Modification of the Scharfetter–Gummel method for calculating the flux of charged particles for simulation of a radio-frequency capacitive coupled discharge

V.S. Zheltukhina*, M.S. Fadeevab**, V.Ju. Chebakovab***

a Kazan National Research Technological University, Kazan, 420015 Russia

b Kazan Federal University, Kazan, 420008 Russia

 E-mail: *vzheltukhin@gmail.com, **manysha-98@mail.ru, ***vchebakova@mail.ru

Received January 15, 2017

Full text PDF

Abstract

A numerical method of solving of nonstationary convection-diffusion equation for charged particle densities in capacitive coupled RF discharge is presented. The method allows us to calculate the particle concentration together with the flux density simultaneously at very rapid changes of particle density and electrical field in the electrode sheath. The method is a modification of the Scharfetter–Gummel algorithm. Firstly, implicit difference approximation of the equation constructed by the integro-interpolation method is used for calculating the charged particles density. Then the flux density of charged particles is calculated. Our modification allows to carry out the calculation at milder restriction on the time step than the Courant condition. In addition, diffusion coefficient may be variable unlike with the original Scharfetter–Gummel algorithm.

Keywords: mathematical modeling, radio-frequency capacitive coupled discharge, modified Scharfetter–Gummel method, convection-diffusion equation, flux density of charged particles

Acknowledgments. The study was supported by the Russian Science Foundation (project no. 16-11-10299).

References

1. Polak L.S., Sinyarev G.B., Slovetsky D.I. et al. Plasma Chemistry. Polak L.S., Lebedev Yu.A. (Eds.). Novosibirsk, Nauka, 1991. 325 p. (In Russian)

2. Berdichevskii M.G., Marusin V.V. Coating, Etching, and Modifying Polymers Using Low-Enthalpy Non-Equilibrium Plasma: Review. Novosibirsk, Inst. Teplofiz., 1993. 107 p. (In Russian)

3. Gadzhiev M.K., Isakaev E.K., Tyuftyaev A.S., Yusupov D.I. A high-power low-temperature air plasma generator with a divergent channel of the output electrode. Tech. Phys. Lett., 2016, vol. 42, no. 1, pp. 79–81. doi: 10.1134/S106378501601020X.

4. Badriev I.B., Zheltukhin V.S., Chebakova V.Yu. On solving of some nonlinear boundary and initial boundary value problems. Materialy XXII Mezhdunar. simpoziuma ``Dinamicheskie i tekhnologicheskie problemy mekhaniki konstruktsii i sploshnykh sred'' im. A.G. Gorshkova [Proc. XXII Int. Symp. ``Dynamic and Technological Problems of Mechanics of Constructions and Continuous Media'' Dedicated to A.G. Gorshkov]. Moscow, TRP, 2016, pp. 31–33. (In Russian)

5. Karengin A.G. Physics and Technology of Low-Temperature Plasma. Tomsk, Izd. Tomsk. Politekh. Univ., 2008. 129 p. (In Russian)

6. Lebedev Yu.A., Tatarinov A.V. Electrodynamics of microwaves in a coaxial non-regular waveguide partly filled with plasma. Plasma Sources Sci. Technol., 2004, vol. 13, no. 1, pp. 1–7. doi: 10.1088/0963-0252/13/1/001.

7. Amirov R.Kh., Isakaev E.Kh., Shavelkina M.B., Yusupov D.I., Shatalova T.B., Emirov R.M. Plasma-jet synthesis of carbon nanostructures and a device for its implementation. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2014, vol. 156, no. 4, pp. 112–119. (In Russian)

8. Lebedev Yu.A., Tatarinov A.V., Titov A.Yu., Epstein I.L. Two-dimensional model of a non-equilibrium strongly non-uniform microwave discharge in a DC external field. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2014, vol. 156, no. 4, pp. 120–132. (In Russian)

9. Wang H.-B., Sun W.-T., Lia H.-P., Bao Ch.-Y. Discharge characteristics of atmospheric-pressure radio-frequency glow discharges with argon/nitrogen. Appl. Phys. Lett., 2006, vol. 89, no. 16, art. 161504, pp. 161504-1–161504-3. doi: 10.1063/1.2362631.

10. Lebedev Yu.A. Comparison of the Non-Equilibrium Plasma of Self-Maintained Electrical Discharges. Radio-Frequency Discharge in Wave Fields. Samara, 1990, pp. 25–62. (In Russian)

11. Abdullin I.Sh., Zheltukhin V.S., Chebakova V.Yu. Radio-frequency capacitive coupled discharge: Modeling (review). Vestn. Kazan. Tekhnol. Univ., 2014, vol. 17, no. 23, pp. 9–14. (In Russian)

12. Tikhonova N.V., Zheltukhin V.S., Chebakova V.Yu., Borodaev I.A. Mathematical model of radio frequency plasma processing of multilayer materials of shoe uppers. Vestn. Kazan. Tekhnol. Univ., 2012, vol.15, no.17, pp. 36–39. (In Russian)

13. Goedheer W.J. Lecture notes on radio-frequency discharges, dc potentials, ion and electron energy distributions. Plasma Sources Sci. Technol., 2000, vol. 9, pp. 507–516. doi: 10.1088/0963-0252/9/4/306.

