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We study thermoelectric transport through a serial double quantum dot (DQD)coupled to two metallic leads with different thermal energies. We take intoaccount the electron sequential and cotunneling effects via different masterequation approaches. In the absence of intra- and inter-dot Coulombinteractions, 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 withstrengths U$_{\rm intra}$, and U$_{\rm inter}$, respectively, avoided crossingsor resonance energies between the intra- and the inter-dot two-electron states,2ES, are found. These resonances induce extra transport channels through theDQD leading to strong side peaks in the thermoelectric and heat currents at $E_{\rm L} \text{-} E_{\rm R} = \pm (U_{\rm intra} \text{-} U_{\rm inter})$ inaddition to the main peak generated at $E_{\rm L} \text{=} E_{\rm R}$. Thecurrent side peaks are enhanced by increased strength of the Coulombinteractions. Interestingly, the current side peaks are enhanced whencotunneling terms are considered in which the resonances of the 2ESs assist theelectron cotunneling process through the system. Furthermore, the issue ofcoherences is carefully checked in the DQD-leads system via differentapproaches to the master equation, which are the Pauli, the Redfield, a firstorder Lindblad, and the first- and second order von-Neumann methods. We realizethat the Pauli method gives a wrong results for the thermoelectric transportwhen the role of the coherences is relevant.

Thermal transport controlled by intra- and inter-dot Coulomb interactions in sequential and cotunneling serially-coupled double quantum dots