressure but different probe diameters should lie on the same straight line, while the values for a different pressure should lie on a parallel line. This is precisely what is shown in figure 2: the figure shows the numbers K i and D λ for the different pressures, calculated from the measurements with three different probe diameters. As we expected, most of the points for a single pressure are indeed located on the same line, which validates the employed collisional probe theory for the interpretation of I-V under our conditions. Disagreement between theory and our measurements is only found for one point, measured using the thickest probe tip, at the lowest pressure. We propose that this discrepancy is caused by the ion current collected by the top surface of the probe, which should be larger for thick probe tips and low pressure, as is the case for this measurement. In this case, the ion current would be larger than predicted by the probe theory, resulting in an overestimation of the ion density leading to a too large Debye number, as is observed here. However, this discrepancy only occurs for the largest probe diameter investigated here. In the following, we will, thus, perform all measurements using the probe diameter of 0.075 mm, for which we consider the probe theory to be adequately validated by the experiment described above.
Having validated the probe theory, the possible contamination of the probe surface by nanoparticles or the growth of a resistive film is another aspect that requires attention. This is a common issue for Langmuir probes operated in dusty plasmas but should be less of a concern in the case of a floating double probe which is always negatively biased compared to the plasma and, thus, repelling the negatively charged nanoparticles. Contamination of the probe would show up as changes to the probe I-V characteristic over time. To examine whether such changes occur, the probe was placed near the grounded electrode of the reactor shown in figure 1 and repeated measurements were performed over time. Figure 3 shows the I-V curves obtained over the course of 10 min in a plasma with 0.71% of silane in a total 35 sccm gas flow. As the figure demonstrates, the I-V curves remain almost identical over the course of 10 min. Thus, we may assume that stable and reproducible measurements with the double probe are possible.

Results
Having ensured that valid and reproducible measurements with the double probe are possible under our conditions, measurements were performed under varying silane mole fractions.

Unexpectedly high electron temperature
The double probe measurements were performed in the center of the discharge, between powered and grounded electrode. Silane mole fractions were varied from 0 (pure Ar plasma), to 0.16%, 0.71% and 1.4% causing the pressure in the reactor to change between 1.4 Torr and 1.9 Torr. The discharge power was kept constant at 10 W. Figure 4 shows the obtained ion densities and electron temperatures as a function of silane mole fraction on a logarithmic scale. The measured ion densities decreased from values around 7 × 10 11 cm −3 at small silane mole fractions to about to 2 × 10 11 cm −3 at 1.4% silane mole fraction. This reduction is likely caused by increased ion losses to the large number of particles created at larger silane concentrations.
In contrast to this rather moderate reduction in ion density, the electron temperatures shows a dramatic increase with silane mole fraction.  Figure 4 shows an increase from T e = 1.7 eV for a pure argon plasma to as high as 12 eV for the 1.4% silane mole fraction. A temperature increase is certainly expected under these conditions: the increased silane mole fraction will lead to a larger number of nanoparticles in the plasma, which will in turn increase the losses of electrons. In order to still sustain the plasma despite the larger electron losses, the reduced electric field E/N will then increase and with it the electron temperature. As such, the trend is to be expected, qualitatively. An electron temperature of 12 eV, however, is more than we would expect under these conditions.
In order to decide whether this large apparent electron temperature is real or rather points towards a problem with our measurement methodology, a simple global model of the plasma is introduced.

Plasma model
To assess whether the measured large electron temperatures are reasonable, we set-up a simple global model of the dusty plasma containing electrons, Ar + ions and negatively charged nanoparticles.
Neglecting collisions, the electron and ion currents collected by a particle in the nanometer regime can be described by the OML probe theory [47,48]. The surface potential of a nanoparticle with radius R p is Φ k = Z k /4πε 0 R p , with the (usually negative) charge of the particle, Z k .
