Study of a collisionless magnetized plasma sheath with nonextensively distributed species

R PAUL ,K DEKA ,G SHARMA ,R MOULICK ,S ADHIKARI ,S S KAUSIK,* and B K SAIKIA

1 Centre of Plasma Physics,Institute for Plasma Research,Assam 782402,India

2 Institute for Energy Technology,Instituttveien 8,Kjeller 2007,Norway

Abstract A weakly magnetized sheath for a collisionless,electronegative plasma comprising positive ions,electrons,and negative ions is investigated numerically using the fluid approach.The electrons are considered to be non-Maxwellian in nature and are described by Tsalli’s distribution.Such electrons have a substantial effect on the sheath properties.The study also reveals that non-Maxwellian distribution is the most realistic description for negative ions in the presence of an oblique magnetic field.In addition to the negative ion temperature,the sheath potential is also affected by the nonextensive parameters.The present research finds application in the plasma processing and semiconductor industry as well as in space plasmas.

Keywords: magnetised plasma sheath,electronegative plasma,non-extensive distribution

1.Introduction

The intricacies of sheaths have made them one of the most researched topics in plasma physics.Although they have been studied and understood by both fluid [1-7] and kinetic approaches [8-12],there is enough scope to investigate the problem further.In the case of the fluid approach,usually,the ions are governed by the hydrodynamic equations and the distribution of electrons are defined by Maxwell-Boltzmann statistics.However,it is often realized that although the Maxwellian approach is simplistic,it overlooks several details[13].Besides,the electrons found in astrophysical plasmas[14,15] possess a high energy tail and hence the distribution deviates from the Maxwellian nature.It is also proven that the electrons found in the processing chambers are also far from being Maxwellian[16,17].Generally,laboratory plasmas are found to have a high energy tail,thus losing their Maxwellian identity[18,19].Moreover,the Boltzmann behaviour restricts the electron distribution to the macroscopic ergodic equilibrium state and is often found to be inadequate to describe long-range interactions[20].Also,the plasma sheath is regarded as a transition region between the plasma and the wall and is considered a non-neutral region besides possessing a non-Maxwellian nature [12,21].Due to such characteristics,the sheath plays a substantial role in influencing the particle and energy transport towards the wall.Therefore,the hunt persists for a more generalized version of Boltzmann-Gibbs statistics that could provide a better description of such non-Maxwellian behaviour.

In 1988,Tsallis proposed an entropy for systems that are no longer in an ergodic equilibrium state and it is defined as[22]

Figure1.Geometry of the system.

where piis the probability of the ith microstate,kBis the Boltzmann constant and q is a real number.In this case,q measures the degree of nonextensivity [22-25].Usually,the entropy for an entire system in equilibrium is the summation of the entropies of their respective parts.However,this is not the case for a nonextensive system.In such systems,the generalized entropy of the system is either greater than the sum of its parts for q ＞1 or smaller than the sum when q ＜1.Boltzmann-Gibbs entropy can be derived from q-entropy[20,25].This distribution is found to be more suitable for those systems where long-range particle interactions are more prevalent.To describe the nonextensive nature of the plasma,it is introduced in the system via electrons.Huibin Qiu et al[26] have measured the electron nonextensive parameter using a nonextensive single electric probe and found it to be 0.775.

Furthermore,the nonextensive distribution studies have provided meaningful insights while investigating complex problems such as ion-acoustic waves[27],dust charging[28]or in understanding solitary structures [29].In addition,the behaviour of nonextensive electrons on the sheath structures has also been widely studied.Gougam and Tribeche [30]studied the Debye shielding phenomenon considering both nonextensive electrons and ions.They found that the debye length in the case of nonextensive electrons is smaller compared to the standard Maxwellian case.Besides investigating the role of the nonextensive parameter,Moulick et al [31]studied the effect of ion-neutral collision in the presheath and sheath region.Moreover,multi-electron species described by nonextensive distribution have been studied by Borgohain et al [32] using the Sagdeev pseudo-potential method and they concluded that the ion entrance velocity depends upon the temperature of the ions,the temperature and the density ratio of the electron species.

