Modelling Chlorine Plasmas

Duarte Rogado Nina

October 2016

Abstract

Chlorine plasmas are widely used in the technological indus-try to create integrated circuits. To improve their use, amore detailed and expansive study of the processes involvedis necessary. There have been several models used in simula-tion work, with different reactions sets, to study the variousregimes of the chlorine plasma. The possibility that the vi-brational distribution of molecular chlorine had a large impactin the system has been raised in recent discussions.

In this article, a global model for an inductively coupledplasma is developed in conjunction with a varied reaction set.The code used is a rewrite on an existing in-house tool withthe successful goals of increasing ease of use and accessibil-ity. The simulation results is compared against experimentalresults, showing good agreement. Various vibrational energytransfer reaction sets are used to conclude that only the lowervibrationally excited energy levels of the molecular chlorinehave a measurable effect in the system. The influence ofthe wall recombination probably is studied, showing that itsincrease reflects a decrease in dissociation of the molecularchlorine and an increase in electronegativity. The creationand destruction processes for atomic chlorine and its nega-tive ion are analysed, discovering that the power had a largeeffect on the excitation of the atomic chlorine and on thecreation of the atomic negative ion by ionization.

1 Introduction

Chlorine is one of the commonly used gases in etching, mainlybecause of two reasons [17, 30]: its low dissociation energyand high electronegativity. Chlorine molecules have a lowdissociation energy, which allows the easy formation of chlo-rine atoms. These chlorine atoms have one of the highestelectronegativities, which means they very easily bond withother components. This means that a chlorine plasma reactshighly with the wafer, resulting in a faster removal of theunneeded sections from the wafer. To improve the efficiencyin chlorine’s usage, it is needed to both test and experimentwith the plasma and to create simulations that bring moredetail to the processes behind it.

Chlorine plasmas started being simulated with fluid modelsby Park and Economou [15] and by Sommerer and Kushner[16]. Detailed global models were later developed by Meekset al [19, 20] and by Lee and Lieberman [18]. One of the mostimpactful parameters is the wall reaction probability, that in-fluences the recombination of atomic chlorine and thereforethe whole system. The wall recombination probability hasbeen measured by various groups, and has been shown thatit depends on atomic/molecular chlorine ratio [28], on thegas temperature [22] and on the wall material [22, 29]. Morerecently, Kemaneci et al [34] have studied the effects of thewall recombination probability on the system, adding excited

states for the atomic chlorine and a small vibrational distri-bution.

In this work, we analyse a more detailed vibrational distri-bution in an inductively coupled chlorine plasma (ICP). Thisarticle starts with a general overview of the global model usedand the non-chlorine unique reaction types in section 2. Sec-tion 3 follows into a detailing of the chlorine reactions andtheir coefficients, and section 5 presents the code behind thesimulations. Section 6 starts with a comparison of the simu-lation results with real world data. Several parameters, suchas different vibrational sets, are tested to gauge their rela-tive importance and effects in the chlorine plasma. Finally,the creation and destruction processes for Cl and Cl− areanalysed.

2 Theoretical Model

Since we consider the plasma is uniform, we can simply modelthe changes in density [30] as

dnidt

=∑j

Rcreationi,j−∑j

Rlossi,j (cm−3 s−1), (1)

where ni is the density for the species i and Ri,j are the rates(creation and loss) between the species i and j. We thenlook for the the steady state, where these changes in densityshould be 0. It is usually assumed that the electrons followa simple Maxwellian EEDF [26, 32], with an electron tem-perature (Te) calculated through the code. The Maxwelliandistribution is defined as

n(E) = n0 exp

(− E

kB T

), (2)

where n0 is an appropriate normalization constant, E is theenergy of the particle and T is its temperature.

We can define these rates as a product of the involvedspecies’ densities and a rate coefficient (kReaction)

Ri = kReaction ·n∏j

nj (cm−3 s−1). (3)

A large fraction of the reactions are collisions with elec-trons (density ne), with the rest being either heavy particlecollisions or diffusions. Most of the reactions involve two dif-ferent species, with some having one reactant or more thantwo. These reaction coefficients can be calculated with dif-ferent expressions, depending on the type of reaction. Wecan use simple experimental laws, theoretical laws or evenintegrate particle cross-sections with the EEDF [23, 36, 37].A very common reaction type is the diffusion to the wall,where a positive ion can receive electrons and return as aneutral or a neutral species can recombine or de-excite. Thiswall diffusion is mostly controlled by the particle transport,which depends on the particle’s charge and state.

