Effect of collisions, ionisation and non-extensivity on sheath formation in an electronegative warm plasma under electron emission from the wall
Subject Areas : Journal of Theoretical and Applied Physics
Yetendra Jha
1
,
Mayank Kumar
2
,
Hitendra K. Malik
3
1 - Plasma Science and Technology Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi, India.
2 - Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India.
3 - Plasma Science and Technology Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi, India.
Keywords:
Abstract :
Effect of collisions, ionisation and non-extensivity in an electronegative warm plasma associated with electron emission from the wall
Yetendra Prasad Jha*1, Mayank Kumar2, Hitendra K. Malik1
1Plasma Science and Technology Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi-110016, INDIA
3Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi-110016, INDIA
Corresponding author: (a)yatit20@gmail.com*
(b) Mayank.Kumar@mech.iitd.ac.in
(c) hkmalik@physics.iitd.ac.in
The effect of collisions, ionisation and non-extensivity is studied numerically in an electronegative warm plasma associated with electron emission from the wall. Electrostatic potential, space charge density, net negative charge density and emitted electron beam density are plotted with the normalised distance to see the effects of aforesaid parameters The negative ion is described with fluid equations to see it’s effect of the mass ratio (negative ion to positive ion) on emitted electron beam density inside the sheath. The three types of electronegative plasma taken are 
, 
and 
. The emitted beam electron density is more in number at the wall for higher collisional and ionisation case and less in higher mass ratio. For super-extensive case the emitted beam electrons is lesser than compared to the Boltzmann distributed electrons. The sheath thickness is found to be more in higher mass ratio, emitted beam electrons and super -extensive case while for higher collisional and ionisation case the sheath thickness is less.
Introduction
The plasma and material interaction results in the formation of a sheath which is very crucial for application point of view in the areas of plasma processing, [1], plasma-based treatment [2,3] etching, sputtering [4-6] electric space propulsion [7], nitriding [8,9] and microelectronics [10]. The sheath characteristics depend on various parameters like electronegativity [11,12], collisions between the ions and neutrals [13-19], ionization [20-22], magnetic field [22], temperature of the charged species [23]. The sheath characteristics also depends upon the electrons emitted from the wall. Electron emission from the wall may takes place due to various reasons like bombardment of electrons with wall this type of emission is basically called secondary electron emission [24, 25]. The primary electrons in the plasma having energy greater then 30eV can produces the secondary electrons [26-28]. The positive ions can also produce secondary electrons from the wall but for this emission the positive ion must have energy have greater than 1keV. There is also in some case the emission is due to heating of electrode due to applying of high voltage to the electrode or due to heating of wall by the plasma itself present in the container as in the fusion plasma [29-32]. The emitted electrons from the wall significantly put impact on electron velocity distribution function (EVDF) and it reduces the electrostatic potential of the sheath which increases the heat flux to the wall and more and more electrons are produced from the wall. The emission affects the ion implantation, particle/energy transport, and sputtering. Various theoretical model [33-38] is given to explain the sheath with emitted electrons. Few of researchers explained the electron emission in the electronegative plasma [39] but they have considered the Boltzmann distribution of electrons and negative ion the plasma. In reality there exist some system where particles deviate from the Boltzmann distribution [40,41]. The use of Maxwellian distribution in Boltzmann-Gibbs statistics is valid till the system is in equilibrium. While for system where a non-equilibrium stationary state exists and the long-range interaction occurs just like plasma and gravitational system, the Maxwellian distribution is inadequate for the description of the system. For such a system, generally the Tsallis or non-extensive statistics is applied and the entropy is given by
(1)
where 
, 
, 
and 
. This entropy is non-additive in nature. Suppose we have two independent systems a and b and if 
and 
be the corresponding probabilities of the two system. The total entropy of their composed system 
is given by
(2)
Which is not just the sum of the individual entropies of the two system. For 
the system is called as sub-extensive and if 
the system is called as super-extensive. As 
in Eq. (1) the Boltzmann-Gibbs entropy is obtained, i.e., now the entropy is extensive in nature.
