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 PhysicsWave nature of electrons becomes important

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Webb et al. 1985

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 getconductance, 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 !!

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Landauer formula

Conductance from transmission

conductance quantum resistance quantum

dissipation and irreversibility

boundary conditions

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Conductance: finite temperature

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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 beamreflected beam

averages:Each particle can only be either transmitted or reflected:

Shot noise power

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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

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Muli-channel conductor: scattering matrix

transmission probabilities

reflection probabilities

Multi-channel conductance, kT = 0, two terminal

unitary

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orthogonal

Total transmission probability

Incident current in channel n

Eigen channels

hermitian matrix; real eigenvalues

hermitian matrix; real eigenvalues

are the genetic code of mesoscopic conductors !!

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Mulichannel = parallel conductance of many single channel conductors

Conductance and shot noise hermitian matrix; real eigenvalues

hermitian matrix; real eigenvalues

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If all Schottky (Poisson)

Fano factor Khlus (1987)Lesovik (1989)Buttiker (1990)

Quantum point contact

2D-electron gas

gate

gate

van Wees et al., PRL 60, 848 (1988)Wharam et al, J. Phys. C 21, L209 (1988)

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Quantized conductance: saddle Saddle-point potential

Transmission probability

Buttiker, Phys. Rev. B41, 7906 (1990)

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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)

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

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Crossover from thermal to shot noise

tunnel junction

H. Birk et al., PRL 75, 1610 (1995)

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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

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Shot-noise: Metallic diffusive wireBeenakker and Buttiker, PRB 46, 1889 (1992)

Henny et al. PRB 59, 2871 (1999)

Shot-noise: Chaotic cavityJalabert, 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 cavitygive no information on long range coherence but short range coherence, quantum diffraction is necessaryDiffraction 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