Free-electron theory
In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the Drude–Sommerfeld model.
Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially
The free model follows the Drude model in which four basic assumptions are taken into account:
The crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification is given by Bloch's Theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass m becoming an effective mass m* which may deviate considerably from m (one can even use negative effective mass to describe conduction by electron holes). Effective masses can be derived from band structure computations that were not originally taken into account in the free electron model.
For a free particle the potential is . The Schrödinger equation for a free particle — in our case an electron — is
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