MOSFET Drain Current in Saturation (Square Law)

Put enough voltage on the gate and an inversion layer forms under the oxide, connecting source to drain. The charge in that layer is proportional to the overdrive $V_{GS} - V_{th}$, and the speed the carriers travel is proportional to the field along the channel, so the current comes out proportional to the square of the overdrive. Once the drain voltage exceeds the overdrive, the channel pinches off at the drain end and the current stops caring about the drain voltage: the device has become a current source controlled by its gate, which is the property every analogue circuit is built on.

The overdrive is the parameter a designer actually thinks in. A large overdrive gives a fast device with plenty of current but poor voltage headroom and low transconductance per amp; a small overdrive is efficient and high-gain but slow, and eventually drops into the subthreshold region where the device behaves like a bipolar transistor with an exponential characteristic. Analogue designers choose an overdrive first — 150 to 250 mV is common — and size the width to reach the current they need.

Take the square law seriously as physics and it will mislead you at any modern node. It assumes carrier velocity rises without limit as the field rises, and it does not: past roughly $10^4$ V/cm the carriers saturate at about $10^7$ cm/s and stop going faster. In a short channel the field reaches that point at ordinary voltages, and the current becomes closer to LINEAR in the overdrive than quadratic. Every device below about a quarter of a micron lives in that regime, along with a family of other short-channel effects — drain-induced barrier lowering, threshold roll-off, mobility degradation from the vertical field — that the model knows nothing about.

So why teach it? Because it is the only model simple enough to reason with, and the relationships it captures survive the corrections: more overdrive means more current, a wider device means proportionally more current, a longer channel means less. It is also the model that makes the subthreshold region visibly wrong, and noticing that is important. Below threshold this equation says the current is zero. It is not zero; it falls off exponentially, and that exponential tail is precisely the leakage that ended Dennard scaling. The square law's most useful lesson is the thing it cannot see.