Kinetic phenomena in electron transport in radio-frequency fields

Applied Surface Science 192 (2002) 1±25

Kinetic phenomena in electron transport in radio-frequency ®elds Z.Lj. PetrovicÂa,b,*, Z.M. RaspopovicÂa, S. Dujkoa, T. Makabeb a

b

Institute of Physics, University of Belgrade, PO Box 68, 11080 Zemun, Yugoslavia Department of Electronics and Electrical Engineering, Keio University, 3-14-1 Hiyoshi, Yokohama 223-8522, Japan

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 ®elds, complex waveforms of transport coef®cients due to phases between electric and magnetic ®elds, 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 ®elds, one has to include effects due to magnetic and time resolved magnetic ®eld 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† ®elds. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Kinetic phenomena; Radio-frequency; Electron

1. Introduction In this paper, we summarize the fundamental properties of electron swarms in radio-frequency (rf) ®elds, or in time varying ®elds that may not be describable on the basis of dc or quasi-dc theory (instantaneous ®eld approximation). Some of these properties may bear the label kinetic phenomena as their basis is in collective motion of swarm particles driven by electric ®elds and dissipating their energy in collisions with molecules of the background gas. In that respect, these * Corresponding author. Present address: Institute of Physics, University of Belgrade, PO Box 68, 11080 Zemun, Yugoslavia. E-mail addresses: [email protected], [email protected] (Z.Lj. PetrovicÂ).

phenomena may not be easily described by motion of single particles without performing the analysis of the transport of the entire ensemble. For the purpose of this paper, we shall de®ne swarm as an ensemble of particles (in this case charged) that is developing in the gas by gaining the energy from the electric ®eld and by dissipating it in collisions with gas molecules. The basis of swarm theory has always been to develop theory for an isolated swarm in an in®nite space [1,2]. This approach was followed by development of the very high accurate experiments to measure well (or hopefully well) de®ned transport properties of particles. Hydrodynamic approximation to the electron energy distribution function (EEDF) was thus implicitly assumed without any complete proof of its general applicability [2±4]. Nevertheless, it was

0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 2 ) 0 0 0 1 8 - 1

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Z.Lj. Petrovic et al. / Applied Surface Science 192 (2002) 1±25

possible to design good quality experiments to meet the requirements of such theory by minimizing the effect of boundaries including the electrodes [1]. However, the very analysis of the experimental observables had to involve some interaction with the surfaces, mostly metallic, and to violate the basic assumptions of the theory. The boundary effects and the failure of hydrodynamic approximation under the circumstances have been investigated extensively by kinetic theories and simulations [5±7], including the case of very high values of E=N when non-locality of transport may extend over the entire discharge [8,9]. Theories in inhomogeneous ®elds [10], especially those attempting to model or simulate the kinetics of cathode fall [11] of a dc or sheath of an rf discharge revealed the non-local nature of electron transport, and this proved to be the most dif®cult problem in early development of plasma models. Time resolved ®elds were also subject of studies of swarm theories. However, temporal non-locality have, until recently, rarely been a subject of special studies or inclusion in plasma models. Mostly, effective dc ®eld or quasi-dc instantaneous ®eld approximations are being used [12,13] in plasma modeling. Another issue that is highly relevant for modeling plasmas for etching of integrated circuits is the transport in crossed electric and magnetic ®elds. In most cases, the effect of magnetic ®eld on electron transport is being neglected in plasma modeling, or at best effective ®eld approximations are being applied [14]. These approximations calculate the effective dc ®eld that may be used to calculate the transport and rate coef®cient that may be adequate replacement for the time dependent coef®cients. In this paper, we propose that one of the critical problems in plasma modeling is that of the temporal non-locality. It is also often associated with the spatial non-locality. The calculations that have been made with accurate theories and Monte Carlo simulations (MCSs) in time varying ®elds point to a large number of effects that are simply neglected by all or at least by most of the techniques that are being employed. Those effects are kinetic in nature, i.e. cannot be predicted on the basis of single particle trajectories. It is important to try to set up a set of tests for such effects that would properly test the performance of the codes and at the same time to discuss which of the observed

phenomena should be taken into account in plasma models and which may be neglected. We shall brie¯y review the relevant developments in the area of rf swarms, including the temporal relaxation and tests of the numerical techniques. One example of a kinetic effect of negative absolute mobility during the temporal relaxation will also be shown. Finally, we discuss the examples of electron transport in purely electric ®eld and transport in crossed electric and magnetic ®elds both operating at rf. The issues of spatial non-locality and dc spatiotemporal relaxation and transport in dc E  B ®elds will be left for other reviews, including those available in the present issue by other authors [15,16]. Thus we shall give just a brief review of the development of time resolved solutions to electron transport, relevant information on electron transport in rf ®elds, in E  B ®elds and on the development of MCS techniques. 1.1. Historical review of the research on swarms in rf ®elds Even the early studies of electron transport in gases were concerned with rf ®elds, mostly because a major motivation in those developments was modeling of interaction of electromagnetic waves with ionosphere. Thus, Holstein [17] discusses application of the Boltzmann equation for rf ®elds in his historical paper. The same is true for the work of some other authors of that period including McDonald and Brown [18] and Margenau [19]. This research was, however, followed by the development of effective ®eld theory, whereby the rf transport is represented by an effective dc value of E=N (where E is the electric ®eld and N the gas number density, and the units for this quantity are Townsend, 1 Td ˆ 10 21 V m2 ). This approximation was reviewed recently [12] and also tested by MCS [20]. The approximation works very well at very high frequencies and it was extended and applied systematically for microwave discharges by Ferreira and Loureiro [21]. Time resolved solutions to the Boltzmann equation with limited number of terms (in development of the distribution function by both spherical harmonics in the velocity space and by number of the terms in Fourier expansion) were ®rst developed in 1970s by

Z.Lj. Petrovic et al. / Applied Surface Science 192 (2002) 1±25

Wilhelm and Winkler [22] and also by Morgan and Penetrante [23] (time resolved solutions to Boltzmann equation were also considered by Napartovich and coworkers [24]). Similar approach was adopted by Goto and Makabe [25] in the late 1980s with an idea to analyze the applicability of time resolved swarm data in interpretation of measurements for rf plasmas. The basic time resolved theory was also applied in studies of kinetics of excited states in plasmas [26]. The rf swarm theory was extended to a larger, but still limited, number of terms in 1980s [27,28]. Solutions to the Boltzmann equation with accuracy that is in principle unlimited were developed in late 1980s and in 1990s. Drallos and Wadehra [29] have developed solutions to the time resolved Boltzmann equation. In particular, we have to mention four groups that have recently developed exact approaches to electron transport in time varying ®eld. The group in Greifswald (Germany) [16] has studied relaxation of EEDF [30] by using their multi-term solution to the Boltzmann equation [31]. These authors have applied kinetic treatment to inhomogeneous ®elds [32] and realistic gas discharges such as positive column [33]. The group at James Cook University (Cairns and Townsville, Australia) [15,34] has developed a detailed multi-term theory of electron and ion [35] swarms in periodic ®eld. They have applied it primarily to study a number of kinetic phenomena [36,37] such as anomalous diffusion, non-conservative effects [38] and also to propose benchmark calculations [39,40] that may be used to test other techniques. At many points their work was used to test the techniques and results presented in this paper and applied in plasma modeling [12]. The group is integrating full kinetic treatment with self-consistent ®eld calculation in order to become able to model discharges and collisional plasmas. They have also used approximate momentum transfer theory (MTT) to study transport in rf ®elds and have made a wide range of studies of electron transport close to the electrodes and transport in E  B ®elds that will be mentioned later. In addition, a very important issue of the applicability of the hydrodynamic approximation has been addressed but only for a very special set of circumstances [37].

