Effect of a dielectric layer on plasma uniformity in high frequency electronegative capacitive discharges

Transcription

1 Effect of a dielectric layer on plasma uniformity in high frequency electronegative capacitive discharges Emi KawamuraDe-Qi WenMichael A. Lieberman and Allan J. Lichtenberg Citation: Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 35, 05C311 (2017); doi: / View online: View Table of Contents: Published by the American Vacuum Society

2 Effect of a dielectric layer on plasma uniformity in high frequency electronegative capacitive discharges Emi Kawamura a) Department of Electrical Engineering, University of California, Berkeley, California De-Qi Wen Department of Electrical Engineering, University of California, Berkeley, California and School of Physics and Optoelectronics Engineering, Dalian University of Technology, Dalian , People s Republic of China Michael A. Lieberman and Allan J. Lichtenberg Department of Electrical Engineering, University of California, Berkeley, California (Received 28 April 2017; accepted 28 June 2017; published 13 July 2017) The authors use a fast 2D axisymmetric fluid-analytical code to study the effect of adding a dielectric layer over the wafer electrode of a high frequency capacitively coupled plasma (CCP) reactor. At higher frequencies and larger areas, the wavelengths of the radially propagating surface waves in the plasma can become significantly shorter than the reactor dimensions, leading to center-high plasma nonuniformities. These wavelengths increase with increasing sheath widths, suggesting that a method to suppress wave effects in a high frequency CCP is to increase the effective sheath width by adding a dielectric layer over the wafer electrode. The authors conducted simulations with and without a dielectric layer and found that the dielectric layer improved plasma uniformity. The authors also studied the effect of adding a thin conducting or resistive silicon wafer above the dielectric layer and found that a conducting silicon wafer shorts out the fields and shields the discharge from the dielectric layer, while the resistive silicon wafer allows the fields to pass through to the dielectric layer. VC 2017 American Vacuum Society.[ I. INTRODUCTION In this work, we study the effect of adding a dielectric layer over the wafer electrode of a high frequency capacitively coupled plasma (CCP) reactor, in order to suppress electromagnetic (EM) effects and thus improve plasma uniformity. High frequency, large area CCP reactors with electronegative feedstock gases are widely used in the semiconductor processing industry for reactive ion etching. Larger area CCPs allow larger wafers and increased production. Increasing the frequency results in the decrease in the sheath widths and voltages, leading to a decrease in the ion bombarding energy, which may be desirable for processing integrated circuits with smaller dimensions. Also, for CCPs at a given power, the fraction of power deposited into electrons (as opposed to ions) increases with increasing frequency, 1 leading to higher plasma density and ion flux at the wafer target. However, at higher frequencies and/or larger areas, the wavelength k p of the EM surface wave in the plasma can become significantly shorter than the reactor radius, leading to standing wave effects which negatively affect plasma uniformity. 2,3 Previous works have modeled EM effects in large area, high frequency CCPs by solving Maxwell s equations either in the frequency domain 4 10 or in the time domain, capturing nonlinear effects. Other studies investigated nonlinear effects that can strongly enhance center-high fields even at frequencies for which the fundamental k p is much larger than the reactor radius Typical CCP reactors used in industry are asymmetric with the powered wafer electrode significantly smaller than a) Electronic mail: kawamura@eecs.berkeley.edu the grounded electrode. Both axially symmetric and antisymmetric radially propagating surface wave modes can exist in asymmetric capacitive discharges. 6 The upper and lower axial sheath fields (E z ) are aligned for the symmetric mode, while they are opposed for the antisymmetric mode. In a previous paper, 10 we used a linear analytic EM model to study the modes of an asymmetric CCP reactor with unequal uniform sheath widths s and w for the wafer and grounded electrodes, respectively. The model was not self-consistent, as it solved Maxwell s equations for the fields assuming known and uniform values for electron density n e andelectrontemperature T e. The fields were solved in both the wafer region (0 < r < R x ) and outside the wafer region (R x < r < R), where R x and R are the radii of the wafer and discharge, respectively. The wavenumber of a radially propagating symmetric surface wave in the wafer region is given approximately by 10 k s x 1=2 l eff ; (1) c s þ w with c being the speed of light, x the applied radian frequency, and l eff an effective discharge width due to the skin effect. The effective width l eff is related to the actual discharge width l by the skin depth d ¼ c/x p,wherex p ¼ n e e 2 1=2 = ð 0 m e Þ is the electron plasma frequency. For s, w l eff l eff ð l 0 e z=d dz ¼ dð1 e l=d Þ: (2) For d l, d and l eff decrease with increasing n e. The wavenumber of an antisymmetric mode surface wave is given approximately by 10 05C311-1 J. Vac. Sci. Technol. A 35(5), Sep/Oct /2017/35(5)/05C311/9/$30.00 VC 2017 American Vacuum Society 05C311-1

