65 th GEC, October 22-26, 2012

Transcription

1 65 th GEC, October 22-26, D Fluid/Analytical Simulation of Multi-Frequency Capacitively-Coupled Plasma Reactors (CCPs) E. Kawamura, M.A. Lieberman, D.B. Graves and A.J. Lichtenberg A fast 2D hybrid bulk fluid/analytical sheath code which was developed to simulate inductively-coupled plasmas ICPs [1] has been applied to multi-frequency 30 and 40 cm wafer CCPs. The code uses Comsol (finite elements PDE Solver) and Matlab. Ion energy distributions (IEDs) at the wafer are obtained by coupling the 2D Fluid/Analytical Comsol-Matlab code with a Matlab executable (MEX) version of a 1D PIC code. 1D Particle-in-cell (PIC) simulations are used to verify the multifrequency analytical sheath models and the IEDs obtained from the fluid/analytical code + sheath PIC. [1] Kawamura et al, PSST 20, (2011). University of California Berkeley 1

2 CCP Reactor Geometry Axisymmetric cylindrical geometry. Wafer chuck insulated from walls by quartz dielectric spacer. Thin vacuum sheath around plasma. Relative dielectric constant of plasma: University of California Berkeley 2 p 1 2 p ( jv m )

3 EM Model Capacitive rf bias at electrode generates transverse magnetic (TM) fields (H f,e r,e z ). Solve for one field in the trio, get the other two via Maxwell s equations. Assume all fields F exp(j t) and solve Helmholtz equation in each region. ( 2 + ( /c) 2 )F + f = 0 Plasma is a lossy dielectric since p has an imaginary (dissipative) part. University of California Berkeley 3

4 Bulk Plasma Power Deposition Solve the TM Helmholtz equation to obtain E. The total plasma current density is given by J T = (s p + j e 0 )E where s p = j e 0 ( p 1) is the plasma conductivity. Time averaged power density profile in the plasma is: p dep = 0.5 Re(J T E*) = 0.5 Re(s p )( E r 2 + E z 2 ). p dep is used as input in the electron energy balance equation of the Bulk Plasma Fluid Model. University of California Berkeley 4

5 Fixed Width Sheath Model Sheath width s depends on E, n e and T e. Computationally inconvenient to adjust position of plasmasheath boundary. Sheath with constant width s 0 and varying dielectric constant s = s 0 /s used to mimic a vacuum sheath with = 1 and varying width s (Lee et al, PSST 17, , 2008). The sheath voltage V sh same for both cases: V sh = (E vac / s ) n s 0 = E vac n s Modified Lee et al model 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 s to account for the electron and ion sheath heating. University of California Berkeley 5

6 Fast 2D CCP Reactor Model Overview A portable, user-friendly Comsol-Matlab platform. A typical argon multi-frequency CCP run on a 2.3 GHz CPU machine with 8GB DRAM was less than an hour. University of California Berkeley 6

7 1D Planar PIC vs. 2D Rectangular Fluid 1D planar PIC tracks (z, v x, v y, v z ) of particles. Poisson s equation solves for E z. No assumptions about particle velocity distributions. 2D rectangular fluid model solves for TM fields (H y, E x, E z ), n(x,z) and T e (x,z). Assumes ambipolar, quasineutral plasma with Maxwellian EEDF. l = discharge gap = 4 cm. Each plate area = 1600 cm 2. d = depth of discharge in y-direction = 40 cm. Multi-frequency current-source driven 20 mtorr argon CCP. University of California Berkeley 7

8 PIC vs. Fluid for multi-freq. 20 mtorr Argon CCP PIC Fluid n mid (m -3 ) 7.04e e16 T emid (V) V 1in (V), f , ,-52.7 V 2in (V), f 2 614, ,-87.5 P source (W) P loss (W) V rfsh1 (V) V rfsh2 (V) V rfsh (V) V dcsh (V) A@2MHz, 30A@60MHz, 15A@162MHz PIC Fluid n mid (m -3 ) 5.85e e16 T emid (V) V 1in (V), f , ,-56.4 V 2in (V), f 2 519, ,-87.2 V 3in (V), f 3 88, ,-70.8 P source (W) P loss (W) V rfsh1 (V) V rfsh2 (V) V rfsh3 (V) V rfsh (V) V dcsh (V) University of California Berkeley 8

9 Coupling fluid/analytical with PIC sheath Maximal sheath width s m, sheath voltage V sh (t), fluxes and temperatures from fluid/analytical code are inputted into PIC. This allows us to obtain the ion energy and angular distributions at the wafer electrode.

10 Full PIC vs. Fluid/Analytical + Sheath PIC 20 mtorr Argon University of California Berkeley 10

11 CCP Gas Pressure (mtorr) 20mTorr, 500 sccm Ar 30 cm CCP 40 cm CCP University of California Berkeley 11

12 CCP Gas Temperature (K) 20mTorr, 500 sccm Ar 30 cm CCP 40 cm CCP University of California Berkeley 12

13 CCP Plasma Density (m -3 ) 1000W@2MHz, 500W@60MHz, 20mTorr, 500 sccm Ar 30 cm CCP 40 cm CCP University of California Berkeley 13

14 CCP Electron Temperature (ev) 20mTorr, 500 sccm Ar 30 cm CCP 40 cm CCP University of California Berkeley 14

15 CCP IEDs 20mTorr, 500 sccm Ar 30 cm CCP 40 cm CCP University of California Berkeley 15

16 Plasma Density (m -3 ) above Wafer 30 cm CCP 40 cm CCP 500W@2MHz 500W@2MHz 1000W@2MHz 1000W@2MHz University of California Berkeley 16

17 Conclusions Hybrid bulk fluid/analytical sheath reactor models allow fast computation of plasmas with flow and can be applied to both ICPs and CCPs. Fluid/analytical code can be coupled to a particle-in-cell (PIC) code to obtain IEDs at wafer. Multi-frequency analytical sheath models tested with PIC code, showed good agreement. IEDs obtained from fluid/analytical + sheath PIC tested with full PIC simulations showed good agreement. 2D fluid/analytical models developed for both 30 and 40 cm wafer multi-frequency CCP reactors. This work was partially supported by the Department of Energy Office of Fusion Energy Science Contract DE-SC and by gifts from Micron Corporation and Lam Research Corporation. University of California Berkeley 17