ORIGINAL PAPER

The Existence of Non-negatively Charged Dust Particlesin Nonthermal Plasmas

M. Mamunuru1 • R. Le Picard2 • Y. Sakiyama1 •

S. L. Girshick3

Received: 30 November 2016 / Accepted: 13 February 2017 / Published online: 22 February 2017� Springer Science+Business Media New York 2017

Abstract Particles in nonthermal dusty plasmas tend to charge negatively. Howeverseveral effects can result in a significant fraction of the particles being neutral or positively

charged, in which case they can deposit on surfaces that bound the plasma. Monte Carlo

charging simulations were conducted to explore the effects of several parameters on the

non-negative particle fraction of the stationary particle charge distribution. These simu-

lations accounted for two effects not considered by the orbital motion limited theory of

particle charging: single-particle charge limits, which were implemented by calculating

electron tunneling currents from particles; and the increase in ion current to particles

caused by charge-exchange collisions that occur within a particle’s capture radius. The

effects of several parameters were considered, including particle size, in the range

1–10 nm; pressure, ranging from 0.1 to 10 Torr; electron temperature, from 1 to 5 eV;

positive ion temperature, from 300 to 700 K; plasma electronegativity, characterized in

terms of n?/ne ranging from 1 to 1000; and particle material, either SiO2 or Si. Within this

parameter space, higher non-negative particle fractions are associated with smaller particle

size, higher pressure, lower electron temperature, lower positive ion temperature, and

higher electronegativity. Additionally, materials with lower electron affinities, such as

SiO2, have higher non-negative particle fractions than materials with lower electron

affinities, such as Si.

Keywords Dusty plasmas � Particle charging � Monte Carlo simulations � Particle chargelimits � Electron tunneling � Non-negative particles

& S. L. Girshickslg@umn.edu

1 Lam Research Corporation, 11355 SW Leveton Drive, Tualatin, OR 97062, USA

2 Lam Research Corporation, 4650 Cushing Parkway, Fremont, CA 94538, USA

3 Department of Mechanical Engineering, University of Minnesota, 111 Church St. S.E.,Minneapolis, MN 55455, USA

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Plasma Chem Plasma Process (2017) 37:701–715DOI 10.1007/s11090-017-9798-6

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Introduction

In many nonthermal plasmas used for materials processing, charging of condensed-phase

particles in the plasma is dominated by collisional attachment of electrons and ions. In that case,

the much higher mobility of electrons compared to ions typically causes particles to charge

negatively. For the same reason,walls that bound the plasma tend to charge tonegative potential

with respect to the plasma. The resulting electric potential profile then confines the negatively-

charged particles in the plasma, preventing their diffusion to the walls. This phenomenon is

beneficial in semiconductor processing, where it is typically desired to avoid deposition of dust

particles onto the wafers being processed, as well as in nanoparticle synthesis, where particle

losses towallsmay limit process efficiency.Additionally, if all particles are charged negatively,

then coagulation is considerably reduced by Coulomb repulsion, which may be beneficial for

deliberate synthesis of monodisperse nanoparticles for applications.

Conversely, phenomena that may cause a non-negligible fraction of particles in a

plasma to be neutral or positively charged can have important consequences. As neutral

particles are not electrostatically confined to the plasma they are free to diffuse out of the

plasma, while positively-charged particles are actually accelerated by the sheath potential

drop toward walls. The resulting deposition of these particles onto surfaces might result in

unacceptable process contamination, particularly in the case of plasmas used for micro-

electronics fabrication. While the criterion for how small the non-negative charge fraction

would have to be for it to be considered negligible is not well understood, it is reasonable

to suppose that relatively small non-negative charge fractions might represent a potential

contamination problem. Additionally, the existence of non-negative dust particles would

promote coagulation, which broadens the particle size distribution and can lead to the

formation of nonspherical agglomerates, effects that may be undesirable for controlled

synthesis of nanoparticles [1], although the non-negative fraction would probably have to

be higher, perhaps at least several percent, for this to become a significant concern. On the

other hand, the existence of very small positively charged particles, with diameters around

2 nm or smaller, could potentially be exploited for controlled deposition of nanocrystalline

films, as has been observed experimentally [2].

