electron-rich sheath dynamics. i. transient currents and sheath-plasma instabilities ·...
TRANSCRIPT
Electron-rich sheath dynamics. I. Transient currents and sheath-plasmainstabilities
R. L. Stenzel,1 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 17 February 2011; accepted 26 May 2011; published online 28 June 2011)
The evolution of an electron-rich sheath on a plane electrode has been investigated experimentally. A
rapidly rising voltage is applied to a plane gridded electrode in a weakly ionized, low temperature,
and field-free discharge plasma. Transient currents during the transition from ion-rich to electron-rich
sheath are explained including the current closure. Time-resolved current-voltage characteristics of
the electrode are presented. The time scale for the formation of an electron-rich sheath is determined
by the ion dynamics and takes about an ion plasma period. When the ions have been expelled from
the sheath a high-frequency sheath-plasma instability grows. The electric field contracts into the
electron-rich sheath which implies that the potential outside the sheath drops. It occurs abruptly and
creates a large current pulse on the electrode which is not a conduction but a displacement current.
The expulsion of ions from the vicinity of the electrode lowers the electron density, electrode current,
and the frequency of the sheath-plasma oscillations. Electron energization in the sheath creates
ionization which reduces the space charge density, hence sheath electric field. The sheath-plasma
instability is weakened or vanishes. The ionization rate decreases, and the sheath electric field
recovers. A relaxation instability with repeated current transients can arise which is presented in a
companion paper. Only for voltages below the ionization potential a quiescent electron rich-sheath is
observed. VC 2011 American Institute of Physics. [doi:10.1063/1.3601858]
I. INTRODUCTION
The physics of sheaths in plasmas is a topic of broad in-
terest ranging from laboratory to space plasmas. While ion-
rich sheaths on floating or negatively charged boundaries have
received most attention, electron-rich sheaths have been stud-
ied less and still have many interesting and unexplored prop-
erties. These are important for many applications such as
Langmuir probes,1 current collectors in space,2–5 virtual cath-
ode oscillators,6,7 and electrodes forming fireballs in discharge
plasmas.8–10 Electron-rich sheaths are subject to various insta-
bilities, high-frequency sheath-plasma oscillations near the
electron plasma frequency,11 low frequency potential relaxa-
tion instabilities,12 and ionization instabilities in the presence
of neutrals leading to unstable fireballs.10,13–19
An area of particular interest is the transition from an ion-
rich to electron-rich sheath which occurs when a positive volt-
age step is applied to an electrode in a plasma. Current over-
shoots have been observed and plausibly explained.2,20–25
In magnetized plasmas the perturbation of the plasma in the
flux tube of the electrode has been studied extensively. It
can couple to ion cyclotron oscillations,26–28 potential
relaxation oscillations,12 and, for unmagnetized ions, to
recursive ion expulsions from the electrode flux tube.29,30
The present experiment deals with transient currents
during the rapid formation of an electron-rich sheath in a
partially ionized field-free plasma. Particular attention is
given to the sheath-plasma instability which is used for diag-
nosing the sheath. It reveals new properties of electron den-
sities and potentials near the electrode. New findings include
large initial transient currents identified as displacement cur-
rents, time-resolved current voltage characteristics showing
the field penetration prior to and after sheath formation, a
new current transient at the time of formation of an electron-
rich sheath, ion ejection from the sheath producing density
depressions, sheath ionization which perturbs the plasma
potential profile near the electrode and triggers potential
relaxation oscillations and current modulations. The present
paper focuses on transients effects, a companion paper31 on
ionization phenomena in electron-rich sheaths. Both phe-
nomena are present in both papers.
The paper is organized as follows: After a brief descrip-
tion of the experimental setup in Sec. II, the observations of
the instability properties are presented in Sec. III and sum-
marized in Sec. IV.
II. EXPERIMENTAL ARRANGEMENT
The experiments are performed in two similar plasma
devices, one at UCLA and one in Innsbruck, shown sche-
matically in Fig. 1(a). These produce unmagnetized dc dis-
charge plasmas in a parameter regime of electron density
ne ’ 109 cm�3, electron temperature kTe ’ 3 eV, at an Ar-
gon gas pressure p ’ 5� 10�4 mbar, discharge voltage
Vdis¼ 20, …,100 V and current Idis ’ 0:1; :::; 1A, and dimen-
sions 45 cm diam, 100 cm length. In this parameter regime
the Debye length is kD ’ 0:6 mm and the sheath thickness
s ’ 5kD ¼ 3 mm.
