Rate Grown Transistor 1954 Initiate Pull Velocity

# Initiate the Pull: Beginning the Velocity-Controlled Segregation Sequence

The seed has been positioned at the melt surface, the meniscus has locked in the interface geometry from step eight's melt-back, and the system has reached thermal equilibrium at the dip point. Now the seed begins to move upward, away from the furnace's hottest zone, at a carefully controlled velocity $v_0$ — the starting pull rate of the pull-rate schedule. As the seed moves, solid crystal grows onto the seed's trailing face at the meniscus, and the melt's boundary layer responds to the motion. In the 1951 germanium series, the pull rate is the operator's primary control lever: changing $v$ changes the thermal gradient that the interface experiences, which changes how much dopant gets incorporated into the growing crystal. In the rate-grown series, the pull rate serves the same role, but the context is fundamentally different — the melt now carries two dopants whose effective segregation coefficients $k_{\text{eff}}(D)$ and $k_{\text{eff}}(A)$ respond independently to changes in $v$. The starting velocity $v_0$ is chosen to set an initial ratio of donor to acceptor incorporation that will form the base dopant profile for the rest of the crystal. Every subsequent velocity change in the schedule will adjust that ratio, encoding compositional transitions into the growing lattice.

## 1. Starting Velocity and the Initial Segregation Ratio

$$k_{\text{eff}}(v_0) = \frac{k_0}{k_0 + (1-k_0)e^{-v_0\delta/D}} \quad \text{and} \quad \text{ratio} = \frac{k_{\text{eff}}(D)}{k_{\text{eff}}(A)}$$

The Burton-Prim-Slichter model links effective segregation coefficient directly to pull rate: $k_{\text{eff}} \propto e^{-v\delta/D}$. A slow pull rate gives a thick boundary layer $\delta$, which increases the exponent and decreases $k_{\text{eff}}$ (more segregation, less dopant incorporation). A fast pull rate thins the boundary layer, increases $k_{\text{eff}}$ (less segregation, more dopant incorporation). The ratio of incorporation rates for donor and acceptor depends on how their individual $k_{\text{eff}}$ values respond to $v_0$. Because donor and acceptor have different diffusion coefficients $D_D$ and $D_A$, they have different $k_{\text{eff}}(v)$ curves. At any given velocity $v_0$, the ratio $k_{\text{eff}}(D, v_0) / k_{\text{eff}}(A, v_0)$ sets the initial doping balance — how much more donor (or acceptor) gets incorporated relative to the other. The choice of $v_0$ is therefore not arbitrary: it determines the net doping of the first grown layers, which in turn determines the effective dopant profile as a function of position along the crystal.

## 2. Real Diagram: Growth Interface Motion and Boundary-Layer Response

Seed Pull Begins: Interface Moves, Boundary Layer Responds pull velocity v₀ controls boundary-layer thickness, which controls k_eff for both dopants independently slow pull (small v₀) melt thick δ, low k_eff (more segregation) slow growth, strongly segregated dopants fast pull (large v₀) melt thin δ, high k_eff (less segregation) fast growth, weaker segregation, more incorporation increasing v₀ v₀ sets the initial balance between donor and acceptor incorporation

## 3. The Velocity-Dependent Segregation Curve: Mapping v₀ to the Initial Junction Type

$$\text{Net doping} = (N_D - N_A) \cdot f(v_0, k_D, k_A, \delta) \quad \text{where} \quad f \text{ is non-linear}$$

The effective segregation coefficients for donor and acceptor depend on pull rate, but they do not scale linearly with $v$. The exponential relationship $k_{\text{eff}} \propto e^{-v\delta/D}$ means that small changes in velocity can produce large changes in the incorporation ratio, especially at low velocities where the exponential is steep. An operator choosing $v_0$ is therefore choosing a point on a velocity-segregation curve that maps pull rate to the initial net doping of the grown crystal. For a given pair of dopants with diffusion coefficients $D_D$ and $D_A$, and for a given melt composition and interface geometry, there exists a unique velocity (or narrow range) that produces $N_D = N_A$ — a perfectly compensated junction where net doping is zero at the start of pulling. Velocities slower than this produce net acceptor doping (p-type). Velocities faster than this produce net donor doping (n-type). The schedule's first step chooses this initial velocity deliberately, knowing that subsequent velocity changes will create compositional transitions from this starting point.

Effective Segregation Coefficient vs. Pull Rate k_eff(D) and k_eff(A) both follow k_eff ∝ exp(−vδ/D), but with different D values pull rate v → k_eff (dopant incorporation efficiency) k_eff(donor) k_eff(acceptor) v₀ chosen here start: donor-heavy slow pull: high segregation (low k_eff, strong net doping) different D values mean different slopes; the ratio of k_eff values encodes the doping balance

## 4. First-Layer Incorporation: Encoding Begins Immediately

Once the meniscus has moved up and solid has begun to form on the trailing face of the seed, the very first grown layer encodes the segregation behavior determined by $v_0$ and the boundary-layer geometry. Because the seed's tip from step eight has (potentially) asymmetric geometry, the boundary-layer thickness may still be non-uniform across the interface. This means the effective segregation may still be spatially heterogeneous in that first layer, even as the overall doping level is determined by $v_0$. As the crystal grows and the interface moves further from the seed's tip, the interface geometry becomes more controlled by the thermal profile and less by the seed's initial asymmetry. But the first layer — grown while the meniscus is still at the seed's tip — carries the permanent signature of steps eight and nine. All subsequent velocity modulation will occur on top of this initial condition.

## Initiate the Pull's Place in the Process Lineage

Initiating the pull is step ten of this project's rate-grown series — the moment the seed transitions from static immersion to active crystal growth. The starting velocity $v_0$ is chosen to set the initial segregation ratio between donor and acceptor incorporation, launching the velocity-modulation sequence that will encode compositional transitions into the grown crystal. The seed's geometric signature from steps eight and nine is now locked into the interface and persists through the first grown layers. Steps eleven through twenty-nine execute the pull-rate schedule, modulating velocity at carefully timed intervals to create the intended dopant profile. Step ten is the beginning of that sequence.

Take rate grown transistor 1954 initiate pull velocity further

Ask the copilot about this term, or have our engineers assess it against your process.

Ask CFSGPT about this → Request a technical assessment → Open the CFS app