In this paper we examine several effects that may lead to the existence of a population

of non-negative particles in processing plasmas, and conduct numerical simulations to

obtain quantitative estimates of the non-negative particle fraction for various process

conditions. As the existence of non-negative particle populations is likely to be most

important for very small nanoparticles, we focus on particles with diameters in the range

1–10 nm. The regime considered involves nonthermal argon plasmas at pressures of

0.1–10 Torr, and with electron temperatures of 1–5 eV. We consider the effects of several

parameters, including pressure, electron temperature, ion temperature, plasma elec-

tronegativity, particle size and particle material.

We do not here consider particle charging by UV-induced photodetachment or sec-

ondary electron emission. These phenomena, which can be important in plasmas with

relatively high fluxes of UV and VUV photons, or with non-Maxwellian electron energy

distributions having overpopulated high-energy tails, respectively, can strongly shift the

particle charge distribution toward positive charging [3–5], but are often relatively

unimportant, compared to collisional attachment, under the conditions examined. Thus,

insofar as our results show that the existence of non-negative particles can be important

under some of the conditions considered, they indicate that this can be so even without

these explicitly electron-emissive effects.

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It should also be noted that the simulations reported here are not self-consistent, in that

the plasma conditions are fixed and varied one at a time, regardless of the particle charge

distribution. In reality the particle density and charge distribution would affect the plasma,

parameters such as pressure would affect properties such as electron and ion temperatures,

and so forth. Such self-consistent simulations of dusty plasmas have been reported (e.g.

[5–7]) and can provide considerable insight. However such calculations can be quite

computationally expensive, especially when performed for spatially non-uniform plasmas

[7, 8], and it is also of interest to explore the independent effects of various fixed plasma

parameters on particle charge distributions, based on computationally inexpensive

numerical simulations, such as the Monte Carlo charging simulations reported here. While

not self-consistent, this allows one to isolate the effects of each of the plasma parameters

on the particle charge distribution, and thus can provide valuable insights as well as useful

guidance to process designers.

Numerical Model

Overview

A well-established theory known as the orbital motion limited (OML) theory is widely

used to predict particle charging in plasmas by collisional attachment [9]. This theory

assumes that dust particle radii are much smaller than the plasma Debye length, that the

electrical sheath around each particle is collisionless, and that particles do not interact with

each other. As particle charging is inherently a stochastic process, the charge of any

particle fluctuates, and a population of particles exhibits a distribution of charge states. A

steady state charge distribution exists when, on average, the positive and negative currents

to a particle balance each other. OML theory allows one to quantitatively predict these

stationary charge distributions [10, 11], which depend on parameters including the particle

diameter dp, the electron and positive ion temperatures Te and T?, and the ratios of the

number densities, n?/ne, and masses, m?/me, of positive ions to electrons, negative ions

being usually neglected in applications of the theory because of their much lower mobility

compared to electrons.

Although the ion and electron currents in OML theory explicitly depend on n? and ne,

respectively, most estimates in the literature of particle charge based on OML theory

assume that these two densities are equal (e.g. [10, 11]). However, in many cases the

plasma may be electronegative [5], meaning that much of the negative charge is carried not

by free electrons but by negative ions and/or by negatively-charged dust particles them-

selves [6, 12]. In such cases plasma quasi-neutrality requires the positive ion density to

exceed the electron density, in some cases by a considerable factor. Therefore, as electrons

have much higher mobility than ions, the average particle charge in electronegative

plasmas can be expected to be less negative than in electropositive plasmas, potentially

leading to an increase in the non-negative charge fraction.

Additionally, two effects that may be important under many conditions are neglected by

OML theory. First, depending on the pressure and the particle size, the assumption that the

electrical sheath around each particle is collisionless may not be valid. Second, the theory

neglects the fact that the number of electrons a dust particle can hold is limited [13]. For

the solid particles and particle sizes considered here, the most important source of this

charge limit is electron tunneling, which causes attached electrons to be emitted from the

Plasma Chem Plasma Process (2017) 37:701–715 703

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particle, and is related to the particle’s material-dependent electron affinity [14–16]. In this

work we find that both of these effects—the effect of pressure on ion currents, and the

existence of charge limits—can under some circumstances strongly increase the fraction of

particles that are not charged negatively.

