Journal of Fusion Energy

, 30:481

Effect of Negative Ion Parameters on Sheath Formation Criterion in Electronegative Plasmas

Authors

  • Hamid Ghomi
    • Laser and Plasma Research InstituteShahid Beheshti University
    • Department of Physics, Boroujerd BranchIslamic Azad University
Original Research

DOI: 10.1007/s10894-011-9412-5

Cite this article as:
Ghomi, H. & Khoramabadi, M. J Fusion Energ (2011) 30: 481. doi:10.1007/s10894-011-9412-5

Abstract

Using a fluid model for three types of particles, the plasma-sheath formation criterion has been analyzed in collisional electronegative plasma, and the effects of the negative ion temperature and density are examined on the positive ion transition velocity. It is shown that in the collisional sheath, there will be an allowable interval for the positive ion velocity between two upper and lower limits as the plasma-sheath formation criterion; by increasing the mean temperature of the negative charge carriers, this velocity interval decreases. To confirm the correction of the allowable interval, the plasma sheath equations are numerically solved, and the negative ion temperature effect for example, is examined on the sheath formation.

Keywords

Ion temperaturePlasma-sheath transitionElectronegative plasmas

Introduction

Identifying the sheath formation criterion and the sheath structure is necessary in surface treatment by plasma [1]. In plasma diagnostics and plasma mass spectrometry the knowledge of the sheath characteristics are also required to deduce the measured data [2, 3]. The electrical specs of the sheath are defined by the plasma-sheath boundary condition. Therefore, a sheath criterion is required to provide the boundary condition for the sheath formation and to calculate the sheath characteristics.

The sheath formation criterion in the plasma depends on the plasma parameters and is different in various plasmas. In a collision-less cold plasma, the Bohm criterion [4] establishes the necessary condition for the sheath formation. Electronegative plasmas are often applied in plasma material processing. The sheath formation criterion in the electronegative plasmas has been examined extensively [58]. It has been well known that in order for the sheath formation in the electronegative plasmas, the positive ion speed at the sheath edge has to be greater than a minimum speed [6].

Das et al. [9] could describe the effect of the finite ion temperature on the necessary condition for the sheath formation in the electropositive plasmas. The sheath formation criterion, in the collisional plasmas has been investigated by Liu et al. [10]. Effect of ion temperature on RF plasma sheath was described by Lei et al. [11]. Alterkop [12] studied the DC plasma sheath formation in the thermal plasmas. Ghomi et al. [13] studied the boundary condition for collisional thermal electropositive plasma and concluded an upper limit in addition to a lower limit for ion transition velocity and sheath formation. Khoramabadi et al. also examined the ion temperature effect on the electropositive plasma-sheath structure [14]. They specially concluded that rising the ion temperature decreases the sheath width.

Here we use the fluid model to study the effects of the negative ion temperature and density on the sheath formation criterion in the collisional electronegative plasma. Indeed, we consider a collisional electronegative plasma-sheath with thermal effects to obtain a general sheath formation criterion. This general criterion will exchange to the simpler ones in cold, collisionless and electropositive plasma-sheath in the special cases. Moreover, in order to confirm the results, we solve the plasma-sheath equations and examine the negative ion temperature effect on the sheath formation criterion in collisional plasmas with constant collision frequency.

After introduction, the hypothesis and sheath formulation are quoted in Sect. 2. In this section using the sheath formation definition, it will be introduced a universal criterion for the ion transition velocity at the sheath edge. For more description of the criterion, an example is represented in Sect. 3. Finally, in Sect. 4 a brief conclusion has been brought.

Basic Equations of Problem

The boundary between a quasi-neutral electronegative plasma and a non-neutral plasma-sheath near a conductive planar wall will be analyzed. The x-axis is selected normal to the boundary surface between plasma and sheath such that, plasma and sheath are placed in x < 0, and x > 0 respectively. The model developed here is the same model of Ghomi et al. [13] in the electronegative plasmas.

