In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in 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 Wiedemann–Franz law which relates electrical conductivity and thermal conductivity; the temperature dependence of the electron heat capacity; the shape of the electronic density of states; the range of binding energy values; electrical conductivities; the Seebeck coefficient of the thermoelectric effect; thermal electron emission and field electron emission from bulk metals.
The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals. The free electron model considers that metals are composed of a quantum electron gas where ions play almost no role. The model can be very predictive when applied to alkali and noble metals.
Ideas and assumptions
In the free electron model four main assumptions are taken into account: Free electron approximation: The interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. The ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily the source of collisions. Independent electron approximation: The interactions between electrons are ignored. The electrostatic fields in metals are weak because of the screening effect. Relaxation-time approximation: There is some unknown scattering mechanism such that the electron probability of collision is inversely proportional to the relaxation time , which represents the average time between collisions. The collisions do not depend on the electronic configuration. Pauli exclusion principle: Each quantum state of the system can only be occupied by a single electron. 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.
The name of the model comes from the first two assumptions, as each electron can be treated as free particle with a respective quadratic relation between energy and momentum.
The crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification was given a year later (1928) by Bloch's theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass me becoming an effective mass m* which may deviate considerably from me (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.
From the Drude model
Many physical properties follow directly from the Drude model, as some equations do not depend on the statistical distribution of the particles. Taking the classical velocity distribution of an ideal gas or the velocity distribution of a Fermi gas only changes the results related to the speed of the electrons.
Mainly, the free electron model and the Drude model predict the same DC electrical conductivity σ for Ohm's law, that is with
where is the current density, is the external electric field, is the electronic density (number of electrons/volume), is the mean free time and is the electron electric charge.
Other quantities that remain the same under the free electron model as under Drude's are the AC susceptibility, the plasma frequency, the magnetoresistance, and the Hall coefficient related to the Hall effect.
Properties of an electron gas
Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the Fermi energy as
where is the reduced Planck constant. The Fermi energy defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of electronvolts above the free electron band minimum energy.
Density of states
The 3D density of states (number of energy states, per energy per volume) of a non-interacting electron gas is given by: