Langmuir Probe

1. Theory

• Definition: A Langmuir probe is a small electrode inserted into a plasma whose current-voltage curve gives electron temperature, plasma density, floating and space potentials and the electron energy distribution (Chen notes; UCI manual).
• Three regions: Ion saturation at negative bias, an exponential electron-retardation region, and electron saturation above the space potential (Chen notes).
• Sheath: A space-charge sheath of a few Debye lengths surrounds the probe and sets which theory applies (Chen notes; background knowledge, Lieberman and Lichtenberg).
• Bohm criterion: Ions enter the sheath at the ion sound speed, so ion current gives density once $T_e$ is known (Chen notes).

2. Working Principle

• Probe: Tungsten or carbon tip of 0.1 to 1 $\mathrm{mm}$ diameter and 2 to 10 $\mathrm{mm}$ exposed length in a thin alumina tube (Chen notes).
• Sweep: The bias is swept and current read across a resistor of about 50 $\Omega$ (Chen notes).
• RF compensation: Resonant chokes and an auxiliary electrode make the probe follow RF swings of the space potential (Chen notes).
• Analysis: Ion-line subtraction, semilog fit, Bohm density and derivative methods (Chen notes; UCI manual).

3. Applications

• Process plasma characterization: Density and $T_e$ maps in etch and deposition reactors (Chen notes; background knowledge).
• Chamber matching: Comparing plasma parameters between tools (background knowledge).
• Model validation: Inputs for actinometry and sheath models (Optical Emission Spectroscopy Endpoint (OES) page).
• Teaching labs: Argon RF discharges near $6\times 10^{-4}$ $\mathrm{torr}$ (UCI manual).

Current-Voltage Characteristic

Name	Formula	Related Formulae	Variables/Constants
Electron retardation region	$I_e=I_{es}\exp \left[\frac{e\left(V_p-V_s\right)}{k_B{T}_e}\right]$	$T_{eV}={\left(\frac{d\ln I_e}{d{V}_p}\right)}^{-1}$	$I_e:electroncurrent$ $I_{es}:electronsaturationcurrent$ $V_p:probevoltage$ $V_s:spacepotential$ $\begin{array}{r}{T}_{eV}:electrontemperatureinvolts;Chen\\ notes:semilogslopegives{T}_e,independent\\ ofprobearea,shapeandcollisions,ion\\ currentsubtractedfirstbyextrapolating\\ through{I}_{sat},exampleslope0.335giving\\ 2.99\mathrm{eV};UCImanual:ionlinesubtracted\\ andregionIVfittedtoaMaxwellian\end{array}$ $(computed:1/0.335=2.99\mathrm{eV})$
Electron saturation current	$I_{es}=enA{\left(\frac{k_B{T}_e}{2\pi m}\right)}^{1/2}$	$\frac{I_{es}}{I_{sat}}=\frac{1}{\alpha }{\left(\frac{M}{2\pi m}\right)}^{1/2}$	$n:plasmadensity$ $A:probearea$ $m,M:electronandionmass$ $\begin{array}{r}\alpha :Bohmcoefficient;Chennotes:I_{es}more\\ than200times{I}_{sat}inargonfor\\ unmagnetizedlow-pressuredischarges,10\\ to20timesinmagneticfields,knee\\ roundedbycollisionsandRF,draw{I}_{es}for\\ nomorethanafewmilliseconds,density\\ from{I}_{es}discouraged\end{array}$ $\begin{array}{r}(computed:215forargonwith\alpha =0.5and\\ 176with\alpha =0.61)\end{array}$
Ion saturation (Bohm) current	$I_{sat}=\alpha neA{c}_s,c_s={\left(\frac{k_B{T}_e}{M}\right)}^{1/2}$	$n=\frac{I_{sat}}{\alpha eA{c}_s}$	$\begin{array}{r}{c}_s:Bohm(ionsound)speed;Chennotes:\alpha =\\ exp(-1/2)=0.61intheory,about0.5as\\ anapproximation,0.6to0.7from\\ microwavecalibration,densityuncertain\\ byabout10percent;UCImanual:n=\\ 2I/(qvA)withv=(k_B{T}_e/M{)}^{1/2},currentread\\ three{T}_{e}belowthestartofionsaturation\end{array}$ $\begin{array}{r}(assumed:argon,T_e=3\mathrm{eV},n=1\times 10^{11}\mathrm{c}{\mathrm{m}}^{-3},\\ 0.25mmby5mmtip;computed:c_s=2690\\ m/s,2.63mA/c{\mathrm{m}}^2,0.10\mathrm{mA}and5.2\mathrm{mV}across\\ 50\Omega )\end{array}$

