Scattering theory of conductance and noise

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Scattering theory of conductance and noise
Multi-probe conductors
Markus Büttiker University of Geneva
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Multi-probe conductors
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Buttiker, PRL 57, 1761 (1986) IBM J. Res.
Developm. 32, 317 (1988)
Quantum Kirchhoff law current conservation
gauge invariance
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Multi-probe conductors scattering matrix
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magnetic field symmetry
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Four-probe resistances
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G has eigenvalue zero!
Current contacts
Voltage probes
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Reciprocity
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From
and
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Reciprocity Benoit et al.
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Benoit, Washburn, Umbach, Laibowitz, Webb, PRL
57, 1765 (1986)
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Reciprocity van Houten et al.
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skipping orbit
van Houten et al. , Phys. Rev. B39, 8556 (1989)
electron focusing
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Skipping orbits
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(semi-classical) quantization
immune to disorder
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Edge states
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Halperin, Phys. Rev. B25, 2185 (1982)
Landau gauge
bulk Landau levels
edge states
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Quantum Hall effect
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Buttiker, Phys. Rev. B38, 9375 (1988)
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Edge states smooth potential
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velocity
equipotential line
Transition from N3 to N 2 edge states
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QPC in high magnetic field
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Buttiker, PRB 41, 7906 (1990)
equipotential lines at the saddle edge states
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QHE Non-ideal contacts
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 The Quantum Hall Effect in Open Conductors ,
M. Buttiker, in  Nanostructured Systems , M.
Reed, ed., Semiconductor and Semimetals, Vol.
35, 191 - 277 (1991).
Edge states a physical reality
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Noise spectral density
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Multi-probe conductance leads
asymptotic perfect translation invariant potential
seprable wave function
channel threshold
energy of transverse motion
energy for transverse and longitudnial motion
scattering channel
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Occupation number and current amplitudes
Buttiker, PRB 46, 12485 (1992)
Incident current at kT 0
Incident current at kT gt 0
Occupation number
lt gt statistical average
Creation and annihilation operators
Incident current 
 Current amplitude 
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Current operator
Buttiker, PRL 65, 2901 (1990)
single channel result
Current in contact
current amplitude
(incoming)
(outgoing)
Current in contact
multi-channel channel result
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Current operator
Buttiker, PRL 65, 2901 (1990)
component vectors
quantum statistical average
average current, conductance
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Conductance finite temperature
quantum statistical average
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Noise spectral density
Spectral density S (noise power)
Use
quantum statistical average of four creation and
annihilation op.
zero-frequency spectrum (white noise limit)
equilibrium
fluctuation-dissipation theorem
shot-noise
non-equilibrium
Buttiker, PRL 65, 2901 (1990) PRB 46, 12485
(1992)
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Equilibrium current fluctuations
Use
with
for all
auto-correlation
cross-correlation
QHE-plateau N
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Experimental test
M. Henny, et al., Science 284, 296 (1999)
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Equilibrium voltage fluctuations
infinite impedance external circuit
Langevin sources with
QHE plateau
for longitudinal resistance
for Hall resistance
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Summary
Conductance of multi-probe structures
Reciprocity (Onsager-Casimir symmetry relations)
Quantum Hall effect
Occupation number fluctuations
Noise spectral density for equilibrium and shot
noise
Johnson-Nyquist noise in multiprobe conductors
Outlook
Shot noise correlations in multiprobe conductors
Two particle interferometers and shot noise
Entanglement
Quantum state tomography
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Review on Shot Noise
 Shot Noise in Mesoscopic Conductors  Ya. M.
Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000)