Sheath stability conditions for PIC simulation

Transcription

1 Plasma_conference (2011/11/24,Kanazawa,Japan) PIC Sheath stability conditions for PIC simulation Osaka Prefecture. University H. Matsuura, and N. Inagaki,

2 Background Langmuir probe is the most old but widely-used tool for plasma monitoring. Many improvement work to monitor many parameters are still conducting all over the world. Double probe, triple probe --- density, Te Plasma absorption probe, Frequency shift probe --- ne E missive probe --- Vs Mach probe, directional probe --- flow Ion sensitive probe, tunnel probe --- Ti Combined force-mach probe --- Pressure, Ti Thermal probe --- Heat flux, Ti, negative ion Understanding of sheath is key issue.

3 Schematic view of kinetic/pic model Plasma side Inflection point (ni=ne) Wall side concave(ni<ne) convex(ni>ne) i e 1D in space, stationary. no collision, particle source(sink) potential is zero at plasma boundary and negatively biased externally at wall boundary ref. R.D.Smirnov; Dr.Thesis,

4 Solution of Boltzmann equation 1D steady collisionless Boltzmann equation v x f j (x, v x ) x + q j m j dφ dx f j (x, v x ) v x = 0, can be transferred so that ɛ jx = 1 2 m jv 2 x + q j Φ is used as independent valuable instead of v x. v x (x, ɛ jx ) f j(x, ɛ jx ) x = 0, So energy distribution functions of particle j is the same as those at plasma source boundary.(ex. Maxwellian)

5 Boundary condition Plasma particles( electrons, positive ions, negative ions ) are supplied from boundary. In order to simulate the sheath, choose particle current intensity( or density ) at source boundary so that Debye length is much smaller than geometry size. choose current( or density) ratio of positive and negative particles so that potential profile has at least one inflection point( that is perfect neutral point n+ = n-) in the geometry. At wall boundaries, ions and electrons are perfectly absorbed.(no reflection/no secondary electrons) Initial velocity distribution is Maxwellian.

6 Velocity distribution function Ion f(v x ) = Electron f(v x ) = n is mi 2πT i exp( 1 T i ( 1 2 m iv 2 x + eφ(x))) 0 (v x > 2e(Φ (v x < s Φ(x)) m i ) 2e(Φ s Φ(x)) m i ) 2e(Φ(x) Φ 0 (v x < w ) n es me 2πT e exp( 1 T e ( 1 2 m ev 2 x eφ(x))) (v x > m e ) 2e(Φ(x) Φ w ) m e ) Number density n j = 0 v x v 2 x + 2Z j e(φ s Φ) m j f js(v x )dv x

7 Plasma parameter Ion density n i = n is exp( e e (Φ s Φ))erfc( (Φ s Φ)) T i T i Electron density n e = n es exp( e e (Φ s Φ)){1 + erf( (Φ Φ w ))} T e T e Particle flux Γ i = n is T i 2πm i, Γ e = n es T e 2πm e exp( e T e (Φ s Φ w )) Heat flux Q i = {e(φ s Φ) + 2T i }Γ i Q e = 2T e Γ e

8 Inflection point of potential From normalized Poisson equation d2 φ ds 2 = ρ(s) = (n i n e )/n 0, potential profile is convex in the sheath(ρ(s) > 0). Though, ρ(s) may be equal to be 0 in presheath of real plasma, we set ρ(s) to be slightly positive ( potential is concave ) at plasma source boundary so as that inflection point of potential ( and flat region around it ) exists middle of the simulation geometry. If n is = n es and T i = T e, Normalized potential depth φ = (Φ s Φ)/T e 1.7 at the inflection point. (In the fluid model, φ = 0.5 at presheath end.) And this value does not depend upon mass ration m i /m e and slightly depends upon T i /T e.

9 potential profile calculation Semi-implicit Runge-Kutta method( Kaps-Rentrop-Shapine formula ) is used to solve stiff Poisson equation. T.Watabe et al., "Numerical software by fortran 77",(Maruzen, Tokyo, 1990)[in Japanese]. phiw=10.0d0 eφ w T e denie=1.0d0 ( n i n e ) s tie=1.0d0 ( T i T e ) s xst = 0.0d0 (x/λ D ) s y(1) = 0.0d0 eφ s T e y(2) = 0.7d0 ( eλ D Te dφ dx ) s The effect of "y(2)"(boundary E-field) and "tie"(ion temperature) is mainly studied.

10 Sheath potential from kinetic model Distance is normalized with Debye length.(ti=te) The legends are the normalized electric field(y(2))at source boundary.

11 Condition for sheath establishment System length/debye length System length from kinetic model No sheath E field/(t e /Debye length) at boundary Absolute value of electric field at source boundary must exceed the critical value. The long flat potential is obtain for lower electric field.

