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Application to transport phenomena. Current through an atomic metallic contact Shot noise in an atomic contact Current through a resonant level Current through a finite 1D region Multi-channel generalization: Concept of conduction eigenchannel . ». m. I. A. V. - PowerPoint PPT Presentation

Application to transport phenomena Current through an atomic metallic contact Shot noise in an atomic contact Current through a resonant level Current through a finite 1D region Multi-channel generalization: Concept of conduction eigenchannel

Current through an atomic metallic contactSTM fabricatedMCBJ techniqued.c. current through the contact

The current through a metallic atomic contact Non-linear generalization Energy dependent transmission coefficientSame single-channel modelLeft leadRight leadperturbation

We use, though, the full energy dependent Green functions of the uncoupled electrodes: previous calculation Then

For a more general calculation it is useful to express the current in terms of the electrodes diagonal Green functions It is also convenient to use the specific Dyson equation for (in terms of )

Problem: derivation of expression: Start from Use for Use for Subtract:

With this expression the tunnel limit is immediately reproduced:lowest ordertunnel expression (low transmission)

Using for the calculation of where Ga and Gr are calculated from Problem

First notice that higher order process in t are included in the denominator Tunnel limit It is possible to identify the energy dependent transmission Landauer-like

Current noise in a metallic atomic contactSame single-channel model We define the spectral density of the current fluctuations: where

The noise at zero frequency will be given by: Remembering that the current operator has the form in this model: The current-current correlation averages contains terms of the form: However in a non-interacting system they can be factorized (Wicks theorem) in the form As the averages of the form are related to

A simple algebra leads to: Wide-band approximation (symmetrical contact): Keldish space Direct unsophisticated attack: Dyson equation in Keldish space

Problem: solve Dyson equation for the Green functions Problem: substituting in expression of noise

Identifying the transmission coefficient:

Shot noise limit: Fano reduction factor Poissonian limit (Schottky) binomial distributioncharge of the carriers (electrons)

Resonant tunneling through a discrete levelresonant levelQuantum Dot

Anderson model out of equilibrium Non-interacting case: U=0

Equilibrium case: L1

stationary current As in the contact case: useful expression in terms of diagonal functions: And now we use the specific Dyson equation for

Problem: substitution in expression of current: Linear conductance As we haveand

For a symmetrical junction: Resonant condition: Irrespective of

A more interesting case: e-e interaction in the level resonant levelQuantum Dot Coulomb blockade and Kondo effects

Coulomb blockade and Kondo effects:Equilibrium spectral densityCoulomb blockade peaksKondo resonance

Current through a finite mesoscopic region As a preliminary problem let us first analyze Current through a finite 1D system

Current (stationary) between L and 1: stationary current In terms of diagonal Green functions in sites L and 1:

Problem: same steps as in the single resonant level case:

Linear conductance:

Self-consistent determination of electrostatic potential profile Oscillations with wave-length

Multi-channel generalization

Even a one-atom contact has several channels if the detailed atomic orbital structure is included s-like N=1simple metalsalkali metalssp-like N=3III-IV groupd-like N=5transition metalsAl atomic contact

Same model than in the 1-channel case: tight-binding model including different orbitals isitesaorbitals

In practice, the effect of a finite central region can be taken into account in a matrix notation :1D chainfinite region

Linear regimeHermitian matrixdiagonalization: eigenvalues & eigenvectorsconduction channels

The PIN code of an atomic contactelectron reservoirsEFEF+eV

Microscopic origin of conduction channelss-like N=1simple metalsalkali metalssp-like N=3III-IV groupd-like N=5transition metals