3

The group at Keio University (Yokohama, Japan) has developed direct numerical procedure (DNP) to solve Boltzmann equation including spatial and temporal dependencies [42,43]. The aim of their work was to understand the kinetic phenomena in rf ®elds [44], to obtain accurate data for plasma modeling, phenomenological interpretations of processes in rf plasmas, transport in E  B ®elds [45] and to model the transport of particles in the sheath towards the surface of silicon wafer or sputtering target. MCS technique was used by the group at the Institute of Physics in Belgrade (Yugoslavia) to study the transport at very high values of E=N [9], effects of electrodes on electron swarm transport [46] and transport of electrons in rf ®elds. They have also developed time resolved MTT. In particular this group analyzed transport in time dependent E…t†  B…t† ®elds. In this paper, we shall give a review of some of the results obtained by the groups at the Institute of Physics in Belgrade and Keio University. One should also mention that for some special circumstances, exact solutions exist even for rf ®elds and those may be used as a basis to test the theories and numerical procedures [12,47]. Number of studies of ion transport in rf ®elds is very limited [48,49]. One should also not failed to mention that swarm experiments were limited only to the low current plasma studies at the lowest power conditions and to the non-thermal Cavelleri's experiment [50]. That is a version of a standard Cavalleri's experiment [51] with the additional continuous rf ®eld (in addition to a rapidly quenched strong rf ®eld that is used for pulses of ionization). Thus the great success in the development of the swarm physics, relied purely on carefully designed dc experiments [1,52] and dc theories and simulations [53±56]. This approach has produced the best sets of the cross-section data that are available [52] and dc theories and simulations [1,57]. One may claim that the need for rf swarm experiments was generated at the time when swarm experiments in general started disappearing, so it is not surprise that rf swarm experiments were not developed. The greatest application of both the swarm based theories, experimental data and the cross-sections was in modeling of the rf plasmas which developed in 1980s and reached its peak in the ®rst half of 1990s. These models [58±61] recognized the issue of spatial

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non-locality but the issue of temporal non-locality was rarely addressed, except for the effective relaxation theories that were employed [61]. The models use the following approaches to treating spatio-temporal behavior of transport coef®cients: 1. Nothing at all, i.e. dc data are employed. This approach involved zero-dimensional chemical kinetic models or ¯uid models with dc transport data [62,63]. 2. Relaxation continuum approximation where dc collision frequencies are employed to determine the spatial and temporal relaxation for momentum and energy of the swarm properties [61]. 3. Hybrid approach [64±66], where MCS is used purely to follow the high energy electrons and map their spatial distribution of ionization. For properties of low energy electrons, dc swarm data are employed and for the high energy electrons usually only the time averaged spatially resolved ionization is employed. If Monte Carlo part is used to determine anything more than pro®le of ionization, approximate formulae are applied to establish transport coef®cients on the basis of sampled total collision frequency. 4. Particle in cell (PIC) [67] simulations where Monte Carlo like procedure is employed to determine the trajectories and the collision rates of effective particles. 5. Monte Carlo techniques [68,69] which are mostly based on electron transport Monte Carlo codes. 6. Numerical solution of kinetic equations [70] with different degrees of approximation involved. One should bear in mind that the term plasma modeling is used in different contexts in this text. When we discuss generation of the transport data we primarily refer to ¯uid and hybrid models that use such data. However, when we refer to the modeling and testing of the models then we refer to all the models, in particular those that may be tested for the kinetics of electrons. In general, we aim our discussion to the comprehensive numerical models aimed at rf plasmas for processing. In principle, techniques (1) and (2) will lead to a varying degree of success but they cannot be tested directly for the effect of kinetic phenomena, perhaps only through comparisons with other plasma models. However, the techniques (3)±(6) may be tested very directly in their performance to determine

the transport coef®cients, though one should also be aware that the implementation of such data may hold some further approximations. These tests may be made by using carefully chosen examples of swarm calculations from dc to ac ®elds. Systematic testing of the swarm codes on benchmarks was the basis of the development of swarm physics. The basic theory employed in the plasma models was based on swarm theories. However, with the exception of the plasma models that were developed directly from the swarm codes, very few comparisons were made [71] between the parts of the plasma models that are used to calculate electron kinetics and the well-de®ned benchmarks. Even then the test was not made for the well-de®ned ``tough'' benchmarks of the numerical techniques. 2. Monte Carlo technique This paper does not have room for a thorough review of the Monte Carlo techniques as applied to electron transport in gases. However, we may state that there were very few studies that involved time resolved sampling of the relaxation even though it is a part of each simulation. Even more importantly there are only a few studies of transport in rf ®elds [72] apart from the work of the Belgrade group. In this work, we follow the space and time development of a swarm of electrons under the in¯uence of electric and magnetic ®elds. It is assumed that the electron density is suf®ciently small so Coulomb interactions between the particles as well as shielding of the ®eld are negligible. The swarm is assumed to develop in an in®nite gas space under a uniform electric ®eld (i.e. Laplace ®eld), and boundary effects are neglected. In analysis of electron motion, one has to be able to follow accurately the trajectory and to determine the moment of the next collision. The probability that the electron will have a collision in the time interval …t; t ‡ dt† is given by  Z t  p…t† ˆ nT …E…t†† exp nT …E…t†† dt ; (1) 0

where nT …E…t†† is the time dependent total collision frequency. One may solve Eq. (1) either by assuming that the collision frequency is constant or by numerical

Z.Lj. Petrovic et al. / Applied Surface Science 192 (2002) 1±25

integration in small time steps. The ®rst method is known as null-collision technique. It requires the addition of the null-collision cross-section to the set of cross-sections so that the total collision frequency (summed over all processes) is constant. Null-collision process does not affect the properties of electrons in any way [73]. However, the null-collision technique does not have an advantage for time varying ®elds as the procedure has to be stopped many times during one period in order to change the electric ®eld. Thus, we use the integration technique where the time steps are determined by the minimum of the three relevant time constants (mean collision time, period of the ®eld and cyclotron period for E  B ®elds) divided by a large number (20±100). Varying the number used to divide the time constant gives a test of the convergence of the technique. A large number of electrons, typically 103 ±106 , is followed over small time steps. Period of the electric ®eld is always divided by 100, and these moments are used to sample the transport properties. In case attachment and ionization are present, we increase (from the ensemble of remaining electrons) or decrease, respectively, the number of electrons as certain limits are reached. This is equivalent to adding a constant collision frequency±zero energy loss ionization and attachment and separate studies without changing externally the number of electrons proved that the procedure does not change any of the properties of the ensemble. The formulae used to sample transport coef®cients were taken from White et al. [39] and Nolan et al. [40] and allow determination of both bulk and ¯ux properties and correct representation of non-conservative processes [74]. In addition, we may also determine the EEDF and electron velocity distribution function (EVDF). Here, we use notation that the axis along the electric ®eld is denoted by E, along the magnetic ®eld by B and perpendicular to both by E  B. The electric ®eld is in the direction of x-axis, while the magnetic ®eld is along the y-axis. All calculations were performed for zero gas temperature. The bulk transport coef®cient may be determined from the mean position of the electron swarm. The number changing reaction rate is de®ned by o…0† ˆ

aˆ

d … ln N†; dt

(2)

5

the drift velocity by o…1† ˆ o ˆ

d hri; dt

(3)

and the diffusion tensor by 1 d 2 hr i; 2! dt where r ˆ r hri. The equation   d dxi oi ˆ hxi i ˆ ˆ hvi i dt dt o…2† ˆ D ˆ

(4)

(5)

does not hold for the non-conservative situation and then the right-hand side of the equation de®nes the ¯ux drift velocity. The ¯ux components of the diffusion tensor are given by DEB ˆ hzvz i

hzihvz i;

(6)

DB ˆ hyvy i; DE ˆ hxvx i

(7) hxihvx i;

1 d…hxzi hxihzi† 2 dt 1 ˆ …hxvz i ‡ hzvx i hxihvz i 2

(8)

