A MINI-COURSE ON THE PRINCIPLES OF PLASMA DISCHARGES

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1 A MINI-COURSE ON THE PRINCIPLES OF DISCHARGES Michael A. Lieberman Department of Electrical Engineering and Computer Science, CA LiebermanShortCourse06 1

2 THE MICROELECTRONICS REVOLUTION Transistors/chip doubling every years since ,000-fold decrease in cost for the same performance In 20 years one computer as powerful as all those in Silicon Valley today CMOS transistors with 4 nm (16 atoms) gate length EQUIVALENT AUTOMOTIVE ADVANCE 4 million km/hr 1 million km/liter Never break down Throw away rather than pay parking fees 3 cm long 1 cm wide Crash 3 aday LiebermanShortCourse06 2

3 OUTLINE Introduction to Plasma Discharges and Processing Summary of Plasma Fundamentals Break Summary of Discharge Fundamentals Analysis of Discharge Equilibrium Break Inductive Discharges Description and Model Inductive Discharges Power Balance and Matching Break Free Radical Balance in Discharges Adsorption and Desorption Kinetics Plasma-Assisted Etch Kinetics ORIGIN 45 hr graduate course at Berkeley = 12 hr short course in industry = 4 hr mini-course LiebermanShortCourse06 3

4 INTRODUCTION TO DISCHARGES AND PROCESSING LiebermanShortCourse06 4

5 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 30cm 1m Frequencies from DC to rf (13.56 MHz) to microwaves (2.45 GHz) LiebermanShortCourse06 5

6 EVOLUTION OF ETCHING DISCHARGES FIRST GEN- ERATION SECOND GENER- ATION THIRD GEN- ERATION?? LiebermanShortCourse06 6

7 RANGE OF MICROELECTRONICS APPLICATIONS Etching Si, a-si, oxide, nitride, III-V s Ashing Photoresist removal Deposition (PECVD) Oxide, nitride, a-si Oxidation Si Sputtering Al, W, Au, Cu, YBaCuO Polymerization Various plastics Implantation H, He, B, P, O, As, Pd LiebermanShortCourse06 7

8 ANISOTROPIC ETCHING {}}{{}}{ Wet Etching Ion Enhanced Plasma Etching Plasma Etching LiebermanShortCourse06 8

9 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 LiebermanShortCourse06 9

10 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 1eV= 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 1V 11, 600 K Pressure unit is pascal (Pa); 1 Pa =1N/m 2 Atmospheric pressure 10 5 Pa 1 bar Often use English units for gas pressures Atmospheric pressure = 760 Torr 1Pa 7.5 mtorr LiebermanShortCourse06 10

11 DENSITY VERSUS TEMPERATURE LiebermanShortCourse06 11

12 RELATIVE DENSITIES AND ENERGIES LiebermanShortCourse06 12

13 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 E i E 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 LiebermanShortCourse06 13

14 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 Ions accelerated to walls; ion bombarding energy E i = plasma-wall potential V p LiebermanShortCourse06 14

15 CENTRAL PROBLEM IN DISCHARGE MODELING Given V rf (or I rf or P rf ), ω, gases, pressure, flow rates, discharge geometry (R, l, etc) Find plasma densities n e, n i, temperatures T e,t i, ion bombarding energies E i, sheath thicknesses, neutral radical densities, etc Learn how to design and optimize plasma reactors for various purposes (etching, deposition, etc) LiebermanShortCourse06 15

16 CHOOSING PROCESSING EQUIPMENT How about inductive? (figure published in 1991) LiebermanShortCourse06 16

17 SUMMARY OF FUNDAMENTALS LiebermanShortCourse06 17

18 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) LiebermanShortCourse06 18

19 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 v e = 1 ( ) 1/2 d 3 8kTe vvf e (v) = n e πm Average electron flux lost to a wall z Γ e y Γ e = dv x x dv y dv z v z f e (v) = 1 4 n e v e [m 2 -s 1 ] 0 Average kinetic energy lost per electron lost to a wall E e =2T e LiebermanShortCourse06 19

