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Modeling and Simulation of a Chemical Vapor Deposition

We are motivated to model PE-CVD plasma enhanced chemical vapor deposition processes for metallic bipolar plates, and their optimization for depositing a heterogeneous layer on the metallic plate. Moreover a constraint to the deposition process is a very low pressure nearly a vacuum and a low temperature about 400K . The contribution of this paper is to derive a multiphysics system of multiple physics problems that includes some assumptions to simplify the complicate process and allows of deriving a computable mathematical model without neglecting the real-life processes. To model the gaseous transport in the apparatus we employ mobile gas phase streams, immobile and mobile phases in a chamber that is filled with porous medium plasma layers . Numerical methods are discussed to solve such multi-scale and multi phase models and to obtain qualitative results for the delicate multiphysical processes in the chamber. We discuss a splitting analysis to couple such multiphysical problems. The verification of such a complicated model is done with real-life experiments for single species. Such numerical simulations help to economize on expensive physical experiments and obtain control mechanisms for the delicate deposition process.


Introduction
We motivate our research on producing high-temperature films by low-pressure deposition processes. Such deposition can only be achieved with plasma-enhanced processes; see 1 . In standard applications based on depositing binary systems such as TiN and TiC, new material classes arose. More delicate to deposit are nanolayered ternary metal-carbide or metal-nitride materials, including MAX-phase materials, while we have to control three species in the deposition process see 2 .
To understand the growth of a thin film done by PE-CVD plasma-enhanced chemical vapor deposition processes, we have to model a complicated multiphysical process; see 3,4 . Based on the process constraints, which are very low-pressure nearly vacuum and a low temperature about 400 K , we can simplify the complicated model.

Mathematical Modeling
In the model we have included the following multiple physical processes, related to the deposition process: i flow field of the ionized plasma: Navier-Stokes equation; see 17 , ii transport system of the species: mobile and immobile gaseous phases with Kinetically controlled adsorption heavier and lighter species ; see 4 , iii electric field: Poisson equation; see 13, 18 . In the following we discuss the three models separately and combine all the models into a multiple physical model. We assume a two-dimensional domain of the apparatus with isotropic flow fields; see 17 .

Flow Field
The conservation of momentum is given by flow field: Navier-Stokes equation where v is the velocity field, p the pressure, v 0 the initial velocity field and the position vector x x 1 , x 2 t ∈ Ω ⊂ R 2, . Furthermore, we assume that the flow is divergence free and the pressure is predefined.

Transport Systems (Multiphase Equations)
We model the ionized plasma as an underlying medium in the chamber with mobile and immobile phases. Here transport in the plasma with gaseous species contain of mobile and immobile concentrations, 3 . For such a heterogenous plasma, we applied our expertise in modeling multiphase transport through a porous medium; see 2 .
To amplify the modeling of the gaseous flow to the chamber which is filled with ionized plasma, we deal with the so-called far-field model based on a porous medium. Here the plasma can be modeled as a continuous flow 17 , that has mobile and immobile phases; see 8 .
We assume nearly a vacuum and a diffusion-dominated process, derived from the Knudsen diffusion, 19 . In such viscous flow regimes, we deal with small Knudsen numbers and a pressure of nearly zero. We discuss a modified model, including new system parameter spaces such as porosity and permeability, which describe the plasma flow through the reactor. In Figure 1, the chamber of the CVD apparatus is shown. It is modeled as a porous medium, where the porosity is given as the ratio between the void space and the total volume of the ceramic or bulk material: where V v is the void space and V b is the volume of the ceramic or bulk material. For the one-phase model, the velocity v of the underlying fluid is calculated by Darcy's law and given by: where κ is the permeability, ∇p is the pressure gradient, φ is the porosity and μ the dynamic viscosity.
To model a retarded gas concentration in the porous medium, we have to consider multiphases of the underlying fluid. The new model includes immobile and adsorbed phases, where the velocity of the fluid is zero and impermeable layers are given.
In the model, we consider both absorption and adsorption taking place simultaneously and with given exchange rates. Therefore we consider the effect of the gas concentrations' being incorporated into the porous medium. We extend the model to two more phases: ii adsorbed phase.
In Figure 2, the mobile and immobile phases of the gas concentration are shown in the macroscopic scale of the porous medium. Here the exchange rate between the mobile gas concentration and the immobile gas concentration control the flux to the medium. In Figure 3, the mobile and adsorbed phases of the gas concentration are shown in the macroscopic scale of the porous medium. To be more detailed in the mobile and immobile phases, where the gas concentrations can be adsorbed or absorbed, we consider a further phase. Here the adsorption in the mobile and immobile phase is treated as a retardation and given by a permeability in such layers.
Journal of Applied Mathematics The parameters are given as: The effective porosity is denoted by φ and declares the portion of the porosities of the aquifer that is filled with plasma, and we assume a nearly fluid phase. The transport term is indicated by the Darcy velocity v, that presents the flow direction and the absolute value of the plasma flux. The velocity field is divergence free. The decay constant of the ith species is denoted by λ i . Thereby, k i denotes the indices of the other species which react with species i.

