  We study thermoelectric transport through a serial double quantum dot (DQD)
coupled to two metallic leads with different thermal energies. We take into
account the electron sequential and cotunneling effects via different master
equation approaches. In the absence of intra- and inter-dot Coulomb
interactions, a small peak in thermoelectric and heat currents is found for
$E_{\rm L} \text{=} E_{\rm R}$ indicating the Coulomb blockade DQD regime,
where $E_{\rm L}(E_{\rm R})$ is the energy of the state of the left(right)
quantum dot. In the presence of intra- and inter-dot Coulomb interactions with
strengths U$_{\rm intra}$, and U$_{\rm inter}$, respectively, avoided crossings
or resonance energies between the intra- and the inter-dot two-electron states,
2ES, are found. These resonances induce extra transport channels through the
DQD leading to strong side peaks in the thermoelectric and heat currents at $
E_{\rm L} \text{-} E_{\rm R} = \pm (U_{\rm intra} \text{-} U_{\rm inter})$ in
addition to the main peak generated at $E_{\rm L} \text{=} E_{\rm R}$. The
current side peaks are enhanced by increased strength of the Coulomb
interactions. Interestingly, the current side peaks are enhanced when
cotunneling terms are considered in which the resonances of the 2ESs assist the
electron cotunneling process through the system. Furthermore, the issue of
coherences is carefully checked in the DQD-leads system via different
approaches to the master equation, which are the Pauli, the Redfield, a first
order Lindblad, and the first- and second order von-Neumann methods. We realize
that the Pauli method gives a wrong results for the thermoelectric transport
when the role of the coherences is relevant.
