Three-terminal thermoelectric transport through a molecule placed on an Aharonov-Bohm ring

The thermoelectric transport through a ring threaded by an Aharonov-Bohm flux, with a molecular bridge on one of its arms, is analyzed. The transport electrons also interact with the vibrational excitations of that molecule. This nano-system is connected to three terminals: two are electronic reservoirs, which supply the transport electrons, and the third is the phonon bath which thermalizes the molecular vibrations. Expressions for the transport coefficients, relating all charge and heat currents to the temperature and chemical potential differences between the terminals, are derived to second order in the electron-vibration coupling. At linear response, all these coefficients obey the full Onsager-Casimir relations. When the phonon bath is held at a temperature different from those of the electronic reservoirs, a heat current exchanged between the molecular vibrations and the transport electrons can be converted into electric and/or heat electronic currents. The related transport coefficients, which exist only due to the electron-vibration coupling, change sign under the interchange between the electronic terminals and the sign change of the magnetic flux. It is also demonstrated that the Aharonov-Bohm flux can enhance this type of conversion.