NARROW GAP ELECTRONEGATIVE CAPACITIVE DISCHARGES AND STOCHASTIC HEATING

Transcription

1 NARRW GAP ELECTRNEGATIVE CAPACITIVE DISCHARGES AND STCHASTIC HEATING M.A. Lieberman, E. Kawamura, and A.J. Lichtenberg Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 9472 Motivation: widely used for thin film etch and deposition Download this talk: lieber LiebermanPSC13 1

2 UTLINE Narrow gap oxygen discharges PIC simulations Equilibrium discharge model Stochastic (collisionless) heating Narrow gap oxygen Two-step density model with argon hmic heating issues Single-step with finite slope LiebermanPSC13 2

3 DISCHARGE CNFIGURATIN xygen at 1 1 mtorr, V rf = 5 2 V 1D plane-parallel geometry ( 1 1 cm gap length L) Usual model is stratified discharge with electronegative (EN) core and electropositive (EP) edge As L is decreased, the EP edge can disappear and new interesting phenomena are found LiebermanPSC13 3

4 PIC SIMULATINS AND EQUILIBRIUM MDELING (Vary gap length L at p=5 mtorr, V rf = 5 V) LiebermanPSC13 4

5 L=4.5 cm (EP EDGE EXISTS) 2e e+16 1e+16 5e+15 (a) Particle Density (m-3) e (c) Temperature (V) Low Te e EN core EP edge Sheath e + 2 (b) Current Density (A/m 2) Large pos (d) Electron Heating (W/m 3) p e Core flux p ohm p stoc Small neg LiebermanPSC13 5

6 L=2.5 cm (N EP EDGE) (a) Particle Density (m-3) (b) Current Density (A/m 2) 2e e EN core No EP edge.4.2 e Core flux 1e+16 5e e e (c) Temperature (V) High Te e Sheath (d) Electron Heating (W/m 3) Negative (A SURPRISE!) p e p ohm p Large pos stoc LiebermanPSC13 6

7 EEDF S AND DENSITY DETAILS (a) EEDF (a.u.) for L = 4.5 cm (b) EEDF (a.u.) for L = 2.5 cm EP edge: bi-maxwellian 1 No EP edge: Maxwellian Energy (ev) Energy (ev) 2e+15 (a) Densities (m-3) for L= 4.5 cm 1e+15 (b) Densities (m-3) for L= 2.5 cm 1.5e+15 1e+15 5e+14 n + - n - n e sheath edge n e+14 6e+14 Sheath begins at end of EP edge 4e+14 2e+14 n + - n - n e n ~ sheath edge Sheath begins inside EN core LiebermanPSC13 7

8 TIME-VARYING DENSITY (L=2.5 cm, N EP EDGE) 1e+15 Electron density (m-3) at T/8 intervals + 2 5e+14 scillating electron cloud uncovers core Core LiebermanPSC13 8

9 MDELING CNSIDERATINS EP edge exists (larger gap lengths L) Bi-Maxwellian EEDF About half the ion flux generated in sheath/ep edge Usual Child law rf sheath Usual positive collisionless heating in sheath No EP edge (smaller gap lengths L) Maxwellian EEDF ver half the ion flux generated in sheath Attachment in sheath is important Unusual rf sheath containing negative ions Negative collisionless heating in sheath, positive in core Models developed: model with some inputs from PIC results self-consistent model LiebermanPSC13 9

10 MDEL WITH SME INPUTS FRM PIC Rate coefficients and collisional energy losses using PIC EEDF Power deposition in core from PIC results Solid lines (2-region model with EP edge); Dashed lines (1-region model without EP edge); circles (PIC results) (a) n e (m 3 ) at 1 mtorr, 5 V L (m) (c) n e (m 3 ) at 5 mtorr, 5 V (b) n e (m 3 ) at 25 mtorr, 5 V L (m) (d) n e (m 3 ) at 1 mtorr, 5 V L (m) (e) n e (m 3 ) at 5 mtorr, 1 V L (m) (f) n e (m 3 ) at 5 mtorr, 2 V Reasonable agreement between model and PIC results (submitted to Physics of Plasmas, 213) L (m) L (m) LiebermanPSC

11 STCHASTIC (CLLISINLESS) HEATING (WRK IN PRGRESS) LiebermanPSC13 11

12 PIC RESULTS FR VARIUS GAPS L (5 mt, 5 V) Stochastic heating small at transition where EP edge disappears EP edge exists positive heating in sheath, negative in core No EP edge negative heating in sheath, positive in 1 core L (m) Large pos (d) Electron Heating (W/m 3) p e L=4.5 cm pohm 1 p ohm p pstoc stoc Small neg LiebermanPSC13 12 S (W/m 2 ) S ohm S stoc A SURPRISE! (d) Electron Heating (W/m 3) L=2.5 cm Negative (A SURPRISE!) p e Large pos

13 S stoc (x) FR VARIUS GAPS L Integrate p stoc (x) from electrode (x = ) toward discharge midplane S stoc (x) Large L n e (core) > n e (sheath) positive heating in sheath, negative heating in core Small L n e (sheath) > n e (core) negative heating in sheath, positive heating in core LiebermanPSC13 13

14 TW-STEP DENSITY MDEL 1. I.D. Kaganovich, Phys. Rev. Lett. 89, 2656 (22). 2. E. Kawamura, M.A. Lieberman and A.J. Lichtenberg, Phys. Plasmas 13, 5356 (26). Use to investigate stochastic heating LiebermanPSC13 14