1

2.1 Diffusion

For neutral particles the diffusion reaction rate can be calcu-lated through [17]

Rni,W = ni

(Λ2

0

Di+

2V (2− γi)Aviγi

)−1

, (4)

where Di is the diffusion coefficient, V is the plasma volume,γi is the wall reaction probability, A is the area of contact and

vi is the mean neutral speed,√

8kBTg

πmi. Λ0 is the diffusion

length and is given by

Λ0 =

[(πL

)2

+

(2.405

R

)2]−1/2

, (5)

where L is the length of the cylinder and R the radius.For the positive ions, we can define the reaction rate as a

product of the species’s density with a velocity and a ratio ofan effective area with the plasma volume

R±i,W = uBi· ni ·

AeffiV

, (6)

where uBiis Bohm’s velocity,

√kBTe

mi, and Aeffi is the ef-

fective area of contact [27]

Aeffi = 2π(R2hL,i + LRhR,i

). (7)

hL,i and hR,i are two parameters (axial, L, and radial, R)defined [30] as

hk,i =√h2a,k,i + h2

c,k,i , (8a)

ha,L,i ≈0.86√

3 + ηL/2λ

1

1 + α0, (8b)

ha,R,i ≈0.8√

4 + ηR/λ

1

1 + α0, (8c)

hc,L,i = hc,R,i ≈1(

√γ− +

√γ+ ·

n1/2i∗ ·ni

n3/2−

) . (8d)

Subscripts − and + mean parameters related to the neg-ative and positive ions, respectively (n− is the negative ion

density, for example). η is 2T+

T++T−, a balance of the different

ion temperatures, usually 1; α0 is the central electronegativ-ity, 3

2n−ne

; γ± are the temperature ratios between the electrons

and the charged particles, Te

T±; and ni∗ is a parameter with

dimensions of density defined as

ni∗ =15

56

vikR,iλi

. (9)

In ni∗, vi is the thermal speed, kR,i is the rate coefficientof the mutual neutralization with negative ions, and λi is theion mean free path given by

1

λi=∑j

njσi,j , (10)

where σi,j is the cross section of the collision between ion iand species j. This mean free path can be approximated by

λi =kBTgPσavg

, (11)

with P being the gas pressure and σavg an average of thecross sections in equation (10).

3 Chlorine Chemistry

Table 2 shows the complete list of reactions used in thiswork. Most coefficients are in units of cm3 s−1 with excep-tions marked in the table. All coefficients, with the exceptionof the ones marked ’Set Vibrational Custom’, come from [34].’Set Vibrational Custom’ includes reactions that have coef-ficients calculated through experimental fits, as explained inthe sections in the coefficient column.

There are several important aspects to note in this table.The first is the existence of reverse reactions, as noted bya double arrow (←→), where the number of reagents doesnot change between the input and output. Reverse, or back-wards, reactions are simply the opposite of the normal, orforwards, reactions, with the inputs and outputs swapped.The coefficients for the reverse reactions are calculated bydetailed balancing.

Also of note are the reactions that point to a section in-stead of having a mathematical expression. These reactionsdo not have a simple coefficient and are either too complexto show in this table, or are calculated through experimentalfits. The sections indicated expand on these coefficients.

At the bottom of the table, there are two separate setsof reactions. These groups refer to the vibrational set used:’Set Vibrational [Kem+14]” is the set of vibrational-relatedreactions used in [34]; and ’Set Vibrational Custom’ is the setcreated as shown in sections 4.1 to 4.3. Simulations in theresults will use ’Set Vibrational [Kem+14]” unless otherwisenoted.

3.1 Species considered and Energies

The species considered in this work include ground-stateatomic and molecular chlorine, two positive ions, one neg-ative ion, two atomic excited states and Cl2 vibrationallyexcited states. There were several factors in choosing thespecies, such as availability of measurements (for the reac-tion coefficients) and fraction of density. Note that in thisdocument, Cl2 is the same as Cl2(v=0) and that Cl(2P3/2)is usually referred as just Cl. The energies and the statisticalweights for the species are shown in table 1.

Species Energy (eV) Ref. Stat. Weight

Cl2 0.03 [2, 25, 38] —Cl+2 11.47 [34] —Cl2(v>0) — [2, 25, 38] —

Cl− -2.33 [34] —Cl(2P3/2) 1.28 [34] 4Cl(2P1/2) 1.38 [34] 2Cl(1P5/2) 10.20 [34] 6Cl+ 14.28 [34] —

Table 1 List of Species and their energies.

