Electron-rich sheath dynamics. II. Sheath ionization and relaxationinstabilities

R. L. Stenzel,1,a) J. Gruenwald,2 C. Ionita,2 and R. Schrittwieser2

1Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA2Institute for Ion Physics and Applied Physics, University of Innsbruck, A-6020 Innsbruck, Austria

(Received 16 February 2011; accepted 1 June 2011; published online 28 June 2011)

Instabilities in an electron-rich sheath on a plane electrode in a discharge plasma have been

investigated experimentally. The high-frequency sheath-plasma instability near the electron plasma

frequency is observed. With increasing dc voltage, the instability exhibits bursty amplitude and

frequency jumps. The electrode current shows spikes and jumps, and the plasma potential near the

electrode shows large fluctuations below the ion plasma frequency. Sheath-ionization has been

identified as the cause for these low frequency instabilities. Electrons energized in the sheath

produce ions which reduce the space charge in the sheath and the electric field and the ionization

rate. Ions are ejected from the sheath which increases the charge density, electric field, and

ionization rate. The positive feedback between these processes leads to a relaxation instability

whose time scale is determined by ion inertia and ionization rates. The associated density and

potential fluctuations affect the amplitude and frequency of the sheath-plasma instability. When the

sheath ionization rate exceeds the ion losses, the sheath expands into an anode plasma or “fireball.”

The potential drop across the sheath decreases and the sheath-plasma instability vanishes. The

electrode current-voltage characteristics develop a region of negative conductance. For short grid

voltage pulses, the ionization effects can be avoided. VC 2011 American Institute of Physics.

[doi:10.1063/1.3601860]

I. INTRODUCTION

Biased electrodes in plasmas have a wide use in plasma

physics such as for diagnostic probes,1 pulsed anodes,2 beam

and rf sources,3 antennas4 and tethered satellites,5–8 and lim-

iters.9 Ion rich sheaths of negatively biased electrodes have

received more attention than electron-rich sheaths because

the latter frequently introduce plasma modifications and

instabilities.10 Electron-rich sheaths have many similar fea-

tures to vacuum diodes whose instabilities have been studied

extensively.11–13 For example, electron inertia in diodes cre-

ates a negative rf resistivity which leads to high frequency

oscillation. Analogous transit-time instabilities have been

observed in electron-rich sheaths14 and fireballs.15

Ionization phenomena can also be a source of low fre-

quency instabilities in sheaths and double layers.6,16,17 When

ionization starts in a sheath, it can expand into a localized

discharge called a “fireball.” Their properties have been

studied extensively but are still under investigation due to

their chaotic behavior.18–25 Similar to discharges, the feed-

back between plasma production and losses can lead to

relaxation instabilities which evolve into chaos.22,26,27

In the present work, ionization-related instabilities have

been observed in an electron-rich sheath. They were also

visible in the companion paper28 but not elaborated on since

the focus was on transient current phenomena. Sheath ioniza-

tion occurs in weakly ionized gases when the electrons gain

sufficient energy for ionization. While the electrons are rap-

idly collected, the ions remain for about an ion plasma period

in the sheath. They decrease the space charge and thereby

the electric field and ionization rate. After the ions are

expelled for a few Debye lengths, the electron-rich sheath

recovers and the process repeats as a relaxation instability.

The potential and density fluctuations strongly modulate the

collected current. The purpose of this investigation is to

explain the often spiky and chaotic current waveforms.

Potential and density fluctuations also modify the high fre-

quency sheath-plasma instability14 which can be used to

diagnose the plasma dynamics at the sheath edge. Strong

sheath ionization expands the sheath into a fireball which

quenches the sheath-plasma instability.

The paper is organized as follows: After a brief review

of the experimental setup, the observations of sheath instabil-

ities are presented in Sec. III, compared with earlier numeri-

cal simulations in Sec. IV, and summarized in Sec. V.

II. REVIEW OF THE EXPERIMENTAL ARRANGEMENT

The experiments have been performed in a low tempera-

ture discharge plasma ( ne ’ 109 cm�3, electron temperature

kTe ’ 3eV, at an argon gas pressure p ’ 5� 10�4 mbar) of

chamber dimensions 45 cm diameter, 100 cm length. The

plasma is unmagnetized, uniform, quiescent, and well suited

for basic studies of waves and instabilities.

A gridded electrode (6 cm diameter) is inserted into the

discharge center and biased positively, either in dc or with a

step-function voltage pulse up to 300 V. A movable rf probe

is used to detect potential fluctuations. Grid voltage, current,

and rf signals are recorded simultaneously on a digital oscil-

loscope. Light from the sheath is detected with a photodiode

or a digital camera. Further details were presented in a com-

panion paper.28a)Electronic mail: stenzel@physics.ucla.edu.