20 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 e p e p e (x) p e (x + dx) Drag force u e Neutrals x x + dx x LiebermanShortCourse06 20

21 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 LiebermanShortCourse06 21

22 DIELECTRIC CONSTANT ɛ p RF discharges are driven at a frequency ω E(t) =Re(Ẽejωt ), etc Define ɛ p from the total current in Maxwell s equations H = J c + jωɛ 0 Ẽ }{{} jωɛ pẽ Total current J Conduction current J c = en e ũ e is mainly due to electrons Newton s law (electric field and neutral drag) is jωmũ e = eẽ mν mũ e Solve for ũ e and evaluate J c to obtain ] ωpe ɛ p ɛ 0 κ p = ɛ 0 [1 2 ω(ω jν m ) with ω pe =(e 2 n e /ɛ 0 m) 1/2 the electron plasma frequency For ω ν m, ɛ p is mainly real (nearly lossless dielectric) For ν m ω, ɛ p is mainly imaginary (very lossy dielectric) LiebermanShortCourse06 22

23 CONDUCTIVITY σ p It is also useful to introduce the plasma conductivity J c σ p Ẽ RF plasma conductivity e 2 n e σ p = 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: jωɛ p = σ p + jωɛ 0 Due to σ p, rf current flowing through the plasma heats electrons (just like a resistor) LiebermanShortCourse06 23

24 SUMMARY OF DISCHARGE FUNDAMENTALS LiebermanShortCourse06 24

25 ELECTRON COLLISIONS WITH ARGON Maxwellian electrons collide with Ar atoms (density n g ) dn e = νn e = Kn g n e dt 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 [m 2 ] for process, over Maxwellian distribution K(T e )= σv Maxwellian Ar Ar LiebermanShortCourse06 25

26 ELECTRON-ARGON RATE COEFFICIENTS LiebermanShortCourse06 26

27 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 λ i = 1 n g σ i Practical formula λ i (cm) = 1, p in Torr 330 p Rate coefficient for ion-neutral collisions K i = v i λ i with v i =(8kT i /πm) 1/2 Ar + Ar + Ar Ar Ar + Ar Ar Ar + LiebermanShortCourse06 27

28 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 E e =2T 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 LiebermanShortCourse06 28

29 COLLISIONAL ENERGY LOSSES LiebermanShortCourse06 29

30 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 i = kt e 2 At the plasma-sheath edge (density n s ), electron-ion pairs are lost at the Bohm velocity ( ) 1/2 kte u i = u B = M LiebermanShortCourse06 30

31 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 /M ν 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 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 LiebermanShortCourse06 31

32 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 n s l/2 0 l/2 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 Applies for pressures < 100 mtorr in argon LiebermanShortCourse06 32

33 AMBIPOLAR DIFFUSION IN LOW PRESSURE CYLINDRICAL DISCHARGE ns = hr n0 R Plasma n e = n i = n 0 ns = hl n0 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 ), h 0.8 1/2 R (4 + R/λ i ) 1/2 Applies for pressures < 100 mtorr in argon l LiebermanShortCourse06 33

34 ANALYSIS OF DISCHARGE EQUILIBRIUM LiebermanShortCourse06 34

35 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 LiebermanShortCourse06 35

36 ELECTRON TEMPERATURE IN ARGON DISCHARGE LiebermanShortCourse06 36

37 ION ENERGY FOR LOW VOLTAGE SHEATHS E i = energy entering sheath + energy gained traversing sheath Ion energy entering sheath = T e /2 (voltage units) Sheath voltage determined from particle conservation in the sheath Γ i Plasma Sheath Insulating wall Γ i Γ e Density n s + V s e V s /T e Γ i = n s u B, Γ e = 1 4 n s v e }{{} with v e =(8eT e /πm) 1/2 Random flux at sheath edge The ion and electron fluxes must balance V s = T ( ) e M 2 ln 2πm or V s 4.7T e for argon Accounting for the initial ion energy, E i 5.2T e LiebermanShortCourse06 37

