Computer Simulation of an Analytical Direct Current Glow Discharge in Argon: Influence of the Cell Dimensions on the Plasma Quantities ANNEMIE BOGAERTS* AND RENAAT GIJBELS Department of Chemistry, University of Antwerp, Universiteitsplein 1, B-2610 Wilrijk-Antwerp, Belgium A set of three-dimensional mathematical models was developed DESCRIPTION OF THE MODELS for describing the behavior of the dierent plasma species in a A number of separate models have been developed for the direct current glow discharge in argon used as an analytical dierent plasma species in a direct current glow discharge in ion source for mass spectrometry.The models were applied to argon with a copper cathode, and these models were combined cylindrical cells (flat cathode and hollow anode) with various to obtain an overall picture of the glow discharge. Table 1 dimensions to study the eect of the dimensions on the gives an overview of the dierent species assumed to be present calculated plasma quantities.The results show that the cell in the plasma and the models used to describe these species. dimensions have no significant influence on the qualitative Fluid models are utilized for plasma species that are more or behavior of the plasma quantities, but they do aect the less in equilibrium with the electric field in the discharge (i.e., absolute values, at least for cell dimensions ranging from 0.5 the energy gained by the electric field is more or less balanced to 2 cm.For larger cells, the absolute values also remain more by the energy lost due to collisions), so that they can be or less constant. The results suggest that, for the selected considered as a fluid and described with continuity and trans- discharge conditions of 1000 V, 1 Torr and about 2 mA and a port equations. Monte Carlo simulations are employed for copper cathode, a cell with both length and radius equal to plasma species that are far from equilibrium, so that their 2 cm is a good choice for analytical mass spectrometry.This behavior has to be described explicitly (i.e., trajectory calcu- paper demonstrates that the models are in principle able to lated with Newton’s laws and collisions treated statistically by predict trends in plasma behavior and performance in random numbers). A short explanation of the dierent models analytical applications and that they can therefore be useful in (i.e., the relevant processes considered in each model) is also developing new cells.included in Table 1. The models are combined and solved iteratively until final convergence is reached. More information Keywords: Glow discharge; modeling; cell design about these models can be found elsewhere.3–12 The models are applied to the cell geometry represented in Fig. 1. We chose a very simple and general cell configuration, so that the results of this study can most easily be applied to Glow discharges are used for a range of applications: as various, more specific types of cell geometries. The length and spectroscopic sources for mass spectrometry or optical spectro- radius of the cell are each varied independently between 0.5 metric techniques,1 for deposition of thin films and for plasma and 4 cm.In the initial ‘standard cell’, the length and radius etching and modification of surfaces in the semiconductor are each taken as 2 cm. A metallic disk of 0.25 cm radius acts industry,2 as plasma displays, as metal vapor ion lasers and as the cathode and is represented by the black rectangle at l= also in the lighting industry. It is to be expected that the 0.All the other cell walls (i.e., the cylindrical portion and the results in the dierent application fields will depend strongly cylinder ends) are at anode potential. The cathode and anode on the cell configuration. One possibility for optimization is are separated by an insulating ring (0.1 cm wide).The fluid to build dierent cells and to investigate which configuration models are developed in two dimensions: due to the cylindrical and which dimensions yield the best results. However, this is symmetry of the cell, the three dimensions could, indeed, be sometimes based on trial and error, and it can be a time- reduced to two dimensions. The Monte Carlo models, however, consuming and expensive approach. It would be much cheaper are completely constructed in three dimensions.to predict the optimum cell configuration by computer simulations prior to building the cells. We have developed a set of mathematical models for describ- RESULTS AND DISCUSSION ing the behavior of the dierent species present in a direct Discharge Current as a Function of Voltage and Pressure current glow discharge in argon used as an ion source for mass spectrometry. These models were first developed in one The calculations were all performed at a 1000 V discharge dimension3–8 and later extended to three dimensions and voltage and 1 Torr argon gas pressure.When the pressure, applied to the standard cell used for analyzing flat samples in voltage and gas temperature are given, the models allow the a VG9000 glow discharge mass spectrometer.9–12 Reasonable self-consistent calculation of the electrical current flowing agreement with experimental results (e.g., based on laser- through the cell. Comparison of the calculated currents with induced atomic fluorescence measurements) could be