Scattering theory of conductance and noise

Markus BüttikerUniversity of Geneva

Multi-probe conductors

Multi-probe conductors

Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)

2

Quantum Kirchhoff law; current conservation; gauge invariance

Multi-probe conductors: scattering matrix

magnetic field symmetry

3

Four-probe resistances

Current contactsVoltage probes

G has eigenvalue zero!

4

Reciprocity

From and

5

Reciprocity: Benoit et al. Benoit, Washburn, Umbach, Laibowitz, Webb, PRL 57, 1765 (1986)

6

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

8

Edge states

bulk; Landau levels edge states

Halperin, Phys. Rev. B25, 2185 (1982)9

Landau gauge

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

11

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

Noise spectral density

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

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)

Current operator

Current in contact single channel result

current amplitude: (incoming) (outgoing)

multi-channel channel resultCurrent in contact

Buttiker, PRL 65, 2901 (1990)

Current operator

component vectors

quantum statistical average

average current,conductance

Buttiker, PRL 65, 2901 (1990)

Conductance: finite temperature

quantum statistical average

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)

Equilibrium current fluctuations

Use

with for all auto-correlation

cross-correlation

QHE-plateau N:

M. Henny, et al., Science 284, 296 (1999)

Experimental test

Equilibrium voltage fluctuations

infinite impedance external circuit

Langevin sources with

QHE plateau: for longitudinal resistance for Hall resistance

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

Review on Shot Noise

« Shot Noise in Mesoscopic Conductors »Ya. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000)