Two types of filamentary cathodes have been used:
standard incandescent tungsten wires (0.2 mm diam, 5 cm
long) and Barium-oxide coated nickel wires (5 cm long).
The former have a temperature-limited emission, i.e., the
cathode current is nearly independent of cathode voltage,
while the latter have a space charge-limited emission where
1070-664X/2011/18(6)/062112/9/$30.00 VC 2011 American Institute of Physics18, 062112-1
PHYSICS OF PLASMAS 18, 062112 (2011)
Idis progressively increases with Vdis or plasma potential. The
discharge can be pulsed so as to study sheath properties in an
afterglow plasma.
The electrode consists of a 6 cm diam plane grid made
of stainless steel grid (0.25 mm wire spacing, 80% optical
transparency) spot-welded on a Ta wire ring. It can be biased
with a dc voltage or, as shown in Fig. 1(b), with a pulsed
voltage Vgrid¼ 0, …,200 V generated with a fast (trise¼ 50
ns, …, 1 ls) FET transistor switch. In order to measure the
time-dependent grid current to ground one has to minimize
stray capacitances of the power supply to ground. This is
accomplished with a floating storage capacitor (200 lF)
charged with a dc-dc converter (Cstray< 50 pF). When the
transistor is conducting the capacitor is connected between
ground and the electrode. The current in this circuit is meas-
ured from the voltage drop on a small series resistor (5 X) to
ground. A positive current flows toward the grid. The grid
voltage to ground is obtained from a resistive voltage di-
vider. When the transistor is not conducting the grid is float-
ing. The best time resolution of transient and rf currents is
obtained by inserting a broadband transformer in series with
the grid connection at the vacuum flange. Although most of
the present experiments were done with the gridded elec-
trode a single wire electrode (1 mm diam, 10 cm length) has
also been used to compare plane and cylindrical sheaths.
A coax-fed rf probe is used to detect high frequency
oscillations near the grid. For proper high frequency measure-
ments the probe voltage is connected via a terminated 50 Xcoax cable to a 4-channel digital oscilloscope (500 MHz,
0.4 ns=sample, 106 samples). The low frequency voltage of
the unbiased probe is useful to infer changes in the plasma
potential.
III. EXPERIMENTAL OBSERVATIONS
A. Transient currents
We start with the observation of the electrode current
for a pulsed voltage. As shown in Fig. 2 the current wave-
form depends on the rise time of the voltage pulse. For a fast
rising grid voltage [Fig. 2(a)] there is a large initial current
overshoot during the voltage rise. This pulse is absent for
slowly rising voltages [Fig. 2(b)]. In both cases a delayed
current pulse is observed.
The transient currents are primarily displacement cur-
rents due to sheath capacitances and time-varying potentials.
This conclusion is obvious when one considers the current
closure. Single probes do not exist, a current circuit needs at
least two electrodes, here the grid and the chamber wall. The
same current which flows from the external circuit into the
probe also flows from the plasma into the chamber wall and
closes externally. For a change we first address the current to
the wall which is pulsed negative with respect to the plasma
potential.
The current overshoot on the wall cannot be produced
by ions since it can be easily an order of magnitude larger
than the ion current to the chamber wall. Furthermore, the
ion flux to the wall cannot change within a fraction of the ion
transit time through the wall sheath. The wall current over-
shoot is due to a displacement of electrons away from the
FIG. 1. Schematic of (a) the experimental setup and (b) the switching
circuit.
FIG. 2. (Color online) Grid current and voltage for different rise times. (a)
Fast rising voltage step producing a large initial displacement current pulse.
The second pulse is due to a plasma potential drop when the electron-rich
sheath has formed. (b) Voltage with slower rise time comparable to the ion
plasma period producing no initial displacement current pulse. Parameters:
Vdis¼ 100 V, Idis¼ 1.1 A, p¼ 3.3� 10�4 mbar.
062112-2 Stenzel et al. Phys. Plasmas 18, 062112 (2011)
wall, i.e. an electron displacement current due to sheath
expansion. This is caused by a rise in plasma potential. The
plasma is located between two sheath capacitors, one at the
grid and the other at the wall. Its potential must rise to a
value between the grid and ground potential, irrespective of
particle collection.
