Numerical calculation of temperature, velocity, and species concentration distributions in a hot filament chemical vapor deposition in a reactor using a CH₄/H₂ mixture

Abstract

This study is a numerical modeling of transport phenomena occurring in the reaction chamber during diamond or amorphous hydrogenated carbon films growth by a hot filament chemical vapor deposition (HFCVD) technique. A two-dimensional model was adopted to study the HFCVD reactor. The equations of heat, momentum, and mass transfer were solved numerically; the simulation was performed using a program in FORTRAN language. All temperature, velocity, and species concentration distributions were similar at the filaments and they were also similar between the filaments. The results show that the gas temperature increases when the number of filaments increases from three to four filaments. We also noted an increase in the production of CH₃ and C₂H₅ radicals near the surface; there was also an increase in the growth rate of the thin film. The concentrations of C₂H₆, C₂H₄, and C₂H₅ were very high. Temperature and concentrations were affected by the distance between filaments and the distance filaments-substrates.

Introduction

Thin films based on diamond or on hydrogenated amorphous carbon (a-C:H) present many technological interests: a-C:H films are commonly used, for example, in organic printable electronic devices [1]. They can be used as protective and antireflection films on optical instruments and on solar cells [2, 3]. They can also be used in optoelectronic device [4] to develop sensors (LED: Light-Emitting Diode), electrochemical detectors, bioelectronic and bioelectrochemical applications [5,6,7,8].

These thin films can be produced by chemical vapor deposition (CVD) using a CH₄/H₂ gas mixture, in which H₂ is dissociated into atomic H and CH₄ undergoes pyrolysis reactions leading to the formation of radicals such as CH₃ and CH₂ and stable species such as C₂H₂, C₂H₄, and C₂H₆ [9]. Activation of chemical reactions can occur at elevated temperatures, up to 3000 K in plasmachemical processes [10]. The radicals H and CH₃ are very important for the mechanisms of deposition [11].

The hot filament chemical vapor deposition (HFCVD) technique is very suitable because it is economic and scalable to large areas [12]. The HFCVD technique is very popular as it allows coating of complex shapes and internal surfaces, with a low operating cost [13]. An advantage of hot filament CVD reactors over the other reactors is the simplicity of the system and the uniform diamond deposition over a large area. HFCVD is a simple and a reproducible way to grow diamond at low pressures. It is also the first method to achieve nucleation and continuous growth of diamond on various substrates [9].

Diamond carbon can be deposited on a substrate made of Si, Mo, or silica, etc., which is mounted at a distance of 0.5 to 2 cm from the glowing filament and kept at 700 to 1000 °C either by the radiation from the filament or by a separate substrate heater [9]. A group of parallel filaments can be arranged in single or multiple parallel planes and in horizontal or vertical orientation [14].

In the HFCVD process, the production of thin films depends on substrate and gas temperatures and the velocity and volume density of gas as well as on complex chemical reactions of the gas phase [15]. Goodwin and Gavillet [16] developed the first one-dimensional numerical model for the diamond growth. They calculated temperature, velocity, and species concentration profiles. Debroy et al. [17] developed the first two-dimensional model of the HFCVD process. The authors showed that diffusion mechanisms are mostly responsible for both heat and mass transfer inside the reaction chamber. Some studies have focused on the concentrations of different species and radicals, in particular CH₃, and the growth rate of thin layers of diamonds [18,19,20]. One-dimensional models [21, 22] were developed to understand the deposition processes and the gas-phase behavior. Other numerical studies about two- and three-dimensional models [17, 19, 23,24,25,26,27] were interested in studying the effect of the parameters such as reactor geometry, the diameter, the number of the filaments, and the separation between hot filaments in the growth process of diamond films. These numerical models allow the creation of simplified models to reduce experiences that are often expensive [10].

Many other works [28,29,30,31,32,33] were focused on several aspects related to HFCVD processes (physical parameters of gas, gas-phase diffusion, chemical reaction rates, equilibrium constants, concentrations of species, filament temperature, filament geometry, filament arrangement, deposition parameters…); these factors can affect the growth of thin films.

In experimental and calculation work, Kondoh et al. [18] studied the effect of some HFCVD parameters on diamond growth in CH₄/H₂ gas mixture. Spatial distributions of the gas temperature, fluid velocity, and chemical species concentrations have been calculated in a two-dimensional HFCVD system for CH₄/H₂ gas mixture [18]. The partial differential equations used were those of conservation of mass, momentum, energy, and chemical species. May et al. [20] studied mechanism of growth in diamond films using A_r/CH₄/H₂ gas mixtures in the HFCVD reactor. They used a two-dimensional computer model to calculate the gas-phase composition within the reactor. Using experimental and calculated data, they showed the competition between H atoms, CH₃ radicals, and other species film morphology. Eckert et al. [34] were interested in the calculation of the sticking coefficients of species CH_x (x: 0–4), C₂H_x (x: 0–6), C₃H_x (x: 0–2), C₄H_x (x: 0–2), H, and H₂ on clean and hydrogenated diamond. They used the molecular dynamic simulation method.

