ADVENTURES IN TWO-DIMENSIONAL PARTICLE-IN-CELL SIMULATIONS OF ELECTRONEGATIVE DISCHARGES PART 1: DOUBLE LAYERS IN A TWO REGION DISCHARGE E. Kawamura, A.J. Lichtenberg, M.A. Lieberman and J.P. Verboncoeur University of California Berkeley, CA 9472 Download this talk: http://www.eecs.berkeley.edu/ lieber LiebEcolePoly9 1
OUTLINE Introduction Experiments 2D particlin-cell (PIC) simulations with rescaled oxygen cross sections Theory 1D collisionless model of double layer (DL) Global (volumaveraged) model of upstream and downstream particle and energy balances Comparisons Reasonable agreement but also some differences Slow and fast waves The DL has timvarying structure LiebEcolePoly9 2
y z 1 cm 1 cm Upstream Chamber 2 E z = E z cos ωt DL INTRODUCTION Downstream Chamber 1 3.5 cm E z = 3. cm 3.5 cm Oxygen Plasma x 3 25 2 15 1 5.4.8 x (m) Timaveraged Potential (V) (2D PIC 6 mtorr O 2 ) DL.12.16.2.2.8.1.6.4 Upstream edge 2 Charge density ρ DL + Downstream edge 1 Potential V = Vs Upstream edge 2 DL Downstream edge 1 Why does a DL form at low pressures? The particle loss rate is greater upstream than downstream due to the smaller upstream radius A higher ionization rate (and T e ) is needed upstream than downstream A DL both insulates the low downstream T e from the high upstream T e, and it accelerates electrons upstream to increase the ionization rate there V= LiebEcolePoly9 3
PIC SIMULATION METHOD Self-consistent results from first principles with no assumptions about electron and ion velocity distributions Upstream heating at 13.56 MHz to the plane of the simulation RF field adjusted to keep number of upstream electrons constant Stability, speed and accuracy require small plasma reactors with low densities and large Debye lengths: n e 4 1 14 m 3, λ D.8 mm Due to low densities, rescaled oxygen cross sections were used: Positivnegative ion recombination 2 Dissociative attachment 5 A typical simulation takes 1 2 weeks The pressure range explored is.5 24 mtorr A DL was observed for 1 24 mtorr LiebEcolePoly9 4
PIC RESULTS FOR 6 mtorr O 2 DISCHARGE Axial Potential (V) Radially averaged T (V) 3 6 25 5 Th 2 15 DL 4 3 Tc 1 2 O- 5 4e+15 3e+15 2e+15.5.1.15.2 O- O + 2 x (m) Axial Density (m-3) n /ne ~ 1 1 215 1.515 115 O2+.5.1.15.2 x (m) K Axial Collision Rates (m 3 /s) iz high Kiz 1e+15.5.1.15.2 x (m) 516 K att low Kiz.5.1.15.2 x (m) LiebEcolePoly9 5
RESULTS FOR 6 mtorr O 2 DISCHARGE (CONT D) 6 4 2-2 -4 Charge Density Components in DL (V/m 2 ) DL charge Transverse ion loss in DL Total ρ/ε Ex/ x Ey/ y Ex/ x + Ey/ y.6.8.1.12.14 x (m) Up edge x-component EEDF (a.u.) 1e+18 1e+17 1e+16 1e+15 3.17 2.53 Up region Up edge 5.13 4.95 Down region 3.95 EEPF (a.u.) Down edge 4.66 1e+14 1 2 3 4 5 ε (ev) Down-edge x-component EEDF (a.u.).1.1 up-directed Accelerated component Tc Th down-directed.1.1 up-directed Tc down-directed Th.1.1 15 15-3 -2-1 1 2 3-3 -2-1 1 2 3 x-energy (V) x-energy (V) LiebEcolePoly9 6