DH ˆ

hzihvx i†;

(9)

where DH is the off-diagonal term also known as ``Hall'' diffusion. The rate coef®cients (including the ionization) may be determined by counting the appropriate events and normalizing the count by the time step and number of electrons. The ionization rate coef®cient may be determined both by this technique and by following the spatial or temporal growth of the number of electrons and as a test of the sampling procedures we performed comparisons. In all cases both procedures to determine the ionization rate agreed very well. The basic set of cross-sections that was used in most calculations is the ramp model devised by Reid [56]: sel …e† ˆ 6  10 20 m2 …elastic cross-section†; sinel …e† ˆ 10  …e 0:2† …inelastic cross-section†: (10) The model representing non-conservative collisions was devised from an ionization model of Lucas and

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Z.Lj. Petrovic et al. / Applied Surface Science 192 (2002) 1±25

Saelee [75] and attachment model of Robson and Ness [76]:

where gk is a tensor of rank k, and denotes a k-fold scalar product. Tensorial functions gk are normalized

sel …e† ˆ 4e1=2  10 20 m2 …elastic cross-section†; 0:1…1 F†…e 15:6†  10 20 m2 ; e > 15:6 eV …inelastic cross-section†; sinel …e† ˆ e < 15:6 eV;  0; 0:1F…e 15:6†  10 20 m2 ; e > 15:6 eV …ionization cross-section†; si …e† ˆ 0; e < 15:6 eV; m p 20 2 ˆ 10 3 : sa …e† ˆ ae  10 m …attachment cross-section†; M We haveperformed calculationsforF ˆ 0 (conservative case), 0.5 and 1 and for attachment model with different values of a and p [41]. In all cases excellent agreement was found with other sources of data for the same models. For real gases appropriate data for the cross-sections were taken from the literature. 3. Direct numerical procedure for solving the Boltzmann equation One of the ways to analyze rf electron swarm transport is to solve the Boltzmann equation. We have applied the density gradient expansion method in order to understand the time dependent phenomena of electron swarms in rf ®elds. Time variation of EVDF, g…r; v; t†, may be obtained from the Boltzmann equation: @ @ @ g…r; v; t† ‡ v
g…r; v; t† ‡ a…t†
g…r; v; t† @t @r @v ˆ J‰g…r; v; t†Š; (12) where r; v and t are the electron position, velocity and time, respectively. J‰gŠ is the collision term, a…t† is equal to eE…t†=m, where e and m are the electron charge and mass, respectively. The external ®eld E…t† is assumed to be uniform in space and varying in time, and its waveform is treated in this paper as p E…t† ˆ 2ER cos …ot† ˆ E0 cos …ot†: (13) Under an rf ®eld, g…r; v; t† may be expanded in terms of spatial gradients of the electron number density n…r; t† as [2±4]:   1 X @ k k g…r; v; t† ˆ g …v; t† n…r; t†; (14) @r kˆ0

according to  Z 1 …k ˆ 0†; k g …v; t† dv ˆ 0 …k ˆ 6 0†:

(11)

(15)

Continuity equation may also be expressed in terms of density gradients with time dependent transport coef®cients xk …t† as   X @ @ k k n…r; t† x …t† n…r; t† ˆ 0; (16) @t @r k o0 …t†; x1 …t† and x2 …t† give the effective ionization rate, drift velocity and diffusion tensor, respectively. In general, these transport coef®cients must have a space dependence, however, we now consider them as the averaged value in real space, i.e., o0 …t† means the total gain/loss of electron number density, o1 …t† the velocity of the mass center of the swarm, o2 …t† the time derivative of dispersion from the mass center. Considering the electron continuity equation (16), the substitution of Eq. (14) into the Boltzmann equation (12) leads a hierarchy of equations. For k ˆ 0; 1, equations may be written as [43,44] @ 0 @ g …v; t† ‡ a…t†
g0 …v; t† ‡ o0 g0 …v; t† @t @v J‰g0 …v; t†Š ˆ 0; @ 1 @ g …v; t†‡ a…t†
g1 …v; t† ‡ o0 g1 …v; t† @t @v ˆ vg0 …v; t† x1 …t†g0 …v; t†:

(17) J‰g1 …v; t†Š (18)

Note that these equations have the same form as Eq. (28a) in [2]. Considering the axial symmetry in the ®eld direction, which is parallel to the z-axis, we decompose the vector function g1 …v; t† to two

Z.Lj. Petrovic et al. / Applied Surface Science 192 (2002) 1±25

components in solving Eq. (18) as g1L …v; t† ˆ jg1 …v; t†j cos y;

(19)

g1T …v; t†

(20)

1

ˆ jg …v; t†j sin y;

where y denotes the polar angle from the vz -axis, and g1L and g1T are the components parallel and perpendicular to the ®eld, respectively. The effective ionization rate o0 and the drift velocity to vz direction vdz …t† are given by Z 0 o …t† ˆ N ‰Qi …v† Qa …v†Švg0 …v; t† dv; (21) x1z …t†

vdz …t† ˆ ˆ Z ‡ N ‰Qi …v†

Z

7

be de®ned under the condition of no ionization and attachment as x2 …t† ˆ

Z

2

DT …t† 6 vg1 …v; t† dv ˆ 4 0

3

0 DT …t†

0 0

0

DL …t†

0

7 5: (23)

We apply DNP of the Boltzmann equation to solve Eqs. (17) and (18) [43,44]. 4. Test of the numerical procedure

vg0 …v; t† Qa …v†Švg1L …v; t† dv;

(22)

where N is the gas density, Qi …v† and Qa …v† are the collision cross-sections for ionization and attachment, respectively, and v denotes jvj. Since there only exists the external ®eld, the other component of drift velocity must be zero. In addition, the longitudinal and transverse diffusion coef®cients DL …t† and DT …t† may be obtained from the diffusion tensor x2 …t† which is to

The codes have been cross-checked [41,44] and also tested against the results of the group from James Cook University [39,40]. For that purpose, we have used the following standard tests for swarm conditions:  Reid's ramp model for dc E and E  B fields.  Ionization (Lucas±Saelee) and attachment (Robson± Ness) models for non-conservative transport in dc E field.

Table 1 Comparison of results obtained for Reid's ramp model obtained by solution to Boltzmann equation [40] (B), MCS [40] (MC) and the present MCS code. Differences between the present results and two other sets of data are given as percentages Ref. [40] (MC) d (MC) (%) Ref. [40] (B) d (B) (%) Present Ref. [40] (MC) d (MC) (%) Ref. [40] (B) d (B) (%) Present Ref. [40] (MC) d (MC) (%) Ref. [40] (B) d (B) (%) Present Ref. [40] (MC) d (MC) (%) Ref. [40] (B) d (B) (%) Present

12

12

24

24

0

0.2693 ‡0.4 0.2689 ‡0.5 0.2703

0.0005 x:x 0 x:x 0.0017

500

0.1124 ‡0.8 0.1123 ‡0.9 0.1133

0.4161 ‡0.3 0.4154 ‡0.4 0.4170

2.3180 0.0 2.3180 0.0 2.3180

0.02412 ‡0.1 0.02405 ‡0.4 0.02415

0

0.4058 ‡1.3 0.4079 ‡0.8 0.4113

0.0009 x:x 0. x:x 0.016

8.878 0.8 8.886 0.9 8.804

0.31944 ‡1.0 0.3192 ‡1.0 0.3228

5.686 1.1 5.688 1.1 5.624

5.509 0.1 5.516 0.2 5.506

200

6.833 ‡0.01 6.838 0.06 6.834

1.133 ‡0.8 1.135 ‡0.6 1.142

1.138 ‡0.2 1.135 ‡0.4 1.140

0.5816 2.2 0.5688 ‡0.04 0.5690

0.065 x:x 0 x:x 0.0022

1.006 1.0 0.997 0.1 0.996

0.03647 ‡1.0 0.03694 0.3 0.03683

0.00630 ‡0.8 0.00674 0.8 0.00642

1.138 0.6 1.134 0.3 1.131

1.141 0.9 1.134 0.3 1.131

0.4684 3.0 0.4609 1.4 0.4546

0.6299 ‡4.6 0.6738 2.1 0.6601

1.184 0.3 1.182 0.2 1.180

0.4463 ‡5.6 0.4669 1.0 0.4714

0.071 x:x 0 x:x 0.001 0.458 ‡17 0.5223 ‡2.8 0.5370

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Table 2 Comparison of results obtained for non-conservative ionization model …F ˆ 1† obtained by solution to Boltzmann equation [77] (B), MCS [39] (MC), and the present MCS code (P). Differences between the present results and two other sets of data are given as percentages e (eV)

