Lorenz number and electronic thermoelectric figure of Merit: Thermodynamics and direct DFT calculations

Abstract

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 ($K_{\mathrm{el}}$) and the electrical conductivity (σ), and can be expressed as $\mathrm{LT}=K_{\mathrm{el}}/\sigma$ where $T$ is temperature. We demonstrate that the Lorenz number simply equals to the ratio of two thermodynamic quantities: the electronic heat capacity ($c_{\mathrm{el}})$ and the electrochemical capacitance ($c_N)$ through $\mathrm{LT}=c_{\mathrm{el}}/c_N$, 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 $L={\pi }^2/3{\left(k_B/e\right)}^2$ at low temperature limit; ii) the Drude value $L=\left(3/2\right){\left(k_B/e\right)}^2$ at the high temperature limit with the free electron gas model, and iii) possible higher values than the Sommerfeld limit for certain semiconductors. Importantly, we demonstrate that the purely electronic contribution to the thermoelectric figure-of-merit can be directly and efficiently computed using high-throughput density functional theory (DFT) calculations, eliminating the need for the computationally intensive Boltzmann transport theory for electronic thermal and electrical conductivities. For thermoelectric materials with low or negligible lattice thermal conductivity, this approach provides a rapid and reliable estimation of the thermoelectric figure-of-merit. These findings highlight the utility of the proposed methodology in high-throughput workflows for thermoelectric material discovery and screening.