Theory of the Electron Sheath and PresheathBrett Scheiner,1, a) Scott D. Baalrud,1 Benjamin T. Yee,2 Matthew M. Hopkins,2 and Edward V. Barnat21)Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242,USA2)Sandia National Laboratories, Albuquerque, New Mexico 87185, USA

(Dated: 14 October 2015)

Electron sheaths are commonly found near Langmuir probes collecting the electron saturation current. Thecommon assumption is that the probe collects the random flux of electrons incident on the sheath, whichtacitly implies that there is no electron presheath and that the flux collected is due to a velocity spacetruncation of the electron velocity distribution function (EVDF). This work provides a dedicated theory ofelectron sheaths, which suggests that they are not so simple. Motivated by EVDFs observed in Particle-In-Cell (PIC) simulations, a 1D model for the electron sheath and presheath is developed. In the model,under low temperature plasma conditions (Te � Ti), an electron pressure gradient accelerates electrons inthe presheath to a flow velocity that exceeds the electron thermal speed at the sheath edge. This pressuregradient generates large flow velocities compared to what would be generated by ballistic motion in responseto the electric field. It is found that in many situations, under common plasma conditions, the electronpresheath extends much further into the plasma than an analogous ion presheath. PIC simulations revealthat the ion density in the electron presheath is determined by a flow around the electron sheath and thatthis flow is due to 2D aspects of the sheath geometry. Simulations also indicate the presence of ion acousticwaves excited by the differential flow between electrons and ions in the presheath which result in sheath edgefluctuations. The 1D model and time averaged PIC simulations are compared and it is shown that the modelprovides a good description of the electron sheath and presheath.

I. INTRODUCTION

Sheaths, which are present at essentially any plasmaboundary, are one of the most fundamental structuresin plasma physics and have been studied extensively1.Sheaths play the important role of maintaining globalcurrent balance, allowing the existence of a quasineutralplasma. At floating boundaries the sheath is ion rich (anion sheath), providing a thin positive space charge layerthat limits the electron losses to the boundary. Not allsheaths need to be ion rich. Sheaths near small elec-trodes, such as those around Langmuir probes, can beelectron rich (electron sheaths) when the electrode is bi-ased positive with respect to the plasma potential. Dueto the requirements of global current balance, electronsheaths are possible only near electrodes that are smallenough that the ratio of their area to the plasma chamberwall area satisfies AE/Aw <

√2.3me/mi where AE and

Aw are the effective surface areas for collecting chargedparticles at the electrode and the wall, respectively2. Theeffect of the electrode-to-wall area ratio on the sheathform has been experimentally verified3.

Electron sheaths are most commonly encounteredaround Langmuir probes collecting the electron satura-tion current4–6, but are also encountered around plasmacontactors7,8, tethered space probes9, and in laser accel-erated plasmas10. Electron sheaths have also been ob-served to play a role in probe induced particle circula-tion in dusty plasmas crystals11 and are also importantfor providing electrons with the energy needed to ionize

a)brett-scheiner@uiowa.edu

neutral atoms in the formation of anode spots12,13. Thepresent understanding of electron sheaths from Langmuirprobe theory is comprised of the following: 1) The elec-tron sheath collects the random flux of electrode-directedelectrons4,14. This flux is given by ΓR = 1

4neAE ve,

where ve =√

8Te/πme is the mean electron velocity,Te is the electron temperature in eV, and me is theelectron mass. 2) Since the flux collected is random,the EVDF is a half Maxwellian at the electron sheathedge5,6. 3) The electron sheath analog of the Bohm cri-terion is trivially satisfied15,16 because the truncation ofthe EVDF at the sheath edge provides the required flowmoment. Presheaths have not been considered. 4) Ionsnear the electron sheath follow a Boltzmann density pro-file ni = no exp(−eφ/Ti), where ni is the ion density, nois a reference density at φ = 0, Ti is the ion temperaturein eV, and φ is the electrostatic potential17. In this paperwe consider a dedicated theory of the electron sheath andfind that each of these assumptions need to be revisited.

In a recently submitted paper Yee et al.18 (here-after YE), it was shown that under low temperatureplasma conditions the electron sheath is accompanied bya presheath where an electron flow of approximately anelectron thermal speed is generated due to pressure gra-dients. This presheath was shown to extend well into thebulk plasma, even extending beyond the range of an anal-ogous ion presheath. In YE, results from particle-in-cellsimulations with direct simulation Monte-Carlo collisions(PIC-DSMC) showed that the EVDF near the sheathedge was a flowing Maxwellian. In the present paper, wepresent a new theoretical model for the electron sheathand presheath based on observations from these recentexperiments and simulations. This new theory shows

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that the electron fluid flow exceeds the electron thermalspeed by the sheath edge, satisfying an electron sheathanalog of the Bohm criterion. The 1D model describesthe electron flow as being pressure gradient driven, thisis significantly different than the electric field driven flowin ion presheaths. This presheath pressure gradient gen-erates large flow velocities over regions with little changein potential. The effect on the bulk plasma can be signif-icant even with small gradients in the plasma potential.

Although the 1D model provides an accurate charac-terization of many aspects of the electron sheath andpresheath, some aspects of the 2D simulations are notcaptured by the 1D theory. The need to satisfy globalcurrent balance usually results in electron sheaths oc-curring only around small electrodes, hence the infiniteplanar picture common to 1D models is not perfect. Inthe simulation the electron presheath causes the ion flowto be redirected around the small electrode, resulting ina significantly different situation than that described bya Boltzmann density profile. Analysis of the PIC sim-ulations reveal that the ion density is only accuratelydescribed when the ion flow is taken into account.

