A Kinetic Theory of Planar Plasma Sheaths Surrounding ...

A Kinetic Theory of Planar Plasma Sheaths Surrounding

Electron Emitting SurfacesJ. P. Sheehan1, I. Kaganovich2,

E. Barnat3, B. Weatherford3, H. Wang2,D. Sydorenko4, N. Hershkowitz1, and Y. Raitses2

1 University of Wisconsin – Madison2 Princeton Plasma Physics Laboratory3 Sandia National Laboratories4 University of Alberta

DOE Plasma Science Center Teleseminar, December 7, 2012 2

Outline

● Fluid theory of emissive sheaths● Kinetic theory of emissive sheaths

● Electrons lost to surface● Temperature of emitted electrons

● Particle in cell simulations● Afterglow of capacitively coupled plasma● Measurements of emissive sheath versus time● Conclusions


Emitted electrons reduce sheath potential and electric field at surface

● Three species

● Plasma electrons● Plasma ions● Emitted electrons

● Plasma fills the - half-plane

● One dimensional

● Emitted electrons reduce:

● The electric field at the surface

● The floating sheath potential

x̂Collecting Sheath

Emissive Sheath


Fluid Theory: a SCL emitting surface floats Tep below the plasma potential

● Collisionless

● Φ = 0 at the sheath edge (definition)

● Plasma electrons

● Maxwellian (temperature Tep)

● Boltzmann relation

● Emitted Electrons

● Zero energy at surface

● Plasma Ions

● One species

● Cold (Ti = 0 eV)

● Singly ionized

● Integrate Poisson's equation

● Bohm's criterion, E = 0 at sheath edge

● E = 0 at surface G. D. Hobbs and J. A. Wesson, Plasma Physics 9 (1), 85 (1967).

Space-Charge Limited Solution

Φw=−eϕwT ep

=1.02 E0=miu0

2

2T ep=0.58


Emissive probes are used tomeasure the plasma potential

● Emissive probes are Langmuir probes that emit electrons

● Usually Joule heated to emit thermionically

● Allows good control over emission current

● Used to measure the plasma potential

● Electrons are emitted when probe bias is above plasma potential, but not when below

● Can be used in plasmas where Langmuir probe measurements fail

● Smaller uncertainty than Langmuir probeJ. P. Sheehan and N. Hershkowitz, Plasma

Sources Science and Technology 20 (6), 063001 (2011).


The floating point method isoften used in Hall thrusters

● Heat probe until floating potential saturates

● Potential at saturation is measure of plasma potential

● Heating voltage swept at 0.1Hz

● Potential measured through a high impedance op-amp

● Potential saturates past peak heating current because probe continues to heat

● Uncertainty ~0.1Te/e


Inflection point in the limit of zero emission attempts to reduce space-charge effects● Typically 7 I-V traces were

taken

● Emission current less than electron saturation current

● Inflection point versus temperature limited emission current approximately linear

● Extrapolate inflection point to zero emission current

● Noise increases uncertainty, but using multiple emission levels reduces it

● Uncertainty ~0.1Te/e


The floating potential of a highly emitting probe in a Hall thruster was ~2Tep below the

plasma potential

J. P. Sheehan, Y. Raitses, N. Hershkowitz, I. Kaganovich and N. J. Fisch, "A comparison of emissive probe techniques for electric potential measurements in a complex plasma," Phys. Plasmas 18, 073501 (2011).


Motivation

● Emissive probe measurement of plasma potential● Floating potential of a highly emitting probe is near the

plasma potential● Knowledge of emissive sheath yields more accurate

measurements

● Secondary electron emission in laboratory plasmas● Significant in determining plasma potential and EVDF in

low temperature plasmas● Increase electron loss to divertors in tokamaks● Modify operation of Hall thrusters, etc....


