Investigation of enhanced thermoelectric response in the extended Lieb lattice using the Hubbard model

We investigate the thermoelectric properties of the extended Lieb lattice under the influence of intrinsic spin-orbit coupling, on-site Coulomb interaction, and an external magnetic field using the Hubbard model and linear response theory. The unique electronic structure of this system, characterized by two tunable flat bands, provides a fertile platform for controlling charge and heat transport. By systematically analyzing the electronic contributions to thermal conductivity, electrical conductivity, Seebeck coefficient, and the thermoelectric figure of merit (ZT_e), we reveal how spin-orbit coupling and strong electron correlations suppress thermal conductivity while enhancing thermoelectric efficiency. Our results show that increasing the on-site Coulomb interaction and magnetic field strength significantly improves ZT_e, particularly at low to intermediate thermal energy. The modulation of flat bands via spin-orbit and interaction effects plays a critical role in enhancing the density of states near the Fermi level, thereby optimizing the Seebeck response. These findings highlight the role of flat-band engineered lattices in shaping thermoelectric response and provide theoretical insights into band-structure engineering of correlated low-dimensional materials with tunable transport behavior.