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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 l.jpg

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


Motivation l.jpg

Motivation

I

V

  • 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?


Outline l.jpg

Outline

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


Slide4 l.jpg

IVR

Nanomachines

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 l.jpg

TL

TR

J

I. Phononic transport


Single mode thermal conduction harmonic model l.jpg

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).


Slide7 l.jpg

Nonlinear interactions:

Truncated phonon spectrum

Asymmetry:

Spin-boson thermal rectifier

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


Single mode heat conduction by photons l.jpg

2006

Single mode heat conduction by photons

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


Slide9 l.jpg

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)


Slide10 l.jpg

J

TL ; L

TR ; R

II. Energy transfer in a fermionic Model

No charge transfer


Slide11 l.jpg

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.

Cos(k)

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

Energy transfer between metals


Slide12 l.jpg

II.1 Linear dispersion limit


Ii 2 nonlinear dispersion case l.jpg

J

n=2..

n=1

n=0

TL ; L

knn+1

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:


Relaxation rates l.jpg

-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 rates15 l.jpg

Relaxation rates

Deviation from linear dispersion


Slide16 l.jpg

TL ; L

TR ; R

J

Single mode heat conduction

Linear dispersion

Nonlinear dispersion

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


Single mode heat conduction nonlinearity l.jpg

Single mode heat conduction: Nonlinearity

No negative differential conductance- Need strong system-bath coupling


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III. Rectification

Nonlinear dispersion relation

Asymmetry

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.


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Rectification


Iv realizations exchange of energy between metals l.jpg

STM tip

Adsorbed molecules

Metal

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.


Slide21 l.jpg

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 l.jpg

e

Other effects…

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


V on the proper definition of the heat current operator l.jpg

V(s-1,s)

V(s,s+1)

js-1

js

hs-10

hs0

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 l.jpg

V(s-1,s)

V(s,s+1)

js-1

js

hs-10

hs0

A more general definition


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J

TL ; L

TR ; R

Energy transfer in a fermionic Model

u

d

Second order, Markovian limit Steady state


Summary l.jpg

Summary

  • 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.


Slide27 l.jpg

Extensions

  • Transport of charge and energy,

  • Thermoelectric effect in low dimensional systems

  • Realistic modeling

Thanks!


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Bosonization

  • 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.


Luttinger model l.jpg

Noninteracting Hamiltonian:

Second quantization:

Spinless fermions

Two species

Linear dispersion

Luttinger Model


Bosonization30 l.jpg

Density operators:

Commutation relations:

Boson operators:

Bosonization


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Interaction Hamiltonian

  • Scattering of same species:

  • Different species:

Note: scattering must conserve momentum


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