experimental values can then be used to test whether the achieved.13–15 These models can in principle be used to predict models present a realistic picture of the glow discharge.the optimum cell design.16 To illustrate this in the present Assuming a gas temperature of 450 K, the calculated currents paper, the models were applied to a simple, cylindrically for the dierent cell dimensions investigated are presented in symmetrical glow discharge cell with a flat cathode.The cell Table 2. They range from 0.7 to 2.6 mA for all cell dimensions configuration was kept constant, but the length and the radius under study. Hence the discharge conditions investigated (1000 of the cell were varied to investigate their influence on the V, 1 Torr and 0.7–2.6 mA) are typical discharge conditions for glow discharge mass spectrometry (GDMS). calculated quantities. Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 (751–759) 751Table 1 Overview of the plasma species assumed to be present in the glow discharge plasma, the models used to describe these species, the relevant processes considered in these models and references giving more information about these models Plasma species Model Relevant processes Ref. Ar atoms at rest Not explicitly calculated Atoms assumed to be uniformly distributed throughout the cell — Fast electrons (i.e. energy Monte Carlo model Elastic collisions with Ar atoms; electron impact ionization of Ar atoms in the ground state 3,4,9 high enough for and in the metastable level, and of sputtered Cu atoms; electron impact excitation of Ar inelastic collisions) atoms in the ground and metastable states Thermalized electrons Fluid model Continuity and transport equations (transport by diusion and migration in the electric field); 4,9 equations coupled to Poisson equation, to obtain self-consistent electric field Ar+ ions Fluid model idem 4,9 Ar+ ions in the CDS* Monte Carlo model Symmetric charge transfer; elastic collisions with Ar atoms; ion impact ionization and 3,5,9 excitation of Ar atoms Ar fast atoms in the CDS Monte Carlo model Elastic collisions with Ar atoms; atom impact ionization and excitation of Ar atoms 3,5,9 Ar metastable atoms Fluid model Balance equation with dierent production terms (electron, ion and atom impact excitation to 6,11 the metastable levels and electron–ion radiative recombination), and loss terms (electron impact ionization and excitation from the metastable levels, electron collisional transfer to the nearby levels, metastable atom–metastable atom collisions, Penning ionization of sputtered Cu atoms, two- and three-body collisions with Ar atoms); moreover, transport is diusion controlled, and subsequent de-excitation at the walls is an additional loss process Cu atoms Monte Carlo model Thermalization immediately after sputtering, due to collisions with Ar gas atoms 10 Cu atoms and Cu+ ions Fluid model Further transport of Cu atoms (diusion controlled), ionization of Cu atoms (by Penning 7,11 ionization by Ar metastable atoms, asymmetric charge transfer by Ar ions and electron impact ionization) and transport of the Cu ions (by diusion and migration in the electric field) Cu+ ions in the CDS Monte Carlo model Elastic collisions with Ar atoms 7,11 * CDS=Cathode Dark Space.Table 2 Calculated quantities for the dierent cell dimensions investigated (at 1000 V and 1 Torr, argon discharge with copper cathode) l=2 cm l=0.5 cm l=1 cm l=3 cm l=4 cm l=2 cm l=2 cm l=2 cm l=2 cm Parameter r=2 cm r=2 cm r=2 cm r=2 cm r=2 cm r=0.5 cm r=1 cm r=3 cm r=4 cm Electric current/mA 2.4 0.65 1.9 2.45 2.5 0.87 1.3 2.5 2.55 Length of the CDS/cm 0.15 0.23 0.16 0.15 0.145 0.20 0.17 0.15 0.15 Max.value of the plasma potential/V 1.9 4.6 2.2 1.9 1.9 9.8 3.8 1.9 1.9 Max. axial electric field strength at cathode/kV cm-1 -20 -15 -19 -20 -20 -15 -17 -20 -20 Max.axial electric field strength at anode 13 240 36 7 4 70 27 12 12 end-plate/V cm-1 Max. radial electric field strength at anode side-walls/ 5 2 3 5 4.5 413 32 1.6 0.8 V cm-1 Max. value of Ar ion and thermalized electron 1.9×1012 1.6×1011 1.3×1012 1.9×1012 2.0×1012 5.7×1011 8.4×1011 2.0×1012 2.1×1012 density/cm-3 Max. value of fast electron density/cm-3 9.6×107 3.4×107 8.7×107 9.9×107 1.1×108 5.7×107 5.8×107 9.9×107 9.9×107 Max.value of sputtered Cu atom density/cm-3 2.1×1013 4.6×1012 1.7×1013 2.1×1013 2.1×1013 2.0×1012 3.5×1012 2.1×1013 2.1×1013 Max. value of Cu ion density/cm-3 4.6×1010 2.5×108 1.8×1010 4.7×1010 5.5×1010 1.4×109 4.1×109 5.7×1010 5.9×1010 Ratio of Cu ion to Ar ion density (%) 2.4 0.16 1.4 2.5 2.8 0.25 0.50 2.9 2.8 Degree of ionization of Cu (%) 2.1 0.009 0.57 2.4 2.7 0.033 0.23 4.1 4.2 752 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12Fig. 1 Schematic diagram of the cylindrical glow discharge cell with various dimensions to which the models were applied (flat cathode and hollow anode).The cell with length (l ) and radius (r) equal to 2 cm is taken as the ‘standard cell’. The lengths and radii were each varied independently from 0.5 cm to 4 cm. It appears that for lengths and radii smaller than 2 cm, the current increases clearly with increasing length and radius. The fast electrons can travel longer in the plasma before they reach the walls and can hence give rise to more ionization collisions, more electron multiplication and hence higher