38 ION ENERGY FOR HIGH VOLTAGE SHEATHS Large ion bombarding energies can be gained near rf-driven electrodes embedded in the plasma s Ṽ rf ~ C large Plasma V s 0.4 Ṽrf V s + + Vs Low voltage sheath 5.2T e Ṽ rf ~ Plasma V s 0.8 Ṽrf V s + The sheath thickness s is given by the Child Law J i = en s u B = 4 ( ) 1/2 3/2 2e 9 ɛ V 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 LiebermanShortCourse06 38

39 POWER BALANCE AND n 0 Assume low voltage sheaths at all surfaces E T (T e ) = E c (T e ) }{{} + 2T e }{{} +5.2T e }{{} Collisional Electron Ion Power balance Power in = power out Solve to obtain 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 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 LiebermanShortCourse06 39

40 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) LiebermanShortCourse06 40

41 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. Then h l h R 0.3 and d eff 0.17 m [pp. 26, 32, 34] From the T e versus n g d eff figure, T e 3.5 V [p. 35] From the E c versus T e figure, E c 42 V [p. 28]. Adding E e =2T e 7 V and E i 5.2T e 18 V yields E T = 67 V [p. 27] Find u B m/s and find A eff 0.13 m 2 [pp. 29, 38] Power balance yields n m 3 [p. 38] Ion current density J il = eh l n 0 u B 2.9 ma/cm 2 Ion bombarding energy E i 18 V [p. 36] LiebermanShortCourse06 41

42 EXAMPLE 2 Apply a strong dc magnetic field along the cylinder axis = particle loss to radial wall is inhibited For no radial loss, d eff = l/2h l 0.5 m From the T e versus n g d eff figure, T e 3.3 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 =70V Find u B m/s and find A eff =2πR 2 h l m 2 Power balance yields n 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 = Significant increase in plasma density n 0 LiebermanShortCourse06 42

43 ELECTRON HEATING MECHANISMS Discharges can be distinguished by electron heating mechanisms (a) Ohmic (collisional) heating (capacitive, inductive discharges) (b) Stochastic (collisionless) heating (capacitive, inductive discharges) (c) Resonant wave-particle interaction heating (Electron cyclotron resonance and helicon discharges) Achieving adequate electron heating is a central issue Although the heated electrons provide the ionization required to sustain the discharge, the electrons tend to short out the applied heating fields within the bulk plasma LiebermanShortCourse06 43

44 INDUCTIVE DISCHARGES DESCRIPTION AND MODEL LiebermanShortCourse06 44

45 MOTIVATION Independent control of plasma density and ion energy Simplicity of concept RF rather than microwave powered No source magnetic fields LiebermanShortCourse06 45

46 CYLINDRICAL AND PLANAR CONFIGURATIONS Cylindrical coil Planar coil LiebermanShortCourse06 46

47 EARLY HISTORY First inductive discharge by Hittorf (1884) Arrangement to test discharge mechanism by Lehmann (1892) LiebermanShortCourse06 47

48 HIGH DENSITY REGIME Inductive coil launches electromagnetic 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 1 ω Im(κp 1/2 ) where κ p = plasma dielectric constant [p. 21] 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 LiebermanShortCourse06 48

49 TRANSFORMER MODEL For simplicity consider 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 LiebermanShortCourse06 49

50 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 [p. 22] mν m = 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 LiebermanShortCourse06 50

51 Model the source as a transformer COUPLING OF COIL TO Coil Plasma 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 LiebermanShortCourse06 51

52 SOURCE CURRENT AND VOLTAGE Put Ṽp = ĨpR p in transformer equations and solve for impedance Z s = Ṽrf/Ĩrf seen at coil terminals Z s = jωl 11 + ω2 L 2 12 R p + jωl p R s + jl 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 From source impedance = V rf P abs = 1 2Ĩ2 rfr s Ṽ rf = ĨrfZ s LiebermanShortCourse06 52

