Characterization and global modelling of low-pressure hydrogen-based RF plasmas suitable for surface cleaning processes

A typical spectrum from a H₂/Ar plasma recorded in a wide range of wavelengths is shown in figure 2. Some hydrogen and argon lines are designated in the plot: lines from different transitions in atomic H and Ar and Fulcher α bands of lines coming from electronic transitions in H₂ molecule.

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In figure 3 the variation of maximal intensity of four lines from optical emission spectra with power is shown. Two lines of atomic hydrogen: Hα—656.3 nm and Hβ—486.1 nm and two Ar lines: 750.4 nm and 811.5 nm all show increase in peak values with power increase. At powers higher than 1400–1600 W, when electron energies increase (see supporting data for electron temperature) Hα line exhibits larger increase in intensity compared to other lines. This is due to the fact that direct and dissociative excitation cross-section of this line is higher compared to other lines, and in addition the cross-section increases with increasing energy. The intensities of these lines were used for actinometric calculations.

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The degree of dissociation of molecular hydrogen is a very important parameter in H₂ and H₂-containing low-temperature plasmas for understanding surface processes. For hydrogen, when an initial concentration of hydrogen molecules $N_{{\rm H}_{2}}^{0}$ dissociates into a certain concentration of hydrogen atoms N_H, the dissociation degree x can be expressed as

$$\begin{equation} x=\frac{1}{2}\cdot \frac{N_{{\rm H}}}{N_{{\rm H}_{2}}^{0}}{\rm =}\frac{N_{{\rm H}}/2}{N_{{\rm H}_{2}}+\left({N_{{\rm H}}}/{2}\right)}. \end{equation} \tag{ 1 }$$

Actinometry is one of the emission spectroscopy methods which relate the relative concentrations of two ground state atom species in the plasma with their emission intensity measurements. The method requires an addition of a small amount of noble gas (actinometer) in the feed gas and subsequent observation of lines belonging to the de-excitation transitions in atoms of both gases. The method proved to be an effective diagnostic technique to measure the densities of the various species [57–62] under conditions that fulfil the following assumptions: (i) the addition of a known small amount of an actinometer, A (Ar or other noble gas) should not affect the emissions of the feed gas atomic species G; (ii) the monitored optical emission should originate from excited states of the actinometer and the atomic species, A* and G*, which are dominantly produced through direct electron-impact excitation from ground states; (iii) the excitation cross-sections of A* and G* should have similar threshold energies and similar shapes as a function of energy; and (iv) the loss of A* and G* species should be dominated by radiative processes. Condition (iii) actually underlines that the same group of electrons in the electron energy distribution will take part in the excitation of A and G radiative states, making the method independent of the energy distribution function. Except in few cases [59, 61], these conditions prove to be over limiting since they neglect many processes such as cascading processes from higher excited states, dissociative excitation, collisions between heavy particles and quenching.

Particularly, in the case of H₂ plasmas, conditions (ii) and (iv) are not fulfilled. Detailed analysis of the kinetics of hydrogen excited species [31, 37] showed that in many conditions one has to include electron-impact dissociation of hydrogen molecules as an origin of excited H atoms and, at pressures higher than 100 Pa, the quenching of excited states by other species [38]. In the case of Ar lines, apart from direct excitation from a ground level, excitation of upper levels may be achieved through a metastable state [31]. However, at the low pressures used in this work the density of Ar metastables is a few orders of magnitude lower than the density of Ar atoms, allowing one to disregard this excitation channel. All reactions taken into account for the actinometry are summarized in table 2. One should note that the reactions and corresponding cross-section data in table 2 consider electron-impact excitation to a particular level, while the reactions used in the global model (table 1) do not consider excited states. The rate coefficients used in all calculations were obtained by taking into account the cross-sections for the direct and dissociative excitation for Hα, Hβ taken from Lavrov and Pipa [63] and direct excitation for Ar lines using a set of Hayashi cross-sections [55] and calculated using Maxwell electron energy distribution function with mean electron energy obtained from Langmuir probe measurements.

