A Langmuir probe is a device used to determine the electron temperature, electron density, and electric potential of a
plasma. It works by inserting one or more electrodes into a plasma, with a constant or time-varying electric potential between the various electrodes or between them and the surrounding vessel. The measured currents and potentials in this system allow the determination of the physical properties of the plasma.
''I-V'' characteristic of the Debye sheath
The beginning of Langmuir probe theory is the
''I–V'' characteristic of the
Debye sheath, that is, the current density flowing to a surface in a plasma as a function of the voltage drop across the sheath. The analysis presented here indicates how the electron temperature, electron density, and plasma potential can be derived from the ''I–V'' characteristic. In some situations a more detailed analysis can yield information on the ion density (
), the ion temperature
, or the electron energy
distribution function (EEDF) or
.
Ion saturation current density
Consider first a surface biased to a large negative voltage. If the voltage is large enough, essentially all electrons (and any negative ions) will be repelled. The ion velocity will satisfy the
Bohm sheath criterion, which is, strictly speaking, an inequality, but which is usually marginally fulfilled. The Bohm criterion in its marginal form says that the ion velocity at the sheath edge is simply the sound speed given by
.
The ion temperature term is often neglected, which is justified if the ions are cold. Even if the ions are known to be warm, the ion temperature is usually not known, so it is usually assumed to be simply equal to the electron temperature. In that case, consideration of finite ion temperature only results in a small numerical factor. ''Z'' is the (average) charge state of the ions, and
is the adiabatic coefficient for the ions. The proper choice of
is a matter of some contention. Most analyses use
, corresponding to isothermal ions, but some kinetic theory suggests that
. For
and
, using the larger value results in the conclusion that the density is
times smaller. Uncertainties of this magnitude arise several places in the analysis of Langmuir probe data and are very difficult to resolve.
The charge density of the ions depends on the charge state ''Z'', but
allows one to write it simply in terms of the electron density as
, where
is the charge of an electron and
is the number density of electrons.
Using these results we have the current density to the surface due to the ions. The current density at large negative voltages is due solely to the ions and, except for possible sheath expansion effects, does not depend on the bias voltage, so it is
referred to as the ion saturation current density and is given by
where
is as defined above.
The plasma parameters, in particular, the density, are those at the sheath edge.
Exponential electron current
As the voltage of the Debye sheath is reduced, the more energetic electrons are able to overcome the potential barrier of the electrostatic sheath. We can model the electrons at the sheath edge with a
Maxwell–Boltzmann distribution
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and use ...
, i.e.,
,
except that the high energy tail moving away from the surface is missing, because only the lower energy electrons moving toward the surface are reflected. The higher energy electrons overcome the sheath potential and are absorbed. The mean velocity of the electrons which are able to overcome the voltage of the sheath is
,
where the cut-off velocity for the upper integral is
.
is the
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
across the Debye sheath, that is, the potential at the sheath edge minus the potential of the surface. For a large voltage compared to the electron temperature, the result is
.
With this expression, we can write the electron contribution to the current to the probe in terms of the ion saturation current as
,
valid as long as the electron current is not more than two or three times the ion current.
Floating potential
The total current, of course, is the sum of the ion and electron currents:
.
We are using the convention that current ''from'' the surface into the plasma is positive. An interesting and practical question is the potential of a surface to which no net current flows. It is easily seen from the above equation that
.
If we introduce the ion
reduced mass , we can write
Since the floating potential is the experimentally accessible quantity, the current (below electron saturation) is usually written as
.
Electron saturation current
When the electrode potential is equal to or greater than the plasma potential, then there is no longer a sheath to reflect electrons, and the electron current saturates. Using the Boltzmann expression for the mean electron velocity given above with
and setting the ion current to zero, the electron saturation current density would be
Although this is the expression usually given in theoretical discussions of Langmuir probes, the derivation is not rigorous and the experimental basis is weak. The theory of
double layers typically employs an expression analogous to the
Bohm criterion, but with the roles of electrons and ions reversed, namely
where the numerical value was found by taking ''T''
''i''=''T''
''e'' and γ
''i''=γ
''e''.
In practice, it is often difficult and usually considered uninformative to measure the electron saturation current experimentally. When it is measured, it is found to be highly variable and generally much lower (a factor of three or more) than the value given above. Often a clear saturation is not seen at all. Understanding electron saturation is one of the most important outstanding problems of Langmuir probe theory.
