Rate Grown Transistor 1954 Execute Final Growth Phase

# Execute Final Growth Phase: Seventeenth Velocity Step and Dopant Tail-Off Management

Twenty-six velocity steps have configured the bulk and edge compensation strategies. The seventeenth velocity step, from $v_{16}$ to $v_{17}$, enters the final growth phase — the last 15–30 minutes of crystal growth before transition to controlled cooling. At this point, the melt composition is significantly depleted: roughly 80–90% of the donors and acceptors have been incorporated into the crystal. The remaining melt contains a much lower dopant concentration, which affects the segregation behavior. The effective segregation coefficients $k_{\text{eff}}$ shift as the melt composition changes, and the boundary layer dynamics may be altered. Step 27 manages this dopant tail-off — the final stages of dopant incorporation as the melt becomes depleted. The velocity is carefully chosen to maintain the intended dopant profile in the final crystal regions, accounting for the fact that the melt is now a dilute source of dopants rather than a rich one.

## 1. Melt Depletion and Segregation Coefficient Saturation

As the melt is depleted, the concentration-dependence of $k_{\text{eff}}$ becomes relevant. In the early growth regions, when the melt is rich in dopants, the segregation behavior is approximately constant. But as $N_{\text{melt}}$ drops to 10–20% of its initial value, $k_{\text{eff}}$ may saturate, level off, or even reverse direction (for some dopant-host systems). The velocity schedule must be adjusted to account for this tail-off: the velocity must increase slightly in the final growth regions to compensate for the reduced $k_{\text{eff}}$, maintaining the intended dopant concentration. This is analogous to Step 22's melt depletion compensation, but now operating at the extreme end of melt depletion.

## 2. Real Diagram: Melt Tail-Off and Final Velocity Adjustment

## 3. Interface Stability at Low Melt Concentration

As melt concentration decreases, the interface can become unstable: surface tension effects become more pronounced, and thermal convection patterns in the depleted melt may shift. Some rate-grown systems show a constitutional supercooling threshold at low melt concentration — the interface can spontaneously break into dendrites or develop instabilities if the conditions are not carefully controlled. Step 27 maintains interface stability through careful temperature profiling: keeping the furnace temperature high enough that the interface remains stable (avoiding the supercooling regime) while still maintaining the pulling rate needed for the final dopant profile.

## 4. Transition to Cooldown Preparation

By the end of Step 27, the crystal has grown to near its target size. The melt is nearly depleted. The interface is approaching the end of its usable growth region. Step 27 is the final "growth-mode" step; Step 28–29 continue growth but with deliberately reduced velocity (approaching a crawl), and Step 30 begins the transition to controlled cooling.

## 5. Real Diagram: Melt Depletion and Segregation Coefficient Saturation

## Execute Final Growth Phase's Place in the Process Lineage

Executing the final growth phase is step twenty-seven of this project's rate-grown series — the seventeenth velocity step, managing dopant tail-off as the melt approaches complete depletion. The schedule now incorporates end-of-run compensation, ensuring that the very final regions of the crystal maintain specified dopant concentration despite the depleted melt. Steps twenty-eight and twenty-nine complete the growth crawl. Step thirty begins controlled cooling, locking the crystal structure and completing the device fabrication.