Table 2. Processes occurring in H₂/Ar low-pressure plasmas used for actinometry calculations. The reference data is given in the far right column.

	Process		Eth (eV)	Rate coef. at 3 eV (10−18 m3 s−1)		Ref.
1a	H atom excitation	H(n = 1) + e → H(n = 3) + e	12.09	$k_{{\rm H}}^{{\rm dir}}$	41.4	[63]
1b		H(n = 1) + e → H(n = 4) + e	12.75		8.86
2	H2 dissociative excitation	H2 + e → H(n = 3) + H(n = 1) + e	16.57	$k_{{\rm H}}^{{\rm dis}}$	0.53	[63]
		H2 + e → H(n = 4) + H(n = 1) + e	17.23		0.01
3	Ar direct excitation	Ar(3p) + e → Ar(4p'[1/2]0) + e	13.48	$k_{{\rm Ar}}^{{\rm dir}}$	10.7	[55]
		Ar(3p) + e → Ar(4p[5/2]3) + e	13.08		11.2
4	Radiative de-excitation		Aij (106 s−1)	∑ Aij
4a		H(n = 3) → H(n = 2) + hv (Hα − 656 nm)	44.1		99.8	[64]
		H(n = 4) → H(n = 2) + hv (Hβ − 486 nm)	8.4		30.2
4b		Ar(4p'[1/2]0) → Ar(4s) + hv (750 nm)	44.5		44.7	[64]
		Ar(4p[5/2]3) → Ar(4s) + hv (811 nm)	33.1		33.1

From the reactions in the table 2, balance equations for atomic H and Ar excited species (H_j and Ar_p, respectively) can be written and are detailed in the online supplementary data section S1 (stacks.iop.org/JPhysD/46/475206/mmedia). From these equations and for steady-state plasma, the density ratios of H and Ar can be calculated as:

$$\begin{equation} \frac{N_{{\rm H}}}{N_{{\rm Ar}}}=\frac{I_{{\rm H}}}{I_{{\rm Ar}}}\gamma \frac{k_{{\rm Ar}}^{{\rm dir}}}{k_{{\rm H}}^{{\rm dir}}(1+ Ds)} \end{equation} \tag{ 2 }$$

where

$$\begin{equation*} \gamma =\frac{C\left( {\lambda}_{{\rm Ar}} \right)\lambda _{H}A_{{\rm Ar}p}(\sum\limits_j {A_{{\rm H}ji})}}{C\left( {\rm \lambda}_{{\rm H}} \right)\lambda _{{\rm Ar}}A_{{\rm H}j}(\sum\limits_p {A_{{\rm Ar}pq})}} \end{equation*}$$

and

$$\begin{equation*} D{\rm s=}\frac{N_{{\rm H}_{2}}k_{{\rm H}}^{{\rm dis}}}{N_{{\rm H}}k_{{\rm H}}^{{\rm dir}}} \end{equation*}$$

A_ji and A_pq are the spontaneous emission coefficients (in s⁻¹) between j and i states for atomic hydrogen and from p to q for Ar. λ_H and λ_Ar are the emission wavelengths corresponding to these two transitions and C(λ_x) is the total optical detection efficiency of the spectroscopic system at λ_x.

Formula (2) shows a relation which links the line emission intensity ratio to the concentration ratios of atomic hydrogen and actinometer. Ds is the ratio of dissociative to direct excitation and the factor (1+Ds) modifies the 'classical' actinometry formula by adding an electron-impact dissociation contributon. Rearranging formula (2) in the way that can fit expression (1), the dissociation degree of hydrogen can be obtained

$$\begin{equation} x=\left( \frac{N_{{\rm Ar}}}{N_{{\rm H}_{2}}^{0}}\frac{I_{{\rm H}}}{I_{{\rm Ar}}}\gamma \frac{k_{{\rm Ar}}^{{\rm dir}}}{k_{{\rm H}}^{{\rm dir}}}-\frac{k_{{\rm H}}^{{\rm dis}}}{k_{{\rm H}}^{{\rm dir}}} \right)\Bigg/\left( 2-\frac{k_{{\rm H}}^{{\rm dis}}}{k_{{\rm H}}^{{\rm dir}}} \right) \end{equation} \tag{ 3 }$$

and thus the H densities.

