numerical simulation of hydrogen microwave …...applied mathematical sciences, vol. 6, 2012, no....

Applied Mathematical Sciences, Vol. 6, 2012, no. 121, 6003 - 6019

Numerical Simulation of Hydrogen Microwave Plasma

Discharge Using a Fluid Model Approach

M. El Bojaddaini1, H. Chatei1, M. Atounti2,M. El Haim3, I. Driouch1 and M. El Hammouti3

1Laboratoire de Physique de la Matiere et de RayonnementFaculte des Sciences

Universite Mohammed Ier, Oujda, [email protected], [email protected], driouch [email protected]

2Departement de Mathematiques et InformatiqueFaculte Pluridisciplinaire de Nador

Universite Mohamed IerB.P 300, Selouane, 62700, Nador, Maroc

[email protected]

3Equipe science de l’environnementFaculte Pluridisciplinaire de Nador

Universite Mohamed IerB.P 300, Selouane, 62700, Nador, Maroc

[email protected], [email protected]

Abstract

Microwave plasma processing technology plays a vitally importantrole in various fields such as electronic engineering and development ofnew materials. Further, it is one of the promising ways to synthesizelarge crystals of diamond in high growth rates. However, physical andchemical phenomena that occur in the plasma are very complex andstrongly coupled. Understanding the correlation between the parame-ters of such phenomena, like electrons and ions density, electric field,plasma potential, microwave power and gas pressure, can significantlycontribute to an efficient use of microwave plasma technology. Numer-ical modeling of plasma can be of great use to optimize the dischargeparameters in order to improve the knowledge of plasma processingtechnology. This paper presents a numerical simulation of the impactof microwave power and gas pressure on the hydrogen microwave plasmadischarge characteristics, using a fluid model approach which solves theelectron and ion continuity equations, momentum transport equation

6004 M. El Bojaddaini et al

and the Poisson’s equation. The simulation results show a strong effectof power density and pressure on the species densities distribution inthe plasma.

Mathematics Subject Classification 2010: 82D10, 74S20, 58C15, 31A30.

Keywords: Microwave plasma, Fluid model, Numerical simulation, finitedifference method

1 Introduction

In recent years there has been a growing interest in microwave discharge plasmabecause of increasing number of its applications. It has been widely appliedas a manufacturing method for etching or deposition [18] because it is cleanand has high chemical reactivity [14]. Thus, microwave plasma created us-ing a gas mixture consisting of primarily hydrogen with small additions ofcarbon-containing gases, has found an application in the chemical vapor de-position (CVD) of diamond films [10]. Indeed, it is one of the promising waysto synthesize large crystals of diamond in high growth rates [24]. Recently, ithas been reported that approximately 10-mm-thick high-quality single-crystaldiamonds have been synthesized by microwave plasma CVDs [23]. The hydro-gen atoms play a key role in diamond CVD since they stabilize the growth ofdiamond and suppress the formation of graphite [6]. Different types of CVD re-actors are used, in which hot filament, RF, DC and microwave discharges, andflames are used. The advantages of microwave discharge reactors are: absenceof electrodes; high specific power contribution; high densities of excited andcharged particles; and a relatively large area and high homogeneity of the film[9]. However, discharge plasma has physicochemical phenomena very complexand strongly coupled. Further, it is difficult to experimentally observe physicalquantities of plasmas inside the reactors. Therefore, the numerical simulationof microwave plasma is a necessity to understand the plasma behavior insidethe reactor, and to improve the knowledge for deposition or etching by meansof plasma technology. The present paper offers a numerical simulation of apure hydrogen discharge characteristics in a cylindrical reactor, using a fluidplasma approach. The hydrogen plasma example was chosen because diamondfilm deposition processes often consist of high percentages of hydrogen in thedischarge. And, as shown by Koemtzopoulos et al, adding small percentages(e.g. 1%) of methane to a hydrogen discharge has only a minimal effect on theelectron energy distribution function [22]. The results found concern mainlythe effect of pressure and microwave power density on the dischrage charac-teristics.This paper is organized as follows: In section 2, the plasma discharge models

Numerical simulation 6005

are presented, while section 3 describes the fluid approach. Section 4 presentsthe discharge parameters. Section 5 gives the numerical simulation. In section6, the simulation results and discussion are presented. Finally, the conclusionis given in section 7.

