Sheath Formation in Multi-Component Plasma

Sheath Formation in Multi-Component Plasma

K. Annou, N. Saoula and R. Tadjine

Centre de Développement des Technologies Avancées, BP 17, Baba Hassen, Algiers, Algeria

Abstract

Using a multi-fluid model, the sheath formation criterion in electronegative plasma is examined. It is shown that in a collisional sheath there are both upper and lower limits for the Bohm entrance velocity ions. The Sagdeev potential in the whole sheath expansion were used. The criterion was specified for several special cases, viz., constant ion mean-free path and constant ion mobility.

The sheath-edge potential as well as the floating potential are also calculated.

Introduction.

The plasma sheath is the localized electric field that separates plasma from a material boundary. Sheath formation is one of the old problems in plasma physics [1].That is why, many authors paid attention to the sheath structure formation in electron-ions plasmas [2].Recently, much effort has been invested in studying the sheath of multicomponent plasmas [3]. Due to electron attachment, a large number of negative ions are generated in the course of plasma discharge. Plasmas containing negative ions usually referred to as electronegative plasmas are of practical interest in plasma processing as well as their significance in theoretical research [4]. A plasma sheath involving negative ions is called an electronegative plasma sheath. Electronegative plasma sheaths have been extensively used in many areas such as material surface treatment, etching and thin-film deposition processing, and plasma chemistry [5,6]. Thus,

investigation of electronegative plasma sheaths is considerably significant. The electrodynamics properties of plasma-sheath boundary are of great importance in a wide range of applications, such as electrostatic plasma probes, electrode phenomena, and surface of materials processed in dc and RF discharges[7]. In these cases, the electrical characteristics and plasma parameter profiles of the sheath are defined by the condition of the plasma- sheath boundary. Therefore, a sheath criterion is needed to provide the boundary condition to calculate the plasma parameter profiles. For an electron-ion plasma, the necessary condition for steady-state charged sheath existence was first obtained by Bohm1 and is commonly known as the Bohm criterion [8]. This criterion determines the ion velocity at the sheath entrance, which is necessary for steady-state sheath existence in two-component plasma. The presence of the negative ions modies the structure of theplasmasheath considerably. Indeed, in electronegative plasmas the negative ion density often exceeds the density of electrons, and the presence of negative ions introduces complications into discharge analysis [9]. In surface treatment by such plasmas, the knowledge of the sheath structure is essential since processes taking place at the surface will depend on the particle distribution at the vicinity of the wall. Additionally, the effect of negative ion in the formation of a sheath is considered to be important for potential control of an electrode in etching and deposition

plasmas [10]. Several authors derived [11]conditions of the sheath existence using a fluid

criterion in a collisional plasma sheath by using a

model of the plasma with Boltzmann-distributed electrons. In all of these works, the conditions for steady-state sheath existence were obtained by expanding solutions for small electric potentials in the sheath. Their validity may be verified by analyzing the solution of Poissons equation in all ranges of the potential. Besides, Das et al.[13] derived the plasma sheath equation, and studied the effect of the finite ion temperature on the required condition for the sheath formation in electropositive plasma. Liu et al.[14] investigated the sheath

1.Model equation.

We consider a one dimensional plasma sheath treating electron and negative ions as thermal Boltzmann particles, but positive ions as cold fluid. The sheath edge at x=0 is the plasma-sheath interface (interface between essentially neutral and non-neutral regions).Both electrons and negative ions in the sheath are assumed to obey the Boltzmann distribution

two fluid model. Ghomiet al. [15] studied the boundary condition for collisional thermal electropositive plasma. Motivated by previous works, in this paper, we use the multi-fluid model to examine the effects of density of negative ions on the positive ion transition velocity in the plasma sheath boundary and on the sheath formation in collisional plasmas. The objective of the present investigation is to derive the steady-state sheath criterion by analyzing the Sagdeev potential throughout the entire sheath.

constant ion mean-free path and = 1for constant ion mobility. Assuming, in this section, the wall is infinitely long in the y and z direction, the quantities changes only in the direction normal to the wall, ie:

.

