Heat is transported by phonons and by free electrons in solids.
For pure metals, however, the electronic contributions dominate in the thermal conductivity.
[citation needed] In impure metals, the electron mean free path is reduced by collisions with impurities, and the phonon contribution may be comparable with the electronic contribution.
[citation needed] Although the Drude model was fairly successful in describing the electron motion within metals, it has some erroneous aspects: it predicts the Hall coefficient with the wrong sign compared to experimental measurements, the assumed additional electronic heat capacity to the lattice heat capacity, namely
per electron at elevated temperatures, is also inconsistent with experimental values, since measurements of metals show no deviation from the Dulong–Petit law.
This problem seemed insoluble prior to the development of quantum mechanics.
This paradox was solved by Arnold Sommerfeld after the discovery of the Pauli exclusion principle, who recognised that the replacement of the Boltzmann distribution with the Fermi–Dirac distribution was required and incorporated it in the free electron model.
When a metallic system is heated from absolute zero, not every electron gains an energy
Only those electrons in atomic orbitals within an energy range of
Electrons, in contrast to a classical gas, can only move into free states in their energetic neighbourhood.
This relation separates the occupied energy states from the unoccupied ones and corresponds to the spherical surface in k-space.
through the relation for the electronic energy when described as free particles,
Using the expressions above the integrals can be rewritten as: These integrals can be evaluated for temperatures that are small compared to the Fermi temperature by applying the Sommerfeld expansion and using the approximation that
The expressions become: For the ground state configuration the first terms (the integrals) of the expressions above yield the internal energy and electron density of the ground state.
Substituting this into the expression for the internal energy, one finds the following expression: The contributions of electrons within the free electron model is given by: Compared to the classical result (
This explains the absence of an electronic contribution to the heat capacity as measured experimentally.
the heat capacity of metals can be written as a sum of electron and phonon contributions that are linear and cubic respectively:
We report this value below:[1] The free electrons in a metal do not usually lead to a strong deviation from the Dulong–Petit law at high temperatures.
The deviation of the approximated and experimentally determined electronic contribution to the heat capacity of a metal is not too large.
For Fe and Co the large deviations are attributed to the partially filled d-shells of these transition metals, whose d-bands lie at the Fermi energy.
However even sodium, which is considered to be the closest to a free electron metal, is determined to have a
Certain effects influence the deviation from the approximation: Superconductivity occurs in many metallic elements of the periodic system and also in alloys, intermetallic compounds, and doped semiconductors.
The entropy change is small, this must mean that only a very small fraction of electrons participate in the transition to the superconducting state but, the electronic contribution to the heat capacity changes drastically.
The calculation of the electron heat capacity for super conductors can be done in the BCS theory.
The entropy of a system of fermionic quasiparticles, in this case Cooper pairs, is: where
The last two terms can be calculated: Substituting this in the expression for the heat capacity and again applying that the sum over
this yields: To examine the typical behaviour of the electron heat capacity for species that can transition to the superconducting state, three regions must be defined: For
and the electron heat capacity becomes: This is just the result for a normal metal derived in the section above, as expected since a superconductor behaves as a normal conductor above the critical temperature.
the electron heat capacity for super conductors exhibits an exponential decay of the form:
At the critical temperature the heat capacity is discontinuous.