PLASMA MODELING TECHNIQUES Mark J. Kushner University of Michigan Dept. Electrical Engineering and Computer Science Ann Arbor, MI 48109-2122 USA mjkush@umich.edu, http://uigelz.eecs.umich.edu ISPC Summer School August 2013 1

University of Michigan Institute for Plasma Science & Engr. AGENDA Part 1: Modeling Low Temperature Plasmas The Multi-Fluid Model Rate Coefficients Simplifications Circuits Integration Techniques Implementation in a 2-D Hybrid Model Technology Example: Large Area CCPs Part 2: Micro-Course on Monte Carlo Methods Acknowledgements: Jerry Wang, Wei Tian, Peng Tian, Yang Yang, Yiting Zhang, Sang-Heon Song, Jean-Paul Booth University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 2

University of Michigan Institute for Plasma Science & Engr. 1991 NRC REPORT: “PLASMA PROCESSING OF MATERIALS” “Currently, computer-based modeling…[is] inadequate for developing plasma reactors.” “There is no fundamental obstacle to improved modeling…nor… creation of computer aided design tools for designing plasma reactors.” Needed: Extensive data bases of experiments for validation of models and fundamental parameters for input. Efficient algorithms and supercomputers for simulating magnetized plasmas in 3 dimensions. Ref: “Plasma Processing of Materials”, NRC, 1991. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

RESEARCH TO TECHNOLOGICAL APPLICATIONS 1995 NRC REPORT: PLASMA SCIENCE: FROM FUNDAMENTAL RESEARCH TO TECHNOLOGICAL APPLICATIONS NRC Decadal Study on Plasma Science (http://books.nap.edu/catalog.php?record_id=4936) Low Temperature Plasmas: A serious problem…has been a lack of reproducible experimental verification of theoretical predictions… [This is also] due to the fact that the models are too limited and qualitative to be of general use…However given adequate funds, more realistic models could be developed to investigate these complex phenomena… Ref: Adapted from “Plasma Science: From Fundamental Research to Technological Applications Knowledge in the National Interest”, NRC, 1995. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

KNOWLEDGE IN THE NATIONAL INTEREST 2007 NRC REPORT: PLASMA SCIENCE: ADVANCING KNOWLEDGE IN THE NATIONAL INTEREST NRC Decadal Study on Plasma Science (http://www.nap.edu/catalog.php?record_id=11960) Low Temperature Plasmas: Extreme challenges face modeling and the allied sciences to develop comprehensive and validated theories, computer models and databases that place predictive capabilities in the hands of technologists. “This represents the highest level of challenge and the highest potential return…to both quantify and advance our understanding of low temperature plasmas, and to leverage that understanding by speeding the develop of society benefiting technologies.” Ref: Adapted from “Plasma Science: Advancing Knowledge in the National Interest”, NRC, 2007. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

DEPT. ENERGY OFFICE OF SCIENCE WORKSHOP Low Temperature Plasma Science: Not only the Fourth State of Matter but All of Them (September 2008) Priorities in Modeling and simulation Expand plasma capabilities to combine theory, simulation, and reacting flow equations to model closely-coupled, stochastic processes. Develop multi-scale methods describing interactions of plasmas with nanoscale features such as nano-particles and nano-textured surfaces. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

MODELS FOR PHYSICS BASED OPTIMIZATION Although always application driven, the marriage of fundamental plasma transport algorithms with design capable codes increased the relevance of modeling. Contributions to lighting, lasers, plasma displays, pollution control, aerospace, materials processing…. Equipotentials and Xe excitation rates. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 7

PHYSICS BASED DESIGN IN INDUSTRY 3D plasma equipment models are now used in the semiconductor fabrication industry as design tools. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 8

COMMERCIAL PLASMA EQUIPMENT MODELS The availability of commercial plasma, multi-physics equipment models is extending the reach of sophisticated simulation into the hands of non-experts. Electron temperature and electric fields in ICP sustained in argon. http://www.comsol.com/ University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 9

COMMERCIAL GENERATION OF FUNDAMENTAL DATA Commercial availability of state of the art algorithms for generation of electron impact cross sections is enabling analysis of complex chemistries on demand. Electron impact excitation of CF2 radical. http://quantemol.com/ University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 10