Electric Field (Distribution)
In addition, in order to find the distribution of the electric field, it is necessary to solve the Poisson equation which relates the divergence of the local electric fields to the charge density 13 , and is given by where e is the charge of the electron, 0 the permittivity of the gas mixture, c i the particle density of species i and Z i the relative charge of species i; see 21 .
A general model for PECVD reactors is very complicated and we therefore simplify the model with some assumptions to obtain tractable problems.
i Predefined electric field E respecting different predefined areas Ω j , we apply: Journal of Applied Mathematics Permeable yers la where we assume that c i ≈ c const,i for Ω j is a constant particle density of species i in domain Ω j with j ∈ 1, . . . , m and m is the number of domains.
ii The plasma chemistry in the reactor is neglected.

Multiphysics Model
We derive the multiphysics model based on the multiphase and electric field model. We start by developing the multiphysics model in the following steps: i transport system multiphase model , ii electrical field is approximated by a permeable layer field model , iii multiphysics model with multiphase transport and embedded electrical field.
We assume that the number of gas particles in the plasma atmosphere Ar Argon is low and the density of the particles can be treated as nearly constant. Therefore, we can derive a constant field in each sector of the apparatus where E Ω j is constant and Ω j ⊂ Ω are sectors in the domain. Such an electric field, see Figure 4, can be assumed in near-field experiments to be nearly homogeneous, where the field intensity is decreasing and can be assumed to be constant in small areas; see 4 .
Next the idea is to embed the electric field into a porous medium model. A simplified model can be derived with given domain-dependent parameters and a retardation factor that includes the influence of the electric field. 8

Journal of Applied Mathematics
First, the mobile phase equation 2.4 can be reformulated with retardation factor given by: where c i is the particle density and F i the flux of species i. R g,i is the reaction term of species i, meaning the right-hand side of 2.4 , all other parameters as in 2.4 -2.8 . R i is defined as a specific and constant Henry isotherm, called the retardation factor; see 22 . Such factors are known as sorption coefficients and model the rate between the mobile and immobile concentration, they are derived from experiments; see 23 .
In a second step, we calculate the retardation factor, which is based on the influence of the electric field.
We subtract 2.4 from 2.14 and obtain We include the assumption of the electric field dependence of the domain parts and consider so we obtain the derivation of R i for a specific domain Ω j ⊂ Ω and we have: In the third step we add the equations 2.6 -2.8 to the modified mobile phase equation 2.14 and we obtain the full coupled system. Here the novel contribution is to derive a full coupled system of a multiphase and multiflow problem to include all the physical processes in the chamber. Such ideas have been developed and considered in fluid dynamical models for many years 24 .
Remark 2.1. For the flow through the chamber, for which we assume a homogeneous medium and nonreactive plasma, we have considered a constant flow 1 . A further simplification is given by the very small porous substrate, for which we can assume the underlying velocity to a first approximation as constant 4 .

Remark 2.2.
For an in-stationary medium and reactive or ionized plasma, we have to take into account the relations of the electrons in thermal equilibrium. Such spatial variation can be considered by modeling the electron drift. Such modeling of the ionized plasma is done with Boltzmanns relation, 3 .

