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Numerical calculation of temperature, velocity, and species concentration distributions in a hot filament chemical vapor deposition in a reactor using a CH_{4}/H_{2} mixture
Journal of Engineering and Applied Science volume 69, Article number: 73 (2022)
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 twodimensional 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 CH_{3} and C_{2}H_{5} radicals near the surface; there was also an increase in the growth rate of the thin film. The concentrations of C_{2}H_{6}, C_{2}H_{4}, and C_{2}H_{5} were very high. Temperature and concentrations were affected by the distance between filaments and the distance filamentssubstrates.
Introduction
Thin films based on diamond or on hydrogenated amorphous carbon (aC:H) present many technological interests: aC: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: LightEmitting Diode), electrochemical detectors, bioelectronic and bioelectrochemical applications [5,6,7,8].
These thin films can be produced by chemical vapor deposition (CVD) using a CH_{4}/H_{2} gas mixture, in which H_{2} is dissociated into atomic H and CH_{4} undergoes pyrolysis reactions leading to the formation of radicals such as CH_{3} and CH_{2} and stable species such as C_{2}H_{2}, C_{2}H_{4}, and C_{2}H_{6} [9]. Activation of chemical reactions can occur at elevated temperatures, up to 3000 K in plasmachemical processes [10]. The radicals H and CH_{3} 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 onedimensional numerical model for the diamond growth. They calculated temperature, velocity, and species concentration profiles. Debroy et al. [17] developed the first twodimensional 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 CH_{3}, and the growth rate of thin layers of diamonds [18,19,20]. Onedimensional models [21, 22] were developed to understand the deposition processes and the gasphase behavior. Other numerical studies about two and threedimensional 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, gasphase 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 CH_{4}/H_{2} gas mixture. Spatial distributions of the gas temperature, fluid velocity, and chemical species concentrations have been calculated in a twodimensional HFCVD system for CH_{4}/H_{2} 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}/CH_{4}/H_{2} gas mixtures in the HFCVD reactor. They used a twodimensional computer model to calculate the gasphase composition within the reactor. Using experimental and calculated data, they showed the competition between H atoms, CH_{3} 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), C_{2}H_{x} (x: 0–6), C_{3}H_{x} (x: 0–2), C_{4}H_{x} (x: 0–2), H, and H_{2} on clean and hydrogenated diamond. They used the molecular dynamic simulation method.
In recent work, Tao Zhang et al. [35] presented a threedimensional 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 gasphase reactions on characteristics of the flow in a heated cylindrical channel of a gas mixture CH_{4}/H_{2} reactor with using the direct simulation Monte Carlo method which is a method based on solving the NavierStokes 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 twodimensional model for HFCVD, for a gas mixture CH_{4}/H_{2}. 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 twodimensional 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: NavierStokes 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 CH_{4}/H_{2} 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 filamentssubstrates. In this fourth section, we also present concentrations of CH_{3}, C_{2}H_{5}, 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 CH_{4}/H_{2} enters the reactor with a constant flow in the x direction. The initial composition of the mixture gas is 5% of methane CH_{4} and 95% of hydrogen H_{2}. The given initial temperature, velocity, and pressure are respectively T_{0}, U_{0}, and P_{0}. The volumetric mass density ρ_{0} is given according to ideal gas state equation as ρ_{0} = P_{0}/RT_{0}, 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.
For the mathematical model, we consider a gas mixture in twodimensional 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 (NavierStokes 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:
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): H_{2}, CH_{4}, H, CH_{3}, C_{2}H_{2}, C_{2}H_{3}, C_{2}H_{4}, C_{2}H_{5}, and C_{2}H_{6}. 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]:
where D_{k} is the diffusivity of species k and R_{k} is its rate production (or consumption) in volume.
Table 2 presents the gasphase reaction mechanism; these reactions were used in works of OlivasMartinez et al. [15] and Mankelevich et al. [23]. The gasphase 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_{r}T^{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}.
The general chemical reaction is expressed as [40]:
The general form of reaction rate is given by:
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]:
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} is in m^{2}/s, C_{tot} = P/K_{b}T 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 LennardJones parameters.
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 LennardJones 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 NavierStokes 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 gasphase 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:
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 GaussSeidel (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 GaussSeidel 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 = U_{0}, W = 0, T = T_{0}; 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 U_{0} (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):
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 P_{0} = 10^{3} Pa (7.5 Torr), U_{0} = 10 mm/s, T_{fil} = 2100 K, T_{0} = 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 CH_{4} and 95% of H_{2}. 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].
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 T_{0} = 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.
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)U_{0}. 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.
