A SHORT COURSE ON THE PRINCIPLES OF PLASMA DISCHARGES AND MATERIALS PROCESSING

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1 A SHORT COURSE ON THE PRINCIPLES OF DISCHARGES AND MATERIALS PROCESSING Michael A. Lieberman Department of Electrical Engineering and Computer Sciences, CA LiebermanShortCourse15

2 TABLE OF CONTENTS Introduction to Plasma Discharges and Processing (Ch. 1) Summary of Plasma Fundamentals (Ch. 2, Secs ) Summary of Discharge Fundamentals (Secs. 3.1, 3.5, , ) Analysis of Discharge Equilibrium (Secs ) Capacitive RF Discharges Symmetric Homogeneous Model (Sec. 11.1) Self Consistent Sheath Results (Sec. 11.2) Simulation and Experimental Results (Sec. 11.3) Example Equilibrium Calculations (Sec. 11.2) Asymmetric Systems (Sec. 11.4) Inductive RF Discharges Transformer Model and Matching (Secs ) Power Balance (Secs ) Capacitive RF Sheaths Transit Time Effects (Sec. 11.5) Ion Energy Distribution (IED) (Sec. 11.6) i LiebermanShortCourse15

3 TABLE OF CONTENTS (CONT D) Chemical Fundamentals Atoms and Molecules (Sec. 3.4, Ch. 8) Gas Phase Kinetics (Sec ) Adsorption and Desorption (Sec ) Chemistry in Discharges Neutral Free Radicals (Sec. 10.2) Negative Ions (Sec. 10.3) Example of Oxygen (Sec. 10.4) Time-Varying Global Models (Sec ) Etching Processes (Secs ) Plasma-Induced Charging Damage (Sec. 15.5) Pulsed Discharges (Sec. 10.6) Dual Frequency Capacitive Discharges High Pressure Discharges and Deposition Kinetics (Ch. 16) ERRATA for textbook About the Instructor Self-Study Problems ii LiebermanShortCourse15

4 THE NANOELECTRONICS REVOLUTION Transistors/chip doubling every years since 1959 Billion-fold increase in performance for the same cost over the last 40 years CMOS transistors with 4 nm (16 atoms) gate length EQUIVALENT AUTOMOTIVE ADVANCE 60 billion miles/hr (90 speed of light!) 20 billion miles/gal 1 cm long 3 mm wide iii LiebermanShortCourse15

5 DAY 1 (ETCHING EMPHASIS) 8:00 AM 8:30 AM: Registration 8:30 AM 10:00 AM Introduction to Plasma Discharges and Processing Summary of Plasma Fundamentals (Undriven) 10:00 AM 10:30 AM: Coffee Break 10:30 AM 12 Noon Summary of Plasma Fundamentals (Driven) Summary of Discharge Fundamentals Analysis of Discharge Equilibrium 12:00 Noon 1:30 PM: Lunch 1:30 PM 3:00 PM Capacitive RF Discharges: Symmetric Homogeneous Model Capacitive RF Discharges: Self-Consistent Sheath Results Capacitive RF Discharges: Simulation and Experimental Results Capacitive RF Discharges: Example Equilibrium Calculations 3:00 PM 3:30 PM: Coffee Break 3:30 AM 5:00 PM Capacitive RF Discharges: Asymmetric Systems Inductive RF Discharges: Transformer Model and Matching Inductive RF Discharges: Power Balance iv LiebermanShortCourse15

6 DAY 2 (ETCHING EMPHASIS) 8:30 AM 9:30 AM Capacitive RF Sheaths: Ion Transit Time Effects Capacitive RF Sheaths: Ion Energy Distribution (IED) 9:30 AM 10:00 AM: Coffee Break 10:00 AM 12 Noon Chemical Fundamentals: Atoms and Molecules Chemical Fundamentals: Gas Phase Kinetics Chemical Fundamentals: Adsorption and Desorption Chemistry in Discharges: Neutral Free Radicals 12:00 Noon 1:30 PM: Lunch 1:30 PM 3:00 PM Chemistry in Discharges: Negative Ions Chemistry in Discharges: Example of Oxygen Chemistry in Discharges: Time-Varying Global Models Chemistry in Discharges: Etching Processes 3:00 PM 3:30 PM: Coffee Break 3:30 AM 5:00 PM Plasma-Induced Charging Damage OR Pulsed Discharges Dual Frequency Capacitive Discharges v LiebermanShortCourse15

7 DAY 1 (DEPOSITION EMPHASIS) 8:00 AM 8:30 AM: Registration 8:30 AM 10:00 AM Introduction to Plasma Discharges and Processing Summary of Plasma Fundamentals (Undriven) 10:00 AM 10:30 AM: Coffee Break 10:30 AM 12 Noon Summary of Plasma Fundamentals (Driven) Summary of Discharge Fundamentals Analysis of Discharge Equilibrium 12:00 Noon 1:30 PM: Lunch 1:30 PM 3:00 PM Capacitive RF Discharges: Symmetric Homogeneous Model Capacitive RF Discharges: Self-Consistent Sheath Results Capacitive RF Discharges: Simulation and Experimental Results Capacitive RF Discharges: Example Equilibrium Calculations 3:00 PM 3:30 PM: Coffee Break 3:30 AM 5:00 PM Capacitive RF Discharges: Asymmetric Systems High Pressure Discharges High Pressure Capacitive Discharges vi LiebermanShortCourse15

8 DAY 2 (DEPOSITION EMPHASIS) 8:30 AM 9:30 AM Alpha-To-Gamma Transition 9:30 AM 10:00 AM: Coffee Break 10:00 AM 12 Noon Chemical Fundamentals: Atoms and Molecules Chemical Fundamentals: Gas Phase Kinetics Chemical Fundamentals: Adsorption and Desorption Chemistry in Discharges: Neutral Free Radicals 12:00 Noon 1:30 PM: Lunch 1:30 PM 3:00 PM Chemistry in Discharges: Negative Ions Chemistry in Discharges: Example of Oxygen Chemistry in Discharges: Time-Varying Global Models Chemistry in Discharges: Etching Processes Chemistry in Discharges: Deposition Kinetics 3:00 PM 3:30 PM: Coffee Break 3:30 AM 5:00 PM Pulsed Discharges Dual Frequency Capacitive Discharges vii LiebermanShortCourse15

9 INTRODUCTION TO DISCHARGES AND PROCESSING 1 LiebermanShortCourse15

10 S AND DISCHARGES Plasmas A collection of freely moving charged particles which is, on the average, electrically neutral Discharges Are driven by voltage or current sources Charged particle collisions with neutral particles are important There are boundaries at which surface losses are important The electrons are not in thermal equilibrium with the ions Device sizes 30 cm 1 m Frequencies from DC to rf (13.56 MHz) to microwaves (2.45 GHz) 2 LiebermanShortCourse15

