Massimiliano Di Ventra Department of Physics, University of California, San Diego

1. Massimiliano Di Ventra Department of Physics, University of California, San Diego Fundamental aspects of transport in nanostructures and atomic physics Collaborators Neil Bushong (NC) Yuriy Pershin (USC) Chih-Chun (LANL) Mike Zwolak (OSU) Roberto D’Agosta (Spain) Giovanni Vignale (U. Missouri)

2. Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid • (large for structures with smaller transmissions) • Properties of steady states and predictions • Theory: Microcanonical picture of transport • Experiments: Atomic gases in optical lattices

3. Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid • (large for structures with smaller transmissions) • Properties of steady states and predictions • Theory: Microcanonical picture of transport • Experiments: Atomic gases in optical lattices

4. Field: nanoscale electronics Atomic point contacts Nanotubes/wires Molecular junctions Tans et al. (1997) Organic electronics from Nitzan et al.(2003) Scheer et al. (1998) Fast DNA sequencing Lagerqvist et al. (2006) Z.Q. Li et al. (2006)

5. What do we want to describe ? R Solved ? Not quite, especially at the atomic level !

6. Major difference with macro/mesoscopic systems forces Large current densities increased e-e, e-ph scattering

7. Why is the problem difficult (and interesting) ? • The system is out of equilibrium • (non-equilibrium statistical mechanics is still an open subject; • do we need to go beyond Hamiltonian dynamics?) • Interactions among electrons • (Coulomb blockade, correlations, non-Fermi liquid behavior) • Interactions among electrons and ions • (e.g., el-phonon scattering, current-induced forces) • Interaction with the environment • (dissipation and dephasing) • Physical properties are quite sensitive to atomic details

8. From experiment to model system Approximation 1: open quantum systems Closed system Open system: dynamical interaction with reservoirs

9. From experiment to model system Approximation 1: open quantum systems Battery dense spectrum No initial correlations Small interaction In general, no closed equation of motion for rs

10. From experiment to model system Approximation 2: ideal steady state Assume existence of at least one steady state solution Still many-body open quantum system !

11. From experiment to model system Approximation 3: “openness” vs boundary conditions Loss of information

12. From experiment to model system Approximation 4: mean-field approximation With or w/o interaction Non-interacting electrons Non-interacting electrons If leads are interacting NO closed equation of motion for the current !

13. From experiment to model system Approximation 5: independent channels and energy filling Non-interacting electrons Non-interacting electrons With or w/o interaction

14. The Landauer current Non-interacting electrons From scattering theory This formula has nothing to do with NEGF !!!

15. Interacting sample Non-interacting electrons Non-interacting electrons With interactions From NEGF Meir and Wingreen, 1992

16. Physical origin of many-body corrections: linear-response theory C3 C2 C1 C4 Current conservation Gauge invariance

17. Physical origin of many-body corrections: linear-response theory “Proper” current-current response function + + … + For a non-interacting system

18. Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid • (large for structures with smaller transmissions) • Properties of steady states and predictions • Theory: Microcanonical picture of transport • Experiments: Atomic gases in optical lattices

19. Interactions in the whole system: the microcanonical picture of transport M. Di Ventra, T.N. Todorov, (J. Phys. Cond. Matt. 2004)

20. Fast relaxation of momentum w momentum relaxation time 1/nc  tc  ħ/E  m w2/p2 ħ2  1 fs Bushong, Sai and M. Di Ventra (Nano Letters 2005)

21. Comparison with Landauer formula Chen, Zwolak and Di Ventra , in preparation

22. Entanglement entropy R L Gaussian Binomial Exact Approximate C(t) = correlation matrix PL = projection operator Klich and Levitov, PRL 2009 Chen, Zwolak and Di Ventra , in preparation

23. Electron flow Quasi-2D electron liquid, TDDFT V= 0.2V Sai, Bushong, Hatcher, and Di Ventra , PRB 2007

24. continuity F=ma Hydrodynamics of the electron liquid Exact! n = density v = j/n Information on all e-e interactions (generally unknown) A hydrodynamic formulation is more natural in QM than in classical physics Anticipates TDDFT by many years ! Martin and Schwinger, Phys. Rev. (1959)

