Download Part 1 - Capri Spring School

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, …..
Conductance from transmission
Heuristic discussion
4
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
Equilibrium noise
linear response
equilibrium fluctuations
thermal noise (Johnson-Nyquist noise)
conductance and equilibrium noise give the same information
Fluctuation dissipation theorem
7
9
Shot noise
occupation numbers:
incident beam
transmitted beam
reflected beam
averages:
Each particle can only be either transmitted or reflected:

Shot noise power
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
9
Muli-channel conductor: scattering matrix
13


Incident current in channel n
reflection probabilities
orthogonal unitary
transmission probabilities
Multi-channel conductance, kT = 0, two terminal
Total transmission probability

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
11
Conductance and shot noise
hermitian matrix; real eigenvalues
hermitian matrix; real eigenvalues
If all

Schottky (Poisson)
Fano factor
Khlus (1987)
Lesovik (1989)
Buttiker (1990)
12
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
13
Quantized conductance: saddle
Buttiker, Phys. Rev. B41, 7906 (1990)
Saddle-point potential
Transmission probability
14
Quantized conductance-magnetic field
Buttiker, Phys. Rev. B41, 7906 (1990)
magnetic field B
15
Shot-noise: Qunatum point contact
16
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
A. Kumar, L. Saminadayar, D. C. Glattli,
Y. Jin, B. Etienne, PRL 76, 2778 (1996)
17
18
Crossover from thermal to shot noise
tunnel junction
H. Birk et al., PRL 75, 1610 (1995)
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)
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