2 Plasma discharge models

A number of different numerical models were published for the plasmas mod-eling used in a variety of applications. The most commonly used models are:fluid models [1, 13, 17], particle-in-cell/Monte Carlo (PIC/MC) models [16, 25]and hybrid models [3, 4]. All these modeling approaches have their specificadvantages and limitations, and therefore, the choice of the model is often dic-tated by the gas discharge and conditions under study. Particle-in-cell/MonteCarlo (PIC/MC) models are generally known as the most accurate approach,because they consider the plasma species on their lowest microscopic level.The trajectory of each species is calculated using Newton’s laws, while thecollisions between the plasma species depend on the cross sections, and aredetermined by random number [2]. However, these models are very time con-suming because of a large number of particles in the plasma. Fluid models arerelatively simple modeling way of plasmas compared with kinetic approaches.They are generally based on solving the continuity and transport equationsfor the various plasma species, in combination with Poisson equation, in orderto obtain a self-consistent electric field distribution. This approach is partic-ularly suitable for describing the detailed plasma chemistry. Indeed, a largenumber of different plasma species and chemical reactions can be included inthe model, without too much computational effort. These models are widelyused for the modeling of different kinds of plasmas [19], their main advantageis that their computational effort is still lower compared to the other numer-ical technique for plasma modeling. Hence, this makes a fluid model a usefultool for the modeling of discharge plasmas. Thus, many of the modeling effortperformed to obtain a better understanding of the discharge mechanisms arebased on a fluid approximation. Further, plasmas can also be described bythe hybrid model which combines several models (e.g., fluid model and MonteCarlo model) into a modeling network. In this way, the advantages of the in-dividual models can be combined, whereas the disadvantages can be avoided.In this study, a fluid plasma model is applied to describe the hydrogen dis-charge characteristics by solving the electron and ion continuity equations,momentum transport equations and the Poisson’s equation.


3 Fluid approach description

The simulation of plasma processes can be based generally on two major ap-proaches. One is the particle approach, which is carried out using a particlesimulation technique that treats the plasma as a combination of particles (elec-tron, ion, neutral). The other approach is the fluid method, which treats theplasma as a fluid and solves the equations obtained from the moments of theBoltzmann transport equation. The Boltzmann equation (BE) is a fundamen-tal equation describing the transport of an ensemble of particles. It is givenby the following form [5]:(

∂

∂t+−→ν ·

−→∇ +

−→F

m·−→∇ν

)f(−→r ,−→ν , t) =

(∂f(−→r ,−→ν , t)

∂t

)coll

(1)

Here, f(−→r ,−→ν , t) is the distribution function, −→r denotes the spatial position,−→ν denotes the velocity, and t denotes the time. m is the mass of the particle,−→F denotes the external forces, and the term on the right side of the equation(1) represents the collision term of the Boltzmann equation, which accountsfor changes of the electron velocity distribution function because of electronscollisions undergo mainly with neutrals but also with other electrons and ions.Equation (1) is a partial integro-differential equation in seven dimensions (threein space, three in velocity and time), and as such is extremely difficult to solve.The fluid model of plasma, reducing the complexities in the kinetic descrip-tion, is based on partial differential equations which describe the macroscopicquantities such as density, flux, average velocity, pressure and temperature.The equations for macroscopic quantities, called fluid equations, are obtainedfrom the Boltzmann equation by taking velocity moments [12]. Thus, zeromoment of the Boltzmann equation

∫ +∞−∞ (BE)d3ν yields continuity equation

for the particle density as:

∂n

∂t+−→∇ · (n−→u ) =

∫ +∞

−∞

(∂f

∂t

)coll

d3ν (2)

Here, the particle density n and the average velocity −→u are defined as n =∫ +∞−∞ fd3ν and −→u = 1

n

∫ +∞−∞−→ν fd3ν. The source term on the right side of

the continuity equation corresponds to the collision term of the Boltzmannequation.The momentum transport equation (equation of motion) can be found as first

moment of the Boltzmann equation(m∫ +∞−∞−→ν (BE)d3ν

)such as:

mn∂−→u∂t

+mn(−→u ·∇)−→u +∇−→P −nq(

−→E+−→u ×

−→B ) = m

∫ +∞

−∞(−→ν −−→u )

(∂f

∂t

)coll

d3ν

(3)


Where−→E and

−→B are the electric and magnetic fields, respectively.

