Plasma boundary sheath in the afterglow of a pulsed inductively coupled RF plasma

Plasma boundary sheath in the afterglow of a pulsed inductively coupled RF plasma

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IOP PUBLISHING PLASMA SOURCES SCIENCE AND TECHNOLOGY

Plasma Sources Sci. Technol. 16 (2007) 355–363 doi:10.1088/0963-0252/16/2/019

Plasma boundary sheath in the afterglowof a pulsed inductively coupled RF plasmaM Osiac1,5, T Schwarz-Selinger2, D O’Connell1,6, B Heil1,Z Lj Petrovic3, M M Turner4, T Gans1,7,8 and U Czarnetzki1

1 Institute for Plasma and Atomic Physics, CPST, Ruhr-University Bochum, Bochum 44780,Germany2 Centre for Interdisciplinary Plasma Science, Max-Planck-Institute for Plasma Physics,Garching 85748, Germany3 Institute of Physics Belgrade, Zemun POB 68 Zemun 11080, Serbia4 National Centre for Plasma Science and Technology, Dublin City University, Dublin 9,Ireland5 Faculty of Physics, University of Craiova, Craiova, Romania6 Institute for Electrical Engineering and Plasma Technology, CPST, Ruhr-UniversityBochum, 44780 Bochum, Germany7 Centre for Plasma Physics, Queen’s University Belfast, Belfast BT7 1NN, UK

E-mail: [email protected]

Received 21 July 2006, in final form 6 February 2007Published 26 March 2007Online at stacks.iop.org/PSST/16/355

AbstractThe sheath dynamics in the afterglow of a pulsed inductively coupledplasma, operated in hydrogen, is investigated. It is found that the sheathpotential does not fully collapse in the early post-discharge. Time resolvedmeasurements of the positive ion flux in a hydrogen plasma, using a massresolved ion energy analyser, reveal that a constant 2 eV mean ion energypersists for several hundred micro-seconds in the afterglow. The presence ofa finite sheath potential is explained by super-elastic collisions betweenvibrationally excited hydrogen molecules and electrons in the afterglow,leading to an electron temperature of about 0.5 eV. Plasma density decaytimes measured using both the mass resolved energy analyser and aLangmuir probe are in good agreement. Vibrational temperatures measuredusing optical emission spectroscopy support the theory of electron heatingthrough super-elastic collisions with vibrationally excited hydrogenmolecules. Measurements are also supported by numerical simulations andmodelling results.

1. Introduction

Inductively coupled plasmas (ICPs) using radio-frequency(RF) excitation are widely used for technological applications.Increasing current interest in pulsed mode discharges hasarisen to enable additional control and more flexibility oftechnological processes. For example, pulsed discharges havebeen found to reduce the effects of surface charging, whichcauses serious problems in devices, in particular in the micro-electronics industry. It has been suggested that suppression ofcharge build-up on substrates is attributed to the sheath collapseand rapid electron cooling in the post-discharge, restrictingnegative ion confinement.

8 Author to whom any correspondence should be addressed.

Pulsing of the discharge not only provides additionalcontrol of various plasma parameters but also reduces the timeaveraged heat strain on the substrate. The process temperatureis a crucial parameter for the treatment of temperature sensitivematerials such as food packaging and bio-medical materials.In particular, pulsed ICPs have recently been used for plasmasterilization of such temperature sensitive materials [1].

Despite the increasing importance of these discharges, thetransition from the plasma on-phase to the afterglow is not yetfully understood. In general, it is commonly assumed that themean electron energy drops rapidly after the plasma is switchedoff. This would result in the collapse of the plasma boundarysheath potential on a time scale of a few micro-seconds in theafterglow [2]. The decay of the plasma density will occur on

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M Osiac et al

Figure 1. An overview of the experimental set up and the appliedtechniques of diagnostics.

a longer time scale due to ambipolar diffusion of the particlesto the walls [3–20].

Pulsed discharges and their afterglows have been studiedboth experimentally and theoretically. However, manyinvestigations have concentrated on rare gas plasmas [4–6], inparticular helium, even though it is predominantly molecularplasmas that are of interest for industrial processes. Herewe concentrate on hydrogen discharges. The collapse ofthe positive space charge sheath has been investigated in theafterglow of a pulsed inductively coupled rf plasma (ICP).The principal diagnostic techniques used are time and massresolved ion energy analysis in conjunction with Langmuirprobe measurements and optical emission spectroscopy (OES).These investigations provide insight into the additional controlof different plasma parameters in pulsed ICPs, in particularelectron heating and the corresponding ion bombardment ofthe substrate.

The structure of the paper is as follows. Firstly,the experimental setup for time resolved measurementsis described. Secondly, the decay of charged particledensities and energies in the afterglow is discussed. Thenmeasurements of the vibrational temperature using opticalemission spectroscopy are presented. Numerical andanalytical modelling results support the measurements andtheir interpretation.

