Fundamental Relations as the Leading Order in Nonlinear Thermoelectric Responses with Time-Reversal Symmetry

In recent years, nonlinear transport phenomena have garnered significant interest in both theoretical explorations and experiments. In this work, we utilize the semi-classical wave packet theory to calculate disorder-induced second-order transport coefficients: second-order electrical ($σ$), thermoelectric ($α$), and thermal ($κ$) coefficients, capturing the interplay between side-jump and skew-scattering contributions in systems with time-reversal symmetry. Using a topological insulator model, we quantitatively characterize the Fermi-level dependence of these second-order transport coefficients by explicitly including Coulomb impurity potentials. Furthermore, we elucidate the relationships between these coefficients, establishing the second-order Mott relation and the Wiedemann-Franz law induced by disorder. This study develops a comprehensive theoretical framework elucidating the nonlinear thermoelectric transport mechanisms in quantum material systems.