Potentials and Sheath

Name	Formula	Related Formulae	Variables/Constants
Floating potential	$V_f-V_s=-\frac{k_B{T}_e}{e}\left[\frac{1}{2}\ln \left(\frac{M}{2\pi m}\right)-\ln \alpha \right]$	$V_f-V_s\approx -\frac{k_B{T}_e}{2e}\ln \frac{M}{2\pi m}\text{(notes’ Eq. 5)}$	$\begin{array}{r}{V}_f:floatingpotential;Chennotes:Eq.5\\ aswritten,quotedvalues-3.5T_{eV}for\\ hydrogenand-5.4T_{eV}forargon,about-4\\ to-5T_{eV}forcylindricalprobes,more\\ negativewithfastelectronsor\\ uncompensatedRF;UCImanual:floating\\ potentialoffsetfromzerobecauseofthe\\ ion-electronmassratio;fullbalancewith\\ \alpha (backgroundknowledge,Liebermanand\\ Lichtenberg)\end{array}$ $\begin{array}{r}(computed:argon4.68,5.17and5.37T_{eV}\\ forno\alpha ,\alpha =0.61and\alpha =0.5;hydrogen\\ 2.84,3.33and3.53)\end{array}$
Space potential and Debye length	${\lambda }_D={\left(\frac{{\epsilon }_0{k}_B{T}_e}{n{e}^2}\right)}^{1/2},{\xi }_p=\frac{R_p}{{\lambda }_D}$	$V_s:\max I_e^{\prime }\left(V\right)\text{or}{I}_e^{\prime \prime }=0$	${\lambda }_D:Debyelength$ ${\xi }_p:proberadiusinDebyelengths$ $\begin{array}{r}{R}_p:proberadius;Chennotes:V_{s}fromthe\\ kneeintersection,thepeakof{I}_e^{\prime },the\\ zeroof{I}_e^{\prime \prime },oranemissiveprobewith{V}_f\\ closeto{V}_s;V_{s}about5k_B{T}_{e}withgrounded\\ walls;examplen=4\times 10^{12}\mathrm{c}{\mathrm{m}}^{-3}at3\mathrm{eV}with{\xi }_p\\ about20;thinsheathaboveabout1\times 10^{11}\mathrm{c}{\mathrm{m}}^{-3}\end{array}$ $\begin{array}{r}(computed:{\lambda }_D=6.4\mu \mathrm{m}at4\times 10^{12}and41\mu \mathrm{m}at\\ 1\times 10^{11}\mathrm{c}{\mathrm{m}}^{-3};{\xi }_p=20means{R}_p=129\mu \mathrm{m};V_{s}about\\ 15\mathrm{V}at3\mathrm{eV})\end{array}$
Child-Langmuir sheath	$J=\frac{4{\epsilon }_0}{9}{\left(\frac{2e}{M}\right)}^{1/2}\frac{V^{3/2}}{d^2}$	$d\propto V^{3/4}$	$J:ioncurrentdensity$ $V:sheathvoltage$ $\begin{array}{r}d:sheaththickness;Chennotes:\\ Child-Langmuirlawasreconstructed,\\ sheaththicknessscalingwith{V}^{3/4}inthe\\ floating-potentialmethod,not\\ theoreticallyjustifiedforthese\\ geometries\end{array}$ $\begin{array}{r}(myreading:athickersheathatmore\\ negativebiasenlargesthecollectingarea\\ ofathinwire,whichiswhytheion\\ branchdoesnotsaturateflat)\end{array}$

