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Bohm criterion for electronegative dusty plasmas
Thin Solid Films 506 – 507 (2006) 637 – 641 www.elsevier.com/locate/tsf
Bohm criterion for electronegative dusty plasmas Zheng-Xiong Wang, Yue Liu *, Li-Wen Ren, Jin-Yuan Liu, Xiaogang Wang State Key Laboratory of Materials Modification by Laser, Electron and Ion Beams, Department of Physics, Dalian University of Technology, Dalian, 116024, China Available online 31 August 2005
Abstract Bohm criterion for complex plasmas which are composed of electrons, negative and positive ions as well as dust grains is investigated with Sagdeev potential, taking into account the self-consistent dust charge variation. The numerical solutions show that the positive ion and dust Bohm velocities increase with the growth of the dust density, while both of them decrease with the growth of negative ion density. Furthermore, the interactions between the two Bohm velocities exist. The results are examined to be reliable by the quantitative analysis of Sagdeev potential. D 2005 Elsevier B.V. All rights reserved. PACS: 52.27.Cm; 52.27.Lw; 52.40.Kh Keywords: Electronegative plasma; Dust grains; Bohm criterion; Sagdeev potential
1. Introduction Dust grains are frequently observed in space and laboratory plasmas [1 –3]. In recent years, much attention has been paid to many various aspects of collective processes in dusty plasmas, such as the dusty sheath [4,5], linear and nonlinear dusty waves and instabilities [6– 8], as well as observation and mechanism of the dust lattice formation [9– 11], etc. These theoretical and experimental investigations have advanced further developments of other fields, such as the potential of spacecraft in the Earth’s magnetosphere and space, and the study of condensed matter systems, etc. In the plasma enhanced chemical vapor deposition (PCVD), multicomponent complex plasma systems are often created, containing electrons, positive and negative ions, as well as charged dust grains. In the low-pressure high-density silane (SiH4) plasmas, the dust density is sometimes up to 107 cm 3 with sizes from tens to hundreds of nanometers [12,13]. These electronegative dusty plasmas are used intensively today for device applications such as
* Corresponding author. E-mail address: [email protected] (Y. Liu). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.08.052
solar cells and thin-film transistor. Some experiments show that the dusts have significant effects on sheath structures as well as on the silicon film quality [14,15]. In order to understand the complicated sheath well and accurately calculate the ion and dust flux to a wall or an electrode, a detailed study of Bohm criterion for the low-pressure highdensity silane (SiH4) plasmas is warranted. The dust particles are often negatively charged and the value of their charge may vary depending on different charged particle densities around dusts, so the variability of the dust charge will lead some new effects in dust-acoustic solitons and dust-ion acoustic waves as well as in the dusty sheath near the wall [16,17]. In this paper, using the Sagdeev potential and self-consistent dust charge variation, we studied the Bohm criterion for the electronegative dusty plasma sheath comprising Boltzmann electrons and negative ions but cold-fluid positive ions and dust grains. It is shown that the dust and negative ion densities have effects on the dust and positive ion Bohm velocities and that both of the two Bohm velocities also affect each other. The paper is organized as follows. In Section 2, we present the steadystate sheath model and dust charging model in the sheath. In Section 3, we obtain the numerical results of Bohm criterion and discuss the physics reasons. Finally, the paper is concluded with a short summary in Section 4.
