† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11475220 and 11405208), the Program of Fusion Reactor Physics and Digital Tokamak with the CAS “One-Three-Five” Strategic Planning, the National ITER Program of China (Grant No. 2015GB101003), and the Higher Education Natural Science Research Project of Anhui Province, China (Grant No. 2015KJ009).

The properties of a collisionless plasma sheath are investigated by using a fluid model in which two species of positive ions and secondary electrons are taken into account. It is shown that the positive ion speeds at the sheath edge increase with secondary electron emission (SEE) coefficient, and the sheath structure is affected by the interplay between the two species of positive ions and secondary electrons. The critical SEE coefficients and the sheath widths depend strongly on the positive ion charge number, mass and concentration in the cases with and without SEE. In addition, ion kinetic energy flux to the wall and the impact of positive ion species on secondary electron density at the sheath edge are also discussed.

Plasma with multiple positive ion species has been found to have a wide range of applications in lab plasmas,^{[1]} space plasmas,^{[2]} and fusion plasma devices.^{[3]} The study of plasma sheath with multiple ion species is an important topic theoretically and experimentally since the sheath plays a very important role in determining the plasma-wall interactions and strongly affects the edge-plasma properties. For a weakly collisional sheath plasma with multiple positive ion species, Riemann pointed out that the ion velocities at the sheath edge must satisfy the Bohm criterion^{[4]}

*c*

_{si}and

*v*

_{i}are the ion sound speed and individual drift velocity and

*n*

_{0e}and

*n*

_{0i}are the electron and ion densities at the sheath edge, respectively. Apparently, an infinite number of combinations of ion velocities satisfy inequality (1). However, two extreme cases are usually considered. One is that each species satisfies its own Bohm velocity at the sheath edge. The other is that all ions reach the sheath edge with the common sound velocity. For the first case, the effects of the mass ratio and the relative concentrations of two ion species on the Bohm criterion are discussed in Refs. [5] and [6]. The ion Bohm velocity increases with the increase of the charge number of positive ions.

^{[4,7]}Though a huge number of contributions have been devoted to the Bohm criterion for multiple ions, the effect of the SEE is not taken into account,

^{[4–13]}since it has been found from simulations that SEE can change the Bohm criterion in a single-ion plasma;

^{[14]}therefore, it is important to study the Bohm criterion of multiple ion species including the effect of SEE.

The characteristics of the sheath containing two single-charged ions with different mass values have been investigated by several authors.^{[13,15–17]} It is shown from these references that the two-ion-species plasma sheath is affected by ion temperature, external applied magnetic field and negative ion. However, the effect of SEE on a plasma sheath containing two ion species is still absent. For a single-ion sheath, SEE can cause the sheath potential drop and sheath width to decrease, it can even give rise to the sheath instability if the SEE coefficient reaches a critical value *γ*_{c} (defined as the value when the electric field equals zero at the wall).^{[18–25]} For *γ* > *γ*_{c}, the sheath structure becomes complex.

In this work, we will investigate a sheath of plasma with two species of positive ions and secondary electrons. In Section 2, the fluid equations are described. In Section 3, the Bohm criterion and the results of the analysis of the model are presented. In Section 4, conclusions are given.

In this section, we consider a collisionless plasma model containing electrons, two species of positive ions, and secondary electrons emitted from the wall. As shown in Fig. *x* = 0 is defined as the plasma-sheath boundary, *x* < 0 is the bulk plasma region and *x* > 0 is the plasma sheath region.

The electron density in the sheath can be described by the Boltzmann distribution in the thermodynamic equilibrium^{[18–23]}

*n*

_{e}and

*T*

_{e}are the density and temperature of primary electrons, respectively.

*n*

_{e0}is the primary electron density at the sheath edge, and

*e*is the elementary electron charge.

In the steady state, for a collisionless plasma sheath, the continuity and momentum transport equations for cold ions are described as follows:

*m*,

_{i}*n*,

_{i}*Z*, and

_{i}*v*are the mass, density, charge number and velocity of the

_{i}*i*-th positive ion, respectively. The subscript

*i*= 1, 2 denotes single- and multi-charged ion, respectively.

When primary electrons impinge on the solid material, secondary electrons can be emitted from the wall. Usually, the ion-inducted SEE is small compared with the electron-induced SEE, and the ion-inducted SEE is ignored unless ion impact energies of *T*_{i} ≥ 1 keV.^{[26,27]} Secondary electrons emitted from the wall are low in energy due to the interplay between primary electrons and the wall. The peak of energy distribution of secondary electrons from the wall is generally on the order of a few electronvolts. Thus secondary electrons in the sheath can be assumed to follow the conservation of flux and energy^{[21]}

*n*

_{s}and

*v*

_{s}are the density and velocity of secondary electrons, respectively. The subscripts “0”and “w” denote the locations at the sheath edge and the wall, respectively.

According to Eqs. (

*j*is the flux density.

*j*

_{i},

*j*

_{e}, and

*j*

_{s}can be given, respectively, by

*γ*is the SEE coefficient.