The density of hydrogen atoms in the chamber centre was also determined using a catalytic probe, from the time dependence of temperature recorded from a thermocouple located at the probe tip and merged to catalytic material. Namely, after plasma ignition the thermocouple registers a temperature increase until it saturates at some constant value. Indeed, due to extensive recombination of hydrogen atoms at the surface of the catalyst, energy dissipation of the reaction (heating term) makes the recorded temperature above the ambient temperature. A typical temperature curve is shown in the online supplementary data (see figure S2 (stacks.iop.org/JPhysD/46/475206/mmedia)). The temperature saturates when the contribution of heating and cooling terms become equal [26]. Immediately after turning the plasma off, the catalytic material of the probe starts to cool down, the temperature drops and dT/dt describes the cooling term. The full physical formalism and precision of the method is discussed elsewhere [65, 66], while details of the calculation are described in the online supplementary data section S3 (stacks.iop.org/JPhysD/46/475206/mmedia). The result of the calculation is an equation for the density of H atoms obtained by equating the heating and the cooling terms at the dropping edge of the temperature curve:

$$\begin{equation} n = \frac{{\rm 8\,m}C_{p}}{{\nu}W_{{\rm D}}{\rm \gamma}_{{\rm cat}}A}\left( \frac{{\rm d}T}{{\rm d}t} \right), \end{equation} \tag{ 4 }$$

where v is the average thermal velocity of H atoms at room temperature (2520 m s⁻¹), W_D is 7.24 × 10⁻¹⁹ J or 4.52 eV. γ_cat is assumed to be constant and equal to 0.18. The probe area A was 21 mm², mass m was 0.0203 g and the specific heat capacity C_p of gold is 130 J kg⁻¹ K⁻¹.

In figure 4, densities of H atom in the centre of the reactor for different powers measured with the catalytic probe are shown (green diamonds). Together with probe measurements, absolute atom densities calculated from actinometry are also plotted (squares and circles). Calculations were performed using the constants shown in table 2, electron temperature of 3 eV measured with a Langmuir probe (see section 4.3) and assuming a gas temperature of T_g = 315 K. In figure 4(a) calculations were performed employing formula (3), while results in figure 4(b) are obtained using the 'classical' actinomery formula, disregarding the dissociation contribution term Ds in equation (2). The densities were calculated using four different actinometric line pairs: (Hα, Ar750), (Hα, Ar811), (Hβ, Ar750) and (Hβ, Ar811). All results show an increase of the density of atomic hydrogen with power. In the case of actinometry including Ds term, comparisons show very good agreement of probe measurements with actinometric results using (Hβ, Ar750) and (Hβ, Ar811) line pairs, while calculations with (Hα, Ar750) and (Hα, Ar811) pairs produced values which are about three times lower than probe measurements.

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Similar results are also observed for the classical actinometry formula (with Ds = 0), with values only slightly increased at low powers (where the Ds term is larger). The accuracy of the catalytic probe is typically about 30%. The actinometric measurements have an accuracy of 15%. The discrepancy between the (Hβ, Ar750), (Hβ, Ar811) and the catalytic probe results falls within the accuracy of both techniques.

The discrepancy in actinometry results obtained with different line pair has been noticed before [67], and it may come from atomic data differences (i.e. cross-sections and/or transition constants) [37] or from the fact that the excited levels are not populated only by direct electron-impact from the ground state [67]. Moreover, when threshold energy for an excitation process lies in the tail of an electron energy distribution function, differences in threshold energies become important since different electron groups take part in the excitation processes of hydrogen and Ar. This may be the reason for better agreement of the catalytic probe data with actinometry calculations with Hβ line, since the difference in threshold energies between hydrogen and argon is smaller in this case (see table 2). Anyhow, small differences between results obtained from the formula including the dissociation term and the one without it, show that in this case the dissociation contribution can be neglected. Despite the differences between absolute values coming from the actinometry and experimental measurements, all data obtained with the actinometric method have the same tendency as the results from the catalytic probe. However, since H densities in actinometry are obtained from integrated line intensities along the full diameter of the chamber and the catalytic probe is collecting atoms from the one point, at the centre of the chamber 5 cm above the wafer holder, some disagreement could be expected.