3 05C311-2 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-2 k a x c! 1=2 1 þ d2 ðs þ wþ : (3) swl eff At higher frequencies and lower sheath widths, both k s and k a are larger so that the corresponding wavelengths k s ¼ 2p/ k s and k a ¼ 2p/k a are smaller, making wave effects more visible. This suggests that a way to suppress the wave effects in a high frequency CCP is to increase the effective sheath width by adding a dielectric layer over the wafer electrode. This paper is divided into the following sections. In Sec. II, we briefly describe the self-consistent 2D linear fluid-analytical model of an axisymmetric cylindrical CCP reactor. In Sec. III A, we present and discuss the results of simulations of a low pressure, high frequency Cl 2 discharge with and without a dielectric layer placed over the wafer electrode. In Sec. III B, we present and discuss the results of simulations of the same discharge when a conducting or resistive silicon wafer is placed over the dielectric layer. Conclusions are given in Sec. IV. II. 2D FLUID-ANALYTICAL CCP MODEL DESCRIPTION As the fast 2D fluid-analytical model used for the CCP simulations has been fully described previously, 9,22 we will only give a brief summary below. The model was developed using the finite element simulation tool COMSOL in the Matlab numerical computing environment. Figure 1 shows the CCP geometry for the three different configurations studied in this paper: (1) the base case with no dielectric layer, (2) the case with a dielectric layer over the wafer electrode, and (3) the case with both a silicon wafer and a dielectric layer over the wafer electrode. In all configurations, we assume an axisymmetric cylindrical geometry with the center of symmetry at r ¼ 0(z-axis). The plasma chamber has a width l of 5 cm and a radius R of 25 cm. There is a 15 cm radius showerhead gas inlet at the top and radial center and a 3 cm width gas outlet at the bottom and radial edge. The wafer electrode has a radius R x of 15 cm and is insulated from other conducting surfaces by a 2.5 cm wide quartz dielectric spacer. A dielectric layer of varying widths z d and relative dielectric constants j d may be placed above the bottom wafer electrode as shown in Fig. 1(b). Also, a thin silicon wafer with variable resistivity q Si can be placed over this dielectric layer as shown in Fig. 1(c). The bulk plasma region is surrounded by a sheath region with a nominal width of s 0 ¼ 0.5 mm. The feedstock gas in the simulations is Cl 2, and we use the chlorine reaction rate set compiled by Thorsteinsson and Gudmundsson. 23 The model treats each region of the reactor as a dielectric slab. The free-space magnetic permeability l ¼ l 0 is assumed everywhere, while ¼ j 0 depends on the relative dielectric constant j of each region. The sheath relative dielectric constant j s is initially set to one but is calculated as a function of the local electric field and plasma parameters as discussed in Refs. 9 and 22. The quartz dielectric spacer has j ¼ 4. In the plasma region, the relative dielectric constant is FIG. 1. Model geometry of the 2D axisymmetric CCP reactor for configurations with (a) no dielectric layer, (b) a dielectric layer over the wafer electrode, and (c) a silicon wafer and dielectric layer over the wafer electrode. j p ¼ 1 x 2 p xx ð j m Þ ; (4) where x is the applied radian rf frequency and m is the electron-neutral momentum transfer collision frequency. Note that j p is complex with a dissipative imaginary component so that the model treats the plasma region as a lossy dielectric. In axisymmetric geometry, the capacitive fields E r, E z, and H / are in the transverse magnetic mode. In this case, the magnetic field is transverse to the axis of symmetry, while the electric field has components both parallel and transverse to the axis of symmetry. The model assumes that all the field components are proportional to e jxt. This eliminates the time-dependence from the field so that the time-independent Helmholtz equation can be used to solve for the fields. The simulation consists of four basic parts: (1) a linear EM model which uses the time-independent Helmholtz equation to solve for the capacitive fields in the linearized frequency domain; (2) an ambipolar, quasineutral bulk plasma model which solves the time-dependent fluid equations for ion continuity and electron energy balance; (3) an analytical sheath model which solves for the sheath parameters (i.e., sheath voltage, sheath width, and j s ); (4) a gas flow model which solves for the steady-state composition, pressure, temperature, and velocity of the reactive gas. The total J. Vac. Sci. Technol. A, Vol. 35, No. 5, Sep/Oct 2017

