Maximum cooling temperature and uniform efficiency criterion for inhomogeneous thermoelectric materials

The maximum cooling temperature of a uniform thermoelectric material is limited by its dimensionless figure of merit $ZT$. Inhomogeneous or graded thermoelectric materials are mainly studied when there is a large temperature gradient and the material composition is typically optimized for maximum local $ZT$. We show that this is not the correct optimization for maximizing the cooling temperature. We give the theoretical limit of maximum cooling temperature for an ideal inhomogeneous material. Surprisingly, the optimum Seebeck profile in the device has three sections with distinct characteristics. As a contrast to the local $ZT$ optimization, the uniform efficiency criterion is proposed for the design of graded thermoelectric materials in cooling applications. This optimization is applied to the practical ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ material which is common in thermoelectric applications. Temperature and electrical conductivity dependences of the material properties are taken into account. The graded material is numerically optimized and it achieves a 27% cooling enhancement compared to the best homogeneous material.