  In a previous work, we predicted that a thermally biased tunnel junction
between two different superconductors can display a thermoelectric effect of
nonlinear nature in the temperature gradient, under proper conditions. In this
work we give a more extended discussion, and we focus on the two main features
of the nonlinear contributions: i) the $linear-in-bias$ thermoelectricity, that
can be associated to a spontaneous breaking of electron-hole symmetry, ii) the
strong contribution at the matching peak singularity, which is typically
associated to the maximum output power and efficiency. We discuss the nonlinear
origin of the thermoelectricity and its relationship with the non-linear
cooling mechanism in superconducting junctions previously discussed in the
literature. Finally, we design and characterize the performance of the system
as a heat engine, for a realistic design and experimental parameter values. We
discuss possible non-idealities demonstrating that the system is amenable to
current experimental realization.