Ref. [39] (MC) Ref. [77] (B) P d (%)

4.97 4.97 4.97 0.0

v (104 m s 1 )

DT (105 m2 s 1 )

DL

Flux

Flux

Flux

7.32 7.33 7.30 0.3

Bulk 9.47 9.48 9.51 ‡0.4

 Relaxation of transport coefficients for ionization model.  Quasi-stationary (relaxed) transport coefficients for sinusoidal E field for ionization model (attachment model and Reid's model).  Anomalous longitudinal diffusion for Reid's model in sinusoidal and step function E fields. Two examples of results are shown for dc ®elds (see Tables 1 and 2) and some of the other examples have been shown later in this paper. In general, there is an excellent agreement between all the techniques employed in this paper and other available results, most notably those from James Cook University. As the agreement extends to time varying ®elds, E  B ®elds and non-conservative transport, one may have high con®dence in both DNP and MCS results if the procedures are executed properly. The code has also been applied to gases such as N2 , SF6, Ar and O2 and many more.

Bulk

2.43 2.43 2.42 0.4

2.72 2.72 2.72 0.0

2.38 2.38 2.37 0.4

a (103 s 1 ) Bulk 2.94 2.93 2.93 0.4

2.42 2.42 2.43 ‡0.4

comparisons with kinetic theories in order to understand the basic processes. In Fig. 1, we show the temporal relaxation of an electron swarm properties (drift velocity and mean energy) for Reid's model gas initiated with two different Maxwellian distributions one with mean energy of 0.1 eV and the other with 0.6 eV. While one may make a more detailed analysis of the overshoot and the transition, it is clear that the ®nal conditions are independent of the initial distribution as requested by the simulation. As DNP technique may be more ef®cient in producing the required data, one may bene®t from making the comparisons with the MCS data. This is required since DNP may be subject to numerical errors due to mesh and step size, while MCS is limited by the statistical properties of the results. In Fig. 2, we show comparisons between the mean energies obtained by the two techniques for Reid's model at three different ®eld frequencies. There is excellent agreement

5. Relaxation of swarm properties Initial properties of electrons gradually relax to the quasi-stationary ®nal state. While it is possible to start quite close to the ®nal distribution and thus speed up the simulation, the actual plasma models, such as the relaxation continuum theory (RCT) [61] may require knowledge of the relaxation times. Thus it may be an advantage to extend the relaxation period in order to calculate the data required for plasma modeling. In general, however, one may state that in all cases MCS passes through a period of relaxation and while these data are usually not recorded in an aim to sample only the relaxed solution, the transition itself may be of interest both as input data for some models or for

Fig. 1. Temporal relaxation of transport properties of electrons in Reid's model gas at 1 Torr and for dc E=N ˆ 12 Td (B=N ˆ 10 Hx). Initial distributions are Maxwellians with an energy of (a) 0.1 eV (dashed line); (b) 0.6 eV (solid line) [78].

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9

Fig. 2. Temporal relaxation of the mean energy of electrons in Reid's model gas at 1 Torr and for rf E0 =N ˆ 14:14 Td. Initial distributions is Maxwellian with an energy of 0.5 eV. The results of DNP are shown as solid points and the MCS results as a line. Data were obtained for (a) 200 MHz; (b) 50 MHz; (c) 10 MHz [20].

between the results during the transition and in quasistationary regime giving a con®rmation to the accuracy of both techniques. 5.1. Relaxation of energy distribution functions In particular, it is interesting to discuss the features in EEDF as the swarm develops. Similar studies have been made extensively by Loffhagen and Winkler [30], mostly under conditions that the dominant energy losses are through transitions to electronically excited states. We have covered a range of conditions by using realistic or model gases [20,78,79] with an aim to study how quite different initial properties relax with time. Thus, we selected mostly monoenergetic beams and Maxwellian distribution functions with different initial angular distributions [20]. In Fig. 3(a), we show the relaxation of the EEDF for electron swarm in rf ®eld at 200 MHz. The gas is Reid's ramp model with one inelastic process with 0.2 eV loss and the initial energy distribution is an isotropic monoenergetic beam with the energy of 0.5 eV. Almost immediately the initial beam forms two other beams, one at 0.3 eV and one at 0.1 eV, which are due to electrons that had one or two inelastic losses. The three beams expand due to the isotropic distribution and merge into ®nal EEDF that oscillates

Fig. 3. Temporal relaxation of the electron EEDF obtained by MCS at 1 Torr in Reid's ramp model with a monoenergetic, isotropic initial distribution of the energy of 0.5 eV. The data were obtained in rf ®elds at (a) 200 MHz [79]; (b) 1 GHz [20].

weakly due to the external rf ®eld. It is interesting to note that at the phases of p and 2p the peaks due to the initial distribution are recovered. While the ®rst one may be expected from the EEDF in the ®rst half of the period, the EEDF in the second half appears to be fully relaxed and yet it still bears a mark of the initial distribution. The formation of the beams may be followed better if the frequency is increased as in Fig. 3(b) where 1 GHz rf ®eld is used. The merging of the beams is diminished due to the limited time and the beams spread and contract independently over the entire period. It was also shown that the addition of a magnetic ®eld increases mixing of the beams while the ultimate mean energy is somewhat reduced [79].

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5.2. Negative absolute conductivity in afterglow One special example of kinetic phenomena has been discussed recently in the literature. That is the quasi-stationary absolute negative conductivity. Transient negative conductivity was observed experimentally in [80] and explained theoretically in [81]. In principle, during the thermalization of electrons with an initial mean energy in the region of rapidly rising cross-section, such as the region from the Ramsauer± Townsend minimum (RTM) to the higher energies, electrons accelerated by the ®eld have an increasing probability of collision and a chance to reverse the direction of their motion. On the other hand, decelerated electrons have a decreasing probability of collisions so there is a chance that most electrons will have motion against the ®eld acceleration. The process ceases when most electrons slow down suf®ciently so the ®eld may accelerate them. Recently, however, Napartovich and coworkers have proposed to use a mixture of Ar with an electronegative gas (NF3 or F2 ) under afterglow conditions, where plasma would decay in a very weak electric ®eld insuf®cient to maintain it. It was observed that a quasi-stationary negative mobility may be achieved [82,83]. In addition to formal explanations, a basic physical explanation was given [83], explaining the phenomenon by the fact that low energy attachment removes the thermalized electrons. Thus one may have continuously a situation such as found in the transient negative mobility, whereby the price of quasi-stationary negative mobility is paid by a rapidly decaying plasma. In Fig. 4, we show the temporal development of the electron drift velocity obtained by a two term solution to the Boltzmann equation and by MCS. A small disagreement between the two sets of results is due to the fact that in MCS zero gas temperature was assumed, while the Boltzmann equation solution was calculated for the room temperature gas [84]. One should also notice that the negative value is observed only for the ¯ux drift velocity which cannot be measured experimentally, but still it is important as it determines the ¯ux of electrons. The bulk drift velocity is positive. Formally, we may explain the phenomenon as attachment heating, or non-equilibrium hole burning in the low energy part of the EEDF. One may argue

