modelling and applications of the hollow cathode …172081/...space-charge sheath in an rf hollow...

ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2008

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 433

Modelling and Applications of theHollow Cathode Plasma

DANIEL SÖDERSTRÖM

ISSN 1651-6214ISBN 978-91-554-7206-1urn:nbn:se:uu:diva-8747

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To Cecilia

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I H. Baránková, L. Bárdoš and D. Söderström. Cold AtmosphericPlasma in Nitrogen and Air Generated by the Hybrid Plasma Source.Journal of Vacuum Science & Technology A, 24:1410–1413, 2006.

II H. Baránková, L. Bárdoš , D. Söderström and L-E. Gustavsson. Char-acterization of Hybrid Atmospheric Plasma in Air and Nitrogen. Pro-

ceedings of the 49th Annual Tech. Conf. of the Society of Vacuum Coat-

ers (SVC), Washington, DC. Paper E-8, pp. 41–43. April 2006.III D. Söderström, H. Baránková, and L. Bárdoš. On dimensions of

atmospheric-pressure hollow cathodes. IEEE Transactions on Plasma

Science, 35(3), 2007.IV D. Söderström, H. Baránková, and L. Bárdoš. Space-charge sheaths in

atmospheric pressure hollow cathodes. Proceedings of the 16th Symp.

on Applications and Plasma Processes on Plasma Processing (SAPP

XVI). Podbanské, Slovakia. January 2007. P69, pp. 261–262.V D. Söderström, H. Baránková, and L. Bárdoš. Time evolution of the

space-charge sheath in an rf hollow cathode. Journal of Physics: Con-

ference Series, 100, 2008.VI D. Söderström, H. Baránková, and L. Bárdoš. Modelling the Dynamics

of a Pulsed Atmospheric Plasma in Hollow Electrode Geometry. Sub-

mitted to Journal of Physics D: Applied Physics, 12 April, 2008.

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Modelling of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Describing the Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Definition of Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Equations of Motion for Many Particles . . . . . . . . . . . . . . . . . . 132.3 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 The Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 The Zeroth Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 The First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 The Second Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.4 Closing the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.5 Some Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 182.5 Equation of Motion for One Particle . . . . . . . . . . . . . . . . . . . . 18

2.5.1 Motion of a charged particle in uniform E and B fields . . . 192.6 Waves in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.1 Electrostatic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6.2 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Plasma Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Plasma Generation by an Electric Field . . . . . . . . . . . . . . . . . . 25

3.1.1 Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 DC Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.3 Pulsed DC Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Plasma Generation by Electromagnetic Fields . . . . . . . . . . . . . 29

3.2.1 Radio Frequency Generated Plasmas . . . . . . . . . . . . . . . . 29

3.2.2 Microwave generated plasmas . . . . . . . . . . . . . . . . . . . . . 303.3 Hybrid Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 The Plasma Sheath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 The Cathode Sheath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Radio Frequency Sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Plasma Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 Electric Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Optical Emission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.1 Spectrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.2 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.3 Spectroscopic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Plasma Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1 Analytical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Fluid Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2.1 Low Pressure Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 466.2.2 High Pressure Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 476.2.3 Rate and Transport Coefficients . . . . . . . . . . . . . . . . . . . . 486.2.4 Solving the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.3 Kinetic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3.1 PIC Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.4 Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Modelling Hollow Cathodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.1 Modelling of the RF Hollow Cathode Sheath . . . . . . . . . . . . . . 547.1.1 Solving the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.1.2 The Potential Distribution . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2 Two-dimensional Self-Consistent Model . . . . . . . . . . . . . . . . . 577.2.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.3 Atmospheric Pulsed Plasma in Hollow Electrode Geometry . . . 637.3.1 Dynamics at Pulse Rise . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 The H-HEAD Plasma Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.1 Cold Nitrogen and Air Plasmas by the H-HEAD . . . . . . . . . . . 708.2 Deposition of Diamond Films by the H-HEAD . . . . . . . . . . . . 708.3 Measuring Vibrational Temperature . . . . . . . . . . . . . . . . . . . . . 71

9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7310 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511 Summary of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7712 Svensk Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Nomenclature

B T Magnetic flux density

D C/m2 Displacement field

E V/m Electric field

H A/m Magnetic field

J A/m2 Current density

F N Force

P Pa Pressure

Q W/m2 Heat flux density

Γ m−2s−1 Particle flux density

v m/s Particle velocity

w m/s Particle random velocity

u m/s Particle mean velocity

k m−1 Wave number

n – Surface unit normal

B T Scalar magnetic flux density

E V/m Scalar electric field

J A/m2 Scalar current density

P Pa Scalar pressure

k m−1 Scalar wave number

Vi J Ionisaion potential

D m2/s Diffusion coefficient

e C Electron charge

q C Charge

m kg Mass

me kg Electron mass

mi kg Ion mass

n m3 Particle number density

ne m−3 Electron number density

ni m−3 Ion number density

n0 m−3 Plasma density

N m−3 Neutral number density

Te K Electron temperature

T K Temperature

S m−3 Electron and ion source term

S ε J/m3 Electron energy source term

j A/m2 Current density

v m/s Particle velocity

u m/s Particle mean velocity

t s Time

νm s−1 Momentum transfer frequency

ν Hz Frequency

φ V Electrostatic potential

ρ C/m3 Charge density

γ – Secondary electron emission coefficient

ω rad/s Angular frequency

ωc rad/s Cyclotron frequency

ωp rad/s Plasma frequency

μ m2/Vs Mobility coefficient

σ A/Vm Conductivity

ε J Electron mean energy

List of Abbreviations

RF Radio Frequency

DC Direct Current

AC Alternating Current

ECR Electron Cyclotron Resonance

DBD Dielectric Barrier Discharge

H-HEAD Hybrid Hollow Electrode Activated Discharge

FD Finite Difference

FE Finite Element

PDE Partial Differential Equation

1. Introduction

The use of plasmas in technological applications is rapidly expanding. Fromthe early days, when gas discharges were produced in evacuated tubes just forthe purpose of finding out the properties of the gas discharge, we have come tohighly advanced technologies that use plasmas for cleaning of exhaust gases,surface treatment, surface deposition, water treatment, lightning, etc.

The development of plasma science is closely related to the developmentof electricity. One of the first scientific observations of a plasma could beJean Picard’s experiments with a mercury barometer in 1675 [1]. If it wasdark enough around him and the barometer, he could observe a glow in theglass tube above the mercury. The glow was also strongest when the mercurywas moving up and down. Later, Francis Hauksbee the Elder made some ex-periments on static electricity, and could in his machines create a dischargeplasma. Since no batteries were available at that time, all experiments wereon static electricity or pulsed discharges (e.g. sparks). Later, when the batterywas invented by Alessandro Volta in 1800, one had a source for continuousdischarges [2].

The first practical use of a plasma was the arc for electrical illuminationused in the 19th century. This required transmission of high current DC, andthe incandescent lamp eventually took over due to its ability to be operated atAC. After Irving Langmuir’s and Lewi Tonk’s work in the 1920’s, the plasmaresearch began to develop rapidly.

Many different discharge configurations have been developed since the firstsimple anode–cathode discharge tubes, for example electron cyclotron reson-ance discharges, dielectric barrier discharges, and hollow cathodes.

Both theoretical and experimental research on plasma discharges for in-dustrial applications is carried out at the Division of Electricity at UppsalaUniversity. The focus of this thesis is on numerical modelling of hollow cath-ode gas discharges and gas discharges in hollow electrode geometries. A partof the thesis is also devoted to characterisation of the hybrid hollow electrodeactivated discharge (H-HEAD) by measurement of vibrational temperaturesof nitrogen molecules.

1.1 Modelling of Plasmas

There exists a whole range of methods to diagnose plasmas, such as differenttypes of probes and spectroscopic methods. There are, however, parametersthat are hard to measure, or the measurements themselves can be hard to in-

11

terpret. Plasma modelling makes an important contribution here. Since onehas complete control of what one puts in a model, one can also discern thecontribution each part makes to the solution. Comparing the models with ex-periments can give valuable information about the plasma properties, and howto optimise the discharge performance.

Modelling of high pressure plasma discharges is particularly important,since the small spatial sizes of these discharges make it very difficult to phys-ically measure the plasma properties.

1.2 Outline

The outline of this thesis is the following:In chapter 2 the fundamental equations that are used to describe a plasma

are presented. They are needed to develop the theory of sheaths etc. in thefollowing chapters.

Chapter 3 describes the generation of plasmas and some common plasmadischarge configurations.

Chapter 4 gives an introduction to plasma diagnostic techniques such aselectric probes, mass spectrometry, and optical emission spectroscopy.

Chapter 5 introduces the concept of the plasma space-charge sheath.Chapter 6 describes some common plasma modelling techniques suitable

for modelling of gas discharge plasmas.Chapter 7 presents the modelling of hollow cathodes and discharges in hol-

low geometry. Three models and results from these are presented.Chapter 8 presents the H-HEAD plasma source and its characterisation.

Results from experiments with this source is also given.Chapter 9 gives a short summary of each paper included in this thesis, em-

phasising the contribution by the author.

12

2. Describing the Plasma

In order to treat the plasma theoretically, one needs to put down equationsfor the motion of the particles in the plasma. We will see, that by starting withthe most complex mathematical description one can make approximations andsimplifications that leads to less and less complex descriptions of the plasma.

2.1 Definition of Plasma

A plasma is a more or less ionised gas. There are, however some criteria thatthe ionised gas has to fulfil in order to be considered a plasma. The air aroundus also has a small fraction of ionised atoms and molecules, but it can still notbe considered a plasma. The first criterion is that the Debye length (see chapter4) should be much less than the dimension of the plasma itself. Second, thenumber of particles within a volume with the radius of a Debye length shouldbe much greater than 1. These first two criteria ensure that the plasma is quasi-neutral. The last criterion is that the plasma particles should be described byelectromagnetic forces, rather than hydrodynamic forces as would be the caseif the particles were highly collisional.

A plasma is sometimes also called the fourth state of matter. It is not truein a strict manner, but comes from the fact that a gas will become a plasma, ifenough energy is supplied to the gas particles.

The word plasma is due to Langmuir [54], who first used it in 1928 todescribe an ionised gas. The word “plasma” comes from the Greek verb“πλασσω” (“plasso”) meaning “to form” or “to mold”. It is not clear whyLangmuir chose this word [71].

2.2 Equations of Motion for Many Particles

One can start the description of the plasma by writing Newton’s second lawfor N (distinguishable) particles,

miRi(t) = Fi(t), i = 1, . . . ,N. (2.1)

To solve this, to get the velocity Ri(t) and the position Ri(t) for the ith particleat a time t, one needs the initial position and velocity of each particle. Forlarge N this is practically impossible. The force Fi(t) also often depends on thevelocities and positions of all particles. So we have one nasty set of coupledequations.

13

2.3 Kinetic Theory

If N is very large, we can, instead of keeping track of every single particle inthe plasma, turn to a statistical description. We can say that the probability toencounter a particle i(i = 1,2, . . . ,N), or a particle indistinguishable from theparticle i, within a small volume in phase space dridvi around (ri, vi) at time t

is

fN

N∏i=1

dridvi, (2.2)

where fN = fN(ri,vi, t) is the generic N-particle distribution function. To getthe probability to find particle i(i = 1,2, . . . ,N), or a particle of the same typeas i, at a certain position and with a certain velocity, one integrates fN over allpositions and velocities of all particles except particle i. Here one must takecare of the permutation of the number of like particles. Often one has only onetype of particle, e.g. electrons. Then∫

fN

N∏i=1

dridvi = N!, (2.3)

since there are N! ways to pick N identical particles. The reduced distribution

function fr is obtained by summing over all but s particles (and normalising).If s = 1 one gets the probability to find a particle within drdv of (r, v).

We can now look at how the reduced distribution function for s = 1, hence-forth called just the distribution function and denoted f , changes over time.We have

d f (r,v, t)

dt=∂ f

∂t+∂ f

∂r

dr

dt+∂ f

∂v

dv

dt. (2.4)

The first term the right hand side is the explicit dependence of f on time.The change in f due to this term is what an observer among the particleswould see if he or she would stand still. The second term is v · ∇ f , which ishow f changes as the observer moves into regions where f is different. Fromequation (2.1) we have that dv/dt = F/m for a particle. Using this, we canwrite the third term as (F/m)(∂ f /∂v). Including forces between like particlesand other particles (e.g. ions in a plasma) and any external fields in F, we getafter some algebra (see [18] for a thorough derivation)

∂ f

∂t+v · ∂ f

∂r+

F

m· ∂ f

∂v=

(∂ f

∂t

)c

, (2.5)

which is called the Boltzmann equation. The term on the right hand sideis the collision term. If the collision term is zero, we have the collisionlessBoltzmann or the Vlasov equation.

14

2.4 The Fluid Equations

The kinetic equations are very complex. In some cases this complexity is notneeded, and much simpler equations are obtained if one averages the distri-bution function over the velocity. The averaged quantities are particle density,average velocity, and energy density. These quantities are called the macro-scopic quantities and they constitute the fluid equations.

From probability theory, we know that the expectation value, or the average,of a quantity α(x) with the distribution function f (x) is

< α >=

∫ ∞

−∞α f (x)dx. (2.6)

If α(x) = 1, we get the zeroth moment, if α(x) = x the first moment andso on. Since an average contains less information, it simplifies things. If wetake different moments of the Boltzmann equation (2.5) by multiplying withdifferent functions of velocity, it will replace the equations for the distributionfunction f (r,v, t) by equations which only depend on r and t. One choosesthe functions on v so that they are related to physical quantities, so that, forexample, the zeroth moment m gives the local mass density, the first momentmv the momentum, and the second moment mv2/2 the energy. In this case wehave

< α(r,v, t) >=1

n(r, t)

∫ ∞

−∞α(r,v, t) f (r,v, t)dv, (2.7)

where

n(r, t) =

∫ ∞

−∞f (r,v, t)dv (2.8)

is the number density of the particles in the small volume element dr aroundposition r at time t.

