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Magnetized plasma sheath plays an important role in semiconductor processing, material surface modification, film deposition, etc. In plasma experiments and discharge applications, multi-ion plasma consisting of more than two kinds of ions often exists. For a long range interacting plasma system, non-Maxwellian electrons can be described by the non-extensive distribution of Tsallis. In this work, a fluid model with one-dimensional spatial coordinates and three-dimensional velocity coordinates is established for the multi-ion plasma sheath. It is assumed that the electron velocity in the sheath follows a non-extensive distribution, and the background helium ions and different kinds of impurity ions are magnetized in a magnetic field with a certain tilt angle. The effects of non-extensive parameters, impurity ions and oblique magnetic field on the number density, velocity, wall potential and kinetic energy of ions in the multi-ion magnetic sheath are investigated by numerical simulation. The results show that in the helium-hydrogen or helium-argon mixed plasma sheath, the ionic velocity along the vertical wall direction decreases with the increase of the non-extensive parameters, the number density of ions and electrons in the sheath, the sheath thickness , and the kinetic energy of ions at the wall decrease. When the concentration of impurity ions increases, the kinetic energy of ions on the wall is independent of the type of ions. With the increase of magnetic field intensity, the number density of helium ions and the velocity along the vertical wall fluctuate along the sheath edge, and the fluctuation amplitude increases with the decrease of non-extensive parameters, while the heavy ions have no obvious fluctuation. In addition, the effects of the types and concentrations of impurity ions on the related properties of the sheath are also analyzed. With the increase of the magnetic field intensity, the number density and the velocity along the vertical wall direction fluctuate at the sheath edge, and the fluctuation amplitude increases with the decrease of the non-extensive parameter, whereas there are no significant fluctuations for heavy ions. In addition, when impurity ions are heavy ions, the absolute value of wall potential increases with the increase of impurity ion concentration and the decrease of non-extensibility parameters, and the kinetic energy of background ions increases at the wall surface. When the impurity ion is a light ion, the absolute value of the wall potential decreases with the increase of the impurity ion concentration and the decrease of the non-extensibility parameter, and the kinetic energy of the background ion at the wall decreases.
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Keywords:
- multi-ion plasma /
- non-extensive distribution /
- magnetized sheath /
- Bohm criterion
[1] Chen F F 1974 Introduction in Plasma Physics (New York: Plenum Press) pp291–296
[2] Langmuir I 1929 Phys. Rev. 33 954
Google Scholar
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[4] Nejman A, Kaminska I, Jasinska I, Celichowski G, Cieslak M 2020 Molecules 25 3476
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[5] Nejman A, Kaminska I, Cieslak M 2019 Plasmas Process. Polym. 16 e1800194
Google Scholar
[6] Qiu H B, Zhou Z Y, Peng X K, et al. 2020 Phys. Rev. E 101 043206
Google Scholar
[7] Suliali N J, Goosen W E, Janse van Vuuren A, et al. 2022 Vacuum 195 110698
Google Scholar