2. Experimental setup

A diagram of the experimental set up is shown in figure 1.The discharge chamber is a GEC reference cell modified toimprove inductive coupling. A planar two-spiral antenna, ofdiameter 12.5 cm, is used and the usual stainless steel housingsurrounding the antenna is replaced by a quartz cylinder withan outer diameter of 14 cm.

The ICP antenna is driven with an excitation frequencyof 13.56 MHz. For impedance matching a standard L-typematching network is used. The RF power supply can be pulsedat frequencies between 3 Hz and 12 kHz, with duty cyclesranging from 1 to 100%.

The stainless steel vessel is pumped using a turbomolecular pump with a Holweck stage and a membranepump. During plasma operation pumping is reduced usinga butterfly valve. Constant gas flow is maintained using massflow controllers. The absolute pressure is measured with acapacitive manometer. The signal of the pressure gauge isused as a feedback to control the pressure.

At a distance of 5 cm below the antenna an aluminiumelectrode, of diameter 15 cm, limits the discharge region inthe vertical direction. The electrode is grounded and watercooled. A mass resolved ion energy analyser (Balzers plasmaprocess monitor, PPM 422) is incorporated into the groundedelectrode. Ions are sampled through a 100 µm diameter orifice.A background pressure of 4×10−3 Pa is maintained in the PPMat a working discharge pressure, in hydrogen, of 10 Pa. Theorifice surface is flush with the grounded aluminium electrodeto avoid any electric field distortion at the point of extraction.The PPM consists of an Einzel lens (EL), a cylindrical mirrorenergy analyser (CMA), a quadrupole mass filter (QMS) anda secondary electron multiplier (SEM) arranged in series. TheSEM is operated in ion counting mode in the plateau regionat high voltage. The energy filter is operated with a constantpass energy of 15 eV corresponding to an energy resolution ofabout 0.5 eV at FWHM. The mass filter is operated at high massresolution. For accurate measurements of the undisturbedion energy distribution function (IEDF) a procedure basedon simulations of ion trajectories and extensive experimentalchecks is applied [21].

Special attention was paid to the calibration of the zeropoint of the CMA energy axis. The first method usesthe ionization chamber of the instrument as a source ofions with a well defined energy. For a minimized electroncurrent and a small ion extraction voltage the ionizationchamber is assumed to be free of fringing fields and thereforeions are produced at the applied potential. Measuring theenergy of the ions produced in the ionization chamber allowsdetermination of the CMA’s energy scale to an accuracy of0.5 eV. In addition, charge exchange collisions of plasmaions are exploited for accurate energy calibration. Timeintegrated IEDF measurements are performed in continuousmode plasmas, generated in argon, helium and hydrogen. TheIEDFs of Ar+, He+ and H+

2 have a distinct low energy peak dueto resonant charge exchange collisions occurring in the sheath.This distinct peak is due to ions generated by charge exchangein the proximity of the extraction hood of the mass resolvedion energy analyser. Since the extraction hood is grounded thisfeature in the IEDF is at 0 eV in the energy scale. For all casesinvestigated, the energy peak of the charge exchange collisionscoincided with 0 eV, supporting the correct energy calibration.

For time resolved measurements the detector pulses areacquired using a multi channel scaler (MCS). The MCSis triggered with the same pulse as the RF power supply.The MCS card (Ortec MCS-PCI) has 65536 channels and atemporal resolution of 100 ns. Measurements are performedby acquiring several thousand plasma scans and accumulatingthe signal, for a given energy set by the CMA, and a givenmass, set by the QMS.

A 2 m spectrograph is installed for OES measurements.Light from the centre of the discharge is imaged with fibreoptics onto the entrance slit of the spectrograph. The spectrum

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Sheath dynamics in the afterglow of a pulsed ICP

of hydrogen molecules in the spectral range between 600 and640 nm are recorded. Spectra are sampled with an intensifieddiode array. The micro-channel plate of the intensifier issynchronized with the pulsed RF power supply for timeresolved measurements.

A Langmuir probe system (Smart Probe, ScientificSystems) [36] is installed and allows temporally resolvedmeasurements. To measure the time resolved Langmuirprobe current–voltage characteristic in the afterglow, dataare acquired over many RF cycles using a delay generatorconnected to the temporal gate of the Langmuir probe. Atungsten probe tip of diameter 0.05 mm and length 5.5 mmis used. The inductors used for passive RF compensationare removed from the probe system. This compensation isunnecessary in the afterglow and only introduces an additionalresistance between the Langmuir probe tip and the electronics.

