Kinetic modeling of the electronic response of a dielectric plasma-facing solid
We present a self-consistent kinetic theory for the electronic response of a plasma-facing dielectric solid. Based on the Poisson equation and two sets of spatially separated Boltzmann equations, one for electrons and ions in the plasma and one for conduction band electrons and valence band holes in the dielectric, the approach gives the quasi-stationary density and potential profiles of the electric double layer forming at the interface due to the permanent influx of electrons and ions from the plasma. The two sets of Boltzmann equations are connected by quantum-mechanical matching conditions for the electron distribution functions and a semi-empirical model for hole injection mimicking the neutralization of ions at the surface. Essential for the kinetic modeling is the ambipolarity inside the wall, leading to an electron-hole recombination condition, and the merging of the double layer with the quasi-neutral, field-free regions deep inside the wall and the plasma. To indicate the feasibility as well as the potential of the approach we apply it to a collisionless, perfectly absorbing interface using intrinsic and extrinsic silicon dioxide and silicon surfaces in contact with a two-temperature hydrogen plasma as an example.
pacs:52.40.Hf, 52.40.Kh, 68.49.Jk, 68.49.Sf
The basic electronic response of a plasma-facing solid (wall) is the accumulation of electrons from the plasma. It acquires a negative charge because impacting electrons are deposited inside the solid more efficiently than electrons are extracted from it by the neutralization of positive ions. The negatively charged wall in turn triggers a positively charged space charge region in front of it: the plasma sheath [1, 2]. The total result of the electronic response of the plasma-wall interface is thus an electric double layer. Yet, ever since the early days of gaseous electronics , the negative part of the double layer–the wall charge–plays no essential role in the studies of the plasma sheath. Typically the focus is either on how the sheath merges with the quasi-neutral bulk plasma (see the reviews [4, 5, 6, 7, 8] and references therein) or on how the emissive properties of the wall, most notably, electron reflection (or absorption) and secondary electron emission, affect the spatial structure of the sheath [9, 10, 11, 12, 13].
Evidently for this type of studies the wall is considered as a reservoir characterized by a geometrical boundary and probabilities for particle reflection, absorption, and emission which may be chosen ad-hoc to make plasma simulations reproduce experimental findings, deduced from quantum-mechanical calculations [14, 15], or–in very rare occasions–obtained from independent measurements [16, 17]. In situations where the length and time scales of the plasma and the wall are well separated this is a viable strategy. Even microdischarges  with a linear extension of can be successfully modeled by such an approach . However, if this is not the case, or if the electrons accumulated in the solid are an integral part of the physical system one is interested in, as it is, for instance, the case for the plasma bipolar junction transistor , this approach is not sufficient. A kinetic description has then to be set up also for the wall and merged to the one of the plasma by suitable matching conditions.
In particular, solid-state based integrated microdischarges [21, 22, 23, 24] can be expected to soon reach the sub-micron range where the electron transit time through the sheath of the discharge approaches the electron energy relaxation time inside the solid. In this case, the electronic subsystem of the wall remains out-of-equilibrium between subsequent electron encounters from the plasma and surface parameters have to be obtained for a solid in strong electronic non-equilibrium. Taking, as an illustration, a wide microdischarge with a screening length of one-tenth of its width, which seems to be feasible , and an electron temperature yields , where is the thermal velocity of the electron. This is only one order of magnitude smaller than the typical electron energy relaxation time in the conduction band of a dielectric solid such as SiO, , assuming a kinetic energy of the injected electron of above the bottom of the conduction band, an effective mass of , and an inelastic scattering length of (from the universal curve ), but already much shorter than the electron-hole recombination time which is on the order of nanoseconds. In addition, the separation between integrated microdischarges can be made small enough to enable crosstalk through the space charge layers inside the wafer opening thereby perhaps opportunities for novel opto-electronic plasma devices. There are thus situations conceivable where the electronic processes in the solid and the plasma cannot be considered independently anymore. Focusing on the positive part of the electric double layer alone will then be of course also no longer sufficient.
