III. Electromagnetic uniformity: finite RF wavelength in large area, VHF reactors: standing waves, and telegraph effect

Transcription

1 III. Electromagnetic uniformity: finite RF wavelength in large area, VHF reactors: standing waves, and telegraph effect IV. Uniformity in time: minimize transients, rapid equilibration to steady-state process parameters - Direct pumping of a plasma reactor. V. So where is the problem? - Causes of non-uniformity. Some recommendations. A. aah Howling III ICANS-23 Utrecht August 23 (29)

2 Motivations for Very High Frequency (VHF: 3-3 MHz) adapted from Jacques Schmitt Orleans ISPC XV 2. Larger power in for: - less sheath voltage - less ion bombardment - better film quality - higher deposition rate - reduced dust build up plasma RF capacitive sheath over-simplified equivalent circuit for plasma C C R RF "For a given RF power in the plasma, require a given RF current irf in the resistive plasma. Sheath capacitances C dominates the total impedance. irf For the current irf, the RF Voltage across a capacitive sheath is VRF =. 2π frfc Therefore higher frequency has lower RF sheath voltage for given RF power" 2 aah III

3 Motivation for VHF - some experimental measurements maximum ion energy (ev) MHz 7 MHz 25 MHz 3 MHz 42 MHz 5 MHz 7 MHz pic 7 W pic 3 W capacitive sheaths resistive sheaths ion energy (ev) and (rf voltage)/2 (V) floating potential ion energy (pic) (ev) ion energy (expt) (ev) rf voltage/2 (V) total ion flux (counts/s) max energy = 29 ev max energy = 39 ev f 2. f rf voltage/2 (V) ion energy ~ half RF voltage amplitude for a symmetric plasma reactor frequency (MHz) For constant RF power: RF voltage decreases with frequency, therefore the ion energy decreases (down to floating potential) frequency (MHz) For constant ion energy: ion flux increases with frequency 3 aah III W. Schwarzenbach, A. A. Howling, M. Fivaz, S. Brunner, Ch. Hollenstein, J. Vac. Sci. Technol. A4 32 (996)

4 Very large plasma capacitor 98 2 cm Reactor diagonal: 4 cm 25 THE NO-LONGER LIST: RF wave length >> reactor size? No First electromagnetic source of non-uniformity: Very High Frequency and the standing-wave problem 4 aah III

5 standing wave effect need reactor dimension less than / of vacuum wavelength (factor 4 for node-antinode, and factor ~2.5 for worsening effect) vacuum wavelength m vac. wavelength / m MHz For m diagonal, - standing wave barely observable at 3 MHz - contributing to profile non-uniformity at 27 MHz - strong non-uniformity at 7 MHz For 2 m diagonal, - even 3 MHz is a problem 5 aah III adapted from Jacques Schmitt Orleans ISPC XV 2.

6 RF coupling and RF voltage homogeneity Examples i) 3.56 MHz edge RF connection 3.56 MHz RF coupling point ii) 7 MHz edge RF connection edge 7 MHz iii) 7 MHz central RF connection centre 7 MHz RF v oltage distribution across the electrode Experiment Calculation (a) measurements (b) calculations depth (cm) depth depth (cm) width (cm).3 depth (cm) depth (cm) 2 2 (cm) depth depth (cm) width (cm).3 depth (cm) depth (cm) depth (cm) width (cm) depth (cm) depth (cm) width (cm) (cm) width (cm) width (cm) width (cm) 35 cm Film thickness interferogram Film thickness uniformity 45 cm 6 aah III L. Sansonnens, A. Pletzer, D. Magni, A. Howling, Ch. Hollenstein, J. Schmitt, Plasma Sources ScienceTechnol. 6, 7 (997)

7 High frequency vacuum capacitor from Feynman lectures : self-consistent solution of oscillating E and B fields an electromagnetic solution is necesary; no longer electrostatic! plates are not equipotential: Bessel function distribution 7 aah III

8 Axisymmetric H φ in a cylindrical reactor wave equation ( 2 2 H + k H ) = φ field equation vacuum TEM mode Bessel function J(k r) Transverse ElectroMagnetic l H φ E z E r = -l 8 aah III