12 Berkeley code(xoopic) 2-Dimensional PIC code developed and distributed by PTSG group( Prof. C. K. Birdsall ) J.P.Verboncoueur et al.;comp.phys.comm.,87(1995)

13 parameter limits for sheath simulation For 1D sheath simulation, spatial mesh size x λ D. time step size t 2π/ω pe. Courant condition v e t < x. Sheath stability( derived from generalized Bohm condition ) L x /λ D < 50.

14 1D Simulation model coordinate Cartesian geometry size : L x 4.0, 6.0, 8.0, 10.0[cm] : L y 10.0[cm] mesh size x 150 y 50 time step t [s] particle weight : g Source current : I 0.30, 0.40, 0.55, 0.70[A] Ion He + Boundary condition : Top & bottom Perfect absorbing dielectric : Left Plasma source( and Exit ) with Φ = 0[V] : right conductor with Φ = 100[V]

15 Sheath stability condition 20 z-range=0.04 z-range=0.06 z-range=0.08 z-range=0.10 z-range= current=0.30[a] current=0.40[a] current=0.55[a] current=0.70[a] current=0.70[a] Potential -40 Potential z[m] If system size/debye length becomes too large, the simulation becomes unstable due to plasma oscillation. z[m] If plasma density increase and Debye length becomes small, similar oscillation occurs.

16 Generalized Bohm condition From Poisson equation ɛ d2 Φ dx 2 = e(n e Z i n i ) we obtain ɛ Φ 2 (E2 x Ex0) 2 = e (n e (Φ ) Z i n i (Φ ))dφ Φ 0 As for electrons, Boltzmann relation is assumed here. (Inclusion of the effect of truncation of electron velocity distribution is straightforward, but rather complex.) n e = n es exp( e(φ s Φ) T es ) e Φ Φ 0 n e (Φ )dφ = n es (exp( e(φ s Φ) T es ) exp( e(φ s Φ 0 ) T es ))

17 Generalized Bohm condition(2) For general ion velocity distribution, similar derivation is possible. Z i e Φ Φ 0 n i (Φ )dφ = 0 ( m i u 2 )((1 + 2Z ie(φ s Φ) m i u 2 ) 1 2 (1 + 2Z ie(φ s Φ 0 ) m i u 2 ) 1 2 )fis (u)du By setting x 0 = x s and expanding around here, we obtain, ɛ 2 (E2 x E2 xs ) = (n ese 2 2T es Z2 i n is 2m i < 1 u 2 >)(Φ s Φ) 2 + So generalized Bohm condition is 1 Z it es m i < 1 u 2 > Unfortunately for some kind of distributions( ex. half Maxwellian), f is (u) 0 for u = 0. For such f is (u), < 1 u >= 1 2 n is 0 u f 2 is (u)du will diverge for Φ Φ s and generalized Bohm condition is not satisfied at plasma boundary(x = x s ). 1

18 half-maxwellian ion distribution For some finite values of E xs, there exists an inflection point (x = x n ) and perfect charge neutrality n e = Z i n i is satisfied. Hereafter we use normalized potential depth φ = e(φ s Φ)/T e and normalized ion temperature at plasma boundary τ = T is /T es. ɛ φ 2 (E2 x E2 x0 ) = T es (n es exp( φ ) Z i n is exp( φ φ φ 0 τ )erfc( τ ))dφ = T es n es (exp( φ) exp( φ 0 )) +T es Z i n is τ(exp( φ φ τ )erfc( τ ) + 2 φ π τ exp( φ 0 τ )erfc( φ0 τ ) 2 φ0 π τ ) Then by setting φ 0 = φ n and using n es exp( φ n ) = Z i n is exp( φ n τ )erfc( we obtain ɛ 2 (E2 x E2 xn ) = T esz i n is (( τ ) exp(φ n τ )erfc( φn τ ) 1 τ π φ n τ ) τ φ n )(φ φ n ) 2 +

19 half-maxwellian ion distribution(2) So Bohm condition for half Maxwell ion distribution 10 φn π(τ + 1) τ exp(φ n τ )erfc( φn τ ) 1 In right figure, Green line is the left hand side of above equation plotted as the function of normalized potential depth at neutral point φ n.(τ = 1 and Z i = 1 case ) Another line (Blue ) is Z i n i /n e as as the function of φ. This graph shows φ n, at which blue line cross unity, satisfies above relation. However this may be trivial. There is no reason that so-called sheath boundary where bohm condition is satisfied coincide with an inflection point (x = x n ) (Phi_s-Phi)/Te

20 half-maxwellian ion distribution(3) Instead of expanding with φ φ n, we set φ = φ s. Since E xn = 0 and φ s = 0, we obtain ɛ φn 2 E2 xs = e (n e (Φ ) Z i n i (Φ ))dφ φ s = T es n es (1 exp( φ n ) +T es Z i n is τ(1 exp( φ n τ )erfc( φn By normalized with T es /eλ D, λ 2 D = e2 n es /m e T es, τ ) 2 π φn τ ) E 2 xs (T es /eλ D ) 2 = 2(1 exp( φ n ) + Z in is τ n es (1 exp( φ n τ )erfc( φn τ ) 2 π φn τ ) This value is also plotted in above figure(red line) as the function of φ n. For τ = 1 and Z i = 1 case, φ n 1.7 and E xs 1.3. There values is well agree with kinetic model.