[8] Manos M D, Flamm D 1989 Plasma Etching: An Introduction (New York: Academic Press) pp143–175
[9] Baba K, Hatada R, Tanaka Y 2006 Surf. Coat. Technol. 201 8362
[10] Brown H L, Thornley S A, Wakeham S J, Thwaites M J, Curry R J, Baker M A 2015 J. Phys. D Appl. Phys. 48 335303
Google Scholar
[11] Shibata K, Ito H, Yugami N, Miyazaki T, Nishida Y 2001 Thin Solid Films 386 291
Google Scholar
[12] Aanesland A, Rafalskyi D, Bredin J, et al. 2015 IEEE Trans. Plasma Sci. 43 321
Google Scholar
[13] Saslaw W C, Arp H 1986 Phys. Today 39 61
[14] Caceres M O 1999 Braz. J. Phys. 29 125
Google Scholar
[15] Tsallis C, Mendes R, Plastino A R 1998 Physica A 261 534
Google Scholar
[16] Hatami M M, Tribeche M 2018 IEEE Trans. Plasma Sci. 46 868
Google Scholar
[17] Singh H, Graves D B 2000 J. Appl. Phys. 88 3889
Google Scholar
[18] Tsallis C 2009 J. Phys. 39 337
Google Scholar
[19] Silva R, Plastino A R, Lima J 2002 Phys. Lett. A 249 401
Google Scholar
[20] Lima J A S, Silva G R, Santos J 2000 Phys. Rev. E 61 3260
Google Scholar
[21] Liu Y, Liu S Q, Zhou L 2013 Phys. Plasmas 20 043702
Google Scholar
[22] Navab Safa N, Ghomi H, Niknam A R 2014 Phys. Plasmas 21 082111
Google Scholar
[23] Hatami M M 2015 Phys. Plasmas 22 023506
Google Scholar
[24] 邹秀, 刘惠平, 张小楠, 邱明辉 2021 物理学报 70 015201
Google Scholar
Zou X, Liu H P, Zhang X N, Qiu M H 2021 Acta Phys. Sin. 70 015201
Google Scholar
[25] Fouial N, Tahraoui A, Annou R 2016 Phys. Plasmas 23 113702
Google Scholar
[26] 陈龙, 孙少娟, 姜博瑞, 段萍, 安宇豪, 杨叶慧 2021 物理学报 70 245201
Google Scholar
Chen L, Sun S J, Jiang B R, Duan P, An Y H, Yang Y H 2021 Acta Phys. Sin. 70 245201
Google Scholar
[27] Chen L, An Y H, Sun S J, Duan P, Jiang B R, Yang Y H, Cui Z J 2022 Plasma Sci. Technol. 24 074011
Google Scholar
[28] Chen L, Yang Y H, An Y H, Duan P, Sun S J, Cui Z J, Kan Z C, W F Gao 2023 Plasma Sci. Technol. 25 035003
Google Scholar
[29] Ishikawa J 2000 Rev. Sci. Instrum. 71 1036
Google Scholar
[30] Sarma B K, Sarma A, Bailung H, Chutia J 1998 Phys. Lett. A 244 127
Google Scholar
[31] Amemiya H 1990 J. Phys. D: Appl. Phys 23 999
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Google Scholar
[33] Lee D, Hershkowitz N, Severn G D 2007 Appl. Phys. Lett. 91 041505
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Google Scholar
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Google Scholar
[36] Tskhakaya D, Kuhn S, Tomita Y 2006 Contrib. Plasma Phys. 46 649
Google Scholar
[37] 李家祺, 崔怀愈, 赵东迪, 安博, 赵永蓬 2022光学学报 42 1134022
Li J Q, Cui H Y, Zhao D D, An B, Zhao Y P 2022 Acta Opt. Sin. 42 1134022
[38] 张雅, 渠宇霄, 赵凯悦, 何寿杰, 赵雪娜, 李庆 2019 真空科学与技术学报 39 237
Google Scholar
Zhang Y, Qu Y X, Zhao, K Y, He S J, Zhao X N, Li Q 2019 J. Vac. Sci. Technol. 39 237
Google Scholar
[39] 洪文玉, 严龙文, 王明旭, 程均, 钱俊 2008 核聚变与等离子体物理 28 17
Google Scholar
Hong W Y, Yan L W, Wang M X, Cheng J, Qian J 2008 Nucl. Fusion Plasma Phys. 28 17
Google Scholar
[40] 孟令义 2022 博士学位论文 (合肥: 中国科学技术大学)
Meng L Y 2022 Ph. D. Dissertation (Hefei: University of Science and Technology of China
[41] Hatami M M, Kourakis I 2022 Sci. Rep. 12 6950
Google Scholar
[42] Hatami M M, Niknam A R, Shokri B, Ghomi H 2008 Phys. Plasmas 15 053508
Google Scholar
[43] Basnet S, Khanal R 2019 Plasma Phys. Control. Fusion 61 065022
Google Scholar
[44] Franklin R N 2003 J. Phys. D: Appl. Phys. 36 34
Google Scholar
[45] Silva J R, Plastino A R, Lima J A S 1998 Phys. Lett. A 249 401
Google Scholar
[46] Trabeche M, Djebarni L, Amour R 2010 Phys. Plasmas 17 042114
Google Scholar
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图 1 等离子体磁化鞘层模型示意图
Figure 1. Schematic diagram of plasma magnetization sheath model.