9 The capacitor in a vacuum chamber: The hand-waving description small skin depth in metal electrodes r WAVE OUT WAVE IN WAVE image RF current flow in a capactive reactor RF power propagates inwards from the edges 9 aah III courtesy of Jacques Schmitt

10 CYLINDRICAL TRANSMISSION LINE APPROACH (6 optional slides) Admittance of cylindrical parallel plates, gap l l r εa Capacitance of parallel plates, C = l ε2πrdr Elemental cylindrical capacitance, dc = l ε2πr Capacitance per unit length in cylinder along r = l Capacitive impedance =, admittance = j ωc. jωc Admittance per unit length (along r) of cyl. parallel plates, Y = ε jω 2 π r l aah III

11 aah III Effective permittivity of plasma-&-sheaths l Plasma short-circuits the transverse electric field: ε A Combined capacitance of sheaths =, 2s εε eff A Effective capacitance of the gap =, l l Effective permittivity of the gap, εeff =. 2s Effective admittance per unit length (along r) of plasma-&-sheaths 2π r in cylindrical parallel plates, Y = j ωεεeff. l s s

12 Series impedance of parallel plates Ampere's law (uniform B between plates): B ds = μ I, Flux B= I B l I W D r ( μ ) B = μ I W, unaffected by plasma =, Φ = BlD = μ IlDW Parallel plate inductance, L= Φ I = μ ld W 2 aah III

13 Series impedance of cylindrical parallel plates, gap l l r μldr Inductance of cylindrical element, dl =, 2π r μldr Impedance of cylindrical element = j ω. 2 π r l Series impedance in cylinder per unit length along r, is Z = j ωμ. 2 π r 3 aah III

14 Consider a general cylindrical transmission line l I V r ~ Uniform voltage and current around the circumference, open circuit on axis, series impedance Z, and parallel admittance Y, per unit length along r. I -I V Z Y di V+dV I+dI -(I+dI) dv di Kirchoff's laws: = IZ, = YV. dr dr Substitute for I (or for V): 2 d V dz dv =, which is the 2 + YZV dr Zdr dr general voltage wave equation. 4 aah III

15 Consider a parallel-plate cylindrical transmission line gap is constant, V is variable. l I V r ~ 5 aah III l 2π r Use the parallel-plate expressions: Z = j ωμ, Y = jωεεeff 2π r l dz = and the general cylindrical transmission line equation is: Z dr r 2 dv dv 2 2 kv dr + + = r dr, the voltage wave equation in cylindrical geometry, ( ) = = ω μ ε ε = Bessel function solution J kr, where k YZ k ε ; eff eff 2π c k =, k = k ε, and the wave phase velocity = =. eff λvac μεε eff εeff (sheath width s, ε, assumed independent of V) eff

16 Effect of plasma on vacuum capacitor Bessel function J (k r ) VACUUM STANDING WAVE k = 2π λ vac k r node = 2.45 J (ε eff /2 k r ) SLOWER WAVE ε eff /2 k r node = 2.45 courtesy of Jacques Schmitt 6 aah III ε eff equivalent dielectric acts as a worsening factor for uniformity

17 Cylindrical reactor experiment optical emission & surface electrostatic probes metre H. Schmidt, L. Sansonnens, A. A. Howling, and Ch. Hollenstein, J. Appl. Phys. 95, 4559 (24) 7 aah III

18 Standing wave effects With plasma, the first node moves inside: dark ring at r =.3 m! EXPERIMENTS AT MHz E-field relative profile at MHz (bench test with scanning probe) probe array ion saturation currents (normalized to the central values).8 2 W.5 Torr mtorr aah III.4 Electromagnetic modeling.2 J (k r) IN VACUUM {J (3.5 k r)} m m metre metre

19 A compensation solution Gaussian shape dielectric lens a = a exp(-ε D k 2 r 2 /4) Plasma gap unchanged, at least one electrode with a lens Solution of Maxwell s equations Constant vertical E-field across the diameter Still valid in presence of a plasma Fig.2 of Patent US (Unaxis) aah III L. Sansonnens and J. Schmitt, Appl. Phys. Lett. 82, 82 (23)

