Silicon Grown Junction 1954 Introduce Donor Dopant

# Introduce Donor Dopant: Overcoming Compounded Gallium in a 1414°C Melt

Dropping the second dopant pellet into the crucible is an operation of double compensation: the newly introduced antimony donor atoms must not only overwhelm the initial antimony that drifted during collector growth, but also completely overpower the massive liquid gallium charge added in Step 13 to flip the semiconductor back to n-type for emitter growth. While the first pellet drop (Step 13) had to clear only the Scheil enrichment of the initial melt, this second addition faces a liquid pool already concentrated with $1.27 \times 10^{19}\ \text{atoms/cm}^3$ of gallium. Furthermore, because a high-performance n-p-n transistor requires an emitter heavily doped relative to the base ($N_D^+ \gg N_A$) to secure high emitter injection efficiency ($\gamma \to 1.0$), the antimony dose must force the solid donor concentration into the $10^{18}\ \text{atoms/cm}^3$ regime. With antimony partitioning at $k_{\text{Sb}} \approx 0.023$, the required liquid antimony concentration reaches nearly $10^{20}\ \text{atoms/cm}^3$—approaching the fundamental solid solubility limit of antimony in silicon at 1414°C.

## 1. Double Compensation and the Emitter Injection Efficiency Mandate

$$n_{\text{solid}} = k_{\text{Sb}} \, C_L^{\text{Sb}} - k_{\text{Ga}} \, C_L^{\text{Ga}} \quad \text{and} \quad \gamma \approx \frac{1}{1 + \frac{D_p \, N_A \, W_B}{D_n \, N_D \, L_p}}$$

The electrical performance of the finished transistor places strict demands on the second dopant addition:
1. The Injection Efficiency Imperative:
Transistor current gain $\alpha = \gamma \cdot \alpha_T$ is bounded by emitter injection efficiency $\gamma$. If emitter donor doping $N_D$ is comparable to base acceptor doping $N_A \approx 1.0 \times 10^{17}\ \text{atoms/cm}^3$, forward-biasing the emitter junction causes holes to flood backward from the base into the emitter, wasting current and collapsing $\gamma$. To ensure $\gamma > 0.98$, the emitter must be heavily doped ($n^+$), requiring a solid donor density of $N_D \ge 2.0 \times 10^{18}\ \text{atoms/cm}^3$—twenty times higher than the base.
2. Overcoming Liquid Gallium:
Because gallium atoms remain dissolved in the molten silicon ($C_L^{\text{Ga}} \approx 1.27 \times 10^{19}\ \text{cm}^{-3}$), the freezing solid continuously incorporates $N_A^{\text{solid}} = k_{\text{Ga}} C_L^{\text{Ga}} \approx 1.0 \times 10^{17}\ \text{cm}^{-3}$ of acceptors. The antimony dose must compensate this entire acceptor population before providing any net free electrons.
3. Calculating the Massive Antimony Payload:
To incorporate a total solid antimony concentration of $N_{\text{Sb}}^{\text{solid}} = N_D + N_A \approx 2.1 \times 10^{18}\ \text{cm}^{-3}$, the liquid concentration in the crucible must reach:

$$C_L^{\text{Sb}} = \frac{N_{\text{Sb}}^{\text{solid}}}{k_{\text{Sb}}} \approx \frac{2.1 \times 10^{18}}{0.023} \approx 9.1 \times 10^{19}\ \text{atoms/cm}^3$$

In the remaining $\sim 45\ \text{g}$ silicon melt ($V_L \approx 19\ \text{cm}^3$), this requires introducing over $1.7 \times 10^{21}$ antimony atoms ($\approx 350\ \text{mg}$ of antimony)—a payload orders of magnitude larger than the trace pellet dropped in Step 13.