DL MODEL Charge density ρ and potential V within the DL are found by solving Poisson s equation Potential V = Vs Upstream edge 2 Six types of particles contribute to ρ: thermal positive ions, negative ions, and electrons positive ions accelerated downstream negative ions and electrons accelerated upstream The particle motions are 1D and collisionless The boundary conditions are that ρ and dρ/dv vanish at the DL edges DL Downstream edge 1 An additional condition is that the sum of positive and negative charge in the double layer vanishes; equivalently, the total force acting on the double layer vanishes A final condition that upstream and downstream-directed electron fluxes nearly balance determines the equilibrium value of V s /T h. V= LiebEcolePoly9 7
PARTICLE AND ENERGY BALANCES Upstream region 2 DL Downstream region 1 We use a 2D rectangular geometry To determine the equilibrium quantities we use global particle balance upstream global particle balance downstream global energy balance downstream Upstream energy balance (which determines the upstream electron density) depends on the input power; we use the PIC simulation value for the comparisons LiebEcolePoly9 8
MODEL (TOP) and PIC (BOTTOM) EEDF S (6 mtorr O 2 ) 1.1.1 Up edge x-component EEDF up-directed Accelerated component Tc Th down-directed 1.1.1.1 Down edge x-component EEDF up-directed Tc/Th=.668 Tc down-directed Th.1-3 -2-1 1 2 3 x-energy (V) Up edge x-component EEDF (a.u.) -3-2 -1 1 2 3 x-energy (V) Down-edge x-component EEDF (a.u.).1.1 up-directed Accelerated component Tc Th down-directed.1.1 up-directed Tc down-directed Th.1.1 15-3 -2-1 1 2 3 x-energy (V) 15-3 -2-1 1 2 3 x-energy (V) LiebEcolePoly9 9
MODEL (WHITE) AND PIC (BLACK) FOR 6 mtorr O 2 Te (V) Up elec bal (#/s) 1 2e+16 8 6 1.5e+16 4 1e+16 2 5e+15 Th Thdn Tc Tcdn Iz Att Gam- Gam+ Wall 8e+17 7e+17 6e+17 5e+17 4e+17 3e+17 2e+17 1e+17 Down energy bal (ev/s) 3.5e+15 3e+15 2.5e+15 2e+15 1.5e+15 1e+15 5e+14 Down elec balance (#/s) No transverse ion loss in DL Edn Eup Idn Nup Coll Wall Iz Att Gam- Gam+ Wall LiebEcolePoly9 1
1.4 1.2.8.6.4.2 MODEL (DASH) AND PIC (CIRCLES) VS PRESSURE 1 Vs/Th 1 1 p (mtorr) 1 8 6 4 2 Th (V) 1 1 p (mtorr) Vs (V) 14 12 1 8 6 4 2 1 1 p (mtorr) ne2/ne1 5 4 3 2 1 1 1 LiebEcolePoly9 p (mtorr) 11
SLOW AND FAST WAVES: INSTABILITIES (6 mtorr O 2 ) 1 Ex(t) at x=1 cm (dn DL edge) time interval = 59 microsecs 1 FFT of Ex(t) at x = 1 cm (dn DL edge) 8.475e4 Hz 9.3225e5 Hz 5 1 1-5 15 25 35 45 55 time (secs) 1 2e+5 4e+5 6e+5 8e+5 1e+6 frequency (Hz) At 2 12 mtorr, the DL coexists with an unstable slow wave that originates downstream and propagates upstream as it grows The wave frequency is 5 1 khz with a wavelength of order 1 cm The wave produces 2% oscillations in the double layer potential and.5 cm oscillations in the DL position We believe the wave is driven by counter-streaming flows of positive and negative ions (Tuszewski and Gary, 23) There is also a fast wave that increases spatially along with the slow wave at the higher pressures, with f 1 MHz and λ 1 cm LiebEcolePoly9 12
MOVIE SHOWING SLOW AND FAST WAVES Red solid line: fast waves averaged over.1475 µs intervals Blue solid line: slow waves averaged over 1.18 µs intervals Slow Fast DL LiebEcolePoly9 13
ADVENTURES IN TWO-DIMENSIONAL PARTICLE-IN-CELL SIMULATIONS OF ELECTRONEGATIVE DISCHARGES PART 2: TRANSPORT IN A MAGNETIZED DISCHARGE E. Kawamura, M.A. Lieberman and A.J. Lichtenberg University of California Berkeley, CA 9472 LiebEcolePoly9 14