(3)
When the particles are not in thermodynamic equilibrium, they usually deviate from the Boltzmann distribution. For example, in space plasmas [42]-[44] there are larger number of particles present in the tail which have higher values in the velocity distribution function. Experimentally [45] and theoretically [46-50] researchers have shown this behaviour. Tsallis [51] have explained the non-extensive case in the system. Also, negative ion in the high pressure do not follow Boltzmann distribution as the nature of negative ion is just similar to that of positive ions i.e., the mobility of negative ions is same as that of positive ion. This behavior of negative ion is considered in the model of Dhawan et al., [52]. Also, in the model [39] they have considered the cold plasma and neglected the collisions between neutrals and ions. Effect of ionization was also not present in their model. In experiments, it is generally found that the positive ions have finite temperature and there is ionization and collisions present in the plasma which affect the sheath properties. These days mostly use of electronegative plasma is preferable for all the plasma processing applications as it helps in to control the potential of electrode. The non-extensivity in the plasma is observed for the system which is distant from thermodynamic equilibrium. Hence, it is very important to take all these parameters in the model for a better understanding of the sheath formation mechanism. In this paper, a mathematical model is developed to study the characteristics of the sheath which takes all the parameters like collisions between neutral and positive /negative ions, ionization, non-extensivity in the plasma and electron emission from the wall. The fluid equation is written for negative ions to see the effect of mass ratio (negative to positive ion) on electrostatic potential, sheath thickness and the density of emitted electrons at the wall. The effect of emitted beam electrons, non- extensivity on the electrostatic potential, net space charge densities, net negatively charged species and sheath thickness is investigated.
Theoretical model
The position of imaginary plane at 
corresponds to the position where the quasi-neutrality holds sheath is started as depicted in figure 1. The continuity equations and momentum transfer equations are written to describe the ions (positive and negative) and beam electrons (emitted from the wall). The plasma electrons are described by their non-extensive distribution. The Poisson’s equation relates the density with electric potential and distance from the edge of the sheath 
towards the wall.
Figure 1. Schematic diagram depicting sheath associated with cathode wall emitting beam electrons.
The flux and energy of the emitted beam electrons inside the sheath are given as follow
(4)
(5)
Eqn. (1) & Eqn. (2) together give Eqn. (3)
(6)
Continuity equation for the positive and negative ions are
(7)
(8)
Momentum transfer equation for both the ions read
(9)
(10)
Non-extensive distribution of the electrons is given by
(11)
Poisson’s equation
(12)
The quasi-neutrality condition reads
(13)
Here the subscripts p and n stand for positive and negative ions, respectively, subscripts b and e stand for beam electrons and plasma electrons, respectively, M is the mass of ions and q is used to denote the non-extensivity parameter. V is potential inside the sheath with respect to edge, 
is the potential applied at the cathode, 
is the attachment rate, 
is the detachment rate, 
is the Boltzmann constant, v is the velocity, n is the density and T is the temperature.
Using normalization parameters as 
, 
, 
, 
, , 
and 
, the various physical quantities used in Eqn. (4) to Eqn. (13) are normalised. The normalized quantities appear to be
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
; 
where 
Here 
and is the Debye length. The normalized equations are as follows
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
In the above equations, the frist derivative is denoted by single prime and second derivative is denoted by double prime. Equations (14) – (20) are coupled equations which are solved by numerical method. Researchers have used numerical methods for solving different existing problems [53-56]. We use Runge-Kutta (ODE 45) to solve these coupled equations. The initial conditions for solving the equations is similar to our previous work [57].
Results and discussion
For different values of 
and 
, the normalised electrostatic potential, net space charge density, positive and net negative charged density for different values of collisional parameter and electron beam current density are plotted as a function of normalised distance in figure 2. As increases the net space charge density gets decreased near the wall (figure 2(b)) which is due to the increase in net negative charged species inside the sheath (figure 2(c)) and hence there is a reduction in the slope of the electrostatic potential (dotted line in figure 1) . Under this situation, the sheath thickness gets increased. This beahvior matches with the work of other researchers [39] where they have obtained the larger thickness with the increase in emitted electron beam current density (from the wall).