3. Results

3.1. Time resolved measurements of the ion velocitydistribution function

Measurements presented here were performed in hydrogenplasmas. For pressures ranging from 5 to 20 Pa the dominantion species in a hydrogen ICP is H+

3 [37], since H+2 ions, formed

through electron impact ionization, in the plasma bulk areefficiently converted into H+

3 ions through collisions with thebackground gas [38]. Figure 2(a) shows the time resolvedH+

3 ion flux, in a pulsed hydrogen plasma measured using themass resolved ion energy analyser. The pulsing frequency is100 Hz and duty cycle 50%. The RF power during the plasmaon-phase is 300 W and the gas pressure 10 Pa. Energy scansare performed with 0.5 eV steps and 1500 scans accumulatedfor each energy. The measured signal intensity for mass 3, H+

3,is displayed as a function of time and energy in a contour plot,with logarithmic scale, in figure 2(a). Only the first 5.5 msof the entire 10 ms cycle are shown. The zero of the abscissacoincides with the ignition of the plasma.

It can be clearly observed that during the ignition phaseion energies exceed the displayed energy ranges. This istypical for ICP discharges igniting in capacitive mode. In thiscapacitive mode, with large sheath potentials, the ion energiesreach 35 eV. Within a few hundred micro-seconds after ignitionthe mean ion energy decreases and the plasma reaches stableinductive mode. The mean ion energy during this inductivephase is around 13 eV. After 5 ms the RF power is switched offand an immediate drop in the ion energy is observed. Thisinitial phase is in accordance with the expectation that thesheath voltage collapses instantaneously when the plasma isturned off. In the post-discharge both the ion energy and theion density are expected to decay. However, in contrast to this,the mean ion energy does not decay fully to zero but stagnatesat a few eV and remains constant for several hundred micro-seconds.

Figure 2(b) shows the afterglow phase with betterresolution for the same plasma conditions. Measurements aretaken with energy steps of 0.2 eV. It can be observed that aftera transient phase between the plasma on- and off- phase, afinite ion energy of about 2 eV is observed for several hundredmicro-seconds after the plasma is switched off. The ion energy

Figure 2. Time resolved H+3 ion flux for a pulsed hydrogen ICP;

fpulsing = 100 Hz, duty cycle 50%, P = 300 W and p = 10 Pa. Onthe right side of the graph is a logarithmic scale indicating theintensity of the ion flux. (a) plasma on and part of the off phase,recorded in steps of 0.5 eV. (b) afterglow recorded with 0.2 eV. TheH+

3 ion energy in the afterglow is around 2 eV.

stays constant while the plasma density decays. The ions gainenergy through the plasma boundary sheath potential as theyare accelerated in the sheath electric field before impacting onthe electrode. A sheath with a potential of about 2 eV survivesin the post-discharge when the RF power is turned off. Thiscorresponds to a plasma boundary sheath potential caused by0.5 eV electrons. This finite electron energy observed in theafterglow can be attributed to either an additional energy sourcefor electrons in the afterglow or the inability of electrons to losetheir energy.

3.2. Decay of the charged particle density in the afterglow

Collisional cooling of electrons, through elastic collisions withthe background gas, can be the expected dominant coolingmechanism in the afterglow. This cooling process typicallytakes a few micro-seconds (τ ≈ M

mν−1 = 3.5 µs), where ν

is the elastic electron molecule collision frequency. In theinitial phase, after the RF power is switched off there is arapid cooling of the electron temperature, evident through therapid decrease in ion energy. The decay of the ion energy,from the measurement, is around 10 µs in the initial switch-off phase, in agreement with the above approximation for

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Figure 3. (a) Decay time of the H+3 ion flux measured with the

plasma monitor in the afterglow for an energy of 2 eV. (b) Decaytime of the electron density measured with the Langmuir probe atdifferent instants in the afterglow. In both (a) and (b) the symbolsrepresent the measurements and the solid line is a linear fit to thedata; the measured decay time is 80 µs.

collisional cooling. However, when the electron temperaturecools down to about 0.5 eV, we see no further decay of theelectron temperature.

The plasma density decay time t0 can be calculated fromthe decay of the measured ion flux. The measured flux signalis the product of density and the species velocity. In theafterglow, both density and energy change in time. However,after the initial transient switch-off phase it can be seen thatthe energy remains constant. Thus the measured flux decaytime is the density decay time. Figure 3(a) shows the signalintensity decay of H+

3 for the same parameters as in figures 2(a)and (b). Note that the intensity is plotted in logarithmic scalefor a given energy of 2 eV. The straight solid line shows that thedecay can be approximated by an exponential function. Fittingan exponential function to the measured decay of the H+

3 ionflux gives a decay time of about 80 µs.

