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Diffusion of low-pressure electronegative plasma in magnetic field
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 EPL 102 55004 (http://iopscience.iop.org/0295-5075/102/5/55004) View the table of contents for this issue, or go to the journal homepage for more
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June 2013 EPL, 102 (2013) 55004 doi: 10.1209/0295-5075/102/55004
www.epljournal.org
Diffusion of low-pressure electronegative plasma in magnetic field Dmitry Levko1(a) , Laurent Garrigues1,2 and Gerjan Hagelaar1,2 1
LAPLACE (Laboratoire Plasma et Conversion d’Energie), Universit´e de Toulouse, UPS, INPT Toulouse 118 route de Narbonne, F-31062 Toulouse cedex 9, France, EU 2 CNRS, LAPLACE - F-31062 Toulouse, France, EU received 20 March 2013; accepted in final form 29 May 2013 published online 26 June 2013 PACS PACS
52.27.Cm – Multicomponent and negative-ion plasmas 52.25.Fi – Transport properties
Abstract – A new one-dimensional analytical model describing the diffusion of low-pressure electronegative plasma in homogeneous magnetic field is developed. The conditions are found when the electron diffusion becomes independent of ion diffusion, while the diffusion of the ion component depends on the electron diffusion. The model demonstrates the influence of various parameters such as power, magnetic field and gas pressure on the diffusion. The results predicted by the model are compared with the results of one-dimensional numerical modeling. c EPLA, 2013 Copyright
Introduction. – Electronegative plasmas are plasmas composed of the positive and negative ions and only a small fraction of electrons. Nowadays this plasma is mainly applied in plasma etching technologies (see [1] and references therein). Recently it was proposed to use electronegative plasma in a new type of thrusters for electric propulsion (PEGASES thruster [2]). The advantage of such thrusters is the possibility to use both positive and negative charges for thrust, which removes the need for additional downstream current and space charge neutralization. There are two main ways to produce electronegative plasmas [1–6]. The first way is the generation during the afterglow of electronegative discharge, when the electron temperature decreases and the rate of electron attachment to electronegative molecules increases. The second way is the use of a magnetic filter, which decreases both the electron mobility and diffusion through the plasma and increases the rate of electron attachment. Despite the great interest in electronegative plasmas only a few studies were devoted to the numerical (Particle-in-Cell and fluid) and theoretical study of the diffusion of the plasma species in magnetic field (see [3–5] and references therein). Franklin and Snell [3] used a fluid model in order to study the one-dimensional (1D) transport of an electronegative plasma in a homogeneous magnetic field perpendic(a) E-mail:
[email protected]
ular to the diffusion direction. A boundary condition consistent with the sheath formation was chosen, namely, the flux of negative ions at the boundary was zero. Processes in plasma sheath were not studied in the model. A similar 1D model was used by Leray et al. [4]. However, this model considered other boundary conditions in order to obtain the sheath-free plasma. Also, the term describing the losses of electrons and positive ions to the side walls was added into the continuity equations. In addition, a homogeneous negative ion density profile was assumed in the model. A similar analytical model was developed by Kawamura et al. in [5], where a constant electronegativity along the whole simulation domain was assumed. In this work we develop a 1D analytical model describing the diffusion of the current free low-pressure electronegative plasma in a homogeneous magnetic field. In this model we assume neither homogeneous negative ion density profile, nor constant ratio between negative ion and electron densities. The boundary conditions for plasma specie densities are taken from the result of a 1D fluid numerical model. The processes inside the plasma sheath are not considered in the analytical model. It is found that the electron diffusion becomes independent of ion diffusion if the magnetic field exceeds some value, while the ion diffusion depends on electron diffusion. It is shown that the negative ion flux towards the walls becomes non-zero even without the application of bias voltage. The influence of various parameters such as power, magnetic
