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J. Phys. D: Appl. Phys. 33 (2000) 375–380. Printed in the UK PII: S0022-3727(00)05647-3 On the possibility of negative electron mobility in a decaying plasma N A Dyatko†, A P Napartovich†, S Sakadžić‡, Z Petrović‡ and Z Raspopović‡ † Troitsk Institute for Innovation and Fusion Research, Troitsk, Moscow Region, 142092 Russia ‡ Institute of Physics, University of Belgrade, PO Box 57, 11001 Belgrade, Yugoslavia Received 1 July 1999, in final form 3 November 1999 Abstract. 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. 1. Introduction In the last few years a number of papers on the negative mobility of electrons in low-temperature plasma have appeared [1–20]. This phenomenon is sometimes called absolute negative conductivity, because the plasma conductivity is mainly governed by the electrons. Several different physical situations are presently known in which negative electron mobility has been studied. At first, negative electron mobility was predicted to occur in weakly ionized relaxing plasma in heavy rare gases on the time scales of the order of the plasma electron thermalization time [1, 2]. The results of the theoretical work [2] initiated the publication of previously unpublished experiments [3] in which transient negative conductivity on a nanosecond time scale was detected in relaxing Xe plasma, ionized by a hard x-ray pulse. In [4, 5] a modification of the experimental conditions was proposed by adding a small amount of Cs to Xe in order to use laser radiation instead of x-ray radiation, since Cs atoms have low ionization energy. Another physical situation was considered in theoretical works [6–10], where the possibility of negative conductivity under steady-state conditions in externally ionized (electron beam discharge) gas mixtures was investigated. In this case gas mixtures with a small amount of electronegative gas were studied: Ar:CCl4 [6], Ar:F2 [7, 10], Xe:F2 [8, 9]. So far no experiments were made to detect steady-state negative electron mobility in these mixtures. We can mention the experimental paper [11] in which the Q-factor of a resonator filled with a Xe–Hg–Cs–CO gas mixture was measured. The Q-factor was found to increase when the gas mixture was irradiated by ultraviolet radiation in order to produce a steady weakly ionized plasma. The authors of [11] explained 0022-3727/00/040375+06$30.00 © 2000 IOP Publishing Ltd this experimental result by the fact that the mobility of the steady plasma was negative, but in the theoretical work [12] this statement was disproved. The possibility of negative electron mobility in Ar:Na and Ar:Na:N2 photoplasma was also theoretically studied in [13, 14]. Finally, negative electron mobility was predicted in Ar:NF3 plasma under steady-state Townsend conditions [15] and in a decaying Ar:NF3 [16–18] and Ar:F2 [19, 20] plasma after a short pulse of ionization. In the latter case the quasisteady-state negative electron mobility exists along with a decreasing of the electron concentration due to attachment processes. All of the theoretical studies mentioned above were made by solving an appropriate Boltzmann equation using the so-called two-term approximation for the electron energy distribution function (EEDF). In fact, the applicability of this approximation for the electron mobility calculations under conditions when the mobility is negative is not clear enough. In particular, we are concerned with the calculations of negative electron mobility in decaying plasma. The purpose of the present work is to clarify the situation by calculation of the electron drift velocity in a decaying Ar:F2 plasma using two different methods: a Monte Carlo (MC) simulation and the solution of the Boltzmann equation (BE) with the twoterm approximation for the distribution function. 2. Statement of the problem The solution of the BE for the electrons in a homogeneous plasma acted upon by a steady electric field is usually based on the expansion of the distribution function f (E v ) in 375 N A Dyatko et al Legendre polynomials Pn (cos θ): f (E v) = ∞ X fn (v)Pn (cos θ) (1) 0 where vE is the electron velocity, θ is the angle between the electron velocity and the electric field directions and the functions fn depend only on the absolute value of the electron velocity. In many cases, when the anisotropy of the distribution function is low, the expansion (1) can be restricted to two terms: f (E v ) = f0 (v) + cos θ f1 (v) (2) where f0 (v) is the symmetrical part of the distribution function and f1 (v) describes the directed motion of the electrons along the electric field. The common condition for the applicability of the two-term approximation is