The previous simulations in YE showed that the elec-tron sheath edge exhibits fluctuations on the order of 1MHz. Two-dimensional FFTs of the ion density showthese fluctuations are ion acoustic waves excited by thedifferential flow between fast electrons, and ions in theelectron presheath. The time dependence of the ion den-sity fluctuations closely correlate with the sheath edgefluctuations. The sheath edge position fluctuations mayexplain the current fluctuations previously observed forprobes biased above the plasma potential3,19,20. In ad-dition, these fluctuations may contribute to an effec-tive electron-ion collision rate in the electron presheaththrough instability enhanced collisions21.

This paper is organized as follows. Section II discussesthe implications of different EVDF models on the elec-tron sheath and presheath, and develops a fluid-basedapproach motivated by PIC simulations. Section IIIA de-scribes the PIC simulations and results. Section IIIB pro-vides a comparison between simulations and the model,Sec. IIIC focuses on the ion behavior in the presheath,and Sec. IIID on the time-dependent aspects as well asinstabilities. Concluding statements are made in Sec. IV.

II. MODEL

A. Conventional kinetic models

The present understanding that a Langmuir probe col-lects the random thermal flux in electron saturation isbased on the assumption that the electron sheath in-terfaces with the bulk plasma without a presheath. Adirect consequence is that the EVDF under this modelis a Maxwellian that has no flow shift, but is truncatedat zero velocity at the sheath edge; see Fig. 1. In thissection, the consequences of the conventional assumption

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

Velocity

VDF

half−Maxwelliantruncated flowing Maxwellianflowing Maxwellian

Student Version of MATLAB

FIG. 1. Three different model velocity distribution func-tions. The half Maxwellian corresponding to a collisionlesselectron-rich sheath with no presheath, the truncated flow-ing Maxwellian corresponding to a collisionless sheath with apresheath, and the flowing Maxwellian corresponding to thecollisional sheath with presheath.

FIG. 2. PIC simulation results showing that the EVDF canbe modeled as a flowing maxwellian in the electron sheathand presheath. The sheath edge is at y ≈ 0.25 cm.

are first explored. This demonstrates that even under theconventional assumption a finite electric field, and hencea presheath should be expected. Since a presheath gener-ates flow, a flowing truncated Maxwellian such as shownin Fig. 1 might be suggested as an appropriate model.However, recent PIC simulations have shown that theexpected flow speed should be very fast, approaching theelectron thermal speed by the sheath edge. Furthermore,the observed distribution was a flowing Maxwellian; seeFig. 2. Motivated by these simulation results, a fluid-based model is developed in Sec. IIB. This provides amodel for the minimum flow speed to be expected atthe sheath edge, a model for pressure driven flow in thepresheath, and a model for electric field driven flow inthe sheath.

Consider, briefly, an implication of the conventionallyassumed picture where an electron sheath interfaces di-rectly with the bulk plasma without a presheath. In thiscase the sheath edge EVDF is a truncated Maxwellian,however the assumed EVDF in the bulk plasma isMaxwellian. This picture does not allow the matchingof the bulk plasma to the sheath since such a transitionwould break flux conservation. Since the bulk plasma

3

EVDF is typically close to maxwellian, stress gradientsand friction terms in the momentum equation are negligi-ble, so any change in electron flux must be due to an elec-tric field. This difficulty of transitioning between the bulkplasma and sheath suggests a need for a plasma regionwhere an electric field allows the the acceleration of elec-trons providing an electron flux at the sheath edge. Twoimplications of this directly hint at the need to revisitthe half-Maxwellian assumption: 1) A non-zero electricfield at the sheath edge implies that the electrons have aflow due to the electric force exerted on them, and 2) afinite electric field also implies that there is a density gra-dient in the quasineutral presheath that can give rise topressure gradient induced flows, e.g., for Boltzmann ionsni = no exp(−eφ/Ti), dni/dz = eniE/Ti. The followingsection will show that the latter effect is the dominantmechanism for electron-flow generation in the presheathwhen Te � Ti.

B. New fluid model

Motivated by PIC simulation results showing thatnot only is there a flow shift, but that the EVDF isMaxwellian at the sheath edge, this section describes afluid-based model of the electron sheath and presheath.The acceleration mechanism is found to be a pressuregradient. The implications of this pressure driven floware that the electrons can achieve large flow velocitieseven over regions where the potential varies by a smallamount. In this section, it is shown that the electronsaccelerated by this presheath must enter the sheath witha flow speed exceeding the electron thermal speed, a re-sult that may be considered an electron sheath analog ofthe Bohm criterion. We will refer to this as the electronsheath Bohm criterion (the speed that must be satisfiedwill be referred to as the electron sheath Bohm speed,denoted veB). A 1D model is developed for the density,flow, and potential profiles in the presheath and sheathregions.

1. Sheath edge

For the purposes of modeling the presheath and sheathedge, consider a model that describes electrons with con-tinuity and momentum equations, assuming that theplasma is generated at a rate proportional to the den-sity. Ions are assumed to obey a Boltzmann density,ni = no exp(−eφ/Ti), where no is the density in the bulkplasma. These equations are supplemented with Pois-son’s equation and an isothermal closure for electrons.Since we are concerned with the presheath, the quasineu-trality condition applies, and the density gradient can bewritten as dne/dy = eniE/Ti. Inserting this into themomentum equation

VedVedy

= − e

meE − Te

mene

dnedy− Ve(νR + νs) (1)

shows that the pressure gradient term is Te/Ti timeslarger than the electric field term. Here, in Eq. (1), Vedenotes the first moment of the EVDF, and νR and νs de-note the collision frequencies due to momentum transfercollisions and particle source rate respectively. In typicallow temperature plasmas Te/Ti ∼ 10 − 100, hence theflow is dominantly pressure driven. This situation makesa significant contrast with ion sheaths, where instead theion pressure gradient term is Ti/Te � 1 smaller than theelectric field term.