2.4.1 The Zeroth Moment

Taking the zeroth moment of (2.5) gives (see [24] or [18] for further details)

∂n

∂t+∇ · (nu) =

(∂n

∂t

)c

, (2.9)

where

u =< v >=1

n(r, t)

∫ ∞

−∞v f (r,v, t)dv. (2.10)

This is called the continuity equation. In the integration procedure, we haveassumed that f → 0 faster than any power of v as |v| → ∞. This is true fore.g. a Maxwellian distribution. The collision term on the right hand side of(2.9) includes processes where particles are created or removed, for exampleionisation and recombination.

15

2.4.2 The First Moment

The first moment of the Boltzmann equation, obtained by multiplying withmv and integrating over v, gives us the momentum equation

mn

[∂u

∂t+ (u · ∇)u

]= qn(E+u×B)−∇ ·P+

+

(∂

∂t(mnu)

)c

−mu

(∂n

∂t

)c

, (2.11)

where it has been assumed that our force is the Lorentz force

F = q(E+v×B). (2.12)

P is the pressure tensor and comes from the peculiar or random velocity w ofthe particles,

w = v−u. (2.13)

2.4.3 The Second Moment

Next, to get the second moment of the Boltzmann equation, which is the flowof energy, we multiply by 1

2 mv2 and integrate. This gives, in three dimensions[15],

3

2

dP

dt+

5

2P∇ ·u+∇ ·Q = C, (2.14)

which is the energy equation. It has been assumed here, that the distributionfunction is an isotropic function of the random velocity w, which makes theoff-diagonal terms of the pressure tensor P vanish, and the diagonal termsequal. We thus get a pressure scalar P =

∑3i

13

Pii. C includes all collision termsand Q is the heat flux.

With the help of gas kinetic theory, we can introduce the kinetic temper-ature T . We know that for each degree of freedom, we have that the meanrandom energy of the gas particles is E = 1

2kBT . So we have that, in our three-

dimensional case,3

2kBT =

1

2m

∫w2 f dv. (2.15)

But

Pi j =m

3

∫wiwj f dv, (2.16)

so thatPii = nkBT. (2.17)

We can now write equation (2.14) as

3

2

d

dt(nkBT )+

5

2nkBT∇ ·u+∇ ·Q =C. (2.18)

16

2.4.4 Closing the Equations

Looking at our equations above, we see that each moment contains a termof the next higher moment (the zeroth moment gives us the density, butalso contains u). We must break this chain at some point. One often doesthis by considering the isothermal or adiabatic limit of equation (2.14). Thetime derivative in this equation has the dimension of 1/time and the spatialderivatives have the dimension of 1/length. It is over this time and this lengththat the quantities changes significantly. Calling this time and length thecharacteristic time tchar and characteristic length lchar, respectively, we canget the velocity of some phenomena as vph ∼ lchar/tchar.

If the thermal velocity of the particles vth � vph, the left hand side can beneglected, since the ratio to the right hand side is vph/vth. If the collisionalterm can be neglected, the heat flux is dominating, which implies that thetemperature will become spatially uniform. This is the isothermal limit. Forthe momentum equation (2.11), we still need an equation relating P to n.From kinetic theory, we know that in the isothermal case, for an equilibriumMaxwellian distribution P = nkBT , where kB is Boltzmann’s constant.

If, on the other hand, vth � vph, the heat flux is negligible. Compared to thelarge t−1

charterm, the collisional term can be neglected (the collision frequency

νcoll � t−1char

). This is the adiabatic limit. One can also close the equations bysetting Q = −κth∇T , where κth is the thermal conductivity [55].

We also need an equation for the electric and magnetic fields. Here Maxwellhelps us with his four equations

∇ ·D = ρ (2.19)

∇×E = −∂B∂t

(2.20)

∇ ·B = 0 (2.21)

∇×H = J+∂D

∂t(2.22)

where D = εE and B = μH. J and ρ are the free current and charge densit-ies, respectively. These equations describe electric and magnetic fields in amedium. In a plasma it is often impractical to use D and H, so one uses thevacuum equations instead [24].

If E is electrostatic, it can be written as the gradient of a scalar potential φ,

E = −∇φ, (2.23)

which could be used to write (2.19) as the Poisson equation

∇2φ = −ρ/ε0. (2.24)

17

Velocity (arb. units)

Den

sity

(arb

.u

nit

s)

80706050403020100

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

Figure 2.1: An example of a distribution function with a large group of low-velocity

particles and a small group of high-velocity particles.

2.4.5 Some Concluding Remarks

Since we take the average of the distribution function over the velocity v toget the fluid equations, a complicated dependence of the distribution functionon the velocity will not enter into the fluid equations. A distribution functionlooking like the one in figure 2.1, with a group of fast particles, concentratedaround some velocity much larger than the velocity of the main part of theparticles, can not be achieved in the fluid model, since there is only one velo-city, the fluid velocity u.

The collision terms in equations (2.9), (2.11), and (2.14) determine thechange of number density, momentum, and energy density in a small volumedr around r. This means that, if a particle interacts with some other particleoutside the volume dr, the number density etc. within dr may not be con-served, though the total density in the total volume is. It is also clear that theenergy density within dr cannot be changed by some process outside dr.

2.5 Equation of Motion for One Particle

In some cases, it is desirable to describe the motion of a single particle inthe plasma, for example to see how the particle behaves in external electricor magnetic fields. For the purpose of this thesis, it will suffice to show themotion of a particle in constant and uniform electric and magnetic fields. Thefields are not affected by the fields from the charged particle. There are alsono collisions that will interfere with the particle motion.

18

2.5.1 Motion of a charged particle in uniform E and B fields

As before, we can start with Newton’s second law, equation (2.1). The forceacting on a charged particle in static electric and magnetic fields is the Lorentzforce,

mR = F = q(E+v×B). (2.25)

We take the magnetic field to be directed along the z-axis, B= Bzz. The E-fieldcan then be resolved into components parallel, E‖, and perpendicular, E⊥, tothe magnetic field, with the perpendicular component defining the y-axis. Wethus have Ex = 0. The components of (2.25) are then

dvz

dt=

q

mE‖, (2.26)

dvy

dt=

q

mE⊥−ωcvx, (2.27)

and

dvx

dt= ωcvy, (2.28)

where ωc ≡ qB/m is the cyclotron frequency. The first equation gives

vz = v‖+q

mE‖t, (2.29)

which is just an acceleration, with initial velocity v‖, along the magnetic field.Without collisions or other means to lose energy, the particle will be acceler-ated to relativistic speeds, which are not covered here.

The other two coupled equations give

vx = v⊥ cos(ωct)+E⊥B

(2.30)

and

vy = −v⊥ sin(ωct), (2.31)

where a zero phase shift has been assumed.We see that the velocity along the y-axis has an oscillatory behaviour, as

do the velocity along the x-axis. But the latter also has an additional velocityE⊥/B, which is called the guiding centre drift velocity. If we remove the ex-ternal electric field, this velocity disappears, and the motion of the particle isa circle around the uniform magnetic field. Since the motion parallel to themagnetic field is unimpeded, a particle with a velocity at some angle to themagnetic field will trace out a helix as it moves along. The radius of the cir-cular motion around the magnetic field can be obtained by solving

dvx

dt= ωcvy, (2.32)

19

which givesx = rL sin(ωct)+ x0. (2.33)

rL ≡ v⊥/ωc is called the Larmor radius. Particles with opposite charges willgyrate in different directions around the magnetic field, since the cyclotronfrequency ωc will change sign with the charge of the particle.

2.6 Waves in Plasmas

As in neutral gases, a compressional wave can also propagate in a plasma,but since the plasma particles are charged, they can interact by electric andmagnetic fields in addition to collisions. This gives rise to a whole range ofdifferent types of waves. We will only look at a few here, which can be derivedfrom the fluid equations and Maxwell’s equations. Even more waves can befound by using kinetic theory instead of the fluid equations.

To be able to solve the equations analytically, we will linearise the equa-tions. This is done by considering the quantities consisting of an unperturbed,equilibrium part, and a small perturbation,

f (x, t) = f0+ f1,

where f1 � f0. When the equations are linearised, it is convenient to Fouriertransform them, to get more simple expressions [41].

2.6.1 Electrostatic Waves

Taking the divergence of (2.11), putting

uk = uk0+uk1, nk = nk0+nk1, E = E0+E1, Pk = Pk0+Pk1,

for species k assuming no magnetic fields, isotropic pressure, so that the pres-sure can be described as a scalar, P = P, and that E0 = 0, uk0 = 0, ∇Pk0 = 0 wearrive at

mknk0∂∇ ·uk1

∂t= −qknk0∇2φ1−∇2Pk1, (2.34)

where second-order terms and higher have been neglected, and −∇φ1 = E1.Linearising (2.9), we get

∂nk1

∂t+nk0∇ ·uk1 = 0, (2.35)

assuming no creation or annihilation of particles. Combining (2.34) and (2.35)yields

mk

∂2nk1

∂t2= qknk0∇2φ1+∇2Pk1. (2.36)

20

We need some expression for the pressure in order to close the equations. Ifwe assume an isothermal plasma, we have that

Pk = nkkBTk,

so that

mk

∂2nk1

∂t2= qknk0∇2φ1+ kBTk0∇2nk1. (2.37)

By taking the Fourier transform of (2.37) according to

f (x, t) =1

(2π)2

∫ ∞

−∞f (k,ω)exp[i(k ·x−ωt)]d3kdω, (2.38)

where ω is the frequency and d3k = dkxdkydkz, we can replace

∂

∂t→−iω,

and∇→ ik.

We now havemkω

2nk1 = qknk0k2φ1 + kBTk0k2nk1. (2.39)

Henceforth, the tilde, indicating the Fourier amplitude, will be omitted to sim-plify the notation. Linearising and taking the Fourier transform of the Poissonequation 2.24,

ε0k2φ1 =∑

k

nk1qk, (2.40)

we can, after some rearrangement, write (2.39) as⎡⎢⎢⎢⎢⎢⎢⎣1−∑k

ω2pk

k2ω2− kBTk0/mk

⎤⎥⎥⎥⎥⎥⎥⎦φ1 = 0, (2.41)

where

ω2pk =

nk0q2k

ε0mk

(2.42)

is the plasma frequency. For (2.41) to have a non-trivial solution, we musthave φ1 � 0, so that

1−∑

k

ω2pk

k2ω2− kBTk0/mk

= 0. (2.43)

21

This is called the dispersion relation and describes the relation between ω andk. Plotting ω against k in one dimension would give a line of allowed ω’s andk’s for this type of wave in a plasma.

2.6.2 Electromagnetic Waves

We now consider waves not only with a perturbed electric field, but also witha perturbed magnetic field, B1 � 0. There is no constant magnetic field, B0 =

0. This corresponds to electromagnetic waves. Taking the time derivative of(2.22) and the curl of (2.20), and linearising, we get

c2∇× ∂B1

∂t=

1

ε0

∂J1

∂t+∂2E1

∂t2, (2.44)

and

∇× (∇×E1) = ∇(∇ ·E1)−∇2E1 = −∇×∂B1

∂t, (2.45)

where c2 = 1/(μ0ε0). Putting (2.44) in (2.45), and doing the same trick withthe Fourier transform as before, we arrive at

−k(k ·E1)+ k2E1 =iω

ε0c2J1+ω2

c2E1. (2.46)

Since we do not want to have electrostatic waves, with the propagation vectork parallel to the electric field, we set k ·E1 = 0, i.e. we look at transversewaves. What is left is now

(ω2− c2k2)E1 = −iωJ1

ε0. (2.47)

The current in the plasma can be described by Ohm’s law with the conductivityσ = nee2/meνe (with νe being the momentum transfer frequency for electrons,i.e. we have collisions),

J1 = σE1. (2.48)

so that

k2 =ω2

c2+ iσω

ε0c2(2.49)

for a non-trivial solution to exist. This suggests that k has an imaginary com-ponent, and we can represent k as an imaginary quantity as

k = α+ i/δ. (2.50)

The real part stands for the oscillating electromagnetic field, and the imaginarypart describes a damping of the wave. The damping comes from the electronmotion and their collisions with neutrals.

22

Putting (2.50) in (2.49), we get, after much hassle, the real and the imagin-ary part of the propagation constant,

α =

√σμ0ω

2

⎡⎢⎢⎢⎢⎢⎣ωε0σ +√

1+(ωε0

σ

)2⎤⎥⎥⎥⎥⎥⎦1/2

(2.51)

and

1

δ=

√σμ0ω

2

⎡⎢⎢⎢⎢⎢⎣√

1+(ωε0

σ

)2

− ωε0σ

⎤⎥⎥⎥⎥⎥⎦1/2

. (2.52)

The damping of the wave will be done over a distance of length δ, which iscalled the skin depth. Since σ = nee2/meνe, we see that

ωε0

σ=ωνe

ω2pe

. (2.53)

At low pressures, it is often true that νeω� ω2pe, so that we can write

δ =

√2

σμ0ω. (2.54)

The existence of a skin depth is not dependent on collisions. Electrons canabsorb energy from the incoming wave and transmit the energy as electro-magnetic radiation. This is for example what happens in the ionosphere, whenradio waves “bounce”, if they have a frequency that is less than the electronplasma frequency in the ionosphere. This situation is described by the sameequations as above, but instead of expressing the current in the plasma byOhm’s law, which includes collisions, one just uses

J1 = −neeue1, (2.55)

and get ue1 from

ue1 =eE1

iω. (2.56)

This yieldsk2 = ω2/c2−ω2

pe/c2, (2.57)

which shows that if ω2pe > ω

2, k becomes purely imaginary, and the wave is

exponentially damped. If, on the other hand, ω2pe < ω

2, k is real and the wavecan propagate through the plasma.

