Silicon Grown Junction 1954 Begin Crystal Pulling

# Begin Crystal Pulling: Reversing the 1414°C Interface via Stefan Heat Balance

Step nine deliberately dissolved the seed tip into the melt; step ten must reverse that exact interface from receding to advancing, and the reversal isn't achieved merely by hoisting the pull rod — it is governed by the rigorous Stefan heat balance at the 1414°C freezing boundary. At the instant melt-back concludes, the solid-liquid meniscus is still in a state of net dissolution ($v_{\text{interface}} < 0$). Initiating crystal pulling requires simultaneously coordinating upward withdrawal and a minor reduction in RF heating power so that heat conducted away through the solid seed exceeds the heat arriving from the liquid melt plus the latent heat released by solidification. Because silicon releases an extraordinary $1787\ \text{kJ/kg}$ of latent heat upon freezing—$3.5\times$ more than germanium—the thermal boundary is far more sensitive to pull velocity. This step produces the first few millimeters of newly grown single-crystal silicon, nucleating cleanly onto the pristine melted-back lattice prepared in Step 9.

## 1. The Stefan Condition Under Silicon's Extreme Latent Heat Release

$$k_s \left(\frac{\partial T}{\partial z}\right)_{\text{solid}} - k_l \left(\frac{\partial T}{\partial z}\right)_{\text{liquid}} = \rho_s \, L_f \, v_{\text{interface}}$$

The physical transition from dissolving seed to growing single crystal is dictated by the thermal continuity equation at the freezing interface:
1. The Competing Heat Fluxes:
* Solid-Side Conduction ($q_s$): Heat conducted upward through the solid seed crystal: $k_s (\partial T/\partial z)_s$, where solid silicon thermal conductivity near the melting point is $k_s \approx 22\ \text{W/(m}\cdot\text{K)}$.
* Liquid-Side Conduction ($q_l$): Heat delivered to the interface from the superheated melt pool: $k_l (\partial T/\partial z)_l$, where metallic liquid silicon conductivity is $k_l \approx 67\ \text{W/(m}\cdot\text{K)}$.
* Latent Heat Generation: As liquid freezes into solid, latent heat is continuously liberated at rate $\rho_s L_f v_{\text{interface}}$, where $\rho_s \approx 2.33\ \text{g/cm}^3$ and $L_f = 1787\ \text{kJ/kg}$, yielding a volumetric release of $\sim 4.16 \times 10^9\ \text{J/m}^3$.
2. The Reversal Mechanism:
In Step 9, $q_l > q_s$, driving $v_{\text{interface}} < 0$ (net melting). To initiate growth, the furnace operator trims the RF induction generator power by $1\text{ to }3\%$ and starts the mechanical pull drive. Lowering melt superheat flattens the liquid thermal gradient $(\partial T/\partial z)_l$, allowing heat extraction through the seed shank to exceed liquid influx. The interface velocity $v_{\text{interface}}$ passes through zero and becomes positive ($v_{\text{interface}} > 0$), initiating crystallization.
3. The Penalty of Excess Pull Speed:
If the operator simply increases mechanical pull speed without establishing the required thermal extraction gradient, the massive latent heat liberated by freezing cannot escape. The interface overheats, the liquid meniscus stretches into an unstable liquid thread, and the seed pulls clear of the melt with zero crystal growth.

## 2. Real Diagram: Dissolution Meniscus Reversing into an Advancing Solidification Front

Silicon Interface Reversal: Dissolution to Solidification satisfying the Stefan condition: axial heat conduction through solid must absorb massive L_f (1787 kJ/kg) step 9: net dissolution (v < 0) <111> seed molten Si (1414°C) liquid heat in > solid conduction seed melts back; interface recedes step 10: net solidification (v > 0) <111> seed new Si crystal molten Si (1414°C) solid conduction > liquid flux + L_f crystal advances; builds new silicon rod power reduction and pull speed together reverse interface velocity from negative to positive

## 3. Pull Speed, Rotation, and Necking: Trapping Dislocations at the Start

$$v_{\text{interface}}\big|_{t \to t_{10}^+} > 0 \quad \text{and} \quad \omega_{\text{seed}} \sim 10\text{--}20\ \text{rpm}$$

Successfully transitioning into steady crystal growth requires managing mechanical and thermal dynamics:
1. Seed Rotation for Axisymmetric Meniscus:
Molten silicon is held under a slight positive argon pressure. To prevent azimuthal thermal asymmetries caused by asymmetric RF induction coil turns, the seed chuck rotates continuously at $10\text{ to }20\ \text{rpm}$. Rotation averages out horizontal thermal variations, maintaining a perfectly circular, stable liquid meniscus.
2. The Initial Necking Procedure:
When withdrawal starts, operators withdraw the seed at a slightly elevated velocity ($v \sim 3\text{ to }4\ \text{mm/min}$) to form a narrow neck ($1\text{ to }2\ \text{mm}$ in diameter). Gordon Teal's team recognized that dislocations generated at the seed interface during thermal contact glide along $\{111\}$ slip planes at $1414^\circ\text{C}$. In a thin neck, these dislocations intersect the exterior lateral surface and terminate, preventing dislocation networks from propagating down into the main ingot body.
3. Inheriting Step 1–9 Preparation:
The newly crystallized silicon is the collector region of the transistor. It atomically replicates the $\langle 111 \rangle$ lattice of the seed (Step 3), maintains the target donor doping ($N_D \approx 10^{15}\ \text{cm}^{-3}$) established in Step 2, and preserves high carrier lifetime through the clean crucible and argon environment (Steps 5 & 6).

Silicon Interface Velocity vs. Seed Withdrawal Rate Stefan condition: crossing point v = 0 marks the transition from dissolution to crystal pulling mechanical withdrawal rate (mm/min) → interface velocity, v (growth > 0, dissolution < 0) reversal threshold, v = 0 step 9: net dissolution step 10 target: stable growth trimming RF power lowers the liquid temperature gradient, shifting the v = 0 crossing to stable speeds

## Begin Crystal Pulling's Place in the Process Lineage

Beginning crystal pulling is step ten of this project's 1954 silicon grown-junction transistor series — following seed tip immersion, and preceding diameter stabilization. It reverses the interface from net dissolution to net crystallization by satisfying the Stefan heat balance at 1414°C, generating the very first newly solidified silicon of the transistor ingot. Step eleven, stabilize rod growth, adjusts pull velocity and furnace power to flare the initial neck into a constant-diameter cylindrical ingot.

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