Collisional sheath dynamics in the intermediate rf frequency regime

Transcription

1 Collisional sheath dynamics in the intermediate rf frequency regime N.Xiang, F.L.Waelbroeck Institute for Fusion Studies, University of Texas Austin,Texas April 16, 4 Abstract A sheath model is proposed for the case when the rf frequency is comparable to or larger than the ion plasma frequency of the bulk plasma and the ion collisionality in the sheath is significant. In this case, the ion momentum equation can be solved easily. We find that the ion velocity in the sheath varies with time and the resulting ion energy distribution is bimodal even though the rf frequency is much larger than the ion plasma frequency in the sheath. The results of the model are compared with the numerical solutions of the fluid equations. Both are in very good agreement. 1

2 1 Introduction Low temperature plasmas are widely used in the semi-conductor industry. In processing plasma, the ion flux, energy distribution and angular distribution on the wafer surface are crucial to the applications. These ion properties depend tightly on the sheath dynamics. Although the majority of plasma processing is operated at low pressure, rf capacitive discharges at intermediate pressure (1- Torr) have been used extensively for high power gas lasers[1, ]. Recently, plasma discharges at an atmospheric pressure have attracted growing attention[3]-[5]. Many theoretical as well as experimental contributions have been made to the understanding of the sheath dynamics[6]-[15]. It has been found that the ion dynamics in a collisionless sheath is characterized by the ratio of the rf frequency and the ion transit frequency crossing the sheath tr. Usually, tr and are used interchangeably in the literature. Kawamura et al. have shown that the ion transit frequency is approximately equal to the ion plasma frequency in the plasma for a collisionless sheath[16]. Many collisionless sheath models in the different rf frequency regimes have been developed[6][1]-[1]. These collisionless sheath models however are only applied to very low pressure discharges where the ion mean free path is much larger than the sheath thickness so that the ion transit frequency across the sheath is much greater than the ion collision frequency. For example, in argon discharges, the ion mean-freepath can be estimated by λ i (33P ) 1, where P is the pressure in torr and λ i is in cm[7]. The typical sheath thickness s m is smaller than 1 cm. Thus, the

3 collisions in the sheath are only negligible when the discharge pressure is lower than 3 mtorr. For higher pressure operations, collisions will play an important role in determining ion dynamics. The collisional sheath model is far less understood than the collisionless model. One possible reason for this is that it is difficult to provide a boundary condition for a collisional sheath model. For a collisionless sheath, the plasma sheath boundary is usually defined at the place where the ion velocity equals the Bohm velocity[17]. However, if the collisions are significant in the sheath, the Bohm criterion no longer holds[17, 18]. There is no conventional way to define the plasma-sheath interface. The presence of the ion collisions adds an extra time scale in the ion dynamic process. In the high frequency regime, Lieberman obtained an analytical solution for a highly collisional sheath[7] by assuming that the ions respond only to the time-average rf field. He found that the sheath capacitance depends only on the sheath thickness and that a sinusoidal sheath current produces a sinusoidal voltage across the sum of two sheaths in a symmetric rf discharges. Regimes of lower collisionality were subsequently investigated numerically in Refs.[8, 19]. In all the above investigations, the discharge parameters were assumed, γ i, here γ i is the ion collision frequency. In the practical operations, however, the discharge parameters are often beyond this range. In an argon plasma with plasma density n i = cm 3, for example, the ion plasma frequency f pi 7.4 MHz, The typical rf frequency f MHz. If the electron temperature T e 4ev, for the discharge pressure 3