1070-664X/2011/18(6)/062113/9/$30.00 VC 2011 American Institute of Physics18, 062113-1

PHYSICS OF PLASMAS 18, 062113 (2011)

An example of light emission from a positively biased

grid is shown in Fig. 1. At low pressures, a thin layer

around the grid emits light when electrons energized in the

sheath excite transitions in Ar neutrals. At higher pressures

[Fig. 1(b)], a fireball is formed. It is a spherical discharge

plasma produced by electrons accelerated in a double layer

at the fireball boundary. The double layer potential drop is

of order of the ionization energy. The potential drop across

the sheath on the grid is small and insufficient to excite the

sheath-plasma instability. Plasma fireballs are also the prin-

ciple of hollow anode devices29 or constricted plasma

sources.30

III. EXPERIMENTAL RESULTS

A. Sheath ionization

A dc discharge in single or double plasma devices pro-

duces usually very quiescent plasmas.31 Thus one would

expect that a positively biased electrode has a stable sheath

and a constant sheath-plasma frequency. However, this is

only observed for very low electrode voltages. Figure 2

shows (a) the waveforms and (b) the time resolved spectra of

the sheath plasma instability for four different grid voltages.

To avoid transient phenomena, the voltage is not pulsed but

is a constant dc voltage. With increasing grid voltage, the rf

waveform changes from a low level but stable oscillation

into large amplitude bursts. Visual inspection shows a glow

around the grid as displayed in Fig. 1(a). Figure 8 in part I

(Ref. 28) showed that the frequency of the sheath-plasma

instability is proportional to the plasma frequency close to

the grid where the instability occurs. From the spectra, one

must therefore conclude that with increasing grid voltage the

average density near the electrode decreases which is rea-

sonable since the ions are repelled from the vicinity of the

grid.

However, within each burst, there are also periods when

the frequency increases, in particular for high grid voltages.

For example, Fig. 2(d) shows a 10% jump in frequency

which implies a 20% jump in density. An explanation for

such large density jumps is sheath ionization. Ions created

inside the sheath are expelled, which temporarily increases

the density at the sheath edge but subsequently falls again.

The event repeats on the time scale of ion plasma periods,

2p=xpi ’ 1 ls.

FIG. 1. (Color online) Excitation of light due to energetic electrons in

(a) the sheath and (b) the fireball of a plane gridded electrode in a low

density discharge plasma. Parameters: Vgrid ’ 80V, (a) p ’ 10�4 mbar, (b)

p ’ 10�3 mbar.

FIG. 2. (Color online) Fluctuations in the sheath-plasma instability in a quiescent plasma. (a) With increasing dc grid voltage, the rf waveforms become bursty.

Time resolved frequency spectra showing frequency fluctuations and a general frequency drop with increasing Vgrid. Parameters: (a) Vdis¼ 65 V, Idis¼ 0.7 A,

p¼ 2.9� 10�4 mbar, 0.8 ns=sample.

062113-2 Stenzel et al. Phys. Plasmas 18, 062113 (2011)

Sheath ionization strongly affects the potential profile

and the instability amplitude. Ion injection into an electron-

rich sheath reduces the electron space charge density which

decreases the potential gradient, i.e., widens the sheath.32

This results in an increase in the electron transit time through

the sheath and a mismatch with the rf period which decreases

the growth rate of the sheath-plasma instability and creates a

bursty waveform. If sheath ionization occurs differently

across the grid or on either side of it, the instability can oscil-

late at several frequencies simultaneously as seen in

Fig. 2(b).

If the dc grid voltage is raised much further, the instabil-

ity vanishes completely. In order to explain this surprising

result, we return to experiments with pulsed grid voltages

which allow us to observe the evolution of the sheath and

plasma.

Figure 3 shows waveforms of Igrid, Vgrid, and Vprobe for

three large grid voltages and the corresponding instability

spectra. As the grid voltage rises, a spike in the grid current

arises which indicates the formation of the electron-rich

sheath. At this time, the ions have been expelled from the vi-

cinity of the electrode, the grid potential drops off mainly

inside the electron-rich sheath, hence decreases elsewhere

near the grid and outside the sheath. The grid current spike is

a displacement current associated with the plasma potential

drop and the sheath capacitance.28 Once an electron-rich

sheath is formed, the sheath instability starts as seen in Figs.

3(a) and 3(b) for Vgrid¼ 100 V. Both frequency and grid cur-

rent gradually decrease indicating a density loss near the grid

due to ion repulsion. The same effect was seen by increasing

the dc grid voltage (Fig. 2).