(a)
(b)
For increase in the collsions between the positive/negative ions with the netural atoms, there is an increase in the slope of electrostatic potential. This is because of the increased postive chrage density in the presence of higher collisions (dotted line in figure 2(c)); this increases the net space charge density inside the sheath (dotted line in figure 2(b)). The sheath thickness gets decreased for this case.
(c)
(d)
Figure 2 (a)Variation of normalised electrostatic potential (b) net space charge density (c) positive and net negative charged species (d) emitted electron beam density as a function of normalized distance for different values of 
and 
when 
, 
, 
, 
, 
q=0.99, A=0.0005, D=0.0015, , 
, 
, 
, 
and 
.
On increasing the value of , the normalised electron beam denisty inside the sheath and at the sheath edge () get increased which results in the enhanced total negative charged species density inisde the sheath (shown in figure 2 (b)). Hence, the net space charge density decreases and the slope of the electrostatic potential also decreases, as shown in figure 2(a). As collsions between positive/negative ions increases, the emitted electrons are more in number near the wall (blue dotted line in figure. 2(d)) because the slope of the potential gets increased (figure 2(a)) i.e., the electric field gets increased and the emitted electrons experience a greater force.
(a)
(b)
The effect of non-extensivity and the ionization rate on electrostatic potential and the net space charge density is plotted in figure 3(a) and figure 3(b) respectively. It is observed that as the value of extensivity changes from q=0.5 to q=0.99 from non -extensive distribution of electrons to Boltzmann distribution there is lesser number of electrons inside the sheath and the net space charge density is more inside the sheath as shown in figure 3(b) and hence the electrostatic potential gradient gets increased. This also depicts that the sheath thickness gets reduced. This result matches with the observation of [48] Also, with the increase in the ionization rate there is more production of positive ions and electrons. The positive ions get attracted towards the wall (since applied potential to the wall is negative) and hence the sheath thickness gets reduced which is observed in the figure 3(a) and the electrostatic potential gradient gets increased. Due to increase of this potential gradient the force on the emitted beam electrons increases and the density gets increase inside the sheath and hence more emitted beam electrons are observed in the sheath (in figure 3 (c)) for the case of Boltzmann distributed electrons (dotted line) and high ionisation (solid line) as compared to the non-extensive distributed electrons (dashed line).
(c)
Figure 3. (a)Normalized electric potential (b) net space charge density (c) emitted electron beam density as a function of normalized distance for different values of q and r when , , 
, , , A=0.0005, D=0.0015, , 
, and .
(a)
The three types of electronegative plasma taken are , and so that the effect of mass ratio of negative ion to positive ion on the electrostatic potential and the density of emitted beam electrons can be understood. As it is shown in the figure 4. The gradient of electrostatic potential decreases with the increase in the negative ion to positive ion (mass ratio) as shown in figure 4 (a). And hence there is increase in the thickness of the sheath. This result is consistent with the result of [52]. The density of emitted beam electrons decreases inside the sheath as shown in figure 4 (b) as the negative ion to positive ion (mass ratio) is increased this is because the electric field is lesser for the higher mass ratio so lesser force is experience by the emitted beam electrons hence lesser density is found.
(b)
Fig. 4 (a) Normalised electric potential (b) normalised electron beam density as a function of normalized distance for different ratio (mass of negative ion to positive ion 
) when 
, 
, A=0.0005, D=0.0015, 
, 
, 
, , 
and q=0.5.
