Free electron model



In Fermi-Dirac statistics. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially

  • the thermal conductivity;
  • the temperature dependence of the heat capacity;
  • the shape of the electronic density of states;
  • the range of binding energy values;
  • electrical conductivities.

Ideas and assumptions

As in the Drude model, electron-electron interactions are completely neglected (they are weak because of the shielding effect).

The crystal lattice is not explicitly taken into account. A quantum-mechanical justification is given by 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 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

E=\frac{\hbar^2 k^2}{2m^*}

(with the free-electron mass m replaced by m*). A lattice electron with a fictitious mass can be seen as a phonons).

Relation with other electron models

The assumption of electrons that move freely through a periodic potential should be contrasted with the band theory.

See also

External articles and references

  • Ashcroft, Mermin: Solid State Physics.
 
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