The same conclusion can be reached by observing
the transient current at the switch-off of Vgrid, visible in
Fig. 3(a). The total current reverses sign which cannot be
explained by particle collection. It would imply ion collec-
tion or electron emission from a positively biased grid which
is unphysical. Thus, the current is due to the displacement of
electrons away from the electrode toward the wall.
After the end of the transient currents (t> 0.2 ls), a
quasi-dc current remains which is a conduction current due
to electron collection at the electrode and ion collection at
the wall. The conduction current scales with applied voltage
as shown in Fig. 4(d) for t¼ 0.3 ls. The conduction current
must also exist during the voltage rise (t< 0.2 ls) and fall
ðt ’ 0:8 lsÞ although at a lower level for the reduced grid
voltage. The measured scaling allows one to find the conduc-
tion current during the voltage rise and fall which is shown
in Fig. 3(b). Subtracting the conduction current from the total
current yields the displacement current. Thus each sheath
carries both conductive and displacement currents.
Bills et al.20 proposed that the overshoot arises from an
enhanced collection of electrons during the phase when ions
FIG. 3. (Color online) Grid current and voltage for a short grid pulse. (a)
Igrid shows a transient displacement current at the fast rise and fall of Vgrid
and a smaller delayed current spike when the sheath-plasma instability
starts. Its frequency decreases as Vgrid decays. The smoothed waveform
Vgrid
� �indicates that the rf and dc amplitudes can be comparable. (b) Sepa-
ration of total grid current into displacement and conduction currents, the
latter being proportional to the grid voltage. Parameters: Vdis¼ 37 V,
Idis¼ 0.93 A, p¼ 3.6� 10�4 mbar, BaO cathode.
FIG. 4. (Color online) Time-resolved current-voltage characteristics of a positively biased electrode. (a) Family of fast-rise voltage waveforms with different
amplitudes. (b) Corresponding grid current waveforms showing two types of transient overshoots. (c) I-V characteristics for the first current overshoot. (d) I-V
characteristics showing the transition from unshielded to Debye-shielded field penetration. Parameters: Vdis¼ 32 V, Idis¼ 0.5 A, p¼ 3.6� 10�4 mbar.
062112-3 Electron-rich sheath dynamics. I Phys. Plasmas 18, 062112 (2011)
are expelled from the sheath. This commonly accepted ex-
planation is in conflict with the present data which show that
the overshoot can be much shorter than the time of ion
expulsion. The overshoot current scales linearly with grid
voltage and rise time and exceeds the ion flux to the wall. If
a charge given by the overshoot current and duration was
collected by the grid there would be a very large charge
imbalance left in the plasma which is not observed.
Figures 2(a) and 3 also show a second smaller current
pulse, even when Vgrid is constant. It is also a displacement
current caused by a drop in plasma potential when the poten-
tial collapses into an electron-rich sheath, which will be dis-
cussed further below. First it is useful to determine the
relation between electrode current and voltage which is pre-
sented in Fig. 4.
A family of fast rising voltage steps with different
amplitudes is shown in Fig. 4(a), and the corresponding grid
currents are given in Fig. 4(b). The current-voltage (I-V)
characteristics depend noticeably on the time of the sheath
evolution. At the initial current overshoot [Fig. 4(c)] the I-V
characteristics is essentially linear as expected for a displace-
ment current, I¼CdV=dt. The data yield a capacitance
C ’ 600 pF which is an upper limit under the assumptions
that V¼Vgrid, and there is no conduction current. It is the
total capacitance between grid and ground which contains
two sheath capacitances in series, the smaller grid sheath and
the larger wall sheath capacitance. In vacuum the capaci-
tance of the circuit and electrode to ground is an order of
magnitude smaller than in plasma. The electron-rich sheath
has also a sheath resistance. From the slope of the I-V char-
acteristic after the capacitive overshoot one finds
Rsheath ’ 50 X. The resistance can be compared with the
sheath reactance xCð Þ�1’ 15 X at f ’ 1=trise ’ 20 MHz to
conclude that the fast transient pulses are predominantly dis-
placement currents. At the sheath plasma frequency
ðf ’ 300 MHzÞ the rf grid current is essentially a displace-
ment current.