This model can be used to determine the conditions onthe electron flow velocity at the sheath edge. Expand-ing the charge density about a position at the sheathedge ρ(φ) = ρ(φ0) + dρ/dφ|φ=φ0

(φ− φ0) + ..., and defin-ing the sheath edge as the location where neutralitybreaks down, gives a common definition of the sheathedge15

∣∣dρ/dφ|φ=φ0

∣∣ > 0. This requirement, whichis known as the sheath criterion, can be rewritten as∑s qsdns/dy ≤ 0 where the sum is over each plasma

species. The Bohm criterion for a fluid model can be ob-tained by inserting the fluid equations into this form ofthe sheath criterion22,23. For the electron sheath, con-sider a thin region near the sheath edge where the sourceand collision terms can be neglected. The electron con-tinuity equation, along with Eq.(1) and the Boltzmanndensity relation for ions, then imply the following elec-tron sheath analog of the Bohm criterion

Ve ≥√Te + Time

≡ veB . (2)

A similar electron sheath Bohm criterion was previouslyfound24, but was not derived from consideration on theEVDF. The electron sheath Bohm speed in Eq.(2) is ap-

proximately√mi/me greater than the ion sound speed,

which is the ion flow generated in an ion presheath. Be-cause this is significantly faster than the ion sound speed,the differential flow between ions and electrons is ex-pected to excite ion acoustic instabilities in the electronpresheath. This will be studied in Sec. IIID. Next wewill consider analytic solutions for the plasma parameterprofiles in the presheath and sheath.

2. Presheath

In this subsection the properties of the quasineutralpresheath are explored. A mobility limited flow equa-tion is derived for the electron fluid. The equations forvelocity and potential profiles are solved in a region inthe vicinity of the sheath edge and analytic solutions arefound for the cases of constant mean free path and con-stant collision frequency. The solutions demonstrate thatlarge flow velocities are obtained over regions in whichthere is a small potential gradient. From these solutionsit is found that in some cases the electron presheath hasan extent that is

√mi/me longer than that of the anal-

ogous ion presheath, and under more typical low tem-

4

perature plasma conditions the presheath is ∼ 6 timeslonger than the ion presheath. This means that the elec-tron sheath can perturbed the bulk plasma over a fewcentimeters under typical laboratory conditions.

Starting with the quasineutrality condition on the den-sity gradient, and using the first two fluid moment equa-tions, an electron mobility limited flow equation is ob-tained,

Ve = −µe(

1− V 2e

v2eB

)E. (3)

This equation is analogous to the ion mobility limitedflow equation, but where µe = e(1+Te/Ti)/[me(νR+2νs)]is the electron mobility. When compared with the the ionmobility in an ion presheath with a common collision fre-quency due to volume ionization of neutrals, the electronmobility greatly exceeds ion mobility µe ≈ Temi

Timeµi.

Next, consider a region in the vicinity of the sheathedge that is thin enough that an assumption of constantflux, neVe = noveB , is accurate. Here no is the density atthe sheath edge. Using this form of the electron densityalong with the Boltzmann density for ions in Poisson’sequation gives(

λ2De

l2

)d2(eφ/Te)

d(y/l)2= −

(e−eφ/Ti − veB

Ve

), (4)

where l is the presheath length scale. Taking thequasineutral limit λDe/l → 0 gives the potential as afunction of flow velocity

φ = −Tie

ln

(veBVe

). (5)

This form of the potential along with the mobility limitedflow in Eq. (3) results in a differential equation for theflow velocity in terms of spatial position,

dy

dVe=

v2eB − V 2e

(νR + νs)V 2e

. (6)

The solution to this differential equation along with Eq.(5) gives the flow and potential profile. This differentialequation has an ion sheath analog25, which has analyticsolutions26 for 1) the case of constant mean free path,ν = Ve/l, and 2) constant collision frequency, ν = vB/l.For the case of constant mean free path the flow velocityis

VeveB

= exp

{1

2− y

l+

1

2W−1

[− exp

(2y

l− 1

)]}(7)

where W−1 is the −1 branch of the Lambert Wfunction27. Eq. (5) gives the potential profile

− eφ

Ti=y

l− 1

2− 1

2W−1

[− exp

(2y

l− 1

)]. (8)

For the constant ν case the flow and potential profilesare

VeveB

= 1− y

2l

(1 +

√1− 4l

y

)(9)

and

− eφ

Ti= arccosh

(1− y

2l

). (10)

The flow velocity and potential profiles for these twocases are shown in Fig. 3. These show that large flowsare obtained over regions with shallow potential gradientsand little change in potential. Flow velocities of thismagnitude are not seen in the ion presheath.

0.4

0.6

0.8

1

VelocityV/veB

Constant νConstant mean free path

−1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.5

0

Distance y/l

Potentialeφ/Ti

Student Version of MATLAB

FIG. 3. The flow velocity (top) and potential profile (bottom)for the presheath for the cases of constant collision frequencyand constant mean free path from Eqs. (7)-(10).

For the case of constant collision frequency the char-acteristic length scale of electron and ion presheaths canbe compared explicitly. Two cases are considered; A) aplasma where volume ionization is the dominant effect,and B) a helium plasma with momentum transfer colli-sions and no volume ionization.

A) Consider a plasma where the dominant collisionprocess is volume ionization so that νs is the same forions and electrons. If the sheath attached to the elec-trode is an ion sheath the presheath length scale wouldbe li = cs/ν, while if the sheath were an electron sheaththe presheath length scale is le = veB/ν. The ratio ofthese two length scales is

leli

=veBcs

=

√Te + Time

mi

Te≈√mi

me. (11)

This suggests that the characteristic length scale of anelectron presheath can be more than an order of magni-tude longer than an ion presheath. A typical ion sheathlength scale in low temperature plasma experiments is∼ 1 cm 28, which means for the case of an argon plasma,where

√mi/me ≈ 270, the implied presheath length

5

FIG. 4. The neutral helium-electron momentum scattering,and neutral helium-helium ion charge exchange and elasticcross sections used for the calculation of the rate constant.Note the extrapolation of the helium cross section, this wasnecessary due to lack of data at low energies.

scale would be le ≈ 270 cm. This is longer than thescale of many plasma experiments, so it would be ex-pected that the presheath would fill approximately halfthe experiment length29.

