This calculator uses the Deal-Grove model to find the thickness of oxide (SiO2) grown when silicon is heated in oxygen or steam. Change between dry and wet oxidation, (100) and (111) surfaces, temperature and time to see how the growth rate and the silicon consumed beneath the film change.
Results
Final oxide thickness
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Thickness grown
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Silicon consumed
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Parabolic rate constant B
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Linear rate constant B/A
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Calculated with the Deal-Grove model x² + Ax = B(t + τ). The coefficients come from Deal and Grove's Arrhenius fits for (111) silicon; for (100), B/A is divided by 1.68. Silicon consumed is taken as 0.44 times the thickness grown. Dry oxidation of very thin films, below about 30 nm, is known to be faster than the model predicts, so treat the results as estimates.
See it as a graph
Other inputs stay at their current values while the chosen input varies from 1/4 to 4 times its current value. The red dot is the current value.
The Deal-Grove equation
In thermal oxidation an oxidant such as O2 or H2O diffuses through the existing oxide and reacts at the silicon interface to form new oxide. Setting the diffusion flux equal to the reaction flux gives x² + Ax = B(t + τ). B is the parabolic rate constant set by diffusion, B/A is the linear rate constant set by the interface reaction, and τ = (xi² + A·xi)/B accounts for an initial thickness xi. With the defaults of dry oxidation, (100), 1000 °C and 1 hour, B ≈ 0.0104 µm²/h, B/A ≈ 0.0449 µm/h and the oxide is about 38.5 nm thick.
Linear and parabolic regimes
When the film is thin and t + τ is much less than A²/4B, x ≈ (B/A)(t + τ), so thickness grows in proportion to time (the linear regime). Here the interface reaction limits growth, and because reactivity depends on orientation, (111) silicon oxidizes faster than (100); the default conditions on (111) give about 54.2 nm. As the film thickens, the oxidant takes longer to diffuse through it, and growth moves to the parabolic regime, x² ≈ Bt. Doubling the thickness then takes about four times as long, which is why the curve flattens.
Dry versus wet oxidation
Wet (steam) oxidation is much faster than dry oxidation because the solubility of H2O in the oxide is several hundred times that of O2. At 1000 °C for 1 hour on (100), dry oxidation grows about 38.5 nm while wet oxidation grows about 388 nm. Dry oxidation is slow enough to control thickness precisely and gives a dense film with few interface traps, so it is used for thin films such as gate oxides. Wet oxidation is used for thick films of hundreds of nanometers or more, such as LOCOS field oxide and masking oxide.
Silicon consumption and caveats
Oxide grows by consuming silicon: forming a film of thickness x removes about 0.44x of the original silicon, so roughly 44% of the film lies below the original surface and 56% above it. This matters for the step height and bird's beak of LOCOS and the shape of isolation structures. In dry oxidation the early growth below about 30 nm is faster than the model predicts, so Deal and Grove corrected for it with an initial thickness (τ) equivalent to about 20 to 25 nm. Gate oxides only a few nanometers thick are tuned by measurement.
In thermal oxidation silicon itself reacts with oxygen to form SiO2. From the ratio of silicon's atomic density (about 5.0×10²² cm⁻³) to SiO2's molecular density (about 2.3×10²² cm⁻³), forming an oxide of thickness x consumes about 0.44x of silicon: 44% of the oxide lies below the original surface and 56% above, a volume expansion of about 2.2×. Oxidized regions therefore form steps, and the interface is always freshly formed and clean.
AThickness is roughly proportional to the square root of time (parabolic law)
BThickness is proportional to the square of time
CThickness is constant regardless of time
Show answer and explanation
Answer: A. Thickness is roughly proportional to the square root of time (parabolic law)
The Deal-Grove model gives x² + Ax = B(t + τ). For thin oxides the interface reaction limits growth and thickness rises nearly linearly with time (x ≈ (B/A)t); for thick oxides oxidant diffusion through the film limits it and thickness follows the square root of time (x ≈ √(Bt)), so doubling thickness takes about four times as long. Thick oxides are therefore sped up with high temperature, steam and high pressure.
AThickness is proportional to the square root of time
BThickness is roughly proportional to time (increases linearly)
CThickness stays constant regardless of time
Show answer and explanation
Answer: B. Thickness is roughly proportional to time (increases linearly)
While the oxide is thin, oxidant passes through it easily and the interface reaction limits growth, so thickness increases nearly linearly with time (rate constant B/A). As the oxide thickens, diffusion through it becomes limiting and thickness follows the square root of time (rate constant B). The crossover occurs around a thickness of A/2—tens of nm to about 100 nm for dry oxidation.
AInitial oxidation proceeds faster than the model predicts
BOxidation does not proceed at all
CThe oxide becomes a conductor
Show answer and explanation
Answer: A. Initial oxidation proceeds faster than the model predicts
The Deal-Grove model describes thick-oxide growth well, but dry oxidation in the initial regime below a few tens of nm proceeds faster than predicted. Proposed causes include alternative oxidant transport through thin films and interfacial stress effects. The model compensates with an effective initial thickness (τ, equivalent to about 20 nm), but setting gate oxides of a few nm still relies on empirical calibration.
FAQ
How much silicon is consumed when 100 nm of oxide is grown?
About 0.44 times that, or 44 nm. Roughly 44 nm of the new oxide lies below the original silicon surface and the remaining 56 nm sits above it. This calculator multiplies the thickness grown by 0.44.
Does doubling the oxidation time double the thickness?
Only in the thin-film linear regime. In the parabolic regime the thickness grows by only √2 ≈ 1.4 times. For wet oxidation of (100) silicon at 1000 °C, for example, 1 hour gives about 388 nm and 2 hours about 610 nm, about 1.6 times, between the two limits.
Why do (100) and (111) silicon oxidize at different rates?
The linear rate constant B/A, which describes the interface reaction, depends on the arrangement of surface atoms. The (111) surface has more bonds available for reaction, so its B/A is about 1.68 times that of (100). The diffusion-limited B does not depend on orientation, so the difference shrinks as the film gets thicker.