Conclusions
In the presence of electron beam current density, the net negative charge density increases while the net space charge density and the slope of electrostatic potential reduce. There is an increase in the sheath thickness for the larger electron beam current density. For increasing collisions between the ions and neutrals, there is an increase in the positive charge density and the net space charge density. These result in larger slope of the electrostatic potential. The sheath thickness gets decreased for the higher collisions. For non-extensive case the gradient in the electrostatic potential, net space charge density and the emitted electron beam density is less while these values get increased with the consideration of Boltzmann distributed electrons and higher ionization rate. The sheath thickness is greater for the non-extensive case while it is lesser for Boltzmann distribution and high ionization. For higher mass ratio (negative to positive) the gradient in the electrostatic potential and the emitted electron beam density is smaller while the thickness in the sheath is larger. Our model is very much useful in the field of plasma processing (etching) where there is presence of negative ions which has a considerable effect. Also, the negative ion’s mass plays a crucial role on the thickness of the sheath and other’s sheath parameters. Also during the plasma processing the effect of collision, ionization and the non-extensivity in the plasma can be explained by using our model. Our model is also useful to control the electrode potential, where the emission of secondary electrons or thermionic electrons takes place from the electrode and the electrostatic potential and the thickness of sheath gets affected. The results may find other interesting applications in negative ion sources and film depositions.
Acknowledgement
The author Yetendra Prasad Jha thanks the UGC-CSIR (Government of India) for supporting this work financially.
References
1. Lieberman, M.A., Lichtenberg, A.J.: Principles of Plasma Discharges and Materials Processing. Wiley, London (2005)
2. Dorranian, D., Abedini, Z., Hojabri, A., Ghoranneviss, M.: Structural and optical characterization of PMMA surface treated in low power nitrogen and oxygen RF plasmas. J. Non-Oxide Glasses 1, 217–229 (2009)
3. Jafari, M., Dorranian, D.: Surface modifcation of PMMA polymer in the interaction with oxygen-argon RF plasma. J. Theor. Appl. Phys. 5, 59–66 (2011)
4. Shibata, K., Nishida, Y., Yugami, N.: Production of Thin-film by Sheet-shaped Plasma. Appl. Phys, 63 559 (1994)
5. Shibata, K., Ito, H., Yugami, N., Miyazaki, T., Nishida, Y.: Production of Nb thin film by ECR sheet plasma. Thin solid films 386 291-294 (2001)
6. San Andrés, E., del Prado, A., Mártil, I., González, G., Martı́nez, F. L., Bravo, D., Fernández, M.: Physical properties of plasma deposited SiOx thin films. Vacuum 67 525-529 (2002)
7. Oudini, N., Tadjine, R., Alim, M. M., Bendib, A.: Electron transport across a magnetic filter: Magnetic field gradient effects on plasma properties. Physics of Plasmas 26 113505 (2019)
8. Singh, O., Dahiya, P., Malik, H.K., Kumar, P., Singh, V.: Investigation of titanium nitride thin flms treated in hot cathode arc discharge plasma system. Appl. Sci. Lett. 2, 37–41 (2016)
9. Singh, O., Malik, H.K., Dahiya, R.P., Kumar, P.: Infuence of negative bias voltage on structural and mechanical properties of nanocrystalline TiNx thin flms treated in hot cathode arc discharge plasma system. Ceram. Int. 42, 18019–18024 (2016)
10. Makabe, T., Petrovic, Z.L.: Plasma electronics: applications in microelectronic device fabrication. CRC Press (2006)
11. Araghi, F., Dorranian, D.: Effect of negative ions on the characteristics of plasma in a cylindrical discharge. Journal of Theoretical and Applied Physics 7 1-8 (2013)
