R N Franklin

The plasma-sheath boundary region

In this review an attempt is made to give a broad coverage of the problem of joining plasma and sheath over a wide range of physical conditions. We go back to the earliest works quoting them, where appropriate, to understand what those who introduced the various terms associated with the structure of the plasma–sheath had in mind. We try to bring out the essence of the insights that have been gained subsequently, by quoting from the literature selectively, indicating how misunderstandings have arisen. In order to make it accessible to the generality of those currently working in low temperature plasmas we have sought to avoid mathematical complexity but retain physical insight, quoting from published work where appropriate. Nevertheless, in clarifying my own ideas I have found it necessary to do additional original work in order to give a consistent picture. In this way I have sought to bring together work in the late 1920s, the 1960s, and now mindful of the commercial importance of plasma processing, work over the past 15 years that adds to the general understanding.

Electronegative plasmas—why are they so different?

This paper seeks to summarize the work that has gone on over the past 20 years in plasmas that has become of great commercial importance but was not studied in depth previously. It points up the differences between conventional electropositive plasmas and those where negative ions dominate. It removes some of the errors and misunderstandings that have occurred and tries to bring the field into a coherent consistent whole.

The Boltzmann relation in electronegative plasmas: When is it permissible to use it?

This paper reports the results of computations to obtain the spatial distributions of the charged particles in a bounded active plasma dominated by negative ions. Using the fluid model with a constant collision frequency for electrons, positive ions and negative ions the cases of both detachment-dominated gases (such as oxygen) and recombination-dominated gases (such as chlorine) are examined. It is concluded that it is valid to use a Boltzmann relation ne = ne0exp(eV/kT) for the electrons of density ne, where the temperature T is approximately the electron temperature Te, and that the density nn of the negative ions at low pressures obeys nn = nn0exp(eV/kTn), where Tn is the negative-ion temperature. However, at high pressure in detachment-dominated gases where the ratio of negative-ion density to electron density is constant and greater than unity, and when the attachment rate is larger than the ionization rate, the negative ions are distributed with the same effective temperature as the electrons. In all other cases there is no simple relationship. Thus to put nn/ne = const, nn = ne0exp(eV/kTe) and nn = nn0exp(eV/kTn) simultaneously is mathematically inconsistent and physically unsound. Accordingly, expressions deduced for ambipolar diffusion coefficients based on these assumptions have no validity. The correct expressions for the situation where nn/ne = const are obtained without invoking a Boltzmann relation for the negative ions.

The active magnetized plasma-sheath over a wide range of collisionality

Recent work by Sternberg and Poggie (2004 IEEE Trans. Plasma Sci. 32 2217), concerned with the structure of an active collisionless magnetized plasma-sheath, is extended to include the effect of collisions on the ion motion. In the collisionless case an expression for the eigenvalue of the problem analogous to that obtained by Franklin and Ockendon (1970 J. Plasma Phys. 4 371) for an active unmagnetized plasma is obtained. As is to be expected with increasing collisionality, increased transport of ions across the magnetic field lines means that the effect of the magnetic field is diminished. Special attention is given to the case when the magnetic field is nearly parallel to the wall.

Modelling discharges in electronegative gases

This paper builds on earlier work to give a consistent treatment of the positive column of discharges in electronegative gases covering the transition from collisionless to collisional. In particular it seeks to elucidate the conditions under which there is an ion-ion plasma core surrounded by an electron-ion plasma, and when there is not. The parameters which describe the processes of ionization, attachment, detachment and recombination are related to the central negative ion density relative to the electron density and, where appropriate, the size of the core. The use, by earlier workers, of the Boltzmann approximation to describe the negative ion distribution and to obtain ambipolar diffusion coefficients at higher pressures is shown not to be justified. This leads to the clarification of an inconsistency in the literature. Where possible, the work is related to other recent treatments of the same problem in order to begin to build a comprehensive picture of such discharges. The need to have results which combine both detachment and recombination as the negative ion loss processes is identified as outstanding. This, when rectified, should lead to a fully comprehensive treatment.

