The structure of an electronegative magnetized plasma sheath with non-extensive electron distribution

Xiu ZOU (邹秀), Huiping LIU (刘惠平), Yongzheng ZHU (朱永政),Xiaonan ZHANG (张小楠) and Minghui QIU (邱明辉)

School of Science, Dalian Jiaotong University, Dalian 116028, People’s Republic of China

Abstract

Keywords:plasma, sheath, non-extensive distribution

1.Introduction

The plasma sheath is a very narrow positive space charge layer that exists between the laboratory plasma and the surrounding wall or electrode.The thickness of the plasma sheath is equal to several electron Debye lengths.Because of chemical reactions or the adsorption of electrons, the sheath often contains negative ions.This sheath is called the electronegative sheath.The structure of the plasma sheath plays an important role in many processes such as plasma etching and material surface modification.Therefore, relevant theoretical and experimental research works are still a special topic of interest [1-14].These works have studied many plasma sheath layers with different physical conditions, such as the collision between ions and neutral particles, ion temperature and applied magnetic field.

In these research works, it is assumed that both the electron and negative ion satisfy the Boltzmann distribution at low pressure [1-14].In 2009, Ghim and Hershkowitz [4]proved that the negative ions follow the Boltzmann distribution through experiments.On this premise, Femandez Palop et al [5, 6], Wang et al [7] and Gong et al [8] studied the Bohm criterion of the electronegative sheath,respectively,and the results showed that the higher the density ratio of negative ions to positive ions and the temperature ratio of electrons to negative ions,the lower the critical velocity of positive ions to the sheath boundary.Zhao et al [9] studied the influence of negative ions on the Bohm criterion of the non-collision plasma sheath and believed that the critical value of the ion Mach number would decrease with the increase of the charge carried by the negative ion.We also have done a lot of research on the electronegative magnetized plasma sheath [7, 10-14].The main conclusion is that the magnetic field has great effects on the plasma sheath structure.Applied magnetic field makes the density distribution of electron, positive ion and negative ion larger than the case

Plasma Sci.Technol.22 (2020) 125001 (7pp)https://doi.org/10.1088/2058-6272/abb3dc without magnetic field.The existence of negative ions also has effects upon the sheath.The ion and electron densities fall more rapidly in the case of strongly electronegative plasma.

However, in these studies of the electronegative plasma sheath [1-14], the electrons are assumed to satisfy the classical Maxwell distribution, while the particles in the plasma have a long-range electromagnetic interaction, and the distribution will deviate from the Maxwell distribution.In order to describe the system, in 1988 Tsallis [15]proposed the extensive entropy,expressed aswhere k is the Boltzmann constant,Piis the probability of theithmicroscopic state, and q is the normalized non-extensive parameter.Based on this theory,Tsallis established the nonextensive statistical mechanics, and found a good description framework for the long-range interaction system.When the parameter q=1,all the results return to the case that the particles satisfy the classical Maxwell distribution.q＜1 and q＞1 correspond to super-extensive distribution and sub-extensive distribution, respectively.Non-extensive statistics is a new statistical mechanics,which has been widely applied in recent years, especially in solving various astrophysics and plasma physics problems, including plasma instability,plasma sheath,nonlinear waves,etc.The study of plasma with non-extensive distribution has become an important issue [16-28].