15 FIXED INS, PIC ELECTRNS (3 mt Ar, MHz) (Models EN plasma with EP edge) (Models EN plasma with no EP edge) n i n i n e n e Sheath oscillation Sheath oscillation Positive Fermi kick? Positive low density kick? Negative high density kick? Negative high density kick? Positive low density kick? Negative Fermi kick? LiebermanPSC13 15

16 ELECTRN HEATING GAS PRESSURE EFFECTS Large device 2:1 step at 6 cm At 2 mt 66% of heating is ohmic; at 3 mt 96% is ohmic error in finding stochastic heating 2 mtorr 3 mtorr LiebermanPSC13 16

17 PIC SIMULATIN HMIC HEATING ISSUES (see Lister, Li and Godyak, 1996) hmic power density p ohm = 1 2 J rf 2 Re[1/(σ p + jωǫ )], where σ p = 4π 3 e 2 m v 3 dv df e jω + ν m (v) dv The usual simple expression (see Margenau, 1946) e 2 n e σ p = m(jω + ν m ) is not correct unless ν m = const Argon is a Ramsauer gas with ν m a function of v The momentum transfer frequency ν m must be distinguished from the total collision frequency ν coll Large errors in calculating stochastic heating for ohmic heating stochastic heating Can use pseudo-argon in PIC simulation ν m = ν coll = const LiebermanPSC13 17

18 VARIUS CLLISIN MDELS (UNIFRM ) Fixed Ion Uniform 5 mtorr Argon with n b = 4.8e15 m 3 V rf =1 V, f=13.56 MHz, L=.5 m, Area=.1 m 2 pstoc (Margenau) pstoc (Margenau) pstoc (Margenau) pstoc(crrect!) Margenau LiebermanPSC13 18

19 STCHASTIC HEATING (5 CM UNIFRM ) (Ramsauer elastic cross section, non-isotropic scattering) Sstoc(x) (W/m 2) for Uniform Profiles -3 mtorr Ar, L= 5 cm, Area =.1 m 2, 1V@13.56MHz 2 3 mtorr 1 mtorr 5 mtorr 2 mtorr 1mTorr.5 mtorr LiebermanPSC13 19

20 STCHASTIC HEATING (2:1 STEP, d edge = 6 CM) (Ramsauer elastic cross section, non-isotropic scattering) 12 3 mtorr 1 2mTorr 1 mtorr 8 5 mtorr 2 mtorr 6 1 mtorr.5 mtorr Sstoc(x) (W/m 2) for d edge = 6 cm, ν (x) coll 2-3 mtorr Ar, L= 3 cm, Area =.1 m 2,.41A@13.56MHz LiebermanPSC13 2

21 STCHASTIC HEATING (2:1 STEP, d edge = 6 CM) (Constant collision frequency, isotropic scattering) Sstoc(x) (W/m 2) for d 12 3 mtorr 1 mtorr 1 5 mtorr 2 mtorr 8 1 mtorr.5 mtorr edge = 6 cm, Iso., const K.5-3 mtorr Ar, L= 3 cm, Area =.1 m 2,.41A@13.56MHz LiebermanPSC13 21 el

22 STCHASTIC HEATING (2:1 STEP, d edge = 1 CM) (Constant collision frequency, isotropic scattering) Sstoc(x) (W/m 2) for d 12 3 mtorr 1 mtorr 1 5 mtorr 2 mtorr 8 1 mtorr.5 mtorr edge = 1 cm, Iso., const K.5-3 mtorr Ar, L= 3 cm, Area =.1 m 2,.41A@13.56MHz el LiebermanPSC13 22

23 ENERGY KICK FR TW-STEP DENSITY + V s E rf (bulk) E rf (sheath) Reflected 5 Transmitted There is a potential drop V s = T e ln(n b /n s ) across the step There are rf fields E rf (bulk) and E rf (sheath) Calculate the phase-averaged energy kick E for transmitted and reflected particles There is no contribution to the kick at the oscillating plasma-sheath edge (2 3) LiebermanPSC13 23

24 SINGLE STEP WITH VARIABLE SLPE (PIC) 3 mtorr Argon fixed Ion Tanh profile with a 1 =1 to 4, a 2 =3, n l /n r =.5, I rf =.3 A, f=27.12 MHz, L=.1 m, Area=.1 m 2, n l =2e15 m 3 (a) a 1 = 1 (b) a 1 = 2 n(x) = n 2 [ ( a1 a 2 + tanh L ( L) )] x 2 Vary slope using a 1 in tanh function and find stochastic heating LiebermanPSC13 24

25 STCHASTIC HEATING WITH VARIABLE SLPE 6 3 Stochastic Heating in Center Region (W/m ) 3 mt Ar, Irf=.3A@27.12MHz, L=.1 m, Area =.1 m^ n =2e15 m, n =4e15 m l -3-3 r (d/l)(a1/a2) d = v e /ω.4 cm = mixing length (1 radian phase change) (v e = electron thermal velocity) La 2 /a 1 = scale length of step (.25 3 cm) Saturation for small scale length; adiabatic for large scale length Electron oscillation amplitude (.4 cm) may be significant LiebermanPSC13 25

26 SUMMARY A transition from a narrow gap EN discharge with an EP edge, to a narrower gap discharge with no EP edge, was investigated with PIC simulations and modeling The effects of a bi-maxwellian EEDF, with an EP edge, and sheath attachment and core uncovering, with no EP edge, need to be taken into account in modeling A transition from sheath to internal stochastic heating after the EP edge disappears is observed, and is being studied with fixed ion, two-step and single-step density, PIC simulations At the higher pressures, the ohmic heating has to be carefully calculated in order to determine the true stochastic heating in the PIC simulations LiebermanPSC13 26