When the peak grid voltage is raised to Vgrid¼ 200 V,

two grid current spikes are observed [Figs. 3(c) and 3(d)].

The first has been described above, the second is spontane-

ous, and the beginning of repeated spikes is discussed further

below. Between the two pulses, the grid current and sheath

plasma frequency are constant. Ionization compensates the

density loss due to ion expulsion.

For a peak grid voltage of Vgrid¼ 300 V, the grid current

rises dramatically [Figs. 3(e) and 3(f)]. The frequency of the

sheath plasma instability also rises in time. Thus, the density

near the sheath increases since the ionization rate exceeds

the ion loss by expulsion. However, sheath ionization

increases the sheath thickness which quenches the sheath-

plasma instability. Thus, no high frequency oscillations are

found in steady state. The sheath expands into a fireball as

shown in Fig. 1(b). More evidence for fireball formation will

be presented further below.

B. Ionization relaxation oscillations

In experiments with a pulsed grid voltage, it has been

noticed that the first current spike is followed by a second

one [Fig. 3(c)]. When the pulse length is increased, one

observes repeated transient phenomena in the grid current

even though the grid voltage is constant. Strong fluctuations

are also observed on the rf probe but depend on the distance

from the grid. The transient phenomena have a strong effect

on the sheath-plasma instability.

Figure 4 shows waveforms of the most important param-

eters, grid current and voltage with sheath-plasma oscilla-

tions and the smoothed rf probe voltage as an indicator of

plasma potential fluctuations. The results are shown for two

different axial probe positions for single shot events which

are generally not reproducible.

The start of the waveforms has been described earlier:

At the beginning, the grid is surrounded by an ion-rich

sheath. The rapidly rising grid voltage creates a large initial

grid current since the electric field is not Debye shielded. Af-

ter about an ion plasma period, the ions are ejected and an

electron-rich sheath forms, the potential drops abruptly,

which creates a displacement current pulse in the grid and

FIG. 3. (Color online) Transients and instabilities for different grid voltages. (a), (b) Igrid, Vgrid, Vprobe and time resolved spectrum Vgrid,rf(f,t) for Vgrid,dc¼ 100

V. The frequency decreases indicating a density drop by ion expulsion. (c), (d) Same data for Vgrid,dc¼ 200 V, where ionization prevents the initial density

drop. A second current transient appears. (e), (f) Waveforms and spectrum for Vgrid,dc¼ 300 V, where enhanced ionization raises the density, hence current and

frequency. Further current rise indicates the transition from sheath ionization to fireball formation which quenches the sheath-plasma instability. Parameters:

Vdis¼ 65 V, Idis¼ 0.5 A, p¼ 3.5� 10�4 mbar, 0.4 ns=sample, single shot traces, Igrid and Vgrid smoothed to eliminate high frequency oscillations.

062113-3 Electron-rich sheath dynamics. II. Phys. Plasmas 18, 062113 (2011)

the beginning of the sheath-plasma instability. By comparing

Figs. 4(a) and 4(c), the probe voltage, i.e., plasma potential,

rises in both locations but the subsequent fluctuations arise

only near the grid.

The subsequent repetitive events have the following

character: The plasma potential near the grid rises and falls

abruptly with an approximately repetition time of a few

microseconds. When the potential is low, the electron-rich

sheath and the sheath instability exist. When the potential is

high, the sheath has been partially neutralized by sheath ioni-

zation which quenches the instability. The grid current rises

when sheath ionization occurs and falls when the potential is

low and the sheath is thin. Neither a narrow nor a wide

sheath is stable. Ionization in the narrow sheath widens it. In

a wide sheath, ionization is reduced since the potential rises

everywhere and the electrons gain little energy. Since ions

are still ejected, the electron-rich sheath near the electrode

recovers and the cycle repeats. Thus, ions created in the

sheath disrupt the sheath potential are ejected and the sheath

regrows. Ion inertia and ionization rates determine the time

scale of the sheath dynamics. An example of ions ejected af-

ter an ionization event is shown in the companion paper28 in

Fig. 6(b) for 7< t< 11 ls.

Figures 4(b) and 4(d) present on an expanded time scale

single cycles which show a difference in the plasma potential

at the two probe positions. Near the grid, the potential drops

when the electron-rich sheath is formed. At 5 cm from the

grid, the potential increases, albeit by a much smaller

amount. Thus, the axial electric field near the grid is

enhanced which produces the displacement current visible in

the grid current. The same displacement current is produced

at the chamber wall where the plasma potential rises. Since

the sheath capacitance to ground is much larger than at the

grid, only a small potential rise is needed to produce the

return current.

