Electron Series Resonant Discharges: Part II: Simulations of Initiation

Transcription

1 Electron Series Resonant Discharges: Part II: Simulations of Initiation K. J. Bowers 1 W. D. Qiu 2 C. K. Birdsall 3 Plasma Theory and Simulation Group Electrical Engineering and Computer Science Department University of California at Berkeley 231 Cory Hall c/o Professor C. K. Birdsall Berkeley, CA Submitted to Plasma Sources Sci. Technol. December 20, kbowers@eecs.berkeley.edu 2 wgqiu@eecs.berkeley.edu 3 birdsall@eecs.berkeley.edu

2 Abstract This article is Part II of a three part simulation study of electron series resonant (ESR) discharges. This article describes the initiation of an ESR discharge. A rapid transition ( lock-on ) from a decaying capacitive-looking discharge to an ESR sustained discharge is observed in simulation. The transition occurs when the sheath of a decaying plasma has expanded sufficently that the ESR frequency and the RF drive frequency coincide. Phenomena related to lock-on are discussed and are presented for various gas pressures and external circuit parameters. Qualitative models for some of these phenomena (phase space bunching and slow time scale ringing) are presented.

3 1 Introduction and Simulation Model This article is Part II of a three part simulation study of electron series resonant (ESR) sustained discharges. Part I presents a general introduction to ESR discharges and contains a more complete list of references to ESR discharge literature. In this article, the initiation of an ESR discharge is studied. Of particular interest here: Initiation of the resonant discharge Heating profile and electron heating physics Sheath dynamics and plasma potential The simulation model is similar to the model described in Part I. For clarity, the model is briefly restated here. For further simulation details, consult Part I. The simulations here model an argon plasma (neutral pressures at 3mT orr and 10mT orr) bound by metal parallel plates. Plate area is 160cm 2 ; plate separation is 6.7cm. The simulations are performed with the electrostatic 1d3v PIC-MCC code PDP1 (Verboncoeur et al [1]). In this article, the RF power supply is modeled by an ideal voltage source (low impedance power supply). 2 Initiation and Lock-On Initial Conditions The simulations start with the diode filled by a spatially uniform warm argon plasma (neutral pressures of 3mT orr and 10mT orr with a plasma density of n cm 3 ). Electrons and ions are initially isotropic Maxwellian with temperatures of T e 2eV and T i 0.026eV = 300K (room temperature) respectively. An ideal RF voltage source V = V RF sin ω RF t is applied to the diode at t = 0. Simulations with V RF at 7.07V, 10V, 14.1V and 20V (zero-to-peak) were conducted. The conditions under which the lock-on phenomeon occurs may be understood by considering the ESR frequency (derived in Part II (Appendix A)): ω r = ω p 2 s L (1) (ω r is the ESR frequency, ω p is the peak electron plasma frequency, 2 s is the combined cycle-averaged sheath wih and L is the diode separation.) For the lock-on phenomenon to happen, ω RF must come into resonance with the ESR. For the given initial profile (uniform), space charge sheaths form and move towards the diode center at roughly the ion acoustic speed at first leading to an ESR frequency increasing with time as 1