THE MULTI-FLUID MODEL ISPC2013_model 11

University of Michigan Institute for Plasma Science & Engr. MULTI-FLUID LTP MODEL The complete model consists of a set of partial differential equations for continuity, momentum and energy for each species, and Poisson’s equation for electric potential. This is known as a multifluid model. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 12

ELECTRON ENERGY EQUATION The electron energy equation for average energy  (electron temperature Te) is the fundamental connectivity between the macroscopic quantities (such as electric field) that heat electrons and electron impact rate coefficients that produce reactivity. Need a way to go from  to kij(Te) University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 13

CONVERTING  to kij(Te) Ultimately, this requires some sort of solution of Boltzmann’s equation. Many ways to solve BE Direct integration in space/time Computationally expensive; would replace energy equation. Can be integrated with Poisson solution. Monte Carlo simulation Quasi-expensive; provides spatial information Stationary solution leading to lookup table Fast; well represents non-Maxwellian features. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 14

CONVERTING  to kij(Te): BOLTZMANN LOOKUP Create lookup table of kij(Te) which is interpolated during execution of code using Te from energy equation. 1. Run stationary Boltzmann code to compute f(), Te, kij(Te) over a large range of E/N. 2. Number of cases (and range of E/N) must provide smooth transition of transport and rate coefficients. 3. Construct a table of [E/N, Te, kij(Te)] 4. Can have additional dimensions to account for excited state density if quantities significantly vary over plasma. 4. Throw away the E/N column – left with table of [Te, kij(Te)] 5. Periodically update table during execution of code as conditions change (e.g., mole fractions, partial ionization). E/N (Td) Te (eV) k1 k2 … 0.1 0.03 10-12 0.3 0.15 10-11 1.0 0.5 10-10 1000 9.0 10-8 University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 15

SIMPLIFICATIONS TO MULTIFLUID MODEL Drift-Diffusion for charged particles mi = momentum transfer collision frequency. Di = Diffusion cofficient = kBTi/q0 i = mobility = q0/(mimi) >> any other frequency in system  Single heavy particle temperature. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 16

SIMPLIFICATIONS TO MULTIFLUID MODEL Eliminate spatial derivatives using a Diffusion length   Global Model Net space charge is very small and plasma is essentially quasi- neutral everywhere  Ambipolar diffusion Do not solve Poisson’s equation for  - obtain from current conservation. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 17

University of Michigan Institute for Plasma Science & Engr. SOURCES OF GAS HEATING In principle, the calculation of Tion differs from Tneutral only by inclusion of Joule heating !! Contribution to Chemical Reactions Rl = rate of reaction l (cm-3s-1) ) fli = fraction of exothermicity into i If –ΔH >> kBTi , then initial and final momentum must sum to zero. Figure: SOGH_P8_Fig1.tif/cdr SOGH_P8_Fig2.tif/cdr University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 18

GAS HEATING: FRANK-CONDON EFFECTS Frank - Condon Heating: In electron impact dissociation, an intermediate (bound) excited state crosses over onto a dissociative curve. The fragments carry of kinetic energy. Δε = Electron impact excitation threshold energy D0 = Binding energy of ground state “-ΔH” = Δε – D0 Dissociative Recombination Figures: SOGH_P9_Fig1.tif/cdr SOGH_P9_Eq1.tif/cdr SOGH_P9_Eq2.tif/cdr SOGH_P9_Eq3.tif/cdr University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 19

University of Michigan Institute for Plasma Science & Engr. SOURCES OF GAS HEATING Charge Exchange: Similar to Frank-Condon heating. Difference in ionization potential, must be dissipated in either excited states or translational energy . Elastic Collisions: This is the “Tj-Ti” term. V-T Collisions Figures: SOGH_P10_Fig1.tif/cdr Sputter Heating and Reflection Neutrals: When energetic ions neutralize on surfaces, they can reflect with many eV of energy. Sputtered atoms leave surface with many eV. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 20

University of Michigan Institute for Plasma Science & Engr. CIRCUIT MODELS All LTP models require a voltage or electric field as a means of heating electrons and accelerating ions and which determines Te. In the steady state, If Te > Teo  Net electron gain If Te < Teo  Net electron loss In the steady state, the operating point, Teo, corresponds to a single electric field or voltage. The circuit model must provide that electric field. Figure file name: fig1_source_sink_Te University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 21