A fully kinetic model of the planar emitted sheath was developed

● Plasma electron loss cone: modification of EVDF due to electrons lost to the boundary

● Kinetic emitted electrons: half-Maxwellian distribution with temperature parameter Tee

● Ions are assumed to be cold● Poisson's equation and the generalized Bohm

criterion solved simultaneously● Highly nonlinear equations were solved

numerically


Kinetic Theory: plasma electrons donot follow the Boltzmann relation

● Boltzmann relation

●

● Assumes fraction of electrons lost to surface is small

● Valid for collecting sheath, not for emissive sheath

● Full kinetic model

●

● Accounts for electrons lost to surface

● Close to surface, lost electrons are significant

● Boltzmann relation over-estimates the plasma electron density in the sheath

● Considering electrons lost to the surface reduces net charge in the sheath, reduces the sheath potential

nep(Φ)

nep(0)=exp(−Φ)

nep(Φ)

nep(0)=exp(−Φ)(1+erf (√Φw−Φ)

1+erf (√Φw) )


Emitted Electrons: account for kinetic effects of non-zero emitted electron temperature (Tee)

● Plasma to emitted electron temperature ratio Tep/Tee = Θe

● Fluid expression

●

● Assumes Θe → ∞

● Kinetic expression

●

● Maxwellian emitted electrons (Tee)

● Fluid equations over estimate emitted electron density in the sheath

● Higher emitted electron temperature reduces emitted electron density in sheath, reduces sheath potential

nee (Φ)

nee(0)=(1− Φ

Φw )−12

nee(Φ)

nee(0)=exp (Θe(Φw−Φ))erfc(√Θe (Φw−Φ))

exp (ΘeΦw)erfc(√ΘeΦw)


EVDFs of emitted and plasmaelectrons are modified Maxwellians

Plasma Electrons Emitted Electrons


Emitted electrons modifythe Bohm criterion

● For arbitrary electron distribution

● Required for positive space-charge in sheath

● Solved for E0

● Since ions are cold in all descriptions, defines simple condition for ion energy


Higher temperature emitted electrons reduce net electron density in sheath


Emissive sheath potential is reducedby the emitted electron temperature


Emitted electrons only slightlyaffect the Bohm criterion


Emissive sheath wassimulated using EDIPIC

● (Performed by Hongyue “Della” Wang)

● Argon

● System length of 5 mm

● Plasma source electron temperature: 1 eV

● Plasma source ion temperature: 0.025 eV

● Collisionless

● No magnetic field

● Simulated time: 100 μs

● At source (x = 0 mm)

● Escaping particles thermalized and reflected

● Zero electric field● At emitter (x = 5 mm)

● Fixed potential of 0 V● Constant emission

current● Emitted electron

temperatures of 0.2 – 0.001 eV


Potential profiles of emissive sheath calculated from PIC simulations


Kinetic theory was confirmed using particle in cell simulations


Enhanced EEDF tail (>eΦw)increases the sheath potential

● Bi-Maxwellian electron energy distribution function (EEDF)

● Two electron temperatures (Tep2/Tep = Θp)

● Hot electron fraction

● Sheath potential normalized to colder electron temperature Tep

● 5% hot electrons in figure

● Hot electrons can significantly affect the sheath potential even at low concentrations

β=nep2(0)

nep (0)+nep2(0)


Emissive sheath potential depends nonlinearly on the hot electron fraction

● Above a certain fraction of hot electrons, the temperature of the hot species begins to dominate

● This break point depends on the plasma electron temperature ratio Θp = Tep2/Tep

● In figure, Θe = Tee/Tep = 10

● In laboratory plasmas, secondary electrons can be source of hot electrons and constitute a significant fraction of the plasma electrons


Sheath potential has a non-monotonic dependance on the hot electron fraction● For data shown

● Θe = Tee/Tep = 10

● Θp = Tep2/Tep = 10

● Sheath potential normalized to colder plasma electron temperature

● The colder electrons define the ion flux via Bohm's criterion

● The hotter electrons dictate the electron flux through the sheath

● Sheath must be large to reduce electron current to maintain current balance through the sheath


