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Scattering theory of conductance and noise

Scattering theory of conductance and noise

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Scattering theory of conductance and noise

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  1. Scattering theory of conductance and noise Multi-probe conductors Markus Büttiker University of Geneva

  2. 2 Multi-probe conductors Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988) Quantum Kirchhoff law; current conservation; gauge invariance

  3. 3 Multi-probe conductors: scattering matrix magnetic field symmetry

  4. 4 Four-probe resistances G has eigenvalue zero! Current contacts Voltage probes

  5. 5 Reciprocity From and

  6. 6 Reciprocity: Benoit et al. Benoit, Washburn, Umbach, Laibowitz, Webb, PRL 57, 1765 (1986)

  7. 7 Reciprocity: van Houten et al. skipping orbit van Houten et al. , Phys. Rev. B39, 8556 (1989) electron focusing

  8. 8 Skipping orbits (semi-classical) quantization immune to disorder

  9. 9 Halperin, Phys. Rev. B25, 2185 (1982) Edge states Landau gauge bulk; Landau levels edge states

  10. 10 Quantum Hall effect Buttiker, Phys. Rev. B38, 9375 (1988)

  11. 11 Edge states: smooth potential velocity equipotential line Transition from N=3 to N= 2 edge states

  12. 12 QPC in high magnetic field Buttiker, PRB 41, 7906 (1990) equipotential lines at the saddle: edge states

  13. 13 QHE: Non-ideal contacts « The Quantum Hall Effect in Open Conductors », M. Buttiker, in « Nanostructured Systems », M. Reed, ed., Semiconductor and Semimetals, Vol. 35, 191 - 277 (1991). Edge states a physical reality

  14. Noise spectral density

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

  16. Buttiker, PRB 46, 12485 (1992) Occupation number and current amplitudes Incident current at kT = 0 Incident current at kT > 0 Occupation number < > = statistical average Creation and annihilation operators «Incident current » « Current amplitude »

  17. Buttiker, PRL 65, 2901 (1990) Current operator single channel result Current in contact current amplitude: (incoming) (outgoing) Current in contact multi-channel channel result

  18. Buttiker, PRL 65, 2901 (1990) Current operator component vectors quantum statistical average average current, conductance

  19. Conductance: finite temperature quantum statistical average

  20. Noise spectral density Spectral density S (noise power) Use quantum statistical average of four creation and annihilation op. zero-frequency spectrum (white noise limit) equilibrium fluctuation-dissipation theorem shot-noise non-equilibrium Buttiker, PRL 65, 2901 (1990); PRB 46, 12485 (1992)

  21. Equilibrium current fluctuations Use with for all auto-correlation cross-correlation QHE-plateau N:

  22. Experimental test M. Henny, et al., Science 284, 296 (1999)

  23. Equilibrium voltage fluctuations infinite impedance external circuit Langevin sources with QHE plateau: for longitudinal resistance for Hall resistance

  24. Summary Conductance of multi-probe structures Reciprocity (Onsager-Casimir symmetry relations) Quantum Hall effect Occupation number fluctuations Noise spectral density for equilibrium and shot noise Johnson-Nyquist noise in multiprobe conductors Outlook Shot noise correlations in multiprobe conductors Two particle interferometers and shot noise Entanglement Quantum state tomography

  25. Review on Shot Noise « Shot Noise in Mesoscopic Conductors » Ya. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000)