Effects of the bulk plasma
The Debye sheath theory explains the basic behavior of Langmuir probes, but is not complete. Merely inserting an object like a probe into a plasma changes the density, temperature, and potential at the sheath edge and perhaps everywhere. Changing the voltage on the probe will also, in general, change various plasma parameters. Such effects are less well understood than sheath physics, but they can at least in some cases be roughly accounted.
Pre-sheath
The Bohm criterion requires the ions to enter the Debye sheath at the sound speed. The potential drop that accelerates them to this speed is called the pre-sheath. It has a spatial scale that depends on the physics of the ion source but which is large compared to the Debye length and often of the order of the plasma dimensions. The magnitude of the potential drop is equal to (at least)
The acceleration of the ions also entails a decrease in the density, usually by a factor of about 2 depending on the details.
Resistivity
Collisions between ions and electrons will also affect the ''I-V'' characteristic of a Langmuir probe. When an electrode is biased to any voltage other than the floating potential, the current it draws must pass through the plasma, which has a finite resistivity. The resistivity and current path can be calculated with relative ease in an unmagnetized plasma. In a magnetized plasma, the problem is much more difficult. In either case, the effect is to add a voltage drop proportional to the current drawn, which
shears
Shears may refer to:
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* ...
the characteristic. The deviation from an exponential function is usually not possible to observe directly, so that the flattening of the characteristic is usually misinterpreted as a larger plasma temperature. Looking at it from the other side, any measured ''I-V'' characteristic can be interpreted as a hot plasma, where most of the voltage is dropped in the Debye sheath, or as a cold plasma, where most of the voltage is dropped in the bulk plasma. Without quantitative modeling of the bulk resistivity, Langmuir probes can only give an upper limit on the electron temperature.
Sheath expansion
It is not enough to know the current ''density'' as a function of bias voltage since it is the ''absolute'' current which is measured. In an unmagnetized plasma, the current-collecting area is usually taken to be the exposed surface area of the electrode. In a magnetized plasma, the projected area is taken, that is, the area of the electrode as viewed along the magnetic field. If the electrode is not shadowed by a wall or other nearby object, then the area must be doubled to account for current coming along the field from both sides. If the electrode dimensions are not small in comparison to the Debye length, then the size of the electrode is effectively increased in all directions by the sheath thickness. In a magnetized plasma, the electrode is sometimes assumed to be increased in a similar way by the ion
Larmor radius.
The finite Larmor radius allows some ions to reach the electrode that would have otherwise gone past it. The details of the effect have not been calculated in a fully self-consistent way.
If we refer to the probe area including these effects as
(which may be a function of the bias voltage) and make the assumptions
*
,
*
*
, and
*
,
and ignore the effects of
*bulk resistivity, and
*electron saturation,
then the ''I-V'' characteristic becomes
,
where
.
Magnetized plasmas
The theory of Langmuir probes is much more complex when the plasma is magnetized. The simplest extension of the unmagnetized case is simply to use the projected area rather than the surface area of the electrode. For a long cylinder far from other surfaces, this reduces the effective area by a factor of π/2 = 1.57. As mentioned before, it might be necessary to increase the radius by about the thermal ion Larmor radius, but not above the effective area for the unmagnetized case.
The use of the projected area seems to be closely tied with the existence of a magnetic sheath. Its scale is the ion Larmor radius at the sound speed, which is normally between the scales of the Debye sheath and the pre-sheath. The Bohm criterion for ions entering the magnetic sheath applies to the motion along the field, while at the entrance to the Debye sheath it applies to the motion normal to the surface. This results in a reduction of the density by the sine of the angle between the field and the surface. The associated increase in the Debye length must be taken into account when considering ion non-saturation due to sheath effects.
Especially interesting and difficult to understand is the role of cross-field currents. Naively, one would expect the current to be parallel to the magnetic field along a
flux tube. In many geometries, this flux tube will end at a surface in a distant part of the device, and this spot should itself exhibit an ''I-V'' characteristic. The net result would be the measurement of a double-probe characteristic; in other words, electron saturation current equal to the ion saturation current.
When this picture is considered in detail, it is seen that the flux tube must charge up and the surrounding plasma must spin around it. The current into or out of the flux tube must be associated with a force that slows down this spinning. Candidate forces are viscosity, friction with neutrals, and inertial forces associated with plasma flows, either steady or fluctuating. It is not known which force is strongest in practice, and in fact it is generally difficult to find any force that is powerful enough to explain the characteristics actually measured.