In addition to multi-electron plasma,the sheath,along with additional components such as negative ions,in the presence of,as well as in the absence of,magnetic field has been studied considering nonextensive electrons [33,34].In all such investigations,negative ions are considered to be Maxwellian in nature.It emerges as an apt choice with the justification that the magnetic field has a negligible effect on the negative ions as compared to the positive ions [35].This provides the balance between the electric field force and the pressure force,therefore,retaining the Boltzmannian behaviour.As the inertia of the negative ions is larger than that of the electrons,this acts as the point of contention for choosing Maxwellian distribution.Also,considering Maxwellian negative ions has resulted in oscillatory structures while studying sheath properties [36,37].Such oscillations are not realized if realistic distributions are considered[38].It is a hint at the possibility that the negative ions might have a non-Maxwellian distribution.Additionally,the negative ion energy is also experimentally found to deviate from the usual Maxwellian nature [39].Besides,there are instances of considering non-Maxwellian negative ions in understanding the dust charging process [40].It is quite well known that the negative ions have a notable effect on the sheath structure.Apart from that,the negative ions also find their presence in many industrial applications as well as in fusion devices[41-45].It is usually favoured over positive ions in the case of neutral beam injection (NBI)as negative ions have greater efficiency in heating the fusion plasma to sustain the desired temperature [46].Thus,negative ion-rich plasmas provide a promising platform for basic as well as application-related studies.

The present paper aims to understand the nature of negative ions in the presence of an oblique magnetic field and nonextensive electrons.Initially,Maxwellian distributed negative ions are considered and the sheath properties are investigated.Later on,negative ions are considered to be nonextensively distributed.It is seen that the dynamics of positive ions change with the nature of the negative ion distribution.It concludes that the negative ions are better described by a non-Maxwellian distribution under the influence of the magnetic field.

The paper has been divided into the following sections.Section 2 discusses the theoretical model and the basic equations.Section 3 analyzes the theoretical findings and a brief conclusion has been presented in section 4.

2.Theoretical model and basic equations

The present study considers a steady-state multi-component hydrogen plasma comprising hydrogen positive ions,electrons and hydrogen negative ions.The plasma is collisionless and is considered to be in contact with a planar wall.An oblique magnetic field B,as shown in figure 1,is considered in the X-Z plane with an angle of inclination,θ,with the Xaxis perpendicular to the wall.The sheath is formed along the positive Z-direction.The cold ions are described by the hydrodynamic equations in a 1D-3V representation given by

where B is the magnetic field intensity,and mi,v are the mass and velocity of the positive ions,respectively.

The first term in the momentum equation (1) represents the electrostatic force,the second term corresponds to the magnetic force and the third term refers to the momentum transfer due to ionization.The ions are usually cold and hence the pressure gradient term may be neglected.However,a finite isotropic ion temperature is considered for the present case.The magnitude of the ion temperature is small enough to avoid the pressure gradient term.Though,in a magnetized plasma,anisotropy in ion temperature is often encountered which results in numerous instabilities in the plasma[5].But,the plasma considered in this paper is stable and is devoid of any such anisotropy.On the other hand,the ion-neutral collisions are neglected and the dominant force in the sheath formation is the Lorentz force.Additionally,the momentum transfer due to ionization is neglected in the present case.However,the momentum transfer due to ionization in the momentum equation of the ions is usually responsible for any increase in the ion velocity.But,as compared to the other terms such as the electric force and magnetic force,the transfer of momentum due to ionization is nominal.Thus,any increase in the ion temperature due to ionization will be insignificant and hence ion-neutral collisions can be neglected [47].

The continuity equation is given by

In this case,an exponential ion source,Siis considered and can be defined as

whereZ is the ionization rate which is assumed to be constant throughout the domain.Sirepresents the rate at which the ions are created per unit volume per unit time [48].

The electrons,being more mobile in nature as compared to the other species,are often represented by the Boltzmann relation,defined as,

where Teand ne0represent the temperature and equilibrium density of the electrons and ϕ is the electric potential.In such a case,the pressure gradient balances the electric field force[2].As mentioned above,electron distributions are often found to deviate from Maxwellian nature.It has been found that the Tsallis distribution is best suited for such scenarios.As the present plasma deals with non-Maxwellian electrons,the distribution of the electrons can be defined as

where q is the nonextensive parameter.For q=1,the above equation behaves similarly to that of the Boltzmann relation.The nonextensive parameter q has been largely distinguished into two sections,namely superextensive and subextensive.Superextensive (q ＜1) corresponds to cases where most of the particles are considered to be fast-moving and possess a high-energy tail.The subextensive cases (q ＞1) contain low energetic particles mostly [49].However,the present study focuses mostly on the presence of high energetic particles and the superextensive case is best suited for such a model.

