{'fraction_non_alphanumeric': 0.028326133318751025, 'fraction_numerical': 0.01607699458631815, 'mean_word_length': 4.855907780979827, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Numerical Modeling of Non-equilibrium Plasma Discharge of Hydrogenated Silicon

doi: 10.5829/ije.2020.33.08b.01


A B S T R A C T
In this work, we model a radiofrequency discharge of hydrogenated silicon nitride in a capacitive coupled plasma reactor using Maxwellian and non-Maxwellian electron energy distribution function.
The purpose is to investigate whether there is a real advantage and a significant contribution using non-Maxwellian electron energy distribution function rather than Maxwellian one for determining the fundamental characteristics of a radiofrequency plasma discharge. The results show the evolution of the non-Maxwellian electron energy distribution function, the mobility and the diffusion coefficient required to determine the fundamental characteristics of the radiofrequency plasma discharge of a hydrogenated silicon nitride deposit at low pressure and low temperature, between the two electrodes of the capacitive coupled plasma reactor. By comparing these results using non-Maxwellian electron energy distribution function with those calculated using the Maxwellian one, we conclude that the use of non-Maxwellian electronic energy distribution function is more efficient for describing the evolution of a radiofrequency plasma discharge in a capacitive reactor, which will improve the quality of the deposition of thin films.

INTRODUCTION
Hydrogenated silicon nitride has attracted the interest of researchers over the last decades because of its wide application in thin films [1,2]. Hydrogenated amorphous silicon nitride films can be deposited in a capacitive coupled plasma (CCP) reactor because of high electron density and low electron temperature, leading to high-rate depositions of silicon films [3,4]. The plasma discharge in CCP reactor combines several phenomena during deposition: creation of electrical charges by radiofrequency (RF) excitation, fluid flow and chemical reactions on the surface.
The RF plasma discharge is a method that allows obtaining a plasma medium with specific parameters, both geometrically and in terms of the fluid itself. Moreover, it is a method well suited to the description of non-linear phenomena. RF discharges attracted the attention of physicists before their practical applications in the field of physics were known [5][6][7][8]. The capacitive RF discharge allows to define a frequency range between 1 and 100 MHz, gas pressure between 1 mTorr and few Torrs with a dimension of about a few cm and a power between 1 and 10 watts. Under these conditions, the RF discharge plasma is weakly ionized and out of thermodynamic equilibrium. The temperature value of the electrons depends in particular on the gas pressure and the discharge conditions.
In order to obtain a high quality of deposition, it is necessary to seek the better characteristics of the plasma discharge in CCP reactor. The fluid model can be used to determine the characteristics of the plasma discharge. The resolution of the fluid model equations is usually done by assuming a Maxwellian electron energy distribution function (EEDF). However, in the sheath regions of the fluid model of a RF discharge, the electron temperature is much higher than the temperature of heavy particles. That is why it is more likely that the real EEDF is non-Maxwellian because of its capability to describe an out-of-equilibrium system with high density and temperature gradients compared to the EEDF Maxwellian one.
Much attention is devoted to the non-equilibrium EEDF, because of their applications in many technological fields [1,9]. Godyac (1990Godyac ( ,1995 has shown the existence of two electron groups, characterized by a weak inter-interaction, a significative different temperatures and their energy distribution can be a bi-Maxwellian EEDF [10,11]. Hageleer and Pitchford (2005) [12] developed a new resolution model of the Boltzmann equation (BE) to obtain transport coefficients and diffusion coefficients applicable for fluid models describing much more general conditions.
In this work, we are interested in studying the fundamental characteristics of the capacitive RF plasma discharge of silicon nitride using a non-Maxwellian EEDF which we calculated based on the Boltzmann model developed by Hageleer and Pitchford [12]. Thus, we intend to validate the choice of using the non-Maxwellian EEDF. Then, we focus on comparing the plasma discharge characteristics obtained using the non-Maxwellian EEDF with those calculated by a Maxwellian distribution function.
This article is organized as follows. In Section 2, we briefly describe the hypothesis and mathematical formulations used for determining non-equilibrium EEDF and transport coefficients. In Section 3, we describe hybrid physical model of plasma discharge using a non equilibrium EEDF. In Section 4, we present and compare the simulation results obtained by using a non-Maxwellian and Maxwellian EEDF. Finally, the conclusion is set out in Section 5.

