PIC-MCC simulations for complex plasmas

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 PIC-MCC simulations for complex plasmas Irina Schweigert Institute of Theoretical and Applied Mechanics, SB RAS, Novosibirsk

Outline GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 1. Introduction - Why we need the kinetic approach for discharge simulation? - Particle in cell Monte Carlo collisions (PIC-MCC) algorithm for discharge plasma simulations. Different models for complex plasma description: - Reactive plasma in CH/Ar mixture - Size dependent influence of nanoparticles on discharge properties - Effect of asymmetrical screening of micrometer size particles 3. Conclusion

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Schematic geometry of a planar capacitive radio frequency discharge chamber Dust - Discharge frequency: 13.56 MHz -Applied voltage: 100-1000 V - Gas pressure: 10-100 mtorr - Dust size: from nanometers to microns - Different gas mixtures

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Discharge structure 100 50 0-50 -100 100 50 0-50 -100 E (V/cm) 4 6 4 6 x (cm) Time dependent ion and electron density profiles 100 50 For electrons: 0 ν m =v e Nσ(v e ) -50 Averaged over rf cycle electrical field, E (V/cm) -100 ν m >> ω rf, fluid approach, ν m ω rf 4 6 x (cm), kinetic approach t/10-7, s 4 3 1 0 100 mtorr (b) 0.5 1.0 1.5 x, cm Electrode sheath

Electron energy distribution function in helium and argon discharge helium EEPF (ev -3/ ) EEPF (ev -3/ ) 10 0 10-1 10-10 -3 10-4 10-5 10 0 10-1 10-10 -3 10-4 10-5 P=1 Torr 0.3 Torr 0.1 Torr 0.03 Torr 10 0 30 40 U e (ev) P=1 Torr 0.3 Torr 0.1 Torr 0.03 Torr 5 10 15 0 U e (ev) argon σ t (10-16 cm ) 10 1 10 0 10-1 elastic in He elastic in Ar ionization excitation in Ar 10-1 10 0 10 1 U e (ev) Ramsauer minimum in the electron elastic cross section in argon Experiment: V. A. Godyak, R. B. Piejak, B. M. Alexandrovich, Plasma Sources Sci. Technol. 1, 36 (199) Simulations: I.V. Schweigert, V.A. Schweigert, Plasma Source Sci Technol., 13(), 315 (004)

Electron energy distribution function in helium and argon discharge helium argon 10 0 10 0 EEPF (ev -3/ ) 10-1 10-10 -3 10-4 10-5 P=1 Torr 0.3 Torr 0.1 Torr 0.03 Torr 10 0 30 40 U e (ev) EEPF (ev -3/ ) 10-1 10-10 -3 10-4 10-5 P=1 Torr 0.3 Torr 0.1 Torr 0.03 Torr 5 10 15 0 U e (ev) T e (ev) 6 4 He 4 3 1 Ar P (Torr) 10-10 -1 10 0 P (Torr)

Particle in cell Monte Carlo collisions algorithm Integration of equations of motion, moving electrons/ions Monte-Carlo collisions Weighting (force) Weighting (charge) Integration of Poisson equation on grid (electrical field) [1] R. Hockney and J. Eastwood, Computer Simulation Using Particles, Adam Hilger, Philadelphia 1988. [] Birdsall C K and Langdon A B 1985 Plasma Physics Via Computer Simulation (New York: McGraw-Hill) [3] Birdsall C K 1991 IEEE Trans. Plasma Sci. PS19 65

Monte Carlo collisions calculation with null collision method Electron cross sections for Ar σ total elastic excitation ionization attachment to dust Collision frequency: ν total =N gas σ total v(ε), Probability of collision in a time step t: P collision,m =1- exp(- tν total ), Collision occurs if P collision,m >P 1, where P 1 is a random number, What kind of collision?

Monte Carlo collisions calculation with null collision method Electron cross sections for Ar σ total elastic excitation ionization σ de = 4πrd exp( ϕ d / Te ) * nd / ϕ σ di = 4πrd d / Ti * nd / N N attachment to dust

.Different models of dusty plasmas What types of dust particles occur in gas discharge? Model I: reactive plasma Model II: plasma with nanoparticles Model III: plasma with microparticles H H H C C H H H H H H Heavy hydrocarbons formation R=10-100 nm n d =10 6-10 7 cm -3 R=1-10 µm n d =10 4 cm -3

Model I: reactive plasma Capacitive 13.56 MHz discharge in a mixture of Ar/CH at P=10 Pa, U 0 =50-100 V for conditions of Bochum experiments E. Kovacevic, I. Stefanovic, J. Berndt, and J. Winter, J. Appl. Phys. 93, 94 (003). I. Stefanovic, E. Kovacevic, J. Berndt, and J. Winter, New J. Phys. 5, 39.1-1 (003). H H H C C H H H H H H Heavy hydrocarbons formation D. Ariskin, A. Alexandrov, I. Schweigert, A. Bogaerts, F. Peeters, 008

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Model I: reactive plasma Kinetic equations for electron distribution functions: Transport equation for positive and negative ions: Poisson equation: Balance equations for neutrals:

Electron neutral collisions Capacitive 13.56 MHz discharge in a mixture of Ar/CH (5.8%) at P=10 Pa, U0=50-100 V

plasma species 8 neutrals, 15 positive ions, 6 negative ions, 11 radicals

A. Acetylene influence for pure Ar 10 10 for Ar/C H cations (a) n (cm -3 ) 10 9 10 8 10 7 anions 4 6 x (cm) Fig. 1: Ion density distributions for pure argon and for argon with 5.8% of CH P/10-3 (W/cm -3 ) 6 4 0 15 (b) Ar ionization C H ionization 4 6 x (cm) E (ev).5.0 for pure Ar for Ar/C H P/10-3 (W/cm -3 ) 10 5 C H excitations C H vibration Ar excitation 1.5 0 4 6 x (cm) 1.0 4 6 x (cm) Fig. : Mean electron energy distributions for pure Ar and for Ar with 5.8% of CH Fig. 3: Power consumed by ionization processes (a) and different excitation processes (b) for Ar with 5.8% of CH

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Neutrals density Ar 10 15 10 14 n (cm -3 ) 10 13 10 1 C H C 4 H C 6 H H C 8 H C 10 H 10 11 C 1 H 10 10 4 6 x (cm)

Comparison with experiment FIG. 1: Experimental positive ion spectrum. FIG. : Positive ion spectrum (only ions with energy near 10 ev were taken).

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Model II: discharge with nanoparticles f=13.56 MHz, P=10 Pa, U 0 =90V - 180V, n d =10 7 cm -3, r d =(10 100) nm Kovacevic, Stefanovic, Berndt, J. Winter, J. Appl. Phys. (003) Discharge with nanoparticles in Ar/CH mixture SEM micrograph of particles collected 10 min after plasma ignition

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Light scattering (solid symbols) and Ar+ ion density (open symbols) as a function of time, when nanoparticles grow up 100 nm ω=13.56 MГц, P=10 Па, U 0 =90V - 180V, n d =10 7 cm -3,r d < 100 нм Kovacevic, Stefanovic, Berndt, J. Winter, J. Appl. Phys. (003)

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Model II: discharge with nanoparticles Kinetic equations for electron distribution functions: Balance equation for electron and ion current on the dust: Continuity equation for dust density: Poisson equation:

The computational algorithm can be separated in a several main steps: a) We set the initial distributions of electrons, ions and nanoparticles b) Calculate the electrical field distribution solving the Possion equation c) Calculate the EEDF, IEDF solving the kinetic equation and find all macroscopic discharge parameters d) Calculate the dust floating potential e) Calculate electrostatic and ion drag forces acting of the nanoparticles f) Calculate of nanoparticle distribution solving the continuity equation Then we return to the point b) electrical field calculation from the Poisson equation with the new dust charge, electron and ion distributions Irina Schweigert

Example of electron, ion and nanoparticle profiles relaxation from initial conditions Nanoparticle density distributions in units 10 7 cm -3, r=100 nm Density distributions n/10 9 cm -3 : for electrons (red curve), for ions (green curve), for nanoparticle charge (blue curve)

Comparison with experiment 00 Acetylene on 150 growth starts 160 U 0 (V) 150 100 10 90 60 30 r d (nm) n e (10 15 m -3 ) 1 calculations experiment 10 80 40 r d (nm) 50 0 5 10 time (min) 0 4 t (min) 6 8 I.V. Schweigert, A.L. Alexandrov, D.A. Ariskin, F.M. Peeters, I. Stefanovic, E. Kovacevic, J. Berndt, J. Winter, Effect of transport of growing nanoparticles on ccrf discharge dynamics, Phys. Rev. E, 008 (accepted).

Ion drag and electrostatic forces F/10-11 (dynes) 10 5 0-5 (a) r d =0 nm F dr F ele Ion drag and electrostatic forces distribution for r=0 nm (a) and r=30 nm (b). F/10-11 (dynes) 10 (b) 5 0-5 r d =30 nm 4 6 x (cm)

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Transition between different modes Ε, V/cm 100 (a) 10 1 EEDF 10-3 (c) r=10 nm 10-1 r=30 nm r=40 nm 10 - r=60 nm 0.1 10-4 5 1 3 (b) 10-5 10 0 εe, ev ε, ev 4 3 1 3 Distribution of electrical filed (a), mean electron energy (b) and EEDF (с) for particles with radii 10, 30, 40 и 60 nm. x, cm Irina Schweigert

Distribution of nanoparticles of different radii Scattering signal (a. u.) 1 8 4 5 min 7 min 10 min (a) Experiments, Bochum n d /10 7 (cm -3 ) 0 6 4 0 nm 30 nm 90 nm x (b) PIC-MCC calculations 0 4 6 x (cm)

Excitation rates of Ar/CH mixture PIC-MCC calculations Experiment, Bochum ν i /10 13, cm -3 s -1 6 4 Emission Intensity (a. u.) 100 1000 800 600 400 00 min, r = 30±15 nm 5 min, r = 70±15 nm 7 min, r = 95±15 nm 10 min, r = 135±15 nm 4 6 x, cm 0 0 30 60 90 10 150 180 10 40 lower electrode Pixel number upper electrode Excitation rates of Ar/CH mixture with nanoparticles of different radii: 10 nm (black line), 30 mn (blue), 45 nm (green) and 60 nm (red).