OUTLINE Introduction Experiments with uniform electron temperature 2D PIC simulations with rescaled oxygen cross sections Model and comparisons New model (differs from Gary et al, to appear in J. Phys. D) Experiments with hot core and cold edge 2D PIC simulations with iodinlike cross sections Two-region model and comparisons LiebEcolePoly9 15
INTRODUCTION Rectangular grounded chamber with insulating walls Spatially varying axial rf current density J x y L=2 cm Jx 2R=3 cm Jx(x,y,t) B = 6 G Plasma (2 mtorr oxygen) Jx Dielectric κ=4.25 cm x For an unmagnetized discharge, negative ions are confined by the (positive) plasma potential For a magnetized discharge with uniform axial B : What electron temperature profiles can be obtained? What is the negative ion flux to the transverse walls? What are the electron fluxes to the axial and transverse walls? What are the transverse diffusion coefficients for positive ions, negative ions, and electrons? LiebEcolePoly9 16
PIC SIMULATION METHOD 13.56 MHz axial RF current density J x Adjust amplitude and spatial variation to obtain T e const J x = 1 sin 4πx πy sin L 2R [A/m2 ] Transverse heating (E y s or E z s) did not work Larger amplitudes or uniform J x s large T e s at the transverse and/or axial edges B = 6 G (sometimes 3 or 12 G) Stability, speed, accuracy, and T e const require low densities and large Debye lengths: n e 2 1 13 m 3, λ D.8 cm (!) Due to low density, rescaled oxygen cross sections were used: Positivnegative ion recombination 2, 1, etc A typical simulation takes 1 2 weeks LiebEcolePoly9 17
PIC RESULTS FOR 1 K rec O 2 DISCHARGE Radial Potential (V) Radial T (ev) 3 3 2.5 2.5 2 2 1.5 1.5 1.5 1.5 1xO- 1xO2+.5.1.15.2.25.5.1.15.2.25 Radial Density (m-3) Axial Density (m-3) 1e+14 O2+ 1e+14 O2+ 8e+13 O- 8e+13 O- 6e+13 6e+13 4e+13 2e+13 4e+13 2e+13.5.1.15.2.25.25.5.75.1.125.15.175 x (m) LiebEcolePoly9 18
PIC RESULTS FOR OTHER O 2 DISCHARGES 2e+14 1.5e+14 Radial Density (m-3) (a) B x =6 G, L=19.5 cm, K recfac=2 Mag. ions O2+ 2e+14 1.5e+14 Radial Density (m-3) (b) B x =12 G, L=19.5 cm, K recfac=2 Unmag. ions O- O2+ 1e+14 1e+14 O- 5e+13 5e+13.5.1.15.2.25.5.1.15.2.25 2e+14 1.5e+14 Radial Density (m-3) (c) B x =6 G, L= 9.5 cm, K recfac =2 Mag. ions O2+ O- 2e+14 1.5e+14 Radial Density (m-3) (d) B x =6 G, L= 9.5 cm, K recfac =2 Mag. ions 1e+14 1e+14 O2+ O- 5e+13 5e+13.5.1.15.2.25.5.1.15.2.25 LiebEcolePoly9 19
CONCLUSIONS FROM PIC SIMULATIONS Cosine transverse electron and negative ion density profiles constant electronegativity α = n n /n e n n n n n e n e Leray et al model Transverse direction What PIC shows Transverse direction Transverse electron/negative ion flux Γ e /Γ n D e /α D n Leray et al model confirmed for axial loss frequency ν L 2D ( i 1 + T ) 1/2 e RL T i Transverse diffusion coefficients D e, D i, and D n are classical LiebEcolePoly9 2
