Geometric formulation of the thermoelectric dimensionless figure of merit within the Onsager framework

This paper presents a geometric formulation of the thermoelectric dimensionless figure of merit zT based on the metric structure of the Onsager matrix. Although zT is conventionally expressed in terms of electrical conductivity, Seebeck coefficient, thermal conductivity, and temperature, we show that it is determined by the intrinsic coupling structure encoded in the Onsager matrix. By first interpreting the Onsager matrix as a Gram matrix and then introducing a rescaled vector representation on a common energy-transport scale, a geometric description on the Onsager plane is constructed, revealing that zT =cot 2 θ P , where θ P is a single angle characterizing electrothermal coupling. Within this framework, the intrinsic dimensionless figure of merit zT , determined by the Onsager matrix for a specified thermodynamic state, is represented geometrically by the relative orientation of the transport vectors. Experimentally reconstructed values may deviate from this intrinsic quantity when the constituent transport coefficients are evaluated under inconsistent measurement conditions. The reversible limit corresponds to the singular boundary at which the metric becomes degenerate, while a simplified transport model based on Fermi–Dirac statistics yields a model-dependent characteristic geometric scale for zT . These results provide a framework for interpreting thermoelectric performance as a geometric balance between coupling and dissipation.