图 2 非广延参量对玻姆速度及鞘层中两种离子x方向的速度变化的影响 ($\theta = 15^\circ $, $B = 0.06$, $\delta = 0.1$) (a) 玻姆速度分布; (b) 两种离子x方向的速度分布
Figure 2. Influence of non-extensive covariates on Bohm’s velocity and velocity changes in the x-direction of two ions in the sheath layer ($\theta = 15^\circ $, $B = 0.06$, $\delta = 0.1$): (a) Bohm velocity distribution; (b) velocity distribution of two ions in the x direction.
图 3 非广延参量和杂质离子浓度对离子流通量的影响($\theta = $$ 15^\circ $, $B = 0.06$) (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$的离子流通量; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$的离子流通量
Figure 3. The influence of non-extensive parameters and impurity ion concentration on ion flux ($\theta = 15^\circ $, $B = 0.06$): (a) The ion flux of ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) the ion flux of ${\mathrm{A}}{{\mathrm{r}}^ + }$.
图 4 非广延参量和杂质离子浓度对${\text{H}}{{\text{e}}^{+}}$数密度和电子数密度分布的影响 ($\theta = 15^\circ $, $B = 0.06$) (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$数密度; (b) 电子数密度
Figure 4. The effect of non-extensive parameter and impurity ion concentrations on ions and electrons number density distribution ($\theta = 15^\circ $, $B = 0.06$): (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$ number density; (b)electron number density.
图 5 非广延参量对鞘层空间净电荷和电势分布的影响($\theta = 15^\circ $, $B = 0.06$, $\delta = 0.1$) (a) 空间净电荷分布; (b) 电势分布
Figure 5. The effect of non-extensive parameters on net charge and potential in sheath space ($\theta = 15^\circ $, $B = 0.06$, $\delta = 0.1$): (a) Space net charge distribution; (b) potential distribution.
图 6 非广延参数对鞘层壁面电势和鞘层厚度的影响($ \theta = $$ 15^\circ $, $B = 0.06$) (a) 壁面电势; (b) 鞘层厚度
Figure 6. The effect of non-extensive parameters on the floating wall potential and sheath thickness ($ \theta = 15^\circ $, $B = $$ 0.06$): (a) Wall potential; (b) sheath thickness.
图 7 磁场强度和非广延参量对离子数密度分布的影响 ($\theta = 15^\circ $, $\delta = 0.1$) (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$
Figure 7. The effect of magnetic field intensity and non-extensive parameters on ion number density distribution ($\theta = 15^\circ $, $\delta = 0.1$): (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$.
图 8 磁场强度和非广延参量对${\text{H}}{{\text{e}}^{+}}$沿x方向速度分布的影响 ($\theta = 15^\circ $, $\delta = 0.1$) (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$
Figure 8. The effect of magnetic field intensity and non-extensive parameters on velocity distribution of ${\text{H}}{{\text{e}}^{+}}$ in x direction ($\theta = 15^\circ $, $\delta = 0.1$): (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$.
图 9 磁场角度对鞘层离子密度分布的影响 ($B = 0.06$, $\delta = 0.1$, $q = 0.7$) (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$
Figure 9. The effect of magnetic field angle on ion density distribution in sheath ($B = 0.06$, $\delta = 0.1$, $q = 0.7$): (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$.
图 10 磁场角度对鞘层离子速度分布的影响 ($B = 0.06$, $\delta = 0.1$, $q = 0.7$) (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$
Figure 10. The effect of magnetic field angle on the distribution of sheath ion velocity ($B = 0.06$, $\delta = 0.1$, $q = 0.7$): (a) ${\mathrm{H}}{{\mathrm{e}}^ + }$; (b) ${\mathrm{A}}{{\mathrm{r}}^ + }$
图 11 杂质离子种类、浓度和非广延参量对鞘层厚度和壁面电势的影响 ($B = 0.06$, $\theta = 15^\circ $) (a) $q = 0.7$; (b) $q = 1.3$
Figure 11. The effect of impurity ion type, concentration and non-extensive covariates on sheath thickness and wall potential ($B = 0.06$, $\theta = 15^\circ $): (a) $q = 0.7$; (b) $q = 1.3$.