GLOBAL MODELS: AVERAGE PLASMA RESISTANCE A global model provides a spatially averaged electron density, collision frequency, temperature → Plasma resistance R With this resistance, the plasma can become part of a circuit. Figure file name: fig2_plasma_volume Figure file name: fig3_plasma_circuit; fig3_plasma_circuit_reduced_file_size.tif University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 22

EXAMPLE: CIRCUIT IN GLOBAL MODEL Pulsed, humid air discharge, 1 atm, Vs = 20 kV. Figure file name: fig5_plasma_current_circiut CS charged (20 kV) at t=0 when switch is closed. CP helps feed current to plasma on “other side” of inductance. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

SPATIALLY DEPENDENT MODELS: CURRENT SOURCES Spatially dependent models usually require a voltage on an electrode for use in solving Poisson’s equation In these models it is difficult (if not impossible) to define an average discharge resistance to place in a circuit model. The plasma is instead treated as a current source. The current flowing into the electrode is Figure file name: fig4_plasma_current is constructed by recording past values of E(t) University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

SPATIALLY DEPENDENT MODELS: CURRENT SOURCES Using plasma generated currents, time derivatives of current are obtained by recording past history. ID ID Figure file name: fig5_plasma_current_circiut In principle, the voltage drop along a current path, VP = V2-V1, should self consistently be reflected in the currents. During a transient (or AC excitation), I1 may not equal I2 due to displacement currents ID to other surfaces. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

CIRCUIT EQUATIONS: WAVE HEATED PLASMAS Inductively coupled plasmas (ICPs) are wave heated - an electromagnetic wave launched from an antenna is absorbed by electron acceleration and collisions. The antenna of the ICP is a circuit element that interacts with the plasma through the plasma conductivity, . Since there is a voltage on the coil, there are also electrostatic effects that may dissipate current. ×  Produces E&M Wave Produces Electrostatic Field University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 26

ELECTROSTATIC EFFECTS – ICPs The electrostatic potential of the coil appears to the plasma as being identical to an “applied voltage” and so can capacitively couple to the plasma through the dielectric window. The division between this capacitive power and the inductively produced power is the origin of the E-to-H transition. The capacitive current “bleeds” off conduction current from coil and so reduces inductively coupled E- field that is proportional to Icoil. Power resulting from IC produces a phase difference between Vcoil and Icoil. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 27

University of Michigan Institute for Plasma Science & Engr. CIRCUIT FOR COIL Plasma Match Box Ri Li Grounded Plane of Chamber Air Sheath Termination Impedance Dielectric Divide coil into sub-units modeled as discrete transmission line. Ri = Resistance of coil for segment I; Li = Inductance Zi = Impedance due to capacitively coupling to ground Write set of differential equations for current and voltage on nodes Determine Zi by Fourier analyzing current at antenna due to Er and Ez fields: If , then Z0i has a real resistance – capacitive losses. The circuit was drawn using a online tool, not editable locally. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 28

EXAMPLE: CAPACITIVE COUPLING IN H-MODE Ar/Cl2=10/90, 20 mTorr Total Power (forward- reflected): 450 W Inductive: 291.4 W Capacitive: 155.4 W Coil heating: 3.2 W Figure file name: fig5_plasma_current_circiut University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

EXAMPLE: MULTI-FLUID NON-EQUILIBRIUM Ar/Cl2=10/90, 20 mTorr Cl is heated by Frank-Condon effects e + Cl2 Cl2* + e  Cl(hot)+Cl(hot)+ e Cl is hotter than Cl2 as collision frequency is not high enough to equilibrate temperatures. Figure file name: fig5_plasma_current_circiut University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

INTEGRATION TECHNIQUES: EXPLICIT, IMPLICIT In most LTP models, a set of partial or ordinary differential equations are integrated. N(t)  N(t+t) The most basic decision that needs to be made is how to integrate the equations. Explicit: Information in the present (t) is used to produce derivatives. Implicit: Information in the future (t+t) is used to produce derivatives. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 31

LIMITS TO EXPLICIT INTEGRATION Explicit integration techniques are far more straight forward and simple to implement. However, the maximum stable integration time step is limited. For drift transport, timestep is constrained by charge transport “flipping” the sign of the electric field. tD = Dielectric relaxation time  10-12 s in low pressure ICPs. t C = Courant limit  10-9-10-8 s in low pressure ICPs. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 32