21 2D Simulation model coordinate Cylinder geometry size : L z 4.5, 4.8, 6.0[cm] : L r 1.0[cm] mesh size x 150 y 50 time step t [s] particle weight : g Source current : left 0.55[A] :top [A] Ion He + Boundary condition : Top Plasma source( and Exit ) : bottom exit(not axis) or dielectric : Left Plasma source( and Exit ) with Φ = 0[V] : right conductor or dielectric

22 2D Sheath stability 2D simulation must obey stability condition simultaneously both in r-direction and in z-direction. lz=20[mm] is somehow too long, and potential oscillation and electron energy burst are observed.

23 2D Sheath stability (2) When gap between probe head and left boundary is reduced to lz=5[mm], instability is suppressed. This length corresponds to about 30 times Debye length.

24 Conclusion A kinetic sheath model with Boltzmann - Poisson equations and XOOPIC simulation are compared. Spatial profile of sheath potential, is determined by characteristic electric field strength, which is assigned in artificial boundary condition. Checking validity of modeling ( or simulation )with only potential profile shape is meaningless. In order to keep stable sheath, the necessary value of potential drop is well agree to the estimation with generalized Bohm condition. Simple Bohm condition or fluid model gives us wrong value.

25 Conclusions(cont.) In our simple geometry, the distance between plasma boundary and solid boundary can not exceed over 50 times of Debye length. 2D simulation must obey this condition simultaneously both in r-direction and in z-direction. We acknowledges Prof. Y.Tomita and Prof. D.Tskhakaya for their advic on sheath kinetic model, and Dr.K.Asano for his advice on XOOPIC. We also thank Plasma Theory and Simulation Group in UC-Berkeley for their excellent PIC codes.

26 Appendix

27 Bug of XOOPIC Current version may still contain this bug.

28 Input file correction Input files for old versions must be used carefully.

29 Source file correction physics/vmaxwelln.cpp

30 "simple" Bohm condition If ion velocity distribution is beam like ( f is (u) = n is δ(u V s ), V s > 0 ) and charge neutrality at x = x s is assumed (n es = Z i n is ), n i = n is (1 + 2Z ie(φ s Φ) m i Vs 2 ) 1 2 Z i e Φ n i (Φ )dφ = n is m i Vs 2 ((1+ 2Z ie(φ s Φ) Φ 0 m i V 2 s ) 1 2 (1+ 2Z i e(φ s Φ 0 ) m i V 2 s and, by setting x 0 = x s and expanding around here, we obtain, ɛ 2 (E2 x E2 xs ) = n ese 2 (1 Z it es 2T es m i Vs 2 )(Φ s Φ) 2 + As in stable sheath E 2 x E 2 xs 0, conventional Bohm condition is obtained. ) 1 2 ) V 2 s Z it es m i

31 cutted Maxwellian ion distribution mi f is (u) = Cn is H(u u c ) exp( m i u 2 ), u c > 0 2πT is 2T is Normalization Constant C is determined from n is = 0 f is (u)du m i u C = 2(erfc( 2 c )) 1, erfc(x) = 2 exp( t 2 )dt 2πT is π ion density x n i = Cn is 2 erfc( x c (Φ)) exp( Z ie(φ s Φ) T is ) x c (Φ) = m iu 2 c 2πT is + Z ie(φ s Φ) T is By setting y = Z ie(φ s Φ) T is and y c = m iu 2 c 2πT is and expanding around y s = 0, Z i e Φ n i (Φ )dφ = ( T is ) Cn is Φ s 2 = T is n is { y y s erfc( y c + y ) exp(y )dy y (1 1 πyc exp(y c )erfc( y c ) )y2 + }

32 cutted Maxwellian ion distribution(2) So Bohm condition for cutted ion distribution 10 ɛ 2 (E2 x E2 xs ) = n ese 2 (Φ s Φ) 2 + Z2 i n ise 2 1 (1 2T es 2T is πyc exp(y c )erfc( y c ) )(Φ s Φ) 2 + And, in order to get stable sheath potential, T is Z i T es πyc exp(y c )erfc( y c ) In right figure, right hand side of above equation is plotted as the function of y c = m iu 2 c 2πT is. If T is = T es and Z i = 1, y c > 0.2 will be necessary. And, if T is decreases, y c will increase. (1) mu^2/2ti

33 example of VDF 0.6 vdist.dat using 1:2 vdist.dat using 1:3 0.6 vdist.dat using 1:2 vdist.dat using 1: vdist.dat using 1:2 vdist.dat using 1:3 vdist.dat using 1: Red line: ion, Blue line: green line:electr e(φ s Φ(x)) T e = 0.0, 1.7, 8.0 n is /n es = 1.0, T i /T e = 1.0, m i /m e =