What electronic structure provides the largest figure of merit for thermoelectric materials? To answer that question, we write the electrical conductivity, thermopower, and thermal conductivity as integrals of a single function, the transport distribution. Then we derive the mathematical function for the transport distribution, which gives the largest figure of merit. A delta-shaped transport distribution is found to maximize the thermoelectric properties. This result indicates that a narrow distribution of the energy of the electrons participating in the transport process is needed for maximum thermoelectric efficiency. Some possible realizations of this idea are discussed.Thermoelectric materials can be used to make refrigerators or power generators (1, 2). These solid state devices have no moving parts and are extremely reliable. Their efficiency is low, so they are used in products where reliability is more important than efficiency. Thermoelectric refrigerators are used to spot cool electronic components such as infrared sensors or computer chips. Power generators are used in space stations and satellites.There has been much effort to find the best thermoelectric materials (1-3). There has also been numerous discussion of the physical limits-i.e., what are the best possible thermoelectrics allowed by nature (4)? Here we provide a new and simple estimate of the maximum efficiency of thermoelectric materials. The efficiency of thermoelectric energy converters depends on the transport coefficients of the constituent materials through the figure of merit (1): a-S2where a-is the electrical conductivity and S is the Seebeck coefficient. The quantity in the denominator is the thermal conductivity; it is given by the sum of contributions from the electronic carriers Ke and the lattice KI. The efficiency is increased by making ZT as large as possible, where T is the mean operating temperature of the device. At room temperature, the best thermoelectric material now known is Bi2Te3, which has ZT 1 (1, 2). With this value, the coefficient of performance of thermoelectric coolers is about one-third the value for conventional compressor systems. At room temperature, with the current design, thermoelectric refrigerators will be competitive with conventional compressor systems if a material is found with ZT 4. However, any small increment in this value (ZT 2 1) will result in many new applications for these devices. This technology is environmentally cleaner and more reliable than traditional compressor systems. Therefore, it is worth exploring the possibility of increasing ZT to find a material with ZT >