  We investigate the nonlinear scattering theory for quantum systems with
strong Seebeck and Peltier effects, and consider their use as heat-engines and
refrigerators with finite power outputs. This article gives detailed
derivations of the results summarized in Phys. Rev. Lett. 112, 130601 (2014).
It shows how to use the scattering theory to find (i) the quantum
thermoelectric with maximum possible power output, and (ii) the quantum
thermoelectric with maximum efficiency at given power output. The latter
corresponds to a minimal entropy production at that power output. These
quantities are of quantum origin since they depend on system size over
electronic wavelength, and so have no analogue in classical thermodynamics. The
maximal efficiency coincides with Carnot efficiency at zero power output, but
decreases with increasing power output. This gives a fundamental lower bound on
entropy production, which means that reversibility (in the thermodynamic sense)
is impossible for finite power output. The suppression of efficiency by
(nonlinear) phonon and photon effects is addressed in detail; when these
effects are strong, maximum efficiency coincides with maximum power. Finally,
we show in particular limits (typically without magnetic fields) that
relaxation within the quantum system does not allow the system to exceed the
bounds derived for relaxation-free systems, however, a general proof of this
remains elusive.
