# Governing Equations of Chemical Vapor Deposition

Since the velocity of the precursors is generally very low and the characteristic length is also very small, the corresponding Reynolds number is under 100 and the Grashof number governing natural convection is under 106. Therefore, the transport phenomena in the CVD process are laminar in nature. The temperature in a reactor varies significantly (typically from 300 to 900K), so the Boussinesq approximation is no longer appropriate. It is necessary to use the compressible model for transport phenomena in CVD processes.

The governing equations for the CVD process can be obtained by simplifying the generalized governing equations in Chapter 3. The following assumptions can be made to obtain the governing equations:

1. The reference frame is stationary.

2. The body force X is gravitational force, which is the same for all components in the precursors.

3. Dilute approximation is valid because the partial pressure of the reactant is much lower than that of the carrier gas.

4. The deposited film is very thin (from nanometers to microns) and its effect on the flow field can be neglected.

The continuity equation is

$\frac{D\rho }{Dt}+\rho \nabla \cdot \mathbf{V}=0 \qquad \qquad(1)$

where the precursor gases are treated as a compressible fluid mixture. The momentum equation is

$\rho \frac{D\mathbf{V}}{Dt}=\nabla \cdot \mathbf{{\tau }'}+\rho \mathbf{g} \qquad \qquad(2)$

where the stress tensor is

$\mathbf{{\tau }'}=-p\mathbf{I}+2\mu \mathbf{D}-\frac{2}{3}\mu (\nabla \cdot \mathbf{V})\mathbf{I} \qquad \qquad(3)$

The energy equation is

$\rho {{c}_{p}}\frac{DT}{Dt}=\nabla \cdot (k\nabla T)+T\beta \frac{Dp}{Dt} \qquad \qquad(4)$

where the effect of viscous dissipation and the Dufour effect have been neglected.

The conservation of species mass in terms of the mass fraction is

$\rho \frac{D{{\omega }_{i}}}{Dt}=-\nabla \cdot {{\mathbf{J}}_{i}}+{{{\dot{m}}'''}_{i}}\begin{matrix} , & i=1,2,...N-1 \\ \end{matrix} \qquad \qquad(5)$

where ωi is the mass fraction of the ith component in the gaseous precursor. The mass flux ${{\mathbf{J}}_{i}}$ includes mass fluxes due to ordinary diffusion driven by the concentration gradient, and thermal (Soret) diffusion. It can be obtained using the approach described in Section 1.3.1. The production rate of the ith species, ${{{\dot{m}}'''}_{i}}$, can be obtained by analyzing the chemical reaction. If the number of chemical reactions taking place in the system is Ng, the mass production rate is (Mahajan, 1996)

${{{\dot{m}}'''}_{i}}=\sum\limits_{j=1}^{{{N}_{g}}}{a_{ij}^{g}{{M}_{i}}\Re _{j}^{g}} \qquad \qquad(6)$

where $a_{ij}^{g}$ is the stoichiometric coefficient for the ith component in the jth chemical reaction in the gas phase, and $\Re _{j}^{g}$ is the net reaction rate of the jth chemical reaction in the gas phase (see Section 3.2.2).

The density of the gas is related to the pressure and temperature by the ideal gas law:

$\rho =\frac{p}{{{R}_{g}}T} \qquad \qquad(7)$

The boundary conditions for the governing equations of a CVD process depend on the geometric configuration of the reactor. It is generally assumed that the nonslip condition is applicable to all solid walls. The normal velocity on a solid wall is zero (no penetration) for all walls except the susceptor where chemical reaction takes place. The total net mass flux of all species can lead to a normal velocity component on the susceptor as (Mahajan, 1996)

${{v}_{n}}=\frac{1}{\rho }\sum\limits_{i=1}^{N}{\sum\limits_{j=1}^{{{N}_{s}}}{a_{ij}^{s}{{M}_{i}}\Re _{j}^{s}}} \qquad \qquad(8)$

where $a_{ij}^{s}$ is the stoichiometric coefficient for the ith component in the jth chemical reaction on the susceptor surface, and $\Re _{j}^{g}$ is the net reaction rate of the jth chemical reaction on the susceptor surface.

The net surface reaction rate $\Re _{j}^{g}$ is a product of the sticking coefficient γj (fraction of product that can be stuck on the substrate) and the effusive flux of the jth species (Mahajan, 1996), i.e.,

$\Re _{j}^{s}={{\gamma }_{j}}\frac{{{x}_{j}}{{p}_{j}}}{\sqrt{2\pi {{M}_{j}}RT}} \qquad \qquad(9)$

The growth rate of the deposit on the susceptor is

$\frac{d\delta }{dt}={{M}_{f}}\sum\limits_{i=1}^{N}{\sum\limits_{j=1}^{{{N}_{s}}}{a_{ij}^{s}\Re _{j}^{s}{{\alpha }_{i,f}}}} \qquad \qquad(10)$

where Mf is the molecular mass of the deposited film, and αi,f is the number of film atoms in the ith species.

## References

Mahajan, R.L., 1996, “Transport Phenomena in Chemical Vapor-Deposition Systems,” Advances in Heat Transfer, Academic Press, San Diego, CA.