Application to transport phenomena

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)(Gdthe ,

LR,

RL 2I

)(tieLRRL

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)(Gdthe ,

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)(Ggdthe ,

RR,LL

,RR

,LL 2

2I

)GI(gGIG aa,rr,

Problem: derivation of expression:

)(Gg)(Ggdthe ,

RR,LL

,RR

,LL 2

2I

Start from )(G)(Gdthe ,

LR,

RL 2I

Use for ,LRG

,rraa,,, GgGggG 1D

Use for ,RLG

,rraa,,, gGgGgG 2D

Subtract: ,LR

,RL GG

)(Gg)(Ggdthe ,

RR,LL

,RR

,LL 2

2I

With this expression the tunnel limit is immediately reproduced:

lowest order ,RR

,RR gG

)(gg)(ggdthe ,

RR,LL

,RR

,LL 2

2I

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

,LL 12

)(f)(f)()(dthe

RLRRLL 2242I

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 from a,ra,ra,ra,ra,r GΣggG a,Dr

tr,aRL

r,aLR

Problem

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

)()(tdheI

RL

RL

22

22

1

42

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

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

)()(tdheI

RL

RL

22

22

1

42

Tunnel limit It is possible to identify the energy dependent transmission

)(f)eV(f)V,(dheI 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:

)()()()()( tttttiet 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 ,Gcc ,G

A simple algebra leads to:

)(Adthe)(S 2220

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

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

,RR

,RR

,LL

,RL

,LR

,RL

,LR

Wide-band approximation (symmetrical contact):

Wi)()(g)(g a

LaR

12112

)(f)(f)(f)(f

Wi)(

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

211

212

2211 2

)(t/)(LRG

2

2

Wt

)(f)(f RL

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

,RL

,RR

,LL

Problem: substituting in expression of noise

)f(f)f(fd)(he)(S LRRL 111402

Identifying the transmission coefficient: 21

4)(

Shot noise limit: eVTkB

eV)(he)(S 1402

Fano reduction factor

Poissonian limit (Schottky)

VheI 22

0

eI)(S

20

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 ttGttGte

LLLI

)()(2 ,0

,0 LLL GGdt

heI

stationary current

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

)()(2 ,00

,,00

,2 GgGgdthe

LLLLLI

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:

)()()()()(422

00222 RL

rRLRL ffGdtt

heI

Linear conductance

2

00222

2

)()()(42 rRLRL Gtt

heG

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

RR ii)(G

000

1

220

2

)(42

RL

RL

heG

For a symmetrical junction: RL

220

22

442)(

heG

Resonant condition: 0

heG2

02)( Irrespective of

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

resonant level

L R

Quantum Dot

00 nnU

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

LDO

S

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)(Gdthe ,

L,LL 11

2I

stationary current

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

)(G)(g)(G)(gdthe ,,

LL,,

LLL 111122I

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

0

L RtL tR

)()()()()(422

00222 RL

rRLRL ffGdtt

heI

0

L R

0 0 0

t t ttL tR

1 2 N

)(f)(f)(G)()(dtthe

RLrNRLRL

2

122242I

Linear conductance:

)(f)(f)(G)()(dtthe

RLrNRLRL

2

122242I

2

1222

2

42 )(G)()(ttheG r

NRLRL

2

1

2

42 )(GheG r

NRL )(t LLL 2

2

14 )(G rNRL

-2 -1 0 1 2

0,0

0,2

0,4

0,6

0,8

1,0tra

nsm

issi

on

/t

1 atom

-2 -1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

3 atoms

trans

mis

sion

/t

-2 -1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

trans

mis

sion

/t

10 atoms

-2 -1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

trans

mis

sion

/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)()(dhe

RLrNRL

2

142I

)(t)( LLL 2

)(f)(f)V,(dhe

RL 2I finite region

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

rNL 1140 GΓGΓ

Linear regime

),(heG 02 2

Hermitian matrix

n

nheG 22

diagonalization: eigenvalues & eigenvectors

conduction channels

The PIN code of an atomic contact

electron reservoirs

EF

EF+eV

S

S1

2

N

n

nheG 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