4 05C311-3 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-3 simulation time for a Cl 2 CCP reactor is about 1 2 h on a moderate workstation with 2.7 GHz central processing unit and 12 GB of memory. III. 2D FLUID-ANALYTICAL SIMULATIONS In this section, we present and discuss the results of 2D fluid-analytical simulations of a 500 sccm, 10 mtorr Cl 2 CCP reactor with the configurations shown in Fig. 1 with l ¼ 5cm, R ¼ 25 cm, and R x ¼ 15 cm. The discharges were driven with a total input power of P rf ¼ 300 W at a frequency of f ¼ 100 MHz. At a high frequency of 100 MHz, most of the P rf values go to electron heating P e, resulting in an n e / P e of about m 3 in the wafer region. At this n e, the effective width l eff from Eq. (2) is about 2.76 cm which is significantly smaller than the actual discharge width of l ¼ 5cm.Theinput parameters for power and frequency resulted in significant center-high nonuniformities for the base case with no dielectric layer. A. Simulations with and without the dielectric layer We initially simulate the base case with no dielectric layer as shown in Fig. 1(a) and a series of simulations with a dielectric layer placed over the bottom wafer electrode as shown in Fig. 1(b). The dielectric layer width z d is initially fixed at 3 mm, and the relative dielectric constant j d is varied from 1 to 9. Since the capacitance per unit area of the dielectric layer is c d ¼ j d 0 /z d, we can define an effective width z eff z d =j d ; (5) such that c d ¼ 0 /z eff. For a fixed applied frequency and wafer area, the capacitance and therefore impedance of the dielectric layer depend only on z eff so that dielectrics with the same z eff should have similar EM effects even if they have different z d or j d. We test the accuracy of this assumption by conducting an additional simulation with z d ¼ 1 mm and j d ¼ 3 to compare with the z d ¼ 3 mm and j d ¼ 9 case, as both have the same z eff ¼ 1/3 mm. In Fig. 2, we show the results of the base case with no dielectric layer for (a) n e (r) at the midplane (solid) and along the bottom (dashed) and top (dotted) sheath edges, (b) the midplane positive and negative ion densities n þ (r) (solid) and n (r) (dashed), (c) the rf sheath voltage amplitudes, and (d) timeaveraged sheath widths at the bottom wafer (solid) and top grounded (dashed) electrodes. All the diagnostics display significant center-high nonuniformity, indicating that the discharge is near a spatial resonance. The first and second spatial resonances for the symmetric mode occur at k s R ¼ v and k s R ¼ v , where v 01 and v 02 are the first and FIG. 2. 2D fluid-analytical results for the base case with no dielectric layer, showing (a) n e (r) at the midplane (solid) and along the bottom (dashed) and top (dotted) sheath edges, (b) the midplane n þ (r) (solid) and n (r) (dashed), and (c) the rf sheath voltage amplitudes and (d) the time-averaged sheath widths at the bottom wafer (solid) and top grounded (dashed) electrodes. JVST A - Vacuum, Surfaces, and Films

