Field-Emission Driven Microplasmas in Metallic MEMS: PIC/MCC
Simulations and Direct Measurements
Ayyaswamy Venkattraman, Anurag Garg, Dimitrios Peroulis and Alina Alexeenko
Purdue University, West Lafayette, IN 47907, USA
Abstract: Particle-in-cell/Monte Carlo collision simulations including field
emission effects are performed with the aim of quantifying the gas charging in
microgaps encountered in metallic MEMS. Simulations are performed for a
range of voltages in both Argon and Nitrogen at 1 atm pressure and comparisons
are made. The net charge in the gap is shown to be positive and increases
exponentially with voltage for both gases. For a given voltage it is shown that the
current density in Argon is almost twice the current density in Nitrogen.
Keywords: PIC/MCC, MEMS, Microdischarges, Microplasma, Field emission
1. Introduction
Micro-Electro-Mechanical-Systems (MEMS) have
emerged as a promising technology for the
development of miniaturized low-cost and lowpower switches, actuators, and sensors. However,
there are a number of physical mechanisms that need
more understanding and this remains as one of the
biggest hurdles to widespread commercial
adaptation of MEMS-based radio frequency (RF)
electronics. The main goal of this work is to study
the phenomenon of gas charging in micro-gaps using
particle-in-cell/Monte Carlo collision (PIC/MCC)
simulations with inputs from experiments thereby
relating it to being one of the possible failure modes
in electrostatic MEMS.
While gas charging in macro-gaps is well
established and is governed by the traditional
Paschen curve[1], the Paschen theory breaks down
when the gap sizes are decreased below ~ 5 μm.
There have been various experiments in the past
which have observed breakdown related phenomena
such as glows, sparks in micro-gaps at voltages
significantly below breakdown voltages predicted by
the Paschen curve with this deviation has been
attributed to the field emission of electrons from the
cathode. Figure 1 shows data from some of the
experiments[2-4] that have observed deviation from
the Paschen curve at micro-gaps and compares it to
the Paschen curve for air at a pressure of 1 atm. In
order to account for this deviation, there have been a
number of analytical models[5] which attempt to
describe the transition from a field emission
dominated breakdown at very small gaps to the
Paschen theory breakdown at macro-gaps leading to
what is referred to commonly as the modified
Paschen curve[6].
Figure 2. Measured breakdown voltages for microgaps in 1 atm
air compared to the traditional Paschen curve and a typical
modified Paschen curve.
2. Theory & Background
Field emission is the phenomenon of extraction of
electrons from metals by the application of very high
electric fields ~ 108 V/m. The current density of
emitted electrons is related to the applied electric
field by the Fowler-Nordheim (F-N) theory[7] based
on quantum mechanics and is given by
j fe 
measurements, fabrication and experimental set-up
explained in detail by Garg et.al[9].
  B 3/2v( y ) 
A 2 E 2
exp


t 2 ( y)
E


where  is the work function of the metal under
consideration, β is the field enhancement factor and
constants A and B are F-N constants given by
A  6.2 106 A / eV
B  6.85 107 V / cm / eV 3/2
The terms v(y) and t2(y) were not part of the original
F-N theory and were corrections included later[8].
The corrections are given by
v( y )  0.95  y 2
t 2 ( y)  1.1
where y  3.79 104  E /  . The value of β
strongly depends on the surface properties such as
roughness which in turn depend on the technique
used to fabricate the devices. It is very hard to
predict a value of β with various experiments[10] in
the past reporting values between 1.5 and 115.
Figure 2. Comparison of measured current for two devices with
corresponding F-N theory curves for various values of β.
3. Results & Discussion
The secondary electron emission coefficient (γse) for
ions striking the electrode was set to a constant value
of 0.05. The electrode at X = 0 is the anode and the
electrode at X = 1 is the cathode. The simulations
divided the domain into 100 cells and a timestep of
10-14 s was used. Results are presented for two
different gases Ar and N2. The default collision
models in XPDP1 for Ar were used and for N2, 3
electron-neutral
collisions
including
elastic
scattering, excitation, and ionization and 2 ionneutral collisions including elastic scattering and
charge exchange were implemented in XPDP1. The
collision cross-sections for the electron-neutral
collisions were taken from databases[12-13] while
the ion-neutral collision cross sections were assumed
to be 10-18 m2 independent of the energy of the
colliding ion. The results presented below for
number density and current density are all timeaveraged quantities averaged over 100,000 timesteps
after the simulation reached state.
