r A SHORT COURSE ON THE PRINCIPLES OF PLASMA DISCHARGES AND MATERIALS PROCESSING Michael A. Lieberman Department of Electrical Engineering and Computer Sciences University of California, Berkeley, CA 94720 Rf or microwave power CF4/02 e, CF3, CF3\ F, 0", 02, CO, SiF4, etc. Plasma Si02 Pump University of California, Berkeley• Lieb er manShort Course 15, -PLA SM A-S /-\ TABLE OF CONTENTS • Introduction to Plasma Discharges and Processing (Ch. 1).........1 • Summary of Plasma Fundamentals (Ch. 2, Sees. 4.1—4.2)......... 11 • Summary of Discharge Fundamentals (Sees. 3.1, 3.5, 5.1-5.3, 6.1-6.3) ... 35 • Analysis of Discharge Equilibrium (Sees. 10-1—10.2)........... 45 • Capacitive RF Discharges — Symmetric Homogeneous Model (Sec. 11.1)............. 54 — Self Consistent Sheath Results (Sec. 11.2) ............. 73 — Simulation and Experimental Results (Sec. 11.3) .......... 77 — Example Equilibrium Calculations (Sec. 11.2)............ 83 — Asymmetric Systems (Sec. 11.4).................. 88 • Inductive RF Discharges — Transformer Model and Matching (Sees. 12.1—12.3)......... 95 — Power Balance (Sees. 12.1-12.2).................. 109 • Capacitive RF Sheaths — Transit Time Effects (Sec. 11.5).................. 115 — Ion Energy Distribution (IED) (Sec. 11.6) ............. 125 i LiebermanShortCoursel5i University of California, Berkeley PL SMA ^ /-\ TABLE OF CONTENTS (CONT'D) • Chemical Fundamentals - Atoms and Molecules (Sec. 3.4, Ch. 8)............... 141 - Gas Phase Kinetics (Sec. 9.1-9.2)................. 164 - Adsorption and Desorption (Sec. 9.3-9.4).............. 169 • Chemistry in Discharges - Neutral Free Radicals (Sec. 10.2)................. 176 - Negative Ions (Sec. 10.3)..................... 182 - Example of Oxygen (Sec. 10.4).................. 187 - Time-Varying Global Models (Sec. 10.4-10.6)............ 196 - Etching Processes (Sees. 15.1-15.2)................ 202 • Plasma-Induced Charging Damage (Sec. 15.5).............. 211 • Pulsed Discharges (Sec. 10.6)..................... 228 • Dual Frequency Capacitive Discharges................. 244 • High Pressure Discharges and Deposition Kinetics (Ch. 16)........ 270 • ERRATA for textbook........................ 304 • About the Instructor ........................ 308 • Self-Study Problems......................... 309 t ii LiebermanShortCourse 15 i ^ University of California, Berkeley PL SMA ^ THE NANOELECTRONICS REVOLUTION • Transistors/chip doubling every 1 J—2 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 x speed of light!) • 20 billion miles/gal • 1 cm long x 3 mm wide University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' 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 LiebermanShort Course 15 j University of California, Berkeley PL SMA 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 Lieb er manShort Course 15 j University of California, Berkeley PL SM A 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 Lieb er manShort Course 15 j University of California, Berkeley PL SM A 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 Lieb er manShort Course 15 j University of California, Berkeley PL SM A INTRODUCTION TO PLASMA DISCHARGES AND PROCESSING University of California, Berkeley i Lieb er man Short Co PL i MA r PLASMAS 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 9 Electrode ^rf© Gas (a) (b) Device sizes ~ 30 cm - 1 m Frequencies from DC to rf (13.56 MHz) to microwaves (2.45 GHz) University of California, Berkeley• Lieber man Short Course 15j -PLA SMA-' r EVOLUTION OF ETCHING DISCHARGES Gas feed Substrate • Vacuum pump Capacitive (Anisotropic etch) Rf Rf bias Inductive Rf 0 Dual frequency capacitive discharge Microwaves Rf bias ECR High frequency rf source Low frequency bias source University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r ANISOTROPIC ETCHING Isotropic etch Directional etch Vertical etch Wet Etching Plasma Etching Ion Enhanced Plasma Etching University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' ISOTROPIC PLASMA ETCHING 1. Start with inert molecular gas CF4 2. Make discharge to create reactive species CF4 —> CF3 + F 3. Species reacts with material, yielding volatile product Si + 4F —► SiF4 T 4. Pump away product 5. CF4 does not react with Si; SiF4 is volatile ANISOTROPIC PLASMA 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 Plasma Ions Mask lllll^ll University of California, Berkeley 5 Lieb er man Short Course 15 -PLASMA-y r UNITS AND CONSTANTS SI units: meters (m), kilograms (kg), seconds (s), coulombs (C) e = 1.6 x 10~19 C, electron charge = —e Energy unit is joule (J) Often use electron-volt 1 eV = 1.6 x 10"19 J Temperature unit is kelvin (K) Often use equivalent voltage of the temperature Te (volts) = kTe (kelvins) where k — Boltzmann's constant = 1.38 x 10 23 J/K 1 V 11,600 K Pressure unit is pascal (Pa); 1 Pa = 1 N/m2 Atmospheric pressure = 1 bar « 105 Pa ~ 760 Torr 1 Pa 7.5 mTorr University of California, Berkeley1 Lieb er man Short Course 