MWP 1.9 MODELING AND SIMULATION OF ELECTROMAGNETIC EFFECTS IN CAPACITIVE DISCHARGES Insook Lee, D.B. Graves, and M.A. Lieberman University of California Berkeley, CA 9472 LiebermanGEC7 1
STANDING WAVES AND SKIN EFFECTS High frequency and large area standing wave effects High frequency high density skin effects 1. M.A. Lieberman, J.P. Booth, P. Chabert, J.M. Rax, and M.M. Turner, Plasma Sources Sci. Technol. 11, 283, 22 2. P. Chabert, J. Phys. D: Appl. Phys. 4, R63, 27 3. Insook Lee, D.B. Graves, and M.A. Lieberman, Modeling of electromagnetic effects in capacitive discharges, submitted to Plasma Sources Sci. Technology, 27 LiebermanGEC7 2
CYLINDRICAL CAPACITIVE DISCHARGE Consider only the high frequency source V h ~ + Sheath z s Plasma r 2d 2l Sheath s 2R Fields cannot pass through metal plates (1) V s excites radially outward wave in top vacuum gap (2) Outward wave excites radially inward wave in plasma LiebermanGEC7 3
SURFACE WAVE MODE Power enters the plasma via a surface wave mode: Surface Wave Mode Decay (weak) Decay δ Standing wave λ Plasma s 2d s Radial wavelength for surface wave (low density limit): λ λ 1 + d/s λ 3 with λ = c/f the free space wavelength Axial skin depth for surface wave: δ c ω p There are also evanescent modes leading to edge effects near r = R LiebermanGEC7 4
EXPERIMENTAL RESULTS FOR STANDING WAVES 2 2 cm discharge p = 15 mtorr 5 W rf power The standing wave effect is seen at 6 MHz and is more pronounced at 81.36 MHz (A. Perret, P. Chabert, J-P Booth, J. Jolly, J. Guillon and Ph. Auvray, Appl. Phys. Lett. 83, 243, 23) LiebermanGEC7 5
FINITE ELEMENT METHOD (FEM), 2D EM SOLUTIONS (with Insook Lee and D.B. Graves) Arbitrary (asymmetric) discharge geometries and materials Transition from global to local power balance Distinguish edge effects (electrostatic) versus EM effects Series resonance stop band FEM single frequency EM solve (plasma + sheath) E-field ~3µs FEM fluid plasma solve Analytical sheath solve.24.2 z (m) Sheath Plasma H φ = const s, n, ν eff Solution Procedure.2.24.2 FEM Mesh (6252 elements) (Analytical model: collisional Child law, variable sheath width, stochastic and ohmic heating in the sheath) LiebermanGEC7 6
2 STANDING WAVES 4 W, 15 mtorr Edge effect 13 MHz 6 MHz Standing wave effect 8 MHz 1 MHz LiebermanGEC7 7
1 2 SKIN EFFECTS 15 mtorr Pr/Pz 1 1 1 1-1 E H V =213 V V =158 V V =1 V V =45 V 1-2..5.1.15 FEM model (with Insook Lee and D.B. Graves) Transmission line model (P. Chabert et al, Plasma Sources Sci. Technol. 15, S13, 26) Transmission line model: collisionless sheaths, no edge effects, purely local power deposition In both cases spatial E to H transitions are seen LiebermanGEC7 8
COMPARE 2 CM AND 4 CM RADIUS REACTORS (15 mtorr, 2 MHz, V rf = 1 V on-axis) 4x1 17 (a) 4x1 17 (b) 3x1 17 3x1 17 R = 2 cm n e (m -3 ) 2x1 17 n e (m -3 ) 2x1 17 2R = 4 cm 1x1 17 1x1 17..1.2.3..5.1.15 Radial plasma profile for (a) 4 and (b) 2 cm radius reactors 2 (a) 2R = 4 cm 2 (b) R = 2 cm 15 15 Power/Area (W/m 2 ) 1 5 P z P tot Power/Area (W/m 2 ) 1 5 P z P tot P r..1.2.3 P r..5.1.15 Radial P r and axial P z power deposition versus radius r, and their sum Edge effect for 2 cm radius reactor, and wave effects, are apparent LiebermanGEC7 9
SERIES RESONANCE 2 MHz, 15 mtorr 11 2 W 4 W Surface wave does not propagate for 4 W case: ω res < ω < ω p ω res = series resonance frequency ω p = plasma frequency LiebermanGEC7 1
ASYMMETRIC (BOTTOM) EXCITATION 15 mtorr.24.2 Sheath z (m) Plasma 13 MHz 4 W.2.24.2 H φ = K/r Reduced edge effect 8 MHz 4 W 1.3 H 2 r(m) r(m) r(m) r(m) LiebermanGEC7 11
ASYMMETRIC VOLTAGE WAVEFORMS.24.2 Sheath z (m) Plasma.2.24.2 H φ = K/r 7 45 2 Voltage asymmetry Voltage asymmetry disappears.2.2 LiebermanGEC7 12
CONCLUSIONS A 2-D axisymmetric model and finite element method (FEM) simulation strategy was developed to determine radial plasma uniformity in large-area, high frequency capacitive discharges Electromagnetic effects and electrostatic edge effects are well captured by the simulations The use of a FEM-based simulation allows for irregular and complex geometries, as well as fluid flow, heat and mass transfer, and chemical kinetics, although we do not include most of these effects here Download this poster: http://www.eecs.berkeley.edu/ lieber LiebermanGEC7 13