In all derivations above, we have assumed that the plasma is cold, i.e. Te =

Ti = 0, and that there is no external magnetic field present, B0 = 0. Removingthese assumptions brings about a whole bunch of new types of waves, for ex-ample Alfvén and whistler waves. As we saw in chapter 2.5, charged particleswill gyrate around magnetic fields. If a wave in a plasma with B0 � 0 has the

23

same frequency as, for example, the electron cyclotron frequency, the wavewill resonate with the electron motion.

We also assumed that the plasma is completely homogeneous. Doing soensured us that the plasma frequency was constant in all spatial directions, al-lowing us to Fourier transform the equations. If we have a density gradient inthe plasma, the plasma frequency will change with position, since it dependson the plasma density. This forbids us from Fourier transform the equationsin the spatial direction. If the density gradient is only in one direction, forexample in the x direction, and the plasma still is uniform in the y and z dir-ections, so that the plasma frequency only depends on x, we are allowed toFourier transform in the y and z directions. The analysis of this situation callsfor a special technique, called WKB after Wenzel, Kramers, and Brillouin.This technique assumes that the solution can be described by a generalisationof the constant relation

E1 ∼ exp(ik ·x) (2.58)

as

E1(x, ky, kz) ∼1√κ(x)

exp

(i

∫ x

κ(x′)dx′), (2.59)

where

κ(x) =

√ω2

c2

⎛⎜⎜⎜⎜⎜⎝1− ω2pe(x)

ω2

⎞⎟⎟⎟⎟⎟⎠− k2y − k2

z , (2.60)

which act as an “effective” wave number in the x direction.The abrupt change from a plasma to vacuum, or some other medium, is an

example of an inhomogeneity. It occurs for example when we have a plasmacolumn in air. This situation gives rise to a new type of wave, called a plasmasurface wave [72].

24

3. Plasma Discharges

To create a plasma, one must supply energy to a gas, in order to ionise it. Thiscan be done in several ways. One can for example thermally heat a gas witha flame, compress the gas adiabatically, or supply energy by electromagneticfields. The most common way in plasma technologies is to create the plasmawith an electric or electromagnetic field.

3.1 Plasma Generation by an Electric Field

To create a plasma, one can take two metal plates, and give one a negativecharge and the other a positive charge. The negative plate is then called acathode, and the positive plate is called an anode. Between the plates, an elec-tric field will be established. If not placed in vacuum, there will be some gasatoms or molecules between the plates. Since there are always free electronsin the gas (from e.g. ionising cosmic rays), the electric field will acceleratethem [25]. When they have gained enough energy in the electric field, theywill start to ionise the gas. An avalanche of electrons will build up towards theanode. After some short time, a plasma discharge will be created. To sustainthe plasma discharge, one has to balance the power input through the electricfield to recombination of charged particles, and to the loss of charged particlesto the surroundings.

3.1.1 Breakdown

The Townsend criterion for breakdown can be derived by assuming thatthe current between the cathode and the anode increases without limit. Thesources of electrons in a discharge are mainly secondary emission from thecathode, where impinging ions knock out electrons, and volume ionisation,where electrons accelerated by the electric field ionise the background neutralgas. The criterion is given by [67]

γexp(αd) = γ+1. (3.1)

In this relation, γ is the secondary emission coefficient, which is the ratio ofthe number of emitted electrons at the cathode to the number of incident ions,α is Townsend’s first ionisation coefficient, which is the number of ionisingcollisions an electron makes on the average as it travels one meter along theelectric field, and d is the cathode-anode distance. Townsend’s first ionisation

25

coefficient is given by [67]

α = APexp(−APxi), (3.2)

where A is a constant which depends on the kinetic temperature and on the gastype, P is the pressure, and xi is the distance an electron must travel before itcan get enough energy from the electric field to be able to ionise a neutralatom in the gas. If the ionisation potential is Vi, the distance an electron musttravel before it can ionise is at least x = Vi/E, where E is the electric field.Since the electrons make elastic and excitation collisions also, the length isusually larger, so that one can introduce an effective ionisation potential Veff ,taking into account the other collisions as well,

xi ≈Veff

E. (3.3)

Thusα = APexp(−APVeff/E). (3.4)

If we take the breakdown voltage to be Vb, we have that

Eb =Vb

d. (3.5)

Taking the natural logarithm of (3.1), inserting the Townsend coefficient (3.4)and the breakdown voltage, one gets

Vb =AVeffPd

ln(APd/ ln[1+ (1/γ)]). (3.6)

The breakdown voltage is apparently a function of the product of the pressureand the distance between the anode and the cathode. This relation is known asPaschen’s law [67].

Lately, there has been some doubts expressed regarding the physics in-volved deriving this relation [77].

3.1.2 DC Discharge

The above description fits the DC (Direct Current) discharge in its most simpleform. Before we have a breakdown, the current through the discharge willincrease with little increase in voltage. This is called a Townsend discharge,see figure 3.1. Further increase in voltage will take us into another region, thecorona discharge, where the voltage drops and the current increases. After thecorona regime comes the breakdown, when the number of electrons that areable to ionise and excite the background gas is high enough to produce a glowvisible to our eyes [67]. This glow, called the normal glow, will be visible closeto the cathode, but will not cover the whole cathode surface. As the currentincreases, at more or less constant voltage, the glow will cover more and moreof the cathode [20]. Eventually, the voltage and current will start to rise, and

26

Figure 3.1: The different discharge regions for a DC discharge. After [67].

we will have an abnormal glow discharge. The cathode will now be heated bythe increasing number of impinging ions, attracted by the negative potential.Thermionic emission of electrons will be the dominating electron source, andthe voltage will drop [20]. The discharge is now called an arc.

DC Discharge Structure

The DC discharge is often shown as divided into distinct regions, as in figure3.2. In reality, the discharge does of course not have these distinct features,but the picture is good to illustrate the structure of the DC discharge.

Close to the cathode, a thin luminous layer is often observed. This cathodeglow is believed to be caused by atoms excited by electrons coming from thecathode or ions accelerated from the plasma towards the cathode [67]. Thedark space between the cathode and the glow is called the Aston dark space.Here the electrons are too slow to cause excitation or ionisation of the neutralgas. Moving towards the anode after the cathode glow, we see a dark region

AnodeCathode

− VC

Cathode dark space Positive columnAnode glowNegative glow

Aston dark space

Faraday dark space Anode dark spaceCathode glow

Figure 3.2: The structure of the DC discharge with dark and luminous regions.

27

called the cathode (or Crookes or Hittorf) dark space. The largest potentialdrop occurs within the region cathode–cathode dark space. This region is oftenreferred to as the cathode sheath or cathode fall. The major part of the cathodesheath has a net positive space charge due to the positive ions attracted by thenegative potential.

The cathode dark space is directly adjacent to the negative glow, which isoften the most light intensive part of the discharge. Electrons have here gainedenough energy to ionise and excite the neutral gas. After the negative glow,another dark space follows: the Faraday dark space. The electrons depositedalmost all their energies in the negative glow, and the Faraday dark space is asign of a lack of electrons with energies sufficiently large to cause ionisation orexcitation. The positive column following the Faraday dark space has a smallelectric field, just large enough to maintain ionisations. Close to the anode,there might be an anode glow followed by an anode dark space, which, asopposed to the cathode dark space, has a negative space charge.

The large potential drop over the cathode sheath region leads to heavy bom-bardment of the cathode by positive ions. The energetic ions will sputter thecathode surface, as well as cause emission of secondary electrons. Secondaryelectrons will be accelerated towards the anode. After some distance, the elec-trons have acquired enough energy to be able to excite or ionise the neutralgas. Reaching the required energy, the electrons will make a collision withina distance depending on the mean free path. The mean free path, in turn, de-pends on the electron energy and the neutral gas density (if we consider col-lisions with neutral gas atoms). It is thus reasonable to assume that the cath-ode sheath structures will depend on the pressure of the neutral gas, and notthe distance between the cathode and anode. Indeed, this is the case [4]. Thelength of the positive column, on the other hand, will depend on the distancebetween the electrodes, since the electrons therein have equilibrated with theelectric field.

Magnetron

If a magnetic field is introduced in the plasma, it will affect the motions ofelectrons and ions, as described in chapter 2.5. Since electrons will have asmall Larmor radius compared to the much more heavy ions in a plasma withan external magnetic field, the path length of the electrons will be much in-creased, if the magnetic field is perpendicular to the electric field. This willenhance the ionisation rate.

Hollow Cathode

Instead of having a planar cathode, one can form it into a hollow cylinder.The discharge will then develop inside the cathode. The negative glow willbe seen inside the cathode if the inner diameter is large enough to hold it.A big advantage with this configuration is that electrons accelerated in thecathode sheath region will meet a cathode sheath at the opposite side if ithas enough energy to traverse the negative glow. It will then be reflected andso oscillate back and forth between the sheaths until it has lost most of its

28

energy through collisions and become thermalized. So instead of going intothe positive column and then be lost to the anode, it will have time to giveup most of its energy in the negative glow, since the path length has beenincreased.

The electric field in the middle of the hollow cathode is directed outwardsalong the axis of the cylinder to the anode placed somewhere outside the hol-low cathode. Thermalized electrons are conducted by the electric field to theanode [31].

Since the plasma is “trapped” inside the hollow cathode, ions, photons, andmetastables will hit the cathode surface with greater probability than with aplanar “open” cathode [3, 10, 9]. This will lead to increased secondary elec-tron emission.

DC hollow cathodes have been used for beam creation [68] and etching[63].

3.1.3 Pulsed DC Discharge

Instead of transferring power to the discharge in a continuous manner, one cando it in pulses with different lengths and duty cycles. The advantages are thatone can deliver higher power and use the properties of the active plasma betterthan in continuous DC. A non-equilibrium continuous DC plasma is sustainedby a balance between power input and losses in collisional processes. Sincethe temperature of the electrons is often in the range 1–3 eV in such plasmas[14, 69], the main part of the electrons are unable to cause ionisations. Bypulsing, one can use much higher power, but for short periods, which willgive the electrons considerably higher energies. In this way the average energyneeded to create one ion-electron pair is reduced [58].

3.2 Plasma Generation by Electromagnetic Fields

The DC discharges can be rather inefficient in transferring energy to the elec-trons in the plasma. A rapidly varying electric field will target the electronsmore efficiently, since they can follow the variations of the electric field. Theions, being much more heavy than the electrons, will not be able to respondto the varying electric field, but will follow a time-average field.

3.2.1 Radio Frequency Generated Plasmas

Radio frequency (RF) discharges operate at frequencies of about 1–100 MHz.This means that the electromagnetic waves generated have a wavelength of300–3 m, generally much larger than the dimensions of the discharge plas-mas. It is therefore possible to look at the RF discharge as electrostatic whenmodelling it.

Since the power is coupled more efficiently to the electrons with RF, thesedischarges often have more efficient ionisation.

29

The structure of the RF discharge is similar to the DC discharge. A sheathwill develop at the cathode, but since the potential is varying, the electrodeswill by turns be the cathode, and thus a sheath will expand and collapse at bothelectrodes. This type of discharge is called a capacitively coupled discharge.

Since the voltage and current are oscillating, the net current to the electrodesshould be equal. If the powered electrode has a smaller area than the groundedelectrode, the potential drop at the powered electrode must be larger than atthe grounded electrode in order to collect the same current [25]. This negativepotential is called the “self-bias” voltage.

Another type of RF plasma is the inductively coupled plasma. An externalcoil, for example wrapped around a dielectric vacuum tube, carries an RF cur-rent, which creates a varying magnetic field, which in turn induces an electricfield that accelerates the electrons.

In a DC discharge, a current must flow through the plasma in order to sus-tain it. The current in an RF discharge is mostly a displacement current, com-ing from the oscillatory motion of the electrons as they follow the electricfield. This makes it possible to have the electrodes separated from the plasmaby a dielectric material. These so-called dielectric barrier discharges (DBD)are used for example in plasma display panels [16] and in applications whereone wants to protect the cathode surface from bombarding ions. DBD’s arealso popular as exhaust gas treatment discharges [47].

Hollow cathodes are often operated at RF, and there exist many varieties ofsuch sources, both at low and high pressures. Hollow cathodes can e.g. be usedfor surface treatment or deposition at low pressures [10]. Special arrangementsof hollow cathodes allow for surface treatment of large areas [6, 11, 59]

3.2.2 Microwave generated plasmas

As we saw in chapter 2.6.2 a wave incident on a plasma can propagate throughthe plasma, be reflected, or absorbed. Microwaves absorbed in a plasma areused to heat the electrons. The absorption occurs within the skin depth, as wehave seen earlier. There is no exact frequency that corresponds to microwaves,but a range of frequencies. One common frequency used is 2.45 GHz, whichcorresponds to a wave length in vacuum of 12.24 cm. An advantage of mi-crowave sources is that no electrodes are needed, so that contamination fromthe electrode material can be avoided. Microwave generated plasmas also of-ten have a higher ionisation fraction than DC and low frequency RF discharges[67].

If a static magnetic field is introduced in the plasma, the microwaves canresonate with the gyromotion of the electrons. These type of discharges arecalled electron cyclotron resonant (ECR) microwave plasma sources.

30

3.3 Hybrid Sources

A combination of different types of sources is also plausible. The type ofsources one combines depends on what properties of the generated plasma oneis looking for. One example of a hybrid source is the combination of a hollowcathode generated plasma and a microwave generated plasma [8] describedlater in this thesis.