4 2 s increases. Over longer time scales, ambipolar diffusion takes over and the bulk plasma decays (unless lock-on has occured) leading to a decreasing ESR frequency with time as ω p decays. Hence, initially V RF must be large enough to allow the sheaths to expand sufficiently such that the drive frequency and the ESR frequency coincide before diffusion causes the plasma to decay away. For the simulations here, the minimum voltage for lock-on (V cr ) was empirically found to be 5V (zero-to-peak), which was larger than the minimum voltage V r required to sustain the resonant discharge. (V r is derived by Godyak [2] and by Cooperberg and Birdsall [3]. It is explicitly given in Part III.) The critical voltage is expected to be dependent on the profile of the plasma before lock-on and also whether or not the ESR frequency approaches from below or above the drive frequency. In this article, lock-on with ω r approaching a constant drive frequency from below is studied. An exact resonant discharge requires V RF = V r. However, with V cr > V r for the initial conditions studied here, an exact resonant discharge cannot be entered into directly. One solution to get to a resonant discharge is to use V RF V cr until the plasma locks-on and then decrease the applied voltage to V r. These results are supported by the hysteresis curves given later in Part III. Explosive Growth A decaying discharge is observed immediately after the RF voltage is applied across the diode (t = 0), with I RF leading V RF by close to 90 (capacitive). As electrons and ions flow to the wall, sheaths form at both electrodes and their cycle-averaged wihs increase with time. Since the peak plasma frequency ω p stays nearly constant, the ESR frequency ω r increases in time due to the increasing sheath wih s (Figure 1e). As the sheath wih increases, what appears to be an explosive event develops with time. 1 The explosive growth of the drive current amplitude, the cycle-averaged power, plasma potential and average kinetic energy per particle is observed. Figure 2 shows the behavior of the sheath wih before lock-on for two different pressures and different RF drive voltages. Least squares fitting to a line the log of sheath wih versus the log of time before lock-on from the data taken at all the RF drive voltages at a given pressure gave α 1/3 at p = 3mT orr and α 1/4 at p = 10mT orr. During this time, the motion of the plasma bulk is in-phase with the voltage drive. If V RF < V cr, the explosive growth does not happen and there is no lock-on. For V RF > V cr, the explosive growth ends at some time t 0, denoted as the transition time. For the same dimensions and the same initial plasma, t 0 decreases with increasing RF voltage (this is a consequence of the 1 A quantity which undergoes explosive growth obeys a scaling law 1/(t 0 t) α, α > 0. Generally, the quantity is limited by other physical effects from becoming infinite at t 0. Explosive growth is faster than exponential growth. 2

5 Figure 1: Lock-On: The plasma is driven by an ideal voltage source with V RF = 7.07V at 3mT orr neutral pressure. Graphs (a), (c) and (d) show the cycle-averaged signal and signal envelope. Lockon occurs at t 0 = 4µs. RF current magnitude, power absorbed, plasma potential, sheath wih and average electron kinetic grow explosively before lock-on; (t 0 t) 1/3 for this case. Slow time-scale ringing is seen after lock-on and the plasma no longer decays. 3

6 Total sheath wih (s l + s r ) (µm) 10 4 Explosive growth in sheath wih (p=3mtorr) V RF = 7.07V V RF = 10V V RF = 14.1V V RF = 20V s (t 0 t) 1/ Time before lock on (µs) Total sheath wih (s l + s r ) (µm) 10 4 Explosive growth in sheath wih (p=10mtorr) V RF = 7.07V V RF = 10V V RF = 14.1V V RF = 20V s (t 0 t) 1/ Time before lock on (µs) Figure 2: Explosive Growth: These graphs show the growth of the sheath wih versus time at two different pressures and several RF drive voltages for the same starting plasma. All curves show behavior similar to 1/(t 0 t) α shortly before lock-on. α appears dependent on the gas pressure (the dashed line in the figures is from a least squares fit of the slope from all the simulations at the respective pressures) with α 1/3 at 3mT orr and α 1/4 at 10mT orr. α is less dependent on the RF drive voltage. 4