Assuming that the electrons satisfy the non-extensive distribution, Hatami [16, 17] studied two sheath models with and without collisions, respectively.The sheath criterion was modified.The research results showed that the velocity of ions at the sheath edge was inversely proportional to the value of the non-extensive parameter q.Recently,Hatami,Tribeche and Mamun [18] revised the Debye length in the magnetized sheath model.Zhao et al[19]studied the influence of the nonextensive parameter q on the emission of secondary electrons in the plasma sheath, and the results showed that with the increase of q, the thickness of the sheath decreased and the number of secondary electrons in the sheath increased.In addition, Liu et al [20, 21] studied Debye length and Bohm criterion of a dusty plasma sheath model with non-extensive distribution of electrons.Tantawy,Tribeche and Moslem[22]studied the nonlinear structure of a plasma system with nonextensive distributed electrons and positrons.Emamuddin et al[23]and Navab Safa et al[24]also studied dusty plasma sheaths containing non-extensive distribution of electrons,respectively.So far, there are a few studies on the electronegative magnetized plasma sheath with non-extensive distribution of electrons.Mehdipoor and Mohsenpour [25]studied the double layer structure of electronegative plasma in a vertical applied magnetic field.In this work, the electron was described by non-extensive distribution.Borgohain et al[26,27]investigated the behavior of two kinds of sheath.One sheath system contains two kinds of the non-extensive distribution of the electrons.Another is an electronegative sheath system with one kind of the non-extensive distribution of the electrons.The particle number density, potential, and the net space charge were numerically studied.The results showed that the thickness of the sheath reduces with the increase of q.Recently, Basnet and Khanal [28] used the fluid model to study the characteristics of a magnetized plasma sheath in the presence of two groups of electrons,namely,low-temperature electrons and high-temperature electrons.The results signify that the non-extensive parameter q and concentration ratio have significant effects on the magnetized plasma sheath parameters.

Figure 1.Geometry of the magnetized plasma sheath model.

In the related work mentioned above,the electronegative plasma sheath of Borgohain’s work[27]is unmagnetized,and the magnetic field in Basnet’s work [28] is parallel to the board.The structure of an electronegative magnetized plasma sheath with non-extensive electron distribution remains unclear.In this work, the external magnetic field is oblique.Using the method of fluid mechanics, we study the structure of an electronegative magnetized plasma sheath with nonextensive electrons.The Bohm criterion of this sheath model is derived and compared with the sheath model with Maxwell distribution [1], and the effects of the non-extensive parameter q on the sheath structure is studied.The corresponding discussions, including sheath potential, ion density, electron density, negative ion density, and the net charge density are presented.Also, the effects of magnetic field on the sheath structure are compared with the sheath model with Maxwell distribution [10].

2.Sheath model

First, a model of the electronegative plasma sheath is established,as shown in figure 1.Considering that the thickness of the sheath is much smaller than the scale of the wall, the plasma sheath model has one-dimensional coordinate space and three-dimensional velocity space.The constant applied magnetic field lies in the()x z, plane, and the angle between the magnetic field and the x-axis is θ.At the sheath edge, the coordinate is =x0, and the space potential isφ= 0.

It is assumed that the plasma sheath contains electrons,negative ions, and positive ions.According to the theory of Tsallis [15], the electrons follow non-extensive distribution,and the velocity distribution function is:

where

is a normalized constant, Γ is the standard gamma function,andme,veandTeare the mass, the velocity and the temperature of the electrons, respectively.ne0is the electron density at the sheath edge.The electron density can be obtained by integrating the velocity distribution function[16, 17, 19, 27]

The negative ions follow the Boltzmann distribution[27]

neandT−are the density and temperature of the negative ions, respectively.

The ions are cold, which satisfy the continuity equation and the momentum equation of the fluid

whereni,viandmiare the density, velocity and mass of the positive ion, respectively.

The sheath potential satisfies the Poisson equation

At the sheath edge, the quasi-neutral condition isni0=ne0+n−0.

To normalize the model equations (2)-(6), we introduce the following variables:Hereis electronic Debye length,is ion cyclotron frequency.Because it is one dimensional coordinate space,The unit vector of the oblique magnetic field is expressed aswhich are used to indicate the density and temperature of negative ions.From equations (2)-(6) we get:

From equations (9) and (10), considering the boundary conditions at the sheath edge:x=0, Φ →0,Ni→1,we get

Substituting equations (7), (8) into equation (13)

The integral of equation(15)leads to Sagdeev potential ()ΦV[2,19].Considering the condition of Sagdeev potential at the sheath edge

There is a pre-sheath region between the plasma and the sheath [2, 3].The ions are accelerated in the pre-sheath region.The electric field in the pre-sheath region cannot be ignored [1-3, 27, 29, 30].Taking into account the ion drift motion,we get velocity components of ions at the sheath edge[14, 31-34],Substitutingu yi0into inequality (17), then we have

Figure 2.Ion Mach number versus θ for different q(α = 0.05, τ = 5.0).

