A Kinetic Theory of Planar Plasma Sheaths Surrounding Electron Emitting Surfaces

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1 A Kinetic Theory of Planar Plasma Sheaths Surrounding Electron Emitting Surfaces J. P. Sheehan1, I. Kaganovich2, E. Barnat3, B. Weatherford3, H. Wang2, D. Sydorenko, N. Hershkowitz, and Y. Raitses 1 University of Wisconsin Madison 2 Princeton Plasma Physics Laboratory 3 Sandia National Laboratories 4 University of Alberta

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 2

3 Emitted electrons reduce sheath potential and electric field at surface Three species Plasma electrons Plasma ions Emitted electrons Plasma fills the -x half-plane One dimensional Emitted electrons reduce: Collecting Sheath The electric field at the surface The floating sheath potential Emissive Sheath 3

4 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 e ϕw Φ w = =1.02 T ep 2 mu E 0= i 0 =0.58 2T ep Emitted Electrons Space-Charge Limited Solution 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). 4

5 Emissive probes are used to measure 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 probe J. P. Sheehan and N. Hershkowitz, Plasma Sources Science and Technology 20 (6), (2011). 5

6 The floating point method is often 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 6

7 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 7

8 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, (2011). 8

9 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... 9

10 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 10

11 Kinetic Theory: plasma electrons do not 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 n ep ( Φ) =exp( Φ) n ep ( 0) ( 1+erf ( Φw Φ ) n ep (Φ) =exp( Φ) n ep ( 0) 1+erf ( Φw ) ) 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 11

12 Emitted Electrons: account for kinetic effects of non-zero emitted electron temperature (Tee) Plasma to emitted electron temperature ratio Tep/Tee = Θe Fluid expression n ee (Φ) = 1 Φ Φw n ee (0) Assumes Θe ( ) 1 2 Kinetic expression n ee (Φ) exp (Θe (Φw Φ))erfc ( Θe (Φ w Φ)) = n ee (0) exp (Θe Φ w )erfc ( Θe Φ w ) 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 12

13 EVDFs of emitted and plasma electrons are modified Maxwellians Plasma Electrons Emitted Electrons 13

14 Emitted electrons modify the 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 14

15 Higher temperature emitted electrons reduce net electron density in sheath 15

16 Emissive sheath potential is reduced by the emitted electron temperature 16

17 Emitted electrons only slightly affect the Bohm criterion 17

18 Emissive sheath was simulated using EDIPIC (Performed by Hongyue Della Wang) Argon System length of 5 mm Plasma source electron temperature: 1 ev Plasma source ion temperature: ev At source (x = 0 mm) No magnetic field Simulated time: 100 μs Zero electric field At emitter (x = 5 mm) Collisionless Escaping particles thermalized and reflected Fixed potential of 0 V Constant emission current Emitted electron temperatures of ev 18

19 Potential profiles of emissive sheath calculated from PIC simulations 19

20 Kinetic theory was confirmed using particle in cell simulations 20

21 Enhanced EEDF tail (>eφw) increases the sheath potential Bi-Maxwellian electron energy distribution function (EEDF) Two electron temperatures (Tep2/Tep = Θp) Hot electron fraction β= n ep2 (0) n ep (0)+n ep2 (0) 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 21

22 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 22

23 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 23

24 Planar dispenser cathode was installed 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 24

25 Dispenser cathode floating potential vs. time at various heating currents 25

26 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 26

27 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 27

28 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 28

29 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 29

30 Electron temperature decay measured 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 30

31 Floating potential of heated electron falls, then rises in afterglow Afterglow: 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 31

32 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 32

33 Electron temperature decays 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 33

34 Normalized emissive sheath potential is greatly reduced at low electron temperatures 34

35 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 35

36 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 36

37 Acknowledgments This work was supported by US Department of Energy grants No. DE-AC02-09CH11466 and No. DE-FG0297ER54437, the DOE Office of Fusion Energy Science Contract DE-SC , 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 37

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