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Electron Series Resonant Discharges: Part II: Simulations of

Initiation

K. J. Bowers1 W. D. Qiu2 C. K. Birdsall3

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 94720

Submitted to Plasma Sources Sci. Technol.

December 20, 2000

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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 su cently 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.

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 pro le 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 brie y

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 160cm2 ; 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 lled by a spatially uniform warm argon plasma (neutral

pressures of 3mT orr and 10mT orr with a plasma density of n 0.1 1010 cm3 ). Electrons and ions

are initially isotropic Maxwellian with temperatures of Te 2eV and Ti 0.026eV = 300K (room

temperature) respectively. An ideal RF voltage source V = VRF sin RF t is applied to the diode at

t = 0. Simulations with VRF 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)):

r = p (1)

( r is the ESR frequency, p is the peak electron plasma frequency, 2 is the combined cycle-averaged

sheath width and L is the diode separation.)

For the lock-on phenomenon to happen, RF must come into resonance with the ESR. For

the given initial pro le (uniform), space charge sheaths form and move towards the diode center

at roughly the ion acoustic speed at rst leading to an ESR frequency increasing with time as

2s increases. Over longer time scales, ambipolar di usion takes over and the bulk plasma decays

(unless lock-on has occured) leading to a decreasing ESR frequency with time as p decays.

Hence, initially VRF must be large enough to allow the sheaths to expand su ciently such that

the drive frequency and the ESR frequency coincide before di usion causes the plasma to decay

away. For the simulations here, the minimum voltage for lock-on (Vcr ) was empirically found to be

5V (zero-to-peak), which was larger than the minimum voltage Vr required to sustain the resonant

discharge. (Vr 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 pro le 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 r approaching a constant drive frequency from below is studied.

An exact resonant discharge requires VRF = Vr . However, with Vcr > Vr 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 VRF Vcr until the plasma locks-on and then decrease the applied

voltage to Vr . 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 IRF leading VRF by close to 90 (capacitive). As electrons and ions ow to the wall, sheaths

form at both electrodes and their cycle-averaged widths increase with time. Since the peak plasma

frequency p stays nearly constant, the ESR frequency r increases in time due to the increasing

sheath width s (Figure 1e).

As the sheath width 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 width

before lock-on for two di erent pressures and di erent RF drive voltages. Least squares tting to

a line the log of sheath width versus the log of time before lock-on from the data taken at all the

RF drive voltages at a given pressure gave 1/3 at p = 3mT orr and 1/4 at p = 10mT orr.

During this time, the motion of the plasma bulk is in-phase with the voltage drive.

If VRF Vcr,

the explosive growth ends at some time t0, denoted as the transition time. For the same dimensions

and the same initial plasma, t0 decreases with increasing RF voltage (this is a consequence of the

1 A quantity which undergoes explosive growth obeys a scaling law 1/(t t), > 0. Generally, the quantity is

limited by other physical e ects from becoming in nite at t0 . Explosive growth is faster than exponential growth.

Figure 1: Lock-On: The plasma is driven by an ideal voltage source with VRF = 7.07V at 3mT orr

neutral pressure. Graphs (a), (c) and (d) show the cycle-averaged signal and signal envelope. Lock-

on occurs at t0 = 4 s. RF current magnitude, power absorbed, plasma potential, sheath width and

average electron kinetic grow explosively before lock-on; (t0 t)1/3 for this case. Slow time-scale

ringing is seen after lock-on and the plasma no longer decays.

Explosive growth in sheath width (p=3mTorr)

V = 7.07V

V = 10V

V = 14.1V

VRF = 20V

Total sheath width (sl + sr) ( m)

1/3

s (t0 t)

2 1 0

10 10 10

Time before lock on ( s)

Explosive growth in sheath width (p=10mTorr)

V = 7.07V

VRF = 10V

V = 14.1V

V = 20V

Total sheath width (sl + sr) ( m)

s (t t) 1/4

2 1 0

10 10 10

Time before lock on ( s)

Figure 2: Explosive Growth: These graphs show the growth of the sheath width versus time at

two di erent pressures and several RF drive voltages for the same starting plasma. All curves show

behavior similar to 1/(t0 t) shortly before lock-on. appears dependent on the gas pressure

(the dashed line in the gures is from a least squares t of the slope from all the simulations at the

respective pressures) with 1/3 at 3mT orr and 1/4 at 10mT orr. is less dependent on the

RF drive voltage.

larger initial sheath oscillations due to the larger applied voltages).

During Lock-On

As t approaches t0, r increases to RF leading to a dramatic transition in just a few RF cycles.

The decay rate of the total number of electrons, Ne, suddenly increases as a burst of electrons

ows to the walls (Figure 1b). Consequently the plasma potential mid roughly triples (Figure 1d).

IRF becomes in-phase with VRF 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 eld in the bulk plasma is opposite

in direction to the electric eld of the vacuum diode and the electric eld 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 vx phase space (Figures 3c and

3d). The unique heating pro le 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 labora-

tory 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 M Hz .

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 o resonance). Such is shown explicitly later in this article.

After the decay, IRF leads VRF 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 pro le has both positive and negative portions (Figure

3d). Observation of the bulk plasma also shows another di erence 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 di erence in the bulk motion is seen from the potential and electric

DC and RF potential (before) DC and RF potential (after)

40 40

30 30

20 20

Potential (V)

10 10

0 0

10 10

DC DC

At V (t)=V At V (t)=V

20 20

ckt RF ckt RF

At V (t)= V At V (t)= V

ckt RF ckt RF

30 30

0 2 4 6 0 2 4 6

Position (cm) Position (cm)

Electron Phase Space (before) Electron Phase Space (after)

20 20

x Kin. Energy (eV) (+ right, left)

15 15

10 10

5 5

0 0

5 5

10 10

15 15

20 20

0 2 4 6 0 2 4 6

Position (cm) Position (cm)

Electron and Ion Heating Profile (before) Electron and Ion Heating Profile (after)

2500 2500

Electrons Electrons

2000 2000

Ions Ions

Power absorbed (W/m3)

1500 1500

1000 1000

500 500

0 0

500 500

1000 1000

1500 1500

0 2 4 6 0 2 4 6

Position (cm) Position (cm)

Figure 3: Before and After Lock-On: This data was taken 3 s before and after the lock-on transition

for the simulation shown in Figure 1. (top) The bulk plasma electric eld after lock-on is 180 out-

of-phase with the applied eld and with the before lock-on bulk electric eld. (middle) The particle

position versus x-directed kinetic energy phase space snapshots were taken when V ckt (t) = VRF .

Electron bunching and bunch acceleration is seen after lock-on. Ionization threshold is 15.8eV .

(bottom) The unique heating pro le forms after lock-on.

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 (4 s), 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.

eld inside the diode (Figure 3b).

It should also be noted that the peak RF potentials at the sheath edges greatly exceed the

applied RF eld 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 acceler-

ated 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

allows the slow time scale evolution of the high frequency spectrum to be monitored. The gure

uses a Hann window to keep spectral leakage low with a window length of 1.13 s and a sample rate

equal to the simulation timestep of 140ps.) In this gure, the dark vertical band at 81M Hz is the

drive frequency. Before lock-on the ESR frequency is seen increasing (from 60M hz at t = 0) as

the space charge sheaths form. At t = 4 s, the ESR interacts with the drive frequency and the

plasma is sustained.

As this discharge is driven slightly o 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 rst Tonks-Dattner resonance is seen decreasing in frequency as the plasma pro le changes

during lock-on holding constant at 200M Hz after lock-on (Tonks-Dattner resonances are the cuto

of thermal waves trapped in the plasma sheath and were rst 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 elds 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 eld 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

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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 p ), the phase of the plasma sloshing and

RF elds 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 di cult 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.