## 2. Real Diagram: Double-Compensated Melt Chemistry and Pellet Release

Step 17: Introducing Donor Dopant — Overcoming Compounded Gallium massive Sb payload drives liquid C_L(Sb) to 9.1 × 10¹⁹ cm⁻³ to secure N_D⁺ >> N_A for emitter injection compounded liquid concentrations Step 2 (Initial Melt): C_L(Sb) = 5.0 × 10¹⁶ cm⁻³ (lightly n-type) Step 13 (First Drop): C_L(Ga) = 1.27 × 10¹⁹ cm⁻³ (inverts to p-type) Step 17 (Second Drop — Current): C_L(Sb) = 9.1 × 10¹⁹ cm⁻³ (heavily n⁺) overwhelms Ga by 7× in liquid, 20× in solid ✓ secures γ > 0.98 emitter efficiency high-payload Sb carrier pellet drop Sb-Si carrier pellet (~350 mg) rapid dissolution at 1414°C intense convection disperses Sb in < 3 s approaching solid solubility ceiling preheated master-alloy bead prevents localized thermal shock and melt freezing upon entry

## 3. Solubility Limits and Thermal Management at 1414°C

$$C_{\text{solid}}^{\text{max}}(\text{Sb in Si}) \approx 6.0 \times 10^{19}\ \text{atoms/cm}^3 \quad \text{vs.} \quad N_{\text{Sb}}^{\text{solid}} \approx 2.1 \times 10^{18}\ \text{atoms/cm}^3$$

Injecting such a concentrated dopant charge into a high-temperature silicon melt introduces severe thermodynamic constraints:
1. Solid Solubility Headroom:
The maximum equilibrium solid solubility of antimony in single-crystal silicon is approximately $6.0 \times 10^{19}\ \text{atoms/cm}^3$ at 1414°C. Our operating target of $N_{\text{Sb}}^{\text{solid}} \approx 2.1 \times 10^{18}\ \text{atoms/cm}^3$ occupies about $3.5\%$ of the solubility limit. Operating safely below the retrograde solubility curve prevents the formation of antimony precipitates, dislocation loops, or constitutional supercooling at the growth front.
2. Thermal Quench Prevention:
Because the antimony payload ($\sim 350\ \text{mg}$) is substantially larger than the tiny gallium pellet used earlier, introducing a cold mass into the 1414°C melt could cause localized freezing or send thermal shockwaves through the delicate liquid meniscus. Texas Instruments engineers solved this by using a high-density, preheated silicon-antimony master alloy bead. Dropping the bead into the outer perimeter of the crucible allowed intense RF induction currents and toroidal thermal convection to dissolve and assimilate the pellet within two to three seconds.
3. Irreversible Finality:
This drop marks the ultimate dopant switch of the crystal pull. With both gallium and massive quantities of antimony now coexisting in the liquid pool, the remaining silicon charge is irreversibly committed. The ingot must promptly establish the second junction (Step 18) and complete the heavily doped emitter region (Step 19).

Net Solid Carrier Concentration Across Ingot Pull Axis Step 17 donor addition drives emitter doping to 2 × 10¹⁸ cm⁻³, completing the n-p-n sandwich axial crystal growth distance z → net carrier concentration (log scale) Collector: n ≈ 10¹⁵ cm⁻³ Base: p ≈ 10¹⁷ Emitter: n⁺ ≈ 2 × 10¹⁸ cm⁻³ Step 17 Drop Junction 2 (E-B) heavy emitter doping ensures injected electrons dominate, achieving high transistor current gain

## Introduce Donor Dopant's Place in the Process Lineage

Introducing the donor dopant is step seventeen of this project's 1954 silicon grown-junction transistor series — following the timed growth of the thin p-type base, and preceding the emergence of the emitter-base junction. It is the second and final active dopant-switch event in the pulling process, injecting an intense antimony dose to overpower the gallium acceptors and invert the melt back to n-type. Step eighteen, form second junction, establishes the emitter-base metallurgical boundary that completes the physical n-p-n sandwich.

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