In equilibrium, and neglecting negative ions, the fluxes of ions and electrons to each nanoparticle must be in balance. Thus, from OML theory: (3) Here, v i = (k B T i /2πm i ) 1/2 is the thermal speed of ions, while n i,e , m i,e and T i,e are the density, mass and temperature of ions and electrons, respectively. Here, we treat the electrons kinetically, by integrating the electron energy probability function f p (EEPF) over the electron total energy ε for those electrons fast enough to reach the particle surface. This kinetic treatment is necessary as the EEPF cannot be expected to be in equilibrium under the present conditions, as explained in more detail in section 4.3.
Equation (3) is valid if the EEDF is nonlocal in the vicinity of each nanoparticle, i.e. the EEDF is a function of the electrons' total (kinetic + potential) energy [49][50][51].
The assumption of nonlocal behavior is valid when the energy relaxation length λ er , is larger than the sheath dimension around the particles. The energy relaxation length λ er ≈ (λ el λ inel ) 1 2 is a function of elastic (λ el ) and inelastic (λ inel ) electron mean free path, while the sheath thickness is a few Debye lengths. Even at the elevated pressure in our experiment, the energy relaxation length was estimated to be 800 µm for electrons with an energy of 15 eV. Since the Debye length is only around 20 µm, the assumption of a nonlocal EEDF should hold true.
Additionally, equation (3) is only valid for negatively charged particles, Z k < 0. However, the assumption of negative charge is justified because of the large difference in mass and temperature between electrons and ions. A positive particle charge can only occur for an unreasonably severe electronegativity δ as low as A further limitation of equation (3) is that OML requires a collisionless probe sheath, a condition that is not completely fulfilled for the ions under our experimental conditions. However, introducing collisional ion currents [52,53] would make the model considerably more complex, without qualitative changes to the results.
The quasineutrality condition of the plasma provides us with a second equation for our model: where n p is the number density of nanoparticles. Again, the contribution of negative ions was neglected, which could lead to a slightly lower electron density than predicted by our model [54]. The third equation used here is the ion balance, accounting for losses by ambipolar diffusion and to the particles, as well as ion production due to ionization by electron impact: Here S = 4πR p 2 is the surface area of particles and ν iz (E/N) is the total ionization frequency calculated by BOLSIG+ [55][56][57][58][59] as a function of the reduced electric field E/N. Λ = R/2.4 is the characteristic length scale of the reactor, where R = 11 mm is the radius of the reactor tube. D + a is the ambipolar diffusion coefficient, derived from the flux balance of ions and electrons: where µ i ,µ e ,D i ,D e are mobilities and diffusion coefficients of ions and electrons and E is the ambipolar electric field strength.
Because of their large mass, dust particles are relatively fixed in the plasma and only ions and electrons are considered in equation (6). From the flux balance, we derive the ambipolar diffusion coefficient, D + a as: Combining equations (4) and (5), we get: which can be solved numerically together with equation (3), as will now be explained in detail.

Model solution
The model is evaluated assuming a constant pressure and ion density of 2 Torr and n i = 5 × 10 11 cm −3 , respectively. Particles are assumed to be 3 nm in diameter [8]. Model results with 1 nm and 2 nm particle diameters were obtained, but are not discussed here, since no qualitative differences were observed. Results from the model are obtained as a function of the electronegativity δ = n e /n i , for values between δ = 0.001 and δ = 1. This corresponds to the conditions of our experiment, where the silane mole fraction was varied: without any silane in the reactor, the plasma is electropositive δ = 1, whereas a strong electronegativity should be expected for the highest investigated mole fraction of 1.4%.
The ion diffusion coefficient D i needed in equation (7) was calculated as where the frequency of ion-neutral collisions ν cx is assumed to be mostly caused by charge-exchange with a cross-section of σ cx = 1.5 × 10 −18 m 2 [60]. For the ion and gas temperatures, T i = T g = 300 K is assumed throughout this work. To solve this model, 500 BOLSIG+ calculations were performed, for E/N between 1 Td and 500 Td. The resulting EEPFs, transport coefficients and ionization rates were saved. Then, equations (3) and (8) were solved together by varying E/N until equation (3) was fulfilled. The gap between two BOLSIG+ calculation results was linearly interpolated, to allow for a smooth variation of E/N, needed by the numerical solver (scipy. optimize. minimize [61]).