151 -PLA SM A-S r PLASMA DENSITY VERSUS TEMPERATURE 25 Solid Si at room temperature Air at STP 20 High pressure arcs Laser plasma Shock tubes 15 Focus Theta pinches Fusion reactor University of California, Berkeley• Lieb er man Short Course 15J -PLASMA-' RELATIVE DENSITIES AND ENERGIES 1016 1012 108 f Etch [. gas f Etch " ^product f Free I radicals Plasma ions 10-2 Capaci ti ve Di scharge Plasma electrons 1 Tor <£>(V) Bombarding ions lO2 1016 (cm-3) 1012 108 Etch gas Etch > ^ product J Free k radicals Plasma ions 10-2 Inductive/ECR Discharge Plasma electrons [ Bombarding ions 1 lO2 Tor <£>(V) University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r NON-EQUILIBRIUM Energy coupling between electrons and heavy particles is weak Input power strong >Wallg strong (Wallg Electrons are not in thermal equilibrium with ions or neutrals Te ^>Ti in plasma bulk Bombarding ion Si ^ Te 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 University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' r ELEMENTARY DISCHARGE BEHAVIOR Uniform density of electrons and ions ne and ni at time t — 0 Low mass warm electrons quickly drain to the wall, forming sheaths Sheaths Bulk plasma is quasi-neutral => ne ^ Ions accelerated to walls; ion bombarding energy Si — plasma-wall potential Vp University of California, Berkeley1 10 Lieb er man Short Course 15 -PLASMA-S r SUMMARY OF PLASMA FUNDAMENTALS 11 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r POISSON'S EQUATION An electric field can be generated by charges v or E-dA = For slow time variations (dc, rf, but not microwaves) E « -V$ E = electric field (V/m), p — charge density (C/m3) $ = potential (V), e0 = 8.85 x 10"12 F/m In ID planar geometry Combining these yields Poisson's equation d2 _P_ dx2 This field powers a capacitive discharge or the wafer bias power of an inductive or ECR discharge y ' xxxxK^ University of California, Berkeley• 12 Lieb er man Short Course 15J -PLA SM A-' r FARADAY'S LAW An electric field can be generated by a time-varying magnetic field VxE = <9B or E.dl = -| B dA A B | -4-1----4----- \ -Lt_ ]C B = magnetic induction vector This field powers the coil of an inductive discharge (top power) I rf e o o o o o 13 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r AMPERE'S LAW Both conduction currents and displacement currents generate magnetic fields H <9E V x H = Jc + e0 dt = J [A/m2] Jc = conduction current density (physical motion of charges) CüdEi/dt — displacement current density (flows in vacuum) J = total current density Note the vector identity V • (V x H) = 0 =>> V • J = 0 In ID lit) //////,(////// v, JSheath__ Nonuniform plasma "Sheath Jit) V7777T, Total current J is independent of x 14 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r REVIEW OF PHASORS Physical voltage (or current), a real sinusoidal function of time V(t) = V0 cos(u;t + (j)) 1-^— Lüt Phasor voltage (or current), a complex number, independent of time Vj V = VQeP* = VR+jVI Note that Hence Vr VR V(t) V (given uj) 15 University of California, Berkeley• Lieb er man Short Course 15 -PLASMA-S r THERMAL EQUILIBRIUM PROPERTIES Electrons generally near thermal equilibrium Ions generally not in thermal equilibrium Maxwellian distribution of electrons fe(v) = ne where v2 — v2 + v2 + v2 ( m 3/2 V 27vkTP fe(Va exp mv 2kTP (kTjm)1/2 Pressure p — nkT For neutral gas at room temperature (300 K) ng(cm-6) « 3.3 x 1016p(Torr) University of California, Berkeley 16 Lieb er man Short Course 15X -PL SM A r AVERAGES OVER MAXWELLIAN DISTRIBUTION Average energy (\mv2) = ^- J d3v^mv2fe(v) = \kTe Average speed (=i/^«/.(«)) Average electron flux lost to a wall ^P [m-V1] r 1 - re = -neve -oo J — oo J 0 dvy / dvzvzfe(v) Average kinetic energy lost per electron lost to a wall &p — 2 f_LL 17 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r FORCES ON PARTICLES For a unit volume of electrons (or ions) mne—— — qnehj — \7pe at mnevm\ie mass x acceleration = electric field force + + pressure gradient force + friction (gas drag) force • m — electron mass ne — electron density ue = electron flow velocity q — — e for electrons (+e for ions) E = electric field pe — nekTe — electron pressure vm — collision frequency of electrons with neutrals [p. 36] Pe (x + dx) Drag force o o o o ° Neutrals o o o x x I djX University of California, Berkeley Lieber man Short Course 1J -PL SMA-' r BOLTZMANN FACTOR FOR ELECTRONS If electric field and pressure gradient forces almost balance 0 « —eneE — Vpe Let E = -V$ and pe = nekTe V$ = —--- e ne Put kTe/e — Te (volts) and integrate to obtain 19 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' UNDERSTANDING PLASMA BEHAVIOR • The field equations and the force equations are coupled Fields, Potentials Maxwell's Equations Charges, Currents 20 University of California, Berkeley• Lieber man Short Course 15, -PLA SMA-S r DEBYE LENGTH ADe 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 ^X ne = m = n0 Electrons -X n ni = n0 _i + ---- 