The Poissons equation relates the electrons and ions to the self-consistent potential,

2

2 = 4.

=

0

0

Where is the space charge density in the sheath and

is given by

= 0

=

.

Where Is the local potential, ()is the electrons (negative ions) temperature and 0 (0)

is the electron (negative ion) density at the sheath

Simplifying the equations, we can use the dimensionless variables as follows:

= , = , = , = , = ,

edge where = 0 = 0.

The cold positive ions satisfy the continuity and

= 0.As well as

, where = in

0

4 20

momentum equations:

= 0.

= .

Where is the ion mass, is the neutral gaz density, and = is the momentum-

the Debye electron length.

Substituting them into the governing Eqs.(4) and

(5)one can obtain:

. = 2+ ()

= 1 0 ()

is the Mach number, is the dimensionless

transferring cross sections for collision between ions 0

and neutrals. = 1 2s the ion acoustic

electric field.

velocity, is the cross section measured at ion acoustic velocity,is an index and set as = 0 for

1. Sagdeev potential.

   Investigation of a collisional unmagnetized

   plasma sheath consisting of electrons, negative ions as well as positive ions can be carried out by solving

   In addition, at the sheath edge ( = 0), Eq.(6) takes the following form:

   00 = 2+ . ()

   0 0

   Eqs.[(1)-(7)]. However, to solve these equations the boundary conditions must be specified. This leads to determine the Bohms criterion. First we derive the generalized the Bohms criterion, and then we reduced it to some special cases.First, integrating Eq.(7) once, we get

   1 2 + , 0 = 1 2. ()

   It is evident that 0 0 due to neutral drag to the positive ion in the plasma. Therefore, the necessary condition of entrance of the positive ions into the sheath region is > 0 which means that there must exist acceleration force to overcome collision drag. Considering Eq.(12) and the above mentioned

   0

   0

   condition, we get

   2 2 0

   where is dimensionless electric field at the sheath

   2+ . ()

   0 0

   edge.Eq.(8) is known as plasa sheath equation for the plasma we have taken into account. The boundary conditions are 0 , which

   In relation (11)-(13), we find the upper and lower limits of Bohms sheath criterion:

   a/ For constant cross section:

   measures the sheath thickness,

   0,

   1 +

   1 2

   1 2

   and determines the wall potential.

   0

   1 + + 1 + 0

   0

   0

   0

   = 0. ()

   b/ For collisional mobility:

   , 0 = 1 +

   + 1

   0 . ()

   2

   2

   1 +

   2

   +

   1 +

   1

   1 +

   0

   0

   , 0 is the Sagdeev potential satisfying

   0

   0

   boundary conditions 0, = 0, 0,0 = 0.

2. Bohm sheath criterion.

   The variation of , 0 yields the basic characteristics of the sheath formation in plasma.Eq.(8) is analogous the conservation law for a classical particle in a well , 0 , with being the analogous total energy. Similar to the analysis of a

   partial well, it requires , 0 < 0 in the sheath,

   40 1 + 1 + 2 0 1 +

   0

   0

   = 1. ()

   Now to investigate the validity of the derived Bohms criterion, we would like to reduce our general Bohms criterion to some special cases;

   i/ For collisionless electrostatic plasma( = 0, = 0, = 0) inequalities[(14)-(15)] becomes0 1, which is the well-known Bohm criterion for a collision-free sheath. It means that the ion flow

   speed at the plasma boundary must be at least as

   2

   2

   which leads to 2 ,0

   =0

   < 0.

   great as the ion sound speed in order for a sheath to

   Then, from Eq.(9), we get

   2 , 0

   2 = + 2

   form.

   ii/ For collisional electrostatic plasma( 0, = 0, = 0) inequalities [(14)-(15)] reduces to

   1 0 2 . ()

   From Eq.(10), by considering the condition for

   1 2

   1 2

   0

   0

   0

   0

   = 0 , and

   maximizing the Sagdeev potential at the sheath

   + 0

   2 2

   1

   2 1 +

   2

   0 0 = 1 .