31

4. The Plasma Sheath

If an object is immersed in a plasma, the electrons and ions in the plasma willcollide with the object. Since the electrons often have much higher velocitiesthan the (positive) ions, they will arrive in greater number than the ions tothe surface of the object. If the electrons are not drained off they will givethe surface a negative charge. The negative charge that is built up will repelelectrons and attract ions, and eventually a balance is reached when the currentto the object is zero. It is said that the object is at a floating potential.

If the object is charged negative by e.g. an external circuit, the electroncurrent will decrease with the potential of the object, until only the ion currentwill arrive at the surface.

This means, that close to a charged surface, the plasma does not containequal numbers of electrons and ions. That region, with a surplus of positive ornegative charges, is called a space charge sheath, or simply just a sheath.

The term sheath to describe the space-charge layer in front of an electrodeimmersed in an ionised gas was first used by Langmuir in 1923 [53].

4.1 Debye Shielding

One of the characteristic properties of a plasma is that it screens or shieldscharges. If a negatively charged object, a test charge, is placed in the plasma,the electrons in the plasma surrounding the object will be repelled by thecharge, leaving the ions behind. The ions in the plasma are usually, due totheir large masses, considered as stationary, forming a homogeneous positivebackground. In a sphere around the negative object there will be more pos-itive ions than electrons, and the surplus of positive charges in this spherewill screen the negative charge. For an observer outside this sphere, it appearsneutral.

If we assume that we have no magnetic field, and that the plasma is invacuum, we have from equation (2.19)

∇ ·E = ρε0=

e(ni−ne)

ε0, (4.1)

where ni and ne is the ion and electron density, respectively. If the electric fieldis static, we can write

E = −∇φ, (4.2)

33

so that we can use the electrostatic potential φ instead of E. We further assumethat the electrons have a Maxwellian energy distribution. The electron densitywill then behave as [76]

ne = n0 exp

(eφ

kBTe

)(4.3)

with n0 being the unperturbed plasma density. Since the ions are assumed tobe a stationary background, we have

ni = n0. (4.4)

Inserting this into (4.1) along with (4.2), we get

∇2φ = −en0[1− exp(eφ/kBTe)]

ε0. (4.5)

The perturbation introduced by the charge, eφ, may be considered much smal-ler than kBTe, eφ � kBTe. Thus, we can expand the exponential term in aTaylor series, which gives us

∇2φ ≈ n0e2

ε0kBTeφ. (4.6)

The plasma may be considered isotropic, so the potential is spherically sym-metric. Taking the origo of our coordinate system to be placed at the negativetest charge, we arrive at

∂2

∂r2(rφ) =

n0e2

ε0kBTe(rφ), (4.7)

which have the solution

φ =φ0

rexp(−r/λD), (4.8)

with λD = (ε0kBTe)/(n0e2), the Debye length.The concept of Debye shielding is perhaps not as easy as one might think

from the description above. For example, if the test charge is positive, elec-trons will be attracted and accelerated towards it, and by particle flux con-servation, their density must decrease towards the test charge. We will endup with a net positively charged plasma close to the test charge, so that theelectrons actually anti-shield it. This has, however, been shown to be a toosimple description [43]. Other effects will remove the anti-shielding so thatthe electrons will shield the positive charge.

4.2 The Cathode Sheath

Since the cathode has a negative potential compared to the plasma, it willrepel electrons and attract the positive ions. The ions will screen the negative

34

potential of the cathode as described above. The description above is, however,simplified. The ions will eventually start to move towards the test charge andin reality, the test charge would be neutralised if the ions are not drained awaywhen they collide with it. A cathode has a surplus of negative charges, so theions will pick up an electron when they come to the cathode surface, but theycan not neutralise the cathode.

If we assume that the cathode is planar and infinite, by symmetry, we needonly look at one direction, the one normal to the cathode surface. The iontemperature is set to zero for simplicity, and we have no collisions or magneticfields. If we look at the stationary state, we get from (2.11), for the ions,

miniuidui

dx= qniE. (4.9)

This is solved to get1

2miu

2i (x) =

1

2miu

2s −

q

miφ. (4.10)

us is the velocity of the ions as they enter the sheath, which is located at x= xs,where we have defined φ=φs = 0. We can call this the plasma-sheath interface.We also have that everywhere in the sheath, the ion flux must be conserved,because we have no creation or annihilation of ions there,

ni(x)ui(x) = ni,sus, (4.11)

where ni,s is the ion density at the plasma-sheath interface. Combining (4.10)with (4.11) and using the Boltzmann relation for the electron density in thesheath (4.3), we get from (4.1) and (4.2)

d2φ

dx2=

ens

ε0

⎡⎢⎢⎢⎢⎢⎣exp

(eφ

kBTe

)−

(1− 2eφ

miu2s

)−1/2⎤⎥⎥⎥⎥⎥⎦ . (4.12)

For this equation to have a solution, we have that [24]

us >

(kBTe

mi

)1/2

, (4.13)

which is called the Bohm sheath criterion [17]. This means that the ion ve-locity as they enter the sheath must be larger than the Bohm velocity uB =

(kBTe/mi)1/2. If the velocity would be smaller than the Bohm velocity, the ion

density would decrease too fast in the sheath and become smaller than theelectron density at some point, which would mean that the electrons are notrepelled from the cathode at all points in the sheath [65].

If the cathode voltage is very negative, the electrons will not penetrate farinto the sheath and we can say that, approximately, the electron density drops

35

ne≈ n

p n

e< n

p

ne≈ n

p

ne=n

sexp(eφ/kT

e)

ns

ne= n

p n

e= 0

ne= n

p n

e= 0

np= n

s

SS

(a) (b)

(c) (d)

Figure 4.1: Simplified sheath models. (a) Continuous plasma-wall model, (b) step-

front electron sheath model, (c) sheath model with diffuse electron boundary, and (d)

matrix sheath model.

from ne = ns to ne = 0 in the sheath. Then (4.12) reduces to

ε0d2φ

dx2= −ens

(1− 2eφ

miu2s

)−1/2

. (4.14)

This is called the step-front electron sheath model. The simplest sheath modelis the matrix sheath model, in which we assume a homogeneous ion distribu-tion ni = n1 and ne = 0. The above equations then reduce to

ε0d2φ

dx2= −ens. (4.15)

The different sheath models are summarised graphically in figure 4.1. Sinceusually Ti � Te, there must be some electric field penetrating into the plasma,causing the acceleration of the ions to the Bohm velocity uB, which is pro-portional to the electron temperature. This region of the plasma is often calledthe presheath. While the sheath width is of order λD, the presheath width is oforder L where L � λD (λD/L = ε → 0), since it would otherwise be includedin the sheath analysis, such that the Bohm criterion must be fulfilled at theentrance of the presheath. The Bohm criterion originates from ion continu-ity and energy conservation, which means that one of these must be brokenin the presheath. Therefore L is usually determined by the ionisation length(L = uB/ν

I , νI being the ionisation frequency) or by the geometry of the sys-tem containing the plasma.

If one looks at the plasma equations, where ne = ni, one can see that theion velocity and the potential is singular and goes to infinity at the Bohmvelocity [33, 65]. The sheath equations, on the other hand, assumes that the

36

potential goes to zero in the plasma. Clearly, the solutions must be matchedat some intermediate layer [65, 35], and one needs to consider three scales inorder to match the solutions. This is, however not accepted by all researchers.There has been suggestions to modify the Bohm criterion to take into accountcollisions, in order to be able to match the plasma and the sheath solutions[38], or by joining the plasma and sheath solutions at a specific value of thepotential [39]. This has been refuted by others [33, 32, 65].

To define a sheath edge is not easy, since one cannot identify a well-definedboundary between the plasma and the sheath. The transition is gradual. Therehave, however, been attempts at defining a sheath edge [34].

4.3 Radio Frequency Sheaths

The analysis of the cathode sheaths is more complicated if we allow for the po-tential at the cathode to vary with time. If the time variation is slow, the sheathcould still be considered a DC sheath. But if the time variation is sufficientlyfast, the ions in the plasma will not be able to respond to the instantaneouspotential, but only to a time-average potential. The sheath can no longer beconsidered a DC sheath. The lighter electrons can still be able to follow thepotential, which is often the case at the RF 13.56 MHz.

Previously published models [56, 22] of RF sheaths in front of an electrodehave involved simplifications in order to get an analytically solvable solution.One popular simplification is to consider the sheath to be a step-front sheath(see figure 4.2). This assumes that the electron density drops fast to zero inthe sheath. Over an RF period, as the sheath edge expands and collapses, theaverage electron density in the sheath looks like the dashed curve in 4.2. Thissimplification assumes that the electron temperature is much lower than thesheath potential. If this assumption is not true, the model gets too complicatedto solve analytically and we have to solve the model numerically. This is thecase, for example in a hollow cathode, where a group of hot electrons, withtemperatures close to the maximum sheath potential, are present.

37

n

n0

ni

ne

ne

sm0 x

PLASMAELECTRODE

s(t)

Figure 4.2: The step-front electron density model. The instantaneous electron density

ne drops fast to zero in the sheath. The ion density ni follows the time average potential

φ. ne is the time average electron density.

38

5. Plasma Diagnostics

There exists a wide variety of plasma diagnostic tools. The most basic ones in-clude measurements of voltage and current through the plasma, photography,temperature measurements, etc. Other, more sophisticated tools are, for ex-ample, probes, optical and mass spectroscopy, and microwave techniques. Itis out of the scope of this thesis to give a thorough introduction to all these.Instead a few, often used diagnostic tools will be presented, however not ingreat detail.

5.1 Electric Probe

One of the first techniques used for plasma diagnostics was the electrostaticprobe. It was first used by Langmuir in 1924. The probe can be just a simplewire connected to a power supply, that can give the wire a negative or pos-itive charge. An ammeter connected to the probe measures the current goingthrough it. As the probe is inserted in the plasma, it will, depending on itsvoltage, attract or repel the electrons in the plasma. Figure 5.1 shows a typicalcurrent-voltage curve for an electrostatic probe inserted in a plasma. At φ= φs,called the space potential, the probe has the same potential as the plasma. Dueto the high mobility of the electrons compared to the ions, the current is carriedmostly by electrons. The current at this point is called the electron saturationcurrent, since if the current collector area is more or less constant, the currentcan not increase beyond this point.

φ

I

φf

φs

Figure 5.1: Schematic illustration of a typical current-voltage curve for an electric

probe in a plasma.

39

If the voltage is lowered, electrons start to be repelled and ions can start tocontribute to the current. At φ = φf , the electron and ion currents to the probeare equal, so that the net current is zero. This potential is called the floatingpotential, since it is the potential an object “floating” in the plasma would get.

As the voltage is decreased further, more and more electrons are repelleduntil we arrive at the ion saturation current region, similar to what we hadbeyond φs.

As we saw in chapter 4, an object in a plasma will be surrounded by aspace-charge sheath. This is of course also true for the electrostatic probe. Thecurrent to the probe is dependent on the surface that collects the charges fromthe plasma, at least when the distance from the surface to the probe is small.This surface is the sheath surface, which depends on the sheath thickness,which in turn depends on the plasma density at the plasma-sheath interface.So to measure the plasma density, one must know the sheath thickness. If thedistance between the sheath surface and the probe is large, i.e. when the sheaththickness is large, not all particles entering the sheath will hit the probe. In thiscase, one must also take into account orbit theory [23].

From the slope between the plateaus of the current-voltage curve in figure5.1, one can determine the electron temperature. If we assume that the elec-trons have a Maxwellian distribution, and that the sheath is collisionless, theirdensity in the sheath will be given by equation (4.3),

ne = n0 exp

(eφ

kBTe

).

Since the current I to the probe is [23]

I = A je = Ane

(kBTe

2πme

)1/2

, (5.1)

where A is the sheath area, we have that

I = An0

(kBTe

2πme

)1/2

exp

(eφ

kBTe

). (5.2)

A plot of ln I against φ should give a straight line, which slope is proportionalto Te, if the electrons are Maxwellian.

The theory of probes becomes much more complicated if one takes intoaccount collisions and effects of magnetic fields. In chapter 2.5 we saw thatcharged particles will gyrate around a magnetic field line. This makes themconstrained to move along the field lines, which of course affects their abilityto get to a probe surface. Since the Larmor radius for electrons is much smallerthan that for ions, electrons are more affected than ions by a magnetic field.

40

5.2 Mass Spectrometry

Since the charged particles in a plasma are affected by electric and magneticfields, as we saw in chapter 2.5, one can analyse the particles by letting thempass through electric and magnetic fields. According to (2.25), the deflectionof the particles will depend on the mass and charge of the particles.

In a magnetic field analyser, with a magnetic field perpendicular to the in-coming ions, the equation of motion for the ions will be

mi

v2i

r= qiviB, (5.3)

where r is the curvature radius of the ion motion. The beam of ions let into theanalyser must be collimated, and then let through a region of known lengthl with a magnetic field perpendicular to the ion velocity [60]. The ions willthen be deflected through an angle θ. The deflected ions will be detected byan array of detectors at different angles.

In electrostatic analysers, one uses the force qE to deflect particles for en-ergy analysis.

To extract charged particles from the plasma, a potential, different from theplasma potential, is put on an electrode with an orifice, through which theparticles can enter the device. If ions are collected, they will then enter an ionoptic, in which they will be focused into a beam. The focused beam then entersthe energy analyser which separates ions with different energies with the helpof electric or magnetic fields. The resulting ion beam then enters a quadrupole,which can separate out ions with a preferred mass by varying electric fields.The ions that make it through the quadrupole are then collected by a detector,for example a secondary electron multiplier.

By changing the potential of the electrode at the entrance of the spectro-meter, ions with different energies can be collected and their distribution func-tion can be measured [29].