Similarly, the energy equation can be found as second moment of the Boltz-

mann equation(

12m∫ +∞−∞ ν2(BE)d3ν

)as :

1

γ − 1

(∂p

∂t+∇ · (p−→u )

)+(−→P ·∇)−→u +∇·

−→Q =

1

2m

∫ +∞

−∞(−→ν −−→u )2

(∂f

∂t

)coll

d3ν

(4)−→Q = 1

2m∫ +∞−∞ (−→ν −−→u )(−→ν −−→u )2fd3ν defines heat flux

−→Q .

−→P = m

∫ +∞−∞ (−→ν −−→u )(−→ν −−→u )fd3ν defines pressure tensor

−→P , and Pij = pδij

defines scalar pressure p. γ is the ratio of specific heats.This simulation consists of the particle and momentum equations for electronsand ions, which are combined with the Poisson’s equation.In the steady state, the governing equations used in this study are given by:

∇2ψ =e

ε0(ne − ni) (5)

−→∇ ·−→Je = nennkion − αrnine (6)

−→∇ ·−→Ji = nennkion − αrnine (7)−→Je = −neµe

−→E −De

−→∇ne (8)

−→Ji = niµi

−→E −Di

−→∇ni (9)

Equation (5) represents the Poisson’s equation, which gives the electric inter-action between electrons and ions [20], where ψ is the electric potential.

The electric field−→E is derived from a scalar potential, ψ, by:

−→E = −

−→∇ψ.

Equations (6) and (7) represent the electron and ion continuity equations,respectively. They are written by balances among convective diffusion, pro-duction/ vanishing due to ionization/ recombination.Thus, the term on the left-hand side of equations (6) and (7) represents theparticles flux variation in position, while the first and second terms on theright-hand side correspond to source terms due to ionization and recombina-tion, respectively.Equations (8) and (9) represent the momentum balances for electrons and ions,respectively, in which the species fluxes are expressed as the sum of drift anddiffusion terms. Indeed, the first term on the right-hand side of these equationsrepresents the migration of the charged particles under influence of an electricfield, while the second term gives the diffusion due to particles concentrationgradient.The drift diffusion approximation reduces the number of partial differentialequations included in the model by the use of the algebraic expression for par-ticle flux (Equations (8) and (9)) instead of full equation of motion [12, 15].In the above equations, ne and ni are the electron and ion densities, respec-


tively;−→J e and

−→J i are the electron and ion fluxes, respectively; kion is the in-

elastic rate constant for ionization; and αr is the recombination rate constant;De,i and µe,i are the electron and ion diffusivities and mobilities, respectively.

4 Discharge parameters

The hydrogen plasma discharge is assumed to be partially ionized and partiallydissociated. Then, the major particle interaction processes are the electron-H2

molecule inelastic collision, electron-H2 molecule elastic collision and electron-hydrogen ion recombination.The electron-H2 inelastic collisions include the H2 molecule ionization process.The rate coefficient in equations (6) and (7) for this collision process can beexpressed using the Arrhenius relationship [21, 22] as:

kion = Aionexp

(−εionKBTe

)where εion is the threshold energy for H2 molecule ionization, Te is the electrontemperature, KB is the Boltzmann constant and Aion is the pre-exponentialfactor, which is obtained by approximating the rate constant data at low elec-tron temperatures to this relationship.The reactions considered in this study are:- Ionization :

e+H2 −→ e+H+2 + e

H+2 +H2 −→ H+

3 +H

- Recombination :e+ ion −→ neutral

It should be noted that the only neutral species considered in the ion andelectron simulations was the H2 species and that the dominant ionic speciesin the plasma is H+

3 [21, 22].The rate and transport parameters that are used in the model are summarizedin the table 1 [21]:


µenn = 1.0× 1024m−1V −1s−1

Denn = 5.0× 1023m−1s−1

µinn = 3.5× 1022m−1V −1s−1

Dinn = 3.5× 1021m−1s−1

Aion = 1.0× 10−14m3s−1

αr = 1.0× 10−13m3s−1

εion = 15.4eV

Table 1: Rate and transport parameters.

