Send carriers across a silicon bar by an electric field (drift) or by their own random motion (diffusion), and compare how long each takes as the bar shrinks and the voltage changes.
A silicon bar of length a: carriers enter at the left and leave at the right
Drift velocity versus field
Transit time versus length (log-log)
What to look for
τdrift = a/vd ≈ a²/(μV), τdiff = a²/(2μφt)
At low field vd = μE, but near 10⁴ to 10⁵ V/cm the velocity saturates at about 10⁷ cm/s, so a short bar at a few volts is limited by vsat, not by mobility.
Drift time grows like a (when saturated) or a² / (μV) (low field). Diffusion time always grows like a², so diffusion wins for long bars at low voltage and loses for short bars.
At low field the two times are equal at V = 2φt ≈ 52 mV, independent of length and mobility: below that voltage diffusion carries the carriers across faster.
Try this
Set a = 1 µm and V = 0.05 V. Which transit time is shorter? Then raise V to 1 V.
Shorten the bar to 10 nm at V = 1 V. Does τdrift still follow a² / (μV)? Which dashed line does the amber curve leave?
Raise the doping from 10¹⁴ to 10²⁰ cm⁻³. How much does μ fall, and how do both transit times respond?
Switch to holes. Why is the drift time longer than for electrons at the same V and a?