25. Fast relaxation of momentum w momentum relaxation time 1/nc  tc  ħ/E  m w2/p2 ħ2  1 fs Bushong, Sai and M. Di Ventra (Nano Letters 2005)

26. Quantum Navier-Stokes equations 1) 2) 3) R. D’Agosta and M. Di Ventra, JPCM (2006)

27. Conductance quantization from hydrodynamics 1D, stationary, non-viscous fluid Bernoulli Quantized conductance is the one of a 1D ideal, non-viscous charged fluid D’Agosta and M. Di Ventra, JPCM (2006)

28. Electron Dynamics Electron dynamics in nanostructures similar to a viscous liquid Predictions turbulence electron heating effect on ion heating

29. b Turbulence in nanoscale systems QPCs molecules Actual atomic structure Approx. potential adiabatic non-adiabatic D’Agosta and M. Di Ventra, JPCM (2006)

30. Turbulence in nanoscale systems Adiabatic constrictions (e.g., QPC), laminar flow Nonadiabatic constrictions (e.g., molecules), turbulent flow laminar turbulent D’Agosta and M. Di Ventra, JPCM (2006)

31. Time-Dependent Current DFT no memory, Markov approx. bulk viscosity Vignale and Kohn PRL 1996; Vignale, Ullrich and Conti (1997)

32. Electron turbulence TDCDFT Closed system, quasi-2D electron liquid, TDCDFT Eddies size NS nanoscale, for fully developed turbulence Bushong, Gamble, and M. Di Ventra (Nano Lett. 2007); D’Agosta and Di Ventra JPCM (2006)

33. 0.02 V (Laminar) Bushong, Gamble and M. Di Ventra (Nano Lett. 2007)

34. 3 V (Turbulent) Bushong, Gamble and M. Di Ventra (Nano Lett. 2007)

35. Possible exp. verification Laminar Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007)

36. Possible exp. verification Turbulent Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007)

37. Possible exp. verification Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007)

38. Electron Dynamics Electron dynamics in nanostructures similar to a viscous liquid Predictions turbulence electron heating effect on ion heating

39. Electron heating: elementary considerations Power in the junction: Heat dissipated in the electrodes: D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006)

40. Electron heating from hydrodynamics Heat equation (no turbulence) Thermal conductivity e.g. Au QPC viscosity D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006)

41. Electron Dynamics Electron dynamics in nanostructures similar to a viscous liquid Predictions turbulence electron heating effect on ion heating

42. Ionic Heating: elementary considerations Power in the junction: Heat dissipated in the electrodes: Y-C Chen, M. Zwolak, M. Di Ventra Nano Letters (2003) T.N. Todorov Phil. Mag, (1998)

43. Effect of heating on shot noise and current ? Chen, Di Ventra, PRB (2003); Phys. Rev. Lett. (2005) Agrait et al., Phys. Rev. Lett. (2002)

44. H2 e--e- e--ph Effect of el. and ion heating on inelastic scattering Exp. Djukic et al PRB (2005) Th. D’Agosta and M. Di Ventra, J. Phys. Cond. Matt. (2008)

45. e--e- e--ph Effect on the ionic temperature: theory D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006)

46. a c Effect on the ionic temperature: exp. Huang et al. Nano Lett. (2006); Huang et al. (Nature Natotech., 2007) Figure 3

47. Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid • (large for structures with smaller transmissions) • Properties of steady states and predictions • Theory: Microcanonical picture of transport • Experiments: Atomic gases in optical lattices

48. Interactions in the whole system: the microcanonical picture of transport M. Di Ventra, T.N. Todorov, (J. Phys. Cond. Matt. 2004)

49. Formation of steady states w ~ G0 The formation of a steady state has nothing to do with the infinite nature of the electrodes momentum relaxation time 1/nc  tc  ħ/E  m w2/p2 ħ2  1 fs Bushong, Sai and M. Di Ventra (Nano Letters 2005)

50. Cold atoms are ideal systems for studying transport phenomena • You can choose: 1.fermions or bosons 2.harmonic trap, optical lattice, or both (Box-potential is coming soon!) 3.single or multi components or species 4.dimensions (3D, 2D, 1D, or mixed) • You can tune: 1.interactions among atoms (via Feshbach resonance) 2.trap depth or lattice constant 3.temperature 4.density / filling factor