Energy exchange between metals: Single mode thermal rectifier

1. 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

2. 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?

3. 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

4. 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

5. TL TR J I. Phononic transport

6. 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).

7. Nonlinear interactions: Truncated phonon spectrum Asymmetry: Spin-boson thermal rectifier D. Segal, A. Nitzan, PRL (2005).

8. 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).

9. 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)

10. J TL ; L TR ; R II. Energy transfer in a fermionic Model No charge transfer

11. 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

12. II.1 Linear dispersion limit

13. 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:

14. -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!

15. Relaxation rates Deviation from linear dispersion

16. TL ; L TR ; R J Single mode heat conduction Linear dispersion Nonlinear dispersion D. Segal, Phys. Rev. Lett. (2008)

17. Single mode heat conduction: Nonlinearity No negative differential conductance- Need strong system-bath coupling

18. 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.

19. Rectification

20. 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.

21. 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

22. e Other effects… J. B. Pendry, J. Phys. Cond. Mat. 11, 6621 (1999)

23. 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).

24. V(s-1,s) V(s,s+1) js-1 js hs-10 hs0 A more general definition

25. J TL ; L TR ; R Energy transfer in a fermionic Model u d Second order, Markovian limit Steady state

26. 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.

27. Extensions • Transport of charge and energy, • Thermoelectric effect in low dimensional systems • Realistic modeling Thanks!

28. 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.

29. Noninteracting Hamiltonian: Second quantization: Spinless fermions Two species Linear dispersion Luttinger Model

30. Density operators: Commutation relations: Boson operators: Bosonization

31. Interaction Hamiltonian • Scattering of same species: • Different species: Note: scattering must conserve momentum