Charge neutrality at the sheath edge *x* = 0 is

*ε*

_{0}is the permittivity of free space.

For convenience, we introduce the dimensionless variables as follows: *ξ* = *x*/*λ*_{De}, *φ* = *eϕ*/*T*_{e}, *N*_{e,i,s} = *n*_{e,i,s}/*n*_{e0}, *u*_{i} = *v*_{i}/*c*_{s1}, *u*_{sw} = *v*_{sw}/*c*_{s1}, and *μ* = *m*_{1}/*m*_{e}, where *λ*_{De} = [*ε*_{0}*T*_{e}/(*n*_{e0}*e*^{2})]^{1/2} is the electron Debye length and

In this section, the sheath characteristics are investigated numerically. For clarity, we assume that the first ion species, Ar^{+} is the main component of positive ion in plasma. At the sheath edge, *ξ* = 0, *φ* = 0 and the edge electric field *E*_{0} = − d*φ*/d*ξ *|_{ξ=0} = 0.01 is set instead of zero due to the more numerical stable solutions since the sheath structure in the case *E*_{0} ≪ 1 is very similar to that in the case *E*_{0} = 0.^{[17,28,29]} At the wall, the normalized velocity of secondary electrons is assumed to be *u*_{sw} = 5.^{[21]} The boundary velocity *u*_{10} can be determined by the sheath criterion.

The ion velocities at the sheath edge, containing two positive ion species, have been investigated by many authors.^{[4–12]} The Bohm criterion for the sheath, is still an open issue. Especially, the two-stream instability has been mentioned in recent years. The two-stream instability is not observed in the parameter range explored and each ion species satisfies its individual Bohm criterion for a collisionless plasma.^{[6–9,30]} So in the absence of ion-ion streaming instability, we can assume

The ion velocity at the sheath edge will be affected in the presence of SEE, from Eq. (

*V*(

*φ*) is called the Sagdeev potential.

^{[31]}At the sheath edge, the Sagdeev potential satisfies the boundary conditions

*V*|

_{φ=0}= 0 and

*∂V*/

*∂φ*|

_{φ=0}= 0. Combining Eqs. (

^{2}

*V*/d

*φ*

^{2}|

_{φ=0}≤ 0, we can obtain

When SEE is neglected, equation (

*N*

_{20}= 0 in inequality (

*u*

_{10}≥ 1, which is the usual Bohm criterion for cold ions.

^{[32]}

When SEE is taken into consideration in two-ion-species plasma, the critical value of ion velocity at the sheath edge *u*_{10} can be obtained from Eqs. (*N*_{20}. Figure *u*_{10} increases monotonically with SEE coefficient *γ* for each of Ar–He, Ar, and Ar–Xe plasmas while *N*_{20} = 0.1. Especially no matter what kind of positive ion is contained in the plasma, the effect of SEE on the boundary ion velocity is very obvious when the SEE coefficient approaches to 1. The different species of ion only slightly modifies the critical ion velocity when *N*_{20} = 0.1. Figure *γ*_{c}. The critical value of ion velocity increases with the rising of the ion mass ratio. Comparing Fig. *γ* < *γ*_{c}; however, the critical ion velocity in the Ar–Xe plasma is the greatest in the three cases when *γ* = *γ*_{c} in Fig.

The effects of ion mass ratio, SEE coefficient and the density and charge number of the second ion species on the wall potential are shown in Fig. *m*_{2}/*m*_{1} and increases with the rising of the charged number of another positive ion *Z*_{2}. In addition, figure *m*_{2}/*m*_{1} > 1) at the sheath edge, the lower the wall potential is. For the lighter ion (*m*_{2}/*m*_{1} < 1), the opposite is the case. The result is the same as that of Ref. [30]. Figure

The sheath potential drop will be reduced in the presence of SEE. The sheath potential drop is no longer monotonic and the sheath structure becomes complex when SEE coefficient *γ* is greater than *γ*_{c}.^{[19,22,23]} In Ref. [18], the critical SEE coefficient is

The critical SEE coefficient, *γ*_{c} can be found from Eqs. (*γ*_{c} increases when the relative ion concentration for *n*_{Xe+}/*n*_{e0} increases in the argon plasma. The critical coefficient decreases with the rising of ion concentration for *n*_{He+}/*n*_{e0}. From Fig. *N*_{20} = 0.2 the critical SEE coefficient reaches values of 0.973 in the Ar–Xe plasma and 0.959 in the Ar–He plasma, respectively. However, the critical coefficient is 0.97 in the pure Ar plasma. It indicates that the critical coefficient in the heavier plasma is greater than that in the lighter plasma. Figure *N*_{20} = 0.2, the critical SEE coefficient has values of 0.966 for *Z*_{2} = 2 and 0.959 for *Z*_{2} = 3, respectively. The fact that the lighter mass ion has a smaller critical SEE coefficient is because it has a smaller wall electric field without SEE. Hence it needs a smaller SEE coefficient to satisfy the condition that the wall electric field equals zero.