2

# Reaction Coefficient (cm3 s−1)

(1) e + Cl2 → 2Cl + e 1.04× 10−7 T−0.29e e−8.84/Te

(2) 2Cl + Cl2 → 2Cl2 3.50× 10−33 e810./Tg cm6 s−1

(3) 3Cl →Cl2 + Cl 8.75× 10−34 e810/Tg cm6 s−1

(4) e + Cl←→Cl(2P1/2) + e 4.55× 10−8 T−0.46e e−2.01/Te+0.001/T2

e

(5) e + Cl←→Cl(1P5/2) + e 7.03× 10−11 T 0.55e e−2.15/Te−1.5/T2

e −2.05/T3e

(6) Cl(1P5/2) →Cl 1.0× 105 s−1

(7) e + Cl →Cl+ + 2e 3.17× 10−8 T 0.53e e−13.29/Te

(8) e + Cl(2P1/2) →Cl+ + 2e 3.17× 10−8 T 0.53e e−13.19/Te

(9) e + Cl(1P5/2) →Cl+ + 2e 4.33× 10−8 T 0.55e e−0.15/Te−0.85/T2

e

(10) e + Cl2 →Cl+2 + e + e 5.12× 10−8 T 0.48e e−12.34/Te

(11) e + Cl+2 → 2Cl 9.00× 10−8 T−0.50e

(12) e + Cl2 →Cl + Cl+ + e 2.14× 10−7 T−0.07e e−25.26/Te

(13) e + Cl2 → 2Cl+ + 3e 2.27× 10−10 T 1.92e e−21.26/Te

(14) e + Cl− →Cl+ + 3e 3.62× 10−9 T 0.72e e−25.38/Te

(15) e + Cl2 →Cl+ + Cl− + e 2.94× 10−10 T 0.19e e−18.79/Te

(16) e + Cl− →Cl + 2e 9.02× 10−9 T 0.92e e−4.88/Te

(17) Cl+2 + Cl− → 3Cl 5.00× 10−8((Tg/300)

−1)0.50(18) Cl+2 + Cl− →Cl + Cl2 5.00× 10−8

(19) Cl+ + Cl− → 2Cl 5.00× 10−8((Tg/300)

−1)0.50(20) e + Cl2 →Cl + Cl− 3.43× 10−9 T−1.18

e e−3.98/Te +3.05× 10−10 T−1.33

e e−0.11/(Te+0.014)

(21) e + Cl2(v=1) →Cl + Cl− 14.06× 10−9 T−1.18e e−3.98/Te +

12.51× 10−10 T−1.33e e−0.11/(Te+0.014)

(22) e + Cl2(v=2) →Cl + Cl− 30.18× 10−9 T−1.18e e−3.98/Te +

26.84× 10−10 T−1.33e e−0.11/(Te+0.014)

(23) e + Cl2(v=3) →Cl + Cl− 46.31× 10−9 T−1.18e e−3.98/Te +

41.18× 10−10 T−1.33e e−0.11/(Te+0.014)

(24) e + Cl2←→Cl2(v=1) + e 3.99× 10−6 T−1.5e e−7.51/Te−0.0001/T2

e

(25) e + Cl2←→Cl2(v=2) + e 3.28× 10−11 T−1.12e e−0.37/Te +

2.86× 10−11 e−(lnTe+0.99)2/2×1.062

(26) e + Cl2←→Cl2(v=3) + e 1.30× 10−11 T−1.24e e−0.41/Te +

6.08× 10−12 e−(lnTe+0.94)2/2×1.022

(27) e + Cl2(v=1-2)←→Cl2(v+1) + e 3.00× 10−10 T−1.00e e−0.37/Te +

4.61× 10−10 e−(lnTe+1.04)2/2×1.102

(28) e + Cl2(v=1)←→Cl2(v=3) + e 1.25× 10−10 T−1.13e e−0.36/Te +

1.06× 10−10 e−(lnTe+1.01)2/2×1.062

(29) Cl2(v=0-3) + Cl+ →Cl + Cl+2 5.40× 10−10

(30) Cl+2 + Wall →Cl2 Sections 2.1 and 3.2(31) Cl+ + Wall →Cl Sections 2.1 and 3.2(32) Cl + Wall → 1/2Cl2 Sections 2.1 and 3.2(33) Cl(2P1/2) + Wall →Cl Sections 2.1 and 3.2(34) Cl(1P5/2) + Wall →Cl Sections 2.1 and 3.2

