Harmonic Content of Electron Impact Source Functions in Inductively Coupled Plasmas using an "On-the-Fly" Monte Carlo Technique*

Arvind Sankaran and
Mark J. Kushner

Department of Electrical & Computer Engineering

University of Illinois at Urbana-Champaign

1406 W. Green St., Urbana, IL 61801

1. Introduction

Electron temperatures in low-pressure (< 10’s mTorr) inductively coupled plasma (ICP) reactors operating at 10s MHz do not significantly vary during the radio frequency (rf) cycle. There can be, however, considerable modulation of electron impact source functions having high threshold energies due to modulation of the tail of the electron energy distributions (EEDs). In many instances, it is convenient to use cycle-averaged values for these quantities in models due to the computational burden of computing and storing spatial and time dependent EEDs. In this work a new “on-the-fly” (OTF) Monte-Carlo technique is described to address these time-dependent plasma parameters. The OTF method directly computes moments of the EEDs during advancement of the trajectories of the pseudoparticles, thereby reducing computational complexities. The method was also used to directly calculate the harmonic components of excitation, which can subsequently be used to reconstruct the time dependent source functions.

2. Computational Model

The OTF method, developed to investigate harmonics of
excitation in ICPs, was implemented into
HPEM. The specifics of the algorithms employed in the EMCS have recently been described in detail in Journal of Applied Physics.. The electron impact rate coefficient (*k _{ml}*) for electron impact process m and location l in the conventional method is given as

The EEDs *f _{il}* are obtained from raw statistics, collected in the EMCS,

The rate coefficient for process m at time s is then obtained from

The OTF method has potential advantages over the conventional EMCS by being more accurate than calculating and storing intermediate values of the EED followed by calculating the rate coefficients. The method is advantageous from a computational standpoint as well. Since moments of the distribution function are less sensitive to statistical noise than the EED itself, fewer particles are required in the simulation. The OTF technique additionally can calculate the Fourier components of the electron transport coefficients. From these components, one can then reconstruct the time dependence of electron impact reactions. The Fourier component for the n^{th} harmonic is obtained by replacing Eq. (2a) with

where [*e*]* _{I-1,l}* and

where *n _{m}* is the number of harmonics computed. The source function for any given electron impact event should always be positive. The maximum function in Eq. (6) is used to account for noise in the EMCS, which might result in the sum of the phase weighted amplitudes being negative.

**3. Results**

For validation purposes, results from the HPEM using the conventional Monte Carlo method and the OTF technique were compared for an ICP reactor using a 5 mTorr Ar/N_{2} =90/10 gas mixture. The final reactor averaged gas temperature was 415 K for a power deposition of 650 W at 13.56 MHz. The resulting electron densities and source functions for electron impact ionization of Ar are shown below.

Fig. 1: Electron density in an ICP reactor obtained using (a) conventional MCS method (b) OTF method; and source functions for electron impact ionization of argon using (c) conventional MCS method and (d) OTF method. |

The rationale for the choice of gases is that vibrational excitation of N_{2} has low threshold energies (< 1 eV) and is produced in large part by the bulk of the EED. Electron impact processes with Ar, particularly ionization, are high threshold reactions and are sensitive to the tail of the EED. The tail of the EED, being more collisional than the bulk is expected to be more modulated than the bulk and so high threshold processes has more modulation. As diagnostics for the differences between bulk and tail behavior, electron source functions for vibrational excitation of N_{2} with threshold energy = 0.29 eV, and ionization of Ar with threshold = 16 eV were examined.

Fig. 2: Source functions for a) electron impact ionization of Ar and b) vibrational excitation of N_{2}. |

**Reference**

1. A. Sankaran and M. J. Kushner, Harmonic Content
of Electron Impact Source Functions in Inductively Coupled Plasmas Using an
“On-the-Fly” Monte-Carlo Technique, J. Appl. Phys. **92**, 736 (2002).

^{*}This work is sponsored by Semiconductor Research Corporation, National Science Foundation (CTS99-74962), Applied Materials and Sandia National Laboratories

Last updated: August 27, 2003.