Silicon Grown Junction 1954 Stabilize Rod Growth

# Stabilize Rod Growth: Flaring to Full Diameter Under 1414°C Radiative Feedback

Reversing the interface in Step 10 only initiates growth on a narrow neck; flaring that neck outward to full ingot diameter and maintaining a uniform cylinder is a continuous thermal control challenge dominated by silicon's intense radiant heat dissipation. Once crystal growth begins, the ingot diameter is not fixed by a physical mold — it is dynamically governed by the balance between heat conducted away up the solid crystal and heat supplied by the 1414°C molten pool. In silicon, flaring the crystal shoulder from a 2 mm neck to a 20 mm body increases the crystal's radiating surface area by an order of magnitude. Because radiant loss scales as $T^4$, the expanding shoulder acts as a powerful heat sink that extracts heat from the freezing front, triggering a self-accelerating freeze-out unless the operator actively balances pull rate and RF generator power. Stabilizing rod growth holds pull speed rock-steady—a non-negotiable prerequisite for the micron-scale base-width timing calculations in the upcoming junction-forming steps.

## 1. Shoulder Flaring Dynamics and Radiative Feedback at 1414°C

$$\frac{dr}{dt} = v_{\text{pull}} \, \tan(\theta_L - \theta_0) \quad \text{and} \quad P_{\text{rad}} \approx \epsilon \, \sigma_{\text{SB}} \, A_{\text{shoulder}} \, T^4$$

The lateral evolution of the crystal radius $r$ depends on the meniscus contact angle and the thermal boundary conditions:
1. Meniscus Contact Angle Dynamics:
Growth proceeds along the $\langle 111 \rangle$ axis with a characteristic equilibrium growth angle $\theta_0 \approx 11^\circ$. If the meniscus contact angle $\theta_L$ matches $\theta_0$, the crystal grows as a straight cylinder ($dr/dt = 0$). To flare the crystal outward from the initial narrow neck ($r \sim 1\ \text{mm}$) to full ingot diameter ($r \sim 10\ \text{mm}$), the operator slightly reduces pull velocity $v_{\text{pull}}$ or lowers furnace power, tipping $\theta_L > \theta_0$ so that the solid-liquid boundary flares laterally ($dr/dt > 0$).
2. The Runaway Radiative Feedback Loop:
At 1414°C ($1687\ \text{K}$), radiant heat loss from the crystal surface into the water-cooled quartz chamber exceeds $100\ \text{kW/m}^2$. As the crystal shoulder expands from 2 mm to 20 mm across, its radiant surface area increases tenfold. This sudden surge in radiant emission chills the solidifying meniscus from above. Left unmanaged, the localized cooling accelerates lateral crystallization, causing the crystal to spread uncontrollably across the melt surface until it freezes against the crucible walls ("freeze-out").
3. Coordinated RF Power Trimming:
To arrest shoulder widening at the target diameter, the operator must preemptively increase RF generator heating power as the crystal expands. By supplying additional heat to the liquid perimeter to counterbalance radiant losses from the shoulder, $\theta_L$ is restored to $\theta_0$, locking the ingot into a uniform, straight-walled cylindrical rod.

## 2. Real Diagram: Coordinated Shoulder Flaring vs. Uncontrolled Freeze-Out / Necking

Silicon Ingot Diameter Stabilization Under 1414°C Radiant Feedback shoulder flaring increases radiant heat loss tenfold — RF power must compensate to prevent freeze-out coordinated control: steady cylinder seed neck 20 mm cylindrical body dr/dt = 0 (constant pull rate) molten Si (1414°C) ✓ stable baseline ready for junction timing uncoordinated: runaway radiative freeze-out runaway shoulder chill severe diameter drift molten Si (1414°C) distorts base width timing; ruins transistors straight vertical sidewalls guarantee that pull rate translates directly into planned base width

## 3. Why Ingot Diameter Stability Dictates Transistor Base Width Precision

$$W_B = v_{\text{pull}} \cdot \Delta t_{\text{interval}} \quad \text{and} \quad f_\alpha \propto \frac{1}{W_B^2}$$

In the grown-junction transistor process, maintaining constant rod diameter is directly tied to device electrical performance:
1. The Base-Width Scheduling Equation:
In upcoming Steps 14 through 16, the ultra-thin p-type base region is produced by dropping a gallium pellet into the melt, allowing it to dissolve, pulling crystal for a short time interval $\Delta t$, and dropping an overriding antimony donor pellet. Because the crystal is pulled continuously during this sequence, the resulting base width $W_B$ is calculated purely as pull velocity times time interval: $W_B = v_{\text{pull}} \cdot \Delta t_{\text{interval}}$.
2. The Threat of Emergency Pull-Rate Corrections:
If the crystal diameter begins flaring or necking unexpectedly, the operator is forced to rapidly adjust $v_{\text{pull}}$ or RF generator power. Any mid-run surge in pull speed directly distorts base thickness. For a target base width of $25\ \mu\text{m}$ pulled at $1.5\ \text{mm/min}$ ($\Delta t \approx 60\ \text{seconds}$), a mere $20\%$ fluctuation in pull rate alters $W_B$ by $5\ \mu\text{m}$.
3. High-Frequency and Breakdown Consequences:
The alpha cutoff frequency $f_\alpha$ depends inversely on the square of base width ($f_\alpha \propto 1/W_B^2$). A swollen base drastically reduces high-frequency response, while an irregularly necked base causes localized collector-emitter punch-through shorts. Stabilizing rod growth into a perfect cylinder in Step 11 is what locks $v_{\text{pull}}$ into an invariant constant before dopant pellets are introduced.

Silicon Ingot Radius vs. Pull Velocity r ∝ v_pull^(-1/2) — high 1414°C radiant emission steepens the sensitivity curve pull velocity, v_pull (mm/min) → ingot radius, r (mm) stable operating band (r = 10 mm ± 0.5 mm) too slow: runaway freeze-out too fast: neck-off detachment holding v_pull invariant in this band guarantees precise base width W_B across the entire bar

## Stabilize Rod Growth's Place in the Process Lineage

Stabilizing rod growth is step eleven of this project's 1954 silicon grown-junction transistor series — following the initiation of crystal pulling, and preceding the growth of the first n-type collector region and the subsequent dopant pellet drops. It transitions the narrow seed neck into a full-diameter, uniform cylindrical crystal under intense 1414°C radiative feedback, locking the pull rate into the rock-solid baseline that governs all downstream junction-timing calculations.

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