Effect of Pressure

Regarding the effect of pressure, the assumption in OML theory that particles are sur-

rounded by a collisionless sheath may be valid at sufficiently low pressure but breaks down

as pressure is increased. At high pressure one is in a fully collisional, hydrodynamic

regime. At intermediate pressures, one is in a collision-enhanced regime where positive

ions may experience a charge-exchange collision with a neutral that occurs within the

particle’s capture radius. While the original ion may have sufficient energy to escape the

attractive potential well of the negative particle, the newly created ion may have less

energy, increasing the probability that it will be collected by the particle. Hence the

positive ion current to the particle increases, causing the particle to be less negatively

charged than it would be otherwise. Khrapak et al. [17] proposed that the pressure-de-

pendent transition between the collisionless and collision-enhanced regimes can be char-

acterized in terms of a particle Knudsen number KnR0 based on a capture radius R0, defined

such that the potential distribution around the particle has a minimum at KnR0 � 1. Gattiand Kortshagen [18] extended this concept over a wide range of pressure and collisionality,

encompassing the collisionless regime (OML), the collisional-enhanced regime (CE), and

the hydrodynamic regime (HY). They expressed the positive ion current to a negatively-

charged particle as a weighted function of the currents in these three regimes [18]:

Iþ ¼ P0IOMLþ þ P1ICEþ þ P[ 1IHYþ ; ð1Þ

where IOMLþ , ICEþ and I

HYþ denote the positive ion currents (s

-1) to a particle in each of the

three regimes, and the terms P0, P1 and P[ 1 represent the corresponding probabilities that

a positive ion experiences zero, one, or more than one collision within a particle’s capture

radius. The probabilities are given by

P0 ¼ exp �1

KnR0

� �; ð2Þ

P1 ¼1

KnR0exp � 1

KnR0

� �ð3Þ

and

P[ 1 ¼ 1� P0 � P1: ð4Þ

The ion currents in Eq. (1) are given by

IOMLþ ¼ pR2nþvþ;th 1�e/pkTþ

� �; ð5Þ

ICEþ ¼ p aR20� �

nþvþ;th; ð6Þ

and

704 Plasma Chem Plasma Process (2017) 37:701–715

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IHYþ ¼ 4pRnþlþ /p�� ��; ð7Þ

where R is the ordinary particle radius, v?,th is the ion thermal velocity, e is the elementary

charge, /p is the particle potential, k is the Boltzmann constant, a is a constant equal to1.22 for a Maxwellian ion velocity distribution, and l? is the ion mobility. For Ar

? ions in

an argon plasma, l? = 0.145 m2 V-1 s-1 [19].

The capture radius depends on the ion mean free path, which itself depends on the ion-

neutral collision cross section. The total Ar? cross section in an argon plasma equals

*10-14 cm2 over a wide range of ion energies, up to *400 eV, which encompasses theconditions considered in our simulations. Then for purposes of estimating the capture

radius, assuming a heavy species temperature of 300 K, the mean free path of Ar? ions in

an argon plasma can be approximated by

kþ ¼1

330p; ð8Þ

where k? is in units of cm and the pressure p is in units of Torr [19].Equations (1)–(4) guarantee that the ion currents for the two limiting cases of the

collisionless regime and the hydrodynamic regime are correctly expressed. Equation (6)

gives the ion current for the intermediate regime where the incoming ion experiences

exactly one charge-exchange collision inside the particle’s capture radius, creating a new

ion. In this case it is assumed that all such newly-created ions are eventually collected by

the particle. As the probability increases that an incoming ion will experience multiple

collisions, the ion current for the hydrodynamic regime is gradually phased in, as given by

Eq. (1).

In simulations reported here the electron current to particles (s-1) is always assumed to

lie in the OML regime, and is therefore expressed as

IOMLe ¼ pR2neve;th 1�e/pkTe

� �: ð9Þ

Anion currents to particles are neglected, because of the much higher mobility of

electrons compared to ions. We checked this assumption by including anion currents in

some of the simulations, assuming that the total negative charge carried by nanoparticles is

negligible so that the anion densities have their highest possible value for given values of

n?/ne. The contribution of the anion current to charging was found to be quite small even at

n?/ne = 1000, and would be even smaller if one accounted for the fact that much of the

negative charge in a dusty plasma would be carried by nanoparticles.

Particle Charge Limits

The maximum number of electrons that can coexist on a single particle is limited. Various

expressions for charge limits are reviewed in [13]. In simulations of particle charging

reported in that work, the charge limit was either taken as an arbitrary parameter or was

based on an expression for the effective electron affinity of a particle [20]. As recently

pointed out, for the small solid nanoparticles of interest here the main source of such

charge limits is the tunneling of electrons attached to the particle through the potential

barrier posed by the particle’s negative charge [15]. The resulting emission of electrons

from the particle, or tunneling current, depends on the particle’s charge and on its electron

affinity, which in turn depends on the electron affinity of the bulk (flat) material and on the

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particle’s size. Here, instead of imposing charge limits we calculate the tunneling current,

which effectively imposes charge limits.

