Rate Grown Transistor 1954 Establish Compensated Melt Composition
# Establish the Compensated Melt's Initial Composition: Two Dopants, Set Once, Never Adjusted Again
This step does something neither the 1951 germanium series nor the 1954 silicon series ever does at this point in the process: it fixes both the donor and the acceptor concentration in the melt simultaneously, before a single centimeter of crystal has been pulled, and it will not revisit that decision later. Both earlier series set one dopant concentration here and leave the other for a mid-pull pellet drop still steps away. This process has no equivalent later step waiting to introduce a second species — donor and acceptor are both already dissolved in the melt the moment loading finishes, which means the melt's net electrical character at any future instant cannot come from changing what is dissolved. It will have to come from something else entirely: how fast the crystal is pulled through a melt whose composition never moves again.
## 1. A Compensated Melt's Net Doping Depends on Two Segregation Coefficients, Not One
In a single-dopant melt, the solid's net doping is set almost entirely by how much of one species is present in the liquid, scaled by that one species' segregation coefficient $k_0$. Here, both a donor concentration $C_D$ and an acceptor concentration $C_A$ are present in the liquid simultaneously, and each freezes into the solid according to its own distribution coefficient — $k_D$ for the donor, $k_A$ for the acceptor. Because gallium's segregation coefficient in germanium ($k_A \approx 0.087$) differs sharply from antimony's ($k_D \approx 0.003$), the two dopants do not incorporate into the growing solid at the same rate even though both are present in the liquid at every instant. This is the first consequence of compensating the melt from the start: net doping is now a difference of two unequal, simultaneously active processes, not the presence or absence of a single one.
## 2. Real Diagram: Setting Two Concentrations Once, With No Later Correction Step
## 3. Choosing C_D and C_A So That Pull-Rate Modulation, Not Composition Change, Can Still Produce a Junction
This is the equation that makes the entire rate-grown approach possible, and it is why $C_D$ and $C_A$ have to be chosen carefully right now rather than adjusted later. The Burton-Prim-Slichter relation shows that a dopant's *effective* segregation coefficient $k_{\text{eff}}$ is not a fixed material constant — it depends on the growth velocity $v$, the boundary-layer thickness $\delta$, and the liquid diffusion coefficient $D$. Raise the pull rate $v$ and $k_{\text{eff}}$ rises toward 1 for any dopant, regardless of its intrinsic $k_0$; lower it and $k_{\text{eff}}$ relaxes back toward the intrinsic value. Because gallium's intrinsic $k_0$ and antimony's intrinsic $k_0$ respond to velocity changes by different amounts, a single pull-rate schedule can be found that makes the net incorporation favor donor at one rate and acceptor at another — without ever touching $C_D$ or $C_A$ again. $C_D$ and $C_A$ must therefore be sized now so that *both* achievable $k_{\text{eff}}$ ranges, across the pull-rate window the furnace can actually sustain, straddle the compensation point in opposite directions.
## Establish the Compensated Melt's Initial Composition's Place in the Process Lineage
Establishing the compensated melt's initial composition is step two of this project's rate-grown series — the step where both donor concentration $C_D$ and acceptor concentration $C_A$ are decided together, with neither species added or adjusted at any later step the way the 1951 and 1954 grown-junction series both still require. What this step actually buys is the separation into two different pull-rate-dependent effective segregation coefficients demonstrated in Section 3, which is what will let later pull-rate modulation substitute for the pellet drop entirely. Step three, preparing the seed crystal, selects the crystallographic orientation this compensated melt will freeze against.