Characteristics of Positive Ions in the Sheath Region of Magnetized Collisional Electronegative Discharges

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1 Plasma Science and Technology, Vol.6, No.6, Jun. 204 Characteristics of Positive Ions in the Sheath Region of Magnetized Collisional Electronegative Discharges M. M. HATAMI, A. R. NIKNAM 2 Physics Department of K. N. Toosi University of Technology, , Tehran, Iran 2 Laser-Plasma Research Institute of Shahid Beheshti University, G. C., , Tehran, Iran Abstract A hydrodynamic model is used to investigate the characteristics of positive ions in the sheath region of a low-pressure magnetized electronegative discharge. Positive ions are modeled as a cold fluid, while the electron and negative ion density distributions obey the Boltzmann distribution with two different temperatures. By taking into account the ion-neutral collision effect in the sheath region and assuming that the momentum transfer cross section has a power law dependence on the velocity of positive ions, the sheath formation criterion (modified Bohm s criterion) is derived and it is shown that there are specified maximum and minimum limits for the ion Mach number M. Considering these two limits of M, the behaviors of electrostatic potential, charged particle density distributions and positive ion velocities in the sheath region are studied for different values of ion-neutral collision frequency. Keywords: electronegative plasma, hydrodynamic model, magnetized plasma sheath, Bohm s criterion PACS: Kh, Cj DOI: 0.088/ /6/6/02 (Some figures may appear in colour only in the online journal) Introduction Today, direct-current (DC) discharges have found many practical applications [,2], and thus many authors have made great efforts to investigate the characteristics of these kinds of discharges [3 8]. One of the most important properties of the electronegative and electropositive discharges is the formation of a positive ion rich region, called the sheath, near metallic surfaces bounding the discharge volume. Many of the well-known applications of discharges are closely related to the formation of this region [ 3]. Therefore, the investigation of the sheath formation criterion and the characteristics of this region, such as the charged particle velocities, density distributions and electrostatic potential, is of great importance. In comparison with the electropositive discharges, the presence of negative ions in a discharge can give rise to a quantitatively new behavior not seen in the electropositive case [,3 6,9 ]. For example, the Bohm s criterion, which determines the minimum allowable velocity of an ion at the sheath edge to enter into the sheath region, takes a different form in electronegative plasmas [0 3]. Many authors have investigated the plasma sheath formation in an electronegative discharge under different conditions [3 8]. For example, Sheridan et al. [3] considered a low-pressure, planar, electronegative discharge and investigated the flux of positive ions existing in this discharge as a function of the negative ion concentration and temperature. Also, in Ref. [4], Sheridan studied the effect of ion-neutral collisions on the structure and ion flux emanating from a steady-state, planar discharge with two negative components. Furthermore, using the fluid model with a constant collision frequency for electrons, positive ions and negative ions, the characteristics of electronegative plasma were examined by Franklin [5]. In the present work, some characteristics of a lowpressure electronegative plasma sheath are investigated under the action of an external constant magnetic field. These plasmas have many applications in plasma processing (material surface treatment by magnetron), etching, plasma chemistry, and thin-film deposition [,2]. As the first step in this investigation, a modified Bohm s sheath criterion for collisional magnetized electronegative plasmas is derived and then the properties of the sheath region in these plasmas are studied numerically. In this derivation, the collision between positive ions and neutrals is discussed in a general form, i.e., it is assumed that the momentum transfer cross section has a power law dependence on the positive ion velocity. Two specific cases of this general form are: (i) constant cross section (constant ion mean-free path) and (ii) constant collisional mobility. Due to the fact that the plasma is low-pressure and is not highly electronegative, to a good approximation the electron and negative ion densities obey the Boltzmann distribution with two different temperatures [3 5,7,8]. It is worth- 552

