Free Electron
A free electron is an electron that is not bound to any particular atom and so can move through a material and carry electric current when a field is applied. In a metal such as copper roughly one electron per atom is free in this sense, forming a dense gas that fills the whole piece of metal; in a semiconductor the free electrons are the comparatively rare electrons that have been lifted into the conduction band. The word free is relative: these electrons are still pushed and pulled by the crystal's atoms, and they are free only in the sense that they belong to the whole solid rather than to one nucleus.
The first successful theory of free electrons came from Paul Drude in 1900, only three years after J. J. Thomson identified the electron. Drude treated the conduction electrons in a metal as a classical gas of tiny charged particles bouncing between heavy, fixed ions, colliding at random intervals. An applied field accelerates them between collisions, and each collision resets their drift, so the average velocity settles to a steady value proportional to the field, which reproduces Ohm's law. With a typical time between collisions of about 10^-14 seconds, the model gave sensible values for the resistivity of metals and explained why good electrical conductors are also good heat conductors, a link known as the Wiedemann-Franz law. It was remarkable for a theory formulated before quantum mechanics existed.
Drude's model failed in instructive ways that quantum theory repaired. If every free electron carried its classical share of thermal energy, metals would have a far larger electronic heat capacity than the measured one, which is only a small percentage of the prediction. Arnold Sommerfeld solved this in 1927 by replacing classical statistics with Fermi-Dirac statistics, built on Wolfgang Pauli's exclusion principle of 1925. Because no two electrons can occupy the same quantum state, the electrons fill states from the bottom up to a maximum energy, the Fermi energy, which for copper is several electron-volts. Only the few electrons within about a thermal energy of that edge can respond to heat or to a field, which explains the small heat capacity and shows that the electrons carrying current move at the very high Fermi velocity, about 1.6 × 10^6 metres per second in copper, even though the net drift is tiny.
Free electrons move quickly at random but drift slowly as a group, and the difference explains a very common misunderstanding. In a 1 mm² copper wire carrying one ampere, the average drift velocity is on the order of a tenth of a millimetre per second, because there are about 10^29 free electrons per cubic metre. Yet a light switch seems to act instantly, since the electric field propagates along the wire at a large fraction of the speed of light and starts every electron moving at once. Electrons do not shuttle from the switch to the bulb carrying energy; the energy travels in the fields around the conductor, and the electrons act as the medium that responds.
In semiconductors the free electrons are the electrons in the conduction band, and their number depends strongly on temperature, doping and light. In pure silicon at room temperature thermal generation, where lattice vibrations break a bond and lift an electron across the 1.12 electron-volt gap, leaves only about 10^10 free electrons per cubic centimetre, against roughly 10^23 in copper. Doping with phosphorus or arsenic can raise the number by a factor of a billion or more, while shining light with sufficient photon energy creates extra electrons and holes in pairs, which is how a solar cell or a camera sensor works. The reverse process, recombination, returns electrons to bonds at the same time, and the equilibrium between generation and recombination sets the carrier population at any instant.
Inside a crystal, free electrons behave as though they were different particles, so the free-electron picture is only an approximation. Inside the lattice an electron responds to a force as if it had an effective mass, which in silicon is a fraction of the true electron mass and differs with direction, because the crystal's periodic potential adds its own forces. The free-electron model is therefore a useful approximation, accurate for simple metals such as sodium and aluminium and instructive for the rest, but not literally true for transition metals or semiconductors. Modern device simulators correct for these effects by replacing the true electron mass with an effective mass, and at the nanometre scale of today's transistors electrons also show quantum confinement and wave behavior that the old gas picture cannot capture.
Read a free electron as an electron released from any single atom to belong to the entire solid, the shared mobile charge that metals have in abundance, that semiconductors must earn from heat, light or dopants, and that quantum theory shows to be both quick and subtle.