In the case of negative ions,they have been defined by the Boltzmann relation as

where T-and n-0represent the temperature and equilibrium density of the negative ions.The reason behind the Boltzmann behaviour for negative ions is that the magnetic field has a negligible effect on them as compared to the positive ions [35].Also,the negative ions are pushed by the electric field towards the centre of the discharge,thereby a decrease in the velocities is observed which aids in balancing the electric field force with the pressure gradient.This validates the reason for choosing the Boltzmann relation [5].

Asserghine et al[34]and Zou et al[50]have studied the effect of the nonextensive electrons on the electronegative plasma sheath in the presence of an oblique magnetic field.However,as the focus of both the studies lies on the Debye sheath,they have equated the sheath edge velocities to the Mach number.This also,in turn,implies that the presheath region is not considered in their study.On the other hand,in the present study,the presheath region is considered and thus the equations are scaled in the ionization length scale.Considering the 1D-3V representation of the system,the equations can be resolved into components along the X,Y and Z directions as

The electrostatic potential is governed by Poisson’s equation

To solve the above set of equations,it is necessary to normalize the quantities by the following dimensionless parameters

Table 1.List of simulation parameters.

ωXand ωZare the respective ion-cyclotron frequencies along the X and Z directions.

The normalized equations are given by

The normalized electron and negative ion densities are given by

2.1.Numerical techniques

The above set of equations(11)-(15)are ordinary differential equations that can be solved using the Runge-Kutta fourth order method.As these are initial valued problems,the initial values are obtained from the Taylor series expansion[4,51,52].The series may be constructed as

where N,V,η represents the density,velocity,and potential,respectively for ions.

The above set of series is used in the governing equations and the first coefficients obtained by the Taylor series are considered the initial values.The first-order coefficients are given as

At exactly ξ=0,all the physical quantities such as density,velocity,and potential become zero.Therefore,the starting point has been slightly right-shifted from the origin,i.e.ξ=0 such that all the quantities have nonzero physical values.

The default parameters chosen for simulation are in accordance with low-pressure plasma discharges and are listed in table 1.

For the considered plasma parameters,the ion-neutral collisional mean free path (λni=0.33 m),electron-ion collisional mean free path (λei=0.14 m),and ion-electron collisional mean free path (λie=29 m) are large as compared to the length scale of the sheath (ξ=9×10-4m).Such collisions do not participate in the sheath formation and are,therefore,neglected in the present model.The wall is considered to be floating and is assumed to be situated where the ion flux balances the random electron flux [4,51,52].It is given as

Figure2.Normalized density of all species for different nonextensive parameters,q and a constant value of τ-=2.

3.Results and discussions

3.1.Negative ions described by Boltzmann distribution

To understand the effect of nonextensive electrons on various sheath parameters,an investigation into spatial variation of the density of the species can provide meaningful insight into the problem.Figure 2 shows the normalized density variation with respect to the different q parameters for an electron-rich plasma (α ＜1) and for that reason α is chosen to be 0.6 [5].For the present case,τ-=2 such that the negative ion temperature lies within the well-established limits [54],and also to avoid any multi-valued plasma potential solution[55].The variation in the q parameter greatly affects the density descent of all the species along the sheath.With the decrease in the q value,the density of the individual species declines rapidly as can be seen in figure 2.The distribution corresponding to lower q values is wider.As a result,such distributions have more high energetic electrons as compared to higher q values.This consequently compels the positive ions to complete the shielding process faster,thereby leading to a quick fall.The negative ions,on the other hand,follow the positive ions and limit themselves near the core region.The density fall becomes rapid for lower q value,which has been earlier observed by Borgohain et al,for an electrostatic case[33].Particularly in the case of the positive ions,they are unable to complete the gyration due to the drift experienced by the guiding centre as a consequence of the rapid motion of the species [47].This implies that the magnetic field has a moderate effect on the ions.