图 12 杂质离子种类、浓度和非广延参量对本底离子动能的影响 ($B = 0.06$, $\theta = 15^\circ $) (a) 杂质离子为${\mathrm{A}}{{\mathrm{r}}^ + }$($q = 0.7$); (b) 杂质离子为${\mathrm{A}}{{\mathrm{r}}^ + }$($q = 1.3$); (c) 杂质离子为${{\mathrm{H}}^ + }$($q = 0.7$); (d) 杂质离子为${{\mathrm{H}}^ + }$($q = 1.3$)
Figure 12. The effect of impurity ion type, concentration and non-extensive covariates on kinetic energy of local ion ($B = 0.06$, $\theta = 15^\circ $): (a) The impurity ion is ${\mathrm{A}}{{\mathrm{r}}^ + }$($q = 0.7$); (b) the impurity ion is ${\mathrm{A}}{{\mathrm{r}}^ + }$($q = 1.3$); (c) the impurity ion is ${{\mathrm{H}}^ + }$($q = 0.7$); (d) the impurity ion is ${{\mathrm{H}}^ + }$($q = 1.3$).
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[1] Chen F F 1974 Introduction in Plasma Physics (New York: Plenum Press) pp291–296
[2] Langmuir I 1929 Phys. Rev. 33 954
Google Scholar
[3] Li S, Han K Q, Rong H P, Li X Z, Yu M H 2013 J. Appl. Phys. 131 40250
[4] Nejman A, Kaminska I, Jasinska I, Celichowski G, Cieslak M 2020 Molecules 25 3476
Google Scholar
[5] Nejman A, Kaminska I, Cieslak M 2019 Plasmas Process. Polym. 16 e1800194
Google Scholar
[6] Qiu H B, Zhou Z Y, Peng X K, et al. 2020 Phys. Rev. E 101 043206
Google Scholar
[7] Suliali N J, Goosen W E, Janse van Vuuren A, et al. 2022 Vacuum 195 110698
Google Scholar
[8] Manos M D, Flamm D 1989 Plasma Etching: An Introduction (New York: Academic Press) pp143–175
[9] Baba K, Hatada R, Tanaka Y 2006 Surf. Coat. Technol. 201 8362
[10] Brown H L, Thornley S A, Wakeham S J, Thwaites M J, Curry R J, Baker M A 2015 J. Phys. D Appl. Phys. 48 335303
Google Scholar
[11] Shibata K, Ito H, Yugami N, Miyazaki T, Nishida Y 2001 Thin Solid Films 386 291
Google Scholar
[12] Aanesland A, Rafalskyi D, Bredin J, et al. 2015 IEEE Trans. Plasma Sci. 43 321
Google Scholar
[13] Saslaw W C, Arp H 1986 Phys. Today 39 61
[14] Caceres M O 1999 Braz. J. Phys. 29 125
Google Scholar
[15] Tsallis C, Mendes R, Plastino A R 1998 Physica A 261 534
Google Scholar
[16] Hatami M M, Tribeche M 2018 IEEE Trans. Plasma Sci. 46 868
Google Scholar
[17] Singh H, Graves D B 2000 J. Appl. Phys. 88 3889
Google Scholar
[18] Tsallis C 2009 J. Phys. 39 337
Google Scholar
[19] Silva R, Plastino A R, Lima J 2002 Phys. Lett. A 249 401
Google Scholar
[20] Lima J A S, Silva G R, Santos J 2000 Phys. Rev. E 61 3260
Google Scholar
[21] Liu Y, Liu S Q, Zhou L 2013 Phys. Plasmas 20 043702
Google Scholar
[22] Navab Safa N, Ghomi H, Niknam A R 2014 Phys. Plasmas 21 082111