7 larger initial sheath oscillations due to the larger applied voltages). During Lock-On As t approaches t 0, ω r increases to ω RF leading to a dramatic transition in just a few RF cycles. The decay rate of the total number of electrons, N e, suddenly increases as a burst of electrons flows to the walls (Figure 1b). Consequently the plasma potential φ mid roughly triples (Figure 1d). I RF becomes in-phase with V RF and the RF power absorbed becomes (partially or totally) real from reactive (Figures 1a and 1c). The sloshing motion of the bulk plasma changes from in-phase with the drive voltage to 180 out-of-phase with the drive voltage. Correspondingly, the electric field in the bulk plasma is opposite in direction to the electric field of the vacuum diode and the electric field in the sheath regions after lock-on. (Figures 3a and 3b) Electrons begin to be alternately bunched and accelerated into the bulk plasma by the left and right sheaths, leading to complicated structure in the electron x v x phase space (Figures 3c and 3d). The unique heating profile of the resonant discharge develops as well (Figures 3e and 3f). The rapidity of the transition suggests that a more realistic power supply model which includes the supply response time would be necessary to accurately depict the transition process in a laboratory device. However, this idealized transition provides an extreme case by which to test theoretical discharge models. After Lock-On After lock-on, many plots in Figure 1 show an envelope, modulated at a low frequency of a few MHz. The amplitude of the modulation decays and, after a relatively long time ( 10 3 RF cycles), the plasma goes to a stable resonant state with no envelope modulation. This ringing can be explained as the beat frequency between the ESR frequency and the RF drive frequency (note that these discharges are driven slightly off resonance). Such is shown explicitly later in this article. After the decay, I RF leads V RF by about 20 to 30 for the low voltage drive, which implies a nearly resistive plasma. Accordingly, it is observed that the cycle-averaged power deposited into the plasma is no longer negligible (Figure 1c). The electron heating as seen in the J E profile has both positive and negative portions (Figure 3d). Observation of the bulk plasma also shows another difference of the capacitive discharge and resonant discharge. The motion of the bulk is not in-phase with the voltage drive; it is almost completely out-of-phase. The difference in the bulk motion is seen from the potential and electric 5

8 40 DC and RF potential (before) 40 DC and RF potential (after) Potential (V) DC At V (t)=v 20 ckt RF At V (t)= V ckt RF Position (cm) Potential (V) DC At V (t)=v 20 ckt RF At V (t)= V ckt RF Position (cm) 20 Electron Phase Space (before) 20 Electron Phase Space (after) x Kin. Energy (ev) (+ right, left) Position (cm) x Kin. Energy (ev) (+ right, left) Position (cm) Electron and Ion Heating Profile (before) Electrons Ions Electron and Ion Heating Profile (after) Electrons Ions Power absorbed (W/m 3 ) Power absorbed (W/m 3 ) Position (cm) Position (cm) Figure 3: Before and After Lock-On: This data was taken 3µs before and after the lock-on transition for the simulation shown in Figure 1. (top) The bulk plasma electric field after lock-on is 180 outof-phase with the applied field and with the before lock-on bulk electric field. (middle) The particle position versus x-directed kinetic energy phase space snapshots were taken when V ckt (t) = V RF. Electron bunching and bunch acceleration is seen after lock-on. Ionization threshold is 15.8eV. (bottom) The unique heating profile forms after lock-on. 6

9 Figure 4: Current Spectrogram: This data is from the same simulation used in Figure 1. The ESR, RF drive (81M Hz), RF harmonics (162M Hz and 243M Hz) and Tonks-Dattner resonant frequencies are visible (200M Hz). At the lock-on time (4µs), the RF drive and the ESR coincide and the discharge enters a resonant state. Lock-on ringing in Figure 1 is seen as the teeth around the RF drive at lock-on. field inside the diode (Figure 3b). It should also be noted that the peak RF potentials at the sheath edges greatly exceed the applied RF field due to the resonant behavior of the plasma slab after lock-on and the RF voltage drop across the bulk plasma is comparable to the RF sheath voltage drop (Figure 3b). This behavior is in stark contrast to capacitive discharge behavior detailed in Lieberman and Lichtenberg [4]. In the electron phase space the formation of high energy (above the ionization threshold) bunches in the sheath region is seen (Figure 3d). During an RF cycle, these bunches are alternately accelerated from the sheath into the bulk plasma. The bunches provide the ionization for ESR discharges at low pressures. This bunching is discussed later in this article. The entire history of the transition is captured by the current spectrogram shown in Figure 4. (A spectrogram is a set of Fourier transforms performed over a window which advances in time; it 7

