Lorenz Number and Electronic Thermoelectric Figure of Merit: Thermodynamics and Direct DFT Calculations

The Lorenz number (L) contained in the Wiedemann-Franz law represents the ratio of two kinetic parameters of electronic charge carriers: the electronic contribution to the thermal conductivity (K_el) and the electrical conductivity (sigma), , and can be expressed as LT=K_el/sigma where T is temperature. We demonstrate that the Lorenz number simply equals to the ratio of two thermodynamic quantities: the electronic heat capacity (c_el) and the electrochemical capacitance (c_N) through LT=c_el/c_N , a purely thermodynamic quantity, and thus it can be calculated solely based on the electron density of states of a material. It is shown that our thermodynamic formulation for the Lorenz number leads to: i) the well-known Sommerfeld value L=pi^2/3(k_B/e)^2 at the low temperature limit, ii) the Drude value L=3/2(k_B/e)^2 at the high temperature limit with the free electron gas model, and iii) possible higher values than the Sommerfeld limit for semiconductors. It is also demonstrated that the purely electronic contribution to the thermoelectric figure-of-merit can be directly computed using high-throughput DFT calculations without resorting to the computationally more expensive Boltzmann transport theory to the electronic thermal conductivity and electrical conductivity.