Double layers are abundant in nature and have been studied in various contexts. They arise at any interface separating positive and negative charges. In solid state physics the most important double layer is the pn-junction  which is at the heart of many electronic devices. Double layers occur also when two different gaseous plasmas approach each other [27, 28, 29, 30]. The double layer at the plasma-wall interface is however special. On the one hand, and in contrast to pn-junctions, it is always far from equilibrium, involving very hot charge carriers. On the other hand, and this distinguishes it from the gaseous double layers, it is spatially pinned by the crystallographic ending of the wall and energetically constrained by the wall’s band structure. Irrespective of the demands arising from the miniaturization of microdischarges studying double layers at plasma-wall interfaces is thus also of fundamental interest.
Up to now electric double layers at plasma-facing solids have not been studied extensively. To the best of our knowledge metallic surfaces have not been investigated at all and there are only a few studies [31, 32] devoted to dielectric surfaces in contact with a plasma. But even for them a satisfactory description is still missing. In our own work  on the subject, for instance, we employed a thermodynamical principle to distribute the electrons missing in the plasma sheath in a graded potential which interpolates between the sheath and the potential inside the wall . The basic assumption, however, that at quasi-stationarity the electrons are thermalized within the wall, is only valid for some of the electrons and not for all. In addition, the approach was based on drift-diffusion equations. Hence, the dynamical variables were from the start particle densities, fluxes and electric potentials. Distribution functions did not appear. The coupling of the positive and negative parts of the double layer was thus entirely due to the matching conditions for the electric potential. It was hence impossible to include quantum-mechanical reflection of electrons by the surface potential and/or electron extraction due to the neutralization of ions.
Below we present a kinetic approach for a dielectric surface which is general enough to overcome these shortcomings. It works with the Poisson equation and two sets of Boltzmann equations operating in disjunct half-spaces. The matching at the interface is performed not only for the electric potential but also for the distribution functions which enables us to keep the ambipolarity of the plasma side (electrons and ions) alive inside the wall (electrons and holes). Eventually, this allows us to formulate a recombination condition for electrons and holes, which in turn limits, in conjunction with further conditions imposed at quasi-stationarity, the continuous influx of electrons and ions from the plasma. For the electron distribution functions the matching conditions are essentially identical with the matching conditions used for solid interfaces [33, 34, 35]. The matching condition for the ion distribution function on the other hand is a semi-empirical model for electron extraction (that is, hole injection) connecting the ion distribution function of the plasma with the hole distribution function of the wall. A thermodynamical principle is no longer used. Instead, we only demand as boundary conditions quasi-neutral, field-free regions far away from the interface.
In the numerical calculations we consider a collisionless perfectly absorbing surface. But the kinetic approach is first described in broader terms so that it becomes clear how to include collisions and how to include quantum-mechanical reflection of electrons. The numerical solution of the complete kinetic model is however rather demanding and beyond the scope of the present work. The numerical calculations yield thus only the self-consistent quasi-stationary potential and density profiles across an idealized interface. For given temperature and mass ratios the incoming flux of electrons and ions is self-consistently determined. Despite the simplicity of the model, the numerical results for SiO and Si surfaces in contact with a hydrogen plasma are very promising and clearly indicate the feasibility and potential of our approach for revealing the rich physics taking place inside the wall.
The outline of the remaining part of the paper is as follows. In Sect. 2 we describe in detail our approach, first, in general terms and then for the special case of a collisionless, perfectly absorbing dielectric surface. Section 3 discusses representative numerical data for idealized intrinsic and doped SiO and Si surfaces exposed to a two-temperature hydrogen plasma focusing on quasi-stationary density and potential profiles. A discussion of the issues which need to be resolved before the approach becomes quantitative for realistic surfaces is given in the concluding Section 4. Mathematical details interrupting the flow of the discussion are relegated to an Appendix.
2 Kinetic theory
2.1 General approach
Before we consider the simpler case of a collisionless, perfectly absorbing dielectric surface in contact with a collisionless plasma we describe in this subsection the kinetic modeling of the electronic response of a dielectric plasma interface in broader terms. A similar formulation could be worked out for a metallic plasma interface. As far as the modeling presented in this subsection is concerned, the main difference between metallic and dielectric plasma-facing solids is that for metals the neutralization of ions does not always lead to the injection of holes into a fully occupied valence band. Instead, if the ion’s ionization energy is small enough holes are injected into the partially filled conduction band, that is, into the same band into which also electrons are injected from the plasma. In this case, only the conduction band is involved, which moreover is partially filled. The hole representation we use in the following is then no longer advantageous and the theory on the solid side of the interface has to be formulated solely in terms of distribution functions for the conduction band electrons.