From the obtained value of E/N, all other model parameters can then be calculated. Figure 5 shows the density (■, left axis) and average charge of particles ( , right axis) from the model, as a function of electronegativity. The particle charge is always negative, due to the The reason for this trend can be found in the flux balance onto the particles, as described by equation (3): larger values of δ imply a higher electron density. Thus, the electron flux onto each particle is larger and the particle assumes a more negative potential to limit this flux. The more negative potential naturally means a larger negative particle charge.
It should be noted that ion-neutral collisions are neglected in the model. Under our conditions, these collisions would increase the ion current towards the particles, thus driving the particle charge closer to zero [52,53]. Figure 5 also shows the nanoparticle density n p , calculated from the quasineutrality condition of the plasma, equation (3). Because δ = n e /n i is used as an independent parameter, n p corresponds to the particle density that is required to achieve a given electronegativity. The nanoparticle density decreases with increasing δ. This is not surprising, since in an electropositive plasma (δ = 1) the density of negatively charged nanoparticles must be zero, by definition. For a strong electronegativity (small δ), n p can exceed the ion density by an order of magnitude. The density of ions and nanoparticles becomes equal around δ = 0.012. If all the added silane was converted to 3 nm diameter silicon particles, the particle density would be 1.5 × 10 12 cm −3 for our 1.4% silane mole fraction, corresponding to δ = 0.004. Figure 6 shows the ion loss rates to the nanoparticles (■) and by ambipolar diffusion (□). The ion loss due to diffusion remains relatively constant for all investigated electronegativities. In contrast, the ion loss rate to particles shows a strong decrease with increasing δ. For most values of δ, the ion loss rate to particles is more than a factor of two larger than ion loss rate by diffusion. Only for plasmas close to electropositivity (δ = 1), diffusion losses dominate since the nanoparticle density is low (compare figure 5). If ion-neutral collisions were included in our model, particle losses would be even larger.  Figure 6 also shows the mean electron energy u mean (right axis, ) obtained from the model. In order to compensate for the elevated ion and electron loss rates at small δ, the average electron energy needs to increase. Thus, u mean decreases from around 7.6 eV at δ = 0.001 to 5.8 eV at δ = 1. This corresponds to an effective electron temperature decrease from 5.1 eV to 3.9 eV, calculated as T eff = 2/3 u mean .
This limited effective electron temperature variation results from the relatively high-pressure condition (2 Torr) in our experiment. Higher pressures reduce electron losses by diffusion, so a lower ionization rate is sufficient to sustain the plasma. Additionally, even the moderate effective electron temperature increase predicted by our model can elevate the ionization rate by a few orders of magnitude, enough to compensate for the electron loss.
However, this limited electron temperature variation obtained from the model is in strong disagreement with the electron temperatures measured by the double probe (compare figure 4). There, we observed a strong temperature increase from T e = 1.7 eV for a pure argon plasma to 12 eV for the 1.4% silane mole fraction. Since the increase in silane mole fraction should lead to a stronger electronegativity (smaller δ), the trend observed in the experiment is only qualitatively reproduced by our model, but at a much weaker temperature increase. One might suspect that we simply did not explore sufficiently strong electronegativities in our model to reproduce the large values of T e . However, this is not the case since a complete conversion of 1.4% silane would lead to an electronegativity of δ = 0.004, as discussed above. Thus, our model covers the entire electronegativity range possible for our experiment.
Instead, the reason for the discrepancy between the effective electron temperatures from our model and the apparent electron temperatures obtained by the double probe can be found in the way the double probe works.