3.3. Langmuir probe measurements of the EVDF in theafterglow

It was illustrated in the previous sections that the finite ionenergy, implying a non-zero electron temperature, exists intothe relatively late afterglow of a hydrogen discharge. Thus, ameasurement of the EVDF in the afterglow is imperative. The

Figure 4. Electron velocity distribution function measured atτ = 40 µs in the post-discharge, solid symbols. The solid linesrepresent a linear fit to the measured EVDF. The plasma parametersare fpulsing = 600 Hz, duty cycle 50%, P = 300 W and p = 10 Pa.

EVDF was measured, using a Langmuir probe, in the afterglow,40 µs after the discharge was switched off, at a pressure of10 Pa and a modulation frequency of 600 Hz. In this casethe dynamic range of the EVDF measurement is three ordersof magnitude. Below 108 cm−3 eV−3/2 the measurements areunreliable due to a low signal-to-noise ratio. Figure 4 showsthe EVDF, in logarithmic scale, obtained from the Langmuirprobe. It can be observed that the EVDF is non-Maxwellianwith a complex structure in the high energy range. The plasmapotential is determined from the zero-crossing of the secondderivative of the current–voltage characteristic. The plasmapotential in the afterglow is 2.5 eV and the electron density109 cm−3. Electron temperatures are determined from theslopes of the exponential parts of the EVDF. The EVDFshows two exponential decays with electron temperatures ofTe = 0.14 eV in the thermal region, at ε < 0.5 eV, and inthe higher energy tail a temperature of Te = 0.27 eV. Thistemperature of the high energy tail determines the finite sheathpotential discussed above in section 3.1.

The electron density decay obtained from the integratedEVDF at different instants in the post-discharge is shown infigure 3(b). The decay time of the electron density was foundto be 80 µs. This agrees well with the measured density decaytime from the ion flux measurements (figure 3(a)) discussed inthe previous section.

The kink in the Langmuir probe measurement of theEVDF in the afterglow is at 0.5 eV; this corresponds to theenergy gap between the ground state and the first vibrationallevel of molecular hydrogen. The ion energy distributions inthe afterglow also correspond to a sheath potential caused by0.5 eV electrons. This motivates the theory of electron heatingin the hydrogen afterglow through super-elastic collisions withvibrationally excited hydrogen molecules.

The plasma density in the afterglow decays exponentiallywith a time constant of τ = 80 µs. This number is obtainedconsistently by the Langmuir probe measurements from theintegral over the EEDF and the ion flux measurements by thePPM. An analytical analysis of the one-dimensional transportincluding the ion inertia term and allowing non-Maxwellian

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Sheath dynamics in the afterglow of a pulsed ICP

Figure 5. Stationary electron temperature in the afterglow obtainedfrom the PIC simulation as a function of vibrational excitation.

distribution functions for all transport coefficients and theBohm criterion provides a simple relation between the decaytime τ and the mean electron energy 〈Ee〉 (see appendix A):

〈Ee〉 = 3M

2

(Lνmi

π√

νmiτ − 4

)2

. (1)

Here M is the ion mass (H+3), νmi = 1.75 × 106 s−1

the ion collision frequency and L = 5 cm the dischargegap. A gas temperature of 450 K is assumed, which wasmeasured previously at similar powers without pulsing thedischarge. This yields a mean energy of 0.33 eV, which isin excellent agreement with the mean energy of the measureddistribution of 0.32 eV. However, the floating potential at thewall is determined only by the higher energetic part of thedistribution function above this potential, i.e. above 2 eV. Thispart could not be measured by the Langmuir probe. The highereffective temperature determined from the ion energy of 0.5 eVis, therefore, an indication that the effective tail temperatureis rising further with increasing energy and finally approachesthe measured vibrational temperature.

3.4. PIC simulation of electron heating through super-elasticcollisions in the afterglow

Simulations were carried out using an electrostatic particle-in-cell simulation, with Monte Carlo collisions. The plasmawas assumed to consist of electrons and H+

3 ions, whoseparticles interacted by collisions with a background gas thatwas assumed to be both rotationally and vibrationally excited.The rotational and translational temperatures were taken tobe identical at 300 K, while the vibrational temperature wasvaried as indicated in figure 5. The vibrational temperaturewas assumed not to relax appreciably during the simulation.Electron–molecule collisions were modelled using the crosssections of Buckman and Phelps, and data from Simkoet al were adopted for ion–molecule processes. Thissimulation relates a vibrational temperature of 5000 K to anelectron temperature of 3100 K or a mean electron energyof 0.4 eV, respectively. This is just slightly higher than themeasured value.

Figure 6. Evolution of the electron temperature in the afterglowobtained from the numerical solution of the Boltzmann equationwith (α = 0.3) and without (α = 0) taking into account vibrationalexcitation.