11 EVOLUTION OF ETCHING DISCHARGES 3 LiebermanShortCourse15

12 ANISOTROPIC ETCHING {}}{{}}{ Wet Etching Ion Enhanced Plasma Etching Plasma Etching 4 LiebermanShortCourse15

13 ISOTROPIC ETCHING 1. Start with inert molecular gas CF 4 2. Make discharge to create reactive species CF 4 CF 3 + F 3. Species reacts with material, yielding volatile product Si + 4F SiF 4 4. Pump away product 5. CF 4 does not react with Si; SiF 4 is volatile ANISOTROPIC ETCHING 6. Energetic ions bombard trench bottom, but not sidewalls (a) Increase etching reaction rate at trench bottom (b) Clear passivating films from trench bottom Mask Plasma Ions 5 LiebermanShortCourse15

14 UNITS AND CONSTANTS SI units: meters (m), kilograms (kg), seconds (s), coulombs (C) e = C, electron charge = e Energy unit is joule (J) Often use electron-volt 1 ev = J Temperature unit is kelvin (K) Often use equivalent voltage of the temperature T e (volts) = kt e(kelvins) e where k = Boltzmann s constant = J/K 1 V 11, 600 K Pressure unit is pascal (Pa); 1 Pa = 1 N/m 2 Atmospheric pressure 1 bar 10 5 Pa 760 Torr 1 Pa 7.5 mtorr 6 LiebermanShortCourse15

15 DENSITY VERSUS TEMPERATURE 7 LiebermanShortCourse15

16 RELATIVE DENSITIES AND ENERGIES 8 LiebermanShortCourse15

17 NON-EQUILIBRIUM Energy coupling between electrons and heavy particles is weak Input power Electrons weak Ions strong Walls Walls weak weak strong Neutrals strong Walls Electrons are not in thermal equilibrium with ions or neutrals T e T i Bombarding ion E i T e in plasma bulk at wafer surface High temperature processing at low temperatures 1. Wafer can be near room temperature 2. Electrons produce free radicals = chemistry 3. Electrons produce electron-ion pairs = ion bombardment 9 LiebermanShortCourse15

18 ELEMENTARY DISCHARGE BEHAVIOR Uniform density of electrons and ions n e and n i at time t = 0 Low mass warm electrons quickly drain to the wall, forming sheaths Bulk plasma is quasi-neutral = n e n i Ions accelerated to walls; ion bombarding energy E i = plasma-wall potential V p 10 LiebermanShortCourse15

19 SUMMARY OF FUNDAMENTALS 11 LiebermanShortCourse15

20 POISSON S EQUATION An electric field can be generated by charges E = ρ or ǫ Ē da = Q encl 0 S ǫ 0 For slow time variations (dc, rf, but not microwaves) E Φ E = electric field (V/m), ρ = charge density (C/m 3 ), Φ = potential (V), ǫ 0 = F/m In 1D planar geometry de x dx = ρ ǫ 0, dφ dx = E x Combining these yields Poisson s equation d 2 Φ dx 2 = ρ ǫ 0 This field powers a capacitive discharge or the wafer bias power of an inductive or ECR discharge V rf E Q encl S ~ E 12 LiebermanShortCourse15

21 FARADAY S LAW An electric field can be generated by a time-varying magnetic field E = B or E dl = B da t C t A I rf A C B B = magnetic induction vector This field powers the coil of an inductive discharge (top power) E I rf ~ E E 13 LiebermanShortCourse15

22 AMPERE S LAW Both conduction currents and displacement currents generate magnetic fields H H = J c + ǫ 0 E t = J [A/m2 ] J c = conduction current density (physical motion of charges) ǫ 0 E/ t = displacement current density (flows in vacuum) J = total current density Note the vector identity ( H) = 0 J = 0 In 1D J(x, t) x = 0 Total current J is independent of x 14 LiebermanShortCourse15

23 REVIEW OF PHASORS Physical voltage (or current), a real sinusoidal function of time V (t) V 0 V (t) = V 0 cos(ωt + φ) 0 2π ωt Phasor voltage (or current), a complex number, independent of time V I Ṽ = V 0 e jφ = V R + jv I V 0 φ V R Note that V (t) = Re (Ṽ ) e jωt Hence V (t) Ṽ (given ω) 15 LiebermanShortCourse15

24 THERMAL EQUILIBRIUM PROPERTIES Electrons generally near thermal equilibrium Ions generally not in thermal equilibrium Maxwellian distribution of electrons ( ) 3/2 m f e (v) = n e exp( mv2 2πkT e 2kT e where v 2 = vx 2 + vy 2 + vz 2 f e (v x ) ) v Te = (kt e /m) 1/2 v x Pressure p = nkt For neutral gas at room temperature (300 K) n g (cm 3 ) p(torr) 16 LiebermanShortCourse15

25 AVERAGES OVER MAXWELLIAN DISTRIBUTION Average energy 1 2 mv2 = 1 n e d 3 v 1 2 mv2 f e (v) = 3 2 kt e Average speed ( ) 1/2 8kTe v e = (= 1ne πm Average electron flux lost to a wall z Γ e [m 2 s 1 ] y ) d 3 v vf e (v) Γ e = 1 4 n e v e (= x ) dv x dv y dv z v z f e (v) Average kinetic energy lost per electron lost to a wall E e = 2 T e 0 17 LiebermanShortCourse15

26 FORCES ON PARTICLES For a unit volume of electrons (or ions) du e mn e = qn e E p e mn e ν m u e dt mass acceleration = electric field force + + pressure gradient force + friction (gas drag) force m = electron mass n e = electron density u e = electron flow velocity q = e for electrons (+e for ions) E = electric field p e = n e kt e = electron pressure ν m = collision frequency of electrons with neutrals [p. 36] p e p e p e (x) p e (x + dx) Drag force u e Neutrals x x + dx x 18 LiebermanShortCourse15

27 BOLTZMANN FACTOR FOR ELECTRONS If electric field and pressure gradient forces almost balance 0 en e E p e Let E = Φ and p e = n e kt e Φ = kt e n e e n e Put kt e /e = T e (volts) and integrate to obtain Φ n e (r) = n e0 e Φ(r)/T e n e x n e0 x 19 LiebermanShortCourse15

28 UNDERSTANDING BEHAVIOR The field equations and the force equations are coupled Fields, Potentials Maxwell's Equations Newton's Laws Charges, Currents 20 LiebermanShortCourse15

29 DEBYE LENGTH λ De The characteristic length scale of a plasma Low voltage sheaths few Debye lengths thick Let s consider how a sheath forms near a wall Electrons leave plasma before ions and charge wall negative n n e = n i = n 0 n + + Φ n i = n 0 + n e Electrons Sheath region ~ few λ De Quasi-neutral bulk plasma x x x Φ 0 Assume electrons in thermal equilibrium and stationary ions 21 LiebermanShortCourse15