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Model III: microparticles in gas discharge sheath boundary Dust particle radius R=1-5µm, potential U=1-3V, charge Z 10 4 e, density n d =10 3-10 4 cm -3, interparticle distance a 500µm, coupling parameter Γ=(e Z /a)/kt 0000 Γ>>Γ * =135-170 lower electrode

Dust particle dynamics Top view top 1 upper lower 3 bottom 4 Equation of dust particles motion r d 1 r r i dri 1 r 1 = F ( ), i ν + Fl U ri dt M dt M M Irina Schweigert

PIC-MCC calculation of potential distribution around dust particles Kinetic equations for ion collisional motion: Electrons obey a Boltzmann distribution ρ Poisson equation: Ion flux ρ = (x +y ) 0.5 I. Schweigert, V. Schweigert, F. Peeters, Phys. Plasmas 1, 113501-113510 (005) z

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Ion trajectories in the collisional case 500 500 400 400 z (µm) 300 00 300 00 100 100 0 00 150 100 50 0 50 100 150 00 r (µm) r (µm) 0

Scaling parameters obtained with linear kinetic analysis With standard linear perturbation approach after Fourier transform ee m k f r r e f k φ r r + ik V + f = f n + ik r V m V = ( k z, k ) k 0 ( ν) k ν g k ; κ Z n ( k) =, n( r) = G / G + κ 0 1 Zn( r / λ, M, κ) = V V t κ 4π n / m M / = / ν, λ e 0 I. Schweigert, V. Schweigert, F. Peeters, Phys. Plasmas 1, 113501-113510 (005)

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Negative-ion drag force Ion flux Diagram of ion drag forces acting on dust particles, for λ D =5, 50, 100 µm, Z=35000e For ion collisional motion, the ion drag force can be directed along the ion motion and in opposite direction. I.V. Schweigert, A.Alexandrov, F.M. Peeters, IEEE TRANS. ON PLASMA SCIENCE 3(), 63-636 (004).

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Acceleration and orbits of dust particle Experiment with Mach corners (Samsonov, Goree,, PRL, 1999). Discharge power: 50 W, ion density in the sheath: 9x10 9 см -3, gas pressure: 5 Па.

. ) ( ) ( = L i p i j i i j i i e i u i p F dt d M U Z U Z dt d M r r r r r r r r r ρ ν ρ ρ ρ ρ ρ ρ ρ. ) ( = L e p e i e l i e p F dt d M U Z dt d M r r r r r r ρ ν ρ ρ ρ ρ For dust particle beneath Equations of motion for monolayer dust particles: Equations of motion for dust particles: ) ( j i i U ρ ρ r r ) ( j i u U ρ ρ r r ) ( j i l U ρ ρ r r

10-1 xu (ev cm) 10-3 10-4 U (ev) 10-4 6 x (/10 - cm) 10-3 z=00µm z=0 z=-00µm 0.0 0.04 0.06 x (cm) Ui Uu r r ρ ρ ) ( i j r r ρ ρ ) ( i j Ul r r ρ ρ ) ( i j

Mechanism of acceleration of a dust particle under the monolayer Particle trajectories (t=0.18 sec) V. Schweigert, I. Schweigert, V. Nosenko, and J. Goree, Phys. Plasma 9, 4465 (00)

GRADUATE SUMMER INSTITUTE "Complex Plasmas August 4, 008 Transition between different regimes of motion of a dust particle under monolayer Dust particle kinetic energy as a function of reduced friction coefficientc 10 3 from the experiment of Samsonov et al 10 E (ev) 10 1 III II I 10 0 10-1 0,1 0, 0,3 0,4 ν/ω p V. Schweigert, I. Schweigert, V. Nosenko, and J. Goree, Phys. Plasma 9, 4465 (00)

Summary Reactive plasma We have developed the hybrid model for simulations of the 13.56 MHz discharge in CH/Ar mixture at the low gas pressure. We considered the formation of heavy hydrocarbons up to 1 carbon atoms. Both negatively and positively charged heavy hydrocarbons can be precursors for nanoparticles formation in the discharge volume, since their densities are sufficiently large ( 107 cm 3). The total density of negative ions reaches a half of the positive ion density. Thus a small fraction of acetylene (5.8%) in the argon discharge makes the mixture electronegative. Plasma with nanoparticles At the initial stage of growth (~30 nm) nanoparticles are placed near the sheath - plasma boundary, where they suppress the ionization due to the absorption of fast electrons. The occurrence of growing nanoparticle near the sheath - plasma boundary initiates the transition between capacitive mode and volume dominated one. Mocroparticle in plasma flux Asymmetrical pair potential of interaction between dust particles determine their dynamics. For the collisional ion motion case, the ion grad force can be directed along ion motion or in opposite direction depending on system parameters.