We consider three types of charged particles; electrons, positive ions and negative ions in the plasma fluidal approximation. The negative ion number density as well as the electron number density is introduced by the Boltzmann relation in low pressure electronegative plasmas [15]. Ghim (Kim) et al. [16] presented the first experimental report to verify that the negative ions in low pressure plasmas are in the thermal equilibrium and obey the Boltzmann distribution function. Therefore, we will have
$$ n_{e} = n_{eo} \,\exp \left( {{\frac{eV}{{k_{B} T_{e} }}}} \right), $$
(1)
and
$$ n_{ - } = n_{ - o} { \exp }\left( {{\frac{eV}{{k_{B} T_{ - } }}}} \right) $$
(2)
where ne and n_ are the electron and negative ion densities in the sheath region respectively, Te and T_ the electron and negative ion temperature respectively, neo and n_o the electron and negative ion densities at the sheath edge respectively, and V the local potential in the sheath region.
In the steady state, ion continuity equation in one-dimensional generation less sheath \( \left[ {d({{n_{ + } v_{ + } } \mathord{\left/ {\vphantom {{n_{ + } v_{ + } } {dx}}} \right. \kern-\nulldelimiterspace} {dx}}) = 0} \right] \) results
$$ n_{ + o} v_{ + o} = n_{ + } v_{ + } , $$
(3)
where n+ and v+ are the density and x-component velocity of the positive ion in the sheath respectively, and n+o and v+o are the respective parameters at the plasma-sheath boundary. On the other hand, using the force balance equation for the positive ion fluid in the steady state we will find
$$ Mv_{ + } {\frac{{dv_{ + } }}{dx}} = - e\frac{dV}{dx} - {\frac{1}{{n_{ + } }}}{\frac{{dp_{ + } }}{dx}} - M\left( {n_{n} \sigma v_{ + } } \right)v_{ + } , $$
(4)
where M is the positive ion mass, nn the neutral gas density, σ = σs(v+/cs)β the momentum-transferring cross section for collision between the positive ions and the neutrals \( [c_{s} = \left( {{{k_{B} T_{e} } \mathord{\left/ {\vphantom {{k_{B} T_{e} } M}} \right. \kern-\nulldelimiterspace} M}} \right)^{{({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})}} \) is the ion acoustic velocity, σs the cross section measured at the ion acoustic velocity and β a dimensionless parameter ranging from β = 0 in the constant mean free path \( (\lambda_{ + } = 1/n_{n} \sigma_{s} ) \) case to β = −1 in the constant collision frequency \( (\nu_{ + } = n_{n} \sigma_{s} c_{s} ) \) case] and \( p_{ + } = k_{B} T_{ + } n_{ + } \) the positive ion pressure with T+ as the positive ion temperature. For a self-consistent plasma-sheath boundary, these equations should be gathered with the Poisson’s equation
$$ {\frac{{d^{2} V}}{{dx^{2} }}} = - {\frac{e}{{\varepsilon_{0} }}}\left( {n_{ + } - n_{e} - n_{ - } } \right). $$
(5)
Introducing the normalized parameters
$$ \eta = - {\frac{eV}{{k_{B} T_{e} }}} , N_{ + } = {\frac{{n_{ + } }}{{n_{e0} }}} , N_{ - } = {\frac{{n_{ - } }}{{n_{e0} }}} , N_{e} = {\frac{{n_{e} }}{{n_{e0} }}} , {{\Updelta}} = {\frac{{n_{ - 0} }}{{n_{e0} }}}, $$
$$ \xi = {\frac{x}{{\lambda_{De} }}} , u = {\frac{{v_{ + } }}{{c_{s} }}} , \alpha = \lambda_{De} n_{n} \sigma_{s} , \tau = {\frac{{T_{ + } }}{{T_{e} }}}, \theta = {\frac{{T_{ - } }}{{T_{e} }}}, $$
