Channel length and short-channel effects: Foundations

This page assumes you have seen nothing. We build every letter from the ground, in an order where each one only uses letters already defined. Whenever you want the bigger picture of the device itself, see MOSFET operation and regions.

Definition Our reference device and sign convention (fix this before any symbol)

Everything below describes an n-channel MOSFET (nMOS) unless stated otherwise. That means:

The body/bulk is p-type silicon; the source and drain are n-type islands in it.
The mobile carriers in the channel are electrons (negative charge).
We take voltages as positive when the named terminal is above the source: $V_{GS} > 0$ and $V_{D S} > 0$ turn an nMOS on. The body is normally tied to the source or to the most negative supply.
Conventional current $I_{D}$ flows into the drain and out of the source (drain → source inside the device). The electrons physically move the opposite way, source → drain — conventional current always points against electron flow. When this page says "carriers walk source → drain" it means the electrons; the current $I_{D}$ counts as drain → source.

pMOS is the mirror image: n-type body, p-type source/drain, holes carry current, and every sign flips ( $V_{GS}, V_{D S}, V_{T}$ all negative). Every formula below holds for pMOS if you read all voltages and charges with their signs reversed. See MOSFET operation and regions.

Layer 0 — The physical device you are looking at

Before any symbol, look at the object. A MOSFET is four things stacked and side-by-side.

Figure 1 — cross-section of the nMOS: the four terminals, the channel, and the length $L$ .

Figure — Channel length and short-channel effects

The channel is the thin sheet of mobile electrons the gate pulls up, directly under the gate, bridging source to drain (see Figure 1). No channel → no current → switch is OFF. Channel present → switch ON.

Layer 1 — Length, width, and the "how far / how wide" symbols

Why the topic needs $L$ : every short-channel effect is defined by $L$ getting small. It is the single knob the whole chapter turns.

Layer 2 — Voltages: the "push" symbols

A voltage is an electrical "push" measured in volts (V). We always measure it between two terminals — it is a difference. Two matter most:

Why the topic needs both pushes: short-channel misbehaviour is exactly the drain ( $V_{D S}$ ) doing part of the gate's ( $V_{GS}$ ) job. You cannot state that without naming both.

Layer 3 — Charge, permittivity, and the capacitor idea

Now the single most important derived symbol — the oxide capacitance.

Layer 4 — Depletion regions: the wedges that steal control

This is the concept the entire chapter hinges on. Build it slowly.

There are two directions a depletion region can grow, and confusing them is the classic error.

Figure 2 — the two perpendicular depletion directions: sideways ( $x_{d S}, x_{d D}$ ) versus straight down ( $x_{d m}$ ).

Layer 5 — Threshold voltage and the "supported charge"

Putting the long-channel threshold together (every symbol now earned), with $V_{S B} = 0$ : $V_{T 0} = starting offset V_{F B} + bend to invert 2 ϕ_{F} + volts to hold Q_{B} \frac{Q _{B}}{C _{o x}}$ Each term is a voltage the gate must spend: an offset, the bending, and the depletion-charge cost converted to volts by $C_{o x}$ .

Layer 6 — Electric field, mobility, velocity

Figure 3 — carrier velocity versus field: linear at low field, flattening to $v_{s a t}$ .

nMOS — p-type body, n-type source/drain, electrons carry the current.

Which way does conventional

I_{D}

point versus electron flow in an nMOS?

I_{D}

is positive drain → source inside the device; electrons drift the opposite way, source → drain.

What is channel length

L

, and in which direction is it measured?

The source-to-drain distance, measured along the direction of current flow.

What does

C_{o x} = ε_{o x} / t_{o x}

physically mean, and its units?

Channel charge the gate builds per volt applied, in F/m²; thinner oxide → larger

C_{o x}

What is absolute vs relative permittivity?

ε = ε_{0} ε_{r}

;

ε_{0} = 8.85 \times 1 0^{- 12}

F/m is fixed,

ε_{r}

is a unitless material multiplier (Si ≈ 11.7, SiO₂ ≈ 3.9).

Why is the surface term in

V_{T 0}

exactly

2 ϕ_{F}

The surface must bend from

- ϕ_{F}

(bulk p-type) to

+ ϕ_{F}

(strong inversion), a total swing of

2 ϕ_{F}

What is

Q_{B}

, its units, and its formula?

The bulk depletion charge per area (C/m²),

Q_{B} = 2 q ε_{s i} N_{A} (2 ϕ_{F} + V_{S B})

What is the difference between

x_{d m}

and

x_{d S}

x_{d m}

is vertical (gate depletion into the bulk);

x_{d S}

is lateral (source depletion into the channel).

Why does reverse body bias

V_{S B}

raise

V_{T}

It widens the bulk depletion region, increasing

Q_{B}

, so the gate must spend more volts.

What are the signs and roles of

λ

η

θ

All positive;

I_{D} (1 + λ V_{D S})

V_{T} - η V_{D S}

μ_{0} / (1 + θ (V_{GS} - V_{T}))

— each grows as

L

shrinks.

Why does the lateral field

E = V_{D S} / L

blow up in a short channel?

Same

V_{D S}

across a tiny

L

gives a huge volts-per-metre push.

What does

v_{s a t}

mean and why does it exist?

A top carrier speed reached above

E_{cr i t}

; collisions with the lattice steal any extra energy so velocity stops rising.

What does subthreshold slope

S

measure?

The

V_{GS}

change (mV) needed to change leakage current by one factor of 10.

Foundations — Channel length and short-channel effects

Layer 0 — The physical device you are looking at

Layer 1 — Length, width, and the "how far / how wide" symbols

Layer 2 — Voltages: the "push" symbols

Layer 3 — Charge, permittivity, and the capacitor idea

Layer 4 — Depletion regions: the wedges that steal control

Layer 5 — Threshold voltage and the "supported charge"

Layer 6 — Electric field, mobility, velocity

Layer 7 — Drain current and the slope symbols (with signs and ranges)

How the foundations feed the topic

Equipment checklist