Measuring Electron Energy Distribution Functions in Low Temperature Plasmas

Transcription

1 Measuring Electron Energy Distribution Functions in Low Temperature Plasmas Steve Shannon Associate Professor NC State University Raleigh NC Presented at 2009 GEC Conference Workshop on Plasma Kinetics Available for Download at

2 EEDF s drive much of what is interesting in low temperature plasmas Dissociation Ionization Low Energy Sheath Dynamics Distribution (1/cm 3 ev) Electron Density (1/cm 3 ev) Energy (ev) Energy (ev) Maxwellian Reconstructed Maxwellian Heating Attachment 2

3 At the same time, accurately measuring EEDF s has been a challenge in the area of LTP s EEDF measurement techniques OES Thompson scattering Langmuir probe Other diagnostics assume a characteristic shape (Maxwellian) and estimate average energy from it Most challenges lie in the limitations of the diagnostics with respect to valid regimes, intrusiveness, time to make a measurement, and complexity 3

4 Optical Emission Spectroscopy Line ratio and CR modeling (Zhu and Pu; Plasma Sources Sci. Technol. 17 (2008) ) TRG OES (Chen et. al.; J. Vac. Sci. Technol. A, Vol. 27, No. 5, Sep/Oct 2009) Recent work at Max Planck Institute that looks very interested, but I have no idea what to call it (Dodt et.al.; J. Phys. D: Appl. Phys. 41 (2008)

5 5 Thomson Scattering The intensity of scattered monochromatic light is measured as a function of wavelength, and the electron energy distribution can be inferred from the shape of this broadening Non invasive and impervious to RF noise and potential fluctuations However, difficult to get a measurable population of scattering events in a reasonable amount of time for low density plasmas S Maurmann et. al.; J. Phys. D: Appl. Phys. 37 (2004)

6 Most EEDF work has centered on Langmuir probe measurements Druyvestian relationship (Druyvesteyn; Physica (1930)). Represented in various forms, but most recognized as: I e Ae p 2 = m 4 e evprobe ( )( + probe ) f E E ev E de or f( E) E 4 mv d I = ev = probe Ae e dv p 2 e probe e probe 6 Challenges RF distortion, integral reconstruction, sheath approximations, collisions, determination of other parameters (plasma potential, ion current, etc.), perturbation effects, and on and on Challenges that we will take a look at today EEDF reconstruction from integral problem Impact of finite probe dimensions and sheaths in EEDF measurement

7 Addressing the integral problem Straight double differentiation of probe characteristic using analog circuits (Boyd Twiddy method) No point by point differentiation helps with noise amplification Established method Good reference: Homfray and Crowley (2007) Fusion Eng. Des Data smoothing using various filter techniques EASY! Works in many cases Easy to incorporate in digitized systems Solution conditioning More accurately addresses the integral problem for digitized data than smoothing Previous efforts in EEDF analysis have centered on Tikhonov Regularization based conditioning methods 7

8 Using Tikhonov Regularization for EEDF extraction Volkova, Devjatov, Kralkina, Sedov, and Scherif; Moscow University Physics Bulletin , 502 (1975) Chegotov; J. Phys. D: Awl. Phys. 27 (1994) Gutiérrez Tapia and Flores LLamas; Phys. Plasmas, Vol. 11, No. 11, November 2004 El Saghir; IEEE Trans. Plasma Sci; Accepted for publication,

9 Why Regularization? It works very well with other integral problems Comparison to typical data smoothing techniques show substantial improvement in accuracy Take a Maxwellian Calculate Ie from Druyvesteyn Add Noise Solve for f(e) Compare 9 El Saghir et. al.; IEEE Trans. Plasma Sci.; Accepted for publication (2009)

10 Integral Reconstruction Using Tikhonov Regularization I e Ae p 2 = m 4 e evprobe ( )( + ) probe f E E ev Let f(e) be represented by a set of step functions, effectively reducing the EEDF to a hystogram with n bins, each with a height of n e. E de 10 This reduces the problem to a system of linear equations Tikhonov solution for f e : I e = Kf f ( α) = ( K T K+ α D T D) -1 K T I e e

11 Problem is that a single regularizing parameter cannot simultaneously capture the entire energy range for an EEDF f ( α) = ( K T K+ αd T D) -1 K T I e 14 x α too small high energy tail is lost Druyvesteyn Reconstructed Druyvesteyn SNR=20 14 x α too big low energy distribution is distorted Druyvesteyn Reconstructed Druyvesteyn SNR=20 Electron Density (1/cm 3 ev) Electron Density (1/cm 3 ev) Energy (ev) Energy (ev) 11 El Saghir et. al.; IEEE Trans. Plasma Sci.; Accepted for publication (2009)

