The solid electric thermal conductivity tensor Ke, in addition to the phonon thermal conductivity tensor (i.e., total conductivity K = Ke + Kp), determines heat conduction in solids through the Fourier law qk = −K · ∇T. The heat capacity of an electron cυ,e when in local thermal equilibrium with the lattice (i.e., phonon) is small, except at very high temperatures. Electrons can also have a net motion under an applied electric field ee, thus creating opportunities for exchange of their gained kinetic energy, e.g., with the lattice through inelastic scattering in Joule heating. The coupling of electronic and thermal transport, known as thermoelectricity, leads to Peltier heating/cooling.
In Section 2.6.5, we examined the electronic energy states of an idealized electron gas by solving the Schrödinger equation for the case of a collection of free electrons while for metals the conduction electron density ne, c does not change with temperature, for semiconductors ne, c(T = 0 K) = 0 for intrinsic (non-doped) semiconductors and increases with temperature. In Section 2.6.6 we also derived the electronic energy states of hydrogenlike atoms, along with the designation of the quantum numbers and atomic orbitals. As atoms gather in a cluster or a bulk phase, their orbiting electrons and their energy states are altered because of various nuclear and electronic interactions (Section 2.2), including representation as interatomic potentials. These interactions may increase or decrease the energy gaps between the electron orbital states of individual atoms.