Google Scholar
[23] Hatami M M 2015 Phys. Plasmas 22 023506
Google Scholar
[24] 邹秀, 刘惠平, 张小楠, 邱明辉 2021 物理学报 70 015201
Google Scholar
Zou X, Liu H P, Zhang X N, Qiu M H 2021 Acta Phys. Sin. 70 015201
Google Scholar
[25] Fouial N, Tahraoui A, Annou R 2016 Phys. Plasmas 23 113702
Google Scholar
[26] 陈龙, 孙少娟, 姜博瑞, 段萍, 安宇豪, 杨叶慧 2021 物理学报 70 245201
Google Scholar
Chen L, Sun S J, Jiang B R, Duan P, An Y H, Yang Y H 2021 Acta Phys. Sin. 70 245201
Google Scholar
[27] Chen L, An Y H, Sun S J, Duan P, Jiang B R, Yang Y H, Cui Z J 2022 Plasma Sci. Technol. 24 074011
Google Scholar
[28] Chen L, Yang Y H, An Y H, Duan P, Sun S J, Cui Z J, Kan Z C, W F Gao 2023 Plasma Sci. Technol. 25 035003
Google Scholar
[29] Ishikawa J 2000 Rev. Sci. Instrum. 71 1036
Google Scholar
[30] Sarma B K, Sarma A, Bailung H, Chutia J 1998 Phys. Lett. A 244 127
Google Scholar
[31] Amemiya H 1990 J. Phys. D: Appl. Phys 23 999
[32] Rotkina L, Xiong Y, Jiang C L, McDougall S 2022 Microsc. Microanal. 28 84
Google Scholar
[33] Lee D, Hershkowitz N, Severn G D 2007 Appl. Phys. Lett. 91 041505
Google Scholar
[34] Severn G D, Wang X, Ko E, Hershkowitz N 2003 Phys. Rev. Lett. 90 145001
Google Scholar
[35] Severn G, Yip C S, Hershkowitz N, Baalrud S D 2017 Plasma Sources Sci. Technol 26 055021
Google Scholar
[36] Tskhakaya D, Kuhn S, Tomita Y 2006 Contrib. Plasma Phys. 46 649
Google Scholar
[37] 李家祺, 崔怀愈, 赵东迪, 安博, 赵永蓬 2022光学学报 42 1134022
Li J Q, Cui H Y, Zhao D D, An B, Zhao Y P 2022 Acta Opt. Sin. 42 1134022
[38] 张雅, 渠宇霄, 赵凯悦, 何寿杰, 赵雪娜, 李庆 2019 真空科学与技术学报 39 237
Google Scholar
Zhang Y, Qu Y X, Zhao, K Y, He S J, Zhao X N, Li Q 2019 J. Vac. Sci. Technol. 39 237
Google Scholar
[39] 洪文玉, 严龙文, 王明旭, 程均, 钱俊 2008 核聚变与等离子体物理 28 17
Google Scholar
Hong W Y, Yan L W, Wang M X, Cheng J, Qian J 2008 Nucl. Fusion Plasma Phys. 28 17
Google Scholar
[40] 孟令义 2022 博士学位论文 (合肥: 中国科学技术大学)
Meng L Y 2022 Ph. D. Dissertation (Hefei: University of Science and Technology of China
[41] Hatami M M, Kourakis I 2022 Sci. Rep. 12 6950
Google Scholar
[42] Hatami M M, Niknam A R, Shokri B, Ghomi H 2008 Phys. Plasmas 15 053508
Google Scholar
[43] Basnet S, Khanal R 2019 Plasma Phys. Control. Fusion 61 065022
Google Scholar
[44] Franklin R N 2003 J. Phys. D: Appl. Phys. 36 34
Google Scholar
[45] Silva J R, Plastino A R, Lima J A S 1998 Phys. Lett. A 249 401
Google Scholar
[46] Trabeche M, Djebarni L, Amour R 2010 Phys. Plasmas 17 042114
Google Scholar
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