In recent work, Tao Zhang et al. [35] presented a three-dimensional model for depositing the diamond films by a hot filament CVD. Their simulation model coupled heat transfer mechanisms by radiation, by conduction, and by convection. The authors showed that the thermal conductivity influences the temperature. Rebrov et al. [36] and Gorbachev et al. [37] studied the impact of gas-phase reactions on characteristics of the flow in a heated cylindrical channel of a gas mixture CH₄/H₂ reactor with using the direct simulation Monte Carlo method which is a method based on solving the Navier-Stokes equations. The study shows the complexity of the influence of the various physical and geometric parameters on the concentrations and on the deposition.

In this work, we present a two-dimensional model for HFCVD, for a gas mixture CH₄/H₂. This model studies distributions of temperature, velocity, and species concentration and the effect of the number of filaments and their distance from substrate. The proposed geometry of the reactor is different from several shapes cited in the literature. Numerical simulation can give extensive information about effect of several factors on gas mixture of HFCVD process.

The proposed reactor has two substrates of horizontal form. Long, equidistant and coplanar hot filaments are parallel to the substrates. The gas mixture enters the reactor with a constant flow. The fluid is considered Newtonian and the flow is laminar. We use the fluid model in the two-dimensional stationary regime to calculate spatial distributions of gas velocity and gas temperature and species concentrations.

In the second section, we present the principal equations of the mathematical model and the proposed calculation method. We use the nonlinear stationary differential equations: Navier-Stokes equations and continuity equation for velocity, heat equation for temperature, and equation of diffusion for concentrations. In the third section, we present the numerical model based on the finite volume method (FVM) with an adequate choice of boundary conditions. The results of this work are presented in the fourth section. Spatial distributions of velocities, temperature, and species concentrations are calculated for a gas mixture CH₄/H₂ with discussion and comments. First, we present distributions of velocity and temperature for three and four filaments. In the next step, we present species concentrations in volume, effects of number of filaments, and effects of distance filaments-substrates. In this fourth section, we also present concentrations of CH₃, C₂H₅, and H near the substrates. As an application, an estimation of the growth rate of deposited thin film is made. The last section includes the conclusion of this work and the future prospects.

Methods and mathematical model

The principal aim is calculation of distributions of temperature, velocity, and species concentration in an HFCVD process. The considered HFCVD reactor has two openings for the inlet and outlet gas. The substrates are horizontal plates; they have a constant temperature T_sub. Let L be the length of the reactor in the x direction and let Z_max be the distance between substrates. Long, equidistant and coplanar hot filaments are parallel to the substrates. In the HFCVD reactor, there are “n” filaments which are heated to a same temperature T_fil. The distance between two filaments is D_ff. A gas mixture CH₄/H₂ enters the reactor with a constant flow in the x direction. The initial composition of the mixture gas is 5% of methane CH₄ and 95% of hydrogen H₂. The given initial temperature, velocity, and pressure are respectively T₀, U₀, and P₀. The volumetric mass density ρ₀ is given according to ideal gas state equation as ρ₀ = P₀/RT₀, where R is the gas constant. A simplified scheme of the HFCVD reactor used in the model is shown in Fig. 1. Table 1 presents values of parameters used for the reactor. A numerical study of the phenomena is required.

Fig. 1

Simplified scheme of the HFCVD reactor

Table 1 Values of parameters used for the reactor

For the mathematical model, we consider a gas mixture in two-dimensional coordinates in stationary regime (independent of time). We assume that the substrates are wide enough (direction y) so that this space dimension does not affect the results. We use the fluid model: the fluid is considered Newtonian and the flow is laminar; the effect of radiation has not been taken into account. The flow gas in the reactor obeys the conservation laws of mass (continuity equation) Eq. (1), momentum (Navier-Stokes equations) (Eq. (2) and Eq. (3)), and energy diffusion equation Eq. (4). To calculate gas velocity and gas temperature, we use the following nonlinear differential equations:

$$\mathit{\operatorname{div}}\left(\rho \overrightarrow{\boldsymbol{V}}\right)=0$$

(1)

$$\rho U\frac{\partial U}{\partial x}+\rho W\frac{\partial U}{\partial z}=-\frac{\partial p}{\partial x}+\mu \left(\frac{\partial^2U}{\partial {x}^2}+\frac{\partial^2U}{\partial {z}^2}\right)+\rho {g}_x$$

(2)

$$\rho U\frac{\partial W}{\partial x}+\rho W\frac{\partial W}{\partial z}=-\frac{\partial p}{\partial z}+\mu \left(\frac{\partial^2W}{\partial {x}^2}+\frac{\partial^2W}{\partial {z}^2}\right)+\rho {g}_z$$

(3)

$$\rho {C}_p\left(U\frac{\partial T}{\partial x}+W\frac{\partial T}{\partial z}\right)=\lambda \left(\frac{\partial^2T}{\partial {x}^2}+\frac{\partial^2T}{\partial {z}^2}\right)$$