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Dmitry Levko et al. field and gas pressure on diffusion is studied. The results predicted by this model are compared with the results of 1D fluid numerical model. Discussion of low-pressure electronegative plasma diffusion in magnetic field. – We consider in this letter a 1D diffusion of low-pressure recombinationdominated electronegative SF6 plasma for PEGASES thruster [2] typical conditions: heating power in the range 10–103 W, gas pressure 0.1–10 Pa, and magnetic field strength 10−2 –10−1 T. At low pressure the main ion species + − in SF6 electronegative plasma are SF+ 5 , SF3 and SF6 (see, for instance, [6]). For instance, at a pressure of 1.3 Pa and with unmagnetized species the mobility of each ion is 2 −1 −1 2 −1 −1 s , μ(SF+ s and μ(SF+ 5 ) ≈ 4.49 m V 3 ) ≈ 4.95 m V − 2 −1 −1 μ(SF6 ) ≈ 4.1 m V s , respectively, while the electron mobility is μe ≈ 1.16 × 10 m2 V−1 s−1 [6]. One can see + that the mobility of SF+ 5 and of SF3 are comparable. Therefore, we suppose in our model that the plasma consists only of one positive and one negative type of ions and electrons. The plasma is located between two parallel conducting grounded walls (the distance is 0.12 m). The magnetic field is homogeneous and perpendicular to the diffusion direction. Electrons are heated in the whole region except in the narrow region near the walls. Both the electron and ion temperatures are supposed homogeneous. The negative ions were generated due to the electron attachment to SF6 and the positive ions were generated due to the electron impact ionization of SF6 . Since the SF6 plasma is a recombination-dominated plasma [7] the negative ions were lost due to recombination with positive ions. In the drift-diffusion approximation the fluxes of plasma components are
Γn =
μn /μe α(De − Dp )∇ne (1 + μp /μe ) + α(μn + μp )/μe −
Γp =
[Dn (1 + μp /μe ) − α(Dn μp + Dp μn )/μe ] · ∇nn , (1 + μp /μe ) + α(μn + μp )/μe
(6)
−μp /μe (1 + α)(De − Dn )∇ne (1 + μp /μe ) + α(μn + μp )/μe −
[Dp (1 + μp /μe ) + (1 + α)(Dn μp + Dp μn )/μe ] · ∇np (1 + μp /μe ) + α(μn + μp )/μe
−
Dp (μn + μp )/μe · ∇np . (1 + μp /μe ) + α(μn + μp )/μe
(7)
Here α = nn /ne is the plasma electronegativity. Lisovskiy studied the diffusion of strongly electronegative plasma in the absence of magnetic field [8]. It was shown that the diffusion of all plasma species becomes free, if the electronegativity is large (in SF6 α > 103 ) and if electrons and negative ions are in Boltzmann equilibrium. In magnetic field there is no Boltzmann equilibrium between electrons and negative ions and the analysis presented in [8] is no longer valid. However, the fluxes (5)–(7) can be simplified, if the electronegative plasma diffuses in magnetic field exceeding some value. As was shown in [6] in low-pressure SF6 plasmas μn ≈ μp and, as a consequence, Dn ≈ Dp . The magnetic field reduces the electron mobility, which could become comparable with the ion mobility. The electron diffusion coefficient is defined as De =
μe Te , 1 + h2
(8)
where μe = mqeeνe is the mobility without magnetic field, and h = μe B is the Hall parameter. Also, the electron mobility in magnetic field is μe = μe /(1 + h2 ). Here qe is (1) the electron charge, me is the electron mass, and νe is the Γe = −De ∇ne − μe ne E, electron momentum transfer frequency. In the considered Γn = −Dn ∇nn − μn nn E, (2) model, ionization compensates attachment for SF , when 6 Te ≈ 4.9 eV [7]. If B = 0.01 T and Te ≈ 4.9 eV one has Γp = −Dp ∇np + μp np E. (3) νe ≈ 7.5 × 107 s−1 [7], h ≈ 23, μe ≈ 2.2 m2 V−1 s−1 and 2 −1 Here Dq , μq and nq are the diffusion coefficients, mobil- De ≈ 11 m s . The Hall parameter for ions is still small Supposing ity and particle number density of plasma components, (h ≈ 0.05) and they remain unmagnetized. 