I
Ua
UU plfl
electrons
ions
Figure 1: VA characteristics of a cylindrical Langmuir
probe.
s
ra
r
v
v1
2
v
α
Figure 2: Trajectory of a charged
particle in a sheath around a probe.
1 Langmuir probe
Langmuir probe is a small conductor inserted into plasma, whose VA characteristics can be used
for determination of some plasma parameters, e.g. concentration and temperature of electrons,
plasma potential and electron energy distribution function. We will summarize here the results
of the probe theory, which calculates probe current by study of trajectories of charged particles
in a collisionless sheath around the probe, determination of the part of the velocity-space that
contains particles that will reach the probe and calculation of the probe current by integration of
radial velocity component and velocity distribution function over this part of the velocity-space,
i.e.
I = qSs
C
v1 g(v) d3
v, (1)
where q is the charge of a particle, Ss is the area of the plasma-sheath boundary, g is the velocity
distribution function. The following text will show formulas obtained for a cylindrical probe.
1.1 Particles repelled from the probe
(i.e. electrons for Ua < Upl and positive ions for Ua > Upl) Particles can reach the probe surface
only if their kinetic energy is sufficiently high
1
2
mv2
≥ q(Ua − Upl)
and if they enter the sheath with sufficiently small angle α
sin2
α ≤
r2
a
r2
s
1 −
2q (Ua − Upl)
mv2
.
1
By integration of the flux of repelled particles over this part of the velocity space we obtain the
VA characteristics
I = qS
1
2
√
2m
∞
−qU
E + qU
√
E
f(E) dE, (2)
where U = Upl − Ua, E is the kinetic energy, f(E) is the energy distribution function and S is
the surface area of the probe. This formula is valid also for planar and spherical probes. If the
studied species have the Maxwell energy distribution function, the formula (2) can be simplified
to
I = qS
1
4
n¯v exp
qU
kT
(3)
¯v =
8kT
πm
The distribution function can be determined from (2) by means of the so-called Druyvesteyn
formula
f(qU) =
2 2m|U|
q5/2S
d2I
d|U|2
(4)
1.2 Particles attracted to the probe
The calculation of electric current carried by species attracted to the probe is more complicated
and in general depends on the dimensions of the sheath around the probe. If the species have
Maxwell EDF, the current flowing to the cylindrical probe can be calculated as
I =
1
4
Sqn¯v
rs
ra
erf
r2
a
r2
s − r2
a
qU
kT
+ 1 − erf
r2
s
r2
s − r2
a
qU
kT
exp
qU
kT
, (5)
where
erf x =
2
√
π
x
0
e−t2
dt,
which can be for rs ra, (so-called OML, orbital motion limited theory) simplified to
I ≈
1
4
Sqn¯v 1 +
qU
kT
. (6)
In practice, thanks to the OML theory the electron current for Ua Upl can be described
by
Ie =
1
4
Sqn¯v 1 +
e (Ua − Upl)
kTe
(7)
and the positive ion current for Ua Ufl by
Ii = Ii0 1 +
e (Upl − Ua)
kTe
κ
. (8)
2
I
Ua
UU plfl
Figure 3: VA characteristics of an (unheated)
Langmuir probe.
I
Ua
UU plfl
Figure 4: VA characteristics of an emissive
probe.
T
U
U
pl
fl
Figure 5: Dependence of the floating potential of an emissive probe on the probe temperature.
2 Emissive probe
Heating of probe results in thermoemission of electrons, which shifts the floating potential to
more positive values. The floating potential will grow when probe temperature will be increased.
But, because emitted electrons can not escape from probe to plasma for Ua > Upl, the floating
potential increase will stop at the value of plasma potential.
3
V oscilloscopeRo Co
probe
V oscilloscopeRo Co
C(U) Ie Ii
probe
sheath
plasma
plasma
sheathC
a) b)
Figure 6: Electric scheme of the uncompensated probe circuit: The real situation (a) and the
approximation of sheath by a capacitor (b).