The grid current fluctuations are due to local potential

fluctuations near the grid. Sheath expansion and contraction

varies the effective surface area for collecting electrons. The

large transient currents are not due to collecting electrons but

to shift them into the electron-rich sheath.

Now, we will investigate the density fluctuations near

the grid, using the sheath-plasma instability as a diagnostic

tool. In order to resolve the high frequency waveform, a fast

sampling rate is needed which results in large files when re-

cording long waveforms. This has been done only in selected

cases one of which is shown in Fig. 5. It displays the wave-

forms of grid current and rf probe voltage in the center of the

grid [Fig. 5(a)] and the time-resolved spectra of the sheath-

plasma oscillations [Fig. 5(b)].

The sheath-plasma oscillations start about an ion plasma

period after applying the grid voltage pulse. The starting fre-

quency f ’ 400 MHz gives a measure for the initial density

FIG. 4. (Color online) Spontaneous transient currents and instabilities. (a) Probe at the grid center shows large amplitude potential fluctuations indicated by

Vprobe. When the potential drops, a large displacement current pulse is induced in the grid, best seen on an expanded time scale (b). (c), (d) Probe axially

displaced from the grid center detects only small fluctuations. The potential away from the grid rises when the potential near the grid drops, implying a

potential well is formed. The sheath-plasma instability is large when the potential near the grid drops, i.e., a large sheath potential drop is present. Parame-

ters: Vdis¼ 100 V, Idis¼ 0.5 A, p¼ 3.3� 10�4 mbar, 10 ns=sample, single shot traces, Vprobe smoothed to eliminate high frequency oscillations.

062113-4 Stenzel et al. Phys. Plasmas 18, 062113 (2011)

since ions have only been moved by a few Debye lengths.

However, ion expulsion from the vicinity of the grid contin-

ues and after t ’ 10 ls, when the instability vanishes, the fre-

quency has dropped to f ’ 250 MHz or the density to 39%

of its initial value. The grid current also has dropped but by a

smaller ratio than the density, indicating that the density per-

turbation in the center of the grid is larger than at the edge.

The coincidence between the loss of the instability, a

rise in the plasma potential, and an end of the current decay

has been explained by ionization in the sheath. Thus, when

the instability reappears, it has a higher frequency than when

it ended. However, the ionization rate is not sufficient to

reestablish the initial density. Each subsequent burst has the

same starting and ending frequency implying the same pro-

duction and losses occur from burst to burst. Without ioniza-

tion, the frequency would only decay.

Sheath ionization is usually accompanied by excitation

of light. Light emission has been detected with a simple pho-

todiode as shown in Fig. 6. Unfortunately, light fluctuations

on the time scale of sheath fluctuations could not be resolved

due to the low light intensity from a thin sheath. Neverthe-

less, light emission from the sheath occurs in the same pa-

rameter regime where the sheath fluctuations are observed.

In the present experiment, all observations were done

for a plane sheath geometry. However, a brief comparison

with a cylindrical sheath was also performed. A straight wire

(1 mm diameter, 10 cm length) was used as electrode instead

of the plane grid. When the same voltage waveform was

applied, the current into the wire also showed a broad over-

shoot and subsequent current spikes as displayed in Fig. 7.

The sheath-plasma instability also occurred in bursts. The

initial frequency is higher than that of the subsequent bursts.

The frequency during the bursts and from burst to burst is

nearly constant since the duration and spacing of the bursts

are on the order of the ion plasma period. The radial expul-

sion of ions from the thin wire causes a much smaller and

more localized density perturbation than that of the 6 cm

wide grid. Nevertheless, a cylindrical electron-rich sheath is

also unstable to high frequency sheath-plasma oscillations

and low frequency ion relaxation instabilities. Modulation of

the high-frequency instability by a low frequency near xpi

had been observed earlier14 and the present observations

explain the mechanism.

C. Sheath expansion into fireball

Sheath ionization is not very efficient since the sheath

thickness is small compared to the mean free path for

FIG. 6. (Color online) Light emission from a grid with sheath ionization as

indicated by fluctuations in grid current and sheath-plasma oscillations. Photo-

diode response does not resolve light fluctuations. Parameters: Vgrid,dc¼ 50 V,

Vdis¼ 96 V, Idis¼ 0.5 A, p¼ 3.3� 10�4 mbar.

