Landau damping and anomalous skin effect in low-pressure gas

discharges: self-consistent treatment of collisionless heating

Princeton Plasma Physics Laboratory

Igor D Kaganovich

In collaboration with Oleg Polomarov, Constantine Theodosiou

Department of Physics and Astronomy, University of Toledo, Toledo, Ohio.

Badri Ramamurthi and Demetre J. Economou

Plasma Processing Laboratory, Department of Chemical Engineering, University of Houston,

Houston, TX

Analytical Solution for Nonlinear Damping Decrement Accounting for

Collisions)(ντγγ Π= Lnl

0.01 1 1000

1

Π(ντ)

ντ

I.Kaganovich, PRL 1999

02eumφΔ≡

1/kuτ≡Δ

Anomalous Skin Effect

x

Erf(x)=E

0exp(-x/δ−iωt)

Normal skin effect

Linear theory nonlinear theory

ωimnEe

j−

=2

jcdx

dBB

c

i

dx

dE πω 4; −==

pωω <<

p

c

ωδ =

ωδ

>>TVωimnEe

janom −<<

2

panom

c

ωδ >>

xVVBE Δ=>+ ][yy VE Δ=> 00yyyemVAVc−==>Δ=

ante

nna

Anomalous skin effect

y

Collisionless Heating in Slab Geometry

L

nVLT xb πωω 2/2 ==

yVΔ

bounce resonances

xVΔ Change resonance

>ΔΔ=< tDMC /: 2ε

)tanh(: ντqlDDTheory =

10-3 10-1 ν/ω10-3

10-1

D

yVΔ No change in resonance

xVΔ

Importance of Nonlinear Effects for Calculation of Surface Impedance

The real part of surface impedance in ohm. The plasma parameters are n=1011 cm-3, Te=5ev, l=4cm.

10-4 10-3 10-2 10-1 ν/ω

ConclusionsThe electron Boltzmann kinetic equation

has been solved analytically for nonlinear Landau damping problem for any value of collision frequency.

γnl = γl tanh(ντr).The efficiency of the collisionless

heating is described by the diffusion coefficient D=Dql tanh(ντr).

Self-consistent System of Equations for Kinetic

Description of Low-pressure Discharges Accounting for nonlocal and collisionless

Electron Dynamics

For more info:Phys. Rev.E 68, 026411 (2003); Plasma Sources Sci. Technol. 12, 170 & 302 (2003),

Overview Calculate nonlocal conductivity in

nonuniform 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.

Inductive Discharge

ε1

ε2

x

Erf(x)

-Φ(x)

The transverse rf electric field is given by 222224[()()]yydEiEjxIxdxccωπωδ+=−+

The electron energy distribution is given by*00(),dfdDSfddεεε−=

Nonlocal Conductivity

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

200()(,')(')'(',)(')' xLeyyyxenJxGxxExdxGxxExdxmæö÷ç ÷ç =+÷ç ÷÷ç ç èøòò

()**121/2 **2120cosh((,))cosh((',))() (',)2,sinh((,))2()(')/ xxxxGxxdvxxvexxmejj¥FFG=F+-ò

0()(')'. fdeeee¥G=ò00(,)',2(('))/ xxixxdxexmwnej+F=-ò

Energy Diffusion Coefficient

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

0000()()()(),keeeekkkkkudfddDDVfuffddduεενεεενεεε∗∗∗∗∗⎡⎤+⎢⎥−+−=++−⎢⎥⎣⎦∑

222220()4()[()]xxynxnbxbxeDdEmnεεεεπνεεεεων∞=−∞−=ΩΩ−+∑∫

,eeeeDV kν∗

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

0()cos(()) ()(). LybxynxxExnxEdxvqeepW=ò

Comparison With Experiment

Comparison between experimental data [V. A. Godyak and R. B. Piejak, J. Appl. Phys. 82, 5944 (1997).] and simulation predictions using a non-local model (a) RF electric field and (b) the current density profiles for a argon pressure of 1 mTorr.

Comparison With Experiment

Comparison between simulated (lines) and experimental (symbols) EEDFs for 1 mTorr. Data are taken from V. A. Godyak and V. I. Kolobov, phys. Rev.

Lett., 81, 369 (1998).

ε (V)

f0(ε)(V

-3/2

cm-3)

10 20 30107

108

109

1010

12 W50 W200 W12 W50 W200 W

1 mTorr

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

Influence of Plasma Potential on rf Heating

Surface impedance for different plasma profiles.

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

φ, V

/x L

φ( )=5 (2/ )x L2( - /2)x L 2

φ( )=5 (2/ )x L4( - /2)x L 4

0 2 4 6 8 10 12 140.0

0.5

1.0

1.5

Re Z (

Ω)

( )L cm

,Maxwellian EDF=5 , Te eV n

e=5 10x 11,

ω=8.52 10x 7,ν=107. Without a potential withφ( )=5 (2/ )x L2( - /2)x L 2

Withφ( )=5 (2/ )x L4( - /2)x L 4

ConclusionThe self-consistent system of equations

is derived for description of collisionless heating and anomalous skin effect in nonuniform plasmas.

The robust kinetic code was developed for fast modeling of discharges, which predicts nonMaxwellian electron energy distribution functions in rf discharges.

ConclusionsA novel nonlinear effect of anomalously deep penetration of an

external radio frequency electric field into a plasma is described. A self-consistent kinetic treatment reveals a transition region between the sheath and the plasma. Because of the electron velocity modulation in the sheath, bunches in the energetic electron density are formed in the transition region adjacent to the sheath. The width of the region is of order VT/ω, where VT is the electron thermal velocity, and ω is frequency of the electric field. The presence of the electric field in the transition region results in a cooling of the energetic electrons and an additional heating of the cold electrons in comparison with the case when the transition region is neglected. Additional information on the subject is posted in

I. Kaganovich, PRL 2002 http://arxiv.org/abs/physics/0203042