Planar 1959 Step 11 Oxide Growth Kinetics Diffusion Mask

# Derive the Oxide Growth Rate That Makes a Diffusion Mask Thick Enough on Schedule: The Chemistry Behind Step 1's Shortcut

## 1. Why a Mask This Series Has Taken for Granted Finally Needs Its Own Kinetics

Every step since Step 1 has treated the masking oxide as a fixed, already-grown fact sitting on the wafer; this step goes back and derives the actual growth law that oxide obeys, because the furnace schedule this series depends on only works if the oxide reaches masking thickness inside the same thermal budget the diffusion itself is scheduled against. Silicon dioxide grows by a reaction at the oxide-silicon interface, consuming silicon and oxidant together, and the rate of that reaction is not constant — it starts fast, limited only by how quickly oxidant reacts at the interface, and slows as the growing oxide itself becomes the bottleneck, forcing oxidant to diffuse through a layer that keeps getting thicker.

$$x_{\text{ox}}^2 + A\,x_{\text{ox}} = B\,(t + \tau), \qquad \frac{dx_{\text{ox}}}{dt}\Big|_{t\to 0} = \frac{B}{A}\ \text{(linear)}, \qquad \frac{dx_{\text{ox}}}{dt}\Big|_{t\to\infty} = \frac{B}{2x_{\text{ox}}}\ \text{(parabolic)}$$

where $A$ and $B$ are the Deal-Grove rate constants set by oxidant species and furnace temperature, and $\tau$ an offset absorbing any oxide already present before this growth step began. Early growth is reaction-limited and nearly linear in time; later growth is diffusion-limited and slows to a parabolic crawl, which is exactly why this series' own furnace schedule budgets a long initial oxidation specifically to get past the fast regime before the diffusion step that follows ever starts.

Two Growth Regimes, One Oxidation Step oxide thickness versus time, the law this series has been relying on since step 1 linear, reaction limited parabolic, diffusion limited regime crossover x t x² + Ax = B(t+τ) — the exact curve step 1's continuous oxide was already riding nobody in this series had to name this law for it to govern every furnace run since

## 2. Real Diagram: Watching the Interface Eat Both Silicon and Oxidant at Once

Oxide growth is not a coating deposited on top of silicon; it consumes the silicon surface it grows on, because the reaction happens at the moving oxide-silicon interface, with oxidant diffusing in through oxide already formed and silicon being incorporated from underneath. That consumption is why this series has always measured oxide thickness as a trade against remaining silicon, not as a simple addition.

The Interface Moves Down as Oxide Grows Up oxidant diffuses in through oxide already grown, consuming silicon at the moving front growing oxide, oxidant diffusing downward through it remaining silicon, consumed from the top as the interface advances interface, moving down over time thicker masking oxide always means less silicon remaining underneath it, not merely a coating on top

## 3. Why This Rigor Was Always Owed to a Step the Series Had Only Gestured At

Step 1 of this series simply assumed a continuous oxide was present and intact; every step since has relied on that oxide's thickness, its masking adequacy, and its freedom from pinholes without ever deriving the growth law that produced it in the first place. The 1957 series that first grew masking oxide for this project measured growth rates empirically, furnace run by furnace run, without separating the linear and parabolic regimes this step now names explicitly. That separation matters for this series specifically because the contact-window etch at Step 4 and the field-oxide integrity Step 7 inspects both depend on knowing, in advance, how thick the oxide will be after a given time at a given temperature — not after the fact, by measurement, but by calculation, before the furnace is ever loaded.

Step 11 does not change anything this series has already built; it finally derives the law that made Step 1's entire premise — a continuous, adequately thick oxide, grown once and never cut through — a schedulable certainty rather than a lucky furnace run.

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