Positive ion flux from a low-pressure electronegative discharge

Plasma Sources Sci. Technol. 8 (1999) 457 462. Printed in the UK PII: S0963-0252(99)03994-8 Positive ion flux from a low-pressure electronegative discharge T E Sheridan, P Chabert and R W Boswell Space Plasma and Plasma Processing Group, Plasma Research Laboratory, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia Received 18 November 1998, in final form 11 March 1999 Abstract. We compute the flux of positive ions exiting a low-pressure, planar, electronegative discharge as a function of the negative ion concentration and temperature. The positive ions are modelled as a cold, collisionless fluid, while both the electron and negative ion densities obey Boltzmann relations. For the plasma approximation, the plasma edge potential is double-valued when the negative ions are sufficiently cold. When strict charge neutrality is relaxed, spatial space-charge oscillations are observed at the edge of the plasma when the flux associated with the low (in absolute value) potential solution is less than that of the high potential solution. However, the flux is always well defined and varies continuously with the negative ion concentration. We demonstrate that the correct solution for the plasma approximation is that having the greater flux. 1. Introduction The physical and chemical processes occurring at a surface in contact with a discharge are to a great extent determined by the ion flux and impact energy at that surface. The ion impact energy is largely determined by the sheath [1, 2], since most of the potential drop occurs there, while the ion flux is determined by the plasma [3 5], since most ions are created there. (Conceptually, a discharge can be divided into two parts: the plasma, a quasi-neutral region many Debye lengths wide, and the sheath, an ion-rich boundary layer a few Debye lengths wide separating the plasma from the wall.) Discharges made using electronegative molecules (e.g. SF 6 or O 2 ) or molecules composed of electronegative atoms (e.g. O, F or Cl) are often encountered in plasma processing [6]. The presence of negative ions in a discharge can significantly reduce the positive ion flux exiting the plasma, with implications both for plasma processing and electrostatic probe diagnostics [7 9]. Braithwaite and Allen [10] have presented a theory for the positive ion saturation current to a spherical probe in an electronegative plasma. Their model is similar to that of Schott [11], who considered a planar discharge with a low-density, energetic electron component (responsible for ionization) and a high-density, low-temperature, Maxwellian component (i.e. like negative ions). In both papers [10, 11], it was found that for the plasma approximation there are parameter regimes for which the potential at the plasma edge, and hence the flux, is double-valued. Braithwaite and Allen [10] proposed that the sheath forms at the Permanent address: Laboratoire PRIAM, UMR 9927 CNRS-ONERA, Fort de Palaiseau, F-91761 Palaiseau Cedex, France. first location where charge neutrality is violated, and were thereby led to the surprising conclusion that the flux is not a continuous function of the negative ion concentration. Although not always explicitly stated, this assumption appears to be widely used [7, 8]. If the plasma approximation is relaxed, Schott [11] demonstrated that these multiplesolution regimes are associated with non-neutral spatial oscillations (quasi-periodic double layers) at the plasma edge. Clearly, to use such theories to interpret the flux from an electronegative discharge, one must first know how to interpret the theories. In this paper, we compute the flux of collisionless positive ions exiting a planar, electronegative discharge as a function of the negative ion concentration and temperature. Our model is solved analytically for the plasma approximation and numerically for the non-neutral case. We find that when multiple solutions exist for the plasma approximation, the correct solution is that which gives the greater flux. Consequently, the positive ion flux is shown to be a continuous function of the negative ion concentration. In section 2 we present the equations that model the discharge and non-dimensionalize them. Solutions to the model are studied in section 3 as a function of the negative ion concentration and temperature. Section 4 is a brief summary. 2. Model equations We consider a plane, symmetric discharge containing cold, positive fluid ions of mass M +, density n + and velocity v +, Boltzmann negative ions with a temperature T and density n, and Boltzmann electrons with a temperature T e and 0963-0252/99/030457+06$19.50 1999 IOP Publishing Ltd 457

