PE2-1.1 MOS Device Physics & Modelling

What this is and why it exists

Every circuit in this course is built from one device. Before assembling anything from it, you need to know what it does and where the simple description stops being true.

Deriving the current expression once yourself is what separates using a transistor from understanding it. The regions of operation then stop looking arbitrary.

The vocabulary

• Threshold voltage — the gate voltage at which a conducting channel forms.
• Triode region — operation where the device behaves somewhat like a controlled resistor.
• Saturation region — operation where the current depends mainly on the gate voltage, not the drain voltage.
• Channel length modulation — the effective channel shortening as drain voltage rises, so current keeps creeping up.
• Body effect — the threshold voltage rising when the source is not at the substrate potential.
• Velocity saturation — carriers reaching a speed limit in a strong field, so current stops rising as the square law predicts.
• Transconductance — how much drain current changes for a change in gate voltage.
• Small signal model — the linear circuit that describes the device near one operating point.
• Short channel device — one small enough that the simple equations no longer describe it.

The mental model

The physics in one sentence. A voltage on the gate attracts carriers to the surface under it. Once it exceeds the threshold there are enough of them to form a channel between source and drain.

The derivation follows from that picture. The gate voltage above threshold sets how much charge is in the channel. The drain voltage sets how fast that charge moves. Multiply the charge by its velocity and you have the current.

Two regions fall out. While the drain voltage is small, raising it raises the current roughly in proportion, because the channel is intact. As the drain voltage rises further, the channel at the drain end runs out of charge and pinches off. Beyond that point the current stops depending much on drain voltage and depends on the gate voltage squared. That is the square law result, and every later design equation contains it.

Now the corrections, and this is where analog design actually lives.

Channel length modulation says the pinch-off point moves further from the drain as the drain voltage rises, shortening the effective channel. So the current keeps creeping up in saturation instead of being flat. That slope is what sets the output resistance of a stage, and therefore its gain.

Body effect says the threshold is not fixed. Raise the source above the substrate potential and the threshold rises with it. That matters in any stage where the source is not at the supply rail. It is why the source follower in the next topic falls short of the ideal equations.

Velocity saturation says carriers cannot go arbitrarily fast. In a short device the field is strong enough that they hit that limit. The current then rises roughly in proportion to gate voltage rather than as its square.

That is the important shift. In a long device these three are corrections. In a short device they are the behaviour. The industry's whole story of shrinking devices is the story of the simple equations ceasing to apply.

Layout is where the abstract width and length become a shape on silicon, and that shape decides the parasitics. Two devices with identical width and length can behave differently if drawn differently.

The capacitances between the terminals close the topic, and they matter more than they look. Every frequency response in the rest of this course is set by them. The gate has capacitance to the channel and overlap capacitance to the source and drain, and the junctions have capacitance to the substrate.

The small signal model is the tool the rest of the course uses. Pick an operating point and linearise about it. The device becomes a current source controlled by the gate voltage, with a resistance across it and the capacitances in place. Commit that model to memory. It pays back on the first circuit of the next topic.

What you should now be able to explain or do

Derive the drain current and say where each region comes from. Explain channel length modulation and connect it to output resistance and gain. Explain body effect and name a stage it damages. Say what velocity saturation changes about the square law. Draw the small signal model and say which capacitance sets which frequency limit.

Check yourself