(4)

where U and W are components of the velocity $\overrightarrow{\boldsymbol{V}}$ on directions x and z respectively; T is the temperature of gas. The constants ρ, μ, λ, and C_p, are the volume density, the dynamic viscosity, the heat transfer efficiency, and the specific heat respectively. In this model, the various species considered in the calculations are nine species (k_max = 9): H₂, CH₄, H, CH₃, C₂H₂, C₂H₃, C₂H₄, C₂H₅, and C₂H₆. To resolve the concentration C_k of each radical or molecule (k = 1, 2, …, k_max), we consider the equation of chemical species [38, 39]:

$$\overrightarrow{\boldsymbol{V}}\overrightarrow{\boldsymbol{\operatorname{grad}}}{C}_k={D}_k{\nabla}^2{C}_k+{R}_k$$

(5)

where D_k is the diffusivity of species k and R_k is its rate production (or consumption) in volume.

Table 2 presents the gas-phase reaction mechanism; these reactions were used in works of Olivas-Martinez et al. [15] and Mankelevich et al. [23]. The gas-phase reaction mechanism consists of 13 reactions. We calculate the variations of chemical rate constants K_r with the temperature T for interactions between molecules and radicals. The general form is K_r = A_rT^β exp(−E_a/RT), where A_r and β are constants, E_a is the reaction activation energy, and R is the gas constant. For the units, A_r is in mol cm⁻³ s⁻¹; E_a is in kJ mol⁻¹, T is in K, and R is in J mol⁻¹ K⁻¹.

Table 2 Gas-phase reaction mechanism and corresponding rate constants

The general chemical reaction is expressed as [40]:

$${a}_1{A}_1+{b}_1{B}_1+..\dots \Rightarrow {a}_2{A}_2+{b}_2{B}_2+..\dots$$

(6)

The general form of reaction rate is given by:

$${R}_k={K}_r(T){\left[{A}_i\right]}^{a_i}{\left[{B}_i\right]}^{b_i}.\dots$$

where K_r(T) is the reaction rate constant, [A_i], [B_i] are the species concentrations, and a_i, b_i,..., are stoichiometric coefficients of chemical reactions. The diffusion coefficient D_k of a particle k in a gas can be calculated from the coefficient D_mm’ of binary collisions between two species m and k using the formulas below [41,42,43]:

$${D}_k=\frac{\left({C}_{tot}-{C}_k\right)}{\left(\sum \limits_{m=1,m\ne k}^{k_{\mathrm{max}}}\frac{C_m}{D_{km}}\right)}$$

(7)

where C_tot is the total concentration and D_mk is the diffusion coefficients of the neutral species k in each of the background gases m.

$$D_{mk}=\frac3{16}\frac{\left(4\pi K_bT/2M_{mk}\right)^{1/2}}{C_{tot}{\pi\sigma^2}_{mk}\Omega_D\left(T^\ast\right)}$$

(8)

D_mk is in m²/s, C_tot = P/K_bT in m⁻³, K_b is the Boltzmann constant (J K⁻¹)

M_mk is the reduced mass ${M}_{mk}=\frac{M_m\times {M}_k}{M_m+{M}_k}$; σ_mk is the binary collision diameter ${\sigma}_{mk}=\frac{\sigma_m+{\sigma}_k}{2}$, and it is in Å.

${T}^{\ast }=\frac{T}{\sigma_{mk}}$ with the binary collision diameter ε_mk = (ε_m × ε_k)^1/2 in K. ε_m and ε_k are Lennard-Jones parameters.

$${\Omega}_D\left({T}^{\ast}\right)=\frac{A_D}{T^{\ast {B}_D}}+\frac{C_D}{e^{D_DT\ast }}+\frac{E_D}{e^{F_DT\ast }}+\frac{G_D}{e^{H_DT\ast }}$$

(9)

with A_D = 1.06036, B_D = 0.15610, C_D = 0.19300, D_D = 0.47635, E_D = 1.03587, F_D = 1.52996, G_D = 1.76474, H_D = 3.89411 [41, 42]:

The used values of Lennard-Jones parameters are cited in [44].

In the calculation, we take for the physical constants ρ, μ, λ, and C_p those of mixture gas where the composition is dominant; they are according to data used in [31]. The details of the method or the numerical modeling for the treatment of the problem are presented in the next section.

Numerical model and boundary conditions

The numerical model is based on the finite volume method (FVM) [45]. We discretize the spatial domain of the reactor at discrete nodes (points). Let be a constant step along x axis and let i be the index of the corresponding node; let be a constant step along z axis, and let j be the index of the corresponding node. As it is known in the FVM, we apply an integral around an elementary volume at the level of the node (i,j).

