Foundations — Subthreshold leakage current

This page assumes you know nothing. We build every letter, ratio, and picture the parent note leans on, in an order where each one only uses things already defined. When you finish, re-read the parent — every symbol will feel like an old friend.

0. The stage: what a MOSFET physically is

Before any symbol, picture the object. A MOSFET is a tiny sandwich sitting on a slab of silicon.

• The source (S) and drain (D) are two terminals we want current to flow between.
• The gate (G) is a metal plate hovering just above the silicon, separated by a paper-thin insulator called the oxide.
• Between source and drain lies the channel — the strip of silicon the current must cross.

We link the geography of these regions to the parent's MOSFET operating regions.

1. Voltage differences: $V_{GS}$ and $V_{DS}$, and the drain current $I_D$

• $V_{GS}$ = gate voltage minus source voltage = how hard we press the valve.
• $V_{DS}$ = drain voltage minus source voltage = how hard we pull current across the channel.

Why the topic needs these. The whole subthreshold story is "how does the drain current $I_D$ (the drip) depend on the two knobs?" The parent's headline formula has exactly these two knobs: $V_{GS}$ inside the exponential (the strong control) and $V_{DS}$ inside a separate bracket (the weak control). You cannot read that formula without knowing which letter is which knob.

2. Threshold voltage $V_{th}$ — the "officially open" line

The key mind-shift: $V_{th}$ is a convention, a line we drew, not a physical cliff. Look at the figure — the current curve is smooth and continuous through $V_{th}$. Nothing snaps at that line. Below it there is still a real, measurable trickle. That trickle is the subthreshold current.

We build on Threshold voltage Vth. The quantity that matters most turns out to be the distance below threshold, $V_{GS}-V_{th}$ (a negative number in the OFF state) — that difference is what sits in the exponential.

3. Drift vs diffusion — two ways charge moves

Current is just charge in motion. There are exactly two reasons charge moves, and the topic swaps from one to the other at threshold.

Fick's law: why diffusion current ∝ the crowding slope

That "flow proportional to the concentration slope" is Fick's law. In symbols the slope is $\frac{dn}{dx}\approx \frac{n(0)-n(L)}{L}$, and multiplying by the electron charge $q$, the flow area $A$, and a proportionality constant $D_n$ (how mobile the carriers are) turns a flow of particles into a current:

$I_{sub}\approx qA{D}_n\frac{dn}{dx}\approx qA{D}_n\frac{n(0)-n(L)}{L}.$

We now unpack every letter in that expression.

The slope $\frac{n(0)-n(L)}{L}$ is just "how steeply the crowd thins out from source to drain" — the steeper the thinning, the stronger the diffusion push. Hold on to the two ends $n(0)$ and $n(L)$: the source end will give the $V_{GS}$ exponential, and the drain end will give the $V_{DS}$ term.

4. Boltzmann statistics — why "exponential"

Here is the single most important idea on the whole page: where does the exponential come from?

That single line is the seed of every exponential on the parent page.

5. Thermal voltage $V_T=kT/q$

The combination $kT/q$ shows up so often we give it a name and a unit of volts.

Using $V_T$, the Boltzmann factor $e^{q{\psi }_s/kT}$ becomes the tidier $e^{{\psi }_s/V_T}$. This is the parent's $V_T$; see Thermal voltage kT-q. Notice $V_T\propto T$ — hotter chip, bigger $V_T$ — which will explain why leakage worsens with heat.

6. Surface potential ${\psi }_s$ — how much the gate bends the silicon

The catch: not all of $V_{GS}$ reaches the surface. Some of your gate push is "wasted" pushing back the depletion layer. That leads directly to the next symbols.

7. Capacitors $C_{ox}$, $C_{dep}$ and the divider factor $n$

They form a capacitive voltage divider — the gate voltage splits between them. Here is why series capacitors split a small voltage change in the ratio the parent uses.

Why the topic needs $n$. It is the "efficiency loss" that makes the exponential lazier: instead of $e^{V_{GS}/V_T}$ you get $e^{V_{GS}/(n{V}_T)}$. That single $n$ is exactly why the subthreshold swing (below) is $\ge 60$ mV/decade, never less. The same capacitor stack drives the Body effect.

8. The drain end and the $V_{DS}$ term — the second knob

We built $n(0)$ (source end) from the gate. What about $n(L)$, the drain end? This is where $V_{DS}$ enters — and where the parent's bracket $(1-e^{-V_{DS}/V_T})$ comes from.

Now feed both ends into the diffusion current from Section 3. The current is proportional to $n(0)-n(L)$, i.e. to $n(0)-n(L)\propto e^{(V_{GS}-V_{th})/(n{V}_T)}(1-e^{-V_{DS}/V_T}).$

The source term $e^{(V_{GS}-V_{th})/(n{V}_T)}$ factors out front; what's left in the bracket is the drain's contribution. That reproduces the parent's boxed formula: $I_{sub}=I_0{e}^{(V_{GS}-V_{th})/(n{V}_T)}(1-e^{-V_{DS}/V_T}).$

Edge cases in $V_{DS}$: linear vs saturation

9. Reading exponentials in "decades": ${\log }_{10}$ and $\ln 10$

The current spans many factors of ten, so we plot ${\log }_{10}I$, not $I$.

Because $I_{sub}\propto e^{V_{GS}/(n{V}_T)}$, taking ${\log }_{10}$ gives a straight line in $V_{GS}$ — its inverse slope is the swing:

10. The prerequisite map

Every arrow is a "you need this before that." Follow any path top-to-bottom and you re-derive a piece of the parent page.

11. Two quick sanity numbers

Equipment checklist

Test yourself — cover the right side. If any fails, re-read that section before the parent page.

Connections

• Subthreshold leakage current — the parent this page equips you for.
• MOSFET operating regions — where subthreshold sits relative to strong inversion.
• Threshold voltage Vth — the reference line of the exponential.
• BJT diffusion current — same diffusion physics.
• Thermal voltage kT-q — the $V_T$ built in Section 5.
• Body effect — same $C_{dep}/C_{ox}$ divider that makes $n$.
• Short-channel effects / DIBL — later refinements that shift $V_{th}$ and worsen leakage.
• Static vs dynamic power dissipation — why this trickle matters for whole chips.