Illustrated Physics
physics, drawn
Concept

Bloch wave — where it appears

A wave in a periodic medium whose amplitude repeats with the period while its phase advances by a fixed amount per cell. That advance is what a band structure plots against frequency, and where no real advance exists the medium carries nothing at all.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as periodic media — the same set of essays touches all of them, so they are one junction rather than several.

The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is.

The gap a repeat opens

Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

waves · Periodic media The one frequency a broken repeat lets through. Transmission through a quarter-wave stack whose repeat is broken once, on a logarithmic scale, against frequency in units of the quarter-wave design frequency. The perfect stack forbids this whole band; adding a single half-wave layer at the centre opens one line inside it, at exactly the middle of the gap, through which the stack transmits everything. With 4 pairs either side the line is 2.50e-3 wide, a quality factor of 400, on a background of 4.5e-4; With 6 pairs either side the line is 1.56e-4 wide, a quality factor of 6410, on a background of 6.3e-6; With 8 pairs either side the line is 1.00e-5 wide, a quality factor of 100000, on a background of 9.2e-8. The line narrows geometrically as the mirrors are made thicker, because the field inside the defect leaks out through a barrier whose transmission falls exponentially with its thickness — so the useful quantity of a band gap turns out to be not what it excludes but how well it can trap what a single flaw is allowed to hold.

The mode that lives in the mistake

A perfect stack of alternating layers refuses a whole band of frequencies — not weakly, not with loss, but not at all. Break the repeat once, by inserting a single layer of the wrong thickness, and exactly one frequency inside that band passes through the whole stack with a transmittance of one. The useful thing about a forbidden band turns out to be not what it excludes but what a single flaw is thereby allowed to hold.

waves · Periodic media 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 The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all.

The frequency a lattice cannot carry

A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

waves · Periodic media

Named alongside it

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

Band gapPeriodic mediaDispersion relationEvanescent waveGroup velocityTransfer matrixCavityConductivityContinuum limitCurvatureCutoffDefect state

All concepts