We apply FVM, to calculate the component U by resolving Navier-Stokes equation Eq. (2); the component W is deduced from continuity equation Eq. (1). We apply FVM, to calculate the temperature T by resolving heat equation Eq. (4). We apply FVM also, to calculate the gas-phase species concentrations by resolving diffusion equation Eq. (5). The parameters K_r and R_k are calculated according to the data of the temperature T and the concentrations C_k of the point of coordinates (x, z).

For a fixed value i and a variable value j, the development of Eqs. (2), (4) and (5) is of the form:

$${\alpha}_{i,j-1}^i{X}_{j-1}^i+{\alpha}_{i,j}^i{X}_j^i+{\alpha}_{i,j+1}^i{X}_{j+1}^i={\beta}_j^i$$

(10)

where parameters ${\boldsymbol{\upalpha}}_{\mathbf{j},\mathbf{j}-\mathbf{1}}^{\mathbf{i}}$, ${\boldsymbol{\upalpha}}_{\mathbf{j},\mathbf{j}}^{\mathbf{i}}$, ${\boldsymbol{\upalpha}}_{\mathbf{j},\mathbf{j}+\mathbf{1}}^{\mathbf{i}}$, and ${\boldsymbol{\beta}}_{\boldsymbol{j}}^{\boldsymbol{i}}$are constants and parameters related to physical properties and numerical values of the discretized system.

It is clear to see the matrix form [A] × [X] = [B] of Eq. (10). The matrix [A] has the elements ${\boldsymbol{\upalpha}}_{\mathbf{j},\mathbf{j}-\mathbf{1}}^{\mathbf{i}}$, ${\boldsymbol{\upalpha}}_{\mathbf{j},\mathbf{j}}^{\mathbf{i}}$, ${\boldsymbol{\upalpha}}_{\mathbf{j},\mathbf{j}+\mathbf{1}}^{\mathbf{i}}$ (which are not zero), the matrix (or vector) [B] has the element ${\boldsymbol{\upbeta}}_{\mathbf{j}}^{\mathbf{i}}$, and the matrix (or vector) [X] presents the unknown variable (${\mathbf{T}}_{\mathbf{j}}^{\mathbf{i}}$, ${\mathbf{U}}_{\mathbf{j}}^{\mathbf{i}}$, or ${\mathbf{C}}_{\mathbf{k},\mathbf{j}}^{\mathbf{i}}$). For a fixed value i, this system of coupled equations is easily solved with the Gauss-Seidel (GS) method [46,47,48], and with the solution convergence test for values i and j, we have iterative calculation. For an iteration p, convergence is reached for absolute value ((X^p+1-X^p)/X^p+1) lower than a limit value that is too small. This general form to solve equations is the Gauss-Seidel iterative method (GSIM).

The boundary conditions for components U and W of velocity vector ($\overrightarrow{\mathrm{V}}$), temperature T, and concentrations C_k are as follows:

• In flow (x = 0): U = U₀, W = 0, T = T₀; C_k = C_k,z = 0;

• Out flow (x = L) : ${\left(\frac{\partial U}{\partial x}\right)}_{x=L}=0$,W = 0, ${\left(\frac{\partial T}{\partial x}\right)}_{x=L}=0,\ {\left(\frac{\partial C}{\partial x}\right)}_{x=L}=0;$  

• At filament $\left(x={X}_{fil},z=\frac{Z_{\mathrm{max}}}{2}\right)$: U = 0, W= 0, T = T_fil; the filaments are considered punctual without dimensions;

• At substrates (z = Z or z = Z_max): U = 0, W = 0, T = T_sub, ${\left(\frac{\partial C}{\partial x}\right)}_{x=L}=0 .$

These boundary conditions give the elements of matrix [A] and matrix [B] corresponding to values of i and j.

To avoid the problems of singularities, we choose continuous functions for the initialization of the values of the components of the velocities and the initialization of the values of the temperatures. We choose a parabolic variation for U₀ (as function of z) and a linear variation for T_sub (as function of x) near the entrance of the reactor. To have the initialization of concentrations, we calculate them for each temperature from 300 to 2100 K; we use GSIM to solve equations for concentrations C_k,0(i,j):

$${R}_k=\sum \left(\pm {K}_r(T){\left[{A}_m\right]}^{\alpha }{\left[{B}_{m\hbox{'}}\right]}^{\beta }..\dots \right)=0$$

(11)

We developed a numerical program using FORTRAN language; the general numerical scheme for calculation of T, U, W, and C_k is summarized in three stages as follows:

• Stage 1: Initialization

- Initialization of T, U, W with continues values and according to boundary conditions

• Stage 2: Calculation of T, U, W without variation of C_k

By a loop on index i, we have to:

• (a) Calculate T by GSIM

• (b) Calculate U by GSIM and correction of flux

• (c) Calculate W from continuity equation

• (d) Return to step (a) until convergence for all values i and j

• Stage 3: Calculation of T, U, W, and C_k

The calculated values on step 2 are initial values for this step. By a loop on index i, we have to:

• (a) Initialize T, U, W, and C_k

• (b) Calculate T by GSIM

• (c) Calculate U by GSIM and correction of flux

• (d) Calculate W from continuity equation

• (e) Calculate K_reac and D_k as a function of z (or index j)

• (f) Calculate C_k for different k particles by GSIM

• (g) Return to step (b) until convergence for all values i and j

• (h) Take final results and calculate different parameters

Results and discussion

In this section, we present temperatures, velocities, and concentrations of molecules and radicals for a reactor with three filaments (n = 3) and four filaments (n = 4). Data values are P₀ = 10³ Pa (7.5 Torr), U₀ = 10 mm/s, T_fil = 2100 K, T₀ = 300 K, and T_sub = 1173 K. Most results are for dimensions of the reactor: Z_max = 20 mm, L = 240 mm, D_ff = 50 mm for n =3 and Z_max = 20 mm, L = 240 mm, D_ff = 33.3 mm for n = 4. The composition of the gas mixture is 05% of CH₄ and 95% of H₂. These values of filament temperature (2100 K) and gas mixture rate (05%, 95%) are according to data used in [15, 16, 31].

Distributions of velocity and temperature

In this subsection, we present distributions of temperature T and distributions of velocity components U and W for the cited values. A few characteristic planes are defined. The plane of the filaments is the horizontal plane carrying the filaments for z = Z_max/2; the plane of a filament is the vertical plane containing a filament (x = 7 cm, 11.99 cm, 16.98 cm or n = 3 filaments and x = 8.66 cm, 11.99 cm, 15.30 cm, 18.61 cm for n = 4 filaments). We also define the vertical planes between filaments (example x = 9.5 cm for n = 3 filaments).

For n = 3, the calculations show that the temperature distributions at the hot filaments are identical; similarly, the temperature distributions between the hot filaments are identical. For n = 4, we have very similar curves and the conclusions are also identical. Figure 2 shows the distribution of the gas temperature as a function of z between filaments and at each filament for three and four filaments. The temperature distributions are symmetrical. The temperature takes a maximum value at the center of the reactor (T_fil = 2100 K, z = Z_max/2) when the gas comes near to the hot filaments along the z direction, and then decreases gradually when the gas goes far away from the filaments (Fig. 2, at filaments). Between the filaments, temperature increases from 3 to 4 filaments by 11.46% (Fig. 2, between filaments). The temperatures in the central regions, around filaments, are more than the temperatures far from the filament region; this result confirms other results [49].

Fig. 2

Distribution of T as a function of z between filaments and at the filaments for n = 3 and for n = 4

Figure 3 shows the distribution of the gas temperature as a function of x for horizontal filament plane (z = Z_max/2) for n = 3 and n = 4. The symmetry at the filaments and between the filaments is confirmed. The zones between filaments have a remarkable periodicity. The difference at the inlet of the reactor and the outlet is due to the initial conditions and the conditions of the fluid flow. Indeed, at the inlet of the reactor, the temperature of the gas is T₀ = 300 K and that of the substrate is T_sub = 1173 K; we have assumed a linear variation as a function of x at the first zone of the substrates. This choice of continuous function avoids the problems of singularities when solving the differential equations.

Fig. 3

Distribution of T as a function of x for z = Z_max/2 and for n = 3 and n = 4

As for the temperature distributions, the calculations show that the distributions of components u and W of velocity as function of z at the filaments are identical. These distributions between the filaments are also identical. Figure 4 shows distribution of the component U of the velocity as a function of z between filaments and at the filaments for three filaments and four filaments. The increased gradient between the filaments was much larger than that in the filaments; the maximum value of velocity U_max is at 15.23 mm/s (for n = 3) and at 14.96 mm/s (for n = 4). Velocity distribution has near parabolic form far from filaments. For exact parabolic form U_max = (3/2)U₀. The velocity reaches zero values on filaments by the severe interaction between filaments and the flow of viscous gas [31]. Between filaments, the value of U_max for 3 filaments is greater than that for 4 filaments by 2.98%. These results were consistent with what was found by Goodwin and Gavillet [16]: velocity distribution has a parabolic form far from filaments.

Fig. 4

Distribution of U as a function of z, between filaments for n=3 (a) and for n=4 (b) and at filaments for n=3 (c) and for n=4 (d)

For three filaments and four filaments, the distribution of the component W of the velocity as a function of z is also symmetrical, between filaments and at the filaments.

Distributions of concentrations

In this subsection, we mainly present the concentrations of molecules and radicals in the reactor for n = 3 and n = 4.

Concentration of CH₃, H, and C₂H₅

Calculations show that the curve of concentration of each radical CH₃, H, or C₂H₅ is symmetrical and perfectly identical at the filaments. Between filaments, the curves of concentrations of these radicals are also symmetrical and identical.

Figure 5 shows the distributions of CH₃ radical concentration as function of z at the filaments and between the filaments for n = 3 and for n = 4. The curves are symmetrical. At the filaments, the comparison shows that the concentration for n = 3 is higher than that for n = 4; the difference varies from 1.59 to 1.79%. Between the filaments, the concentrations for n = 4 is higher than that for n = 3, and it is from 3.95 to 2.24%.