4 P 1 mt, the ion mean free path λ i.3cm is a few Debye lengths and much smaller than the sheath thickness. The ion collision frequency can be estimated by γ i V B /λ i 1MHz and is comparable to the rf frequency and the ion plasma frequency. To our knowledge, little research has been done in this parameter range. In this paper, we will discuss the sheath dynamics when the rf frequency is comparable to the ion plasma frequency and ion collision frequency γ i max(, ). We will show that the Liberman collisional sheath model[7] is only valid for the case γ i. If the collision frequency is significant so that γ i max(, ), the ion velocity will vary with time even though the ion density is nearly time-independent. The paper is organized as follows. In Sec.II, the sheath model is described. The ion velocity and ion energy distribution at the electrode are calculated for (i) constant ion collision frequency, and (ii) constant ion mean-free-path. The analytical solutions are presented in Sec.III. The comparisons of the analytical results and the numerical solutions of the fluid equations are shown in Sec.IV. Lastly, the results are summarized in Sec.V. Model Description The ion dynamics in both the plasma and sheath can be described by the following cold fluid model. 4

5 n i t + x (n iv i ) = R i n e, (1) v i t + v v i i x = e φ m i x γ iv i R i v i, () Here n i and v i are ion density and velocity respectively, R i is ionization rate, φ is the electrical potential, m i is the ion mass. γ i is collision frequency. Since the rf bias frequency is usually much smaller than electron plasma frequency, we will use the drift-diffusion approximation to describe the electrons dynamics [, 1]. The continuity equation for the electrons thus becomes n e t + x (n eµ φ x D n e x ) = R in e. (3) Where n e is the electron density, and µ and D are electron mobility and diffusion coefficient respectively. These two coefficients are related by T e = ed µ, where T e is the electron temperature. The system of equation is closed with Poisson s equation, ɛ φ x = e(n e n i ). (4) For the convenience of the numerical solution, we nondimensionalize the Eqs.(1)- (4) by introducing the following dimensionless quantities, τ = t, z = x d, u i = V i V B, N i = ni n, N e = ne n, Φ = φ T e, E = Φ z, ɛ = λ D d, γ = (γi+ri)d V B, ν = R id V B. τ = d V B, α = τ, β = d D. 5

6 Here n is the maximum plasma density, λ D is the Debye length, d is the plasma ɛ length. (λ D = T e T e n ), V B is the Bohm velocity(v B = e m i ), Then Eqs.(1) - (4) become α N i τ + (N iu i ) = νn e, (5) z α N e τ α u i τ + u u i i z = E γu i, (6) + α β z ( N ee N e z ) = νn e, (7) ɛ E z = N e N i. (8) The ion and electron continuity equations imply that the total current is conserved. ɛ E τ + Nu i α N NE + z + β = I (τ). (9) Here I is the total current. The first term in Eq.(9) represents displacement current, while the second and the third represent the ion and electron currents respectively. 3 Analytical solutions In the sheath, we define the ion transit frequency tr = v i x, the spatial derivative terms in the ion continuity and momentum equations thus are proportional to tr. In the collision-dominated case, tr can be estimated by taking e m i E γ i v i, which gives tr pis. Here s is the ion plasma frequency in γ i the sheath and s = ns n. Here n and n s is the ion density in the plasma center and sheath region respectively(see fig.1). Since the ion density drops rapidly in the sheath, n s n away from the sheath boundary. We consider 6

7 1.9 ion density electron density.8 n at center ns(sheath) z Figure 1: Schematic representation of the ion density in the plasma and sheath regions. the case γ i max(, ). Hence, tr < s γ i, which means the convection term in the ion momentum equation can be neglected. The ion momentum equation thus becomes, α u i τ = E γu i, (1) Since the ionization rate is much smaller than the rf frequency, the ionization can be neglected in the sheath. If is comparable to, then tr. The leading term in the ion continuity equation gives N i τ =, (11) Taking the time average of the first order terms in the ion continuity equation, we obtain N i u i = J i = const. (1) 7