'Sheath region) [~ few XDe J ^Quasi-neutrať [bulk plasma -X -X Assume electrons in thermal equilibrium and stationary ions University of California, Berkeley 21 Lieb er man Short Course 151 -PLASMA-' r DEBYE LENGTH ADe (CONT'D) Newton's laws [p. 18] ne(x) = n0 e$/Te, m = n0 Use in Poisson's equation [p. 12] d2 eno^_e*/Te^ dx2 co Linearize e$/Te « 1 + $/Te d2& eno Solution is dx2 eoTe In practical units Te in volts, no in cm" ADe(cm) = 740^/^, Example At Te = 1 V and n0 = 1010 cm"3, XDe = 7.4 x 10"3 cm => Sheath is ~ 0.15 mm thick (Very thin!) 22 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r ELECTRON PLASMA FREQUENCY cj pe The fundamental timescale for a plasma Consider a plasma slab (no walls). Displace all electrons to the right a small distance xeo, and release them Charge/area eriQX + t— * 0 Xf "Ions" E Electrons" "•r Charge/area '-eriQXe E(x) ■x Maxwell's equations (parallel plate capacitor) [p. 12] eUQXeit) E = 23 University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-S r ELECTRON PLASMA FREQUENCY cjpe (CONT'D) Newton's laws (electron motion) [p. 18] d2xe(t) m- — —eE — — 6 n° xe(t) dt2 eo Solution is electron plasma oscillations Xe(t) = Xe0 COSCJpet, Practical formula is fpe(Rz) — 9000^/no1 no in cm 3 microwave frequencies (> 1 GHz) for typical plasmas 24 University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' r ID SIMULATION OF SHEATH FORMATION Particle-in-cell (PIC) simulations with uniform fixed ion density; 4000 electron sheets; solve Newton's laws + Maxwell's equations Te = 1 V (random), ne = m = 1013 m~3 (low), l = 0.1m Electron vx-x phase space at t — 0.77 fis Phase Space 2e + 06 -2e + 06 Electron Note absence of electron sheets near the walls 25 University of California, Berkeley• 0.1 Lieb er man Short Course 15 -PLASMA-S r ID SIMULATION OF SHEATH FORMATION (CONT'D) Electron number J\f versus t Number 4000.0 M(t) 3660.0 Time 4.68e-0.7 Note 340 electron sheets lost to walls to form sheaths 26 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-^ r ID SIMULATION OF SHEATH FORMATION (CONT'D) Electron density ne(x) at t — 0.77 fis Density 1.5e + 13 n(x) /Electrons Note sheath width is a few Debye lengths Sheath thickness 0.1 27 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r ID SIMULATION OF SHEATH FORMATION (CONT'D) Electric field E(x) at t = 0.77 fis E field 800 EJx) -800 0 x 0.1 Note electric field retards electrons, accelerates ion into walls 28 University of California, Berkeley1 LiebermanShortCoursel5j -P MA r ID SIMULATION OF SHEATH FORMATION (CONT'D) Potential 4>(rc) at t — 0.77 fis 0(x) Potential 0 x 0.1 Note plasma potential builds up to a few Te with respect to wall 29 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ ID SIMULATION OF SHEATH FORMATION (CONT'D • Right hand potential $(x = /) versus t 0.0289 RHS Potential -0.133 Time 4.68e-0.7 Due to asymmetric electron initial conditions, a small oscillation the right hand potential is excited at the plasma frequency 01 University of California, Berkeley • 30 LiebermanShortCoursel5j --P SMA r PLASMA DIELECTRIC CONSTANT e, RF discharges are driven at a frequency uo E(t) = Rje(Ěe?U3t), etc. [p. 15] Define ep from the total current in Maxwell's equations [p. 14] V x H — Jc + jojcqE = juJ6pE S-v-ť Total current J Conduction current Jc — —eneue is due to electrons Newton's law (electric field and neutral drag) is [p. 18] juomúe — —eE — mvmůe Solve for ue and evaluate Jc to obtain ~-p 1 pe with ujpe — (e2ne/eom)1^2 the electron plasma frequency [p. 23] For uj ^ z/m, ep is mainly real (nearly lossless dielectric) For ^ cj, ep is mainly imaginary (very lossy dielectric) 31 University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-S r RF FIELDS IN LOW PRESSURE DISCHARGES Consider mainly lossless plasma (uj ^ vm) For rf discharges, upe ^> u =>> ep is negative (ep — —1000 eo) RF current density J is continuous across the discharge [p. 14] Plasma J (continuous) E= J JG)£E JC080 Sheath E= J JC080 Electric field in plasma is 1000 x smaller than in sheaths! Although field in plasma is small, it sustains the plasma! University of California, Berkeley• 32 Lieber man Short Course 15i -PL SMA-' r PLASMA CONDUCTIVITY aT It is useful to introduce rf plasma conductivity Jc = apE Find Jc to be a linear function of E [p. 31] ap = e2ne m(iym +juj) DC plasma conductivity (cj The plasma dielectric constant and conductivity are related by jCJCp = (7p + JUJ€0 RF current flowing through the plasma heats electrons (just like a resistor) 33 University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-S r- OHMIC HEATING POWER • Time average power absorbed/volume pd = (3(t) ■ E(*)) = X- Re (J • E*) [W/m3] Here E* — complex conjugate of E • Since J is the same everywhere in the discharge [p. 32], put E — J/jüüCp to find pd in terms of J alone • For discharges with uj uope (all rf discharges) Pd = \\A2— [W/m3] 34 University of California, Berkeley1 Lieb er man Short Course 151 -PLA SM A-S r SUMMARY