Universal Scaling Formalism and Analytical Optimization Criterion for Multiscale Geometric Design of Thermoelectric Metamaterials

Abstract

Thermoelectric (TE) generators can directly convert heat into electricity, but their performance is often constrained by limited temperature gradients. Here it is shown that width-modulated metamaterials with constrictions and expansions (constricted geometries) enhance temperature difference DT by reduced Transmissivity (Tr), a geometry-based parameter defined by the ratio of constriction to expansion cross-sections. A universal scaling behavior of transport and key TE efficiency metrics with Transmissivity is demonstrated, spanning from the nanoscale to the macroscale. Analytical formalism validated through finite element calculations for a range of modulation geometries reveals that DT, electrical and thermal resistances, efficiency, and power output are governed by a single scaling function, g(Tr), independent of carrier type, material, or operating conditions. This function represents the conductance of a constricted geometry relative to a uniform-width counterpart. The developed framework yields TE Performance Design Maps and an analytical criterion for optimal TE performance, with the maximum power density achieved at an optimal Transmissivity Tr_opt, determined by the condition that the functional g(Tr_opt) equals the Biot number, the dimensionless ratio hL/k of the convection coefficient h, the structure length L and the material thermal conductivity k. Transmissivity is established as a robust, multiscale design parameter - analogous to nature's hierarchical structures for optimized functionality. This work provides the 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.