2 −1 = T = 0.125 eV one has D ≈ 0.56 m s and Dn ≈ T p n p respectively, and E is the electric field. Further the Ein2 −1 s , i.e., D D and D D , while the electron 0.5 m e p e n stein relation Dq = Tq μq is supposed to hold. Indexes e, n mobility is comparable with the ion mobility. and p denote, respectively, electrons, negative and positive If electronegativity satisfies the condition ions. Let us suppose Tp = Tn = Ti but μn = μp and, as a consequence, Dn = Dp . Using the quasineutrality and μe + μp α , (9) the requirement of zero total current in the bulk of the μn + μp plasma, one obtains the ambipolar electric field: the first term in denominator of eqs. (5)–(7) can be ne(Dp − De )∇ne + (Dp − Dn )∇nn E= . (4) glected. Also, using that α 1, μe ∼ μp , μe ∼ μn , (μe + μp )ne + (μn + μp )nn De Dp , De Dn , eqs. (5)–(7) can be reduced to Substitution of eq. (4) into eqs. (1)–(3) gives the fluxes Γe =
[−Te μp − (μn + μp )Te α − Dp ] · ∇ne − (Dp − Dn )∇nn , (1 + μp /μe ) + α(μn + μp )/μe (5)
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Γe ≈ −De ∇ne , μn De ∇ne ≈ Γn ≈ −Dn ∇nn + μn + μp 1 − Dn ∇nn + De ∇ne , 2
(10)
(11)
Diffusion of low-pressure electronegative plasma in magnetic field μp De ∇ne ≈ μn + μp 1 − Dp ∇np − De ∇ne . 2
Γp ≈ −Dp ∇np −
(12)
One can see that the diffusion of electrons is independent of ion diffusion, while the ion diffusion depends on electron diffusion. If the electron density profile is more uniform in comparison with the ion density profiles, the last term in eqs. (11), (12) can be neglected and ion diffusion becomes also independent of electron diffusion. Otherwise, the ion diffusion depends on electron diffusion. Electron flux is positive, i.e., electrons leave the plasma, if ∇ne < 0. Equation (11) shows that the negative ion flux is positive only if ∇nn < 0 and if the first term exceeds the second term, i.e., if Dn ∇nn > 0.5De ∇ne . It is important to note that for a given magnetic field the approximation (10)–(12) is only valid, if electronegativity satisfies the condition (9). For instance, at B = 0.01 T one obtains α 1, i.e., the negative ion density has to be larger than the electron density. This requirement is weaker than the one obtained in [8] without magnetic field. If the mobility of positive and negative ions is equal, the lower limit of magnetic field at the given α is B
m e νe qe
μe 2αμp
− 1. For instance, supposing α ≈ 40
one obtains B 10−3 T. The latter relation shows that if one has a gas with more mobile ions, for instance, O2 instead of SF6 , both the lower limit of the magnetic field and α (9) decrease. Substituting (10) in the electron steady-state continuity equation and taking into account that the electron diffusion coefficient does not depend on x, one obtains d2 ne [ki (Te ) − ka (Te )] · ng ne = 0. + 2 dx De (Te )
(13)
Here ng is the gas density, ki and ka are the rate constants of ionization and attachment, respectively. The boundary conditions in eq. (13) are dne (x = 0) = 0, dx ne (x = 0) = ne0 , ne (x = R) = 0.
(14)
Using the dependence of the electron diffusion coefficient on B and Te (8), and taking into account that h 1, one obtains from eq. (16) ki (Te ) − ka (Te ) π 2 me 1 1 ≈ . Te · km (Te ) 2 qe R 2 B 2
(17)
Here km is the rate constant of the electron momentum transfer reaction. Equation (17) defines the dependence of Te on magnetic field. Also, eq. (17) shows that Te does not depend on the gas pressure. Using eqs. (11) and (13) and the steady-state continuity equation of negative ions, one obtains the nonlinear differential equation describing the negative ion density kr (Te )ne nn kr (Te ) 2 d2 nn − − n = dx2 Dn Dn n [ki (Te ) + ka (Te )] ng ne − , 2Dn
(18)
where kr is the recombination coefficient. The boundary conditions are the values of nn at x = 0 and at the entrance to the plasma sheath. Also, nn has zero first derivative at x = 0. In the bulk of plasma the negative ion density is mainly defined by the balance between attachment and recombination. Therefore, the negative ion balance equation gives (ki + ka )ng ne0 · cos(πx/2R). (19) nn (x) ≈ 2kr (ki +ka )ng One can see that nn (x = 0) = nn0 ≈ ne0 . 2kr Equation (19) allows one to understand the dependence of the electronegativity on power W . It shows that √ α ∝ 1/ ne . Assuming that the introduced power goes into the electron component of plasma, the electron energy balance equation [9] gives that ne0 is proportional to √ W . Thus, one has α ∝ 1/ W , i.e., the increase in power leads to the increase in both ne and nn but leads to the decrease in electronegativity. In order to analyze the dependence of plasma potential on magnetic field and electronegativity, let us suppose equal mobility and diffusion coefficients of ions: μp = μn = μi and Dp = Dn = Di . Then the electric field (4) becomes
