# Scattering Theory of Conductance and Shot Noise

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```					   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

Mesoscopic Physics
Wave nature of electrons becomes important

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 get
conductance, noise, …..
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
Conductance from transmission

conductance quantum               resistance quantum

dissipation and irreversibility

boundary conditions
Conductance: finite temperature                                           6


current of left movers

current of right movers

net current

linear response


conductance         Transmission probability evaluated in the equilibrium potential
7
Equilibrium noise
linear response

equilibrium fluctuations

thermal noise (Johnson-Nyquist noise)

conductance and equilibrium noise give the same information
Fluctuation dissipation theorem
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

asymptotic perfect translation invariant potential

separable wave function

energy of transverse motion        channel threshold
energy for transverse and longitudnial motion
         scattering channel
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
Eigen channels                                11

hermitian matrix; real eigenvalues
hermitian matrix; real eigenvalues

are the genetic code of
mesoscopic conductors !!

Mulichannel = parallel conductance of many single channel conductors
Conductance and shot noise                      12

hermitian matrix; real eigenvalues
hermitian matrix; real eigenvalues

If all            
Schottky (Poisson)
Fano factor                        Khlus (1987)
Lesovik (1989)
Buttiker (1990)
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
Buttiker, Phys. Rev. B41, 7906 (1990)

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

magnetic field B
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
Shot-noise: Quantum point contact            17

A. Kumar, L. Saminadayar, D. C. Glattli,
Y. Jin, B. Etienne, PRL 76, 2778 (1996)
18
Crossover from thermal to shot noise

tunnel junction

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

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         
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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