HIGH FREQUENCY CAPACITIVELY COUPLED PLASMAS: IMPLICIT ELECTRON MOMENTUM TRANSPORT WITH A FULL-WAVE MAXWELL SOLVER* Yang Yang a) and Mark J. Kushner b) a) Department of Electrical and Computer Engineering Iowa State University, Ames, IA 50011, USA b) Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109, USA October 2009 YY_MJK_GEC2009 * Work supported by Semiconductor Research Corp., Applied Materials and Tokyo Electron Ltd.

AGENDA  Introduction – High Frequency, large diameter CCPs  Description of the model  Comparison of CCP Properties: With and Without Electron Inertia  Concluding Remarks University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

MULTI-FREQUENCY PLASMA ETCHING REACTORS  State of the art plasma etching reactors use multiple frequencies to create the plasma and accelerate ions into the wafer.  Applied voltage propagates around metal electrodes (not through them).  Ref: S. Rauf, AMAT University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

WAVE EFFECTS IN HF-CCP SOURCES  As frequency and wafer size increase, the plasma shortened wavelength approaches the wafer diameter.  Wave effects (i.e., propagation, constructive and destructive interference) can significantly affect the spatial distribution of power deposition.  As sheath speed approaches electron thermal speed, electron inertia at can effect power deposition at high frequencies. G. A. Hebner et al., PSST, 15, 879 (2006). University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

INVESTIGATION OF ELECTRON INERTIA AT HIGH FREQUENCY, LARGE DIAMETERS  In this talk, report on investigation of role of electron inertia in high frequency, dual frequency CCPs.  Electron momentum equations implicitly included with full wave Maxwell solver in 2-d plasma equipment model.  Comparison of plasma properties using drift-diffusion and electron momentum formalisms over a range of frequencies ( MHz). University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

HYBRID PLASMA EQUIPMENT MODEL (HPEM)  Electron Energy Transport Module:  Electron Monte Carlo Simulation provides EEDs of bulk electrons  Separate MCS used for secondary, sheath accelerated electrons  Fluid Kinetics Module:  Heavy particle and electron continuity, momentum, energy equations  Maxwell’s Equation  Plasma Chemistry Monte Carlo Module:  IEADs onto wafer E, N Fluid Kinetics Module Fluid equations (continuity, momentum, energy) Maxwell Equations Te,S, μ Electron Energy Transport Module Plasma Chemistry Monte Carlo Module University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

 Addressing electron momentum transport with full Maxwell’s equations is challenging, partly due to coupling between electromagnetic (EM) and sheath forming electrostatic (ES) fields.  EM fields are generated by rf sources and plasma currents while ES fields originate from charges.  Separately solve for EM and ES fields, implement the solution in a time-slicing fashion:  Solve wave equations to obtain EM field.  Then solve Poisson’s equation together with electron continuity and momentum equations in the same matrix to obtain ES field, electron flux and density. NUMERICAL APPROACH University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

 Launch rf fields where power is fed into the reactor.  For cylindrical geometry, TM mode gives E r, E z and H .  Solve EM fields using FDTD techniques with Crank-Nicholson scheme on a staggered mesh:  Mesh is sub-divided for numerical stability. EM SOLUTION University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

ES FIELD, ELECTRON FLUX AND DENSITY  Momentum  Continuity  Poisson  After normalization, implicitly solved in the same matrix using sparse matrix techniques. University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

 Spatial discritization of the convection term :  = 0: Central differentiation.  = 1: donor-cell method (employed in this work).  Temporal discritization: numerically evaluated Jacobian elements by perturbing potential and density.  Linearization of the Faraday force, where, to obtain 2 nd order precision: NUMERICAL TECHNIQUES University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

REACTOR GEOMETRY  2D, cylindrically symmetric.  Ar, 50 mTorr, 400 sccm  Sources:  HF upper electrode: MHz, 300 W  LF lower electrode: 10 MHz, 300 W  Specify power, adjust voltage. University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

MOMENTUM SOLUTION: EFFECT OF HF  Ar, 50 mTorr  HF: MHz/300 W  LF: 10 MHz/300 W  HF = 50 MHz, Max = 1.1 x cm -3  HF = 150 MHz, Max = 1.8 x cm -3  Momentum solution captures changes in [e] profile with increasing HF.  50 MHz: Edge peaked due to electrostatic field enhancement and inductive skin effect.  150 MHz: Center peaked due to constructive interference of finite wavelengths. University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009

