a quantum wire
Think of an incredibly thin thread of material — a wire only a few nanometres across but long in one direction, like a strand of spaghetti shrunk until it is just a handful of atoms wide. An electron in that thread is boxed in across its narrow cross-section but can still slide freely along its length. That is a quantum wire: confinement in two directions, freedom in one. Because a single direction of motion survives, it is called a one-dimensional structure — the middle case between the fully caged quantum dot and the flat quantum well.
In practice a quantum wire is a semiconductor nanowire (silicon, indium phosphide, gallium nitride) or any filament thin enough that the two confined directions squeeze the electron's energies apart, following the same 1 over size squared box rule. The energy levels form a series of one-dimensional sub-bands: within each, electrons still move like free particles along the wire, but jumping between sub-bands costs a confinement-set energy. Nanowires are grown by catalysed vapour-liquid-solid growth (a metal droplet feeds atoms into a lengthening crystal), by templated deposition, or by etching them out of a film.
A carbon nanotube is a natural, beautifully perfect quantum wire — a rolled graphene cylinder about a nanometre across in which electrons are confined around the circumference but free along the tube. Quantum wires and nanowires are studied as channels for tiny transistors, as sensors (their huge surface-to-volume ratio makes them exquisitely sensitive), and as one-dimensional testbeds for confinement physics. The dimensional label, once more, is just a count: two directions shrunk to the nanoscale, one left free.
A silicon nanowire 5 nm across but a micron long, grown by the vapour-liquid-solid method with a gold droplet at its tip, behaves as a quantum wire: electrons flow along its length like a one-dimensional current, while across the wire only a few confined energy levels are allowed. Coat its surface with molecular receptors and a single binding event changes the current — a nanowire biosensor.
A 1D structure: confined across the cross-section, free along the length — a carbon nanotube is the ideal example.
The term nanowire describes any very thin wire; quantum wire specifically means one thin enough that confinement noticeably splits the electron energies. Every quantum wire is a nanowire, but a thick nanowire (tens of nanometres) may show little confinement and behave essentially like bulk along and across it.