Rate Grown Transistor 1954 Second Velocity Step

# Continue Pull-Rate Modulation: Second Velocity Step and Compositional Layering

The first compositional transition front has been created by the velocity change in step 11, establishing a p-n junction in the crystal. Now the pull-rate schedule executes a second velocity change, from $$v_1$$ to $$v_2$$ . This second step can serve different purposes depending on the device design: it can widen the depletion region by shifting the doping balance further, it can create a graded junction with a controlled compositional gradient, or it can reset the doping balance to grow a second junction further up the crystal for a bipolar or heterojunction structure. In the 1951 germanium series, multiple junctions required pausing the pull, changing the melt composition by dropping a pellet, and resuming with new segregation conditions. In the rate-grown series, the second and all subsequent junctions are encoded by pure velocity modulation: the melt composition is fixed, dopant ratios are changed only by changing how fast the crystal grows through it.

## 1. Velocity Step Sequence and Compositional Stacking

v(z) = \begin{cases} v_0 & \text{for } 0 < z < z_1 \\ v_1 & \text{for } z_1 < z < z_2 \\ v_2 & \text{for } z_2 < z < z_3 \\ \vdots \end{cases}

The pull-rate schedule is, at its simplest, a piecewise function: different velocities for different axial regions of the crystal. Step 12 takes the velocity from $$v_1$$ to $$v_2$$ at position $$z_2$$ , beginning the third compositional region. If $$v_2 > v_1$$ , the crystal grows faster, the boundary layer thins further, and $k_{\text{eff}}$ increases for both dopants — but again at different rates, so the incorporation ratio shifts. If the first junction (from step 11) was donor-heavy on one side and acceptor-heavy on the other, a third velocity can create a third region with a new doping balance: p-type again, or n-type stronger, or nearly intrinsic depending on whether $$v_2$$ is chosen to equal $$v_0$$ , exceed it, fall between the two, or approach an intermediate value. The compositional profile becomes a multi-layer structure, with each layer encoding the segregation behavior of its velocity. For a basic p-n junction, only two velocities suffice. For a more complex device — a graded junction with controlled doping gradient, a Zener diode with narrow depletion region, or a bipolar transistor with separate base and collector regions — the schedule specifies many more velocity steps.

## 2. Real Diagram: Multi-Layer Compositional Profile from Velocity Stepping

## 3. Diffusion and the Transition Width: Sharpness vs. Grading

\text{transition width} \propto \sqrt{D_{\text{dopant}} \cdot t_{\text{transit}}} \quad \text{where} \quad t_{\text{transit}} = \frac{\Delta z}{v}

The compositional transition created at step 11 (from $$v_0$$ to $$v_1$$ ) is not infinitely sharp. The width of the transition — the distance over which doping changes from predominantly donor to predominantly acceptor — is set by diffusion: dopants diffuse both in the liquid (in the boundary layer) and in the solid as it cools after being grown. A transition grown at a slow pull rate gives dopants time to diffuse, broadening the transition into a graded junction — a region where doping changes gradually rather than abruptly. A transition grown at a fast pull rate freezes dopants quickly into their incorporation positions, producing a sharp junction — a narrow region where doping changes steeply. Step 12's velocity $$v_2$$ influences not just the doping of the region it controls, but also indirectly affects how sharply step 11's transition front appears in the final dopant profile, because the thermal and composition gradients the crystal experiences while growing region 2 affect how much dopants diffuse into or out of region 1 beneath it.

## 4. Encoding Device Structure Through Velocity Modulation

The rate-grown series' fundamental achievement is this: dopant profile control through velocity modulation alone, with a fixed melt composition. A slow-fast-slow velocity sequence (e.g., $$v_0 < v_1 < v_2$$ then back down) can create a graded junction with a well-defined depletion region width. A faster-slower sequence ( $$v_0 > v_1 < v_2$$ ) can create a narrow, sharp junction suitable for Zener breakdown or tunnel effects. A sequence with a long plateau of intermediate velocity creates a thick, highly graded base region for a transistor. The entire dopant profile — which regions are n-type, which are p-type, how sharply they transition, how thick each is — is encoded not by changing the melt, but by the time-varying pull rate. This gives the rate-grown approach its power and flexibility: the same melt, with the same pair of dopants, can be used to grow many different device types simply by changing the pull-rate schedule.

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

Continuing pull-rate modulation is step twelve of this project's rate-grown series — the second velocity step in the schedule. It creates the second compositional region and the second transition front, adding complexity to the dopant profile beyond a simple two-region p-n junction. The profile is now a multi-layer structure, with each layer's doping set by its velocity and each transition's width shaped by diffusion. Steps thirteen through twenty-nine continue the pull-rate schedule, layering more compositional regions and building the complete dopant profile for the finished device. Step thirty cools the crystal to room temperature, fixing the dopant profile in place.

Rate Grown Transistor 1954 Second Velocity Step

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