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.
History The technique of using a probing electrode to explore gas discharges predates Langmuir; William Crookes employed such an electrode, which he termed an "idle pole", to explore the distribution of potential in a gas discharge. In 1923-1924, Irving Langmuir and Harold M. Mott-Smith invented the diagnostic method for measuring both temperature and density with an electrostatic probe. Langmuir's research was driven by the post-WWI industrial demands for reliable electronic devices, and especially improved vacuum tubes. Following the initial invention in the early 1920s, significant refinements to sheath theory emerged in the 1930s and 1940s through collaborative work between Lewi Tonks and Langmuir.
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 ( n i {\displaystyle n_{i}} ), the ion temperature T i {\displaystyle T_{i}} , or the electron energy distribution function (EEDF) or f e ( v ) {\displaystyle f_{e}(v)} .
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
c s = k B ( Z T e + γ i T i ) / m i {\displaystyle c_{s}={\sqrt {k_{B}(ZT_{e}+\gamma _{i}T_{i})/m_{i}}}} . The ion temperature term is often neglected, which is justified if the ions are cold. Z is the (average) charge state of the ions, and γ i {\displaystyle \gamma _{i}} is the adiabatic coefficient for the ions. The proper choice of γ i {\displaystyle \gamma _{i}} is a matter of some contention. Most analyses use γ i = 1 {\displaystyle \gamma _{i}=1} , corresponding to isothermal ions, but some kinetic theory suggests that γ i = 3 {\displaystyle \gamma _{i}=3} . For Z = 1 {\displaystyle Z=1} and T i = T e {\displaystyle T_{i}=T_{e}} , using the larger value results in the conclusion that the density is 2 {\displaystyle {\sqrt {2}}} 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 quasineutrality allows one to write it simply in terms of the electron density as q e n e {\displaystyle q_{e}n_{e}} , where q e {\displaystyle q_{e}} is the charge of an electron and n e {\displaystyle n_{e}} 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
j i m a x = q e n e c s {\displaystyle j_{i}^{max}=q_{e}n_{e}c_{s}} where c s {\displaystyle c_{s}} 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, i.e.,
f ( v x ) d v x ∝ e − 1 2 m e v x 2 / k B T e {\displaystyle f(v_{x})\,dv_{x}\propto e^{-{\frac {1}{2}}m_{e}v_{x}^{2}/k_{B}T_{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
⟨ v e ⟩ = ∫ v e 0 ∞ f ( v x ) v x d v x ∫ − ∞ ∞ f ( v x ) d v x {\displaystyle \langle v_{e}\rangle ={\frac {\int _{v_{e0}}^{\infty }f(v_{x})\,v_{x}\,dv_{x}}{\int _{-\infty }^{\infty }f(v_{x})\,dv_{x}}}} , where the cut-off velocity for the upper integral is
v e 0 = 2 q e Δ V / m e {\displaystyle v_{e0}={\sqrt {2q_{e}\Delta V/m_{e}}}} .
Δ V {\displaystyle \Delta V} is the voltage 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
⟨ v e ⟩ = k B T e 2 π m e e − q e Δ V / k B T e {\displaystyle \langle v_{e}\rangle ={\sqrt {\frac {k_{B}T_{e}}{2\pi m_{e}}}}\,e^{-q_{e}\Delta V/k_{B}T_{e}}} . With this expression, we can write the electron contribution to the current to the probe in terms of the ion saturation current as
j e = j i m a x m i / 2 π m e e − q e Δ V / k B T e {\displaystyle j_{e}=j_{i}^{max}{\sqrt {m_{i}/2\pi m_{e}}}\,e^{-q_{e}\Delta V/k_{B}T_{e}}} , 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:
j = j i m a x ( − 1 + m i / 2 π m e e − q e Δ V / k B T e ) {\displaystyle j=j_{i}^{max}\left(-1+{\sqrt {m_{i}/2\pi m_{e}}}\,e^{-q_{e}\Delta V/k_{B}T_{e}}\right)} . 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
Δ V = ( k B T e / q e ) ( 1 / 2 ) ln ( m i / 2 π m e ) {\displaystyle \Delta V=(k_{B}T_{e}/q_{e})\,(1/2)\ln(m_{i}/2\pi m_{e})} . If we introduce the ion reduced mass μ i = m i / m e {\displaystyle \mu _{i}=m_{i}/m_{e}} , we can write
Δ V = ( k B T e / q e ) ( 2.8 + 0.5 ln μ i ) {\displaystyle \Delta V=(k_{B}T_{e}/q_{e})\,(2.8+0.5\ln \mu _{i})}
Since the floating potential is the experimentally accessible quantity, the current (below electron saturation) is usually written as
j = j i m a x ( − 1 + e q e ( V 0 − Δ V ) / k B T e ) {\displaystyle j=j_{i}^{max}\left(-1+\,e^{q_{e}(V_{0}-\Delta V)/k_{B}T_{e}}\right)} .