10 allows the slow time scale evolution of the high frequency spectrum to be monitored. The figure uses a Hann window to keep spectral leakage low with a window length of 1.13µs and a sample rate equal to the simulation timestep of 140ps.) In this figure, the dark vertical band at 81MHz is the drive frequency. Before lock-on the ESR frequency is seen increasing (from 60Mhz at t = 0) as the space charge sheaths form. At t = 4µs, the ESR interacts with the drive frequency and the plasma is sustained. As this discharge is driven slightly off resonance, the ESR frequency is seen above the drive frequency after lock-on (compare with the Figure 1e) settling at 100M hz well afterwards. A strong third harmonic of the drive is seen (as can a weak second harmonic). The slow time scale modulation immediately after lock-on is visible as the teeth about the RF drive frequency and the third harmonic. The first Tonks-Dattner resonance is seen decreasing in frequency as the plasma profile changes during lock-on holding constant at 200MHz after lock-on (Tonks-Dattner resonances are the cutoff of thermal waves trapped in the plasma sheath and were first quantitively explained in Parker et al [5]). The second Tonks-Dattner resonance is not distinguishable from the third harmonic of the drive frequency after lock-on. Interactions between drive harmonics and Tonks-Dattner resonances may contribute to hysteresis and mode-jumping phenomena seen in resonant discharges (discussed in Part I and Part III). These interactions should be most pronounced at lower pressures when the Tonks-Dattner resonances are the least collisionally damped. 3 Phase Space Bunching and Heating The electron phase space bunching is qualitatively explained by examining the fields associated with the ESR. Figure 5 graphically shows how the ESR RF potential interacts with the cycle-averaged device potential to generate the bunching seen in Figure 3d. The potentials in the bottom sketches of Figure 5 correspond to the simulation potential snapshots shown in Figure 3b. When the peak potential is at the left sheath, electrons are attracted to the high voltage and are bunched. Half an RF cycle later, the peak potential is at the right sheath and the bunched electrons at the left sheath are accelerated into the plasma bulk by the strong dipole field of the ESR. The same process occurs out-of-phase at the right sheath. This process accelerates electrons to energies well above the thermal energy of electrons in the bulk plasma. It appears to allow for discharge operation at low neutral pressures such that ohmic heating is negligible and at low electron temperatures such that the bulk electron energy distribution contributes little to ionization processes. In typical capacitively-coupled and inductively-coupled 8

11 9 [ 9 [ 9 [ W W! "# * + * + * + $&%'(! )(# * + Figure 5: Electron Bunching Mechanism: The cycle-averaged potential and the ESR RF potential (top left and top right respectively) interact during an RF cycle to bunch and launch electrons alternately from the sheaths into the bulk. In the bottom left graph, electrons are bunching at the left sheath. Half a cycle later (bottom right), the bunch is launched into the bulk. The same process occurs at the right sheath 90 out-of-phase. See also Figure 3b. discharges, the low frequency of operation (compared to ω p ), the phase of the plasma sloshing and RF fields at the sheaths are not conducive to this bunching process. Besides the generation of bunched electrons, this process also is observed in the spectrum of the plasma potential (measured at the middle of the diode) as spikes at even harmonics of the RF drive frequency and as spikes associated with the transit time of electron bunches across the diode (not shown). It may be that the bunching process shown in the simulations is further enhanced by a resonance of the bunch transit time with the RF drive frequency but this is difficult to ascertain given the energy spread of the electron bunches. 4 Lock-On Ringing Using the time domain equation of motion for the circuit parameters developed in Part II (Appendix A), a simple model for the slow time scale ringing already seen in Figures 1 and 4 may be developed. 9