12. Dorranian, D., & Alizadeh, M.: Effect of negative oxygen ions on the characteristics of plasma in a cylindrical DC discharge. Journal of Theoretical and Applied Physics: 8 1-6 (2014)
13. Crespo, R. M., Ballesteros, J., Palop, J. F., Hernandez, M. A., Lucena-Polonio, M. V., Diaz-Cabrera, J.M.: Study of the electropositive to electronegative sheath transition in weakly ionized plasmas. Plasma Sources Science and Technology 20 015019 (2011)
14. Yasserian, K., Aslaninejad, M.: On the space-charge formation in a collisional magnetized electronegative plasma. Physics of Plasmas 19 073507 (2012)
15. Yasserian, K., Aslaninejad, M., Borghei, M., Eshghabadi, M.: Electronegative plasma-sheath characteristics over a wide range of collisionality. (2010)
16. Moulick, R., Mahanta, M. K., Goswami, K.S.: Effect of collision parameters in electronegative plasma sheath with two species of positive ions. Physics of Plasmas 20 094501 (2013)
17. Hatami, M. M.: Charged particle density distributions in multi-component plasmas with two species of warm positive ions. Physics of Plasmas 20 013509 (2013)
18. Borgohain, D. R., Saharia, K.: Behavior of collisional sheath in electronegative plasma with q-nonextensive electron distribution. Physics of Plasmas. 25 032122 (2018)
19. Hatami, M. M., Niknam, A. R.: Characteristics of Positive Ions in the Sheath Region of Magnetized Collisional Electronegative Discharges. Plasma Science and Technology. 16 552 (2014)
20. Valentini, H. B.: Sheath formation in low-pressure discharges. Plasma Sources Science and Technology, 9 574 (2000)
21. Yasserian, K., Aslaninejad, M., Ghoranneviss, M.: Structure of presheath-sheath in magnetized electronegative plasma. Physics of Plasmas, 16 023504 (2009)
22. Malik, H.K., U. Stroth.: Nonlinear solitary waves (solitons) in inhomogeneous magnetized warm plasma with negative ions and nonisothermal electrons. Plasma Sources Sci. Technol. 17 035005 (2008)
23. Morales Crespo, R.: Positive ion temperature effect on the plasma-wall transition. Physics of Plasmas, 25 063509 (2018)
24. Lafleur, Trevor, Pascal Chabert, and Jean-Paul Booth.: Secondary electron induced asymmetry in capacitively coupled plasmas. Journal of Physics D: Applied Physics 46 135201 (2013)
25. Nakamura, K., Mitsuaki T., Hideo S.: Energy measurements of sheath-accelerated secondary electrons in plasma immersion ion implantation. Surface and Coatings Technology 169 57-60 (2003)
26. Ryutov, D. D.: Axial electron heat loss from mirror devices revisited. Fusion science and technology 47 148-154 (2005)
27. Raitses, Y., Kaganovich, I. D., Khrabrov, A., Sydorenko, D., Fisch, N. J., Smolyakov, A.: Effect of Secondary Electron Emission on Electron Cross-Field Current in $ E\times B $ Discharges. IEEE Transactions on Plasma Science 39 995-1006 (2011)
28. Lafleur, Trevor, Pascal Chabert, and Jean-Paul Booth.: Secondary electron induced asymmetry in capacitively coupled plasmas. Journal of Physics D: Applied Physics 46 135201 (2013)
29. Modinos, A.: Theory of thermionic emission. Surface Science 115 469-500 (1982)
30. Sheehan, J. P., Raitses, Y., Hershkowitz, N., Kaganovich, I., Fisch, N. J.: A comparison of emissive probe techniques for electric potential measurements in a complex plasma. Physics of Plasmas, 18 073501 (2011)
31. Sitnov, V. I., Énder, A. Y., Yakovlev, E. V.: Current-voltage characteristics of underneutralized Knudsen thermionic converters with electron emission from the collector. Technical Physics 44 373-380 (1999)
32. Campanell, M. D.: Possible mitigation of tokamak plasma–surface interactions using thermionic divertor plates with inverse sheaths. Physics of Plasmas 27 042511 (2020)
33. Stangeby, P. C.: Plasma sheath transmission factors for tokamak edge plasmas. The Physics of fluids 27 682-690 (1984)
34. Sizonenko, V. L.: The effect of high secondary-electron emission on plasma-layer. ZHURNAL TEKHNICHESKOI FIZIKI 51 2283-2291 (1981).