OF DISCHARGE FUNDAMENTALS 35 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r ELECTRON COLLISIONS WITH ARGON Maxwellian electrons collide with Ar atoms (density ng) # collisions of a particular kind -o-= vne — KnQ ne s-ur v — collision frequency [s_1], K(Te) — rate coefficient [m3/s] Electron-Ar collision processes e + Ar —> Ar+ + 2e (ionization) e + Ar —> e + Ar* —> e + Ar + photon (excita^on) e + Ar —> e + Ar (elastic scattering) Rate coefficient K(Te) is average of cross section ct(vr) [m2] for process, over Maxwellian distribution K(Te) = {& Vr)Maxwellian vr — relative velocity of colliding particles 36 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ r 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 gi ~ 10 14 cm2 Ion-neutral mean free path (distance ion travels before colliding) Practical formula A* (cm) 330 p p in Torr Ion-neutral collision frequency with ví = (8/cTi/TrM)1/2 Vi. = 38 Vi A, University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-S r THREE ENERGY LOSS PROCESSES 1. Collisional energy Sc lost per electron-ion pair created KizSc = KizSiz + Kex£ex + Kei(2m/M)(3Te/2) => Sc(Te) (voltage units) £iz, ^ex, and (3m/M)Te 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 — 2 Te 3. Ion kinetic energy lost to walls is mainly due to the dc potential Vs across the sheath Si ~ Vs • Total energy lost per electron-ion pair lost to walls T 39 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' BOHM (ION LOSS) VELOCITY uB uB Plasma Sheath Density n, Wall Due to formation of a "presheath", ions arrive at the plasma-sheath edge with directed energy kTe/2 -2Mu% = — Electron-ion pairs are lost at the Bohm velocity at the plasma-sheath edge (density ns) 41 University of California, Berkeley1 Lieb er man Short Course 15 -PLASMA-S r AMBIPOLAR DIFFUSION AT HIGH PRESSURES Plasma bulk is quasi-neutral (ne & m = ri) and the electron and ion loss fluxes are equal (re ~ 1^ ~ T) Fick's law T = -DaVn with ambipolar diffusion coefficient Da — kTe/Mvi Density profile is sinusoidal Loss flux to the wall is Twall nsuB = hiriQUB From diffusion theory, edge-to-center density ratio is hi = — — - — Applies for pressures > 100 mTorr in argon University of California, Berkeley1 42 Lieb er man Short Course 1 f -P M -' 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 wall\^ ] J>0 -1/2 The edge-to-center density ratio is ^ ns s x 311 1/2 where = ion-neutral mean free path [p. 38] Applies for pressures < 100 mTorr in argon 43 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r AMBIPOLAR DIFFUSION IN LOW PRESSURE CYLINDRICAL DISCHARGE nSR - hR no nsi = hi n0 For a cylindrical plasma of length / and radius R, loss fluxes to axial and radial walls are Taxial — hiTlQUB- Tradial — ^fi^O^B where the edge-to-center density ratios are hi 0.86 (3 + Z/2Ai) 1/2 h 0.8 r (4 + i?/Ai)1/2 Applies for pressures < 100 mTorr in argon 44 University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-S r ANALYSIS OF DISCHARGE EQUILIBRIUM 45 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r PARTICLE BALANCE AND Te Assume uniform cylindrical plasma absorbing power Pabs Particle balance Plasma ne=nl= n0 Production due to ionization = loss to the walls Kizngr/o7TR2l — (27rR2hir/o + 27rRlhRnj())uB Solve to obtain where ^eff — - Rl 2 Rhi + lhR is an effective plasma size Given ng and de^ =^ electron temperature Te Te varies over a narrow range of 2 5 volts 46 University of California, Berkeley1 Lieb er man Short Course 15i -PLASMA-y ELECTRON TEMPERATURE IN ARGON DISCHARGE r ION ENERGY FOR LOW VOLTAGE SHEATHS Si — energy entering sheath + energy gained traversing sheath Ion energy entering sheath — Te/2 (voltage units) [p. 41] Sheath voltage determined from particle conservation i Density n [p. 41] Ti = nsuBl Plasma i Sheath i * ~ 0.2 mm I ^sulating §\ wall [p. 17] with ve = (SeTe/Trm)1/2 s ue G flux at sheath edge [p. 19] -Vs/Te The ion and electron fluxes at the wall must balance or Vs ~ 4.7 Te for argon Accounting for the initial ion energy, Si 5.2 X. University of California, Berkeley• 48 Lieberman rselj r ION ENERGY FOR HIGH VOLTAGE SHEATHS Large ion bombarding energies can be gained near rf-driven electrodes embedded in the plasma ^rf Claj-ge Plasma K + + K Vs ~ 0.4 Vrf Plasma Fs4 Low voltage sheath ~ 5.2 TP vs ~ 0.8 Vrf The sheath thickness s (~ 0.5 cm) is given by the Child Law Ji 4 (2e\^vT ™=9e° M IS" Estimating ion energy is not simple as it depends on the type of discharge and the application of bias voltages University of California, Berkeley• 49 Lieberman rseli r POWER BALANCE AND n0 Assume low voltage sheaths at all surfaces [p. 40] [p. 17] [p. 48] ST(Te) = Sc(Te) + 2% + 5^ Collisional Electron Ion Power balance Power in = power out ^kbs = (hin027rR2 + hRn027rRl) uB eST Solve to obtain n0 = abs AeSuBe£T [V] [W] where Aet? = 2irR2hi + 27rRlhR is an effective area for particle loss Density no is proportional to the absorbed power Pabs Density no depends on pressure p through