12 For clarity, the equation of motion is given here: ( d ν d ) ( ) d 3 + ω2 r I = C v 3 + ν d2 2 + d ω2 p V (2) The sudden change in sheath wih at lock-on (Figure 1e) may be treated as a step from the pre-lock-on value to the post-lock-on value while the plasma density is assumed constant through lock-on. The equation of motion may then be solved before lock-on and after lock-on. Assuming a diode voltage of the form Ṽ = V s ĨR s where V s and R s are given and substituting into (2) yields: d [R 3 s C v 3 + (1 + R sc v ν) d2 2 + (ν + R sc v ωp) 2 d ] ( d 3 + ω2 r Ĩ = C v 3 + ν d2 2 + ω2 p d ) V s (3) For a simple RF drive of the form, V s = V RF sin ω RF t, the entire right hand side of (3) is a known sinusoidal function. Before the sheath step, all transients are assumed to have died out, leaving only the equilibrium solution which may be quickly obtained from (3). The equilibrium solution after lock-on may be similarly obtained. The sudden change in sheath wih (and hence in ω 2 r) introduces short transients after lock-on. A root finder may be used to compute the three characteristic complex frequencies associated with the homogeneous part of (3). Requiring Ĩ to be continuous through the second derivative across the change in ω 2 r fixes the magnitude of these three transient signals. Figure 6 demonstrates the ringing model applied to the lock-on shown in Figure 1. Qualitative agreement between the post-lock-on ringing in the simulated current and the model current is seen. The damping time is associated with the electron collision frequency ν and source resistance R s. The ringing frequency is approximately ω r ω p at the low pressure here and is seen as the teeth separation in Figure 4. The ringing model parameters corresponding to the simulation are R s = 0Ω, C v = 2.11pF, ν ω p /300, ω p 2π(284MHz). The resonant frequencies before and after lock-on were taken from the spectrogram in Figure 4. While the model produces qualitative agreement, it requires several input parameters to compute the ringing and is sensitive to those parameters. Also, the plasma resistance is greatly underestimated at lower pressures in the model as the model does not account for the collisionless bunching mechanism discussed in the previous section (this higher resistance from non-collisional heating does not seem to result in a faster ringing damping time though in the simulation). However, if this model were coupled with particle and power balance equations (including the non-collisional resistance) it may have much more predictive power regarding transients and hysteresis in low-pressure low-temperature discharges. 10

13 1 Simulated Lock on Ringing (p=3mtorr, V RF =7.07V, R s =0Ω) Current into electrode (A) Time (µs) 1 Homogeneous Model Fit (step in sheath wih) Current into electrode (A) Time (µs) Figure 6: Ringing Model Compared to Simulation: The simulation data corresponds to the lock-on shown in 1. The ringing model matching solutions to (3) across a step in sheath wih. Qualitative agreement with the simulated data is seen. 11

14 5 Summary In this study the initiation of planar resonant discharges was shown via 1d3v PIC-MCC simulation. The abrupt transition ( lock-on ) of a decaying capacitive-looking plasma into a resonantly sustained plasma was discussed and some of the phenomena of lock-on were qualitatively explained. The transition to a resonant discharge studied here requires V RF > V cr and a ω p > ω RF. V cr (which depends on how the initial plasma was formed) may be above the resonant discharge voltage in which case a resonant discharge must be approached from a sustained plasma at a higher discharge voltage. The sheath wih and other discharge parameters appear to grow explosively before lock-on. The heating profile observed in simulation shows different characteristics from non-resonant discharges. The heating is greatest at the sheaths edges, but spikes of electron cooling are seen just inside the sheaths. Electron bunching is observed in phase space and produces non-thermal high energy electrons above the ionization threshold. The bunching mechanism may be understood through consideration of the interaction between the cycle-averaged device potential and the RF potential. At low pressures and low bulk electron temperatures, the bunch mechanism is responsible for producing the ionizing electrons. Acknowledgments Thanks to Dr. V. A. Godyak, Dr. H. Smith and Dr. J. Verboncoeur for helpful discussions. This work was supported by DOE contract DE-FG03-97ER54446, AFOSR contract FDF , ONR contract N G001 and the Fannie and John Hertz Foundation Fellowship Program. 12

