An exciton is a Coulomb correlated electron/hole pair in a semiconductor. It is the elementary excitation state in a semiconductor.
A vivid picture of an exciton formation is as follows: a photon collides into a semiconductor, which effect is to excite an electron of the valence band into the conduction band. The missing electron in the valence band leaves a hole behind it, of opposite electric charge and consequently to which it is attracted by Coulomb force. The exciton results as the binding of the electron with its hole. The wavefunctions of the bound state are of the hydrogenoid type (an "exotic atom" state akin to that of a hydrogen atom), with however a binding energy much smaller and a size much bigger than hydrogen atoms.
The probability of the electron falling into the hole is limited by the difficulty of losing the excess energy, so that the exciton may have a relatively long lifetime. (Lifetimes of up to several milli-seconds have been observed in Cuprous Oxide.) Another limiting factor to the recombination probability is the spatial overlap of the electron and hole wave-functions (roughly the probability for the electron to run into the hole). This overlap is smaller for lighter electrons and holes and for higher excited hydrogenic states.
The existence of exciton states may be inferred from the absorption of light associated with their excitation. Alternatively, an exciton may be thought of as an excited state of an atom or ion, the excitation wandering from one cell of the lattice to another.
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