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 contact

STM fabricated MCBJ technique

AI

V

d.c. current through the contact

The current through a metallic atomic contact

Non-linear generalization

Energy dependent transmission coefficient

Same single-channel model

L R

tLeft lead Right lead

eVRL

)(t LRRLRL

ccccHHH perturbation

)(G)(Gdth

e ,LR

,RL

2I

)(tie

LRRL

ccccI

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

)(g),(g rRR

rLL previous calculation

Then

)(f)(g)(g)(g LrLL

aLL

,LL

)(G)(Gdth

e ,LR

,RL

2I

For a more general calculation it is useful to express the current in terms of the electrodes diagonal Green functions

,RR

,LL G,G

It is also convenient to use the specific Dyson equation for (in terms of )

,Gra G,G

)(Gg)(Ggdth

e ,RR

,LL

,RR

,LL 2

2I

)GI(gGIG aa,rr,

Problem: derivation of expression:

)(Gg)(Ggdth

e ,RR

,LL

,RR

,LL 2

2I

Start from )(G)(Gdth

e ,LR

,RL

2I

Use for ,

LRG

,rraa,,, GgGggG 1D

Use for ,

RLG

,rraa,,, gGgGgG 2D

Subtract: ,

LR,

RL GG

)(Gg)(Ggdth

e ,RR

,LL

,RR

,LL 2

2I

With this expression the tunnel limit is immediately reproduced:

lowest order ,RR

,RR gG

)(gg)(ggdth

e ,RR

,LL

,RR

,LL 2

2I

)(f)(i)(g LL,LL 2 )(f)(i)(g LL

,LL 12

)(f)(f)()(dth

eRLRRLL 224

2I

tunnel expression (low transmission)

Using for the calculation of ,RR

,RR G,G

)GI(gGIG aa,rr,

3D )GI(gGIG aa,rr,

where Ga and Gr are calculated froma,ra,ra,ra,ra,r GΣggG a,Dr

tr,aRL

r,aLR

Problem

)(f)eV(f)(g)(gt

)()(td

h

eI

RL

RL

22

22

1

42

First notice that higher order process in t are included in the denominator

)(f)eV(f)(g)(gt

)()(td

h

eI

RL

RL

22

22

1

42

Tunnel limit It is possible to identify the energy dependent transmission

)(f)eV(f)V,(dh

eI 2

Landauer-like

22

2

1

4

)(g)(gt

)()(t)V,(

RL

RL

Current noise in a metallic atomic contact

Same single-channel model

L R

tLeft lead Right lead

eVRL

We define the spectral density of the current fluctuations:

)t()()()t(dte)(S ti IIII 00

where )t(I)t()t( II

The noise at zero frequency will be given by:

)t()()()t(dt)(S IIII 000

Remembering that the current operator has the form in this model:

)()()()()( tttttie

t LRRL ccccI

The current-current correlation averages contains terms of the form:

cccc

However in a non-interacting system they can be factorized (Wick’s theorem) in the form

cccccccc

As the averages of the form are related to cc ,G

cc ,G

A simple algebra leads to:

)(Adth

e)(S 2

220

)(G)(G)(G)(G

)(G)(G)(G)(G)(A,LL

,RR

,RR

,LL

,RL

,LR

,RL

,LR

Wide-band approximation (symmetrical contact):

W

i)()(g)(g a

LaR

121

12

)(f)(f

)(f)(f

W

i)(

LL

LLL

g Keldish space

Direct “unsophisticated” attack: Dyson equation in Keldish space

GggG ),....(G),(G),(G),(G ,LR

,RL

,RR

,LL

t

tRLLR 0

0ΣΣ

Problem: solve Dyson equation for the Green functions

)()()(

)( RRLLLL

ggG

21

1

212

221

1 2

)(

t/)(LRG

2

2

W

t

)(f)(f RL

),....(G),(G),(G),(G ,LR

,RL

,RR

,LL

Problem: substituting in expression of noise

)f(f)f(fd)(h

e)(S LRRL 1114

02

Identifying the transmission coefficient: 21

4

)(

Shot noise limit: eVTkB

eV)(h

e)(S 14

02

Fano reduction factor

Poissonian limit (Schottky)

Vh

eI

22

0

eI

)(S

2

0

binomial distribution

charge of the carriers (electrons)