T E Sheridan et al density n e. Positive ions are created at a rate proportional to n e and then accelerated to the walls by the self-consistent potential φ. The centre of the discharge is at x = 0 and the sheath plasma boundary is at x = L. On the centre plane: φ = 0, n e = n e0, n + = n +0 and n = n 0. Quasi-neutrality requires n + n e + n in the plasma, and n +0 n e0 + n 0 at the centre of the discharge. The steady-state fluid equations of continuity and motion for the positive ions in a slab geometry are [12] d(n + v + )/dx = ν iz n e (1) M + n + (v + d/dx)v + = en + E M + v + ν iz n e (2) where ν iz is the net ionization rate. The second term on the right of equation (2) represents a decrease in ion fluid momentum due to ions born at rest. Fully-collisional models [6] neglect ion inertia and therefore cannot describe the lowpressure case considered here. Both the electron and negative ion densities obey Boltzmann relations, so that and n e = n e0 exp[eφ/(kt e )] (3) n = n 0 exp[eφ/(kt )]. (4) That is, the confined species are considered massless. This assumption is valid for the negative ions so long as wall losses dominate volume recombination [6]. Finally, φ and E are determined self-consistently from Poisson s equation ε 0 de/dx = e(n + n e n ) E = dφ/dx. (5) To non-dimensionalize the model equations, we introduce the following variables [4] ξ = x/ ñ = n + /n e0 u = v + /c se η = eφ/(kt e ) ε = e E/(kT e ) (6a) where the ion acoustic speed in the absence of negative ions is c se = (kt e /M + ) 1/2, the ionization length is = c se /ν iz and the non-neutrality parameter is q = λ e0 / (6b) (6c) with the electron Debye length λ e0 = [ε 0 kt e /(e 2 n e0 )] 1/2. The negative ion component is characterized by its concentration α and temperature γ, α = n 0 /n e0 γ = T e /T (7) where we assume that γ > 1. The flux of positive ions exiting the plasma and entering the sheath is Ɣ s, which is non-dimensionalized by n e0 c se,so Ɣ s = n +sv +s =ñ s u s (8) n e0 c se n e0 c se where the subscript s refers to values at the sheath plasma boundary. Note that Ɣ s is nearly constant in the sheath as almost no ions are created there. Using the variables given above, the dimensionless equations describing the discharge are (ñu) = e η (9a) (ñuu) =ñε (9b) q 2 ε =ñ e η αe γη (9c) η = ε (9d) where the prime symbol denotes differentiation with respect to ξ. Thus, we obtain a system of four first-order, nonlinear, ordinary differential equations in four unknowns: ñ, u, η and ε, and three physical parameters: the plasma non-neutrality q, the negative ion concentration α and the negative ion temperature γ. For the plasma approximation q = 0 (or λ e0 /L = 0), equations (9a) (9d) describe a charge-neutral plasma with a sheath of zero thickness. The plasma solution becomes singular at the sheath plasma boundary since the sheath is demonstrably not charge neutral. If the source terms are neglected and q>0, then equations (9a) (9d) represent the sheath [2], and their solution requires assumptions about the values of the ion velocity and electric field at the sheath edge [12, 13]. Consequently, for q > 0 the equations describe two distinct behaviours with disparate length scales a quasineutral plasma and a positive space-charge sheath. 3. Results and discussion 3.1. Solutions for the plasma approximation We are interested in investigating the dependence of the ion flux on the negative ion parameters α and γ. We first consider the case q = 0 (i.e. the plasma approximation) to gain some insight into the nature of solutions to equations (9a) (9d). In this case, the equations reduce to (ñu) = e η (ñuu) =ñη ñ = e η + α e γη. (10) From the third equation, we have ñ = ñ(η). We can then eliminate ñ from the second equation and integrate to find u(η), [ u 2 = 1 e η + α ]/ γ (1 e γη ) (e η +αe γη ) (11) which can then be substituted into the first equation, giving { dη (e η + α e γη ) 2 (e η +αγ e γη ) dξ [ 1 e η + α ]} γ (1 e γη ) ( =2e [(e η η +αe γη ) 1 e η + α 1/2 ))] γ (1 e γη. (12) This equation can be solved by separation of variables to find ξ(η), although the integration would probably have to be done numerically. For the plasma approximation, the desire to form a sheath is thwarted by the requirement of charge neutrality and dη/dξ becomes infinite at the plasma edge. Using this condition, we 458

A low-pressure electronegative discharge can calculate the potential at the plasma edge η s, and then the flux there since we know both u(η) and ñ(η). At the plasma edge, the right-hand side of equation (12) can be neglected, so that η s must obey (e ηs + α e γηs ) 2 (e ηs +αγ e γηs ) [ 1 e ηs + α ] γ (1 e γηs ) =0. (13) This condition is equivalent to equating the ion velocity in equation (11) to the Bohm speed for a plasma with two negative Boltzmann components [10] u 2 B = (e ηs + α e γηs )/(e ηs + αγ e γηs ). For α = 0, η s = ln 1/2 and the positive ion density at the sheath edge [5] is ñ s = 1/2. From the more accurate kinetic model [4], it is found that ñ s = 0.4871. Thus, the fluid and kinetic models give nearly the same results. We can approximate