Fig. 4. Temporal development of the electron drift velocity in a decaying F2 (0.5%)/Ar mixture plasma. Calculations were made for dc E=N ˆ 0:1 Td, N ˆ 0:26  1026 cm 3 [83].

that negative mobility is contradiction with the laws of thermodynamics. However, the non-equilibrium situation is maintained by the source of energy producing the plasma and the whole system (plasma) decays rapidly as a result. However, it is a virtue of nonequilibrium plasmas that the kinetics of electrons may be disconnected from the rest of the system which may even lead to some applications. The role of electron attachment in this example is that of the Maxwell's daemon, as attachment selects and removes electrons with particular properties. The key mechanism for achieving the effect is the non-conservative nature of the electron attachment. In Fig. 5, we show how the mean energy relaxes under the in¯uence of nonconservative collisions. The non-conservative process yields a higher mean energy and an overshoot in relaxation, while the treatment of the attachment as a conservative process with attachment threshold of X eV yields a quite different transition and the ultimate state. In case when conservative treatment of attachment is used, drift velocity is positive at all times. One should bear in mind that in Fig. 4 we show two types of drift velocities. The ¯ux value (mean electron velocity) shows a negative value and indicates that indeed electrons effectively move in the direction against the acceleration by the electric ®eld. On the other hand, the bulk value, the velocity of the center of mass of electron swarm has a positive value, i.e. the motion is as one would normally expect. This means that there must be a spatial separation of lower and

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Fig. 5. Relaxation of the mean energy of electrons [85] from the initial Maxwellian distribution with 1 eV mean energy (in gas F2 (0.5%)/Ar): (a) attachment is treated as a conservative process; (b) attachment is treated as a non-conservative process. Calculations were made for rf ®eld of frequency of 200 MHz for gas number density of N ˆ 2:4  1019 cm 3 and for E=N ˆ 0:141 Td.

higher energy electrons, and a wave of attachment ``eats'' the spatial distribution of the electron swarm giving a motion of the center of mass along the ®eld direction. At the same time, majority of the remaining electrons have their velocities directed against the acceleration by the electric ®eld. Practical implementation of this effect would require a means to sustain the plasma, i.e. to compensate for the losses. The obvious approach would be to add the third component with a very low threshold for ionization (such as cesium vapor). Another approach was to use photo-excited gas mixture (again containing alkaline vapors), and this approach has already shown some results [86]. Finally, we shall discuss using the rf ®elds for this purpose, later in this paper. 6. Unique characteristics of electron transport in rf ®elds Once the EEDF has relaxed to quasi-stationary state, we may begin sampling the properties of electron swarms in rf ®elds. The results may be presented over a single period but may be sampled over many periods. The results that were obtained by MCS for fully relaxed swarm properties have been tested by

Fig. 6. Time resolved mean energies for non-conservative Lucas± Saelee model at E=N ˆ 10 cos ot Td, p ˆ 1 Torr [34,41] obtained by MCS and from solution to the time dependent Boltzmann equation.

comparisons with DNP data and data obtained from multi-term solution to the Boltzmann equation [41]. One example of comparisons of MCS results with numerical solution to time dependent Boltzmann equation [34] is shown in Fig. 6. The good agreement between MCS results and the Boltzmann equation solution has bearing on the still open question of the applicability of hydrodynamic approximation in time varying ®elds which has been answered satisfactorily only for some special cases [12,37]. Similar results with similar agreement were obtained for drift velocities and diffusion coef®cients (both ¯ux and bulk). Results for very rough approximations such as eE mn for drift velocity, and

vd ˆ

Dˆ

vd kB T 2vd hei ˆ ; eE 3eE

(24)

(25)

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the basic (non-corrected) Einstein relation for diffusion tensor are applied very often in plasma modeling. A relatively good agreement between these approximations and the MCS ¯ux data [41] may be just a coincidence in the case shown there and should not be regarded as a general trend. It, however, indicates that these approximations may be used to represent only the ¯ux data with accuracy that is insuf®cient for swarm measurements but which may be satisfactory for plasma modeling. 6.1. Basic properties of electron transport in rf ®elds In general, the characteristics of electron swarms in rf ®elds may be explained by considering the instantaneous ®eld approximation and adding temporal relaxation of momentum and energy [35]. Thus the ®rst and the very basic special features occur at rf frequencies for standard range of pressures, when energy relaxation fails to take place on the time scale of ®eld change. Another basic feature is the delay that may occur between the actual change in the ®eld and the change in time pro®le of sampled transport properties [44]. As frequencies become higher, failure to achieve momentum relaxation become apparent. Most importantly the effects of the limited times of relaxation become observable for some parts of the periods and also for some parts of the EEDF and the time pro®les of sampled properties may become quite complex. One example of the frequency dependence of swarm properties is given in Fig. 7. The mean energy is maintained as frequency of the external ®eld increases, while the dissociation rate decreases rapidly. This is caused by the fact that at high frequencies the high energy tail of the EEDF disappears due to inability of high energy electrons to follow the electric ®eld. 6.2. Negative differential conductivity in rf ®elds Negative differential conductivity (NDC) has been studied for some time in electron swarm physics [87,88], as it proved to be an interesting effect from the point of view of fundamental understanding and also a very dif®cult test of transport theories. In principle, the terminology used by swarm physicists is not exactly accurate, as conductivity in weakly

Fig. 7. Frequency dependence of mean energy and dissociation collision frequency in 1 Torr of CF4 [20] obtained by MCS at E0 =N ˆ 282 Td.

ionized gas involves a product of electron density and drift velocity. While it is possible to modify such conductivity by an attachment process that is strongly dependent on E=N, this aspect of NDC is trivial and we shall limit ourselves to the dependence of the drift velocity on E=N only. NDC is thus achieved when the drift velocity decreases with an increasing E=N. The basic explanation may be reached [89±91] on the basis of balance of momentum and energy relaxation and all the observed instances of NDC fall well within those explanations. One should bear in mind that it was sometimes erroneously assumed that the NDC is the necessary outcome of the sudden increase of drift velocity [93] due to the thermalizing effect of inelastic collisions. However, it is by no means necessary that a decrease of drift velocity will occur under those circumstances and special features of both elastic and inelastic cross-sections are required, in principle, to achieve the basic NDC. Once the conditions that favor NDC are set, then it is possible to induce it by other effects such as:  non-conservative nature of transport leading to modified definition of transport coefficients [91];  effect of non-conservative collisions on EEDF [91];  electron±electron collisions [92] and  anisotropic scattering [93]. At this point, however, we discuss mainly the features in time resolved drift velocities under conditions that lead to NDC in dc case. On the basis of a

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Fig. 8. Time dependence of drift velocity during one period for different rf frequencies for electrons at E=N ˆ 50 Td and 1 Torr in CH4 [95] by MCS.

simple extrapolation from the dc E=N dependence of the drift velocity, one may expect symmetric nonsinusoidal pro®les of drift velocity with two or even three peaks. The actual calculations of drift velocities con®rm such expectation [37,94,95]. However, with increasing frequency the second peak in the drift velocity disappears and the ®rst peak is skewed and also disappears (but at a much higher frequency) as the pro®le assumes sinusoidal shape as shown in Fig. 8. The signi®cance of the temporal drift velocity pro®les shown in Fig. 8 is that convolution between the electric ®eld and the drift velocity, which is equivalent to the power deposited into electron swarms by external ®eld, strongly depends on the shape of the drift velocity which may be quite different from the expected sinusoidal pro®le. One should also bear in mind that the early studies employing limited number of terms in the solution of the Boltzmann equation did not show time dependent detailed structure in NDC [25]. The reason is that the limited number of Fourier components that were employed could not represent the rapidly changing pro®le of time dependent NDC. On the other hand, the effect of NDC was imprinted on the effective maximum value of the drift velocity harmonics, which showed NDC like behavior as a function of the peak value of E…t†=N. In fact, the behavior is exactly the opposite, the peak value of drift velocity is constant while the shape of the peak changes with frequency and E…t†=N [95]. This shows the importance of performing calculations of highest degree of accuracy in studies of transient kinetic phenomena, and the MCS results

