Foundations — Body effect and substrate bias

This page builds every symbol the parent topic note leans on, starting from things a curious 12-year-old already knows. Read top to bottom; nothing appears before it is earned.

0. The stage: what a MOSFET physically is

Before any symbol, picture the device. Imagine a sandwich lying flat:

• a slab of silicon at the bottom (the body or substrate or bulk — three names, one thing),
• a very thin glassy layer of oxide on top of it,
• a metal plate (the gate) on top of the oxide,
• and two wells dug into the silicon at the left and right ends, called source and drain.

The channel is the thin strip of silicon directly under the oxide, between source and drain. When we "turn the transistor on" we are pulling mobile electrons into that strip so current can flow source → drain. The gate does the pulling; the oxide is the insulator that lets the gate push electrically without touching.

1. Charge, and its symbol $q$

The whole topic is a story about charge, so we start there.

Why the topic needs it: the frozen charge under the channel is made of individual ionised atoms, each missing/holding one $q$. To count total charge we multiply "number of atoms" by $q$.

2. Doping and the symbol $N_A$

Pure silicon barely conducts. We deliberately sprinkle in foreign atoms — doping. For an nMOS body we add acceptor atoms (like boron).

Each acceptor, once it grabs an electron, becomes a fixed negative ion — it cannot move; it is bolted into the crystal. Hold this thought: these bolted-down ions are the "hard dry rocks" of the parent's water-pipe story.

Why the topic needs it: higher $N_A$ means more bolted ions per volume, which (we'll see) makes the body effect stronger — this is exactly why $N_A$ sits inside $\gamma$.

3. Mobile vs. fixed charge — the depletion picture

Two kinds of charge live in the body:

• Mobile carriers — free to wander (in p-type, the mobile carriers are "holes").
• Fixed ionised dopants — the bolted-down acceptor ions, charge $-q$ each, that cannot move.

When the gate pulls, it first chases the mobile carriers away from the region under the channel. What's left behind is a zone containing only the fixed ions — no mobile charge. That swept-clean zone is the depletion region.

Why the topic needs it: back-biasing the source widens $W$, exposing more fixed ions, so $Q_{dep}$ grows. That growth is the body effect. Everything else is figuring out how fast $Q_{dep}$ grows with voltage.

4. Voltage and potential — $V$, $\psi$, ${\varphi }_F$

Voltage is "electrical push per unit charge" — how many joules it costs to move one coulomb. We measure it between two points.

4a. The source-to-body voltage $V_{SB}$

4b. Band bending / surface potential $\psi$

The gate's pull bends the electrical potential inside the silicon near the surface. We call the amount of bend $\psi$ (psi).

4c. Fermi potential ${\varphi }_F$ and the magic number $2{\varphi }_F$

The parent uses the combination $2{\varphi }_F$. Here is why that specific amount of bending:

Why the topic needs it: $2{\varphi }_F$ is the baseline "dent depth" the gate must always create. Back-bias adds $V_{SB}$ on top, so the total dent becomes $2{\varphi }_F+V_{SB}$ — that sum is what sits inside the master equation's square root.

5. The tools that turn the picture into numbers

5a. Permittivity $\epsilon$

5b. Poisson's equation — WHY this tool

The parent derives $W$ from Poisson's equation. Why this equation and not something simpler?

Because the charge density $\rho =-q{N}_A$ is constant across the depletion region, integrating a constant curvature twice gives a parabola in $\psi$. That is where the square-root shows up:

$\psi =\frac{q{N}_A{W}^2}{2{\epsilon }_{si}}\Rightarrow W\propto \sqrt{\psi }\Rightarrow Q_{dep}=q{N}_{A}W\propto \sqrt{\psi }$

Why the topic needs it: this  ​ is the reason $V_T$ rises fast at first then flattens — the single most important qualitative fact of the whole topic. (See parent's "not linear!" mistake box.)

5c. Oxide capacitance $C_{ox}$ — WHY this tool

More depth on this component lives in Oxide capacitance Cox.

6. Threshold voltage $V_T$ and its baseline $V_{T0}$

Deeper structure of $V_T$ itself (including $V_{FB}$, the flat-band voltage) is covered by MOSFET threshold voltage and Flat-band voltage and work function.

7. Putting the symbols together — a preview

Now every symbol in the parent's master equation is defined:

$V_T=V_{T0}+\gamma \left(\sqrt{2{\varphi }_F+V_{SB}}-\sqrt{2{\varphi }_F}\right),\gamma =\frac{\sqrt{2q{\epsilon }_{si}{N}_A}}{C_{ox}}$

Read it aloud with your new vocabulary: "New threshold = baseline threshold, plus the body coefficient times (the widened depletion cost minus the baseline depletion cost)." The coefficient $\gamma$ packs together every ingredient — charge $q$, doping $N_A$, silicon permittivity ${\epsilon }_{si}$, and the gate's efficiency $C_{ox}$.

8. How the foundations feed the topic

Equipment checklist

Cover the right side; can you state each from memory before opening the parent note?