In order to evaluate the approximate limits of validity of actinometry for all combination of line pairs used, an analysis of actinometric formula (equation (3)) was performed and is presented in the online supplementary data section S2 (stacks.iop.org/JPhysD/46/475206/mmedia). For the Hα line, the dissociation term can be neglected (i.e. Ds ⩽ 0.1) for density ratios larger than 5% (dissociation > ∼2.5%) for any electron temperature (figure S1a (stacks.iop.org/JPhysD/46/475206/mmedia)). In the case of Hβ line, neglecting the term Ds (i.e. Ds ⩽ 0.1) is justified only for electron temperatures lower than 3 eV and density ratios larger than 10% (dissociation > ∼5%) (figure S1b (stacks.iop.org/JPhysD/46/475206/mmedia)). Nevertheless, for our experiments it seems that even for Hβ line it is safe to neglect dissociative excitation except at low powers (below 1200 W see figure 4). Since Hβ lines better compare with experiments if the conditions discussed above and detailed in the online supplementary data are met, it is best to use the Hβ line.

Finally, in table 3 we present values of the actinometric conversion coefficient, i.e. the coefficient to convert intensity ratios to density ratios in the case of 'classical actinometry' for several electron temperatures. Using the conversion coefficient allows simple conversion of actinometric line intensity ratio to absolute H density, assuming that the dissociation term can be neglected. Additionally, spectrometric constants of the system used in our experiment and ratios of atomic transition probabilities are also given. Spectral efficiency constants are obtained using a standard calibration lamp. Rate coefficients entering (3) are calculated using reference data given in table 2 and assuming a Maxwellian energy distribution of electrons. It is hoped that this table can be of use to people doing H actinometry as an approximate semi-quantitative guide to convert actinometric data to densities.

Table 3. Coefficients used for actinometric calculations.

Actinometry ratio used	λ-involved (nm)	Detector response ratio in our system	Ratio of A coefficients	γ	Actinometric conversion coefficient $\gamma ^\ast k_{{\rm Ar}}^{{\rm dir}}/k_{{\rm H}}^{{\rm dir}}$ to convert intensity to concentration ratio of NH/NAr.
					2 eV	2.5	3 eV	3.5	4 eV	4.5
Hα/Ar750	656/750	0.7	2.2	1.4	0.25	0.31	0.36	0.41	0.44	0.48
Hα/Ar811	656/811	0.4	2.3	0.8	0.19	0.20	0.22	0.22	0.23	0.23
Hβ/Ar750	486/750	1.7	3.6	3.9	3.55	4.16	4.70	5.19	5.60	5.99
Hβ/Ar811	486/811	1.1	3.6	2.2	2.62	2.71	2.78	2.83	2.85	2.87

We used a cylindrical Langmuir probe to measure the I–V characteristics of the plasma and extract the ion and electron saturation current, the plasma potential, the electron temperature and ion flux using the ESPion software [68].

In figure 5, in the left hand-side axis, the change in densities of electrons and ions for different powers forwarded to plasma is presented. Measured densities exhibit slow increase towards higher powers, with nearly parallel slopes. However, for powers higher than 1600 W electron density seems to saturate and slightly decrease, while ion density continues to increase. Measured electron densities are in the range 0.4–0.7 × 10¹⁶ m⁻³ and measured ion densities are 2–4 times higher ((0.9–1.7) × 10¹⁶ m⁻³) depending on the power forwarded to the plasma. One has to take into account the fact that determination of the ion densities by using ion saturation current can lead to errors that could easily go up to 20–30% [5].

