Condensed Matter Ep 11: Quantum Dots

A metre of copper wire obeys a rule that never lets you down: halve the length, halve the resistance. Cut it to a centimetre and the resistance falls by a hundred. Cut it to a micron and it falls again, faithfully. Now swap the copper for a sheet of electrons buried a hundred nanometres under the surface of a gallium arsenide chip, cooled to a few hundredths of a degree above absolute zero, and squeeze the channel shut with a pair of gates. The resistance stops falling smoothly. It comes down a staircase — and the treads are at values built out of nothing but the electron's charge and Planck's constant. This conversation is about what happens when a conductor gets small enough that Ohm's law stops being true, and it starts by having the answer guessed before it is derived. Conductance is current over voltage; that is charge squared divided by energy times time; and energy times time is an action. So the natural unit of conductance is the electron charge squared over Planck's constant, and you can get that far with no physics at all — which is the point, and also the limit. It gives you the currency and not the amount. The amount is Landauer's, and it inverts what resistance means. Conductance is not an obstruction to be overcome; it is a **transmission probability**. A perfect channel does not have zero resistance — it has about thirteen thousand ohms, and that resistance does not live in the channel at all. It lives in the join, where a wide reservoir has to squeeze its current into a countable number of ways through. Which is also why the number is universal: every one of those experiments got the same step height out of a different chip. Then the hero: pinch two constrictions nearly shut and the puddle between them becomes a **quantum dot** — an artificial atom, and the word is meant literally. It has discrete levels for the same particle-in-a-box reason an atom does, three orders of magnitude larger in size and getting on for five orders smaller in energy. Its shell closings come at two, six and twelve, and it obeys Hund's rule. Put one more electron on and you pay a charging energy; that price is why the current comes in sharp peaks separated by dead zones, and why sweeping gate against bias draws diamonds you can read the addition energy off with a ruler. The last act is the Kondo effect, one of very few strong-correlation problems that is genuinely solved — you can compute the resistivity, susceptibility and specific heat at every temperature, each a universal function of one measured number. In a bulk metal a magnetic impurity makes resistance *rise* as you cool. In a dot it does the opposite, because the impurity is not beside the road, it *is* the road — and the conductance climbs to the maximum a single channel is allowed. Ends where every tool in the hour fails: the zero point seven anomaly, which has been argued about since nineteen ninety-six. ——— Series: Physics - Tutorial · Grad level. Every equation spoken in full, no chalkboard required. Built by Fermi AI.