that the momentum transfer frequency νm (u) should be considerably larger than the inelastic collision frequency νin (u) at each energy u from the energy interval under consideration. The relation νm (u) νin (u) leads to an almost perfect description of the EEDF with approximation (2)†. In the case of the two-term approximation, the function f1 (u) is defined as (in the following we will use variable u = 0.5mv 2 instead of v) f1 (u) = eE df0 (u) NQm (u) du 3. Solution methods (3) and the electron drift velocity W is calculated from the formula [22] r Z 1 2 ∞ uf1 du W =− 3 m 0 r Z ∞ eE 2 u df0 (u) = − du (4) 3 N m 0 Qm (u) du where e and m are the charge and mass of the electron, E is the electric field strength, Qm (u) is the momentum transfer cross section and N is the number of atoms and molecules per unit volume. The normalization condition for the function f0 may be taken in the form Z ∞ √ uf0 (u) du = 1 0 Expression (4) shows that, in order for the electron drift velocity to be negative, the derivative of f0 (u) should be positive (df0 /du > 0) in a certain energy range. Such a function has a local maximum at some energy u = u∗ and is usually called ‘the inverse function’. According to the expression (3), f1 (u) is equal to zero at u = u∗ and is small at energies near u = u∗ . Our estimations also showed that, if the calculated drift velocity is negative, its value is small with respect to two parts (positive, where df0 /du < 0, and negative, where df0 /du > 0) of integral (4). For these reasons it is not obvious that the contribution of the higher-order terms of expansion (1) to the electron drift velocity is negligible even if the relation † Even under these conditions, the coefficient of transverse diffusion was found to be inadequately calculated by the two-term approximation in the case of argon due to a rapidly increasing cross section [21]. 376 νm (u) νin (u) is satisfied. In order to clarify the situation, it is necessary to compare results of the BE (using the two-term approximation) calculations with results of a more accurate method. For this purpose, a MC simulation method was chosen. Calculations were made for decaying plasma conditions similar to those described in [19, 20]. We considered a uniform weakly ionized Ar:F2 = 1:0.005 plasma in an external electric field, assuming that the initial densities of electrons and positive ions are equal. A plasma with such initial conditions can be created by a short ionizing pulse. If the applied electric field is not too strong, then the electric loss rate caused by electron attachment to fluorine molecules will be higher than the ionization rate, so plasma will start to decay and, simultaneously, the EEDF will begin to deviate from its initial shape. The shape of the EEDF, the electron drift velocity and the mean electron energy were calculated as functions of time in a decaying Ar:F2 plasma. All calculations were carried out for a gas temperature T = 300 K and at atmospheric pressure. The initial EEDF was considered to be Maxwellian with a prescribed temperature Te0 . The degree of the initial plasma ionization was assumed to be fairly low, so we did not incorporate electron–electron and electron–ion collisions and electron recombination processes in calculations. 3.1. Boltzmann equation We numerically solved the time-dependent BE for the spherically symmetric part of the EEDF, F (u, t) = n(t)f0 (u, t), with the initial condition F (u, t) = n(0)f0 (u, 0), where n(t) is the electron concentration and t is the time. The corresponding BE has the form √ dF = IE + St (F ) u dt (5) where the quantity IE describes electron heating in an external electric field applied to the plasma and St (F ) is the collision integral incorporating both elastic and inelastic collisions. In the computations, we took into account the following processes: the elastic scattering of electrons by Ar atoms and F2 molecules; the excitation of the vibrational levels and electronic states of F2 molecules; and the electron attachment to F2 molecules. The set of cross sections for the interaction between electrons and F2 molecules was chosen in accordance with [23]. The transport cross section for electron scattering by Ar atoms was taken from [24]. The time-dependent equation (5) was solved by the method that was used in [18]. Some of the cross sections used in the calculations are shown in figure 1. It is easy to estimate that the relation νm (u) νin (u) is satisfied, taking into account the fact that we considered a low concentration of the F2 admixture. 