how to control and predict plasma parameters in plasma sources

3/24/20041

How to Control and Predict Plasma Parameters in Plasma Sources

Igor Kaganovich Princeton Plasma Physics Laboratory

2

Outline

Observed dependences of ne,Te on p, L

Global model for ne,Te (p,L)

Evaporating electron cooling in afterglow

Fast kinetic code for plasma sources modeling

May you have the hindsight to know where you've been, the foresight to see where you're going and the insight to know when you're going too far...

3

Outline






4

19.8 cm

10.5 cm

Godyak’s Experiment is Benchmark.

The Inductively Coupled Plasmas (ICP) in a cylindrical stainless steel chamber as shown in Fig.1.

Measurements were made at

f=0.45-13.56 MHz

in argon gas pressures of 0.3-300 mTorr and

rf power dissipation in the plasma 6-400 W.

5

Electron Density and Plasma Potential Versus Power

ne φpl (~6 Te) in the discharge center as a function of power (P) at f=6.78 MHz

ne scales linear with P, whereas φpl and Te ~const.

6

Electron Density and Plasma Potential Versus Pressure

ne,Te in the discharge center at fixed power 50W, f=6.78MHz

More gas, more plasma and smaller Te

7

Electron Density, Temperature and Potential Profiles

1mTorr 10mTorr

8

Outline






9

Plasma is Partially Ionized

Plasma density ne = 109 - 1013 cm-3

Gas density ng= 3.5 1013 cm-3 p(mTorr)

– Small degree of ionization ne/ng < 10-3

– Collisions with gas atoms are dominant

10

Particle Balance Determines Te

Quasi-Steady-State =>

– Rate of plasma production = rate of plasma loss, or

– Ionization frequency = loss frequency to the wall

( ) ( )iz e loss eT Tν ν=

11

Calculation of Ionization Frequency

where I is ionization potential 15.8eV,VT is the electron thermal velocity.

2

3

/ 2| | ( ) ( ) ,iz e g izm I

n n f dν σ∞

>= ∫ v

v v v v

/ ,eI Tiz g T izn V eν σ −=

12

Loss Frequency

Electrons are confined

( ) (0)exp( / )e e en x n e Tφ=

/s eC T M=

/loss sC Lν γ=

Potential of order Teaccelerates ions

13

Particle Balance: Te (PL)

( ) ( )iz e loss eT Tν ν=

More ngL less Te

/ /eI T e

g T iz

T Mn V e

Lσ γ− =

14

Experiment (left) and Global Model (right) Give Close Agreement

0.1 1 10 100 1.1030

2

4

6

8

10

gas pressure, mTorr

elec

tron

tem

pera

ture

, eV

10

0

Te

10000.1 Pgr

Te as a function of gas pressure at f=6.78 MHz

15

Outline






16

Evaporative Electron Cooling in Afterglow

Experiment: Kortshagen, et al. Appl. Surf. Sci. 192, 244 (2002)Similar to Godyak’s, but ID=14cm, L=10cm

17

ε

f(ε)

ε1

ε2

x

-Φ(x)

0.15610, 72ττττ,µµµµs

inelasticneTe

Inelastic Collisions (Ar Excitation) cool tail quicklyThan evaporated cooling as fast electrons leave fastτ(Te)~ τ(ne)

Evaporative Electron Cooling in Afterglow

Electron temperature can cool to 30K! Biondi (1954)

18

Experimental Temperature Decay Time and Theoretical Estimate For Density Decay

Experiment Te decay time in microseconds.

Theoretical estimate for n decay in microseconds.

10 20 30 40 50 60 70

60

80

100

120

Pressure, mTorr

Loss

Tim

e, m

icro

s

120

50

τexp

701 Pexp

10 20 30 40 50 60 70

60

80

100

120

Pressure, mTorrLo

ss T

ime,

mic

ros

120

50

τloss106⋅

705 Pgr

19

Outline


Global model for ne,Te (p,l)




20

BibliographyI. D. Kaganovich et al “Anomalous skin effect for anisotropic electron

velocity distribution function”, to be published in Phys. Plasmas (2004).

I. D. Kaganovich, et al , “Landau damping and anomalous skin effect in low-pressure gas discharges: self-consistent treatment of collisionless heating”, to be published in Phys. Plasmas (2004).

I. D. Kaganovich and O. Polomarov, “Self-consistent system of equations for kinetic description of low-pressure discharges accounting for nonlocal and collisionless electron dynamics”, Phys. Rev. E 68, 026411 (2003).

B. Ramamurthi, et al " Effect of Electron Energy Distribution Function on Power Deposition and Plasma Density in an Inductively Coupled Discharge at Very Low Pressures", Plasma Sources Sci. Technol. 12, 170 and 302 (2003).