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 CH_{3}, H, and C_{2}H_{5}
Calculations show that the curve of concentration of each radical CH_{3}, H, or C_{2}H_{5} 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 CH_{3} 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%.
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 C_{2}H_{5} 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 CH_{4}, H_{2}, and other species C_{2}H_{6}, C_{2}H_{4}, C_{2}H_{2}, C_{2}H_{3}
Our calculations show that the distributions of concentration of C_{2}H_{6}, C_{2}H_{4}, C_{2}H_{2}, and C_{2}H_{3}, 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 CH_{3}, H, and C_{2}H_{5}.
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 (U_{0} and P_{0}) 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 H_{2} and CH_{4} are dominant. For CH_{4} 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%.
The third species is ethane C_{2}H_{6}, in the order of 10^{18} m^{−3}. For the fourth and fifth order, there are C_{2}H_{4} and C_{2}H_{5} with close concentration values (10^{16} m^{−3} at 10^{15} m^{−3}). Contrary to the plasma enhanced chemical vapor deposition (PECVD) processes where the concentration of methyl CH_{3} is high, here in HFCVD processes, the concentration of ethyl C_{2}H_{5} is higher than that of CH_{3}. Other works [18,19,20] give concentration of CH_{3} higher than those of C_{2}H_{6}, C_{2}H_{4}, and C_{2}H_{5}; 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 (C_{2}H_{6}, C_{2}H_{4}, C_{2}H_{5}) are more important. Indeed, chain reactions will allow consumption of CH_{3} radicals by the reactions R6 and R7. We have in the sixth order CH_{3} with a concentration less than 10^{15} m^{−3}. Although the hydrogen molecule H_{2} is dominant, the atomic hydrogen H has a low concentration.
The initial concentrations of H_{2} and CH_{4} are respectively 2.29 × 10^{23} m^{−3} and 1.20 × 10^{22} m^{−3}, so 95% and 5% respectively. Inside the reactor, at the filaments’ plane (z = Z_{max}/2), we note approximate rate of H_{2} and CH_{4} of 98.1% and 1.9% respectively; there is production of H_{2} and consumption of CH_{4}. We note also approximate productions of C_{2}H_{6} and C_{2}H_{4}, about 2.3 × 10^{−3}% and 3.6 × 10^{−5}% respectively. It is in the seventh order with a concentration of around 10^{15} m^{−3}. The concentration of C_{2}H_{3} varies from 1.28 × 10^{8} to 9.48 × 10^{8} m^{−3}. In the last position, we have the radical C_{2}H_{2} 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).
For real experiments, at the outlet of the reactor, the radicals C_{2}H_{5}, CH_{3}, and C_{2}H_{3} and H should give stable molecules (H_{2}, CH_{4}, C_{2}H_{6}, C_{2}H_{4}, and C_{2}H_{2}). 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 filamentssubstrates
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, CH_{3}, and C_{2}H_{5}. 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 CH_{3} and C_{2}H_{5} near the surface increases and at the same time the growth rate of the thin film will increase.
Table 5 presents the effect of distance filamentssubstrates Z_{max}/2 on concentrations of radicals H, CH_{3}, and C_{2}H_{5} for n = 4. The values of U_{0} 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 filamentssubstrates increase, the average concentrations of CH_{3} and C_{2}H_{5} increase and the average concentrations of H decreases. The effects of chemical reactions and temperature in volume are the most important.
Concentrations of CH_{3}, C_{2}H_{5}, 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 CH_{3} are practically constant for n = 3 and n = 4; the concentration ratio C_{H}/(C_{H} + C_{CH3}) is 20%. The concentration of C_{2}H_{5} is 4.5 greater than that of H and CH_{3} for n = 3, and it is 5.1 greater than that of H and CH_{3} 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, CH_{3}, and C_{2}H_{5} 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]:
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:
where, for radicals H, CH_{3}, and C_{2}H_{5}, C_{k,av} is the average concentrations, S_{k} is the sticking coefficient, and m_{kd}is 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 U_{0} and some values of distance filamentssubstrates Z_{max}/2.
For average value d = 3.51 g/cm^{3} and with two radicals H and CH_{3}, for the same values Z_{max} = 20 mm U_{0} = 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 C_{2}H_{5}, for same values Z_{max} = 20 mm U_{0} = 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]:
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 (C_{2}H_{6}, C_{2}H_{4}, and C_{2}H_{2}) would make it possible to better understand the mechanisms and correct the different models.
Conclusions
In this work, we have presented a twodimensional model for HFCVD for CH_{4}/H_{2} 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 CH_{3} and C_{2}H_{5} near the surface increases and the growth rate of the thin film increases. The calculations show that concentrations of C_{2}H_{6}, C_{2}H_{4}, and C_{2}H_{5} 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 (C_{2}H_{6}, C_{2}H_{4} and C_{2}H_{2}) 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/cm^{3}
D_{ff} Distance between filaments, mm
D_{k} Diffusivity of species, m^{−3}
D_{mk} Diffusion coefficients of the neutral species k, m^{2}/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, m^{3}/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
P_{0} Initial pressure, Pa
R Universal gas constant, J mol^{−1} K^{−1}
R_{k} Rate production, m^{−3} s^{−1}
S_{k} Sticking coefficients
T_{0} Initial temperature, K
T Temperature of gas, K
T_{fil} Filament temperature, K
T_{nS} Temperature near substrate, K
T_{sub} Substrate temperature, K
U_{0} 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}
ρ_{0} Volume density, m^{−3}
λ Heat transfer efficiency, W m^{−1} K^{−1}
σ_{mk} Binary collision diameter, Å
ε_{m},ε_{k} LennardJones 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 GaussSeidel 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
Availability of data and materials
All data generated or analyzed during this study are included in this published article and are available from the corresponding author.
Abbreviations
 aC:H:

Amorphous hydrogenated carbon
 HFCVD:

Hot filament chemical vapor deposition
 CH_{4} :

Methane gas
 H_{2} :

Hydrogen gas
 LED:

Lightemitting diode
 CVD:

Chemical vapor deposition
 PECVD:

Plasma enhanced chemical vapor deposition
 FVM:

Finite volume method
 av:

Average
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Acknowledgements
Our acknowledgments go to the responsible in charge of the Computer Resources Center (CRI) of Kasdi Merbah Ouargla University (UKMO) where part of the calculations were made. The authors acknowledge also the Directorate General of Scientific Research and Technological Development (DGRSDT) and the Thematic Science and Technology Research Agency (ATRST) who have given their support to the Laboratory LRPPS.
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FK proposed the subject of this work which is the principal axis NK’s doctoral thesis under the supervision of FK. The modelization, the interpretation, and the redaction of the manuscript were done by NK, FK, and OB. The numerical methods and the calculations are realized by NK. All authors read and approved the final manuscript.
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Khelef, N., Khelfaoui, F. & Babahani, O. Numerical calculation of temperature, velocity, and species concentration distributions in a hot filament chemical vapor deposition in a reactor using a CH_{4}/H_{2} mixture. J. Eng. Appl. Sci. 69, 73 (2022). https://doi.org/10.1186/s44147022001214
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DOI: https://doi.org/10.1186/s44147022001214