B) In this case the collision frequencies are different,using νs = ngKs the ratio of presheath length scales is

leli

=veBcs

νiνe≈√mi

me

Ki

Ke(12)

where Ks is the rate constant for collisions between neu-tral helium and species s. When the temperature is smallthe rate constant is

Ks(U) ≈ Uσs(U). (13)

Using this approximation the rate constants were es-timated using flow speeds representative of typicalpresheath velocities, U = veB/2 for the electronpresheath and U = cs/2 for the ion presheath. For thecalculation of Ke the total momentum cross section fore− + He collisions was obtained from LXcat30, while forthe calculation of Ki the cross section for He+ + He elas-tic and charge exchange collisions were considered31. Forthe He+ + He cross sections the values at 4 eV were ex-trapolated to 0 eV as has been previously done32, thiswas due to a lack of data within this range of energies.The cross sections used are shown in Fig. 4. The ratioof presheath length scales shown in Fig. 5 suggest that,for this case, the electron presheath is approximately sixtimes longer than the ion presheath. These values arein good agreement with the estimated presheath lengthsdetermined from density measurements in YE where theelectron and ion presheaths were measured to be approx-imately 25mm and 6mm respectively.

These results differs substantially from the conven-tional picture of electron sheaths, which are thought tobe local phenomena. Instead, this suggests that electronsheaths often influence a plasma globally. This sugges-tion will be considered further in Sec. IIIB with PICsimulations where there is no presheath volume genera-tion of plasma and hence νs = 0.

FIG. 5. The ratio of presheath length scales as a function ofTe.

3. Sheath

For the electron sheath, the sheath-presheath transi-tion is a region where the flow switches from being pres-sure driven to electric field driven. In the thin sheathregion, the collision and source terms can be neglected33.This provides a way to determine a relation between theflow velocity and potential, which shows that at smallpotentials the pressure represents a significant correctionto the electron ballistic motion, while at high potentialsthe 3/4 power law scaling of the Child-Langmuir law34 isrecovered.

Under the assumptions mentioned above, combiningthe continuity and momentum equations, integrating andmatching the sheath edge conditions results in(

VeveB

)2

− 2 ln

(VeveB

)=

2eφ

Te+ 1. (14)

The second term on the left hand side is the contribu-tion due to the electron pressure. The solution to thisequation can be written in terms of the Lambert W func-tion, however the logarithmic electron pressure term is atmost a correction of ≈ 20% in the sheath and drops offat higher flow velocity. In the asymptotic limit this termis negligible (see Appendix A).

Using the asymptotic solution for Ve, enforcing thatthe electron density within the sheath obeys flux conser-vation (ne(φ)Ve(φ) = noveB), and neglecting the ion den-sity, which decreases exponentially with increasing poten-tial, Poisson’s equation can be written as

d2φ

dy2=

4πeno√1 + 2eφ

Te

. (15)

Integrating twice with respect to y gives

yflowingλDe

= 0.79

(e∆φ

Te

)3/4

(16)

which is the same as what is obtained for the ionsheath14. A different relation for the electron sheath hasbeen previously given1435, here the sheath scaling wasgiven as

ytruncatedλDe

= 0.32

(e∆φ

Te

)3/4

. (17)

6

This different numerical factor is due to the randomflux assumption. Comparing Eq. (16) and (17) gives thecorrection to the sheath scale

yflowingytruncated

= 2.47, (18)

which suggests that the electron sheath is more thantwice as thick as previously thought. In Sec. IIIB thisrelation is found to be in excellent agreement with simu-lations.

III. SIMULATIONS

The model in the previous section assumed a 1D planarelectron sheath. In this section the model is tested using2D PIC simulations. These show that the electron sheathhas some inherently 2D features that are not accuratelycaptured by the model. In particular, the ion density isfound to be determined by a 2D ion flow velocity profilearound the electron sheath. Nevertheless, basic featuresof the 1D model, such as the minimum electron flow speedat the sheath edge, are found to accurately represent thesimulations. Modifications of the 1D theory to address2D ion flow are found to lead to improvements in thepredicted presheath profiles. The PIC simulations alsoexhibit fluctuations in sheath thickness. In Sec. IIID,evidence is shown that these fluctuations are ion acousticwaves excited in the presheath.

A. Aleph

The simulations were performed using the PIC-DSMCcode Aleph. Aleph is an electrostatic PIC code thatutilizes direct simulation Monte Carlo (DSMC) kinetictechniques36 for interparticle collisions. The algorithmrepresents a plasma by evolving electrostatically coupledcomputational particles in time and computes the parti-cle positions and velocities on an unstructured mesh in1D, 2D, or 3D, each with three velocity components37.In our simulations a 2D triangular mesh with a reso-lution of approximately 0.7λDe was used. The simula-tions utilized a 7.5 cm ×5 cm domain with one reflectingand three grounded absorbing boundaries. An electrodeof length 0.25 cm was embedded in an absorbing walladjacent to the reflecting boundary and was separatedfrom the wall by a gap filled with a dielectric of length0.2 cm, see Fig. 10 for an image of the simulation do-main. The domain, which was set up to resemble ex-periments on a reduced scale, was filled with a heliumplasma that was continuously generated in a source re-gion ∼4 cm above the electrode, and expanded to fill thedomain. Plasma in this region was sourced at a rate of2.35× 109 cm−3 µs−1 resulting in a bulk plasma densityof approximatey ne ≈ 5×108cm−3. The particle weightswere 4 × 109, 1.6 × 104, and 2 × 103 for neutral helium,helium ions, and electrons respectively. The plasma was

sourced with an ion temperature of 0.086 eV and electrontemperature of 4 eV. The 2D electron temperature, whichis the relevant temperature for comparison with the the-ory, is defined as Te = ne

∫d3vme(v

2r,x+v2r,y)fe/2, where

vr,i = (v−Ve) · i. The 2D electron and ion temperatureshad a value of 1.64 eV and 0.048 eV near the sheathgiving an electron sheath Bohm speed of 54.4 cm µs−1.Only elastic collisions between ions and neutrals with abackground pressure of 1 mTorr were included, and therewas no volume generation of plasma. A 1× 10−4µs timestep was chosen to resolve the local electron plasma fre-quency throughout the domain. The simulation ran for5× 106 time steps resulting in 50µs of physical time.

