Single-molecule-mediated heat current between an electronic and a bosonic bath

1. Single-molecule-mediated heat current between an electronic and a bosonic bath Yuval Vinkler Racah Institute of Physics, The Hebrew University In Collaboration with: Avi Schiller, The Hebrew University Natan Andrei, Rutgers University

2. An Experimental Motivation: Nanodevices The vibrational mode of the molecule is coupled to the tunneling electrons and to the substrate phonons, and relaxes by both. Substrate phonons may have a different temperature than the tunneling electrons. Electronic leads Liang et. al. (2002) Substrate

3. Relevant Systems Several systems display similar physical phenomena of a vibrational mode coupled to both a bosonic and an electronic source: A molecule adsorbed on a surface A single molecule transistor An Aharonov-Bohm interferometer with a molecular device on one of its arms Entin-Wohlman and Aharony (2011)

4. The Heat Current The above mentioned systems can be mapped onto an effective model for the coupling of the vibrational mode to the electrons in the lead And the coupling of the vibrational mode to the bosonic bath What happens if the phononic and electronic baths are held at different temperatures? We expect a heat current mediated by the phononic mode.

5. Solution of the Model By applying Abelian bosonization one can map the Hamiltonian onto a bosonic one. In the limit where , and are small with respect to the effective cutoff, the bosonic Hamiltonian takes the form This Hamiltonian is quadratic in bosons and correlation functions can be calculated exactly, relying on the Keldysh formalism for nonequilibrium steady state. The heat current operator is given by

6. The Electronic case – Landauer formula In the case of a purely electronic system without interactions the heat current is given by an appropriate variant of the Landauer formula is the transmission coefficient, which for small quantum dots shows a typical resonance structure. For temperatures low with respect to the resonance and for small temperature difference the heat current has the behavior For temperatures high with respect to the resonance width and a large temperature difference the heat current saturates to a constant value.

7. Results for an Ohmic bath In the case of an ohmic bath our results are markedly different than the electronic case. The heat current is given by the expression Here is a spectral function of the local phonon with a resonance near . At low temperatures and small temperature difference the heat current displays a much stronger temperature dependence At high temperatures the heat current does not saturates but rather converges to a linear dependence on the temperatures difference

8. Results for an Ohmic bath Considering an ohmic bath we hold and sweep through different values of

9. Master Equation Approach Lack of saturation follows from the bosonic nature of the vibrational mode which allows for an arbitrarily high excitation energy. The degree of excitation can be quantified by an effective temperature. This picture can be demonstrated by a master equation approach, with rate equations written for the occupancy probabilities of the energy levels of the vibrational modes: Solving for steady state we find the effective temperature

10. Summary We considered several systems where heat transfer between two reservoirs of different nature is mediated by a molecular device. These systems can be mapped onto a single quadratic bosonic Hamiltonian, where certain nonequilibrium steady state quantities can be calculated exactly. Dependence of heat current on the temperature difference is markedly different than in a purely electronic system, displaying a cross-over from quartic dependence to a linear one at high temperatures. The behavior of the system at high temperatures was elucidated by a master-equation approach, revealing, in particular, an effective temperature for the vibrational mode.