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
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Mesoscopic Physics
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Wave nature of electrons becomes important
Webb et al. 1985
Heiblum et al. 1996
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Scattering Theory of Electron Transport
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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, ..
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Conductance from transmission
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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
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Conductance from transmission
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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
Transmission probability evaluated in the
equilibrium potential
conductance
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Equilibrium noise
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linear response
equilibrium fluctuations
thermal noise (Johnson-Nyquist noise)
conductance and equilibrium noise give the same
information
Fluctuation dissipation theorem
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Shot noise
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occupation numbers
incident beam
transmitted beam
reflected beam
averages
Each particle can only be either transmitted or
reflected
Shot noise power
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Multi-channel conductance leads
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asymptotic perfect translation invariant potential
separable wave function
channel threshold
energy of transverse motion
energy for transverse and longitudnial motion
scattering channel
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Muli-channel conductor scattering matrix
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unitary
orthogonal
Incident current in channel n
transmission probabilities
reflection probabilities
Multi-channel conductance, kT 0, two terminal
Total transmission probability
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Eigen channels
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hermitian matrix real eigenvalues
hermitian matrix real eigenvalues
are the genetic code of mesoscopic conductors !!
Mulichannel parallel conductance of many single
channel conductors
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Conductance and shot noise
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hermitian matrix real eigenvalues
hermitian matrix real eigenvalues
If all
Schottky (Poisson)
Fano factor
Khlus (1987)
Lesovik (1989)
Buttiker (1990)
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Quantum point contact
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van Wees et al., PRL 60, 848 (1988)
Wharam et al, J. Phys. C 21, L209 (1988)
gate
2D-electron gas
gate
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Quantized conductance saddle
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Buttiker, Phys. Rev. B41, 7906 (1990)
Saddle-point potential
Transmission probability
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Quantized conductance-magnetic field
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Buttiker, Phys. Rev. B41, 7906 (1990)
magnetic field B
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Shot-noise Qunatum point contact
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• 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
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Shot-noise Quantum point contact
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A. Kumar, L. Saminadayar, D. C. Glattli, Y. Jin,
B. Etienne, PRL 76, 2778 (1996)
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Crossover from thermal to shot noise
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tunnel junction
H. Birk et al., PRL 75, 1610 (1995)
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Fermions versus Bosons
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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 wire
Beenakker and Buttiker, PRB 46, 1889 (1992)
Henny et al. PRB 59, 2871 (1999)
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Shot-noise Chaotic cavity
Jalabert, Pichard and Beenakker, Europhys. Lett.
27, 255 (1994)
for symmetric cavity with
Oberholzer et al., PRL 86, 2114 (2001)
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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
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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