15 " + +! # $ * # % +, -. & ' ( ) 0 / Figure 7: Homogenous Model of a Bounded Plasma: The electrons are cold, uniform and bordered by matrix sheaths. The ions uniformly fill the device and have the same density as the electrons. A Time Domain Equations of Motion of a Cold Homogenous Bounded Plasma Slab Many key phenomena such as the electron series resonance (ESR) in bounded plasmas may be qualitatively demonstrated with the homogeneous model employed by Godyak [2] and by Lieberman and Lichtenberg [4]. In this model, stationary ions uniformly fill a parallel plate capacitor while a uniform cold electron slab neutralizes the ion charge in the center. Matrix sheaths exist to the left and right of the electron slab. The potential due to the space charge of the sheaths confines the electrons in the center. The homogeneous model is shown graphically in Figure 7. A.1 Derivation from Device Impedance Considerations The time domain relationship between the voltage across the capacitor plates and the current flow into plates for high frequencies (well above the ion plasma frequency) is desired. The simplest way to find this is to treat the left sheath, bulk plasma and the right sheath as capacitors in series. The dielectric constant of the bulk plasma is given by the high frequency dielectric constant for cold electrons: ε 0 [1 ω 2 p ω (ω jν) ] (4) (ε 0 is the permittivity of free space; ω p is the electron plasma frequency, e 2 n/ε 0 m; ω is the frequency being examined; ν is the effective electron collision frequency; e is the magnitude of the electron charge; m is the electron mass.) The impedance of the device is thus: Z(ω) = s l jωε 0 A + d [ jωε 0 A 1 ω2 p ω(ω jν) [ ] ] + s 2 s 1 ω2 p r jωε 0 A = ω(ω jν) + d [ ] jωε 0 A 1 ω2 p ω(ω jν) 13

16 Figure 8: High Frequency Equivalent Circuit: This circuit is for a planar cold homogenous bounded plasma. The ESR occurs when the device reactance is zero. A is the area of the electrodes, s l and s r are the left and right sheath wihs respectively, d is the wih of the bulk plasma, ω p is the electron plasma frequency, ν is the electron-neutral collision frequency and ε 0 is the permittivity of free space. The impedance of this circuit is the same as (5). = [ ] L 2 s L ω2 p ω (ω jν) jωε 0 A ωp 2 ω (ω jν) (5) 2 s is the combined wih of the matrix sheaths (s l + s r ). With V = V 14 being the voltage across the plates, I being the current flowing into the left plate and Z(ω) = V (ω)/i(ω), rearranging terms gives: ( ω 2 + jων + 2 s ) L ω2 p I(ω) = jωε 0A ( ω 2 + jων + ω 2 ) p V (ω) (6) L As I(t) I(ω)e jωt dω and V (t) V (ω)e jωt dω, the time domain differential relationship may be written by inspection: ( d ν d ) ( ) d 3 + ω2 r I(t) = C v 3 + ν d2 2 + d ω2 p V (t) (7) C v is the vacuum capacitance of the parallel plates (ε 0 A/L) and ω r is the electron series resonance frequency (ω p 2 s/l). An equivalent derivation takes the circuit corresponding to Z(ω) (shown in Figure 8) and finds the relationship between I(t) and V (t). The result is the same. Breaking down the equivalent circuit component-wise: the left and right capacitors represent the electron depleted sheaths, the center capacitor represents displacement current through the plasma bulk. The resistor represents the electron-neutral collisions and the inductance represents electron inertia (not a magnetic effect). It should be noted that the above derivations implicitly assume ω p, ν, s l and s r are constant in time and consequently ignore non-linearities in the sheath capacitances. These non-linearities 14