Set Vibrational [Kem+14]

(35) Cl2(v=1-3) + Cl →Cl2(v-1) + Cl 1.3× 10−11((Tg/300)

1)0.5(36) Cl2(v=1-3) + Wall →Cl2(v-1) Sections 2.1 and 3.2

Set Vibrational Custom

(37) Cl2(v>0) + Cl←→Cl2(w<v) + Cl Section 4.1(38) Cl2(v>0) + Cl2←→Cl2(v-1) + Cl2 Section 4.2(39) Cl2(v>0) + Cl2(w<v)←→Cl2(v-1) + Cl2(w+1) Section 4.3(40) Cl2(v>0) + Wall →Cl2(v-1) Sections 2.1 and 3.2

Table 2 Complete list of reactions and their coefficients.

3

These parameters are mostly used for the detailed balanc-ing, where the coefficient for a reverse reaction is calculatedfrom the forwards reaction and the energies of the interveningspecies [1],

kreverse = kforwardΠ goutΠ gin

exp

(ΣEin − ΣEout

kBTi

), (12)

where the ki are the coefficients for the reactions, gi are thestatistical weights and Ei are energies of the species in eachof input, in, and output, out, according to the backwardsreaction. Meaning that if the forwards reaction involves anincrease of energy this energy difference will be positive andthe backwards coefficient will be higher. Ti is the electrontemperature (Te) when it is a collision with an electron, orthe gas temperature (Tg) if otherwise.

The vibrational levels energy can be calculated using amodified harmonic oscillator equation [2, 25],

E(v) = ωe(v +1

2)− ωexe

(v +

1

2

)2

+ ωeye

(v +

1

2

)3

+ ... , (13)

where v is the vibrational level and ωe, ωexe and ωeye arevibrational constants [38]. Terms of higher power of thevibrational level are neglectable since their constants decreasemuch faster [2, 25].

3.2 Diffusions

The diffusions presented in section 2.1 have several param-eters that depend on the species being diffused. For theneutral particles, there are two such parameters in equation(4): the diffusion coefficient and the wall reaction probability.We can calculate the diffusion coefficient, Di, using

Di =D0i

n

√Tg300

= D0i

kB · T 3/2g

P√

300, (14)

with D0ibeing a constant coefficient. A value of 0.149 cm2

s−1 at 300K and 1 atm [13] was used for DCl, which signifiesa value for D0Cl

around 3.682 ×1024 cm−1 s−1. Due to thelack of data for the diffusion coefficient of Cl2, the same valueas for Cl was used.

Species Wall Reaction Probability

Cl 0.25Cl2(v > 0) 1.0 [34]Cl(2P1/2) 1.0 [34]Cl(1P5/2) 1.0 [34]

Table 3 Wall reaction probabilities for the different species(for equation (4)).

Table 3 shows which values of wall recombination werechosen for each species. Note that for the excited states ofCl, the value of 1 was used in [34], see section 6 for thediscussion.

For the positive ions, there are also two parameters tonote: the rate coefficient of mutual neutralization kR,i, andthe average cross section σ, both entering in the calculationof ni∗ in equation (9). The mutual neutralization reactionsin table 2 are numbered 17 to 19. The corresponding coeffi-cients were summed to obtain the Cl+2 neutralization rate tobe used in equation (9). A value of 70 ×10−16 cm2 was usedfor the average cross section in the simulations [34], basedon values for the different species [30].

4 VT and VV Energy Transfers

Now that we have seen the chlorine parameters of the generalequations, we can look at the expressions for the chlorine vi-brations energy transfers. There are two kinds of vibrationalenergy transfers: vibration-translation (VT) where a particlegives/receives translational energy so that one molecule canraise/decrease its vibrational level; and vibration-vibration(VV) where two vibrationally excited molecules exchange vi-brational quanta, resulting in one of them raising its level andthe other decreasing it.

Note that vibrational levels start at 0, the ground state;and the symbol used to denote the vibrationally excited stateis usually v, with the exception of when two different vi-brational levels (or ranges) appear, where w is used for thesecond level or range.

