This simulation calculates the steady-state growth rate of a silicon film deposited by chemical vapor deposition (CVD) from silane, SiH4(g) → Si(s) + 2H2(g), on a wafer. Silane diffuses across a stagnant boundary layer (mass-transfer coefficient hg) and reacts on the surface (first-order rate constant ks). The two steps act as resistances in series.
The left plot is an Arrhenius plot of the log of the growth rate versus 1000/T. The solid blue curve is the actual growth rate; the dashed curves are the reaction-limited (red) and mass-transfer-limited (green) growth rates. The black dot marks the temperature set by the slider; you can also drag the dot or click on the plot to change the temperature.
The right figure shows the silane concentration profile across the boundary layer, the fluxes, and the fraction of the total resistance due to each step. Use the sliders to change temperature, total pressure, silane mole fraction, boundary-layer thickness, and activation energy. Compare atmospheric-pressure (APCVD, about 1 atm) and low-pressure (LPCVD, about 10−3 atm) operation.
Silicon is deposited from silane:
At steady state the flux of silane across the gas boundary layer, , equals the rate of the surface reaction, :
where and are the silane concentrations (molecules/cm3) in the bulk gas and at the surface, is the gas-phase mass-transfer coefficient, and is the first-order surface rate constant (both cm/s). Solving for the surface concentration:
The film growth rate is the flux divided by the atom density of solid silicon, atoms/cm3:
The two steps are resistances in series, so the total resistance is the sum of the individual resistances. The fraction of the resistance due to mass transfer is .
The surface rate constant follows the Arrhenius equation, with cm/s:
The mass-transfer coefficient is the diffusivity divided by the boundary-layer thickness ; the gas diffusivity is inversely proportional to total pressure and increases with temperature ( cm2/s for SiH4 in H2 at 300 K and 1 atm):
The bulk gas concentration is calculated from the ideal gas law, where is the silane mole fraction, is total pressure, and is Boltzmann's constant:
Limiting cases. When (low temperature), the process is reaction-limited, , and the growth rate increases strongly with temperature; the slope of the Arrhenius plot is . When (high temperature), the process is mass-transfer-limited, , and the growth rate depends only weakly on temperature ().
Because and at constant mole fraction, the mass-transfer-limited growth rate does not depend on total pressure, but the reaction-limited rate is proportional to pressure. Lowering the pressure therefore shifts the transition to higher temperature, so LPCVD operates in the reaction-limited regime. In that regime the growth rate depends on surface temperature, which is easy to keep uniform, rather than on local gas flow, so many closely stacked wafers can be coated uniformly. APCVD reactors give higher growth rates but usually operate in the mass-transfer-limited regime, where uniformity depends on reactor flow design.
The regime label uses (reaction-limited) and (mass-transfer-limited). The model assumes a first-order surface reaction, dilute-gas diffusion, and a stagnant-film boundary layer whose thickness is independent of pressure.
Nomenclature
| silane concentration at the wafer surface (molecules/cm3) | |
| silane concentration in the bulk gas (molecules/cm3) | |
| diffusivity of silane in the gas (cm2/s) | |
| diffusivity at 300 K and 1 atm (cm2/s) | |
| activation energy of the surface reaction (kJ/mol) | |
| silane flux to the surface (molecules/(cm2 s)); at steady state | |
| flux of silane across the boundary layer (molecules/(cm2 s)) | |
| rate of the surface reaction per area (molecules/(cm2 s)) | |
| film growth rate (cm/s in the equations; displayed in nm/min) | |
| gas-phase mass-transfer coefficient (cm/s) | |
| Boltzmann's constant, 1.381 × 10−23 J/K | |
| Arrhenius pre-exponential factor (cm/s) | |
| first-order surface reaction rate constant (cm/s) | |
| atom density of solid silicon (atoms/cm3) | |
| total pressure (atm) | |
| ideal gas constant, 8.314 J/(mol K) | |
| temperature (K) | |
| mole fraction of silane in the bulk gas | |
| boundary-layer thickness (cm) |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.