Fig. 5

Distribution of concentration of CH₃ as function of z; at the filaments and between the filaments

The calculation of concentrations of H, as a function of z at the filament for n = 3 and n = 4, shows that the concentrations have approximate values; the concentration for n = 4 is higher than that for n = 3, and the difference is from 1.31 to 0.86%. Between the filaments, the concentration for n = 3 is higher than that for n = 4; the difference is from 3.13 to 2.95%.

For C₂H₅ radical, the concentration for n = 3 is higher than that for n = 4; the difference is from 5.66 to 1.25% at the filaments. Between the filaments, the concentration for n = 4 is higher than that for n = 3; it varies from 61 to 17%.

Concentrations of CH₄, H₂, and other species C₂H₆, C₂H₄, C₂H₂, C₂H₃

Our calculations show that the distributions of concentration of C₂H₆, C₂H₄, C₂H₂, and C₂H₃, as a function of x, for filaments’ plane (z = Z_max/2) and substrates (z = 0, z = Z_max) for n = 4 have the same remarks concerning forms and the symmetry as distributions of CH₃, H, and C₂H₅.

As it has shown before, the temperature curves are identical and superimposable at the level of the vertical planes containing the filaments (at filaments). At the median vertical planes between two successive filaments, the temperature curves or distributions are identical and superimposable. We have homogeneity or periodicity of the zones between two successive filaments. The distributions of velocity components U and W have the same shapes respectively component to component.

The distributions of the concentrations C_k of all the species (k = 1,…, 9), as a function of z, have respectively identical curves at vertical planes containing the filaments (at filaments). We note the same remark for median vertical planes between filaments; there is homogeneity or periodicity of the zones between two successive filaments. This result is also linked to the choice of entry conditions for the gas mixture (U₀ and P₀) relative to the dimensions of the reactor (L, Z_max, D_ff). This choice gives a periodicity mainly for T, U, and W. These results have the same remarks for the concentrations which results are mainly dependent on the reaction constants K_reac and the diffusion coefficients D_k; the effect of temperature (filaments and substrates) is the most important. Debroy et al. [17] concluded that diffusion mechanisms are mostly responsible for both heat and mass transfer inside the HFCVD reaction chamber.

Figure 6 shows the concentrations C_k of the different molecules and radicals as a function of z, four n = 4, between the filaments (Fig. 6a–c) and at the filaments (Fig. 6b–d). There is an order in the presence of different species in the reactor; the molecules of H₂ and CH₄ are dominant. For CH₄ and between the filaments, the concentration for n = 3 is higher than that for n = 4; the difference is from 4.10 to 2.31%. At the filament, for n = 3 and n = 4, the concentrations have approximate values; the difference is about 1.21 to 0.73%.

Fig. 6

Concentrations C_k of molecules and radicals as a function of z, for n = 4, a, c between the filaments, b, d at the filaments

The third species is ethane C₂H₆, in the order of 10¹⁸ m⁻³. For the fourth and fifth order, there are C₂H₄ and C₂H₅ with close concentration values (10¹⁶ m⁻³ at 10¹⁵ m⁻³). Contrary to the plasma enhanced chemical vapor deposition (PECVD) processes where the concentration of methyl CH₃ is high, here in HFCVD processes, the concentration of ethyl C₂H₅ is higher than that of CH₃. Other works [18,19,20] give concentration of CH₃ higher than those of C₂H₆, C₂H₄, and C₂H₅; this result is different from ours. We have shown in a conference presentation that in HFCVD processes, temperature is too high and the reaction rates which give heavy elements (C₂H₆, C₂H₄, C₂H₅) are more important. Indeed, chain reactions will allow consumption of CH₃ radicals by the reactions R6 and R7. We have in the sixth order CH₃ with a concentration less than 10¹⁵ m⁻³. Although the hydrogen molecule H₂ is dominant, the atomic hydrogen H has a low concentration.

The initial concentrations of H₂ and CH₄ are respectively 2.29 × 10²³ m⁻³ and 1.20 × 10²² m⁻³, so 95% and 5% respectively. Inside the reactor, at the filaments’ plane (z = Z_max/2), we note approximate rate of H₂ and CH₄ of 98.1% and 1.9% respectively; there is production of H₂ and consumption of CH₄. We note also approximate productions of C₂H₆ and C₂H₄, about 2.3 × 10⁻³% and 3.6 × 10⁻⁵% respectively. It is in the seventh order with a concentration of around 10¹⁵ m⁻³. The concentration of C₂H₃ varies from 1.28 × 10⁸ to 9.48 × 10⁸ m⁻³. In the last position, we have the radical C₂H₂ with a very low concentration (less than 10⁻² m⁻³).