Fig. 4(d) shows I-V characteristics when the grid voltage
has become constant. After approximately an ion plasma pe-
riod ( 2p=xpi ’ 1 ls in Ar at ne¼ 109 cm�3) the electron
current begins to saturate with voltage, i.e., the sheath forms
and the electric field penetrates less into the plasma. The lin-
ear I-V characteristic in the early phase implies that there is
little shielding and the electrons are collected from an equi-
potential surface whose area increases with voltage. The
bump in the I-V trace at t¼ 0.6 ls is due to second current
spike. Its timing varies with voltage, hence appears as a peak
vs voltage for a fixed time. Since just after the first current
overshoot the I-V characteristics is linear one can readily
separate the rising total current into displacement and con-
duction current which is presented in Fig. 3(b). The current
is a conduction current when both Vgrid and Vplasma are con-
stant. At turn-off ðt ’ 0:75 lsÞ the conduction current scaling
is obtained from I-V curves as shown Fig. 4(d).
Initially the electrode was floating, hence had an ion-
rich sheath. Upon application of the positive voltage pulse
the ions, which stream at the Bohm speed toward the elec-
trode have to be accelerated away from the electrode to es-
tablish the electron-rich sheath. This process occurs on the
time scale of about an ion plasma period, but also depends
on grid voltage and rise time. The second current spike
occurs at the time when the electron-rich sheath has formed.
However, it is not due to an increased electron conduction
current, as frequently assumed,20 but it is a again a displace-
ment current, now created by a sudden drop in the plasma
potential near the electrode.
The potential near the electrode must drop when the
sheath is formed. Prior to this time there was no shielding
and the potential increased over large distances. But when
the electron-rich sheath is formed the electric field is concen-
trated in the sheath and highly shielded from the plasma.
One might expect that the field contraction occurs gradually
on the time scale of the ion expulsion 2p=xpi ’ 1 ls� �
but
by observation it occurs much faster ðDt ’ 50 nsÞ which is
not easy to explain. Computer simulations5 show that the
sheath edge is a non-equilibrium plasma with counterstream-
ing electrons and ions, double layer formation and instabil-
ities, some of which will be shown in the companion
paper.31 At a constant grid voltage a rapid drop in plasma
potential induces a displacement current in the grid. The
same displacement current flows from the plasma to the wall
where the rising plasma potential widens the sheath.
Evidence for the potential changes is obtained from the
rf probe data. The rf probe, designed to study high-frequency
signals, is connected via a 50 X termination to ground and
draws a small “dc” current to ground. The polarity of the sig-
nal indicates an ion current which implies that the floating
potential is above ground and that the probe at 5 mm from
the grid is outside the electron-rich sheath. The ion current
increases when the plasma potential increases and decreases
for a potential drop. Density changes on the time scale short
compared to an ion plasma period cannot explain the fast
probe current drop. Figure 5 shows simultaneous traces of
(a) the grid current and (b) the probe voltage on a 50 X ter-
mination for different radial probe positions from the grid
center at the time when the second grid current spike occurs.
The grid voltage rises slowly as in Fig. 2(b). With rising grid
voltage the plasma potential, hence the probe voltage
increases too at all positions. The same observations are
made in axial direction, which will be shown in the compan-
ion paper.31
Suddenly, at t ’ 0:8 ls, the potential drops abruptly in
the center of the grid which coincides with the current pulse
on the grid. It must be a displacement current since electron
collection would raise the local plasma potential. The poten-
tial decreases in the center of the grid while it increases out-
side the grid. At the wall the potential increase relative to
ground produces a displacement current through the wall
sheath which closes the current to ground.
The current spike is frequently followed by oscillations
arising in the center of the grid. The frequency range
(10, …, 20 MHz) lies between the electron and ion plasma
frequencies, hence the oscillations excite neither sound
waves nor electron plasma waves but are close the Buneman
instability frequency, xB ’ 0:7xpeðme=miÞ1=3 ’ 7 MHz
(Ref. 25). The potential also drops axially, typically by a fac-
tor 3 in 3 cm. The potential drop lasts only for a few micro-
seconds, recovers gradually, and often repeats intermittently
062112-4 Stenzel et al. Phys. Plasmas 18, 062112 (2011)
resulting in potential relaxation instabilities presented in the
companion paper.31
Secondary electron emission may also change the sheath
potential near a positive electrode as observed in steady-state
experiments.32 Secondary electrons released at the grid
potential are trapped in the electron rich sheath and steepen
the potential drop near the grid but contribute neither to the
grid current nor to the sheath-plasma instability. Trapped
secondary electrons cannot be responsible for the potential
drop well outside the sheath. Secondary electron emission
cannot explain the delay of an ion plasma period and should
be negligible for low grid voltages (30 V) where the transient
current is still observed.
