  Quantum dots (QDs) can serve as near perfect energy filters and are therefore
of significant interest for the study of thermoelectric energy conversion close
to thermodynamic efficiency limits. Indeed, recent experiments in [Nat. Nano.
13, 920 (2018)] realized a QD heat engine with performance near these limits
and in excellent agreement with theoretical predictions. However, these
experiments also highlighted a need for more theory to help guide and
understand the practical optimization of QD heat engines, in particular
regarding the role of tunnel couplings on the performance at maximum power and
efficiency for QDs that couple seemingly weakly to electronic reservoirs.
Furthermore, these experiments also highlighted the critical role of the
external load when optimizing the performance of a QD heat engine in practice.
To provide further insight into the operation of these engines we use the
Anderson impurity model together with a Master equation approach to perform
power and efficiency calculations up to co-tunneling order. This is combined
with additional thermoelectric experiments on a QD embedded in a nanowire where
the power is measured using two methods. We use the measurements to present an
experimental procedure for efficiently finding the external load $R_P$ which
should be connected to the engine to optimize power output. Our theoretical
estimates of $R_P$ show a good agreement with the experimental results, and we
show that second order tunneling processes and non-linear effects have little
impact close to maximum power, allowing us to derive a simple analytic
expression for $R_P$. In contrast, we find that the electron contribution to
the thermoelectric efficiency is significantly reduced by second order
tunneling processes, even for rather weak tunnel couplings.
