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Energy exchange between metals: Single mode thermal rectifier. T L ;  L. T R ;  R. Dvira Segal Chemical Physics Theory Group University of Toronto. Definition of the heat current operator Lianao Wu. Motivation. I. V. Nonlinear transport: rectification, NDR

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Energy exchange between metals: Single mode thermal rectifier

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Energy exchange between metals: Single mode thermal rectifier

TL ; L

TR ; R

Dvira Segal

Chemical Physics Theory Group

University of Toronto

Definition of the heat current operator

Lianao Wu




  • Nonlinear transport: rectification, NDR

  • Transport of ENERGY:Heat conductionin bosonic/fermionic systems.

  • Nanodevices: Heat transfer in molecular systems. Radiative heat conduction.

  • Bosonization: What happens when deviations from the basic picture exist? What are the implications on transport properties?


I. Phononic thermal transport (bosonic baths)

II. Energy transfer between metals (fermionic baths)

(1) Linear dispersion case

(2) Nonlinear dispersion case

III. Rectification of heat current

IV. Realizations

V. On the proper definition of the current operator

VI. Conclusions



Fourier law in 1 D.

carbon nanotubes

Molecular electronics

Heating in nanojunctions.

C. Van den Broeck, PRL (2006).

I. Vibrational energy flow in molecules




I. Phononic transport

Electrical rectifier

Reed 1997

Asymmetry + Anharmonicity

Thermal Rectification

Single mode thermal conduction: harmonic model

D. Segal, A. Nitzan, P. Hanggi, JCP (2003).

* M. Terraneo, M. Peyrard, G. Casati, PRL (2002);

*B. W. Li, L. Wang, G. Casati, PRL (2004);

*B. B. Hu, L. Yang, Y. Zhang, PRL (2006).

Nonlinear interactions:

Truncated phonon spectrum


Spin-boson thermal rectifier

D. Segal, A. Nitzan, PRL (2005).


Single mode heat conduction by photons

D. R. Schmidt et al., PRL 93, 045901 (2004). Experiment: M. Meschke et al., Nature 444, 187 (2006).

Exchange of information

Radiation of thermal voltage noise

The quantum thermal conductance is universal, independent of the nature of the material and the particles that carry the heat (electrons, phonons, photons) .

K. Schwab Nature 444, 161 (2006)


TL ; L

TR ; R

II. Energy transfer in a fermionic Model

No charge transfer

For a 1D system of noninteracting electrons with an unbounded strictly linear dispersion relation, k= F+vF(k-kF) the Hamiltonian can be bosonized to yield a bosonic Hamiltonian with equivalent properties.


E. Miranda, Brazilian J. of Phys. (2003).

Energy transfer between metals

II.1 Linear dispersion limit





TL ; L


TR ; R

II.2 Nonlinear dispersion case

Assuming weak coupling, going into the Markovian limit, the probabilities Pn to occupy the n state of the local oscillator obey

Steady state heat current:

-F ( ) Em,n

Relaxation rates

The key elements here are:

(i) Energy dependence of F() (ii)Bounded spectrum

Breakdown of the assumptions behind the Bosonization method!

Relaxation rates

Deviation from linear dispersion

TL ; L

TR ; R


Single mode heat conduction

Linear dispersion

Nonlinear dispersion

D. Segal, Phys. Rev. Lett. (2008)

Single mode heat conduction: Nonlinearity

No negative differential conductance- Need strong system-bath coupling

III. Rectification

Nonlinear dispersion relation


We could also assume L R, LR

Relationship between the bosonic and fermionic models:

We could also bosonize the Hamiltonian with the nonlinear dispersion relation and obtain a bosonic Hamiltonian made of a single mode coupled to two anharmonic boson baths.


STM tip

Adsorbed molecules


IV. Realizations: Exchange of energy between metals

  • (1) Phonon mediated energy transfer

  • Strong laser pulse gives rise to strong increase of the electronic temperature at the bottom metal surface. Energy transfers from the hot electrons to adsorbed molecule. Energy flows to the STM tip from the molecule.

  • No charge transfer

  • Only el-ph energy transfer from the molecule to the STM, ignore ph-ph contributions.

TL ; L

TR ; R

D. R. Schmidt et al., PRL 93, 045901 (2004). Experiment: M. Meschke et al., Nature 444, 187 (2006).

  • 2. Photon mediated energy transfer

  • Two metal islands:

  • No charge transfer

  • No photon tunneling

  • No vibrational energy transfer


Other effects…

J. B. Pendry, J. Phys. Cond. Mat. 11, 6621 (1999)







V. On the proper definition of the heat current operator

Lianao Wu, DS, arXiv:0804.3371

J. Gemmer, R. Steinigeweg, and M. Michel, Phys. Rev.

B 73, 104302 (2006).







A more general definition


TL ; L

TR ; R

Energy transfer in a fermionic Model



Second order, Markovian limit Steady state


  • We have studied single mode heat transfer between two metals with nonlinear dispersion relation and demonstrated thermal rectification.

  • In the linear dispersion case we calculated the energy current using bosonization, and within the Fermi Golden rule, and got same results.

  • The same parameter that measures the deviation from the linear dispersion relation, (or breakdown of the bosonization picture), measures the strength of rectification in the system.

  • In terms of bosons, the nonlinear dispersion relation translates into anharmonic thermal baths. Thus the onset of rectification in this model is consistent with previous results.

  • We discussed the proper definition of the heat flux operator in 1D models.


  • Transport of charge and energy,

  • Thermoelectric effect in low dimensional systems

  • Realistic modeling



  • Representing 1D Fermionic fields in terms of bosonic fields.

  • The reason is that all excitations are particle-hole like and therefore have bosonic character.

  • A powerful technique for studying interacting quantum systems in 1D.

Noninteracting Hamiltonian:

Second quantization:

Spinless fermions

Two species

Linear dispersion

Luttinger Model

Density operators:

Commutation relations:

Boson operators:


Interaction Hamiltonian

  • Scattering of same species:

  • Different species:

Note: scattering must conserve momentum

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