Rate Grown Transistor 1954 Immerse Seed Tip Melt Back

# Immerse the Seed Tip: Creating the Initial Boundary Layer That Encodes the Junction's Starting Condition

The seed crystal arrives at the melt with a cut, handled tip — and that tip's outermost layer carries mechanical damage from whatever saw or cleave produced it, dislocations and disturbed lattice that would propagate directly into the growing crystal if growth started unchanged. Immersing the seed tip into the stabilized melt does two things at once in this process: it wets the tip into intimate contact with the liquid, and it deliberately melts back a controlled amount of the seed's own material, dissolving away exactly the damaged surface layer. But in a compensated melt, this dissolution step does more than clean the seed — it *creates* the initial boundary-layer geometry that the pull-rate schedule will modulate later. An uneven melt-back, one face dissolving faster than others, produces a non-uniform boundary-layer thickness $\delta(x)$ across the seed's tip, and that non-uniformity is indistinguishable from a spatially-heterogeneous initial condition for the dopant incorporation that pulling depends on to encode the junction.

## 1. Melt-Back Depth and the Two-Dopant Dissolution Profile

$$d_{\text{meltback}}(t) \approx \sqrt{D_{\text{eff}} \, t} \quad \text{where} \quad D_{\text{eff}} = f(D_D, D_A, \rho_D, \rho_A)$$

The seed's dissolution into the melt is a diffusion-limited process, with melt-back depth growing roughly as the square root of immersion time — just as in 1951. But the diffusion coefficient governing that melt-back is no longer purely germanium self-diffusion in a single-dopant liquid. In a compensated melt carrying both donor $N_D$ and acceptor $N_A$, the dissolution rate at the interface is coupled to both dopants' diffusion into the liquid. The effective diffusion coefficient $D_{\text{eff}}$ responds to the presence and concentration of both species. If one dopant diffuses faster than the other into the surrounding melt, the dissolution rate at different faces of the seed tip can become non-uniform, even if the furnace temperature is perfectly stable. The target window for melt-back depth remains narrow on both sides: immersion has to continue long enough that $d_{\text{meltback}}$ exceeds the depth of mechanically damaged material left by the cutting step, but it can't continue so long that an excessive fraction of the prepared seed dissolves away. Now, though, the *uniformity* of that melt-back across the seed's surface has become a third constraint: uneven dissolution leaves a seed tip with unequal exposed-tip depths at different azimuthal positions.

## 2. Real Diagram: Uneven Melt-Back and the Initial Boundary-Layer Non-Uniformity

Boundary-Layer Formation During Seed Immersion uneven melt-back creates non-uniform δ, mimicking spatial heterogeneity in dopant incorporation uniform melt-back (ideal) melt surface boundary layer, uniform δ symmetric interface, uniform δ(θ) ready for pulling uneven melt-back (problem) melt surface boundary layer, non-uniform δ(θ) asymmetric interface mimics spatial k_eff heterogeneity both donor and acceptor interdiffusion control dissolution rate; uneven dissolution → uneven δ at start of pulling

## 3. Non-Uniform Boundary-Layer Thickness Reads as a Spatially-Dependent Segregation Error

$$\delta(x, y) = \text{const} + \Delta\delta(x, y) \quad \Rightarrow \quad k_{\text{eff}}(v; \delta(x,y)) \text{ spatially heterogeneous}$$

When the pull starts, the growth velocity $v$ is the same everywhere on the advancing interface. But if the boundary-layer thickness $\delta$ varies from point to point on the seed's circular tip — thicker on one azimuthal face, thinner on another — then the effective segregation coefficients $k_{\text{eff}}(D)$ and $k_{\text{eff}}(A)$ will vary spatially across the interface, even though the pull rate is uniform. The Burton-Prim-Slichter model shows that $k_{\text{eff}} \propto e^{-v\delta/D}$; a variation in $\delta$ across the interface translates directly into a variation in how much donor and acceptor get incorporated at different points along the circular junction perimeter. The pull-rate schedule, carefully designed to modulate $v$ and thus control the ratio $k_{\text{eff}}(D) / k_{\text{eff}}(A)$ as a function of position along the crystal *growth axis*, cannot compensate for a spatial heterogeneity that is already frozen in by the seed's uneven dissolution geometry.

This is a hidden initial condition: the schedule sees a uniform pull rate and assumes a uniform interface, but the interface itself carries a spatial signature from the immersion step that will read, at parametric test later, as if the dopant profile is not what the pull-rate plan predicted.

Effective Segregation vs. Boundary-Layer Thickness at Seed Tip k_eff ∝ exp(−v δ / D) — uneven δ across seed face creates spatial variation in incorporation boundary-layer thickness δ (azimuthal position on seed tip) → k_eff at that position k_eff(donor), expected at uniform δ k_eff(acceptor), expected at uniform δ uniform δ: all azimuth at this k_eff uneven δ: k_eff scattered across the curve spatial dopant heterogeneity frozen in by seed immersion only a uniform δ lets pull-rate modulation work as designed

## 4. Controlling Uniformity: Thermal Stability is the Constraint

To ensure that melt-back is uniform across the seed's tip, the furnace temperature — especially at the melt surface and in the boundary layer immediately adjacent to the seed — must be rock-stable. This was also true in 1951. But now, with two dopants present, an additional constraint emerges: if the melt surface temperature oscillates, the effective diffusion coefficient $D_{\text{eff}}$ at the interface will oscillate too (because $D_{\text{eff}}$ depends on local temperature and local dopant concentration). That oscillation, even at the slow timescale of the immersion hold, can create oscillations in the dissolution rate around the seed's perimeter. The seed is sitting passively in the melt for the duration of the immersion hold — typically minutes — and any convective asymmetry in the furnace can produce a slightly hotter zone on one face of the seed and a slightly cooler zone on another. In a single-dopant system, that would just lengthen the immersion on one face and shorten it slightly on another; the resulting mild thickness variation would be re-equilibrated during the dip phase (step nine, which will follow). Here, a thickness variation in the boundary layer is a geometric boundary condition that persists through the dip and into the pull, becoming a permanent signature in the crystal's dopant profile. The stabilization achieved in step 7 — with the convective currents settled and the melt surface at a steady temperature — must be maintained throughout this immersion hold. The seed dip begins only when thermal conditions have stabilized completely.

## Immerse the Seed Tip's Place in the Process Lineage

Immersing the seed tip is step eight of this project's rate-grown series — after the melt has been stabilized and is ready to accept a seed, and before the seed dips deeper into the liquid to form the first growth interface. In the 1951 germanium series, this step removes mechanical cutting damage from the seed's surface by controlled dissolution. Here, it serves that same purpose, but it also *creates* the initial boundary-layer geometry that the pull-rate schedule will later modulate to encode the junction. An uneven melt-back produces a spatially non-uniform boundary-layer thickness, which the Burton-Prim-Slichter model shows will translate into spatially-variable effective segregation coefficients — a hidden initial condition that the pull-rate schedule cannot correct. Step nine, dipping the seed deeper, is where the first growth interface forms and the seed and melt first enter the pulling apparatus.

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