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 v e 0 = 0 {\displaystyle v_{e0}=0} and setting the ion current to zero, the electron saturation current density would be
j e m a x = j i m a x m i / π m e = j i m a x ( 24.2 μ i ) {\displaystyle j_{e}^{max}=j_{i}^{max}{\sqrt {m_{i}/\pi m_{e}}}=j_{i}^{max}\left(24.2\,{\sqrt {\mu _{i}}}\right)}
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
j e m a x = q e n e k B ( γ e T e + T i ) / m e = j i m a x m i / m e = j i m a x ( 42.8 μ i ) {\displaystyle j_{e}^{max}=q_{e}n_{e}{\sqrt {k_{B}(\gamma _{e}T_{e}+T_{i})/m_{e}}}=j_{i}^{max}{\sqrt {m_{i}/m_{e}}}=j_{i}^{max}\left(42.8\,{\sqrt {\mu _{i}}}\right)}
where the numerical value was found by taking Ti=Te and γi=γe. 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)
Φ p r e = 1 2 m i c s 2 Z e = k B ( T e + Z γ i T i ) / ( 2 Z e ) {\displaystyle \Phi _{pre}={\frac {{\frac {1}{2}}m_{i}c_{s}^{2}}{Ze}}=k_{B}(T_{e}+Z\gamma _{i}T_{i})/(2Ze)}
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 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 A e f f {\displaystyle A_{eff}} (which may be a function of the bias voltage) and make the assumptions
T i = T e {\displaystyle T_{i}=T_{e}} ,
Z = 1 {\displaystyle Z=1}
γ i = 3 {\displaystyle \gamma _{i}=3} , and
n e , s h = 0.5 n e {\displaystyle n_{e,sh}=0.5\,n_{e}} , and ignore the effects of
bulk resistivity, and electron saturation, then the I-V characteristic becomes
I = I i m a x ( − 1 + e q e ( V p r − V f l ) / ( k B T e ) ) {\displaystyle I=I_{i}^{max}(-1+e^{q_{e}(V_{pr}-V_{fl})/(k_{B}T_{e})})} , where
I i m a x = q e n e k B T e / m i A e f f {\displaystyle I_{i}^{max}=q_{e}n_{e}{\sqrt {k_{B}T_{e}/m_{i}}}\,A_{eff}} .
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 V b i a s {\displaystyle V_{bias}} is the voltage applied between two identical electrodes, the current is given by;
I = I i m a x ( − 1 + e q e ( V 2 − V f l ) / k B T e ) = − I i m a x ( − 1 + e q e ( V 1 − V f l ) / k B T e ) {\displaystyle I=I_{i}^{max}\left(-1+\,e^{q_{e}(V_{2}-V_{fl})/k_{B}T_{e}}\right)=-I_{i}^{max}\left(-1+\,e^{q_{e}(V_{1}-V_{fl})/k_{B}T_{e}}\right)} , which can be rewritten using V b i a s = V 2 − V 1 {\displaystyle V_{bias}=V_{2}-V_{1}} as a hyperbolic tangent:
I = I i m a x tanh ( 1 2 q e V b i a s k B T e ) {\displaystyle I=I_{i}^{max}\tanh \left({\frac {1}{2}}\,{\frac {q_{e}V_{bias}}{k_{B}T_{e}}}\right)} . 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 A 1 {\displaystyle A_{1}} is the area of the larger tip then:
I = A 1 J i m a x [ coth ( q e V b i a s 2 k B T e ) + ( A 1 A 2 − 1 ) e − q e V b