Alloy Junction 1952 Cool the First Assembly

# Cool the First Assembly: A Cooling Rate Window, Not Just a Cooling Rate

The tack bond step fourteen just formed is still vulnerable in two completely different ways during cooldown, and they pull in opposite directions — cool too slowly and the bond keeps dissolving germanium it was never supposed to touch; cool too quickly and indium's own thermal contraction, mismatched against germanium's, can crack the interface that just formed. This isn't the same thermal-stress problem step four solved for the whole crystal during ingot cooldown — that was a bulk, whole-wafer concern about uneven cooling across a large piece. This is a small, local problem confined to the disk's own footprint, driven by two different materials trying to shrink by different amounts as they cool through the same temperature range together.

## 1. Mismatched Contraction Rates Build Stress Right at the Bonded Interface

$$\sigma_{\text{interface}} \approx \frac{E_{\text{Ge}}}{1-\nu}\,(\alpha_{\text{In}} - \alpha_{\text{Ge}})\,\Delta T$$

As the assembly cools through a temperature range $\Delta T$, indium and germanium contract at different rates set by their own thermal expansion coefficients $\alpha_{\text{In}}$ and $\alpha_{\text{Ge}}$, and that mismatch has nowhere to go except into stress concentrated right at the interface where the two materials are actually bonded. This stress scales directly with how far the assembly has cooled — the faster and further the temperature drops before this step is done, the larger $\Delta T$ becomes and the more stress the still-fresh bond has to absorb, which is why cooling this assembly is not simply a matter of getting it below the melting point as efficiently as possible.

## 2. Real Diagram: Stress Concentrates at the Edge of the Bond, Not the Middle

Two Materials, Two Contraction Rates, One Interface the mismatch is worst at the bond's own edge, where the constraint is sharpest germanium wafer, α_Ge indium disk, α_In contracts more as it cools interface edge interface edge the center of the bond is far less constrained than its own perimeter

## 3. Cooling Rate Has to Stay Inside a Window, Bounded on Both Sides

$$\dot{T}_{\min} \le |\dot{T}| \le \dot{T}_{\max}$$

The lower bound $\dot{T}_{\min}$ exists for the same reason step fourteen kept its firing brief — a cooldown that lingers too long at elevated temperature continues dissolving germanium into the still-molten or still-hot indium, quietly consuming more of step eleven's dissolution budget than the tack bond was ever supposed to spend. The upper bound $\dot{T}_{\max}$ exists because of the interfacial stress this step's first equation describes — cooling too fast drives $\Delta T$ through its full range before any stress can relax, concentrating the full mismatch at the bond's edge all at once. Neither bound is optional, and unlike many single-sided limits elsewhere in this process, violating either one in either direction produces a real, distinct failure.

Two Failure Modes, One Acceptable Window Between Them this is a bounded-rate problem, not a slower-is-always-safer one cooling rate, |Ṫ| → risk too slow — extra dissolution too fast — interface cracks acceptable window this step's whole job is landing inside that narrow band, not minimizing one risk alone

## Cool the First Assembly's Place in the Process Lineage

Cooling the first assembly is step fifteen of RCA's forty-two-step alloy-junction manufacturing sequence — immediately after the first firing bonded the collector disk, and before the emitter disk is placed on the opposite face. It is the step that solidifies step fourteen's shallow tack bond against two opposing risks at once, continuing to guard the germanium dissolution budget step eleven established while also protecting the fresh interface from the thermal-expansion mismatch stress a too-rapid cooldown would concentrate right at its edge. Step sixteen, placing the emitter disk, begins only once this bond is cooled solid and intact.

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