Source of the unexpectedly high electron temperature
Comparison with our global model shows that the electron temperature increases with silane mole fraction, as measured with the double probe, is much larger than should be possible for our plasma. The reason is to be found in how the electron temperature was obtained from the measurement.
A double probe only evaluates the slope of the EEDF around the floating potential: where f p (ε) is the EEPF, connected to the EEDF f E (ε) as: For a Maxwellian EEDF, this temperature determined from the slope is equal to the electron temperature. However, the EEDF in our experiment is far from Maxwellian, as our BOLSIG+ calculations reveal. Thus, the probe only measures an apparent electron temperature. Figure 7 illustrates this concept of the apparent electron temperature. We assume the EEDF to be nonlocal in the vicinity of the probe tips. With zero applied voltage, both probe tips are at the floating potential and, thus, biased negatively compared to the surrounding plasma. Hence, only electrons with a total energy larger than the energy corresponding to the floating potential, ε fl = e |V fl | , will be able to reach the probe tips. Accordingly, only the red shaded part of the EEPF shown in figure 7 is sampled by the probe. As a small bias is applied to the probes, the probe tip potentials move away from their floating potentials and sample the slope of the EEPF around ε fl .
As figure 7 shows, the decrease of EEPF with electron energy becomes steeper for higher energies because highenergy electrons loose energy due to ionization and excitation, which is not possible for low-energy electrons. As the double probe measures the slope of the EEPF at the position of floating potential, any shift in floating potential would change the part of the EEPF sampled by the probe and would, thus, change the obtained apparent electron temperature, even if the EEPF  (10)) that might be measured by a double probe, depending on the floating potential.
did not change at all. Such a floating potential shift is indicated in the figure by the blue arrow.
Such a shift in the floating potential of the probe could be induced by a RF voltage across the probe sheath, which can occur in the presence of an oscillating plasma potential, as is the case in RF discharges [23,35]. However, this effect depends on the stray capacitance of the measurement system, as well as the amplitude of the plasma potential oscillations, both of which should be mostly independent of the silane mole fraction in the plasma. Thus, this effect cannot explain the observed increase in apparent electron temperature for larger silane mole fractions.
Instead, we propose that the large apparent electron temperature increase is caused by the increase in electronegativity, when silane is added to the system. We will now try to determine if this hypothesis can explain our measurements. To this end, figure 8(a) shows the EEPFs for different electronegativities, obtained from the model. Because of insufficient electron-electron collisions, the EEPFs of our plasma are far from Maxwellian, which would be a straight line on the graph. The high energy tail (ε > 11.5 eV) of EEPFs varies strongly with electronegativity due to changes in ionization and excitation of the working gas (argon). However, the much more populated low energy part of all EEPFs is always similar, resulting in the small variation in average electron energy predicted by the model (compare figure 6).
Since the double probe obtains the apparent electron temperature only from the slope at a single point on the EEPF, the position of this point (the floating potential) largely determines the measurement. Thus, figure 8(b) shows the apparent electron temperature that would be obtained in this fashion, as a function of electron energy, or floating potential, respectively, for different electronegativites. Under any plasma electronegativity condition, the apparent T e sharply decreases with increasing electron energy (or floating potential). For example, for an electropositive plasma (n e /n i = 1), the apparent T e decreases from about 15 eV at low electron energy to less than 1 eV at an electron energy of 15 eV (or a floating potential of 15 V). Additionally, the apparent T e also follows the trend of average energy observed in figure 6 in that it increases with decreasing n e /n i .
Whether the dependence of apparent T e on n e /n i or on the floating potential will dominate the measurement results depends on how strongly the floating potential is actually changing with electronegativity over the range of our experiment.