4.1 Atomic VT

The atomic vibrational-translation transfers (reaction 37) arethe reactions where a chlorine molecule collides with a chlo-rine atom, transferring energy from one to the other. Usingthe coefficients presented in [9], it is possible to create a fitthat depends on the gas temperature (Tg) and the vibrationallevel (v),

Pv(T ) ≈ A(v) log Tg +B(v)

6.022× 1010, (15)

with the parameters A(v) and B(v) defined as

A(v) = −0.017v2 + 0.5396v + 0.127 , (16a)

B(v) = 0.071v2 − 2.4322v − 1.2726 . (16b)

Assuming a linear decrease with ∆v = v−w (w < v) [11],we can calculate the reaction coefficients, kv,v′ , using

kv,w =Pvv

+ s(v) ·(v

2+

1

2−∆v

)(cm3 s−1) , (17)

where s(v) is a fit on the vibrational level,

s(v) =

(18.797

6.022× 1011

)v−1.332 . (18)

This fit was calculated for v ≤ 14 and a limited range ofgas temperatures, so behaviour with vibrational levels above15 or gas temperature above 1500K is not predicted. Whilev=15 was not considered in the experimental fit, it is stillacceptably predicted.

4.2 Molecular VT

The next set of reactions is the molecular collisions, where theenergy of one Cl2 transfers to another molecule, increasingor decreasing the energy level of one of them. One changesits translational energy while the other alters its vibrationalstate. We start with the relaxation time [5] given by,

Pτ = exp[A(T−1/3g − 0.015µ1/4

)− 18.42

]= A exp

(A · T−1/3

g

), (19)

where P is again the gas pressure, τ is the relaxation time,A is an experimental constant related to the species, in thiscase A = 58; and µ is the reduced mass of the collision,

4

µ = Πmi

Σmi, which for the Cl2 - Cl2 collision is 35.453 u. This

gives us A = 1.197 × 10−9 atm·s. We can then use thisrelaxation time to calculate the base coefficient, k1,0,

1

τ= ν〈k1,0〉 , (20a)

k =ν

n〈k〉 , (20b)

with the collision frequency, ν, the reaction probabil-ity, 〈k1,0〉, and the density, n. Combining equations (20)and (19), we obtain an expression for the base coefficient,

k1,0 = 3.413× 10−11

(Tg300

)exp

(−58 T−1/3

g

). (21)

After k1,0 is known, it is possible to use the first orderperturbation theory SSH (Schwartz, Slawsky, Herzfeld) [3,4] as modified by Bray [8] and Carrier and Keck [6]. TheSSH theory allows to create a scaling law based on the basecoefficient. It is important to note that, since it is a per-turbation theory, the scaling law may overestimate the ratecoefficients for high temperatures or high vibrational levels.The molecular VT coefficients can then be calculated with,

kv,v−1 = v1− xe1− xev

F (yv,v−1)

F (y1,0)k1,0 , (22)

where xe is a vibrational constant [38], used in equation (13),and F (y) is the adiabacity factor [6] given by

F (y) =

{12

[3− exp

(− 2

3y)]

exp(− 2

3y)

0 ≤ y ≤ 20

8(π3

)1/2y7/3 exp

(−3y2/3

)20 < y

,

(23)with y being defined as

yv,v−1 = g (1− 2 xev) , (24a)

g =

(1

2

)3/2(θ′

Tg

)1/2

, (24b)

θ′ =4π2ω2

eL2µ

kB. (24c)

ωe is a vibrational frequency [38]; µ is again the reducedmass of the collision; and L is a length characterizing theshort range exponential of the intermolecular potential. Thevalue of 1

4.1 A = 0.244A was used for L [7].

4.3 Molecular VV

For the vibration-vibration transfers, instead of one of the Cl2giving/receiving translational energy, both molecules changetheir vibrational levels. The calculation for the coefficientfollows a similar path as for the molecular VT (section 4.2),starting with the calculation of k0,1

1,0. Using the transition

probability 〈k0,11,0〉(ε) from Bergeron et al [10, 12], where

ε = E~ωe

is the relative energy of collision in units of the har-monic oscillator of Cl2, we can integrate with a Maxwelliandistribution of the velocities

〈k0,11,0〉 =

∫ ∞Emin

[〈k0,1

1,0〉(E)]· e−βEd(βE) , (25a)

Emin = 0 , (25b)

β =1

kBTg, (25c)

E =µv2

2. (25d)

With 〈k0,11,0〉(ε) = 0.0441ε+0.0276 [12], we obtain 〈k0,1

1,0〉 =

5.5× 10−5 Tg + 0.0276. We can now use equation (20b) to

calculate k0,11,0. Since [14]