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I ¼ pr2 eð8T =pm Þ1=2 n expðeqd =rT Þ;
2. Analytical models We consider a one-dimensional plasma sheath treating electrons and negative ions as thermal Boltzmann particles but positive ions and dust grains as cold fluids. The sheath edge at x = 0 is the plasma-sheath interface (interface between essentially neutral and nonneutral regions). Then x < 0 is the plasma region and x > 0 is the sheath region. The spatial potential is taken zero / = 0 at the sheath edge where the quasi-neutrality condition is n0þ n0e n0 Z0d n0d ¼ 0;
ð1Þ
with n 0e, n 0 , n 0+, and n 0d representing number densities of electron, negative ion, positive ion, and dust grain at the sheath edge, respectively; Z 0d = q 0d/e, e is the electron charge, and q 0d is dust charge at the sheath edge. Both electrons and negative ions in the sheath are assumed to obey the Boltzmann distribution [18,19],
Iþ ¼ pr2 enþ vþ 1 2eqd =rmþ v2þ ;
ð9Þ ð10Þ
where m e and m are electron and negative ion masses, respectively; q d is the dust charge in the sheath, and thus the dust surface potential is expressed as / d = q d/r. In the equilibrium state, the dust charge conservation is Ie þ I þ Iþ ¼ 0:
ð11Þ
Finally, the Poisson’s equation relates the electron, negative and positive ion as well as dust densities in the sheath to a self-consistent potential d2 /=dx2 ¼ 4p½eðnþ ne n Þ þ qd nd :
ð12Þ
nþ vþ ¼ n0þ v0þ ;
ð4Þ
Then we write dimensionless variables d e = n 0e /n 0+ , d = n 0 /n 0+, d d = n 0d/n 0+ as the density ratio of electrons, negative ions, dust grains to positive ions, respectively. Furthermore, we introduce other dimensionless quantities U = e//Te, U d = e/ d/Te denoting the normalized dust surface potential with z = rTe/e 2 (= 695rTe), then the relation Z d = |zU d| represents the charge number of dust grain measured in units of e. Also we define the temperature ratio of negative ions to electrons c = T /Te, the dimensionless spatial coordinate X = x/k D with respect to the Debye length k D = (Te/4pe 2n 0+)1/2. The dimensionless positive and dust velocities in the sheath are M + = v +/c s+ and M d = v d/c sd, respectively, where c s+ = (Te/m +)1/2 and c sd = (Z d0Te/m d)1/2 are the positive ion and dust acoustic velocities, respectively. Thus M 0+ = v 0+/c s+ and M 0d = v 0d/c sd denote the positive ion Bohm velocity and dust Bohm velocity, respectively. Thus the quasi-neutrality at the sheath edge, Eq. (1), can also be written a dimensionless form as
mþ vþ ðdvþ =dxÞ ¼ eðd/=dxÞ;
ð5Þ
de ¼ 1 d dd Z0d ;
ne ¼ n0e expðe/=Te Þ;
ð2Þ
n ¼ n0 expðe/=T Þ;
ð3Þ
where n e and n are the electron and negative ion number density in the sheath, respectively, and each negative particle species is assumed to possess a charge e, Te and T are electron and negative ion temperatures, respectively. The positive ions are accelerated in the sheath, so the cold fluid model for positive ions is a good approximation [18,19]. The positive ions in the sheath of low-pressure plasmas are ruled by the continuity and the steady-state momentum equations
where n + and v + are the positive ion density and velocity in the sheath, respectively, v 0+ is the positive ion enteringsheath velocity towards the wall and m + is the positive ion mass. In this paper the dust grains are so small (100 nm) that the gravitational force is negligible. Therefore supposing the dust grain has a wall-wards velocity v 0d at the sheath edge, then one can write continuity and momentum equations for dust grains as nd vd ¼ n0d v0d ;
ð6Þ
md vd ðdvd =dxÞ ¼ qd ðd/=dxÞ;
ð7Þ
where n d, v d, q d and m d are the density, fluid velocity, charge, and mass of the dust grain in the sheath, respectively. When a dust grain of radius r is immersed in the complex plasma, the charging currents from electrons, negative and positive ions can respectively be written as [20] Ie ¼ pr2 eð8Te =pme Þ1=2 ne expðeqd =rTe Þ;
ð8Þ
ð13Þ
where the Z 0d is the dust charge number at the sheath edge. Substituting these dimensionless variables into Eqs. (8) – (12), we obtain 2Ud 8mþ 1=2 M0þ 1 2 ¼ de expðUd UÞ pme Mþ 8cmþ 1=2 Ud U þ d exp ; c pm ð14Þ
d2 U U ¼ de expð UÞ d exp dX 2 c 1=2 2U 2W 1=2 þ 1þ 2 dd Zd 1 2 ; Mþ Md
ð15Þ
U where WðUÞ ¼ X0 Z˜ dU with Z˜ = Z d/Z d0. Integrating Eq. (15) once, we obtain
1 1 ðdU=dX Þ2 þ V ðUÞ ¼ E02 ; 2 2
ð16Þ
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639
for electropositive plasmas. If we set d d = 0, through Eq. (18) we can obtain M + > [1 d (1 1/c)]1/2 = M 0+, the positive ion Bohm criterion for electronegative plasmas [23].