3.2. Monte Carlo simulation The Monte Carlo code used in this paper is essentially the same as the one used in our previous studies of electron Negative electron mobility in a decaying plasma 1E+2 2 1E+1 1E+1 1 1E +0 f 0 (u ), eV Q (u ), 1 0 -1 6 -3 /2 cm 2 3 1E+0 1 E -1 1 E -2 5 1 E -3 3 4 1E -4 4 2 1 1 E -1 1E -5 1 E -2 1 E -1 1E+0 1E -6 1E+1 u , eV 1E -7 Figure 1. Cross sections used in our study: 1, the transport cross sections for Ar; 2, the transport cross sections for F2 ; 3, the cross sections for electron attachment to F2 molecules; 4, excitation of the vibrational levels of F2 molecules. transport in time varying electric fields [25, 26]. In order to determine the moment of the collision, small time steps are followed with the length of a step being equal to a small part of the mean time between collisions. The number of steps is chosen in such a way that further reduction in the length of the time steps does not affect the results. We calculated all the properties of electron swarms including the EEDF, mean electron energy and electron drift velocity. Two drift velocities were determined [27–30]: Wb = and Wf = d hri i dt dxi = hvi i dt bulk drift velocity flux drift velocity (6) (7) where xi and vi are the position and velocity of the ith electron. It follows from equation (7) that the flux drift velocity is the mean velocity of electrons. If we are considering the region of a uniform plasma, this value should be compared with the drift velocity obtained from BE analysis (4). The bulk drift velocity (6) characterizes the motion of the total ensemble of electrons and is the velocity of displacement of the mean position of the electron swarm. The bulk and the flux drift velocities were shown to be identical for ‘conservative’ (number conserving) collisions while they differ in the case of ‘non-conservative’ (number changing collisions such as attachment or ionization) transport. As non-conservative collisions dominate the transport in the present case, it is important to note that our code includes both attachment and ionization correctly, as was confirmed by benchmark calculations and comparisons with other available results [31, 32]. The code was designed in such a way to maintain a constant number of electrons throughout the simulation (106 ) by replacing each electron lost by another 0 1 2 3 4 5 6 7 u , eV Figure 2. The established distribution function f0 (u) in a decaying Ar:F2 = 1:0.005 plasma. 1, E/N = 0.04 Td; 2, E/N = 0.07 Td; 3, E/N = 0.1 Td; 4, E/N = 0.5 Td; 5, E/N = 1 Td. electron from the remaining electrons. The technique was shown to be in excellent agreement with the results obtained without such replacement [32] for general benchmarks. The code employs the technique of keeping the number of test electrons fixed by effectively introducing a constant collision frequency ionization without energy loss, which is a standard procedure in Monte Carlo simulations [33, 34]. Such a process allows us to keep the number of test particles fixed and to maintain an acceptable statistical uncertainty. In addition this procedure implies that each electron represents a different number of real electrons at different times. Since we do not calculate the field distribution or since, under swarm conditions, there is no interaction between the swarm particles, we do not need to follow explicitly the changing number of real particles (which we do but it has no consequence on the conditions of the simulation). The decaying swarm is thus described only by the changing properties of the electrons starting with the initial conditions. In our case the electron number, for the entire period of decay, drops by eight orders of magnitude so the technique without compensation could not be easily applied. In the region where both techniques were applied, results were in excellent agreement. Generally, the calculations were quite lengthy in order to achieve reasonably good statistics over the entire period of relaxation. Simulations were carried out on a Origin 2000 Silicon Graphics computer and usually lasted between 24 hours and several days. 4. Results and discussion Let us consider first the results of BE calculations. Our computations show (see also [17–20]) that in a decaying plasma the EEDF approaches its steady-state profile f0 (u) 377 N A Dyatko et al 3E+5 W , cm s -1 1 2E+5 5 1E+5 0E+0 4 2 3 -1 E + 5 1 E -3 1 E -2 1 E -1 1E+0 E /N , T d Figure 3. The established electron drift velocity as a function of E/N in a decaying Ar:F2 = 1:0.005 plasma. The numbered circles on the curve correspond to the numbers of the curves in figure 2. and the electron density becomes an exponentially decreasing function. In other words, after a certain time interval, the function F (u, t) takes the form F (u, t) = n(t)f (u) where n(t) ∼ exp(−νt) and ν corresponds to the attachment rate constant rate