η s for the cases α 1 and α 1 from equation (13), and then calculate the flux. We find 1 Ɣ s 2 + 1 2γ α α 1 =ñ s u s = [( ) n e0 c se 1 1 1/γ + 1 ] γ 2 2 α α 1. (14) For γ = 1 (i.e. the two negative species are indistinguishable), both expressions reduce to the same correct expression. The α-dependence of the flux when α is small and γ is large is quite weak, making measurement of the negative ion concentration using electrostatic probes difficult in this regime. (Similar expressions can also be derived for a spherical presheath [10].) When γ is large enough, equation (13) admits two physical solutions for the plasma edge potential, as shown in figure 1(a). The same behaviour was also noted in [10] and [11] for similar models. For our model, multiple solutions exist for γ 9.90, in agreement with the analytic result of 5 + 24 given in [10]. (This is related to the fact that our model and that of [10] have the same Bohm velocity.) Consequently, the flux at the plasma edge calculated using ñ and u from equations (10) and (11) is double-valued (figure 1(b)). (The model investigated here has also been considered by Franklin and Snell [14], although they did not consider the implications of multiple solutions.) It has been proposed [10] that the sheath always forms at the first singularity encountered (i.e. the smaller value of η s ), leading to the conclusion that the flux changes discontinuously at some value of α (see figure 4 in [10]). This assumption is reasonable in the context of the plasma approximation, as charge neutrality cannot be twice violated. However, as we show in the next section from a consideration of non-neutral solutions, the physically correct solution for η s is that which gives the greater flux. Consequently, the flux is found to vary continuously with the negative ion concentration. 3.2. Non-neutral solutions In the previous section, we considered the properties of solutions for the plasma approximation, q = 0. In this section we relax that assumption and integrate the governing Figure 1. (a) The potential at the sheath plasma boundary and (b) the positive ion flux there as a function of α = n 0 /n e0 for γ = T e /T = 20. In (a) we show that equation (13) admits multiple solutions over a finite range of α. Here the line labelled BA give the transition proposed in [10], while the line labelled SCB gives that found in this paper. In (b) we plot the flux calculated using the solution in (a), and compare it to numerical solutions of the model equations with q>0 (open diamonds and circles). Open circles indicate values of α for which the numerical solutions are oscillatory. The numerically computed flux agrees with the larger value of the flux found for the plasma approximation. ordinary differential equations numerically to calculate the flux at the sheath plasma boundary as a function of negative ion concentration and temperature. The discharge equations (9a) (9d) are solved as an initial value problem rather than as a boundary value problem (BVP). (When posed as a BVP this is an eigenvalue problem, further complicating matters.) That is, given initial values at the centre of the discharge, we integrate equations (9a) (9d) towards the wall. The solution is completely determined by the upstream conditions, since the positive ion flow is solely outwards. To carry out this program, we require a set of consistent initial conditions. In particular, on the centre plane we know u 0, η 0, ε 0 = 0. For q = 0, we have ñ 0 = 1+α, while for q>0, ñ 0 is determined by the cubic equation ñ 2 0 (ñ 0 1 α) = 2q 2. (15) When q 2 1, ñ 0 1+α+2q 2 the space charge imbalance in the plasma is of order q 2. Having found the initial conditions it might seem that we need merely integrate the governing equations using any 459

T E Sheridan et al Figure 3. (a) Positive ion velocity and (b) plasma potential profiles corresponding to the three cases shown in figure 2. Figure 2. Positive ion, negative ion and electron density profiles with q = 10 3, γ = T e /T = 20 and for negative ion concentrations (a) α = n 0 /n e0 = 0.2, (b) 2.6 and (c) 5. In (b), spatial oscillations are seen near the plasma edge. standard numerical technique, for example, the fourth-order Runge Kutta method. However, within the equations lurks the abrupt transition from the plasma to the sheath, so that the differential equations are stiff and an implicit integration scheme is required. We use a fully-implicit, second-order scheme. For example, equation (9a) is written in finitedifference form as ñ i+1 u i+1 ñ i u i ξ = exp( η i+1) +exp( η i ) 2 (16) which is equivalent to the trapezoid rule. After writing each equation in finite-difference form, a system of four nonlinear equations in four unknowns (the values at the next step) results, which is solved iteratively using Newton s method. The scheme is further enhanced by using a variable step size [15] to limit the local