hi, Hr, and Te 50 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r- PARTICLE AND POWER BALANCE • Particle balance =>> electron temperature Te (independent of plasma density) • Power balance =>> plasma density no (once electron temperature Te is known) 51 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' EXAMPLE 1 • Let R = 0.15 m, I = 0.3 m, ng = 3.3 x 1019 m"3 (p = 1 mTorr at 300 K), and Pahs = 800 W • Assume low voltage sheaths at all surfaces • Find Xi = 0.03 m [p. 38]. Then hi « hR « 0.3 [p. 44] and 4ff ~ 0.17 m [p. 46] • Te versus ngde^ figure gives Te « 3.5 V [p. 47] • £c versus Te figure gives £c « 42 V [p. 40]. Adding £e = 2Te « 7 V and £ « 5.2Te « 18 V yields £T = 67 V [p. 39] • Find uB « 2.9 x 103 m/s [p. 41] and find Aeii « 0.13 m2 [p. 50] • Power balance yields no ~ 2.0 x 1017 m-3 [p. 50] • Ion current density Ju — ehinoUß ~ 2.9 mA/cm2 [p. 46] • Ion bombarding energy Si ~ 18 V [p. 48] 52 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r- EXAMPLE 2 • Apply a strong dc magnetic field along the cylinder axis => particle loss to radial wall is inhibited • Assume no radial losses, then de^ — l/2hi ~0.5m • From the Te versus ngde^ figure, Te « 3.3 V (was 3.5 V) • From the Sc versus Te figure, Sc ~ 46 V. Adding Se — 2Te « 6.6 V and Si « 5.2Te « 17 V yields St — 70 V • Find uB « 2.8 x 103 m/s and find Aeii = 2^R2hi « 0.043 m2 • Power balance yields no ~ 5.8 x 1017 m-3 (was 2 x 1017 m-3) • Ion current density Ju — ehinoUB ~ 7.8 mA/cm2 • Ion bombarding energy Si ~ 17 V => Slight decrease in electron temperature Te => Significant increase in plasma density no EXPLAIN WHY! What happens to Te and no if there is a sheath voltage Vs — 500 V at each end plate? 53 LiebermanShortCourse 15, University of California, Berkeley PL SMA ^ CAPACITIVE RF DISCHARGES SYMMETRIC HOMOGENEOUS MODEL University of California, Berkeley 54 Lieber man Short Co PL IMA 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 Jrf (1-10 mA/cm2) Driving frequency uj (2 13.56 MHz) Neutral gas density ng (1014 1016 cm-3) Electrode separation / (1-10 cm) • Discharge parameters to find Plasma density n (109 1010 cm-3) Electron temperature Te (2-4 V) Discharge voltage Vri (100 1000 V) Discharge power Prf (50 500 W) Ion bombarding energy Si (50 500 V) 55 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-^ r- CONFIGURATIONS • Multi-wafer parallel plate and "hex" configurations (1980's) Substrate pump • Modern configurations are single wafer parallel plate, sometimes driven at multiple rf frequencies 56 University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' r CURRENT-DRIVEN HOMOGENEOUS MODEL sit) V< Sheath n University of California, Berkeley1 57 X LiebermanShortCoursel5. -P MA-y 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 rii(z) = n0 (We will correct this later) 58 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r ELECTRON SHEATH EDGE MOTION n v \ Electrode ä Irfft)? \ \ ni-n Iap(t) Plasma p -»-z 0 Sa(t) Sm The electric field is found by integrating the charge density in the sheath [p. 12] dE en Z < Sa(t) dz to obtain en E(z,t) = —(z-sa(t)) sa(t) ->-z 59 University of California, Berkeley • Lieb er man Short Course 15, -PLA SM A-' r ELECTRON SHEATH EDGE MOTION (CONT'D) The displacement current in the sheath is [p. 14] Iap ~^Adt - en A dt Let Ivt(t) — Iq cos out and integrate to obtain sa(t) = sq - so sin(jjt The oscillation amplitude of the sheath motion is so, but what is the "constant of integration" sq? 60 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ r CONDUCTION CURRENT Assume a steady loss of ions to electrode a Ii — enusA The time-average total conduction current to the electrode is zero Hence electrons must be lost to the electrode The sheath thickness sa(t) must then collapse to zero at some time during the rf cycle =>> Sq — Sq sa(t) = s0(l - smut) sm=2s0 Ii -Ie(t) Ii University of California, Berkeley1 Lieber man Short Course 15 -PL SMA-^ r VOLTAGE ACROSS THE SHEATH n v \ Electrode ä Irf(t)J —► ■ \ \ ni-n Iap(t) Plasma p Ö ša(t) & m -»>z Vap -sa(t) V Vap(tH sa(t) V(z,t) The voltage is found by integrating the electric field in the sheath dV dz -E 62 University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' r VOLTAGE ACROSS THE SHEATH (CONT'D) Integrating the electric field in the sheath [p. 12] -E we obtain dz fSa(t) Vap(t) = / E(z,t)dz = Jo Using sa(t) [p. 61] en sl(t) e0 2 Vap{t) = - — s0(l - smut) ZCq Vap(i) is a nonlinear function of 7rf; there are second harmonics 63 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r VOLTAGE ACROSS BOTH SHEATHS By symmetry Sb(t) — sq(1 + sincjt); since sa(t) — sq(1 — sin out) sa(t) + Sb(t) — 2 so — const Vab(t) Irf(t)J $5*-* 1 ^ Irf(t) a p b There is a rigid bulk electron cloud oscillation 64 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' VOLTAGE ACROSS BOTH SHEATHS (CONT'D) Voltage across sheath b is en Vbp(t) = - —Sq(1 + Sirica) Zeo Voltage across the plasma is small because 7rf = juoepEA and ep is large =>> E across bulk plasma is small Discharge voltage is VTf = Vap + Vpt Each sheath is nonlinear, but the combination of both sheaths is linear 65 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ r DC VOLTAGE ACROSS ONE SHEATH n v \ Electrode ä Irfft)? \ \ ni-n Iap(t) Plasma p Ö ša(t) šn7 -»-z en Vpa\t) — -š0Í1 — 2sincji + sin ut) 2cq Take time average — 3 en c Compare to rf voltage across discharge [p. 65] — 3~ ^Vs = -Vrf o We can think of Vvf as divided equally across the two sheaths 4 Ä with Vs = ^Vri 66 University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' SHEATH VOLTAGES VERSUS TIME v Sheath voltages Vap(t), Vpb(t), and their sum Vab(t) — VTf(t)\ the time average Vs of Vpb(t) is also shown Sinusoidal — cot University of California, Berkeley• 67 Lieb er man Short Course 15J -PLASMA-' r SHEATH POTENTIAL VERSUS POSITION AT VARIOUS TIMES Spatial variation of the total potential $ (solid curves) for the homogeneous model at four different times during the rf cycle. The dashed curve shows the spatial variation of the time-average potential <£> = Vs 68 University of California, Berkeley• 1/4 0 1 3/8 & 1/4 Vd(t) = sin cot -£>- hf(t) cot = 0 2^ 3x 2 Lieb er man Short Course 151 -PLA SM A-' r SHEATH CAPACITANCE Define total discharge capacitance by Ivi (t) — Iq COSCJt, which yields Vri(t) = sin out [p. 65] We can think of each sheath as having a capacitance We now have a lossless discharge model a p b 69 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r OHMIC AND STOCHASTIC HEATING Ohmic heating in the bulk plasma [p. 34] Pn = \\Xi\2"^^A [watts] e2n Stochastic heating by oscillating sheaths Electrode \ us(t) with v -v + 2u8(t) — uq cos cut with Uq — CJSq Average energy transferred is A£e= 1 m(-u + 2Ms(t))--mv' — mu o ASe is positive, so the oscillating sheath heats electrons 70 LiebermanShortCoursel5i University of California, Berkeley PL SMA ^ r- STOCHASTIC HEATING POWER • For a Maxwellian distribution, the electron flux incident on a sheath is [p. 17] 1 e = —nve 4 with ve = (8eTe/7rm)1/2 the mean electron speed • The time-average stochastic heating power is found to be Psa — TeA • 2ASe — -mnveuü2s^A [watts] • This is a powerful electron heating mechanism in a capacitive discharge 71 University of California, Berkeley1 Lieb er man Short Course 151 -PLA SM A-S r HEATING POWERS VERSUS DRIVING VOLTAGE For stochastic heating [p. 71] 1 2 Psa = -mnveu2šlA Using Vvt — 2ensg/eo [p. 65] we obtain for the two sheaths For bulk ohmic heating [p. 70] Pn = -j\Jvf\ —^—A e2n Using Jrf = enujso and Vvf given above Ohmic and stochastic heating powers depend on VTf 72 University of California, Berkeley• L ieber man Short Course 15, -PLA SMA-' CAPACITIVE RF DISCHARGES SELF-CONSISTENT SHEATH RESULTS University of California, Berkeley 73 Lieber man Short Co PL IMA r HOMOGENEOUS AND CHILD LAW SHEATHS Homogeneous Model n Electrod^ 1 ni | Bulk neÍ plasma S(t) Sm Self-Consistent Child Law n Electrode Bulk plasma s(t) Sm Child law ion density decreases and sheath width increases compared to homogeneous model 74 University of California, Berkeley• Lieb er man Short Course 15 -PLASMA-S r COLLISIONLESS CHILD LAW SHEATH + Va 1 x^Wall Plasma x' Sheath -4 = ensuB ] ne 0 Larger sheath width =>> larger sheath oscillation velocity us oc uos m Stochastic heating is larger than for homogeneous model Collisionless ion motion =>> Child law relating sm to ns and V. Ji — ensUB — 0.82 eo MJ si RF voltage Vs across sheath =>> dc voltage Vt V, « 0.83 V, 75 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' r- SUMMARY — SELF-CONSISTENT MODEL irf ^ 1 Sheath Vrf=2Vs , (^rasma) l^heatji; Irf 0 Sm Si = V, = 0.83 V, Ji — ensUB — 0.82 e0 2ey/2yf M Iri^ 1.23 juj^Vs m m Ps = 1.12 ~-uj2e0veVsA Ze Pn = 1.73—^u2€oumd(TeVs)^2A no Ze 76 University of California, Berkeley• Lieber man Short Course 151 -PLA SMA-' CAPACITIVE RF DISCHARGES SIMULATION AND EXPERIMENTAL RESULTS University of California, Berkeley 77 Lieber man Short Co PL IMA r PIC SIMULATION OF DENSITIES Symmetric rf discharge with right hand electrode grounded, Vrf = 1 kV at 10 MHz, 20 mTorr hydrogen gas (Thesis of D. Vender, Australian National University, ~ 1990) c o Q Note electron motion in sheath J_I_ 10 Position (cm) 20 University of California, Berkeley• 78 Lieb er man Short Course 15J -PLASMA-' r PHASE SPACE AND POTENTIALS VERSUS POSITION Symmetric rf discharge with right hand electrode grounded, yrf = 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) University of California, Berkeley • o -4—» "o _o 0.4 -0.4 79 Electrons Note stochastic - heating Instantaneous irl2 2s Note plasma always + w.r.t. electrodes. 