Alloy Junction 1952 Grow Uniform Single Crystal

# Grow a Uniform Single Crystal: Fighting the Drift Normal Freezing Introduces on Its Own

Nobody has to deliberately change this melt's composition during the pull — step two already fixed a single target resistivity, and this step's entire job is to deliver a crystal that still has that resistivity everywhere along its length. That sounds like it should be the easy case compared to a process that deliberately times composition changes into the melt mid-growth, but ordinary solidification works against uniformity on its own: as a Czochralski pull consumes the melt, any solid-liquid segregation coefficient other than exactly one enriches or depletes the remaining liquid in dopant, so the crystal solidifying later in the pull forms from a melt that is no longer at the concentration the pull started with. Holding the crystal uniform means fighting a drift normal freezing introduces for free, not simply avoiding the mistake of adding a second dopant.

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

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

As fraction solidified $g$ increases from zero toward one, the solid composition $C_s$ drifts away from the melt's starting concentration $C_0$ purely as a consequence of the segregation coefficient $k_0$ — for germanium's common donor species, $k_0 < 1$, so solute is rejected into the shrinking melt faster than it's consumed, and the melt's own concentration rises as the pull proceeds. A crystal pulled from a melt obeying this relation does not have one resistivity; it has a resistivity that creeps from one end to the other, entirely independent of any intentional process error, which is exactly why "grow a uniform crystal" is a real control problem rather than a default outcome of simply not adding anything extra to the melt.

## 2. Real Diagram: The Melt Concentrates Itself as the Pull Consumes It

The Remaining Melt Enriches Itself as the Rod Grows no second dose is added — the drift comes from segregation at the freezing interface alone crucible, melt early: close to C₀ shrinks and concentrates as g rises growing rod seed-end material tail-end material near target ρ ρ drifted off target same melt, same intended dose — the drift is the freezing process itself, not an error

## 3. The Pull Schedule Has to Keep the Drift Inside One Resistivity Band

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

Because step two's target resistivity applies to the whole crystal, not just its seed end, the practical control problem is choosing a pull schedule — rotation rate, thermal gradient, and most directly how much of the melt is allowed to be consumed before pulling stops — that keeps $C_s(g)$ inside an acceptable tolerance band $\delta_{\text{tol}}$ out only as far as $g_{\max}$, rather than pulling the melt to exhaustion and accepting whatever drift results. This directly foreshadows the very next operations in this sequence: cooling and inspecting the crystal, then selecting usable sections, are the steps that actually measure where $C_s(g)$ left the tolerance band and draw the cutoff line this step's schedule was only trying to anticipate.

Solid Composition Drifts With Fraction Solidified the tolerance band sets how far the pull can run before the crystal drifts out of spec fraction solidified, g → solid composition, C_s tolerance band, δ_tol g_max — pull stops here out of band if pulled further growth doesn't stop because the melt runs out — it stops because the tolerance band does

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

Growing a uniform single crystal is step three of RCA's forty-two-step alloy-junction manufacturing sequence — immediately after N-type doping has fixed a single target resistivity, and before the crystal is cooled and inspected. It is the step that confronts the fact that ordinary solidification drifts composition on its own, with no deliberate dose change required to cause it, and manages that drift against a tolerance band rather than against a scheduled change the process actually wants. Step four, cooling and inspecting the crystal, is the first point where this step's success or failure at holding that band actually gets measured.

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