currents.Moreover, the ions and (slow) electrons will not arrive so rapidly at the walls, where they would be neutralized by electron–ion recombination, so that their density in the plasma is higher and they can carry more current. At lengths and radii larger than 2 cm, the calculated current does not appear to Fig. 2 Calculated potential distribution for the standard cell (l=r= increase further.This indicates that at the selected discharge 2 cm) at 1000 V, 1 Torr and 2.4 mA in an argon discharge with copper cathode. conditions the range in which fast electrons can produce ionization collisions is more or less limited to about 2 cm from the cathode (see below). the thin plasma sheath in front of the walls stays behind with a positive space charge. This leads to a potential increase in Potential and Electric Field Distributions the plasma with respect to the walls, which gives rise to a positive plasma potential, i.e., the plasma is the most positive Fig. 2 presents the potential distribution throughout the discharge for the cell of length and radius 2 cm. The cathode is body in the discharge. It can be understood that in a small discharge cell, the perturbation by the walls is more significant, represented by the black rectangle at z=0. The potential is -1000 V at the cathode, and increases rapidly immediately in and the plasma has to react more to oppose the potential change at the walls, yielding a higher (more positive) plasma front of the cathode.It goes through zero at about 0.15 cm from the cathode, and is slightly positive (about 1.9 V, the potential. The position where the plasma potential reaches its maximum is slightly closer than about 1 cm from the cathode plasma potential) in the rest of the plasma. The position where the potential goes through zero defines the interface between for the cells with lengths and radii 2 cm (hence it is not always in the middle of the discharge cell, as one might expect), Cathode Dark Space (CDS) and negative glow (NG), and it is therefore indicated with a thicker line.The potential distri- and it was found to be at about 0.4–0.5 cm from the cathode for the cells with smaller dimensions. The cell dimensions butions calculated for the other cell dimensions are qualitatively the same. The length of the CDS is always more or less investigated here do not yet appear to be large enough to give rise to the formation of a Faraday dark space or positive similar, as can be seen from Table 2, except at the smallest lengths and radii investigated, since the current is lower there; column, although when l=2 cm and r=4 cm the calculated potential again becomes slightly negative (about -0.2 V) at a hence the electrons cannot give rise to so much ionization, and the CDS therefore has to be longer to sustain the discharge. radial position of about 3 cm from the cell axis, which indicates the beginning of a Faraday dark space being formed.The NG always fills up the rest of the discharge cell, thus being small for the small dimensions and large for the larger From the potential distributions, the electric field strengths throughout the discharge can also be calculated. The axial dimensions. The value of the plasma potential was calculated to be roughly constant for lengths and radii 2 cm, but electric field is extremely negative at the cathode (due to a large potential drop over a small distance), it increases rapidly increased for shorter lengths and radii (see Table 2).The reason for this is found in the phenomenon of ‘sheath formation’ in the CDS to small negative values at the interface with the NG, it crosses the zero-line always at about 0.5–1.2 cm from (Debye shielding):2 if the potential in the plasma is perturbed, the plasma reacts to oppose that change.Since the electrons the cathode (called ‘field-reversal;’ it occurs at the position where the potential reaches its maximum) and it increases to have a much higher mobility than the argon ions, they will diuse more rapidly to the walls, where they will be lost, and small positive values at the anode end-wall. The radial electric Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 753field is negative in the CDS at the edges of the cathode and it is slightly positive in the NG, increasing slightly towards the anode side-walls.These electric field distributions are qualitatively similar to those calculated for the VG9000 glow discharge cell,9 and are therefore not illustrated here again. Table 2 presents the maximum values of the axial electric field strengths at the cathode and at the anode end-wall, and the maximum values of the radial electric field strengths at the anode sidewalls for the dierent cell dimensions investigated.It appears that the axial electric field at the cathode increases for lengths and radii increasing from 0.5 to 2 cm (because the CDS becomes shorter and the potential has to fall over a shorter distance, giving rise to a higher electric field), and is approximately constant for the larger dimensions (because the CDS length was also found to be more or less constant). The axial electric field at the anode end-wall increases considerably for decreasing cell lengths, which is attributed to the shorter distance over which the potential has to drop to zero at the wall.The eect is most pronounced for the small values of