(De − Di )∇ne . E≈− 2μi nn Equation (13) shows that for homogeneous Te the electron flux is directed towards the walls only if ki > ka . Points Substituting the electron density from (15) and the negx = 0 and x = R correspond to the center and boundary of ative ion density from (19) one obtains the profile of the the system, respectively. The value of ne0 is defined by the potential with respect to the grounded walls: energy balance equation. Since ki , ka and De are functions πx of Te , i.e., are not free parameters of the model, one needs (De − Di )ne0 ≈ cos ϕ(x) = to fix the electron density at x = R. The solution of μi nn0 2R eq. (13) combined with eq. (14) gives πx Te . (20) cos μe B 2 μi α0 2R ne (x) = ne0 · cos(πx/2R), (15) Here α0 is the electronegativity in the center of the system. π 2 De (Te ) ki (Te ) − ka (Te ) = . (16) One can see that φ(x) is proportional to B −2 and to α−1 . 2 2ng R 55004-p3
Dmitry Levko et al.
Fig. 1: (Colour on-line) Profiles of the densities (a) and fluxes (b) of electrons and ions obtained in the numerical model at 50 W, 1.3 Pa and 0.01 T.
Thus, the plasma potential decreases, when α or magnetic field increases. In order to check the predictions of the analytical model, the 1D numerical fluid simulations using the model described in detail in [10] were carried out. In this model the system of fluid equations for three-component plasma + (electrons, SF− 6 and SF6 ) was solved in the x-direction. Namely, the electron and ion continuity and balance equations, Poisson equation and electron energy balance equation were solved. Ionization of SF6 by the electron impact, attachment of electrons to SF6 and recombination + between SF− 6 and SF6 were considered in the model as the main processes. The rate constants of these reactions were taken from [7]. Both ki and ka were the sum of rate constants of possible ionization and attachment processes of SF6 . The momentum transfer of electrons and ions as well as the inelastic energy losses of electrons in collisions with SF6 were taken into account. Also, the homogeneous Te and equal negative and positive ion temperatures Tn = Tp = Ti = 0.125 eV were supposed in the model. The analytical model shows that the small difference in the mobility of positive and negative ions does not influence significantly the character of ion diffusion. Therefore, it was supposed in the numerical model that μp = μn = μi and Dp = Dn = Di . Two sets of rate constants were considered in [7], namely, the rate constants obtained for Druyvesteyn and Maxwellian electron energy distribution function (EEDF). The Druyvesteyn function takes into account electron en-
Fig. 2: (Colour on-line) (a) Dependence of electron density and electronegativity in the center of the system at 1.3 Pa and 0.01 T on power obtained in the numerical model. (b) Dependence of electron temperature and potential at 1.3 Pa and 0.01 T on power obtained in the numerical model. (c) Dependence of electron temperature and potential at 1.3 Pa and 50 W on magnetic field obtained in the numerical model and from eq. (17).
ergy losses due to the electron attachment to electronegative molecules and describes adequately EEDF at large gas pressure. At low gas pressure in magnetized plasma EEDF becomes Maxwellian due to the electron-electron collisions [11]. However, it is important to note that the use of either Druyvesteyn or Maxwellian EEDF does not influence the qualitative predictions of our analytical model. The profiles of the densities of plasma components and potential distribution are shown in fig. 1(a). One can see that ne nn and ne np in the whole simulation domain, i.e., there is no separation of plasma in electron-ion and ion-ion regions in homogeneous magnetic field. Since the electronegativity is large under the considered conditions, the plasma potential according to eq. (20) is small.