we can normalize the Eqs. (15). Here \( \lambda_{De} = \sqrt {\varepsilon_{0} k_{B} T_{e} /n_{e0} e^{2} } \) is the electron Debye length and α is a dimensionless collision parameter. Considering charge neutrality in the plasma-sheath boundary \( \left( {n_{ + 0} = n_{e0} + n_{ - 0} } \right) \) we will get \( n_{ + 0} /n_{e0} = 1 + {{\Updelta}} \). Rewriting the basic equations versus these new parameters and variables, and omitting N+, N_ and Ne among them, we find the positive ion equation of motion and the Poisson’s equation respectively,
$$ uu^{'} = u{\frac{du}{d\xi }} = \left( {{\frac{d\eta }{d\xi }} - \alpha u^{2 + \beta } } \right)\left( {1 - {\frac{\tau }{{u^{2} }}}} \right)^{ - 1} , $$
(6)
and
$$ \eta^{''} = {\frac{{d^{2} \eta }}{{d\xi^{2} }}} = \left( {1 + {{\Updelta}}} \right){\frac{{u_{0} }}{u}} - e^{ - \eta } - {{\Updelta}}e^{ - \eta /\theta } . $$
(7)
Solving these two recent equations, and finding η and u versus ξ, one can find the normalized density of the particles, using the relations
$$ N_{ + } = \left( {1 + {{\Updelta}}} \right){\frac{{u_{0} }}{u}}, $$
(8)
$$ N_{e} = {\text{e}}^{ - \eta } , $$
(9)
$$ N_{ - } = \Updelta {\text{e}}^{ - \eta /\theta } . $$
(10)
Integrating of the Poisson’s equation from the sheath edge into the sheath and using the well-known boundary conditions \( \eta_{0}^{\prime } \ne 0 \) and η0 = 0 at the plasma-sheath interface, we obtain
$$ \mathop \int \limits_{{\eta '_{0} }}^{\eta '} \eta 'd\eta ' = \mathop \int \limits_{0}^{\eta } (1 + {{\Updelta}}){\frac{{u_{0} }}{u}}d\eta - \mathop \int \limits_{0}^{\eta } e^{ - \eta } d\eta - \mathop \int \limits_{0}^{\eta } {{\Updelta}}e^{ - \eta /\theta } d\eta , $$
(11)
or,
$$ \frac{1}{2}\eta^{\prime 2} = \frac{1}{2}\eta_{0}^{\prime 2} - S\left( {\eta ,u_{0} } \right) , $$
(12)
where \( \eta_{0}^{\prime } \) is the normalized electric field at the plasma-sheath boundary, and S is called the Sagdeev potential defined by
$$ S\left( {\eta ,u_{0} } \right) = 1 - e^{ - \eta } + \theta {{\Updelta}}\left( {1 - e^{ - \eta /\theta } } \right) - (1 + {{\Updelta}})\mathop \int \limits_{0}^{\eta } {\frac{{u_{0} }}{u}}d\eta . $$
(13)
S function results to the boundary conditions \( S\left( {0,u_{0} } \right) = 0 \) and \( \partial S(0,u_{0} )/\partial \eta = 0 \) at the sheath edge, and this means that the sheath edge is an extremum point for the Sagdeev potential. From Eq. (13), one can find
$$ \begin{aligned} {\frac{{\partial^{2} S\left( {\eta ,u} \right)}}{{\partial \eta^{2} }}} = & & - e^{ - \eta } - {\frac{\Updelta }{\theta }}e^{ - \eta /\theta } + (1 + \Updelta ){\frac{{u_{0} }}{{u^{2} }}}\;{\frac{du}{d\eta }} \\ = & & - e^{ - \eta } - {\frac{\Updelta }{\theta }}e^{ - \eta /\theta } + (1 + \Updelta ){\frac{{u_{0} }}{{u^{2} }}}\;{\frac{u'}{\eta '}} . \\ \end{aligned} $$
(14)
The condition for maximizing the S function at the sheath edge, and attaining the negative values for S in the sheath region is:
$$ {\frac{{\partial^{2} S\left( {0,u_{0} } \right)}}{{\partial \eta^{2} }}} = - 1 - {\frac{{{\Updelta}}}{\theta }} + (1 + {{\Updelta}}){\frac{1}{{u_{0} }}}\,{\frac{{u_{0}^{'} }}{{\eta_{0}^{'} }}} < 0 $$
(15)