FIG. 5. (Color online) Spontaneous current fluctuations and high frequency bursts. (a) Waveforms of Igrid and Vrf,probe for a long grid voltage pulse, sampled

rapidly to resolve the high frequency oscillations. (b) Time resolved spectrum of Vrf,probe. During each burst, the frequency decays due to ion losses. Between the

bursts sheath, ionization increases the density so that the starting frequency is higher than the ending frequency. Parameters: (a) Vgrid,dc¼ 45 V, Vdis¼ 32 V,

Idis¼ 0.9 A, p¼ 3.8� 10�4 mbar, 0.8 ns=pt, 105 pts, single shot trace.

062113-5 Electron-rich sheath dynamics. II. Phys. Plasmas 18, 062113 (2011)

ionizing collisions. Periods of density increase are followed

by periods of plasma ejection such that only local density

fluctuations are produced. The ionization rate scales with

electron density, electron energy, and neutral density. When

the ion production exceeds the ion ejection, the sheath

expands into an anode discharge called a plasma “fireball.”

In a pulsed experiment, this is indicated by a rapid growth of

the grid current, as shown in Fig. 3(e). As the density rises,

the frequency of the sheath-plasma instability increases [Fig.

3(f)]. Unfortunately, the sheath expansion into a double layer

quenches the sheath-plasma instability such that one cannot

follow the entire density rise.

However, an alternate approach is to use a double pulse

technique. A long voltage pulse is broken up into two pulses,

each of which briefly produces the instability at the voltage

turn-on. By comparing the frequencies, the density increase

from the first pulse is obtained.

An example of this double pulse technique is shown in

Fig. 8. On a long time scale, Fig. 8(a) displays Vgrid, Igrid,

and Vprobe,rf for both pulses. The second voltage pulse has

been made shorter than the first one since the rf burst is also

of short duration. The first grid current shows an initial rise

with the grid voltage step, followed by another rapid rise due

to the growth of a fireball. The current increase is partly due

to the growing surface area for electron collection and partly

FIG. 7. (Color online) Instabilities in the cylindrical sheath of a 1 mm diam-

eter, 10 cm long wire. (a) Pulsed voltage, current and sheath-plasma instabil-

ities. (b) Time resolved spectra of the bursty rf emissions. Parameters:

Vdis¼ 32 V, Idis¼ 0.9 A, p¼ 3.8� 10�4 mbar, 0.4 ns=sample.

FIG. 8. (Color online) Double pulse experiment with fireball formation. (a) Waveforms of grid current, voltage, and rf oscillations. The delayed current rise is

due to fireball formation. (b), (c) Similar waveforms for the first and second pulses on an expanded time scale. (d), (e) Time-resolved spectra of the sheath-

plasma oscillations showing a large frequency increase of the second pulse due to the density increase of the first fireball. Parameters: Vdis¼ 24 V, Idis¼ 0.6 A,

p¼ 4.3� 10�4 mbar, BaO cathode, 10 ns=sample, Vgrid and Igrid are 20 MHz bandwidth limited.

062113-6 Stenzel et al. Phys. Plasmas 18, 062113 (2011)

due to the density increase. The second current pulse starts at

a higher value; hence ionization occurs faster since the first

pulse has increased the density.

Figures 8(b) and 8(c) show the same waveforms on an

expanded time scale, albeit from two repeated events. The

first rf waveform is larger and less erratic than that of the

second pulse. However, the main difference lies in their

spectra displayed in Figs. 8(d) and 8(e). The first pulse starts

at 300 MHz and decays due to ion expulsion and little ioni-

zation. The second rf spectrum starts around 500 MHz with

multiple lines, indicating a density rise by a factor

ð5=3Þ2 ’ 2:8 with significant density gradients across the

grid. The pronounced beat waveform is likely due to a den-

sity gradient normal to the grid since fireballs generally de-

velop only on one side of a grid [see Fig. 1(b)].

By varying the spacing between the pulses, the density

decay can be mapped. By varying the pulse width of the first

pulse, the density growth can be measured. If two grids were

used, the spatial density profile could be mapped. Thus, the

sheath-plasma frequency can be used for time and space

resolved density diagnostics.

The fireball formation also produces a strong change in

the current-voltage relation of an electrode. As in Fig. 4 of

the companion paper,28 Fig. 9 displays time-resolved I-V

characteristics of the grid, although for larger voltages and

longer time scales. A set of voltage step functions has been

applied to the grid [Fig. 9(a)]. Since the voltage source is a

charged capacitor, its voltage decays for large grid currents.

The measured grid currents for each voltage are displayed in

Fig. 9(b). After the initial capacitive overshoot, the current

first decreases due to ion expulsion, then increases due to

ionization which increases with Vgrid. The grid current can

rises to an order of magnitude above the original “saturation”

current. The BaO coated cathode is partly responsible for the

efficient ionization. The fireball raises the plasma potential

in the ambient plasma, thereby increasing the space-charge

limited cathode emission and energy of the primary electrons

which in turn produces a denser plasma.

