Semiconductor Fabrication, CMOS Front-End & Furnace Oxidation
Deal-Grove Silicon Thermal Oxidation Thickness & Tau Formula
Calculate Pre-Existing Oxide Time Shift tau (hr), Final SiO2 Gate/Field Oxide Thickness x_o (nm), Net Oxide Grown (nm), and Consumed Silicon Substrate Depth (0.46 * dx nm).
Why thin dry oxides grow linearly (reaction-limited B/A) while thick wet field oxides slow to SQRT(B*t) diffusion—and eat 46% into the silicon wafer!
During high-temperature (900°C–1,100°C) dry O2 or wet H2O steam oxidation of crystalline silicon wafers, oxidizing species must BOTH diffuse across the already-grown SiO2 layer AND react chemically at the Si/SiO2 interface. Under the foundational Deal-Grove quadratic law x_o^2 + A * x_o = B * (t + tau), Pre-Existing Native/Pad Oxide x_i (nm) maps to equivalent prior furnace time tau = (x_i^2 + A * x_i) / B (hr), giving exact Final Oxide Thickness x_o(t) = (A / 2) * (SQRT(1 + (t + tau) / (A^2 / (4 * B))) - 1)! Furthermore, molar volume expansion from Si to amorphous SiO2 consumes 0.46 nm of silicon substrate for every 1.00 nm of SiO2 grown—creating the LOCOS bird’s-beak step! Simply adding x_i + SQRT(B * t) double-counts diffusion resistance and severely overpredicts thickness!
Try this 1,000°C Wet Oxidation Furnace Recipe first—Initial_Pad_Oxide x_i = 20.0 nm (B2), Linear_Length_Constant A = 100.0 nm (B3), Parabolic_Rate_Constant B = 20,000.0 nm²/hr (0.020 um²/hr, B4), and Furnace_Oxidation_Time t = 2.00 hr (B5): Pre-Oxide Time Shift tau equals (400 + 2,000) / 20,000 = 0.120 hr (E2), Final Oxide Thickness hits 161.90 nm (E3), Net Oxide Grown equals 141.90 nm (E4), and Silicon Wafer Consumed Below Original Surface equals 0.46 * 141.90 = 65.27 nm (E5)! In a shared Arte Sheets fab-process workbook, a `suggester` can test an extended 8-hour field-oxide run in Suggest Mode while asking a hint-first AI prompt ("Why must pre-existing oxide x_i be converted into equivalent shift time tau inside the Deal-Grove square root instead of added outside?") before a `contributor` accepts the recipe.
| Deal-Grove Oxidation KPI (xi=20 nm, A=100 nm, B=20,000 nm²/hr, t=2 hr) | Formula | Verified Output |
|---|---|---|
| Equivalent Time Shift tau (hr) | (xi^2 + A * xi) / B | 0.120 hr |
| Final SiO2 Thickness x_o (nm) | (A/2) * (SQRT(1 + (t+tau)/(A^2/(4*B))) - 1) | 161.90 nm (356.08 @ t = 8 hr) |
| Net New SiO2 Grown dx (nm) | Final_Oxide_xo - Initial_Oxide_xi | 141.90 nm (336.08 @ t = 8 hr) |
| Silicon Substrate Consumed (nm) | 0.46 * (Final_Oxide_xo - Initial_Oxide_xi) | 65.27 nm (154.60 @ t = 8 hr) |
Exact Deal-Grove Linear-Parabolic Oxide Thickness Formula
Validate Initial_Oxide x_i >= 0, Linear_Constant A > 0, Parabolic_Constant B > 0, and Oxidation_Time t > 0!
=LET(xi_n, B2, a_ln, B3, b_pr, B4, tm_h, B5, IF(OR(xi_n < 0, a_ln <= 0, b_pr <= 0, tm_h <= 0), "Invalid Deal-Grove Input", LET(ta_u, ((xi_n ^ 2) + a_ln * xi_n) / b_pr, ch_t, (a_ln ^ 2) / (4 * b_pr), ROUND((a_ln / 2) * (SQRT(1 + (tm_h + ta_u) / ch_t) - 1), 2))))Parabolic Diffusion Slowdown Check (Quadruple Time t from 2.0 hr to 8.0 hr)
Quadruple Furnace_Time B5 from 2.00 hr to 8.00 hr: because thick oxide enters the diffusion-limited parabolic regime, Final Oxide Thickness E3 grows only 2.20x (not 4x!) from 161.90 nm to 356.08 nm, consuming 154.60 nm of silicon in E5!
=IF(ISTEXT(E3), E3, LET(ta_u, ((B2 ^ 2) + B3 * B2) / B4, ch_t, (B3 ^ 2) / (4 * B4), xo_f, (B3 / 2) * (SQRT(1 + (B5 + ta_u) / ch_t) - 1), ROUND(0.46 * (xo_f - B2), 2)))Try the example
Use the original values or change them to test your understanding.