  We present a study on inelastic thermoelectric devices, wherein charge
currents and electronic and phononic heat currents are intricately
interconnected. The employment of double quantum dots in conjunction with a
phonon bath positions them as promising candidates for quantum thermoelectric
diodes and transistors. Within this study, we illustrate that quantum coherence
effects yield significant charge and Seebeck rectification effects. It's worth
noting that, while the thermal transistor effect is observable in the linear
response regime, especially when phonon-assisted inelastic processes dominate
the transport, quantum coherence does not enhance thermal amplification. Our
work provides valuable insights for the optimization of general thermoelectric
devices.
