Optimal efficiency and power and their trade-off in three-terminal quantum thermoelectric engines with two output electric currents

We establish a theory of optimal efficiency and power for three-terminal thermoelectric engines which have two independent output electric currents and one input heat current. This set-up goes beyond the conventional heat engines with only one output electric current. For such a set-up, we derive the optimal efficiency and power and their trade-off for three-terminal heat engines with and without time-reversal symmetry. The formalism goes beyond the known optimal efficiency and power for systems with or without time-reversal symmetry, showing interesting features that have not been revealed before. A concrete example of quantum-dot heat engine is studied to show that the current set-up can have much improved efficiency and power compared with previous set-ups with only one output electric current. Our analytical results also apply for thermoelectric heat engines with multiple output electric currents, providing an alternative scheme toward future high-performance thermoelectric materials.