This is the Bohm criterion for the electronegative magnetized plasma sheath with non-extensive electron distribution.After considering the ion drift motion, the ion Mach number does not depend on the electric field and the magnitude of the magnetic field.The ion Mach number is related to the nonextensive parameter q, the angle of the magnetic field, the density and temperature of negative ions.Many studies [14,31-34]obtained similar results after considering the drift motion of ions.The angle of magnetic field has an effect on the Mach number.For an electron-ion magnetized plasma sheath with Maxwell distribution (q=1,θ≠0,α=0), from inequality(18)we getMi≥cosθ.It was reported by Chodura[1].For an electron-ion sheath with non-extensive distribution (q≠1,θ=0,α=0),from inequality(18)we getThis result is similar to those results in the literature [16, 17, 19,21, 26, 27].

3.Results and discussion

In this paper, we focus on the effect of the non-extensive parameter q on the sheath structure.Figures 2-4 intuitively show the variation of the ion Mach number with the angle of the magnetic field, the density, and the temperature of the negative ions, respectively.The effect of the parameter q is revealed in each figure.As shown in figure 2,the increase of the external magnetic field angle will reduce the value of the ion Mach number.According to the work of Chodura[1],the potential drop in the presheath region becomes larger for increasing angles, which also causes the decrease of the ion Mach number.Compared with the classical Maxwell distribution sheath (q=1), the ion Mach number increases forq＜1,and the ion Mach number decreases forq＞1.Whenq＞1, the average velocity of electrons is relatively slower.As a result,less kinetic energy is needed for ions to enter the sheath, or to say, less kinetic energy is needed for ions to countering the accumulation of electrons.The structure of sheath with different q is discussed later.In figures 3 and 4,we also find that the ion Mach number decreases with the increase of the density and the temperature of the negative ions.This result is the same as that in the sheath with the classical Maxwell distribution (q=1).The two figures also reveal the influence of the non-extensive parameter q on the ion Mach number.It can be seen that when the density of the negative ions increases or the temperature of the negative ions increases, the influence of the parameter q on the ion Mach number is weakened.

In the numerical simulation of equations (7)-(13), the commonly used parameters in the experiments are taken, for example, the density of argon plasma is 5 × 1014m−3,and the electron temperature is 3.48 ×104K.To prevent the divergence of the numerical results, we assume that the electric field at the sheath edge takes an infinitesimal value[16, 17, 26, 27, 29, 30, 32].Figures 5-10 have the same parameters withE0=0.01,α=0.05andτ= 5.0.

Figure 5 shows the comparison of space potential distribution between the electronegative magnetized plasma sheath with non-extensive distribution and with Maxwell distribution.It shows the influence of the parameter q on the space potential of the electronegative magnetized plasma sheath.Compared with the case of Maxwell distribution(q=1), whenq＜ 1, the fall-off of the space potential is slower, and whenq＞1, the fall-off of the space potential is more rapid.Compared with Maxwell distribution(q=1),for super-extensive distribution (q＜1), the average velocity of electrons is relatively faster,and for sub-extensive distribution(q＞1),the average velocity of electrons is relatively slower.The velocity of the electron affects the potential.

Figures 6 and 7 show the distributions of three kinds of particles between the electronegative magnetized plasma sheath with non-extensive distribution and with Maxwell distribution,respectively.In figure 5,compared with the case ofq=1, whenq＜1, the space potential falls more slowly.Therefore, in figure 6, the electron density falls relatively slowly,and the density distributions of ions and negative ions also fall slowly.Suppose the position of the wall is set at the zero-electron density [17, 27, 32]; as a result, the sheath will be thicker at lower values of q.Similar results were reported by Hatami [17] and Zhao et al [19].Whenq＞1, the space potential drops rapidly.Correspondingly in figure 7, the electron density drops to zero faster, as well as the density distribution of ions and negative ions.Figure 8 shows the space charge density distribution of the electronegative magnetized plasma sheath with different q values.Compared with the electronegative magnetized plasma sheath with the Maxwell distribution (q=1), whenq＜1, the electron density distribution drops down slowly,so the peak of the net charge curve is smaller and closer to the wall.In this case,more energetic electrons penetrate into the sheath region;as a result, the net space charge density decreases.Whenq＞1,accordingly the electron density decreases rapidly,which also leads to a higher peak of the net charge curve,and the peak is far away from the wall.