Resonant tunneling through a discrete level

resonant level

L R

Quantum Dot

M M

Anderson model out of equilibrium

cccc

nnnHHH R

00

0000

t

UL

RL,

Non-interacting case: U=0

0

Lt Rt

L R

eVRL

Equilibrium case: L1

RR ii)(G

000

1

0

Lt Rt

L R

0 RL

)(

)(2

2

RRR

LLL

t

t

000

1

)(g

),(),( ,0

,0 ttGttGt

eLLL

I

)()(2 ,

0,

0 LLL GGdth

eI

stationary current

As in the contact case: useful expression in terms of diagonal

functions:

)()(2 ,

00,,

00,2 GgGgdt

h

eLLLLLI

And now we use the specific Dyson equation for )(),( ,00

,00 GG

)GI(gGIG aa,rr,

3D )GI(gGIG aa,rr,

Problem: substitution in expression of current:

)()()()()(42 2

00222 RL

rRLRL ffGdtt

h

eI

Linear conductance

2

00222

2

)()()(42 r

RLRL Gtth

eG

As we have )(2 LLL t )(2 RRR t and

RR ii)(G

000

1

220

2

)(

42

RL

RL

h

eG

For a symmetrical junction: RL

220

22

4

42)(

h

eG

Resonant condition: 0

h

eG

2

0

2)( Irrespective of

A more interesting case: e-e interaction in the level

resonant level

L R

Quantum Dot

00nnU

Coulomb blockade and Kondo effects

Coulomb blockade and Kondo effects:

-0.5 0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

LD

OS

U / 10

U

Equilibrium spectral density Coulomb blockade peaks

Kondo resonance

Current through a finite mesoscopic region

As a preliminary problem let us first analyze

Current through a finite 1D system

0

L R

0 0 0

t t ttL tR

1 2 N

Current (stationary) between L and 1:

0

L R

0 0 0

t t ttL tR

1 2 N

)t,t(G)t,t(Gte ,

L,LL 11

I

)(G)(Gdth

e ,L

,LL 11

2I

stationary current

In terms of diagonal Green functions in sites L and 1:

)(G)(g)(G)(gdth

e ,,LL

,,LLL 1111

22I

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

0

L RtL tR

)()()()()(42 2

00222 RL

rRLRL ffGdtt

h

eI

0

L R

0 0 0

t t ttL tR

1 2 N

)(f)(f)(G)()(dtth

eRL

rNRLRL

2

12224

2I

Linear conductance:

)(f)(f)(G)()(dtth

eRL

rNRLRL

2

12224

2I

2

1222

2

42

)(G)()(tth

eG r

NRLRL

2

1

2

42

)(Gh

eG r

NRL )(t LLL 2

2

14 )(G rNRL

-2 -1 0 1 2

0,0

0,2

0,4

0,6

0,8

1,0tr

ansm

issi

on

/t

1 atom

-2 -1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

3 atoms

tran

smis

sion

/t

-2 -1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

tran

smis

sion

/t

10 atoms

-2 -1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

tra

nsm

issi

on

/t

30 atoms

L

R

eVRL

Self-consistent determination of electrostatic potential profile

Oscillations with wave-length 2/F

Multi-channel generalizationelectron reservoirs

EF

EF+eV

M

mesoscopic region

left lead right lead

Even a one-atom contact has several channels if the detailed atomic orbital structure is included

s-like N=1simple metalsalkali metals

sp-like N=3III-IV group

d-like N=5transition metals

Al atomic contact

000 RLRL HHHHHH

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

left lead right lead

LH RH

0H0LH 0RH

ji,ij

j,ii

ii t ccnH i sites

orbitals

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

1D chain

)()()()(Tr)V,( aNR

rNL 114 GΓGΓ

)(f)(f)(G)()(dh

eRL

rNRL

2

142

I

)(t)( LLL 2

)(f)(f)V,(dh

eRL 2

I finite region

)()()()(Tr),( aNR

rNL 1140 GΓGΓ

Linear regime

),(h

eG 0

2 2

Hermitian matrix

n

nh

eG

22

diagonalization: eigenvalues & eigenvectors

conduction channels

The PIN code of an atomic contact

electron reservoirs

EF

EF+eV

S

S1

2

N

n

nh

eG

22

PIN code n

Microscopic origin of conduction channels

s-like N=1simple metalsalkali metals

sp-like N=3III-IV group

d-like N=5transition metals