3 Measurement of RF components
Langmuir probes usually contain RF compensation, which blocks RF probe currents in order
to prevent the RF components from distortion of the probe VA characteristics. Unfortunately,
the compensation usually disables measurement of RF components of plasma potential. Consequently,
the RF components can be measured by means of probes without any RF compensation,
which can be made as a simple wire connected to a high-impedance probe of an oscilloscope.
However, the measured probe potential waveform differs from the plasma potential waveform,
because the probe is separated from the plasma by a sheath. Therefore, it is necessary to
determine the waveform of the voltage on the sheath around the probe (U) from the measured
waveform of the probe current (I). The electric current flowing to the probe can be described
as a sum of the displacement (Id), electron (Ie) and ion (Ii) current
I = Id + Ie + Ii (9)
Id = Sε0
dE
dt
(10)
Ie = −
enS
4
8kTe
πme
exp −
eU
kTe
, (11)
where S denotes the surface area of the probe, ε0 the vacuum permitivity, E the electric field
intensity at the probe surface, e the elementary charge, n the electron concentration in the bulk
plasma, k the Boltzmann constant, Te the electron temperature and me the electron mass. The
ion current can be assumed to be constant during the whole discharge period, since the ion
plasma frequency is usually significantly smaller than the frequency of the applied electric field.
The total probe current (I) can be easily obtained from the measured probe voltage waveform
and from the known input impedance of the oscilloscope. When the values of ion current (Ii),
sheath voltage in one moment of the RF period (U(t = 0)), electron concentration (n) and
electron temperature (Te) are known and when the relation between electric field intensity (E)
and sheath voltage (U) is defined (see fig. 7), it is possible to use the equations (9) – (11) for
numerical calculation of the temporal evolution of the sheath voltage: From known values of I,
Ii and Ie in the initial (t = 0) moment, the actual displacement current and the actual temporal
4
-400 -300 -200 -100 0 100 200
E [kV/m]
-5
0
5
10
15
20
25
30
U[V]
Figure 7: An example of the relation between
electric field intensity at the probe surface and
sheath voltage.
probe
probe
probe
U
(t)
U
Figure 8: Scheme of the numerical calculation
of the plasma potential.
derivative of electric field intensity are calculated. The temporal derivative of electric field
intensity is converted to the temporal derivative of the sheath voltage. This derivative is used
for the determination of the sheath voltage in the forthcoming moment. Now it is possible to
calculate the value of electron current in this second moment and the described procedure can
be repeated until the whole period of the sheath voltage is acquired, see scheme 8. After that,
it is possible to find the plasma potential waveform as the sum of the probe voltage measured
by the oscilloscope and the sheath voltage
Upl = Ua + U. (12)
The values of ion current and initial sheath voltage can be found by means of two requirements:
First is the periodicity of the sheath voltage. The second claims that the mean value of plasma
potential waveform calculated by equations (9) – (11) must be equal to the value measured by
compensated Langmuir probe.
In some cases, the sheath around the probe can be modeled as a capacitor. In this approximation,
the electric circuit drawn in the fig. 6b is solved and each frequency component of
plasma potential (upl) is calculated according to
upl = ua 1 +
Co
C
−
i
ωCRo
, (13)
where ua are the frequency components of the probe voltage measured by the oscilloscope,
Co and Ro are the capacity and resistance of the probe to ground, respectively, and C is the
5
0 10 20 30 40 50 60 70
t [ns]
-5
0
5
10
15
20
25
30
35
Uo
,Upl
[V]
20 ×
plasma
high-Z probe
low-Z probe
Figure 9: Waveforms of plasma (black),
high-impedance probe (red) and low-impedance
probe (blue) potentials measured in a capacitively
coupled discharge ignited in argon at
6 Pa.
0 10 20 30 40 50 60 70
t [ns]
0
5
10
15
20
25
30
35
Upl
[V]
17 Pa
10 Pa
5 Pa
3.5 Pa
Figure 10: Plasma potential waveforms measured
in a nitrogen capacitively coupled discharge
at various pressures.
capacity of the sheath around the probe. In this laboratory work, the values Co = 3.9 pF,
R = 108 Ω and C = 0.65 pF can be used.
6