8 We thus have n i n i (x) + O( tr ): the ion density is nearly time-independent. In the sheath, the ratio of the displacement current and the ion current can be estimated by J d J i = ɛ E t en i v i n γ n s. If γ pi i pi, then J d J i displacement current is dominant in the sheath. We assume that the total current is sinusoidal so that 1. This means the ɛ E τ = J sin τ (13) Here J is constant. Integrating the above equation as in Ref.[6], we obtain the electric field, E (cos τ cos ϕ), for s(τ) < z, E =, for s(τ) > z. (14) Where E = J ɛ and s represents the electron step front. At τ = ϕ, z = s(τ). We consider the following two cases. 3-A Constant ion collision frequency γ i Since E = 1 ɛ z s N i(z)dz, the electron sheath front is governed by N i (z) dz dϕ = J sin ϕ (15) Since N i (z) = J i /u i = (γj i )/E and the time-average electric field is E = E π (sin ϕ ϕ cos ϕ). Thus, dz dϕ = J E πγj i sin ϕ(sin ϕ ϕ cos ϕ) (16) 8

9 The sheath voltage is then given by Φ = s z Edz = π τ eq.(16 ), we obtain for < τ < π, E dz dϕdϕ. By using Φ = A [ π π cos τ 1 4 τ cos τ 1 1 τ cos τ cos (τ) + τ cos (3τ) cos τ sin (τ) 1 sin (3τ)] (17) 9 Here A = J E πγj i. Expand Φ = k= Φ k cos kτ. We obtain, Φ = ( π π )A, Φ 1 = 9πA 3, Φ = 3A 75π, Φ 3 = πa 96. The second harmonic is 15.4 percent of the fundamental, and the third harmonic is 3.3 percent of the fundamental. For a symmetrically driven discharge, the phase difference between the grounded and powered sheaths is π. Hence, the drop across the two sheath Φ s Φ 1 cos τ. The effective capacitance C sh, given by C sh = I, is the constant. dφs dτ The ion energy at the cathode is one of the parameters we are interested in. Inserting Eq.(14) into eq.(1), we have u i = u i (τ )e γ α (τ τ) + E α [ γ α cos τ+sin τ γ 1+( γ α cos τ+sin τ + α ) 1+( γ α ) ( γ α ) e γ α (τ τ ) E γ cos ϕ(1 e γ α (τ τ ) )] for s(τ) < z, (18) u i (τ )e γ α (τ τ), for s(τ) > z. In the cathode, ϕ = π. We assume at τ = τ, u i = u i is the time-average velocity. Note γ α = γi, after a few rf periods, the exponential term can be 9

10 neglected, The ion velocity at the cathode is then given by u i = 1 + γ γi i u i [ cos τ + sin τ 1 + ( γ ] (19) i ) Setting η = γi and sin θ = η, eq.(19) can be rewritten as 1+η Where u i = E γ. u i η = 1 + sin (τ + θ) () u i 1 + η By eq.(), we see that only when γ i, u i u i. Clearly, Liberman s collisional sheath model is valid only in this limit. If the ion collision frequency is comparable to the rf frequency, the ion velocity varies with time and the normalized amplitude of the fluctuation of the ion velocity is less than 1 for any collision frequency. At high pressure, elastic scattering dominates over charge exchange collisions, the ion energy distribution(ied) P (ε) is given by Here J i = π J i (τ)dτ and ε = 1 u i By eq.(19), we obtain P (ε) = J i(τ) dε J i dτ 1 (1) is the normalized ion energy. P (ε) = 1 u i du i dτ 1 () P (ε) = ( γ ) 1 E ( ε max ε)( ε ε min ) (3) The IED is bimodal with its two peaks corresponding to the minimum and maximum ion energies given by ε max = 1 (u η η ) ( η ), (4) 1

11 ε min = 1 (u η η ) (1 1 + η ). (5) Where u = E α. Clearly, ε max and ε min decrease with the collisional frequency. The width of the IED is ε = u η 1 + η. (6) We see that the IED width decreases with η. If we define the time-average ion energy ε = 1 u i, then ε = u η +η 1+η, thus, ε = 4 η 1+η +η ε. For η 1, the width is about four times of the ion time-average energy. 3-B Constant ion mean free path λ i For constant ion mean free path γ = ui λi λ. Here λ = d is the normalized ion mean free path. The sheath thickness can be estimated from the collisional form of the Child law[], s m λ 4/5 Ds λ1/5 i Φ 3/5 ( v B vs ) /5, here Φ is the sheath voltage, λ Ds is the Debye length in the sheath, and v s is the ion velocity in the sheath. Hence, when λ i < Φ 3/4 λ Ds, collisions are important. Eq.(1) then becomes α u i τ = E u i λ. (7) If αλ 1, we can obtain ui τ = and u i λ = E. This corresponds to Lieberman s model[7]. However, if the rf frequency is smaller and the ion collisions are significant, so that αλ 1, the ion velocity is time-dependent. The electric field is still given by Eq.(14). At the cathode, (7) becomes α u i τ = E (1 + cos τ) u i λ. (8) 11