53 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, and d eff 34 cm [pp. 26, 32, 34] Particle balance (T e versus n g d eff figure) yields T e 2.1 V [p. 35] Collisional energy losses (E c versus T e figure) are E c 110 V [p. 28]. Adding E e + E i =7.2T e yields total energy losses E T 126 V [p. 27] u B cm/s and A eff 185 cm 2 [pp. 27, 38] Power balance yields n e cm 3 and n se cm 3 [p. 38] Use n se to find skin depth δ p 2.0 cm [p. 47]; estimate ν m = K el n g (K el versus T e figure) to find ν m s 1 [p. 25] Use ν m and n se to find σ dc 61 Ω 1 -m 1 [p. 22] Evaluate impedance elements R s 23.5 Ω and L s 2.2 µh; Z s ωl s 190 Ω [p. 51] Power balance yields Ĩrf 7.1A; from impedance Ṽrf 1360 V [p. 51] LiebermanShortCourse06 53

54 MATCHING DISCHARGE TO POWER SOURCE Consider an rf power source connected to a load + Z T Ĩ + Ṽ T ~ Ṽ Z L Source Load Source impedance Z T = R T + jx T is given Load impedance Z L = R L + jx L Time-average power delivered to load P abs = 1 2 Re (Ṽ Ĩ ) For fixed source ṼT and Z T, to maximize power delivered to load, X L = X T R L = R T Maximum time-average power delivered to load: P abs = 1 ṼT 2 8 R T LiebermanShortCourse06 54

55 MATCHING NETWORK Insert lossless matching network between power source and coil Power source Matching network Discharge coil Continue EXAMPLE with R s =23.5 Ω and ωl s = 190 Ω; assume R T = 50 Ω (corresponds to conductance of 1/50 mho) Choose C 1 such that the conductance seen looking to the right at terminals AA is 1/R T =1/50 mho = C 1 =71pF 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 ) = I rf R T = 245 V LiebermanShortCourse06 55

56 PLANAR COIL DISCHARGE Magnetic field produced by planar coil RF power is deposited in 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 LiebermanShortCourse06 56

57 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) LiebermanShortCourse06 57

58 INDUCTIVE DISCHARGES POWER BALANCE LiebermanShortCourse06 58

59 RESISTANCE AT HIGH AND LOW DENSITIES Plasma resistance seen by the coil ω 2 L 2 12 R s = R p Rp 2 + ω 2 L 2 p High density (normal inductive operation) 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 1 ne n e δ p plasma size n e LiebermanShortCourse06 59

60 POWER BALANCE WITHOUT MATCHING Drive discharge with rf current Power absorbed by discharge is P abs = 1 2 Ĩrf 2 R s (n e ) Power lost by discharge P loss n e [p. 38] Intersection gives operating point; let Ĩ1 < Ĩ2 < Ĩ3 P loss Power P abs = 1 2Ĩ2 3 R s P abs = 1 2Ĩ2 2 R s P abs = 1 2Ĩ2 1 R s Inductive operation impossible for Ĩrf Ĩ2 n e LiebermanShortCourse06 60

61 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 z A small capacitive power is absorbed = low density capacitive discharge P loss Power Cap Ind P abs = 1 2Ĩ2 3 R s P abs = 1 2Ĩ2 1 R s Cap Mode Ind Mode n e LiebermanShortCourse06 61

62 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 LiebermanShortCourse06 62

63 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 1 ne n e 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 LiebermanShortCourse06 63