HIGHER ORDER EXPLICIT TECHNIQUES “Physics” can only provide first order time derivatives. Higher order explicit integration techniques attempt to overcome the limitation on timestep. Higher order explicit integration techniques are ALL implementations of Taylor’s expansion. Runge-Kutta or predictor-corrector integration techniques (2nd -4th order) are methods which synthesize higher order terms in the Taylor expansion given only the “physics” provided dN(t)/dt. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 33

3rd ORDER RUNGE-KUTTA PREDICTOR-CORRECTOR Physics: Densities at any time t provides derivatives dN(t)/dt Integrate from t t + t 1. Use known values at t to predict values at t + t/2 2. Use Np(t) to estimate dN/dt(t + t/2). 3. Use both derivatives to correct slope at t and predict value at t + t. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 34

3rd ORDER RUNGE-KUTTA PREDICTOR-CORRECTOR 4. Use Np(t+t) to estimate dN/dt(t + t). 5. Use all three derivatives to correct initial slope at t to make full integration to t+t. This synthesizes the first 3 terms in Taylor’s expansion. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 35

University of Michigan Institute for Plasma Science & Engr. IMPLICIT INTEGRATION In principle, implicit integration techniques do not have a limit on timestep t since future information is being used. In practice, the Jacobian elements (dN/dM) that are used in solution are linearized based on present values, N(t), and so set a practical limit to t. Challenge – How do you obtain dN(t+t)/dt when you do not know N(t+t)? Usually use a matrix technique. This is far more work than an explicit technique. However, if you can increase t by a factor of 10-1000, then the extra work pays off. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 36

University of Michigan Institute for Plasma Science & Engr. IMPLICIT INTEGRATION PDE to integrate Discretize onto mesh for spatial location i Fig. ISPC_Peng.lay/tif is Jacobian element that expresses change in S for a change in N. This is usually numerically derived. This linearized correction to S limits the timestep. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 37

University of Michigan Institute for Plasma Science & Engr. IMPLICIT INTEGRATION Solve for Ni(t+t) Express as Matrix Fig. ISPC_Peng.lay/tif University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 38

University of Michigan Institute for Plasma Science & Engr. IMPLICIT INTEGRATION In matrix form: Solve for vector of N vs position at future time. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 39

SEMI-IMPLICIT SOLUTION OF POISSON EQ. In most LTP models, Poisson’s equation cannot be solved explicitly; ΔtD is too small. Including Poisson’s equation in “the matrix” for an implicit solution requires that all charged species also be in the matrix. This often results in a huge matrix that is expensive to solve. Semi-implicit solutions predict charge densities at future time, and works surprisingly well: Δt > 103 ΔtD Example: Predicted charge densities at future time with drift dominated transport University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 40

SEMI-IMPLICIT SOLUTION OF POISSON EQ. Solve for potential  by bringing all  terms onto the “left side”. Effective  that “warps” numerical mesh to enclose charge convected by electric field. At this point, solve Poisson’s equation as you would for an explicit solution – and provides (t+t) University of Michigan Institute for Plasma Science & Engr. ISPC2013_model 41

DESCRIPTION OF IMPLEMENTATION OF MULTI-FLUID ALGORITHMS IN 2-D MODEL ISPC2013_model 42

University of Michigan Institute for Plasma Science & Engr. DESCRIPTION OF HPEM Modular simulator that combines fluid and kinetic approaches. Resolves cycle-dependent phenomena while using time-slicing techniques to advance to the steady state. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. HPEM-EQUATIONS SOLVED - Maxwell’s Equations – Frequency Domain Wave Equation Azimuthal antenna currents – retain only E, Brz Plasma currents Collisional ion currents Kinetically derived non-local electron currents capture nonlocal effects. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. ELECTRON ENERGY TRANSPORT S(Te) = Power deposition from electric fields L(Te) = Electron power loss due to collisions  = Electron flux (Te) = Electron thermal conductivity tensor SEB = Power source source from beam electrons All transport coefficients are tensors: University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. HPEM-EQUATIONS SOLVED - Electron Energy Distributions – Electron Monte Carlo Simulation Cycle dependent electrostatic fields Phase dependent electromagnetic fields Electron-electron collisions using particle-mesh algorithm Phase resolved electron currents computed for wave equation solution. Captures long-mean-free path and anomalous behavior. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. HPEM-EQUATIONS SOLVED - University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

