Scattering theory of conductance and noise
-
Upload
nyssa-tillman -
Category
Documents
-
view
55 -
download
1
description
Transcript of Scattering theory of conductance and noise
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)
7
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)
10
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)
12
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).
13
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)