3. Numerical results and discussions We take some typical plasma parameters which are representative of low-pressure silane discharges [12]: Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3, n 0 = 0.05 –0.4 1010 cm 3, n 0d = 0.2 –1.4 107 cm 3 (the positive and negative ions being SiH 3+ and SiH3). Adopting these above parameters, we numerically solve Eqs. (14) and (18). The results are displayed in the following Figs. 1 –3. Fig. 1 reveals the effects of dust and negative ion densities as well as dust Bohm velocity on the positive ion
Fig. 1. Positive ion Bohm velocity M 0+ versus the density ratio of dust to positive ion d d with Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3: (a) d = 0.0, (b) d = 0.05, (c) d = 0.2, (d) d = 0.4.
where E 0 = (dU/dX)0 å 0 is the electric field at the sheath edge, and Sagdeev potential V(U) is V ðUÞ ¼ de ð1 expð UÞÞ þ cd ð1 expð U=cÞÞ h 1=2 i þ Mþ2 1 1 þ 2U=Mþ2 h 1=2 i þ dd Zd0 Md2 1 1 2W=Md2 :
ð17Þ
Eq. (16) is analogous to the energy conservation law for a classical particle in a potential well V(U), with E 02/2 being the analogous total energy. Analogous to the analysis of a particle in the potential well, it requires the potential V(U) < 0 in the sheath [21,22], which leads to 2 B V < 0: ð18Þ BU2 U¼0 If we set d = 0 and d d = 0, through Eq. (18) we can obtain M + > 1 = M 0+, the well-known positive ion Bohm criterion
Fig. 2. Dust Bohm velocity M 0d versus the density ratio of dust to positive ion d d with Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3: (a) d = 0.0, (b) d = 0.05, (c) d = 0.2, (d) d = 0.4.
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point. Detailed physical understanding of Sagdeev potential approach is in Ref. [22].
4. Summary
Fig. 3. Sagdeev potential V(U) versus the dimensionless spatial potential U with d d = 0.001, Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3; two satisfying solutions: curve 1 is for d = 0.0, M 0d = 1.5, M 0+ = 1.5, and curve 2 for d = 0.2, M 0d = 0.8, M 0+ = 0.8; two unsatisfying solutions: curve 3 is for d = 0.05, M 0d = 0.6, M 0+ = 0.6, and curve 4 for d = 0.4, M 0d = 0.3, M 0+ = 0.3.
Bohm velocity M 0+. First, one can notice that M 0+ increases if the dust density continues to increase. Especially, in Fig. 1(a), the positive ion velocity entering sheath must exceed its sound velocity due to the dust electrostatic drag (DED) force whose direction is in x. Also, one can see that the growth of dust Bohm velocity M 0d will make M 0+ decrease. This is due to that the relative velocity between positive ion and dust become small, and that the effect of dust on positive ion becomes weak, so the DED force becomes small. From Fig. 1(a) –(d), finally, one may notice that increasing the fraction of negative ion can effectively lower M 0+ value. If the negative ion density increases, electron density will decrease consequently, so the mobility of the negative charges can be significantly reduced. It is the modified conditions at the sheath edge that lower the positive ion Bohm velocity. However, when the negative ion density is sufficiently large d > 0.5, the effect of finite positive ion temperature on M 0+ should be taken into account. Fig. 2 shows the effects of dust and negative ion densities as well as the positive ion Bohm velocity M 0+ on the dust Bohm velocity M 0d. It is found that the growth of dust density d d will increase M 0d value. This is because that there exists a positive ion electrostatic drag force on dust which makes the dust entering-sheath velocity larger. This drag force will become strong with the growth of d d, therefore M 0d value will also become large. When M 0+ value increases, the drag force will become small, so the M 0d value will become small consequently. In the same way, increasing d value will also reduce the dust Bohm velocity M 0d owing to the modified conditions of the sheath edge. Finally, we validate the reliability of results in Figs. 1 and 2 by Sagdeev potential approach, which is usually used to obtain solutions in plasma nonlinear waves. We can notice from Fig. 3 that the curves 1 and 2 bend down, which indicates that there may exist the ‘‘potential well’’ near the U = 0 point. But the curves 3 and 4 bend up, indicating that there is no possibility of the ‘‘potential well’’ near the U = 0
In this paper, we have systemically studied the Bohm criterion for dusty electronegative plasmas composed of electrons, negative and positive ions, as well as dust grains. The dynamical sheath model of the complex plasmas is presented and the self-consistent dust charge variation is also taken into account. Sagdeev potential has been developed to determine positive ion and dust Bohm velocities. It is found that both positive ion and dust Bohm velocities increase with the growth of dust density due to the interaction between positive ions and dust grains, while both of them decrease with the increscent negative ion density owing to the sheath edge conditions modified by negative ions. Furthermore, changing one of the two Bohm velocities also can have effects on the other one. These results are examined to be reliable by the quantitative analysis of Sagdeev potential. The conclusion of this work should be adopted as valuable boundary conditions of sheath in electronegative dusty plasmas to calculate the dust and positive ion flux to a wall or to a negative electrode.
