The compatibility approach in the classical theory of thermoelectricity seen from the perspective of variational calculus

The compatibility approach introduced by Snyder and Ursell opens a new pathway for the improvement of thermoelectric (TE) device performance. It has been shown that sufficient compatibility is - besides an increase of the averaged figure of merit Z - essential for efficient operation of a TE device, and that compatibility will facilitate rational materials selection, device design, and the engineering of functionally graded materials (FGMs). In this paper, the authors give an overview of the fundamental results and present a new approach from the perspective of variational calculus. A particular focus is on the role of ideal self-compatibility, i.e., adjusting compatibility locally at any position along a TE leg.