We consider a planar interface at separating a dielectric solid residing in the subspace from a plasma in the subspace. The theoretical treatment of the interface is based on the Poisson equation and two sets of Boltzmann transport equations operating respectively in the positive and negative half-spaces and describing in total four species: electrons and ions on the plasma side and conduction band electrons and valence band holes on the wall side. We use a species index to denote electrons, ions, conduction band electrons, and valence band holes. The interface is assumed to be homogeneous in the lateral directions and so that the spatial dependence of all quantities is reduced to a dependence on . For a quasi-stationary electric double layer at a homogeneous interface the distribution functions of the various species depend thus only on the spatial coordinate and the three-dimensional wave vector .
Anticipating the quantum-mechanical derivation of the matching conditions for the distribution functions [33, 34] as well as a possibly iterative numerical treatment of the Boltzmann equations along the lines employed in the transport theory of semiconductor heterostructures [36, 37] we replace the set of independent variables by where is the total energy and the lateral momentum. The species’ distribution functions are thus written as .
If not stated otherwise, we give all equations in atomic units measuring energy in Rydbergs and length in Bohr radii. The zero of the energy scale is chosen to coincide with the electron affinity of the dielectric, that is, we set .
The model on which our calculations rest is shown in Fig. 1. As it is drawn it already assumes an electric double layer with a negative and positive space charge, respectively, inside the wall and inside the plasma. The double layer is driven by a source at releasing an electron flux and an ion flux . We treat only the quasi-stationary case. The fluxes are thus equal to each other and exactly balanced by loss processes inside the wall. Far away from the interface, at , the wall becomes a reservoir for conduction band electrons and valence band holes.
On the wall side Fig. 1 shows a valence and a conduction band. Their edges are given by
with the energy gap between the valence and the conduction band and the electric potential energy given by the solution of the Poisson equation,
is the charge density and is the dielectric constant, both split, with the help of the Heaviside function , into a wall and a plasma part. The connection between the solutions of the Poisson equation in the two half-spaces of the interface is given by the matching condition for the electric potential energy,
The way is defined in (4), the charge densities on the wall and the plasma side are given by
where and are, respectively, the concentration of donors and acceptors while are the species’s densities obtained by integrating the distribution functions over the independent variables and ,
where is the absolute value of the component of the species’ velocity. Since the distribution functions depend via the Boltzmann equation (see below) on , the Poisson equation is in general a highly nonlinear integro-differential equation.
The physical meaning of the band edges is as follows: gives the lowest energy a conduction band electron can have at . Likewise gives the highest energy a valence band electron can have at location . At the plasma-wall interface valence band electrons per se are not directly relevant. What matters are the electrons in the valence band which have been extracted from it by neutralizing an ion impinging on the interface . It is thus natural to describe the valence band in terms of missing electrons, that is, in terms of holes . Instead of using electrons with a negative charge and a negative effective mass the hole picture describes the valence band by quasi-particles with a positive charge and a positive effective mass. The energy a hole can have at location is always larger then
indicated by the dashed blue line in Fig. 1. It is the edge for the motion of valence band holes.
On the plasma side the model contains ions and electrons. Their energies are given by
with the solution of the Poisson equation in the positive half-space. The energies for ions and electrons at position are always larger than, respectively, and . Figure 1 also shows the ionization level of the ion and its broadening , both taken at the turning point of the ion trajectory. These two energies are needed in the hole injection model to be described later. The ionization energy determines at what energy the hole is injected into the valence band and the broadening gives the energy range over which injection may occur. Notice, it is not the kinetic energy of the ion’s center of mass motion which matters for hole injection but the ion’s internal potential energy .
At the bottom of Fig. 1 we illustrate how distribution functions at particular locations, for instance, and can be determined if the distribution functions are known at the boundaries and and matching conditions are available connecting distribution functions across the interface at . For each species we distinguish particles moving to the left from particles moving to the right. Hence, we write
with and characterized, respectively, by and , where is the absolute value of the -component of the species’ velocity. Assuming for simplicity parabolic bands inside the dielectric, the velocities can be written as
where is the electron mass and is the species’s (effective) mass.