From the time-varying current and voltage waveforms,

the instantaneous I-V characteristics have been constructed.

For low voltages (Vgrid< 80 V), the grid current shows a sat-

uration value Igrid ’ 0:2 A once the sheath has been formed

(t> 2 ls). Prior to that, the I-V characteristics are essentially

nearly linear since the unshielded electric field penetrates

into the plasma and collects electrons from an area which

increases with voltage.

At higher voltages (Vgrid> 120 V) and later times

(t> 3 ls), the I-V curves first develop a jump, then a region

with dI=dV< 0, followed by a steep positive slope in the I-V

characteristics. The negative slope is due to the discharging of

FIG. 9. (Color online) Effect of fireball formation on the current voltage characteristics of a pulsed electrode. (a) Family of grid voltage pulses. (b) Grid cur-

rent waveforms for different voltages. Some corresponding curves are labeled a-i. (c) Instantaneous Igrid vs Vgrid characteristics at early times. Ionization causes

the large current increase at Vgrid> 120 V. (d) Igrid�Vgrid characteristics at late times showing a negative conductance when fireballs are formed. Parameters:

Vdis¼ 32 V, Idis¼ 0.5 A, p¼ 3.6� 10�4 mbar.

062113-7 Electron-rich sheath dynamics. II. Phys. Plasmas 18, 062113 (2011)

the storage capacitor with increasing grid current. A negative

differential conductance has also been seen when a dc voltage

is applied via a series resistor.22

IV. COMPARISONS OF LAB EXPERIMENTS WITHEARLIER SIMULATIONS

Several numerical simulations have been performed on

positively pulsed spherical electrodes with sheath ionization.

It is interesting to point out the common observations as well

as differences.

The simulation model of Singh and Jaggernauth6 con-

sists of a sphere with large positive dc bias in an unmagne-

tized collisionless plasma. At t¼ 0, a shell of neutral gas is

pulsed on which starts sheath ionization. Ma and Schunk8

and Borovsky33 simulate the case of a pulsed voltage but do

not include ionization. All simulations are done for spherical

sheaths while the lab experiment investigates plane sheaths.

In the first-mentioned simulation, Poisson’s and hydro-

dynamic equations are solved in space and time. Ionization

causes the spherical sheath to expand, temporarily forming a

double layer, then a positive potential hill, and finally it col-

lapses into a thin sheath. The potential hill is due to ions cre-

ated inside the sheath. In time, the hill is neutralized by

ionization-produced electrons. The ionization-produced

plasma expands radially outward but is disrupted by wave

turbulence leading to the collapse of the expanded sheath.

The wave turbulence is due to counter-streaming electrons

and ions and maximizes at the outer edge of the sheath. Ioni-

zation in the thin sheath restarts sheath expansion and a

relaxation oscillation evolves. The collected current

increases after the radial sheath expansion and exhibits a

very strong peak due to the collection of trapped electrons

upon sheath collapse.

Common features between the laboratory experiment

and the numerical model are the sheath dynamics due to ion-

ization. Ionization causes sheath expansion followed by ion

ejection and sheath collapses, which restarts the ionization

and leads to relaxation oscillations. Spiky currents are seen

in both the simulations and the lab experiments. Both lab

experiments and models predict an optimum range of neutral

gas pressure for sheath ionization to occur (10�4–10�3

mbar). Sheath ionization is negligible for pressures an order

of magnitude below this range and fireball formation and

plasma resistivity dominates at an order of magnitude higher

pressure.

Differences are that the lab experiment uses a pulsed

voltage in a dc plasma while the model considers a pulsed

gas cloud and a constant voltage, hence does not include the

initial transient current overshoot. The latter was studied by

Ma and Schunk8 who explained it by an electron conduction

current. No theory discusses how the current is closed and

how transient return currents arise. As pointed out earlier, a

large current overshoot on the positive electrode is also pres-

ent on the negative return electrode and has to be explained.

It cannot be provided by ion collection hence must be a dis-

placement current, which requires a global plasma potential

rise. In the present experiment, a collection of electrons at

the overshoot current and duration would create a huge

charge imbalance since neither sufficient ions can be

extracted nor electrons resupplied, which is obvious for an

afterglow plasma. Thus, the overshoot current cannot be due

to electron collection.

The fluid simulations do not indicate high frequency

sheath-plasma oscillations but clearly show two-stream

instabilities near the ion plasma frequency at the sheath

edge. Similar oscillations are also seen in the present labora-

tory experiment (Figs. 5(b) and 9(a) and 9(b) in the compan-

ion paper28). However, the observed frequencies (f¼ 10–20

MHz) are well above the ion plasma frequency which calls

for further studies of the actual wave mode.