Table 3 gives the average values of the concentrations of the different species near the substrates (z = 0 and z = Z_max) and at the plane of the filaments (z = Z_max/2); the average values are taken on the x range from 50 mm to 230 mm (to avoid the effects of the edges).

Table 3 Average concentrations of species at the substrates and at the plane of the filaments

For real experiments, at the outlet of the reactor, the radicals C₂H₅, CH₃, and C₂H₃ and H should give stable molecules (H₂, CH₄, C₂H₆, C₂H₄, and C₂H₂). We think that this effect will not have too much influence on the results of our calculations nor the generality of the model used.

Effects of number of filaments and distance filaments-substrates

As we mentioned, the calculation shows the homogeneity and periodicity between the regions for n = 3 and for n = 4.

Table 4 shows the effect of number of filaments on concentrations of radicals H, CH₃, and C₂H₅. As the number of filaments increases, the distance between filaments decreases; heating energy input increases and the temperature of the gas increases. We observe that the production of CH₃ and C₂H₅ near the surface increases and at the same time the growth rate of the thin film will increase.

Table 4 Effect of the number of filaments on concentrations

Table 5 presents the effect of distance filaments-substrates Z_max/2 on concentrations of radicals H, CH₃, and C₂H₅ for n = 4. The values of U₀ are taken to have similar flow velocities (U_max is of the order of 15 mm/s). The maximum values of the speeds and the minimum values of the temperatures are calculated at Z_max/2 and between filaments; the values of the concentrations are the average values taken for x varying from 50 to 230 mm. If the distance filaments-substrates increase, the average concentrations of CH₃ and C₂H₅ increase and the average concentrations of H decreases. The effects of chemical reactions and temperature in volume are the most important.

Table 5 Effect of distance filaments-substrates on concentrations for n = 4

Concentrations of CH₃, C₂H₅, and H near the substrates

In our calculations of the concentrations in volume, we did not consider the deposition and the interaction of particles with the surface. This assumption will not significantly influence concentrations in the volume, the amount deposited is too low compared to the initial amount: this is the effect of the bath in physics.

The temperature of the substrate is T_sub = 1173 K. Near the substrates, the average concentrations of H and CH₃ are practically constant for n = 3 and n = 4; the concentration ratio C_H/(C_H + C_CH3) is 20%. The concentration of C₂H₅ is 4.5 greater than that of H and CH₃ for n = 3, and it is 5.1 greater than that of H and CH₃ for n = 4.

The type of substrates plays an important role in the growth of thin film. A growth rate can be estimated by proposing a diamond layer growth with a given sticking coefficient. We consider sticking coefficients S_H = 0.01, S_CH3 = 0.11, and S_C2H5 = 0.07 of H, CH₃, and C₂H₅ respectively [16, 32, 50]. For particles k of mass m_k at temperature near substrate T_nS = T_sub, we calculate the most probable thermal velocity [51]:

$${V}_k-=\sqrt{\frac{2{K}_b{T}_{ns}}{m_k}}$$

(12)

For diamond carbon thin film, we can estimate the value of growth rate G_k resulting from the radical k. An approximate formula of G_k is:

$${G}_k=\frac{1}{d}{C}_{k, av}\times \left(\frac{1}{6}{V}_k\right)\times {S}_k\times {m}_{kd}$$

(13)

where, for radicals H, CH₃, and C₂H₅, C_k,av is the average concentrations, S_k is the sticking coefficient, and m_kdis the mass of the deposited radical (m_kd is not necessarily m_k, because of the hydrogen atoms which release sites from the surface). d is volumetric mass density of diamond carbon.

Table 6 presents values of growth rate G, from some values of U₀ and some values of distance filaments-substrates Z_max/2.

Table 6 Values of growth rate G for some values of Z_max

For average value d = 3.51 g/cm³ and with two radicals H and CH₃, for the same values Z_max = 20 mm U₀ = 10 mm/s, calculated maximum values of growth rate G_max are 0.0022 μm/h for n = 3 and 0.00221 μm/h for n = 4. For a given number of filaments (example n = 4), when Z_max decreases, there is not necessarily an increase in the growth rate. These results are smaller than other theoretical and experimental works. The values of growth rate are about 0.4 μm/h in work [18] and less than 0.2 μm/h in [20]. Taking into account the presence of C₂H₅, for same values Z_max = 20 mm U₀ = 10 mm/s, calculated deposition rates (0.0135 μm/h for n = 3 and 0.0151 μm/h for n = 4) are lower than the values recorded in the experiments.

We can use the Hertz- Knudsen equation [52, 53] to estimate the flux of particles which collide with surface. For a particle of mass m, at temperature T, and in pressure P, the flux φ is [52]:

$$\varphi =\frac{P}{\sqrt{2\pi \times m\times {K}_b\times T}}$$

(14)

From values of flux, growth rate can be estimated for a sticking coefficient S_k and for diamond carbon density d.