Far away from the interface we assume the distribution functions to be local Maxwellians,
normalized to the density
The temperatures in (14) are input parameters whereas the values the densities approach at the boundaries, the boundary densities at and , are considered as variables. Hence, , , , and have to be determined in the course of the calculation. This adjustment of the densities is required to make the plasma source consistent with the losses and the reservoir inside the wall mimicking the self-consistent response of the plasma to the wall and vice versa as it takes place in reality. On the plasma side this leads to the sheath modeling of Schwager and Birdsall  which has been also used in a number of particle-in-cell simulations [10, 12].
In order to determine in the way sketched at the bottom of Fig 1 we need matching conditions at . Using the methods employed to match distribution functions across solid interfaces, originally developed by Falkovsky  for charge transport inside metallic surfaces, but subsequently applied also to solid heterostructures [33, 35], the distribution functions for electrons in the plasma and electrons in the conduction band of the wall are for at connected by
the quantum-mechanical reflection probability for a three-dimensional potential step of height . For energies just above the potential step is close to unity while for large it vanishes.
It should be noted that describes only the quantum-mechanical specular reflection on the potential step. Inelastic processes inside the wall which may bring the electron back to the interface, and hence also possibly back to the plasma, if it successfully passes the potential step in the reverse direction, are not included in . Hence, in (16) and (17) the reflection coefficient cannot be replaced by with the electron sticking coefficient obtained, for instance, by the method of invariant embedding  because accounts for inelastic processes which in the present modeling of the interface’s particle kinetics have to be incorporated into the collision integrals of the Boltzmann equations. The same holds for secondary electron emission due to impacting electrons with energies above the band gap of the wall. It also arises from inelastic processes which have to be accounted for by the collision integrals of the Boltzmann equations. Secondary electron emission due to Auger neutralization of ions  or other heavy particle collisions with the surface, on other hand, could be incorporated into the matching conditions by augmenting the right hand side of (16) by a source function describing the probability of the surface to emit an electron with total energy and lateral momentum because the kinetic equations we set up do not track the occupancies of the internal electronic states of the ion (or other heavy projectiles). The particular form of this function depends on the elementary surface process. It is best worked out semi-empirically along the lines we use now for the resonant neutralization of ions at a surface.
The matching of the ion and hole distribution functions differs from the matching of electron distribution functions because ions cannot enter the wall and most importantly the center of mass motions of the ions and the valence band holes are not coupled via energy and momentum conservation laws. In our model ions are resonantly neutralized at the interface whereby they inject holes into the wall’s valence band. Hence, the ion and hole distribution functions are for at connected by
with a source function to be constructed by other means, the probability for wall neutralization, which we assume here to be independent of the energy of the ion’s center of mass motion. If the ions are neutralized with probability less than unity some ions have the chance to come back to the plasma, having thus velocity in negative -direction and contributing thereby to . Only for , that is, for perfect neutralization, no ions are coming back to the plasma, as it is often assumed in the modeling of the plasma sheath.
A model for hole injection is required to complete the description of the matching condition (19). The source function is the probability to inject an hole with total energy and lateral momentum into the valence band. It is important to realize that in (19) and are not the total energy and lateral momentum of the impinging ion responsible for the injection. To connect the source function with the ion distribution function at the interface, , we recall that the total flux of injected holes has to be identical to the total flux of impinging ions multiplied by . Hence, with
we obtain by using (20) the condition
where the integration variables on the left (right) belong to the center of mass motion of the ion (hole). To proceed one either determines from a microscopic model for ion neutralization at a surface or one makes plausible assumptions about the overall behavior of this function. In the next section, where we discuss the simple collisionless, perfectly absorbing interface, we assume, for instance, holes to be injected with uniform probability over the relevant energy and momentum range.
The matching conditions (16), (17), (19), and (20) are essential for our approach. A comment about their impact on the distribution functions is thus in order. The distribution functions react freely to the matching conditions, that is, the values they assume at are determined self-consistently by the interplay of the plasma with the solid. Close to the interface the distribution functions deviate from the distribution functions far away from it in precisely such a way as it is dictated by the matching conditions. Since the matching occurs predominately in the tails of the distribution functions, that is, at high energies, where charge carriers cross the interface, collisions have enough phase space to efficiently heal the perturbation due to the interface making a merging of the solid and plasma distribution functions possible.
