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

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

Scattering Theory of Electron Transport 3 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, …..

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

Conductance from transmission conductance quantumresistance quantum dissipation and irreversibility boundary conditions 5

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

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

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

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

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

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

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

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

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

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

Shot-noise: Qunatum point contact Ideally only one channel contributes 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) 16

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

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

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 19

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

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

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) Quantum corrections to Drude conductance (weak localization, UCF) Drude conductance 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

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