3.5. Numerical solution of the Boltzmann equation in theafterglow

We have also attempted to analyse thermalization in theafterglow using a Monte Carlo code. In principle it correspondsto an unbounded plasma with no particle losses and with zeroelectric field. The code has been developed independentlyand used to study spatial and temporal relaxation of electronand ion swarms in non-hydrodynamic conditions [28]. It isan improved version of our previous codes based on timeintegration of the collision probability that have been testedto the uncertainty of less than parts of a per cent, both fordc and RF fields in benchmark calculations and comparisonswith other codes [29,30]. Most importantly the code has beenwritten to include the gain of energy in electron–moleculecollisions at non-zero gas temperature and to include super-elastic collisions.

The number of electrons in the simulation was 10 000to reduce statistical fluctuations. Test of the code, howeverwas performed with up to 106 electrons. Elastic as well asinelastic vibrational and rotational collisions are taken intoaccount. Thermal distribution of rotational levels is assumedand the super-elastic collisions involving rotationally excitedmolecules are included. As for vibrational excitation wehave assumed a constant density of molecules in the firstvibrationally excited state during the process of relaxation.Figure 6 compares the cases with and without vibrationalexcitation. In the calculation, the degree of vibrationalexcitation is set at a fixed value of α = 0.3. It can be reasonablyassumed that the degree of excitation does not change withinthe afterglow since the density of electrons is about four ordersof magnitude lower than the density of vibrationally excitedmolecules. In principle the present model should be exactfor the afterglow where swarm coefficients for free electronsshould be applicable [31] for the bulk of the plasma. The resultclearly demonstrates the effect of vibrational heating and alsothe effective temperature of a plateau at 0.2 eV, correspondingto a mean energy of 0.35 eV. This is in excellent agreementwith the measurement. The temporal development shows a

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M Osiac et al

very fast thermalization due to inelastic and elastic collisionsfrom the initial energy of 2 eV.

It should be noted that the effect of super-elastic collisionson electron energy distribution functions in hydrogen has beenconsidered in numerous papers by Capitelli and coworkers [32]and also by other authors [33]. It is shown to be considerableand that high degrees of vibrationally excited molecules mayexist in hydrogen afterglows.

3.6. Simple analytical estimation of the effective electrontemperature in the afterglow

An investigation of the inelastic collision terms of theBoltzmann equation that takes into account only vibrationalcollisions reveals immediately the following result (seeappendix B):

f (E + E0) = αf (E). (2)

Here f (E) is the velocity distribution function as a function ofenergy E. E0 is the energy gap between the vibrational statesv = 0 and v = 1 in hydrogen. This means that the vibrationalcollisions ‘copy’ the distribution function from one energyinterval to the next. The ‘efficiency’ of this process is givenby the degree of vibrational excitation α. The discontinuitiesat the interval borders can be expected to be smeared outby elastic collisions. One can now define an effective tailtemperature T

(hot)e by identifying the ratio of the distribution

in two subsequent intervals with a Boltzmann term. Sinceα can be expressed by a Boltzmann term with a vibrationaltemperature Tvib, one finds T

(hot)e = Tvib. The effective tail

temperature is found from the ion energy to be about 0.5 eV.Therefore, one would expect a vibrational temperature of about5500 K. This is consistent with the measurement as shown inthe next subsection.

3.7. Measurement of the vibrational temperature during theplasma on-phase

The existence of energetic electrons in the afterglow,explained by electron heating through super-elastic collisionswith vibrationally excited hydrogen molecules, prompts themeasurement of vibrationally excited molecular hydrogen inthe discharge. The excited vibrational levels of hydrogenmolecules in the electronic ground state are long lived[22]. The vibrational excitation in the plasma on-phase isinvestigated using OES. The degree of vibrational excitation ofhydrogen molecules is described by a vibrational temperatureTvib. If a Boltzmann distribution is assumed, it is defined bythe ratio of the population density of the vibrationally excitedlevels relative to the ground state:

N�v

N�0= exp

(−E�v − E�0

kTvib

). (3)

The analysis of the intensities of diagonal Q-branch lines ofthe Fulcher-α bands (d 3�−

u → a 3�+g ) in the 600–640 nm

spectral range allows us to determine the relative populationsof hydrogen molecules in the ground state, up to v = 4. Inorder to determine the vibrational temperature in a hydrogenplasma the method reported in [23, 24] was applied.

In non-equilibrium plasmas the ro-vibrational levelpopulations in the electronic excited states are connected

with those in the electronic ground state by the excitation–de-excitation balance equation for electronically excitedstates. The main relaxation mechanism is depopulation byspontaneous emission. Secondary processes like cascades andrecombination are neglected since, under our conditions, theyonly play a minor role [25, 26]. The rate coefficients fordirect electron impact excitation for the transition involvedhere are assumed to have non-zero values, only for the samerotational transitions. They are considered to be independentof the rotational quantum number and are proportionalto the corresponding Franck–Condon factors. Momentumtransfer due to electron impact excitation and the rather smalldependence of the rate coefficients on the EEDF is neglected.The values for the Franck–Condon factors for the transitiond 3�−

u ← X 1�+g have been taken from [23].

