How do you define the temperature of an electron gas? Like particles in general, electrons in a metal prefer to occupy energy levels in a way that minimizes their total energy. As fermions, they obey the Pauli exclusion principle and at zero temperature completely fill all the states below the Fermi energy. At nonzero temperatures, some electrons are found above the Fermi level by an amount k B T, the thermal energy of the system. More quantitatively, electrons at a particular temperature occupy energy levels with a probability given by the Fermi–Dirac distribution f(E) = 1/[1 + exp(E/k B T)]. Electrons reach this distribution by inelastic scattering—exchanging energy between themselves or with the surrounding crystal lattice.
Any probe that is sensitive to the width of the electron distribution function can work as a thermometer. By the same token, any device that narrows...