Let us now turn to the (quasi-stationary) Boltzmann equations to be satisfied by the distribution functions of the four species. They can be cast into the form
where is either positive or negative, depending on the species, and and are collision integrals which also depend on the species. For instance, ions may suffer charge-exchange collisions whereas electrons may be collisionless on the plasma side but subject to intra- and interband Coulomb and phonon collisions on the wall side. Similarly holes may also suffer intra- and interband Coulomb and phonon collisions. In addition, electron-hole recombination may take place inside the wall, for instance, via Auger or radiative processes. All of it has to be included in the collision integrals and . Since in the following we numerically treat only the collisionless interface we do not give explicit formulae for and but they can be worked out in all cases using the techniques of semiclassical kinetic theory .
Once the collision integrals are specified the set of equations for the theoretical description of the electric double layer is complete. It contains the modifications of the band structure inside the dielectric wall as well as the modification of the electric potential in front of the wall due to the permanent influx of electrons and ions. The fluxes are released from a plasma source and annihilated inside the wall. How the fluxes are annihilated, radiatively or non-radiatively, is beyond the present model. Solving the Poisson equation(3) together with the Boltzmann equations (23) for the four species subject to the matching conditions (5), (16), (17), (19), and (20) gives the species’ distribution functions and eventually the density and potential profiles of the double layer across the interface. The source and the reservoir can be made self-consistent by enforcing additional conditions to fix the boundary densities , , , and , which depend however on what kind of collisions are included. This in turn determines the way the system of Boltzmann-Poisson equations is solved numerically. In the next subsection we treat the simplest case–a collisionless interface–in full detail. It thus becomes apparent what additional conditions are needed and how the Boltzmann-Poisson equations are actually solved.
2.2 Collisionless, perfectly absorbing surface
So far the description of our approach was quite general. We now specialize the treatment to the simplest possible case: a collisionless plasma in contact with a perfectly absorbing collisionless surface. The reason is a practical one. In this particular case the Boltzmann equations become first order ordinary differential equations from which the two-dimensional lateral momentum can be eliminated. Since also drops out from the perfect absorber matching conditions, one no longer deals with distribution functions depending on four independent variables but only on two . Without collision integrals the Boltzmann equations can be furthermore easily integrated yielding, in addition, charge densities which do not explicitly depend on but only on . The Poisson equation can thus be integrated once analytically and the remaining numerical task is rather modest. In Sect. 3 we will discuss how realistic the collisionless theory is.
The lateral momentum can be eliminated as follows. First, the collisionless Boltzmann equations are integrated over the lateral momentum before switching from to as independent variables. They then become equations for
Switching then from to , with now the total energy without the kinetic energy in the lateral directions, leads to collisionless Boltzmann equations of the form
from which the distribution functions can be determined. Since no longer contains the kinetic energy in the lateral directions, the velocities in (25) are given by
with still defined as before.
Next, setting for a perfectly absorbing surface, the matching conditions for the electron distribution functions, (16) and (17), can be transformed back to -space and integrated there over leading to conditions for . Changing now the independent variables to as in the Boltzmann equations leads to conditions for with again the total energy without the kinetic energy in lateral directions. Obviously, this procedure is only applicable if is assumed to be independent of . In general this is not the case. Hence, to reduce in the general matching conditions the number of independent variables requires additional assumptions about the energy and momentum dependence of . Only for the perfect absorber model, where from the start, no further assumptions are necessary. For , as again postulated by the perfect absorber model, the matching conditions for ions can be handled similarly. Hence, the lateral momentum can be also eliminated from them.
Altogether, in terms of the functions the matching conditions for the collisionless, perfectly absorbing interface become
The first equation indicates that for electrons not to come back to the plasma, as postulated by the perfect absorber model, we have to assume not only but in addition for . This thermalization condition is necessary because is the same in Eqs. (16) and (17). Quantum mechanical reflection by a potential step does not depend on the direction the step is crossed. Conditions (29) and (30) describe a surface perfectly annihilating ions by injecting holes into the valence band. Assuming the source function to be a uniform probability for hole injection, it is given by
with the normalization
to ensure, at the interface, the equality of ion and hole flux. In the normalization condition for we already anticipated for the range of integration discussed in Fig. 2 and set .