30 DEBYE LENGTH λ De (CONT D) Newton s laws [p. 18] n e (x) = n 0 e Φ/T e, n i = n 0 Use in Poisson s equation [p. 12] d 2 Φ ( ) dx 2 = en 0 1 e Φ/T e ǫ 0 Linearize e Φ/T e 1 + Φ/T e d 2 Φ dx 2 = en 0 Φ ǫ 0 T e Solution is ( ) 1/2 ǫ0 T Φ(x) = Φ 0 e x/λ e De, λ De = en 0 In practical units λ De (cm) = 740 T e /n 0, T e in volts, n 0 in cm 3 Example At T e = 1 V and n 0 = cm 3, λ De = cm = Sheath is 0.15 mm thick (Very thin!) 22 LiebermanShortCourse15

31 ELECTRON FREQUENCY ω pe The fundamental timescale for a plasma Consider a plasma slab (no walls). Displace all electrons to the right a small distance x e0, and release them Charge/area + en 0 x e Ions + Electrons E(x) x e E Charge/area en 0 x e Maxwell s equations (parallel plate capacitor) [p. 12] E = en 0x e (t) ǫ 0 x 23 LiebermanShortCourse15

32 ELECTRON FREQUENCY ω pe (CONT D) Newton s laws (electron motion) [p. 18] m d2 x e (t) dt 2 = ee = e2 n 0 x e (t) ǫ 0 Solution is electron plasma oscillations x e (t) = x e0 cosω pe t, ω pe = Practical formula is f pe (Hz) = 9000 n 0, ( e 2 ) 1/2 n 0 ǫ 0 m n 0 in cm 3 = microwave frequencies (> 1 GHz) for typical plasmas 24 LiebermanShortCourse15

33 1D SIMULATION OF SHEATH FORMATION Particle-in-cell (PIC) simulations with uniform fixed ion density; 4000 electron sheets; solve Newton s laws + Maxwell s equations T e = 1 V (random), n e = n i = m 3 (low), l = 0.1 m Electron v x x phase space at t = 0.77 µs Note absence of electron sheets near the walls 25 LiebermanShortCourse15

34 1D SIMULATION OF SHEATH FORMATION (CONT D) Electron number N versus t Note 340 electron sheets lost to walls to form sheaths 26 LiebermanShortCourse15

35 1D SIMULATION OF SHEATH FORMATION (CONT D) Electron density n e (x) at t = 0.77 µs Note sheath width is a few Debye lengths 27 LiebermanShortCourse15

36 1D SIMULATION OF SHEATH FORMATION (CONT D) Electric field E(x) at t = 0.77 µs Note electric field retards electrons, accelerates ion into walls 28 LiebermanShortCourse15

37 1D SIMULATION OF SHEATH FORMATION (CONT D) Potential Φ(x) at t = 0.77 µs Note plasma potential builds up to a few T e with respect to wall 29 LiebermanShortCourse15

38 1D SIMULATION OF SHEATH FORMATION (CONT D) Right hand potential Φ(x = l) versus t Due to asymmetric electron initial conditions, a small oscillation of the right hand potential is excited at the plasma frequency 30 LiebermanShortCourse15

39 DIELECTRIC CONSTANT ǫ p RF discharges are driven at a frequency ω E(t) = Re (Ẽ ejωt ), etc. [p. 15] Define ǫ p from the total current in Maxwell s equations [p. 14] H = J c + jωǫ 0 Ẽ }{{} jωǫ pẽ Total current J Conduction current J c = en e ũ e is due to electrons Newton s law (electric field and neutral drag) is [p. 18] jωmũ e = eẽ mν mũ e Solve for ũ e and evaluate J c to obtain ǫ p ǫ 0 κ p = ǫ 0 [1 ω 2 pe ω(ω jν m ) with ω pe = (e 2 n e /ǫ 0 m) 1/2 the electron plasma frequency [p. 23] For ω ν m, ǫ p is mainly real (nearly lossless dielectric) For ν m ω, ǫ p is mainly imaginary (very lossy dielectric) ] 31 LiebermanShortCourse15

40 RF FIELDS IN LOW PRESSURE DISCHARGES Consider mainly lossless plasma (ω ν m ) ( ) ǫ p = ǫ 0 1 ω2 pe ω 2 For rf discharges, ω pe ω = ǫ p is negative (ǫ p = 1000 ǫ 0 ) RF current density J is continuous across the discharge [p. 14] Sheath ǫ 0 J Plasma ǫ p (continuous) Sheath ǫ 0 E= ~ J ~ E= ~ ~ J ~ E= jωεp J ~ jωε 0 jωε 0 Electric field in plasma is 1000 smaller than in sheaths! Although field in plasma is small, it sustains the plasma! 32 LiebermanShortCourse15

41 CONDUCTIVITY σ p It is useful to introduce rf plasma conductivity J c σ p Ẽ Find J c to be a linear function of Ẽ [p. 31] σ p = e 2 n e m(ν m + jω) DC plasma conductivity (ω ν m ) σ dc = e2 n e mν m The plasma dielectric constant and conductivity are related by jωǫ p = σ p + jωǫ 0 RF current flowing through the plasma heats electrons (just like a resistor) 33 LiebermanShortCourse15

42 OHMIC HEATING POWER Time average power absorbed/volume p d = J(t) E(t) = 1 2 Re( J Ẽ ) [W/m 3 ] Here Ẽ = complex conjugate of Ẽ Since J is the same everywhere in the discharge [p. 32], put Ẽ = J/jωǫ p to find p d in terms of J alone For discharges with ω ω pe (all rf discharges) p d = 1 2 J 2 1 σ dc [W/m 3 ] 34 LiebermanShortCourse15

43 SUMMARY OF DISCHARGE FUNDAMENTALS 35 LiebermanShortCourse15

44 ELECTRON COLLISIONS WITH ARGON Maxwellian electrons collide with Ar atoms (density n g ) # collisions of a particular kind s-m 3 = νn e = Kn g n e ν = collision frequency [s 1 ], K(T e ) = rate coefficient [m 3 /s] Electron-Ar collision processes e + Ar Ar + + 2e (ionization) e + Ar e + Ar e + Ar + photon (excitation) e e + Ar e + Ar (elastic scattering) e Rate coefficient K(T e ) is average of cross section σ(v R ) [m 2 ] for process, over Maxwellian distribution K(T e ) = σ v R Maxwellian v R = relative velocity of colliding particles Ar Ar 36 LiebermanShortCourse15

