Spin dependent thermoelectric transport in a multiterminal quantum dot hybrid including a superconductor and ferromagnets

We investigate the thermoelectric response of a hybrid system consisting of two ferromagnetic electrodes and one superconducting lead coupled to a single-level quantum dot with finite Coulomb repulsion. Using the non-equilibrium Green's function technique within the Hubbard-I approximation, local and non-local thermoelectric coefficients, along with their spin counterparts, such as electrical and thermal conductance, and the Seebeck coefficient are calculated up to linear order with respect to generalized forces. Here, we present a derivation of spin-dependent thermoelectric coefficients for a three-terminal system, extending the existing theory which allowed to describe only cases independent of spin-bias voltage, i.e. when spin accumulation is irrelevant. In the considered system, four competing processes- single particle tunneling, quasiparticle tunneling, direct and crossed Andreev reflection make the system highly adaptable for tuning charge and heat currents. A full analysis of their impact on thermoelectric effects is provided. Moreover, the output power and efficiency of the system operating as a heat engine are evaluated. The extensive goal of this work is to demonstrate how the presence of an additional terminal modifies the hybrid QD-based device's performance and under which conditions non-local thermoelectric effects become significant.