   2

   2

   edge, ie., 0,0 < 0, we have

   40

   2 0

   1 +

   <

   , ()

   The result is in agreement with Jin-Yun Liu et al. As

   mentioned by Liu, in collisional plasma ion Mach

   0 1 + 0 0

   number is between upper and lower limits. The lower limit of the collisional sheath criterion shows

   0

   0

   the reduction of the ion entering speed 0 due the existence of neutral drag > 0, on the other hand, the upper limit shows the balance between the driving (electric field ) and the drag (the

   We confine ourselves to investigate stationary solutions that depend on space and time in thefollowing way: = .

   0

   0

   collisionality 2+ ).

   iii/ For collisional electronegative plasma sheath ( 0, 0, 0), the range of the Mach is determined by inequalities[(14)-(15)]. To discuss the result, we consider a quasineutral argon plasma with following parameters: = 0.134 (corresponding to neutral pressure of P=0.1 Torr at temperature 290K), the density of argon plasma is 5.0 × 108 3 and

   the electron temperature = 3 .For the above- mentioned parameters, Figure1 shows the upper and lower limits of Bohms criterion of the collisional electronegative plasma sheath varying with the normalized electric field at the sheath edge,

   0

   0

   for = 0 and = 1.Using inequalities (14- 15) and assuming = 10, = 0.01 = 0.1, it is found that the range of allowable ion Mach number is 0.86 0 1.29 for = 0 and

   0.74 0 1.71 for = 1 .A comparison with

   ion temperature is made in figure 2.

3. Floating potential.

In this section we shall be looking for analytically exact solutions of the nonlinear equations systems Eqs. [3-5].The normalized governing equations of the plasma evolution are given by

+

Figure1.The lower and upper limits of Bohms sheath criterion as a function of the normalized electric field

0

0

For ( = 0.134, = 10, = 0.01)(a) = 0 and (b) = 1

0

0

Figure2. The lower and upper limits of Bohms sheath criterion vs.

For = 0.134, = 0.01for = 0.

x

= 0.

+ = .

x x

Figure3. Floating potential vs..

In the stationary frame, we obtain from Eqs.[(3)-(5)]

2

2

2 =

x

.

the density as

Normalized electrons and negative ions are given by

= ,

= 1 +

2

1 2

= .

= 0 0.

Then the Poissons equation becomes:

2

2

1 2

2 = + 1 +

The normalized space charge density will be defined as

= 1 + 2 1 2 .(23)

So the exact Sagdeev potential reads as,

, = 1 + 1

+ 2 1 1 + 2

2

From the equation of integral energy, the normalized Maxwell stress reads as

1 2 = 1 + 1

2

2

2

+ 2 1 1 + 2 .

Consequently, at the cathode = it will be of the form

1 2 = 1 + 1

the plasma composition in the quasi-neutral region.The effect of the ion densities ratio and masses on the floating potential is given in Figure 3. It is seenthat for large concentration of negatively charged ion (larger value of) the floating potential will be more negative. In addition, the heavier positive ions have the same effect. Indeed, the same figure reveals that the potential is reduced for

heavier ions ( ). Thus our theoretical model may be applicable in RF plasma to decrease the potential difference between an electrode and the plasma.

Acknowledgments

Reviewers comments and suggestions are gratefully acknowledged by the authors, without which this paper would not have been in the present form.

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   2

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   At the floating electrode, the current due to the positive ions is balanced at the electrode by that of the electrons and negative ions . Thus we have

   + = ,

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   = ,

   2 2 3 2

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   = 20,

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   = 1 1 1 + 1 + (31)

   2 2

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