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Scattering Theory of Conductance and Shot Noise

This paper discusses the scattering theory of electron transport in nanostructures, specifically focusing on conductance and shot noise. It explores the wave nature of electrons, transmission and reflection probabilities, quantum resistance, and the effects of temperature and magnetic fields. The analysis includes multi-channel conductance and the behavior of shot noise in quantum point contacts and metallic diffusive wires.

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Scattering Theory of Conductance and Shot Noise

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  1. Scattering Theory of Conductance and Shot Noise Markus Büttiker University of Geneva The Capri Spring School on Transport in Nanostructures April 3-7, 2006

  2. 2 Mesoscopic Physics Wave nature of electrons becomes important Webb et al. 1985 Heiblum et al. 1996

  3. 3 Scattering Theory of Electron Transport Conductor = Scattering potential for electrons Contacts = Emitters and absorbers of electrons From scattering data r,t and statistical assumptions of the emitters and absorbers get conductance, noise, …..

  4. 4 Conductance from transmission Heuristic discussion Fermi energy left contact Fermi energy right contact applied voltage transmission probability reflection probability incident current density density of states independent of material !! Landauer formula

  5. 5 Conductance from transmission conductance quantum resistance quantum dissipation and irreversibility boundary conditions

  6. 6 Conductance: finite temperature current of left movers current of right movers net current linear response Transmission probability evaluated in the equilibrium potential conductance

  7. 7 Equilibrium noise linear response equilibrium fluctuations thermal noise (Johnson-Nyquist noise) conductance and equilibrium noise give the same information Fluctuation dissipation theorem

  8. 9 Shot noise occupation numbers: incident beam transmitted beam reflected beam averages: Each particle can only be either transmitted or reflected: Shot noise power

  9. 9 Multi-channel conductance: leads asymptotic perfect translation invariant potential separable wave function channel threshold energy of transverse motion energy for transverse and longitudnial motion scattering channel

  10. 13 Muli-channel conductor: scattering matrix orthogonal unitary Incident current in channel n reflection probabilities transmission probabilities Multi-channel conductance, kT = 0, two terminal Total transmission probability

  11. 11 Eigen channels hermitian matrix; real eigenvalues hermitian matrix; real eigenvalues are the genetic code of mesoscopic conductors !! Mulichannel = parallel conductance of many single channel conductors

  12. 12 Conductance and shot noise hermitian matrix; real eigenvalues hermitian matrix; real eigenvalues If all Schottky (Poisson) Fano factor Khlus (1987) Lesovik (1989) Buttiker (1990)

  13. 13 Quantum point contact van Wees et al., PRL 60, 848 (1988) Wharam et al, J. Phys. C 21, L209 (1988) gate 2D-electron gas gate

  14. 14 Buttiker, Phys. Rev. B41, 7906 (1990) Quantized conductance: saddle Saddle-point potential Transmission probability

  15. 15 Quantized conductance-magnetic field Buttiker, Phys. Rev. B41, 7906 (1990) magnetic field B

  16. 16 Shot-noise: Qunatum point contact • A. Kumar, L. Saminadayar, D. C. Glattli, • Y. Jin, B. Etienne, PRL 76, 2778 (1996) M. I. Reznikov, M. Heiblum, H. Shtrikman, D. Mahalu, PRL 75, 3340 (1996) Ideally only one channel contributes

  17. 17 Shot-noise: Quantum point contact A. Kumar, L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, PRL 76, 2778 (1996)

  18. 18 Crossover from thermal to shot noise tunnel junction H. Birk et al., PRL 75, 1610 (1995)

  19. 19 Fermions versus Bosons Fermions: upper sign, f(E) Fermi distribution function Bosons: lower sign, f(E) Bose distribution function Remember: Partition enhances noise of Fermions but reduces noise of Bosons Shot noise probes two particle properties: Later we use this property of shot noise to violate a Bell inequality

  20. Shot-noise: Metallic diffusive wire Beenakker and Buttiker, PRB 46, 1889 (1992) Henny et al. PRB 59, 2871 (1999)

  21. Shot-noise: Chaotic cavity Jalabert, Pichard and Beenakker, Europhys. Lett. 27, 255 (1994) for symmetric cavity with Oberholzer et al., PRL 86, 2114 (2001)

  22. Is shot noise quantum or classical? metallic diffusive wire Scattering approach: Beenakker and Buttiker, PRB 46, 1889 (1992) Langevin approach: Nagaev, Phys. Lett. A 169, 103 (1992) Drude conductance Quantum corrections to Drude conductance (weak localization, UCF) Shot noise spectrum Quantum correction to shot noise Fano factors for metallic diffusive wire or for chaotic (many) channel cavity give no information on long range coherence but short range coherence, quantum diffraction is necessary Diffraction can be switched off in chaotic cavities Ehrenfest time

  23. Summary Conductance and shot noise of two-probe conductors Eigenchannels Quantum point contact Outlook Conductance and shot noise of multi-probe conductors Integer quantum Hall effect Voltage probes Dephasing probes

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