5 05C311-4 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-4 second zeroes of the zero-order Bessel function J 0 (v). The first and second spatial resonances for the antisymmetric mode occur at k a R ¼ v and k a R ¼ v , where v 11 and v 12 are the first and second zeroes of the first-order Bessel function J 1 (v). The average values of the sheath widths and electron density in the wafer region are s 0.54 mm, w 0.48 mm, and n e m 3. Substituting these values into Eqs. (1) and (3), weobtaink s R 2.64 and k a R 6.9. The former is fairly close to the first symmetric mode resonance, while the latter is very close to the second antisymmetric mode resonance. In a previous work, 20 we found that for the present CCP configuration with n e m 3,thefirst antisymmetric mode resonance occurs at about f ¼ 72.5 MHz, while the first symmetric mode resonance occurs at about f ¼ 90 MHz. From Fig. 2, we see that the smaller wafer electrode sheath does not dominate as it would near a symmetric mode resonance. Instead, the peak sheath voltages and sheath widths at the radial center are similar for both electrodes, indicating that the antisymmetric mode resonance dominates. Recall that E z values for the top and bottom electrodes are aligned for the symmetric mode, while they are opposed for the antisymmetric mode. Let E zs and E za be the axial fields for the symmetric and antisymmetric modes, respectively, and let E zb and E zt be the fields at the bottom wafer and top grounded electrodes, respectively. Then, E zb ¼ E zs þ E za and E zt ¼ E zs E za so that E zs ¼ E zb þ E zt ; (6) 2 and E za ¼ E zb E zt : (7) 2 In Fig. 3, we show the base case results for E zb (r) (solid) and E zt (r) (dashed), as well as their symmetric and antisymmetric mode components E zs (r) (dotted) and E za (r) (dot-dashed) in the wafer region in four different phases / ¼ xt ¼ p=4; p=2; 3p=4, p of an rf half-cycle. (The second half-cycle results would just be a reflection across the E z ¼ 0 axis of the first half-cycle results.) We see that for most of the rf cycle, E zb and E zt are opposed and the antisymmetric mode dominates over the symmetric mode. At / ¼ p=4 and p/2, the symmetric and antisymmetric modes are of similar magnitudes, while at / ¼ 3p=4 and p, where the fields are at their maxima, the antisymmetric mode is much larger than the symmetric mode. Since the base case is close to an antisymmetric mode resonance, we observe significant wave effects in all the FIG. 3. 2D fluid-analytical results for the base case showing E zb (r) (solid), E zt (r) (dashed), E zs (r) (dotted), and E za (r) (dot-dashed) in four different phases / ¼ xt ¼ p=4; p=2; 3p=4, p of an rf half-cycle. J. Vac. Sci. Technol. A, Vol. 35, No. 5, Sep/Oct 2017

6 05C311-5 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-5 diagnostics in Fig. 2, especially the sheath voltages and sheath widths. We expect the sheath width variation to closely follow that of the sheath voltage since our analytical sheath model assumes a Child Law relation s / V 3=4 sh.atthe low discharge pressure of 10 mtorr, the high diffusion rate counteracts the center-high power deposition so that the densities are not as centrally peaked as the sheath voltages. The discharge is also highly electronegative with an electronegativity a ¼ n /n e 10. In highly electronegative discharges with high ion densities, the recombination rate may exceed the ionization rate at the radial center where the densities are peaked. This can cause a dip in the ion densities at the radial center, which can also counteract the center-high power deposition. We also note that there is little axial variation in n e, as the midplane and sheath edge densities are all close in value. In a highly electronegative plasma with n n e, quasineutrality in the bulk is maintained mostly by the negative ions rather than the electrons. Since the less mobile negative ions have a much smaller temperature than the electrons, only a small ambipolar field with j/ a ðþjt z e is required to confine the negative ions to the plasma core. The electrons are in Boltzmann equilibrium with this small field so that n e ðþ¼ z n e0 exp / a ðþ=t z e, where n e0 is the midplane electron density. The zero of the potential is at the axial center and / a ðþ< z 0. Since j/ a ðþjt z e ; n e ðþ z n e0. Also, since the negative ions are confined to the core, n (r) 0andn þ (r) n e (r) along the sheath edges. In Fig. 4, we show the same diagnostics as in Fig. 2 for the case of a dielectric layer with z d ¼ 3 mm and j d ¼ 1 placed over the bottom wafer electrode. In this case, z eff ¼ 3 mm and the effective sheath width along the bottom electrode is s 0 ¼ s þ z eff. For this case, we have s 0.6 mm, w 0.4 mm, s 0 3:6 mm, and n e m 3 in the wafer region. If there were no change in n e, we would expect from Eqs. (1) and (3) that k s R would decrease as (s þ w) 1=2 from 2.64 to 1.33 and k a R would approximately decrease as [(s þ w)/(sw)] 1/2 from 6.9 to 5.8. However, adding the dielectric layer reduced n e by about 20% from 2.5 to m 3, resulting in d increasing from 3.36 to 3.75 cm and l eff increasing from 2.60 to 2.76 cm. From Eqs. (1) and (3), we see that k s R increases as l 1=2 eff, while k ar increases approximately as d=l 1=2 eff so that a 20% reduction in n e results in a slight increase of about 3% in k s R and a modest increase of about 8% in k a R. The combined effect of reducing n e and increasing the effective sheath width resulted in an overall decrease in the wavenumbers such that k s R 1.37 and k a R 6.2, neither of which are near a spatial resonance. Adding the dielectric layer increased the effective sheath width and the wavelengths of the radially propagating surface modes, moving the discharge away from spatial resonances. Thus, the wave effects are suppressed, and we FIG. 4. 2D fluid-analytical results for the case of a dielectric layer with z d ¼ 3 mm and j d ¼ 1 placed over the bottom wafer electrode, showing the same diagnostics as in Fig. 2. JVST A - Vacuum, Surfaces, and Films

