Modelling Low Pressure, Low Temperature Discharges using Particle-in-cell Techniques


Principal Investigator

Helen Smith

Plasma Research Lab,

Research School of Physical Sciences and Engineering




This project involves kinetic modelling of low pressure, low temperature, radio-freqency plasma
discharges, using a well established technique, known as Particle-in-cell (PIC). These types of plasmas are widely used in laboratory experiments to study basic plasma phenomena, and also have important applications in the microelectronics and materials processing industries.
The project uses electrostatic codes, which model the plasma in one spatial dimension. The current area of particular interest involves studying both short and long time-scale changes occuring in non-equilibrium discharges, including plasma breakdown and evolution to steady-state when power is turned on, and plasma decay in the afterglow when the power is turned off. These studies are being used to understand physical phenomena important in time-modulated, or pulsed, rf discharges, which are becoming increasingly important for industrial applications.

A very important aspect of understanding the plasma evolution requires modelling the interaction between the evolving plasma impedances, and the external circuit used to "match" the power from the generator to the plasma. The matching network is very important in determining the magnitude and shape of the voltage waveform which appears on the "powered" electrode, and consequently on the plasma development. For optimum power transfer from the source to the plasma the circuit + plasma impedance must equal the generator impedance (typically 50 Ohms). Experimental systems typically accomplish this by using an L-type matching network which includes two variable capacitors (the "load" and "tune" capacitors) which can be adjusted to compensate for the plasma impedance.


Bonar Dickson

Rod Boswell

Plasma Research Laboratory,

Research School of Physical Sciences and Engineering

Henry Gardner

Department of Computer Science and




w03 - VPP


Appendix A -


What are the results to date and the future of this work?

Most of the work carried out during 1997 involved porting the existing PIC codes modelling pulsed systems to the VPP. This required extensive re-writing of many of the major algorithms to improve the vectorisation and performance of the code. The code was orginally written in Fortran 77 on a DEC workstation, and when initially imported to the VPP test runs showed very poor performance, with only 2% vectorisation and longer run-times than on the workstation. After substantial re-writing of routines (described in the following section) vectorisation reached 78% and run- times improved by a factor of 50. The consequent improvement in speed allowed the investigation of plasma behaviour for a large range of input parameters. Furthermore, with the new code it will be feasible to simulate plasma evolution times of the order of milliseconds, which are 10 to 100 times longer than previously possible.

The second part of the project required developing a model of an L-type matching network, as used in experimental systems. This was included in the vectorised code and various test runs were carried out to evaluate the performance of the modelled matching network. Comparison with experimental results led to the development of a second generation model, completed in November 97, which has produced some very interesting initial results.

In the future a parameter "scan" of the matching circuit inputs will be carried out to determine their effect on plasma evolution. Different matching network models may be trialed in order to produce the optimum transfer of power to the plasma (this will be directly applicable to experimental systems used in the Plasma Research Lab, RSPhysSE). This work will be incorporated in a wider study of pulsed rf discharges, in which it is intended to study the effect of time and spatial variation in the background gas and look at more complex plasma chemistry.

A longer term goal is to develop a self-consistent code incorporating two spatial dimensional dimensions. This will allow the investigators to model much more realistic system geometries and allow studies of intrinsically multi-dimensional plasma behaviour which cannot be observed using the 1D code.

What computational techniques are used?

The code uses a technique known as Particle-in-Cell, which is commonly utilised for low density, non-equilibrium plasma models. In this method up to 50,000 individual electron and ion "macro-particles" are modelled. Since these plasmas have densities of the order of 10^16 particles /m^3, each of these "macro-particles" represents up to 10^10 real particles in the plasma. Particles are moved under the influence of self-consistently calculated electric fields via Newton's equations. An artificial spatial grid is introduced, to which particles are weighted according to their position. This allows straight-forward calculation of the potentials and electric fields in the plasma, but also introduces numerical noise and spatial aliasing into the field calculations. The model of the external matching circuit is included in the plasma simulation via boundary condition calculations.

Collisions between charged species and a background neutral density (assumed to be

- Appendix A


unvarying) are calculated using Monte-Carlo techniques. Energy-dependant collision cross-sections are determined empirically from experimental measurements and used to determine the collision frequencies.

The main subroutines include:

1) electron movement (including collisions with neutrals)

2) ion movement (including collisions with neutrals)

3) wall collisions

4) density and field calculations

The main difficulty with the particle movement routines, in terms of vectorisation, was that the routines operated on single particles at a time and looped over the particle number (which has a value up to 50,000) with conditional statements involving collisions within the loop. Consequently, by moving the collision mode conditional out of the loop, and updating particle positions and velocities as an ensemble a factor of 12 increase in speed was attained. A similar method were used to vectorise the routines used to check for collisions with the walls.

Density and field calculations were partially vectorised by using memory in the form of auxiliary arrays to remove recursive referencing to particles. However, this entails a tradeoff betwen vectorisation of the weighting process, and memory and additional processing required to accumulate and add vectors. Typically vector lengths of 256 particles were found to be most effective. Together the listed modifications produced an increase in measured vectorisation from 2% to 78%, and a speed increase of a factor of 50.

Appendix A -