2 M. M. HATAMI et al.: Characteristics of Positive Ions in the Sheath Region while mentioning that by considering the effects of external magnetic field and positive ion-neutral collision as well as power law dependence of momentum transfer cross section on positive ion velocity, the present work is different from other similar studies on electronegative discharges [3 8,9 ]. Firstly, our results show that there are upper and lower limits for the sheath criterion (the ion Mach number M [,8,9] ) and the characteristics of the negative ion, such as density (at the sheath edge) and temperature, affect only the lower limit. And then, it is shown that the increase of ion-neutral collision frequency causes the electron density distribution to decrease faster in the depth direction. Moreover, the electrostatic potential of the sheath region increases as the magnitude of the collision frequency increases. Furthermore, by going toward the wall, the curve of the net charged particle densities N net decreases nonmonotonically and the amplitude of the fluctuations produced in this curve increases with increasing. Finally, the three-dimensional velocity of positive ions in the sheath region is depicted for different collisional regimes (cases (i) and (ii)), and the results are compared with each other. In section 2 we describe the model, define appropriate dimensionless variables and derive the modified Bohm s sheath criterion for collisional magnetized electronegative plasmas. In section 3 the results obtained are discussed and conclusions are drawn in section 4. 2 Sheath model and basic equations We use the fluid model to investigate a low-pressure, steady-state, planar electronegative plasma sheath. We assume that the external magnetic field is weak and the low-pressure plasma is not highly electronegative. Therefore, the negative ion density obeys the Boltzmann distribution [5,4]. Also, the wall potential is specified by a given value ϕ w, which can be the floating potential or a more negative value for which Boltzmann distribution holds, i.e., ϕ w > 3T e /e [7,5,6]. Therefore, the electron and negative ion density distributions are written as follows: n e = n 0e exp( eϕ T e ), () n = n 0 exp( eϕ T ), (2) where e, n 0e, n 0 and ϕ are the unit electric charge, the electron and negative ion densities at the sheath edge and the electrostatic potential, respectively. Also, n e, n, T e and T are the electron and negative ion densities and temperatures, respectively. The quasineutrality condition of the electronegative discharge at the plasma-sheath edge is written as follows: n 0+ = n 0e + n 0, (3) where n 0+ is the positive ion densities at the sheath edge. Since the discharge is low-pressure and the positive ions are cold, the steady-state continuity and momentum transfer equations for positive ions are expressed as follows [3,4,8,9] : (n + V + ) = 0, (4) m + (V + ) V + = e (E + V + B 0 ) F c, (5) where m +, n + and V + are the mass, density and velocity of the positive ions, respectively. Also, E, B 0 and F c = m + ν + V + are the electric field, the external magnetic field and the drag force that positive ions experience as they travel through the sheath, respectively. Considering that the electron energy is not very high, it is reasonable to neglect ionization in Eq. (4). Also, as seen in Fig., it is assumed that the external magnetic field lies in the x-z plane and forms an angle θ with the x coordinate (the depth direction). Fig. Schematic geometry of the sheath model The collision frequency of positive ions with neutral particles ν + is expressed as follows [] : ν + = n n σv +, (6) where n n and σ are the neutral gas density and the momentum transfer cross section for collisions of positive ions with neutral particles, respectively. In a general case, we can consider the power law dependence of σ on V + as follows [7] : ( ) γ V+ σ = σ s, (7) c s where γ is a dimensionless parameter ranging from (constant collisional mobility) to 0 (constant cross section), σ s is the cross section measured at that velocityand c s = (T e /m + ) /2 is the ion acoustic velocity. Therefore, the ion collision frequency can be expressed as ( ) γ V+ ν + = n n σ s V +. (8) c s The Poisson s equation for such an electronegative discharge consisting of singly charged negative ions, positive ions and electrons is written as follows: 2 ϕ = e ε 0 (n + n n e ), (9) where ε 0 is the electric permittivity of free space. 553

3 Here, we assume that the physical quantities such as density of positive ions and electrons change only along the x direction (normal to the wall). Therefore, is replaced with / x in the above equations. To normalize the model equations, (Eqs. ()-(9)), we introduce the following variables u = V + /c s, N + = n + /n 0+, N e = n e /n 0e, ξ = x/λ De, N = n /n 0, η = eϕ/t e, ε = T e /T, where λ De = ( ε 0 T e /n 0 e 2) /2 is the electron Debye length. Using these variables, the dimensionless model equations describing the sheath region of an electronegative discharge can be written as follows: N e = exp( η), (0) N + = M, () ξ = η ξ + ρu y sin θ u γ+, (2) u y ξ = ρ [ sin θ + u z cos θ] u γ+ u y, (3) u z ξ = ρu y cos θ u γ+ u z, (4) 2 η ξ 2 = [N + δn ( δ) N e ], (5) δ where M = V x+ (x = 0)/c s is the ion Mach number, δ = n 0 /n 0+, N = exp( εη), = n n σ s λ De is a dimensionless parameter that characterizes the degree of collisionality in the sheath, ρ = λ De /r where ρ is proportional to the magnitude of the external magnetic field and r = ( m + T e /e 2 B0) 2 /2 is the positive ion gyroradius. 