Two cases were considered, one where the electrodewas biased +20 V with respect to the grounded walls,and the other with the electrode bias at -20 V. For the+20 V electrode an electron sheath was allowed to formsince the electrode satisfied AE/AW <

√2.3me/mi and

the probe was biased above the plasma potential2. Thepotential gradient and electron and ion current vectorsare shown in Fig. 6 for an electron sheath biased at +20V and an ion sheath biased at -20 V. In the electronsheath case, the electron current indicates that the bi-ased electrode has an effect on the bulk plasma that issignificantly grater than that of the ion sheath. The im-portance of the 2D nature of the electron sheath can beseen in two effects. First, the current vectors of the re-pelled population (ions for the electron presheath andelectrons for the ion presheath) are almost absent in thecase of the ion presheath near the electrode, while thosefor the electron presheath indicate a significant flow ve-locity. For an infinite planar boundary it is not possibleto have flow around the boundary. This is difficult toachieve for an electron sheath because the electrode musthave a dimension that is small compared to the chambersize in order to be biased positive with respect to theplasma. The second effect is the convergence of the elec-tron current into the electrode, even for distances greaterthan 1 cm away. This convergence is not seen for the ionpresheath. The importance of these 2D effects will beexplored in Sec. IIIC.

7

[ niV i(2

x1010

cm−2µ

s−1 )]

Dis

tanc

e (c

m)

Electron Sheath +20V

0 0.2 0.4 0.6 0.8 10

0.5

1

[ neV e(1

011cm

−2µ

s−1 )]

Electron Sheath +20V

0 0.2 0.4 0.6 0.8 10

0.5

1

Distance (cm)

[ niVi(2x1010cm−2µs−1)]

[ niV i(2

x1010

cm−2µ

s−1 )]

Dis

tanc

e (c

m)

Ion Sheath −20V

0 0.2 0.4 0.6 0.8 10

0.5

1

Distance (cm)

[ neVe(1011cm−2µs−1)]

[ neV e(1

011cm

−2µ

s−1 )]

Ion Sheath −20V

0 0.2 0.4 0.6 0.8 10

0.5

1

Pote

ntia

l (V)

−20

−15

−10

−5

0

5

10

15

20

FIG. 6. The current flow vectors plotted on top of the potential for niVi (left column) and neVe (right column) for an electronsheath (top row) biased +20 V and ion sheath (bottom row) biased -20 V. The electrode is between x = 0 and x = 0.25 cm, seeannotations on Fig. 10 for more details of the simulation domain. All potentials are measured relative to the grounded wall.The greatest differences between these are that the electron presheath has a much greater effect on the electrons in the bulkplasma than the ion presheath has on the ions, and that the electron presheath redirects the ions, while the ion presheath haslittle effect on the electrons. Note the difference in scale for vectors in the left and right columns.

B. Electron fluid

The simulations have shown that the electron sheathinterfaces with the bulk plasma through a presheath.In this subsection simulations are compared to thepresheath description given in the 1D model. Fig. 7shows a comparison of the potential profile from the elec-tron sheath PIC simulations and the models given for thepresheath and sheath. For the presheath, the models forconstant mean free path given in Eq.(8) and constantcollision frequency in Eq.(10) were compared by fixingthe value φ = 0 at the location where the electron sheathBohm speed is attained, from here moving out some dis-tance y into the plasma the potential profile was plotted.In the sheath, starting at the electrode, the sheath thick-ness as a function of potential from Eq.(16) was plottedfor an argument ∆φ = φE − φ(y) from φ(y) = φE outto φ(y) = 0, the potential at which the electron sheathBohm speed was attained. Here φE is the electrode po-tential. For comparison, the conventional model fromEq.(17) is also plotted. The potential profiles withinthe sheath are in excellent agreement with Eq.(16) in-dicating that the numerical factor corresponding to theflowing Maxwellian is the correct value. This result is

significant since it indicates that the electron sheath isapproximately twice as thick as was previously thoughtunder the random flux assumption. The presheath po-tential profiles are plotted with a presheath length scaleof l = 0.3 cm, which approximately corresponds to theregion in which Eq.(21) accurately describes the ion den-sity in Fig. 7, as well as the region in Fig. 8a where thepressure gradient dominates over the electric field. Thepresheath potential profiles from the theory were shal-lower than that from the simulations near the sheath,however the slopes are in better agreement further away.This is possibly due to matching the simulation dataat the theory’s singular point. Simulation results onlymatch the theory in a region where the electron presheathis dominant, however the model does not consider the in-terface of the presheath with a nonuniform bulk plasmasuch as the one in the simulations.

The flow profiles of Eq.(7) and Eq.(9) are also com-pared to the simulations in Fig. 7. The flow profilesshow that the sheath and presheath are in good agree-ment with theory. Due to presheath ion density fluctu-ations (see Fig. 14), the sheath edge is difficult to lo-cate in the time averaged simulation data. To comparethe electron flow velocity at the sheath edge two defini-tions are utilized, 1) the sheath thickness given by Eq.