13

were con®rmed independently by numerical solutions of the Boltzmann equation [37]. It is worth noting also that the very nature of NDC both in its dc and rf forms is revealed best by studying the temporal development of the velocity distribution functions such as those shown in [95]. There, one may observe the role of processes controlling the energy relaxation and the momentum relaxation ®rst in achieving an increased drift velocity due to anisotropy induced by energy losses and the role of momentum relaxation in achieving a decreasing drift velocity with an increasing electric ®eld. Having in mind such results would help avoid repetitions and misconceptions. 6.3. Negative absolute conductivity in rf ®elds As mentioned earlier, the effect of the negative absolute mobility occurs in rapidly decaying afterglow plasma. Any practical application would require maintenance of plasma. We have discussed possibilities to use low ionization threshold components of the mixture or to have photoexcited gas with the effect of superelastic collisions on the EEDF [86]. Another option would be to use time varying (rf) electric ®eld. We have performed simulations for the same mixture that was used for the afterglow calculations, i.e. F2 (0.5%)/Ar. This time, however, the rf ®eld was used with peak value of the reduced electric ®eld of E=N ˆ 0:141 Td. Before proceeding to analyze the results, one should bear in mind that the ¯ux and bulk values of drift velocities have, respectively, negative and positive values in the standard test example showing the effect of negative mobility. In pure Ar, the two types of drift velocities coincide since there are no non-conservative processes [85]. However, when we make calculations for the mixture, the ¯ux drift velocity is exactly in the opposite phase to the electric ®eld, while the bulk property is mostly in phase with the ®eld (see Fig. 9(a)). If we treat the attachment in the mixture as an energy loss process, the two drift velocities coincide and have positive values (see Fig. 9(b)). It is interesting to note that the ¯ux drift velocity is almost exactly in the opposite phase to the electric ®eld while there is a signi®cant delay for the bulk property in the non-conservative case with attachment. That is due to the fact that it takes more time to establish the front of attachment which makes a hole in the low energy part of the

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been observed independently by groups at James Cook University [36] and at Keio University [44]. The detailed physical mechanism of the anomalous diffusion is in Ref. [44]. These two groups mostly studied model systems and the basic phenomenology so the

Fig. 9. The temporal pro®le of drift velocity in F2 (0.5%)/Ar mixture in rf ®elds with E0 =N ˆ 0:141 Td at 200 MHz. The results of MCS are shown for treatment of attachment as (a) nonconservative; (b) conservative process [85].

spatial pro®le, when the direction of ®eld changes sign. On the other hand, the relaxation of the direction of motion is much faster and ¯ux property changes sign very rapidly. Thus we may conclude that negative absolute mobility occurs also in rf ®elds, and it may be of relevance for certain regions of rf plasmas, where effective ®eld may be low and at the same time typical mixtures used in processing are those that show the negative mobility effect. 6.4. Anomalous diffusion in rf ®elds Another well-documented feature of rf transport of electrons is the anomalous behavior of the longitudinal component of the diffusion tensor. This effect has

Fig. 10. Time dependence of the transverse and longitudinal components of the diffusion tensor for electrons in Reid's model gas (E0 =N ˆ 14:14 cos ot Td). The data obtained by DNP are shown as lines, the data obtained by MCS by points for different frequencies [44]. (a) Instantaneous ®eld approximation; (b) 10 MHz; (c) 50 MHz.

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effect was also shown to occur for a number of realistic gases [96]. In Fig. 10, we show the temporal pro®le of transverse and longitudinal diffusion coef®cients for Reid's model gas. The instantaneous ®eld approximation gives minima equal to thermal values for both components of the diffusion tensor. Transverse component follows the expected behavior with the addition of the limited time of energy and momentum relaxation, which does not allow full relaxation of the diffusion coef®cient to the thermal value and the minimum occurs with some phase delay. However, the longitudinal component shows two unexpected aspects. When electric ®eld changes sign, a peak, rather than minimum occurs in the longitudinal diffusion coef®cient transients. Sometimes that peak even exceeds the value of the transverse diffusion coef®cient which is also unexpected. The importance of the effect of anomalous diffusion is that it is a kinetic effect occurring for rapidly changing ®elds that cannot be predicted on the basis of any effect for dc ®elds. At the same time the effect has a clear relevance in heating of rf plasmas, and is neglected by the present day models of an rf plasma. 7. Electron transport in crossed E  B ®elds Following electron trajectories in E  B ®elds is not trivial, as most of the basic numerical procedures based on ®nite differences may lead to departure from the real trajectories. For example, if one wants to follow motion of electrons in magnetic ®eld only, most of the simple numerical procedures will yield a trajectory that does not close on itself. In case when collisional frequency is greater than the cyclotron frequency, such techniques yield good results assuming that proper testing with varying time steps is performed. However, in order to maintain proper trajectory and, at the same time, to have accurate determination of the velocity in a more general set of circumstances, one has to apply the so-called Boris-rotation in its basic or modi®ed forms [41,67]. We have made through testing of the numerical procedure for calculating trajectories in E  B ®elds, even in cases when cyclotron frequency exceeds very much the collisional frequency [41]. Very few MCSs exist even for dc E  B ®elds [40,97,98].

15

Electron transport in dc E  B (crossed electric and magnetic ®elds) usually at 90 [1,77,98] is characterized by the new properties, the perpendicular drift velocity vEB (here, we adopt the notation for the axes: E, along the electric ®eld; B, along the magnetic ®eld and E  B perpendicular to the plane de®ned by electric and magnetic ®elds), reduction of the mean energy as the value of the normalized magnetic ®eld …B=N† increases (though this trend may be reversed under some special circumstances [99]), and freezing of electron motion perpendicular to the magnetic ®eld. A very special feature of electron diffusion was discussed by Raspopovic et al. [100]. They have observed that the NDEB component of the diffusion tensor at low B=N behaves like the B component. However, as B=N increases it makes a very rapid transition and its values begin to coincide with NDE . In other words, as shown in Fig. 11 the acceleration in the E  B direction acts in a similar way as the electric ®eld in inducing the anisotropy of the diffusion tensor. Inductively coupled plasmas [101] are very frequently used for plasma processing. In addition, magnetrons and magnetically enhanced reactive ion etching are becoming increasingly important, and in all these cases values of B=N are quite high due to operation at low pressures. Typical peak values exceed few thousand Huxley's (Huxley, 1 Hx ˆ 10 27 T m3 ). However, in majority of models (including mostly ¯uid and hybrid models) magnetic ®elds are not taken into account. This means that in determination of the transport data, neither dc nor rf magnetic ®elds are considered. We shall discuss some aspects of rf

Fig. 11. B=N dependence of the component of the diffusion tensor for a ®xed dc E=N ˆ 12 Td and for Reid's ramp model.