THE TIMESCALE CHALLENGE Technological plasmas have vastly different timescales that must be addressed in models. Integrating timestep (required for numerical stability): t Dynamic timescale (to resolve the evolution of phenomena): T Plasma transport: Dielectric relaxation t = /  1 ps – 10 ns T = ns - ms Surface chemistry: t = s, T = 10 s University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

THE FUNDAMENTAL APPROACH Virtually all processes in plasmas are interdependent. Electron kinetics depend on circuitry, gas flow, surfaces…. Radiation transport depends on electron energy distributions, heaving particle quenching… Plasma harmonics depend on cable lengths…. The most fundamental and accurate of modeling is capturing all interdependencies in a massive set of partial differential equations (or equivalent PIC). Integrate, integrate, integrate….Far too time consuming. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

University of Michigan Institute for Plasma Sci. & Engr. HYBRID APPROACH Use your knowledge of the physics to compartmentalize the problem into modules. Optimize modules for the physics they address. Exchange information between modules on frequency dictated by the physics. Let conditions determine the physical models used in modules. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

University of Michigan Institute for Plasma Sci. & Engr. THE HYBRID APPROACH: I Virtually all processes in plasmas are interdependent…but most processes are hierarchical. Electron transport rapidly comes into equilibrium with slowly evolving heavy particle fluid dynamics. Surfaces evolve on long time scales compared to plasmas. Circuit parameters change on timescales long compared to plasmas but short compared to surfaces. Goal I: Seek our hierarchical relationships University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

THE HYBRID APPROACH: II The interdependence of physical processes often rely on ensemble averaged quantities and not fine level details. Electromagnetic field propagation depends on conductivity but not the details of N2(v) distribution. Electron dynamics in harmonic systems depend on phase-dependent quantities whereas surface kinetics do not. Some quantities can be integrated over harmonic cycles with little loss in accuracy. Goal II: Understand physics relationships between phenomena. What does process A need from process B? In math terms….Know the orders of magnitude of the Jacobian elements A/B. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

THE HYBRID APPROACH: III Once relationships are established: B needs N1 from A A needs N2 from B ….It doesn’t matter how N1 is provided, as long as it is available when needed. Process A Process B N1 N2 University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

HIERARCHICAL RELATIONSHIP AND LOCAL CONDITIONS With hierarchy, “local conditions” determine physical model. Electromagnetics require conductivities from electron transport. Electron transport requires fields from electromagnetics. Physical models, method of solution should not affect hierarchy as long as required quantities are produced. Electromagnetics Electron Transport E(r,t) (r,t) Freq, harmonics, anomalous sheaths determine model… Frequency domain Time domain Non-local conductivity Local field Pressure, frequency, energy determine model… Continuum energy eqn. Monte Carlo simulation Beam-bulk Local field University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

University of Michigan Institute for Plasma Sci. & Engr. MODULES: COMPARTMENTALIZE, INPUT - OUTPUT Compartmentalize physical processes into modules having minimum of overlap. Determine physics that are “inputs” and “outputs”. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

ESTABLISH MODULE RELATIONSHIPS Modules may accept input and provide output from and to multiple modules. Establish relationships, timescales for exchange, degree of lock-stepping vs time-slicing. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

FLEXIBILITY OF APPROACH FOR DIFFERENT SYSTEMS Consider 2 modules: Electron Energy Transport Module (EETM) and Fluid Kinetics & Poisson Module (FKPM). Path through modules will be different depending on pressure, current density…. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

TIME SLICING vs LOCKSTEP Certain phenomena are so tightly coupled they must be integrated in lockstep. Those less coupled (usually having vastly different timescales) can be addressed using time slicing techniques. Integrate modules using t appropriate for its physics; exchange information on T commensurate with meaningful change. Leverage short T rapidly coming into dynamic equilibrium with phenomena having longer T University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

TIME SLICING APPROACH AND LIMITATIONS Example: Plasma phenomena requires t = 0.1 ns, comes into dynamic equilibrium in T = 100 ns. Hold plasma properties constant while integrating fluid with t = 10 s, T = 1 ms. Transient of system determines maximum T…To resolve true transient behavior at all scales, all t, T are lock-stepped. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