Z =ν

n= πd2

√8kBTgπµ

, (26)

where d = 4.83 A, we obtain νn = 1.789 × 10−11 T 1/2 cm3

s−1.Finally, we use the SSH theory [3, 4, 6, 8], as in equa-

tion (22), modified for 2 vibrational ranges,

kw−1,wv,v−1 = v w

(1− xe)2

(1− xev) (1− xew)

F (yw−1,wv,v−1 )

F (y0,11,0)

k0,11,0 , (27)

where the adiabacity factor F (y) is the same of equa-tion (23), y is [14] given by

yw−1,wv,v−1 =

πL

~

(µ

2kBTg

)1/2

|∆Ew−1,wv,v−1 | , (28)

and ∆Ew−1,wv,v−1 is simply the change in energy given by (Ev +

Ew−1)− (Ev−1 + Ew). Note that w > v > 0.

5 Implementation

Now that we know the theory and all the necessary equations,we can then start to create the simulation. The languageof choice was MATLAB [35] and the code was a rework ofan existing codebase, called LoKI (LisbOn KInetics), an in-house tool created at IPFN. The LoKI code is an in-housesimulation tool that couples two main calculation blocks tosolve: (i) the homogeneous two-term electron Boltzmannequation (for a pure gas or a gas mixture, including first andsecond-kind collisions, as well as electron-electron collisions);(ii) the system of zero-dimensional (volume averaged) ratebalance equations for the most relevant charged and neutralspecies in the plasma [23, 36, 37]. For this work, only module(ii) was used, replacing module (i) with a fixed MaxwellianEEDF.

The new, rewritten code has the advantage of flexibilityand ease of use. It adds several types of input files to defineand change the different requisites and settings of the sim-ulation. This means that to create a new system, or to adda new gas to a mixture, it is simply a question of creatingthe correct files and adding them to the right settings file.The program considers only one gas temperature, with nodifference between the temperature for positive and negativeions.

The system of equations is solved using an ODE solverfrom MATLAB [21, 24, 35], and repeats the time simulationuntil the required pressure and quasi-neutral balanced arereached.

6 Results & Discussion

6.1 Comparisons with real world data

Figure 1 shows four comparisons of the simulation to exper-imental results from Booth, Sirse et al [31, 33] and to thesimulation results of Kemaneci et al [34], for a large rangeof operating conditions. The first thing to note is that the

5

top two graphs show a varying power, and the bottom twoshow a fixed power, while the pressure is the other variable,constant in the top graphs and varying in the bottom. Gastemperature depends on both power and pressure, but sincethis simulation receives as input the gas temperature andnot power, a relation between gas temperature and powerand pressure is shown in table 4 [34].

For points with same power and pressure as in the LPP ex-periments, both experimental gas temperature and electrondensity were used as inputs for the code. A linear interpola-tion between each experimental point inputs was done to runmore simulations, which explains the ”bumps” in the densi-ties.

With a fixed power, the simulations show good agreementwith the experimental results and other simulation, with theCl− having the largest discrepancy. It follows the generaltrend for all species shown. For the results with a fixed pres-sure, the simulation still agrees well with the experimentaldata, with again a large discrepancy in the Cl−. Due tothe lack of experimental measurements of the rest of the in-tervening species, it becomes more difficult to analyse thereason for this difference. A possible explanation is the fixedelectron temperature and density instead of a given power,which means the quasi-neutral balance may be different, thuschanging the densities of the ions.

7 Sensitivity analysis

7.1 Vibrational Set

Now that we confirmed that the simulations yield good re-sults, we can explore the different parameters and processesinvolved in the plasma. The most important process to lookinto is the vibrational distribution. Figure 2 shows the vibra-tional distribution of three different sets of vibration-relatedreactions at two operating conditions. On the left, figure 2(a)shows the distribution at 200 W and 10 mTorr; and on theright, figure 2(b), the inputs were 500 W and 50 mTorr. Tocompare the results, fits for a Boltzmann distribution (equa-tion (2)) were done, resulting in particle temperatures. Twodifferent fits were done for each set, one for all points (Tall)and one for just the first two levels (T10). The fits are shownas lines, though the fit for T10 is only shown for ’Set Vi-brational [Kem+14]”, since for the other sets, both fits aresimilar.