The expulsion of ions during the formation of the elec-
tron-rich sheath can be readily observed. Figure 6(a) shows
the ac component of the probe signal at different axial dis-
tances from the grid center. Following a stationary transient
response to the potential rise there are traveling pulses whose
propagation characteristics are obtained from the time-
of-flight diagram shown in Fig. 6(b). The first perturbation
propagates at a supersonic speed and can be interpreted as a
ballistic signal of ions with 47 eV energy. This value is smaller
than the peak grid voltage since the plasma potential rises with
Vgrid. The ions are accelerated up to 1 cm from the probe, indi-
cating that the time-dependent electric field is not confined
to the initial Debye sheath. From the half-width of the ion sig-
nal, Dt ’ 0:25 ls, and the streaming velocity one obtains the
width of the region from where the ions originate, Dz ’ 3 mm.
It corresponds to about 5 D length at ne ’ 109 cm�3 and
kTe ’ 3 eV, the typical width of the initially ion-rich sheath.
The second perturbation travels at a much lower velocity and
has the opposite sign as the first perturbation, hence may not
be an ion burst but a fast ion wave phenomenon. Both these
traces originate from the grid at the start of the voltage ramp.
A third perturbation also leaves the grid but starts at a
much later time when Vgrid¼ const, hence must be produced
by the above-mentioned plasma potential relaxations near
the sheath. The traveling ion density perturbations are small
compared to the ambient ion density, hence produce no sig-
nificant axial density gradients.
Finally, the role of the electron source in the electron
collection has been investigated. Since the grid current is
comparable to the discharge current one might think that the
electrons collected are those emitted from the cathode. How-
ever, this is demonstrably not the case since there is little dif-
ference in Igrid during the discharge and in the early
afterglow of a pulsed discharge. Furthermore, the cathode
current shows no transient enhancements as the grid voltage
is applied.
Figure 7 shows the waveforms of grid current, voltage,
discharge current, and rf instabilities (a) before and (b) after
the switch off of the discharge current. The grid current
exhibits the same overshoots and current spike irrespective
of whether electrons are emitted from the cathode or not. No
spike is observed in the cathode current for case (a). This
FIG. 6. (Color online) Time-of-flight of ion bursts. (a) Probe signal at differ-
ent axial distances z from the grid center. (b) Time-of-flight diagram of dif-
ferent perturbations in Vprobe. Note that last perturbation is not triggered at
turn-on of Vgrid. Parameters: Vgrid¼ 110 V, Vdis¼ 100 V, Idis¼ 0.11 A,
p¼ 2.4� 10�4 mbar.
FIG. 5. (Color online) Radial variation of the potential drop in front of the
grid (Dz¼ 0.5 cm) after grid current spike. Parameters: (a) Vdis¼ 32 V,
Idis¼ 0.53 A, W cathode, (b) Vdis¼ 100 V, Idis¼ 1.1 A, BaO cathode,
p¼ 3.3� 10�4 mbar.
062112-5 Electron-rich sheath dynamics. I Phys. Plasmas 18, 062112 (2011)
observation shows that the electrode collects predominantly
the secondary, low temperature electrons in the discharge. In
the afterglow the electron and ion currents cancel at the wall,
but when the grid voltage is applied the electron flux is redir-
ected to the grid, the ion flux continues to flow into the wall.
Electrode and wall form a “double probe” with unequal
areas. The steady-state electron current to the grid is equal to
and limited by the ion current to the wall. The initial dis-
placement current is however much larger than the steady-
state current. Its value is limited by the sheath capacitance
and grid voltage to which the plasma potential can maxi-
mally charge,Ð
dIdt ¼ Q ¼ CVgrid . In the present case
dI ’ 0:1 A with half width Dt ’ 0:4 ls, C ’ 600 pF such
that V ’ Vgrid ¼ 67 V.