12 We can show that the solutions for Eq.(8) are periodic(see appendix). Although this equation can not be solved analytically, we can still gain some useful information. The width of the IED can be obtained immediately by eq.(8), ε = λe cos τ max (9) Here τ max is the time when u i reaches its maximum. If ε represents the timeaverage ion energy, then ε = λe /. Thus ε = ε cos τ max (3) This means the IED width is less than the twice ion average energy. Eq.(8) is a first order ordinary differential equation and can be readily solved numerically. We are interested in the case when the ion velocity in the sheath is greater than the Bohm velocity. If λ i pi λ D, then α u i /λ, the collision term is then dominant. Thus, we have u i = λe (1 + cos τ) (31) Hence, the minimum ion energy ε min = and maximum energy ε max = λe. The IED is P (ε) = J i 1 (3) J i 1 (1 ε ε max ) Two peaks appear at ε min and ε max. 4 Comparison with the numerical results To evaluate the assumptions used in the analysis of the previous section, we have solved Eqs.(5)-(8) numerically for a plasma contained between two electrodes 1

13 Figure : Schematic representation of the plasma and rf circuit model. and separated from each electrode by a sheath. Fig. represents schematically the plasma and rf circuit model we used for the numerical calculations. The grounded electrode is then placed at z= while the biased electrode is at z=1. A weak constant ionization rate which is much smaller than the ion collision frequency is assumed to maintain the plasma. The numerical method and the boundary conditions are the same as used in Ref.[3]. 4-A Calculation for a constant collision frequency We choose =. and the bias voltage Φ = 1. This is beyond the range of the Liberman collisional sheath model. Usually, the ion plasma frequency in the sheath is about one order smaller than the ion plasma frequency in the bulk plasma. Thus it is expected that s. If the ion collision frequency 13

14 (a) / =.,η=.5 (b) η=1 8 J i /α J e /β J d 8 J i /α J e /β J d current x 1 3 current x Figure 3: results for the ion current, electron current and the displacement current for =.. is high enough so that γ i, then the ion transit frequency tr and J d J i γ pis 1 and our model is valid. The corresponding results are shown in figs.-7. Fig.3 illustrates the numerical results for the ion, electron and displacement currents for η =.5 and 1. Correspondingly, γ i =.5 and respectively. We see that the ratio of the displacement current and the ion current increases with the collision frequency. Note that the ion current varies with time. This is because the ion velocity depends on time as shown by eq.(19). Clearly, the larger η is, the better the total current can be approximated by the sinusoidal function. Even though the displacement current is not described well by a sinusoidal function in the case η =.5, the electric field still can be approximated 14

15 1 9 (a) / =.,η=.5 E (1+cosτ) 1 9 (b) η=1 E (1+cosτ) Figure 4: and sinusoidal electric fields at the cathode for =.. as a sinusoidal function. Fig.4 compares the numerical and sinusoidal electric fields at the cathode. Again, the larger η is, the better agreement we obtain. The normalized ion density at the cathode is given in fig.5 for η =.5, 5 and 1. It can be seen that the larger η is, the smaller the amplitude of the fluctuation of the ion density is. This is because the ion transit frequency decreases as the collision frequency increases. Since the ion density n i n i (x)+ O( tr ), the ion density becomes less dependent of time with a higher collision frequency. Note that even with a significant time-variation of the ion density, the ion velocity at the cathode obtained by eq.(19) is still in good agreement with the numerical result. Fig.6 shows the comparisons betweens the numerical and analytical ion velocities at the cathode. The agreements are excellent for 15