The relation between the population density of the d 3�−u

vibrational state, after summation over the rotational levels,and the vibrational temperature, is given by

1

P(Tvib)

∑v

qdv′Xv exp

(−�EXv

kTvib

)∝ Ndv′

τdv′, (4)

where qdv′Xv are the Franck–Condon factors, �EXv the

vibrational energy difference, P(Tvib) the vibrational partitionfunction, Tvib the vibrational temperature, Ndv′ the populationdensity of the upper vibrational level and τdv′ the radiativelifetime. This method of determining the vibrationaltemperature is based on a sufficiently simple kinetic modeland intensity measurements.

Time resolved measurements of the vibrational temper-ature, within the on-phase of the pulsed plasma, are shownin figure 7(a). The power is 300 W, pressure 10 Pa, pulsingfrequency 100 Hz and duty cycle 50%. The intensity for asingle plasma pulse is insufficient; therefore integration overmany pulse cycles is performed. Experimental errors wereestimated to be around 15%, by repeating several times forthe same plasma conditions. The vibrational temperature risessharply at the beginning of the pulse and then reaches a stablevalue of about 4000–5000 K at the end of the pulse. Whenthe discharge is turned on the electron temperature establishesrapidly. However, the vibrationally excited molecules build upon a longer time scale of several hundred micro-seconds, evi-dent in figure 7(a). Therefore, the vibrational population canbe controlled by varying the plasma on-time. Thus, if vibra-tionally excited molecules are responsible for the finite energyin the afterglow a correlation between the vibrational popula-tion and ion energy in the afterglow can be made by varyingthe plasma on-time.

Figure 7(b) shows a plot of the measured ion energy inthe afterglow as a function of plasma on-time. The conditionsare the same as in figure 7(a) and the on-time was varied usingthe duty cycle. The ion energy initially increases with plasmaon-time and reaches a constant value within 1000 µs. It isclear from figures 7(a) and (b) that there is a strong correlationbetween the ion energy in the afterglow and the vibrationalexcitation of hydrogen molecules in the plasma on-phase. Asthe plasma on-time is increased, the vibrational excitation ofmolecules increases and thus there is more electron heatingin the afterglow. At 1 ms, the vibrational temperature ofmolecules in the discharge has completely built up and reached

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Sheath dynamics in the afterglow of a pulsed ICP

Figure 7. (a) Evolution of the vibrational temperature during theplasma on-phase, determined from the optical emissionmeasurements, solid symbols. The errors are estimated to be 20%.The plasma parameters are P = 300 W, fpulsing = 100 Hz,p = 10 Pa and duty cycle 50%. The minimum pulse gate is about20 µs. (b) Evolution of the ion energy H+

3 in the afterglow obtainedas a function of plasma phase on-time determined from the plasmamonitor measurements. The errors are estimated to be 0.4 eV. Theplasma parameters are P = 300 W, p = 10 Pa, fpulsing = 100 Hz andduty cycle ranging from 1–80%.

a steady state. This is another strong indication of the energytransfer from vibrationally excited hydrogen molecules toelectrons in the afterglow through super-elastic collisions.

4. Discussion

In order to explain the high vibrational excitation observed twopossibilities are discussed. Excited hydrogen molecules mightbe generated through electron–molecule inelastic collisions inthe plasma bulk during the on-phase or generated at the surfacethrough recombination of atomic hydrogen.

Generation of excited molecular hydrogen throughinelastic collisions with electrons and through wall processescan be described by the following equation:

dN∗

dt= NKe + NKw − N∗

τ, (5)

where N∗ and N are number densities of molecules in excitedstates and the ground state, τ the effective lifetime of N∗, Ke =

〈σ(ve)ve〉ne, the rate coefficient describing the collision ofhydrogen molecules with electrons, where ne is the numberdensity of electrons, Kw = γ ·β ·δ · 1

4vth1

leff, the rate coefficient

describing the wall processes with hydrogen molecules, leff =V/A the effective length, V and A are the volume and area ofthe discharge, vth the thermal velocity of molecular hydrogen,γ the degree of dissociation of molecular hydrogen, β thecoefficient regarding the wall interaction and δ the coefficientregarding the distribution of excited molecules over variousvibrational states. The ratio of these production processes isgiven by

Ke

Kw= 4leff · 〈σ(ve) · ve〉 ne

γ · β · δ · vth. (6)

