9
Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities R. L. Stenzel, 1 J. Gruenwald, 2 C. Ionita, 2 and R. Schrittwieser 2 1 Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA 2 Institute 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. V C 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, 25 virtual cath- ode oscillators, 6,7 and electrodes forming fireballs in discharge plasmas. 810 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. 10,1319 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. 2,2025 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, 2628 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 paper 31 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 n e 10 9 cm 3 , electron temperature kT e 3 eV, at an Ar- gon gas pressure p 5 10 4 mbar, discharge voltage V dis ¼ 20, …,100 V and current I dis 0:1; :::; 1A, and dimen- sions 45 cm diam, 100 cm length. In this parameter regime the Debye length is k D 0:6 mm and the sheath thickness s 5k D 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 V C 2011 American Institute of Physics 18, 062112-1 PHYSICS OF PLASMAS 18, 062112 (2011)

Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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Page 1: Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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)

Page 2: Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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)

Page 3: Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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)

Page 4: Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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)

Page 5: Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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)

Page 6: Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities · 2016-10-09 · Electron-rich sheath dynamics. I. Transient currents and sheath-plasma instabilities

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)

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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)

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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)

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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)