Introduce Donor Dopant

# Introduce Donor Dopant: Overwhelming a Melt That Has Now Drifted Twice

By the time the antimony pellet goes in, the melt's chemistry has moved twice since its last deliberately-set state — once during region one's growth, enriching it in donor, and again during the base's growth, depleting it of the gallium that drop thirteen worked to establish. Step thirteen only had to account for a single drift, region one's donor enrichment, because the melt hadn't yet been through an active dopant-switch event. This dose has no such luxury: it has to overwhelm whatever net acceptor concentration actually remains after two full growth intervals have each pulled the melt's composition in a different direction, not the concentration drop thirteen's pellet nominally produced.

## 1. Two Compounding Drifts, Not One

$$N_A^{\text{melt}}(t_{17}) = N_A^{\text{melt}}(t_{13}^+) \cdot \left[1 - (1-k_0^{\text{Ga}})\, \frac{V_{\text{base}}}{V_{\text{melt}}(t_{13}^+)}\right]^{-1}$$

Exactly as region one's growth enriched the melt's donor concentration by the Scheil mechanism step thirteen had to account for, the base region's own growth depletes the melt's gallium concentration by the same mechanism, now acting on the acceptor species instead. The antimony pellet's required mass has to clear this twice-drifted concentration — first pushed up by step twelve's quiet interval, then pushed down again by step sixteen's — and a dose sized only against the melt state immediately after step thirteen's drop, without accounting for the base-growth depletion on top of it, risks undershooting a target that has moved a second time since the last time anyone measured it directly.

## 2. Real Diagram: Tracking the Melt Through Both Growth Intervals

Melt Composition History Across Three Intervals the antimony dose has to clear wherever this history actually ends up, not a single lookup value elapsed pull time → region 1 growth N_D drifts up step 13: Ga drop base growth N_A drifts down step 17: Sb drop, here target: twice-drifted N_A the pellet mass from step four was computed before either drift was known in combination

## 3. Antimony's Own Segregation Coefficient Still Governs Incorporation

$$C_s^{\text{Sb}} = k_0^{\text{Sb}} \, C_l^{\text{Sb}}$$

Once the melt's net type flips back to donor-dominant, the newly-frozen solid's actual donor concentration is set by antimony's segregation coefficient — the same relationship step one used to describe zone-refining and step thirteen used to track region one's drift, now applied a third time to close the loop on the n-p-n sandwich. This closes a structural parallel the process has followed since step fourteen: every dopant switch has two separate questions attached — did the melt's net type actually flip (a threshold problem, already covered in the "Second Pellet Drop" discussion of base-width timing), and what concentration does the solid that freezes afterward actually carry (a segregation problem, distinct from timing and governed by $k_0^{\text{Sb}}$ alone).

Melt Acceptor Concentration, Two Growth Intervals the pellet-four assumption only matches reality at one single point on this curve elapsed pull time → melt acceptor concentration, N_A step 13 assumption, single point actual N_A at step 17, drifted down the gap between these two points is exactly what step 17's dose has to still clear

## Introduce Donor Dopant's Place in the Process Lineage

Introducing donor dopant is step seventeen of the 1951 grown-junction transistor's full manufacturing sequence — the second and final active dopant-switch event, after the thin p-type base has finished growing, and before the second junction forms and the collector region begins. It is the step where the pellet prepared back in step four meets a melt that has drifted through two separate growth intervals in two different directions, and where "sufficient" has to be measured against that compounded, twice-moved target.

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