In Ref. [15] it is assumed that attached electrons bounce around a particle with an

average velocity that is based on their being in thermal equilibrium at the particle tem-

perature. The electron tunneling current from a particle is then estimated as

Ie;tunnel ¼ qj jffiffiffiffiffiffiffiffiffiffi2kTp

me

r1

2R� T; ð10Þ

where q is the particle charge, Tp is particle temperature, and T is the tunneling probability,

given by

T � exp � 4ph

Z rtR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2me / rð Þ � / rtð Þ½ �

pdr

� �: ð11Þ

Here h is the Planck constant, / rð Þ is the electric potential of an electron at distance r fromthe center of the particle, and rt is the location where the particle’s electron affinity equals

zero, which is the location to which the electron must tunnel to escape the particle. At this

location the electron’s potential energy is given by

/ rtð Þ ¼ / rð Þ � An; ð12Þ

where An, the electron affinity of the neutral particle, is given by

An ¼ A1 �5

8

e2

4pe0R: ð13Þ

Here A? is the electron affinity of the bulk (flat) material, and e0 is the permittivity of freespace.

The exponential form of Eq. (11) results in a tunneling current that increases by many

orders of magnitude for very small increases in the magnitude of the negative particle

charge. Since charge is an integer quantity, this effectively imposes a sharp charge limit.

From Eqs. (10)–(13), one finds that this limit is related to the bulk electron affinity of

which the particle is composed, with lower values of bulk electron affinity implying more

severe charge limits. Thus in the present work we compare two related materials with quite

different bulk electron affinities: Si, with A? = 4.05 eV, and SiO2, with A? = 1.0 eV.

Numerical Method

The Monte Carlo charging model used in this work is based on Ref. [21], a model

originally developed to calculate particle charge distributions due to electron and ion

attachment in low-pressure plasmas. We extend that work by considering electron tun-

neling as well as the effect of gas pressure on the ion current to a particle. The model

calculates the transient evolution of particle charge, starting from a neutral state at t = 0 s.

The time for the next collision charging event between the particle and a positive ion or

electron is chosen randomly based on the currents given by Eqs. (1) and (9). A random

number r1 that lies between 0 and 1 is generated, and is compared to the ratio

r2 ¼Iþ

Iþ þ Ie: ð14Þ

If r1[ r2, a positive ion attaches to the particle, and the particle charge is incremented byone at the time t ¼ t þ I�1þ . Otherwise an electron attaches to the particle and the particle

706 Plasma Chem Plasma Process (2017) 37:701–715

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charge is decreased by one at the time t ¼ t þ I�1e . The next time for an electron to beemitted by tunneling from the particle is based on the tunneling current given by Eq. (10).

Only the ratio n?/ne not the value of n? itself affects the stationary particle charge

distribution. However, in all of the simulations presented here we assume that

n? = 109 cm-3. The reason for this is that the ion and electron densities affect the

characteristic particle charging times and hence the time required to reach steady state.

Based on our choice of n?, the typical collision/emission time is on the order of a

microsecond. The time step of the simulation was set to 10-9 s, much smaller than the time

associated with the greatest charging frequency. In the simulations presented, the number

of discrete charging events per simulation ranged from *2 to 5 9 106, providing statis-tically meaningful data for the stationary charge distribution.

Similarly, under the assumption that particles are electrically noninteracting, the sta-

tionary particle charge distribution is not directly affected by the particle number density.

The charge distribution can, however, be affected indirectly, because the charge accu-

mulated on dust particles affects the ratio n?/ne.

Results and Discussion

We first consider a base case that involves SiO2 particles with given values of particle

diameter dp, pressure p, cation density n?, cation-to-electron density ratio n?/ne, electron

temperature Te and cation temperature T?. We then examine the effects of varying each of

these parameters, with the exception of n?, as well as the effect of particle material, where

we compare results for SiO2 and Si.

Note that we characterize the plasma electronegativity, usually defined as n–/ne, n–being the anion density, in terms of n?/ne. This is because, as noted above, anions make

only a small contribution to particle charging under the conditions considered, and are

neglected in the simulations.

