ANGULAR DEPENDENCE OF ELECTRON VELOCITY DISTRIBUTIONS IN LOW-PRESSURE INDUCTIVELY COUPLED PLASMAS 1

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1 ANGULAR DEPENDENCE OF ELECTRON VELOCITY DISTRIBUTIONS IN LOW-PRESSURE INDUCTIVELY COUPLED PLASMAS 1 Alex V. Vasenkov 2, and Mark J. Kushner Department of Electrical and Computer Engineering Urbana, IL vasenkov@uiuc.edu mjk@uiuc.edu October Work supported by CFD Research Corp., NSF, Applied Materials, Inc. and the SRC 2 Present address: CFD Research Corp., Huntsville, AL 35805

2 AGENDA Angular dependent electron velocity distributions (AEVDs) in inductively coupled plasmas (ICPs) Description of the Monte Carlo (MC) algorithm for modeling of AEVDs Plasma properties, Legendre coefficients, and AEVDs for ICPs in Ar/c-C 4 F 8 Linear and nonlinear forces in ICPs Variations of AEVDs from collisionless to collisional conditions Summary GEC

3 AEVDs IN ICPs ICPs are the workhorse of the microelectronics industry for etching and deposition of materials. Electron kinetics is usually non-equilibrium at typical operating conditions (< 10s mtorr, < 10s MHz, 100s W kw). Little is known about the angular dependence of the AEVDs in and near the skin layer, particularly, when the skin layer is anomalous. The angular anisotropy of the AEVD in ICPs is often assumed to be small so that a two-term spherical harmonic expansion is valid. GEC

4 MC model for calculating Legendre expansion coefficients and AVED was incorporated into Hybrid Plasma Equipment Model (HPEM). HPEM is a two-dimensional, plasma equipment model consisting of an Electromagnetic Module, an Electron MC Simulation Module and Fluid Kinetics Module. Method was developed in which AEEDs is given by r r f ( ε,, φ ) = al( ε, ) P l(cos φ ) l where ε = electron energy r = spatial location P l = l th Legendre polynomial a l = l th Legendre polynomial coefficient φ = angle of the electron trajectory with respect to a reference direction, φ 0. GEC DESCRIPTION OF THE MODEL

5 r r r 1 1 Φ ikj = w j t jδ(( ε i ± εi ) ε j ) αmδ(( m+ k ± m+ k ) j ) 2 where ε, r = energy and location of the i th energy bin µ n = cos(φ n ) ε, r = spatial and energy widths of k th spatial mesh cell r a ( ε,r ) are obtained from the raw statistics as a l l A ( ε l i k i r,r DESCRIPTION OF THE MODEL (CONT.) Raw statistics A l, from which a l are computed, are updated as GEC k ) A ( ε l r,r Φikj j n ) + 2l+ 1 δ(( µ n ± 2 1 r r r ( εi,rk ) = A l ( εi,r k) / A0 ( εi,r k) i i k m P0 dµ µ n ) µ 2 j n ) )P l ( µ.

6 DOMINANT FORCES IN ICP CELL An ICP reactor patterned after Oeherlein et al. was used for investigation. There are at least five dominant forces that act upon electrons: 1. Electrodynamic force F θ (1) = - q E θ. GEC Electrostatic force F θ s = - q E s. 3. Second order non-linear r Lorentz ( 2 ) r force (NLF) F. z = v Brf r r r 4. Force F sh resulting from F ~ E B (Makabe et al, 1998). sh 5. Third order non-linear force F θ (3) acting on the electrons in the θr plane. s rf

7 PLASMA PROPERTIES FOR THE BASE CASE CONDITIONS The peak [e] results from the drift of thermal electrons towards the peak of the plasma potential (Φ pot ). The lower [e] in Ar/C 4 F 8 is due to the higher rate of power dissipation in the molecular mixture. Due to higher rates of loss by attachment to C 4 F 8 and its fragments a higher T e is required. Ar and Ar/C 4 F 8 = 90/10 at 3 mtorr, 400 W, 3.39 MHz. GEC

8 LEGENDRE COEFFICIENTS IN RZ PLANE Large odd a n in the skin layer imply high energies electrons are accelerated out of the skin layer into the bulk plasma by the NLF. In the bulk plasma, NLF is small and electrons experience a large number of collisions. Odd a n are large only for e > 30 ev. Near the substrate, the even a n are large for e > 20 ev producing an AEVD stretched in -v z and +v z. 3 mtorr, 400 W, 3.39 MHz. GEC

9 LEGENDRE COEFFICIENTS IN Rθ PLANE Even a n dominate implying that the AEVDs are stretched in +v θ and -v θ. Harmonic acceleration by E θ should produce symmetric anisotropy (even a n ). Even a n are smaller in the bulk plasma and near the substrate due to collisions reducing the anisotropy. 3 mtorr, 400 W, 3.39 MHz. GEC

10 AEVDs IN V θ V r PLANE IN THE MIDDLE OF SKIN LAYER 3 mtorr, 400 W, 3.39 MHz. AEVDs are obtained with seven a n. AEVDs are anisotropic at low and high energies as they include electrons from different portions of the rf cycle. The angular anisotropy of AEVDs results in large v θ (1.3x10 8 cm/s). GEC

11 AEVDs IN V z V r PLANE IN THE MIDDLE OF SKIN LAYER AEVD of low-energy electrons is nearly isotropic and shifted to -v z due to the energy pooling. The anisotropy of the AEVDs increases with energy owing to increased NLF. GEC mtorr, 400 W, 3.39 MHz.