The electron diffusivity and mobility are written, respectively, as:

De =KBTemeνen

µe =e

meνen

where νen is the electron-neutral (in this case, electron-H2) momentum trans-fer frequency, e is the elementary charge and me is the electron mass. Thecollision frequency for electron-H2 molecule momentum transfer is relativelyindependent of the electron temperature, so it can be written as [22]:

νen(H2) = 1.44× 1012 × Presuure(Torr)

Tn(K)

where Tn is the neutral temperature, which can be represented by the trans-lational temperature of H2 gas.When working with pure hydrogen discharges, there are two key parametersthat govern the process. These are the input microwave power and the pres-sure in the deposition reactor [11].One of the best ways to increase hydrogen atom density and then growthrates is to increase the power density of the plasma. This can be done eitherby increasing simultaneously pressure and power keeping constant the plasmavolume [8] or by increasing only pressure while keeping constant the power, inthis latter case the plasma volume decreases [7].The empirical equations used for the translational temperature of H2 gas, and


the discharge volume, are [22]:

Translational temperture(K) = 228.6 + 374.3× Incident power(KW )

+ 16.5× Pressure(Torr)± 94.2

Plasma V olume(cm3) = 449.7 + 116.2× Incident power(KW )

− 18.1× Pressure(Torr)+ 57.1× [Incident power(KW )]2

+ 0.25× [Pressure(Torr)]2

− 5.4× Pressure(Torr)× Incident power(KW )

± 15.4

In this study, it is assumed that 100% of the microwave power coupled intothe reactor is absorbed by the plasma.

5 Numerical simulation

The simulation region in this work has a cylindrical form; therefore, the dis-charge behavior can be assumed to be φ symmetric and the discretization ofthe equations reduced to a two-dimensional problem. Thus, the simulationregion remains in the r-z plane only.The governing equations of this problem can be rewritten in cylindrical coor-dinates. Thus, the Poisson’s equation (5) can be given as:

∂2ψ

∂r2+

1

r

∂ψ

∂r+∂2ψ

∂z2=

e

ε0(ne − ni) (10)

The equations (6) and (8) give the electron continuity equation as:

− De

[∂2ne∂r2

+1

r

∂ne∂r

+∂2ne∂z2

]+ µe

[∂ψ

∂r

∂ne∂r

+∂ψ

∂z

∂ne∂z

]+ µe

[∂2ψ

∂r2+

1

∂r

∂ψ

∂r+∂2ψ

∂z2

]ne + αrnine − kionnnne = 0 (11)

Similarly, the equations (7) and (9) give the ion continuity equation as followingform:

− Di

[∂2ni∂r2

+1

r

∂ni∂r

+∂2ni∂z2

]− µi

[∂ψ

∂r

∂ni∂r

+∂ψ

∂z

∂ni∂z

]− µi

[∂2ψ

∂r2+

1

∂r

∂ψ

∂r+∂2ψ

∂z2

]ni + αrnine − kionnnne = 0 (12)

We discretize these equations in two-dimensional cylindrical coordinates (r andz directions as shown in figure 1), by using the finite difference method with


centred scheme. Hence, the discretization of the Poisson’s equations (10) isgiven by:

ψ(i+ 1, j)− 2ψ(i, j) + ψ(i− 1, j)

(∆r)2+ψ(i+ 1, j)− ψ(i− 1, j)

2i(∆r)2

+ψ(i, j + 1)− 2ψ(i, j) + ψ(i, j − 1)

(∆z)2− e

ε0[ne(i, j)− ni(i, j)] = 0 (13)

The discretization of the electrons continuity equation (11) is given as:

− De

[ne(i+ 1, j)− 2ne(i, j) + ne(i− 1, j)

(∆r)2+ne(i+ 1, j)− ne(i− 1, j)

2i(∆r)2

]− De

[ne(i, j + 1)− 2ne(i, j) + ne(i, j − 1)

(∆z)2

]+ µe

[ψ(i+ 1, j)− ψ(i− 1, j)

2(∆r)

ne(i+ 1, j)− ne(i− 1, j)

2(∆r)

]+ µe

[ψ(i, j + 1)− ψ(i, j − 1)

2(∆z)

ne(i, j + 1)− ne(i, j − 1)