Distribution, Theory Limits and RF

Name	Formula	Related Formulae	Variables/Constants
Druyvesteyn EEDF	$f\left(\epsilon \right)\propto \frac{d^2{I}_e}{d{V}_p^2}$	$\text{dither }{V}_p\Rightarrow d{I}_e/d{V}_p\text{directly}$	$\begin{array}{r}f(\epsilon ):electronenergydistribution;Chen\\ notes:validforisotropicdistributions\\ andconvexprobes,secondderivatives\\ noisy,low-frequencyditheringofthe\\ probevoltagegivesthefirstderivative\\ soonlyonefurtherderivativeisneeded\end{array}$ $\begin{array}{r}(myreading:anon-Maxwelliantailshows\\ ascurvatureinthesemilogplot,which\\ thesecondderivativeresolves)\end{array}$
Orbital-motion-limited current	$I^2\propto V_p$	$\text{valid if sheath}\gg R_p$	$\begin{array}{r}OML:orbitalmotionlimitforcylindrical\\ probes;Chennotes:currentscalingwith\\ thesquarerootofvoltage,Isquared\\ linearin{V}_p,validatlowdensitywitha\\ sheathmuchlargerthantheprobe,often\\ wronglyassumedathigherdensity\end{array}$ $\begin{array}{r}(computed:at1\times 10^{11}\mathrm{c}{\mathrm{m}}^{-3}the41\mu \mathrm{m}Debye\\ lengthisathirdofa129\mu \mathrm{m}radius,so\\ OMLismarginal)\end{array}$
RF compensation	$Z_{ck}\gg \left|Z_{sh}\right|,Z=\frac{1}{\omega C}$	$\left|e\delta V_s\right|\lesssim k_B{T}_e$	$Z_{ck}:chokeimpedance$ $\begin{array}{r}{Z}_{sh}:sheathimpedanceofprobeand\\ auxiliaryelectrode\end{array}$ $\begin{array}{r}\delta V_s:RFfluctuationofthespace\\ potential;Chennotes:uncompensated\\ probesgivefalselyhigh{T}_{e}andmore\\ negative{V}_f,designwith{V}_{rf}/T_e=30,chokes\\ of250\mathrm{k\Omega }at2\mathrm{MHz},a2squarecm\\ auxiliaryelectrode,chokesresonantat\omega \\ and2\omega ,1cmofwireabout0.25\mathrm{pF}giving\\ about330\mathrm{k\Omega }at2\mathrm{MHz}\end{array}$ $\begin{array}{r}(computed:0.25\mathrm{pF}is318\mathrm{k\Omega }at2\mathrm{MHz}and\\ 47\mathrm{k\Omega }at13.56\mathrm{MHz})\end{array}$

Important Questions

Ques: How is electron temperature read from a probe curve?

From the slope of the semilog electron current. Below the space potential, electrons are repelled by the probe and the collected current falls exponentially, $I_e=I_{es}\exp [e(V_p-V_s)/k_B{T}_e]$, so a plot of the logarithm of current against voltage is straight for a Maxwellian, with slope $1/T_{eV}$ (Chen notes). The ion current is first subtracted by extrapolating a line through the ion-saturation branch, as both sources describe. Chen’s example slope of 0.335 gives 2.99 $\mathrm{eV}$ (computed). He notes that this is the easiest quantity to obtain because it does not depend on probe area, shape or collisions. Curvature of the semilog plot indicates a non-Maxwellian distribution, which the Druyvesteyn second derivative resolves. Reference: F. F. Chen, ‘Lecture Notes on Langmuir Probe Diagnostics’, mini-course, IEEE ICOPS, Jeju, Korea (2003) (fetch-tool summary; some symbols garbled in extraction); ‘Plasma Physics Advanced Lab’ manual, Rev. 2022 (physics.uci.edu advanced lab; fetch-tool summary).

Ques: Should the Bohm coefficient be 0.5 or 0.61?

The sources differ. The UCI manual uses $n=2I/(qvA)$, a coefficient of 0.5, and Chen gives 0.5 as an approximation, but the theoretical value is $e^{-1/2}=0.61$ and microwave-interferometer calibration gave 0.6 to 0.7 (Chen notes). The calibrated range makes 0.61 more likely. Using 0.5 instead overestimates density by 22 per cent (computed). For argon at an assumed 3 $\mathrm{eV}$ and $1\times 10^{11}$ $\mathrm{c}{\mathrm{m}}^{-3}$, the ion sound speed is 2690 m/s and the current density with 0.61 is 2.63 $mA/c{\mathrm{m}}^2$, so a 0.25 by 5 mm wire collects 0.10 $\mathrm{mA}$, 5.2 $\mathrm{mV}$ across 50 $\Omega$ (assumed; computed). Chen adds that density is uncertain by about 10 per cent even on a good day. Reference: ‘Plasma Physics Advanced Lab’ manual, Rev. 2022 (physics.uci.edu advanced lab; fetch-tool summary); F. F. Chen, ‘Lecture Notes on Langmuir Probe Diagnostics’, mini-course, IEEE ICOPS, Jeju, Korea (2003) (fetch-tool summary; some symbols garbled in extraction).

Ques: Why do sources quote -5.4 Te or about -5 Te for the argon floating potential?

Because they use different ion-current coefficients. Equating ion and electron currents gives $V_f-V_s=-T_{eV}[\frac{1}{2}\ln (M/2\pi m)-\ln \alpha ]$ (background knowledge, Lieberman and Lichtenberg). For argon this is 4.68 $T_{eV}$ without $\alpha$, 5.17 with 0.61 and 5.37 with 0.5 (computed); for hydrogen 2.84, 3.33 and 3.53. Chen’s quoted -5.4 and -3.5 match the $\alpha$ = 0.5 values, while his Eq. 5 as extracted omits $\alpha$. The 0.61 value of about -5.2 $T_{eV}$ is more likely correct for a planar sheath, and cylindrical probes give about -4 to -5 (Chen notes). The same coefficient explains his statement that electron saturation exceeds 200 times ion saturation in argon: 215 with 0.5, 176 with 0.61 (computed). Reference: M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed., Wiley 2005 (background knowledge; not opened); F. F. Chen, ‘Lecture Notes on Langmuir Probe Diagnostics’, mini-course, IEEE ICOPS, Jeju, Korea (2003) (fetch-tool summary; some symbols garbled in extraction).