5.3 Optical Emission Spectroscopy

In plasma spectroscopy, one is generally not interested in examining the in-terior structure of the atoms, as in conventional spectroscopy. The atom or ionin a plasma can not be considered as isolated from its surroundings. It will beaffected by the electric fields in the plasma, caused by the other particles. Thus,by doing plasma spectroscopy, one can get an insight into plasma processesand plasma parameters. Plasma spectroscopy is also a non-invasive method,since no probes etc., which disturb the plasma, are needed.

The wavelengths often used for plasma spectroscopy are between 200 nmand 1 μm. Below 200 nm, quartz glass, which is often used for windows inplasma chambers, is opaque, and the oxygen molecules in the air starts toabsorb light. Above 1 μm thermal noise starts to disturb the measurements.

41

5.3.1 Spectrographs

Basically, a spectrograph has an entrance slit, a dispersive element (grating),mirrors for reflecting the light, and an exit slit. At the exit some kind of de-tector is placed, for example a photomultiplier or a CCD camera. The lightfrom the light source can be imaged onto the entrance slit by imaging optics,or optical fibres can transport the light. Using optical fibres is very convenient,since one can easily get close to the plasma with the end of the fibre.

The resolution of the spectrograph depends highly on the grating used.Gratings are characterised by the number of lines per millimetre. Also thefocal length and the width of the entrance slit has an effect on the resolution.Narrower entrance slit gives higher resolution, but decreases the light intens-ity and vice versa. The final result will of course also depend on the detectorquality.

5.3.2 Diagnostics

One of the first things that come to mind when one thinks of plasma spectro-scopy as a diagnostic tool is the detection of species in a plasma. Since eachatom and molecule has its spectral fingerprint, one can determine which ele-ments are present in the plasma, provided they emit light, by comparing thespectral lines with tables in the literature.

Particles that move and emit light will cause a Doppler shift of the emittedlight. If the particles emitting the light have a thermal motion, with temperat-ure T , there will be a broadening of the spectral lines proportional to T [19].Thus, it is possible to determine the temperature of the emitting particles bymeasuring the broadening of the spectral lines. This requires spectrographswith high resolution.

Since in low temperature plasmas electrons are the main group of particlesthat cause ionisations and excitations, the population of the higher electronicstates in the atoms and molecules will depend on the electron temperature. Itis therefore possible to determine the electron temperature by looking at theintensity of the spectral lines. Other mechanisms are, however, also respons-ible for population of higher electronic states in atoms and molecules [30], soone has to be careful.

Vibrational and Rotational Temperatures

Diatomic molecules have two additional modes of motion compared to singleatoms: rotation and vibration. The molecule can rotate about an axis passingthrough the centre of mass, perpendicular to the line joining the two atoms(internuclear axis), and the atoms can vibrate relative to eachother along theinternuclear axis. These two modes cause structures in the spectra of the di-atomic molecules not seen in the spectra of single atoms. So-called bands ofspectral lines appear when electrons make transitions between different elec-tronic, vibrational, and rotational states.

42

It can be shown [44] that the sum of the emission intensities Iem of a vibra-tional band system with the same upper state is proportional to the number ofmolecules, Nv′ , in the upper state v′. With the lower state denoted as v′′, wethus have ∑

v′′

Iv′v′′em

ν4∝ Nv′ , (5.4)

with ν being the frequency of the emitted line.If we have thermal equilibrium in the gas emitting the band systems, we

have that the number of molecules in the state with energy Ev have a Maxwell-Boltzmann distribution

NEv =N0

Ze−Ev/kBTvib , (5.5)

where N0 is the total number of molecules in the upper vibrational level, Z isthe partition function, or state sum, kB is the Boltzmann constant and Tvib isthe vibrational temperature. Combining equations (5.4) and (5.5), we get

ln∑v′′

Iv′v′′em

ν4= C− Ev′

kBTvib, (5.6)

where C is a constant. Thus, by plotting the logarithm of the sum of the bandstrengths as a function of the energy Ev′ , we will get a straight line which slopeis 1/kBTvib, from which we can determine the vibrational temperature. If onecannot determine the intensities of the lines that should be included in the sumin (5.6), one can still estimate the vibrational temperature by calculating theFranck-Condon factors for each band and divide the band intensity by these.

In practice, one must also account for the sensitivity of the measuring ap-paratus etc. [62].

5.3.3 Spectroscopic Notation

The electronic configuration of molecules have their spectroscopic notation,which is commonly written as follows:

2S+1Λ+,−Ω g,u. (5.7)

S is the total spin quantum number (the sum of spins of all electrons in the mo-lecule), Λ is the projection of the total electronic orbital angular momentumon the internuclear axis, and Ω is the total angular momentum along the in-ternuclear axis. The letters g and u stand for “gerade” and “ungerade” andindicate if the electronic wave function for homonuclear diatomic moleculesis symmetric or asymmetric upon reflection in the internuclear midpoint plane,and + and - if the electronic wave function for a diatomic molecule is sym-metric or asymmetric in the plane of the internuclear axis [19].

43

6. Plasma Modelling

There exists several popular plasma modelling techniques. The plasma mod-elling has been revolutionised by the arrival of the personal computer. In thebeginning of the 1990’s, computers were still rather slow, but simple modelscould be solved in a few days. Today, such models are solved on an ordinarydesktop computer, or even laptop, in a few minutes. It could be tempting touse today’s super computers to solve complex plasma models. However, onecan contribute a lot to the understanding of discharge plasmas with a simplemodel and an ordinary computer.

6.1 Analytical Modelling

Contrary to what we did in chapter 2, where we went from the most complexdescription of a plasma to the more simple, we will here start with the mostsimple modelling technique: analytic modelling. One should perhaps be care-ful with the word “simple”. Analytic modelling is by no means simple in themeaning it is easy. Here “simple” refers to the more simple description of theplasma. Irrespective of using the kinetic, fluid, or particle description of theplasma, as presented in chapter 2, simplifying assumptions must be used inorder to get equations that are analytically solvable

Analytic approaches have been used extensively to model the sheath regionsin plasmas, both for DC [65, 51, 66, 48] and RF [56, 57, 64], but also formodelling of electromagnetic effects in capacitive discharges [21].

The big advantage with analytical models is that they, once you have got anexpression, are extremely fast. The drawback is that they must be obtained bymaking simplifying assumptions. Still, they can give a qualitative descriptionof the plasma, useful for quick estimates.

6.2 Fluid Modelling

Fluid modelling is perhaps the most popular plasma modelling technique.Fluid models are widely used by many authors and exist in a number of dif-ferent shapes.

45

6.2.1 Low Pressure Modelling

At very low pressures, collisions between particles are infrequent. If one wantsto do a time dependent model, one begins with equation (2.9), the continuityequation, which describes the number of particles in a volume element withsides (dx, dy, dz). This number is dependent on the creation or annihilationof particles inside the volume over time, described by the source term on theright hand side, and the flux of particles in and out of the volume, describedby the second term on the left hand side. If one does not have any particlesources or sinks, and a net flux of particles in and out of the volume equal tozero, then

∂n

∂t= 0 (6.1)

so that∇ · (nu) = 0. (6.2)

Second, one has equation (2.11), the momentum equation. In many dis-charge models, it is assumed that there is no magnetic field present (except,of course, in discharges such as magnetrons, where the magnetic field plays acrucial role). The force on charged particles is then

F = qnE. (6.3)

As noted in chapter 2, one needs an expression for the pressure P. It is oftenassumed that the pressure is isotropic and that the plasma is isothermal, so that

P = P = nkBT, (6.4)

where kB is the Boltzmann constant and T is the particle temperature. Thus,the particles are assumed to behave as an ideal gas, so that they are Maxwelldistributed. In some models, the particles are assumed to be cold, so that T = 0.

The last term on the right hand side describes the change in momentum dueto creation or loss of particles. It is generally small, and can be neglected [55].The next-to-last term describes the change in momentum due to collisions.The collisional term can be determined by taking into account every type ofcollision a particle makes. However, one often express it using a collisionfrequency νm that takes into account all momentum-changing collisions, andwrite the collision term as (

∂

∂t(mnu)

)c

= −mnνmu. (6.5)

The energy equation (2.14) can then be used to solve for the particle energydensity. The collisional term on the right hand side includes all collisions thatchange the energy density, e.g. ionisation, excitation, elastic collisions, andfrictional heating. One often makes the assumptions leading to (2.18) and usesthat equation.

46

The equations used above are not self-consistent, since we need the electricfield. The electric field, which is modified by space charges in the plasma, isdescribed by equation (2.19).

Some of the assumptions that are commonly used in low-pressure fluidmodelling, and their influence on the solution, are presented in [27].

6.2.2 High Pressure Modelling

Going to higher pressures means that electrons and ions will collide with neut-rals more frequently. It is of course possible to use the same equations asdescribed above, but one can, due to the frequent collisions, use some simpli-fications that make the equations less complex to solve.

The first equation (2.9) can not be simplified more. Some work can, how-ever, be done on equation (2.11). Looking at the left hand side of that equa-tion, we see a term that describes the change of the mean velocity u over timeat a specific point in space, and a term that describes the change in u overdistance. The term ∇u tells how fast u changes over distance, and when multi-plied with u describes how fast the fluid changes its velocity due to the spatialvariation of the velocity. At high pressures collisions are so frequent, that u

does not change much over the time between collisions. Thus, ∂u/∂t is verysmall over this time, and can be neglected. Similarly, the distance the particlestravel between collisions is so small, that u does not change much over thisdistance, and u · ∇u can also be neglected. What is left is

q

mE− kBT

mn∇n− νmu = 0, (6.6)

where (6.5) and the discussion above have been used. Solving for the meanvelocity u gives

u =q

mνmE− kBT

mνmn∇n. (6.7)

The coefficients in front of the electric field E and the density gradient overdensity (∇n)/n are called the mobility and diffusion coefficients, respectively,

μ =|q|

mνm(6.8)

and

D =kBT

mνm. (6.9)

We can now writeΓ ≡ nu = ±μnE−D∇n, (6.10)

47

where Γ is the flux of the species, and ± stands for positive and negativecharges, respectively. We can now also write equation (2.9) as

∂n

∂t+∇ ·Γ =

(∂n

∂t

)c

. (6.11)

This is commonly called the drift-diffusion approximation. This coupled withthe energy equation (2.18) and (2.19) gives a self-consistent model.

6.2.3 Rate and Transport Coefficients

In order to solve the above equations, one needs expressions or values for thesource terms and the transport coefficients μ and D. The source term in thecontinuity equation (2.9) must be expressed in (m−3s−1) for the dimensionto match the other terms. These so-called rate coefficients and the transportcoefficients can be calculated from the particle distribution function. A simpleapproach is to assume a Maxwellian distribution. More accurate coefficientsmay be calculated by solving the Boltzmann equation (2.5) with all wantedcollisional processes present [42]. The coefficients are then given as functionsof the reduced electric field E/N, where N is the neutral gas density, or asa function of the particle mean energy. In some models, the diffusion andmobility coefficients are also assumed to be constant [40, 61].

What does it mean to give the coefficients as functions of the reduced field?It involves the assumption, that the particles are in equilibrium with the localelectric field, so that the energy given to the particles is balanced by the en-ergy lost in collisions [49]. This means, that, for example, the ionisation ratecoefficient is zero at a point, if the electric field is zero there.

Since the coefficients can be given as functions of the reduced electric field,the energy equation (2.18) is not needed, since one does not use the particlemean energy for anything (unless, of course, one wants the energy anyway). Itis even possible to express the drift velocity u of the particles as a function ofthe local electric field [36]. This makes the momentum equation superfluous.

If the rate and transport coefficients are given as functions of the mean en-ergy, one must solve the energy equation to get the mean energy. This ap-proach gives some nonlocal effects due to the transport of energy from onepoint to another [46].

The local field approximation is suitable for high pressures, since the energyrelaxation time is low [36]. However, in places with very strong electric fields,it might break down, and in that case, the drift-diffusion approximation mightalso be incorrect [28].

6.2.4 Solving the Models

There exist several methods to solve the set of fluid equations. One simplemethod is the Runge-Kutta method, and variations of this. This method is anexample of a finite difference method (FD), where the computational domain

48

is split into a structured grid and the derivatives are approximated by small dif-ferences. Due to large gradients often present in a plasma, the FD method mustbe stabilised. A popular stabilisation technique is the flux-corrected transport(FCT). FD methods are rather computationally heavy, and are basically lim-ited to one or two dimensions with no complex geometries [36].

A more efficient method to approximate partial differential equations (PDE)is the finite element (FE) method. The FE method approximates the solutionby a linear combination of orthogonal basis functions. The computational do-main is discretised into grid elements, in two dimensions most often eitherrectangular or triangular (see figure 6.1). This makes it possible to discretisecomplex geometries. The equations are solved at each node. Since one oftenhas large gradients at some places in a plasma (e.g. in the sheath), and smoothbehaviour in other places, it is most efficient if one uses an adaptive grid,so that there are more nodes where there are large gradients, and less nodeswhere the behaviour is smooth. This is shown in figure 6.1, where the meshis denser close to the boundary to the left. Various stabilisation techniques arealso available for the FE method [45].

6.3 Kinetic Modelling

To get the rate and transport coefficients above, the Boltzmann equation issolved for a few values of the local reduced electric field (E/N). Why notsolve it directly for the conditions in the plasma, instead of going throughthe fluid equations? The question has been answered earlier: it is harder tosolve. Nevertheless, it is possible, and has been done [31, 70]. The advantagewith solving the Boltzmann equation is that one will get accurate distributionfunctions and are able to treat nonlocal effects [50].

Another kinetic approach is to treat the individual particles in a plasma.This is done in the so-called particle-in-cell (PIC) method.