20 World's Simplest Lens Solution (4 optional slides) adapted from P. Chabert et al, Phys. Pl., 48 (24) gap is variable, V p is constant. Dielectric lens d(r) e(r) l V(r) V lens (r) V p We keep a constant plasma gap, l, but add a variable thickness lens er ( ). Total electrode gap is now dr ( ) = l+ er ( ). 2 aah III The plasma voltage is V, which we define constant, the voltage across the lens thickness is V ( r). p p lens The total electrode voltage is V() r = V + V (). r lens

21 World's Simplest Lens Solution; introduction RF current flow in a capactive lens reactor; design the lens so that the plasma current j z & field E z are uniform along r. lens j z j z j z j z 2 aah III I(r) electrode surface current I() =

22 World's Simplest Lens Solution. total radial conduction current =I(r) along the electrode surface constant vertical electric field E z & displacement current density, j z 22 aah III Vp ) We require a constant vertical electric field in gap, Ez = = const. l jωεεeffvp j ωεε rvlens() r 2) require constant vertical current density jz = =. l e() r 3) The radial current in electrodes, Ir ( ) = jzπr (all the current entering radius r radially, must leave vertically, because I() =, and j = constant over all r.) z 2

23 World's Simplest Lens Solution. 4) To calculate V( r), use Ohm's law radially: dv( r) = I( r) Z( r) dr, d( r) where Z( r) = j ωμ is the series impedance of the gap, per unit radius. 2π r dv ( r) 2 j ωεεrvlens( r) 2 d( r) 5) Substitute: = jzπr Z = πr jωμ, dr e( r) 2π r dvlens ( r) 2 d( r) r = kε rvlens ( r), since V( r) = Vlens ( r) + Vp. dr e( r) 2 ε V eff p 6) Substitute for Vlens ( r) using 2): Vlens ( r) = e( r) so that ε r l 2 r 2 r e ( r) = kεrd( r), and since d( r) = e( r) + l, d ( r) = kεrd( r) kε rr 7) d( r) = ( l + e() ) exp, a Gaussian lens. 4 Shape depends on εr, not εeff, i.e. lens shape is independent of the plasma! Also, I( r) and V ( r) ε for given V. 23 aah III lens eff p

24 Experimental demonstration of a compensation lens Probe array: ion current normalized to center {J (ε eff.5 k r)} 2 Parallel Plates for 69 MHz 9 mm mtorr mtorr m Very non uniform a factor aah III Teflon Lens for 69 MHz Recovery uniformity > %

25 Very large plasma capacitor 98 2 cm 25 Reactor diagonal: 4 cm THE NO-LONGER LIST again RF plasma potential is constant? No Second electromagnetic aspect : Current flow over long, resistive plasma 25 aah III

26 Edge perturbation due to a sidewall Small area reactor plasma RF plasma potential averaged over the whole reactor - unequal sheath voltages plasma Large area reactor Symmetric centre equal sheath voltages Non-symmetric edge unequal sheath voltages RF plasma potential is not the same everywhere how does the perturbation propagate? 26 aah III

27 The telegraph equation V exp(iωt) 27 aah III The plasma equivalent circuit is similar to a lossy transmission line The propagation of the rf plasma potential perturbation V is described by the telegraph equation 2 V 2 V = LC 2 t V + RC t for Bessel fetishists The first kind Bessel function for complex wavenumber k, Re{J (kr )}, is a damped standing wave which is almost equivalent to its inward-propagating damped traveling wave component Re{H () (kr )} (a Hankel function) E z top sheath E z bottom sheath sidewall sheath coupling

28 Telegraph effect: probep measurements Rectangular reactor 47 x 57 cm 2 8 surface probes for DC voltage and current measurements Theory: A. A. Howling, L. Sansonnens, J. Ballutaud, Ch. Hollenstein, J. P. M. Schmitt, J. Appl. Phys. 96, 5429 (24) Experiment: A. A. Howling, L. Derendinger, L. Sansonnens, H. Schmidt, Ch. Hollenstein, E. Sakanaka, J. P. M. Schmitt, J. Appl. Phys. 97, 2338 (25) 28 aah III