Figure3.Normalized space charge profile for different nonextensive parameters,q and for constant values of α=0.6 and τ-=2.

Figure4.Normalized electric field profile for different nonextensive parameters,q and a constant values of α=0.6 and τ-=2.

The nonextensive electrons have an effect on another important sheath parameter,namely the space charge profile,which can be seen in figure 3.The space charge profile is often considered a way to visualize the sheath thickness in plasma.The charge deposition in this case decreases near the wall with the decrease in the q parameter,as well as the peak shifts towards the sheath edge.This can be justified as the rapid fall of the negative species with a lower q value occurs near the edge region.This results in the screening of the negative species by the ions at the entrance edge which creates a non-neutral region far away from the wall.The electric field profile depicted in figure 4 also confirms that the net positive charge distribution inside the sheath increases with the decrease in the q parameter.For q=0.9,the charge separation is localized near the wall and this lowers the electric field.With the gradual decrease in the q value,this localized nature gradually disappears and the overall broadening of the positive space charge region occurs which is visible from figure 3.Additionally,the ion entrance velocity along the Z component also decreases with the lower q value,which can be observed in figure 5.This affects the space charge amplitude and is,therefore,found to be least for q=0.6,as observed in figure 3.However,the velocity increases monotonically for lower q values.

Figure5.Normalized velocities along the X,Y and Z directions for different nonextensive parameters,q and for constant values of α=0.6 and τ-=2.

Figure6.Normalized space charge profile for different electron nonextensive parameters,qe and for constant values of negative-ion nonextensive parameter,qn,α=0.6 and τ-=2.

3.2.Negative ions described by q nonextensive distribution

Now,considering a more realistic situation where negative ions can also deviate from the usual Maxwellian nature,here a nonextensive distribution is used to describe them.It is defined as

Figure7.Normalized space charge profile for different electron nonextensive parameters,qe,and for constant values of negative-ion nonextensive parameter,qn,α=0.6 and τ-=2 for a bulk density of 1020 m-3.

Figure8.Normalized density of all species for different nonextensive parameters,qe,and for constant values of qn,α=0.6 and τ-=2.

The normalized density is given by

However,the parameter q for negative ions and electrons is chosen to be different.Thus,the normalized density equations (4) and (18) become

Figure9.Normalized density of all species for different nonextensive parameters,qe at different magnetic field inclinations,θ and for constant values of qn,α=0.6 and τ-=2.

where qe,qnare the q parameters for electrons and negative ions,respectively.

Here the sheath parameters are investigated in the presence of nonextensive electrons as well as nonextensive negative ions.On the onset,the space-charge profile for different qevalues corresponds to a particular value of qn,α=0.6 and τ-=2 is plotted in figure 6.It is seen that the non-neutral region appears broader in this case,indicating a lower charge separation between the bulk region and the wall.Moreover,the numerical noise has been reduced to a certain extent by choosing non-extensive negative ions.For a highdensity plasma(n0=1×1020m-3)the noise further reduces,indicating that the non-Maxwellian nature dominates in such scenarios.Although,the density of the majority charge contributor is low,as seen in figure 7.However,the overall negative charge density has increased inside the sheath.This can be very well observed from the density profile,as seen in figure 8.The negative ions have a slower fall for lower qevalues as compared to the Maxwellian negative ions and the overall particle density has increased inside the sheath.

Also,while choosing the distribution for negative ions,it is always considered that the magnetic field influence is negligible on the negative ions.However,the density distributions in figures 9(a) and (b) show quite an interesting behaviour.The magnetic field inclination is believed to have an impact on the spatial distribution of the species towards the wall.Thus,a consequential consideration of magnetic field inclination becomes essential.

Figure10.Normalized space charge profile for different magnetic field inclinations,θ.

Figure11.Normalized negative ion density profile for different magnetic field inclinations,θ.

