random potential
/ RAN-dum puh-TEN-shul /
Imagine a marble rolling not over a smooth tabletop but over a crumpled bedsheet — full of random dips and bumps with no pattern at all. Where the marble can go, where it speeds up, where it gets stuck in a hollow, all depend on this rumpled landscape. A random potential is the physicist's name for exactly such a bumpy energy landscape that an electron has to move through inside a disordered material.
A potential, in physics, is an energy landscape — a map of how much energy a particle has at each point, with valleys it is drawn into and hills it must climb. A random potential is one where those hills and valleys are scattered irregularly, with no repeating pattern, usually because the material has impurities, defects, or amorphous disorder. An electron crossing such a landscape is jostled this way and that by the random ups and downs. If the bumps are small the electron threads through; if they are deep and dense, it can get pinned in a low spot — the very mechanism behind Anderson localization.
The random potential matters because it is the standard way physicists model the effect of disorder on the electrons in a material, turning a messy real solid into something they can calculate. Many of the field's deepest results — localization, the mobility edge, how conductivity dies as disorder grows — come from studying electrons in a random potential. The caveat is that 'random' here describes the spatial pattern, fixed once and for all in a given sample; it does not mean the landscape jitters in time. For a single piece of material, the bumps stay put.
Scatter a few hundred phosphorus atoms at random through a block of pure silicon and each one tugs slightly on the nearby electrons, creating a little valley in the energy landscape. Taken together, these scattered valleys and the bumps between them form a random potential. The electrons must navigate this rumpled terrain, and how rough it is decides whether they flow freely or get trapped.
Randomly placed dopant atoms carve a bumpy, irregular energy landscape — a random potential — that the electrons must cross.
A random potential is the disordered cousin of the periodic potential of a perfect crystal. In a crystal the energy landscape repeats with the lattice, and electrons glide through as smooth waves. Make the landscape random instead, and that smooth gliding can break down — which is exactly why disorder changes everything about how electrons travel.