For clarity, the equation of motion is given here:

d2 d3 d2

d 2d

+ + r I = C v + 2 + p V (2)

dt2 dt3

dt dt dt

The sudden change in sheath width 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 V = Vs IRs where Vs and Rs are given and substituting into (2) yields:

d3 d2 d3 d2

2d 2d

Rs Cv + (1 + Rs Cv ) 2 + ( + Rs Cv p ) + r I = Cv + 2 + p Vs (3)

dt3 dt 3

dt dt dt dt

For a simple RF drive of the form, Vs = VRF sin 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 width (and hence in r ) introduces short transients after lock-on.

A root nder may be used to compute the three characteristic complex frequencies associated with

the homogeneous part of (3). Requiring I to be continuous through the second derivative across the

change in r xes 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 and source resistance R s .

The ringing frequency is approximately r 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 Rs = 0, Cv = 2.11pF,

p /300, p 2 (284M Hz ). 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 underesti-

mated 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 resis-

tance) it may have much more predictive power regarding transients and hysteresis in low-pressure

low-temperature discharges.

Simulated Lock on Ringing (p=3mTorr, V =7.07V, Rs=0 )

0.8

0.6

Current into electrode (A)

0.4

0.2

0.4

0.6

0.8

3 3.5 4 4.5 5 5.5 6 6.5 7

Time ( s)

Homogeneous Model Fit (step in sheath width)

0.8

0.6

Current into electrode (A)

0.4

0.2

0.4

0.6

0.8

3 3.5 4 4.5 5 5.5 6 6.5 7

Time ( s)

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 width. Qualitative

agreement with the simulated data is seen.

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 VRF > Vcr and a p > RF . Vcr

(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 width and other discharge parameters appear to grow explosively before lock-on.

The heating pro le observed in simulation shows di erent 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 consid-

eration 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-49620-96-1-

0154, ONR contract N00173-98-1-G001 and the Fannie and John Hertz Foundation Fellowship Pro-

gram.

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Figure 7: Homogenous Model of a Bounded Plasma: The electrons are cold, uniform and bordered

by matrix sheaths. The ions uniformly ll 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 ll 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 con nes 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 ow

into plates for high frequencies (well above the ion plasma frequency) is desired. The simplest way

to nd 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:

0 1 (4) e2 n/ 0 m; is the

( 0 is the permittivity of free space; p is the electron plasma frequency,

frequency being examined; is the e ective electron collision frequency; e is the magnitude of the

electron charge; m is the electron mass.) The impedance of the device is thus:

2 1 + d

sl d sr

Z = + + =

2 2

p p

0 A A 1 0 A 0 A 1 0 13

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

sr are the left and right sheath widths respectively, d is the width of the bulk plasma, p is the

electron plasma frequency, is the electron-neutral collision frequency and 0 is the permittivity of

free space. The impedance of this circuit is the same as (5).

2s 2

L p L

= (5)

p 0 A

2 is the combined width of the matrix sheaths (sl + sr ). With V = V14 being the voltage across

the plates, I being the current owing into the left plate and Z = V /I, rearranging terms

gives:

2 2

s 0 A

2 + + 2 + + p V 2

I = (6)

Lp L

I e t d and V (t) V e t d, the time domain di erential relationship may be

As I (t)

written by inspection:

d2 d3 d2

d 2d

+ + r I (t) = Cv + 2 + p V (t) (7)

2 3

dt dt dt dt dt

Cv is the vacuum capacitance of the parallel plates ( 0 A/L) and r is the electron series resonance

frequency ( p 2s/L).

An equivalent derivation takes the circuit corresponding to Z (shown in Figure 8) and nds

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 e ect).

It should be noted that the above derivations implicitly assume p,, sl and sr are constant

in time and consequently ignore non-linearities in the sheath capacitances. These non-linearities

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 di erential 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 sr 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 signi cant in ESR sustained plasmas and the derivation is done strictly

in the time domain.

The derivation begins with Poisson s equation in the device:

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