Physical Experiments
The base of the experimental setup is the plasma reactor chamber of a NIST GEC reference cell. The spiral antenna of a hybrid ICP/CCP-RF plasma source was replaced by a double spiral antenna 25 . This reduces the asymmetry of the magnetic field due to the superposition of the induced fields of both antennas. Also, the power coupling to the plasma increases and enhances the efficiency of the source. A set of MKS mass flow controllers allows any defined mixture of gaseous precursors. Even the flows of liquid precursors with high vapour pressure is controlled by this system. All other liquid and all solid precursors will be directly transported to the chamber by a controlled carrier gas flow. Besides the precursor flow, the density can also be changed by varying the pressure inside the recipient. Controlling the pressure is achieved with a valve between the recipient and vacuum pumps. In addition, a heated and insulated substrate holder was mounted. Thus, a temperature up to 800 • C and a bias voltage can be applied to the substrate. While the pressure and RF power determine the undirected particle energy plasma temperature , the bias voltage adds, only to the charged particles, energy directed at the substrate. Apart from the pressure and RF power control, the degree of ionization and number as well as the size of the molecular fractions can be controlled.
Altogether, this setup provides as free process parameters: During all experiments, the process was observed by optical emission spectroscopy OES and mass spectroscopy MS . The stoichiometry of the deposited films was analyzed ex situ in a scanning electron microscope SEM by energy dispersive X-ray analysis EDX .

Realization of Physical Experiments
The following parameters were used for the physical experiments. Such a reduction allows of concentrating on the important flow and transport processes in the gas phase. Further, we apply the underlying mathematical model far-field model, see Section 2.2 so that we can switch between physical and mathematical parameters see Table 1 .
3 Film at substrate: SiC x .

10
Journal of Applied Mathematics We apply the parameters in Table 2 for interpolation of the substrate temperature.
For the substrate temperature and plasma power, we use the parameters in Table 3.
Remark 3.1. In the process, the temperature and power of the plasma is important and experiments show that these are significant parameters. Based on these parameters, we initialize the mathematical model and interpolate the flux and reaction parts.

Discretization and Solver Methods
We distinguish between the mobile and immobile phases. Here the mobile phases are parabolic partial differential equations and the immobile phases are ordinary differential equations. So for the space-discretization of the PDE's we apply finite volume methods as mass-conserved discretization schemes and for the time-discretization of the PDE's and ODE's we apply Runge-Kutta methods. So first we discretize in space, while we then apply a splitting method, taking into account the different matrices.

Spatial Discretization
We deal with the transport part of the mobile phase, see 2.14 : For the convection part, we use a piecewise constant finite volume method with upwind discretization; see 26 . For the diffusion-dispersion part, we also apply a finite volume method and we assume the boundary values are denoted by n · D e i ∇c i x, t 0, where x ∈ Γ is the boundary Γ ∂Ω; compare with 27 . The initial conditions are given by c i x, 0 c i,0 x . We integrate 4.1 over space and obtain The time integration is done later in the decomposition method with implicit-explicit Runge-Kutta methods. Further the diffusion-dispersion term is lumped; compare with 12 . Equation 4.3 is discretized over space using Green's formula where Γ j is the boundary of the finite volume cell Ω j and V u j is the volume of the cell j. We use the approximation in space; see 12 .

Journal of Applied Mathematics
The spatial integration for the diffusion part 4.4 is done by the mid-point rule over its finite boundaries and the convection part is done with a flux limiter and we obtain: where |Γ e jk | is the length of the boundary element Γ e jk . The gradients are calculated with the piecewise finite-element function φ l .
We decide to discretize the ux with an up-winding scheme and obtain the following discretization for the convection part: where v j,e e v · n j,e ds. We obtain for the diffusion part: We get, using difference notation for the neighbour points j and l; compare with 28 , the full semi-discretization: Remark 4.1. For higher-order discretization of the convection equation, we apply a reconstruction which is based on Godunov's method. We apply a limiter function that fulfills the local min-max property. The method is explained in 26 . The linear polynomials are reconstructed by the element-wise gradient and are given by ∇c dx, with j 1, . . . , I.