45 ELECTRON-ARGON RATE COEFFICIENTS 37 LiebermanShortCourse15

46 ION COLLISIONS WITH ARGON Argon ions collide with Ar atoms Ar + + Ar Ar + + Ar (elastic scattering) Ar + + Ar Ar + Ar + (charge transfer) Total cross section for room temperature ions σ i cm 2 Ion-neutral mean free path (distance ion travels before colliding) λ i = 1 n g σ i Practical formula Ion-neutral collision frequency λ i (cm) = 1, p in Torr 330 p ν i = v i λ i with v i = (8kT i /πm) 1/2 38 LiebermanShortCourse15 Ar + Ar + Ar Ar Ar + Ar Ar Ar +

47 THREE ENERGY LOSS PROCESSES 1. Collisional energy E c lost per electron-ion pair created K iz E c = K iz E iz + K ex E ex + K el (2m/M)(3T e /2) = E c (T e ) (voltage units) E iz, E ex, and (3m/M)T e are energies lost by an electron due to an ionization, excitation, and elastic scattering collision 2. Electron kinetic energy lost to walls [p. 17] E e = 2 T e 3. Ion kinetic energy lost to walls is mainly due to the dc potential V s across the sheath E i V s Total energy lost per electron-ion pair lost to walls E T = E c + E e + E i 39 LiebermanShortCourse15

48 COLLISIONAL ENERGY LOSSES 40 LiebermanShortCourse15

49 BOHM (ION LOSS) VELOCITY u B u B Plasma Sheath Wall Density n s Due to formation of a presheath, ions arrive at the plasma-sheath edge with directed energy kt e /2 1 2 Mu2 B = kt e 2 Electron-ion pairs are lost at the Bohm velocity at the plasma-sheath edge (density n s ) Γ wall = n s u B, u B = ( kte M ) 1/2 41 LiebermanShortCourse15

50 AMBIPOLAR DIFFUSION AT HIGH PRESSURES Plasma bulk is quasi-neutral (n e n i = n) and the electron and ion loss fluxes are equal (Γ e Γ i Γ) Fick s law Γ = D a n with ambipolar diffusion coefficient D a = kt e /Mν i Density profile is sinusoidal n 0 Γ wall Γ wall l/2 0 l/2 n s x Loss flux to the wall is Γ wall = n s u B h l n 0 u B From diffusion theory, edge-to-center density ratio is h l n s = π u B n 0 l ν i Applies for pressures > 100 mtorr in argon 42 LiebermanShortCourse15

51 AMBIPOLAR DIFFUSION AT LOW PRESSURES The diffusion coefficient is not constant Density profile is relatively flat in the center and falls sharply near the sheath edge n 0 Γ wall Γ wall l/2 0 l/2 n s x The edge-to-center density ratio is h l n s n (3 + l/2λ i ) 1/2 where λ i = ion-neutral mean free path [p. 38] Applies for pressures < 100 mtorr in argon 43 LiebermanShortCourse15

52 AMBIPOLAR DIFFUSION IN LOW PRESSURE CYLINDRICAL DISCHARGE n sr = h R n 0 R Plasma n e = n i = n 0 n sl = h l n 0 For a cylindrical plasma of length l and radius R, loss fluxes to axial and radial walls are Γ axial = h l n 0 u B, Γ radial = h R n 0 u B where the edge-to-center density ratios are 0.86 h l (3 + l/2λ i ) 1/2, h 0.8 R (4 + R/λ i ) 1/2 Applies for pressures < 100 mtorr in argon l 44 LiebermanShortCourse15

53 ANALYSIS OF DISCHARGE EQUILIBRIUM 45 LiebermanShortCourse15

54 PARTICLE BALANCE AND T e Assume uniform cylindrical plasma absorbing power P abs R P abs Plasma n e = n i = n 0 Particle balance l Production due to ionization = loss to the walls K iz n g n/ 0 πr 2 l = (2πR 2 h l n/ 0 + 2πRlh R n/ 0 )u B Solve to obtain K iz (T e ) u B (T e ) = 1 n g d eff where d eff = 1 Rl 2 Rh l + lh R is an effective plasma size Given n g and d eff = electron temperature T e T e varies over a narrow range of 2 5 volts 46 LiebermanShortCourse15

55 ELECTRON TEMPERATURE IN ARGON DISCHARGE 47 LiebermanShortCourse15

56 ION ENERGY FOR LOW VOLTAGE SHEATHS E i = energy entering sheath + energy gained traversing sheath Ion energy entering sheath = T e /2 (voltage units) [p. 41] Sheath voltage determined from particle conservation Γ i Density n s Plasma Sheath ~ 0.2 mm Insulating wall Γ i Γ e + V s [p. 41] [p. 17] [p. 19] Γ i = n s u B, Γ e = 1 4 n s v e }{{} e V s/t e with v e = (8eT e /πm) 1/2 Random flux at sheath edge The ion and electron fluxes at the wall must balance V s = T ( ) e M 2 ln 2πm or V s 4.7 T e for argon Accounting for the initial ion energy, E i 5.2 T e 48 LiebermanShortCourse15

57 ION ENERGY FOR HIGH VOLTAGE SHEATHS Large ion bombarding energies can be gained near rf-driven electrodes embedded in the plasma s V rf ~ C large Plasma V s ~ 0.4 V rf + + V s V s Low voltage sheath ~ 5.2 T e V rf ~ Plasma V s ~ 0.8 V rf V s + The sheath thickness s ( 0.5 cm) is given by the Child Law J i = en s u B = 4 ( ) 1/2 2e 9 ǫ V 3/2 s 0 M s 2 Estimating ion energy is not simple as it depends on the type of discharge and the application of bias voltages 49 LiebermanShortCourse15

58 POWER BALANCE AND n 0 Assume low voltage sheaths at all surfaces [p. 40] [p. 17] [p. 48] E T (T e ) = E c (T e ) }{{} + 2 T e }{{} T e }{{} Power balance Solve to obtain Collisional Electron Ion Power in = power out P abs = (h l n 0 2πR 2 + h R n 0 2πRl) u B ee T P abs n 0 = A eff u B ee T where A eff = 2πR 2 h l + 2πRlh R is an effective area for particle loss [V] [W] Density n 0 is proportional to the absorbed power P abs Density n 0 depends on pressure p through h l, h R, and T e 50 LiebermanShortCourse15

59 PARTICLE AND POWER BALANCE Particle balance = electron temperature T e (independent of plasma density) Power balance = plasma density n 0 (once electron temperature T e is known) 51 LiebermanShortCourse15