It is also likely that the magnetic field plays a decisive role in determining the level of electron saturation, but no quantitative theory is as yet available.
Electrode configurations
Once one has a theory of the ''I-V'' characteristic of an electrode, one can proceed to measure it and then fit the data with the theoretical curve to extract the plasma parameters. The straightforward way to do this is to sweep the voltage on a single electrode, but, for a number of reasons, configurations using multiple electrodes or exploring only a part of the characteristic are used in practice.
Single probe
The most straightforward way to measure the ''I-V'' characteristic of a plasma is with a single probe, consisting of one electrode biased with a voltage ramp relative to the vessel. The advantages are simplicity of the electrode and redundancy of information, i.e. one can check whether the ''I-V'' characteristic has the expected form. Potentially additional information can be extracted from details of the characteristic. The disadvantages are more complex biasing and measurement electronics and a poor time resolution. If fluctuations are present (as they always are) and the sweep is slower than the fluctuation frequency (as it usually is), then the ''I-V'' is the ''average'' current as a function of voltage, which may result in systematic errors if it is analyzed as though it were an instantaneous ''I-V''. The ideal situation is to sweep the voltage at a frequency above the fluctuation frequency but still below the ion cyclotron frequency. This, however, requires sophisticated electronics and a great deal of care.
Double probe
An electrode can be biased relative to a second electrode, rather than to the ground. The theory is similar to that of a single probe, except that the current is limited to the ion saturation current for both positive and negative voltages. In particular, if
is the voltage applied between two identical electrodes, the current is given by;
,
which can be rewritten using
as a
hyperbolic tangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
:
.
One advantage of the double probe is that neither electrode is ever very far above floating, so the theoretical uncertainties at large electron currents are avoided. If it is desired to sample more of the exponential electron portion of the characteristic, an asymmetric double probe may be used, with one electrode larger than the other. If the ratio of the collection areas is larger than the square root of the ion to electron mass ratio, then this arrangement is equivalent to the single tip probe. If the ratio of collection areas is not that big, then the characteristic will be in-between the symmetric double tip configuration and the single-tip configuration. If
is the area of the larger tip then:
Another advantage is that there is no reference to the vessel, so it is to some extent immune to the disturbances in a
radio frequency
Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the up ...
plasma. On the other hand, it shares the limitations of a single probe concerning complicated electronics and poor time resolution. In addition, the second electrode not only complicates the system, but it makes it susceptible to disturbance by gradients in the plasma.
Triple probe
An elegant electrode configuration is the triple probe,
consisting of two electrodes biased with a fixed voltage and a third which is floating. The bias voltage is chosen to be a few times the electron temperature so that the negative electrode draws the ion saturation current, which, like the floating potential, is directly measured. A common rule of thumb for this voltage bias is 3/e times the expected electron temperature. Because the biased tip configuration is floating, the positive probe can draw at most an electron current only equal in magnitude and opposite in polarity to the ion saturation current drawn by the negative probe, given by :
and as before the floating tip draws effectively no current:
.
Assuming that:
1.) The electron energy distribution in the plasma is Maxwellian,
2.) The mean free path of the electrons is greater than the ion sheath about the tips and larger than the probe radius, and
3.) the probe sheath sizes are much smaller than the probe separation,
then the current to any probe can be considered composed of two partsthe high energy tail of the Maxwellian electron distribution, and the ion saturation current:
where the current ''I
e'' is thermal current. Specifically,
,
where ''S'' is surface area, ''J
e'' is electron current density, and ''n
e'' is electron density.
Assuming that the ion and electron saturation current is the same for each probe, then the formulas for current to each of the probe tips take the form
.
It is then simple to show
but the relations from above specifying that ''I
+=-I
−'' and ''I
fl''=0 give
,
a transcendental equation in terms of applied and measured voltages and the unknown ''T
e'' that in the limit ''q
eV
Bias = q
e(V
+-V
−) >> k T
e'', becomes
.
That is, the voltage difference between the positive and floating electrodes is proportional to the electron temperature. (This was especially important in the sixties and seventies before sophisticated data processing became widely available.)
More sophisticated analysis of triple probe data can take into account such factors as incomplete saturation, non-saturation, unequal areas.