16 .3.5 Normalized n i (/ =.) η=.5 η=5 η= Figure 5: Normalized ion density at the cathode for =...5 (a): / =.,η=.5 Analytical.5 (b) : η=5 Analytical.5 (c) : η=1 Analytical u i /u Figure 6: Comparisons of the numerical and analytical ion velocities at the cathode for =.. 16

17 .5 (a): / =,η=.5 Analytical.5 (b) : η=.5 Analytical.5 (c) : η=1.5 Analytical u i /u Figure 7: Comparisons of the numerical and analytical ion velocities at the cathode for =. 17

18 (a) / =.,η=1 Analytical (b) / =,η=1.5 Analytical Figure 8: Comparisons of the numerical and analytical sheath voltages for =., η = 1 and =, η =

19 large η. With rf frequency increasing, the ion density becomes less dependent of time while the ion velocity however varies with time if the ion collision frequency is high enough. Fig.7 shows the comparisons of the analytical and numerical ion velocities at the cathode with = for η =.5,.5 and 1.5. The agreements are very good for η =.5 and 1.5. The discrepancy is obvious for η =.5, this is because the convection term is important in this case. For =, the rf frequency is much larger than the ion plasma frequency in the sheath s ( s 15 ). It is expected that the ion density is nearly time-independent, which is confirmed by the numerical results. As long as the collision frequency is comparable to or higher than the rf frequency, however, the ion velocity varies with time instead of being constant as predicted by the collisional high frequency sheath model[7]. The analytical and numerical sheath voltages are plotted in fig.8 for γi =.5 and. The numerical sheath voltages are obtained by defining the plasmasheath interface at (N i N e )/N i =.1. We found that the value of (N i N e )/N i used to define the interface does not affect the sheath voltage significantly. The reasonable agreements between the analytical and numerical results verify the sheath model. Lastly, the ion energy distribution is plotted in fig.9 for different ion collision frequencies(eq.(3)). Note the energy width decreases rapidly with the ion collision frequency. 19

20 3 5 Ion energy distribution for u = 5 η= η=4 η=6 η= Ion energy Figure 9: The ion energy distribution ( taking u = 5 ) for different η. 4-B Calculation for a constant ion mean free path We assume the constant ion mean free path is related to the gas pressure by λ i (33P ) 1, Fig.1-13 illustrate the results for a constant ion mean free path. The bias voltage Φ = 1. Fig.1 shows the numerical results for the ion, electron and displacement currents for different pressures (P = 1 mt and 4 mt, correspondingly, λ D λi.5 and.98). As the pressure increases, the ratio of the displacement current and the ion current increases. Although the displacement current is poorly described by a sinusoidal function, the electric field can be approximated by a sinusoidal function. The ion density at the cathode is plotted in fig.11 for P = 1 mt, mt and 4 mt. Clearly, the ion density varies with time. The

21 4 3 / =.,P=1mT J i /α J e /β J d 4 3 (b) P=4mT J i /α J e /β J d 1 1 current x 1 3 current x Figure 1: results for the ion current, electron current and the displacement current for =.. amplitude of the fluctuation decreases with the pressure. Note that compared to the constant frequency case, the ion density at the cathode is higher. For P = 4 mt, the collision frequency in the sheath is much larger than the rf frequency, the collisional term is dominant in the ion momentum equation and the ion velocity is given by eq.(31). The ion velocities obtained by eq.(31) are compared with the numerical results in fig.1 for P = 1 mt, mt and 4 mt. The agreement becomes better for higher pressure. The discrepancies exist for small u i. This is because the collision frequency is lowered for small u i and the time-derivative terms becomes important in this time interval. For higher rf frequency, the ion density depends less on time and the displace- 1

22 ..18 Normalized n i (/ =.) P=1mT P=mT P=4mT Figure 11: Normalized ion density at the cathode for =.. 6 / =.,P=1mT Analytical 6 (b) : P=mT Analytical 6 (c) : P=4mT Analytical u i /u Figure 1: Comparisons of the numerical and analytical ion velocities at the cathode for =..