Planar dispenser cathode wasinstalled in GEC reference cell

● Working gas: Helium

● Neutral pressure: 25 mTorr

● Electron density: ~109 cm-3

● RF frequency: 10 MHz

● Pulse frequency: 60 Hz

● Afterglow time: 2.5 ms

● Barium tungsten dispenser cathode


Dispenser cathode floating potential vs. time at various heating currents


Langmuir Probe

● 1 cm long, 250 μm diameter● Positioned 3 cm above the edge of the

dispenser cathode● Aluminum tube protected against displacement

currents● I-V traces to measure electron temperature


Emissive Probe

● 1 cm long, 76 μm

● Thoriated tungsten wire was secured by crushing the ends of copper tubes around it

● Aluminum tube reduced displacement currents for emissive probe, as well

● I-V traces to measure plasma potential using inflection point in the limit of zero emission


Slow-sweep emissive probe method measured Vp versus time in afterglow

● Measured current vs time at many probe biases

● Transpose to determine I-V trace vs time

● Easy, inexpensive to execute

● Used for both Langmuir probe and emissive probe I-V traces

● First time inflection point in the limit of zero emission technique was used to measure temporally varying plasma potential


Measuring Te

● Slope of semilog Langmuir probe I-V trace● Requires good signal to

noise ratio● Number averaging and

smoothing may be necessary

● Approximated by sheath potential of floating Langmuir probe● Many assumptions for

this method


Electron temperature decaymeasured versus time

● RF ring down affected measurements tens of μs into the afterglow

● Langmuir probe could not be used for Te measurements later than ~250 μs into afterglow

● Collecting sheath potential was used to approximate the electron temperature

● Remarkable agreement between these two measurements


Floating potential of heated electron falls, then rises in afterglow

● Afterglow: 0 – 2.5 ms

● Floating potential initially drops as plasma cools and loses density

● Increases as emitted electrons begin to dominate the discharge

● Only data before the minimum (870 μs) is relevant to compare to theory


Plasma potential decreases monotonically in afterglow

● Plasma potential drops to a few volts in the first 100 μs

● Decays slowly through afterglow

● Becomes negative at 1150 μs, after emitted electrons begin to dominate discharge


Electron temperaturedecays in afterglow

● Electron temperature decays rapidly once RF heating is turned off

● Monotonic decay in afterglow

● Measurement become negative after 1240 μs when floating potential exceeds plasma potential

● “Negative temperature” measurements excluded since it is in the emission dominated discharge


Normalized emissive sheath potential is greatly reduced at low electron temperatures


Data qualitatively follows trend predicted by theory

● Cannot directly compare: experimental measurements include presheath

● Sheath disappears when plasma electron temperature equals emitted electron temperature

● For intermediate temperatures, measured sheath is larger than expected from kinetic theory

0.1 1 10 100 1000 10 4


Conclusions

● Kinetic theory of emissive sheaths

● Considering the plasma electrons lost to the surface reduces the emissive sheath potential by 10%

● Considering the non-zero emitted electron temperature reduces the emissive sheath potential by up to 50% for some low temperature plasmas

● Validated with particle in cell simulations

● Measurements of emissive sheath in afterglow

● Confirms that as plasma electron temperature approaches emitted electron temperature emissive sheath disappears

● Emissive sheath was larger than expected for intermediate electron temperatures


Acknowledgments

● This work was supported by US Department of Energy grants No. DE-AC02-09CH11466 and No. DE-FG02-97ER54437, the DOE Office of Fusion Energy Science Contract DE-SC0001939, and the Fusion Energy Sciences Fellowship Program administered by Oak Ridge Institute for Science and Education under a contract between the US Department of Energy and the Oak Ridge Associated Universities

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