B. The sheath-plasma instability
Electron-rich sheaths are subject to the sheath-plasma
instability. It is a transit-time instability due to the electron
inertia similar to that in vacuum diodes, klystrons, vircators,
and inverted fireballs.33 The electron transit time through the
sheath is of the order of the electron plasma frequency. If the
sheath oscillates near the plasma frequency the electron cur-
rent is delayed with respect to the electric field which can lead
to a positive feedback and absolute instability. The sheath ca-
pacitance and the inductive response of the plasma below the
plasma frequency provide a natural resonance to be excited.
The instability properties depend sensitively on the properties
of the sheath and plasma near the electrode. Thus, the instabil-
ity can be used to diagnose sheaths. Using probes to diagnose
sheaths causes perturbations due to their size, own sheaths and
the I-V traces are difficult to interpret in non-neutral plasmas
with strong electric fields. Unfortunately, no analytic theory
exists for the sheath-plasma instability, but the instability has
been seen in numerical simulations.34
The sheath-plasma instability can be observed in the
grid current, grid voltage [see Fig. 4(a)] and on the rf probe
near the grid [see Fig. 7]. Phase shift measurements with the
movable rf probe show that the oscillations are not propagat-
ing waves but evanescent fields near the electrode. The oscil-
lations are observed for pulsed and dc grid voltages where
the former is more interesting since it reveals the evolution
of the electron-rich sheath and explains the frequently seen
unstable steady-state behavior.
The basic parameter dependences of the instability fre-
quency are summarized in Fig. 8. For a constant dc grid volt-
age and discharge the frequency has been measured as a
function of discharge current, varied by the cathode tempera-
ture. The discharge current is proportional to the density; its
square root varies as the plasma frequency. Figure 8(a)
shows that the sheath-plasma frequency is proportional to
the electron plasma frequency.
In Fig. 8(b) the grid voltage has been varied at a con-
stant discharge current. The frequency is nearly independent
FIG. 7. (Color online) Current transients and sheath plasma oscillations (a)
during the discharge and (b) in the early afterglow. The electrons collected
at the grid are not those emitted by the cathode. Parameters: Vgrid¼ 60 V,
Vdis¼ 100 V, Idis¼ 0.14 A, p¼ 2.7� 10�4 mbar.
FIG. 8. (Color online) Frequency dependence on (a) the plasma frequency,
proportional to the square-root of density or discharge current, varied by
cathode temperature. (b) Frequency and amplitude dependence on grid volt-
age Vgrid at a constant density. Parameters: Vdis¼ 108 V, Idis¼ 0.11 A,
p¼ 2.5� 10�4 mbar.
062112-6 Stenzel et al. Phys. Plasmas 18, 062112 (2011)
of grid voltage. This observation is explained by the fact that
both the electron velocity and the sheath thickness increase
as V1=2grid and hence the transit time and frequency remain
nearly constant.
However, the instability amplitude strongly depends on
grid voltage. The instability has a threshold for low grid vol-
tages, an optimum voltage and disappears for high voltages.
The optimum may correspond to the best match of transit
time and rf period. The decay at higher voltages is due to
sheath ionization and fireball formation which raises the
plasma potential so as to lower the sheath potential drop
below the threshold.
A very interesting result is the relatively long delay of
the instability onset with respect to the start of a pulsed grid
voltage. Figure 9 reviews transient processes and shows the
instability properties as the ion-rich sheath changes into an
electron-rich sheath. Figure 9(a) displays the waveforms of
the grid voltage and current. The former is unchanged while
the grid current increases with discharge current, i.e., den-
sity. The waveforms have been recorded at 0.8 ns=sample
which fully resolves the high frequency but have been
smoothed so as to show the current spikes clearly. For a
slowly rising voltage [see also Fig. 2(b)] the current spike
occurs during the voltage ramp. The onset time is nearly in-
dependent of density except toward low values where the
delay increases since the ion transit time increases as the
sheath widens.
The smoothed probe voltage is displayed in Fig. 9(b).
The density increase lowers the plasma potential which
decreases the probe voltage prior to the grid voltage. Upon
the start of the grid voltage the plasma potential rises and so
does the probe voltage until the potential collapses which
creates the displacement current pulse as described above.