4.9
The piecewise linear functions are denoted by u jk c j ψ j ∇u| V j x jk − x j , with j 1, . . . , I, 4.10 where ψ j ∈ 0, 1 is the limiter function and it fulfills the discrete minimum-maximum property, as described in 26 .

Splitting Method to Couple Mobile, Immobile, and Adsorbed Parts
The motivation of the splitting method is the following observations.
i The mobile phase is semidiscretized with fast finite volume methods and can be stored in a stiffness matrix. We achieve large time steps, if we consider implicit Runge-Kutta methods of lower order e.g., implicit Euler as a time discretization method.
ii The immobile, adsorpted, and immobile-adsorpted phases are pure ordinary differential equations and each is cheap to solve with explicit Runge-Kutta schemes.
iii The ODEs can be seen as perturbations and are solved explicitly in the iterative scheme.
For the full equation we use the following matrix notation: Furthermore Q, . . . , Q im,ad are the spatially discretized source vectors. Now we have the following ordinary differential equation: where C c, c im , c ad , c im,ad T and the right-hand side is Q Q, Q im , Q ad , Q im,ad T .
For such an equation we apply the decomposition of the matrices ∂ t C AC Q, Journal of Applied Mathematics

4.14
This system of equations is numerically solved by an iterative scheme.
Algorithm 4.2. We divide our time interval 0, T into subintervals t n , t n 1 , where n 0, 1, . . . N, t 0 0 and t N T . We start with n 0.
1 The initial conditions are given by C 0 t n 1 C t n . We start with k 0. 2 Compute the fixed-point iteration scheme given by where k is the iteration index; see 29 . For the time integration, we apply Runge-Kutta methods as ODE solvers; see 30, 31 .
3 The stop criterion for the time interval t n , t n 1 is given by where · is the maximum norm over all components of the solution vector. err is a given error tolerance, for example, err 10 −4 . If 4.16 is fulfilled, we have the result C t n 1 C k t n 1 .

4.17
If n N which means we reach the maximum iterative steps , then we stop and are done. If 4.16 is not fulfilled, we do k k 1 and go to 2 .
The error analysis of the schemes is given in the following theorem.

Experiments for the Multiple Phase Model
In the following subsections, we present our experiments based on the mobile, immobile and adsorbed gaseous phases. We contribute ideas to obtain an optimal layer deposition, which is based on the PE-CVD process, while different additional phases are considered, for example, plasma and precursor media.
The main contributions are an optimal collection of point sources, line sources or moving sources to cover the deposition area, with respect to the remainder concentration in the immobile and adsorbed phases.
We simulate the deposition process with our boundary value solver algorithms and could deal with many different conditions that might be impossible for physical experiments. Such simulation results may benefit the physical experiment and give new ideas to optimize such deposition problems of a complicated physical process.

Verification of the Code with Analytical Solutions
In the verification of our software code UG; see 32 , we apply the multiphase model 2.4 and 2.6 with the following parameters.
We use ascending parameters for the retardation factors. The retardation factors are given by R 1 16, R 2 8, R 3 4, R 4 . The reaction factors are given by The initial conditions for the mobile phase are given by The mobile-immobile exchange rate is g 0.5. The initial conditions for the immobile concentrations are c im 0 1, 1, 1, 1 t . For the time integration method we apply a fourth-order Runge-Kutta method and we apply k 1, . . . , 4 iterative steps for the solver scheme.

Verification of the Model with Physical Experiments
For the next experiments, we have the following parameters of the model, discretization, and solver methods. We apply interpolation and regression methods to couple the physical parameters that were discussed in Section 3, to the mathematical parameters, see Figure 7 and Table 4. Parameters of 2.4 -2.8 In Table 8, we list the parameters for our simulation tool UG; see 32 . The software toolbox has a flexible user interface to allow a large number of numerical experiments and approximations using the known physical parameters.

Discretization Method
Finite-volume method of 2nd order is shown in Table 9.

Time Discretization Method
Crank-Nicolson method 2nd order .

Solver Method
In Tables 10 and 11, we deal with test examples which are approximations of physical  experiments. 18

Numerical Experiment
In the test example, we deal with the following reaction: 2SiC 4H−→ λ SiC CH4 Si.