We can see that both ’Set Vibrational Custom’ followmore closely a Boltzmann distribution than ’Set Vibrational[Kem+14]”, which features a large drop in density for theupper half of its vibrational levels. The particle tempera-tures are also interesting to look at. With the exception ofthe ’Set Vibrational [Kem+14]” at high input parameters, allT10 are close to the gas temperature. This is due to mostof the relevant reactions being heavy particles collisions. Asthe system is given more power, the gas, and thus the vi-brational, temperature raises, meaning that the whole distri-bution increases. The higher levels have a larger error, soTall differ more of the gas temperature, with ’Set Vibrational[Kem+14]” having the largest discrepancy.

Also to note is the rapid decrease in density with the vi-brational level, which means that only the first two or threelevels have a noticeable effect on the other species. This factis also evident in figure 3, where densities from both setsare shown, with the dashed lines being from the ’Set Vibra-

tional [Kem+14]” and dotted lines from the ’Set VibrationalCustom’. The difference between both is only truly relevantin the densities of Cl2, which decreases with the addition ofmore vibration levels, and Cl2(v>0), which increase with theaddition of more vibration levels.

In figure 3 a comparison of densities of ’Set Vibrational[Kem+14]” and ’Set Vibrational Custom’ is shown. Notethat for ’Set Vibrational Custom’, only the set with v=5 wasused, since the differences to the results of v=15 are too smallto notice in the plots. The dotted line is the custom set, whilethe dashed is the set from [34]. Points in figure 3(a) are thesame experimental results from section 6.1 [31, 33]; whilethe points in figure 3(b) and figure 3(c) help differentiatethe different species. The two sets follow similar trends, withthe largest differences being, with no surprise, in the densitiesof Cl2(v≥0), with more of the ground state transferring intothe following vibrational levels.

7.2 Wall Reaction Probability

After testing the different vibrational sets, the next step wasto experiment the different values of the wall recombinationprobability to gauge its impact. A method to analyse theeffects of the wall reaction probability is to simulate in arange of values for the probability.

Figure 4 shows densities varying with the wall reactionprobability, for two different pressures. The gas temperatureand electron density were set as the equivalent of 200 W forthe particular pressure. There are two distinct partitions ac-cording to the change in γrec. The first is the atomic chlorineand its excited states, with the exception of Cl−; while thesecond is composed the the molecular chlorine and its excitedstates, including Cl−. The two groups follow the distributionof their group state, Cl and Cl2, respectively. As the wallrecombination probability increases, the fraction of Cl/Cl2decreases, meaning all Cl excited states also decrease. Theexception, Cl−, has a different behaviour since it is formedfrom the dissociative attchment of the molecules.

We can quantify the effects of the wall reaction prob-ability on the densities through two parameters: dissoci-ation and electronegativity. The degree of dissociation,Kd = nCl/(nCl + 2nCl2), and the degree of electronega-tivity, α = nCl−/ne, are shown in figure 4(b), with dashedand dotted lines respectively. As expected, the degree ofdissociation decreases drastically with the increase in wall re-combination probability. The electronegativity has an inversebehaviour, increasing at a similar rate of the decrease of thedissociation. Note that at very low recombination probabili-ties, the dissociation is close to 100%, meaning that most ofCl2 is broken into Cl.

The wall recombination probability also has a noticeableeffect in the electron temperature, as shown in figure 4(c).The negative ion, Cl−, is created from the molecules, andto maintain the quasi-neutrality balance, both positive ionsneed to increase, and since they depend on both atoms andmolecules, the reaction coefficients need to be higher, requir-ing an higher electron temperature.

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(a) Pressure fixed at 10 mTorr, power varies. (b) Pressure fixed at 50 mTorr, power varies.

(c) Power fixed at 200 W, pressure varies. (d) Power fixed at 500 W, pressure varies

Figure 1 Comparison to real world data. Lines are the results of the simulation; points are experimental results [31, 33];hexagons are simulation results [34]. Electron density was an input of the code, so the values are the same.

Tg (K)````````````P (W)

p (mTorr)2 5 10 20 50

100 400 450 500 600 600200 450 550 650 800 500500 550 700 900 1200 1100

Table 4 Gas temperature from power and pressure [34].

(a) Input values were the equivalent to 200 W of power anda pressure of 10 mTorr.

(b) Input values were the equivalent to 500 W of power anda pressure of 50 mTorr.

Figure 2 Comparison of the vibrational distribution for the three used vibrational sets.

