Silicon Grown Junction 1954 Incorporate Composition

# Incorporate Changed Composition: Convective Mixing and Gallium Partitioning at 1414°C

Overwhelming the liquid melt with a gallium pellet in Step 13 does not instantly produce a uniform, p-type solid crystal — the dissolving dopant must disperse through the liquid pool, and the newly freezing solid partitions gallium according to its own equilibrium distribution coefficient ( $k_{\text{Ga}} \approx 0.008$ ), which is vastly different from that of antimony ( $k_{\text{Sb}} \approx 0.023$ ). The acceptor concentration actually incorporated into the growing silicon lattice is determined by the local liquid-phase concentration at the freezing boundary multiplied by $k_{\text{Ga}}$ . If thermal convection and seed rotation homogenize the molten charge rapidly compared to the growth rate ( $\tau_{\text{mix}} \ll \tau_{\text{growth}}$ ), the transition from n-type collector to p-type base is sharp and well-defined. If convective mixing lags behind crystal withdrawal, the dopant incorporates as a broad, graded transition layer that alters junction capacitance, base thickness, and transistor high-frequency cutoff.

## 1. Disparate Segregation Coefficients and Net Carrier Partitioning

p_{\text{solid}}(z) = k_{\text{Ga}} \, C_L^{\text{Ga}}(z) - k_{\text{Sb}} \, C_L^{\text{Sb}}(z), \qquad k_{\text{Ga}} \approx 0.008 \ll k_{\text{Sb}} \approx 0.023

The electrical transformation of the freezing crystal is dictated by differential solute partitioning at the solid-liquid interface:
1. Unequal Segregation Dynamics:
While the liquid melt was flooded with gallium in Step 13 to flip its chemical composition, solid silicon does not copy liquid stoichiometry directly. Because antimony has a distribution coefficient of $k_{\text{Sb}} \approx 0.023$ and gallium has $k_{\text{Ga}} \approx 0.008$ , solid silicon incorporates antimony nearly three times more readily than gallium per atom in the melt.
2. The Solid-Phase Crossover Condition:
Even after the liquid melt becomes chemically dominated by gallium, the freezing crystal remains n-type until the liquid gallium concentration satisfies:

C_L^{\text{Ga}}(z) > \frac{k_{\text{Sb}}}{k_{\text{Ga}}} \, C_L^{\text{Sb}}(z) \approx 2.88 \, C_L^{\text{Sb}}(z)

Only when local liquid gallium density surpasses this threshold does the net carrier concentration in the solid flip from negative (excess donors) to positive (excess holes), establishing p-type silicon.
3. The Base Doping Plateau:
Once full dissolution and mixing are complete, the liquid gallium concentration stabilizes at $C_L^{\text{Ga}} \approx 1.27 \times 10^{19}\ \text{atoms/cm}^3$ . Partitioning into the freezing crystal at $k_{\text{Ga}} = 0.008$ yields a solid acceptor concentration of $N_A \approx 1.01 \times 10^{17}\ \text{cm}^{-3}$ . Subtracting the compensated solid antimony donors ( $N_D \approx 1.3 \times 10^{15}\ \text{cm}^{-3}$ ) leaves a net p-type base carrier density of $p \approx 1.0 \times 10^{17}\ \text{cm}^{-3}$ .

## 2. Real Diagram: Fast Convective Homogenization vs. Sluggish Graded Transition

## 3. Convective Mixing Time vs. Growth Velocity in 1414°C Silicon

\tau_{\text{mix}} \approx \frac{R_{\text{crucible}}}{u_{\text{conv}}} \quad \text{and} \quad \Delta z_{\text{transition}} = v_{\text{pull}} \cdot \tau_{\text{mix}}

The physical width of the collector-base junction transition layer depends on the interplay between fluid mechanics and crystal pulling speed:
1. Molten Silicon Fluid Dynamics:
Molten silicon at 1414°C is an inviscid liquid metal with low kinematic viscosity ( $\nu \approx 3 \times 10^{-7}\ \text{m}^2/\text{s}$ ) and substantial surface tension ( $\gamma \approx 730\ \text{mN/m}$ ). Radial temperature gradients between the hot crucible wall and the cooler seed interface generate intense natural buoyant convection cells ( $u_{\text{conv}} \sim 2\ \text{cm/s}$ ).
2. Forced Convective Stirring via Seed Rotation:
Axial seed rotation ( $\omega \sim 20\ \text{rpm}$ ) acts as an Ekman boundary-layer pump directly beneath the crystal. Liquid silicon is drawn vertically upward toward the center of the growth interface and driven radially outward along the meniscus. In a standard $50\ \text{g}$ silicon charge ( $R_{\text{crucible}} \approx 2.5\ \text{cm}$ ), this combined natural and forced convection homogenizes the dissolved gallium pellet within $\tau_{\text{mix}} \approx 2\text{ to }4\ \text{seconds}$ .
3. Minimizing Transition Thickness:
At a continuous pull speed of $v_{\text{pull}} = 1.5\ \text{mm/min}$ ( $25\ \mu\text{m/s}$ ), the crystal advances by $50\text{ to }100\ \mu\text{m}$ during a 3-second mixing interval. To sharpen the junction further, Gordon Teal's Texas Instruments protocol called for momentarily retarding or pausing the pull rate upon pellet drop. Slowing the pull during the 3-second dissolution window compressed the spatial transition width $\Delta z_{\text{transition}}$ to under $10\ \mu\text{m}$ , ensuring an abrupt collector-base transition.

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

Incorporating the changed composition is step fourteen of this project's 1954 silicon grown-junction transistor series — immediately following the introduction of the acceptor dopant, and preceding the formal emergence of the collector-base junction. It is the hydrodynamic and thermodynamic bridge where liquid gallium disperses through the 1414°C melt and partitions into the freezing solid according to its specific segregation coefficient. Step fifteen, form first junction, marks the exact crystallographic plane where solid acceptor density overtakes donor density, freezing the first p-n junction into the silicon ingot.

Silicon Grown Junction 1954 Incorporate Composition

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