Scattering theory of conductance and noise Markus Büttiker University of Geneva Multi-probe...
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Scattering theory of conductance and noise
Markus BüttikerUniversity of Geneva
Multi-probe conductors
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Multi-probe conductors
Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)
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Quantum Kirchhoff law; current conservation; gauge invariance
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Multi-probe conductors: scattering matrix
magnetic field symmetry
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Four-probe resistances
Current contactsVoltage probes
G has eigenvalue zero!
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Reciprocity
From and
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Reciprocity: Benoit et al. Benoit, Washburn, Umbach, Laibowitz, Webb, PRL 57, 1765 (1986)
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Reciprocity: van Houten et al.
skipping orbit
electron focusing
van Houten et al. , Phys. Rev. B39, 8556 (1989)
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Skipping orbits
immune to disorder
(semi-classical) quantization
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Edge states
bulk; Landau levels edge states
Halperin, Phys. Rev. B25, 2185 (1982)9
Landau gauge
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Quantum Hall effect
Buttiker, Phys. Rev. B38, 9375 (1988)
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Edge states: smooth potential
equipotential line
Transition from N=3 to N= 2 edge states
velocity
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QPC in high magnetic field
equipotential lines at the saddle:edge states
Buttiker, PRB 41, 7906 (1990)
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QHE: Non-ideal contacts« The Quantum Hall Effect in Open Conductors », M. Buttiker, in « Nanostructured Systems », M. Reed, ed., Semiconductor and Semimetals, Vol. 35, 191 - 277 (1991).
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Edge states a physicalreality
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Noise spectral density
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Multi-probe conductance: leads
asymptotic perfect translation invariant potential seprable wave function
energy of transverse motion
energy for transverse and longitudnial motion
scattering channel
channel threshold
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Occupation number and current amplitudes
Incident current at kT = 0
Incident current at kT > 0
Occupation number
< > = statistical average Creation and annihilation operators
«Incident current » « Current amplitude »
Buttiker, PRB 46, 12485 (1992)
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Current operator
Current in contact single channel result
current amplitude: (incoming) (outgoing)
multi-channel channel resultCurrent in contact
Buttiker, PRL 65, 2901 (1990)
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Current operator
component vectors
quantum statistical average
average current,conductance
Buttiker, PRL 65, 2901 (1990)
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Conductance: finite temperature
quantum statistical average
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Noise spectral density
zero-frequency spectrum (white noise limit)
equilibrium
fluctuation-dissipation theorem
Spectral density S (noise power)
Use
quantum statistical average of four creation and annihilation op.
non-equilibrium shot-noise
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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M. Henny, et al., Science 284, 296 (1999)
Experimental test
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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
Outlook
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
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)