To find out if this is the case, we calculated the floating potential for the different electronegativities from the current balance onto the probe surface. The ion current is obtained from the modified Talbot and Chou theory described before while electron current is calculated as [14]: Here, V is the potential of the probe with V = 0 at the plasma potential, I e (V) the electron current collected by the probe and A p is the probe surface area. Figure 9 shows the calculated currents onto the probe as a function of V. A larger probe potential repels more electrons, decreasing the electron current. Additionally, the electron current at low energies increases with n e /n i since a larger electron density causes more current. For the ion current, the figure shows an increase with probe potential due to the expansion of the probe sheath [15]. The floating potential is found at the intersection of ion and electron currents. Comparison of the currents in the figure reveals that the floating potential increases with increasing n e /n i .
The resulting floating potentials for the different electronegativities are shown in figure 10. The floating potential (■) of the double probe increases from about 3 V for strongly electronegative plasmas (small n e /n i ) to around 13 V for an electropositive plasma. Considering the results from figure 8(b), we find that such a floating potential increase with n e /n i should lead to smaller apparent electron temperatures. This is indeed the case, as figure 10 reveals, which also shows the apparent electron temperature as a function of electronegativity. The apparent T e ( ) increases from 1 eV for the electropositive plasma, n e /n i = 1, to 27 eV for highly electronegative plasma, n e /n i = 0.001. This strong increase in apparent T e is consistent with the observed strong variation of the apparent electron temperature from our double probe for different silane mole fractions shown in figure 4. In particular, for n e /n i = 0.004, which our model predicts for the complete conversion of silane to particles at our highest mole fraction 1.4%, our model predicts an apparent T e of ∼6.4 eV, which is in reasonable agreement with the measured T e of 12 eV.
This comparison provides compelling evidence that the trend in apparent electron temperature obtained with the double probe is indeed dominated by the changes in floating potential and is much less indicative of any real trends in the mean electron energy. We expect this problem to arise for any double probe measurement in strongly electronegative plasmas. Thus, electron temperatures obtained from double probes in electronegative plasmas should only be used in combination with a detailed analysis of the plasma, as was done here. Otherwise, the results would be highly misleading.

Conclusions
In this work, a double probe was utilized to measure the ion densities and apparent electron temperatures of an argon:silane plasma containing nanoparticles. The reliability of the double probe under these chemically reactive conditions was confirmed by comparison of I-V characteristics obtained over a period of 10 min without noticing significant changes. Collisional probe theory was employed to obtain ion densities from the double probe I-V characteristics. This theory was validated by comparing the results obtained for different probe diameters, finding good agreement with the theoretical predictions for different collisional conditions. Double probe measurements were performed at different silane mole fractions up to 1.4% leading to particle formation. These measurements revealed unexpectedly larger electron temperatures of up to 12 eV at the highest silane mole fraction. To test whether these large temperatures are simply the result of an increase of the mean electron energy to compensate for enhanced electron losses to the particles, a simple global model of the plasma was implemented. The model predicted only a moderate increase of the mean electron energy with increasing particle concentration, inconsistent with the results of the probe measurements.
To explain the large variations of the electron temperature observed by the double probe, we argued that the double probe measures an apparent electron temperature that is characteristic of the logarithmic slope of the EEDF around the floating potential of the probe. Our model showed that a strong electronegativity, expected at large silane mole fractions, causes the floating potential to move closer to the plasma potential. Hence, the part of the EEDF which is sampled by the double probe shifts to lower electron energies. Due to the non-Maxwellian EEDF, such a shift of the probe floating potential causes a strong increase in the measured apparent electron temperature, independent of the actual changes in average electron energy. We found this explanation to be consistent with our measured electron temperatures for different silane mole fractions.
Based on these results, we propose that double probe measurements can be used as a simple diagnostic to measure ion densities in electronegative plasmas containing nanoparticles, even at rather high gas pressures (2 Torr). However, the apparent electron temperatures obtained from the I-V characteristic do not follow the actual trends in mean electron energy and should be used with great caution and only in combination with a detailed analysis of the plasma, as was done here. This limitation could potentially be overcome by using more realistic EEDFs obtained from a Boltzmann solver in the analysis of the probe data. We will attempt to do so in the future.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.