| Silicon is to be deposited on wafers in a LPCVD reactor. We want to obtain an analytical solution for the silicon deposition rate and reactant concentration profile for the simplified version of the LPCVD reactor just discussed. Analytical solutions of this type are important in that an engineer can rapidly gain an understanding of the important parameters and their sensitivities, without making a number of runs on the computer. The reaction that is taking place is | |||
|
|
|||
| Sections of the reactor are shown in Figures CDE12-10.1 and CDE12-10.2. | |||
|
|
|||
| 1. Balances. In forming our shell balance on the annular region we shall assume that there are no radial gradients in the annulus and include the outer tube walls and the boat, which consume some of the reactant by deposition on the walls in the balance. In addition, we shall neglect any dispersion or diffusion in the axial direction. Balance on reactant A: | |||
|
|
(CDE12-10.1) | ||
Dividing through by and taking the limit as 0 gives
|
|||
|
|
(CDE12-10.2) | ||
|
Mole balance |
2. Rate laws. The rate of silicon deposition, (mol/ dm 2 s), is equal to the rate of depletion of SiH 2
.
|
||
|
|
(CD12-101) | ||
| where the units of C A and k are mol/dm 3 and dm/s, respectively. Deposition takes place on the reactor walls, the support, and on the wafer surfaces. The corresponding depletion of reactant gas on each of these surfaces is | |||
|
(CDE12-10.3) |
||
|
Radial |
Concentration profile and effectiveness factor. From Example CD12-2 we derived the radial concentration profile between the wafers as | ||
|
|
(CDE12-7.9) | ||
| The corresponding effectiveness factor was | |||
|
|
(CDE12-9.12) | ||
| 4. Concentration profile in the annular region. Combining Equations (CDE12-10.2) and (CDE12-10.3) yields | |||
|
|
(CDE12-10.4) | ||
| Writing F Az and C AA in terms of conversion, we have | |||
|
Axial |
|
(CDE12-10.5) | |
| where C A0 and F A0 refer to the reactant concentration and molar flow rates at the entrance to the reactor. | |||
| Combining Equations (CDE12-10.4) and (CDE12-10.5) gives | |||
|
|
(CDE12-10.6) | ||
| Collecting terms, we can obtain an expression involving the Damköhler number, Da: | |||
|
|
(CDE12-10.7) | ||
|
where
|
|||
| Solving for conversion as a function of distance along the length of the reactor yields | |||
|
|
(CDE12-10.8) | ||
| or, in terms of concentration, | |||
|
|
(CDE12-10.9) | ||
The deposition rate as a function of r and z can now
be obtained as follows. The deposition rate at a location r and is
|
|||
|
|
(CDE12-10.10) | ||
| First, using Equation (CDE12-2.9) to relate C A (r, z) and C AA (z), we obtain | |||
|
|
|||
| Next, we use Equation (CDE12-3.9) to determine the rate as a function of distance down the reactor. | |||
|
|
(CDE12-10.11) | ||
| The thickness,T, of the deposit is obtained by integrating the deposition rate with respect to time, | |||
|
|
|||
| where r is the molar density of the material deposited, g mol/cm
3 . The 2 accounts for deposition on both sides of the wafer. Integrating, we obtain |
|||
|
|
(CDE12-10.12) | ||
The reactant concentration profile and deposition thickness along
the length of the reactor are shown schematically in Figure CDE12-10.3 for the case
of small values of the Thiele modulus
|
|||
|
|
|||
Figure CDE12-10.1
Figure CDE12-10.2
Figure 12-10.3