29 29 aah III Telegraph effect... ΔVplasma ΔVplasma ΔVplasma ΔVplasma

30 ...demonstrat demonstrated ed ΔVplasma ΔVplasma ΔVplasma ΔVplasma 3 aah III

31 Asymmetric electrodes odd mode, H φ (r,-z) = -H φ (r,z) z even mode, H φ (r,-z) = H φ (r,z) EVEN in z EVEN & ODD in z the first even & odd quasi-tem modes give the complete solution (higher modes are evanescent ) quasi-tem first even mode 3 aah III l d -d -l H φ, E z, E r, z e e e top sheath plasma bottom sheath r quasi- TEM first odd mode l d -d -l Ho φ, Eo z, Eo r, top sheath r plasma bottom sheath

32 Physical origin: quasiq uasi-tem even mode the dispersion relation of the even mode reproduces the intuitive equivalent circuit of the standing wave effect: electrode gap self-inductance L plasma Y sheath C U RF plasma as lossy dielectric in a transmission line RF current wave J δ J RF current wave J 32 aah III

33 Physical origin: quasiq uasi-tem odd mode the dispersion relation of the odd mode reproduces the intuitive equivalent circuit of the telegraph effect: Z sq dx plasma I(x) U RF I(x) C s dx C wall sidewall electrode asymmetry is the e/m source of the telegraph effect. plasma as lossy conductor in a transmission line RF current ΔJ RF current ΔJ RF current ΔJ 33 aah III

34 Nonuniformity dominated by telegraph effect Sheath electric field top low frequency 3 MHz ~ grounded reactor box bottom excitation electrode asymmetric sidewall Edge Localized Modes (not the same as in a tokamak!) - fringing fields due to discontinuities (in permittivity and in geometry) 34 aah III

35 Nonuniformity of standing wave effect only Sheath electric field top and bottom high frequency 67.8 MHz ~ symmetric sidewalls edge localized modes - fringing fields due to discontinuities 35 aah III

36 Standing wave & telegraph effect combined Sheath electric field top bottom high frequency 67.8 MHz ~ grounded reactor box excitation electrode asymmetric sidewall edge localized modes - fringing fields due to discontinuities 36 aah III

37 time series : ωt = EVEN MODE standing wave effect observed for symmetric reactor. ODD MODE telegraph effect x (not observable in isolation) PLASMA RESPONSE combined effect observed for asymmetric reactor aah III

38 time series : ω t = π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect standing wave oscillation damped wave propagation combination aah III

39 time series : ω t = 2π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect aah III

40 time series : ω t = 3π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect aah III

41 time series : ω t = 4π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect aah III

42 time series : ω t = 5π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect aah III

43 time series : ω t = 6π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect aah III

44 time series : ω t = 7π 4 EVEN MODE standing wave effect ODD MODE telegraph effect x PLASMA RESPONSE combined effect aah III

45 time series : ω t = 8π 4 EVEN MODE standing wave effect observed for symmetric reactor. ODD MODE telegraph effect x (not observable in isolation) PLASMA RESPONSE combined effect observed for asymmetric reactor aah III

46 .2 m 2 film thickness profiles - individual effects standing wave; parallel plates telegraph effect: experiment standing wave; lens telegraph effect: model 46 L. Sansonnens, aah III H. Schmidt, A. A. Howling, Ch. Hollenstein, Ch. Ellert, A. Buechel, J. Vac. Sci. Technol. A24, 425 (26) L. Sansonnens, B. Strahm, L. Derendinger, A. A. Howling, Ch. Hollenstein, Ch. Ellert, J.P. M. Schmittl, J. Vac. Sci. Technol. A23, 922 (25)

47 Electromagnetic summary L. Sansonnens, A. A. Howling and Ch. Hollenstein, Plasma Sources Sci. Technol (26) A. A. Howling, L. Sansonnens and Ch. Hollenstein, Thin Solid Films (27) 47 aah III

48 Intermediate Conclusions Require uniform RF voltage. If an electrode dimension is > (RF vacuum wavelength/), then special precautions are necessary to avoid the standing wave effect. This is therefore more likely to be a problem at Very High Frequency. Use symmetric electrodes to avoid telegraph effect (RF current injection from asymmetric sidewalls). Avoid edge fringing fields due to discontinuities in permittivity and geometry. Refs. L. Sansonnens, A. A. Howling and Ch. Hollenstein, Plasma Sources Sci. Technol (26) A. A. Howling, L. Sansonnens and Ch. Hollenstein, Thin Solid Films (27) 48 aah III