Energy Balance and Plasma Potential in Low-Density Hot-Filament Discharges
Scott Robertson, Scott Knappmiller, and Zoltan Sternovsky
Abstract—Electron energy balance is shown to play an important role in determining the plasma potential in low-density hot-filament discharges. The confined electrons that are lost to the walls are those with energy just above the plasma potential, thus the electron energy loss rate is the product of the electron loss rate and the height of the potential barrier. The sources of the electron energy are the energy at creation plus the energy gained from equilibration with energetic, unconfined electrons. An experiment in a soup-pot plasma device demonstrates that the plasma potential has values that satisfy the energy balance equation. The ion loss rate affects the electron loss rate through the quasineutrality condition, thus collisions of ions play a role in determining the plasma potential by reducing the particle loss rates.
Index Terms—Plasma devices, plasma measurements, plasma sheaths
S. Robertson and S. Knappmiller are with the Center for Integrated Plasma Studies and the Department of Physics, University of Colorado, Boulder, CO 80304-0390 USA. (e-mail: firstname.lastname@example.org).
Z. Sternovsky is with the Laboratory of Atmospheric and Space Physics, University of Colorado, Boulder, CO 80304-392 USA.
A recent model for particle balance and energy balance in low-density, hot-
1filament discharges gives values for electron temperature and plasma potential that agree
2(to within about 25%) with values measured in a soup-pot type of plasma device
operated at neutral gas pressures of 0.1-1 mTorr. In this work, the energy balance model is extended to higher pressure (8 mTorr) by including the effect of charge-exchange collisions of the ions. An important dimensionless parameter is the ratio of the plasma radius r to the charge-exchange mean free path ~. At the higher pressure of 8 mTorr, r/~
( 30 and charge-exchange collisions significantly reduce the ion loss rate. The model with collisions is in agreement with experimental data at higher pressure.
The energy balance model can be summarized as follows. The confined electrons are those with energy below the plasma potential ；. Electrons are lost from the confined p
population by diffusion in velocity that allows some of the confined electrons to pass over the potential barrier. If there are no other significant energy loss mechanisms, the rate at which the population of confined electrons loses energy is simply the product of the electron loss rate and q； where q is the elementary charge. In a steady state, this p
energy loss is balanced by two sources. The first is the equilibration of the confined electrons with the more energetic unconfined electrons that include the primary electrons
3,4as well as the more numerous secondary electrons from the walls. The second energy
source is the energy with which the electrons are created. This work and ref.  builds
3,4,5,6upon earlier work that enumerated the processes affecting energy balance in soup-
pot types of devices. These devices have not previously been modeled with the level of
7,8,9detail that has been used for DC discharges at higher pressure.
Charge-exchange collisions replace ions that have been accelerated toward the wall with ions that move more slowly, thus the ion loss rate is decreased by collisions. The electron and ion loss rates are coupled by quasineutrality, thus an effect of ion-neutral collisions is a reduction in both the ion and electron loss rates. A reduction in the electron loss rate results in the electrons gaining more energy through equilibration and thus a greater plasma potential (or a greater electron loss rate) is required to satisfy energy balance. We observe in the experiment an increase in plasma potential with neutral gas pressure. This increase is approximately equal to that predicted by the energy-balance model when the effect of charge-exchange collisions is included.
In Sec. II, the model for electron energy balance is developed. The rate of ion loss to the wall is derived, including the effect of collisions, and an expression for energy balance is derived from which the plasma potential can be calculated. In Sec. III, the experiment is described and the measured plasma potentials are compared with those calculated using the energy balance model. Section IV is a conclusion.
II. THE ENERGY BALANCE MODEL
A. Ion production and loss
;;R Electron-ion pairs are assumed to be created at the rate where is the RV
volume-average of the rate of ionization and V is the plasma volume. The loss rate of ions
is ， A, where ， is the flux of ions to the walls and A is the wall area. The volume-ii
averaged rate of ionization may then be found from
;A，i R. (1) ，V
In the absence of collisions, the flux of ions to the walls in cylindrical geometry is ~0.42 n c, where n is the electron density, is the ion sound speed, T is the c，T/meseesei
10,11temperature of confined electrons in energy units, and m is the ion mass. This ion i
12,1314flux is reduced by charge-exchange collisions. Sternovsky has used a kinetic model
to calculate ， for a range of r/~, where r is the plasma radius and ~ is the charge-i
exchange mean free path. The numerical results for r/~ < 1000 can be fit to the function
0.42nces , (2) ，，i1？0.18(r/~)
and the ionization rate can then be found from more easily measured quantities using
0.42nAT/meeiˆ~ R(n,T,)，. (3) eeV1？0.18(r/~)
B. Electrons from ionization
Only a fraction of the electrons from electron impact ionization have energy sufficiently low to be confined by the plasma potential. The distribution in energy S(E) of
secondary electrons from ionization, Fig. 1A, has been measured for several gases by
15Opal et al. and is approximately
??21，? , (4) S(E)(2，??W1(E/W)？??
where E is the energy of the secondary electron and W = 10 eV for Ar. A more accurate
normalization to unity can be made by cutting off the distribution at the maximum energy of a secondary electron. This energy is P = (E ， E)/2, where E is the energy of the priipri
primary electrons from the filaments and E is the ionization energy. The fraction F(；) ip
16of the source distribution that is confined by the plasma potential is then
arctanq；Wq；；？pp . (5) F(；)，(p；？arctanPWW arctan(P/W)
The final expression is valid for q； << W. The rate at which confined electrons are p
;created and lost in a steady state is . R(n,T,~)F(；)eep
C. Electron heating from equilibration with secondaries
Figure 1B illustrates the electron distribution function that is found from probe
17measurements. The electrons with energy below q； are typically confined for many p
electron-electron collision times, thus this part of the electron distribution is nearly Maxwellian. Ionization is from energetic primary electrons which usually have energy in
the range 40，100 eV. The primaries release secondary electrons from the walls that have a distribution in energy that is approximately Maxwellian with a temperature of T ) 2 se
eV. The secondary electrons from the wall are accelerated through the sheath potential
3,4and are thus shifted upward in energy by q；. The distribution in velocity of wall p
secondaries within the plasma is then
2 = 0, v！2q；/mpe
where n is the number density of wall secondaries that would be observed at the wall. se
This density is found consistently from the probe data using the random current of secondaries collected by the probe when it is at zero potential relative to the wall.
