Rate Grown Transistor 1954 Begin Pull Rate Modulation

# Begin Pull-Rate Modulation: The First Velocity Change Encodes the First Compositional Transition

The crystal is now growing at the starting velocity $$v_0$$ , the first layer has been laid down with its segregation behavior set by that initial rate, and the pull-rate schedule now commands the first change: a shift in velocity to $$v_1$$ , deliberately different from $$v_0$$ . This velocity change is not a drift or an accident — it is the intentional first step of a carefully designed modulation sequence. As the pull velocity changes, the boundary-layer thickness responds, the effective segregation coefficients $k_{\text{eff}}(D)$ and $k_{\text{eff}}(A)$ change at different rates (because they depend differently on the boundary layer), and the ratio of dopant incorporation shifts immediately. Where the crystal was growing n-type under $$v_0$$ , it now grows p-type under $$v_1$$ , or vice versa. The interface between the region grown at $$v_0$$ and the region grown at $$v_1$$ is the first compositional transition — a boundary marking a change in net doping and, potentially, a p-n junction if the transition crosses from one dopant type to the other.

## 1. Velocity Step and the Segregation Response

v_1 = v_0 + \Delta v \quad \text{and} \quad k_{\text{eff}}(v_1) \neq k_{\text{eff}}(v_0)

The pull-rate schedule specifies $$v_1$$ as a function of position along the crystal (or time, which translates to position). This is typically either a discrete step change (pull rate jumps from $$v_0$$ to $$v_1$$ instantly, as much as mechanical constraints allow) or a ramped change (velocity transitions gradually from $$v_0$$ to $$v_1$$ over a specified distance). Either way, the effective segregation coefficient for each dopant responds to the new velocity. Using the Burton-Prim-Slichter model, $k_{\text{eff}} = k_0 / [k_0 + (1-k_0) e^{-v\delta/D}]$ : a faster pull rate (larger $$v_1 > v_0$$ ) increases $\delta$ response time, decreases the exponent magnitude, and increases $k_{\text{eff}}$ (less segregation, more incorporation). A slower pull rate (smaller $$v_1 < v_0$$ ) has the opposite effect. But here is the key: donor and acceptor have different $$D$$ values, so their $k_{\text{eff}}$ curves shift at different slopes. A velocity increase that boosts both $k_{\text{eff}}(D)$ and $k_{\text{eff}}(A)$ will boost one more than the other, tilting the incorporation ratio toward the dopant with the steeper $k_{\text{eff}}(v)$ curve.

## 2. Real Diagram: Velocity Change and the Compositional Transition Front

## 3. The Transition Front and Junction Formation

N_{\text{net}}(z) = N_D(z) - N_A(z) = \int_0^z [k_{\text{eff}}(D, v(z')) - k_{\text{eff}}(A, v(z'))] \, dz'

As the crystal grows, each incremental layer freezes in the net doping determined by the velocity at that moment. If the velocity is $$v_0$$ for the first $$z_0$$ of crystal, then changes to $$v_1$$ and remains there for the remaining growth, the doping profile consists of two regions: a donor-heavy region (grown at $$v_0$$ ) and an acceptor-heavy region (grown at $$v_1$$ ). The transition front is the plane where the doping shifts from one sign to the other — where $N_{\text{net}}$ crosses zero. At that plane, $$N_D = N_A$$ , and the material is intrinsic (electrically neutral). If the schedule is designed correctly, this intrinsic plane is where the p-n junction forms: the boundary between p-type (acceptor-doped) material on one side and n-type (donor-doped) material on the other. The sharpness of the transition depends on diffusion: dopants diffuse in the solid and the liquid, so the compositional change is not infinitely sharp but spread over a distance determined by the diffusion length during growth. A faster pull rate produces sharper transitions (less time for diffusion). A slower pull rate produces broader transitions.

## 4. Modulation of the Segregation Ratio: The Pull-Rate Schedule at Work

\text{schedule} = \{(z_1, v_1), (z_2, v_2), \ldots, (z_N, v_N)\}

The pull-rate schedule is a sequence of (position, velocity) pairs that specifies the intended velocity at each point in the crystal. Step 11 is the execution of the first change in that sequence. Steps 12–29 will continue the sequence, each velocity change creating a new compositional transition. For a simple p-n junction, only two velocities are needed: $$v_0$$ (producing one dopant type) and $$v_1$$ (producing the opposite type). For more complex dopant profiles — graded junctions, heterojunctions, or devices with multiple junction regions — the schedule contains many more changes, each carefully timed to position the compositional transitions where they will produce the desired electronic structure. In the rate-grown series, the ability to modulate the pull rate with precision is the entire point: two dopants, loaded once, never adjusted again, their incorporation controlled entirely by velocity-dependent segregation.

## Begin Pull-Rate Modulation's Place in the Process Lineage

Beginning pull-rate modulation is step eleven of this project's rate-grown series — the moment the pull-rate schedule executes its first velocity change. Up to now (steps 1–10), everything has been preparation: dopant selection, melt composition, seed geometry, thermal equilibration, choice of starting velocity. Now the schedule takes over, and velocity changes begin to encode compositional transitions into the crystal. The transition front created by this first velocity change is the beginning of the dopant profile. Steps twelve through twenty-nine continue the pull-rate sequence, creating additional transitions that define the junction structure. Step thirty, cooling, quenches the finished crystal to room temperature.

Rate Grown Transistor 1954 Begin Pull Rate Modulation

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