Equations (25) can be solved by a trajectory analysis taking matching and boundary conditions into account as illustrated in Fig. 2 . In the collisionless model the boundary conditions at and are given by the Maxwellians
as follows: At we enforce for whereas at we require for . This is illustrated in Fig. 2 by the vertical blue lines. The densities at and , the boundary densities, given by , , , and are as pointed out in the previous section variables to be determined by the kinetic model. In addition to the boundary conditions the solution of (25) requires the matching conditions at symbolized in Fig. 2 by the vertical red lines. Since the trajectory analysis is standard  it is not explicitly given here. The principle of the calculation is described in the caption of Fig. 2.
It turns out that the densities obtained from the solutions of (25) depend only on and not on explicitly. This greatly simplifies the further processing of the densities. In particular, it allows us to obtain the first integral of the Poisson equation analytically. What remains to be done numerically is only the calculation of the second integral of the Poisson equation and the solution of four nonlinear algebraic equations. This is of course much easier than a numerical solution of a set of collisional Boltzmann-Poisson equations. In our view this justifies working out the collisionless, perfectly absorbing interface. Despite its idealistic nature the numerical solution may lead to insights useful for attacking the full problem.
for the fluxes on the plasma and the wall side of the double layer. Hence, the matching conditions and the hole injection model preserve by construction the fluxes across the interface. At quasi-stationarity, the electron and ion fluxes satisfy moreover the flux balance condition,
leading to a first condition involving densities at the system’s boundaries
The densities are obtained from (8) using (12), again with the obvious modifications arising from the elimination of . With the expressions for given in A the source for the Poisson equation on the plasma side becomes
while on the wall side the source is given by
In the formulas given above, the densities describe conduction band electrons and valence band holes which are thermalized/trapped within the wall while the densities originating from (27) and (28) describe carriers coming from the continuing influx of electrons and ions to the interface after quasi-stationarity has been reached. This influx does not stop after the double layer is fully developed. Quasi-stationarity makes the electron and ion fluxes coincide, rather than vanish. In the expressions for we employed already (36) to replace the fluxes and by the ion density .
Since the Poisson equation’s sources depend only on it can be integrated once in each half-space. Let us first consider the plasma side. Multiplying (3) by , where the prime indicates here and in the formulae to follow a derivative with respect to , and using , together with , which forces the double layer to be field-free at , yields
In our model we assume to act inside the wall as the only source for the electric potential energy . It is this part of the wall’s charge density which balances the positive charge density on the plasma side of the interface. The density acts in our model not as a source. Instead it will be made to vanish (physically due to electron-hole recombination inside the wall, see below). Thereby it yields an additional condition which in conjunction with the other conditions to be satisfied at the interface enables us to calculate also the continuing influx of electrons and ions. Using thus inside the wall as the only source for and a procedure similar to the one employed to derive (42) leads us to
and . The function is given in A. In deriving these expressions we used which guarantees continuity of at , as required by the first condition in (5). In addition we forced the double layer to be field-free at by setting . The potential profile on the wall side is given by integrating (45) from to resulting in
In order to incorporate into the formalism the role we want the densities to play, we now take a closer look at them. As already mentioned they arise from the continuing influx of electrons and ions after the quasi-stationarity of the double layer has been reached. Hence, for a quasi-stationary double layer cannot act as a source for an electric field as we already anticipated in (45). It is thus reasonable to assume that they recombine nearby the interface, perhaps in a spatial zone stretching from to , where is a distance from the interface not yet specified. Below we will assume to coincide with the inflection point of inside the wall which we need to match the double layer to a quasi-neutral, field-free region, as it happens in reality. On the plasma side an inflection point has to be implemented for the same reason. There it is required to match the sheath to a quasi-neutral, field-free plasma .
In view of what we just said, we hence postulate the recombination condition,
with the electric potential energy at the inflection point . In the second equality we used (45) to replace the integration by an integration over . The last condition we finally have to enforce is the jump condition for the derivative of the electric potential energy at as stated in (5). Using (42) and (45) it becomes
Now, we have all the ingredients together to formulate a self-consistent model for the electric double layer at a collisionless, perfectly absorbing plasma-wall interface.