Figure 6.1: Typical grid structure in the FE method.

49

6.3.1 PIC Modelling

The particle-in-cell (PIC) method provides a self-consistent kinetic descrip-tion of a plasma. The basic thought behind the PIC model is rather simple:use Newton’s second law (2.1) along with the Lorentz force (2.12) to describethe path of the particles. The source terms for the field equations are calculatedfrom the particle locations. Particles are defined in a continuum space, whilethe fields are defined at discrete points in space. The particles and fields arethen advanced in time in discrete steps Δt. If the positions and velocities arecalculated with a difference of a half time step (Δt/2), it is possible to use theso-called leapfrog scheme in finite difference form (here in one dimension),

ut+Δt/2−ut−Δt/2

Δt=

q

m

(Et +

ut+Δt/2+ut−Δt/2

2B

), (6.12)

xt+Δt− xt

Δt= ut+Δt/2. (6.13)

Similar equations are constructed for the fields [74]. The sources for the fieldsare then interpolated to the discrete points (mesh). These are then used to cal-culate new positions for the particles according to the Newton-Lorentz equa-tion, and then one starts over.

The PIC method does not treat every single particle in a plasma. Instead so-called superparticles are used, which represent a high number of real plasmaparticles.

Is is common to include a Monte Carlo description of particle collisions,Monte Carlo collisions (MCC). By knowing the cross section for a specifictype of collision, one can use the Monte Carlo method to decide if a collisionwill occur or not. This is checked each time step. A PIC model with MCC is,surprisingly, called a PIC-MCC model.

The PIC models give a very realistic kinetic description of a plasma, sincethey trace individual particles. At the same time, they demand much computerpower. The time steps must be sufficiently small in order not to “jump over”collision events. In a high pressure plasma, collisions are frequent, so thatvery small time steps must be taken, leading to lengthy computations. ThePIC scheme also under-estimates the Coulomb interaction between particlesat close range, within a mesh cell [74]. This means that the cell width shouldbe less than a Debye length in order to treat these interactions reasonablywell. Another problem in the PIC model is the unphysical numerical heatingof particles appearing in some cases due to the discretisation of the equationsinvolved [73].

6.4 Hybrid Models

The fluid models are, as we noted earlier, unable to treat kinetic effects in theplasma. They are, however, efficient in describing other parts of the plasma,

50

and do it quite accurately. The kinetic models describe the kinetic effects in aplasma well, but are very time-consuming to solve. An alternative is to com-bine the two into a so-called hybrid model. Use the fluid model for particlesthat the fluid model is known to be able to handle, and use the PIC-MCCmodel to model kinetic effects. Fast particles can be tracked with the PIC-MCC model parallel to the solution of the fluid model. One then gets benefitsfrom both models.

Hybrid models are very popular, since they are in some extent able to extendthe fluid models to the kinetic regime. Plasmas with kinetic particles, such ashollow cathodes [5, 52, 26], and beam plasmas [75] are popular to model withthe hybrid approach.

51

7. Modelling Hollow Cathodes

In a previous chapter, we saw that the sheath in a hollow cathode might notbe described by the step-front sheath model due to the presence of a groupof high energy electrons, which are able to penetrate deep into the cathodesheath. These electrons originate at the cathode surface, where they are cre-ated by secondary processes, such as ion, photon, and metastable impact, etc.They are subsequently accelerated by the cathode sheath potential into theplasma. Since the structure of a hollow cathode plasma is sheath–negative

glow–sheath, accelerated electrons, which are able to traverse the plasma, willmeet the opposite sheath, where they will be reflected back into the plasma.So they oscillate until they have lost most of their energy in collisions in theplasma.

The effect of different electron temperatures on the electron density in frontof a charged surface is showed in figure 7.1, where the exponential functionthat governs the electron density is plotted for different exponents.

We will now add these fast electrons, and in addition secondary electronscreated at the cathode surface, in the analysis of the hollow cathode sheath.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1The function exp(−α) for different α

α=1α=5

α=10

Figure 7.1: The function exp(−α) for different α = eφ/kBTe. As α→∞, we see that

the function looks more and more like the step-front model in figure 4.2. But when

α ≈ 1, we cannot approximate the function with the step-front model. Note that in this

figure the surface is to the right and that the plasma is to the left, with the plasma-

sheath boundary at x = 0.

53

7.1 Modelling of the RF Hollow Cathode Sheath

A fundamental property of the hollow cathode is that it must be at least two-dimensional, since two cathode surfaces must oppose each other. It is notenough to have two cathodes; one must include some point with a relative po-tential to the cathodes. This calls for two dimensions. We can, however, solvefor the cathode sheath in one dimension, assuming that the other cathode isremote, and that the potential falls to zero in the middle of the two cathodes.This point is then our “ground”, or reference potential. A drawback of thissystem is that we need to provide boundary conditions at the plasma-sheathboundary, for example for the ion velocity and the electric potential. It will,however, give us a qualitative look at how the fast electrons affect the sheath.

To begin with, the most basic equation, which governs the shape of thepotential in the sheath, is the Poisson equation (here in one dimension),

d2φ

dx2= − e

ε0(ni−ne), (7.1)

where φ is the potential, e is the elementary charge and ni and ne are theion and electron densities, respectively. In the RF case, the potential can bemodelled as φ= λsin(ωt), where λ is a constant andω is the angular frequency.

We now have to find expressions for the ion and electron density in thesheath. Assuming that there are no ionisations or collisions in the sheath andthat the ions are cold (Ti ≈ 0), we get the ion continuity equation, equation(2.11),

1

2mi(u

2i −u2

s ) = −eφ, (7.2)

where us is the directed ion velocity at the plasma-sheath edge and mi is theion mass. Additionally, we have the continuity of the ion flux to the electrode,equation (2.9), assuming steady state,

niui = nsus, (7.3)

where ns is the ion density at the plasma-sheath edge (which is equal to theelectron density there, since the plasma is quasi-neutral). Equations (7.2) and(7.3) give us the ion density in the sheath

ni = ns

(1− 2eφ

miu2s

)−1/2

.

However, when the potential φ is changed with RF, the ions are too heavy tofollow the instant change in potential. They will instead follow a time averagepotential φ, so that the ion density in the RF sheath is

ni = ns

(1− 2eφ

miu2s

)−1/2

. (7.4)

54

In a hollow cathode, the secondary electrons created at the cathode surfaceare accelerated in the cathode sheath. If there are no collisions in the sheath,the secondary electrons will enter the plasma with the energy eφ. The elec-trons can undergo some collisions in the plasma and then enter the oppositesheath. It is reasonable to assume that the electrons do not lose all the energyduring collisions in the plasma, especially not when the width of the hollowcathode is of the same order as the inelastic collision mean free path. Thusthe secondary electrons will enter the opposite sheath with a considerable en-ergy, and the assumption that the electron temperature is much smaller thanthe cathode potential is not valid. Eventually these electrons will lose most oftheir energy due to inelastic collisions in the plasma and become members ofthe thermalized “bulk” group of electrons. Apparently, we should have twotypes of electrons in a hollow cathode plasma: one large group of thermalizedelectrons and one smaller group of “fast” electrons accelerated to a potentialclose to the total potential drop in the sheath. Such a group of electrons hasbeen observed in experiments [37].

The electron density is made up of three groups of electrons: slow electrons(ne1), fast electrons (ne2), and secondary electrons (ne3), so that ne = ne1+ne2+

ne3. Thus the Poisson equation looks like

d2φ

dx2= − e

ε0(ni−ne1−ne2−ne3). (7.5)

The slow and fast electrons are assumed to follow the Boltzmann relation

ne1 = ne1s exp(−eφ/kBTe), (7.6)

and

ne2 = βns exp

(− eφ

kBTe2

)= βns exp

(− eφ

ΘkBTe1

)(7.7)

in the sheath. Thus, the fast electrons are assumed to have a temperatureΘ times higher than the slow electron temperature, and their density at theplasma-sheath edge is a fraction β of the total electron density, ns, there.

For the secondary electrons, we have that they must be conserved as theytraverse the sheath. Thus

ne3 =je3

eve3, (7.8)

where je3 is the current density of secondary electrons at any point in thesheath. This, together with the energy conservation for the secondary elec-trons,

1

2mev2

e3 = −e(φw−φ)+wc, (7.9)

55

where wc = mev2c/2, yields

ne3 =je3

e

(− 2e

me(φw−φ)+wc

)−1/2

. (7.10)

The slow electron density can finally be written

ne1 = ns(1−β−ne3s/ns) exp

( −eφ

kBTe1

), (7.11)

where ne3s = ( je3/e)[−(2e/me)φw +wc]−1/2 is the secondary electron densityat the plasma-sheath boundary.

To get more simple-looking equations, we introduce some normalisedquantities,

η =eφ

kBTe, (7.12a)

η =eφ

kBTe, (7.12b)

ξ =x

λD, (7.12c)

M = vs

vB, (7.12d)

Vi =vi

vB, (7.12e)

Ni,e1,e2,e3 =ni,e1,e2,e3

ns, (7.12f)

Wc =2wc

kBTe, (7.12g)

J = je3/ jc =je3

nse(

kBTe1me

)1/2(−2ηw)3/2

, (7.12h)

where λD = (ε0kBTe1/nse2)1/2 is the Debye length and vB = (kBTe1/mi)

1/2 isthe Bohm velocity. The Poisson equation (7.5) now looks like

− d2η

dξ2= Ni − (1−β−Ne3s)exp(−η)

−βexp(−η/Θ)− J(−2ηw)3/2

(−2ηw+2η+Wc)1/2, (7.13)

56

which, along with

Ni =

(1− 2η

M2

)−1/2

(7.14)

and the boundary conditions η(0) = ηw, η(ξ →∞) = 0 are the equations wewant to solve.

7.1.1 Solving the Equations

To solve the equation system numerically, one first sets an initial ion distri-bution (i.e. guesses the form of φ and puts that into equation (7.14) to get theion density), then solves the equation for the time dependent potential φ fora certain number of steps (e.g. 100 steps). Then one takes these 100 potentialprofiles and makes an average, which gives a new φ. A new ion density profileis calculated from this new φ, and is inserted again in the Poisson equation.These steps are iterated with a Newton scheme until the relative changes inthe time-averaged potential are below a wanted level.

The maximum potential was set to 1 kV and the Bohm velocity for the ionsis satisfied with M = 1.05. The slow electron temperature was 5 eV and thesecondary electron initial temperature was 3 eV.

7.1.2 The Potential Distribution

By solving the equation system given above, it is shown that the sheath thick-ness increases with fast electron temperature and density. The sheath thicknessalso increases with secondary electron current. An example of how the sheaththickness changes is given in figure 7.2. The dashed line is the (negative) po-tential distribution in the sheath without fast and secondary electrons. The fullline shows the potential distribution with fast and secondary electrons withparameters Θ = 80, β = 0.08, and J = 0.008. This shows that the sheath widthmay be of the order of 100 μm, and that it will be wider if there are fast andsecondary electrons present.

Figure 7.3 shows a photograph of a slit-type RF hollow cathode dischargein nitrogen. The sheaths close to the cathode surfaces are visible, indicatingthat their width is of the order of 100 μm [6, 7].

7.2 Two-dimensional Self-Consistent Model

As discussed earlier, the one-dimensional description of the hollow cathodesheath has its drawbacks. Solving the equations in two dimensions the ion ve-locity and potential in the plasma comes out self-consistently. One only needsto provide boundary conditions at the surfaces bounding the two-dimensionalarea.

To further remove simplifications, one can solve for the time-dependent ionand electron densities.

57

0 100 200 300 400 500 6000

100

200

300

400

500

600

Distance (μm)

Ave

rage

pot

entia

l (V

)

Figure 7.2: The (negative) sheath potential with no fast or secondary electrons, β = 0,

J = 0 (dashed) and with fast and secondary electrons, β = 0.08, J = 0.008, Θ = 80

(full).

To account for the high number of collisions between ions and neutrals inan atmospheric pressure plasma, one can model the ions by the so-called drift-diffusion equations [36],

∂ni

∂t+∇ ·Γi = S , (7.15)

where S is the ion source term, ni is the ion density, and Γi is the ion fluxvector

Γi = −Di∇ni+μiniE. (7.16)

E is the electric field, Di is the ion diffusion coefficient, and μi is the ionmobility coefficient.

As before, the slow and fast electrons are modelled as Boltzmann distrib-uted but with different temperatures and densities (see the previous section).The secondary electrons are also modelled as before, but with the secondaryelectron current emanating from the cathode determined by the ion flux to thecathode times a secondary emission coefficient γ,

je3 = γnievw, (7.17)

where vw is the velocity with which they are born at the cathode surface.

58

S lit hollow cathode

C ounter electrode (not seen)

Hollow C athode Discharge 110 W , 13.56 MHz, 20 s lm N2

Upper cathode ring

0 1 2 3 mm

Lower cathode ring

0 1 2 3 mm

Slit hollow cathode

Upper cathode ring

Hollow Cathode Discharge110 W, 13.56 MHz, 20 slm N2

Counter electrode(not seen) Lower

cathode ring

Figure 7.3: Side view of the slit-type radio frequency hollow cathode discharge in

nitrogen.

7.2.1 Modelling

The equation system was solved with the the commercial software ComsolMultiphysics, which uses the Finite Element Method (FEM). The FEM putsa mesh over the two-dimensional domain and then solves the equations at themesh nodes for each time step.

The boundary condition for the ion flux at the surfaces is zero density gradi-ent, ∇ni = 0, and so Γi = −niμi∇φ there. If the ion velocity is away from thesurface, −niμi∇φ = 0.