The rate at which a single energetic electron transfers energy to the confined
YnmdUee , (7) ，dtv
where n is the density of the confined electrons, v is the relative velocity of the collisions, e
22；？ln：Y，4?q/4?；mln： and is the Coulomb logarithm. The energy loss rate can 0e
be integrated over the distribution of wall secondaries to find the rate at which energy is
1transferred from the wall secondaries to the confined population
3?2??m2e?，?；？ ，Ynmnexpq；/Texp；？，mv/2T4vdv , (8) eesepseese?，??2Tse??w
where the angled brackets denote an average over the distribution function of wall secondaries and w，2q；/m is the minimum velocity of wall secondaries within the pe
plasma. The final result for the heating rate is independent of the potential ； and the p
minimum velocity of the secondaries w because the factors containing these variables
D. Electron energy balance
The energy balance equation for confined electrons must include both the energy from equilibration dQ/dt and the mean energy that the electrons from ionization have initially. The distribution of initial energy, Eq. (4) and Fig. 1A, is nearly flat from zero
1energy to q；, thus the mean initial energy is approximately q；. The rates of energy pp2
input are set equal to the rate at which energy is carried to the wall to obtain the energy balance relation
;dQ1ˆ , (9) ；？；？；？；？？Rn,T,~Fq；q；，Rn,T,~Fq；q；eeppeepp2dt
which can be rearranged to give
dQ2dt . (10) ；，;p；？；？qRn,T,~Fq；eep
Equations (3) and (5) can then be used to find the relationship between the plasma potential and other parameters that can be found from probe data
1/2：?9.3 W arctanP/WYmnV1？0.18r/~；？1ese ；(n,T,T,~)，. (11) ，?pseseeqA2?TT/mm，?eseie??
;This equation shows that the plasma potential is not explicitly dependent upon n or R e
and is only weakly dependent upon T. The strongest dependence is upon the density of e
secondary electrons which are the source of heating for the confined electrons. The energetic primaries have not been included as a source of heating because their transfer of energy is much smaller as a consequence of their higher velocity and lower density.
In ref. , three variables (n, T, and ；) were treated as unknowns and were eep
found by solving simultaneously ion particle balance, electron particle balance, and electron energy balance. In this work, the electron particle balance equation is omitted and the number of unknowns is reduced by taking T from probe data. Having a measured e
value for T rather than a model value for T removes any questions that might arise about ee
the accuracy of the model value for T. e
III. THE EXPERIMENT
A. The apparatus
The experiments are performed in a soup-pot type of plasma device, Fig. 2. The vacuum chamber is of aluminum with an inner diameter of 31 cm and a length of 70 cm. The chamber has a stainless steel liner to cover contamination and to make the surface
20potential more uniform. A turbomolecular pump creates the vacuum and the base
-6pressure is < 10 Torr. The plasma is generated by primary electrons from four filaments located on the end walls. The filament bias potential is –80 V and the emission current is
20–160 mA. The working gas is argon at pressures of 0.1– 8 mTorr. The pressure is
measured by an ionization gauge with an extended range. The mean free path for scattering of the primary electrons is about 5 cm at the highest pressure, thus the ionization rate is higher near the filaments than at the center of the chamber. The confined electrons (with energies below ~0.5 eV) have a mean free path comparable to the chamber diameter, thus these electrons should fill the chamber nearly uniformly except in the sheath region at the walls. Probe measurements indicate that the density of confined electrons at the center of the chamber is about 10% lower than the density 10 cm from the end walls.
The plasma parameters are determined by means of a cylindrical probe of stainless steel with a diameter of 190 ?m and a length of 27 mm. The probe is discharge-
cleaned before data are taken. A digital data acquisition system with 16-bit resolution averages 25 current readings at each probe voltage. A subset of the probe data from ，40
to ，10 V are fit to a model for the ion current and this current is subtracted from the probe current to obtain the electron current alone, Fig. 3. This current shows two electron distributions: a low energy (<0.5 eV) distribution that is the confined electrons and a higher energy (>0.5 eV) distribution that is the wall secondaries. Orbit-motion-limited
21,22theory is used to find the densities and temperatures of the two populations as described in ref. 17. The electrons contributing to the probe current for voltages from 0 to ，2 volts is identified as secondaries from the wall because the slope of the semilogarithmic plot corresponds to the expected temperature of 2-3 eV. This part of the probe current cannot be primary electrons because primaries have energy near 80 eV and would create a probe current with a much smaller slope. The density of the wall secondaries is typically a few percent of the total electron density.
The analysis locates the effective wall potential by finding the probe voltage at which the current of confined electrons begins to rise above the current of wall secondaries. This point is typically within 0.1 volt of the ground potential. The plasma
23potential is located at the maximum in the slope of a function fit to the probe data. The
potential ； is found from the difference between the probe voltages at the plasma p
potential and at the wall potential. These are each uncertain by 0.1 V as a consequence of the spacing of the data points. These two uncertainties are added in quadrature to obtain the uncertainty of 0.14 V in ；. p