l, since these are also characterized by a higher plasma potential that has to drop o. The eect of the increasing radius is only small, since the length (and hence the distance over which the potential has to fall in the axial direction) stays constant. The higher axial electric field at the anode end-wall in the case of the small radii is due to the higher plasma potential, but for radii ranging from 2 to 4 cm the axial electric field is constant, since both the plasma potential and the axial distance are the same.On the other hand, the values of the radial electric fields at the anode side-walls change considerably for the dierent radii investigated, as can be seen from Table 2, because (i) the distance over which the plasma potential has to fall to zero varies widely and (ii) at small values of r, a higher plasma potential has to drop o.Also, on decreasing the lengths, the radial electric field at the anode side-walls increases, but the eect is only visible at small values of l, owing to the higher plasma potential that has to drop o to zero, and it is absent at l2 cm, since both the value of the plasma potential and 2E+011 2E+011 2E+011 2E+011 1E+011 1E+012 1E+012 1E+011 1E+011 1E+011 Fig. 7 Calculated density profile of the argon metastable atoms for the cell with l=0.5 cm and r=2 cm, at 1000 V, 1 Torr and 0.65 mA in Fig. 6 Calculated density profile of the argon metastable atoms for an argon discharge with copper cathode. the standard cell under the same discharge conditions as in Fig. 2. in the model, or that further processes would have to be incorporated. Nevertheless, the overall agreement with experi- in these larger cells, owing to the higher slow electron densities ment was satisfactory, and therefore we believe that at least (see above).It was found that in the small cells, diusion and the trend observed in the present results will be correct. de-excitation at the walls are more or less the dominant loss process for the argon metastable atoms (i.e. about 54% at l= 0.5 and r=2 cm and about 41% at l=2 and r=0.5 cm, whereas Density of Sputtered Copper Atoms the values for this loss process at l=4 and r=2 cm and at l= 2 and r=4 cm are only about 5 and 20%, respectively), and Fig. 8 shows the sputtered copper atom density profile for the cell with l=2 and r=2 cm. The density is at its maximum in the larger cells electron collisional transfer to the nearby resonant levels is most important (i.e., about 88 and 70% for close to the cathode and decreases towards the cell walls. This sputtered atom density profile, calculated for tantalum, for a l=4 and r=2 cm and l=2 and r=4 cm, respectively, compared with about 35 and 50% for l=0.5 and r=2 cm and l=2 and comparable cell (six-way cross glow discharge cell) and discharge conditions, was in excellent agreement with results from r=0.5 cm, respectively).The relative contributions of the other loss processes were found to be of minor importance and were laser induced fluorescence experiments,13 which also supports the present results. The calculated sputtered copper atom comparable for all the cells investigated. The most important production process was in all cases electron impact excitation density is clearly higher than that calculated for tantalum, since copper has a much higher sputtering yield than tantalum.2 to the metastable levels although, especially in the small cells, argon ion and atom impact excitation were not negligible.For The copper atom density profiles, calculated for the other cell dimensions investigated, are similar to that in Fig. 8: a maxi- the cell of l=0.5 and r=2 cm, the importance of argon atom impact excitation was even found to be comparable to electron mum is always reached at about 0.05–0.1 cm from the cathode, whereafter the density decreases towards the cell walls.Table 2 impact excitation. It should be mentioned, however, that these calculated data concerning the argon metastable atoms have presents the maximum values for the dierent cell dimensions studied. The density appears to increase for lengths and radii to be considered with caution. The argon metastable atom density profile, calculated for a similar (so-called six-way cross) between 0.5 and 2 cm, but remains constant on further increasing the dimensions.This indicates again that the eect of the glow discharge cell, with comparable discharge conditions, was found not to be in complete agreement with results of laser cell walls is only important in the small cells, and that the cell with both length and radius equal to 2 cm is large enough to induced fluorescence measurements,14 which may indicate that the production and/or loss processes are not correctly described give high concentrations of sputtered atoms in the discharge. 