# Presentation: 449851

## 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

- 0.01

- 1

- 100

- 0

- 1

- ()

- 

- I.Kaganovich, PRL 1999

## Anomalous Skin Effect

- Normal skin effect

- Linear theory

- nonlinear theory

- antenna

- Anomalous skin effect

- y

## Collisionless Heating in Slab Geometry

- L

- bounce resonances

- Change resonance

- 10-3                    10-1              /

- 10-3

-   10-1

*D*

- No change in resonance

## 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      /

## Conclusions

- The electron Boltzmann kinetic equation has been solved analytically for nonlinear Landau damping problem for any value of collision frequency.
-    nl = l tanh(r).
- The efficiency of the collisionless heating is described by the diffusion coefficient *D=D**ql **tanh(****r**).*

## 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

- The transverse rf electric field is given by

- The electron energy
- distribution is given by

## Nonlocal Conductivity

- Nonlocal conductivity is a function of the EEDF *f**0* and the plasma potential *(x).*

## Energy Diffusion Coefficient

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

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

## 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).

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

## Influence of Plasma Potential on rf Heating 

- Surface impedance for different plasma profiles.

## Conclusion

- The 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.

## Conclusions

- A 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/, where VT is the electron thermal velocity, and  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