8

0

5

10Pote

ntial(V

)

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

VelocityV/veB

Distance (cm)

PICConstant ν PresheathConstant λmfp presheathFlowing Maxwellian sheathRandom flux sheath

Student Version of MATLAB

FIG. 7. Top: The potential profile from simulations comparedto the model for the sheath and presheath. Bottom: Compar-ison of the flow velocity profile from simulation to the modelsfor the sheath and presheath.

(16) for ∆φ = φ(0 cm) − φ(1 cm), and 2) the locationwhere the average difference between electrons and ions,2(ne−ni)/(ne+ni) is greater than 30%. The sheath edgeis difficult to locate in the time averaged data. The cho-sen value of 30% corresponds well with the typical sheathedge position in the time dependent data in Fig. 14 shownin Sec. IIID. By these two definitions the sheath edge isbetween 0.213 cm and 0.265 cm, the corresponding flowvelocities are 1.21veB and 0.85veB . Fig. 8a shows the ionand electron density, while Fig. 8b shows the correspond-ing terms in the electron momentum equation. Here thetwo dashed lines indicate the two sheath edge locations.In the region bounded by these two sheath edge defi-nitions, the electric field overtakes the pressure gradientand the sheath begins. The location at which the electricfield becomes the dominant driving term in the electronmomentum equation closely coincides with the locationat which the electron sheath Bohm velocity is achieved.

Previously, in Sec. IIB, a comparison of presheathlength scales was made for sheaths dominated by a com-mon source of plasma generation between electrons andions and no other collisions. For this situation it wasconcluded in Eq.(11) that the electron presheath was√mi/me longer than the ion presheath. In the sim-

ulations, no particles are sourced in the presheath sothe dominant mechanism for determining the electronpresheath length scale is expected to be electron-ion col-lisions. In the PIC simulations the electrons are col-lisionless in the Coulomb collision sense since electron-particle interactions are not considered within the com-putational cells. Another possible mechanism for suchcollisions are those due to particle wave interactions inan unstable plasma21. The collision rate due to elec-tron interactions with ion acoustic waves has been impor-tant for explaining the anomalous scattering of electronsnear the ion sheath38, a phenomenon known as Lang-muir’s Paradox39. The EVDFs in Fig. 2 suggest a sim-ilar anomalous scattering mechanism may be important

here since at the sheath edge and within the sheath elec-trons with velocities directed towards the bulk plasmaare still present. Evidence for the presence of electroncollisions can be obtained by adding up the terms in themomentum equation which are calculated from the sim-ulations. In fact, one can see that the terms in Fig. 8bdo not exactly cancel. Using PIC plasma quantities, theresidual

Re = VedVedy

+e

meE +

Temene

dnedy

(19)

was also plotted. An increase in the residual as the elec-trode is approached suggests that other neglected terms,(i.e. stress gradients, perpendicular velocity gradients,and friction) may be important. In particular a frictionterm may be due to wave particle interactions and couldplay an important role in determining the presheathlength scale since it would determine the value of νRin Eq.(6). Instabilities will be discussed further in Sec.IIID.

1014

10151016

1017

Distance (cm)

cm/s

2

0 0.5 1 1.50

12

34

Distance (cm)

Ve/veB

ni (108 cm−3)

ne (108 cm−3)

E Field TermPressure Gradient Term−Y Flow TermResidual

Student Version of MATLAB

b)

a)

FIG. 8. a) Evaluation of the electron momentum equationterms using PIC simulation results. b) PIC simulation resultsfor electron flow speed, ion density, and electron density. Thetwo vertical lines indicate the sheath edge calculated as thelocation where ni and ne differ by 30% (left) and by the ChildLangmuir law (right).

C. Ions

1. Ion density

Plots of the ion current, in Fig. 6, show that ions flowaround the electrode and are collected by the adjacentwall. Here the ion density will clearly be dominatedby the flow profile around the electrode. This flow is a2D effect that is absent in the description of ion sheathsnear planar boundaries. Size limitations on the electron

9

sheath from global current balance prevent it from beingwell described by an infinite 1D planar geometry.

To model how the 2D flow affects the ion density pro-file, consider the 2D steady state ion momentum equationalong a 1D cut perpendicular to the electrode center,

mini

(VxdVydx

+ VydVydy

)= −eni

dφ

dy− d

dy(niTi). (20)

Here, the stress gradient and friction terms have beenneglected. In Sec. IIB it was found that the electronpresheath has weak potential gradients. Dropping theelectric field term and integrating from the sheath edgeback into the presheath results in

ni(y)

ni(yo)= exp

[−∫ y

yo

mi

Ti

(VxdVydx

+ VydVydy

)dy

], (21)

where yo denotes the sheath edge position. In this formthe ion flow is balanced by the pressure gradient. Thiscan be contrast with the Boltzmann relation where theelectric field and pressure gradient balance. The exactform of the pressure gradient is dependent on the electricfield, after all it is the field that causes the density gradi-ent. Determining the exact pressure gradient would in-volve solving the full 2D momentum equation with Pois-son’s equation using all the boundary conditions. Forthis section, numerical values from PIC simulations areused to test the relation in Eq. (21). Fig. 9 shows thepresheath densities from PIC simulations compared tothe evaluation of Eq. (21). These two quantities are ingood agreement. For comparison the Boltzmann rela-tion, with initial values in the presheath, is also shownin Fig. 9, demonstrating that Eq. (21) is a vast improve-ment in the description of the ion density.

0 0.1 0.2 0.3 0.4 0.50

2

4

6

Distance (cm)

Den

sity

(108 c

m−3

)

Ion DensityModelElectron DensityBoltzmann Density

Student Version of MATLAB

FIG. 9. The electron and ion density for a 1D cut in thesimulation domain perpendicular to the electrode. The iondensity integral of Eq. (21) evaluated for PIC velocity andtemperature profiles (blue triangles) agrees well.