2(∆z)

](14)

+ µe

[ψ(i+ 1, j)− 2ψ(i, j) + ψ(i− 1, j)

(∆r)2+ψ(i+ 1, j)− ψ(i− 1, j)

2i(∆r)2

]ne(i, j)

+ µe

[ψ(i, j + 1)− 2ψ(i, j) + ψ(i, j − 1)

(∆z)2

]ne(i, j)

− nnkionne(i, j) + αrni(i, j)ne(i, j) = 0

Similarly, the discretization of the ions continuity equation (12) is given as:

− Di

[ni(i+ 1, j)− 2ni(i, j) + ni(i− 1, j)

(∆r)2+ni(i+ 1, j)− ni(i− 1, j)

2i(∆r)2

]− Di

[ni(i, j + 1)− 2ni(i, j) + ni(i, j − 1)

(∆z)2

]− µi

[ψ(i+ 1, j)− ψ(i− 1, j)

2(∆r)

ni(i+ 1, j)− ni(i− 1, j)

2(∆r)

]− µi

[ψ(i, j + 1)− ψ(i, j − 1)

2(∆z)

ni(i, j + 1)− ni(i, j − 1)

2(∆z)

](15)

− µi

[ψ(i+ 1, j)− 2ψ(i, j) + ψ(i− 1, j)

(∆r)2+ψ(i+ 1, j)− ψ(i− 1, j)

2i(∆r)2

]ni(i, j)

− µi

[ψ(i, j + 1)− 2ψ(i, j) + ψ(i, j − 1)

(∆z)2

]ni(i, j)

− nnkionne(i, j) + αrni(i, j)ne(i, j) = 0

Here, i and j denote the grid indices in the r and z directions respectively,while ∆r and ∆z denote the space steps.


The boundary conditions for the fluid plasma model at the substrate and theedge of the plasma volume; figure 1; are:

ne = ni = 0

ψ = 0

and at the centerline (r = 0):

∂ne∂r

=∂ni∂r

=∂ψ

∂r= 0.

Figure 1: Active Plasma Zone

In order to solve the system of nonlinear discretized equations, we applied inthis simulation the Newton-Raphson iteration method.

6 Simulation results and discussions

A numerical simulation of a hydrogen plasma discharge using a fluid modelapproach in cylindrical geometry has been performed. The main input param-eters for this model include the pressure and microwave power.The spatial distribution of plasma density at a given microwave power densityis determined as shown in Figure 2 and Figure 3. The results show that theplasma density is maximal in the plasma volume near the center of the dis-charge (r = 0cm), and decreases in the edges and near the substrate region.


0 , 00 , 5

1 , 01 , 5 2 , 0 2 , 5 3 , 0 3 , 5 4 , 0

0 , 51 , 01 , 52 , 02 , 53 , 03 , 54 , 0

1 , 0 0 E + 0 1 7

2 , 0 0 E + 0 1 7

3 , 0 0 E + 0 1 7

4 , 0 0 E + 0 1 7

5 , 0 0 E + 0 1 7

6 , 0 0 E + 0 1 7

7 , 0 0 E + 0 1 7

P l as m

a de n

s i ty (

m- 3)

r ( c m )z ( c m )

Figure 2: Two-dimensional distribution of plasma densityat a power density of 18.88 W/cm3.

0 , 00 , 5

1 , 01 , 5 2 , 0 2 , 5 3 , 0 3 , 5 4 , 0

0 , 51 , 01 , 52 , 02 , 53 , 03 , 54 , 0

1 , 0 0 E + 0 1 72 , 0 0 E + 0 1 73 , 0 0 E + 0 1 74 , 0 0 E + 0 1 75 , 0 0 E + 0 1 76 , 0 0 E + 0 1 77 , 0 0 E + 0 1 78 , 0 0 E + 0 1 79 , 0 0 E + 0 1 7

P l as m

a de n

s i ty (

m- 3

)

r ( cm )

z ( c m )

Figure 3: Two-dimensional distribution of plasma densityat a power density of 26.20 W/cm3.

By increasing simultaneously power and gas pressure keeping constant theplasma volume, the evolution of axial and radial profiles of electron density,for different power densities, is also calculated and presented in Figures 4-7.Thus, Figures 4 and 5 present the evolution of electron density along the axialdirection in the hydrogen discharge at a fixed radial position (r=0 and r=3.5cm respectively), for different power densities, where the substrate is situatedat the position z = 0.