Borovsky used particle simulations to address high fre-

quency sheath oscillations. He considers voltage pulses with

rise times of one electron plasma period and observes strong

ringing at the electron plasma frequency. In our experiment,

no plasma ringing is seen for our “fast” pulses with

trise ’ 30 x�1pe . The simulation does not consider that true

step function pulses cannot be realized experimentally

because they create extremely large displacement currents.

Calder et al.34 have also done particle simulations for

spheres with positive voltage pulses of zero rise time.

Besides the transient oscillations at the electron plasma fre-

quency they have also observed long-lasting oscillations

below xpe which could be the sheath-plasma instability. To

our knowledge, no other simulations of the sheath-plasma

instability have been performed even though it is the domi-

nant high frequency sheath instability.

Cooke and Katz5 simulated the formation of spherical

fireballs in space, whose physical principles are well known

from laboratory experiments. But in the laboratory, fireballs

are usually not concentric to a spherical electrode and

become unstable at high electrode voltages.24

V. CONCLUSION

The dynamics of electron-rich sheaths has been studied

experimentally. When a floating electrode is rapidly pulsed

to a large positive voltage, the current to the electrode starts

with a large overshoot which is a displacement current due

to the sheath capacitance. No oscillations at the electron

plasma frequency are excited as predicted by several com-

puter simulations.4,33,34 However, after an ion plasma period,

the electron-rich sheath forms and the sheath-plasma insta-

bility is excited. Since its frequency depends on density and

its amplitude on the sheath potential, it can be used to diag-

nose the subsequent sheath dynamics. In addition, a small rf

probe is used to measure rf fields, potentials, and ballistic

ions. The probe indicates that the high-frequency sheath-

plasma oscillation is an evanescent field, peaking at the

sheath and decaying on the scale of the electrode dimen-

sions. The Debye sheath does not limit the penetration of the

high frequency fields transient electric field during the volt-

age ramp when shorter than an ion plasma period. Therefore,

the probe I-V characteristics are not described by Langmuir

probe theory but is observed to be nearly linear.

After about an ion plasma period, the electric field con-

tracts abruptly into the electron-rich sheath, which results in

a sudden potential drop outside the sheath. A capacitive

062113-8 Stenzel et al. Phys. Plasmas 18, 062113 (2011)

displacement current pulse is induced on the electrode whose

voltage is constant. The transient current closes to the cham-

ber wall where a potential rise produces a capacitive current

in the ion-rich sheath.

Once the electron-rich sheath has been established, it

does not remain stable due to two competing processes: (i)

ions continue to be expelled from the vicinity of the elec-

trode as evident from the frequency decay of the sheath-

plasma instability and (ii) ions periodically enter the sheath

which reduces the space charge and electric field as evident

from the loss of the sheath-plasma instability. Backstreaming

of ions into a sheath with potential eU� kTe is not possible.

Thus, ions must be created inside the sheath by ionization.

The electrons inside the sheath are energetic enough to pro-

duce light excitation and ionization. After ions have been

embedded inside the sheath, the same processes start as in

the beginning: ions are ejected until the electric field collap-

ses again into an ion-free sheath. The result is a relaxation

instability governed by ion inertia and ionization rates. The

instability differs from other repetitive current instabilities

on positive electrodes in magnetized plasmas where acceler-

ated ions return after one cyclotron period35 or after an ion

sound transit across the flux tube of the electrode.36,37

However, the physics is similar to that of unstable fireballs

or discharge devices.22,26,27 Here, the sheath electric field

destabilizes the balance between ion production and losses

which creates a relaxation oscillation. Many of these

features have also been observed in earlier computer

simulations.6

Finally, sheath ionization can expand into fireballs

which can be readily seen by a highly nonlinear I-V charac-

teristics, by the absence of the sheath-plasma instability or

by visual inspection.

ACKNOWLEDGMENTS

One of the authors (R.L.S.) gratefully acknowledges

support and hospitality while staying as guest professor at

the University of Innsbruck in June=July 2010. This work

was supported in part by the Austrian Science Funds (FWF)

under Grant No. P19901 and in part by NSF=DOE Grant

DE-SC0004660.

1H. M. Mott-Smith and I. Langmuir, Phys. Rev. 28, 727 (1926).2C. V. Budtz-Joergensen, J. Boettiger, and P. Kringhoej, Surf. Coat. Tech-

nol. 137, 104 (2001).3V. Bochkov, V. Dyagilev, V. Ushich, O. Frants, Y. Korolev, I. Sheirlya-

kin, and K. Frank, IEEE Trans. Plasma Sci. 21, 802 (2001).