2. Ion VDFs

The effect of the electron sheath on ions can also beexplored from a kinetic point of view. Fig. 10 shows ion

heating in the electron presheath. This heating is ex-plained as a result of ion interaction with the presheath.This interaction generates a flow moment in the ionVDFs (IVDFs) in the transverse direction when ap-proaching the electrode. The 2D IVDFs, are shown inFig. 11. These demonstrate that the majority of ions areredirected away from the boundary and collected by theadjacent grounded wall. It is this redirection that is pri-marily responsible for the heating, however there is alsoa small population of ions that are reflected back into theplasma.

Distance (cm)

Dis

tanc

e (c

m)

T i (K)

0 2 4 60

1

2

3

4

5

0

200

400

600

800

1000

1200

1400

Electrode

ReflectingBoundary

Plasma Source Region

V=0!!

Dielectric!!

FIG. 10. The ion temperature throughout the simulation do-main. The plasma source region, electrode, reflecting bound-ary, grounded walls, and dielectric are marked. Note the heat-ing in the region just above the electron sheath in the lowerleft corner of the domain.

The 2D IVDFs in Fig. 11 were computed in thepresheath using individual particle positions and veloci-ties over 30 µs and were averaged over 0.1 cm × 0.1 cmboxes starting at the sheath edge around 0.25 cm movingback into the plasma 0.85 cm. The averaging boxes alsoextend the length of the electrode in the x direction, withthe last box including the electrode wall boundary. Farfrom the electrode the IVDFs are flow shifted towardsthe boundary, as would be expected for an expandingplasma, and show little modification apart from a smallpopulation of reflected ions. As the ions approach theelectrode some of their flow velocity is diverted from the-y to x direction since the ions are repelled by the 2Dpresheath electric field which has x and y components.

The modification of the IVDF shape near the electrodecan be described by the flow around the electrode. Con-sider the IVDFs halfway between the plasma source andthe boundary containing the electrode, each starting atthree different locations in the x direction, see the loca-tion marked A in Fig. 12(a). At the starting locationeach IVDF will have a flow due to the plasma expansion,so the distribution will have a flow shift in the directionof the electrode or wall, this is represented in Fig. 12(b).Due to the flow around the electrode each of these distri-butions will end at the location marked B in Fig. 12(a).

10

0.05$cm$ 0.15$cm$ 0.25$cm$

0.25$cm$

0.35$cm$

0.45$cm$0.55$cm$0.65$cm$

0.75$cm$

0.85$cm$

Log 1

0($Par5cle$Num

ber)$

FIG. 11. Ion VDFs near the electron sheath biased +20Vabove ground shown in Fig. 6. The IVDFs were averagedover 0.1 cm x 0.1cm boxes. The labels in the x and y axesindicate the coordinate of the center of the box, the electrodeis on the x axis at y=0 between x=0 and 0.25 cm (averagingstarts 0.2 cm above the electrode), since further below thereare not enough ions for meaningful IVDFs.

Now consider the distribution with flow incident on theelectrode. Since the flow is redirected the flow shift ofthis distribution will be transferred from the -y direc-tion to the x direction as it approaches the electrode.Likewise, a distribution incident to the edge of the an-ode will also have its flow diverted from the -y directionto the x direction, although to a lesser extent. Finally,a distribution incident to the grounded wall will remainunchanged. The final position of these three distribu-tions is shown in Fig. 12(c), although a more realisticexpectation would be smeared out, such as the distribu-tion shown in Fig. 12(d), due to a continuum of startingpositions.

The basic expectations of the model shown inFig. 12(d) are borne out in the simulated IVDFs nearthe boundary in Fig. 11. It is important to note that thephysical picture illustrated in Fig. 12 is not exact sincenot every particle flows along a stream line, but experi-ences diffusion as well. There are small scale features not

Electrode) Wall)

1) 2) 3)A)

B)

Y)Distance)

1,2,3)Vx)

Vy)a)) b)) c))

Vx)

Vy)

IVDF)Loca?on)A) IVDF)Loca?on)B)

12)

3)

Vx)

Vy)d))

FIG. 12. Schematic drawing describing the time-averagedIVDFs at different locations in the plasma. a) The flow linesof particles in 3 different VDFs starting at A and ending atB. b) The VDFs at location A. c) The VDFs at location B,distributions incident on the electrode have their -y velocityredirected in the x direction. d) A realistic IVDF at locationB due to a continuum of starting positions along line A.

explained by the picture in Fig. 12. For instance, in someIVDFs there is a small secondary maximum to the rightof the primary. This situation may be due to time aver-aging of the particle positions and velocities over 30 µsin combination with fluctuations in the presheath causedby instabilities.

D. Fluctuations and instabilities

The simulated electron sheath, shown in the 1 cm×1cm panels of Fig. 13, exhibits fluctuations of the sheathedge position on the order of 0.05 cm on a time scale ofapproximately 1 µs. Fluctuations were not observed forthe ion sheath with the electrode biased at -20 V. Thepresence of a differential flow, approaching the electronthermal speed, between electrons and ions in the electronpresheath is expected to give rise to ion-acoustic insta-bilities for the present values of Ti/Te. In this subsectionthe effect of these waves on the fluctuations is explored.Two-dimensional FFTs of the ion density confirm thatthe sheath fluctuations are due to ion acoustic waves.

The dielectric response for a plasma where the elec-trons are Maxwellian with flow Ve and stationaryMaxwellian ions is40

ε(k, ω) = 1−ω2pe

k2v2Te

Z ′(ξe)−ω2pi

k2v2Ti

Z ′(ξi) (22)

where ξe = ω−k·Ve

kvTeand ξi = ω

kvTi, and Z ′ is the deriva-

tive of the plasma dispersion function41. The dispersionrelation is determined by the zeros of the dielectric func-tion. The approximate solution is

ω

ωpi≈ kλDe√

k2λ2De− 1

2Z′(− Ve

vTe

) , (23)

11

30.0$μs$ 30.5$μs$ 31.0$μs$

31.5$μs$ 32.0$μs$ 32.5$μs$

FIG. 13. The fluctuation of the 2D electron sheath boundaryplotted in a 1 × 1 cm region at 0.5µs intervals.The colorindicates charge density, with red being electron rich, andblue being ion rich.

which was determined by expanding the ion term usingthe large argument expansion of the plasma dispersionfunction42 and by using the approximations ω/k ∼ csand Ve � cs in the electron term.