17 cancel in this simple model as is shown in the following section. Essentially, the model here treats slab motion as a layer of surface charge at points 2 and 3; this is further discussed at the end of the appendix. A.2 Derivation from First Principles The differential relationship for I and V may be more convincingly derived by solving Poisson s equation and the equation of motion for the electron slab in terms of the time domain voltage and current. This makes the device dynamics clearer. Also, the restrictions that s l and s r are constant in time are relaxed. The initial derivation below loosely follows the derivation in Lieberman and Lichtenberg [4]. However, the assumption that the voltage drop across the bulk plasma is negligible is not made as this drop is significant in ESR sustained plasmas and the derivation is done strictly in the time domain. The derivation begins with Poisson s equation in the device: Solving for E gives: x E(x, t) = ρ = ε 0 en ε 0 0 < x < s l (t) 0 s l (t) < x < L s r (t) L s r (t) < x < L en ε 0 E p + en ε 0 (x s l ) E(x, t) = E p E p + en ε 0 (x L + s r ) 0 < x < s l s l < x < L s r L s r < x < L E p (t) is the instantaneous electric field in the bulk plasma. (The above equation is also useful for measuring the sheath wih in PIC-MCC simulations as n and E(x, t) are explicitly known.) As the sheaths are devoid of electrons and the ions are stationary (for the time scales of interest here), I(t) is carried solely by displacement current in the sheaths. Furthermore, in a planar system, the net current (convection plus displacement) is a constant in space. Thus, in the sheaths: { I(t) = ε 0 A E t = ena ds l + ε 0A dep 0 < x < s l ena dsr + ε 0A dep L s r < x < L As I(t) is the same in both sheaths by Kirchhoff s current law: (8) (9) (10) ena ds l + ε 0A de p = ena ds r + ε 0A de p (11) Consequently: d (s l + s r ) = 0 (12) With s l + s r and d as constants, the sheath wihs may be expressed as: s l (t) = s l + s(t) (13) s r (t) = s r s(t) (14) 15

18 s l and s r are the time average left and right sheath wihs, respectively. I(t) now may be written as: I(t) = ena d s + ε 0A de p It should be noted that as v slab = d s/, this expression is also valid in the bulk plasma. Consider the voltage drops across the sheaths and the bulk plasma: (15) V 12 (t) = V 23 (t) = V 34 (t) = E(x, t) dx = ens2 l (t) 2ε 0 + E p (t)s l (t) (16) E(x, t) dx = E p (t)d (17) E(x, t) dx = ens2 r(t) 2ε 0 + E p (t)s r (t) (18) Summing these gives (V = V 14 ): V (t) = V 12 + V 23 + V 34 = en 2ε 0 [ s 2 r (t) s 2 l (t) ] + E p (t)l (19) Noting that s 2 r s 2 l = (s r + s l )(s r s l ) = ( s r + s l )( s r s l 2 s) = ( s 2 r s 2 l ) 4 s s where s = (s l + s r )/2 = ( s l + s r )/2, V (t) may be broken into a DC bias ( V ) and a zero mean time varying term (Ṽ ). Thus: Above, Ē p V = en 2 ( s 2ε r s 2 ) l 0 (20) Ṽ (t) = 2en s ε 0 s(t) + E p (t)l (21) = 0 was used (the DC electric field in the bulk plasma) as the slab shields out all electric fields of sufficiently low frequency. Equivalently, the equilibrium (DC) slab position must have Ēp = 0 otherwise it would not be the equilibrium position. Substituting (21) into (15): ( I = ena d s + ε 1 dṽ 0A L + 2en s ε 0 L The equation of motion for the electron slab is given by: ) d s dṽ = C v enad L d s (22) a slab (t) = e m E p(t) νv slab (t) (23) Here, ν is a collisional drag term. With a slab = dv slab / and v slab = d s/: Using (21) again: d 2 s 2 = e m E p ν d s d 2 s 2 = e ( 1 m LṼ + 2en s ) ε 0 L s ν d s (24) (25) 16