or
$$ u_{0} u_{0}^{'} < u_{0}^{2} \eta_{0}^{'} {\frac{{1 + {{\Updelta}}/\theta }}{{1 + {{\Updelta}}}}}. $$
(16)
On the other hand, Eq. (6) in the boundary is
$$ u_{0} u_{0}^{'} = {\frac{{\eta_{0}^{'} - \alpha u_{0}^{2 + \beta } }}{{1 - \tau /u_{0}^{2} }}}. $$
(17)
Due to a neutral drag force to the ions directed at the plasma region, the relation \( \eta_{0}^{'} > 0 \) will be the ion transition condition into the sheath region. In other words, an accelerating electric field is required to overcome the collisional drag. Then one can find:
$$ {\frac{{\eta_{0}^{'} - \alpha u_{0}^{2 + \beta } }}{{1 - \tau /u_{0}^{2} }}} \ge 0 . $$
(18)
Inequality (18) means that in non-hot plasmas with \( \tau < \left( {\eta_{0}^{'} /\alpha } \right)^{{2/\left( {2 + \beta } \right)}} \), we will have
$$ \sqrt \tau \le u_{0} \le \left( {\eta^{'}_{0} /\alpha } \right)^{{1/\left( {2 + \beta } \right)}} . $$
(19)
and in hot plasmas with \( \tau > \left( {\eta_{0}^{'} /\alpha } \right)^{{2/\left( {2 + \beta } \right)}} \), will have
$$ \left( {\eta_{0}^{'} /\alpha } \right)^{{1/\left( {2 + \beta } \right)}} \le u_{0} \le \sqrt \tau . $$
(20)
Applying the Eq. (17) in the inequality (16) one can find
$$ {\frac{{\eta_{0}^{'} - \alpha u_{0}^{2 + \beta } }}{{1 - \tau /u_{0}^{2} }}} \le u_{0}^{2} \eta_{0}^{'} {\frac{{1 + {{\Updelta}}/\theta }}{{1 + {{\Updelta}}}}}. $$
(21)
By solving the inequality (21) for the ion transition velocity u0, and with account the limiting values of β we attain
$$ \sqrt {{\frac{{1 + {{\Updelta}} + \tau (1 + {{\Updelta}}/\theta )}}{{\left( {1 + {{\Updelta}}} \right)\alpha /\eta '_{0} + 1 + {{\Updelta}}/\theta }}}} \le u_{0} , \quad \left( {\beta = 0} \right), $$
(22)
and,
$$ \left[ {\sqrt {{\frac{{\alpha^{2} }}{{4\eta_{0}^{\prime 2} }}}\left( {{\frac{{1 + {{\Updelta}}}}{{1 + {{\Updelta}}/\theta }}}} \right)^{2} + {\frac{{\left( {1 + {{\Updelta}}} \right)}}{{\left( {1 + {{\Updelta}}/\theta } \right)}}} + \tau } - {\frac{{\alpha (1 + {{\Updelta}})}}{{2\eta_{0}^{\prime } (1 + {{\Updelta}}/\theta )}}}} \right] \le u_{0} , \left( {\beta = - 1} \right). $$
(23)
Since both of these two limitation values for u0 are greater than \( \sqrt \tau \), in non-hot plasmas \( \left[ {\tau < ({{\eta_{0}^{\prime } } \mathord{\left/ {\vphantom {{\eta_{0}^{\prime } } \alpha }} \right. \kern-\nulldelimiterspace} \alpha })^{{{2 \mathord{\left/ {\vphantom {2 {2 + \beta }}} \right. \kern-\nulldelimiterspace} {2 + \beta }}}} } \right] \), using relation (19) we can find out the new generalized sheath criterion:
$$ \sqrt {{\frac{{1 + {{\Updelta}} + \tau (1 + {{\Updelta}}/\theta )}}{{\left( {1 + {{\Updelta}}} \right)\alpha /\eta_{0}^{'} + 1 + {{\Updelta}}/\theta }}}} \le u_{0} \le \sqrt {{\frac{{\eta_{0}^{'} }}{\alpha }}} ,\quad (\beta = 0), $$
(24)
and,
$$ \left[ {\sqrt {\left( {{\frac{\alpha }{{2\eta_{0}^{'} }}}\,{\frac{{1 + {{\Updelta}}}}{{1 + {{\Updelta}}/\theta }}}} \right)^{2} + {\frac{{1 + {{\Updelta}}}}{{1 + {{\Updelta}}/\theta }}} + \tau } - {\frac{\alpha }{{2\eta_{0}^{'} }}}\,{\frac{{1 + {{\Updelta}}}}{{1 + {{\Updelta}}/\theta }}}} \right] \le u_{0} \le {\frac{{\eta_{0}^{'} }}{\alpha }}, \left( {\beta = - 1} \right). $$
(25)