1. EEDF Calculation
In general, the transport of charged species in the plasma can be described by the Boltzmann equation: which is a continuity equation in phase space (r, v). The right side of equation (1) is the collision operator, which describes the interaction between the particles in the system. In our adopted model, we have three approximations: -The initiating electric field and the probability of collision are uniform.
-The distribution of the electrons is then symmetrical around the direction of the electric field.
-In position, the electron distribution space can only change in the direction of the applied electric field.
Using the spherical coordinates in velocity space and choosing the direction z we obtain: The function f in Equation (2) depends on four variables: z, v, θ and t. Using the two-term approximation [12][13][14] we can simplify the dependence of the function f on θ to obtain Equation (3): The function f is the sum of a part isotrope f0 and a small anisotropic perturbation f1 [12,15].
According to the work [13,15] we can separate the energy dependence of f from its dependence on time and space by assuming that : A further simplification on f in Equation (4) can be made with respect to time. Indeed, the electric field and the distribution of electrons can be considered to be stationary or oscillating at high frequency.
Based on the previous simplifications and the collision term calculation, Hageleer and Pitchford [12] have shown Equation (5).
The term to the left of Equation (5) represents a convection part with negative flow velocity W and a diffusion part with coefficient D .
The source term S represents the energy loss due to inelastic collisions. It can be broken down into three parts: attachment, excitation and ionization.
The term λ is a scalar renormalization factor, which ensures that the EEDF has the following property: The left side of Equation (3) is discretized by the exponential scheme [15].

Mobility and Electrons Diffusion
The knowledge of the distribution function has allowed us to determine a number of important parameters, such as the ionization rate by collision of electrons, the mobility of electrons, the diffusion rate, the average energy of electrons, etc.
For the electrons transport, the mobility and diffusion of electrons are given by Equations (7) and (8): For the energy transport, the mobility and diffusion of electrons are given by Equations (9) and (10):

Hybrid Physical Model
Like other reactors used in industry [16][17][18][19][20][21], the CCP reactor involves a set of fairly complex physico-chemical processes. The physical model must describe the dynamics of the plasma medium. In addition, it must take into account chemical reactions that best describe the source terms. We used a macroscopic-microscopic hybrid model where the fluid equations describe the kinetics of the system and the source terms are computed from the non-equilibrium EEDF.
The plasma dynamics is described by taking into account excitation phenomena, radical, ion dissociation, and their collisions with electrons. Efficient crosssections were considered over a wide range of energy. This allows us to better describe the chemical reactions in volume of the species generated by the dissociation and recombination of neutral and radical particles, as well as the surface reactions on the substrate. The fluid flow rate describes the transfer of heat and mass into the reactor. All these phenomena are modeled by a system of equations, non-linear and dispersive, described in the following sections. The differential equations are solved using the finite element method.
To summerize the research methodology adapted in this article, we present this flowchart in Figure 1. Figure 2 shows a synoptic scheme of a symmetrical CCP reactor which is composed of two parallel conductive electrodes, a gas pump and a substrate. The working gases are introduced at the inlet, and then brought into the reaction chamber,  The plasma in the CCP reactor are at low pressure, low electronic temperature and low ionization rate. In a mixture of ammonia and hydrogen silicon, the gas pressure between 0.2 and 0.5 Torr, the frequency is about 13.56 MHz, the voltage is 130 V, and the space between electrodes varies between 2.7 and 6 cm.