The transition from collisionless to collisional active plasma in the fluid model and the relevance of the Bohm criterion to sheath formation

The fluid equations describing the plasma–wall sheath in an electropositive gas are solved numerically with parameters, δi the ratio of the ion collision frequency νi to the ionization rate Z, and α the ratio of the ionization rate to the ion plasma frequency ωpi. From the behavior of the ion speed and from the plasma balance equation relating the plasma dimension L to Z it is possible to identify regions in α−δi space in which the plasma is (a) collisionless, (b) transitional, and (c) collisional and in which the boundary layer is (i) collisionless and (ii) collisional. The Bohm criterion ceases to have significance when the layer matching plasma and wall becomes collisional.

Plasmas with more than one species of positive ion and the Bohm Criterion

The fluid equations for an active plasma generated by electron impact containing more than one species of positive ion are examined in detail when the ionization frequencies are assumed to be spatially constant with a view to generalizing the Bohm Criterion. It is shown that in this specific situation the various ion spatial distributions and transverse ion speed distributions are geometrically similar and thus each species arrives at the sheath with its own Bohm speed, when the plasma is collisionless. The collisional situation is also examined with the conclusion that in that case as well, ion densities and ion speeds are again geometrically similar. The transition between the collisionless and collisional cases is discussed but not examined in detail.

The plasma-sheath transition with a constant mean free path model and the applicability of the Bohm criterion

The fluid model for positive ions is used together with the assumption of a constant ion mean free path to examine the structure of the plasma-wall sheath at low and medium pressures in plasmas where ionization is by electron impact and the ion temperature is set to zero. It is shown that in the near collisionless regime there is the same structure as was found with the constant collision frequency for momentum transfer model with the Bohm speed playing the same role and similar scaling of the transitional layer. At higher pressures there is again no transitional layer, only plasma joining smoothly to a collisional sheath, thus the Bohm criterion has no significance, but the scaling is with a different exponent than for the mobility case. No fundamental differences are found between the two models. An expression is derived to describe the boundary between the parameter region where the Bohm criterion applies and that where it does not.

A critique of models of electronegative plasmas

This paper seeks to bring together into a common format work on electronegative plasmas, carried out over several years by different authors, as well as indicating how the differences have arisen. In the case of oxygen at higher pressures the dominant loss mechanism is detachment rather than recombination and the difference arose because the importance of associative detachment had not been recognized in some work. More generally, the concept of ambipolar diffusion does not carry over to electronegative plasmas. In particular the expressions given by Thompson (Thompson J B 1950 Proc. Phys. Soc. 73 818-21), quoted by Massey (Massey H S W 1974 Negative Ions (Cambridge: Cambridge University Press)) and again by Lieberman and Lichtenberg (Lieberman M A and Lichtenberg A J 1994 Principles of Plasma Discharges and Materials Processing (New York: Wiley-Interscience)), are wrong because they are based on the mathematical inconsistency of taking the negative ion and electron densities to be proportional, while simultaneously obeying Boltzmann relations with different temperatures. In general, electronegative plasmas are structured and there is agreement on this point; furthermore, the ratio of attachment rate to ionization rate is an important parameter in determining the structure. It is shown that a relationship exists between the central negative ion/electron density ratio and the generation and loss processes, namely ionization, attachment, detachment and recombination, over a wide range of pressures. The means of determining the plasma balance equation and so relating the ionization rate to the plasma dimension, the pressure and other discharge parameters are indicated.

There is no such thing as a collisionally modified Bohm criterion

The conventional equations describing plasma and sheath are examined over a full range of collisionality to determine the influence of collisionality on the ion equation of motion—that is, when the ion motion is essentially inertial and when it is collisional. When it is inertial in the sheath the Bohm criterion has been satisfied, while when the sheath is collisional the ions remain in equilibrium with the field all the way to the wall.