truncation error. In particular, many small steps are needed to accurately navigate the plasma sheath transition. Density profiles are shown in figure 2 for q = 10 3 with γ = 20 and α = 0.2, 2.6 and 5. Distances are normalized to the plasma width L. Forα=0.2 (figure 2(a)), we find that the discharge parameters (e.g. the Debye length and ion acoustic speed) are determined primarily by the electron temperature. Qualitatively, these solutions resemble those seen in fullycollisional models [6, 16] the negative ions are confined in the centre of the discharge, with the outer edge of the plasma consisting almost entirely of positive ions and electrons. That is, there is an internal pre-sheath that accelerates positive ions, decreasing their density due to continuity, and confines negative ions, so that the negative ion density at the edge of the plasma is negligible. When the negative ion concentration is large (figure 2(c)), the discharge parameters are determined mostly by the cold, negative ion component, with the electron density remaining nearly constant to the edge of the plasma. Here the negative ions assume the role of electrons in a two-component plasma, with the ion-acoustic velocity and Debye length determined largely by the negative ion temperature. In this case the Bohm speed at the sheath plasma boundary is significantly reduced (figure 3(a)), the sheath thickness decreases (figure 3(b)) and the potential in the plasma is quite flat since the potential drop in the pre-sheath need only be of order kt /e. Between these two cases, we observe a double-layer (i.e. an internal sheath) followed by spatial space-charge oscillations [11], as shown in figure 2(b). In this case, the discharge attempts to form a sheath as in figure 2(c), but fails. As shown in figure 1(b), these oscillatory solutions occur when multiple solutions to equation (13) exist and the flux corresponding to the smaller value of η s is less than that corresponding to the larger. As shown in figure 3(a), at this internal sheath the positive ion flux has not attained the required value and spatial space-charge oscillations result as discussed in [11]. These oscillations are the fluid model analogue of the single double-layer predicted from kinetic 460

A low-pressure electronegative discharge Figure 4. Positive ion flux at the sheath plasma boundary versus negative ion concentration α for negative ion temperatures γ = 5, 10, 20 and 50. Data were computed with the non-neutrality parameter q = 10 3. The dashed lines give the approximate expressions in equation (14). Successive curves are shifted upwards by one unit. theory [17 19]. After a number of oscillations, the Bohm criterion is finally satisfied and a terminal sheath forms. The average potential continues to increase during the oscillations because plasma is being created. To form a sheath two conditions must be met: charge neutrality must be violated and the ion flux must exceed the required Bohm flux. In this case, the former is satisfied, but not the latter. Furthermore, as q approaches zero, the oscillations persist, although their wavelength decreases, while the flux is nearly independent of q. That is, in the oscillatory regime solutions with q>0 differ qualitatively from those for which q = 0. Finally, since the negative ion density is negligible near the wall, the flux at the sheath plasma boundary in the oscillatory regime is a continuation of the small-α (i.e. electron-dominated) regime. The α-dependence of the ion flux exiting the plasma is shown in figure 1(b), where we compare the predictions of the q = 0 theory to values computed numerically for γ = 20. We find that the correct criterion for calculating Ɣ s for the plasma approximation is to choose the larger predicted value of the flux. This differs from the previous proposal [10], which was to choose the flux corresponding to the smaller value of η s. Using our prescription, Ɣ s varies continuously with α, in agreement with the numerical solutions. In fact, the expressions in equation (14) are found to give good approximations to the flux for all α when we choose the flux to be the larger of the two predicted values, and the transition from the electron-dominated to the negative ion-dominated regime occurs very near where the two lines intersect. The dependence of the numerically-computed flux on negative ion concentration and negative ion temperature is shown in figure 4. For α = 0, the flux has a value of 0.506, in good agreement with the analytic (q = 0) value of 1/2. The (normalized) flux depends only weakly on α in both the electron-dominated and oscillatory regimes. (The flux normalized by the positive ion density is decreasing.) In the negative-ion-dominated regime, the flux increases approximately linearly with α (i.e. with the positive ion density). In all cases, the flux is a continuous function of the negative ion concentration. The