Set of supplementary notes.

Chapter 2
Sommerfeld theory – electrons as a
degenerate quantum gas
2.1 The problems with Drude theory
Just as the Lorentz oscillator model is successful at describing the optical response of insulators,
Drude theory works surprisingly well in modelling the optical and transport properties
of metals. Both theories fail dramatically, however, when the thermodynamic properties are
considered. Applying the equipartition theorem to the dipole oscillator model, we would expect
a contribution of k B to the heat capacity of each oscillator. Similarly, within the Drude model,
which treats the conduction electrons like a classical ideal gas, we would expect a contribution
to the heat capacity of 3 2 k B per conduction electron. In reality, the measured heat capacities
for both insulators and metals are far beyond those values (Fig. 2.1).
The reason for this is the same as the reason why the heat capacity due to lattice vibrations
falls below the Dulong-Petit limit at low temperature: electronic motion is largely frozen out,
because in a quantum mechanical model for the electrons, the energy required to excite them
exceeds the thermal energy available.
More specifically, in insulators, the atomic energy levels are separated by large energy gaps
of the order of electron Volts (≃ k B × 11, 000 K). Therefore, the specific heat contribution due
to electronic excitations in insulators will only become noticeable at temperatures of thousands
of Kelvin.
In metals, on the other hand, low energy excitations are always possible, but only for a small
fraction of the electrons. As we will see (and has been shown in the Thermal and Statistical
Physics course), the conduction electrons form a degenerate Fermi gas, in which only the fraction
∼ k B T/E F (where the Fermi energy E F ∼ eV) are close enough to the chemical potential so
that they can contribute to the heat capacity, which is therefore proportional to temperature.
Within the Fermi gas picture, most of the electrons travel at very high speeds, dictated by the
wavevector of the quantum state they occupy. These speeds can reach 10 6 m/s, in contrast
to far lower velocities which appear in the Drude model. This contrast emphasises, again, the
need to interpret the velocity v = ˙u in the Drude model as a drift velocity, averaged over many
particles.
By using quantum statistic, as introduced in the Thermal and Statistical Physics course, we
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