64 FREE RADICAL BALANCE IN DISCHARGES LiebermanShortCourse06 64

65 ONE-DIMENSIONAL SLAB MODEL N 2 discharge with low fractional ionization (n g n N2 ) Determine T e : Assume R l = plate separation. Then ion particle balance is 2n is u B A K iz n g n i la where n is = h l n i. This yields K iz (T e ) u B (T e ) 2h l n g l = T e Determine n i and Γ is : The overall discharge power balance is P abs 2AeE T n is u B This yields n is P abs 2eE T u B A and Γ is n is u B Determine ion bombarding energy E i =5.2T e LiebermanShortCourse06 65

66 DISSOCIATION OF REACTIVE DIATOMIC GAS Example of nitrogen: K e+n diss 2 2N+e Assume K diss and K iz have Arrhenius forms K diss = K diss0 e E diss/t e K iz = K iz0 e E iz/t e Eliminating T e from these equations yields K diss = C 0 K E diss/e iz iz where C 0 = K diss0 /K E diss/e iz iz0 Using ion particle balance to eliminate K iz yields ) Ediss /E iz ( 2hl u B K diss = C 0 n g l In this form, K diss depends only weakly on T e, and the dependance of K diss on n g l is made explicit LiebermanShortCourse06 66

67 FREE RADICAL BALANCE Assume low fractional dissociation and that the only loss of N atoms is due to a vacuum pump S p (m 3 /s) Al dn N = Al 2K diss n g n i S p n NS =0 dt Solving for the free radical density at the surface n NS = 2Aln g K diss n i S p Using n i and K diss found previously yields ( P abs ng l n NS =2C 0 ee T S p 2h l u B Typically, E diss /E iz The flux of N atoms is where v N =(8kT N /πm N ) 1/2 Γ NS = 1 4 n NS v N ) 1 Ediss /E iz LiebermanShortCourse06 67

68 RECOMBINATION, REACTION AND LOADING EFFECT Recall that: P abs n NS =2C 0 ee T S p ( ng l 2h l u B ) 1 Ediss /E iz Consider a recombination coefficient γ rec for N atoms on the walls and a reaction coefficient γ reac for N atoms on a substrate. For M substrates: 1 S p S p + γ rec 4 v 1 N2A + γ reac 4 v NMA subs Note that n NS is reduced due to recombination and reaction losses n NS depends on the number of wafers M, aloading effect LiebermanShortCourse06 68

69 ADSORPTION AND DESORPTION KINETICS LiebermanShortCourse06 69

70 ADSORPTION Reaction of a molecule with the surface A+S K a A: S Physisorption (due to weak van der Waals forces) U 1 3 Å Kd V x Chemisorption (due to formation of chemical bonds) U Å V x LiebermanShortCourse06 70

71 Adsorbed flux STICKING COEFFICIENT Γ ads = sγ A = 1 4 s v An AS s(θ, T) = sticking coefficient θ = fraction of surface sites covered with absorbate Langmuir kinetics s(θ, T) = s 0 (T)(1 θ) s 0 s Langmuir θ Zero-coverage sticking coefficient s s T LiebermanShortCourse06 71

72 DESORPTION Rate constant A: S A+S where E desor = E chemi or E physi K desor = K 0 e E desor/t K s s 1 physisorption chemisorption LiebermanShortCourse06 72

73 ADSORPTION-DESORPTION KINETICS Consider the reactions A+S K a A: S Kd The adsorption flux is Γ ads = K a n AS n 0(1 θ) n 0 = area density (m 2 ) of adsorption sites n AS = the gas phase density at the surface 1 K a = s 0 4 v A/n 0 [m 3 /s] The desorption flux is Γ desor = K d n 0θ K d = K d0 e E desor/t [s 1 ] LiebermanShortCourse06 73

74 LANGMUIR ISOTHERM Equate adsorption and desorption fluxes Γ ads =Γ desor θ = where K = K a /K d Kn AS 1+Kn AS 1 θ A n A K ads Note that T K θ log ER θ as T θ=1 with K r as T Example: 2XeF 2 (g) + Si(s) SiF 4 (g) + 2Xe 1/T LiebermanShortCourse06 74

