2D Hybrid Fluid-Analytical Model of Inductive/Capacitive Plasma Discharges

Transcription

1 63 rd GEC & 7 th ICRP, D Hybrid Fluid-Analytical Model of Inductive/Capacitive Plasma Discharges E. Kawamura, M.A. Lieberman, and D.B. Graves University of California, Berkeley, CA This work was supported by gifts from Lam Research Corp. and Micron Corp., in part by the UC Discovery Grant ele under the IMPACT program, and in part by the Dept. of Energy Office of Fusion Energy Science Contract DE-SC University of California Berkeley 1 GEC10

2 Overview A fast 2D TCP Reactor Model was developed: EM Model with inductive and capacitive coupling of source coils to plasma, and capacitive bias option for wafer electrode. Quasi-neutral Bulk Plasma Fluid Model which solves ion continuity and electron energy balance coupled to an Analytical Sheath Model with electron and ion sheath heating. Gas Flow Fluid Model with reactive gas chemistries. (Hsu et al, J. Phys. D. 39, 3272, 2006) A portable, user-friendly Comsol-Matlab platform. Good agreement with experimental Cl 2 TCP reactor study. University of California Berkeley 2 GEC10

3 TCP Reactor Geometry Axisymmetric cylindrical geometry. Outer surface is a perfect conductor Wafer chuck insulated from walls by quartz dielectric spacer (κ = 4). 4-turn stove-top coil set on top of quartz dielectric window (κ = 4). Thin vacuum sheath around plasma. Relative dielectric constant of plasma: κ p =1 ω p 2 ω ω jv m Coil 1 is attached to an rf current source while Coil 4 is grounded. University of California Berkeley 3 GEC10

4 EM Model Coils produce inductive TE fields (E φ,h r,h z ) and capacitive TM fields (H φ,e r,e z ). Solve for one field in each trio, get other two via Maxwell s equations. Assume all fields F exp(jωt) and solve Helmholtz in each region. ( 2 + (ωκ/c) 2 )F + f = 0 Plasma is a lossy dielectric since κ p has an imaginary (dissipative) part. University of California Berkeley 4 GEC10

5 Inductive TE Simulations (E φ,h r,h z ) Dependent variable is V=2πrE φ, the loop voltage at radius r. Solve Helmholtz Eq. for V with boundary conditions: V = 0 on all outer surfaces and center of symmetry V = V n on the perimeter of the nth coil V n are determined for a given input current I i n to the coil set. University of California Berkeley 5 GEC10

6 Capacitive TM Simulations (H φ,e r,e z ) Dependent variable is I=2πrH φ, the z-directed current intercepting the cross-sectional area of radius r. Solve Helmholtz Eq. for I with boundary conditions: n I = 0 on all outer surfaces I = 0 on center of symmetry since I r Each coil is supplied by an axial feed current i n. i n are determined for a given input current I i n to the coil set. University of California Berkeley 6 GEC10

7 Capacitive Bias Option for Wafer Electrode Solve capacitive Helmholtz Eq. for I=2πrH φ n I = 0 on all conducting walls I = 0 on center of symmetry with I = const = I r f on bottom surface of dielectric spacer TCP coils are floating (coil currents i n = 0). rf voltage supplied by the bias is: V r f = E r dr rf power is P r f = 0.5Re(I r f V r f * ) spacer bottom University of California Berkeley 7 GEC10

8 TCP Coil Circuit Inductance matrix L relates V L =(V 1,V 2,V 3,V 4 ) with I L =(I 1,I 2,I 3,I 4 ): V L = jω L I L Capacitance matrix C relates i c =(i 1, i 2, i 3, i 4 ) to v c =(v 1, v 2, v 3, v 4 ): i c = jω C v c Find L by conducting 4 orthonormal TE simulations in which V k =1 for the nth coil while V k =0 for other 3 coils. Take line integral of H around mth coil to get currents I m n L m n. Find C by conducting 4 orthonormal TM simulations in which i k =1 for the nth coil while i k =0 for other 3 coils. Take line integral of E from mth coil to ground to get voltages v m n C m n. For a given I i n, can solve circuit to get all i k, v k, I k,v k. University of California Berkeley 8 GEC10