7t 3tt/2 Ions / Average 10 20 0 10 Position (cm) 20 > C 4—> O 0.5 Lieb er man Short Course 15J -PL SM A-' r TIME VARIATION OF VOLTAGES AND CURRENTS 1.0 - V(kV) -1.0 - Ä< X 1 1 ♦> 10 MHz I \ \ Vvb / / 1 \ J * * • » t > • > » t • > « t » > « t » > \ / ^i(xl0)^^-^\ (Thesis of D. Vender, ANU, 1990) 20 7(A/m2) -20 271 Phase cuf 47r Note plasma always positive with respect to both electrodes Note steady ion current [p. 61] Note pulsed electron current when Vpa — Vpb — VV£ —>• 0 [p. 61] 80 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' 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, VTf — 1 kV at 10 MHz, 20 mTorr hydrogen gas (Thesis of D. Vender, Australian National University, - 1990) 4?r Phase cof 2tt Sheath motion Stochastic heating / - '" - < .... ivu,.^Ä~Sr">"j-.^-TN\-^ - " V-, v :' 7 : ' ' ' ■ ;,t.'e^--: ✓^45"'x-v"" 'V':"' "'V ..V.*"' 2 1 1 1 81 10 Position (cm) 20 University of California, Berkeley• Lieb er man Short Course 15 -PL/ SM A-S r MEASUREMENTS OF STOCHASTIC HEATING ioy Effective collision frequency z/eff versus pressure 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) V IS~ 10a 107 10e 10" I" Theory «- Experiment 10"3 10"2 p (Torr) 10 rl 82 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' CAPACITIVE RF DISCHARGES EXAMPLE EQUILIBRIUM CALCULATIONS University of California, Berkeley 83 Lieber man Short Co PL IMA r POWER BALANCE Electron power balance El. col + kin Pn + 2PS = ensuB 2A (Sc + 2Te) where PQ = 1.73/i^o;2eo(z/mrf)(Teys)1/2A ze m Ps = 0.56--u;26oveVsA ~ ze Specify Vs =>> ns • Total power balance El. col.+kin. Ion kin. Pabs = ensuB 2A (£c + 2Te + 0.83 Vs) Eliminate ns from electron and total power balance Pabs ~ (Pn + 2PS) 1 + 0.83ys Sc + 2Te Specify Pahs =>> Vs In this case, electron or total power balance =>> n. 84 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r- EXAMPLE 1 • Let p — 3 mTorr argon at 300 K, / = 10 cm, A — 1000 cm2, / = 13.56 MHz (lj = 8.52 x 107 s"1), and Vri = 500 V • Start with estimate sm pa 1 cm • Ion mean free path = l/ng(Ji pa 1.0 cm [p. 38] • With bulk plasma thickness d — I — 2sm — 8 cm, Xi/d pa 0.125 • hi = ns/n0 Pa 0.325 [p. 43] and des = d/2ht = 12.3 cm [p. 46] • With rigdefi pa 1.23 x 1019 m~2, the Te versus ngdeR figure [p. 47] yields Te « 3.1 V • Bohm velocity ub ~ 2.7 x 103 m/s [p. 41] • Sc versus Te figure [p. 40] yields Sc pa 47 V and Sc + 2Te pa 53 V. • Use the Ke\ versus Te figure [p. 37] to find vm pa Ke\ng pa 1.0 x 107 s"1 85 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ /- EXAMPLE 1 (CONT'D) • Evaluate ohmic and stochastic electron heating [p. 76 or 84] Pn « 0.0145 V}/2 [W] Ps « 0.0121 ys [W] • Use Va « yrf/2 = 250 V in above to find Pfi » 0.229 W and Ps » 3.03 W • Electron power balance [p. 84] yields ns « 1.37 x 1015 m-3 • Since /iz « 0.325, n0 « 4.23 x 1015 m"3 • Using ft « 0.83 ys [p. 76] yields « 208 V • J• = ensiiß « 0.59 A/m2 [p. 76] • The Child law [p. 76] gives sm « 0.90 cm • Jrf « 1.23a;eo^/sm ~ 25.8 A/m2 [p. 76] • Total power balance [p. 84] gives Pabs ~ 30.8 W • sm reasonably close to the initial estimate => iteration over d is not useful ,. . .. , _ .., „ , . 86 LiebermanShortCpur University of California, Berkeley —P SMA — r-* EXAMPLE 2 Let p — 3 mTorr argon at 300 K, / = 10 cm, A — 1000 cm2 / = 13.56 MHz and Pabs = 200 W As before, hi « 0.325, Te « 3.1 V, uB « 2.7 x 103 m/s, and £c + 2Te « 53 V. Because np and Te are the same, Pq and Ps are the same functions of Vs as in EXAMPLE 1 Using Pabs = 200 W, we obtain the equation for the rf sheath voltage Vs [p. 84] 0.SSVa 200 = (0.0145V;1/2 + 0.0242ys) 1 + 53 A numerical solution gives Vs — 687 V Then Vri « 2VS « 1374 V and Si = 0.83 ys « 570 V Use this in total power balance [p. 84] to find ns ~ 3.72 x 1015 m~ and no ~ 1.14 x 1016 m~3 We then find Ji « 1.6 A/m2, sm « 1.16 cm, and Jrf « 54.9 A/m2 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' CAPACITIVE RF DISCHARGES ASYMMETRIC SYSTEMS University of California, Berkeley 88 Lieb er man Short Co PL IMA r ASYMMETRIC RF DISCHARGE Powered electrode area Aa smaller than grounded area Ai Clarge Powered a Sheath a —*— Va Vrf0 Plasma Sheath b Grounded b + + Vb Vbi; Va Vb Va- vb-- - dc sheath voltage from plasma to powered electrode a dc sheath voltage from plasma to grounded electrode b rf voltage amplitude across sheath a rf voltage amplitude across sheath b Va = 0.83 Vr a i Vh = 0.83^, Vrf = Va + Vb A negative DC bias voltage Vbias — Vb — Va appears 89 University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-s r DEPENDENCE OF VOLTAGES ON AREAS Given VTf, Aa and Ab, what are Va, Vb and Tobias? Voltage ratio Va/Vb depends on area ratio Ab/Aa (Aby UJ q — area ratio scaling exponent Experiments show q ~ 1 -2.5 Collisionless Child law gives q — 4 90 University of California, Berkeley1 Lieb er man Short Course 151 -PLA SM A-s r COLLISIONLESS CHILD LAW ANALYSIS +Vb- -Va+ Irf S Plasma (good ! conductor); nb Sa Sb Irf rv Electrodes and plasma are good conductors, so Va is the same everywhere along electrode a Capacitive sheath [p. 76] irf oc VnA a-t *-a Child law [p. 76] Eliminating sa V 7a 3/2 a Si irf OC vTnT^a University of California, Berkeley• 91 Lieb er man Short Course 15, -PLA SM A-' COLLISIONLESS CHILD LAW ANALYSIS (CONT'D) Similarly b '% Set currents at sheaths a and b equal and solve for Va/Vb V a nb 4 