Zoran Raspopovic

Kinetic phenomena in electron transport in radio-frequency fields

Abstract We discuss the application of swarm physics based techniques to study the time resolved kinetic phenomena that may be of interest to modeling of radio-frequency (rf) plasma, the relevance of such studies for the data that are being used and models that are being developed. Kinetic phenomena may not be predicted by extrapolation of the dc data or single particle trajectories and require full kinetic treatment or detailed simulations. Our exact solutions to the Boltzmann equation by direct numerical procedure (DNP) and Monte Carlo simulations (MCSs) revealed such phenomena as: anomalous longitudinal diffusion, time resolved negative differential conductivity, absolute negative mobility in afterglow and in rf fields, complex waveforms of transport coefficients due to phases between electric and magnetic fields, due to cyclotronic motion and all the comparisons performed so far between the two techniques and with results of the exact solutions to the Boltzmann equation by the group from James Cook University have showed excellent agreement. While the data used nowadays in plasma modeling are basically for dc fields, one has to include effects due to magnetic and time resolved magnetic field and also the kinetic phenomena in plasma modeling as these may affect the electron transport in real collisional processing plasmas. Thus we believe that kinetic and Monte Carlo based codes should be tested against swarm transport benchmarks, including results for time resolved E ( t ) and E ( t )× B ( t ) fields.

Benchmark calculations for Monte Carlo simulations of electron transport

Benchmark calculations have been performed for electron transport coefficients with an aim to produce a body of data required to verify the codes used in plasma modeling. The present code for the time resolved Monte Carlo simulation (MCS) was shown to represent properly DC transport coefficients in a purely electric field, in crossed electric and magnetic fields, and in the presence of nonconservative collisions, ionization, and attachment. In addition, we have suggested tests of the time dependent solutions. Relaxation of the initial transport coefficient may serve as an accurate test of the code as well as the input data for some fluid codes. In this paper, we show only one example, but several different sets of conditions and cross sections should be used as well. Finally, we propose application of the quasi-steady state results in RF fields. As an example we suggest calculation of the components of diffusion tensor showing anomalous longitudinal diffusion and calculations made with nonconservative collisions (ionization in this case). We also check the application of approximate formulas to determine drift velocity on the basis of total collision frequency and to determine a diffusion coefficient by using the Einstein relation. Other tests required to verify the transport data calculations are discussed as well.

Monte Carlo studies of electron transport in crossed electric and magnetic fields in CF4

Electron transport in crossed electric and magnetic dc fields in CF4 is considered by employing an exact Monte Carlo simulation technique. Emphasis is placed on explicit and implicit effects of the combination of both the shape of cross-sections and magnetic field on relaxation processes and steady-state transport data. It has been shown that the application of a magnetic field changes the relaxation processes by increasing the relaxation times as well as, by introducing the oscillatory behaviour of some transport quantities during the relaxation process itself. The electron transport parameters studied here are collision frequency, mean energy, drift velocity, diffusion tensor and collisional rates for 1 E/N 1000 Td and 0 B/N 1000 Hx (1 Td = 10 −21 Vm 2 ,1 Hx= 10 −27 Tm 3 ). Special attention is paid to the study of sensitivity of transport data in E × B fields on the energy dependence and nature of the cross-sections. The validity of the effective reduced electric field concept through Tonks’ theorem is also investigated for CF4 in the evaluated range of mean energies.

On the possibility of negative electron mobility in a decaying plasma

The electron energy distribution function and the electron drift velocity are studied numerically in a decaying plasma in the external electric field in the mixture Ar:F2 = 1:0.005 at atmospheric pressure. For a reduced electric field strength over the range 0.055-0.55 Td the negative electron mobility was predicted earlier by the solution of the Boltzmann equation using the two-term approximation for the velocity distribution function. The applicability of this approximation for calculations of negative mobility is discussed. The Monte Carlo simulation method is used to verify results obtained from Boltzmann equation analysis. It is shown that there is a reasonable agreement between the Boltzmann equation and Monte Carlo results.