7 05C311-6 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-6 FIG. 5. 2D fluid-analytical results for the case of a dielectric layer with z d ¼ 3 mm and j d ¼ 1 placed over the bottom wafer electrode, showing the same axial fields as in Fig. 3 in four different phases / ¼ xt ¼ p=4; p=2; 3p=4, p of an rf half-cycle. observe improved uniformity in the wafer region for all the diagnostics compared to the base case with no dielectric layer. There is also a significant reduction in the rf sheath voltage at the wafer electrode compared to the base case due to a voltage divider effect. The voltage between the plasma edge and the wafer electrode no longer lies solely across the sheath as in the base case, but it is now divided between the sheath and the dielectric layer. Figure 5 shows the same axial electric fields as in Fig. 3 but for the z d ¼ 3 mm and j d ¼ 1 case. Note that the magnitude of the fields is much smaller compared to the near resonant base case. (The vertical scale in Fig. 5 is about half that in Fig. 3.) Also, since the addition of the z eff ¼ 3 mm dielectric layer moved the system away from an antisymmetric resonance, the antisymmetric mode no longer dominates as it did in the base case. Instead, for most of the rf cycle, the antisymmetric and symmetric modes are of fairly equal magnitudes. Figure 6 shows the rf sheath voltage amplitudes V sh (r) at the bottom wafer electrode for the base case with no dielectric layer (solid black), the dielectric layer cases with z d ¼ 3 mm and j d ¼ 1 (dotted), 4 (dashed), and 9 (dotdashed), and an additional case with z d ¼ 1 mm and j d ¼ 3 (solid gray). For these cases, z eff varies from z eff ¼ 0 for the base case shown in Fig. 2 to z eff ¼ 3 mm for the z d ¼ 3mm with j d ¼ 1 case shown in Fig. 4. Recall that the corresponding k a R values are near resonant k a R ¼ 6.9 for the base case and nonresonant k a R ¼ 6.2 for the z eff ¼ 3 mm case. For our simulated cases, larger z eff should lead to larger s 0, k s, and k a FIG. 6. 2D fluid-analytical results showing rf sheath voltage amplitude V sh (r) at the bottom wafer electrode for the base case with no dielectric layer (solid black) and for the dielectric layer cases. J. Vac. Sci. Technol. A, Vol. 35, No. 5, Sep/Oct 2017

8 05C311-7 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-7 and a greater suppression of wave effects in the wafer region, as we move further away from the antisymmetric mode resonance. Larger z eff also means larger dielectric layer impedances, leading to a reduction in the sheath voltage as more of the voltage between the plasma edge and the wafer electrode falls across the dielectric layer. Both these trends are seen in Fig. 6. Note that the z d ¼ 1 mm with j d ¼ 3 case and the z d ¼ 3 mm with j d ¼ 9 case have the same z eff ¼ 1/3 mm and therefore give similar results. For the largest z eff case of z d ¼ 3 mm with j d ¼ 1, there is a small peak in the sheath voltage at the wafer edge at r ¼ R x ¼ 15 cm, which is not observed in the lower z eff cases. (The peak observed near but not at the interface for the base case with no dielectric layer is due to the shorter wavelength.) The tangential electric field is continuous across the boundary between two different media. The interface between the dielectric layer with j ¼ j d and the quartz dielectric spacer with j ¼ 4 occurs at r ¼ R x ¼ 15 cm. For a layer with small z eff, the tangential field E z will be small at the interface, and so, we do not expect to see a peak in the sheath voltage at the interface. For a dielectric layer with large z eff, E z can be high at the interface, leading to a peak in the sheath voltage at the interface. Even with relatively high z eff as in the j d ¼ 4 case, there will be no spike if the dielectric layer and the quartz spacer have the same dielectric constant since in this case there is no interface between different media. B. Simulations with the silicon wafer over the dielectric layer In this subsection, we discuss the simulation results with the configuration shown in Fig. 1(c) in which a silicon wafer and a dielectric layer are placed over the wafer electrode. For typical applications, a conducting wafer has a resistivity q Si 0:005 X cm, while a high resistivity wafer can have a q Si of up to 90 X cm. As above, we conduct simulations of a 500 sccm, 10 mtorr Cl 2 reactor with P rf ¼ 300 W at f ¼ 100 MHz. We set z d ¼ 3 mm and j d ¼ 1 for the dielectric layer and set the thickness of the silicon wafer to z Si ¼ 1 mm. We simulate both a conducting wafer with q Si ¼ X cm and a highly resistive wafer with q Si ¼ 90 X cm. Figures 7 and 8 show the same diagnostics as Fig. 2 for the q Si ¼ X cm and q Si ¼ 90 X cm cases, respectively. The simulation results of the conducting wafer case shown in Fig. 7 are similar to the base case results shown in Fig. 2, as both display significant center-high nonuniformity and wave effects. The simulation results of the highly resistive wafer case shown in Fig. 8 resemble the dielectric layer case with z d ¼ 3 mm and j d ¼ 1 shown in Fig. 4, as both show improved uniformity and suppression of wave effects compared to the base case. Thus, a conducting top layer shields the plasma from the dielectric layer, while a highly resistive top layer provides little shielding. The explanation for this contrasting behavior can be found from the skin depth FIG. 7. 2D fluid-analytical results for the case of a conducting silicon wafer with q Si ¼ X cm and a dielectric layer with z d ¼ 3 mm and j d ¼ 1 placed over the bottom wafer electrode, showing the same diagnostics as in Fig. 2. JVST A - Vacuum, Surfaces, and Films