2. Equations
The general equations for a plasma discharge [15,22] are calculated taking into account the following hypotheses: -The gas is weakly ionized. -The pressure is low. In our model we solve the system of equations (11)-(16) corresponding to a set of low temperature plasma evolution equations related to the self-coherent field E by Poisson's equation (17).

Chemical Reactions
The calculation of the source term equations (18)- (21), in the fluid model is based on the distribution function calculated by the model of Boltzmann F0 plus gross sections (v) r  which takes into account the most reactive elastic and inelastic excitation and ionization collisions, presented in Table  1.
The cross sections data used in simulation, SiH4 and NH3 collisions are given by Hayashi [23].
The chemical species considered in this work are presented in Table 2. These species are very important for plasma chemistry and have an influence on the deposition process.

4. Boundary Conditions
The boundary conditions used in this section are similar to those presented by Bavafa et al. [2] and Xia et al. [24].
-The boundary condition for Poisson's equation is the potential value on the electrodes: 0 V = electrical potential at the cathode.

RESULTS AND DISCUSSION
To investigate the plasma discharge characteristics in the CCP reactor, the fluid approach are considered. Using the Boltzmann equation [12] we calculate the non-Maxwellian EEDF, the mobility and diffusion coefficients. Based on these factors we proceed to the resolution of the fluid model equations. Thus, we obtain the plasma discharge characteristics that will be used in the start to validate the choice of the non-Maxwellian EEDF, then to compare them with the plasma discharge characteristics obtained using Maxwellian EEDF. In our simulations, the CCP reactor is driven by a sinusoidal voltage with a frequency of 13.56 MHz at a temperature of 573 K. Silicon diluted with hydrogen and ammoniac mixture is used as a deposition precursor gas. The RF voltage taken has an amplitude of 130 V, applied to the anode for an inter-electrode distance d and a variable pressure p . The results are presented at a final relaxation calculation time of around 7.37 μs. Figure 3 shows the evolution of non-Maxwellian EEDF for SiH4/NH3/H2 gases in energy range between 3 and 10 eV. The non-Maxwellian EEDF is calculated from the reactions presented in Table 1, the initial energy is 3eV  = . This non-Maxwellian EEDF will be used in the source terms of the fluid model to calculate the density and the energy electrons.
For the validation of the non-Maxwellian EEDF, we will discuss Figures 4-9. Therefore, Figures 5-7 show the variation of electron density, temperature and electric field as a function of pressure, taking a pressure margin between 0.2 and 0.5 Torr for an inter-electrode distance of 2.7 cm. Keeping the pressure at 0.3 Torr and varying the distance between 3 and 6 cm, Figures 8-10 show respectively the variation of the electron density, the temperature and the electric field as a function of the distance. Finally, Figures 11-13 show the comparison of density, temperature and electric field calculated by a Maxwellian and non-Maxwellian EEDF.  By increasing the pressure in the deposition system, the density of radicals grows as the density of precursors increases ( Figure 5). Another effect of the increase in the pressure of the deposition chamber is the fact that it leads to the accumulation of radicals in the area between ducts, which reduces the temperature in this area ( Figure 6).
An electric field is introduced and added to the field created by the charges in the plasma. Figure 7 shows that it becomes important when the pressure increases, causing a significant displacement of electrons and ions. The field perceived by the electrons is very strong and allows the electrons to accelerate towards the anode. The migration of electrons from the cathode to the anode produces a higher temperature at the anode and therefore the temperature attenuation at the cathode, which is verified in Figure 6. These results are consistent with Daoxin et al. [25]. The influence of the inter-electrode distance on the discharge is shown in Figures 8-10. When the distance becomes larger, collisions between electrons decrease and therefore the creation of secondary electrons will be low. In addition, as the distance increases, there is an expansion of the density towards the electrodes. These results are the same to that of Bavefa et al. [2] and Samir et al. [26] with respect to electron density, electron temperature and electric field distributions.
Summing up, according to these results and based on the comparison with theoretical [15,22] and practical literature works [2,9,[27][28][29], we validate the choice of the non-Maxwellian distribution function. Indeed, the plasma caracteristics resulting from the use of the non-Maxwellian EEDF satisfy the normal evolution of RF plasma discharge.
For an inter-electrode distance of 5 cm, and a pressure of 0.3 Torr, Figures 11-13 show respectively the variation in electron density, temperature and electric field, determined using the Maxwellian and non-Maxwellian EEDF. Figure 11 shows that the density calculated based on a non-Maxwellian EEDF is much higher than that calculated using a Maxwellian EEDF. Indeed, the non-Maxwellian EEDF provides an electron density of about 3.9 x10 9 cm -3 while the Maxwellian EEDF barley reaches 2.5 x10 9 cm -3 . This significant difference in electron density means that there is another type of faster electrons added which couldn't be shown by using Maxwellian EEDF, while the use of the non-Maxwellian EEDF has succeed to reveal it. Figure 12 shows that the electron temperature calculated using a non-Maxwellian EEDF (about 13 eV) is higher than the calculated one using a Maxwellian EEDF (about 9 eV). This result is due to the high speed of electrons (about 3.4 x10 8 cm/s) obtained using non-Maxwellian EEDF compared to that using Maxwellian EEDF (about 2.5x 10 8 cm/s) shown in Figure 13. These electrons have a self-coherent field source, hence the increase in the amplitude of the electric field at the cathode ( Figure 14). The field allows electrons to accelerate towards the anode. This migration produces a higher temperature at the anode, and hence temperature attenuation at the cathode ( Figure 12).

