  We study thermoelectric effects in Coulomb-coupled quantum-dot (CCQD) systems
beyond lowest-order tunneling processes and the often applied wide-band
approximation. To this end, we present a master-equation (ME) approach based on
a perturbative $T$-matrix calculation of the charge and heat tunneling rates
and transport currents. Applying the method to transport through a
non-interacting single-level QD, we demonstrate excellent agreement with the
Landauer-B{\"u}ttiker theory when higher-order (cotunneling) processes are
included in the ME. Next, we study the effect of cotunneling and
energy-dependent lead couplings on the heat currents in a system of two
Coulomb-coupled QDs. Overall, we find that cotunneling processes (i) dominate
the heat currents at low temperature and bias, and (ii) give rise to a
pronounced reduction of the cooling power achievable with the recently
demonstrated Maxwell's demon cooling mechanism. Furthermore, we demonstrate
that the cooling power can be boosted significantly by carefully engineering
the energy dependence of the lead couplings to filter out undesired transport
processes. Our findings emphasize the importance of considering higher-order
cotunneling processes as well as the advantage of engineered energy-dependent
lead couplings in the optimization of the thermoelectric performance of
Coulomb-coupled QD systems.
