1. Thermoelectric Effects: Semiclassical and Quantum Approaches from the Boltzmann Transport Equation

The thermoelectric efficiency of a material depends on its electronic and phononic properties. It is normally given in terms of the dimensionless figure of merit Z T = σ S ² T ∕ κ. The parameters involved in Z T are the electrical conductivity σ, the Seebeck coefficient S, and the thermal conductivity κ. The thermal conductivity has two contributions, κ = κ _e + κ _L , the electron thermal conductivity κ _e and the lattice thermal conductivity κ _L . In this chapter all these parameters will be deduced for metals and semiconductors, starting from the Boltzmann transport equation (BTE). The electrical conductivity, the Seebeck coefficient, and the electronic thermal conductivity will be obtained from the BTE for electrons. Similarly, the lattice or phonon thermal conductivity will be given from the BTE for phonons. The ab initio approaches to obtain both the electronic and phononic transport via the BTE will also be analyzed. All the theoretical studies are based on the relaxation time approximation. The expressions for the relaxation times for electrons and phonons will be discussed. The results will be particularized to nanostructures whenever is possible.