According to the relations (24) and (25) the transition velocity of the positive ions has two upper and lower limits in the collisional sheath. As it can be seen, the lower limit of u0 is a function of the whole plasma parameters, while its upper limit is a function of α and \( \eta_{0}^{'} \). The relationship between these limits and β, τ, α and η0 has been investigated by Ghomi et al. [13]. If we set Δ = 0 (when there is no negative ion in the plasma) or θ = 1 (when there is no difference between the negative ions and electrons) in the recent relations, the results in that paper are renewed again. Here we want to examine the effects of Δ (the negative ion to electron density ratio at the sheath edge) and θ (the negative ion to electron temperature ratio in the plasma) on the positive ion transition velocity.

Relations (24) and (25) in a non-collisional plasma (α = 0) are simplified to
$$ \sqrt {{\frac{{1 + {{\Updelta}}}}{{1 + {{\Updelta}}/\theta }}} + \tau } \le u_{0} . $$
(26)

It means that the positive ions do not need the initial electric field to enter the sheath region in the absence of collision. By setting Δ = 0 the relation (26) is converted the well-known ion transition condition \( \sqrt {1 + \tau } \le u_{0} \) in the isothermal electropositive plasma.

In hot plasmas \( \left( {\tau > \left( {{{\eta_{0}^{\prime } } \mathord{\left/ {\vphantom {{\eta_{0}^{\prime } } \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right){2 \mathord{\left/ {\vphantom {2 {(2 + \beta )}}} \right. \kern-\nulldelimiterspace} {(2 + \beta )}}} \right) \) it can be shown that the sheath formation condition will be the same as Eqs. (24) and (25) with the exception of their upper limits that should be replaced by \( \sqrt \tau \).