1D MODEL Electron balance D e d 2 n e dy 2 = (ν iz ν att )n e ν L (n e + n n ) Negative ion balance D n d 2 n n dy 2 = ν attn e K rec n n (n e + n n ) D e and D n are the transverse ambipolar diffusion coefficients, obtained from Γ i = Γ e + Γ n in terms of the species mobilities and diffusion coefficients Cosine density profiles with n n = α n e (from PIC) Boundary conditions n e = n n = at transverse wall good agreement with PIC results LiebEcolePoly9 21
SCALINGS FOR 1D MODEL Transverse wall negative ion current = 2π L R D nα n e Transverse wall electron current = 2π L R D e n e End wall particle current = 2D i (1 + α )(1 + T e T i ) 1/2 n e Negative ion balance: 4R π ν att = π R D nα + K rec n e α (1 + α ) Range of solutions: < α < 4 π 2 R 2 D n LiebEcolePoly9 22
TWO-REGION IODINE SIMULATIONS Goal was to obtain a high T e core with a cold T e periphery Rectangular grounded chamber with insulating walls 13.56 MHz J x peaked at transverse midplane NOTE Jx! y 2R=3 cm Jx(x,y,t) Jx L=2 cm B =6 G Plasma (2 mtorr iodine) Rescaled oxygen cross sections were used: Positivnegative ion recombination 2 Attachment 5 (w/wo 4.2 V threshold energy) Iodine masses were used Dielectric κ=4.25 cm x LiebEcolePoly9 23
PIC RESULTS FOR IODINE DISCHARGE Radial Density (m ) -3 Radial T (ev) 4 3 2 1 2 mt 2xKrec, 5xKatt(No Thr.) Dielectrics, 3x2 cm, B= 6G x=5 to 15 cm 5xI+.5.1.15.2.25.3 2 mt Iodine 2xKrec, 5xKatt(No Thr.) Discharge 1e+15 1e+14 1e+13 1e+12 Dielectrics, 3x2cm, B= 6 G 1e11α I+ I- Radial Density (m ) -3 Radial T (ev) 4 3 2 1 2 mt Iodine 2xKrec, 5xKatt(4.2V Thr.) Dielectrics, 3x2 cm, B= 6G x=5 to 15 cm 5xI- 5xI- 5xI+.5.1.15.2.25.3 2 mt Iodine 2xKrec, 5xKatt(4.2V Thr.) Discharge 1e+15 1e+14 1e+13 1e+12 Dielectrics, 3x2cm, B= 6 G 1e11α I+ I- 1e+11.5.1.15.2.25.3 No attachment threshold 1e+11.5.1.15.2.25.3 4.2 V attachment threshold LiebEcolePoly9 24
TWO-REGION 1D MODEL Electron balance D e d 2 n e dy 2 = (ν iz ν att )n e ν L (n e + n n ) (ν iz in the core, ν iz = in the periphery) Negative ion balance D n d 2 n n dy 2 = ν attn e (Neglect recombination compared to negative ion transverse loss) Match densities and fluxes at the corperiphery transition Boundary conditions n e = n n = at transverse wall This linear system can be solved exactly LiebEcolePoly9 25
COMPARISON OF MODEL WITH PIC 1 15 unmagions with no thr, Ri=.5 cm 1 15 unmagions with thr, Ri=.5 cm n n, n i n n, n i 1 14 1 14 1E11α n e 1 13 1 13 n e 1E11α 1 12 1 12 Radial Density (m ) -3 1e+15 1e+14 1e+13 1e+12 1 11.5.1.15.2.25.3 2 mt Iodine 2xKrec, 5xKatt(No Thr.) Discharge Dielectrics, 3x2cm, B= 6 G I+ I- 1e11α Radial Density (m ) -3 1e+15 1e+14 1e+13 1e+12 1 11.5.1.15.2.25.3 2 mt Iodine 2xKrec, 5xKatt(4.2V Thr.) Discharge Dielectrics, 3x2cm, B= 6 G I+ I- 1e11α 1e+11.5.1.15.2.25.3 No attachment threshold 1e+11.5.1.15.2.25.3 4.2 V attachment threshold LiebEcolePoly9 26
CONCLUSIONS 2D PIC simulations can be powerful tools to study the physics of electronegative discharges Experimental parameters can be easily varied; various physics can be turned on and off These simulations can provide diagnostics that would be very difficult to do in laboratory experiments The models, verified by PIC, can give important scalings that can be used to design new PIC and experimental configurations Download this talk: http://www.eecs.berkeley.edu/ lieber LiebEcolePoly9 27