expressions in equation (14) are plotted for comparison and agree well with the numerical results, particularly for the larger values of γ that one might expect to encounter in plasma processing discharges. (Of course, determining γ accurately might be difficult.) The existence of the oscillatory solutions raises several questions. First, are the oscillations real? They appear to be a general feature of fluid models, having been found in previous work [11], and to represent real physics within the context of the model (i.e. they are not numerical artefacts). However, if kinetic ions are used [17], a single double-layer is formed [18, 19] kinetic ions born at rest (or suffering collisions) will fill the potential valleys that are allowed by the fluid model. Second, do the oscillations effect the flux? The model considered here is widely used for two-component plasmas (i.e. positive ions and electrons) and gives results in good agreement with kinetic theory. The model also produces good results when negative ions are the dominant species. The only question then is how are the electron-dominated and negative-ion-dominated regimes to be joined when the negative ion temperature is low? In this work, we have found that the flux varies continuously across the transition, and the same result has also been found for the collisionless kinetic model [19]. That is, since the flux is an integral of the positive ion source over the entire discharge, it is not effected by the presence (or absence) of space-charge oscillations when q 1. Although both the fluid and kinetic models [17] can be solved for the collisionless limit, it may prove simpler to extend the fluid model to moderately collisional regimes [5]. 4. Summary We have solved a model for a low-pressure plane, symmetric discharge that includes two negative species, each obeying a Boltzmann relation (e.g. electrons and negative ions). Charge neutrality was not assumed, so that the model includes the transition from a quasi-neutral plasma to a non-neutral sheath. Using this model, the positive ion flux exiting the plasma was computed as a function of the negative ion concentration and temperature. When the negative ions are sufficiently cold, we observe spatial space-charge oscillations at the edge of the plasma when two physical solutions for the plasma edge potential exist, and the flux associated with the low potential solution is less than that of the high potential solution. (There is an analogous criterion [19] for the existence of doublelayers [18] in the kinetic model.) These oscillations occur because quasi-neutrality is violated while the positive ions do not yet satisfy the Bohm criterion [11]. However, it is possible to integrate across the oscillations to recover a well defined value for the ion flux. We find that in the oscillatory regime the correct solution for the plasma approximation is that which gives the larger flux. This is contrary to a previous, and widely-used, proposal that the correct solution is that with 461

T E Sheridan et al the smaller value of the potential at the sheath edge. Using our new criterion, we find that the flux varies continuously with the negative ion concentration, in agreement with numerical solutions of the governing equations for the non-neutral case. Simple approximate expressions (14) for the positive ion flux were derived and shown to be in good agreement with numerical solutions of the full set of equations for all values of the negative ion concentration. References [1] Jurgensen C W and Shaqfeh ESG1988 J. Appl. Phys. 64 6200 [2] Sheridan T E and Goree J 1991 Phys. Fluids. B 3 2796 [3] Harrison E R and Thompson W B 1959 Proc. Phys. Soc. (London) 72 2145 [4] Self S A 1963 Phys. Fluids 6 1762 [5] Self S A and Ewald H N 1966 Phys. Fluids 9 2486 [6] Lichtenberg A J, Kouznetsov I G, Lee Y T, Lieberman M A, Kaganovich I D and Tsendin L D 1997 Plasma Sources Sci. Technol. 6 437 [7] BoydRLFandThompson J B 1959 Proc. R. Soc. A 252 102 [8] Amemiya H 1990 J. Phys. D: Appl. Phys. 23 999 [9] Chabert P, Sheridan T E, Boswell R W and Perrin J 1999 Plasma Sources Sci. Technol. submitted [10] Braithwaite N St J and Allen J E 1988 J. Phys. D: Appl. Phys. 21 1733 [11] Schott L 1987 Phys. Fluids 30 1795 [12] Godyak V and Sternberg N 1990 IEEE Trans. Plasma Sci. 18 159 [13] Riemann K-U 1991 J. Phys. D: Appl. Phys. 24 493 [14] Franklin R N and Snell J 1998 J. Phys. D: Appl. Phys. 31 2532 [15] Press W H, Flannery B P, Teukolsky S A and Vetterling W T 1988 Numerical Recipes in C: The Art of Scientific Computing (Cambridge: Cambridge University Press) pp 305 9 [16] Lichtenberg A J, Vahedi V, Lieberman M A and Rognlien T 1994 J. Appl. Phys. 75 2339 [17] Ingram S G and Braithwaite N St J 1988 Mater. Res. Soc. Symp. Proc. 117 281 [18] Sato K and Miyawaki F 1992 Phys. Fluids B 4 1247 [19] Sheridan T E, Braithwaite N St J and Boswell R W Phys. Plasmas at press 462