23 5 4.5 (a) / =,P=1mT by Eq.(.4.19) (b) P=mT by Eq.(.4.19) u i Figure 13: Comparisons of the ion velocities from numerical simulation and solving eq.(.5.19) for =. 3

24 ment becomes dominant. The ion velocity at the cathode is described by eq.(8) for λ i S m. Fig.13 shows the ion velocity given by eq.(8) is in good agreement with the numerical results. Again, the ion velocity varies with time. 5 Conclusions We have presented an analytical solution for the sheath dynamics in the intermediate frequency regime when the rf frequency is comparable to the ion plasma frequency and the ion collisions are significant so that γ i max(, ). In this case, the ion transit frequency across the sheath is much smaller than the collision frequency and the convection term in the ion momentum equation can be neglected. Furthermore, the displacement current is much larger than the conduction current. Our analytical calculations and numerical simulations show that a sinusoidally biased voltage produces a sinusoidal sheath current in this parameter range. Hence, the electric field also has a sinusoidal form. The ion momentum equation at the cathode becomes a first order ordinary differential equation which is easily solved. If the ion collision frequency is assumed to be a constant, the ion velocity in the cathode can be obtained analytically. We have shown that if γ i, the ion velocity varies with time and the resulting IED is bimodal with a width up to four times of the time-averaged ion energy. If the ion mean free path is assumed to be constant, the ion velocity can be obtained numerically by solving the first-order ordinary differential equation. When the ion mean free 4

25 path is much shorter than the sheath thickness, the ion velocity at the cathode depends on time and the resulting IED is bimodal with a width up to twice the time-averaged ion energy. We compared the results of the model with the numerical results over a wide frequency range from / =. to, they are in a good agreement. Our results show that even though the rf frequency is much larger than the ion plasma frequency in the sheath, the ion velocity varies with time provided the ion collision frequency is high enough. This conclusion indicates the assumption that the ions respond only to the time-average field in the high frequency regime[7, 8, 18], is valid only for the case γ i,. Appendix: Making the transformation τ = T and u i = αλ y τ y, Eq.(8) becomes y T + (a q cos (T ))y =. (A1) Here a = 4 E o α λ and q = E o α λ. Eq.(A1) is the Mathieu equation. The general solution is[4] y(t ) = Ae µt ϕ(t ) + Be µt ϕ( T ). (A) Here A, B and µ are constants, ϕ is a periodic function with period π. If µ is pure imaginary, y(x) oscillates aperiodically. Generally, µ has a real part, thus, after a sufficient long time, the unstable mode is dominant. Takeing y(t ) = Ae µt ϕ(t ) for T 1, we find u(τ) = αλ d dϕ log y = αλ(µ + dt dt ). (A3) Clearly u(τ) is periodic with period π. 5

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28 Captions: Fig.1 Schematic representation of the ion density in the plasma and sheath regions. Fig. Schematic representation of the plasma and rf circuit model. Fig.3 results for the ion current, electron current and the displacement current for =.. Fig.4 and sinusoidal electric fields at the cathode for Fig.5 Normalized ion density at the cathode for =.. =.. Fig.6 Comparisons of the numerical and analytical ion velocities at the cathode for =.. Fig.7 Comparisons of the numerical and analytical ion velocities at the cathode for =. Fig.8 Comparisons of the numerical and analytical sheath voltages for., η = 1 and =, η = 1.5. Fig.9 The ion energy distribution ( taking u = 5 ) for different η. = Fig.1 results for the ion current, electron current and the displacement current for =.. Fig.11 Normalized ion density at the cathode for =.. Fig.1 Comparisons of the numerical and analytical ion velocities at the cathode for =.. Fig.13 Comparisons of the ion velocities from numerical simulation and solving eq.(8) for =. 8