Ques: How is the space potential found?

From the knee of the curve, its derivatives, or an emissive probe. The traditional method intersects straight lines through the retardation and electron-saturation regions; alternatives take the peak of the first derivative or the zero of the second (Chen notes). Collisions, magnetic fields and RF round the knee, so these estimates can disagree, and Chen calls a hot emissive probe, whose floating potential approaches the space potential, the only sure method. With grounded metal walls the space potential is about 5 $k_B{T}_e$, roughly 15 $\mathrm{V}$ at 3 $\mathrm{eV}$ (computed). Electron saturation current should be drawn only for a few milliseconds because it can damage the probe. Reference: F. F. Chen, ‘Lecture Notes on Langmuir Probe Diagnostics’, mini-course, IEEE ICOPS, Jeju, Korea (2003) (fetch-tool summary; some symbols garbled in extraction).

Ques: Why must a probe be RF compensated?

Because RF oscillation of the space potential smears the exponential region. An uncompensated probe averages the current over the swing, giving a falsely high $T_e$ and a more negative floating potential (Chen notes). Compensation makes the probe tip follow the RF: high-impedance chokes resonant at the drive frequency and its second harmonic isolate the tip from the measuring circuit, and an auxiliary electrode lowers the sheath impedance. Chen’s design example uses 250 $\mathrm{k\Omega }$ chokes at 2 $\mathrm{MHz}$ and a 2 square cm auxiliary electrode for an RF amplitude of 30 $T_e$. Stray capacitance limits this: 0.25 $\mathrm{pF}$ is 318 $\mathrm{k\Omega }$ at 2 $\mathrm{MHz}$ but only 47 $\mathrm{k\Omega }$ at 13.56 $\mathrm{MHz}$ (computed). If only density is needed, biasing in ion saturation avoids the problem. Reference: F. F. Chen, ‘Lecture Notes on Langmuir Probe Diagnostics’, mini-course, IEEE ICOPS, Jeju, Korea (2003) (fetch-tool summary; some symbols garbled in extraction).

Ques: When is orbital-motion-limited theory valid?

When the sheath is much larger than the probe. In the OML limit, ions orbit the probe and the current of a cylinder grows as the square root of voltage, so $I^2$ is linear in bias (Chen notes). This needs low density: at $1\times 10^{11}$ $\mathrm{c}{\mathrm{m}}^{-3}$ and 3 $\mathrm{eV}$ the Debye length is 41 $\mu \mathrm{m}$, about a third of a 129 $\mu \mathrm{m}$ probe radius, so OML is already marginal, and at $4\times 10^{12}$ $\mathrm{c}{\mathrm{m}}^{-3}$ it is 6.4 $\mu \mathrm{m}$ (computed). Chen’s example of $4\times 10^{12}$ $\mathrm{c}{\mathrm{m}}^{-3}$ corresponds to a probe radius of about 20 Debye lengths, a thin-sheath case, and he warns that $I^2$ proportional to $V_p$ at high density is often mistaken for orbital motion. In that regime sheath-expansion models such as Child-Langmuir scaling are used instead. Reference: F. F. Chen, ‘Lecture Notes on Langmuir Probe Diagnostics’, mini-course, IEEE ICOPS, Jeju, Korea (2003) (fetch-tool summary; some symbols garbled in extraction).

References

1. F. F. Chen, “Lecture Notes on Langmuir Probe Diagnostics,” mini-course, IEEE International Conference on Plasma Science (ICOPS), Jeju, Korea, 2003. Available: https://www.seas.ucla.edu/~ffchen/Publs/Chen210R.pdf
2. “Plasma Physics Advanced Lab,” laboratory manual, Rev. 2022. Available: https://www.physics.uci.edu/~advanlab/plasma.pdf
3. M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed., Wiley, 2005 (cited from background knowledge).

Related pages in this vault: Optical Emission Spectroscopy Endpoint (OES), SMU and Probe Station, Kelvin Probe Force Microscope (KPFM), Hydrodynamic and Energy-Balance Transport, Electrostatics and Capacitance, Vector Network Analyzer (VNA), Residual Gas Analyzer (RGA), Vacuum Gauges, Plasma and Radical Oxidation Tool, Plasma Immersion Ion Implanter, DC and RF Magnetron Sputter System, PECVD Reactor, Parallel-Plate Reactive Ion Etcher (RIE), Inductively Coupled Plasma Etcher (ICP-RIE).