Thermoelectric Efficiency and Compatibility

The intensive reduced efficiency ${\eta }_r$ is derived for thermoelectric power generation (in one dimension) from intensive fields and currents, giving ${\eta }_r=\frac{E·J}{-\nabla T·J_S}$. The overall efficiency is derivable from a thermodynamic state function, $\Phi =1/u+\alpha T$, where we introduce $u=\frac{J}{\kappa \nabla T}$ as the relative current density. The method simplifies the computation and clarifies the physics behind thermoelectric devices by revealing a new materials property $s=(\sqrt{1+zT}-1)/(\alpha T)$, which we call the compatibility factor. Materials with dissimilar compatibility factors cannot be combined by segmentation into an efficient thermoelectric generator because of constraints imposed on $u$. Thus, control of the compatibility factor $s$ is, in addition to $z$, essential for efficient operation of a thermoelectric device, and thus will facilitate rational materials selection, device design, and the engineering of functionally graded materials.

Figure 1

Plot of reduced efficiency [Eq. (4)] as relative current density, $u$, varies at a constant temperature. $p$-type materials are compared at temperatures where they would contribute to an efficient segmented system: $(\mathrm{B}\mathrm{i},\mathrm{S}\mathrm{b}{)}_2{\mathrm{T}\mathrm{e}}_3$ ($100°\mathrm{C}$), ${\mathrm{Z}\mathrm{n}}_4{\mathrm{S}\mathrm{b}}_3$ ($300°\mathrm{C}$), ${\mathrm{C}\mathrm{e}\mathrm{F}\mathrm{e}}_4{\mathrm{S}\mathrm{b}}_{12}$ ($550°\mathrm{C}$), SiGe ($800°\mathrm{C}$). Maximum efficiency at the compatibility factor, $u=\mathrm{s}$, is indicated with filled circles.