This article discusses the optimal electronic structure for thermoelectric materials, aiming to maximize the figure of merit (ZT). The authors derive a mathematical function for the transport distribution, which gives the largest ZT. They find that a delta-shaped transport distribution maximizes thermoelectric properties, indicating that a narrow distribution of electron energy levels is needed for maximum efficiency. Thermoelectric materials can be used for refrigeration or power generation, with current best materials having ZT ≈ 1. To improve efficiency, materials with ZT ≈ 4 are needed. The authors analyze the transport distribution function and find that the Dirac delta function is the optimal transport distribution. This function represents a narrow energy level distribution near the chemical potential, which is close to the actual electronic structure of some materials like YbAl3. The study shows that the best thermoelectric materials have a sharp singularity in the density of states near the chemical potential. The results suggest that materials with narrow energy level distributions and high carrier velocities in the direction of the electric field are ideal for thermoelectric applications. The authors also note that adding a constant background to the delta-shaped transport distribution reduces the figure of merit, emphasizing the importance of a narrow energy level distribution. The study concludes that searching for materials with narrow energy level distributions is crucial for improving thermoelectric efficiency.This article discusses the optimal electronic structure for thermoelectric materials, aiming to maximize the figure of merit (ZT). The authors derive a mathematical function for the transport distribution, which gives the largest ZT. They find that a delta-shaped transport distribution maximizes thermoelectric properties, indicating that a narrow distribution of electron energy levels is needed for maximum efficiency. Thermoelectric materials can be used for refrigeration or power generation, with current best materials having ZT ≈ 1. To improve efficiency, materials with ZT ≈ 4 are needed. The authors analyze the transport distribution function and find that the Dirac delta function is the optimal transport distribution. This function represents a narrow energy level distribution near the chemical potential, which is close to the actual electronic structure of some materials like YbAl3. The study shows that the best thermoelectric materials have a sharp singularity in the density of states near the chemical potential. The results suggest that materials with narrow energy level distributions and high carrier velocities in the direction of the electric field are ideal for thermoelectric applications. The authors also note that adding a constant background to the delta-shaped transport distribution reduces the figure of merit, emphasizing the importance of a narrow energy level distribution. The study concludes that searching for materials with narrow energy level distributions is crucial for improving thermoelectric efficiency.