21

Electron Distribution Function is not Maxwellian

Measured EDF shows departure from a Maxwellian: Cold electrons are trapped in a small rf electric field.

ε1

ε2

x

Erf(x)

-Φ(x)

22

Concept of Screening Temperature

EDF is not Maxwellian,introducing Tes , so that

11/ 2

02 ( )esT n f dε ε ε

−∞ − = ∫

ln( ) /es eeE T d n dx= −

3/ 2

0

2 2 ( )3 3eT f d

nε ε ε ε

∞= < >= ∫

23

In collaboration with

Oleg Polomarov, Constantine TheodosiouUniversity of Toledo, Toledo, Ohio

Badri Ramamurthi, Demetre J. EconomouUniversity of Houston, Houston, TX

Kinetic Code for Calculation of EDF and Plasma Parameters

24

Overview

Calculate nonlocal conductivity innonuniform plasma

Find a nonMaxwellian electron energy distribution function driven by collisionless heating of resonant electrons

What to expect: self-consistent system for kinetic treatment of collisionless and nonlocal phenomena in inductive discharge

25

Inductive Discharge

ε1

ε2

x

Erf(x)

-Φ(x)

The transverse rf electric field is given by2 2

2 2 2

4 [ ( ) ( )]yy

d E iE j x I xdx c c

ω π ω δ+ = − +

The electron energy distribution is given by*0

0( ) ,dfd D S fd dεε ε

− =

26

Nonlocal Conductivity

Nonlocal conductivity G(x,x’) is a function of the EEDF f0 and the plasma potential ϕ(x).

20

0

( ) ( , ') ( ') ' ( ', ) ( ') 'x L

ey y y

x

e nJ x G x x E x dx G x x E x dxm

= + ∫ ∫

•PIC code is inefficient: limited by electron time step,

•while discharge develops at ion time scale =>

•implicit description of the rf electric field•solved by spectral method

27

Kinetic Equation Is Averaged over Fast Electron Bouncing in Potential Well

Dee Vee are from the electron-electron collision integral, ν* is inelastic collision frequency, upper bar denotes space averaging with constant total energy.

0

00 0

( )( ) ( ) ( ) ,k

ee ee k k k kk

udfd dD D V f u f fd d d uε

εν ε ε ε ν

ε ε ε

∗∗ ∗ ∗ ∗

+ − + − = + + −

∑

Energy diffusion De coefficient is a function of the rf electric field Ey and the plasma potential ϕϕϕϕ(x).

28

Comparison with Experiment

Experimental data (symbols) and simulation (lines) (a) RF electric field and (b) the current density profiles for a argon pressure of 1 mTorr.

Te

ep

Vc δω ω

< <

29

Comparison with Experiment

EEDF simulated (lines) and experimental data (symbols) for 1mTorr. Data are taken from V. A. Godyak and V. I. Kolobov, Phys. Rev. Lett., 81, 369 (1998).

ε (V)

f 0(ε)

(V-3

/2cm

-3)

10 20 30107

108

109

1010

12 W50 W200 W12 W50 W200 W

1 mTorr

R=10cm,L=10cm,antenna R=4cm

30

Predictions of the Code for PPPL Plasma Source

0 1 2 3

3

4

5

6

7

Te=2/3 <ε> full model global model

T e (eV

)

Pressure (mTorr)4 5 6 7

0

2

4

6

8

400W Plasma Density 0.3 mTorr 1mTorr 3mTorr

n ex10

12 (c

m-3)

0<x<L (cm)

31

Predictions Using Power Balance for PPPL Plasma Source

Power deposited into plasma:P=0.5nCs εεεε S

ε ≈ ε* +2I=43eV

N=4 1012 cm-3 at 1mTorr needed power 0.4kW vs 2kW

/s eC T M=

32

Predictions of the Code for PPPL Plasma Source

0 20 40 60

10-5

10-4

10-3

10-2

10-1

EED

F (e

V-3/2)

ε (eV)

400W Normalized EEDF 0.3 mTorr 1mTorr 3mTorr

0 1 2 3 4 5 6 70

1

2

3

4

5

400W Electric field (Volt/cm) 0.3 mTorr 1mTorr 3mTorr

E (V

/cm

)0<x<L (cm)

EDF is Maxwellian due to high degree of ionizationSkin layer of 0.5 cm

33

Conclusions

Te 3-5eV for steady-state, P in mTorr range

Te decay time scale is 10-50 microseconds

N=4 1012 cm-3 correspond to 0.4kW

Skin layer about 1cm

EDF is Maxwellian

how to control and predict plasma parameters in plasma sources

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