60 EXAMPLE 1 Let R = 0.15 m, l = 0.3 m, n g = m 3 (p = 1 mtorr at 300 K), and P abs = 800 W Assume low voltage sheaths at all surfaces Find λ i = 0.03 m [p. 38]. Then h l h R 0.3 [p. 44] and d eff 0.17 m [p. 46] T e versus n g d eff figure gives T e 3.5 V [p. 47] E c versus T e figure gives E c 42 V [p. 40]. Adding E e = 2T e 7 V and E i 5.2T e 18 V yields E T = 67 V [p. 39] Find u B m/s [p. 41] and find A eff 0.13 m 2 [p. 50] Power balance yields n m 3 [p. 50] Ion current density J il = eh l n 0 u B 2.9 ma/cm 2 [p. 46] Ion bombarding energy E i 18 V [p. 48] 52 LiebermanShortCourse15

61 EXAMPLE 2 Apply a strong dc magnetic field along the cylinder axis = particle loss to radial wall is inhibited Assume no radial losses, then d eff = l/2h l 0.5 m From the T e versus n g d eff figure, T e 3.3 V (was 3.5 V) From the E c versus T e figure, E c 46 V. Adding E e = 2T e 6.6 V and E i 5.2T e 17 V yields E T = 70 V Find u B m/s and find A eff = 2πR 2 h l m 2 Power balance yields n m 3 (was m 3 ) Ion current density J il = eh l n 0 u B 7.8 ma/cm 2 Ion bombarding energy E i 17 V = Slight decrease in electron temperature T e = Significant increase in plasma density n 0 EXPLAIN WHY! What happens to T e and n 0 if there is a sheath voltage V s = 500 V at each end plate? 53 LiebermanShortCourse15

62 CAPACITIVE RF DISCHARGES SYMMETRIC HOMOGENEOUS MODEL 54 LiebermanShortCourse15

63 BASIC PROPERTIES Simplicity of concept RF rather than microwave powered Inherent high sheath voltages No independent control of plasma density and ion energy Control parameters RF current Ĩrf (1 10 ma/cm 2 ) Driving frequency ω ( MHz) Neutral gas density n g ( cm 3 ) Electrode separation l (1 10 cm) Discharge parameters to find Plasma density n ( cm 3 ) Electron temperature T e (2 4 V) Discharge voltage V rf ( V) Discharge power P rf ( W) Ion bombarding energy E i ( V) 55 LiebermanShortCourse15

64 CONFIGURATIONS Multi-wafer parallel plate and hex configurations (1980 s) Modern configurations are single wafer parallel plate, sometimes driven at multiple rf frequencies 56 LiebermanShortCourse15

65 CURRENT-DRIVEN HOMOGENEOUS MODEL 57 LiebermanShortCourse15

66 HOMOGENEOUS MODEL ASSUMPTIONS No transverse variations (along the electrodes) Electrons respond to instantaneous electric fields Ions respond to only time-average electric fields Electron density is zero in the sheath regions Ion density is constant in the plasma and sheath regions n i (z) = n 0 (We will correct this later) 58 LiebermanShortCourse15

67 ELECTRON SHEATH EDGE MOTION n Electrode a I rf (t) I ap (t) n e n i =n Plasma p 0 s a (t) s m z The electric field is found by integrating the charge density in the sheath [p. 12] de dz = en, z < s a (t) ǫ 0 to obtain E(z, t) = en ǫ 0 ( z sa (t) ) E + V ap sa (t) E(z,t) z 59 LiebermanShortCourse15

68 ELECTRON SHEATH EDGE MOTION (CONT D) The displacement current in the sheath is [p. 14] I ap = ǫ 0 A E t = enads a dt Let I rf (t) = Ĩ0 cosωt and integrate to obtain s a (t) = s 0 s 0 sinωt s 0 = Ĩ0 enωa The oscillation amplitude of the sheath motion is s 0, but what is the constant of integration s 0? 60 LiebermanShortCourse15

69 CONDUCTION CURRENT Assume a steady loss of ions to electrode a I i = enu B A The time-average total conduction current to the electrode is zero Hence electrons must be lost to the electrode The sheath thickness s a (t) must then collapse to zero at some time during the rf cycle = s 0 = s 0 s a (t) = s 0 (1 sinωt) ~ s m =2s 0 s(t) I i I e (t) 0 0 I i I e 0 0 ωt 2π 61 LiebermanShortCourse15

70 VOLTAGE ACROSS THE SHEATH n Electrode a I rf (t) I ap (t) n e n i =n Plasma p 0 s a (t) s m z E + V ap sa (t) E(z,t) z V ap (t) V s a (t) V(z,t) The voltage is found by integrating the electric field in the sheath dv dz = E z 62 LiebermanShortCourse15

71 VOLTAGE ACROSS THE SHEATH (CONT D) Integrating the electric field in the sheath [p. 12] dv dz = E we obtain sa (t) V ap (t) = E(z, t) dz = en s 2 a(t) ǫ 0 2 Using s a (t) [p. 61] 0 V ap (t) = en s 2 ( ) sinωt 2ǫ 0 V ap (t) is a nonlinear function of I rf ; there are second harmonics 63 LiebermanShortCourse15

72 VOLTAGE ACROSS BOTH SHEATHS By symmetry s b (t) = s 0 (1 + sinωt); since s a (t) = s 0 (1 sinωt) s a (t) + s b (t) = 2 s 0 = const + V ab (t) I rf (t) s a (t) Rigid Electron Cloud s b (t) I rf (t) a p b There is a rigid bulk electron cloud oscillation 64 LiebermanShortCourse15

73 VOLTAGE ACROSS BOTH SHEATHS (CONT D) Voltage across sheath b is V bp (t) = en s 2 ( ) sinωt 2ǫ 0 Voltage across the plasma is small because I rf = jωǫ p EA and ǫ p is large = E across bulk plasma is small Discharge voltage is V rf = V ap + V pb V rf (t) = 2en s2 0 ǫ 0 sinωt Each sheath is nonlinear, but the combination of both sheaths is linear 65 LiebermanShortCourse15

74 DC VOLTAGE ACROSS ONE SHEATH n Electrode a I rf (t) I ap (t) n e n i =n Plasma p 0 s a (t) s m z V pa (t) = en s 2 ( sinωt + sin 2 ωt ) 2ǫ 0 Take time average V s = 3 en s 2 0 = E i 4 ǫ 0 Compare to rf voltage across discharge [p. 65] = V s = 3 8Ṽrf We can think of Ṽrf as divided equally across the two sheaths V s = 3 4Ṽs with Ṽ s = 1 2 V rf 66 LiebermanShortCourse15

75 SHEATH VOLTAGES VERSUS TIME Sheath voltages V ap (t), V pb (t), and their sum V ab (t) = V rf (t); the time average V s of V pb (t) is also shown 67 LiebermanShortCourse15

76 SHEATH POTENTIAL VERSUS POSITION AT VARIOUS TIMES V rf (t) = sin ωt Spatial variation of the total potential Φ (solid curves) for the homogeneous model at four different times during the rf cycle. The dashed curve shows the spatial variation of the timeaverage potential Φ V s Φ 1/4 0 Φ 1 3/8 I rf (t) Φ ω t = 0 π 2 Φ 1/4 π Φ 3π LiebermanShortCourse15