Base Case

As shown in Table 1, our base case involves SiO2 particles with the following conditions:

dp = 2 nm, p = 1 Torr, n? = 109 cm-3, n?/ne = 10, Te = 2 eV, T? = 300 K,

Tp = 300 K. Figure 1 shows the calculated stationary particle charge distribution with and

without considering the effect of charge limits, i.e. with and without including the electron

tunneling current, which effectively establishes the limit. Without accounting for tunnel-

ing, charge fractions exceeding 10-4 exist out to a charge of -5. However, accounting for

tunneling, the particle charge limit in this case is seen to equal -2. As a result, the neutral

and positive particle fractions in this case both increase by about 70% compared to the

Table 1 Base case conditionsParticle material SiO2

Particle diameter 2 nm

Pressure 1 Torr

n?/ne 10

Electron temperature 2 eV

Ion temperature 300 K

Ion density 109 cm-3

Plasma Chem Plasma Process (2017) 37:701–715 707

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simulation without tunneling. We find that accounting for tunneling makes a much larger

difference under some of the other conditions examined, consistent with the results pre-

sented in [13]. As tunneling is a real phenomenon that limits particle charge at the small

sizes considered here, it is included in all of the other simulations discussed below.

Effect of Particle Size

With all other conditions the same as in the base case, we ran simulations for particle

diameters ranging from 1 to 10 nm. The results for the fractions of neutral and positive

particles are shown in Fig. 2. As can be seen, the fractions of both neutral and positive

particles are quite strong functions of particle size. Approximately 50% of 1-nm particles

are predicted to be neutral, while at 10 nm the neutral fraction equals only a few ppm. As

neutral nanoparticles are not trapped in the plasma and can diffuse to walls, this implies

Fig. 1 Effect of particle chargelimits (i.e., electron tunneling),on particle charge distribution forbase case conditions (Table 1)

Fig. 2 Effect of particle size onfraction of particles that areneutral or positively-charged.Refer to Table 1 for otherconditions

708 Plasma Chem Plasma Process (2017) 37:701–715

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that particle deposition will be dominated by the smallest nanoparticles, assuming that a

range of particle sizes exists in the plasma. Moreover this behavior will be amplified by the

strong size-dependence of the particle diffusion coefficient D, which, from kinetic theory

for particles in the free molecule regime, scales with particle diameter dp approximately as

D / d�2p [22]. Hence this analysis suggests that quite significant fluxes to walls of verysmall nanoparticles can be expected in plasmas in which nanoparticles nucleate. Indeed

this has been reported in experimental studies of growth by plasma-enhanced chemical

vapor deposition of ‘‘polymorphous’’ silicon films, which consist of very small nanopar-

ticles embedded in an amorphous silicon matrix [23]. Once nanoparticles that nucleate in a

plasma grow beyond a certain size—in the base case, about 6 or 7 nm, but it depends on

the plasma conditions and the particle material—the neutral fraction becomes negligibly

small, but charge fluctuations for the smallest particles, combined with their high diffu-

sivity, can be expected to result in significant fluxes of these particles to walls and film

substrates.

The positive charge fraction for all sizes seen in Fig. 2 lies about two orders of mag-

nitude below the neutral fraction. However this does not necessarily imply that the positive

charge fraction is unimportant, as positive particles are accelerated to walls by the wall

sheath potential, reaching a drift velocity that is governed by ambipolar diffusion in the

sheath electric field, thereby increasing their flux to walls compared to neutral particles

with the same number density [24].

Effect of Pressure

With all other conditions the same as in the base case, we conducted simulations with

pressures ranging from 0.1 to 10 Torr, and for particle diameters of 2, 5 and 10 nm.

Figure 3 shows the results in terms of the non-negative particle fraction, i.e. the sum of the

neutral and positive particle fractions. As noted above, the non-negative fraction is very

close to the neutral fraction.

For given particle size, increasing pressure over the range 0.1–10 Torr is seen to

strongly increase the non-negative particle fraction. Since the ion and electron densities

here are fixed, and do not scale with pressure, this behavior can be attributed to the fact that

Fig. 3 Effect of pressure onnon-negative particle fraction, forvarious particle diameters. Referto Table 1 for other conditions

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increasing pressure shifts the positive ion current from the collisionless toward the colli-

sion-enhanced regime discussed above. For 2-nm-diameter particles the non-negative

particle fraction ranges from about 0.4% at 0.1 Torr to over 20% at 10 Torr. For 5-nm

particles the non-negative fraction, compared to 2-nm particles, ranges from about three

orders of magnitude lower at 0.1 Torr to one order of magnitude lower at 10 Torr.