9 Bulk Plasma Power Deposition Inductive TE and capacitive TM equations are solved simultaneously to obtain all the fields. The total plasma current density is given by J T = (σ p + jωε 0 )E where σ p = jωε 0 (κ p 1) is the plasma conductivity. Time averaged power density profile in the plasma is: p dep =0.5 R e J T E * =0.5 R e σ p E φ 2 E r 2 E z 2 in d u c tiv e c a p a c itiv e p d e p is used as input in the electron energy balance equation of the Bulk Plasma Fluid Model. University of California Berkeley 9 GEC10

10 Fixed Width Sheath Model Sheath width s depends on local E, n e and T e. Computationally inconvenient to adjust position of plasma-sheath boundary. Sheath with constant width s 0 and varying dielectric constant κ s = s 0 /s used to mimic a vacuum sheath (κ = 1) with varying width s. (Lee et al, PSST 17, , 2008) The sheath voltage V s h same for both cases: V s h = (E v a c /κ s ) n s 0 = E v a c n s Modified Lee et al by (i) treating electron sheath heating as an incoming energy flux to the electron energy balance equation, and by (ii) adding a dissipative (imaginary) term to κ s to account for the electron and ion sheath heating. University of California Berkeley 10 GEC10

11 Outline of TCP Model University of California Berkeley 11 GEC10

12 TCP Reactor Simulations 10 mtorr, MHz, 100 sccm Cl 2 plasma with I i n =15 70A (P a b s = 5.3 to 813 W). Cl 2 reaction set from Thorsteinsson & Gudmundsson, PSST 19, (2010). Each simulation on a moderate 2GHz CPU, 4GB RAM PC ~ 70 min. As I i n, low density capacitive high density inductive mode. University of California Berkeley 12 GEC10

13 Molar Fraction of Cl (10 mt, 100 sccm Cl 2 ) P a b s = 6.0 W P a b s = 763 W University of California Berkeley 13 GEC10

14 Gas Temperature (K) (10 mt, 100 sccm Cl 2 ) P a b s = 6.0 W P a b s = 763 W University of California Berkeley 14 GEC10

15 Inductive vs. Capacitive (763 W, 10 mt, 100 sccm Cl 2 ) Inductive Power Density (W/m 3 ) Capacitive Power Density (W/m 3 ) University of California Berkeley 15 GEC10

16 n e and T e (763 W, 10 mt, 100 sccm Cl 2 ) Electron Density (m - 3 ) Electron Temperature (V) University of California Berkeley 16 GEC10

17 Model vs. Experiment (10 mt, 100 sccm Cl 2 ) Electron density (m- 3 ) vs. P a b s (W) Cl density (m- 3 ) vs. P a b s (W) P a b s /P r f = 0.75 for Malyshev & Donnelly reactor. (Hopwood, PSST 3, ) Cl recombination coefficient = 0.02 at walls. (Corr et al, J. Phys. D. 41, , 2008) University of California Berkeley 17 GEC10

18 Uniformity in a Two Coil Set TCP Reactor 10 mtorr, 200 sccm two coil set Cl 2 TCP reactor with I 1 i n = ±I 2 i n I 1 i n = 13 A = I 2 i n, P a b s = 923 W, P c a p = 116 W, P i n d = 807 W I 1 i n = 13 A = I 2 i n, P a b s = 431 W, P c a p = 81 W, P i n d = 350 W Cl 2 ion flux at wafer (m - 2 /s) Reverse currents control uniformity by suppressing inductive mode University of California Berkeley 18 GEC10

19 Conclusions Hybrid fluid-analytical TCP Reactor Model allows fast computation of chemically active plasmas with flow and calculates both the inductive and capacitive fields. Capacitive fields sheath width and voltage electron and ion sheath heating etch rate predictions. As P a b s rises, inductive coupling, plasma density, gas heating, and Cl 2 dissociation all rise. TCP Reactor Model shows good agreement with Malyshev and Donnelly s experimental data. Next steps include more chemistries (currently Ar, O 2, Cl 2 ), multi-frequency sheath and matching network models, and coupling with particle codes to get IEDs and IADs at the wafer. University of California Berkeley 19 GEC10