Triple probes have the advantage of simple biasing electronics (no sweeping required), simple data analysis, excellent time resolution, and insensitivity to potential fluctuations (whether imposed by an rf source or inherent fluctuations). Like double probes, they are sensitive to gradients in plasma parameters.
Special arrangements
Arrangements with four (tetra probe) or five (penta probe) have sometimes been used, but the advantage over triple probes has never been entirely convincing. The spacing between probes must be larger than the
Debye length of the plasma to prevent an overlapping
Debye sheath.
A pin-plate probe consists of a small electrode directly in front of a large electrode, the idea being that the voltage sweep of the large probe can perturb the plasma potential at the sheath edge and thereby aggravate the difficulty of interpreting the ''I-V'' characteristic. The floating potential of the small electrode can be used to correct for changes in potential at the sheath edge of the large probe. Experimental results from this arrangement look promising, but experimental complexity and residual difficulties in the interpretation have prevented this configuration from becoming standard.
Various geometries have been proposed for use as ion temperature probes, for example, two cylindrical tips that rotate past each other in a magnetized plasma. Since shadowing effects depend on the ion Larmor radius, the results can be interpreted in terms of ion temperature. The ion temperature is an important quantity that is very difficult to measure. Unfortunately, it is also very difficult to analyze such probes in a fully self-consistent way.
Emissive probes use an electrode heated either electrically or by the exposure to the plasma. When the electrode is biased more positive than the plasma potential, the emitted electrons are pulled back to the surface so the ''I''-''V'' characteristic is hardly changed. As soon as the electrode is biased negative with respect to the plasma potential, the emitted electrons are repelled and contribute a large negative current. The onset of this current or, more sensitively, the onset of a discrepancy between the characteristics of an unheated and a heated electrode, is a sensitive indicator of the plasma potential.
To measure fluctuations in plasma parameters, arrays of electrodes are used, usually onebut occasionally two-dimensional. A typical array has a spacing of 1 mm and a total of 16 or 32 electrodes. A simpler arrangement to measure fluctuations is a negatively biased electrode flanked by two floating electrodes. The ion-saturation current is taken as a surrogate for the density and the floating potential as a surrogate for the plasma potential. This allows a rough measurement of the turbulent particle flux
Cylindrical Langmuir probe in electron flow
Most often, the Langmuir probe is a small sized electrode inserted into a plasma which is connected to an external circuit that measures the properties of the plasma with respect to ground. The ground is typically an electrode with a large surface area and is usually in contact with the same plasma (very often the metallic wall of the chamber). This allows the probe to measure the
I-V characteristic
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of the plasma. The probe measures the characteristic current
of the plasma when the probe is biased with a potential
.
Relations between the probe
I-V characteristic
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and parameters of isotropic plasma were found by the
Irving Langmuir
Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry.
Langmuir's most famous publication is the 1919 ar ...
and they can be derived most elementary for the planar probe of a large surface area
(ignoring the edge effects problem). Let us choose the point
in plasma at the distance
from the probe surface where electric field of the probe is negligible while each electron of plasma passing this point could reach the probe surface without collisions with plasma components:
,
is the
Debye length and
is the electron free path calculated for its total
cross section with plasma components. In the vicinity of the point
we can imagine a small element of the surface area
parallel to the probe surface. The elementary current
of plasma electrons passing throughout
in a direction of the probe surface can be written in the form
where
is a scalar of the electron thermal velocity vector
,
is the element of the solid angle with its relative value
,
is the angle between perpendicular to the probe surface recalled from the point
and the radius-vector of the electron thermal velocity
forming a spherical layer of thickness
in velocity space, and
is the electron distribution function normalized to unity
Taking into account uniform conditions along the probe surface (boundaries are excluded),
, we can take double integral with respect to the angle
, and with respect to the velocity
, from the expression (), after substitution Eq. () in it, to calculate a total electron current on the probe
where
is the probe potential with respect to the potential of plasma
,
is the lowest electron velocity value at which the electron still could reach the probe surface charged to the potential
,
is the upper limit of the angle
at which the electron having initial velocity
can still reach the probe surface with a zero-value of its velocity at this surface. That means the value
is defined by the condition
Deriving the value
from Eq. () and substituting it in Eq. (), we can obtain the probe
I-V characteristic
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(neglecting the ion current) in the range of the probe potential