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transport in time varying electric and magnetic ®elds which warrant implementation of kinetic theory or MCS for time varying E  B ®elds in plasma models. 7.1. Electron transport in time varying E  B ®elds In this section, we shall present some of the results obtained by MCS for time varying E…t†  B…t† ®elds over the past 5 years [78,100,102]. It is worth noting that, recently, White and Robson [103] have performed similar calculations based on numerical solutions to the Boltzmann equation and in all cases the preliminary results give excellent qualitative and quantitative agreement with the MCS data. Anomalous transport in ICP, in¯uenced by both E…t† and B…t†, was experimentally pointed out [104]. The following conditions are used (unless stated otherwise): Reid's ramp model gas at 1 Torr with E…t†= N ˆ E0 sin …ot†=N ˆ 14:14 Td, frequency 50 MHz and the phase between magnetic and electric ®elds is 90 (B…t†=N ˆ B0 cos …ot†). The mean electron energy seems to be unaffected by the magnetic ®elds up to 100 Hx, and at higher B0 =N the mean energy drops down gradually (see Fig. 12). The reduction of mean energy is signi®cant at 500 Hx and the shape becomes triangular (rather than sinusoidal). At higher B0 =N even some small, but observable oscillations become superimposed [78]. The longitudinal drift velocity …vE † does not change its peak value as magnetic ®eld increases (see Fig. 13). However the shape of the time dependence becomes more and more triangular, with a slow increase and a

fast decrease (during the period when magnetic ®eld increases), leading even to oscillations of the drift velocity. One should bear in mind that these oscillations, which occur when cyclotron resonance exceeds collisional frequency [78,102] are the imprints of the individual cyclotron motion on the collective averaged property, the drift velocity in this case. Special characteristic of electron transport in E  B ®elds is the perpendicular drift velocity …vEB †, which should be modulated at twice the ®eld frequency as it follows the product E…t†  B…t†. A very speci®c and unexpected feature of vEB is the asymmetry or in other words (as can be seen from Fig. 14) the mean value of this drift velocity is not zero. At the same time it still bears the undulations due to cyclotron motion at the highest B0 =N.

Fig. 12. Mean energy of electron swarm in E…t†  B…t† ®eld under standard conditions, for different values of B0 =N [78,102].

Fig. 14. Perpendicular drift velocity …vEB † of electron swarm in E…t†  B…t† ®eld under standard conditions for different values of B0 =N [78,102].

Fig. 13. Longitudinal drift velocity …vE † of electron swarm in E…t†  B…t† ®eld under standard conditions for different values of B0 =N [78,102].

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Fig. 15. Longitudinal diffusion coef®cient …NDE † of electron swarm in E…t†  B…t† ®eld under standard conditions for different values of B0 =N [78,102].

Transverse component of the diffusion tensor, along the magnetic ®eld NDB is modulated very little with the electric ®eld and the increasing magnetic ®eld, makes a change that is hardly observable and affects only the peak value [78,102]. The longitudinal component NDE shows all the signs of anomalous diffusion at zero magnetic ®eld and this feature is only increased as B0 =N increases. However, at the highest values of B0 =N the phase of the maximum diffusion is shifted to the phase of zero magnetic ®eld, while diffusion is practically zero at all other times with some small modulations due to cyclotron motion (Fig. 15). The transverse diffusion along the (E  B)-axis …NDEB † follows, at zero and small values of B0 =N, the same dependence as NDB . When magnetic ®eld increases to 50 Hx the phase of the NDEB changes by p=2 and for higher magnetic ®elds this component of the diffusion tensor has all the features of anomalous diffusion, just like NDE (see Fig. 16) [78,100]. The behavior of other transport coef®cients, such as Hall diffusion, collision rates and features of the EEDF have been established both for model (Reid) and realistic (Ar, CF4, SiH4 , Si2 H6 , etc.) gases [105,106]. In addition, analysis of non-conservative transport has been completed [107].

17

Fig. 16. Transverse diffusion coef®cient …NDEB † of electron swarm in E…t†  B…t† ®eld under standard conditions for different values of B0 =N [78,102].

peaks when the electric ®eld passes through zero. Thus, the maximum effect of the acceleration by the electric ®eld occurs when there is no or little in¯uence by the magnetic ®eld. As one may ®nd any phase angle between p=2 and zero in rf plasmas it would be interesting to check how the phase angle affects the transport properties. When phase is zero one may expect even smaller mean energy and a stronger effect of magnetic ®eld on drift velocities and diffusion coef®cients. In Fig. 17, we show the effect of the variable phase on the longitudinal and in Fig. 18 on the perpendicular component of drift velocity (one should note that in these ®gures we keep the position of the electric

7.2. The effect of the phase angle on rf transport One particularly interesting feature of transport arises from the phase angle between magnetic and electric ®elds. When phase is p=2, the magnetic ®eld

Fig. 17. Longitudinal drift velocity …vE † of electron swarm in E…t†  B…t† ®eld under standard conditions, for B0 =N ˆ 200 Hx as a function of the phase between the two ®elds [78,102].

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Fig. 18. Perpendicular drift velocity …vEB † of electron swarm in E…t†  B…t† ®eld under standard conditions, for B0 =N ˆ 200 Hx as a function of the phase between the two ®elds [78,102].

®eld ®xed). The drift velocity vE is reduced as phases between two ®elds are reduced and for p=6 and 0, a minimum in the temporal dependence of drift velocity occurs, which is similar to that caused by the NDC. The perpendicular drift velocity shows an even more interesting behavior. Its time dependence is not affected very much by the phase between the ®elds, but the mean value is. The mean value changes from slightly negative to a positive value which is so large that for the zero phase the drift velocity vEB is positive at all times (see Fig. 18). 8. Applications of results for electron transport data Development of the physics of ionized gases was always associated with and driven by some of the applications. The discovery of electron in the late 19th century by J.J. Thompson (and to a large degree J. Townsend) was preceded and followed by the development of techniques from the electrochemistry in order to study the properties of charges in gas phase. These techniques gave birth to early swarm studies with the goal to determine transport properties of charged particles in very weakly ionized gases. However, a major application emerged from Townsend's attempts to measure the elementary charge of electron and that is the development of techniques to detect elementary particles. This application remained as one

of the key motivations for development of gas discharge physics, and starting from Wilson's cloud chamber (a direct derivative from Townsend's experiment), over Geiger±MuÈller counters to drift chambers and other gas phase detectors, numerous advances were made leading even to Nobel prizes in physics. Unfortunately, the two communities have lost their contact over the years and many parallel studies were made and also the elementary particle detector developments did not bene®t from the advances in the physics of swarms so it is welcome to change that recently there has been a strong joint effort in both directions [108,109]. During the 1930s up to 1950s of the 20th century perhaps the main motivation for the development (both fundamental and applied) were the studies of ionosphere and atmospheric electricity [110]. This motivated an interest in rf transport theory that was so important in the early developments such as those of Holstein and coworkers [17,19]. The interest that appeared to have vanished in the past 30 years seems to have returned with an increased interest in elves, sprites and other products of above the cloud thunderstorms [111] and in problems associated with charging of satellites [112]. These applications may have potential to extend the lifetime and pour new interest in studies of swarms and physics of ionized gases in general. The biggest source of motivation and funding in 1960s and 1970s of the 20th century came from the need to develop and model gas lasers. At the same time, computers have become available which allowed numerical solution to the Boltzmann equation for realistic cases, so it became possible to make comparisons of the cross-sections with those obtained by binary collision experiments or theoretical techniques. In mid-1980s it became evident that further advances in plasma technologies for integrated circuit production, plasma etching in particular, cannot be made without:  detailed understanding of sustaining mechanisms [61];  understanding of plasma chemical reactions [113];  understanding of surface reactions and processes [114];  development of predictive and realistic multidimensional models [115];

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 improvement of diagnostic techniques and verification of models [116];  a large body of data which are not available [117];  application of the diagnostics and models for in situ control and computer aided design of the new plasma tools [118]. In parallel with plasma processing in microelectronics, a number of other applications were developed and all of them bene®ted from the major developments in non-equilibrium plasma physics of that period. These applications include: plasma displays [115]; plasma propulsion; surface treatment of polymers, textile and even organic matter; novel light sources; sources of ion, electron and radical beams and many others. While one may not be able to point out a single major application, which would drive the development of the whole area in the near future in the same way as development of gas lasers and plasma technologies in integrated circuit production did in the past (and present), one may single out development of techniques for detection and removal of atmospheric gas pollutants as a major application that will become increasingly important.