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Measurements of plasma potential are shown in figure 5, in the right hand-side axis. For powers below 1700 W, the potential in the centre of the chamber shows almost no change with power, staying around 29 V. At higher powers V_p rises up to 32 V. At the same time measured electron temperature at the centre of the chamber is almost constant (3 eV) up to 1700 W of forwarded power, when it increases and reaches 4.5 eV at the maximum power of 1900 W (see figure S3 in the supplementary data (stacks.iop.org/JPhysD/46/475206/mmedia)). At the same time, for the same powers, there is a decrease in electron density. Sudit and Chen [69] stated that mean electron energy and density of electrons are coupled (in order to have constant pressure (N_ekT_e)). The coupling of these two values can be seen also in our case.

The increase in the mean electron energy for the higher powers is also consistent with the increase of the Hα and Hβ line intensities (figure 3) as well an increase in hydrogen atom density (figure 4).

The flux of ions at the centre of the chamber is shown in figure 6. In the right axis, values of ion current density values are presented. The flux rises with an increase in power, as expected.

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Therefore,it is evident that all plasma parameters presented (N_i, V_p, I_flux, N_H) exhibit similar dependency changes versus RF power.

The obtained experimental results allowed us to make a comparison with the results from a zero-dimensional (0d), global model, supplied with the gas reaction set for hydrogen and the realistically calculated surface recombination coefficient (section 3). We focus our discussion on the results for the electron temperature (T_e), the electron density (N_e), and the density of H (N_H).

The electron temperature is calculated 3.58 eV from the global model; it does not change versus power. It is 19% greater than the measured electron temperature for power from 800 to 1700 W.

In figure 7 the densities of charged particles coming from the model are compared to the experimental results. The calculated density for ${\rm H}_{3}^{+}$ and electron density is (2–4) × 10¹⁷ m⁻³ which is approximately 2 to 4 times higher than the experimental values. Electron density calculated by the model is around 6 times greater than the measured one. However, the relative increase of calculated densities with power follows the experimental results.

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In figure 8 the density of H obtained from the model are shown together with the catalytic probe data measured at the chamber centre and actinometry results from (Hβ, Ar750) and (Hβ, Ar811) lines. The results attained with a surface recombination coefficient (k_r) equal to 0.097 (black line) agree well with the experimental data: the calculated density of H is 1.2–2 times greater than the measured one. The recombination coefficients for quartz glass [48] and stainless steel [49] are weighted according to the contribution of these materials to the surface area in the reactor chamber. The obtained value of 0.097 for k_r depicts a physically realistic situation, since it takes into account both materials present inside the reactor (quartz and stainless steel) with their respective areas. All density curves in figure 8 have the same slope, thus the change of H atoms is independent of the recombination coefficient. Densities of Ar neutrals, metastables and ions were also calculated in the model. Since initial Ar atom density is low, densities of produced ions and metastables are a few orders of magnitude lower.

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When comparing modelling with experimental results one should have in mind that modelling results are coming from a global, 0d or volume-averaged model aiming to capture the complex phenomena occurring in 3d in a plasma reactor. There are works referring to global models where the plasma generation region is treated separately from the processing (diffusion) chamber (see for example [70]). In the global model used here, there was no separate treatment of the two regions; this may be an additional source for differences between experimental and simulation results.

In order to further investigate the potential origin of differences in uncertainties of the model parameters, a sensitivity analysis was performed (see supplementary data). All model parameters (reaction rate coefficients, recombination coefficient and cross-section for ion–neutral collisions) were increased 5 times and decreased 5 times. When a model parameter changed, all others were kept constant. It should be noted that we changed the pre-exponential factor of the reaction rate coefficients. The reaction rate coefficients are functions of T_e, thus a change of the pre-exponential factor, through its effect on the balances, may result into a change of T_e. The sign of the change of the rate coefficient depends on the changes of both the pre-exponential factor and T_e. It is not impossible that an increase of the pre-exponential factor will finally result in a decrease of the reaction rate coefficient through a decrease of T_e. We focused on the effect of model parameters on T_e, N_e, and N_H. A parameter was identified as important if the absolute value of the % change of T_e, N_e, or N_H is greater than 50% or less than −33%, i.e. when the value of T_e, N_e, or N_H is multiplied or divided by 1.5. The results for the important parameters are summarized in table 4.