Figure 3.Ion Mach number versus q and α (θ = °30 , τ = 5.0).

Figure 4.Ion Mach number versus q and τ (θ = °30 , α = 0.05).

In earlier work[10],we have investigated the structure of an electronegative magnetized plasma sheath with the Maxwell distribution of electrons (q=1).The results show that the magnetic field and negative ions have great effects on the plasma sheath structure.In contrast to a sheath without a magnetic field, the density distribution curves of electrons,positive ions and negative ions of the magnetized sheath are relatively higher.It is due to Lorentz force of the external magnetic field on the positive ions.Both the strength and angle of the magnetic field affect the ion velocity.That also causes the change of density of all the particles.When the electrons follow non-extensive distribution (q≠1), the effects of magnetic field on the structure of the sheath are like the results of the previous research work [10].Take super-extensive distribution (q＜1) as an example; the effects of magnitude of magnetic field and the angles on the sheath are shown in figures 9 and 10, respectively.In figure 9, comparing the two cases ofB=0.08 T andB=0.8T, the density curves fall slowly under the stronger magnetic field.When the angle increases,the z-axis component of the magnetic field increases.Consequently, the density curves fall slowly, as shown in figure 10.All these effects are due to the Lorentz force of the magnetic field.The similar conclusions of the case of subextensive distribution (q＞ 1) are not repeated here.

Figure 5.Variation of the sheath potential for different values of q( B =0.15 T,θ = °30).

Figure 6.Variation of the normalized density distribution of charged particles (B =0.15 T,θ = °30).

Figure 7.Variation of the normalized density distribution of charged particles (B =0.15 T,θ = °30).

Figure 8.The normalized net charge density for different values of q (B =0.15 T,θ = °30).

Figure 9.Various of the normalized density distribution of charged particles for different magnitudes of magnetic field( =q 0.9,θ = °30).

Figure 10.Various normalized density distributions of charged particles for different angles of magnetic field( =q 0.9, B =0.15 T).

4.Conclusions

In this paper, an electronegative magnetized plasma sheath model with non-extensive distribution of electron is established, and the Bohm criterion affected by the non-extensive parameter q is theoretically derived.Besides the angle of magnetic field,the density and the temperature of the negative ions, the ion Mach number is also affected by the nonextensive parameter q.Compared with the electronegative magnetized plasma sheath with Maxwell distribution(q=1),whenq＜1, the value of the ion Mach number increases.Whenq＞ 1, the value of the ion Mach number decreases.So, the velocity distribution state of electrons affects the sheath structure,and the energy needed for the ion to enter the sheath is dependent on the parameter q.Through numerical simulation, it is concluded that compared with the case ofq=1, whenq＜1, the space potential falls more slowly, as well as the ion, negative ion and electron density, and the peak of the net charge density curve is closer to the wall.Whenq＞ 1, the space potential falls faster, as well as the density of the ion, negative ion and electron, especially the electron density drops to zero faster, and the peak of the net charge density curve is far away from the wall.These results are also due to the velocity distribution of electrons,due to the kinetic energy of electrons.For an electronegative magnetized plasma sheath with non-extensive electron distribution, the effects of the magnetic field are like the results of the sheath with Maxwell electron distribution.The above simulation results show that the non-extensive parameter q has a nonnegligible influence on the structure of the electronegative magnetized plasma sheath.These conclusions may be useful in the related plasma boundary problem or plasma processing.

Acknowledgments

This work is supported by National Natural Science Foundation of China (No.10605008) and the Scientific Research Foundation of the Education Department of Liaoning Province, China (Nos.L2011069, JDL2017012).