7.2.2 Results

The addition of fast electrons was shown to increase the sheath thickness. Atatmospheric pressure (760 Torr) the increment was around 10 μm, see figure7.4. Most pronounced was the increment at lower pressures (1 Torr), where itwas around 300 μm for certain values ofΘ and β, see figure 7.5. The reason forthe difference in the sheath thickness increment between this model and theprevious one could be that the ion velocity is solved for self-consistently in thismodel, whereas it was given in the previous model. If the ions make frequentcollisions in the sheath, the Bohm criterion does not have to be fulfilled [65].Thus, at higher pressures, the velocity of the ions at the sheath entrance isreduced, giving a smaller increment in sheath thickness compared to the low-pressure case.

Figure 7.6 shows the potential at atmospheric pressure (760 Torr) for onerf period without any secondary or fast electrons. The plotted values are from

59

0 50 100 150 200 250 300 350 400 450 500

x (μm)

~10 μm

Figure 7.4: The potential distribution at maximum cathode potential for the cases

without (—) and with secondary and fast electrons (- - -) at atmospheric pressure

(β = 0.08 , Θ = 130), n0 = 1013 cm−3.

a cross-section in the middle of the computational two-dimensional domain.The sheath thickness is, as seen, about 200 μm.

The time evolution of the electron and ion density during one rf period canbe seen in figures 7.7 and 7.8, without and with fast and secondary electrons,respectively. The inclusion of fast and secondary electrons elevates the elec-tron density in the sheath, as can be seen by comparing the two figures. Theion density during one rf period hardly changes.

60

x (mm)

~300 μm

Figure 7.5: The potential distribution at maximum cathode potential for the cases

without (—) and with secondary and fast electrons (- - -) at 1 Torr (β= 0.08 ,Θ= 130),

n0 = 1011 cm−3. Note the change of scale.

0 50 100 150 200 250 300 350 400 450 500

x (μm)

(0, π)

(π, 2π)

Figure 7.6: The potential distribution at the mid-point cross-section of the computa-

tional domain at 760 Torr, for a whole rf period, without secondary and fast electrons.

n0 = 1013 cm−3.

61

0 50 100 150 200 250 300 350 400 450 500

x (μm)

ni , n

e (

m-3)

Figure 7.7: The ion (- - -) and electron (—) density at the mid-point cross-section of

the computational domain at 760 Torr, for a whole rf period, without secondary and

fast electrons (β = 0.08 , Θ = 130), n0 = 1013 cm−3.

0 50 100 150 200 250 300 350 400 450 500

x (μm)

ni , n

e (

m-3)

Figure 7.8: The ion (- - -) and electron (—) density at the mid-point cross-section of

the computational domain at 760 Torr, for a whole rf period, with secondary and fast

electrons (β = 0.08 , Θ = 130), n0 = 1013 cm−3.

62

7.3 Atmospheric Pulsed Plasma in Hollow ElectrodeGeometry

The above model was extended with rate coefficients calculated from the dis-tribution function, obtained by solving the Boltzmann equation with the freesoftware BOLSIG+ [42], and equations for the electron motion and mean en-ergy,

∂ne

∂t+∇ ·Γe = S , (7.18)

and∂nε

∂t+∇ ·Γε = −eE ·Γe+S ε , (7.19)

respectively, withΓe = −De∇ne−μeneE, (7.20)

andΓε = −Dε∇nε −μεnεE. (7.21)

The ions were described as before by equation (7.15). The electron andelectron energy diffusion and mobility coefficients De, Dε and μe, με , respect-ively, were also calculated from the distribution function as functions of theelectron mean energy ε = nε/ne. The boundary conditions used were

Γe =1

4

(8kBTe

πme

)1/2

ne−γΓi (7.22)

with me the electron mass, kB Boltzmann’s constant, and Te the electron tem-perature, and for the ions

Γi = μiniE if n ·E > 0, otherwise Γi = 0, (7.23)

where n is the unit normal to the surface, directed outward from the domain.For the electron energy, we have

Γε = ε1

4

(8kBTe

πme

)1/2

ne−γεsecΓi. (7.24)

The secondary electrons were assumed to be born with an average energyεsec, and as before we accounted for all different secondary electron emissionprocesses in the constant γ.

The equations were used to model an atmospheric pulsed plasma in a hol-low electrode geometry. The geometry is shown in figure 7.9. Two differentspacings between the cathodes were used, 500 μm and 100 μm. At the openboundaries, the flow of electrons and ions is free. The potential at the anodeswas set to φa = 0, and the potential at the cathodes was modelled as a pulse ofwidth 10 ns, with a rise time of 1 ns, and an amplitude of φ0 = 2 kV. The pres-

63

Figure 7.9: The geometry of the model, which is infinite in the ±z direction and sym-

metric in the xy plane, so that we only need to consider a cross-section in the same

plane. The distance between the cathode plates is 500 or 100 μm.

sure was set to 0.1 MPa. The secondary electron emission coefficient usedin the model was γ = 0.3, and the secondary electron initial average energyεsec = 1.25 eV. The equations were then solved with the commercial FEMsoftware Comsol Multiphysics.

7.3.1 Dynamics at Pulse Rise

Figure 7.10 shows the electron density for different times during pulse risefor the cathode plate separation 500 μm. Large density peaks are formed bythe strong electric field created at the cathode ends. At the centre, betweenthe cathodes, density waves created by the cathode potential meet, producinga localised density enhancement there. The same, but compressed, type ofdynamics can also be seen with a cathode plate separation of 100 μm, seefigure 7.11. The smaller density in the 100 μm case is probably due to thesmaller volume to particle loss rate compared to the 500 μm case.

Figure 7.12 shows the electron mean energy at different times during pulserise for the cathode separation 500 μm. Similar structures as in the electrondensity can be seen, with large energy peaks at the cathode ends and energywaves travelling from the cathodes towards the centre. Comparing with theelectron mean energy for the 100 μm case, shown in figure 7.13, we see thatthe mean energy in the latter is higher, especially the energy peak heights.This is probably due to the larger electric fields in the anode–anode direction.

64

Time = 1.1 ns

Time = 0.5 ns Time = 0.7 ns

Time = 0.9 ns

1000

500

1000

500

1000

500

1000

500

Max: 2.5×1016

0

0.5

1

1.5

2

2.5×1016

0

500

0.5×1016

1×1016

1.5×1016

μm 0

500

μm

0.5×1016

1×1016

1.5×1016

2×1016

2.5×1016

n e (m-3

)

n e (m-3

)

0

500

μm

0.5×1016

1×1016

1.5×1016

2×1016

0

500

μm

0.5×1016

1×1016

1.5×1016

n e (m-3

)

n e (m-3

)

2.5×1016

2×1016

2.5×1016

Cathode

Cathode

Cathode

Cathode

Figure 7.10: The electron density ne over the two-dimensional domain for the times

0.5, 0.7, 0.9, and 1.1 ns for the cathode distance 500 μm. The colour indicates the

amplitude, and has the same scale in all four figures, given by the colour bar. The

height also indicates the amplitude, but the scale is different for all figures. In order to

reveal small structures, the high peaks have been cut.

65



Max: 7.91×1016

0

1

2

3

4

5

6

7

×1016

0100

μm

500

1000

0100

μm

500

1000

0100

μm

500

1000

0.5×1016

1×1016

n e (m-3

)

1×1016

2×1016

3×1016

n e (m-3

)

0100

μm

500

1000

2×1016

4×1016

6×1016

8×1016

n e (m-3

)

0.5×1016

1×1016

1.5×1016

n e (m-3

)

Cathode

Cathode

Cathode

Cathode

Figure 7.11: The electron density ne over the two-dimensional domain for the times

0.5, 0.7, 0.9, and 1.1 ns, with the cathode distance 100 μm. The colour indicates the

amplitude, and has the same scale in all four figures, given by the colour bar. The

height also indicates the amplitude, but the scale is different for all figures, in order to

reveal small structures. The lines over the surface are contour lines.

66

Time = 1.1 ns


Time = 0.9 ns

00

0 0

Max: 6.7

0

1

2

3

4

5

6

0

500

μm

1

ε (e

V)

2

3

4

5

500

1000

0

500

μm

500

1000

1

ε (e

V)

2

3

4

5

0

500

μm

500

1000ε

(eV

)

2

4

6

0

500

μm

500

1000

1

2

3

4

ε (e

V)

Figure 7.12: The mean electron energy ε over the two-dimensional domain for the

times 0.5, 0.7, 0.9, and 1.1 ns for the cathode distance 500 μm. The colour indicates

the amplitude, and has the same scale in all four figures, given by the colour bar. The


reveal small structures.

67

Time = 0.5 ns

0

Time = 0.7 ns


Max: 6.4

0

1

2

3

4

5

6

ε (e

V)

100

μm

500

1000

1

2

3

4

0

ε (e

V)

100

μm

500

1000

1

2

3

4

0

ε (e

V)

100

μm

500

1000

2

4

0

ε (e

V)

100

μm

500

1000

2

4

6 6

Figure 7.13: The mean electron energy ε over the two-dimensional domain for the

times 0.5, 0.7, 0.9, and 1.1 ns, with the cathode distance 100 μm. The colour indicates

the amplitude, and has the same scale in all four figures, given by the colour bar. The


reveal small structures.

68

8. The H-HEAD Plasma Source

The hybrid hollow electrode activated discharge (H-HEAD) is a combinationof two types of plasma sources: a microwave antenna and a hollow cathode[8, 12]. It has the advantage of being able to generate long plasma plumes at at-mospheric pressure in molecular gases and air with very low flow rates. Sinceit generates a “cold”, non-thermal plasma by selectively heating the electrons,it is suitable for surface processing. A big advantage over low-pressure dis-charges is that one does not need any expensive vacuum systems to operateit. Other atmospheric pressure non-thermal plasmas often need a flow ratearound 5000 sccm (standard cubic centimetres per minute) in order to create astable discharge and to extend the length of the plasma. The H-HEAD sourcehas been shown to create several cm long plasma plumes at atmospheric pres-sure with only 120 sccm of gas flow, making it much cheaper to operate. Aschematic picture of the H-HEAD source is shown in figure 8.1.

The microwave antenna serves as the inlet of the gas into a steel nozzle witha diameter of 400 μm, which serves as the hollow cathode. This is excited bya separate generator (radio frequency, DC, or pulsed DC).

Figure 8.1: Schematic picture of the hybrid hollow electrode activated discharge (H-

HEAD).

69

8.1 Cold Nitrogen and Air Plasmas by the H-HEAD

Experiments were conducted with the H-HEAD source with both nitrogen andair. For the experiments, the hollow cathode was powered by pulsed DC, withpulses of 500 ns at 250 kHz. The microwave frequency was 2.4 GHz. Actingas a counter electrode when the hollow cathode was activated was a ther-moprobe with an integrated K-type thermocouple. This allowed temperaturemeasurements of the plasma plume created by the H-HEAD.

For a flow rate of 120 sccm the air plasma plume extends up to 70 mmat a microwave power of 350 W. The behaviour of nitrogen is similar. Theminimum microwave power needed to sustain a stable plasma in air and ni-trogen was about 200 W. Fixing the microwave power at 250 W, the longestplume, for both air and nitrogen, was found at a flow rate of 200 sccm, whereit extended to about 65 mm. The length starts to drop at flow rates exceed-ing 240 sccm due to the onset of turbulent flow. Since the microwave poweris transported by surface waves along the interface between the plasma andthe ambient air, with successive power absorption in the plasma, a turbulentinterface will enhance the absorption.

The steady-state temperature on the thermoprobe at a distance of 1 cm and2.5 cm from the gas nozzle was shown not to exceed 800◦C as it interactedwith the plasma plume generated with microwaves only. When the hollowcathode was turned on, a 25◦C increase in temperature was observed at DCpulsed current 0.2 mA in air.

The onset of the hollow cathode also increases the temperature of the nozzletip. As an effect of this, metal perticles are released from the nozzle, and tans-ported by the plasma to the substrate.

8.2 Deposition of Diamond Films by the H-HEAD

The H-HEAD source was tested for deposition of diamond films. For this, anair plasma was generated by the source, with a microwave power of 300 W,and an average DC current of 0.2 A. A substrate made of stainless steel wascleaned with acetone and ethanol and placed on the thermocouple, so that theplasma plume from the source came in contact with the substrate. From theside, an ethanol and air mixture was delivered to the plasma.

After one hour treatment of the substrate with the air and ethanol mixtureplasma a gray spot was formed on the substrate. An ESCA analysis of thecoating revealed contents of carbon, oxygen, and a trace of nitrogen. The car-bon particles could also be seen on SEM images. A typical thickness of thecarbon coating, after one hour deposition, was 200 nm. A Raman spectra ofthe coating showed a well-pronounced peak of diamond, with no other closepeaks (e.g. from graphite or glassy carbon), see figure 8.3.

70

Figure 8.2: Photographs of microwave plasma columns created by the H-HEAD

source for different flow rates. They are operated in ambient air at a microwave power

of 250 W.

8.3 Measuring Vibrational Temperature

The optical emission spectra of the nitrogen molecule N2 was measured withan EG&G Princeton Applied Research OMA III 1460 multichannel analyser.The vibrational structure of the second positive system (C3Πu → B3Πg) of theN2 molecule was used to calculate the vibrational temperature. The band headintensity was assumed to be the intensity of the band [13]. The vibrationaltemperature Tvib was calculated as (see also chapter 5) [62]

lnIν′ν′′

em (λ)λ3

S rel(λ)q(ν′ → ν′′) = −Eν′

kBTvib+ ln

n0

CZ, (8.1)

where q is the Franck-Condon factor, λ is the wavelength of the transition, n0

the density of all molecules in the upper electronic state, Z the partition func-tion of the vibrational levels in the upper state, and S is the relative spectralsensitivity of the detection system.