For the investigated discharge typical plasma parametersare ne = 4 × 1016 m−3, vth = 2 × 103 m/s, 〈σ · ve〉 =2×10−15 m3 s−1, leff = 5×10−2 m. Typical values of β are formetals ∼10−1 and for insulators ∼10−3 [27]. Typical valuesfor the degree of dissociation are γ = 0.01–0.1. It should benoted that there is a positive feedback between the degree ofvibrational excitation and the degree of dissociation since thedissociation cross section is increasing dramatically with thevibrational state. No data are known to us for the distributionof vibrational excitation and the related coefficient δ. Theuncertainties in the coefficient do not allow an unambiguousjudgement. However, the resulting values for the ratio ofKe/Kw, reveal that for metals wall processes can play a majorrole, while for insulators generation of excited molecules at thesurface can be neglected. Thus, in the investigated dischargewall processes at the metal surfaces can be significant in theproduction of vibrationally excited molecules.

5. Conclusion

Investigations of the sheath dynamics in the afterglow of apulsed ICP show that the plasma boundary sheath potentialdoes not fully collapse in the early afterglow. It stays at aconstant value of about 2 eV for a few hundred micro-seconds.This is revealed through IEDF measurements of H+

3 ions usinga mass resolved ion energy analyser. This sheath potentialis caused by electrons with a temperature of approximately0.5 eV. Measurements of the plasma density decay, determinedby ambipolar diffusion, also give similar electron energies inthe afterglow. The decay is measured using the ion energyanalyser and time resolved Langmuir probe measurements.The Langmuir probe also allows for direct measurements ofthe EVDF in the afterglow. The EVDF is bi-Maxwellian, witha kink at 0.5 eV, separating the ‘cold’ and the ‘hot’ parts. Theelectron temperature in the ‘hot’ part, determining the sheathpotential, is also in agreement with the previous observationsof a finite electron temperature in the afterglow.

This finite electron temperature can be explained throughsuper-elastic collisions with vibrationally excited molecules.Vibrational excitation of molecules builds up in the plasmaon-phase and can be considered temporally constant on thetime scale of the plasma decay in the afterglow. Theserelatively long-living vibrationally excited molecules act asan energy reservoir for electrons in the afterglow.

These observations are supported by PIC simulations andnumerical solutions of the Boltzmann equation.

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The vibrational excitation was measured by OES usingFulcher band transitions. The vibrational population builds upwithin a few hundred micro-seconds and reaches relativelyhigh values of vibrational temperatures (several thousandKelvin) corresponding to several ten per cent vibrationalexcitation. This relatively high vibrational excitation resultsfrom electron impact excitation during the plasma on-phaseand production of vibrationally excited states can occurthrough recombination of atomic hydrogen in wall processes.The degree of vibrational excitation can therefore be influencedby varying the plasma on-time. Measurements of the sheathpotential with various degrees of vibrational excitation arein full agreement with the above-mentioned explanation ofelectron heating through super-elastic collisions.

Acknowledgments

The authors express their gratitude to K-U Riemann and V AKadetov for helpful discussions. The project was funded bythe DFG in the frame of the SFB 591 and the GRK 1051.

Appendix A

The decay of the plasma density in the afterglow can bedescribed easily in the frame of the diffusion equation.For the fundamental diffusion mode this treatment gives ananalytical expression that relates the decay time to the electrontemperature Te in the afterglow. It is assumed that thetemperature stays constant over the time interval consideredhere. This is in fact motivated by the experimental results.In our case, the aspect ratio between the axial and the radialdimension leads to diffusion losses predominantly in the axialdirection. This motivates a one-dimensional treatment. In thediffusion approximation, the inertia term in the momentumbalance equation is neglected. However, in one dimensionthis term can be considered explicitly and leads to certaincorrections to the calculated decay time.

The coupled equations are the continuity equation andthe momentum conservation equation for the ions and theambipolar field Fa for the case of cold ions:

∂n

∂t+

∂(nu)

∂z= 0, (A1)

∂u

∂t= −u

∂u

∂z+

eFa

M− νmiu ≈ 0, (A2)

Fa = −De

µe

1

n

∂n

∂z(A3)

where νmi is the ion–neutral collision frequency and M is theion mass. The collision frequency is assumed to be constant.At low velocities typically found in the bulk this is well justifiedfor hydrogen and also for noble gases. Equation (A2) isapproximately zero since u relaxes on a timescale 1/νmi andthe plasma density relaxes on a timescale that is longer by afactor (L/�i)

2 where L is the discharge gap and �i � L

the ion mean free path at the Bohm velocity. The ratio of theelectron diffusion and mobility constants is in the frame of thetwo-term approximation [3]:

De

µe= C

2〈Ee〉3e

, (A4)

C =3

⟨v2

νme

⟩⟨

1

v2

∂

∂v

v3

νme

⟩〈v2〉

(A5)

where νme is the elastic electron–neutral collision frequency.For a constant collision frequency one finds C = 1,independent of the particular form of the distribution function.In hydrogen the cross section is approximately constant up toan energy of about 4 eV. Therefore, we have investigated C alsofor the case of constant mean free path and a bi-Maxwelliandistribution function. In this case the result depends on theratio of the hot temperature Th to the cold temperature Tc inthe distribution. For Tc/Th > 0.4 one finds 1 � C � 0.9independent of the relative contribution of the hot part to thedistribution that is varied between 0 and 1. The experimentalratio of Tc/Th = 0.5 is well above the discussed ratio of 0.4.Therefore, one can conclude that for the case investigated hereone can very well approximate C by 1.

