Alloy Junction 1952 Grind or Lap the Slices

# Grind or Lap the Slices: Making Step Six's Ceiling the Same Number Everywhere on the Wafer

A sawn slice doesn't have one thickness — it has a thickness that wanders across its own area, because the saw that cut it wandered too, and step six's ceiling equation only means what it says if that number is uniform everywhere the later alloying steps will ever touch. Lapping is the step that takes a rough-cut slice with real, measurable thickness variation across its surface and removes material in a controlled way until both faces are flat, parallel, and consistent — converting step six's single target thickness from a nominal average into an actual, nearly uniform physical reality the rest of the process can rely on at every point on the wafer, not just wherever a spot-check happened to measure it.

## 1. Material Removal Rate Follows a Pressure-Velocity Law, Not a Fixed Time Budget

$$\frac{dt}{d\tau} = -k_p\, P\, v_{\text{rel}}$$

Lapping removes material at a rate set by the local contact pressure $P$ between the slice and the abrasive-charged lapping plate, the relative sliding velocity $v_{\text{rel}}$ between them, and a process constant $k_p$ — a relationship with the same Preston-equation form that governs controlled material removal throughout semiconductor processing. Because removal rate depends on local pressure, any high spot on the slice — a region left slightly thicker by the saw — naturally experiences higher contact pressure against a flat reference plate and gets removed faster than the surrounding material, which is the physical mechanism that makes lapping self-correcting rather than simply a fixed-duration polish applied uniformly regardless of the slice's actual starting shape.

## 2. Real Diagram: A Rough-Cut Slice, Lapped Toward Uniform Thickness

High Spots Wear Faster Because They Press Harder this is why lapping converges toward flat, rather than just thinning everything equally lapping plate, reference flat rough-cut slice, before lapping high spot, higher P, faster removal lower spot, lower P, slower removal after lapping — flat, uniform thickness the same pressure-removal relationship runs on the opposite face at the same time

## 3. Non-Parallel Faces Make Step Six's Single Ceiling Into a Position-Dependent One

$$t_{\text{slice}}(x, y) = t_0 \pm \Delta t_{\text{wedge}}(x, y)$$

Grinding and lapping has a second job beyond uniform thickness: making the two opposing faces genuinely parallel, not just individually flat. A wedge-shaped slice — thicker on one edge than the other — means step six's ceiling equation, $W_B \approx t_{\text{slice}} - x_E - x_C$, is no longer a single number for the whole wafer; it becomes a function of position, $t_{\text{slice}}(x,y)$, and every device later cut from a thinner region of that wedge inherits a lower ceiling than a device cut from a thicker region, even though both were alloyed under the identical furnace recipe. This step's parallelism tolerance exists specifically to keep $\Delta t_{\text{wedge}}$ small enough that step six's ceiling can still be treated as one number for practical purposes, rather than a map that varies bar to bar.

Thickness Across the Wafer, Before and After the goal is a flat line, not just a lower average position across wafer → local thickness as-sawn, before lapping after lapping — Δt_wedge minimized a flat line here means step six's ceiling is one number, not a position-dependent map

## Grind or Lap the Slices's Place in the Process Lineage

Grinding or lapping the slices is step seven of RCA's forty-two-step alloy-junction manufacturing sequence — immediately after the germanium has been sliced, and before machining damage is etched away. It is the step that converts step six's nominal thickness target and its ceiling equation from a single intended average into an actual, nearly uniform physical reality across the whole wafer, by removing material fastest wherever local pressure is highest and holding the two opposing faces parallel enough that the ceiling stays one meaningful number rather than a map that varies with position. Step eight, etching machining damage, works on a wafer whose geometry this step has already fixed.

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