5.3
Here, we have physical experiments and approximate temperature parameters T 400, 600, 800.
We compute the SiC : C ratio at a given temperature T 400, 600, 800 with the UG program and fit to the parameter λ.
We use a Lagrangian formula to compute λ for the new temperatures T 500, 700 and apply the ratio of the simulated new parameters Table 5 . These numerical results are used to initialise the physical experiments in Table 12.

One Source
See Table 6.
In Figure 8, we present the concentration of one point source at 50, 20 with number of time steps equal to 25.
In Figure 9, we show the deposition rates of one point source at 50, 20 , with number of time steps equal to 25.

Remark 5.2.
In the numerical experiments, we could approximate the physical experiments and obtain the same deposition rates for the Si: C deposition. Therefore, the model allows of making predictions about future deposition rates with various parameters. Such numerical experiments can replace expensive physical experiments and allow of restricting them to more special cases.       We have the following outline of the experiment. The exchange in the following experiments of the carbon C species between the mobile and immobile concentrations is very low, it is about g 8 · 10 −14 , we assume less activities in the plasma environment. Further the exchange between the mobile and adsorbed mobile concentrations is also very low, it is about α 4·10 −14 , also the exchange rates between the immobile and adsorbed immobile concentration is the same as in the mobile and adsorbed mobile phases.

Experiments with Full
In previous experiments; see 33 , we obtained optimal deposition results by combining multiple point sources which can be moved in the spatial directions moving sources . Further we could approximate the physical results, while using mobile and immobile phase models.
In this experiment we increase the model to four phases in order to predict the delicate adsorped regions. Physicists called such regions lost deposition areas, where the deposition material vanishes into the apparatus; see 25 .
The concentration of each source in each step is equal to 1. The starting movement of X is given with 50 → 35 → 20 → 35 → 50 → 65 → 80 → 65 → 50.
In Figure 10, we present the experiment with 11 moving sources at the positions in four phases. We obtain high depositions in the mobile phase, while we have only marginal depositions in the immobile and adsorbed phases.
Here the important observations to give a measurement of the deposition are the various deposition rates in the different phases.
In Figure 11, we present the deposition rate of mobile concentration of 11 moving sources moving in the X-direction in steps of 15: X moves from 50 → 35 → 20 → 35 → 50 → 65 → 80 → 65 → 50, with the number of time steps equal to 25. In see Table 12, we discuss the CPU time for the spatial refinements. We consider the L 2 -norm of the solution and obtain convergent results of the solutions.

Remark 5.3.
With moving sources we gain improved deposition rates, this was also observed in 33 . Nevertheless the remaining concentration in the immobile and adsorbed phases are important for seeing the lost areas. In Figure 11, the maximum deposition rate of carbon in the mobile phase is 1.2·10 8 mol , while the maximum lost deposition rate in the other phases are at least 18000 mol . So in the immobile and adsorbed phases we lost only 0.00018 percent concentration of the deposition material. This may seem to be neglectable, but we have considered only 1 sec deposition time, while a full cycle of deposition could be 1000 sec . Therefore we lost about 10% of the concentration in the apparatus, which is immense. Due to this fact, such models are important for foreseeing the percentage of lost concentration. One of the simulation results is the usage of higher amount of depositon material, that is, the higher amount of the underlying precursor gases are used to overcome the lost of concentrations in the apparatus. The second result is to consider more detailed models to predict the influence of the multiphases of the underlying gaseous species.

Conclusions and Discussions
We have presented a continuous model for the multiple phases, we assumed gaseous behaviour with exchange rates to adsorbed and immobile phases at very low-pressure and low temperature while dealing with catalyst processes, for example, plasma environment and precursor gases. We have to take into acount the remaining gas concentrations in each processes. Such detailed modelling allows of seeing the delicate retardation and sorption processes of the underlying plasma medium. From the methodology side of the numerical simulations, the contributions were to decouple the multiphase problem into single-phase problems, where each single problem can be solved with more accuracy. The iterative schemes allows of coupling the simpler equations and for each additional iterative step, we could reduce the splitting error. Such iterative methods allow of accelerating the solver process of multiphase problems.