The important model parameters are the rate coefficients of reactions (G1) (electronic excitation leading to H₂ dissociation, see table 1), (G2) (electronic excitation) and (G9) (ionization of H₂), the surface recombination coefficient (reaction (S1), see table 1) and the cross-section for ion–neutral collisions in plasma (reaction (G30), see table 1); the latter parameter plays an important role to the losses of ions at the wall surfaces by affecting the parameter h (see section 3).

When the rate coefficient for the dissociation reaction (G1) (k₁) increases (decreases), the density of H increases (decreases) and the electron density decreases (increases), while the electron temperature remains constant. The increase (decrease) of the rate coefficient for the electronic excitation reaction (G2) (k₂) causes a decrease (small increase) of both the electron and H density; the electron temperature is not affected by the change of k₂. The increase (decrease) of the rate coefficient for the ionization reaction (G9) (k₉) induces an increase (decrease) of the electron density and a decrease (increase) of the electron temperature; the density of H is slightly decreased with the change of k₉.

Enriching the sensitivity analysis, the effect of the surface recombination coefficient (k_r) on the density of H is shown in figure 8 for the whole power range investigated. Figure 8 includes the densities of H obtained from the model for different surface recombination coefficient together with the catalytic probe data measured at the chamber centre. The results of the model obtained with k_r = 0.485 (0.097 · 5) are far below the experimental data (dots in figure 8). In this case, the value of the coefficient is high, well above the value for stainless steel [49], the recombination seems to be overestimated, and the density of H is lower by a factor of 2 compared to the measured density. The model results acquired with lower recombination coefficient, k_r = 0.019, yielded 3 to 4 times higher H densities compared to the base case (k_r = 0.097, compare the dashed line with the black line in figure 8). Such low recombination coefficients could be present in the case of glass-type surfaces [48].

Finally, the increase (decrease) of the cross-section for ion–neutral collisions (σ_ion-neutral) causes an increase (decrease) of the electron density and a small decrease (increase) of the electron temperature; the change of σ_ion-neutral slightly affects the density of H.

The main discrepancy between the modelling and the experimental results is in the value of the electron density; the electron density is calculated to be 6 times greater. The sensitivity analysis shows that this discrepancy could have been reduced if k₁ and/or k₂ were greater. It could have been also reduced if k₉ was lower; however, in the latter case the electron temperature would have been increased (see table 4).

Regarding k₁ (rate coefficient for the dissociation of H₂), it is rather greater than the value we used: some paths to dissociation coming from vibrational excitations [45, 71] were not taken into account. Sawada and Fujimoto [72] calculated an effective rate coefficient for the H₂ dissociation which is greater than the one used in this work; in particular, for an electron temperature equal to 3.5 eV, they report an effective rate coefficient which is about 4 times greater (figure 5 of [72]) than the one we used. Indeed, the sensitivity analysis shows that a multiplication of k₁ with 5 will result in an electron density 3 (and not 6) times greater than the experimental result.

Regarding k₂ (rate coefficient for electronic excitation reaction), we have no evidence that it has an underestimated value. However, what the sensitivity analysis shows us is that omitted excitation reactions (even if they do not join the mass balances as reaction (G2)) potentially affect the electron density; the addition of more excitation reactions in the reaction set of table 1, will increase the energy loss rate per electron which will subsequently result in a decrease of the electron density. Regarding the reaction set of table 1, the omission of excitation reactions, e.g. vibrational excitations [45, 71] or metastable atoms [43], could result into an overestimation of the electron density. We are working on a 2d plasma model for H₂ discharges, where all these potential sources of discrepancies will be further investigated in conjunction with a more detailed model. In addition, we will also check the effect of using the experimentally obtained EEDF (and not a Maxwellian EEDF) on the rate coefficients of electron-impact reactions and as a consequence on the model results.