2. Bohm s criterion Here, we obtain the Bohm s criterion for magnetized electronegative plasmas consisting of cold and collisional positive ions. The first integral of Eq. (5) with the boundary conditions of η 0 = 0 and η 0 0 at the sheath edge (ξ = 0) leads to η 2 = η 2 0 2φ(η, M), (6) where the prime symbol denotes differentiation with respect to ξ and φ(η, M) = exp( η) + δ [ exp( εη)] ε( δ) M η dη. (7) δ 0 Here, η 0 and φ are the dimensionless electric field at the sheath edge and the Sagdeev potential satisfying the boundary conditions of φ(0, M) = 0 and φ(0, M)/ η = 0, respectively. From Eq. (7), we have 2 φ (η, M) η 2 = exp( η) ( εδ δ ) exp( εη) Plasma Science and Technology, Vol.6, No.6, Jun ( M δ ) u x η u 2. (8) x Considering the condition of maximizing the Sagdeev potential at the sheath edge, i.e., 2 φ(0, M)/ η 2 < 0, one can easily obtain u 0x M < η 0[ + δ (ε )]. (9) In addition, at the sheath edge, Eq. (2) can be written as follows: Mu 0x = η 0 + ρu 0y sin θ M(M 2 + u 2 0y + u 2 0z) (γ+)/2. (20) Here, we assume that the positive ions enter into the sheath region normally to the wall. Therefore, Eq. (2) takes the following form: Mu 0x = η 0 M γ+2. (2) It is evident that u 0x 0 due to neutral drag to the positive ions in the plasma [8]. Therefore, the necessary condition for positive ions to enter into the sheath region is η 0 > 0, which means that there must exist an accelerating force to overcome the drag force. Therefore, from Eq. (2), the upper limit of the Bohm s criterion (ion Mach number) can be found as: M ( ) η /(γ+2) 0. (22) From inequality Eq. (9) and Eq. (2), the lower limit of the Bohm s criterion is determined as: ( 2η 0 + δ(ε ) for γ = and ) 2 + ( ) + δ(ε ) 2η 0 + δ(ε ) M, (23) + M, (24) η + δ(ε ) 0 for γ = 0. Similar to collisional electropositive plasma [6], it is seen that there are upper and lower limits for the Bohm s criterion (ion Mach number) in collisional magnetized electronegative plasmas. For those values of M that do not satisfy the Bohm s criterion, the plasma sheath region, i.e., the positive ion-rich region between the plasma and the wall does not form. To compare this work with electropositive ones, it is sufficient to set δ = 0 in Eqs. (22) and (23). Since the upper limit of the Bohm s criterion is independent of δ, this limit is the same for collisional magnetized electropositive and electronegative plasmas while the lower limit of the Bohm s criterion (M min ) increases with decreasing δ. Therefore, in collisional magnetized electropositive plasmas M min is greater than that in an electronegative ones. 554

4 M. M. HATAMI et al.: Characteristics of Positive Ions in the Sheath Region 3 Results and discussions In this section, we solve Eqs. (0)-(5) numerically and investigate important characteristics of the sheath region by considering the collision effect. For clarity, we take ρ = 2, θ = 30, ε = 0 and δ = 0.2. The effects of different values of ρ, θ, ε and δ on the sheath characteristics have been discussed previously [5,8,9,9]. Also, using the definition of the sheath region [,2], the following initial conditions are used to solve Eqs. (0)(5): u = (M, 0, 0), η = 0, η00 = 0.. In this work, we investigate the case γ = 0. Therefore, using the above initial values and parameters, the allowable values of M can be determined from the following relation: r r M. (25) Fig. 2 shows the variation of the upper and lower limits of Bohm s criterion with for γ = 0. As seen from this figure, both upper and lower limits of the Bohm s criterion decrease with increasing. According to relation (25) and Fig. 2, we choose M = 0.68 which is an allowable value of the ion Mach number for = Fig.3 Normalized density distribution of positive ions (a) and electrons (b) in the sheath region for different values of Fig.4 Variation of the normalized electrostatic potential in the sheath region for different values of Fig.2 The lower (Mmin ) and upper (Mmax ) limits of the Bohm s sheath criterion for γ = 0 versus ion-neutral collision frequency The variation of net density of the charged particles Nnet = n+ + n ne in the sheath region is shown in Fig. 5. This figure shows that the increase of causes the amplitude of fluctuations of Nnet curve near the sheath edge to increase. Moving toward the wall, it is seen that for ξ > 5 these fluctuations disappear and Nnet curves approach to a constant limit. Also, in spite of the same initial behavior of Nnet curves for different values of, it is seen that these curves become different as the distance from the sheath edge increases. The reason for this behavior is as follows: near the sheath edge( ξ <.5), the positive ion velocity is small in comparison with its values for greater ξ. Therefore, the collision force and so the right-hand side of Eq. (2) does not change considerably for different values of near the sheath edge. Considering Eq. (), it is deduced that in the vicinity of the sheath edge the behavior of the curves of N+ has to be the same for different values of (see Fig. 3 (a)). On the other hand, for ξ <.5, the electron density distribution does not change significantly for different values of due to the fact that near the sheath edge the electrostatic potential of the sheath region, η, has nearly the same behavior for different values of (see Figs. 4 and 3(b)). Fig. 3 shows the behavior of the positive ion and electron density distributions for different values of the collision frequency (). In Fig. 3(a), it is shown that the increase of the collision frequency of positive ions with neutrals leads to increase of the normalized density distribution of the positive ions N+ near the sheath edge (ξ <.5). Furthermore, it is seen that for ξ.5, the slope of N+ curve deceases with increasing. Moreover, our numerical results show that an increase of the magnitude of ion-neutral collision frequency can affect the normalized density distribution of electrons in the sheath region; and as this frequency increases, the density distribution of the electrons decreases faster (see Fig. 3(b)). Another important and practically useful characteristic of the sheath region studied here is the electrostatic potential of the sheath region η. Fig. 4 shows the variations of η in the depth direction for different values of. From this figure, one can see that the normalized electrostatic potential of the sheath region increases with increasing. 555

5 Plasma Science and Technology, Vol.6, No.6, Jun. 204 transfer cross section on the positive ion velocities. In this case, using hydrodynamic model, it was found that when the momentum transfer cross section is constant the electron density distribution decreases faster in the depth direction as the magnitude of the collision force increases. Furthermore, it was shown that as the magnitude of the collision force increases, the net density of the charged particles in the sheath region, Nnet, and the electrostatic potential of the sheath region, η, increase nonmonotonically and monotonically, respectively. Finally, the three-dimensional velocity of positive ions is depicted for both the case of constant cross section and the case of constant collisional mobility, and the results are compared with each other. And it was found that in the case of constant cross section the z component of the positive ion velocity is larger than that in the case of constant mobility. Also, it was shown that in the case of a constant cross section the positive ions gyrate on a larger radius. Fig.5 Variation of the net charged particle density distributions in the sheath region for different values of Finally, we would like to show the effect of collision on the motion of positive ions into the sheath region of magnetized electronegative plasmas in two different regimes i.e., γ = 0 and γ =. To do this, the three-dimensional velocity of positive ions in the sheath region is sketched for two specific cases of constant cross section (γ = 0) and constant collisional mobility (γ = ) in Fig. 6(a) and (b), respectively. It is observed that for γ = the gyroradius of the positive ions is smaller than that for γ = 0. Also, in the case γ = the z component of the positive ion velocities is larger than that in the case γ = 0. References Fig.6 The three-dimensional velocity of positive ions in the sheath region with = 0. for (a) γ = 0 and (b) γ = Conclusion The Bohm s sheath criterion and the characteristics of the sheath region of low-pressure, magnetized electronegative plasmas consisting of electrons, positive and negative ions were investigated. The electrons and negative ions were assumed to obey the Boltzmann relation with different temperatures. Also, it was assumed that there is a power law dependence of the momentum 9 Liebermann M A, Lichtenberg A J. 994, Principles of Plasma Discharges and Materials Processing. Wiley, New York Roth J R. 995, Industrial Plasma Engineering. IOP Publishing, Philadelphia Sheridan T E, Chabert P and Boswell R W. 999, Plasma Sources Sci. Technol., 8: 457 Sheridan T E. 999, J. Phys. D: Appl. Phys., 32: 76 Franklin R N, Snell J. 2000, J. Plasma Phys., 64: 3 Franklin R N. 2005, J. Phys. D: Appl. Phys., 38: 2790 Fern andez Palop J I, Ballesteros J, Hern andez M A, and Morales Crespo R. 2007, Plasma Sources Sci. Technol., 6: S76S86 Hatami M M, Niknam A R, Shokri B. 2008, Phys. Plasmas, 5: 2350 Hatami M M, Niknam A R, Shokri B and Ghomi H. 2008, Phys. Plasmas, 5: Boyd R L F and Thompson J B. 959, Proc. Roy. Soc. London A, 252: 02 Braithwaite N St J and Allen J E. 988, J. Phys. D: Appl. Phys., 2: 733 Duan P, Wang Z, Wang W, et al. 2005, Plasma Sci. Technol., 7: 2649 Wang Z X, Liu J Y, Liu Y, and Wang X. 2005, Phys. Plasmas, 2: 0204 Fernsler R F and Slinker S P. 2005, Phys. Rev. E, 7: Davoudabadi M and Mashayek F. 2005, Phys. Plasmas, 2: Self S A. 963, Phys. Fluids, 6: 762 Sheridan T E, Goree J. 99, Phys. Fluid B, 3: 2796 Liu J Y, Wang Z X, Wang X. 2003, Phys. Plasmas, 0: 3032 Franklin R N and Snell J. 999, J. Phys. D: Appl. Phys., 32: 03 (Manuscript received 5 March 203) (Manuscript accepted 7 June 203) address of M. M. HATAMI: m hatami@kntu.ac.ir 556