14. Raizer Y.P., Shneider M.N., Yatsenko N.A. Radio-Frequency Capacitive Discharges. Boca Raton, Fla., CRC Press, 1995. 304 p.

15. Raizer Yu.P. Gas Discharge Physics. Dolgoprudnyi, Intellekt, 2009. 736 p. (In Russian)

16. Chebert P., Braithwaite N. Physics of Radio-Frequency Plasmas. Cambridge, Cambridge Univ. Press, 2011. 386 p.

17. Badriev I.B., Chebakova V.Yu. Mathematical simulation of low-temperature RF plasma in argon. Matematicheskoe modelirovanie i kraevye zadachi: Trudy Desyatoi Vseros. nauch. konf. s mezhdunar. uchastiem [Mathematical Modeling and Boundary Value Problems: Proc. 10th All-Russ. Sci. Conf. with Int. Participation]. Vol. 2. Samara, 2016, pp. 17–21. (In Russian)

18. Badriev I.B., Chebakova V.Y., Zheltukhin V.S. Capacitive coupled RF discharge: Modelling at the local statement of the problem. J. Phys.: Conf. Ser., 2017, vol. 789, no. 1, art. 012004, pp. 1–4. doi: 10.1088/1742-6596/789/1/012004.

19. Badriev I.B., Zheltukhin V.S., Chebakova V.Yu. Numerical study of low-temperature RF plasma in argon at elevated pressures. Materialy XI Mezhdunar. konf. po neravnovesnym protsessam v soplakh i struyakh (NPNJ'2016) [Proc. XI Int. Conf. on Non-Equilibrium Processes in Nozzles and Jets (NPNJ'2016)]. Moscow, 2016, pp. 477–479. (In Russian)

20. Scharfetter D.L., Gummel H.K. Large-signal analysis of a silicon Read diode oscillator. IEEE Trans. Electron Devices, 1969, vol. 16, no. 1, pp. 64–77. doi: 10.1109/T-ED.1969.16566.

21. Kulikovsky A.A. A more accurate Scharfetter–Gummel algorithm of electron transport for semiconductor and gas discharge simulation. J. Comput. Phys., 1995, vol. 119, no. 1, pp. 149–155. doi: 10.1006/jcph.1995.1123.

22. Degtyarev L.M., Favorskii A.P. Flow variant of the sweep method for difference problems with strongly varying coefficients. USSR Comput. Math. Math. Phys., 1969, vol. 9, no. 1, pp. 285–294. doi: 10.1016/0041-5553(69)90021-4.

23. Golshtein E.G., Tretyakov N.V. Modified Lagrangians. Moscow, Nauka, 1989. 400 p. (In Russian)

24. Badriev I.B., Zadvornov O.A. Iterative methods for solving variational inequalities of the second kind with inversely strongly monotone operators. Russ. Math., 2003, vol. 47, no 1, pp. 18–26.

25. Badriev I.B., Zadvornov O.A. On the convergence of the dual-type iterative method for mixed variational inequalities. Differ. Equations, 2006, vol. 42, no. 8, pp. 1180–1188. doi: 10.1134/S001226610608012X.

26. Glowinski R., Le Tallec P. Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. Philadelphia, PA, SIAM, 1989. x, 292 p.

27. Chebakova V.Yu. Numerical simulation of the high-frequency capacitive discharge. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, vol. 157, no. 2, pp. 126–140. (In Russian)

28. Chebakova V.Yu. Modeling of radio-frequency capacitive discharge under atmospheric pressure in argon. Lobachevskii J. Math., 2017, vol. 38, no. 6, pp. 1165–1178. doi: 10.1134/S1995080217060154.

29. Samarskii A.A., Gulin A.V. Numerical Methods. Moscow, Fizmatlit, 1989. 432 p. (In Russian)

30. Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Convection-Diffusion Problems. Moscow, Editorial URSS, 1999. 248 p. (In Russian)

31. Lisovskii V.A. Features of the α - γ transition in a low-pressure rf argon discharge. Tech. Phys., 1998, vol. 43, no. 5, pp. 526–534. doi: 10.1134/1.1259033.



For citation: Zheltukhin V.S., Fadeeva M.S., Chebakova V.Ju. Modification of the Scharfetter–Gummel method for calculating the flux of charged particles for simulation of a radio-frequency capacitive coupled discharge. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 444–457. (In Russian)


The content is available under the license Creative Commons Attribution 4.0 License.