75 -ASSISTED ETCH KINETICS LiebermanShortCourse06 75

76 ION-ENHANCED ETCHING 1. Low chemical etch rate of silicon in XeF 2 etchant gas 2. Tenfold increase in etch rate with the addition of argon ion bombardment of the substrate, simulating plasma-assisted etching 3. Very low etch rate due to the physical sputtering of silicon by the ion bombardment alone LiebermanShortCourse06 76

77 STANDARD MODEL OF ETCH KINETICS O atom etching of a carbon substrate O + CO K a K i K d Y i K i 1 θ θ C(s) CO(s) Let n 0 = active surface sites/m 2 Let θ = fraction of surface sites covered with C : O bonds O(g) + C(s) K a C : O C:O K d CO(g) ion+c:o Y ik i CO(g) (O atom adsorption) (CO thermal desorption) (CO ion-assisted desorption) LiebermanShortCourse06 77

78 SURFACE COVERAGE The steady-state surface coverage is found from dθ dt = K an OS (1 θ) K d θ Y i K i n is θ =0 n OS is the neutral gas density near the surface n is is the ion density at the plasma edge K a is the rate coefficient for O-atom adsorption K d is the rate coefficient for thermal desorption of CO K i is the rate coefficient for ions incident on the surface Y i is the yield of CO molecules desorbed per ion incident on a fully covered surface Typically Y i 1 and Y i ( E i E thr ) (as for sputtering) = θ = K a n OS K a n OS + K d + Y i K i n is LiebermanShortCourse06 78

79 ETCH RATES The flux of CO molecules leaving the surface is Γ CO =(K d + Y i K i n is ) θn 0 (m 2 -s 1 ) with n 0 = number of surface sites/m 2 The vertical etch rate is E v = Γ CO n C (m/s) where n C is the carbon atom density of the substrate The vertical (ion-enhanced) etch rate is E v = n 0 1 n C K d + Y i K i n is K a n OS The horizontal (non ion-enhanced) etch rate is E h = n 0 1 n C K d K a n OS LiebermanShortCourse06 79

80 NORMALIZED ETCH RATES 10 8 Y i K i n is =5K d En C K d n Vertical (E v ) Horizontal (E h ) K a n OS K d High O-atom density highest anisotropy E v /E h =1+Y i K i n is /K d Low O-atom density low etch rates with E v /E h 1 LiebermanShortCourse06 80

81 SIMPLEST MODEL OF ION-ENHANCED ETCHING In the usual ion-enhanced regime Y i K i n is K d 1 = n C 1 1 E v Y i K i n is n + 0 K a n OS n 0 }{{}}{{} Γ is Γ OS The ion and neutral fluxes and the yield (a function of ion energy) determine the ion-assisted etch rate The discharge parameters set the ion and neutral fluxes and the ion bombarding energy LiebermanShortCourse06 81

82 ADDITIONAL CHEMISTRY AND PHYSICS Sputtering of carbon Γ C = γ sput K i n is n 0 Associative and normal desorption of O atoms, C:O C + O(g) 2C:O 2C+O 2 (g) Ion energy driven desorption of O atoms ions + C : O C + O(g) Formation and desorption of CO 2 as an etch product Non-zero ion angular bombardment of sidewall surfaces Deposition kinetics (C-atoms, etc) LiebermanShortCourse06 82

83 CONCLUSIONS Plasma discharges are widely used for materials processing and are indispensible for microelectronics fabrication The discharge parameters (power, pressure, geometry, gas feed, etc) set the ion and neutral fluxes and the ion bombarding energy The charged particle balance determines the electron temperature and ion bombarding energy to the substrate = Y i (E i ) The energy balance determines the plasma density and the ion flux to the substrate = Γ is A transformer model determines the relation among voltage, current, and power for inductive discharges The neutral radical balance determines the flux of radicals to the surface = Γ OS The ion and neutral fluxes and the yield (a function of ion energy) determine the ion-assisted etch rate LiebermanShortCourse06 83