Single-molecule-mediated heat current between an electronic

and a bosonic bath

In Collaboration with:Avi Schiller, The Hebrew UniversityNatan Andrei, Rutgers University

Yuval Vinkler

Racah Institute of Physics, The Hebrew University

An Experimental Motivation: Nanodevices

Liang et. al. (2002)

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

Substrate

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)

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

bbdxxxivH xF†

0†

bathd Hbbg :00: ††

nnnnnnbath bbH †††

What happens if the phononic and electronic baths are held at different temperatures? We expect a heat current mediated by the phononic mode.

In the limit where , and are small with respect to the effective cutoff, the bosonic Hamiltonian takes the form

Solution of the Model

g

By applying Abelian bosonization one can map the Hamiltonian onto a bosonic one.

d02/g

kkkkkk aabbgbbaaH †††0

†

bathkkkd Haa †

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

nnnnbathbathQ bbiHHiHJ ††,

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

dTTfTfJQ 21 ;;

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.

is the transmission coefficient, which for small quantum dots shows a typical resonance structure.

T

TTTTJQ 22

21

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

dATnTnJ ebQ3;;

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

TTTTJ ebQ 344

ebQ TTJ

Here is a spectral function of the local phonon with a resonance near .

A0

Results for an Ohmic bathConsidering an ohmic bath we hold and sweep through different values of eT bT

001.0 eT

Master Equation Approach

Lack of saturation follows from the bosonic nature of the vibrational mode which allows for an arbitrarily high excitation energy.

111111 nnnnnnnnnnnn WWPWPWPP

Solving for steady state we find the effective temperature

eb

ebeff TT

TTT

22

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:

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.