35. Ou, J., Zhao, X., Gan, C.: Plasma-wall interaction in an electrostatic sheath of plasma containing a monoenergetic electron beam. Physics of Plasmas 23 043510 (2016)
36. Taccogna, F., Longo, S., Capitelli, M.: Plasma-surface interaction model with secondary electron emission effects. Physics of Plasmas 11 1220-1228 (2004)
37. Schwager, L. A.: Effects of secondary and thermionic electron emission on the collector and source sheaths of a finite ion temperature plasma using kinetic theory and numerical simulation Physics of Fluids B: Plasma Physics 5 631-645 (1993)
38. Ordonez, C. A.: Fully kinetic plasma‐sheath theory for a cold‐electron emitting surface. Physics of Fluids B: Plasma Physics 4 778-783 (1992)
39. Amemiya, H., Annaratone, B. M., Allen, J. E.: The double sheath associated with electron emission into a plasma containing negative ions. Journal of plasma physics 60 81-93 (1998)
40. Ghim, Y. C., Hershkowitz, N.: Experimental verification of Boltzmann equilibrium for negative ions in weakly collisional electronegative plasmas. Applied Physics Letters 94 151503 (2009)
41. Franklin, R. N., Snell J.: The Boltzmann relation in electronegative plasmas: When is it permissible to use it? Journal of Plasma Physics 64 131-153 ((2000))
42. Bacal, M., McAdams, R., Lepetit, B.: The negative ion mean free path and its possible implications. In AIP Conference Proceedings 1390 13-21 American Institute of Physics (2011)
43. Meige, A., Boswell, R. W.: Electron energy distribution functions in low-pressure inductively coupled bounded plasmas. Physics of plasmas 13 092104 (2006)
44. Leubner, M. P., Vörös, Z.: A nonextensive entropy path to probability distributions in solar wind turbulence. Nonlinear Processes in Geophysics 12 171-180 (2005)
45. Singh, H., Graves, D. B.: Measurements of the electron energy distribution function in molecular gases in an inductively coupled plasma. Journal of Applied Physics 87 4098-4106 (2000)
46. Arghand-Hesar, A., Esfandyari-Kalejahi, A., Akbari-Moghanjoughi, M.: Effects of a monoenergetic electron beam on the sheath formation in a plasma with a q-nonextensive electron velocity distribution. Physics of Plasmas 24 063504 (2017)
47. Borgohain, D. R., Saharia, K., Goswami, K. S.: Behavior of plasma sheath with nonextensively distributed two-temperature electrons and isothermal ions. Physics of Plasmas 23 122113 (2016)
48. Safa, N. N., Ghomi, H., Niknam, A. R.: Effect of the q-nonextensive electron velocity distribution on a magnetized plasma sheath. Physics of Plasmas 21 082111 (2014)
49. Shahmohammadi, N., Dorranian, D., Hakimipagouh, H.: Effect of superthermal electrons on the characteristics of dust acoustic solitary waves in a magnetized hot dusty plasma with dust charge fluctuation. Canadian Journal of Physics 93 344-352 (2015)
50. Farhad Kiyaei, F., Dorranian, D.: Effects of external magnetic field on oblique propagation of ion acoustic cnoidal wave in nonextensive plasma. Physics of Plasmas 24 012107 (2017)
51. Lyra, M. L., & Tsallis, C.: Nonextensivity and multifractality in low-dimensional dissipative systems. Physical review letters 80 53 (1998)
52. Dhawan, R., Malik, H. K.: Sheath characteristics in plasma carrying finite mass negative ions and ionization at low frequency. Chinese Journal of Physics 66 560-572 (2020)
53. Malik, L., Escarguel, A., Kumar, M., Tevatia, A., Sirohi, R.S.: Uncovering the remarkable contribution of lasers peak intensity region in holography. Laser Physics Letters, 18, p.086003 (2021).
54. Malik, L., Rawat, S., Kumar, M., Tevatia, A.: Simulation studies on aerodynamic features of Eurofighter Typhoon and Dassault Rafale combat aircraft. Materials Today: Proceedings, 38, pp.191-197 (2021).
55. Malik, L., Kumar, M., Singh, I.V.: A Three-Coil Setup for Controlled Divergence in Magnetic Nozzle. IEEE Transactions on Plasma Science, 49, pp.2227-2237 (2021).
56. Malik, L., Tevatia, A.: Comparative Analysis of Aerodynamic Characteristics of F16 and F22 Combat Aircraft using Computational Fluid Dynamics. Defence Science Journal, 71 (2021.).
57. Jha, Y. P., Malik, M., Kumar, M., Malik, H. K.: Electronegative warm plasma and material interaction under secondary electron emission. Materials Today: Proceedings 65 3483-3487 (2022)
.
