MOSFET Scaling 1970s Confront Velocity Saturation Limit

# MOSFET Scaling (1970s): Step 13 — Confronting the Velocity Saturation Limit

The assumption Dennard's own paper flagged as eventually breaking. Dennard-style scaling assumes carrier drift velocity in the channel is proportional to the longitudinal electric field (constant mobility), which lets delay scale cleanly with channel length. At the high fields that short channels and compact scaling produce, carrier velocity instead saturates at a roughly field-independent maximum value, and the clean proportionality the whole scaling argument depends on starts to fail. Step 13 covers why, and what it costs.

## 1. From Linear Mobility to Velocity Saturation

At low longitudinal field, drift velocity v = mu * E (mobility mu is roughly constant), so current scales with field as the simple device equations assume. As field rises, scattering mechanisms in silicon cause velocity to level off toward a saturation velocity v_sat (on the order of 10^7 cm/s for electrons in silicon) regardless of further field increases. A short channel with a fixed drain voltage sees a much higher average longitudinal field than a long channel at the same voltage, simply because the same voltage drops across less distance — so short devices run into velocity saturation long before long devices do, even without touching mobility-degrading doping changes.

## 2. Real Diagram: Carrier Velocity vs. Field — Linear Region and Saturation

## 3. What Velocity Saturation Costs the Scaling Argument

Once a device operates in the saturated-velocity regime, drain current becomes less than proportional to the square of gate overdrive (the long-channel current equation's dependence) and current density gains from further channel-length shrink diminish — a device that is scaled shorter without changing field conditions does not speed up as much as the long-channel delay formula predicts, because the carriers are already moving as fast as they are going to move. This is precisely the correction Dennard's own 1974 paper flagged as a longer-term limit on the clean k-for-k scaling law.

## 4. Interaction With Heavier Channel Doping

Step 12's partial fix for threshold rolloff — raising channel doping — makes velocity saturation worse, not better, because higher doping increases impurity scattering and lowers the field at which velocity saturation sets in. The two mitigations this series has introduced (heavier doping against rolloff, thinner oxide and shorter channel against density) now visibly pull against each other, foreshadowing why late-1970s processes increasingly treated scaling as a multi-variable optimization rather than a single formula to apply mechanically.

## 5. Companion Diagram: Stage Sequence and Trend Summary

## Confronting the Velocity Saturation Limit's Place in the Process Lineage

This is step 13 of 20 in the MOSFET Scaling (1970s) series, tracing planar NMOS scaling from early 1970s field pressure through the fixed-voltage-scaling break that opens the adjacent HMOS (1978) series.