The potential drop increases with density or discharge cur-
rent, and it can approach the potential rise due to Vgrid. The
potential drop develops relaxation oscillations on a time
scale of a few ion plasma periods. These relaxation oscilla-
tions will be discussed later. At high densities there are also
bursts of oscillations with frequencies between the ion and
electron plasma frequencies, possibly due to the Buneman
instability, but the present emphasis is on the sheath-plasma
instability.
Fig. 9(c) displays waveforms of the high frequency
oscillations on the grid voltage for different discharge cur-
rents, offset for purpose of display. The onset coincides
with the timing of the grid current spike, giving support for
the interpretation that at this time the electron-rich sheath
has been formed, which is a necessary condition for the
growth of the sheath-plasma instability. The high frequency
oscillations become modulated by the low frequency
FIG. 9. (Color online) Transients and instabilities as a function of discharge current. (a) Grid voltage and current waveforms for 0.3< Idis< 1.1 A, varied in
steps of 50 mA. At high densities low frequency rf bursts (10< f< 30 MHz) are observed. (b) Probe voltage, smoothed to clearly show potential drops and low
frequency signals. The high frequency oscillations can exceed the dc probe voltage. (c) Sheath-plasma oscillations observed on the grid voltage. The instability
onset coincides with the current spike in Igrid which indicates the transition from ion-rich to electron-rich sheath. (d). Frequency of the sheath plasma instability
at onset varies proportional to the electron plasma frequency. (e) At a constant discharge current the frequency decays in time indicating a density drop near
the electrode due to ion ejection from the electrode. Parameters: (a) Vdis¼ 100 V, Idis varied via W cathode temperature, Vgrid¼ 95 V, p¼ 3.3� 10�4 mbar.
062112-7 Electron-rich sheath dynamics. I Phys. Plasmas 18, 062112 (2011)
potential oscillations when both instabilities arise together.
Just after onset of the instability ðt ’ 2 lsÞ the frequency
has been determined by a fast Fourier transform (FFT) of
Vgrid,rf over a time span of 0.5 ls. Figure 9(d) shows again
that the instability frequency at t ’ 2 ls varies proportional
to the electron plasma frequency [see Fig. 8(a)]. Thus fre-
quency changes provide useful information about density
changes.
After the sheath has been established it is by no means
constant. Plasma potential, grid current, the instability ampli-
tude, and frequency all vary in time. For one density the fre-
quency change in time has been displayed in Fig. 9(e). The
time-resolved spectrum is obtained by performing FFTs in
successive 0.5 ls intervals. The frequency drops within 6 ls
from 350 to 270 MHz, implying a density decrease by 60%
in front of the electrode. The density drop is due to the ejec-
tion of ions from the vicinity of the grid. A frequency
increase indicates sheath ionization which will be shown in
the companion paper.31
As shown in Fig. 8(b) the rf amplitude strongly depends
on the sheath potential drop. The large amplitude seen at the
onset of the instability arises from the drop in plasma poten-
tial which for a given electrode potential yields a large
sheath drop. Vice versa, the instability vanishes when the
plasma potential rises close to the electrode voltage. This
arises from sheath ionization and explains why for large dc
voltages the instability amplitude is usually small compared
to the initial transient oscillation.
On the other hand, short high voltage pulses avoid ion-
ization and can produce very large rf amplitudes as shown
in Fig. 10(a). By rapidly switching the grid voltage off the
decay of the sheath-plasma oscillations provides informa-
tion about the lifetime of the electron-rich sheath. Since
the oscillations are observed to continue till complete volt-
age turn-off the electron-rich sheath does not vanish
instantly. It takes again about an ion plasma period to con-
vert the electron-rich sheath into an ion-rich sheath. Thus
for pulses shorter than an ion plasma period a sheath can-
not adjust its sign of charge. During the turn-off the fre-
quency drops which indicates the expulsion of electrons
from the grid. For comparison, Fig. 10(b) shows a slowly
varying voltage pulse where the sheath adjusts during the
voltage decay and the instability stops at its dc threshold
value.
Both the high frequency ðf ’ 300 MHzÞ and the low
frequency (10< f< 30 MHz) rf oscillations occur only in the
vicinity of the electrode. The rf probe has been scanned radi-
ally across the grid surface (Dz¼ 0.5 cm), and the observed
amplitude profiles are shown in Fig. 11. For a finite time
span [3< t< 7 ls in Fig. 9(b)] the average dc voltage shows
a drop in the center of the grid. Both the high-frequency
sheath-plasma oscillations and the lower frequency instabil-
ity maximize in the central region of the grid. No oscillations
are observed when the probe is about one grid diameter radi-
ally (and axially) away from the grid. This implies that the rf
current closure to the chamber wall occurs close to and
coaxial with the bias wire to the grid.
