Using field emission to control the electron energy distribution in high-pressure
microdischarges at microscale dimensions
Yingjie Li, David B. Go*
Department of Aerospace and Mechanical Engineering,
University of Notre Dame, Notre Dame, Indiana 46556, USA
.
*Author to whom the correspondence should be addressed, electronic mail: dgo@nd.edu
Supplemental Material
1. Discrete EED Peaks Corresponding to Single Excitation Reactions
Electrons that experience a single excitation reaction will lose a fixed amount of energy, and
these electron populations will correspond to discrete peaks in the EED. Table S.1 lists all the
excitation reactions used in this study1, the corresponding location of the discrete peak in the
EED, and the subsequent proportion of the electron population with this energy based on the
conditions presented in Fig. 1 (d = 5 µm, p = 380 torr, and  = 50 V).
Table S1: Detailed excitation types and their proportion in the EED based on Fig.1a
Energy loss (eV)
11.55
11.62
11.72
11.83
12.91
13.08
13.09
13.15
13.17
13.27
13.30
13.33
13.48
13.84
13.90
13.98
14.01
14.06
14.09
14.15
14.21
14.23
14.30
14.71
15.20
Peak
location in
Fig. 1a (eV)
38.45
38.38
38.28
38.17
37.09
36.92
36.91
36.85
36.83
36.73
36.70
36.67
36.52
36.16
36.10
36.02
35.99
35.94
35.91
35.85
35.79
35.77
35.70
35.29
34.80
1
Percentage of
electron population
(%)
0.16
0.11
0.33
0.29
0.03
0.12
0.11
0.11
0.12
0.11
0.09
0.07
0.07
0.18
0.17
0.21
0.20
0.19
0.19
0.19
0.16
0.14
0.06
0.14
0.23
2. Characterization of Discrete Peaks in EED for Reduced Excitation List
In order to ensure the discrete peaks observed in Fig. 1a in the main manuscript were in fact due
to discrete collision events, a reduced set of excitation reactions were used. Instead of using 25
excitation reactions, a single excitation reaction was used with cross-sections equivalent to the
combined effect of the 25 different reactions, per Yamabe et al.2 Fig. S1 shows the EED with
these reactions for the equivalent case in Fig. 1a in the main manuscript. Here, the band of peaks
between 35-39 eV is reduced to a single peak at 38.5 eV corresponding to a single excitation
event with an energy loss of 11.5 eV. Further, peaks for two and three discrete excitation
collisions are also distinguishable. The broadened band due to a single ionization event is
unchanged, however, and similar to that discussed in the main manuscript.
FIG. S1: EED at the anode for p = 228 torr at a fixed gap size of d = 5 µm and an applied
potential of Φ = 50 V.
3. Reduced Electric Field Analysis
The reduced electric field can be defined as E/p = Φ/pd. Fig. S2 shows a series of three
simulations all at identical E/p conditions, but varying pd conditions. The respective EEDs at the
anode for each simulation are quite different, suggesting that E/p is not a critical scaling
parameter for field emission-driven microdischarges.
2
FIG. S2: Normalized EED at the anode for different pd values at a fixed reduced electric field of
E/p = 1315.8 V/torr-cm.
4. Curve Fitting of the EED
The Maxwell-Boltzmann energy distribution is:
𝜀
1
3/2
𝑓(𝜀)𝑑𝜀 = 2√𝜋 (𝑘 𝑇)
𝑏
𝜀
exp (− 𝑘 𝑇) 𝑑𝜀 ,
𝑏
(S.1)
where kb is the Boltzmann constant. The EED extracted from the simulation has been curve fitted
as shown in Fig. S3. The existence of the high-energy tail introduces some disagreement in this
curve fit, reducing the coefficient of determination to r2 ~ 0.9, while the fitting result without the
high energy tail has an r2 > 0.95, which is acceptable agreement with the Maxwell-Boltzmann
distribution. The electron temperature Te is the primary fitting parameter in (S.1), and from this
the ensuing mean energy <> can be determined by
3
< 𝜀 >= 2 𝑘𝐵 𝑇𝑒 .
3
(S.2)
FIG. S3: A Maxwellian curve fit to the EED as extracted from a particle simulation at an applied
potential of 200 V; the population in the high-energy tail was not included in the curve fit.
5. Mean Electron Energy Analysis
As shown in the inset of Fig. 4 in the main manuscript, the mean electron energy scales with pd
as ⟨𝜀⟩⁄𝜀𝛷 ~exp(−𝑝𝑑). To explore if there is a dependence on E or E/p in this relationship, two
series of simulations were conducted – one at constant E = 40 V/m and on at constant E/p =
1052.6 V/torr-cm. The data are essentially identical showing that this scaling is not dependent
on E or E/p. Small deviations arise because to obtain constant E/p simulations we vary Φ. This
slightly affects the high energy tail of the EED, which in turn slightly impacts the curve fitting
procedure whereby we extract <>. However, we believe that this does not affect the conclusion
that <>/Φ scales with pd regardless of whether E or E/p is maintained constant.
4
FIG. S4: Comparison of normalized mean electron energy as a function of pd for constant E = 40
V/m and constant E/p = 1052.6 V/torr-cm.
References
1
M. Hayashi, NIFS-DATA-72 (2003).
C. Yamabe, S. J. Buckman, and A. V. Phelps, “Measurement of free–free emission from lowenergy-electron collisions with Ar,” Phys. Rev. A, vol. 27, no. 3, pp. 1345–1352, Mar. 1983.
2
5