Effect of Electron Temperature

Higher electron temperatures lead to higher electron currents to particles and higher

average negative particle charge, and hence to lower fractions of non-negative particles.

This is shown in Fig. 4, for electron temperatures ranging from 1 to 5 eV, and for pressures

ranging from 0.1 to 10 Torr. Reducing Te from 2 to 1 eV approximately doubles the non-

negative charge fraction over the pressure range considered.

Effect of Positive Ion Temperature

As with increasing Te, increasing T? causes the non-negative charge fraction to decrease,

as seen in Fig. 5. This perhaps counterintuitive behavior was previously noted by Mat-

soukas and Russell [10]. As T? increases, the ion has enough energy to escape the

attractive field of a negatively charged particle, causing the cross section for ion capture to

decrease, resulting in a higher average negative charge and thus a lower non-negative

charge fraction. In Fig. 5, an increase in ion temperature from 300 to 500 K causes the

non-negative particle fraction to decrease by a factor ranging from*5 at 0.1 Torr to*2 at10 Torr.

Note that in all these simulations we assumed a fixed particle temperature of 300 K.

However, insofar as ion temperatures are often close to the neutral gas temperature, one

might expect that particle temperatures would tend to be close to the ion temperature. From

Eq. (10), higher particle temperatures would result in higher tunneling currents, and hence

more severe charge limits, potentially leading to higher non-negative charge fractions.

Moreover, especially for the smallest nanoparticles considered here, a recent body of work

Fig. 4 Effect of electrontemperature on non-negativeparticle fraction, over a range ofpressure. Refer to Table 1 forother conditions

710 Plasma Chem Plasma Process (2017) 37:701–715

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indicates that particle temperatures may exceed the gas temperature by up to several

hundred K [7, 25, 26].

Effect of Plasma Electronegativity

Because of the accumulation of negative charge on dust particles, dusty plasmas are

inherently electronegative, with studies finding positive ion densities exceeding the elec-

tron density by factors ranging from several [5, 12] to several hundred [8]. Additionally,

aside from the depletion of electrons due to attachment on nanoparticles, in electronegative

gases anions can be relatively abundant carriers of negative charge, which also causes the

positive ion density to exceed the electron density.

Figure 6 shows the effect of electronegativity, as characterized here by n?/ne, on the

fractions of particles that are either neutral or positive, with all other conditions the same as

in the base case. As expected, increasing electronegativity strongly increases the neutral

and positive charge fractions, with the neutral charge fraction exceeding 10% for values of

n?/ne greater than about 30. The positive charge fraction rises even more steeply with

increasing electronegativity. Thus, as n?/ne increases, the ratio of positive to neutral

particles increases, ranging from a few tenths of a percent at n?/ne = 1 to *25% at n?/ne = 10

3.

The effect of electronegativity on the non-negative particle fraction depends on the

pressure, as seen in Fig. 7. At very high values of n?/ne, *103, the electronegativity

dominates, as the non-negative particle fraction is quite high regardless of the pressure,

exceeding 50% for pressures greater than 1 Torr, and falling only slightly at pressures

below 1 Torr. At more modest values of n?/ne, in the 1–10 range, both higher elec-

tronegativity and higher pressure are positively correlated with increasing values of the

non-negative particle fraction.

Effect of Particle Material

The material of which the particle is composed affects the particle charge limit, via the

electron tunneling current, which depends on the material’s bulk electron affinity. As noted

Fig. 5 Effect of positive iontemperature on non-negativeparticle fraction, over a range ofpressure. Refer to Table 1 forother conditions

Plasma Chem Plasma Process (2017) 37:701–715 711

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above in the section on particle charge limits, Si has a bulk electron affinity approximately

four times higher than that of SiO2. Hence Si nanoparticles can be expected to have less

severe charge limits than SiO2, potentially leading to smaller non-negative charge fractions

for Si than for SiO2.

Figure 8 shows a comparison of the stationary particle charge distribution calculated for

4-nm-diameter particles composed of either Si or SiO2. All other conditions are the same as

in the base case. Because of the difference in electron tunneling, the effective charge limit

for the Si particles equals –6, while it equals –2 for the SiO2 particles. As a result, the

neutral charge fraction for the SiO2 particles is seen to exceed that for the Si particles by

more than one order of magnitude.