77 SHEATH CAPACITANCE Define total discharge capacitance by I rf (t) = Ĩ0 cosωt, V rf (t) = 2en s2 0 ǫ 0 sinωt [p. 65] which yields I rf = C s dv rf dt with C s = ǫ 0A 2 s 0 We can think of each sheath as having a capacitance C a = C b = ǫ 0A s 0 We now have a lossless discharge model a p b I rf (t) C a C b 69 LiebermanShortCourse15

78 OHMIC AND STOCHASTIC HEATING Ohmic heating in the bulk plasma [p. 34] P Ω = 1 2 J rf 2mν md e 2 n A Stochastic heating by oscillating sheaths with Electrode a n n i n e v v 0 s a (t) z v = v + 2 u s (t) [watts] u s (t) u s (t) = ds a dt = ũ 0 cosωt with ũ 0 = ω s 0 Average energy transferred [ is 1 E e = 2 m( v + 2 u s (t) ) ] mv2 = mũ 2 0 E e is positive, so the oscillating sheath heats electrons 70 LiebermanShortCourse15 v v

79 STOCHASTIC HEATING POWER For a Maxwellian distribution, the electron flux incident on a sheath is [p. 17] Γ e = 1 4 n v e with v e = (8eT e /πm) 1/2 the mean electron speed The time-average stochastic heating power is found to be P sa = Γ e A 2 E e = 1 2 mn v eω 2 s 2 0 A [watts] This is a powerful electron heating mechanism in a capacitive discharge 71 LiebermanShortCourse15

80 HEATING POWERS VERSUS DRIVING VOLTAGE For stochastic heating [p. 71] P sa = 1 2 mn v eω 2 s 2 0 A Using Ṽrf = 2en s 2 0/ǫ 0 [p. 65] we obtain for the two sheaths P s = P sa + P sb = m 2e ǫ 0 v e ω 2 Ṽ rf A For bulk ohmic heating [p. 70] P Ω = 1 2 J rf 2mν md e 2 n A Using J rf = enω s 0 and Ṽrf given above P Ω = m 4e ǫ 0ν m d ω 2 Ṽ rf A Ohmic and stochastic heating powers depend on Ṽrf 72 LiebermanShortCourse15

81 CAPACITIVE RF DISCHARGES SELF-CONSISTENT SHEATH RESULTS 73 LiebermanShortCourse15

82 HOMOGENEOUS AND CHILD LAW SHEATHS Electrode Electrode Homogeneous Model n n i n e s(t) s m Bulk plasma Self-Consistent Child Law n n ine s(t) n e s m z Bulk plasma Child law ion density decreases and sheath width increases compared to homogeneous model z 74 LiebermanShortCourse15

83 COLLISIONLESS CHILD LAW SHEATH + V s n i Plasma Sheath Wall J i = en s u B n e n i s m n e 0 Larger sheath width = larger sheath oscillation velocity u s ωs m Stochastic heating is larger than for homogeneous model Collisionless ion motion = Child law relating s m to n s and V s ( ) 1/2 2e V 3/2 s J i = en s u B = 0.82 ǫ 0 M s 2 m RF voltage Ṽs across sheath = dc voltage V s V s 0.83 Ṽs 75 LiebermanShortCourse15

84 SUMMARY SELF-CONSISTENT MODEL + ~ ~ V rf =2V s ~ I rf 0 J i n 0 n s Sheath Plasma Sheath V s + + V s s m ~ I rf E i = V s = 0.83 ( Ṽs ) 1/2 2e V 3/2 s J i = en s u B = 0.82 ǫ 0 M s 2 m Ĩ rf = 1.23 jω ǫ 0A Ṽ s s m P s = 1.12 m 2e ω2 ǫ 0 v e Ṽ s A P Ω = 1.73 n s n 0 m 2e ω2 ǫ 0 ν m d (T e Ṽ s ) 1/2 A 76 LiebermanShortCourse15

85 CAPACITIVE RF DISCHARGES SIMULATION AND EXPERIMENTAL RESULTS 77 LiebermanShortCourse15

86 PIC SIMULATION OF DENSITIES Symmetric rf discharge with right hand electrode grounded, V rf = 1 kv at 10 MHz, 20 mtorr hydrogen gas (Thesis of D. Vender, Australian National University, 1990) 78 LiebermanShortCourse15

87 PHASE SPACE AND POTENTIALS VERSUS POSITION Symmetric rf discharge with right hand electrode grounded, V rf = 1 kv at 10 MHz, 20 mtorr hydrogen gas; left panels show electron and ion phase space; right panels show potentials (see [p. 68] for comparison to theory) (Thesis of D. Vender, Australian National University, 1990) 79 LiebermanShortCourse15

88 TIME VARIATION OF VOLTAGES AND CURRENTS V (kv) J e 10 MHz V pb J(A/m 2 ) 0 0 V rf J i ( 10) 1.0 (Thesis of D. Vender, ANU, 1990) π 4π Phase ωt Note plasma always positive with respect to both electrodes Note steady ion current [p. 61] Note pulsed electron current when V pa = V pb V rf 0 [p. 61] 80 LiebermanShortCourse15

89 SPACE-TIME DISTRIBUTION OF IONIZING COLLISIONS The darkness of each square is proportional to the number of ionizing collisions within that square of time and position intervals; symmetric rf discharge with right hand electrode grounded, V rf = 1 kv at 10 MHz, 20 mtorr hydrogen gas (Thesis of D. Vender, Australian National University, 1990) 81 LiebermanShortCourse15

90 MEASUREMENTS OF STOCHASTIC HEATING Effective collision frequency ν eff versus pressure p, for a mercury discharge driven at 40.8 MHz; the solid line shows the collision frequency due to ohmic dissipation alone (Popov and Godyak, 1985) 82 LiebermanShortCourse15

91 CAPACITIVE RF DISCHARGES EXAMPLE EQUILIBRIUM CALCULATIONS 83 LiebermanShortCourse15

92 Electron power balance where Specify Ṽs = n s Total power balance POWER BALANCE El. col + kin {}}{ P Ω + 2P s = en s u B 2A (E c + 2T e ) P Ω = 1.73 h l m 2e ω2 ǫ 0 (ν m d)(t e Ṽ s ) 1/2 A P s = 0.56 m 2e ω2 ǫ 0 v e Ṽ s A El. { col.+kin. }}{{ Ion}} kin. { P abs = en s u B 2A (E c + 2T e Ṽs) Eliminate n s from electron and total power balance ( ) P abs (P Ω + 2P s ) Ṽs E c + 2T e Specify P abs = Ṽs In this case, electron or total power balance = n s 84 LiebermanShortCourse15