12 Similar trends can be seen in previous regularization efforts to extract EEDF s Gutiérrez-Tapia and Flores-Llamas; Phys. Plasmas.11(11) pp ; November Volkova, L.M., Deviatov, A.M., Kralkina, E.A., Sedov, N.N., Sherif, M.A.; Vestnik Moskovskogo universiteta seriia III, fizika, astronomia. 16 pp ; August

13 However all is not lost! Advanced reconstruction algorithms can significantly enhance accuracy over entire energy range Plots show three reconstructions using same data set: Maxwellian Reconstructed Maxwellian 10 9 Maxwellian Reconstructed Maxwellian Electron Density (1/cm 3 ev) ev) Electron Density (1/cm 10 8 Electron Density (1/cm 3 ev) Energy (ev) Insert shameless plug for student presentation here: Energy (ev) Tikhonov Regularization TSVD Hybrid TSVD with modified regularizer El Saghir et. al.; Evaluation of Advanced Algorithms to improve EEDF extraction from Langmuir Probe data Using Tikhonov Regularization Methods Tomorrow, Low Pressure Plasma Diagnostics, 10:30am, Ballroom 3

14 Cylindrical probes present a slightly different problem then Druyvestyen addressed Druyvesteyn derivation is for a planar probe type configuration Cylindrical probe geometry tweaks integral relationship need to include an angular momentum term to account for electron deflection due to repulsive term e - v e s(v p ) b r p 14 b r 2 V = 1 2 p probe E

15 f(e) Calculating the collection efficiency for an arbitrary distribution E L E E H g(e rθ ) E rθ h(e rθ,b) b E rθ 15 r p

16 Integrate over range of E rθ and b that is valid for electron collection (as opposed to deflection) 0 E r p b 0 16 ( ) 2 π I e V p ev p r p 1 E rθ e r p Λ ev p 0 2E rθ ( ( )) 2 b 2 m e r p + sv p ( ( )) r p + sv p ( ) he rθ, b cos π 2 b r p + sv p ( ( )) db de rθ

17 What does this do to a VI characteristic? Test take a 3eV Maxwellian, calculate current using cylindrical equation, compare to current from Druyvesteyn relation: Normalized Current ratio = 0.1 ratio = 1 ratio = 10 Thin r = 0.1 Thin, r = 1 Thin, r = Slight underestimate of Te Departure from Maxwellian at high energy Not much difference with sheath thickness Voltage (V)

18 What does this do to a distribution if Druyvesteyn is used with a cylindrical probe? Caveat week old data work in progress 1.00E V1 V2 V E-03

19 Corrections are possible Knappmiller et. al.; Phys. Rev. E. 73, (2006) Planar Probe Cylindrical Probe 19

20 Moving forward developing similar regularized solutions for cylindrical probes Vs. Energy b Energy b 20 And do we need to worry about sheath thickness? Initial results suggest no this would really simplify things

21 Fixing the distortion Modification of integral problem combined with data smoothing (Knappmiller) Modification of integral problem combined with advanced regularization (Shannon) Boyd Twiddy methods if this is a simple correction function, it can be applied after EEDF acquisition to fix distortion. It is still unclear how this could be implimented here, but that s not to say it isn t possible. More accurately, formulation has not been attempted yet 21

22 Closing remarks 22 EEDF s are vital sources of information with respect to low temperature plasmas they drive plasma chemistry, sheath dynamics, ionization, and attachment. They are an excellent indication of heating methods in RF systems. EEDF measurement is one of the more challenging endeavors in LTP diagnostics due to it s integral nature and the practical limitations of typical diagnostics. Langmuir probe measurement of EEDF s can benefit from advanced deconvolving techniques (remember see talk tomorrow!) Cylindrical probes present additional challenges not fully accounted for in the Druyvesteyn equation that may contribute to measurable EEDF distortion in LTP measurements.

23 Acknowledgements and references Chegotov; J. Phys. D: Awl. Phys. 27 (1994) Chen et. al.; J. Vac. Sci. Technol. A, Vol. 27, No. 5, Sep/Oct 2009 Dodt et.al.; J. Phys. D: Appl. Phys. 41 (2008) Druyvesteyn; Physica (1930) El Saghir et. al.; IEEE Trans. Plasma Sci.; Accepted for publication (2009) Gutiérrez Tapia and Flores Llamas; Phys. Plasmas.11(11) pp ; November 2004 Homfray and Crowley (2007) Fusion Eng. Des Knappmiller et. al.; Phys. Rev. E. 73, (2006) S Maurmann et. al.; J. Phys. D: Appl. Phys. 37 (2004) Volkova, L.M. et. al.; universiteta seriia III, fizika, astronomia. 16 pp ; August 1975 Zhu and Pu; Plasma Sources Sci. Technol. 17 (2008) A world of thanks to Ahmed El Saghir (who has done most of the work presented here), Elijah Martin, Chris Kennedy, Valery Godyak, Hany Abdel-Kalik, David Lalush, Mohamed Bourham, and John Gilligan This work is supported by the State of North Carolina and by a generous gift from Applied Materials Inc. 23