Illustrated Physics
physics, drawn
Concept

Hall effect — where it appears

The appearance of a voltage across a conductor carrying a current at right angles to a magnetic field, as the field pushes the carriers sideways. Its sign shows whether electrons or holes carry the current and its size their density; in two dimensions at low temperature it is quantised.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.

The mass a curve decides

An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.

quantum · Bands A resistance counted in whole numbers. The Hall resistance of a gallium-arsenide layer with 3 × 10¹⁵ electrons per square metre, against the perpendicular field (solid), beside the classical B/ne (dotted); and the resistance along the current, on an arbitrary scale (red). Classically the Hall resistance rises in proportion to the field. In the layer it climbs in steps and stays on flat plateaus at h/νe² — 25813, 12906, 8604, 6453 ohms — each centred where the classical line crosses it, at integer filling. On each plateau the resistance along the current falls to zero: the current flows with no dissipation at all. Between plateaus both change. The model drawn here gives each Landau level a width 0.18 of the level spacing and lets only the middle 0.16 of it carry current across the sample.

The resistance counted in whole numbers

Pass a current along a thin layer of electrons in a strong magnetic field and measure the voltage across it. Classically the ratio rises steadily with the field. In fact it climbs in steps, and on each step it sits at the fundamental constant h/e² divided by a whole number — 25,812.807 ohms, then half that, then a third — the same to better than a part in a thousand million in every sample, every material and every laboratory, however dirty the layer and however irregular its shape. The steps exist because of the dirt, and they are exact because of a count that disorder cannot change.

quantum · Bands

Named alongside it

The objects these essays reach for when they reach for this one.

Band gapBloch waveConductivityCurvatureDispersion relationEdge statesEffective massFilling factorGroup velocityHoleLandau levelsLocalisation

All concepts