Introduce Acceptor Dopant

# Introduce Acceptor Dopant: Overwhelming a Melt That Has Already Drifted

By the time the acceptor pellet goes in, the melt is no longer at the donor concentration step two deliberately established — growing the entire length of the first N-type region has already enriched it, and the acceptor dose has to clear that drifted, higher concentration, not the original target. Segregation during crystal growth doesn't just set the solid's doping profile the way step one's zone-refining article described; it runs in reverse for the liquid left behind. Because the segregation coefficient for the melt's donor species is less than one, the growing solid incorporates donor less readily than the liquid it came from, which means every bit of solid that freezes during step twelve's quiet interval leaves the remaining liquid slightly richer in donor than it started. The "sufficient acceptor" instruction isn't a fixed recipe — it's sized against wherever the melt's donor concentration actually sits the moment this pellet is dropped.

## 1. The Threshold Target Has Moved Since Step Two

$$N_D^{\text{melt}}(t_{13}) = N_D^{\text{melt}}(t_2) \cdot \left[1 - (1-k_0)\, \frac{V_{\text{solidified}}}{V_{\text{melt,0}}}\right]^{-1}$$

This is the same Scheil-type enrichment mechanism from step one's zone-refining discussion, applied here to the opposite side of the interface: the melt's donor concentration rises as a growing fraction of the total charge solidifies into region one, because each increment of frozen solid rejects more donor into the shrinking liquid than it retains. By the time region one has finished growing to its scheduled length, $N_D^{\text{melt}}$ sitting in the crucible is measurably higher than the $N_D^{\text{melt}}$ step two originally established — which means the acceptor pellet's required mass, computed back in step four against an assumed melt concentration, has to account for this drift rather than the original number, or the dose undershoots a target that quietly moved while region one was growing.

## 2. Real Diagram: Introducing the Dose Into an Already-Drifted Melt

Melt Donor Concentration Drifts Up Before the Acceptor Ever Arrives region one's own growth enriches the remaining liquid, raising the bar this dose must clear melt at step two, N_D(t₂) N_D set deliberately here this is the number step four's pellet mass assumed melt at step thirteen, N_D(t₁₃) N_D now higher — enriched by region 1's growth Ga pellet, dropping in the pellet mass only overwhelms the melt if it's sized against this drifted number, not the original

## 3. Why This Drift Is Invisible Unless Explicitly Tracked

$$\Delta N_D = N_D^{\text{melt}}(t_{13}) - N_D^{\text{melt}}(t_2) > 0$$

Nothing about the furnace readout, the rod's diameter, or the melt's visible appearance changes as this drift accumulates — it's a pure consequence of how much solid has frozen out of a melt with $k_0 < 1$, calculable in advance from the scheduled length of region one, but invisible to any instrument watching the process in real time. A process that treats the step-four pellet mass as a fixed number, reused unchanged regardless of how long region one actually grew, risks a systematically undersized acceptor dose on any ingot where region one ran longer than the nominal schedule — not because the pellet was weighed wrong, but because the target it was weighed against had already moved. This is the same class of quiet, instrument-invisible error as the crucible contamination discussed in step five, surfacing only in the finished device's junction depth or base width.

Melt Donor Concentration Rises as Region One Grows the pellet mass from step four has to account for where this curve ends, not where it started fraction of melt solidified into region 1 → melt donor concentration, N_D step 2 target, N_D(t₂) step 13, N_D(t₁₃) — actual target a longer region 1 means a bigger gap between these two points

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

Introducing acceptor dopant is step thirteen of the 1951 grown-junction transistor's full manufacturing sequence — the first active dopant-switch event, after the first N-type region has finished growing, and before that changed composition solidifies into the base region. It is the step where the pellet prepared back in step four finally meets a melt whose donor concentration has already drifted upward from region one's own growth, and where "sufficient" has to be measured against that drifted target, not the original.

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