Surface Barrier 1953 Grow Uniform Single Crystal

# Grow a Uniform Single Crystal: The Same Segregation Drift, Now Feeding Two Multiplied Numbers Instead of One

Nothing about pulling this crystal differs mechanically from the alloy-junction route this project has already documented — the same Czochralski pull, the same single fixed target resistivity from step two, the same absence of any deliberate mid-pull composition change. What differs is what a drift away from that target now costs. In the alloy-junction route, a resistivity that creeps along the rod's length only ever threatens one downstream number: base body resistance. Here, step two already established that the same resistivity also feeds base-spreading resistance once a confined etch geometry multiplies it later — so any uncorrected drift in this step does not threaten one spec, it threatens two, simultaneously, out of a single uncontrolled process.

## 1. Normal Freezing Still Drifts Concentration With No Deliberate Dose Change

$$C_s(g) = k_0\, C_0\, (1-g)^{\,k_0 - 1}$$

As fraction solidified $g$ rises, solid composition $C_s$ still drifts away from the melt's starting concentration $C_0$ purely through the segregation coefficient $k_0$ — identical physics to the alloy-junction route, because nothing about this step's crystal-pulling apparatus knows yet that the finished device will be etched rather than alloyed. The drift is a property of ordinary solidification, not of what happens to the crystal afterward.

## 2. Real Diagram: One Drift, Doubled Downstream Exposure

The Same Drift, a Wider Downstream Target identical segregation physics, but this process now has two specs riding on one number crucible, melt early: close to C₀ concentrates as g rises, same as alloy-junction growing rod seed-end tail-end body R and spreading R both near target body R and spreading R both drift together one uncontrolled drift, now read out through two multiplied specs instead of one

## 3. The Same Tolerance Band Now Guards Two Specs at Once

$$\left|\frac{C_s(g) - C_0}{C_0}\right| \le \delta_{\text{tol}} \quad \text{for } g \in [0, g_{\max}]$$

The pull schedule still has to hold $C_s(g)$ inside the tolerance band $\delta_{\text{tol}}$ out to whatever $g_{\max}$ the schedule allows, exactly as in the alloy-junction route — the control problem and its governing inequality are unchanged. What changes is what crossing that band actually risks: an alloy-junction device crossing the band ships with a base-resistance deviation alone, while a surface-barrier device crossing the same band risks that deviation showing up twice, once directly and once multiplied by whatever confined etch geometry a much later step settles on. This step cannot see that geometry yet — it has no way to tighten $\delta_{\text{tol}}$ in anticipation of a number not yet decided — so the only available response is to hold the band as tightly as the apparatus allows and let later steps inherit whatever margin remains.

Solid Composition Drift, Read Out Twice Downstream the same tolerance band now protects base body resistance and base-spreading resistance together fraction solidified, g → solid composition, C_s tolerance band, δ_tol g_max — pull stops here out of band — costs two specs here, not one same curve, same band — the stakes behind it are what this series has changed

## Grow a Uniform Single Crystal's Place in the Process Lineage

Growing a uniform single crystal is step three of Philco's surface-barrier process, immediately after N-type doping has fixed a target resistivity and before the crystal is cooled and inspected — mechanically indistinguishable from the equivalent step in this project's alloy-junction series. The distinguishing fact is not in this step's own control problem but in what step two already attached to its outcome: the same uncorrected segregation drift that cost one spec in the alloy-junction route now costs two here. Step four, cooling and inspecting the crystal, is where this step's success at holding the tolerance band first gets measured, exactly as it does in the alloy-junction route.

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