The non-negative particle fraction comparing Si with SiO2 for a range of particle sizes

is shown in Fig. 9. For 1- and 2-nm particles, the non-negative particle fraction is about

twice as high for SiO2 as for Si. For particles that are around 4 nm and larger, the non-

negative SiO2 particle fraction is more than an order of magnitude larger than for Si. This

Fig. 6 Effect of plasmaelectronegativity, characterizedby n?/ne, on fractions of neutraland positive particles. Refer toTable 1 for other conditions

Fig. 7 Effect of n?/ne on non-negative particle fraction, over arange of pressure. Refer toTable 1 for other conditions

712 Plasma Chem Plasma Process (2017) 37:701–715

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behavior is a direct consequence of the quite different charge limits for each material at

each size. Based on inspection of the calculated particle charge distributions, for SiO2 the

charge limits for particles of 1, 2, 4, and 6 nm diameter are given by -1, -2, -2, and -3,

respectively; for Si the corresponding charge limits are given by -2, -4, -6, and -9. The

fact that the charge limit for SiO2 particles is the same, -2, for both 2- and 4-nm particles

is the reason that the curve in Fig. 9 appears non-smooth in that region. Essentially, this

behavior is related to the integer nature of charge.

Summary and Conclusions

In this work we conducted Monte Carlo charging simulations to calculate stationary particle

distributions for a variety of conditions in dusty plasmas. We focused specifically on the

fraction of particles that are not charged negatively, as these particles are not electrostatically

Fig. 8 Effect of particle materialon charge distributions of 4-nm-diameter particles. Refer toTable 1 for other conditions

Fig. 9 Effect of particle materialon non-negative particle fractionover a range of particle sizes.Refer to Table 1 for otherconditions

Plasma Chem Plasma Process (2017) 37:701–715 713

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confined in the plasma and can freely diffuse (in the case of neutral particles) or be elec-

trostatically attracted (in the case of positive particles) to surfaces bounding the plasma.

The simulations accounted for two deviations from orbital motion limited theory—the

existence of single-particle charge limits due to electron tunneling, and the effect of

pressure on the positive ion current to particles of given size due to charge-exchange

collisions that occur with the particle’s capture radius.

The effect of several parameters on the non-negative particle fraction was considered,

including particle size, in the range 1–10 nm; pressure, ranging from 0.1 to 10 Torr;

electron temperature, from 1 to 5 eV; positive ion temperature, from 300 to 700 K; plasma

electronegativity, characterized in terms of n?/ne ranging from 1 to 1000; and particle

material, either SiO2 or Si.

Within the parameter space examined, higher non-negative particle fractions are

associated with smaller particle size, higher pressure, lower electron temperature, lower

positive ion temperature, and higher electronegativity. Additionally, materials with lower

electron affinities, such as SiO2, have higher non-negative particle fractions than materials

with lower electron affinities, such as Si. This is caused by the more severe charge limits

that result from lower electron affinities.

In conclusion, we find that under many conditions that are pertinent to microelectronics

fabrication and other processing plasmas the fraction of nanoparticles that are not nega-

tively charged is high enough to imply significant fluxes of these particles to walls and

material substrates, which may be either a serious contamination concern or a feature that

could deliberately be exploited for synthesis of nanostructured materials. It should also be

noted that these simulations assumed that particle charging is dominated by collisional

attachment of electrons and ions, and did not consider explicitly electron-emissive effects

such as UV and VUV photodetachment and secondary electron emission. These latter

effects, if significant, would increase the non-negative particle fraction even further.

Acknowledgements This work was partially supported by the Lam Research Foundation, the U.S. NationalScience Foundation (CHE-124752), and the U.S. Dept. of Energy Office of Fusion Energy Science (DE-SC0001939).

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The Existence of Non-negatively Charged Dust Particles in Nonthermal PlasmasAbstractIntroductionNumerical ModelOverviewEffect of PressureParticle Charge LimitsNumerical Method

Results and DiscussionBase CaseEffect of Particle SizeEffect of PressureEffect of Electron TemperatureEffect of Positive Ion TemperatureEffect of Plasma ElectronegativityEffect of Particle Material

Summary and ConclusionsAcknowledgementsReferences