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Diffusion of low-pressure electronegative plasma in magnetic field One can see a satisfactory agreement. In addition, the analytical model predicts that Te does not depend on the power and gas pressure. A similar dependence was obtained in the numerical model (see fig. 2(b)). The comparison between the potentials obtained in the numerical model (see fig. 3(a)) shows a qualitative agreement. The comparison between ne (x) obtained in the numerical model and that described by eq. (15) is shown in fig. 3(b). Figure 3(c) presents the comparison among nn (x) obtained in the numerical model, in the numerical solution of eq. (18) and described by eq. (19). The boundary conditions for the continuity equations of electrons and negative ions were taken from the numerical model. Equation (18) was integrated up to the plasma sheath since there is no quasineutrality in the sheath and the analytical model does not describe that region. Figure 3(c) shows that the numerical solution of (18) agrees with the results of the numerical model in the bulk of the plasma, while approximation (19) agrees only near the center of the system. Conclusions. – The one-dimensional analytical model of diffusion of electronegative plasma at low-pressure typical conditions was developed (heating power in the range 10–103 W, gas pressure 0.1–10 Pa, and magnetic field strength 10−2 –10−1 T). It was obtained that the increase in power leads to the decrease in the plasma electronegativity, while both the electron and ion densities increase. Also, the plasma potential increased with the increase in power, while the electron temperature did not change noticeably. The model showed that the increase in magnetic field causes the asymptotic behavior of all plasma parameters, namely, the electron temperature tends to the value where ionization is compensated by attachment. In addition, the analytical model predicted that the electron temperature does not depend on gas pressure. Fig. 3: (Colour on-line) (a) Comparison between profiles of poThe analysis of the electron and ion fluxes in the large tential obtained in the numerical model and described by (20). magnetic field in the drift-diffusion approximation showed (b) Comparison between the results of the numerical model that the electron diffusion becomes independent of ion difand (15). (c) Comparison between the profiles of the negative fusion, while the ion diffusion depends on the electron difions density obtained in the numerical model, described by (19) fusion. This resulted in the non-zero flux of negative ions and obtained from the solution of (18). towards the walls. The results predicted by the analytical model are compared with the results of the one-dimensional numerical Figure 1(b) shows that the fluxes of the plasma compomodel. The comparison between models showed a qualinents are not equal to zero at the walls although the value tative agreement. of the electron flux is the smallest. Therefore, the nega∗∗∗ tive ions can be extracted from such plasma without the application of bias voltage. Numerical model showed that This work is supported by the EPIC “Strongly ElectroDn ∇nn > 0.5De ∇n, i.e., negative ion flux (11) is posinegative Plasmas for Innovative Ion Acceleration” project tive and, as a consequence, negative ions leave the plasma funded by ANR (Agence Nationale de la Recherche) under volume. Contract No. ANR-2011-BS09-40. Figure 2(a) shows that the increase in power leads to the decrease in α and, as a consequence, to the increase in φ (see fig. 2(b)). According to eq. (20) one has ϕ → 0 at REFERENCES large B and α, which agrees with the results presented in [1] Economou D. J., Appl. Surf. Sci., 253 (2007) 6672. fig. 2(c). Also, fig. 2(c) shows the comparison between Te [2] Aanesland A., Meige A. and Chabert P., J. Phys.: obtained in the numerical model and described by eq. (17). Conf. Ser., 162 (2009) 012009. 55004-p5
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[3] Franklin R. N. and Snell J., J. Phys. D: Appl. Phys., 32 (1999) 1031. [4] Leray G., Chabert P., Lichtenberg A. J. and Lieberman M. A., J. Phys. D: Appl. Phys., 42 (2009) 194020. [5] Kawamura E., Lichtenberg A. J. and Lieberman M. A., J. Appl. Phys., 108 (2010) 103305. [6] Kono A., Appl. Surf. Sci., 192 (2002) 115. [7] Kokkoris G., Panagiotopoulos A., Goodyear A., Cooke M. and Gogolides E., J. Phys. D: Appl. Phys., 42 (2009) 055209.
[8] Lisovskiy V. and Yegorenkov V., EPL, 99 (2012) 35002. [9] Lieberman M. A. and Lichtenberg A. J., Principles of Plasma Discharges and Materials Processing, 2nd edition (Wiley-Intersciences, New York) 2005. [10] Hagelaar G. J. M., Fubiani G. and Boeuf J.-P., Plasma Sources Sci. Technol., 20 (2011) 015001. [11] Aanesland A., Bredin J., Chabert P. and Godyak V., Appl. Phys. Lett., 100 (2012) 044102.
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