vb n a A a Simplest assumption for plasma is equal densities at the sheath edges a and b (Aby UJ 92 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ 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 .3/2 V nn oc a Ja ~ 5/2 So! This leads to va _ (Aby-5 Ua) The weaker scaling q — 2.5 is more in agreement with experiments 93 University of California, Berkeley1 Lieber man Short Course 151 -PLA SMA-^ r- VARIATION OF DENSITIES Low density High density Plasma density near powered electrode is usually larger than near grounded electrode This leads to additional modifications in scaling 94 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' INDUCTIVE DISCHARGES TRANSFORMER MODEL AND MATCHING University of California, Berkeley 95 Lieber man Short Co PL IMA r- 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 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r CYLINDRICAL AND PLANAR CONFIGURATIONS Cylindrical coil Planar coil c Dielectric ■ "XT Dielectric i Substrate co / Skin depth layer Substrate -r Rf Rf bias Rf Rf bias 97 University of California, Berkeley• Lieb er man Short Course 151 -PLA SM A-' r EARLY HISTORY First inductive discharge by Hittorf (1884) ( \\Y\X ) • Arrangement to test discharge mechanism by Lehmann (1892) 98 University of California, Berkeley • Lieber man Short Course 15 j -P SMA r HIGH DENSITY REGIME Inductive coil launches decaying wave into plasma Decaying wave Window Wave decays exponentially into plasma Ě = E0e-z/5p, Op = — v Im(«J/2) where kp — plasma dielectric constant [p. 31] 1 - ÜÜ pe P ( \ For typical high density, low pressure (z/m - A transformer 100 University of California, Berkeley1 Lieb er man Short Course 151 -PLA SM A-s r PLASMA RESISTANCE AND INDUCTANCE Plasma resistance R 1 R where with n es circumference of plasma loop p <7dc average cross sectional area of loop adc = ^ [p. 33] density at plasma-sheath edge ^ KP - -77 (7dcl0, P Plasma inductance L p Lp — magnetic flux produced by plasma current plasma current Using magnetic flux = 7rR2fjboIp/l University of California, Berkeley• 101 Lieb er man Short Course 15l -PLASMA-' r COUPLING OF COIL TO PLASMA Model the source as a transformer Zs=ja>Ls+Rs o- + + Coil Plasma Rr Vri = jvLuIrf + jcjLi2Ip Vp = jujL2iírí + jujL22lp Transformer inductances magnetic flux linking coil HQiib2N2 L 12 - L21 - coil current I magnetic flux linking plasma hqtvR2N coil current fiQirR2 I L22 — Lp — I 102 University of California, Berkeley• Lieb er man Short Course 15, -PLA SM A-' r SOURCE CURRENT AND VOLTAGE Put Vp — —IpRp in transformer equations and solve for the impedance Zs — VTf/ITf seen at coil terminals Zs = jwLn + cu2L2 12 Rp + jujL = Rs + juL, p Equivalent circuit at coil terminals ttR yg — M * - Rq = AT2 L. l^07TR2J\i2 Power balance I R2 - 1 From source impedance =>> VYf Vri = IvfZ& rf • Re University of California, Berkeley• 103 Lieberman^hor^Iourse 1 r EXAMPLE Assume plasma radius R — 10 cm, coil radius 6 = 15 cm, length / = 20 cm, J\f — 3 turns, gas density ng — 6.6 x 1014 cm-3 (20 mTorr argon at 300 K), lu = 85 x 106 s_1 (13.56 MHz), absorbed power Pabs = 600 W, and low voltage sheaths At 20 mTorr, A* « 0.15 cm, hi « hR « 0.1, deff « 34 cm [p. 38, 44, 46] Particle balance (Te versus ngde^ figure [p. 47]) yields Te & 2.1 V Collisional energy losses (£c versus Te figure [p. 40]) are Sc ~ 110 V. Adding £e + £i = 7.2 Te yields total energy losses St ~ 126 V [p. 39] uB ~ 2.3 x 105 cm/s [p. 41] and Aeff « 185 cm2 [p. 50] Power balance yields Tip ~ 7.1 x 10 cm and 7.4 x 10 cm [p. 50] Use nse to find skin depth Sp ~ 2.0 cm [p. 99]; estimate vm — Ke\ng (Ke\ versus Te figure [p. 37]) to find vm ~ 3.4 x 107 s_1 Use vm and nse to find plasma size) Rs oc number of electrons in the heating volume oc nt Low density / High density dp ~ plasma size 110 University of California, Berkeley1 Lieb er man Short Course 15, -PLA SM A-S r POWER BALANCE WITHOUT MATCHING Drive discharge with rf current Power absorbed by discharge is Pabs = \ \Ivf\2Rs{ne) [p. 110] Power lost by discharge Pioss oc ne [p. 50] Intersection (red dot) gives operating point; let I\ < I2 < I3 Power D _ 1 j 2 r> ^abs 2 3 s --■—£^bs — 1 f2 p -^abs Inductive operation impossible for jrf < I2 University of California, Berkeley 111 Lieber man Short Course 15i -PL SMA-' r CAPACITIVE COUPLING OF COIL TO PLASMA For /rf below the minimum current 72, there is only a weak capacitive coupling of the coil to the plasma Vrt .-© ® ® Capacitive coupling /© ® Plasma i A small capacitive power is absorbed => low density capacitive discharge Power Cap Mode Ind Mode 112 University of California, Berkeley• Lieb er man Short Course 15i -PLASMA-y r- MEASURMENTS OF ARGON ION DENSITY 15 E o o 1—1 10 0 1 ll 1 1 (a) 0.5 x 10~3 Torr argon i (o) 1.0 x 10-3 Torr (o) 5.0 x 10~3 Torr (•) 15 x 10"3 Torr y Inductive y discharge ^"^z > maximum Rs 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 114 University of California, Berkeley• Lieb er man Short Course 15, -PLASMA-y