Fig. 19. Cross-section set for electron scattering on CF4 as proposed in [124].

8.1. Data for ¯uid and hybrid models

shown in Fig. 19 was modi®ed by rescaling the dissociation cross-section in order to achieve a very good ®t of the ionization and attachment data [124] (see Fig. 20). At higher mean energies (5±20 eV), data for ionization and attachment are measured most accurately, while the data for drift velocities and

In the last 10 years, we witnessed a gradual disappearance of the swarm experiments and signi®cant reduction in production of the cross-section sets based on the transport data, the maintenance of the activity in simulations and improvement of swarm theories and yet one may claim that the word swarm has been used more often than ever before [119]. The primary reason is that the basic theories for modeling of collision dominated plasmas are based on the standard swarm theory with the addition of ®eld calculation, the data used in kinetic, PIC-MC and Monte Carlo models are the cross-sections that have to be normalized by applying the swarm techniques, and that the basic data for continuum models (¯uid and hybrid) require tabulations of electron transport data. While quite often the input data for plasma modeling are just compiled, it is essential to perform critical evaluation and swarm based normalization [120±122]. One such example is the analysis of the transport data and the cross-sections for electrons in CF4 and in CF4 /Ar mixtures [123,124]. The set of cross-sections

Fig. 20. Attachment and ionization rate coef®cients for electrons in CF4 calculated from the derived set of cross-sections. The results [124] are compared to the recommended data from the compilation by Christophorou et al. [123].

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For continuum models, the input data for electron kinetics are the collision rates (see Fig. 21) and for the PIC, kinetic and Monte Carlo (hybrid) codes these are the cross-sections. Additional information useful in establishing techniques to compensate for the nonlocal electron transport may be obtained from the calculated relaxation time constants for the energy and momentum (see Fig. 22). 9. Conclusion

Fig. 21. The inelastic collision rates in pure CF4 for dc ®elds [124].

characteristic energies (D=m, where m is the electron mobility) provide the basis for lower energies (0± 5 eV). Additional information that may be essential in order to establish the cross-sections and also use directly in models may be obtained from the excitation coef®cients, which are measured with reasonable accuracy and which may provide unique data for the excitation cross-sections [125].

Fig. 22. The time constants for energy …tene † and momentum …tm † relaxation for electrons in pure Ar (dotted line), pure CF4 (dashed line) and in the CF4 (5%)/Ar mixture (solid line) for dc ®elds.

The most useful conclusion that could arise from a paper such as this one would be to present some quantitative criteria on the effects of the kinetic phenomena that were discussed. That may not be possible at the moment in a general way, calculations have to be made for each set of conditions. In general one may claim that at very high frequencies effective ®eld approximation will be effective and at very small frequencies instantaneous ®eld approximation will give good results. However, for a very wide range of frequencies and pressures and also distances between electrodes and walls, the kinetic effects may take place and standard approximations may fail. Knowledge of the conditions which may lead to these phenomena may be obtained from the collision frequencies for momentum and energy relaxation. In general, we may draw two sets of conclusions. The ®rst is being associated with the need to consider kinetic phenomena in plasma modeling. The effects such as anomalous diffusion, time dependent NDC, effects of the phase between two ®elds, whole set of effects due to the magnetic ®elds, and their frequency dependencies may be of considerable importance in rf plasma kinetics. For example, power coupled into plasma may be affected by the anomalous diffusion and time resolved NDC. Even when effective ®eld approximation provides accurate values of the total number of ionizations or average diffusion coef®cient and drift velocity it may be important to know exact phase when ionization occurs and local peaks may occur due to the kinetic phenomena. One should also be aware that kinetic phenomena will occur in parallel with other processes and that their effect on the plasma kinetics may be small. For example in high density plasmas collisions with

Z.Lj. Petrovic et al. / Applied Surface Science 192 (2002) 1±25

excited gas particles and Coulomb collisions may be more signi®cant in establishing electron kinetics. Yet it was established that these two processes may both diminish [94,95] and induce [86,92] kinetic phenomena. In addition, one may argue that temporal development of the plasma may be determined in most cases by ion inertia which is absolutely correct. However, electron kinetics has its own temporal development which will determine the heating of electrons and mechanism that sustains the plasma. Thus, we propose that models that include calculation of electron kinetics based on Monte Carlo or Boltzmann equation codes should be tested against standard swarm-benchmarks and the phenomena presented in this paper. Even if the kinetic phenomena affect plasma very little such tests would verify the accuracy of the electron kinetics calculation for the situations when the effects could be larger. The second set of conclusions should actually regard the present day position of swarm physics in relation to plasma modeling and in general. In addition to a major role they play in modeling of plasmas for processing, the future of swarm studies, and in particular, of rf swarm studies may rest in development of a new generation of experiments. That would give motivation for further studies and improvements of the theory. Further advances may be expected from analytic or semi-analytic theories such as MTT. Also further advances may be expected in studies of spatially inhomogeneous systems, electrode effects and spatio-temporal relaxation. Modeling of realistic systems that may be based on swarm techniques and further advances in obtaining the required data are also of great importance and will remain a major motivation in future. It is also worth noting that swarms and low current discharges are prototypes of non-equilibrium systems, they may show non-linear behavior, even self-organization. As those systems are simple to operate in experiments, there is almost complete theoretical understanding of the details of kinetics, one may use them to understand principles of systems in non-equilibrium and may develop some applications. Studies of negative absolute mobility may be a good step in that direction. Yet in all of the cases mentioned here, a new generation of experiments would be essential in giving new ideas and basis for theoretical advances. On the other hand, simulation techniques have been developed suf®ciently to

21

provide detailed pictures, provided that all the relevant data are available. The present results on kinetic phenomena in rf swarms and similar results that may be found for spatially dependent ®elds should be regarded as a warning to plasma modeling community that some additional complexity may be required in their models in order to take into account such effects. On the other hand, one may claim that new fundamental and unexpected results were obtained thanks to a very practical application that motivated research of rf plasmas and rf swarms. At the same time, the advances in understanding of physics were largely based on increased available computing power that became possible as a result of plasma processing technology. This synergism may not last much longer but some other applications may prove to be equally two-directional. Acknowledgements Authors are grateful to our coworkers: S. BzenicÂ, S. SakadzÆicÂ, N. Nakano, N. Shimura, K. Maeda, M. Kurihara, who participated in research presented in this paper. We are also grateful to Dr. S. Vrhovac, for useful discussions and help in formatting of this paper. This research was partly supported by MNTRS, STARC, Monbusho International Scienti®c Research Program, and Keio University Special Grant-in-Aid for Innovative and Collaborative Research Project. References [1] L.G.H. Huxley, R.W. Crompton, The Diffusion and Drift of Electrons in Gases, Wiley, New York, 1974. [2] K. Kumar, H.R. Skullerud, R.E. Robson, Aust. J. Phys. 33 (1980) 343; K. Kumar, Phys. Rep. 112 (1984) 320; M.J. Brennan, K.F. Ness, Nuovo Cimento D 9 (1992) 933. [3] K. Kumar, J. Phys. D 14 (1981) 2199; K. Kondo, Aust. J. Phys. 40 (1987) 367; B.M. Penetrante, J.N. Bardsley, L.C. Pitchford, J. Phys. D 18 (1985) 1087. [4] S.B. Vrhovac, Z.Lj. PetrovicÂ, Aust. J. Phys. 52 (1999) 999. [5] G.L. Braglia, J.J. Lowke, J. Phys. D 12 (1979) 1831. [6] J.J. Lowke, J.H. Parker Jr., C.A. Hall, Phys. Rev. A 15 (1977) 1237. [7] R.E. Robson, B. Li, R.D. White, J. Phys. B 33 (2000) 507; Â uk B. Li, R.E. Robson, R.D. White, in: N. KonjevicÂ, M. C

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