4T. W. Chevalier, U. S. Inan, and T. F. Bell, Radio Sci. 45, RS1010

(2010).5D. Cooke and I. Katz, J. Spacecr. Rockets 25, 132 (1988).6N. Singh and A. Jaggernauth, J. Geophys. Res. 101, 17229, doi: 10.1029/

96JA00925 (1996).7N. H. Stone, W. J. Raitt, and J. K. H. Wright, Adv. Space Res. 24, 1037

(1999).8T. Z. Ma and R. W. Schunk, Plasma Phys. Controlled Fusion 31, 399

(1989).9T. Uckan, R. Isler, T. Jernigan, J. Lyon, P. Mioduszewski, M. Murakami,

D. Rasmussen, J. Wilgen, S. Aceto, and J. Zielinski, Nucl. Fusion 34, 295

(1994).10B. E. Cherrington, Plasma Chem. Plasma Process. 2, 113 (1982).11F. B. Llewellyn, Electron Inertia Effects (Cambridge University Press,

New York, NY, 1941).12J. R. Pierce, J. Appl. Phys. 15, 721 (1944).13C. K. Birdsall and W. B. Bridges, Electron Dynamics in Diode Regions

(Academic, New York, NY, 1966).14R. L. Stenzel, Phys. Fluids B 1, 2273 (1989).15R. L. Stenzel, J. Gruenwald, B. Fonda, C. Ionita, and R. Schrittwieser,

Phys. Plasmas 18, 012104 (2011).16J. C. Johnson, N. D’Angelo, and R. L. Merlino, J. Phys. D: Appl. Phys. 23,

682 (1990).17A. Aanesland, C. Charles, M. A. Lieberman, and R. W. Boswell, Phys.

Rev. Lett. 97, 075003 (2006).18M. Sanduloviciu and E. Lozneanu, Plasma Phys. Controlled Fusion 28,

585 (1986).19B. Song, N. D.’Angelo, and R. L. Merlino, J. Phys. D: Appl. Phys. 24,

1789 (1991).20T. Gyergyek, M. Cercek, R. Schrittwieser, C. Ionita, G. Popa, and V.

Pohoata, Contrib. Plasma Phys. 43, 11 (2003).21L. Conde and L. Leon, Phys. Plasmas 1, 2441 (1994).22C. Ionita, D. Dimitriu, and R. Schrittwieser, Int. J. Mass Spectrom. 233,

343 (2004).23D. G. Dimitriu, M. Afliori, L. Ivan, C. Ionita, and R. Schrittwieser, Plasma

Phys. Controlled. Fusion 49, 237 (2007).24R. L. Stenzel, C. Ionita, and R. Schrittwieser, Plasma Sources Sci. Tech-

nol. 17, 035006 (2008).25S. D. Baalrud, B. Longmier, and N. Hershkowitz, Plasma Sources Sci.

Technol. 18, 035002 (2009).26P. Y. Cheung, S. Donovan, and A. Y. Wong, Phys. Rev. Lett. 61,

13601363 (1988).27T. Klinger, C. Schroeder, D. Block, F. Greiner, A. Piel, G. Bonhomme,

and V. Naulin, Phys. Plasmas 8, 1961 (2001).28R. L. Stenzel, J. Gruenwald, C. Ionita, and R. Schrittwieser, Phys. Plasmas

18, 062112 (2011)29V. Miljevic, Rev. Sci. Instrum. 55, 931 (1984).30A. Anders and M. Kuehn, Rev. Sci. Instrum. 69, 1340 (1998).31R. Limpaecher and K. R. MacKenzie, Rev. Sci. Instrum. 44, 726

(1973).32Y. Nakamura, Y. Nomura, and R. L. Stenzel, J. Appl. Phys. 52, 1197

(1981).33J. E. Borovsky, Phys. Fluids 31, 1074 (1988).34A. C. Calder, W. Hulbert, and J. G. Laframboise, Phys. Fluids B 5, 674

(1993).35S. Iizuka, P. Michelsen, J. Rasmussen, R. Schrittwieser, R. Hatakeyama,

K. Saeki, and N. Sato, J. Phys. Soc. Jpn. 54, 2516 (1985).36J. M. Urrutia and R. L. Stenzel, Phys. Rev. Lett. 57, 715 (1986).37R. L. Stenzel and J. M. Urrutia, Phys. Plasmas 4, 26 (1997).

062113-9 Electron-rich sheath dynamics. II. Phys. Plasmas 18, 062113 (2011)