9 05C311-8 Kawamura et al.: Effect of a dielectric layer on plasma uniformity 05C311-8 FIG. 8. 2D fluid-analytical results for the case of a highly resistive silicon layer with q Si ¼ 90 X cm and a dielectric layer with z d ¼ 3 mm and j d ¼ 1 placed over the bottom wafer electrode, showing the same diagnostics as in Fig. 2. sffiffiffiffiffiffiffiffi 2q d Si ¼ Si xl 0 of the silicon wafers. For q Si ¼ X cm, d Si 0.36 mm z Si ¼ 1 mm. So, the conducting wafer shorts out most of the fields and shields the discharge from the dielectric layer. On the other hand, for q Si ¼ 90 X cm, the skin depth d Si 4.8 cm z Si ¼ 1 mm. So, the fields mainly pass through the highly resistive wafer and penetrate the dielectric layer. A critical resistivity q c is obtained by setting d Si ¼ z Si q c ¼ xl 0z 2 Si : (9) 2 For z Si ¼ 1 mm, q c 0.04 X cm. When q Si q c, the silicon wafer is conducting and shorts out most of the fields. When q Si q c, the silicon wafer is highly resistive and most of the fields mainly pass through to the dielectric layer. IV. CONCLUSIONS A fast axisymmetric 2D fluid-analytical code was used to simulate a 500 sccm 10 mtorr Cl 2 CCP reactor for the three different configurations shown in Fig. 1: (1) no dielectric layer, (2) dielectric layer on the wafer electrode, and (3) silicon wafer and dielectric layer over the wafer electrode. In (8) each configuration, the input power and frequency were set to P rf ¼ 300 W at f ¼ 100 MHz, resulting in an average n e over the wafer region of about m 3. For the base case with no dielectric layer, the system was near an antisymmetric mode resonance and exhibited significant wave effects and center-high nonuniformity. For the second configuration with a dielectric layer over the wafer electrode, the dielectric layer increased the effective sheath thickness and thereby the wavelengths of the radially propagating surface waves. This moved the system away from the near resonant conditions of the no dielectric layer case and lessened the wave effects. The higher the z eff ¼ z d /j d of the dielectric layer, the larger the increase in s 0 ¼ s þ z eff, k a and k s, and the greater the suppression of the wave effects. For the third configuration with a silicon wafer and dielectric layer over the wafer electrode, there is a critical resistivity q c given by Eq. (9) such that the skin depth is equal to the wafer thickness. A conducting wafer with q Si q c shorts out most of the fields and shields the discharge from the dielectric layer, while a highly resistive wafer with q Si q c lets the fields pass through to the dielectric layer. ACKNOWLEDGMENTS This work was supported by the Department of Energy Office of Fusion Energy Science (Contract No. DE-SC ), J. Vac. Sci. Technol. A, Vol. 35, No. 5, Sep/Oct 2017

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