Deal-Grove Thermal Oxide Thickness Calculator
The Deal-Grove model is the standard first-order description of thermal oxidation of silicon. This calculator solves it for oxide thickness, silicon consumed and instantaneous growth rate.
Your numbers
Example: about 0.287 for wet oxidation at 1000 C and about 0.0117 for dry oxidation at 1000 C (111 silicon values from standard references). Use your own calibrated values.
Example: about 1.27 for wet and about 0.071 for dry oxidation at 1000 C.
Native oxide or earlier oxide on the wafer.
Your results
The Deal-Grove model is poor below roughly 25 nm of dry oxide and ignores effects such as chlorine, pressure, orientation and doping. Use your furnace's calibrated recipe data for thin gate oxides.
Oxide thickness vs time (nm)
Sweeps Oxidation time from 0.1 to 6, holding your other inputs fixed.
Furnace and recipe records, searchable privately
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How to use the calculator
Enter the parabolic constant B and linear constant B/A for your temperature, ambient (dry or wet) and silicon orientation, then the oxidation time and any oxide already on the wafer. The tool returns the oxide thickness, the silicon consumed and the instantaneous growth rate. The chart shows thickness over time so you can see where growth slows from the linear regime into the parabolic regime. Treat the example constants as starting points and replace them with values calibrated on your furnace.
- Enter the initial oxide to include the time offset tau.
- Use separate runs for each temperature step in a ramp.
- Check results against a monitor wafer and an ellipsometer.
The linear-parabolic model
Deal and Grove describe oxide thickness x with x squared plus A x equals B times (t plus tau). Solving for x gives x = (A/2) times [sqrt(1 + 4B(t + tau)/A squared) minus 1]. For thin oxides the growth is linear with rate B/A, and for thick oxides it becomes parabolic with x squared roughly equal to B t. The time offset tau accounts for oxide that is already present: tau = (xi squared + A xi)/B. About 44% of the oxide thickness is consumed silicon.
Worked example
For wet oxidation at 1000 C with B of 0.287 and B/A of 1.27, A is 0.226 micrometres. After 1 hour from bare silicon the oxide is about 435 nm and 191 nm of silicon has been consumed. The instantaneous rate at that point is about 330 nm per hour, down from 1270 nm per hour at the start. For dry oxidation at 1000 C with B of 0.0117 and B/A of 0.071, one hour gives about 54 nm, though real furnaces show an initial fast regime that the model underestimates.
Common mistakes
Applying the model to thin gate oxides is the most common error, as the early fast growth in dry oxygen is not captured. Using constants for the wrong ambient or silicon orientation also shifts results significantly. Do not forget to include tau when an oxide is already present. Finally, remember that the model describes ideal growth: recipe ramps, purge steps and loading effects make the real thickness differ, so confirm with measurement.
Frequently Asked Questions
Where do I get B and B/A values?
Textbooks and papers give values by temperature, ambient and orientation, and your furnace group will have calibrated values from monitor wafers. Use calibrated values for production decisions, since small differences in constants change the thickness noticeably.
Why is 44% of the oxide from silicon?
Silicon dioxide is less dense in silicon atoms than the silicon it replaces, so growing oxide consumes a layer of silicon about 0.44 times the final oxide thickness, and the oxide surface rises above the original silicon surface.
Does it work for very thin oxides?
Not well. Below about 25 nm in dry oxygen, measured growth is faster than the model predicts. Use empirical fits or calibrated recipes for gate oxides and other thin films.
Can I model a multi-step oxidation?
Yes. Run the calculator for the first step, then use the resulting thickness as the initial oxide for the next step with the new constants. This is how the tau offset is meant to be used.
If furnace recipes and monitor wafer results are tracked in spreadsheets or an MES, Netray can help set up a private assistant that finds the right recipe and past run data.
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