  Increasing the maximum cooling effect of a Peltier cooler can be achieved
through materials and device design. The use of inhomogeneous, FGM
(functionally graded materials) may be adopted in order to increase maximum
cooling without improvement of the zT (figure of merit), however these systems
are usually based on the assumption that the local optimization of the zT is
the suitable criterion to increase thermoelectric performances. In the present
paper, we solved the heat equation in a graded material and performed both
analytic and numerical analysis of a graded Peltier cooler. We find a local
criterion that we used to assess the possible improvement of graded materials
for thermoelectric cooling. A fair improvement of cooling effect is predicted
for semiconductor materials (up to $36\%$) and the best graded system for
cooling is described. The influence of the equation of state of the electronic
gas of the material is discussed, and the difference in term of entropy
production between the graded and the classical system is also described.