19 Rearranging terms yields: d 2 s 2 + ν d s + ω2 r s = e mlṽ (26) This demonstrates that the electron series resonance is the natural oscillation frequency for sloshing motion of a bounded electron slab. Taking the first and second time derivatives of (22) and substituting (26): di = enad ( e L mlṽ ν d s ) ω2 r s ( ) d 2 = C v 2 + ω2 pd Ṽ + enad L L ( ) d ω2 pd d Ṽ + enad L L ( ) d 3 = C v 3 + ω2 pd d L νω2 pc v d L d 2 I 2 = C v d 2 Ṽ + C v 2 ( ν d + ω2 r ) s (27) [ ( ν e mlṽ ν d s ) ω2 r s + ωr 2 Ṽ + enad L ] d s [ (ωr 2 ν 2 ) d ] νω2 r s (28) In the sum d 2 I/ 2 + νdi/ + ωri, 2 all terms involving s cancel, leaving: ( d ν d ) [ ( ) ] + d 3 ω2 r I = C v 3 + ν d2 ω p d L + d ω2 r Ṽ (29) Noting that ωpd/l 2 + ωr 2 = ωp(d s)/l = ωp 2 and dṽ / = dv/: ( d ν d ) ( d 3 + ω2 r I = C v 3 + ν d2 2 + ω2 p d ) V (30) This is the same equation derived from device impedance considerations. However, no assumptions regarding motion or position of the electron slab were made (other that the implicit assumption that the slab does not oscillate greater than the smallest time average sheath wih). Thus, (30) holds for stronger oscillations than the device impedance derivation implies. To understand better the calculations here, the instantaneous potential and RF potential are shown in Figure 9 for the ESR. The homogeneous model agrees qualitatively very well with the simulation potentials shown in Figure 3. A.3 Equivalence of Slab Motion and Surface Charge The first method treats the system as a set of stationary dielectric loaded capacitors connected in series. However, the second method explicitly shows the plasma slab moves. The first method is a much easier calculation but initially it appears that this method is inappropriate as the electron slab is mobile. Fortunately, the agreement between the two methods is not a lucky accident. The two methods are equivalent for linear (infinitesimal) slab motions; the first method treats slab motion as a layer of surface charge at the sheath edge. 17

20 Q [ W W Q [ W W GW Q Q H 6ODE H 6ODE V V [ V V [ 9 [ W W 9 [ W W G W [ [ 9 [ W W 9 [ W W G W [ [ Figure 9: Electron Series Resonance: As the electron slab sloshes (top), a strong dipole field at the ESR frequency is present across the plasma bulk as may be seen in the instananeous potential (middle) and the RF potential (bottom). The RF voltage drop across the bulk plasma cancels the RF sheath drop. Compare the above potential with Figure 3. 18

21 Q[W Q[ GW Q Q Q Q V V V W [ [ V W GW GU [ W GU [ W GW V V [ [ Figure 10: Slab Motion and Surface Charge: The first method (impedance calculation) treats the motion of the electron slab (top) as a surface charge layer located the the sheath edge (bottom). The second method (electron force calculation) explicitly calculates the sheath motion. Both methods give the same result. That the first method has a layer of surface charge at the sheath edge may be quickly understood by considering the usual electrostatic boundary conditions on field components. In the first method with all the electrons in the slab accounted for in the cold plasma dielectric, there are effectively no free charges in the system. In such a system, the normal electric flux is continuous across interfaces (that is, D x is continuous at the sheath edges). As usual, D x = ε(x)e x. But, ε is discontinuous across the sheath edges (vacuum dielectric in the sheath, cold plasma dielectric in the plasma). Consequently, E x is also discontinuous there. A discontinuity in E x corresponds to a surface charge layer. The relation of this surface charge layer to slab motion is graphically shown in Figure 10. For linear excitations, the perturbed charge resulting from the electron slab motion is localized about the sheath edges and may be effectively treated as a layer of surface charge. Thus, for linear (infinitesimal) slab motions, the first methods yields the same resonant frequencies as the more difficult second method. For finite amplitude (non-linear) motions in different device configurations, it is possible two methods could yield different results (in which case the second method is the appropriate technique). 19

22 References [1] J. P. Verboncoeur, M. V. Alves, V. Vahedi, and C. K. Birdsall, Simultaneous potential and circuit solution for 1d bounded plasma particle simulation codes, J. Comp. Phys., vol. 104, pp , February [2] V. A. Godyak, Soviet Radio Frequency Discharge Research. Falls Church, VA: Delphic, [3] D. J. Cooperberg and C. K. Birdsall, Series resonance sustained plasma in a metal bound plasma slab, Plasma Sources Sci. Technol., vol. 7, pp , May [4] M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing. New York, NY: John Wiley and Sons, [5] J. V. Parker, J. C. Nickel, and R. W. Gould, Resonance oscillations in a hot nonuniform plasma, Phys. Fluids, vol. 7, p. 1489,