0 , 0 0 , 5 1 , 0 1 , 5 2 , 0 2 , 5 3 , 0 3 , 5 4 , 00 , 0 0 E + 0 0 0

1 , 0 0 E + 0 1 7

2 , 0 0 E + 0 1 7

3 , 0 0 E + 0 1 7

4 , 0 0 E + 0 1 7

5 , 0 0 E + 0 1 7

6 , 0 0 E + 0 1 7

7 , 0 0 E + 0 1 7

8 , 0 0 E + 0 1 7

9 , 0 0 E + 0 1 7

Electr

on de

nsity

(m- 3

)

A x i a l p o s i t i o n ( c m )

2 6 . 2 0 W / c m 3

2 2 . 5 5 W / c m 3

2 0 . 7 4 W / c m 3

1 8 . 8 8 W / c m 3

Figure 4: Axial profile of electron density for differentpower densities at r =0 cm.

0 , 0 0 , 5 1 , 0 1 , 5 2 , 0 2 , 5 3 , 0 3 , 50 , 0 0 E + 0 0 01 , 0 0 E + 0 1 62 , 0 0 E + 0 1 63 , 0 0 E + 0 1 64 , 0 0 E + 0 1 65 , 0 0 E + 0 1 66 , 0 0 E + 0 1 67 , 0 0 E + 0 1 68 , 0 0 E + 0 1 69 , 0 0 E + 0 1 61 , 0 0 E + 0 1 71 , 1 0 E + 0 1 71 , 2 0 E + 0 1 71 , 3 0 E + 0 1 7

Electr

on de

nsity

(m- 3

)

A x i a l p o s i t i o n ( c m )

2 6 . 2 0 W / c m 3

2 2 . 5 5 W / c m 3

2 0 . 7 4 W / c m 3

1 8 . 8 8 W / c m 3

Figure 5: Axial profile of electron density for differentpower densities at r =3.5 cm.

It is shown that the electron density increases significantly above the substratewith the axial position until it reaches its maximum value at the center of thedischarge where z = 1.75cm, and then it diminishes to vanishing at the edgeof the plasma.


The radial profiles of electron density in the discharge for different powerdensities are shown in Figures 6 and 7.It can be seen, from these figures, that the electron density decreases from itsmaximum value at the center (r=0) to a minimum value at the edge of theplasma.

0 , 0 0 , 5 1 , 0 1 , 5 2 , 0 2 , 5 3 , 0 3 , 5 4 , 00 , 0 0 E + 0 0 0

1 , 0 0 E + 0 1 7

2 , 0 0 E + 0 1 7

3 , 0 0 E + 0 1 7

4 , 0 0 E + 0 1 7

5 , 0 0 E + 0 1 7

6 , 0 0 E + 0 1 7

Electr

on de

nsity

(m- 3

)

R a d i a l p o s i t i o n ( c m )

2 6 . 2 0 W / c m 3

2 2 . 5 5 W / c m 3

2 0 . 7 4 W / c m 3

1 8 . 8 8 W / c m 3

Figure 6: Radial profile of electron density for differentpower densities at z =1 cm

0 , 0 0 , 5 1 , 0 1 , 5 2 , 0 2 , 5 3 , 0 3 , 5 4 , 00 , 0 0 E + 0 0 0

1 , 0 0 E + 0 1 7

2 , 0 0 E + 0 1 7

3 , 0 0 E + 0 1 7

4 , 0 0 E + 0 1 7

5 , 0 0 E + 0 1 7

6 , 0 0 E + 0 1 7

Electr

on de

nsity

(m- 3

)

R a d i a l p o s i t i o n ( c m )

2 6 . 2 0 W / c m 3

2 2 . 5 5 W / c m 3

2 0 . 7 4 W / c m 3

1 8 . 8 8 W / c m 3

Figure 7: Radial profile of electron density for differentpower densities at z =3 cm

It is also seen from the figures 4 - 7, that the increasing in microwave power


density from 18.88 to 26.20 W/cm3 under constant plasma volume leads to animportant increasing of electron density in the center of the discharge. Thisresult evidences the strong effect of microwave power density which is due toa coupled action of pressure and power.The variation of the maximum density of electrons in the pure hydrogen dis-charge can be represented versus power density as shown in Figure 8. Its profileis seen to vary linearly in the plasma volume from 5.5 1017m−3 to 8. 1017m−3

as the power density is increased from around 16W/cm3 to 26W/cm3.