calculated from the established f0 (u). For a given E/N value, f0 (u) and ν do not depend on the initial conditions (Te0 ). Figures 2 and 3 show the established f0 (u) and drift velocity as functions of the electric field strength. We note an increase of the drift velocity in the range E/N = 0.001– 0.053 Td, followed by an abrupt transition from a positive value at 0.053 Td to a negative value at 0.055 Td. In the range 0.055–0.55 Td the drift velocity is negative and its minimum value is about −5×104 cm s−1 . For E/N > 0.55 Td the drift velocity again becomes positive and increases with E/N . Such an unusual dependence of the electric drift velocity on the electric field is due to the shape of the attachment cross section. An explanation of this phenomenon was given in previous papers [18, 20] in terms of two groups of electrons. Here we briefly repeat this explanation. The plasma electrons can be divided into two groups concerning: (i) electrons with energies u < um , where um = 0.08 eV is the energy at which the cross section for electron attachment is maximum (see figure 1), and (ii) electrons with energies u > um . Particle exchange between these groups is hindered, because they are separated by a barrier whose role is played by electron attachment. As the plasma decays, the electron density of both groups decreases, and the shape of the established distribution function f0 (u) is governed by the ratio between the decay rates of these electron populations. If the electric field is weak, the rate of electron loss in the first group is low, because, in the low-energy range, the attachment cross section is small. Due to elastic and inelastic collisions, the energy of electrons from the second group falls in the range u < 0.4 eV, in which the attachment cross section 378 is large and, hence, the electron loss rate is very high. As a result, the established f0 (u) is governed by electrons from the first group and is a monotonously decreasing function (see curve 1 in figure 2). The corresponding value of the electron drift velocity is positive. As the electric field increases, the mean electron energy grows and the attachment for the electrons from the first group rises, since in the range u < um the attachment cross section increases with energy. For E/N > 0.55 Td the rate of electron loss for the first group is higher than that for the second group. In this case the established f0 (u) is governed primarily by the electrons from the second group and has inverse form (see curves 2 and 3 in figure 2). The corresponding drift velocity becomes negative. As the electric field increases further, the electrons from the first group acquire sufficient energy (during their lifetime) to overcome the attachment barrier in the energy space. In this case, a weakly inverse or even monotonous function f0 (u) forms (see curves 4 and 5 in figure 2) and the drift velocity becomes positive. A physical explanation of the negative drift velocity may be given in the following manner. The shape of the EEDF (see curves 2 and 3 in figure 2) is such that the majority of the electrons have an energy above the Ramsauer minimum. The momentum transfer cross section for Ar grows rapidly with increasing energy in this energy range. The group of electrons that are moving in such a way to gain energy from the field has an increased probability of elastic collisions (which lead to randomization of the velocity direction). The electrons that are moving against the field lose their energy and the probability of elastic collisions decreases. As a result, a flux of electrons against the field is generated. This effect would be lost if a large group of low-energy electrons was formed below the Ramsauer minimum. In the case studied here, the attachment at low energies prevents the formation of the low-energy group. Thus the effect, which would otherwise be transient [2], becomes stationary. The comparison of MC and BE results was carried out for E/N = 0.1 Td and Te0 = 0.666 eV. Figure 4 shows the EEDFs calculated at different times. It follows from figure 4 that both methods of calculation give the same scenario of EEDF variations with time: a fast decrease of the low-energy part and formation of an inverse distribution function with a maximum in the energy range 1–1.5 eV. At a time of about 75 ns, the EEDF is nearly established. Such a scenario is in agreement with the qualitative explanation of the EEDF formation presented above. From figure 4, one can also see that at early times the MC and BE distribution functions are graphically indistinguishable, but their established forms are slightly different. Figure 5 shows the calculated electron drift velocity as a function of time. It