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Tutorial notes for the evaluation of thermoelectric quantum bounds in ideal nanostructures

The wave-like nature of electrons leads to the existence of upper bounds on the thermoelectric response of nanostructured devices [R. S. Whitney, Phys. Rev. Lett. 112, 130601 (2014); Phys. Rev. B 91, 115425 (2015)]. This fundamental result, not present in classical thermodynamics, was demonstrated exploiting a two-terminal device modelled by non-linear scattering theory. In the present paper, we consider non-linear quantum transport through the same type of device working both as thermal machine and as refrigerator. For both operations, starting from charge and heat current expressions, we provide analytic quantum bounds for power exchanged, thermal currents and device efficiencies. For this purpose, we adopt a transmission function that maximizes the engine efficiency for given power output. For the optimal boxcar- or theta function-transmission shapes, we provide in a tutorial way an explicit deduction of the quantum bound expressions reported in the above cited papers.