IV. CONCLUSION
A gridded electrode in an unmagnetized discharge
plasma is pulsed positively so as to study the growth and
decay of an electron-rich sheath. Initially the electrode is
floating, i.e., it has an ion rich sheath. The voltage rises in a
FIG. 10. (Color online) Sheath-plasma instability during transition from
ion-rich to electron-rich sheath (voltage rise) and the reverse (turn-off). (a)
Fast pulse shows the delay in the instability onset due to removing ions from
the sheath, while at switch-off the return of ions is delayed, the sheath
remains electron-rich and the instability continues Vgrid¼ 0. (b) For a slowly
varying grid voltage the sheath adjusts during rise and fall such that the
instability stops at a finite threshold voltage, Vgrid ’ 30V. Parameters:
Vdis¼ 100 V, Idis¼ 0.3 A, Vgrid¼ 90 V, p¼ 3.8� 10�4 mbar.
FIG. 11. (Color online) Radial amplitude variation of the probe dc voltage,
the high frequency sheath plasma oscillations, and the low frequency oscilla-
tions visible in Figs. 9(a) and 9(b) for 3< t< 7 ls. The instabilities only
exist near the grid of radius 3 cm. Parameters: Vdis¼ 100 V, Idis¼ 1.1 A,
Vgrid¼ 90 V, p¼ 3.3� 10�4 mbar.
062112-8 Stenzel et al. Phys. Plasmas 18, 062112 (2011)
time short compared to the ion plasma period but long com-
pared to the electron plasma period. The voltage change
across the sheath capacitance creates a displacement current
pulse. The current closes to the chamber wall, hence the
same current flows through the sheath at the wall, which
requires a plasma potential rise throughout the plasma. From
a family of voltage steps time-resolved I-V characteristics
have been measured. Prior to sheath formation the I-V curves
are linear implying no shielding and an increased collection
area with increasing voltage. The total overshoot current has
been decomposed into conduction and displacement cur-
rents. Conduction currents are supplied by electrons from the
plasma, not from the cathode, as shown by measurements in
an afterglow plasma.
A second current transient is observed when the elec-
tron-rich sheath is formed. Since the grid voltage is constant
the plasma potential in front of the grid decreases to produce
a predominantly capacitive current pulse. The potential drop
indicates that the electric field no longer penetrates into the
plasma but is concentrated in a Debye sheath. The I-V char-
acteristics begin to show a saturation. The strong electric
field in a pure electron sheath gives rise to a strong sheath-
plasma instability. Its frequency decreases in time indicating
a density depletion near the grid due to the ejection of ions
in an electron “pre-sheath.” The density depletion also
causes a current decay.
The current “overshoot” prior to the formation of the
electron-rich sheath has previously been interpreted as the
result of the collection of excess electrons while ions are in
the sheath.20 The present observations suggest that the large
current is mainly a displacement current and partly an elec-
tron conduction current due to the penetration of the electric
field well beyond the Debye sheath.
The decay of the sheath-plasma frequency indicates a
density loss in front of the electrode which is only reversed
in the presence of sheath ionization. However, sheath ioniza-
tion increases the sheath width which lowers the electric
field, quenches the sheath plasma instability, and reduces the
ionization rate. Just like at the first voltage rise, the ions
must be removed from the sheath to reform the electron-rich
sheath. A repetitive relaxation process arises which is dem-
onstrated in a companion paper.31
A fast rising voltage pulse has been adjusted long
enough to establish the electron-rich sheath and then rapidly
switched off. The sheath-plasma instability is observed to
continue oscillating till V¼ 0. The sheath remains electron-
rich for about an ion plasma period which is again the time
scale to reestablish an ion-rich sheath.
ACKNOWLEDGMENTS
R.L.S. gratefully acknowledges support and hospitality
while staying as guest professor at the University of Inns-
bruck 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.
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062112-9 Electron-rich sheath dynamics. I Phys. Plasmas 18, 062112 (2011)