Acknowledgments This work is supported by the National Natural Science Function China (Grant Nos. 40390150, 10175013, 10010760807, 10160420799).
References [1] E.C. Whipple, Rep. Prog. Phys. 44 (1981) 1198. [2] J. Chu, I. Lin, Phys. Rev. Lett. 72 (1994) 4009. [3] H. Thomas, G.E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, D. Mohlmann, Phys. Rev. Lett. 73 (1994) 652. [4] H. Luo, M.Y. Yu, Phys. Rev., E 56 (1997) 1270. [5] J.X. Ma, J.Y. Liu, M.Y. Yu, Phys. Rev., E 55 (1997) 4627. [6] N.N. Rao, P.K. Shukla, M.Y. Yu, Planet. Space Sci. 38 (1990) 543. [7] X.G. Wang, Z.X. Wang, C.H. Wang, B. Guo, Phys. Plasmas 9 (2002) 4396. [8] a Y. Nakamura, H. Bailung, P.K. Shukla, Phys. Rev. Lett. 83 (1999) 1602; b P.K. Shukla, Phys. Plasmas 7 (1999) 139. [9] J.B. Pieper, J. Goree, R.A. Quinn, J. Vac. Sci. Technol., A 14 (1996) 519. [10] A.A. Mamun, P.K. Shukla, G.E. Morfill, Phys. Rev. Lett. 92 (2004) 095005. [11] S. Lizuka, M. Okazi, T. Gohda, Phys. Plasmas 11 (2004) L5. [12] K. Ostrikov, I.B. Denysenko, S.V. Vladimirov, S. Xu, H. Sugai, M.Y. Yu, Phys. Rev., E 67 (2003) 056408. [13] I.B. Denysenko, K. Ostrikov, S. Xu, M.Y. Yu, C.H. Diong, J. Appl. Phys. 94 (2003) 6097. [14] L. Boufendi, A. Bouchoule, Plasma Sources Sci. Technol. 11 (2002) A211.
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[21] F.F. Chen, Introduction to Plasma Physics, Plenum Press, New York, NY, 1974, p. 249. [22] R.Z. Sagdeev, Reviews of Plasma Physics, vol. 4, Consultants Bureau, New York, 1966, p. 23. [23] M.A. Lieberman, A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, Wiley Press, New York, NY, 1994, p. 168.