By integrating Poisson’s equation (

*G*(

*φ*), a function of the sheath potential, is expressed as

*φ*

_{w}can be found from Eqs. (

*γ*and

*N*

_{20}. Then the sheath width

*ξ*

_{w}can be obtained from Eq. (

Figure *γ* = 0, *γ* = 0.5, and *γ* = 0.8. The sheath width increases with the rising of ion concentration for *n*_{Xe+}/*n*_{e0} in the argon plasma and decreases with the rising of ion concentration for *n*_{He+}/*n*_{e0}. It means that the presence of heavier ion increases the sheath width in lighter ion plasma. The heavier ion can cause the wall potential to decrease (see Fig.

Secondary electrons emitted from the wall will move to the bulk plasma due to the action of the sheath electric field. The effective electron temperature in the bulk plasma will reduce when the density of secondary electrons increases. From Figs. ^{+} and decreases in the presence of Xe^{+} as compared with the scenario in the pure argon plasma. Also it can be seen from Fig. *N*_{20} = 0), the secondary electron density has increased rates of 18% for *Z*_{2} = 2 and 50% for *Z*_{2} = 3 respectively when the relative concentration of multiple charge ions is *N*_{20} = 0.2. The reason for this phenomenon is as follows: the sheath potential increases in the presence of lighter ions (see Fig.

The ion kinetic energy flux to the wall plays an important role in plasma processing.^{[33,34]} The ions are accelerated by the electric field of the sheath. Thus the ions arriving at the wall will have very large energy. The kinetic energy flux of ions *Q* can be expressed as

*F*

_{1}and

*F*

_{2}are the particle fluxes of two ion species, respectively. The normalized total ion kinetic energy flux to the wall is given by

*n*

_{e0}

*c*

_{s1}

*T*

_{e}, and we have

The normalized ion kinetic energy fluxes are plotted in Figs. *γ* = 0, the ion kinetic energy flux has reduced rates of 13% for *γ* = 0.5 and 30% for *γ* = 0.8 respectively. The effect of SEE on the ion kinetic energy flux to the wall stems from the sheath potential drop. The sheath potential drop decreases with the rising of SEE coefficient (see Fig. ^{+} in the argon plasma as shown in Fig. ^{+} reduces the kinetic energy flux. As one sees from Fig.

By using a simple sheath model, we study a collisionless sheath structure of plasma consisting of electrons, two species of positive ions and secondary electrons. Based on the ion wave approach, a Bohm criterion including the effect of SEE is obtained theoretically by introducing the Sagdeev potential. It is shown that the critical ion velocity at the sheath edge increases with the SEE coefficient and the tendency is independent of ion species. Since some sheath parameters can be obtained from the plasma parameters at the sheath edge, we investigate the effects of SEE coefficient, ion species concentration and charge number on the sheath parameters such as sheath width and ion kinetic energy flux to the wall without calculating plasma parameters profiles inside the sheath. Our results show that SEE can reduce the sheath potential drop, the sheath width and ion kinetic energy flux to the wall in plasmas with multi-charged ions. Meanwhile, in the presence of lighter ion species, secondary electron density at the sheath edge and ion kinetic energy flux to the wall increase, while the critical emission coefficient and the sheath width decrease. The more the lighter positive ion concentration, the more obvious the variation is. For the presence of the heavier ion species, their variation is opposite. In addition, the increase in the charge number of ions will reduce the critical SEE coefficient, the ion kinetic energy flux to the wall and the sheath width and increase the secondary electron density at the sheath edge.

**Reference**

1 | |

2 | Rev. Sci. Instrum. 80 041301 |

3 | |

4 | IEEE Trans. Plasma Sci. 23 709 |

5 | Appl. Phys. Lett. 91 041505 |

6 | |

7 | Plasma Sci. Technol. 13 385 |

8 | Plasma Sources Sci. Technol. 10 162 |

9 | J. Phys. D: Appl. Phys. 36 R309 |

10 | Phys. Rev. Lett. 90 145001 |

11 | Phys. Rev. Lett. 103 205002 |

12 | Phys. Rev. Lett. 104 225003 |

13 | J. Plasma Phys. 79 267 |

14 | |

15 | Phys. Plasmas 15 053508 |

16 | Phys. Plasmas 20 013509 |

17 | Phys. Plasmas 22 043510 |

18 | Plasma Phys. 9 85 |

19 | Phys. Fluids B 5 631 |

20 | Phys. Plasmas 9 4340 |

21 | Contrib. Plasma Phys. 50 121 |

22 | Phys. Rev. Lett. 111 075002 |

23 | Chin. Phys. B 20 125201 |

24 | Chin. Phys. B 20 065204 |

25 | Chin. Phys. B 23 075203 |

26 | |

27 | |

28 | Plasma Phys. Control Fusion 50 055003 |

29 | Chin. Phys. Lett. 30 085202 |

30 | Contrib. Plasma Phys. 50 909 |

31 | |

32 | |

33 | Plasma Sources Sci. Technol. 18 025009 |

34 | Phys. Plasmas 22 040702 |