Free electron model
[Ashcroft & Mermin 1][Kittel 1] Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals.
In the free electron model four main assumptions are taken into account:[Ashcroft & Mermin 2] 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.
Effective masses can be derived from band structure computations that were not originally taken into account in the free electron model.
[citation needed] Many physical properties follow directly from the Drude model, as some equations do not depend on the statistical distribution of the particles.
For a three-dimensional electron gas we can define the Fermi energy as[Ashcroft & Mermin 5] where
This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the conduction band.
For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy.
) can also be calculated by integrating over the phase space of the system, we obtain[Ashcroft & Mermin 8] which does not depend on temperature.
Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by[Ashcroft & Mermin 8] where
is the total energy, the derivative performed at temperature and chemical potential constant.
According to the Bohr–Van Leeuwen theorem, a classical system at thermodynamic equilibrium cannot have a magnetic response.
The magnetic properties of matter in terms of a microscopic theory are purely quantum mechanical.
The latter contribution is three times larger in absolute value than the diamagnetic contribution and comes from the electron spin, an intrinsic quantum degree of freedom that can take two discrete values and it is associated to the electron magnetic moment.
One open problem in solid-state physics before the arrival of quantum mechanics was to understand the heat capacity of metals.
In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity.
Nevertheless, such a large additional contribution to the heat capacity of metals was never measured, raising suspicions about the argument above.
Evidently, the electronic contribution alone does not predict the Dulong–Petit law, i.e. the observation that the heat capacity of a metal is still constant at high temperatures.
The free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice.
With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,[Ashcroft & Mermin 10] where
Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the mean free path should be infinite.
[Ashcroft & Mermin 11] This implies that the ratio between thermal and electric conductivity is given by the Wiedemann–Franz law, where
is the Lorenz number, given by[Ashcroft & Mermin 12] The free electron model is closer to the measured value of
The close prediction to the Lorenz number in the Drude model was a result of the classical kinetic energy of electron being about 100 smaller than the quantum version, compensating the large value of the classical heat capacity.
However, Drude's mode predicts the wrong order of magnitude for the Seebeck coefficient (thermopower), which relates the generation of a potential difference by applying a temperature gradient across a sample
[4] The free electron model presents several inadequacies that are contradicted by experimental observation.
We list some inaccuracies below:[Ashcroft & Mermin 14] Other inadequacies are present in the Wiedemann–Franz law at intermediate temperatures and the frequency-dependence of metals in the optical spectrum.
[Ashcroft & Mermin 14] More exact values for the electrical conductivity and Wiedemann–Franz law can be obtained by softening the relaxation-time approximation by appealing to the Boltzmann transport equations.
[Ashcroft & Mermin 14] The exchange interaction is totally excluded from this model and its inclusion can lead to other magnetic responses like ferromagnetism.
[Ashcroft & Mermin 14] Adding repulsive interactions between electrons does not change very much the picture presented here.
More exotic phenomena like superconductivity, where interactions can be attractive, require a more refined theory.