University of Michigan Institute for Plasma Sci. & Engr. TIME SLICING APPROACH Depending on time scales, there may be “sub-time slicing” between modules. Example: Tight coupling between Fluid Kinetics & Poisson Module and Electron Energy Transport Module. Loose coupling between FKPM and Surface Chemistry Module. University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

CHOOSING PROPER PHYSICS MODEL Choosing the correct physics model is important in accurately predicting plasma tool performance Example: Representing electron energy transport in an Inductively Coupled Plasma. Electron energy equation (continuum) Monte Carlo Simulation (kinetic-linear) Monte Carlo Simulation (kinetic with electron-electron collisions: non-linear) University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

CHOOSING PHYSICS MODEL Spatial distribution of plasma and excitation rates depend on physics model. Electron energy equation Monte Carlo Simulation without e-e collisions MCS with e-e collisions ICP Ar, 10 mTorr University of Michigan Institute for Plasma Sci. & Engr. ISPC2013_model

CONTROLLING PLASMA PROPERTIES IN LARGE DIAMETER CCPs TECHNOLOGY EXAMPLE: CONTROLLING PLASMA PROPERTIES IN LARGE DIAMETER CCPs ISPC2013_model

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. Voltage finds its way into the plasma propagating around electrodes (not through them). University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

WAVE EFFECTS CHALLENGE SCALING As wafer size and frequencies increase - and wavelength decreases, “electrostatic” applied voltage takes on wavelike effects. Plasma shortened wavelength:  = min(half plasma thickness, skin depth), s = sheath thickness Lieberman, et al PSST 11, 283 (2002) http://mrsec.wisc.edu University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

ELECTROSTATIC TO ELECTROMAGNETIC TRANSITION (CAPACITIVE TO INDUCTIVE) Wave effects are observed when increasing… Frequency (smaller /D) Power (shorter skin depth) University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

CONSTRUCTIVE INTERFERENCE Constructive interference occurs when waves approach center the center of the wafer from the edges, possibly creating standing waves. Plasma emission collapses to center as the frequency increases and constructive interference occurs. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. REACTOR GEOMETRY 2D, cylindrically symmetric. Base conditions Ar/CF4 =90/10, 50 mTorr, 400 sccm HF upper electrode: 10-150 MHz, 300 W LF lower electrode: 10 MHz, 300 W Specify power, adjust voltage. Main species in Ar/CF4 mixture Ar, Ar*, Ar+ CF4, CF3, CF2, CF, C2F4, C2F6, F, F2 CF3+, CF2+, CF+, F+ e, CF3-, F- University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

ELECTRON DENSITY vs FREQ: Ar Increasing HF transitions from edge electrostatic dominated to center electromagnetic dominated. Ar 50 mTorr, 400 sccm HF: 10-150 MHz/300 W LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

ELECTRON DENSITY vs FREQ: Ar/CF4 = 90/10 HF = 50 MHz, Max = 5.9 x 1010 cm-3 HF = 150 MHz, Max = 1.1 x 1011 cm-3 Increasing HF changes produces higher plasma density due to more efficient heating. Shorter wavelength, smaller skin depth produces mix of electrostatic and wave coupling. Ar/CF4=90/10 50 mTorr, 400 sccm HF: 10-150 MHz/300 W LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

AXIAL E-FIELD IN HF AND LF SHEATH: 10/150 MHz |EZ| in HF (150 MHz) Sheath, Max = 1500 V/cm HF Electrode LF Electrode |EZ| in LF(10 MHz) Sheath, Max = 1700 V/cm Significant change of |Ez| across HF sheath in form of traveling wave. HF source also modulates E-field in LF sheath (which should be electrostatically uniform). ANIMATION SLIDE-GIF Ar/CF4=90/10 50 mTorr, 400 sccm HF: 150 MHz/300 W LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. ELECTRON ENERGY DISTRIBUTIONS: HF SHEATH The “higher” the EED in the tail, the more excitation and ionization occurs. 50 MHz: Populated tails for r  7 cm due to electrostatic edge effect. 150 MHz: elevated tails in the center where sheath field is largest. From 50 MHz to 150 MHz: 1 temperature EED transits to 2 temperature. Ar/CF4=90/10, 50 mTorr, 400 sccm HF: 10-150 MHz/300 W, LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. ELECTRON ENERGY DISTRIBUTIONS: LF SHEATH With 10 MHz on the lower electrode, electrostatic effects should dominate. 50 MHz: Edge effect produce high energy tail. 150 MHz: Modulation from upper sheath extends to lower electrode and lifts towards center of wafer. Ar/CF4=90/10, 50 mTorr, 400 sccm HF: 10-150 MHz/300 W, LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