Towards that end let us first discuss the necessity of implementing inflection points into the potential profile . It can be most clearly seen by setting hypothetically and , that is, by making the source and the reservoir charge-neutral. Such a choice would however not lead to and as one would perhaps naively expect. Hence, by initially assuming at and distribution functions confronts us at the end with charge non-neutral boundaries if at the same time we force the net charge of the boundary densities to be zero. Hence, cannot describe the quasi-neutral regions into which the double layer should be embedded.
The reason is of course that some particles are, depending on their energy and/or type, either absorbed or emitted by the interface. Hence, they are lost from or gained by a half-space of the double layer preventing thereby to re-establish at the boundaries. In reality the distribution functions react self-consistently to the presence of the interface making thereby the double layer also charge-neutral far away from the interface. By postulating the form of the distribution functions at and we destroyed this mechanism. Alternatively we could have enforced charge-neutrality at the boundaries. But then we could not have known the distribution functions making the solution of (25) much more complicated.
To mimic the self-consistent reaction of the distribution functions far away from the interface we follow Schwager and Birdsall  and consider the boundary densities, , , , and , appearing in at and , respectively, as variables to be determined from the calculation. This requires to incorporate two inflection points into the potential profile, one at inside the wall and one at inside the plasma. The conditions for the inflection points are charge-neutrality
and the vanishing of the electric field
with and . Notice, on the wall side only appears.
On the wall side, however, charge-neutrality is more involved since in addition to (50) we also have to satisfy
the intrinsic charge density  of the wall at temperature . Using the formulae for the densities given in A and solving Eqs. (50) and (55) simultaneously, assuming either (intrinsic wall), (n-doped wall), or (p-doped wall) we obtain two conditions for the boundary densities and :
where is the error function, see A. For an intrinsic wall and , for a p-type wall leading to and with , while for a n-type wall yielding and with . From the jump condition (49) we finally obtain an equation relating to the density ratios and making the approach self-consistent.
The description of our approach is now complete. The modeling we propose for a quasi-stationary electric double layer at a collisionless, perfectly absorbing plasma-wall interface contains eight parameters: Four energies , , , and and four densities , , , and . Eight equations are available to determine them: The three conditions (37), (57), and (58) for the boundary densities, the quasi-neutrality condition on the plasma side (54), the two conditions forcing the double layer to be field-free around and , (52), (53), the recombination condition (48) and the jump condition (49) guaranteeing at the end that the double layer is globally charge neutral between its physically relevant boundaries and . It should be noticed that the wall provides an absolute scale via the reference density and the band structure parameters and , as does the ionization energy and its broadening . The approach produces thus absolute numbers.
In this section we use parameters applicable to Si and SiO surfaces in contact with a two-temperature hydrogen plasma to obtain numerical results for the electric double layer forming at a collisionless, perfectly absorbing plasma-wall interface. Before discussing the results we give some details about the numerical treatment of the equations derived in the previous section.
For the numerics we normalized energies on both sides of the interface to the thermal energy of the electrons emitted from the plasma source. Lengths, in contrast, are normalized, depending on which side of the interface is considered, to the electron Debye length of the wall,
or the electron Debye length of the plasma
After rewriting the equations in normalized form we replace the boundary density ratios , , and by (37), (57), and (58). We then obtain four equations for the four (normalized) potential energy drops , , , and . They factorize into two sets of two equations each, one for and one for .
More specifically, the equations for arise from the quasi-neutrality condition (54) and the field-free condition (53). After replacing the boundary ratio by (37) they become nonlinear equations for alone and can be casted into the form
Except for the difference arising from the different choice of the zero of the energy axis these two equations are identical to the ones given by Schwager and Birdsall . Replacing the boundary density ratios and by (57) and (58) in the field-free condition (52) and the recombination condition (48) leads to two nonlinear equations for . They can be casted in the same form,
We solve (61)–(64) graphically as explained in Fig. 3 below. The solutions on the plasma side and on the wall side are linked to each other by the jump condition for the electric field (49) which in normalized form becomes a condition containing all four boundary density ratios. Since the boundary ratios , , and are known from the potential drops the fourth ratio can be determined from (49) making thereby the wall side of the double layer consistent with the plasma side. At this point the recombination condition (48) turned out to be essential. Without it the collisionless theory had not enough equations to determine all the unknown parameters.