This section presents results of the particle-incell/Monte
Carlo
collision
(PIC/MCC)[9]
simulations performed to predict the discharge
structure, gas charging in microgaps of Ar and N2
and also direct current measurements in microgaps
of air performed using MEMS structures fabricated
using techniques described in [10]. The 1dimensional code XPDP1[11] developed at the
University of Berkeley was used to perform these
simulations. The original version of XPDP1 does not
include the effects of field emission but the
simulations reported in this work were performed
with a version of XPDP1 that was augmented to
include field emission effects using the F-N theory.
The simulations were performed for a gap size of 1
μm and various applied voltages. The field
enhancement factor (β ) was taken as 55 which is a
typical value from direct current measurements
presented in Figure 2 with the details of the
The cathode material was taken as Ni and hence the
corresponding work function of 5.15 eV was used to
compute the field emission current using F-N theory.
Figure 3 and Figure 4 compare the steady state ion
and electron number densities for various applied
voltages across the 1 μm Argon gap at a pressure of
1 atm. Both ion and electron number densities attain
a maximum near the cathode. This is due to the
cathode being a source of electrons due to field
emission as a result of which most of the ion
production happens near the cathode thereby leading
to an increase in ion number density near the
cathode.
The simulations performed for Ar gas were repeated
for N2 with the same set of simulation parameters.
The 3 electron-neutral collisions implemented for N2
were the 3 most important collisions as reported by
Zhang et.al[3] in their simulations for prediction of
ion generation in air. The threshold energy for
excitation and ionization of Ar are 11.55 eV and
15.76 eV respectively as used in XPDP1.
Figure 4. Comparison of electron number density obtained
using PIC/MCC simulations for various voltages applied across
a 1 μm gap filled with Argon at 1 atm pressure.
Figure 3. Comparison of ion number density obtained using
PIC/MCC simulations for various voltages applied across a 1
μm gap filled with Argon at 1 atm pressure.
The emission of electrons at the cathode leads to a
local increase in number density of electrons. The
number density profiles also indicate a net positive
charge in the gap with the ion densities exceeding
the electron densities by about two orders of
magnitude. Simulations performed for a range of
voltages between 50 V and 59 V indicate an
exponential increase in the net charge in the gap. As
the voltage is increased, the current density also
increases and Figure 5 shows the current density
distribution across the Ar gap for a voltage of 59 V.
It can be seen that most of the current in the gap is
being carried by the electrons. The current density of
ions is higher near the cathode and vice versa for the
electrons. The constant current density across the
gap confirms that the simulations have reached
steady state and this was one of the criteria to
establish convergence.
These values are very close to the electronic
excitation and ionization threshold energies of N2
given by 11.03eV and 15.6 eV respectively. Figure
6 compares the current density in the gap for Ar, N2
gases in the gap as a function of the applied voltage.
Also shown is the current density due to field
emission as predicted by F-N theory. Any difference
between the actual current density and that due to
field emission is due to the gas ionization. It can be
seen that the total current density for Ar is larger
than N2 even though the excitation and ionization
energies of Ar and N2 are comparable.
The differences in the degree of ionization and
hence the current densities for Ar and N2 can be
attributed largely to the higher ionization cross
section for Ar. As a result of the higher cross
section, the number of ionization collisions are
significantly higher for Ar than N2. This trend is also
seen in the breakdown voltages[5] predicted by
Paschen theory with the breakdown voltage for Ar
significantly lower than that of N2 even for macrogaps.
charge at steady state with this charge and the
current density increasing exponentially with
voltage. For any given voltage, it was shown that
the current density obtained for N2 was lower than
that of Ar even though the excitation and ionization
energies of Ar and N2 are comparable. The
differences are largely due to the higher ionization
collision cross section for Ar leading to more
frequent ionization collisions thereby producing a
higher number density of ions and current density.
References
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Figure 5. Comparison of electron, ion, and total current density
profiles across a 1 μm gap filled with Argon at 1 atm pressure
and an applied voltage of 59 V.
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Figure 6. Comparison of current density in Ar, N2 gaps with the
Fowler-Nordheim field emission current density.
4. Conclusions
PIC/MCC simulations including the effects of field
emission were performed for a 1 μm gap of Ar and
N2 with the aim of predicting the charge
accumulated for application to electrostatic MEMS
structures. The simulations show the existence of
stable microdischarges at voltages as low as 55 V.
The simulations performed for Ar for a range of
voltages from 50 V to 59 V indicated a net positive
[10] A. Garg, A. Venkattraman, A. Alexeenko, D.
Peroulis,
IEEE
Int.
Conf.
on
Microelectromechanical Systems (MEMS 2011),
Cancun, Mexico, Jan 23-27, 2011.
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