Energy exchange between metals: Single mode thermal rectifier

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Energy exchange between metals Single mode
thermal rectifier
TL ?L
TR ?R
Definition of the heat current operator Lianao Wu

• Dvira Segal
• Chemical Physics Theory Group
• University of Toronto

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Motivation
I
V

• Nonlinear transport rectification, NDR
• Transport of ENERGY Heat conduction in
  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?

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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

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I. Vibrational energy flow in molecules
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I. Phononic transport
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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).
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Spin-boson thermal rectifier
D. Segal, A. Nitzan, PRL (2005).
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Single mode heat conduction by photons

D. R. Schmidt et al., PRL 93, 045901 (2004).
Experiment M. Meschke et al., Nature 444, 187
(2006).
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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)
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II. Energy transfer in a fermionic Model
No charge transfer
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Energy transfer between metals
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II.1 Linear dispersion limit
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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
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Relaxation rates
The key elements here are (i) Energy
dependence of F(?) (ii) Bounded
spectrum Breakdown of the assumptions behind the
Bosonization method!
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Relaxation rates

Deviation from linear dispersion
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Single mode heat conduction
Linear dispersion
Nonlinear dispersion
D. Segal, Phys. Rev. Lett. (2008)
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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
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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.

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• 2. Photon mediated energy transfer
• Two metal islands
• No charge transfer
• No photon tunneling
• No vibrational energy transfer

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Other effects
J. B. Pendry, J. Phys. Cond. Mat. 11, 6621 (1999)
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V. On the proper definition of the heat current
operator
Lianao Wu, DS, arXiv0804.3371
J. Gemmer, R. Steinigeweg, and M. Michel, Phys.
Rev. B 73, 104302 (2006).
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A more general definition

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Energy transfer in a fermionic Model
u
d
Second order, Markovian limit Steady state
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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.

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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.

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Luttinger Model

• Noninteracting Hamiltonian
• Second quantization

• Spinless fermions
• Two species
• Linear dispersion

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Bosonization

• Density operators
• Commutation relations

• Boson operators

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Interaction Hamiltonian

• Scattering of same species
• Different species

Note scattering must conserve momentum