3.2.1.Effect of magnetic field inclination,θ.As the species are found to have a dependence on the magnetic field,the problem has been further investigated by varying the angle of inclination.Here,both the nonextensive qeand qnare kept fixed.First and foremost,the space charge profile for various magnetic field inclinations has been obtained as shown in figure 10.With the increase in θ,the span of the sheath decreases while the total positive charge deposition increases.The corresponding response of the negative ion distribution across the sheath has been portrayed in figure 11.The nature of the density profile for θ=15° is found to differ from the other three cases.This behaviour can be purely attributed to the resistive nature of the magnetic force on the motion of the species towards the wall.For increasing θ,the Z-component of the magnetic field decreases,which enables more positive ions to gather near the wall,resulting in higher charge deposition.Consequently,a smaller length is required to shield the sheath electric field.Additionally,due to higher charge deposition,the total ion flux as seen in figure 12 also increases with increasing θ.As negative ions follow the positive ions while traversing towards the wall,this affects the negative ions’ motion to a great extent.As a result,it establishes the argument that the role of the magnetic field in the dynamics of the negative ions can not be ignored.Hence,a non-Maxwellian description of the negative ions proves to be the most suitable in such a magnetised scenario.

Figure12.Normalized ion flux profile for different magnetic field inclinations,θ.

3.2.2.Discussion on electric potential(η).Another important parameter to investigate the sheath structure is the electric potential profile.It can be well stated that the potential at the wall is sensitive to the change in the value of the q parameter.For a fixed qnvalue,figure 13(a) shows the sheath potential variation for different values of qeand τ-.It can be observed that as qe→1,changes in η are less prominent.However,as qebecomes more nonextensive,visible changes in η are observed with the change in τ-values.Additionally,it is observed that η increases as τ-decreases.For lower values of qe,the higher energetic tail in the electron distribution function is quite prevalent which increases the mobility of the electrons towards the wall,which in turn increases the net electric field in the sheath.Nevertheless,as qe→1,the high energetic tail of the distribution function disappears,thereby lowering the sheath electric field and hence lower values of potential are recorded.Furthermore,for a fixed qevalue,the potential variation for different values of qnand τ-is shown in figure 13(b).Here,as τ-decreases,for any qnvalues,η is observed to be minimum.Higher τ-values indicate cold negative ions hence,any change in the qnvalue has a negligible effect on it and thus lower values of electric potential are observed.On the contrary,for lower τ-i.e.the higher temperature of the negative ions,the overall electric potential increases as qn→1.As qnbecomes more nonextensive the potential lowers as compared to qn→1,which increases the overall particle density inside the sheath.

As the potential is affected by the temperature of the species,therefore,further investigation into the effect of the electronegativity on the sheath electric potential becomes essential.First and foremost,the wall potential is varied for different electronegativity (α) and qevalues at a constant qnvalue and is shown in figure 14(a).It can be observed here that as qe→1,the change in α has minimum effect on the potential.However,this scenario changes for lower qevalues.Figure 14(b),on the other hand,shows that the wall potential is varied for different electronegativity (α) and qnvalues at a constant qevalue.However,similar observations can be made to that of figure 14(a).This shows that the wall potential is more inclined to the changes in the temperature ratio rather than the electronegativity.

Figure13.Normalized wall potential variation with temperature ratio(τ-)for(a)constant qn and different qe and(b)constant qe and different qn values for α=0.6.

Figure14.Normalized wall potential variation with electronegativity(α)for(a)constant qn and different qe and(b)constant qe and different qn values for τ-=2.

4.Conclusion

The present work investigates an electronegative magnetized plasma sheath numerically using the hydrodynamic equations.The electrons are initially considered to be nonextensive while the negative ions are considered to be Boltzmann distributed.As the distribution of electrons shifts away from the Boltzmann,the species tend to fall faster near the core region.This further shifts the charge deposition region away from the wall.It also shows that the magnetic field has a moderate effect on the dynamics of the positive ions.However,in the case of nonextensive negative ions,the scenario changes.The magnetic field inclination plays a major role in the motion of the species.The negative ions also travel a considerable distance inside the sheath.This confirms that the magnetic field influences the negative ions as it affects the positive ions.Also,the numerical noise often observed in the space charge profile is decreased by choosing nonextensive negative ions.Accordingly,it can be concluded that a generalized distribution is a good candidate for describing the negative ions in a multi-component plasma.Additionally,the wall potential is susceptible to changes in the q parameter as well as the temperature of the negative ions.The observations made in the present study might be extended in understanding any plasma surface or material modification fields.