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IDEAS AND ASSUMPTIONS


As in the Drude model, Valence Electron s are assumed to be completely detached from their Ion s ("electron gas"). As in an Ideal Gas , Electron-electron Interactions are completely neglected (they are weak because of the Shielding Effect ).

  • '' which may deviate considerably from ''m'' (one can even use negative effective mass to describe conduction by Electron Hole s). Effective masses can be derived from Band Structure computations. While the static lattice does not hinder the motion of the electrons, they can well be scattered by impurities and by Phonon s; these two interactions determine electrical and thermal conductivity ( Superconductivity requires more refined theory than the free electron model).


According to the Pauli Exclusion Principle , each Phase Space element (Δk)3(Δx)3 can be occupied only by two electrons (one per Spin Quantum Number ). This restriction of available electron states is taken into account by Fermi-Dirac Statistics (see also Fermi Gas ). Main predictions of the free electron model are derived by the Sommerfeld Expansion of the Fermi-Dirac occupancy for energies around the Fermi Level .


TECHNICALITIES



Effective mass

A Band Structure computation actually yields a Dispersion Relation ''E''(k) between electron Wave Vector k and energy ''E''. An effective mass is obtained by approximating the true dispersion relation in the limit of small ''k'' by the free-electron form
  • }

  • ''). A lattice electron with a fictitious mass can be seen as a Quasiparticle (though there is a one-to-one correspondence to the real particle which is not the case for other quasiparticles such as Phonon s).



Relation with other electron models


The assumption of electrons that move freely through a periodic potential should be contrasted with the Tight-binding Model , which uses the opposite simplification of treating the electrons as tightly bound to the atomic cores. (Coulomb interactions between electrons are still neglected.) The predictions of these two complementary models are reassuringly similar. Taking into account the specifities of the potential in a real, three-dimensional crystal lattice leads to more complicated dispersion relations and to Band Theory .


SEE ALSO



EXTERNAL ARTICLES AND REFERENCES

  • Ashcroft, Mermin: Solid State Physics.