follows from figure 5 that the time evolution of the drift velocity is the same in both cases and the established values are negative. In the case of the MC simulation the drift velocity reveals appreciable statistical fluctuations in spite of the large number of electrons involved in the calculations. This is, obviously, because of the small value of the drift velocity with respect to the thermal electron velocity. The averaged MC drift velocity has an established value of about −4.26 × 104 cm s−1 . The BE calculations give the value −4.74 × 104 cm s−1 , so the discrepancy Negative electron mobility in a decaying plasma 1 .8 1 1 E -1 1 .6 2 U m e an , e V f 0 (u ), eV -3 /2 1E+0 1 1 E -2 2 1 E -3 1 E -4 1 .4 1 .2 1 .0 0 1 2 3 4 5 u, eV 0 .8 1E+0 0 f 0 (u ), eV -3 /2 4 1 E -1 3 1 E -3 1 E -4 1 2 3 u, eV Figure 4. Comparison of BE (full curves) and MC (full circles) distribution functions f0 (u) in a decaying Ar:F2 = 1:0.005 plasma at different times for E/N = 0.1 Td. 1, t = 2 ns; 2, t = 5 ns; 3, t = 25 ns; 4, t = 75 ns. W , W f , cm /s 4E+5 3E+5 2E+5 1E+5 0E+0 -1 E + 5 0 20 40 60 t, n s 80 100 Figure 6. Comparison of the time evolution of BE (full curve) and MC (full circles) electron mean energies in a decaying Ar:F2 = 1:0.005 plasma for E/N = 0.1 Td. 1 E -2 0 20 40 60 t, n s 80 100 Figure 5. Comparison of the time evolution of the BE drift velocity (full curve) and MC flux drift velocity (full circles) in a decaying Ar:F2 = 1:0.005 plasma for E/N = 0.1 Td. is about 10%. The time variations of the mean electron energy, Umean , in a decaying plasma are sufficiently close in both cases (see figure 6), the discrepancy of established values being about 6%. The non-monotonous behaviour of curves seen in figure 6 is explained by rapid attachment of slow electrons at an earlier stage of decay, followed by a slower deceleration of electrons in collisions with atoms and molecules. It is important to note that we have found that the bulk drift velocity differs considerably from the flux drift velocity and that it is always positive. This supports the claim that, overall, electrons move in a direction determined by the electric field. The positive drift is brought about by the fast moving, but short-lived, electrons so on average we have a negative drift velocity and positive spatial motion of the swarm itself. As a result, the fast and slow electrons may be resolved spatially and it may also be possible to form an explanation based on the diffusion of two different types of spatially-resolved particles. In an attempt to verify the MC procedure of keeping the number of particles constant we have performed several simulations with a changing number of particles and could follow it only to 0.07 µs where number of electrons drops to one or less for initial number of 1 000 000. Even with 10 times more initial electrons we could not extend the results much further but still at 0.05 µs the number of particles is sufficient to confirm the negative drift velocity and at that time the drift velocity reaches the constant value. These results were not shown here as the uncertainty increases rapidly with decaying number of electrons but they may be used to confirm our procedure. Further improvement may be achieved if we average the drift velocity over times after the swarm has decayed to its quasi-steady state form, and these results again confirm our procedure. Thus we may conclude that the Monte Carlo procedure was sufficiently accurate to confirm the observation of the phenomenon of the negative quasi-steady state drift velocity and that possible inadequacy of the two-term approximation is not the cause of the observed effect. 5. Conclusion The main conclusion is that the Monte Carlo calculations confirmed the predictions of the existence of negative electron mobility in a decaying Ar:F2 plasma, which were made from solutions of the Boltzmann equation in a two-term approximation for the EEDF. The small difference in the 379 N A Dyatko et al obtained results is probably due to the inaccuracy of twoterm approximation. The observation of the quasi-steady state negative drift velocity may be associated with the observation of the transient negative conductivity [1–3] which was also experimentally confirmed [3]. The decaying swarm is a good model for the decaying plasma which may be used for a possible experimental verification of our result. Our result may also be of interest for modeling of rf plasmas. Acknowledgments Three of the authors (ZR, SS and ZP) are grateful to MNTRS project 01E03 and to the Institute of Physics Computer Facility for support. References [1] Rokhlenko A V 1978 Zh. Exp. Teor. 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