Quantum confinement and low-dimensional materials

Shrink a piece of semiconductor until it is comparable to the size an electron-hole pair naturally occupies, and the continuous bands of the bulk crystal break up into discrete levels whose spacing grows as the crystal gets smaller — so the same material emits blue at two nanometres and red at six, with no change of chemistry. Restricting motion in one, two or three directions gives quantum wells, wires and dots, and each changes not just the number of available states but the shape of the density of states, which is why a quantum-well laser reaches threshold at a far lower current than a bulk one and why these structures are engineered rather than merely observed.

Practise this concept

A particle in a box has discrete energy levels, and the smaller the box the further apart they are. A bulk crystal is an enormous box, so its levels are packed so densely that they merge into continuous bands. Shrink the crystal towards a few nanometres, though, and the box becomes small enough for the spacing to matter again: the levels near the band edges separate out, the gap between the highest filled and lowest empty state widens, and the material starts to behave partly like a very large molecule. This is quantum confinement, and the scale at which it sets in is the size an electron-hole pair would naturally occupy in that material — the exciton Bohr radius — which is a few nanometres in common semiconductors and is why it is a nanoscale phenomenon specifically.

The visible consequence is that colour becomes a matter of size. Cadmium selenide dots of two nanometres emit blue, and the same compound at six emits red, with no chemical difference between the vials. That tunability is the point of quantum-dot displays and of the fluorescent labels used in biology, and it is also a useful diagnostic: if a nanoparticle's colour tracks its size, confinement is at work, whereas gold nanoparticles get their colour from a plasmon resonance instead and behave quite differently.

Confining motion in only some directions gives the other members of the family: a quantum well confines one direction and leaves a two-dimensional sheet of free motion, a wire leaves one, a dot leaves none. What matters about each is less the number of states than how they are distributed in energy. The bulk density of states climbs smoothly from the band edge; the well's rises in steps, the wire's spikes at each subband, and the dot's collapses into sharp lines. Concentrating states at the band edge rather than spreading them over a range is exactly what a laser wants, which is why essentially every semiconductor laser made today is a quantum-well device, and it is the same reasoning that drives interest in two-dimensional materials like graphene and the transition-metal dichalcogenides.

Key ideas

Requires: Particle in a box, Band theory of solids, Semiconductors and doping

Common misconceptions

Assuming that a nanocrystal emitting a different colour must be a different material, or chemically altered at the surface. The composition is identical: what changed is the boundary condition. Squeezing the same wavefunction into a smaller box raises its energy, so the gap between the lowest empty and highest filled state widens as the crystal shrinks, and the emitted photon moves towards the blue.
Treating "nanoparticle" and "quantum-confined" as the same statement. Confinement matters only once the particle is comparable to the length an electron-hole pair naturally occupies in that material — a few nanometres in many semiconductors — so a 50 nm crystal of the same substance behaves essentially as bulk. Colloidal gold is the classic trap: its colour is a plasmon resonance, a collective oscillation of free electrons, not a confinement level shift.
Reading lower dimensionality as simply having fewer states available. The count is not the point — the distribution is. In bulk the density of states rises smoothly from the band edge, in a quantum well it jumps to a plateau at each subband, in a wire it spikes, and in a dot it is a set of sharp lines. Piling states up right at the edge instead of spreading them out is what lets a quantum-well laser reach threshold with far less injected current.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Nanoelectronic materialsLOFY.02.045