Bohm criterion for electronegative dusty plasmas Zheng-Xiong Wang, Yue Liu *, Li-Wen Ren, Jin-Yuan Liu, Xiaogang Wang State Key Laboratory of Materials Modification by Laser, Electron and Ion Beams, Department of Physics, Dalian University of Technology, Dalian, 116024, China Available online 31 August 2005
Abstract Bohm criterion for complex plasmas which are composed of electrons, negative and positive ions as well as dust grains is investigated with Sagdeev potential, taking into account the self-consistent dust charge variation. The numerical solutions show that the positive ion and dust Bohm velocities increase with the growth of the dust density, while both of them decrease with the growth of negative ion density. Furthermore, the interactions between the two Bohm velocities exist. The results are examined to be reliable by the quantitative analysis of Sagdeev potential. D 2005 Elsevier B.V. All rights reserved. PACS: 52.27.Cm; 52.27.Lw; 52.40.Kh Keywords: Electronegative plasma; Dust grains; Bohm criterion; Sagdeev potential
1. Introduction Dust grains are frequently observed in space and laboratory plasmas [1 –3]. In recent years, much attention has been paid to many various aspects of collective processes in dusty plasmas, such as the dusty sheath [4,5], linear and nonlinear dusty waves and instabilities [6– 8], as well as observation and mechanism of the dust lattice formation [9– 11], etc. These theoretical and experimental investigations have advanced further developments of other fields, such as the potential of spacecraft in the Earth’s magnetosphere and space, and the study of condensed matter systems, etc. In the plasma enhanced chemical vapor deposition (PCVD), multicomponent complex plasma systems are often created, containing electrons, positive and negative ions, as well as charged dust grains. In the low-pressure high-density silane (SiH4) plasmas, the dust density is sometimes up to 107 cm 3 with sizes from tens to hundreds of nanometers [12,13]. These electronegative dusty plasmas are used intensively today for device applications such as
* Corresponding author. E-mail address: [email protected] (Y. Liu). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.08.052
solar cells and thin-film transistor. Some experiments show that the dusts have significant effects on sheath structures as well as on the silicon film quality [14,15]. In order to understand the complicated sheath well and accurately calculate the ion and dust flux to a wall or an electrode, a detailed study of Bohm criterion for the low-pressure highdensity silane (SiH4) plasmas is warranted. The dust particles are often negatively charged and the value of their charge may vary depending on different charged particle densities around dusts, so the variability of the dust charge will lead some new effects in dust-acoustic solitons and dust-ion acoustic waves as well as in the dusty sheath near the wall [16,17]. In this paper, using the Sagdeev potential and self-consistent dust charge variation, we studied the Bohm criterion for the electronegative dusty plasma sheath comprising Boltzmann electrons and negative ions but cold-fluid positive ions and dust grains. It is shown that the dust and negative ion densities have effects on the dust and positive ion Bohm velocities and that both of the two Bohm velocities also affect each other. The paper is organized as follows. In Section 2, we present the steadystate sheath model and dust charging model in the sheath. In Section 3, we obtain the numerical results of Bohm criterion and discuss the physics reasons. Finally, the paper is concluded with a short summary in Section 4.
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Z.-X. Wang et al. / Thin Solid Films 506 – 507 (2006) 637 – 641
I ¼ pr2 eð8T =pm Þ1=2 n expðeqd =rT Þ;
2. Analytical models We consider a one-dimensional plasma sheath treating electrons and negative ions as thermal Boltzmann particles but positive ions and dust grains as cold fluids. The sheath edge at x = 0 is the plasma-sheath interface (interface between essentially neutral and nonneutral regions). Then x < 0 is the plasma region and x > 0 is the sheath region. The spatial potential is taken zero / = 0 at the sheath edge where the quasi-neutrality condition is n0þ n0e n0 Z0d n0d ¼ 0;
ð1Þ
with n 0e, n 0 , n 0+, and n 0d representing number densities of electron, negative ion, positive ion, and dust grain at the sheath edge, respectively; Z 0d = q 0d/e, e is the electron charge, and q 0d is dust charge at the sheath edge. Both electrons and negative ions in the sheath are assumed to obey the Boltzmann distribution [18,19],
