Incorporate the Changed Composition

# Incorporate the Changed Composition: Why the Solid Doesn't Just Copy the Melt

Overwhelming the melt's net carrier type in step thirteen doesn't automatically mean the next solid to freeze is cleanly p-type at a known concentration — the solid that incorporates this new composition partitions gallium according to its own segregation coefficient, and that coefficient is not the same number governing the donor species this process has relied on in every prior step. Antimony's segregation coefficient in germanium is one specific small value; gallium's is a different one. The acceptor concentration actually built into the newly-frozen solid is the melt's local gallium concentration multiplied by gallium's own $k_0^{\text{Ga}}$, not a direct copy of whatever concentration the pellet produced in the liquid. This step is where the melt's chemistry, already flipped, finally gets translated into the solid's actual electrical composition — and that translation has its own, separate physics.

## 1. Two Different Segregation Coefficients, Two Different Translations

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

Every prior step's segregation discussion — zone refining in step one, the melt-drift calculation in step thirteen — relied on the donor species' segregation coefficient. Gallium's $k_0^{\text{Ga}}$ in germanium is a materially different number, which means the newly-incorporated p-type solid's actual acceptor concentration can't be predicted by simply assuming it mirrors the liquid-phase concentration computed for the melt-flip threshold. The base region's doping level — a quantity that matters as much to the finished device's electrical behavior as its width does — depends on this second, independent segregation relationship, evaluated fresh for the acceptor species rather than inherited from the donor-species calculations the process has used so far.

## 2. Real Diagram: Mixing Time Determines Whether the Junction Is Sharp or Graded

Growth Continues While the Melt Is Still Mixing whether incorporation is abrupt or graded depends on mixing time vs. growth time fast mixing, τ_mix ≪ τ_growth n region (before) p region, uniform C_s^Ga sharp boundary, well-defined junction slow mixing, τ_mix ≳ τ_growth n region (before) graded, rising C_s^Ga p region, now uniform diffuse boundary, uncertain junction depth crucible convection and pull rate together decide which side of this comparison the process sits on

## 3. Why This Step's Outcome Hides Inside an Apparently Normal-Looking Junction

$$\tau_{\text{mix}} \ll \tau_{\text{growth}} \;\Rightarrow\; \text{sharp junction}; \qquad \tau_{\text{mix}} \gtrsim \tau_{\text{growth}} \;\Rightarrow\; \text{graded junction}$$

A graded transition doesn't necessarily look wrong under casual inspection — the crystal still grows, still looks like ordinary rod material, and the finished device may still function. But a graded acceptor profile instead of an abrupt one changes the effective junction depth and the local electric field distribution right where the base region begins, parameters the device's gain and breakdown characteristics depend on. Crucible convection, melt volume, and the pull rate already governing base width (step eleven's diameter-control loop and the base-width scheduling equation) jointly determine which regime this step actually falls into — a process run at the same nominal pull rate on a larger melt volume, with weaker convective mixing, can produce a measurably more graded junction than the same recipe on a smaller, better-stirred melt, without any single parameter obviously being "wrong."

Incorporated Acceptor Concentration vs. Elapsed Growth Time C_s^Ga = k₀^Ga · C_l^Ga(t) — the solid lags the melt’s own mixing elapsed time since pellet drop → incorporated acceptor conc., C_s^Ga fast mix — abrupt step slow mix — graded rise both curves reach the same final concentration — only the shape of the transition differs

## Incorporate the Changed Composition's Place in the Process Lineage

Incorporating the changed composition is step fourteen of the 1951 grown-junction transistor's full manufacturing sequence — immediately after the acceptor dopant has been introduced, and before the first junction boundary is formally established. It is the step where the melt's already-flipped chemistry is translated into the solid's actual acceptor concentration through gallium's own segregation coefficient, and where the race between melt mixing and continued growth determines whether that translation produces a sharp junction or a graded one.

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