IEADs INCIDENT ON WAFER: 10/10 MHz Center Edge Many factors may produce center-to-edge differences in IEADs: sheath thickness, sheath potential, mixing of ions … 10/10 MHz is dominated by electrostatics. Subtle differences in IEADs to wafer between center-and-edge, but otherwise uniform. Total Ion CF3+ Center Edge Center Edge Ar/CF4=90/10, 50 mTorr, 400 sccm HF: 10 MHz/300 W LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

IEADs INCIDENT ON WAFER: 10/150 MHz Center Edge With 150 MHz excitation, sheath thickness and potential vary from center to edge. IEADs at edge are downshifted in energy, broadened in angle. Total Ion CF3+ Center Edge Center Edge Ar/CF4=90/10, 50 mTorr, 400 sccm HF: 150 MHz/300 W LF: 10 MHz/300 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. 450 mm REACTOR GEOMETRY 2D, cylindrically symmetric. Ar, 50 mTorr, 600 sccm Base conditions HF upper electrode: 10-150 MHz, 450 W LF lower electrode: 10 MHz, 450 W Specify power, adjust voltage. Note: Increase in gap compared to 300 mm. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

450 mm: ELECTRON DENSITY vs FREQUENCY Compared with plasmas sustained in 300 mm tools: More prominent finite wavelength effect. Center and edge peaks for HF  50 MHz due to the coupling of wave and electrostatic effects. Ar 50 mTorr, 600 sccm HF: 10-150 MHz/450 W LF: 10 MHz/450 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

450 mm: EM PROPERTIES IN HF SHEATH Total Electric Field (Normalized by Em at r = 0) Relative Phase (With respect to the electrode edge) With increasing HF, EM field is more radially dependent. Diminishing phase change in the reactor center due to formation of standing wave. Ar 50 mTorr, 600 sccm HF: 150 MHz/450 W LF: 10 MHz/450 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

ION FLUX AND IEADs INCIDENT ON WAFER Center Middle Edge Ion Flux Center Middle Edge Ultimately, E&M finite wavelength effects translate to differences in IEADs and ion flux onto wafer. Ar, 50 mTorr, 600 sccm HF: 150 MHz/450 W, LF: 10 MHz/450 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

CAN WE INNOVATE AROUND THE PHYSICS? For uniform processing at 450 mm, the electrical distance from rf feed-to-plasma must be equal. Segmented electrodes are used in large area plasma processing for LCD panels and solar cells. Has their day come for microelectronics? Let modeling answer that question… The model….segmented electrodes. University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. PROPAGATION OF EM WAVE Segmented Electrode ANIMATION SLIDE-GIF EM Amplitude Max = 650 V/cm Solid Electrode EM Amplitude Max = 950 V/cm EM fields propagate along the sheath and into the bulk plasma with deeper penetration in low [e] region due to less absorption. More radially uniform penetration with segmented electrode. Ar, 50 mTorr, 600 sccm HF: 150 MHz/450 W, LF: 10 MHz/450 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

University of Michigan Institute for Plasma Science & Engr. PLASMA DISTRIBUTION Segmented Electrode [e], Max = 3.6 x 1010 cm-3 Solid Electrode [e], Max = 2.7 x 1011 cm-3 Center and peaked plasma density with segmented electrode: still need some tuning, segmentations with different widths? Ar, 50 mTorr, 600 sccm HF: 150 MHz/450 W, LF: 10 MHz/450 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

ION FLUX AND IEADs INCIDENT ON WAFER Center Middle Edge Ion Flux Center Middle Edge Ion fluxes – need some tuning (real-time-control?) of electrode structures for uniformity. IEADs – Electrical symmetry provides more uniformity. Ar, 50 mTorr, 600 sccm HF: 150 MHz/450 W, LF: 10 MHz/450 W University of Michigan Institute for Plasma Science & Engr. ISPC2013_model

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THE MONTE CARLO METHOD FOR f() (and Radiation Transport) MICRO-COURSE: THE MONTE CARLO METHOD FOR f() (and Radiation Transport) 85 85