Iþ ¼ pr2 enþ vþ 1 2eqd =rmþ v2þ ;
ð9Þ ð10Þ
where m e and m are electron and negative ion masses, respectively; q d is the dust charge in the sheath, and thus the dust surface potential is expressed as / d = q d/r. In the equilibrium state, the dust charge conservation is Ie þ I þ Iþ ¼ 0:
ð11Þ
Finally, the Poisson’s equation relates the electron, negative and positive ion as well as dust densities in the sheath to a self-consistent potential d2 /=dx2 ¼ 4p½eðnþ ne n Þ þ qd nd :
ð12Þ
nþ vþ ¼ n0þ v0þ ;
ð4Þ
Then we write dimensionless variables d e = n 0e /n 0+ , d = n 0 /n 0+, d d = n 0d/n 0+ as the density ratio of electrons, negative ions, dust grains to positive ions, respectively. Furthermore, we introduce other dimensionless quantities U = e//Te, U d = e/ d/Te denoting the normalized dust surface potential with z = rTe/e 2 (= 695rTe), then the relation Z d = |zU d| represents the charge number of dust grain measured in units of e. Also we define the temperature ratio of negative ions to electrons c = T /Te, the dimensionless spatial coordinate X = x/k D with respect to the Debye length k D = (Te/4pe 2n 0+)1/2. The dimensionless positive and dust velocities in the sheath are M + = v +/c s+ and M d = v d/c sd, respectively, where c s+ = (Te/m +)1/2 and c sd = (Z d0Te/m d)1/2 are the positive ion and dust acoustic velocities, respectively. Thus M 0+ = v 0+/c s+ and M 0d = v 0d/c sd denote the positive ion Bohm velocity and dust Bohm velocity, respectively. Thus the quasi-neutrality at the sheath edge, Eq. (1), can also be written a dimensionless form as
mþ vþ ðdvþ =dxÞ ¼ eðd/=dxÞ;
ð5Þ
de ¼ 1 d dd Z0d ;
ne ¼ n0e expðe/=Te Þ;
ð2Þ
n ¼ n0 expðe/=T Þ;
ð3Þ
where n e and n are the electron and negative ion number density in the sheath, respectively, and each negative particle species is assumed to possess a charge e, Te and T are electron and negative ion temperatures, respectively. The positive ions are accelerated in the sheath, so the cold fluid model for positive ions is a good approximation [18,19]. The positive ions in the sheath of low-pressure plasmas are ruled by the continuity and the steady-state momentum equations
where n + and v + are the positive ion density and velocity in the sheath, respectively, v 0+ is the positive ion enteringsheath velocity towards the wall and m + is the positive ion mass. In this paper the dust grains are so small (100 nm) that the gravitational force is negligible. Therefore supposing the dust grain has a wall-wards velocity v 0d at the sheath edge, then one can write continuity and momentum equations for dust grains as nd vd ¼ n0d v0d ;
ð6Þ
md vd ðdvd =dxÞ ¼ qd ðd/=dxÞ;
ð7Þ
where n d, v d, q d and m d are the density, fluid velocity, charge, and mass of the dust grain in the sheath, respectively. When a dust grain of radius r is immersed in the complex plasma, the charging currents from electrons, negative and positive ions can respectively be written as [20] Ie ¼ pr2 eð8Te =pme Þ1=2 ne expðeqd =rTe Þ;
ð8Þ
ð13Þ
where the Z 0d is the dust charge number at the sheath edge. Substituting these dimensionless variables into Eqs. (8) – (12), we obtain 2Ud 8mþ 1=2 M0þ 1 2 ¼ de expðUd UÞ pme Mþ 8cmþ 1=2 Ud U þ d exp ; c pm ð14Þ
d2 U U ¼ de expð UÞ d exp dX 2 c 1=2 2U 2W 1=2 þ 1þ 2 dd Zd 1 2 ; Mþ Md
ð15Þ
U where WðUÞ ¼ X0 Z˜ dU with Z˜ = Z d/Z d0. Integrating Eq. (15) once, we obtain
1 1 ðdU=dX Þ2 þ V ðUÞ ¼ E02 ; 2 2
ð16Þ
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639
for electropositive plasmas. If we set d d = 0, through Eq. (18) we can obtain M + > [1 d (1 1/c)]1/2 = M 0+, the positive ion Bohm criterion for electronegative plasmas [23].
3. Numerical results and discussions We take some typical plasma parameters which are representative of low-pressure silane discharges [12]: Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3, n 0 = 0.05 –0.4 1010 cm 3, n 0d = 0.2 –1.4 107 cm 3 (the positive and negative ions being SiH 3+ and SiH3). Adopting these above parameters, we numerically solve Eqs. (14) and (18). The results are displayed in the following Figs. 1 –3. Fig. 1 reveals the effects of dust and negative ion densities as well as dust Bohm velocity on the positive ion
Fig. 1. Positive ion Bohm velocity M 0+ versus the density ratio of dust to positive ion d d with Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3: (a) d = 0.0, (b) d = 0.05, (c) d = 0.2, (d) d = 0.4.