CONCLUSION
In this work, we used a one-dimensional fluid model based on a non-Maxwellian electron energy distribution function to describe the characteristics of capacitive coupled plasma excited by an RF source using silane diluted with ammoniac and hydrogen (SiH4/NH3/H2). The discharge operates at 13.56 MHz with RF voltage amplitude of 130 V and simulated at variable pressures and inter-electrode distances. The aim was to validate the choice of the non-Maxwellian EEDF and to compare the evolution of the plasma discharge characteristics obtained using the non-Maxwellian EEDF with those calculated by using a Maxwellian one.
By varying the pressure and distance between electrodes, we validated the choice of the non-Maxwellian EEDF and the type of the used gas. Indeed, the results show a significant increase in density when the gas pressure grows in the range of 0.2 to 0.5 Torr, similarly, when the inter-electrode distance increases in the range of 4 to 6 cm.
The use of a non-Maxwellian EEDF allowed us to reveal the existence of another faster type of electrons, which was not detected when a Maxwellian EEDF was used. Indeed, in the non-Maxwellian case an average electron velocity of about 3.4 x10 8 cm/s and an electron temperature of 13 eV were obtained at the cathode, whereas in the Maxwellian case an average electron velocity of about 2.4 x10 8 cm/s and an electron temperature of 9 eV were barely reached at the cathode. In addition, compared with the Maxwellian EEDF the plasma discharge obtained using the non-Maxwellian EEDF has a higher electron density of about 3.8 x10 9 cm -3 which increases the deposition rate, and a reduced temperature at the anode of about 4.8 eV which results in a better distribution of species at the surface.
We deduce that the non-Maxwellian approximation of the electron energy distribution function is more effective to describe the evolution of an RF plasma discharge in a capacitive reactor than the Maxwellian one. Thus, in order to ensure better deposition of thin films, it is recommended to define the electron transport properties by an electron energy distribution function using a non-Maxwillian EEDF, which is a closer assumption to the real conditions of the physical model. Consequently, the application of this function in our fluid model for the discussion of gas type selection, pressure and inter-electrode distance, will provide better characteristics to enhance the quality of CCP reactor.
Finally, it will be interesting in a future work to seek optimal deposition of thin films by studying a twodimensional simulation using a non-Maxwellian EEDF.