1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7

5 , 0 0 E + 0 1 7

5 , 5 0 E + 0 1 7

6 , 0 0 E + 0 1 7

6 , 5 0 E + 0 1 7

7 , 0 0 E + 0 1 7

7 , 5 0 E + 0 1 7

8 , 0 0 E + 0 1 7

8 , 5 0 E + 0 1 7

Maxim

um de

nsity

of el

ectro

ns (m

- 3 )

P o w e r d e n s i t y ( W / c m 3 )

Figure 8: Maximum density of electrons versus power density.

3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0

5 , 0 0 E + 0 1 6

1 , 0 0 E + 0 1 7

1 , 5 0 E + 0 1 7

2 , 0 0 E + 0 1 7

2 , 5 0 E + 0 1 7

3 , 0 0 E + 0 1 7

3 , 5 0 E + 0 1 7

Maxim

um de

nsity

of el

ectro

ns (m

- 3 )

P r e s s u r e ( T o r r )

Figure 9: Maximum density of electrons as a functionof pressure at constant power.

Figure 9 shows the dependence of the maximum density of electrons onpressure for hydrogen discharge at an absorbed power of 1900W . the most


prevalent feature is that the electron density increases as the pressure increases.Indeed, the maximum density of electrons rises from 5. 1016m−3 to around3.3 1017m−3 when the pressure is increased from 35 Torr to 49 Torr. Thisresult shows clearly the important influence of pressure on the electron density.The enhancement in the production of electron density can be related to theincrease in the gas temperature which correspond to the increase in pressure.

7 Conclusion

In this paper, a fluid plasma model is presented to describe the hydrogenmicrowave plasma discharge characteristics by solving the electron and ioncontinuity equation, momentum transport equation and the Poisson’s equa-tion.The governing equations are discretized and solved in two-dimensional cylindri-cal coordinates using the finite difference method. Moreover, Newton-Raphsoniteration is applied in order to solve the nonlinear equations.The characteristics of discharge, including in particular the plasma densitywere investigated.Because plasma distribution could be one of the most important factors whichdetermine power efficiency of the diamond growth rate, we have focused ondisributions of electrons number density to provide information on the keyplasma parameters that control the processes of diamond deposition.As a result of the calculations using this model, distribution of electrons den-sity is obtained for various conditions of power and pressure.The simulation results show a strong effect of gas pressure and power density,on the discharge characteristics, such as plasma density.

References

[1] K. Bera, B. Farouk, Y. H. Lee, Effects of design and operating variables onprocess characteristics in a methane discharge: a numerical study, PlasmaSources Sci. Tech., 10, 211-225 (2001).

[2] A. Bogaerts, E. Bultinck, M. Eckert, V. Georgieva, M. Mao, E. Neytsand L. Schwaederle, Computer Modeling of Plasmas and Plasma-SurfaceInteractions, Plasma Process. Polym., 6, 5 (2009), 295-307.

[3] A. Bogaerts, R. Gijbels, W. J. Goedheer, Hybrid Monte Carlo fluid modelof a direct current glow discharge, J. Appl. Phys,78, 4 (1995), 2233-2241.


[4] A. Bogaerts, R. Gijbels, W. Goedheer, Two-Dimensional Model of a Di-rect Current Glow Discharge: Description of the Electrons, Argon Ions,and Fast Argon Atoms, Anal. Chem., 68, 14 (1996), 2296-2303.

[5] D. J. Economou, Modeling and Simulation of Plasma etching reactors formicroelectronics, Thin Solid Films, 365, 2 (2000), 348-367.

[6] M. Funer, C. Wild and P. Koidl, Simulation and development of optimizedmicrowave plasma reactors for diamond deposition, Surface and CoatingsTechnology, 116-119 (1999), 853-862.

[7] A. Gicquel, N. Derkaoui, C. Rond, F. Benedic, G. Cicala, D. Monegerand K. Hassouni, Quantitative analysis of diamond deposition reactorefficiency, Chemical Physics, 398 (2011), 239-247.