93 EXAMPLE 1 Let p = 3 mtorr argon at 300 K, l = 10 cm, A = 1000 cm 2, f = MHz (ω = s 1 ), and V rf = 500 V Start with estimate s m 1 cm Ion mean free path λ i = 1/n g σ i 1.0 cm [p. 38] With bulk plasma thickness d = l 2s m = 8 cm, λ i /d h l = n s /n [p. 43] and d eff = d/2h l = 12.3 cm [p. 46] With n g d eff m 2, the T e versus n g d eff figure [p. 47] yields T e 3.1 V Bohm velocity u B m/s [p. 41] E c versus T e figure [p. 40] yields E c 47 V and E c + 2T e 53 V. Use the K el versus T e figure [p. 37] to find ν m K el n g s 1 85 LiebermanShortCourse15

94 EXAMPLE 1 (CONT D) Evaluate ohmic and stochastic electron heating [p. 76 or 84] P Ω Ṽ s 1/2 [W] P s Ṽs [W] Use Ṽs V rf /2 = 250 V in above to find P Ω W and P s 3.03 W Electron power balance [p. 84] yields n s m 3 Since h l 0.325, n m 3 Using E i 0.83 Ṽs [p. 76] yields E i 208 V J i = en s u B 0.59 A/m 2 [p. 76] The Child law [p. 76] gives s m 0.90 cm J rf 1.23 ωǫ 0 Ṽ s /s m 25.8 A/m 2 [p. 76] Total power balance [p. 84] gives P abs 30.8 W s m reasonably close to the initial estimate = iteration over d is not useful 86 LiebermanShortCourse15

95 EXAMPLE 2 Let p = 3 mtorr argon at 300 K, l = 10 cm, A = 1000 cm 2, f = MHz and P abs = 200 W As before, h l 0.325, T e 3.1 V, u B m/s, and E c + 2T e 53 V. Because n g and T e are the same, P Ω and P s are the same functions of Ṽs as in EXAMPLE 1 Using P abs = 200 W, we obtain the equation for the rf sheath voltage Ṽs [p. 84] 200 = ( Ṽ 1/2 s Ṽs ) ( Ṽs 53 A numerical solution gives Ṽs = 687 V Then V rf 2Ṽs 1374 V and E i = 0.83 Ṽs 570 V Use this in total power balance [p. 84] to find n s m 3 and n m 3 We then find J i 1.6 A/m 2, s m 1.16 cm, and J rf 54.9 A/m 2 ) 87 LiebermanShortCourse15

96 CAPACITIVE RF DISCHARGES ASYMMETRIC SYSTEMS 88 LiebermanShortCourse15

97 ASYMMETRIC RF DISCHARGE Powered electrode area A a smaller than grounded area A b ~ V rf ~ C large Powered a Sheath a Plasma Sheath b Grounded b V a + + V b + V bias V a = dc sheath voltage from plasma to powered electrode a V b = dc sheath voltage from plasma to grounded electrode b Ṽ a = rf voltage amplitude across sheath a Ṽ b = rf voltage amplitude across sheath b V a = 0.83 Ṽa, V b = 0.83 Ṽb, Ṽ rf = Ṽa + Ṽb A negative DC bias voltage V bias = V b V a appears 89 LiebermanShortCourse15

98 DEPENDENCE OF VOLTAGES ON AREAS Given Ṽrf, A a and A b, what are V a, V b and V bias? Voltage ratio V a /V b depends on area ratio A b /A a V a V b = q = area ratio scaling exponent Experiments show q Collisionless Child law gives q = 4 ( ) q Ab A a 90 LiebermanShortCourse15

99 COLLISIONLESS CHILD LAW ANALYSIS I rf a +V b V a + Plasma (good conductor) s a n a Electrodes and plasma are good conductors, so V a is the same everywhere along electrode a Capacitive sheath [p. 76] Child law [p. 76] n b I rf V aa a s a n a V 3/2 a s 2 a s b b I rf Eliminating s a I rf V 1/4 a n 1/2 a A a 91 LiebermanShortCourse15

100 COLLISIONLESS CHILD LAW ANALYSIS (CONT D) Similarly I rf V 1/4 b n 1/2 b A b Set currents at sheaths a and b equal and solve for V a /V b ( ) 2 ( ) 4 V a nb Ab = V b n a A a Simplest assumption for plasma is equal densities at the sheath edges a and b V a V b = ( ) 4 Ab A a 92 LiebermanShortCourse15

101 COLLISIONAL CHILD LAW SHEATH SCALING For pressures above 3 10 mtorr, ions suffer collisions with neutrals in the sheaths The Child law is modified to the scaling n a V 3/2 a This leads to s 5/2 a V a V b = ( ) 2.5 Ab A a The weaker scaling q = 2.5 is more in agreement with experiments 93 LiebermanShortCourse15

102 VARIATION OF DENSITIES Low density High density b a ~ V rf ~ C large Plasma density near powered electrode is usually larger than near grounded electrode This leads to additional modifications in scaling 94 LiebermanShortCourse15

103 INDUCTIVE DISCHARGES TRANSFORMER MODEL AND MATCHING 95 LiebermanShortCourse15

104 MOTIVATION High density (compared to capacitive discharge) Independent control of plasma density and ion energy Simplicity of concept RF rather than microwave powered No source magnetic fields 96 LiebermanShortCourse15

105 CYLINDRICAL AND PLANAR CONFIGURATIONS Cylindrical coil Planar coil 97 LiebermanShortCourse15

106 EARLY HISTORY First inductive discharge by Hittorf (1884) Arrangement to test discharge mechanism by Lehmann (1892) 98 LiebermanShortCourse15

107 HIGH DENSITY REGIME Inductive coil launches decaying wave into plasma Decaying wave Ẽ H δ p Plasma z Coil Window Wave decays exponentially into plasma Ẽ = Ẽ0 e z/δ p, δ p = c ω 1 Im(κ 1/2 p ) where κ p = plasma dielectric constant [p. 31] ωpe 2 κ p = 1 ω(ω jν m ) For typical high density, low pressure (ν m ω) discharge δ p c ω pe = ( m e 2 µ 0 n e ) 1/2 1 2 cm 99 LiebermanShortCourse15

108 TRANSFORMER MODEL For simplicity consider a long cylindrical discharge N turn coil Ĩ rf b R δ p Ĩ p Plasma z l Current Ĩrf in N turn coil induces current Ĩp in 1-turn plasma skin = A transformer 100 LiebermanShortCourse15

109 RESISTANCE AND INDUCTANCE Plasma resistance R p R p = 1 circumference of plasma loop σ dc average cross sectional area of loop where σ dc = e2 n es mν m [p. 33] with n es = density at plasma-sheath edge = R p = πr σ dc lδ p Plasma inductance L p magnetic flux produced by plasma current L p = plasma current Using magnetic flux = πr 2 µ 0 Ĩ p /l = L p = µ 0πR 2 l 101 LiebermanShortCourse15