In Fig. 1 the upper and lower limits of the positive ion velocity (u0) are shown as functions of the initial electric field \( \left( {\eta_{0}^{\prime } } \right) \) in a cold plasma (τ = 0). This figure has been plotted for a collisional cold plasma (non-hot plasma) with the constant collision frequency (β = −1) for θ = 0.1 and some values of ∆(∆ = 0,1,10,100). It can be seen from this figure, the lower limit of u0 is an increasing function of \( \eta_{0}^{\prime } \), and decreases with ∆; the upper limit of u0 is an increasing function of \( \eta_{0}^{\prime } \), and does not depend on ∆. Increasing of ∆ and reduction the lower limit of u0 increases the allowable interval of u0.
https://static-content.springer.com/image/art%3A10.1007%2Fs10894-011-9412-5/MediaObjects/10894_2011_9412_Fig1_HTML.gif
Fig. 1

The upper and lower limits of the positive ion velocity at the plasma-sheath edge as functions of the initial electric field at the sheath edge for \( \alpha = 0.1,\,\beta = - 1, \tau = 0,\,\theta = 0.1 \) and different amounts of ∆(∆ = 0,1,10 and 100)

The profile of the lower and upper limits of u0 as functions of \( \eta_{0}^{\prime } \) has been shown in Fig. 2 for ∆ = 1 and some values of θ(θ = 0.01,0.1,0.5 and 1). In this figure the plasma parameters are the same as Fig. 1. Here the upper and lower limits of u0 are increasing functions of \( \eta_{0}^{\prime } \) the upper limit is independent of the negative ion temperature (θ) while the lower limit is an ascending function of θ. Increasing of θ and growth the lower limit of u0 decreases the allowable interval of u0.
https://static-content.springer.com/image/art%3A10.1007%2Fs10894-011-9412-5/MediaObjects/10894_2011_9412_Fig2_HTML.gif
Fig. 2

The upper and lower limits of the positive ion velocity at the plasma-sheath edge as functions of the initial electric field at the sheath edge for ∆ = 1 and different amounts of θ(θ = 0.01,0.1,0.5 and 1). The other parameters are the same as Fig. 1

In description of these variations of u0, it is necessary to mention the difference between the electron and negative ion as the negative charge carriers in the plasma. The different temperature of the electron and negative ion is the only difference of them here. Since the negative ion is too much heavier than the electrons, the negative ion temperature is usually lower than the electron temperature (θ < 1). So in the limiting case θ = 1 there is no difference between them. Therefore, decreasing of Δ and increasing of θ increase the mean temperature of the negative charge carriers. On the other hand, we know that decreasing of Δ and increasing of θ rise the lower limit of u0.

Consequently, by increasing the mean temperature of the negative charge carriers, the lower limit of u0 increases too. The upper limit of u0 is independent of the negative charge carriers temperature in the electronegative plasmas.

More description

In order to explain the results and to review the accuracy of the sheath formation criterion [Eqs. (24) and (25)], we numerically solve Eqs. (610) and investigate the ion transition velocity effect on the sheath formation. When a positive space charge is formed around a negative biased electrode, and positive ion velocity becomes an increasing function towards the electrode, it can be said the sheath formation criterion has been correctly satisfied near the electrode (plasma wall). In Figs. 3 and 4, the normalized densities of the negative and positive charge carriers have been shown as functions of the normalized distance from the sheath edge towards the wall.
https://static-content.springer.com/image/art%3A10.1007%2Fs10894-011-9412-5/MediaObjects/10894_2011_9412_Fig3_HTML.gif
Fig. 3

The number densities of the negative charge carriers \( N_{e} + N_{ - } \), and positive ions \( N_{ + } \), normalized by \( \Updelta + 1 \), as functions of normalized distance from the sheath edge ξ for\( \theta = 0.1 \). The other parameters are \( \eta_{{_{0} }}^{'} = 0.1,\,\alpha = 0.1,\,\beta = - 1,\,\tau = 0,\,{{\Updelta}} = 1 \) and \( 0.34 < u_{0allow} < 1 \)

https://static-content.springer.com/image/art%3A10.1007%2Fs10894-011-9412-5/MediaObjects/10894_2011_9412_Fig4_HTML.gif
Fig. 4

The number densities of the negative charge carriers \( N_{e} + N_{ - } \), and positive ions \( N_{ + } \), normalized by \( \Updelta + 1 \), as functions of normalized distance from the sheath edge ξ for \( \theta = 0.5 \). The other parameters are the same as Fig. 3 and \( 0.55 < u_{0allow} < 1 \)