[8] A. Gicquel, K. Hassouni, F. Silva and J. Achard, CVD diamond films:from growth to applications, Current Applied Physics, 1, 6 (2001), 479-496.

[9] A. M. Gorbachev, V. A. Koldanov and A. L. Vikharev, Numerical model-ing of a microwave plasma CVD reactor, Diamond and Related Materials,10 (2001), 342-346.

[10] T. A. Grotjohn, J. Asmussen, J. Sivagnaname, D. Story, A. L. Vikharev,A. Gorbachev and A. Kolysko, Electron density in moderate pressurediamond deposition discharges, Diamond and Related Materials, 9 (2000),322 - 327.

[11] K. Hassouni, F. Silva and A. Gicquel, Modelling of diamond deposition mi-crowave cavity generated plasmas, J. Phys. D: Appl. Phys.,43, 15 (2010).

[12] E. Havlickova, Fluid Model of Plasma and Computational Methods forSolution, WDS’06 Proceedings of contributed Papers, Part III (2006),180-186.

[13] D. Herrebout, A. Bogaerts, M. Yan, R. Gijbels, W. Goedheer and A.Vanhulsel, Modeling of a capacitively coupled radio-frequency methaneplasma: Comparison between a one-dimensional and a two-dimensionalfluid model, Journal of Applied Physics, 92, 5 (2002), 2290-2295.

[14] A. Kobayashi, Y. Takao, K. Komurasaki, Fundamental characteristics ofmicrowave discharge type plasma source under atmosphere pressure, 28thICPIG, Prague, (July 15-20, 2007).

[15] M. Meyyappan and J. P. Kreskovsky, Glow discharge simulation throughsolutions to the moments of the Boltzmann transport equation, J. Appl.Phys., 68, 4 (1990), 1506-1512.


[16] K. Nagayama, B. Farouk, Y. H. Lee, Particle simulation of radio-frequencyplasma discharges of methane for carbon film deposition, IEEE Transac-tions on Plasma Science, 26, 2 (1998), 125-134.

[17] G. J. Nienhuis, W. J. Goedheer, Modelling of a large scale reactor forplasma deposition of silicon, Plasma Sources Science Technology, 8, 2(1999), 295-298.

[18] J. Paraszczak and J. Heidenreich, Applications of microwave plasmas inmicrocircuit fabrication, chapter 15, Microwave Excited Plasmas, editedby Michel Moisan, Jacques Pelletier, Plasma Technology, 4 (1992).

[19] J. D. P. Passchier and W. J. Goedheer, A two-dimensional fluid model foran argon rf discharge, J. Appl. Phys., 74, 6 (1993), 3744-3751.

[20] P. Scheubert, P. Awakowicz, R. Schwefel and G. Wachutka, Fluid dy-namic modelling and experimental diagnostics of an inductive high den-sity plasma source (ICP), Surface and Coatings Technology, 142-144(2001), 526-530.

[21] M. Surendra; D. B. Graves and L. S. Plano, Self-consistent dc glow-discharge simulations applied to diamond film deposition reactors, J.Appl. Phys., 71, 10 (1992), 5189-5198.

[22] W. Tan, T. A. Grotjohn, Modelling the electromagnetic field and plasmadischarge in a microwave plasma diamond deposition reactor, Diamondand Related Materials, 4 (1995), 1145-1154.

[23] H. Yamada, A. Chayahara, Y. Mokuno, Y. Horino and S. Shikata, Numeri-cal analysis of power absorption and gas pressure dependence of microwaveplasma using a tractable plasma description, Diamond and Related Ma-terial, 15 (2006), 1389-1394.

[24] H. Yamada, A. Chayahara, Y. Mokuno, Y. Soda, Y. Horino and N. Fu-jimori, Modeling and numerical analyses of microwave plasmas for opti-mization of a reactor desing and its operating condition conditions, Dia-mond and Related Material, 14 (2005), 1776 - 1779.

[25] M. Yan, W. J. Goedheer, Particle-in-cell/Monte Carlo simulation of radiofrequency SiH4/H2 discharges, IEEE Transactions on Plasma Science,27, 5 (1999), 1399-1405.

Received: June, 2012

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