The boundary condition is

u(z = 0) = −uB and u(z = L/2) = 0, (A6)

u2B = 2

M

⟨1

Ee

⟩ . (A7)

The Bohm velocity uB is determined by the inverse of themean electron energy for the case of mono-energetic ions [35].The density is now split into a spatial and temporal dependentfunction n( z.., t) = H(z)T (t). This leads after integration to

T (t) = exp

(− t

τ

), (A8)

νmiL

2

√3M

2C〈Ee〉 = 1 + η2

η

[π

2− arctan

(η

γ

)]− γ, (A9)

η2 = 1

τνmiand γ = (C〈1/Ee〉〈Ee〉)−1/2 (A10)

The density decays exponentially with a time constant τ .The dimensionless constant γ is investigated again for the caseof a bi-Maxwellian distribution. For Tc/Th > 1/3 one finds1 � γ � 0.7 for all relative contributions of the hot part in therange between 0 and 1. Since η � 1, equation (A9) can nowbe expanded up to first order in η:

νmiL

2

√3M

2C〈Ee〉 ≈ π

2

(1

η+ η

)− 1 + γ 2

γ. (A11)

For γ in the range given above, the second summand inequation (A11) can be well approximated by 2.0. Further, onecan neglect the term η compared with 1/η and the constant.Then the final result is

〈Ee〉 ≈ 3M

2C

(νmiL

π√

τνmi − 4

)2

. (A12)

As already pointed out above, for the case consideredhere C can be well approximated by 1. In the diffusionapproximation only the first term in the denominator wouldoccur. The second term is a correction due to the inertia

362

Sheath dynamics in the afterglow of a pulsed ICP

term that has a non-negligible contribution at the edge ofthe bulk. Although one could already use equation (A9) forthe calculation of the mean energy, the expanded form (A12)provides a better insight and the loss in accuracy is negligiblehere. For our discharge conditions, the correction to the meanelectron energy by inclusion of the inertia term is in the rangeof 20–40%.

Appendix B

In order to develop a basic understanding of how super-elastic collisions with vibrationally excited molecules canaffect the electron distribution function a simple model wasdeveloped. This model is analogous to the work of Claaßenon the effect of metastable states [34]. Here a homogeneousand stationary situation in a field free region is consideredwhere only collisions coupling the two lowest vibrational states(v = 0 and 1) are taken into account. Collisions involvinghigher vibrational states and elastic and rotational collisionsare neglected. In this case the Boltzmann equation can bewritten as

−ν(ε)g(ε) + ν(ε + ε0)g(ε + ε0) + α[−ν∗(ε)g(ε)

+ν∗(ε − ε0)g(ε − ε0)] = 0, (B1)

where ν(ε) is the energy dependent collision frequency forvibrational excitation from v = 0 to v = 1 and ν∗(ε) isthe corresponding quantity for the reverse process. Note thatν(ε) = 0 for ε < ε0 where ε0 is the energy gap between thetwo vibrational states. α is the ratio between the populationin v = 1 and v = 0. It is assumed that α develops tosome fixed value within the on-phase of the discharge andstays constant in the afterglow while the plasma density is stillconsiderable. g(ε) is the EEDF. The four terms in the equationhave the following meaning: the first term describes the loss ofan electron through inelastic collisions with v = 0, the secondterm takes into account the reappearance of this electron at anenergy reduced by ε0, the third term is similar to the first termexcept for v = 1 and the fourth term is similar to the secondterm but here the electron gains an energy ε0. The collisionfrequencies ν are related to the cross sections σ by

ν(ε) = σ(ε)

√2

mε, (B2)

ν∗(ε) = σ ∗(ε)

√2

mε. (B3)

Equation (B1) must also hold in thermal equilibrium.Then α must be expressed by a Boltzmann factor. Since thecross sections depend only on molecular quantities one derivesafter a short calculation:

σ ∗(ε) = ε + ε0

εσ (ε + ε0). (B4)

By inserting this result in equation (B1) and taking intoaccount the relation between the energy distribution functiong and the velocity distribution function g(ε) ∝ √

εf (ε) itfollows that

f (ε + ε0) = αf (ε). (B5)

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