12 THIRD ORDER NON-LINEAR FORCE The third order equation of motion in the collisionless limit is (Chen, 2001) r ( 3 ) r dv { v r = = } ( 3 ) r ( 2 ) ( 2 ) v F m q ( z )E θ + vz Bre θ dt v ( ) Substituting for r 2 v ( ) and with their expressions a 3 rd order z 2 z force is directed along the tangent r F 3 ( 3 ) v θ r 2 q 0 = e E 2 2 θ θ 4m ω [ ] 0 2 ( 2 2 B ω t + 2ωt + cos( 2ωt ) / 2 sin( ωt )) With the base case conditions (3 mtorr, 400 W ) F θ (3) / F θ (1) = 5 at MHz = 100 at 1.13 MHz. GEC

13 EFFECT OF FREQUENCY ON a n /a 0 in RZ plane a n /a 0 at 1.13 MHz are larger than a n /a 0 at MHz as NLF increases with decreasing frequency. The odd a n /a 0 are larger than even a n /a 0 in the skin layer at 1.13 MHz and are commensurate with the even a n /a 0 at MHz. In the bulk plasma and at the edge of the skin layer (z = 8 cm and z = 5 cm), the even a n /a 0 at 1.13 MHz are larger than odd a n /a 0 at ε < 25 ev, producing AEEDs elongated in +v z and v z and symmetric with respect to r axis. Ar, 3 mtorr, 400 W, 3.39 MHz. GEC

14 EFFECT OF FREQUENCY ON a n /a 0 in Rθ plane Only even a n /a 0 are shown as odd a n /a 0 are small. a n /a 0 are the largest in the skin layer, where F θ (1) and F θ (3) peak. a n /a 0 for MHz are larger than or commensurate with a n /a 0 at 1.13 MHz for e < Φ pot due to increased E rf. At e > Φ pot, a n /a 0 at 1.13 MHz are commensurate with or larger than a n /a 0 for MHz implying that F θ (1) exceeds F θ (3). Ar, 3 mtorr, 400 W, 3.39 MHz. GEC

15 EFFECT OF B rf ON a n /a 0 in RZ plane Without B rf, the NLF is zero and the anisotropy of the AEVDs in the rz plane is due only to the thermal diffusion of electrons. In the skin layer, high energy electrons are lost from the plasma leaving only those directed downward to contribute to odd a n /a 0. In the bulk plasma and at the edge of the skin layer (z = 8 and z = 5 cm) small a n /a 0 at low ε implies that without B rf the AEEDs are fairly isotropic. Ar, 3 mtorr, 400 W, 3.39 MHz. GEC

16 EFFECT OF B rf ON a n /a 0 in Rθ plane F θ (1) is not affected by B rf, whereas F θ (3) is directly proportional to B rf. a n /a 0 with B rf are similar to a n /a 0 without B rf for e < Φ pot as AEVD at these energies is determined by F θ (1). At e > Φ pot, a n /a 0 with B rf are larger than a n /a 0 without B rf implying that anisotropy is determined by F θ (3). Ar, 3 mtorr, 400 W, 3.39 MHz. GEC

17 High-order a n in rz plane decrease with increased pressure due to diminished NLF and warm plasma effects. Even a n in the rθ plane are large at both 1 mtorr and 50 mtorr as they originate from both F θ (1) and F θ (3). Even a n in the rθ plane at 1 mtorr are larger than a n at 50 mtorr, due to decreased F θ (3) with increased pressure. EFFECT OF PRESSURE ON a n Ar, 3 mtorr, 400 W, 3.39 MHz. GEC

18 The anisotropy of AEVDs in low-pressure ICPs was investigated using MC techniques by sampling the trajectories of the electrons and computing Legendre coefficients. The anisotropy in the rz plane is largest at higher energies, favoring the high-order odd coefficients due largely to NLF. The anisotropy in the θr plane is due to electron acceleration by linear electrodynamic and nonlinear third-order forces. Anisotropy in the rz plane dominantly occurs when the skin layer is anomalous, whereas anisotropy in the θr plane persists to higher pressures. For operating conditions typical of plasma processing reactors, higher Legendre coefficients in both rz plane and θr plane have significant values. GEC Summary