A small program was written to read all variables and calculate the vibra-tional temperatures, as well as presenting the data fit in a graph. A typicalexample is seen in figure 8.4. The data points could be fitted with a straightline, which shows that the excited states are Boltzmann distributed.

71

Figure 8.3: Raman spectrum of the carbon film deposited with the hybrid hollow

electrode activated discharge (H-HEAD) air and ethanol mixture plasma.

The results from this measurement was that, as the microwave power in-creased over 300 W, the vibrational temperature declined both for the air andnitrogen plasma (from 6500 K to 3200 K, and from 4050 K to 3300 K, respect-ively). This could be due to metal particles being released from the hollowcathode, so that electron energy could go into the excitation of metal atoms,which have low excitation potentials.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.822.2

22.4

22.6

22.8

23

23.2

23.4 λ = 357.69 nm, (0, 1)

λ = 380.49 nm, (0, 2)

λ = 375.56 nm, (1, 3)

λ = 399.7 nm, (1, 4)

λ = 371.05 nm, (2, 4)

Vibrational Energy Level Eν [eV]

ln(P

opul

atio

n D

ensi

ty (

arb.

uni

ts)

Tvib

∼ 5430 K

Figure 8.4: A typical graph showing the data fit and the vibrational temperature Tvib.

72

9. Conclusions

This thesis is concerned with non-equilibrium atmospheric pressure plasmas,both in experiments and in numerical modelling. The main focus is on atmo-spheric pressure hollow cathode discharges.

It has been shown, that a hollow cathode in combination with a microwaveantenna, the so-called H-HEAD source, can produce long air and nitrogenplasma plumes in ambient air. This can be achieved at very low carrier gasflow rates compared to other sources for the same purpose.

This thesis also presents original models of hollow cathodes at atmosphericpressure, and a model of a pulsed atmospheric plasma in hollow electrodegeometry. It was shown, that the inclusion of fast and secondary electrons ina sheath-only model increased the sheath width. The sheath width was shownto be of the order of 100 μm. In an improved model, which added collisionsfor the ions to the previous model, and extended it to two dimensions, thisincrement was shown to be less pronounced at atmospheric pressures than atlower pressures. Still, the sheath thickness was shown to be of the order of100 μm. This is in good agreement with experimental work. This could betaken in consideration when designing hollow cathode plasma sources, wherethe distance between opposite walls in the cathode must be at least the doublesheath thickness.

A model of an atmospheric pressure pulsed plasma in a 2D hollow elec-trode geometry is also presented as a pre-study to the modelling of pulsed DChollow cathodes. It is built on the fluid drift-diffusion approximation with rateand transport coefficients calculated from the Boltzmann equation. This showsthe dynamics of the plasma during the rise time of a pulse with cathode plateseparations of 500 μm and 100 μm. It was shown that large electron densityand mean energy peaks were created at the ends of the cathode plates, and thatthese quantities also were enhanced in the space between the cathodes. Theelectron density was observed to be lower in the smaller dimension than in thelarger, due to decreased volume. The electron mean energy was higher in thesmaller dimension than in the larger.

The models presented have contributed to the understanding of the phys-ics behind atmospheric hollow cathode plasmas. They also build a foundationfor future modelling of atmospheric pressure plasmas, especially hollow cath-odes. The goal is, together with experiments, to get a better picture of thephysics behind these types of discharges, so that one may optimise their per-formance.

73

10. Future Work

Some of the future work, that could be studied with a development of the mod-els presented in this thesis are: A model should be developed, which builds onthe last model presented in this thesis and in the articles enclosed, but adds thekinetic effects of fast electrons. These should be essential in the modelling ofthe hollow cathode. Such a model should be a hybrid model, which combinesa fluid model with a particle-in-cell scheme with Monte Carlo collisions. Thismodel should then be used for RF, DC, and pulsed hollow cathode modelling.The RF model should be tested with different dimensions in order to reveal apossible optimal dimension. The DC hollow cathode should be compared tothe RF hollow cathode to reveal fundamental differences. The pulsed hollowcathode model should be tested with different pulse length and dimensions.This would yield interesting results, from which one perhaps could tell some-thing about an optimal width and/or pulse length.

Both the RF and the pulsed hollow cathode should also be tested with othergases (e.g. molecular gases) or gas mixtures.

A continuity and momentum equation for neutrals should be included inthe fluid model to reveal the spatial distribution of these, particularly in RFhollow cathodes. Gas flow could also be accounted for by giving the neutralsa directed flow.

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11. Summary of Papers

Paper I

Cold atmospheric plasma in nitrogen and air generated by the hybrid

plasma source This paper describes the generation and characters of longplumes of cold atmospheric pressure plasmas in nitrogen and air by the hybridhollow electrode activated discharge (H-HEAD) source, which combines amicrowave antenna and a hollow cathode to create long plasma plumes withvery low gas flow. The author calculated the vibrational temperatures fromthe optical spectra of the plasma. Published in Journal of Vacuum Science and

Technology A, Volume 24, Number 4, 2006.

Paper II

Characterization of Hybrid Atmospheric Plasma in Air and Nitrogen

(Reviewed conference paper) In this paper, the hybrid hollow electrode ac-tivated discharge (H-HEAD) source was used to synthesise nanocluster dia-mond films at steel substrates using an air plasma with a mixture of ethanol.The author calculated the vibrational temperatures from the optical spectra,and assisted in the collection of the Raman spectrum. Published in Proceed-

ings of the 49th Annual Tech. Conf. of the Society of Vacuum Coaters (SVC),

Washington, DC, April 2006.

Paper III

On Dimensions of Atmospheric-Pressure Hollow Cathodes The purposeof this paper was to investigate how a group of fast and secondary electrons,characteristic of the hollow cathode plasma, would affect the sheath thick-ness. Planning and interpretation was done with co-authors. All modelling,and most of the writing of the paper was done by the author. Published in

IEEE Transactions on Plasma Science, Volume 35, Number 3, 2007.

Paper IV

Space-charge sheaths in atmospheric pressure hollow cathodes (Confer-

ence paper) This paper is largely a summary of paper III, presenting model-

77

ling of the space-charge sheath in a hollow cathode. Planning and interpret-ation was done with co-authors. All modelling, and most of the writing ofthe paper was done by the author. Published in the Proceedings of the 16th

Symp. on Applications and Plasma Processes on Plasma Processing (SAPP

XVI). Podbanské, Slovakia. January 2007. P69, pp. 261–262. Presented by

the author at the conference.

Paper V

Time evolution of the space-charge sheath in an rf hollow cathode Themodel published previously is further developed to two dimensions, and withcollisions, in this paper. The time evolution of the potential and densities in theplasma for atmospheric and reduced pressures is presented. Planning and in-terpretation was done with co-authors. All modelling and most of the writingof the paper was done by the author. Published in Journal of Physics: Confer-

ence Series 100, 2008. Presented orally by the author at the IVC-17/ICSS-13

and ICN+T2007 conference in Stockholm, July, 2007.

Paper VI

Modelling the Dynamics of a Pulsed Atmospheric Plasma in Hollow Elec-

trode Geometry The fluid drift-diffusion approximation is used to model ananosecond pulsed hollow cathode. The dynamics of the plasma during thepulse rise time is described. Rate and transport coefficients are retrieved fromthe Boltzmann equation as a function of electron mean energy. Planning andinterpretation was done with co-authors. All modelling and most of the writ-ing of the paper was done by the author. Submitted to Journal of Physics D:

Applied Physics, 12 April, 2008.

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12. Svensk Sammanfattning

Gasurladdningar har varit kända sedan mycket långt tillbaka i tiden, men det ärbara under det senaste århundradet som man börjat förstå fysiken bakom dem.Den amerikanske fysikern Irving Langmuir gjorde tillsammans med LewiTonks banbrytande upptäckter inom gasurladdningsfysiken. Det var Langmuirsom 1928 myntade ordet plasma för att beskriva en joniserad gas.

Tiden efter Langmuirs och Tonks arbete har fyllts av upptäckter inom plas-mafysiken och dess tillämpningar. Från de första gasurladdningarna har vikommit till mycket avancerade tekniker som utnyttjar plasma för att till exem-pel rena avgaser, behandla ytor, belägga ytor, etsa, m.m. Flera olika typer avtekniker för att skapa plasma, så kallade plasmakällor, har utvecklats genomåren. Några exempel på sådana tekniker är skapande av plasma med hjälp avkonstanta elektriska fält, radiofrekventa fält och mikrovågor. Dessa källor ex-isterar också i olika geometrier, till exempel plana elektroder och hålkatoder.Hålkatoderna har den fördelen, att de omsluter plasmat, så att elektronernasoch andra partiklars energi kan tas tillvara i större utsträckning än med vanligaplana elektroder. Vilken typ av källa man väljer beror på vad man ska användaden till, vilka egenskaper man vill att plasmat, som skapas med dem, ska ha.

De första försöken med gasurladdningar skedde i kammare med lågt tryck.Det är nämligen enklare att skapa ett plasma vid låga tryck. Nackdelen medlåga tryck är att man måste ha dyr utrustning för att kunna skapa de lågatrycken. Plasman som kan skapas vid atmosfärstryck har fördelen att de intekräver vakuumsystem. Nackdelen är att de ofta är svåra att få stabila. Teknikerhar därför utvecklats för att kunna skapa stabila plasman vid atmosfärstryck.

I den här avhandlingen presenteras bland annat experiment med en såkallad hybridkälla, hybrid hollow electrode activated discharge (H-HEAD),som är en kombination av en hålkatod och en mikrovågsantenn. Dennahar visat sig kunna skapa långa plymer av plasma vid gasflöden så lågasom 120 sccm (standard kubikcentimeter per minut). Viktig kunskap omdenna källas egenskaper kan bland annat fås ur vibrationstemperaturen hosmolekyler, vilken återspeglar elektronernas temperatur i viss utsträckning.I denna avhandling presenteras sådana mätningar på kvävemolekyler. Delånga plymer av plasma som denna källa kan skapa möjliggör behandlingav tredimensionella objekt vid atmosfärstryck. Experiment som presenterashär visar även att källan kan användas för syntetisering av diamant, då enblandning av etanol och luft tillsätts ett luftplasma.

Mätningar på plasma är inte alltid lätta att utföra, eller så är mätresultateni sig svåra att tolka. Teoretisk modellering av plasma kan hjälpa till att ökaförståelsen för den fysik som styr plasmats beteende. Eftersom man vet vad

79

man stoppar in i en modell, kan man också relativt enkelt se vad olika delar aven modell bidrar med till plasmats egenskaper. Särskilt viktig är modellerin-gen av plasman vid atmosfärstryck, eftersom dessa ofta skapas med källormed små dimensioner, varför mätningar är svåra att utföra. I den här avhand-lingen presenteras unika modeller av hålkatoder och hålelektroder vid atmos-färstryck. Vi har visat att de så kallade rymdladdningslagren, som alltid skiljerett plasma från objekt, till exempel en elektrod, i kontakt med ett plasma, blirtjockare då snabba och sekundära elektroner adderas till de övriga långsammaelektronerna i plasmamodellen. Snabba elektroner är den främsta källan tillden så kallade hålkatodeffekten. Modellen visade att rymdladdningslagretstjocklek är av storleksordningen 100 μm. I en vidareutvecklad tvådimension-ell modell, som även tar hänsyn till jonernas kollisioner med neutrala atomer,visar vi att denna ökning inte är lika stor då vi har atmosfärstryck, som när vihar lägre tryck. Tjockleken visar sig ändå vara av storleksordningen 100 μm.Dessa resultat kan man i framtiden ta hänsyn till, då man konstruerar hålkat-oder.

Avhandlingen presenterar även dynamiken hos ett pulsat plasma vid atmos-färstryck i hålelektrodgeometri. Koefficienter för elektroners elastiska och in-elastiska kollisioner tillsammans med transportkoefficienter beräknas ur för-delningsfunktionen för elektronerna, som fås genom att lösa Boltzmannekva-tionen. Modellen visar att, under pulsens stigtid, som är 1 nanosekund, skapashöga toppar i elektrondensiteten och i elektronernas medelenergi. Den visaräven att dessa kvantiteter förhöjs i området mellan elektroderna.

Modellerna har givit insikter i fysiken hos hålkatodplasman och de utgören grund för framtida studier av plasman, och då särskilt i hålkatoder, vid at-mosfärstryck. Målet är att, tillsammans med experiment, förstå fysiken bakomdessa för att kunna optimera deras prestanda.

80

13. Acknowledgements

First I would like to thank my supervisors Dr. Hana Baránková and Dr. Ladis-lav Bárdoš for giving me the opportunity to pursue Ph.D. studies, and forgiving me guidance during these years.

Many thanks to Claes Weyde for all lunches, discussions, and for proofread-ing the thesis. The same goes for Lars Norin. Thanks for all Friday lunchesthrough the years. Thank you Lars-Erik Gustavsson, former member of theApplied Plasma Group, for all discussions during lunches and coffee breaks.

Further, I would like to thank all the people at the Division of Electricity,where I spent the last 1.5 years of my Ph.D. It was great knowing you! Aspecial thank you to Thomas Götschl for helping me with my computer prob-lems, never afraid of compiling strangely behaving outdated codes, and Gun-nel Ivarsson, who makes life on the Division easier. Special thanks also toMats Leijon, head of the Division for Electricity, for providing a stimulatingatmosphere to work in.

I send many thanks to my former division, Solid State Electronics, where Ispent the first years of my Ph.D. I would especially like to thank MarianneAsplund for all help with all the practical things.

Thank you Pelle Nilsson for showing such enthusiasm, for our discussions,and for commenting on the thesis.

Sist vill jag tacka min familj. Ni betyder mycket för mig. Cecilia! Tack för attdu finns här hos mig och för att du stått ut med mig under stressade veckor.Ditt stöd är ovärderligt.

81

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