silicon grown junction 1954 cut rectangular bars

# Cut Rectangular Bars: Longitudinal Dicing and Radial Dopant Variation

Until this moment, every dopant segregation and melt-compensation formula tracked variation along a single spatial dimension: the axial crystal pull direction $z$. Slicing the cylindrical core block into dozens of discrete rectangular transistor bars shatters that one-dimensional simplification, exposing radial dopant non-uniformity across the circular cross-section. In the high-temperature (1414°C) Czochralski furnace, thermal convection, Marangoni surface tension gradients, and crystal rotation create a distinctly curved (convex) solid-liquid freezing interface. Consequently, a transistor bar cut from the center of the ingot core contains a subtly different base width, sheet resistance, and dopant compensation profile than a bar sliced from the outer perimeter. Slicing the block longitudinally into an orthogonal grid of $0.75 \times 0.75 \times 10\ \text{mm}$ 'matchsticks' converts a single monolithic crystal into a population of discrete devices, introducing bar-to-bar parameter dispersion that will govern downstream electrical sorting and yields.

## 1. Radial Non-Uniformity and the Convex Freezing Interface in Silicon

Radial dopant variation arises from convective transport dynamics across the rotating solidification boundary:
1. Curvature of the Freezing Interface:
Heat flows out of the freezing crystal primarily by conduction upward through the seed rod and radiation from the ingot periphery. Consequently, the solidification isotherm is not a flat planar sheet; it bulges convexly into the 1414°C melt. Because the center solidifies slightly ahead of the periphery, the metallurgical junction planes ($x_{j,1}$ and $x_{j,2}$) exhibit a parabolic sag of $10\text{ to }30\ \mu\text{m}$ across a $12\ \text{mm}$ diameter ingot.
2. Radial Segregation Gradients (BPS Model):
Crystal rotation drives a forced vortex that thins the stagnant liquid diffusion layer $\delta(r)$ near the crystal center and thickens it toward the outer rim. According to the Burton-Prim-Slichter (BPS) equation, effective segregation coefficients $k_{\text{eff}}$ for antimony ($k_0 = 0.023$) and gallium ($k_0 = 0.008$) vary with radial radius $r$. Dopants incorporate slightly more efficiently at the center than at the periphery, creating a radial concentration gradient $\Delta C(r)/C \approx 10\text{ to }20\%$.
3. Bar-to-Bar Gain Dispersion:
Because minority-carrier injection and base transit factor depend exponentially on base width and doping ($\alpha \approx \text{sech}(W_B/L_n)$), bars diced from different radial coordinates inherit distinct electrical personalities: center-cut bars feature tighter base widths and higher gain, while perimeter-cut bars exhibit slightly thicker bases and lower transit cutoffs.

## 2. Real Diagram: Orthogonal Dicing Grid and Radial Variation Across the Core

## 3. Precision Wax Mounting, Gang Slicing, and Kerf Loss Economics

Dicing brittle silicon requires rigid fixturing and careful kerf loss management:
1. Wax Arbor Mounting:
To prevent the diamond wheel from tearing through unconstrained bars as cuts intersect, the core block is adhered to a ground glass plate using thermoplastic glycol phthalate wax ($T_{\text{melt}} \approx 90^\circ\text{C}$). The molten wax embeds the entire crystal, providing continuous hydrostatic support to all four faces of every bar during sawing.
2. Two-Axis Longitudinal Slicing:
Mounted on a precision indexing dicing saw, a thin diamond wheel ($150\ \mu\text{m}$ thick) steps longitudinally along the crystal pull axis at a pitch of $1.0\ \text{mm}$ ($750\ \mu\text{m}$ bar thickness + $250\ \mu\text{m}$ kerf loss). After completing all parallel cuts in the $X$-direction, the fixture rotates 90 degrees on a rotary indexing table, and a second set of parallel cuts is made in the $Y$-direction, cutting slightly into the glass mounting plate.
3. Kerf Consumption vs. Usable Yield:
Because the saw blade kerf ($250\ \mu\text{m}$) represents $25\%$ of the pitch in each dimension, two orthogonal passes turn approximately $44\%$ of the high-purity silicon core into saw dust. From a $12\ \text{mm}$ diameter core, peripheral cuts intersect the circular boundary and produce incomplete fragments, leaving approximately 40 to 60 structurally sound rectangular bars ($0.75 \times 0.75 \times 10\ \text{mm}$).
4. Wax Dissolution:
The dicing plate is placed in a hot bath of trichloroethylene ($60^\circ\text{C}$), melting and dissolving the wax. The freed silicon matchsticks are gathered in stainless steel mesh baskets and degreased with isopropyl alcohol, ready for Step 24.

## Cut Rectangular Bars's Place in the Process Lineage

Cutting rectangular bars is step twenty-three of this project's 1954 silicon grown-junction transistor series — following the extraction of the junction-bearing core block, and preceding the visual and dimensional inspection of the individual bars. It is the mechanical transition point where a single bulk crystal ingot multiplies into dozens of discrete semiconductor devices, exposing radial segregation non-uniformities that will govern individual device ratings. Step twenty-four, inspect the bars, subjects each matchstick to optical microscopes and electrical probes to reject chipped, cracked, or shorted specimens before lead assembly.