where E 0 = (dU/dX)0 å 0 is the electric field at the sheath edge, and Sagdeev potential V(U) is V ðUÞ ¼ de ð1 expð UÞÞ þ cd ð1 expð U=cÞÞ h 1=2 i þ Mþ2 1 1 þ 2U=Mþ2 h 1=2 i þ dd Zd0 Md2 1 1 2W=Md2 :
ð17Þ
Eq. (16) is analogous to the energy conservation law for a classical particle in a potential well V(U), with E 02/2 being the analogous total energy. Analogous to the analysis of a particle in the potential well, it requires the potential V(U) < 0 in the sheath [21,22], which leads to 2 B V < 0: ð18Þ BU2 U¼0 If we set d = 0 and d d = 0, through Eq. (18) we can obtain M + > 1 = M 0+, the well-known positive ion Bohm criterion
Fig. 2. Dust Bohm velocity M 0d versus the density ratio of dust to positive ion d d with Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3: (a) d = 0.0, (b) d = 0.05, (c) d = 0.2, (d) d = 0.4.
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point. Detailed physical understanding of Sagdeev potential approach is in Ref. [22].
4. Summary
Fig. 3. Sagdeev potential V(U) versus the dimensionless spatial potential U with d d = 0.001, Te = 1.2 eV, T = 0.035 eV, r = 100 nm, n 0+ = 1 1010 cm 3; two satisfying solutions: curve 1 is for d = 0.0, M 0d = 1.5, M 0+ = 1.5, and curve 2 for d = 0.2, M 0d = 0.8, M 0+ = 0.8; two unsatisfying solutions: curve 3 is for d = 0.05, M 0d = 0.6, M 0+ = 0.6, and curve 4 for d = 0.4, M 0d = 0.3, M 0+ = 0.3.
Bohm velocity M 0+. First, one can notice that M 0+ increases if the dust density continues to increase. Especially, in Fig. 1(a), the positive ion velocity entering sheath must exceed its sound velocity due to the dust electrostatic drag (DED) force whose direction is in x. Also, one can see that the growth of dust Bohm velocity M 0d will make M 0+ decrease. This is due to that the relative velocity between positive ion and dust become small, and that the effect of dust on positive ion becomes weak, so the DED force becomes small. From Fig. 1(a) –(d), finally, one may notice that increasing the fraction of negative ion can effectively lower M 0+ value. If the negative ion density increases, electron density will decrease consequently, so the mobility of the negative charges can be significantly reduced. It is the modified conditions at the sheath edge that lower the positive ion Bohm velocity. However, when the negative ion density is sufficiently large d > 0.5, the effect of finite positive ion temperature on M 0+ should be taken into account. Fig. 2 shows the effects of dust and negative ion densities as well as the positive ion Bohm velocity M 0+ on the dust Bohm velocity M 0d. It is found that the growth of dust density d d will increase M 0d value. This is because that there exists a positive ion electrostatic drag force on dust which makes the dust entering-sheath velocity larger. This drag force will become strong with the growth of d d, therefore M 0d value will also become large. When M 0+ value increases, the drag force will become small, so the M 0d value will become small consequently. In the same way, increasing d value will also reduce the dust Bohm velocity M 0d owing to the modified conditions of the sheath edge. Finally, we validate the reliability of results in Figs. 1 and 2 by Sagdeev potential approach, which is usually used to obtain solutions in plasma nonlinear waves. We can notice from Fig. 3 that the curves 1 and 2 bend down, which indicates that there may exist the ‘‘potential well’’ near the U = 0 point. But the curves 3 and 4 bend up, indicating that there is no possibility of the ‘‘potential well’’ near the U = 0
In this paper, we have systemically studied the Bohm criterion for dusty electronegative plasmas composed of electrons, negative and positive ions, as well as dust grains. The dynamical sheath model of the complex plasmas is presented and the self-consistent dust charge variation is also taken into account. Sagdeev potential has been developed to determine positive ion and dust Bohm velocities. It is found that both positive ion and dust Bohm velocities increase with the growth of dust density due to the interaction between positive ions and dust grains, while both of them decrease with the increscent negative ion density owing to the sheath edge conditions modified by negative ions. Furthermore, changing one of the two Bohm velocities also can have effects on the other one. These results are examined to be reliable by the quantitative analysis of Sagdeev potential. The conclusion of this work should be adopted as valuable boundary conditions of sheath in electronegative dusty plasmas to calculate the dust and positive ion flux to a wall or to a negative electrode.
Acknowledgments This work is supported by the National Natural Science Function China (Grant Nos. 40390150, 10175013, 10010760807, 10160420799).
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