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(a) Pressure fixed at 10 mTorr, powervaries.

(b) Pressure fixed at 50 mTorr, powervaries.

(c) Power fixed at 500 W, pressure varies.

Figure 3 Comparison of the species’ densities with two vibrational sets. Dashed line - Set Vibrational [34]; Dotted line - SetVibrational Custom. Points in (a) and (b) are experimental results [31, 33]; points in (c) and (d) differentiate species.

(a) Densities with pressure fixed at 50mTorr.

(b) Degree of dissociation (dashed lines)and electronegativity (dotted lines).

(c) Electron temperature.

Figure 4 Effects of variation on wall recombination probability. Gas temperature and electron density were set as theequivalent of 200 W for the particular pressure.

(a) Processes of creation for Cl. (b) Processes of destruction for Cl.

(c) Processes of creation for Cl−. (d) Processes of destruction for Cl−.

Figure 5 Processes of creation and destruction for Cl and Cl−. The specific reactions for the process are included inparentheses, and reverse equations are signaled with an ’R’. Dashed lines are from ’Set Vibrational [34]’, dotted lines arefrom ’Set Vibrational Custom’ with v=5. Pressure was fixed at 10 mTorr.

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7.3 Relative Importance of Creation and De-struction Processes

Finally, to understand more of what is happening with thespecies, it is interesting to take a look to the underlying pro-cesses of creation of destruction of each species. For thissection, the species Cl and Cl− were chosen, since they areextremely important in the overall picture.

In figure 5, the most important processes for the creationand destruction of both Cl and Cl− are shown, with a fixedpressure of 10 mTorr and a range of powers. The processesare normalized with the total rate of creation/destruction tofacilitate the comparison. The specific reactions associatedwith the processes are included in parentheses and reversesare marked with an ’R’.The two vibrational sets are compared, ’Set Vibrational[Kem+14]” in the dashed lines, and ’Set Vibrational Cus-tom’, with v=5, in the dotted lines. There are many inter-esting things to look at in these plots. One of them, and tocall back section 7.1, is the large difference seen in figure 5(c)of the dissociative attachments for higher vibrational levels.As we had seen in figure 2, with the addition of the VT andVV coefficients, the populations for v=2 and v=3 increasedrastically, increasing every rate that depends on those den-sities.

Looking now at the evolution of the fractions, we see dif-ferent behaviours when the power changes. For the creationof Cl, when the power rises, the de-excitation of its excitedstates (Cl(2P1/2) and Cl(1P5/2) and their diffusion (withthe inclusion of Cl+) increase rapidly, and at high temper-ature, have similar importance as the Cl2 dissociation, thatdecreased. While the majority of Cl losses are to the wall, athigher temperatures the excitations to higher energy statesincrease, which explains the rise of the de-excitations in theCl creation.

The creation of Cl− shows an interesting effect. The co-efficients for the dissociative attachment of higher energy vi-brational level are higher than for the ground state, so whenthe temperature rise, and the populations of the higher vi-brational levels increase, the dissociative attachment of v=1increases enough to become the largest creator of negativeions. This effect is increased with the use of the VT and VVcoefficients, as would be expected. While the largest fractionof the Cl− destruction, the neutralizations with Cl+2 , remainsmostly constant, it is interesting to see the increase of theneutralization with Cl+, due to the higher electron density,which becomes a larger fraction than the electronic detach-ment at around 200W.

8 Conclusion

In this document, we have looked at the global model andthe general expressions involved. We expanded on those ex-pressions, analysing the different reaction types available, in-cluding the VT and VV energy transfers. We finished thetheoretical chapters with an overview of the code used in thesimulation and its algorithms.

We then moved on to the simulation results, by first look-ing at relevant issues with the code and what their solutionscould teach us, including the problems with experimental fitsand complex dependences for the coefficients. Simulationresults were compared to real world data, to assess the ac-curacy of the simulations, and were confirmed to have good

agreement. We then observed the effects that several pa-rameters had in the system. The vibrational distribution wasfound to be relevant only for the lower energy levels for vibra-tions, but with better accuracy using the VT and VV model.We observed the effects of the wall recombination probability,confirming that its increase produces a decrease in the degreeof dissociation and an increase of both the electronegativityand electron temperature. Finally, the processes for creationand destruction of Cl and Cl− were studied, and was foundthat the power had a strong positive effect on the excitationsof Cl and on the dissociative attachments that create Cl−.

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