Diffusion of electrons in time-dependent E(t) × B(t) fields

We present results of Monte Carlo simulations for the components of electron diffusion tensor in time-varying E×B fields. In particular we follow the development of anomalous behaviour due to the change of the sign of the electric field which was found for longitudinal diffusion in purely electric rf fields (K Maeda et al 1997 Phys. Rev. E 55 5901). Similar behaviour was found to be present in crossed fields for the longitudinal NDE component and absent for one perpendicular NDB component. It turns out that anomalous behaviour for the NDE×B component is induced even for a relatively small magnetic field. Reids ramp model was used for calculations but the conclusions are quite general as these effects occur for most real gases. The transient kinetic phenomena are usually not taken into account in continuum models of rf plasmas. Our results show that it is important to include such processes in modelling and to verify the electron kinetic codes for their ability to represent the transient phenomena.

Relaxation of electron swarm energy distribution functions in time-varying fields

Monte Carlo simulation code for RF fields was used to study the transition from the initial conditions to the quasi-steady state energy distribution functions. Signatures of the initial conditions were found to be recreated even when the distribution function appeared to have reached the relaxed shape. The inclusion of magnetic field changes the transition to the relaxed distribution as well as the final mean energy and other data.

Drift Velocities of Electrons in Time Varying Electric Fields

Monte Carlo simulations of electron transport have been performed in methane for time resolved rf fields under conditions where negative differential conductivity (NDC) occurs. It was found that the effect of NDC occurs for low frequencies as expected from the quasi stationary (temporally local) model. At higher frequencies the NDC disappears gradually but the NDC behavior is significantly different when field increases and when field decreases. In the latter case the NDC disappears while in the former it blends into one maximum occurring at the point of the maximum of the electric field with an intermediate frequency region with asymmetric time dependence.

On the existence of transiently negative diffusion coefficients for electrons in gases in E ? B fields

A recent study (Raspopovic Z et al 2000 J. Phys. D: Appl. Phys. 33 1298) reported the existence of negative diagonal diffusion tensor elements for electrons in a neutral gas under the influence of crossed radiofrequency electric and magnetic fields. In this paper we demonstrate, using a time-dependent multi-term solution of the Boltzmann equation and time-resolved Monte Carlo simulation, that this phenomenon is a transient relaxation effect obtainable under dc crossed field conditions.

Transport coefficients for electrons in argon in crossed electric and magnetic rf fields

Monte Carlo simulations of electron transport have been performed in crossed electric and magnetic rf fields in argon. It was found that a magnetic field strongly affects electron transport, producing complex behaviour of the transport coefficients that cannot be predicted on the basis of dc field theory. In particular, it is important that a magnetic field, if it has sufficiently high amplitude, allows energy gain from the electric field only over a brief period of time, which leads to a pulse of directed motion and consequently to cyclotron oscillations being imprinted on the transport coefficients. Furthermore, this may lead to negative diffusion. The behaviour of drift velocities is also interesting, with a linear (sawtooth) dependence for the perpendicular drift velocity and bursts of drift for the longitudinal. Non-conservative effects are, on the other hand, reduced by the increasing magnetic field.

Negative mobilities of electrons in radio frequency fields

In this paper, we perform calculations of mobilities (represented by drift velocities) of electrons in mixtures of F/sub 2/ and Ar in radio frequency (RF) fields. The conditions were chosen which correspond to those that led to observation of negative absolute mobility in dc fields in decaying plasmas. In RF fields, a similar effect is observed as the mobility corresponding to the flux drift velocity is negative. At the same time, it was found that the mobility corresponding to the bulk drift velocity is mainly positive in respect to the expected value, but that it has a large phase delay. The effect is caused by nonconservative nature of the attachment and in the absence of nonconservative collisions both mobilities are positive and in phase with the field.