Plasma diagnostics and simulations
Frequency analysis of electrical
measurements in low pressure capacitively
coupled plasma
Tasks
1. Realize the measurements of plasma potential waveform by means of a cylindrical probe in
a low-pressure capacitively coupled discharge ignited in mixtures of N2 and O2.
2. Study the oscillations of plasma potential or current that spontaneously develop in capacitively
coupled plasma and analyze how their frequency depends on electron concentration.
Introduction
The properties of capacitively coupled discharges are to a large extent caused by sheaths, areas
with high spatial electric charge, which spontaneously develop at boundaries between plasma and
any solid surface. Since electric field expels electrons form the sheath area, concentration of positive
ions exceeds the concentration of electrons in sheaths and sheaths are positively charged, see fig. 1.
Because capacitive discharges are driven by voltage with high-frequency, both the electric field in
sheaths and the sheath width strongly vary in time. The electric current flowing through a sheath
at a planar electrode can be expressed as a sum of displacement, electron and ion current:
I = e Se ni(s)
ds
dt
−
en0Se
4
8kTe
πm
e−eU/(kTe)
+ en0Se
kTe
mi
(1)
U =
e
ε0
s
0


ξ
0
ni(x) dx

 dξ, (2)
where I denotes the electric current, Se the electrode area, ni the concentration of positive ions,
s the actual sheath width, n0 the electron concentration in the bulk plasma, Te the electron
temperature, m and mi the electron and ion mass, respectively. U is the sheath voltage and ε0
is the vacuum permitivity. The nonlinear nature of sheaths that is shown in equations (1) and
(2) manifests itself in generation of oscillations of plasma voltage and current with frequency that
differs from frequency supplied to the discharge from the RF generator. These new electric signals
tend to oscillate with the eigenfrequency of the discharge, which is described in the following
paragraph.
The bulk-plasma conductivity
σ =
n0e2
m(ν + iω)
+ iωε0 (3)
(where ν is the electron-neutral collisional frequency for the momentum transfer and ω is the
angular frequency of the driving voltage) has an inductive character. On the other hand, sheaths
with their low concentration of mobile electrons have capacitive character. The serial connection
Laboratory guides (verze 5. června 2018) 2
n
s
nn
e
i
sheath
x
electrode
plasma
Obrázek 1: Concentration of positive ions (ni)
and electrons (ne) in a sheath.
VRo Co
C(U) Ie Ii sheath
plasma
probe
oscilloscope
Obrázek 2: Electric schema of the probe with
no RF compensation.
of inductive bulk plasma and capacitive sheaths needs to have a resonance, which is called the
series resonance or plasma-sheath resonance. Its angular frequency can be estimated by
ωsr = ωpe
stot
l
, (4)
where ωpe = e n/mε0 is the plasma frequency of electrons, l is the distance between electrodes
and stot is the total with of both sheaths at the powered and grounded electrodes. ωsr is the
eigenfrequency of capacitively coupled discharge. Thus, we can expect that waveforms of potential
and current of capacitively coupled discharges will contain oscillations with frequency close to the
frequency ωsr. Since these eigenoscillations will be damped by collisions between electrons and
neutrals, it can be expected that their amplitude will be high at low pressure [1, 2].
Probe measurements of plasma potential waveforms
The problematic of probe measurements of plasma potential waveforms is described in detail in [3].
The measurement is complicated by the fact that a small sheath develops around the probe and
this sheath separates plasma from the probe. Consequently, the sheath voltage needs to be added
to the measured probe voltage in order to find the searched plasma potential
Upl = Uprobe + Ush. (5)
Upl denotes plasma potential, Uprobe the measured probe voltage and Ush the voltage on the sheath
around the probe. The calculation of the sheath voltage waveform is based on the knowledge of
the probe current waveform, which can be measured: From the probe current (Ip) the temporal
derivative of electric field intensity at the probe surface can be acquired when the electron and ion
current components are subtracted from the total probe current in order to obtain the displacement
current
Id = ε0S
dE
dt
= Ip − Ii −
en0S
4
8kTe
πm
eeUsh/kTe
, (6)
where S denotes the probe surface area. Since the applied frequency is significantly higher than
the ion plasma frequency, the ion current (Ii) is expected to be constant during the whole RF
period. In order to calculate the sheath voltage Ush from the waveform of the electric field intensity,
which can be obtained by eq. (6), it is necessary to develop a theoretical model of the sheath.
When such a model is made and solved, both the sheath and plasma potential waveforms can be
obtained at the requirement that we know the value of the ion current (Ii) and the initial value of
the electric field intensity. Fortunately, these two unknown initial conditions can be determined
from the following two requirements: the obtained waveform of the plasma potential is periodical
Laboratory guides (verze 5. června 2018) 3
and its mean value is equal to the real value that can be measured by means of RF-compensated
Langmuir probe. The outlined calculation of plasma potential waveform can be realized by means
of the matlab function “plasma_potential.m”, which can be downloaded from the study materials
of the course.
Experimental
The measurement will be carried out in a spherical stainless-steel reactor with two round electrodes,
each of them with diameter 8 cm. The upper electrode is powered with RF voltage (frequency
13.56 MHz), the bottom electrode is grounded. Capacitively coupled discharge will be ignited in
nitrogen, oxygen and their mixtures in the pressure range 1 – 100 Pa.
Electron concentration, temperature and distribution function together with the mean (DC)
value of plasma potential can be measured with RF compensated Langmuir probe. This probe
blocks the RF signals and, therefore, can not be used for measurements of RF components of
plasma potential.
RF components of plasma potential can be measured with a simple probe made from a wire
that is directly connected to a high-impedance probe of an oscilloscope. The probe length and
diameter are 20.9 mm and 0.27 mm, respectively. The electric schema of the probe, sheath around
the probe (which can transmit displacement, electron and ion current) and the probe connection
to the oscilloscope (the input impedance of the oscilloscope is described by 108 Ω and 3 pF) is
shown in the fig. 2. The probe current can be calculated from measured probe voltage by means
of the known input impedance of the oscilloscope.
References
[1] U. Czarnetzki, T. Mussenbrock, R.P. Brinkmann: Physics of Plasmas 13 (2006), 123503.
[2] P. Dvořák: Plasma Sources Sci. Technol. 22 (2013), 045016.
[3] P. Dvořák, M. Tkáčik, J Bém: Plasma Sources Sci. Technol. 26 (2017), 055022.