Illustrated Physics
physics, drawn
Concept

Hole — where it appears

In a semiconductor, the absence of an electron from an otherwise filled band, which moves and responds to fields like a positively charged particle. Holes carry current in p-type material, and an electron and a hole can bind into an exciton, a hydrogen-like pair inside the crystal.

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 Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero.

The hydrogen atom inside a crystal

Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.

quantum · Bands

Named alongside it

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

Band gapEffective massAbsorptionBloch waveBohr radiusConductivityCurvatureDielectric constantDispersion relationExcitonGroup velocityHall effect

All concepts