Using the polarization scattering model [17] for elastic collision between positive ion and neutral, one can find the collision parameters. For Argon gas in 10 mTorr pressure and 25°C temperature \( (n_{n} = 3.3 \times 10^{14} {\text{cm}}^{ - 3} ) \) with ionization fraction 2 × 10−7 and electron temperature \( T_{e} = 2 {\text{eV}} \gg T_{ + } \) we can find \( \sigma_{s} = 3.1 \times 10^{ - 15} {\text{cm}}^{2} \), \( \lambda_{De} = 0.1 {\text{cm}} \) and \( \alpha = 0.1 \). So the normalization factors are; \( k_{B} T_{e} /e = 2 {\text{V}} \), \( k_{B} T_{e} /e\lambda_{De} = 20 {\text{V}}/{\text{cm}} \), and \( c_{s} = 2200 {\text{m}}/{\text{s}} \). Consequently, the common parameters in these figures are \( \eta_{0}^{'} = 0.1 \) (according to Takizawa et al. [18]), α = 0.1, τ = 0 (assuming cold plasma), \( \Updelta = 1 \) and β = −1. The negative ion temperature is θ = 0.1 and the positive ion transition velocity u0 = 0.5, 0.8, and 1.2 in Fig. 3a, b and c respectively, while in Fig. 4, θ = 0.5 with the same values for u0 in Fig. 4a, b, and c respectively.

According to Fig. 2 [Eq. (25)], for \( \eta_{0}^{'} = 0.1 \) and \( \theta = 0.1 \) the allowable interval of the ion transition velocity in order for the sheath formation is \( u_{0l} = 0.34 < u_{0allow} < 1 = u_{0u} \), and for \( \eta_{0}^{'} = 0.1 \) and θ = 0.5 one can find out \( u_{0l} = 0.55 < u_{0allow} < 1 = u_{0u} \). Thus, as can be seen from Fig. 3a, with \( u_{0} = 0.5 \) and \( \theta = 0.1 \) the sheath formation criterion is satisfied because u0 is inside its allowable interval. On the other hand, Fig. 4a shows that with the same u0 and \( \theta = 0.5 \) the sheath formation criterion is not fulfilled because u0 is outside its allowable interval. Indeed, increasing of θ transfers \( u_{0} = 0.5 \) from the inside of the allowable interval in Fig. 3a to the outside of the allowable interval in Fig. 4a.

In Figs. 3b and 4b with \( u_{0} = 0.8 \) (near the upper limit of the allowable interval) and \( \theta = 0.1 \) and 0.5 respectively, the sheath formation criterion is fulfilled because \( u_{0} = 0.8 \) is in the allowable interval and increasing of θ does not affect on its situation. Moreover, in Figs. 3c and 4c, \( u_{0} = 1.2 \) is out of the allowable interval, and rising of θ from 0.1 to 0.5 does not change its location. Therefore, the sheath is not formed around the negative biasing electrode correctly in this case.

Conclusion

In this paper we examine a collisional electronegative plasma with taking into account the effect of the positive and negative ion temperature and the negative ion density, and attain the ion transition velocity for the sheath formation. We found that there is an allowable interval between two upper and lower limits for the positive ion transition velocity in the plasma-sheath boundary. In a non-hot plasma, the lower limit depends on the all of the plasma parameters, while the upper limit depends on plasma pressure (collision frequency) and initial electric field at the sheath edge.

We conclude that, in thermal electronegative plasma, increasing the mean temperature of the negative charge carriers (increasing the negative ion temperature and decreasing the negative ion density) raises the lower limit of the positive ion transition velocity, and decreases the allowable interval of the positive ion transition velocity.

Copyright information

© Springer Science+Business Media, LLC 2011