110 Model the source as a transformer COUPLING OF COIL TO Ṽ rf = jωl 11 Ĩ rf + jωl 12 Ĩ p Ṽ p = jωl 21 Ĩ rf + jωl 22 Ĩ p Transformer inductances magnetic flux linking coil L 11 = = µ 0πb 2 N 2 coil current l magnetic flux linking plasma L 12 = L 21 = = µ 0πR 2 N coil current l L 22 = L p = µ 0πR 2 l 102 LiebermanShortCourse15

111 SOURCE CURRENT AND VOLTAGE Put Ṽp = ĨpR p in transformer equations and solve for the impedance Z s = Ṽrf/Ĩrf seen at coil terminals Z s = jωl 11 + ω2 L 2 12 R p + jωl p R s + jωl s Equivalent circuit at coil terminals R s = N 2 πr σ dc lδ p L s = µ 0πR 2 N 2 ( ) b 2 l R 2 1 Power balance = Ĩrf P abs = 1 2Ĩ2 rf R s From source impedance = V rf Ṽ rf = ĨrfZ s 103 LiebermanShortCourse15

112 EXAMPLE Assume plasma radius R = 10 cm, coil radius b = 15 cm, length l = 20 cm, N = 3 turns, gas density n g = cm 3 (20 mtorr argon at 300 K), ω = s 1 (13.56 MHz), absorbed power P abs = 600 W, and low voltage sheaths At 20 mtorr, λ i 0.15 cm, h l h R 0.1, d eff 34 cm [p. 38, 44, 46] Particle balance (T e versus n g d eff figure [p. 47]) yields T e 2.1 V Collisional energy losses (E c versus T e figure [p. 40]) are E c 110 V. Adding E e + E i = 7.2T e yields total energy losses E T 126 V [p. 39] u B cm/s [p. 41] and A eff 185 cm 2 [p. 50] Power balance yields n e cm 3 and n se cm 3 [p. 50] Use n se to find skin depth δ p 2.0 cm [p. 99]; estimate ν m = K el n g (K el versus T e figure [p. 37]) to find ν m s 1 Use ν m and n se to find σ dc 61 Ω 1 -m 1 [p. 33] Evaluate impedance elements R s 23.5 Ω and L s 2.2 µh; Z s ωl s 190 Ω [p. 103] Power balance yields Ĩrf 7.1A; from source impedance Z s = 190 Ω, Ṽ rf 1360 V [p. 103] 104 LiebermanShortCourse15

113 MATCHING DISCHARGE TO POWER SOURCE Consider an rf power source connected to a discharge + Z T Ĩ + Ṽ T ~ Ṽ Z L Source Discharge Source impedance Z T = R T + jx T is given Discharge impedance Z L = R L + jx L Time-average power delivered to discharge P abs = 1 2 Re (Ṽ Ĩ ) For fixed source ṼT and Z T, maximize power delivered to discharge X L = X T R L = R T Maximum time-average power delivered to discharge P abs = 1 ṼT 2 8 R 105 T LiebermanShortCourse15

114 MATCHING NETWORK Insert lossless matching network between power source and coil Power source Matching network Discharge coil Continue EXAMPLE [p. 104] with R s = 23.5 Ω and ωl s = 190 Ω; assume R T = 50 Ω (corresponds to a conductance 1/R T = 1/50 S) Choose C 1 such that the conductance seen looking to the right at terminals AA is 1/50 S = C 1 = 71 pf Choose C 2 to cancel the reactive part of the impedance seen at AA = C 2 = 249 pf For P abs = 600 W, the 50 Ω source supplies Ĩrf = 4.9 A Voltage at source terminals (AA ) = ĨrfR T = 245 V 106 LiebermanShortCourse15

115 PLANAR COIL DISCHARGE Magnetic field produced by planar coil RF power is deposited in a ring-shaped plasma volume δ p Ĩ rf N turn coil Primary inductance Coupling inductance Ĩ p Plasma Plasma inductance z As for a cylindrical discharge, there is a primary (L 11 ), coupling (L 12 = L 21 ) and secondary (L p = L 22 ) inductance 107 LiebermanShortCourse15

116 PLANAR COIL FIELDS A ring-shaped plasma forms because { 0, on axis Induced electric field = max, at r 1 2 R wall 0, at r = R wall Measured radial variation of B r (and E θ ) at three distances below the window (5 mtorr argon, 500 W, Hopwood et al, 1993) 108 LiebermanShortCourse15

117 INDUCTIVE DISCHARGES POWER BALANCE 109 LiebermanShortCourse15

118 RESISTANCE AT HIGH AND LOW DENSITIES Plasma resistance seen by the coil [p. 103] ω 2 L 2 12 R s = R p Rp 2 + ω2 L 2 p High density (normal inductive operation) [p. 103] R s R p 1 1 σ dc δ p ne Low density (skin depth > plasma size) R s number of electrons in the heating volume n e R s n e Low density High density δ p ~ plasma size n e 1 n e 110 LiebermanShortCourse15

119 POWER BALANCE WITHOUT MATCHING Drive discharge with rf current Power absorbed by discharge is P abs = 1 2 Ĩrf 2 R s (n e ) [p. 110] Power lost by discharge P loss n e [p. 50] Intersection (red dot) gives operating point; let Ĩ1 < Ĩ2 < Ĩ3 P loss Power P abs = 1 2Ĩ2 3R s P abs = 1 2Ĩ2 2R s P abs = 1 2Ĩ2 1R s Inductive operation impossible for Ĩrf Ĩ2 111 LiebermanShortCourse15 n e

120 CAPACITIVE COUPLING OF COIL TO For Ĩrf below the minimum current Ĩ2, there is only a weak capacitive coupling of the coil to the plasma + Ṽ rf Capacitive coupling Ĩ p Plasma A small capacitive power is absorbed = low density capacitive discharge z P loss Power Cap Ind P abs = 1 2Ĩ2 3R s P abs = 1 2Ĩ2 1R s Cap Mode Ind Mode n e 112 LiebermanShortCourse15

121 MEASURMENTS OF ARGON ION DENSITY Above 100 W, discharge is inductive and n e P abs Below 100 W, a weak capacitive discharge is present 113 LiebermanShortCourse15

122 SOURCE EFFICIENCY The source coil has some winding resistance R coil R coil is in series with the plasma resistance R s Power transfer efficiency is η = High efficiency = maximum R s R s R s + R coil R s n e Low density n 1 e High density δ p ~ plasma size Power transfer efficiency decreases at low and high densities Poor power transfer at low or high densities is analogous to poor power transfer in an ordinary transformer with an open or shorted secondary winding n e 114 LiebermanShortCourse15

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