Universal scaling formalism and analytical optimization criterion for multiscale geometric design of thermoelectric...

Thermoelectric (TE) generators directly convert heat into electricity, yet their performance is often limited by small temperature gradients. Width-modulated metamaterials with constrictions and expansions (constricted geometries) sustain larger temperature differences ΔT than constant-width counterparts due to reduced Transmissivity (Tr) - the geometric ratio of constriction to expansion cross-sections. A scaling behavior of transport and key TE performance metrics with Transmissivity is demonstrated from the nanoscale to the macroscale using analytical formalism and simulations across single- and multiple-constriction profiles. It is shown that ΔT, electrical and thermal resistances, efficiency, and output power are governed by a single scaling function g(Tr) - the conductance of a constricted geometry relative to a uniform-width counterpart-independent of carrier type, material, or operating conditions. Universal scaling formalism leads to Performance Design Maps and an analytical optimization criterion: maximum TE performance occurs at an optimal Transmissivity Tr_opt, where g(Tr_opt) = Bi, with Bi = hL/k denoting the Biot number and h, L, and k the convection coefficient, length, and thermal conductivity, respectively. Compared with the uniform geometry, the optimal constricted geometry produces a maximum output power reduced under fixed ΔT, by a factor of Bi/4, and enhanced under identical convective operating conditions by a factor of (1 + Bi)²/(4Bi). This work provides a theoretical framework for multiscale design and optimization of constricted geometries, thereby enabling systematic exploration of design strategies for next-generation TE modules based on advanced thermoelectric metamaterials analogous to nature's hierarchical structures for optimized functionality.