EM FIELD IN HF SHEATH  HF = 50 MHz, Max = 437 V/cm  HF = 150 MHz, Max = 280 V/cm  |E m | = Magnitude of EM first harmonic at HF (no electrostatic component)  Edge high to center high transition from 50 to 150 MHz due to constructive interference and standing wave with finite wavelength.  Sheath width decreases due to increase in [e].  Ar, 50 mTorr  HF: MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. Axial Distance Not in Scale. YY_MJK_GEC2009

MOMENTUM vs DRIFT-DIFFUSION: 10 MHz, [e]  Momentum (MM) solution, Max = 7.0 x cm -3  Drift-diffusion (DD) solution, Max = 1.1 x cm -3  Ar, 50 mTorr  HF: 10 MHz; 300 W  LF: 10 MHz 300 W University of Michigan Institute for Plasma Science & Engr.  Electron momentum transfer collision frequency ~ ≥ , a regime where MM and DD should provide similar results.  Spatial distributions similar – DD density larger partly due to a more negative dc bias (V dc, DD = V, V dc, MM = - 40 V), likely caused by the different differential schemes employed in two solutions (donor-cell for MM and central differentiation for DD). YY_MJK_GEC2009

MOMENTUM vs DRIFT-DIFFUSION : 10 MHz, e-FLUX  Drift Diffusion  Ar, 50 mTorr  HF: 10 MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr.  Axial electron flux in one rf cycle (r = 5 cm).  Largely same temporal evolution during the rf cycle.  Some differences in magnitude due to the differences in [e].  Momentum YY_MJK_GEC2009 ANIMATION SLIDE-GIF

MOMENTUM vs DRIFT-DIFFUSION: 50 MHz, [e]  Momentum (MM) solution, Max = 1.1 x cm -3  Drift-diffusion (DD) solution, Max = 8.2 x cm -3  Ar, 50 mTorr  HF: 150 MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr.  Both edge peaked at 50 MHz with beginning of wave propagation to center of reactor. Similar magnitudes of [e]. YY_MJK_GEC2009

MOMENTUM vs DRIFT-DIFFUSION : 150 MHz, [e]  MM solution, Max = 1.8 x cm -3  DD solution, Max = 1.5 x cm -3  Ar, 50 mTorr  HF: 150 MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr.  For HF = 150 MHz, [e] profile from MM solution is less diffusion controlled in the axial direction.  With MM electrons unable to “back-fill” sheath as rapidly as sheath retreats, and so sheath thickness increases. YY_MJK_GEC2009

MOMENTUM vs DRIFT-DIFFUSION: 150 MHz, e-FLUX  Ar, 50 mTorr  HF: 10 MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr.  Axial electron flux in one LF rf cycle (r = 5 cm)  Axial electron flux of MM solution has smaller amplitude regardless of larger gradient in density - Partly due to electron inertia.  Momentum  Drift Diffusion YY_MJK_GEC2009

LF PHASE RESOLVED RESPONSE OF E-FLUX  Drift Diffusion  Ar, 50 mTorr  HF: 10 MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr.  Axial electron flux in one LF rf cycle (r = 5 cm)  Factors responsible for differences between MM and DD:  Electron inertia.  Interaction between axial flux and radial flux through convection is neglected by DD approximation.  Momentum YY_MJK_GEC2009

PROPAGATION OF EM FIELD  Ar, 50 mTorr  HF: 150 MHz/300 W  LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr.  MM solution: less deep penetration into the bulk plasma.  Partly results from the out-of-phase motion of electrons.  MM Solution, V/cm  DD solution, V/cm ANIMATION SLIDE-GIFLENGTH NOT IN SCALE YY_MJK_GEC2009

CONCLUDING REMARKS  An implicit algorithm for electron momentum transport coupled with a full set of Maxwell’s equations was developed and incorporated into the HPEM.  MM solution captures effects of electron inertia and the interaction between axial and radial fluxes though convection.  For dual frequency CCPs sustained in Ar mixture at 50 mTorr  Similar to DD solution, MM solution captures the edge high to center high transition with increasing HF.  Profile of [e] is less diffusion controlled for MM solution – inertia prevents electrons from “back filling” sheath quickly and so sheath is thicker.  At HF = 10 MHz, electron fluxes from MM and DD solutions have the same temporal responses to electric field.  At HF = 150 MHz, different temporal responses due to the inertia of electrons and the interaction between axial and radial fluxes. University of Michigan Institute for Plasma Science & Engr. YY_MJK_GEC2009