The comparison of growth rate is not so reliable because of the differences in volume concentrations (gas temperature) and velocities of radicals near surfaces (substrate temperature). The expression of growth rate G used here and in other expressions do not respond to the complexity of the phenomena of deposition. Knowledge of the sticking coefficients of radicals is not sufficient, nor even the coefficients of all radicals and molecules. It would be important to know the rate of coverage of the surface with hydrogen atoms. Otherwise, it is necessary to know the free sites occupied by the hydrogen atoms. A calculation taking into account the recombination coefficients and sticking coefficients of all the radicals and molecules, as well as the occupancy rate in H, would give a more reliable result and would be close to the experimental results. Experimental measurements of the production of stable molecules (C₂H₆, C₂H₄, and C₂H₂) would make it possible to better understand the mechanisms and correct the different models.

Conclusions

In this work, we have presented a two-dimensional model for HFCVD for CH₄/H₂ gas mixture in a reactor with three and four filaments. This model describes the phenomena that occur in the reaction chamber of a reactor and the effect of the number of filaments and their position in relation to the substrate. The proposed reactor gives identical distributions of temperatures, velocities, and concentrations at the filaments and also between the filaments; the different regions between filaments show great homogeneity and remarkable periodicity. The results show that if the temperature of gas increases, the production of CH₃ and C₂H₅ near the surface increases and the growth rate of the thin film increases. The calculations show that concentrations of C₂H₆, C₂H₄, and C₂H₅ are very high. The diffusion phenomena related to temperatures and the rates of chemical reactions also related to temperatures constitute the most important aspects of the dynamics of concentrations in gas.

The study of the growth of the thin film is very complex. In fact, it requires more detailed studies on the coefficients of sticking and recombination of the species and on the rate of occupation of the sites by the hydrogen atom at the deposit surface. Experimental measurements of the production of stable molecules (C₂H₆, C₂H₄ and C₂H₂) would make it possible to better understand the mechanisms.

Nomenclature

[A_i], [B_i] Species concentrations, m⁻³

a_i, b_i Stoichiometric coefficients of chemical reactions

A_r  Constant, mol cm⁻³ s ⁻¹

C_k Concentration of each radical or molecule, m⁻³

C_k,av Average concentrations, m⁻³

C_p Specific heat, J kg⁻¹ K⁻¹

C_tot Total concentration, m⁻³

d Volumetric mass density, g/cm³

D_ff Distance between filaments, mm

D_k Diffusivity of species, m⁻³

D_mk Diffusion coefficients of the neutral species k, m²/s

E_a Activation energy, K_J mol⁻¹

G_k Growth rate, μm/h

g_x Gravity force component in the x direction, m s⁻²

g_z Gravity force component in the z direction, m s⁻²

k Number radical or molecule

K_b Boltzmann constant, J K⁻¹

K_r Chemical rate constants, m³/s

L Length of the reactor in the x direction, mm

m_kd Mass of the deposited radical, g

M_mk Reduced mass, kg

n Number of filaments

N_a Avogadro constant, mol⁻¹

P Gas pressure, Pa

P₀ Initial pressure, Pa

R Universal gas constant, J mol⁻¹ K⁻¹

R_k Rate production, m⁻³ s⁻¹

S_k Sticking coefficients

T₀ Initial temperature, K

T Temperature of gas, K

T_fil Filament temperature, K

T_nS Temperature near substrate, K

T_sub Substrate temperature, K

U₀ Initial velocity, mm/s

U Velocity component in the x direction, mm/s

$\overrightarrow{\mathrm{V}}$ Gas velocity vector, mm/s

V_k Most probable thermal velocity of particle k

W Velocity component in the z direction, mm/s

x Cartesian axis directions, mm

X_fil Filament position

z Cartesian axis directions, mm

Z_max Distance between substrates, mm

Greek symbols

β Constant

μ  Dynamic viscosity, N s m⁻²

ρ₀ Volume density, m⁻³

λ Heat transfer efficiency, W m⁻¹ K⁻¹

σ_mk Binary collision diameter, Å

ε_m,ε_k Lennard-Jones parameters K

φ Flux of the gas molecules, m⁻² s⁻¹

Special symbols

i Unitary vector in the x direction

j Unitary vector in the z direction

$\overrightarrow{\operatorname{grad}}$ Differential operator

div Differential operator

$\frac{\partial }{\partial x},\frac{\partial }{\partial z}$ Partial derivatives

∇ Nabla mathematical operator

[A],[X],[B] Matrix for Gauss-Seidel calculation

${\upalpha}_{\mathrm{j},\mathrm{j}-1}^{\mathrm{i}}$, ${\upalpha}_{\mathrm{j},\mathrm{j}}^{\mathrm{i}}$, ${\upalpha}_{\mathrm{j},\mathrm{j}+1}^{\mathrm{i}}$,${\upbeta}_{\mathrm{j}}^{\mathrm{i}}$ Elements of matrix [A] and the matrix [B]

Ω_D(T^*), A_D, B_D,C_D, D_D, E_D, F_D, G_D, H_D Dimensionless collision integral function of the temperature and its parameters