Fig. 14 shows the ion density along a line extending 1cm perpendicular to the electrode over a 5 µs interval.The figure shows that there are ion density fluctuationsthat propagate towards the sheath edge as time increases.The figure also shows that the sheath edge position fluc-tuations closely follow the propagation of the ion densityfluctuations, meaning that these are likely responsible forthe sheath edge and resulting current fluctuations whichare associated with positively biased probes19,20. The 2DFFT of the ion density shown in Fig. 15 was computedover a line extending 1 cm from the electrode. TheseFFTs were examined to determine whether or not the iondensity fluctuations are ion acoustic waves. The FFTsare in fair agreement with the expected dispersion rela-tion determined from Eq. (23) indicating that the densitydisturbances, which are responsible for the sheath edgefluctuations, are in fact ion acoustic waves. The figurealso indicates that nonlinear effects may be producing acascade to shorter scales.

IV. SUMMARY

In this paper the conventional picture that the electronsheath collects a random flux of electrons was shown to beincomplete. Based on the EVDFs of 2D PIC simulations,a model was developed using the electron momentumand continuity equations where the EVDF is a flowingMaxwellian. In this model the electron sheath interactswith the bulk plasma through a presheath where the elec-tron velocity approaches the electron sheath Bohm speed,√

(Te + Ti)/me. In this presheath there are shallow po-tential gradients that drive a large pressure gradient. It

FIG. 14. Top: The absolute difference between the elec-tron and ion density is plotted to show the sheath edge posi-tion fluctuations, measured along an axis perpendicular to theelectrode, as a function of time over a 5µs interval. Bottom:The ion density over the same time interval. The ion densityfluctuations correspond to the sheath edge fluctuations.

is this pressure gradient that is primarily responsible forthe acceleration of electrons.

The 1D model was compared to the 2D simulations us-ing the time averaged values from the simulation. Withinthe sheath the potential profiles and flow velocities are in

12

FIG. 15. The 2D FFT of the ion density shown in Fig. 14.The solid and dashed red lines corresponds to the real partof the approximate dispersion relation given in Eq. (23) forelectron flows of 0.5veB and 0.9 veB . The yellow and dashedyellow lined show the imaginary part of Eq. (23).

excellent agreement with the flowing Maxwellian model,which results in an electron sheath that is approximatelytwice as thick as the one described by the commonlyassumed random flux model. The simulations are consis-tent with the electron flow velocity attaining the electronsheath Bohm speed by the sheath edge, and this flowvelocity was verified to be the result of acceleration ina pressure driven electron presheath. Comparison withthe simulations also revealed the inherent 2D nature ofthe electron sheath. Due to it’s small size, the electronpresheath does not resemble the presheath of an infiniteplanar boundary, instead there is a divergence of the ionflow around the sheath-presheath region. This flow ne-cessitates a new description of ions where the ion flow isbalanced by the presheath pressure gradients.

Finally, the simulations revealed the existence of iondensity fluctuations in the electron presheath. Thesedensity fluctuations are expected; The theory predictsa large differential flow between ions and electrons in thepresheath which excite ion acoustic instabilities. FFTsof the 2D ion density indicate that these density fluctua-tions are ion acoustic waves. Inspection of the sheathedge position revealed that these ion acoustic wavesare responsible for sheath edge fluctuations, and hencesheath collection area fluctuations, which in turn causefluctuations in the collected electron saturation current.

ACKNOWLEDGMENTS

This research was supported by the Office of FusionEnergy Science at the U.S. Department of Energy un-der contract DE-AC04-94SL85000. The first author wasalso supported by the U.S. Department of Energy, Officeof Science, Office of Workforce Development for Teach-ers and Scientists, Office of Science Graduate StudentResearch (SCGSR) program. The SCGSR program isadministered by the Oak Ridge Institute for Scienceand Education for the DOE under contract number DE-

AC05-06OR23100.

APPENDIX A: EXACT SOLUTIONS TO EQ. (15)

The solution to an equation of the form

y2 − 2 ln(y) = z (24)

can be written in terms of the Lambert W function as

y = exp

{− 1

2W

(− 1

ez

)− z

2

}. (25)

The Lambert W function has two branches, the W0(z)branch and the W−1(z) branch. For the electron sheathproblem we are interested in the asymptotic limit as z →∞. For this limit the W0(z) branch provides unphysicalsolutions because W0(0) = 0 and an accelerating flowvelocity cannot correspond to y → 0, instead we choosethe W−1 branch. The asymptotic limit of the W−1(z)branch as z → 0− is27

W−1(z) = ln(−z)− ln(− ln(−z))

+O(

ln(− ln(−z))ln(−z)

). (26)

Using the asymptotic limit in the solution Eq. (24) givesy =√z, the same result as if the logarithmic term were

dropped. To quantify the error involved in this approxi-mation we plot Eq. (24) against

√z in Fig. 16, and see

that the error is ≈ 20% at small z and decreases at largez.

0 5 10 15 20 25 301

2

3

4

5

6

z

f(z)

f (z ) = exp(−W

−1(−e−z)/2 − z /2)

f (z ) =√

z

0 10 20 30 400

10

20

30

z

% d

iffer

ence

Student Version of MATLAB

FIG. 16. Exact solution to (24) and the solution using theasymptotic approximation of W−1(z) as z → 0−. The subplotshows the percent difference of the two curves.

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