756 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12Fig. 12 Calculated relative contributions of the fast argon atoms, Fig. 10 Calculated relative contributions of asymmetric charge trans- argon ions and copper ions to the sputtering process at the cathode fer, Penning ionization and electron impact ionization to the ionization as a function of the cell dimensions at 1000 V and 1 Torr (argon of the sputtered copper atoms as a function of the cell dimensions at discharge with copper cathode).The influence of the length (l ) is given 1000 V and 1 Torr (argon discharge with copper cathode). The by the full lines and that of the cell radius (r) by the dashed lines. influence of the length (l ) is given by the full lines and that of the cell radius (r) by the dashed lines. not dominant, but these processes are nevertheless not negli- role of asymmetric charge transfer becomes increasingly gible.5 In Fig. 11, the relative contributions of the ionization important for larger cells. It seems to be the dominant process mechanisms for argon are depicted for the dierent cell dimen- for all cell dimensions investigated, except for the two smallest sions under study. Electron impact ionization in the NG was cells (l=0.5 or r=0.5 cm), where Penning ionization is calcu- found to be clearly dominant for all cases, but, since the NG lated to be the most significant process.The argon metastable shrinks accordingly when the cell dimensions decrease (and, atom density was found to be relatively higher in the small moreover, the CDS becomes larger; see Table 2), the role of cells than in the larger cells (see above). Electron impact electron impact ionization in the CDS, and also of argon atom ionization appears to be of minor importance in all cases. It and ion impact ionization, becomes slightly more significant should be noted that the exact contributions of these three in the smaller cells.processes have to be considered with caution, since the rate coecients for Penning ionization, and especially for asymmet- Sputtering at the Cathode ric charge transfer, are not well known in the literature, and the values used (see ref. 11) are, therefore, subject to consider- Finally, since the ratio of copper ion flux to argon ion flux at able uncertainties.Nevertheless, the general trend is expected the cathode increases when the cell becomes larger for cells to be correctly predicted. with dimensions 2 cm, the relative contribution of copper ions to the cathode sputtering (i.e., self-sputtering) is also expected to rise. The relative contributions of copper ions, Ionization of Argon argon ions and fast argon atoms to sputtering are illustrated Whereas for the sputtered copper atoms asymmetric charge in Fig. 12 for the dierent cell dimensions studied. The role of transfer and also Penning ionization are more important than the fast argon atoms appears to be dominant in all cases; the electron impact ionization, the situation is completely dierent argon ions take second place and the copper ions contribute for the ionization of argon. Indeed, asymmetric charge transfer only a few percent. However, it can be noted that their role by argon ions and Penning ionization by argon metastables increases slightly when the cell dimensions change from 0.5 cm play, of course, no role in the ionization of argon, and electron (i.e., about 0.75%) to 2 cm (i.e., about 2.5–3%).On further impact ionization is the dominant process. Two other processes, increasing the cell dimensions, the relative contributions of the i.e., argon ion and atom impact ionization, come into play, copper ions, argon ions and fast atoms remain more or however. Since these ionization mechanisms are only important less constant.in the CDS, close to the cathode, their final contributions are CONCLUSION A number of three-dimensional models, developed for a direct current glow discharge in argon, have been applied to dierent cells with lengths and radii varying from 0.5 to 4 cm to investigate the influence of the cell dimensions on the typical quantities calculated by the models, e.g., the electric current as a function of voltage and pressure, the potential distributions and electric fields in the discharge and the densities of the plasma species.Special emphasis is placed on the use of the glow discharge as an ion source for mass spectrometry (GDMS) (i.e., calculation of the importance of dierent ionization processes and the role of the sputtered atoms and ions in the discharge), but the results of the present investigation can also be extended to other applications of glow discharges. Fig. 11 Calculated relative contributions of electron impact ioniz- It was found that the calculated results are qualitatively the ation in the NG and in the CDS and of fast argon atom and argon same for the dierent cell dimensions investigated, but the ion impact ionization to the ionization of the argon atoms as a absolute values are aected. The most important results are function of the cell dimensions at 1000 V and 1 Torr (argon discharge the following: on increasing the cell dimensions from 0.5 to with copper cathode).The influence of the length (l ) is given by the full lines and that of the cell radius (r) by the dashed lines. 2 cm, the electric current at the same voltage and pressure 758 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12increases slightly, and the same applies to the densities of the REFERENCES plasma species. Moreover, the ionization degree of copper 1 Marcus, R. 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The authors also acknowledge Paper 6/08262I financial support from the Federal Services for Scientific, Received December 9, 1996 Technical and Cultural Aairs (DWTC/SSTC) of the Prime Minister’s Oce through IUAP-III (Conv. 49). Accepted February 28, 1997 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 759